Advances in Applied Mechanics Volume 13
Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAULGERMAIN L. HOWARTH WILLIAM PRAGER T. Y. Wu HANSZIEGLER
Contributors to Volume 13 H. L. Kuo GEORGE VERONIS JOHN
V. WEHAUSEN
ADVANCES IN
APPLIED MECHANICS Edited by Chia-Shun Yih COLLEGE OF ENGINEERING THE UNIVERSITY OF MICHIGAN ANN ARBOR, MICHIGAN
VOLUME 13
1973
ACADEMIC PRESS
New York and London
COPYRIGHT 0 1973, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. N O PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY F OR M OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION I N WRITING FROM T HE PUBLISHER.
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CONGRESS CATALOG CARD
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PRINTED I N TH E UNITED STATES O F AMERICA
Contents
vii
LISTOF CONTRIBUTORS
PREFACE
ix
Large Scale Ocean Circulation George Veronis Introduction I. The Equilibrium Figure of a Self-Gravitating, Rotating, Homogeneous Mass of Fluid 11. Transformations of the Equations of Motion of a Fluid 111. The Coriolis Acceleration IV. Thermodynamic Simplifications-the Boussinesq Approximation V. Scaling of ,the Equations VI. Geostrophic Flow VII. Frictional Dissipation VIII. Modeling of Current Systems IX. The Thermohaline Circulation X. Abyssal Circulation XI. Laboratory Simulation of Large Scale Circulation (with C. C. Yang) References
2 3 6 14 18 28 33 36 42 56 72 75 90
The Wave Resistance of Ships John V. Wehausen I. Introduction 11. The Measurement of Wave Resistance 111. The Analytical Theory of Wave Resistance Bibliography References V
93 96 131 229 230
Contents
vi
Dynamics of Quasigeostrophic Flows and Instability
Theory El. L . Kuo I. Introduction 11. Tendency Toward Geostrophic Balance in Rotating Fluids 111. Simplified Hydrodynamic Equations for Large Scale Quasigeostrophic Flow IV. Permanent-Wave Solutions of the Nonlinear Potential Vorticity Equation in Spherical Coordinates V. Stability of Zonal Currents for Small Amplitude Quasigeostrophic Disturbances VI. General Stability Theory-Integral Relations and Necessary Conditions for Instability VII. Stability Characteristics of Barotropic Zonal Currents and Rossby Parameter VIII. Pure Baroclinic Disturbances IX. Finite Amplitude Unstable Disturbances X. Instability Theory of Frontal Waves XI. Concluding Remarks References
AUTHORINDEX SUBJECT INDEX
248 250 257 265 272 276 281 291 306 316 327 328
331 336
List of Contributors
Numbers in parentheses indicate the pages on which the authors’ contributions begin.
H. L. Kuo, Department of Geophysical Sciences, The University of Chicago, Chicago, Illinois (247) GEORGE VERONIS, Department of Geology and Geophysics, Yale University, New Haven, Connecticut (1)
V. WEHAUSEN, Department of Naval Architecture, University of California, Berkeley, California (93)
JOHN
vi i
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Preface
I n this volume Professor H. L. Kuo treats instabilities of the atmosphere, Professor George Veronis discusses the dynamics of the ocean, and Professor John Wehausen reviews wave resistance of ships. Thus two of the three authoritative articles are studies of our fluid environment, and the third is a testimony of one aspect of man’s successful adaptation to it. This volume should therefore appeal to meteorologists, oceanographers, and naval architects, as well as to fluid dynamicists in general. I n view of recent concerns with the environment and with the relevance of scientific work to human activities, the three articles presented herein are perhaps timely. From another point of view, it can be persuasively argued that scientific work itself is an important element of the quality of life, because it bears upon the human spirit. It is hoped that the excellence and the degree of permanence achieved in these articles will lend support to this now somewhat forgotten point of view.
CHIA-SHUN YIH
ix
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Large Scale Ocean Circulation GEORGE VERONIS Department of Geology and Geophysics Yale University. New Haven. Connecticut
Introduction
......................................................
I . The Equilibrium Figure of a Se1f.Gravitating. Rotating. Homogeneous
Mass of Fluid ................................................... I1 . Transformations of the Equations of Motion of a Fluid .................. I11. The Coriolis Acceleration ........................................... A . Conservation of Potential Vorticity ................................ B. An Angular Momentum Argument for the Neglect of 2 a cos 4 ........ IV . Thermodynamic Simplifications-the Boussinesq Approximation .......... A . The Adiabatic Hydrostatic Field .................................. B. The Boussinesq Equations ....................................... C . Use and Limitations of the Boussinesq Approximation ............... V . Scaling of the Equations ............................................ A . Small Scale Motions-the f-plane ................................. B. Motions of Intermediate Scale-the /%Plane ........................ C . Large Scale Motions ............................................ VI . Geostrophic Flow .................................................. VII . Frictional Dissipation ............................................... A . Ekman Layers .................................................. B . Ekman Layers in the Ocean ...................................... VIII . Modeling of Current Systems ........................................ A. Wind-Driven Ocean Circulation ................................... B . Extensions of Stommel’s Model ................................... C . Inertial Effects .................................................. IX . The Thermohaline Circulation ....................................... A. The Pressure Equation .......................................... B. Boundary Conditions at the Top and Bottom ....................... C . Solutions by Means of a Similarity Transformation .................. D . Generalization of the Similarity Solutions .......................... E . Solutions to the Ideal-Fluid Thermocline ........................... F. The Effect of a Barotropic Mode .................................. G . The Role of Diffusion ........................................... H . Remarks about Thermohaline Circulation Models ...................
1
2 3 6 14 15 17 18 19 21 23 28 30 31 32 33 36 36 40 42 43 49 53 56 57 58 58 60 63 65 68 71
2
George Veronis X. Abyssal Circulation.. ............................................... XI. Laboratory Simulation of Large Scale Circulation (with C. C. Yang) . . . . . . A. The Basis for the Simulation ..................................... B. Discussion of More Complete Solutions.. .......................... C. The Flow Due to a Source of Dense Water.. ....................... References ........................................................
72 75 75 78 81 90
Introduction Our theoretical picture of large scale ocean circulation has grown mostly out of the development of simple models which isolate the particular phenomenon to be analyzed. In this sense dynamical oceanography differs substantially from dynamical meteorology which has progressed hand-in-hand with the amount and types of observational data that have been accumulated. The difficulties and costs of gathering oceanographic data preclude the same type of development of oceanographic theories. The present paper contains a discussion of some of these simple theoretical models together with an attempt to extend a few of them to take into account additional features which are not normally included in the models. The presentation is necessarily selective and another author would no doubt have emphasized other models or other approaches. At the outset the plan was to discuss steady state models as well as those which include transient behavior. However, as the work progressed it became necessary to restrict attention to steady models only. The opening section includes a simple model for deriving the ellipticity of the earth. I t is followed by the derivation of the equations of fluid motion in elliptical coordinates and the approximation involved in the use of a spherical coordinate system to analyze oceanographic motions. The latter is included because the errors associated with the use of spherical coordinates are normally referred to in a casual fashion and no real attempt is made to quantify them. The approximations encountered in theoretical studies of large scale flows are then discussed and the stage is set for introducing theoretical modeling. Simple geostrophic flows and their significance are presented next because they form the basis for the remainder of the paper. The study of Ekman layers and the role that they play in large scale circulation are followed by a brief discussion of turbulent transport.
Large Scale Ocean Circulation
3
T he remaining portion of the article is devoted to a discussion of three types of studies. Th e first is wind-driven ocean circulation beginning with Stommel’s (1948) model exhibiting westward intensification of ocean circulation. Some of the difficulties in extending his model are pointed out in the subsequent development which includes an attempt to incorporate density variations into the model. Theories of the thermohaline circulation (the flows driven by sources and sinks of heat and salt) are then reviewed. T h e paper ends with a section, authored jointly with C. C. Yang, on laboratory modeling of ocean circulation. T he presentation concentrates on the evolution of theoretical oceanography by means of simple models. Because this philosophy was adopted here, the important contributions of numerical studies of ocean circulation have been referred to only in passing. It is obvious that future work in the field will probably rely more and more heavily on numerical computations, particularly as the complex phenomena associated with nonlinear interactions become the focus. Th e reason for restricting attention here to simple analytical and laboratory models is that the latter still provide the theoretician with the clearest understanding of the state of the art and what steps one must take to extend that understanding.
I. The Equilibrium Figure of a Self-Gravitating, Rotating, Homogeneous Mass of Fluid Consider a mass of incompressible, homogeneous fluid which is rotating with a constant angular velocity, S2, about a given axis. T h e velocity, v, of a fluid element at distance, R,from the axis of rotation is
and the acceleration is
and is directed inward toward the axis of rotation. At a point within the fluid the equation for the conservation of momentum is
where p is the (constant) density, p is the pressure, and F is the body force
George Veronis
4
per unit mass. Taking F = -V @ where 0 is the Newtonian gravitational potential and making use of (1.2)in (1.3) yields S2 X (S2
x R)= -V(Q2R2)/2 = -V(p/p) - V@
or
+
V[(PIP)
@ - (Q2R2/2)] = 0.
(1.4)
The first integral of (1.4) yields
( p / p )+ 0 - (Q2R2/2)= const.
(1.5)
For a finite mass such as the earth the pressure at the outer (equilibrium) surface must be a constant and (1.5) becomes
CD - (Q2R2/2)= const.
(1.6)
on the outer surface. Since the gravitational potential, @, depends on the shape of the body, the problem is an implicit one. The earth is very nearly spherical and it is convenient to express the radius as the sum of the radius, a, of the smallest inscribable sphere plus the deviation, 5, due to the bulge which results from the rotation (Fig. 1) r=a+c.
(1.7)
Then the gravitational potential, @, on the surface can be written as the sum of the potential due to the mass within the inscribed sphere and the potential due to the mass in the bulge. The potential at an external point due to the spherical mass can be expressed as the potential due to a point with all of the mass at the center. Thus, @=-
where 4 is the potential associated with the bulge and G = 6.67 x dyn cm2 gm-2 is the gravitational constant. Then (1.6) becomes 1 2
- Q2R2= const.
or, making use of (1.7),
+---4
mpa3
1
G - - Q2(a+ 5)" sin2 8 = const. 3a+5 2
where 6 is the colatitude (Fig. 1).
Large Scale Ocean Circulation
5
A first approximation to the figure of the earth is to neglect 4 in (1.9) and make use of the fact that 5 < a. Since the term involving the rotation in (1.9) is small, the lowest order expression involving 5 is $rrpa2G(5/a) - +Q2a2sin2 % = const.
(1.10)
FIG.1. A sketch of a cross section (through the polar axis) of an elliptical earth with an inner inscribed sphere of radius a. The bulge due to the rotation is denoted by 5.
The difference between ( / a evaluated at the equator (0 = ~ 1 2 )and the pole (0 = 0) yields the ellipticity (C/a)eq.-
( 5 / a ) p o l e = +Q2/9,~G
= iCPa/g M
1/580,
(1.11)
where g = 4rrGpa/3 = 980 cm sec-2 is the gravitational acceleration at the earth’s surface. The value of the ellipticity given by (1.11) is about half the known value. Higher order approximations in ( / a cannot alter the result substantially and the only correction that may yield an improved estimate must come from 4. A second approximation may be obtained by retaining 4 in (1.9) while still retaining the approximation 5 < a. Then (1.9) becomes
4 + Q rrpaG[ - ;Q2a2 sin2 % = const.
(1.12)
Now the term involving 5 incorporates the geometrical effect of the bulge. On the surface the potential due to the mass of material in the inscribed sphere will be smaller where 5 > 0 and larger where M < 0. However, it is more convenient to choose an alternative manner of accounting for this geometrical effect. The potential is evaluated on the spherical surface Y = a and the geometrical effect is taken into account by assuming a surface distribution of mass at Y = a with negative mass where 5 > 0 and positive mass where 5 < 0. The surface distribution of mass at Y = a due to the bulge associated with 5 can be written in terms of a jump condition at Y = a in the radial gradient of the potential (Ramsey, 1964) as
(1.13)
6
George Veronis
where a + and a - correspond to the value of a approached, respectively, from outside and from inside the spherical boundary. Substituting (1.13) into (1.12) yields
$
+ Qa[a4/ar]:zg? = @Pa2
sin2 0
+ const.
(1.14)
The potential due to the spherical mass is included in the constant term and will not contribute to the ellipticity. If the constant term is neglected, the problem reduces to one in which there is no mass except for the surface distribution on r = a so that Laplace's equation is satisfied inside and outside the surface, i.e.
V+=O
for r < a
and r > a .
(1.15)
Equation (1.15) can be solved by separation of variables, and the pertinent solution for present purposes is
+
= K(r/a)2(1 - 3
cos2 0) = K ( ~ / Y1)~ (3 C O S ~0)
Y
< a,
r > a.
(1.16)
The application of condition (1.14) yields
K = -R2a2/4,
(1.17)
and (1.13) then gives
[a$/a~];rgT = - 5Q2a(l - 3 cos2 0)/4=
--47TGp5.
(1.18)
Hence,
C/U = 5Q2(1- 3 COS'
0)/1&Gp = 5Q2a(1- 3 cos2 0)/12g.
(1.19)
The ellipticity is therefore evaluated as
(1.20) [(/a]& = 5Q2a/4gM 1/232, i.e. the gravitational attraction of the material in the bulge serves to increase the ellipticity by a factor of 2.5. The result given in (1.20) was derived by Newton by another method. Much more exhaustive treatments of the problem have been presented by various authors (e.g., Jeffreys, 1962). The most accurate estimate of the ellipticity (1/298) is based on measurements of orbits of artificial satellites.
II. Transformations of the Equations of Motion of a Fluid T o a very good approximation the shape of the earth can be taken to be an oblate spheroid (sometimes called a planetary ellipsoid) with the minor axis of the ellipsoid along the axis of rotation. The equations of motion
7
Large Scale Ocean Circulation
can be written in terms of oblate spheroidal coordinates with gravity taken as a constant on the surface of the ellipsoid. However, oblate, spheroidal coordinates are never used for the analysis of oceanic or atmospheric currents. T he usual procedure is to work with the equations on a sphere. We shall go through the derivation of the equations in oblate spheroidal coordinates and then show the approximations involved in the use of spherical coordinates. T h e vector form of the equations for the conservation of momentum is aV
at
+V
1 vv f 2 a x v = - - vp - v@, P
(2.1)
where the centripetal acceleration terms are incorporated into the gravitational potential 0.Th e conservation of mass can be written as
5 + v . v p f p v. v = at
0.
T o write the equations in terms of any curvilinear coordinate system we introduce the generalized coordinates (ql, q 2 , q3)where the qi can be related to the rectangular Cartesian coordinates by the relations qi
= qi(X1,
~2
9
~
3
)
or
Xi
= xi(q1,
q2
43).
(2.3)
Then (Margenau and Murphy, 1949) the derivative along any direction, si , can be written as
where the Q i are defined by 3
Qi2=
C
j=l
ax (2)
In general, the following relations will be used 3
v= xvjij,
(2.6a)
j=1
V=
C
l a ij--,
]=I
Q j a q j
(2.6b) (2 .6 ~ )
8
George Veronis
where ii are the unit vectors along the three coordinate directions and the last relation follows from the first two. From ( 2 . 6 ~ )we have
where the last term in (2.8) is necessary because the unit vectors can change direction. Now the surface of the earth is, to a good approximation, an oblate spheroid for which the orthogonal surfaces are (Fig. 2) (1) oblate spheroids,
FIG.2. Oblate spheroidalcoordinates with orthogonal surfaces given by oblate spheroids with q1 = constant, hyperboloids of one sheet with q2 = constant, and planes (one is the plane of the figure) through the xp axis with q 3 = constant. Also shown are the polar coordinates in the equatorial plane.
q1 =const., (2) hyperboloids of one sheet, q2=const., and (3) planes through the x3 axis, q3 = const., where the x3 axis is taken as the minor axis of the ellipse and the xl, x2 plane is orthogonal to the x3 axis. I n the following the coordinates (x,y , z ) will replace (xl, x 2 , x3). In addition, it is useful to define the polar coordinates R, X in the xy plane. Note that A-q3. T h e intersection of the surface of the earth and a plane through the x axis is the ellipse given by
where the values of a and q1 on the surface of the earth are determined by noting that at z = 0, re = R = a cosh ql, where re is the earth’s equatorial
9
Large Scale Ocean Circulation
radius, and at R = 0, rp = z = a sinh ql, where rp is the earth’s polar radius. The curves orthogonal to the ellipses are the hyperbolas R2 a2 sin2 q2
-
2 2
a2 cos2 q2
= 1.
(2.10)
The relations between the coordinates of the different systems are x = a cosh q1 sin q2 cos q 3 ,
y = a cosh q1 sin q2 sin q 3 , z = a sinh q1 cos q 2 ,
(2.11)
R = a cosh q1 sin q2 , y/x = tan X = tan q 3 , x 2 + y 2 = R2.
(2.12)
The values of the Q, in (2.5) take the form
Q1 = Q2= a(sinh2 q1 + cos2 q2)1/2, Q3 = a cosh q1 sin q2 .
(2.13)
Furthermore, the derivatives of the unit vectors can be calculated.
.
ai, - a2 cos q2 sin q2 -_ 12, 841
ail
- tc2 sinh
-392
Qi2
-ail _ - a sinh q1 sin q2 1.3 ,
Qi2
ai,
8 1
ai, -Go,
-ai,_ -0,
-ai,_ -
841
842
a43
.,
12
-a91 _-
393
q1 - cosh q1
M
u2 cos
q2 sin q2 .
(2.14)
11,
Qi2
sinh q1 sin q2 Qi
.
u
11 -
cosh q1 cos q2 . 12 * Qi
The equations of motion expressed in terms of oblate spheroidal coordinates become dv, v1v2 - _ _ _ _ a cos q2 sin q2 -
dt
Qi
VZ2
81
u2 sinh
2Qa . 1 ap 1 a@ sinh q1 sin q2 v3 = - --- --, Qi PQi Qi 841
-__
v z
q1 cosh q1 - 3a sinh q1 sin q2
81Q3
(2.15)
George Veronis
10
v32
91--
a cosh q1 cos q2
Q1Q3
(2.16)
and the conservation of mass equation is
(2.18) where
d dt
- -= _
a +I-3 v i a
at
j = l
(2.19)
Q j aqj
The gravity potential @ appears only in Eq. (2.15) since we have assumed that the equipotential surface is an ellipsoid. Next suppose that the equipotential surface of the earth is taken to be a spherical surface. For spherical coordinates we have (91, q z , 43) = (r, 8, A),
Qi =
sin 8,
(2.20)
z = r sin 8.
(2.21)
1, Q z = r, Q3
=r
and x = r cos 8 cos A,
y = Y cos 8 sin A,
The equations of motion take the form
11
Large Scale Ocean Circulation
where d/dt is defined by (2.19) and the velocity components (v,., v a , vA) are along the directions (r, 8, A). Now, what error is involved in replacing the set (2.15) to (2.18) by the set (2.22) to (2.25)? We answer this question by relating: (a) the velocities (vI, v a , vA) in spherical coordinates to (vl, c u 2 , v 3 ) in oblate spheroidal coordinates; (b) the total time derivatives in the two systems; (c) the respective pressure gradients in the two systems; and (d) the velocity divergences in the two systems. It has already been assumed that @ is constant on a particular q1 surface in the oblate spheroidal system and on the spherical surface (r = a, say) in the spherical system. Furthermore, we make the following identification i.e. the radius, u, of our spherical earth is the mean of the polar and equatorial radii for the ellipsoid. With the foregoing relations the only difference between the two systems lies in the different metric terms associated with the variations of the unit vectors. From Eq. (2.11) we derive r = (x2 + y 2
+ z2)112= a(cosh2 q1 sin2q2 + sinh2 q1 cos2 q 2 ) 1 / 2 = a(sinh2 ql
+ sin2 q2)1/2
= a(Cosh2 q1 - cos2 q2)1/2,
(2.27)
and from (2.12) and (2.27) sin 6' = R/r = cosh q1 sin q2/(sinh2q1
+ sin2q,)lI2.
(2.28)
Equation (2.28) can be solved for sin q2 in terms of sin 8 sin q2 = sin 6'
(2.29)
1 - sin2 8/cosh2 q1
Now Eq. (2.9) for the ellipse may be rewritten as
R2 a2 cosh2 q1
+ 2 cosh2 ql[lz2-l/cosh2
-
ql] -
(2.30)
But the general expression for an ellipse with ellipticity e is written as 2 2 R2 = 1, b2 + b2(1 - e)2
(2.31)
where b is the major radius. Thus from (2.30) and (2.31) we can make the following identification between cosh q1 and the ellipticity l/cosh2 q1 = 2e - e2.
(2.32)
12
George Veronis
At the earth's surface the ellipticity is 1/298 so that a good approximation to (2.32) is 1/cosh2 q1 w 2e. (2.33) Making use of (2.32) yields (2.29) in the form sin q2 = sin 8[l - e cos2 O
+ - -1,
(2.34)
where terms of order e2 are omitted. In a similar fashion we may write cos q2 = cos 8[l
+ e sin2 8 + .
sinh q1 = -(1 (2e)u2
5
(2et'!2+):
cash q1 = -(1
.I,
.)
+ .. ' ,
(2.35)
Q1 = Q2= u cosh ql[l - e sin2 O + * * .I, Q3= u cosh q1 sin 8[l - e cos2 0 + . -1. Straightforward substitution of the foregoing into the metric terms of Eqs. (2.15) to (2.17) yields, to first order in e, a2 cos
q2 sin q2 - e cos 8 sin 8 +... -
a
Q13
a2sinh q1 cosh Q13
q1 - 1 - $e cos 28 + a
u sinh q1 sin q2 - 1 - e(1 -
a
+ e(1 - 4 cos 28) +
81 u cosh q1 cos
=sin O[l - e(1 - cos 20)
q2
= cos
Qi
Qi
*
8[l
+
.], (2.36)
*
+ e(1 + cos 28) + - - .],
2 - e(1- 2 cos 28) + . . . , a
Qi
q1 - u2 sin qz cos q2 Q13
QlQ3
-
cot 0[l
. -1
a
u sinh q1 sin q2
u cos q2 cosh
9
*.
QlQ3
+
--
+ + cos 28) + -
Q1Q3
u cosh q1 cos q2 - cot 8[1
3
+ e(l a
cos 26)
+ . . -1- _e _ sin 28 _ 2a
+..a.
13
Large Scale Ocean Circulation Hence, (2.15) to (2.18) become dt
1 - 2 e cos 20) -
a
$ [I
- e(1
-y)] 5 1 ap
dv2 -+ v1v2 (I - e cos 28) dt a 2 ~
- 252
cos 8 v3[l
e sin 20
-
1
a@
(2.37)
e vI2 e sin 28 a [ l + e(1 - cos 28)] + 2a v32
1 aP , + e( 1 - cos 28)] = - -
(2.38)
PQZ 842
+ 252 cos 8 v2[1 + e(1 - cos 28)] + 252 sin 8 v,[l
- e(l
+ cos 28)] (2.39)
3 1 avj dP -dt+ P I Cj = 1 Q j a q j
--+
~
a
e
2-41 - 2 ~ 0 ~ 2 8 ) Vl a
[l + e (1-- co;2e)1
v2-e-
2a
(2.40)
The set (2.37) to (2.40) differs from the set (2.22) to (2.25) only by terms of order e, once the identifications between velocity components, gradients, divergences, and total rates of change have been made. I t is instructive to observe that the largest discrepancies are 3e/2 so that neglecting the metric terms arising from the elliptical correction involves an error of 1/200. This error is substantially larger than the error made in oceanographic studies when the spherical radius r is replaced by its mean value, r = a, in the spherical equations of motion. Furthermore, it is worth noting that the metric terms proportional to sin28 in Eqs. (2.37), (2.38), and (2.40) have no counterpart in the spherical system. Since the first two are associated with the radial velocity, it is difficult to visualize a physical situation when they could contribute materially. The last term in Eq. (2.40) would simply add a small contribution to the preceding term. And finally, one should note that differentiated forms of the equations, such as the vorticity equation or the divergence equation, will involve larger errors when the elliptical metric terms are neglected.
14
George Veronis
I n oceanographic studies the equations are more often written in terms of spherical coordinates with latitude instead of colatitude and with the mean radius, a, replacing r in the coefficients. These equations take the form du dt
uw
uvtan+
a
a
-+---
+2Q cos
+ w - 2Q sin q5 v = - pa cos + ax’ ~
wv u2tan+ -+a++2!2 sin + u =
dv dt
a
-dw _u 2 + v~2 dt
a
--+--+ 1
1 dp pdt
- cos 2!2
au
acosq581
+ u=
1 av a
a+
a
--1 aP Pa a+ ’
1 aP
-- --g, P
(2.42) (2.43)
at-
t an
(2.41)
aw 2w + ++= 0, ar a
(2.44)
where
d --_a _
dt-at
u a +--+--+w-
acosq5 ah
v a
a
a+
a7
(2.45)
and gravitational acceleration has been assumed constant. T h e velocities u, v, w are now in the directions of increasing h, r.
+,
111. The Coriolis Acceleration In almost all studies of oceanographic phenomena the horizontal component of the Coriolis acceleration can be neglected. A crude but simple way of showing that this assumption is justified for large scale flows is to make use of the fact that the hydrostatic equation is known to be a very good approximation in this case. Then (2.43) becomes
+
and the term 2Q cos u is neglected. Now the kinetic energy equation of the system does not contain a term involving the rotation Q because v (2S2 x v) vanishes. Hence, for consistency, if 2Q cos u is neglected in (2.43), it is necessary to neglect 2!2 cos w in (2.41) since the latter term would otherwise contribute to the kinetic energy. Two additional justifications of the neglect of 2Q cos in the equations are given in this section. First the conservation of potential vorticity is
+
+
+
15
Large Scale Ocean Circulation
derived and from scale analysis the neglect of 2 Q c o s 4 is shown to be justified except in the immediate vicinity of the equator. Then the same conclusion is drawn from an angular momentum argument proposed by Phillips (1966). I t is important to observe that the neglect of the horizontal component of the Coriolis acceleration means that the direction determined by gravity, i.e., the local vertical direction, is singled out as a dominant direction in large scale flows. If seawater were homogeneous, the direction parallel to the axis of rotation would be dominant. Hence, the neglect of 2Q cos 4 automatically includes the strong constraint of stratification. In homogeneous models of ocean circulation only the vertical component of the Coriolis acceleration is kept. Therefore, the effect of stratification is included even though it may never appear in the actual model. This point will be mentioned again later.
A. CONSERVATION OF POTENTIAL VORTICITY I n large scale oceanic flows the vorticity of the fluid is an important variable because of the inherent vorticity which each fluid particle has as a result of the earth’s rotation. A quantity, called potential vorticity, is conserved in the absence of dissipation. Its derivation follows. Consider the equation of motion in vector form av -+(VXV)XV+V at
1 +252~~=--Vp-V@,
(3.1)
P
9 +pv dt
v = 0,
and suppose that some state variable, s, is conserved so that dv dt
-=-
as
at
+v.
vs=o,
(3.3)
where s = s(p, p). T he curl of (3.1) yields do
_- o.vv+wv.v=-v dt
where 0
= 2s2
+ v x v.
(3 xvp, -
16
George Veronis
Eliminating V . v from (3.2) and (3.4) yields
T h e gradient of (3.3) yields d
- vs f V v . vs= 0. dt
(3.7)
Then the scalar product of (3.6) with Vs plus the scalar product o f o / p with (3.7) leads to the expression
f
(Y) =O,
where use has been made of the fact that s is a state variable so that Vs- Vp x V p vanishes. T h e quantity, Vs-w/p, which is conserved along a trajectory, is the potential vorticity. Th e derivation given here is that of Ertel (1942). T h e principal use of conservation of potential vorticity for present purposes is to note that in spherical coordinates (Vs. w)/p takes the form
and that the component of the Coriolis term parallel to the earth's surface, 2!2 cos 4, is negligible compared to the normal component, 2!2 sin 4, if
(3.10) Denoting characteristic vertical and horizontal scales of variation of s by D and L , respectively, we can rewrite (3.10) in terms of D and L as
DIL 4 tan 4.
(3.11)
For large scale oceanic motions (i.e. when potential vorticity is a useful variable) the ratio D / L is not larger than Hence, the horizontal component of the earth's rotation is negligible at latitudes beyond a fraction of a degree from the equator.
17
Large Scale Ocean Circulation
B. AN ANGULAR MOMENTUM ARGUMENT FOR
THE
NEGLECT
OF 2 Q C O S 4
A more general argument for the neglect of the horizontal component of the Coriolis term has been proposed by Phillips (1966), who writes the set (2.41)-(2.43)before the “ shallow ” approximation is invoked as
du dt
-= F ,
+ (2Q +A)(. sin + - w cos +) r cos +
dv - = F B - (2Q+ 5 ) u sin dt Y cos 4 dw dt
2Q +-)u Y
U
cos
+
+
(3.12)
wv
-
-
(3.13)
Y
cos
+ +-
V2
(3.14)
Y
where
dh ~=h,-, dt h, = Y cos 4,
d4 ~=h,;Zi, h,
=I ,
dr ~ = h , dt ’ h, = 1 ,
and F,, F , , F, include pressure gradients and frictional forces per unit mass. These equations satisfy the angular momentum balance d -dM __ -
dt
dt
[Y
cos
+ (u + Qr cos +)]
=Y
cos+F,
.
(3.15)
If the “ shallow ” approximation is made in the form h, = a cos +,
h, = a
(3.16)
then (3.12) to (3.14) are written with a replacing Y, i.e., the set (2.41) to (2.43) is obtained. However, as Phillips points out, the set (2.41) to (2.43) does not have the angular momentum balance which is obtained by replacing r in (3.15) by a. In fact, there is no function A(+,Y) such that A F , = dM/dt, in general. If the terms 2Q cos w + (uw/a) were absent from (2.41), the equation would give the proper (approximate) angular momentum balance. If the momentum equations are recast into the vector invariant form
+
av _ F +g - V ( i at -
v2)
+ v x curl(v +Rr
cos
+ i,)
(3.17)
18
George Veronis
and if the relations (3.16) are now used in all curvilinear operators, the final set of component equations is du
-dt= F A
+ (22 +La cos) 4 v sin 4
(3.18)
--
) u sin 4
(3.19)
dt
dw
-=
dt
F, -g .
(3.20)
Hence, the horizontal component of the Coriolis parameter as well as the metric terms which gave rise to the angular momentum difficulty are now absent. Consistent with this approximation,the term 2w/a in (2.44) should be neglected. Although Phillips’ argument is correct, there are obvious cases where the final set (3.18) to (3.20) is inapplicable. Even with the “shallow” approximation, flow between two concentric spheres in a laboratory cannot be analyzed by this set of equations. In other words, some justification for the use of the simplified set of equations must be provided by the particular physical system under consideration. For large scale oceanic flows, the set (3.18) to (3.20) is consistent and represents a good approximation.
IV. Thermodynamic Simplifications-The Boussinesq Approximation The equations of motion inherently contain descriptions of a wide variety of processes ranging from high frequency, small scale phenomena such as sound waves, to low frequency, large scale motions which describe the general circulation. For the study of a particular class of phenomena it is very helpful to “filter ” the equations so that a simplified set of equation is available. For example, for investigations of large scale flows it is common practice to consider seawater as essentially incompressible. The ensuing simplification is an enormous help for it reduces the equations to a much more tractable form. In the present section the analysis for deriving this so-called Boussinesq set of equations is given and some of the limitations of the simplified equations are pointed out. The starting point is a brief discussion of the thermodynamic properties of seawater. Seawater contains many dissolved salts (Sverdrup et al., 1942) the most abundant of which is sodium chloride. Because these salts appear in ap-
Large Scale Ocean Circulation
19
proximately the same relative concentrations in seawater,* for dynamical purposes it suffices to lump them together and to define the salinity, s, as a composite measure of the concentrations of the salts. T h e density of seawater is, therefore, a function of salinity as well as of temperature and pressure. p = p(s,
T , PI.
Empirical formulas for determining p in terms of s, T , and p are summarized by Sverdrup et al. (1942) and by Fofonoff (1962a). Surface water in regions of low salinity and high temperature has a density as low as 1.02 gm/cm3. Water in the deepest trenches is subjected to the highest pressures and has a density as high as 1.07 gm/cm3. Hence, the maximum range of density is only about 5% so that where p appears as a coefficient it can be replaced by its mean value, pm, and a maximum error of only 2.5% is incurred. However, when variations in density are important as driving terms for the motion field, replacing p by pm means that the dynamic effects of density variations are ignored. T h e Boussinesq approximation, which is developed below, allows one to ignore the density variation when it contributes only a small quantitative correction and to keep the variation where it is dynamically significant.
A. THEADIABATIC HYDROSTATIC FIELD I n the absence of motion the conservation of momentum reduces to the hydrostatic equation
vp = -gp. Hence, surfaces of constant density and pressure are level (horizontal) and the only admissible variations in p and p are in the direction of gravity,t z. To complete the description of the motionless state it is necessary to specify the equation of state, the thermodynamic process which controls the state variables, and known values of the state variables at some level. Even in this case, however, there will be an infinite number of systems which can satisfy these constraints because seawater is a multicomponent fluid (salt and temperature can vary). I n order to specify a unique static state the salinity is here taken to be constant, with the value s = 34.85%. Hence,
* Carritt and Carpenter (1958) discuss the variability of relative concentrations. t If diffusion of momentum and heat are included, there can be no relatively static state for a rotating system because the equations describing eyuipotential surfaces are, in general, not solutions to the steady diffusion equations. The slow flow which results is called Sweet-Eddington flow. The implicit assumption here is that it is negligible compared to the motions generated by driving forces of the system.
20
George Veronis
variations in s will occur only via interactions with the bounding media (the atmosphere, the bottom etc.) and will normally be associated with motions of the fluid. With s a known constant the density is a function of T and p only. Now, the first law of thermodynamics is
6 g = T dq = C , dT
+ T(aq/ap),dp,
(4.3)
where q is specific entropy, c, is specific heat at constant pressure and Sq is the specific quantity of heat added. For an adiabatic process S q = 0 and it follows from (4.3) that the vertical variations of the state variables are related by
where the hydrostatic equation has been used to give the right-hand term. T h e thermodynamic relation (%IaP)T = - ( a v / a T ) ~ 7
(4.5)
where v is specific volume, can be used to write (4.4)in the form
( a T / a ~= ) , -g a TIC, (4.6) where o! = I/v(av/aT), is the coefficient of thermal expansion. Values of o! and c, as functions of s, T , and p are tabulated by Fofonoff (1962a) who also gives an empirical formula for (aT/ap),as determined by Fofonoff and Froese (1958). For present purposes it suffices to note that (aT/az),for s = 34.85 % has values ranging from O.O16"C/1000m at the surface with T = -2°C to 0.209"C/1000m at 10,000 m depth with T = 4°C. A typical range for the adiabatic temperature is 0.6"Cfrom the surface to 4000 m depth. Hence, the adiabatic temperature gradient is small relative to observed vertical temperature gradients in the upper layers of the ocean but the two are comparable in deep water. Formulas for the calculation of density as a function of s, T , and p are summarized by Fofonoff (1962a). Hence, from the hydrostatic relation (4.2), the adiabatic temperature gradient (4.6), and the formula for p as a function of T and p (s = 34.85 yo),one can calculate the adiabatic density field given p and T at one level. As stated earlier, the observed range of values of p is from 1.02 gm/cm3 at the surface to 1.07 gm/cm3 at a depth of 10,000 m. T h e range of adiabatic density variations is therefore comparable to the observed range. Hence, in contrast to temperature variations the density variations do not differ much from the adiabatic density variations, i.e. the principal contribution to changes in density comes from the pressure field. T h e adiabatic, hydrostatic state will be denoted by variables with subscript a. 9
21
Large Scale Ocean Circulation
B. THEBOUSSINESQ EQUATIONS T he conservation of momentum for a fluid is now expressed as
dv dt
p-+2pQ
x
v = -0j-gj5,
(4.7)
where the pressure and density have each been divided into an adiabatic part and a perturbed (tilde) part, and p = p a ri; and p = p a +fi. T h e adiabatic, hydrostatic field has been subtracted from the right-hand side. On the left-hand side the density appears as a coefficient. Using the fact that the total variation of density is small relative to the mean value enables one to approximate (4.7) by
+
dv -++a )( v = - -1 v p - g - ,P (4.8) dt Pm Pm where pm rn 1.035 is a mean value of the density for the ocean. T h e formal requirement for (4.8) to be valid is
< 1.
(4.9) T h e equation of state is linearized about the adiabatic state, as discussed below. Thus, (IP-Pm(/Pm)=6
P = Pa
+
(g)*,s T + ($) b + ($) T. s
P.T
5
+-
*
..
(4.10)
Hence, the perturbation density can be written as P/pa M P/Pm = - aT
+K$ +Y?,
(4.11)
where
If (4.11) is substituted into the right-hand side of (4.Q the vertical component of (4.8) has the terms
(4.13) on the right-hand side. For oceanic motions the vertical scale of variation,
H , is lo5 cm so that the first term has magnitude 1) I / H . T h e isothermal compressibility, K , has a magnitude of 4.5 x 10-l1 in cgs units (Sverdrup et al., 1942) so that the t e r m g K j has magnitude 1fi1 / H swhere H , = l/gK is the scale height for seawater. Its value is 2 x lo7cm, i.e., substantially larger
22
George Veronis
than the vertical scale H , so that the pressurefluctuation term in the equation of state is dynamically insigniJcant. This means that the equation of state (4.11) can be approximated by
+
p"/pm Fz: 03 ys".
(4.14)
T h e equation for the conservation of mass takes the form (4.15)
T h e first and third terms are O(6)compared to the last term so that both can be neglected. T h e second term involves a time scale which must be specified. If velocity and length scales, V and L, respectively, are chosen, and if a/& is taken to be O( V / L )(i.e., local changes are due to convection), the second term is the same order as the third. An alternative procedure is to specify the time scale on the basis of some other physical process, such as buoyancy oscillations. I n either case the second term can be neglected. T h e reason why +/at must be considered separately is that it is the term which is associated with acoustic phenomena and one must specify the physical process in order to filter out acoustic waves. Hence, to lowest order (4.15) takes the form v*v=o. (4.16) T h e equation for the conservation of salinity may be used in its primitive form dfldt = 0. (4.17) T h e first law of thermodynamics is p (deldt)= -pV
v,
(4.18)
where e is specific internal energy. For the hydrostatic, adiabatic state the first law was expressed in the form (4.3) to give the adiabatic temperature gradient (4.6). It is convenient at this point to express the change in internal energy in terms of increments in specific volume, temperature, and salinity for a process at constant pressure as de dt
dF dt
p - = c, --p v
.v +
ds" c, -, dt
(4.19)
where c, is the specific heat per unit change of salinity. Even though the actual value of c, is not available for seawater, the form (4.19) suffices for the purpose of this development. Substituting (4.19) into (4.18) yields dp dt
ds" dt
c -+cc,--0.
(4.20)
Large Scale Ocean Circulation
23
In view of the conservation of salinity (4.17), Eq. (4.20) reduces to
dF/dt = 0.
(4.21)
Since variations in salinity occur only in the perturbed state and since the adiabatic temperature gradient (4.6)has already been evaluated, Eqs. (4.17) and (4.21) can be used for the perturbation salinity and temperature. In summary, the equations take the following form when the Boussinesq approximation is employed. For the adiabatic, hydrostatic state,
S = const.,
aT/az = -gaT/c,, For the perturbed state
V p = -gp, p = p( T , p , S).
dv 1 -+2Q x v = - - v p - g e , dt Pm V . v 0, Ps/pmz= -a8 =z
dsldt = 0,
d8ldt = 0,
(4.22)
(4.23) Pm
+ YS,
dps/dt = 0,
where 8 is the potential temperature, the adiabatic temperature gradient having been accounted for in the last of Eqs. (4.22);s is the deviation of the salinity from some constant value; and is the potential density, expressed as a linear combination of 8 and s.
C. USE AND LIMITATIONS OF THE BOUSSINESQ APPROXIMATION The set of equations (4.22) and (4.23) is sometimes called a Boussinesq set because it was derived with the use of the Boussinesq approximation. In laboratory experiments and other very shallow layers of fluid the Boussinesq set is a very good approximation to the primitive set of equations for certain classes of motions. For shallow layers, molecular processes (conduction, diffusion, etc.) can be added to the system in a straightforward fashion. For oceanic flows of even moderate scale it is necessary to treat mixing processes by some approximate technique (eddy coefficients, mixing length theory, or some analogously crude method). The great advantages of the Boussinesq system are (a) the filtering out of acoustic waves and other phenomena associated directly with compressibility, (b) the linearization of the equation of state for the perturbation variables, and (c) the relatively simple form of the energy equation. Potential density is the dynamically significant part of the density field, the remaining part being assumed to adjust adiabatically when an element is displaced vertically. The widespread use of the Boussinesq approximation
24
George Veronis
in analyses for single-component fluids and the straightforward interpretation of the adiabatic temperature and density for such fluids would lead one to expect an equivalent simplification for the oceanic case. However, there are limitations on the use of the Boussinesq system for large scale ocean circulation problems. T h e problem is immediately apparent upon examination of observed Auid properties in the deep ocean. Table 1 shows typical values of T and S TABLE 1 CALCULATED PROPERTIES OF TWO SAMPLES OF SEAWATER PRESSURE OF 4000 dB WITH THE SAME DENSITY BUT DIFFERENT TEMPERATURES AND SALINITIES
AT A
T("C) ~ ~ O i o o )
u(gm/liter) &"C) u@(gm/liter)
Sample 1
Sample 2
2.30 34.90 45.93 1.95 27.92
2.09 34.85 45.93 1.74 27.89
and calculated properties for two elements of seawater at 4400 m depth. The two elements have a common value of density" of 1.04593 but different temperatures and salinities. Observe that the two elements have different valuest of a, but the same value of a. Since the in situ density determination states that the two elements have the same density, study of the potential density alone would lead one to the incorrect conclusion that sample 1 is heavier than sample 2 and that it would, therefore, seek a lower level according to inviscid gravitational stability theory. Table 2 contains typical values of T and S and calculated properties for two elements of water, also at 4000 m depth, but with slightly different densities and substantially different values of T and S. In this case, sample 2 is actually heavier than sample 1 according to the in situ density determination but sample 1 appears heavier according to a, values. On the basis of
* To avoid unnecessary decimal digits oceanographers customarily use the quantity sigma, defined by u = 100(p - 1) where p is expressed in cgs units. For example, seawater with a density of 1.04593 has a value of u equal to 45.93. t T h e quantities u, 8, and were calculated with the computer subroutines of the Woods Hole Oceanographic Institution. Density (or a) is obtained from Ekman's (1908) empirical formula. Potential temperature (8) is calculated from the polynomial expression by Fofonoff and Froese (1958). Potential density (no)is derived from the Knudsen (1901) formula for u with p = 0 and with the temperature replaced by 8.
25
Large Scale Ocean Circulation
in situ density, inviscid gravitational stability theory predicts that sample 2 should sink and sample 1 should rise (relative motions) whereas the same criterion, based on potential density, indicates a sinking of sample 1 and a rising of sample 2. The difference between the densities of the fluid parcels in this example is small (a difference of 0.01 in ue is close to the limit of reliability) but it is typical of observed differences in abyssal waters. TABLE 2
CALCULATED PROPERTIES OF T w o SAMPLES OF SEAWATER PRESSURE OF 4000 dB WITH DIFFERENT DENSITIES, TEMPERATURES, AND SALINITIES
AT A
WC) S(O/OO)
u(gm/liter) wc) ae(gm/liter)
Sample 1
Sample 2
2.21 34.90 45.95 1.86 27.92
1.89 34.85 45.97 1.55 27.91
Figure 3 shows the distribution of 0 0 for a vertical section in the western Atlantic Ocean as calculated by Lynn and Reid (1968) from observed data. According to the distribution of potential density the bottom kilometer or so of water is unstably stratified over a large range of latitudes south of the equator. I n situ stability calculations show that the water is, in fact, gravitationally stable. One must conclude, therefore, that the use of potential density for determining the gravitational stability of water in the deep ocean can lead to incorrect results. This apparently paradoxical instability of abyssal water can be explained as follows. In Fig. 4 isopycnals for seawater are shown for depths of 0, 2000, and 4000m. The curves slope more steeply upward to the right as the depth (or pressure) is decreased, i.e. the entire pattern of isopycnals rotates counterclockwise as p is decreased. Hence, a layer of fluid which is neutrally stable or even slightly stable near 4000 m depth appears to be unstable if the reference surface is the top surface as it is for potential density. It should be observed that the apparent instability is present even when the equation of state is highly nonlinear (the complete expression for u at each level was used to generate the curves shown in Fig. 4).T h e difficulty can be avoided only by allowing the various coefficients, whether they be the coefficients in the complete expression for (T or in the linearized Boussinesq form, to be functions of pressure. Allowing a and y in (4.14) to be
8p0S
I
60° I
I
40' I
I
2po
I
00 I
I
2p0
I
4P0
I
I
8p0N
FIG.3. A plot of potential density, ug, as a function of latitude and depth in a longitudinal section west of the Mid-Atlantic Ridge, according to Lynn and Reid (1968). (Courtesy of Deep-sea Research.)
Large Scale Ocean Circulation
27
functions of pressure would make the equations analytically intractable. However, with negligible error, one could approximate the pressure effect by taking u and y to be functions of depth. T h e equation of state would then be more complicated than the form with u and y constant but it would at least be linear.
S FIG.4. Curves of constant density plotted in the potential temperature-salinity plane for depths of 0 (solid curves), 2000 m (dashed curves), and 4000 m (dash-dot curves).
The issue discussed in this section is not important for phenomena such as internal waves where the particle motions are restricted to local regions. However, for large scale circulation problems, where fluid at the surface in polar regions may sink and eventually make its way to the bottom, the use of the Boussinesq approximation in its simplest form would preclude the possibility of direct verification of particle trajectories by comparing theory with observation. For example, water mass analysis shows that the water near the bottom between 30"s and the equator in Fig. 3 originates in the Antarctic Ocean (Weddell Sea) and the tongue of water with ua > 27.9 comes from the Greenland Sea. At the points of origin, water from the Greenland Sea is denser than the water from the Antarctic. This feature is reflected in the values of a,. In the abyss these waters stratify with Antarctic water lying below the water from the Greenland Sea. This inversion is a direct consequence of the effect of pressure on the densities of water of different temperatures and salinities. T h e simple Boussinesq system could not lead to the observed distribution because, according to the Boussinesq system, water which is densest at the point of origin would, in the absence of mixing, end up in the deepest part of the ocean.
28
George Veronis
V. Scaling of the Equations The equations of motion can be scaled to exhibit important balances for large scale flows. By “large scale” we mean flows whose characteristic horizontal scales are substantially larger than the vertical scale (or depth). The scaling procedure makes use of observed (or perhaps only plausible) magnitudes of quantities to reduce the equations to a simpler form. Results from subsequent analyses of these equations can be compared to observations of appropriate phenomena or features in order to obtain an a posteriori check on the validity of the model equations. For this purpose the variables and operators in the Boussinesq equations of motion will be represented as follows :
a
i ----a cos
-=TQ6,, at
u = VU’,
a
i
l a
+ aA - L aa,
v = VV’,
a
w = WW’,
a
1
1 H”,
a+-
-=-
p = (Ap)p’,
p = (AP)p’,
ar
where the prime quantities and the 6 operators are nondimensional and of order unity and the scales T , L, H , V , W, Ap, and AP are to be determined by restricting attention to certain chosen magnitudes which reflect processes of interest. T h e Boussinesq equations on a sphere then take the following nondimensional form : 7St u
T
+ R[u 6, u + v 6, u +(p/s)w 6, u + vpuw - ~ u tan v $1
6,v
+2p cos 4 w - 2 sin 4 v = - PS,p,
(5.1)
+ R[u6,v + z, 6, w + ( ~ / E ) 6,v w + qpvw +7u2 tan 41 + 2 sin 4 u = - P 6,p,
(54
+
+
6 , ~R(u 6 , ~ Z, 6 , +~( P I & ) 6,w - r p 2 - qv2)]- 2 s cos = -P 6,p - (Fs2/R)p,
F ~ [ T
7atp
S,u
+ R[u a a p + v +
+ S,v
6
- qv tan
6
~ ( ~ 1 s S,P]= ) ~
0,
+ + 2rpw + ( p / s ) 6,w = 0,
+u (5.3)
(5.4) (5.5)
where the primes have been dropped and the following nondimensional parameters appear
R = V/QL, p = W/V, 7 = L/a, E =H / L , F =g HQ2 Ap/pm N 2 / Q 2 , P = AP/QVLpm.
(5.6)
T h e parameter R is called the Rossby number and F is a ratio of frequencies.
Large Scale Ocean Circulation
29
Formal expansions of the variables in powers of each of the small parameters in (5.6) would be messy. It is possible to make some simplification by means of two assertions which lead to balances of primary interest. The first assertion is that the horizontal scale is always much larger than the vertical scale so that E < 1. This assertion is taken to be a dominant one in the sense that all conclusions which result from imposing it as a restriction are valid for the flows to be considered. Since the parameter p = W/V is divided by E in several places, it is necessary to make a statement about the amplitude of p. Hence, the second assertion is that the vertical divergence is always upper bounded by individual horizontal divergence so that p 5 E or, equivalently, W < VH/L. Several important consequences follow from these two assertions. First, in the vertical equation of motion the Coriolis term is O ( E )and the remaining acceleration terms are O(2). The bouyancy term is ([Fe2/R]).I n the ocean F 9 1 and R < 1 almost everywhere so that the parameter F@/R may be O(1). Since large scale flow is is known to be essentially hydrostatic, the parameters P and Fe2/Rmust be of the same order. In the following, the parameter P will be set equal to one and F 2 / R must then be of order unity also. An important consequence of this argument is that the lowest order $ow will be hydrostatic and geographic. T h e latter balance follows directly when P = 1 and both T and R are small. Second, the vertical convection term in each of the first four equations is at most of the same magnitude as the horizontal convection terms. Third, the metric terms involving the radial velocity, w, are at most O ( E )and hence can be neglected to lowest order. Fourth, consistent with the hydrostatic relation and the second assertion ( p 5 E ) above, the horizontal component of the Coriolis terms, 2p cos 4 w, in (5.1) can be neglected to lowest order. Therefore, the equations simplify to the following set 7
6,u
+ R(u 6,u + v 6,u + w 6,u
T
6,v
+ R(u 6,v + v 6,v + w 6 , v +qu2 tan 4) -+ 2 sin 4 u = -6,p, 7- 6,p
+ R ( u 6,p
6,u
+ 6,v
-
~ u tan v 4) - 2 sin 4 v = -S,p,
+ v 6,P
- ~v tan
+ w 6,P) = 0.
4 + 6,w = 0,
(5.7) (5.8)
(5.10) (5.11)
where p has been set equal to its upper bound, e, and Q = F 9 / R . Even though Q is order unity it will be retained as a tracer. The last set of equations is the conventional starting point for almost all studies of large scale flows (with frictional and diffusive processes normally introduced by means of parameterizations of small scale phenomena).
30
George Veronis
Further simplification of the set (5.7)-(5.11) requires additional restrictions imposed by the particular phenomenon under investigation. For example, for flows whose horizontal length scale is much smaller than the radius of the earth the condition 7 < 1 leads to a simplification because some of the geometrical distortion terms associated with the spherical geometry can be ignored. Another type of simplification results for relatively weak flows for which the condition R < 1 is valid. For flows which satisfy both of these latter conditions, it is necessary to compare the relative magnitude of R and q in order to derive the appropriate ordering of the equations. Large scale flows in the ocean can be divided into several types, depending on the horizontal scales which are involved. Phillips (1963) has designated two of these by analogy to atmospheric motions as: (a) motions of type 1, in which q 5 R and R < 1, and (b) motions of type 2, in which R < 1 and q 1. T h e specific conditions and simplifications are developed below and a set of equations is presented for three different scales of motion.
-
A. SMALLSCALEMOTIONS-THEPLANE T h e Gulf Stream meanders and the eddies observed by Swallow and Hamon (1960) are examples of type 1 motion. The horizontal length scale, L, of these motions is of the order of 100 km; hence, it is large compared to the depth ( E < 1) but small compared to the scale of the oceanic basins, or equivalently, of the radius of the earth (6 1). Other typical magnitudes for Gulf Stream meanders are
H-105 cm,
Ap/pm-10-3,
V-102 cm sec-'.
(5.12)
Hence, the following magnitudes obtain for the parameters in Eqs. (5.6) ~=10-',
Q-1,
P-1,
R-IO-l,
~-10-',
(5.13)
where AP, and therefore P, is determined by the hydrostatic balance condition. An additional simplification is obtained by making use of the fact that q is small and expanding the trigonometric coefficients about the latitude near which the motion is to be studied. Thus choose a (mid-) latitude, and write 4 = d o 4' = d o ( Y l 4 = d o qy', (5.14)
+
+
where the linear distance, y , has its origin at yields sin 4 and cos 4 about sin 4 = sin do
+
+ = 4,.
Then expanding
+ qy' cos do + . . . , cos 4 = cos do- qy' sin do4 . . . . (5.15)
31
Large Scale Ocean Circulation From the original east-west derivative we can write
1
a
cos40
a
1
-
cos4,
a (5.16)
so that the nondimensional rectangular Cartesian coordinates, defined to replace A. Analogously,
XI,
has been
(5.17) so that x, y , and z form a local rectangular coordinate system. T o lowest order in 7 the equations are those for a fluid in a uniformly rotating system
au + RV
r-
at
T
aP VU-f v = -ax ’
(5.18)
av + RV VV f f u = --,aP -
(5.19)
aPlaz = -Qp,
(5.20)
at
aY
a, + RV -
(5.21)
- + +-
(5.22)
Vp = 0, at au av aw = 0. ax ay az Here, the primes have been dropped T
.
a ax
a +w ii az
v V -siU - f v ay
and f = 2 s i n + , = c o n s t . T h e set (5.18) to (5.22) is sometimes called an f-plane system and is appropriate for the study of smaller scale properties of large scale phenomena but it is inadequate to describe larger scale flows where the sphericity of the earth is important.
B. MOTIONS OF INTERMEDIATE SCALE-THEP-PLANE Theoretical studies of wind-driven ocean circulation of intermediate scale or larger must take into account the nonuniform vertical component of the Coriolis parameter. A set of equations for ocean basins which lie on the equatorial side of 45”latitude can be derived with geometrical considerations. Again choose a latitude 4, as in (5.15) and assume that 7 = L / a 1 so that sin 4,cos 4,and 8, can be approximated as in Eqs. (5.15) and (5.16).
+
George Veronis
32
Now compare the terms 6,v and r]v tan q5 in the continuity equation (5.11) 6,V=-
av
7vtan+=qtanq5,v
"'
sin q50 cos q50
aY '
+-
-1
. (5.23)
T he conditions 7 < 1 and do< 45" require 7 tan q5, < 1 and the metric term can be neglected to lowest order. Similarly the metric terms in the substantial derivatives can be neglected. Th e Coriolis term in (5.7) can be written as 2 sin q5 v = 2 sinq4, v(l qy' cot q5, + -). (5.24)
+
-
Now even though one assumes 7 < 1, the term qy'cot 4, cannot be neglected because q50 < 45" and cot q5, > 1. Indeed, with decreasing q50 the term 7y' cot 4,may dominate. Th e equations in this case reduce to the 2q set (5.18)-(5.22) with the Coriolis parameter written as 2 sin q5, cos d o y -fo 8 y and they are called the 8-plane equations. They are used quite extensively in theoretical oceanography and almost exclusively in studies of wind-driven ocean circulation (Stommel, 1948; Munk, 1950; Fofonoff, 1954; Bryan, 1963). In cases where the 8-plane is used for basins poleward of 45" latitude an obvious error is incurred in neglecting the metric term in the continuity equation while the 8 term is retained.
+
+
C. LARGE SCALE MOTIONS When the motion to be investigated has truly large length scale, e.g., the global circulation of the oceans brought about by differential incident solar radiation between equatorial and polar regions, the equations take on a different form. Here, L = a = 6 x1O8crn, V-1 cmsec-', 7-1, and R If gravity waves or other high frequency phenomena are to be incorporated into the analysis, the time scale is small and the time derivatives must be retained, i.e., T 1. For long period quasi-steady motions, T < 1 and the use of R < 1 simplifies the equations at lowest order to the form 2 sin 4v = (l/cos q5) ap/aA, (5.25a)
-
-
2 sin q5 u = -ap/+,
(5.25b) (5.25~)
dpldt 1 cos q5
= 0,
aua/\+ a42 (v cos q5) ] + -
:
- = 0.
(5.26a) (5.26b)
Large Scale Ocean Circulation
33
When dissipation processes are included, either via parameterization of boundary layer effects or heuristic treatments of internal dissipation processes, Eqs. (5.25) and (5.26) are used for general circulation models of the ocean. Many of the so-called thermohaline circulation studies in which large scale circulation is driven by imposed surface density conditions (Welander, 1959, Robinson and Welander, 1963 ; Needler, 1967; Kozlov, 1966; Veronis, 1969) make use of amodifiedor extended formof (5.25)and
(5.26). Although the three sets of equations discussed above are applicable for the.study of suitably restricted phenomena, it is obvious that the phenomena are not isolated from each other and that some interaction will occur and indeed may be of great importance. Thus, even though some properties of the motions of type 1 may be studied by the equations (5.18) to (5.22) the generation of such motions depends on the overall circulation of the ocean. For example, the Gulf Stream is a necessary part of the overall circulation of the oceans and the dynamical balances which describe this smaller scale feature must be appended to the equations (5.25) and (5.26) via either a boundary layer or some other asymptotic procedure when large scale circulation is studied.
VI. Geostrophic Flow Steady, linear, frictionless flow in a rotating system is geostrophic and hydrostatic. Because of the fundamental nature of geostrophic flow in rotating fluid theory, a simple example of geostrophic balance is given here. This example is essentially that which was first presented by Phillips (1963) but a simple geometrical argument is used. Consider a right cylindrical basin partially filled with water and rotating about its axis of symmetry with constant angular velocity, R. The parboloidal free surface (Fig. 5a) represents a constant pressure surface and can be calculated as a consequence of the hydrostatic relation and the exact balance between the radial pressure force and the centripetal acceleration
( l h ) ap/az= -g ,
(1/p) ap/ar = n2r.
(6.1)
The solution, subject to the condition that p = 0 at z = h, is
where h is the height of the free surface above the bottom of the vessel and h, is the height at Y = 0. If the container were rotating with constant
34
George Veronis
FIG.5 . The free surface height, h, is shown as a function of the radial coordinate, Y, for rotation rates of (a) R, (b) R IARl, and (c) R everywhere except 0 lhnl between the dashed lines.
+
+
+
angular velocity Q AQ (Fig. 5b) the same equations with Q replaced by Q As2 would hold. The free surface would have a steeper slope for AQ >O. Now consider a thought experiment in which the container rotates with angular velocity Q but a portion of the fluid, shown by the region between dashed lines in Fig. 5c, rotates with angular velocity Q AQ. The free surface would have the shape given by (6.2) everywhere except above the special region where S2 must be replaced by AQ in (6.2). Now subtract Eqs. (6.1) from the corresponding equations with Q replaced by Q AQ to obtain the following balance of forces of the second system (the fluid in the special region) relative to the rotating frame of the first system
+
+
+
+
(l/p) 8 A p / & = 2Q AQr +(AQ)%.
(6.3)
Large Scale Ocean Circulation
35
<< Q, (6.3) can be approximated by ( l i p ) api/ar = 2 ~ , where p' E Ap and A!& = v'. Th e relative velocity, v', and the perturbation pressure, p', are said to be in geostrophic balance. T h e flow is parallel If AQ
to the isobars. Large scale flows in the ocean are in quasigeostrophic balance, i.e., the balance is geostrophic only to lowest order and higher order terms are required to form a closed problem. However, geostrophic flow at lowest order introduces such a simplification that otherwise intractable or extremely complicated systems can be reduced to a much mqre tractable form. T he conditions for quasigeostrophic flow are that: (a) the time scale be substantially larger than the rotation time (T << 1 ) ; (b) the flow be relatively weak (R < 1 ) ; (c) the length scale be large so that the hydrostatic relation is satisfied; and (d) the flow be outside frictional boundary layers (to be discussed below). Under the foregoing conditions the equations of lowest order are
fk X v = -Vp v
*
(6.5a)
- QPk,
(6.5b)
(pv)= 0,
+
where k is the unit vector in the vertical direction and f =fo , f,, /?y,or 2 sin 4 depending on whether the scale of the flow is small, intermediate, or large in the sense used in the preceding section. When the Boussinesq approximation is used, p refers to the fluctuation density and the equation for the conservation of mass can be divided into the continuity equation and the incompressibility condition. When the pressure is eliminated by cross-differentiation of the vertical equation and each horizontal equation of motion, the results in spherical coordinates are (6.6a) (6.6b) and the radial (or locally vertical) shear of the density-weighted velocity can be obtained from horizontal variations in the density field. Equations (6.6) are the so-called thermal wind equations and are used extensively for obtaining vertical shears from measured horizontal density variations. In theoretical studies they form the primary balance for flow in the interior of basins. T o determine the flow explicitly it is necessary to specify the arbitrary functions which result when Eqs. (6.6) are integrated vertically.
36
George Veronis
In field studies such a determination may be made by obtaining measured values of u and v at a single depth for each location. For theoretical work it is necessary to incorporate boundary layer processes to evaluate the arbitrary functions. An additional relation which emerges from the geostrophic-hydrostatic system is obtained by eliminating the pressure field from the horizontal equations of motion to obtain
a ax
- (fpu)
a +(fpv cos 4) = 0. a4
Use of the equation for conservation of mass yields
where /3 = 2 cos 4. Hence, the meriodional velocity, v , is proportional to the vertical divergence of the density-weighted vertical velocity. If (6.8) is integrated betwen two levels where pw is known, the meridional mass transport is determined. Finally, the vertical derivative of (6.8) combined with (6.6a) yields
which expresses the vertical curvative of the density-weighted vertical velocity in terms of the change of the density along a latitude circle. This relation, as well as (6.8), is important for deducing properties of the vertical velocity since the latter cannot be measured. T h e geostrophic relations given in this section are good approximations for large scale flows. Those relations which involve differences between horizontal derivatives and subsequent cancelling of terms are less reliable because the individual terms which cancel are large and only relatively small residuals remain. Hence, it is necessary to exercise some care in the ordering procedure of the more complete equations in order to determine the validity of the derived relations.
VII. Frictional Dissipation A. EKMANLAYERS Transfer of momentum between neighboring fluid particles in the ocean takes place by means of turbulent processes. T h e inadequate development of the theory of turbulence has resulted in approximate treat-
37
Large Scale Ocean Circulation
ments of turbulent transfer. It is customary to draw an analogy between molecular and turbulent processes and to assume that the latter are similar to the former, the only difference being that eddy or Austansch coefficients are used instead of molecular coefficients. This procedure is still by far the most common even though it is known that substantial errors, qualitative as well as quantitative, can result. However, in spite of the inadequacies of the detailed theory it is possible to avoid some of the errors by making use of certain integral features of the processes. It is helpful to consider an exact model first because it provides a useful guide for subsequent considerations. Consider a homogeneous, rotating fluid in the region -co 5 x 5 0 which is driven by steady, imposed stresses or velocities at x = 0 and suppose that the boundary values, hence all quantities, are independent of the horizontal coordinates. The exact equations for such a flow can be expressed in nondimensional form as 26 = E
a2u"/az2,
(7.la)
2u" = E
a2qaz2,
(7.lb)
where the velocities for this model are denoted by a tilde, E = v/fiDais called the Ekman number, D is a vertical scale length, and v is the kinematic viscosity. The solution, subject to the conditions u", d -+0 as x + - co and (u", d),= = ( C 0 , do) is
5 - do sin OeC, d = (do cos 5 + Go sin 5)ec,
(7.2a)
C = (Go cos
(7.2b)
where 5 = Z / E " ~ This . solution was first derived by Ekman (1905). The e - l depth (dimensional) is x, = z/v/fi = DE1J2and is called the Ekman layer depth. In the hodograph (u, v)-plane the solution traces out a spiral so that the velocity vector decreases in magnitude and rotates with increasing distance from the boundary. Hence, the effect of surface processes is confined by the rotation to a limited region near the boundary. It is evident from Eq. (7.2) that the nondimensional stress, To(=aV/az), at the boundary is O ( E - 1 / 2 )If . the stress instead of the velocity vector Go is prescribed at x = 0, the solution to (7.1) is
+ 5+ v" = + E 1 / 2 [ ( ~+ o zTOY) sin 5 +
u" = Q E 1 / 2 [ ( ~ 0 z-r0Y) cos
(T~' - T ~ Y )sin
{let,
(7.3a)
cos 5]e(,
(7.3b)
( T ~ Y- T ~ ' )
where T~~ and -ray are O(E-Il2)quantities. If the region of interest is -1 5 z 5 CO, the above solutions are valid provided that x is replaced by -( 1 x). The choice of x = - 1 for the lower boundary is convenient in this case as will be seen shortly.
+
38
George Veronis
There are several points to be emphasized here. First, the Ekman layer solution is an exact solution to the equations of motion. Second, the surface velocity in solution (7.3) is directed at an angle of 45" to the right of the prescribed surface stress for C2 > 0. Third, the net transport of fluid is to the right of the applied surface stress. The latter result can be verified by integrating Eq. (7.3) from z = - 00 to z = 0 but it is simpler and more straightforward to integrate Eqs. (7.1) directly. The result is (7.4a) (7.4b) and it is easily seen that the transport is to the right of the surface stress. Fourth, the transport given by (7.4) is independent of the details of the flow. I n anticipation of the situation for oceanic flow (7.1) may be written in the more general form (7.5a)
a
(7.5b)
2ii= - ( ETY), az
where E may be a function of z (because of V) and T is the vertical stress in the surface layer. Th e result (7.4) follows directly, the only requirement being that cc vanish at great depth. An important consequence of the foregoing considerations emerges when the prescribed horizontal velocities (or stresses) have horizontal variations of a scale much larger than the Ekman layer depth (Charney, 1955a). Then the Ekman layer solutions are locally valid near the surface and a vertical flow out of the Ekman layer results when the continuity equation is integrated from z = - 00 to z = 0 with the boundary condition zZo=Oatz=O. Thus
or t%-m=
[-
i a (&$) 2 ax
-
a
--
aY
(ET,')]= 1k . v 2
X
(ET,),
(7.6)
where Eqs. (7.4) have been used to evaluate the vertical integrals of u and v. T he parameter E is included with T~ within the del operator in anticipation of oceanic considerations.
Large Scale Ocean Circulation
39
An extension of the theory of pure Ekman layers is necessary in order to make the theory useful for oceanic (and laboratory) flows. T h e interior region, i.e. the fluid outside the Ekman layer, is normally in motion. T h e usual procedure for linear theory (Greenspan, 1968) is to write the dependent variables as the sum of an Ekman layer contribution (denoted by a tilde) and an interior contribution (denoted by subscript I) which does not decay at the edge of the Ekman layer. This approach, or an analogous one, is necessary for treating the flow in the vicinity of a rigid boundary where the total velocity must vanish. For example, consider the flow in the vicinity of a bottom boundary, z = - 1, where the total velocity must vanish. As noted earlier the solution (7.2) is valid there with t; = -( 1 z)E- 112 but the boundary values of the Ekman layer components are (7.7a)
+ -=
,uO
-%b
1
zjo = -q,, ,
(7.7b)
where subscript b corresponds to interior velocities evaluated at the bottom. Hence, the solution in the bottom Ekman layer can be written in terms of the interior velocity as
ii = -(uIb cos 5 - q,,sin t;)et,
(7.8a)
5 + GI,, sin <)er,
(7.8b)
I5 = -(GIb
cos
where t; = -(z + l)E-1’2. Now the tangential stresses at the bottom can be expressed as T ~ = aG/azl,=-,and the use of (7.8) yields
E1”T,”
(7.9a)
= u,, - Z.’Ib 1
Ell2 71, Y~
UIb
f
(7.9b)
OIb?
so that the stress at the bottom is expressed directly in terms of the boundary values of interior velocities. Another important modification of pure Ekman layer theory emerges when the boundary condition on the vertical velocity at the upper surface is written as toI +G = 0 at z = 0. Here, it is necessary to consider the Ekman layer contribution to vanish with increasing distance from z = 0 so that Eq. (7.6) takes the form Go= - &k V x
(ET,).
(7.10)
Then the upper boundary condition on wr is wIo = &k V x (ET,) at
x
= 0.
(7.11)
Greenspan and Howard (1963) call this condition on the interior vertical velocity “ Ekman pumping (or suction)”. It should be noted that w -El”.
40
George Veronis
Physically, the origin of Ekman pumping is clear as can be seen by referring to Fig. 6. Consider the flow generated by the surface stress shown as single arrows in Fig. 6. The transport in the Ekman layer is to the right of the applied stress as shown. Thus, in a vertical section the horizontal Ekman layer convergence requires a vertical divergence of flow out of the Ekman layer and a “pumping” into the interior results.
1
I
FIG.6 . A schematic picture illustrating the mechanism of Ekman pumping. The transport in the Ekman layer is to the right of the surface stress, T. With the distribution of T shown here fluid will converge in the Ekman layer and must flow downward into the interior (broad arrow).
At the bottom the total velocity field must vanish. Integrating the continuity equation for the Ekman layer variables from z = - 1 to z = 03 yields
and the use of Eq. (7.8) and the boundary condition w, = -wIb yields WIb =
+[v,
(J?”~V,~)
+ k v X (E1”VIb)].
(7.12)
Hence, horizontal divergence and vertical vorticity of the interior field evaluated at the bottom boundary determine the Ekman pumping into the interior.
B. EKMANLAYERS IN
THE
OCEAN
Even though the foregoing simple models are not directly applicable to the ocean, some of the consequences of the theory have been observed in nature. Indeed, Ekman’s (1905) analysis was undertaken to explain the observation by Fridtjof Nansen that ice floes on the surface of the Arctic Ocean moved to the right of the wind. Although it is fairly easy to generate Ekman spirals in the laboratory, almost all attempts to observe them in the ocean have failed. An exception is the observation by Hunkins (1966) of an
Large Scale Ocean Circulation
41
Ekman spiral beneath the ice in the Arctic Ocean. T h e observed spiral was consistent with laminar theory when an eddy coefficient of viscosity was used. I n general, the surface layers of the ocean are turbulent (because of thermal convection, wave motions, and/or various instabilities) and the resultant detailed motions can be expected to differ substantially from those predicted by laminar theory (Deardorff, 1972). In oceanography it is common practice to write T = pvv av/az where v, is defined as the vertical component of a highly anisotropic eddy viscosity and has the range of values 1 cm2 sec- 5 v 5 200 cm2 sec- l. Hence, the Ekman layer depth near the surface takes on values between 10 and 100 m at mid-latitudes and frictional effects are normally considered to be confined to a boundary layer in the top 100 m. Just as in laboratory studies the Ekman layer can then be treated as a true boundary layer and the integrations over an infinite range of x can be replaced by an integration over several Ekman depths. Even though the details of the Ekman layer are not applicable to oceanic flows, the integrated results, such as the transport to the right or left of the wind and Ekman pumping, can be used. However, some modification of ideal fluid theory in a uniformly rotating system is necessary in order to make the foregoing results applicable to the ocean. Since the vertical component of the Coriolis parameter is a function of latitude for large scale flows, Eqs. (7.1) through (7.5) are locally valid if E is defined as E = ./(asin do2),i.e. if SZ is replaced by SZ sin +. T h e equations for Ekman layer pumping involve horizontal. derivatives of the horizontal velocities and this requires that E be treated as avariable. That is the reason for including the parameter E in the horizontal derivatives of Eqs. (7.6), (7.10), (7.11), and (7.12). Th e significance of variable rotation can be seen more clearly if the results are written in dimensional form ZE-
OD
WIO
.
= -(l/pln)k
= (l/pnl)k
WIb = [OH (VIb(V/f)1’2
+
v
x (T31f),
v x (Wn, x (vIb(vlf)1’2)],
(7.6‘) (7.11‘) (7.12‘)
where the variation o ff( =2SZ sin +) appears explicitly. In some studies of wind-driven circulation (Munk, 1950; Bryan, 1963) frictional effects are assumed to take place primarily through lateral transfer of momentum. In this case, it is customary to use a horizontal eddy coefficient of friction, pvH, and to draw an analogy between the statistics of eddy and molecular motions to write the frictional term in the horizontal equations of motion as p v H V H 2 v where OHz is the horizontal Laplacian. Typical values of vH for large scale flows range from lo6 to lo8 cm2 sec-’ (Stommel, 1955; Munk, 1950). Although the use of a horizontal eddy coefficient of viscosity admits the existence of a boundary layer which is
42
George Veronis
interpreted as the western boundary current (Gulf Stream, Kuroshio, etc.) there is some evidence that the traditional use of an eddy viscosity to model momentum flux from mean to perturbation flows may not be appropriate for oceanic flows. Indeed, Webster (1961) has cast doubt on the sign of v, when he showed from an analysis of data taken near the Gulf Stream that the flux of momentum may be from the fluctuations to the mean flow. T he fluctuations in this case would have to draw their energy from the potential energy associated with the statification in the ocean. T h e mechanism for transferring heat and salt (or density when the appropriate combination of T and S is used) into the ocean from the boundaries is also assocated with turbulent processes. Anisotropic eddy coefficients are used for the state variables just as they are used for momentum. T he difference between momentum and heat transfer is that the lowest order equations for momentum balance do not include eddy transfer processes whereas for heat, salt etc. the diffusive processes may be basically important. Even though the use of an eddy viscosity is based on heuristic reasoning, the fact that v, can be related to processes associated with an Ekman layer puts a firmer base under the concept of a vertical component of eddy viscosity. Turbulent processes and variable density alter the detailed structure of the Ekman layer but some quantitative measure of integral effects of the Ekman layer cn still be made. For this reason the following development will make use of v, in preference to v, when dissipation must be included. It will be seen, however, that it is not always possible to make such a restriction and that under certain conditions a horizontal eddy viscosity must be introduced.
VIII. Modeling of Current Systems There is no obvious ordering for deducing a hierarchy of oceanic models which successively incorporate processes and features in order of decreasing importance. I n fact, even the forces which give rise to currents in the ocean are not known sufficiently well to be so ordered. Differential heating and cooling of the surface waters lead to vertical density stratification of the ocean. Winds apply stresses to the surface and as a result the current systems tend to separate into more or less well-defined gyres associated with these wind systems (Munk, 1950). Yet the interaction of motions driven by thermal differences and those resulting from wind stresses is sufficiently complex that very few studies have been undertaken to understand the interaction. Even in those cases where such studies have been
Large Scale Ocean Circulation
43
made the models are heuristic ones with no real attempt to establish the relative importance of the mechanisms. Knowledge of dynamical processes has developed primarily through the use of simple, heuristic models which isolate one or another feature of the circulation relating it to a known or measurable process. Stommel (1965) has been particularly successful in developing simple models to explain important large scale features.
A. WIND-DRIVEN OCEANCIRCULATION Stommel’s (1948) elegantly simple explanation of the westward intensification of the wind-driven ocean circulation provides an excellent example of the value of the simple-model approach as well as an example of the difficulty in using a heuristic model as a base for extending the study to include more complex processes. Beginning with Eqs. (5.18) to (5.22) with f = f o ,6y, Stommel made the additional assumptions that the fluid is homogeneous, the depth of the ocean is constant, the flow is steady, and R = 0. He then wrote down the following equations which correspond roughly to a linear, vertically averaged system
+
au
av
ax
ay
-+--0.
Here, the terms EV are dissipation terms which are assumed to be proportional to the average velocity and E is a nondimensional drag coefficient. The surface stress terms T enter as apparent body forces. Equation (8.3) admits the definition of a stream function
and if h is eliminated from (8.1) and (8.2), the following vorticity equation is obtained
+ ,qa+/ax) = k. v x T.
&v2$
The ocean basin is rectangular with boundaries at x = 0, T and y where # = 0. T h e form of the stress term is
k.V
XT=
-sin x siny.
(8.5) = 0,
T
(8.6)
44
George Veronis
The solution of (8.5) and (8.6) with the given boundary conditions is
D,=
- 1 - (1
+ 4&2)1’2 1
2E
D,=
-
1 +(1 2E
,
(8.7)
and it is characterized by the following features (Fig. 7):
1. For E < 1 the meridional flow is southward everywhere except in a thin boundary layer near x = 0 where it is large, O ( E - ~and ) , northward. 2. T h e interior region of southward flow represents a balance between the p term and the curl of the wind stress. This is called the Sverdrup (1947) transport balance. 3. I n the boundary layer region on the west the frictional term (actually, E a2i,b/ax2or E av/ax and the B term are large and balance each other and the wind stress term is negligible. 4. T h e stream function pattern is symmetric about the mid-latitude of the basin. It is particularly informative to write the term as df/dt. Hence, the Sverdrup balance df/dt= k * V x T means that the rate of vorticity input by the wind stress is balanced by the rate of change of planetary vorticity, f,brought about by the meridional flow. In the boundary layer the balance E &/ax = - df/dt means that the rate of change of the planetary vorticity is balanced by the rate of dissipation of vorticity. Also, comparison of the streamline and h (or pressure) patterns shows that the flow is nearly geostrophic, although there is a discernible Ekman wind drift in the vicinity of the boundaries. For the purpose of extending Stommel’s model it is more instructive to derive the results in terms of the physical balances described earlier. The analysis will be done for a sphere. Again start with Eqs. (5.18)-(5.22) and consider the fluid to be homogeneous,* the depth of the ocean to be constant, the flow to be steady, and R = 0. Under these conditions the equations of motion for the interior flow are
fk x
VI
= -Qp,
- Qk
(8.8)
* Wote that a great deal of oceanic physics has already gone into deriving the approximate set of equations (5.18)-(5.22). The fact that the ocean is stratified is implicit. The assumption of a homogeneous ocean at this point is an analytical convenience.
Large Scale Ocean Circulation
f
45
02
Y
77
X
FIG 7. Calculated contours of the stream function (top) and the height of the free surface (bottom) for the linear, frictional model of wind-driven ocean circulation ( E = 0.05).
where subscript I specifies that the variables are interior ones, i.e. the boundary layer contributions are not included. On a spherical earth the ocean is assumed to be bounded on the east and west by parallels of longitude denoted by h = 0 and h = A,. For a homogeneous system the hydrostatic equation ap,/az = -Q can be integrated from the top surface, z = ~ ( xy,) , where p , = 0 downward to give (8.9) PI = arl - 4. Substituting (8.9) into the horizontal components of (8.8) and integrating from the bottom ( z = - 1) to the free surface yields
fk x VI = -QVW
+.I),
(8.10)
46
George Veronis
where V, is the horizontal volume (or mass) transport. T h e curl of (8.10) yields
fvH*v,+pv,=o.
(8.11)
A vertical integration of the continuity equation gives v€l * vI
+
wIO-
WIb=
O,
(8.12)
which, when substituted into (8.1l), yields
BVI =f (WIO - WIb).
(8.13)
Now since the interior flow is hydrostatic and geostrophic, it follows immediately that uI and vI are independent of depth for a homogeneous fluid. Hence V, can be replaced by (1 7)vI w vIand Eq. (8.13) becomes
+
BvI = f ( W I O
- WIb).
(8.14)
Equation (8.14) contains an important part of the physics of all models for steady, large scale ocean circulation. Each particle of fluid has a vorticity, f, called planetary vorticity, because it is on the rotating earth. Since oceanic flow in the interior of the ocean is slow and its scale is large, the planetary vorticity is much larger than the relative vorticity. (This point is directly reflected in the linearity of the equations of motion.) Sincef depends only on latitude, Eq. (8.14) can be written in the alternative form
d f P t =f (WIO - WIb).
(8.15)
Hence, the rate of vertical divergence of a column of fluid is accompanied by a rate of change in the planetary vorticity of the column of fluid, i.e., the fluid will move to the north or south at such a rate as to comply with the rate of change of vertical divergence or convergence of the column. Hence, the meridional motion of the column of fluid is completely determined by the vertical divergence. T h e vertical divergence can be evaluated by substituting wIo and wIb from Eqs. (7.11) and (7.12) to give
( 2 P / f ) ~ , = k * V ~ ( 2 3 ~ ~V )H-* ( ~ I E 1 ‘ 2 ) - k . xV( v , E ~ ” ) , (8.16) where q b and vIb have been replaced by uI and vI because the latter are independent of depth, as noted earlier. Consider first the case for /3 = 0. Here, it follows directly from the geostrophic equations that V,. vI= 0 and that k . V x V, = - &VH2pI. Hence, (8.16) becomes
-
VH2pI= -2k V x ( T ~
(8.17)
Large Scale Ocean Circulation
47
Therefore, if the wind-stress curl is symmetric about a mid-longitude, the solution to the Poisson equation gives a corresponding symmetry for p , and, consequently, for the interior streamline field. An additional consequence of @ = 0 comes from Eq. (8.13), viz., wIo = WIb so that Ekman suction near the bottom is adsorbed by Ekman pumping near the top surface and vice versa. T h e case p # O yields substantially different results. T h e lowest order balance in Eq. (8.16) is vI= (f/ZP)k * V x
(7,
El ').
(8.18)
Thus, as noted below Eq. (8.15), the interior flow is everywhere determined by the rate of curl of the wind stress and it is O(E1I2)of the flow for the case with @ = 0. Comparison of (8.18) and (8,lZ) also shows that wIb is negligible under the present conditions. Therefore, at lowest order the wind stress at the surface generates an interior vertical velocity whose magnitude must decrease linearly to zero at the bottom. The total meridional transport, vT, at lowest order is given by the interior transport (8.18) plus the transport (7.4a) in the top Ekman layer, and is easily verified to be VT =
(fE/2@)k . V x 7 0 ,
(8.19)
a result which can be derived directly without the involved Ekman layer argument. The present method of derivation makes it easier to look at additional details of the flow and to derive corresponding features for the stratified system as will be seen shortly. Equation (8.19) takes the dimensional form VT
= (k
'
X
TO)/k$,
(8.20)
where V , is the Sverdrup transport, referred to earlier. Since VT vanishes at the latitude where k * V x T~ vanishes, the Sverdrup transport satisfies the condition that there be no normal transport at latitudes of zero windstress curl. This is the justification for treating isolated " basins " whose north-south extent is determined by latitudes where the wind-stress curl vanishes. The meridional flow derived above cannot be the complete solution because a region of the ocean in which the wind-stress curl is of one sign would have a undirectional meridional flow everywhere. T h e simplest argument to complete the flow pattern is to note that if the wind-stress curl imparts vorticity of one sign to the ocean the overall circulation must exhibit that vorticity. Thus, if k .V x T~ is negative the total meridional transport is everywhere to the south. An overall negative vorticity of the circulation exists if an intense northward current exists near the western
48
George Veronis
boundary. Analogously, a positive wind-stress curl requires a northward flow everywhere and an accompanying, intense, southward flow near the western boundary. Hence, intense currents will exist near the western boundary. T h e simplest, formal procedure for deriving the intense current is to incorporate bottom frictional processes from Eq. (8.16) by boundary layer techniques. I n this case, derivatives with respect to the longitudinal coordinate must be large to offset the small coefficient Furthermore, it is evident that of the last two terms on the right-hand side of Eq. (8.16) the dominant one contains the longitudinal derivative of the meridional velocity since a/ah and vI are both large near h = 0. Hence, the lowest order equation in the western boundary layer is
(8.21) where 6 denotes the boundary layer contribution to vI . T h e solution is
6 = A(+)exp [(-2/3 cos +lfE1/2)h].
(8.22)
+
T h e quantity A(+)can be evaluated by satisfying the condition J:(e vT) dh = 0, where vT is given by (8.19). [The Ekman layer contribution to 6 is O(E1I2)and does not enter at this order.] Hence,
I (k. A0
A(+)=
V x ro) cos
0
+ dX.
(8.23)
For the wind stress which drives the gyre containing the Gulf Stream the simplified form
k -V x
so= - Wsin r(+ - +o)/+l
+o
2 4 <+o
+
+1,
(8.24)
is often chosen and A(+)can be evaluated explicitly. Since in this case -c0 is independent of A, the complete meridional transport u = uT 6 can be written as
+
k -V x
(8.25) It is interesting to note that thickness of the western boundary layer I n the increases northward as sin1l24 coss2 4 between +o and +o + /3-plane solution the thickness is constant. T h e continuity equation for the total mass transport (including the Ekman layer contribution) is, au a (8.26) - +- (v cos +) = 0.
ax a+
49
Large Scale Ocean Circulation
S ~ V
Since dh vanishes at each latitude it is necessary to satisfy the condition u = 0 on only the east or the west side because the other boundary condition automatically follows. Hence, integrating westward from the eastern boundary, h = A,, where u = 0, gives (8.27) where v is given by Eq. (8.25) and may be replaced by vT except in the vicinity of the western boundary. Also,
W E sin + sin (4 += - 2 where E sin + is independent of + and it is easily seen that the solution in vT cos
+ O b
1
dl
the interior is (8.28) i.e. uT is antisymmetric about 4 = (+o + +J/2. An important aspect of the present solution can be deduced from the east-west structure of the northward velocity. In the interior v is small and there is no Ekman layer contribution near the bottom. However, in the western boundary layer v is large and decreases eastward. Since adlax is negative and large, to lowest order there is an Ekman suction into the bottom boundary layer in this region. I t is by this mechanism that the vorticity input in the interior is dissipated in the boundary layer. In order to satisfy the no-slip condition at h = 0 and A,, it is necessary to add lateral friction to the model. It is easy to see that the effect of such a subboundary layer would be to generate an additional Ekman pumping. In this region 6 is positive and increases eastward, thereby forcing an Ekman pumping from the bottom boundary layer upward.
B. EXTENSIONS OF STOMMEL’S MODEL Given the results of the foregoing model it is natural to ask whether the model can be expanded to take into account additional mechanisms, processes, and features characteristic of the ocean. The analyses of Fofonoff (1954), Morgan (1956), Carrier and Robinson (1962), Bryan (1963), and Veronis (1963, 1966) are attempts to extend the model to take into account nonlinear processes but Stommel’s model still forms the basis for the extension. In particular, stratification is ignored and the ocean is considered to have constant depth. Charney (1955b), Fofonoff (1962b), and
50
George Veronis
Welander (1968) idealized the stratification with a two-layer system and studied extensions of Stommel’s model. Niiler et al. (1965) used an idealized, basically linear stratification to determine the resultant wind-driven ocean circulation. Bryan and Cox (1967) and Bryan (1969) have carried out extensive numerical calculations of a more complete system, including thermal driving forces, and have deducedZhe resulting stratification as part of the analysis. T h e approach to be taken here is to determine whether the results of the simple, idealized model already discussed can be used as a basis for a more extended model. I n particular, the assumptions R = 0 and slat = 0 are retained but the ocean is taken to be stratified and topographic variations may be present. T h e hydrostatic equation includes density variations and the interior flow is now described by the thermal wind relations (6.6). Ekman layers near the top and bottom boundaries are assumed to provide the required momentum transfer to the surface and bottom. T h e introduction of variable density requires either an additional equation for the determination of p or a specification of some part of the density field with an appropriate equation to determine the residual field. Since the present purpose is to make use of the results of the simple wind-driven model, it is clear that the density cannot be calculated ab initio. The geographical range of the model is specified by the wind-stress field and one cannot hope to obtain a realistic density distribution without including the thermal driving forces which will, in general, have larger scales and a different geographical distribution from that of the wind field. Hence, the density field must be specified in some sense. Since the degree of specification depends on other assumptions which can be made, this point will be postponed until it is clear which part of p is required. Now from the thermal wind equations, the hydrostatic equation, and the continuity equation for a Boussinesq fluid, one can derive the planetary vorticity equation pvI =f aw,laz (8.29) and the equation for the vertical curvature of w , (8.30) Equation (8.29) can be integrated in the vertical from z to the surface to give (8.31) V , = (wIo- w,) tan 4. T h e difference between Eqs. (8.31) and (7.6) is that the present integration is taken from an arbitrary level rather than from the bottom. First, consider the case w, -+ 0 as z decreases so that at some depth the vertical velocity is reduced to zero by the stratification. If this depth does
Large Scale Ocean Circulation
51
not intersect any ridges or seamounts, the interior meridional transport is given by wIo or, equivalently, by the wind stress. Then VI is determined as in the homogeneous problem, the principal difference being that the stratification confines V, to a layer bounded below by the depth where W, essentially vanishes. From (8.30) it can be seen that ap,/aA must vanish at great depth ; hence the thermal wind relation
+ av,/az
+) ap,/aA (8.32) allows an integration upward for the determination of the vertical structure of v , once pI is known. Therefore, in this case a knowledge of p, is necessary for the vertical structure of ZI, as well as for w I , but not for the transport
2 sin
= -(Q/cos
v,.
A vertical integral of the continuity equation from z to the surface yields
a (VCOS +) = 0, +ah a+
au
-
(8.33)
where U and V incorporate the Ekman fluxes as well as the geostrophic interior flow and U can be determined to within an arbitrary function of +. Anticipating that a boundary layer will exist on the western side only suggests an integration westward from the eastern boundary where U vanishes, and U is then known explicitly. T h e corresponding simplification in the boundary layer, viz. & + 0 as z decreases, means that bottom frictional processes are not important in the western boundary layer. Hence, the system cannot be closed as in the homogeneous model and it is necessary to incorporate lateral frictional processes near X = 0 in order to satisfy the condition of no net meridional transport. A vertical integration of the continuity equation is again possible and the entire analysis can be carried out in terms of transports rather than velocities. T h e solution depends on the parameterization of the lateral frictional processes and specifically on uH if lateral friction is assumed to have the form uH VH2v.This is essentially the approach taken by Hidaka (1949) and Munk (1950). T h e results of Stommel’s model and that of Hidaka and Munk differ only in the details of the western boundary layer, although it is important to keep in mind that the physical processes in the boundary layers are very different. If the vertical velocity in the western boundary layer does not vanish near the bottom, the problem becomes substantially more complicated. Indeed, if bottom topography is included, the details of the density distribution must also be determined and one is then faced with the general circulation problem, i.e., thermal forcing must also enter and the geographical range of the domain must include the sources and sinks of heat and salt. However, with a few simplifying assumptions it is possible to
George Veronis
52
obtain results for the case where z6 does not vanish at great depth near A = 0. T o derive these results note that under the present assumptions the flow in the western boundary layer is geostrophic (except in the Ekman layers); hence (8.29) is valid. Now neglect bottom topography. Then a vertical integration of (8.29) from bottom to top yields 2(/3/f)8=k*
v
X ( 7 0 E ) - v V H . ( ~ b E 1 ’ 2 ) - k * VX(0bE1’2),
(8.34)
where 8 is the transport in the western boundary layer and 0, is the horizontal velocity in the boundary layer evaluated at the bottom. As in the homogeneous case derivatives with respect to longitude are large and at lowest order the contribution of the wind-stress term can be neglected. T h e principal balance is (2/3p/f)
= -((E”2/Cos
$)(a&/aA).
(8.35)
Because of stratification 6 varies in the vertical and the stratification must be known in order to relate 6, to 8.Suppose that the mean density field in this region is known, either by observation or from a numerical model. Then 6, can be related to V . T h e particular assumption made here is that ag/aA decays exponentially from the surface downward. (Better functional approximations to a$/aA in the western boundary layer can be obtained from observed data but the simple exponential form serves present purposes.) Then, from the geostrophic relation one can write
6 = 6,(1 + cekz),
(8.36)
where, in keeping with observations, K is substantially greater than 1 (so that the stratification does not extend to the bottom) and K may depend on 4. The bottom velocity, 6,,may be either positive or negative. For the Gulf Stream 6 must be positive and large near the surface so that c > O if u, > 0 and c 4 - 1 if u, < 0. T h e transport P can now be evaluated from (8.26) as
P=/
0
6dzmd6,(1+c/k).
(8.37)
-1
Then Eq. (8.35) becomes
y6,
= -2
where y = 2/3 cos 4 (1
a.;,/aA,
+
(8.38)
C/K)/fE”2,
and the solution is
6,= 6,,e-yi,
(8.39)
53
Large Scale Ocean Circulation
where Gbo( =6,) o) can be evaluated as before so as to yield zero net meridional flux across the entire basin. For 6, > 0 the thickness of the boundary layer i ~ f E l ’ ~ / cos2 ( 2 q5( 1 c/K)), i.e. it is thinner than the corresponding boundary layer of the homogeneous system. The reason for this is that the entire vorticity of the interior must be dissipated in the bottom Ekman layer by means of a relatively weaker meridional flow, 6,. Hence, the longitudinal gradient of the meridional flow must be larger, or equivalently, the boundary layer must be thinner. It is interesting to note that the thinner the western boundary layer is near the bottom the more likely it is that some other process such as lateral friction or nonlinearity may become more important and eventually dominate the flow. For example, it was pointed out above that 6,= 0 (this is equivalent to zero boundary layer thickness) requires that lateral friction be introduced. Of course, effects of bottom topography or of nonlinearity or of a more realistic density distribution could alter the argument dramatically. For 6, < 0 the present solution is not valid because it grows exponentially with increasing longitude. T h e reason is that negative bottom velocity cannot lead to the required dissipation of negative vorticity unless 6, becomes even more negative; hence the solution does not boundary layer. Observational evidence (Swallow and Worthington, 1961) indicates that south of Cape Hatteras the deep western boundary layer flow is indeed southward. This southward flow appears to be related to the thermally forced circulation (Stommel, 1965) and because of bottom topography it may be displaced eastward relative to the Gulf Stream. Hence, it is possible that the bottom frictionalmodel may not beinapplicable even in this case. T h e issue is by no means clear, therefore, and will probably not be resolved until success is achieved with models which incorporate both bottom topography and stratification. If wI does not vanish with depth in the interior, effects of bottom topography and stratification must be incorporated everywhere and the winddriven circulation cannot really be separated from the general circulation of the oceans.
+
C. INERTIAL EFFECTS Several different methods have been employed to study nonlinear effects in wind-driven ocean circulation. These include perturbation analyses (Munk et al., 1950; Veronis, 1966), numerical studies (e.g., Bryan, 1963 ; Veronis, 1966), and studies which accept the interior flow as given and which treat nonlinear effects in the boundary layers (Fofonoff, 1954; Charney, 1955b; Morgan, 1956; Carrier and Robinson, 1962).
54
George Veronis
Perhaps the principal benefit from these investigations has been an increased understanding of the nonlinear process rather than an advance in our knowledge of the structure of the wind-driven gyres. The reason is that the models tend to be too idealized to admit incorporation of additional fcatures ; consequently, one cannot build an ordered hierarchy of successively more complicated models. T h e present discussion will be restricted to nonlinear effects in Stommel’s original model on the ,&plane although some remarks will be made about variations on the model. T h e nondimensional vorticity equation for the vertically averaged mass field is
R v . V C $ / ~ V + E C = ~ *XV T,
(8.40)
where
or, in terms of the stream function,
T h e nonlinear terms here are only crude approximations to the actual nonlinear terms because the vertical average of the product v . V( is not generally given by the product of the vertically averaged quantities. The Rossby number, R , can be defined in terms of the amplitude of the windstress since the velocity amplitudes are determined by 7. However, for present purposes it suffices to note that R is so small that nonlinear effects in the interior of the ocean are negligible unless the boundary current penetrates the interior. T h e effect of the inertial terms can be treated by perturbation methods (Veronis, 1966) in a straightforward manner. However, it is easy to see the qualitative effect of nonlinearity by rewriting (8.40) for the western boundary layer region as v * V(R5
+f)=
-E<
(8.42)
since the wind stress can be neglected there. Now the linear ( R = 0) solution states that the rate of dissipation of the intense negative vorticity in the boundary layer is balanced by the advection of planetary vorticity, f, northward. When nonlinear effects are included in the boundary layer south of mid-latitude there is also an advection of negative vorticity, R(, northward and that will tend to decrease the rate of dissipation. The latter is possible if EC is reduced, i.e. if aa/ax is decreased. Hence, the boundary layer must thicken in the formation region of the Gulf Stream. At some
55
Large Scale Ocean Circulation
point north of mid-latitude the relative vorticity must begin to increase from its large negative value (it must essentially vanish in the interior). Therefore, v . V ( R [ )must be positive and the result is that - F [ or - E &/ax is increased. Hence, the boundary layer becomes narrower in this region, a north-south asymmetry is introduced by the inertial terms, and more of the dissipation takes place in the northern half of the boundary layer. As R is increased this north-south asymmetry is intensified, the point of maximum 151 is shifted northward, and the dissipation of vorticity takes place closer to the northwest corner of the basin. For sufficiently large R the western boundary current impinges on the northern boundary and must turn and flow eastward near the northern boundary. In this region the variation of planetary vorticity is small (the eastward current is narrow) and the balance must be between nonlinear advection and dissipation. As it turns out, the boundary layer thickens in this region and the intense downstream flow cannot be maintained. Hence, streamlines leave the northern boundary region and return to the interior. Inertial effects thereby penetrate into the interior. An example of the flow described above is given in Fig. 8 where the streamline pattern shown was derived from a numerical integration of (8.41) (Veronis, 1966) with E = 0.05 and R = 0.01 and k - V x T = -sin x sin y. If lateral friction is included as well, the tangential velocities must vanish at the boundaries and the possibility of a shear flow instability arises as the Rossby number is increased. Blandford (1971) integrated the system (8.41) with sV4$ included as well and with the added conditions a$lax = 0 TI
Y
n X
FIG.8. Calculated contours of the stream function for the nonlinear model of winddriven ocean circulation (Veronis, 1966). E = 0.05, R = 0.01.
56
George Veronis
on x = 0, rr and a2#/ay2= 0 on y = 0, T. He found that for R above a certain value the western boundary layer becomes unstable, presumably to a Rayleigh-type of shear flow instability, and transient eddy motions are generated in the western boundary current north of mid-latitude. If the wind-stress curl is assumed to vanish at a latitude south of the northern boundary, the linear solution looks much like the one that would be obtained by placing a rigid boundary at the latitude where the windstress curl vanishes, i.e., the northern “ boundary” is determined by the Sverdrup balance. However, the nonlinear solution shows that the western boundary current penetrates into the region of zero wind-stress curl. Hence, the presence of boundaries to the north and south is required to contain the oceanic gyre. Presumably in nature these “ boundaries ” are provided by fluid masses of substantially different density so that the intense currents can flow to sea along the region of sharp density gradient. Variations in bottom topography also seem to steer intense currents, at least partially, so that they penetrate into the open ocean. Warren (1963) has shown that the meanders of the Gulf Stream after it leaves the coast are associated with bottom topography. In a numerical calculation for an idealized model Holland (1967) exhibited the same steering mechanism for intense currents.
IX. The Thermohaline Circulation When sources and sinks of density are included as forcing terms, the resulting circulation is called the thermohaline circulation. Boundary conditions for the density are, of course, associated with heat and salt transfer at the boundaries. T h e problem is simplified substantially by the assumption that exchanges in heat and salt are described by identical eddy transport processes. For a Boussinesq fluid temperature and salinity can then be combined to describe the density and the problem can be formulated in terms of the density. (Sometimes an “ effective ’’ temperature, into which temperature and salinity are incorporated, is used in place of the density.) Regions of mean evaporation and precipitation are associated with the wind-driven gyres and must be at least quantitatively significant in the thermohaline circulation of the oceans. However, these specific boundary processes have been ignored and the usual procedure has been to force the circulation by thermal or density exchanges which have global scales at the surface. The problem is described by geostrophic and hydrostatic balance with vertical (eddy) diffusion terms incorporated into the density equation.
Large Scale Ocean Circulation
57
In dimensional form the Boussinesq equations, with p corresponding to the potential density, are 2Q sin +v = (1/a cos +)PA, (9.la)
2 0 sin +u = -( 1/a)P, , p, = -gp,
(9.lb) (9.1~)
v'VP=Kp,z,
(9.2a)
v.v=o,
(9.2b)
where p = po/pm, P = -PIPm, and subscripts A, 4,and z correspond to partial derivatives. Th e assumed global scaling is included in the complete latitudinal dependence of the Coriolis term. Since no diffusion of momentum is included in (9.1) the equations describe only the interior (geostrophic-hydrostatic) flow and the effect of Ekman layers at the top and bottom must be introduced via boundary conditions. Lateral boundary conditions on velocity can be satisfied only along a single boundary. For a basin with lateral boundaries one must append boundary layers at the sides and require the boundary layer and interior solutions to match in the region of overlap. The importance of lateral boundary layers is obvious because cold or warm boundary currents will alter the surface boundary conditions in the interior and the effect can, and probably will, be very important. Bottom topographic variations are also important because of the strong constraint that they impose on the motions of the nearly homogeneous abyssal waters. These, too, have customarily been ignored, or at least not incorporated into a complete quantitative analysis. Although one cannot really justify the use of an open system in which the various features mentioned above have been neglected, some progress has been made on the problem without these features. The discussion will be confined to the analysis of the simplified system (9.1) and (9.2). A discussion of various aspects of the model (the use of vertical diffusion, the boundary conditions which can be satisfied, etc.) will be given at the end of this section.
A. THEPRESSURE EQUATION The system (9.1) and (9.2) can be expressed in terms of the pressure by writing z, = (1/2Qa cos 4 sin 4)PA, (9.3a) u = -(1/2Qa sin +)P,, (9.3b) p = -(l/g)Pa , (9.3c)
George Veronis
58
using the expression for the vertical velocity obtained from (9.2a), i.e., w = [KPZZ
and finally, substituting divergence relation
z,
-
(u/acos + ) P A - ( ~ / ~ ) P d l / P Z 7
(9.4)
from (9.3a) and w from (9.4) into the planetary w, = cot
4 v/a.
(9.5)
T h e result is the pressure, or P, equation derived by Needler (1967): K
sin 4cos 4 ( P , Z p,,,, - p,,,P,,,)
+
=
+
PdPZ,, PA, - p,,PA,,) PA(P,, PdBZ - p,,,PdZ cot 4 p,,P,,), (9.6) where K = 2QKu2. It is a generalization of Welander’s (1959) M equation, where M is essentially the vertical integral of P when the barotropic mode is included in (9.6).
B.BOUNDARY CONDITIONS AT
THE
TOPAND BOTTOM
Even a cursory examination of the density structure of the oceans shows that there is a distinct variation of the density with latitude at any given level. Furthermore, the subsurface density is related to the density near the surface and, in particular, the effect of wind-driven gyres is reflected in the density field. Hence, for surface boundary conditions it is common to specify the density as known at the base of the Ekman layer and to assume that the vertical velocity (Ekman pumping) is given by the known, mean wind-stress distribution. At the bottom the obvious condition that there be no normal velocity should be satisfied. I n addition, since the bottom is essentially a nonconductor (geothermal heat flux is ignored), the bottom is treated as insulated. As will be seen below, it is not possible to satisfy all of these conditions with the solutions presently available, even though the model described by (9.6) would appear to contain sufficient arbitrariness to do so. This point will be taken up again after the solutions have been presented.
C. SOLUTIONS BY MEANSOF
A
SIMILARITY TRANSFORMATION
Some solutions to (9.6) can be obtained by a similarity transformation proposed by Robinson and Welander (1963). Although such a procedure imposes restrictions on the form of the solution, it turns out that the solutions can be generalized so that some of the restrictions can be removed.
Large Scale Ocean Circulation
59
Since the prineipal boundary conditions are to be satisfied on the surface z = 0, this suggests that the following form for P be tried
PO, 4 = q ( k $)G(7), $7
7 =zkO,+).
(9.7)
When (9.7) is substituted into (9.6), it is found to be a solution provided that the following conditions are satisfied
k = (sin $)"[A
+ E($)]",
+ E($)I2"+
,[A
q = (sin $),"+
l,
where m and n are real constants and E is an arbitrary function of G(7)must satisfy the following equation
(9.8)
+. Then
K(G'"G"- GfrfGtff) = (2n - m)GG'G"' + (2m + 1 - 2n)GG"G"
- (2n - m)7G'G'G"' + (3n - m)rlG'G"G", (9.9) where primes denote differentiation with respect to 7. Hence, the nonlinear partial differential equation has been reduced to a nonlinear ordinary differential equation. Boundary conditions on P which are consistent with the form (9.7) can be transformed into boundary conditions on G. Equation (9.9) contains three parameters, K , m, and n, and can be integrated numerically for ranges of values of these parameters. It turns out, however, that a few exact solutions of (9.9) can be obtained by dividing the equation into two parts each of which vanishes separately. This method was proposed by Kozlov (1966) and it gives the only exact solutions which have been derived so far. T h e method of solution of (9.9) is to let the terms multiplied by 7 balance each other. Then (9.10) The remaining terms then yield
K(G'"G"- G"'G"')- (2n - m)GG'G'"
+ (2m + 1
-
2n)GG"G" = 0. (9.1 1)
One solution, given by G' = 0, corresponds to a barotropic fluid and is of no interest in the present context. If G' # 0, and if (3n - m)/(2n- m) # 0, 1, or 00, the general solution of (9.10) is
G = y(7 -
)(min) - 1
+ 6,
(9.12)
where y , q 0 , and 6 are constants. Substituting (9.12) into (9.11) yields
6=0,
m=-n ,
y=4K.
(9.13)
60
George Vuonis
[Another possibility is (3n - m)/(2n - rn) = 0 or 00 but then (9.12) is not valid and one must treat these special cases by going back to (9.9).] Then (9.12) takes the form (9.14) G = 4K/(77 f 70)~. This solution, with q o = 0 and E ( 4 ) = constant in (9.8), was first derived by Fofonoff (1962b) by inspection. The complete solution for P is
q = (sin +)2m+2(h+ E)1-2m,
k = [sin +/(A
+ E)]",
(9.15) (9.16)
When (3n - m)/(2n - m)= 1, i.e. n = 0, the bracketed term of (9.10) gives the solution G = d been, (9.17)
+
where b, c, and d are constants. Substituting (9.17) into (9.11) yields m = -1. Hence, (9.17) is a solution to (9.9) and the complete solution for P is k = l/sin 4, q = h E, (9.18)
+
P = ( A + E)(d
+ beczisin@).
(9.19)
Welander (1959) first presented an exponential solution to the thermocline problem for K = 0. He did not make use of the similarity transformation but, as Needler (1967) has pointed out, the exponential solution even with K # O need not be restricted by the similarity transformation. Blandford (1965) derived a simpler version of (9.19).
D. GENERALIZATION OF THE SIMILARITY SOLUTIONS Since the similarity solutions can be generalized, it is preferable to present the more general forms before discussing the properties of these solutions. Needler (1967) has observed that the exponential form deduced from the similarity transformation can be made more useful by substituting the form P = A ( h , 4) M(A, +)eak(l. 6) (9.20)
+
into (9.6) and determining the conditions for (9.20) to be a solution. The resulting restrictions are k , = 0, (9.21)
A,(k,
+ k cot +) = 0,
(9.22)
M,(kd
+ k cot 4) = 0.
(9.23)
61
Large Scale Ocean Circulation
These cbnditions are all satisfied if k , A, and M are independent of A. Although such a restriction leads to current and density distributions which are zonally uniform, a first approximation to observed patterns can be described in this fashion and a correction, incorporating dependence on longitude, can be sought. However, this approach has not been pursued. Instead, the alternative solution to (9.21)-(9.23)) viz.
k = c sin 4)
(9.24)
where c is a constant, has been investigated. The form (9.24) provides a z dependence in (9.20) which is identical to the one given by Welander (1959) and Blandford (1965). Welander considered a conservative system whereas Blandford restricted his attention to similarity forms. Once (2.24) is satisfied, there are no restrictions on either A or M and (9.20) becomes the most general exponential solution for the thermocline problem with diffusion. A crude picture of the density structure can be obtained from (9.20) by choosing c in (9.24) to give the observed depth for the temperature to decay to e - l of its surface value at some particular location. Needler chose c = (1500 m)-l and C$ = 30"N which gives an e-l depth of 750 m. This is associated with the thermocline depth in the ocean. Then choosing a surface temperature proportional to cos(4 10') and a (bottom) reference temperature of 2.45'C, Needler constructed the isotherm pattern shown in Fig. 9. (His isotherm pattern is equivalent to the density pattern in the present analysis.) T h e qualitative agreement with the observed potential density in the western Atlantic (Fig. 3) is good although there are
+
O - -220
5000
I
10
20
30
40
50
60
ON LATITUDE
FIG.9. Calculated isotherms (or isopycnals) for thermohaline circulation according to Needler (1967).
62
George Veronis
differences. No wind-stress effect is included in the pattern of Fig. 9 so that the gyre (between 20"N and 45"N)associated with the Bermuda High is not brought out so sharply in the calculated profile. An additional feature of the exponential solution can be seen by observing that with A = 0 the vertical velocity takes the form
(9.25)
+=
where p o / ( h + E ) is the surface value of p at 30". T h e vertical velocity, therefore, approaches the value w , = cK/sin q5 at great depth. Here, the contribution of w , exactly balances the vertical diffusion term in the density equation. Hence, if one were to work with w - w, instead of w , the original model could be reduced to a conservative one (the terms w, i3Plaz and K i32P/az2balance exactly). This accounts for the fact that the exponential solution has the same form as the one obtained by Welander for the case where K = 0. A similar generalization can be carried out for the inverse power law solution (9.14). Thus, choose
(9.26) Th e conditions to be satisfied by B, C, and q for P to be a solution to (9.6) reduce to
B = 4 K h sin2 4
qJ3B cot d - Bml
+ E(d),
+ qm B,
= 0,
q A c@ - q@c,4 = O.
(9.27) (9.28) (9.29)
Equation (9.28) requires that surfaces of constant q be given by
dd 4 K sin2d - 4 ~ sin h 4 cos 4
dh 3 cot +E - dE/d+ '
+
(9.30)
T he more general solution in this case is nearly as constrained as the similarity solution. Th e particular advantage of the power law solution is that it provides a slower decay of density with depth and one can match the observed vertical profile of density better than with the exponential solution which yields too large a decay with depth. In this solution the entire vertical structure of P,hence of p and v, is proportional to K whereas in the exponential solution the value of vertical diffusion is not critical.
Large Scale Ocean Circulation
63
E. SOLUTIONS TO THE IDEAL-FLUID THERMOCLINE Welander (1971) has extended his earlier solutions to the ideal-fluid problem by making use of three conservative properties. The latter can be derived from the general steady equations of motion for an ideal Boussinesq fluid p ( v - VV+2Q x v ) = -vp-pvo, (9.31)
v , v=o,
(9.32)
Vp=Q,
(9.33)
V.
where Q, is the force potential. It has already been shown (in Section 111) that the potential vorticity H=(2Q+V
xv>.vp
(9.34)
is conserved. Furthermore, it is a straightforward procedure to show that the Bernoulli function
+ +
(9.35) B = p pQ, BV * v is also conserved. Now, since p is also conserved (by assumption in 9.33), B, p must be related (they are all constant along trajecthe quantities tories) and we can write
n,
IT= F ( p, B),
(9.36)
where F is a general function of p and B. It should be observed that (9.36) represents a first integral of a very general, conservative system. However, the result is rather too general to be informative and it is necessary to turn to more specific models together with suitable, simplifying assumptions on F in order to obtain more explicit information. For a geostrophic, hydrostatic system with Q, =gz the quantities II and B reduce to = 2R sin q5 ap/az, B = p +gpz, (9.37a,b) and (9.36) becomes sin
+
ap/az = F(f,p +gp+
(9.38)
One simple solution for this problem can be derived by assuming that F has the linear form (9.39) F = ap b(P + g P ) c,
+
+
where a, b, and c are constants. Then (9.38) takes the form sin 4 ap/az = ap
+ b(p + g p ) + c
(9.40)
64
George Veronis
and differentiation with respect to x together with the hydrostatic equation yields sin 4 a2p/az2= ( a bgx) +/ax.
+
The latter can be integrated twice to yield
where po and C are arbitrary functions of h and 4,and zo= a/bg, D = (-2/bg)lI2. The constant b must be negative for the solution to decay with depth. If the constant a is positive the profile is monotonic with depth. For a < 0, an inflection point appears in the density profile at the constant depth, x = z,, . A second depth, D sin1’24,varies with latitude and is interpreted as the depth of the thermocline since it describes the decrease of density variations with depth. Hence, the solution allows for a two-scale description of the density with depth and, in particular, there is a sharp thermocline at the equator. With appropriate choices of a, b, and D, Welander was able to reproduce some of the overall features in the meridional density distribution in the South Pacific as shown by Reid (1965). Th e theoretically deduced distribution is shown in Fig. 10. For present purposes it suffices to compare it with the corresponding ranges in latitude and longitude shown in Fig. 3. Th e inflection point in Welander’s solution provides a better fit with the data than does the exponential profile which he had derived earlier and which is much like Needler’s, shown in Fig. 9. Neither solution shows the rapid meridional variations which are observed
Fig. 10. Calculated isopycnals from the ideal-fluid thermocline model of Welander (1971). (Courtesy of J . Mar. Res.)
Large Scale Ocean Circulation
65
at higher latitudes. Th e latter are associated with intense currents that are at least partially generated by wind stresses and the simpler dynamics of the thermocline models do not reproduce these features. Welander presents other special solutions to the ideal-fluid model. He points out in his concluding section that the solutions which emerge from such a model are restricted in the sense that they cannot satisfy all of the obvious boundary conditions required by a satisfactory model of the thermocline, a point to which we shall return presently. However, subject to the limits set by the model itself, Welander’s analysis is quite general and further study of these ideal fluid thermocline solutions should serve to provide us with a clearer picture of the possibilities and the shortcomings of this approach.
F. THEEFFECTOF
A
BAROTROPIC MODE
T he model described by (9.1) and (9.2) contains vertical diffusive as well as convective processes in the equation for the conservation of density. The presence of vertical diffusion should allow for the possibility of satisfying boundary conditions on density at the top and the bottom surfaces (a density flux or a fixed density at each of the two). I n addition, the continuity equation provides an additional degree of freedom which is normally used to specify the effect of wind stress in the form of a given vertical velocity (Ekman pumping) at the base of the Ekman layer. A fourth condition is available from the hydrostatic pressure equation because a barotropic current, independent of the vertical coordinate, can be included by specifying the pressure field at some level. Because of the constrained nature of the analytic solutions proposed thus far, it has not been possible to satisfy all of these conditions. A similarity solution is inherently constrained, of course, because once the form is chosen the boundary conditions must conform to the choice. Thus, satisfying one or two boundary conditions normally suffices to specify the behavior of the system throughout. An example of this constraint was given in Section IX, D where it was shown that the exponential solution proposed by Needler (1967) satisfies the diffusive model only because the vertical velocity is separated into two parts, one part balancing the diffusion term exactly and the other part contributing to the convection of density. In this sense, the exponential solution is simply a superposition of a solution for an ideal-fluid model and a solution for a model in which vertical diffusion and vertical convection balance exactly. Th e constraint enters because the density field for the two must be the same.
66
George Veronis
An informative alternative to the similarity-type solutions has been proposed recently by Needler (1972). Th e approach which he has taken is to ask: under what conditions can a solution P(h, 4, x), of the pressure equation be generalized to include a barotropic mode, A(h, +), so that P(h, +,x) A(h, +) is also a solution for any A(h, +)? The generalizations of the similarity solutions discussed earlier show that barotropic modes are indeed possible ; moreover, the exponential solution is not appreciably restricted by adding an arbitrary barotropic component to the pressure field. Physically, of course, the addition of a barotropic mode will be an added constraint on the possible solutions. At any given location it is possible to divide the pressure field into an internal part, which depends explicitly on depth, and an external part which is independent of depth. However, requiring such a division to be valid for an entire region is to impose a restriction on the fields which must satisfy the nonlinear density equation. Nevertheless, the results are very instructive. If P(h, 4, z ) is a solution to (9.6), under what conditions is
+
P(h, +,z> 3- B ( A 41,
(9.42)
with B arbitrary, also a solution? Substituting (9.42) into (9.6) and making use of the fact the P is a solution yields
Bb[PzzaP,,
-
P,a P,m] - Ba[Pzzrn P,,
- P, 6 P,,,
+ cot 4P.a P z z ]
= 0.
(9.43) Now divide by (PZJ2and make use of ( 9 . 3 ~ to ) obtain
B,-a
a,
["I
- B A -a P + x c o t + ] = O . ax p z
Pz
(9.44)
But p = -g aP/az so p is independent of B(h, 4) and the terms involving a/& in (9.44) must vanish individually. This requires that PaIpz = K a 9
( P @ / f Z )+ z c o t 4 = s i n 4 L , ,
(9.45a)
(9.45b)
where K and L are arbitrary functions whose forms in (9.45) have been chosen for convenience. Th e first equation requires that surfaces of constant p satisfy the equation or that p = constant along surfaces satisfying the condition
z
+ K(X,4) = const.
(9.47)
Large Scale Ocean Circulation Hence, p = p(z
67
+ K , +).
(9.48)
Proceeding in a similar fashion with (9.45b) and making use of (9.48) yields the condition that p be constant along surfaces for which (z/sin +)
+ L = const.
(9.49)
Hence, the two conditions are consistent if p varies along surfaces q = constant where K = L sin +. (9.50) 7 = (z/sin +) L(h, +),
+
For convenience choose (9.51)
P = -(1/g)Q'(7L
where prime denotes d/d7. Then
P = sin 4Q(7)
+ A(h, +),
(9.52)
where A ( h , +) is an arbitrary function. For the ideal-fluid model ( K = 0), (9.52) is the most general solution that includes an arbitrary barotropic mode. T h e restricted conditions (9.45) for an arbitrary barotropic mode can be manipulated to yield p d s i n 4 pz)m
=pdsin
4P J A
= pz/(sin
4
Pz)
.
(9.53)
Hence, surfaces of constant density and surfaces of potential vorticity coincide. Thus the requirement that the solution for P include an arbitrary barotropic mode imposes an additional restriction to the class of idealfluid solutions found by Welander and (9.52) is included as a subset of Welander's solutions. For the diffusive system ( K # 0) the left-hand side of (9.6) must vanish separately and the requirement on Q(7) becomes (9.54) so that
Q = aecn+ b7
+ d,
(9.55)
where a, b, and c are constants. This form for Q is equivalent to the exponnential solution given by Needler (1967). It was shown earlier (Veronis, 1969) that if zonal diffusion of density is included as a term, K,p,,/a2 cos2 4, the exponential solution is still valid. Hence, (9.55) is a solution for this expanded system. Needler points out that the more general solution (9.52) is valid if the density field is time-dependent so that ap/at is included in (9.2a). In this case L and A in (9.50) and (9.52) are arbitrary functions of time as well as of and A.
+
68
George Vwonis
Needler's solution (9.52) is limited in its usefulness as are the other analytical solutions found for the thermohaline circulation because it cannot satisfy all of the physical boundary conditions. T h e restrictive character of a solution of the type given by (9.52) is that it ties the vertical and horizontal dependences to a particular form. In general, one would hope to be able to satisfy conditions on w and T at the surface, z = 0, as well as the condition of zero normal velocity at the bottom, z = --h(X, 4). However, once any two of these conditions are chosen the other is essentially determined. Needler (1967) and Veronis (1969) have discussed this inadequacy of the analytical solution in some detail and in his latest paper Needler shows that the same inadequacy is still present inthesolution given by (9.52). Nevertheless, the value of such models as an aid for increasing our understanding of oceanic processes cannot be denied. I n the process of trying to fit the results to observed distributions new insights into the more general system are generated and, even though these models may eventually be discarded, they serve as an important step toward a more satisfactory model.
G. THEROLEOF DIFFUSION In developing analytical thermocline models it is possible to obtain density distributions which exhibit gross features of the observed field. The results obtained by Welander (1971) and by Needler (1972) exhibit many similar features even though the former restricted his attention to ideal-fluid models. In the formulation of the problem with diffusion it would appear that one ought to be able to satisfy a more complete set of boundary conditions than with the ideal-fluid model. Yet the restricted character of the analytical solutions does not admit this possibility. The question which is naturally raised at this point is: what role does diffusion play in the thermohaline circulation? Welander (1972) has attempted to answer this question by means of a scale analysis of (9.1) and (9.2). He defines a variable M(X, 4, z ) as
M
=s:P
dz
+ 2Qu2 sin2+ /:wo dh,
(9.56)
where wo = w(h, 4,0) is the (presumed known) vertical velocity at the upper surface. Then u = -(1/2Qa sin w = (1/252u2sinz + ) M A ,
'u =
(1/2Qu sin
+ cos +)MAZ (9.57)
P = M , ,p = - (l / g ) M Z 2 ,
Large Scale Ocean Circulation
69
and the following equation in M can be derived by substituting (9.57) into (9.2a) and (9.2b) cot +MAM,,,
= 0.
(9.58)
Welander’s M equation (9.58) is simpler in form than Needler’s P Eq. (9.6), the difference being that Welander has essentially incorporated the barotropic mode into his definition of M . Th e boundary conditions can be written as
M
=
zcw
sin2
+ fw od ~ ,
M,,
= -gpo(h,
+)
at z = o (9.59)
0
and cot +MA= a(h, M,)/a(A, +),
M,,
=0
at z = -h(h,
4).
(9.60)
If the equations are scaled so that the momentum and mass equations contain no parameters, an advective vertical scale depth defined by 6,
= (2fla2W/gpo)1’2
(9.61)
emerges, where W is the amplitude of w o. A (diffusive) vertical scale depth defined by
6,
= K /W
(9.62)
also enters into the problem, and the M equation takes the form
where E =S,/S,. Welander makes the implicit assumption that E is small. If M is now scaled to be O( 1) at the top surface and if an undetermined scale depth 6 is introduced, then M , AM16 (Welander incorporated a barotropic velocity into his discussion but his assumptions relegate it to an unimportant role in his argument) and the boundary conditions at the top surface require M w 1, A M / P w 1. (9.64)
-
Keeping the diffusion term in order to meet the diffusive boundary conditions requires that some other term must balance the diffusion term in (9.63). It follows readily that if the first two terms balance, then E=
P,
A M = 6’.
(9.65)
But then the last term in (9.63) is larger than either of the two terms kept so this balance is inconsistent.
George Veronis
70 Balancing
EM^^^^ and cot +MAM,,, (with cot 6 w 8,
AM
+
1) yields (9.66)
M E ~ .
The neglected term is O ( P )compared to the terms kept and this balance is therefore consistent. In this case the scale depth is E , or 6, in dimensional terms, and the change in M is small. Hence, the vertical velocity is essentially undiminished over the depth of the diffusive region. I n downwelling regions (w < 0) the balance between vertical diffusion and vertical advection would mean that the density increases with depth. (With constant w and K , it is easily seen that p grows exponentially with depth.) Hence, one is forced to abandon the emphasis on vertical diffusion and to seek a new balance. It is easy to verify that an ideal-fluid region is consistent and that the conditions on 6 and AM are 87-1,
AMml.
(9.67)
Here, the dimensional scale depth is 8, the advective depth. T he conclusion is that in downwelling regions the surface density is simply convected downwards and laterally and that a particle of fluid which sinks in this fashion is eventually advected to a region where upwelling occurs. Near the surface of the upwelling region diffusive processes are important and serve to adjust the density to the surface value. Because of the planetary vorticity relation, aw/ax = (v cot +)/a, the vertical convergence generated in a downwelling region is associated with an equatorward transport. Hence, a flow pattern, shown qualitatively in Fig. 11, is set up. Below the advective region diffusive processes may again be important as indicated in the figure. However, the important consequence of Welander's argument is that the thermocline layer may be an ideal fluid regime bounded above and below by diffusive layers.
\
Equator
Polar region
FIG. 11. Qualitative flow pattern for thermohaline circulation according to the scale analysis given by Welander (1972). In regions of upwelling near the surface (denoted by D) diffusive processes are important. Elsewhere, the density is determined by an advective balance (denoted by A). In deep water diffusion may be important.
Large Scale Ocean Circulation
71
H. REMARKS ABOUT THERMOHALINE CIRCULATION MODELS There are several features of thermohaline circulation models that merit more detailed study. First, the use of a constant eddy diffusion coefficient in the density equation surely requires a thorough examination. Although such constant eddy coefficients have been used to advantage in many inquiries, the experimental and theoretical framework on which eddy coefficients are based is shaky. We know, for example, that vertical mixing in a stratified fluid is less intense in more stably stratified regions than it is in less stably stratified regions. Even in the case of homogeneous, rotating fluids, numerical construction of a macroscopic eddy diffusion coefficient shows that the latter varies with distance from the boundary (Deardorff, 1972). Furthermore, the physical processes that are responsible for mixing in surface waters are quite different from those which induce mixing in deep and abyssal waters. If Welander’s conclusion is correct, i.e., if the thermocline is an ideal-fluid layer sandwiched between upper and lower diffusive regions, there is additional evidence that the use of a single, constant, eddy coefficient is suspect. Recent observations of the distribution of tritium in the oceans indicate that mixing takes place more or less along surfaces of constant density (Rooth and Ostlund, 1970). Although this idea has long been accepted by observational oceanographers, it has been disregarded to a large extent in modeling the thermohaline circulation. If the observed vertical distribution of tracers is attributed primarily to vertical mixing, the importance of the latter process is grossly exaggerated and even qualitative results which emerge from the use of an exaggerated vertical mixing coefficient must be viewed with suspicion. Observations of chemical tracers near the bottom of the ocean (Craig et al., 1970) show that a constant eddy mixing coefficient in that region is sometimes remarkably accurate. However, the same authors have shown that in some cases it is not possible to model the mixing process in that fashion. T h e mixing coefficient that arises from these studies may be larger than 200 cm2 sec- l. Since the values used for studies of the thermocline cluster around 1 cm2 sec-l, it is evident that a constant K is not appropriate if it is to account for diffusion in both boundary layers. Even if the bottom boundary layer is disregarded, it is difficult to justify the use of a constant K for near surface waters and for deeper water (Veronis, 1969). A second feature that merits further study has to do with the boundary conditions to be satisfied by the variables. It was stated earlier that surface and bottom conditions on density plus a condition of zero normal flow at
72
George Veronis
the bottom are to be satisfied. A fourth condition is available from the hydrostatic pressure equation since the pressure is determined only to within an arbitrary function of X and 4. Most investigators have tried to specify the vertical velocity at the upper surface as the fourth condition. Even though the restrictive nature of the similarity-type solutions has made it impossible to satisfy all four of the boundary conditions, it should be possible to do so for the general problem. As the problem is formulated, one looks for solutions which are forced by specified conditions of global scale. T h e vertical velocity below the Ekman layer generated by the wind stress is not of global scale, as we have seen. Therefore, one may ask if the thermohaline model should be expected to include the effects of the winds. I t is perfectly possible to specify the density (or density flux) at the surface, the density and the density flux at the bottom, and the condition of zero normal flow at the bottom. Hence, the fourth condition in this case would be associated with the density at the bottom rather than the vertical velocity at the top. Specifying the density at the bottom is equivalent to specifying the velocity shear at the bottom (through the thermal wind equations). The problem thus becomes one in which the fields are completely determined by specified conditions on density and a reasonable kinematic condition. T h e wind stress must then be taken into account by means of a perturbation (or other process) of this basic state. This approach has not been tried but it represents a sensible alternative to the models that have been worked out.
X. Abyssal Circulation T h e circulation driven by wind stress at the surface appears to be confined largely to waters in the top kilometer (or less). T h e thermohaline circulation discussed in the preceding section is oriented toward determining the structure of the thermocline although the associated flow pattern reaches from the surface to the bottom. Given the existence of the thermocline it is a straightforward matter to derive a flow for the deep waters in the open ocean if one makes two plausible assumptions. Since the thermocline is a region in which the mean density decreases upward, any mechanical process which induces vertical exchanges will serve to bring water of lower density to greater depth. In order to maintain the density of the water at a fixed value, it is necessary that dense water be brought upward. Hence, the first assumption for an abyssal model is that across some level (say at 2000 m depth) there is an upward flux of water, w = woo.If this upward flux is taken to be uniform everywhere in
73
Large Scale Ocean Circulation
the open ocean, it generates a horizontal circulation in the deep, homogeneous water which can be readily calculated (Stommel, 1957; Stommel and Arons, 1959). The simplest model will be considered here. T h e ocean is taken to have constant depth, the bottom is at z = 0 and the top of the deep homogeneous layer is at z = r ) + H where r ) @ H . The amplitudes of the velocities are small so the equations for a steady, inviscid flow reduce to the geostrophic, hydrostatic equation (9.1) together with the continuity equation (9.2b). An integration of the hydrostatic equation from z to r ) yields
P =gPm(r) - 4,
(10.1)
and the geostrophic equations become 252v sin 4 = (g/a cos +)ar)/aA,
2Qu sin # = -(g/a)
(10.2a) (10.2b)
ar)/a#.
Eliminating the pressure from (10.2) gives the planetary vorticity equation aw/az = (cot +/+.
(10.3)
Finally, a vertical integral of (10.3) and use of the conditions ~ I , = ~ + ~ = w , ( = c o n s t ) , wI,=,=O,
r)
+HwH
(10.4)
yield the relation
V H= aw, tan 4.
(10.5)
Hence, the meridional velocity is determined. For a positive upward flow across the surface, z = r ) H , the meridional velocity is poleward. The vertically integrated continuity equation (9.2b) together with (10.5) and (10.4) provides an equation for u,
+
au/aA = - ( 2 ~ , a / H ) cos 4.
(10.6)
Hence, if these equations are assumed to be valid up to an eastern boundary, say a meridion, A= A,, where u vanishes, (10.6) can be integrated to give u = (2w, a/H)(A,- A)
cos
4.
(10.7)
Stommel (1957) assumed that a western boundary could be added to provide total mass continuity. With this single condition, the flow for a basin can be specified completely although only the total transport in the western boundary layer (not the detailed velocity distribution) is given explicitly. Since a source of deep water must be added in order to supply the upward flux of water through the surface z = r ) H , it is necessary to specify
+
George Veronis
74
the location of the source and to equate the flux from the source to the total upward flux through the surface. The major oceans (Atlantic, Indian, and Pacific) are connected by the Antarctic Circumpolar Current. The latter is not bounded by meridional boundaries in an essential manner so that the (observed) eastward transport of this current must be specified as an added, imposed feature in the present model. Particle trajectories for an idealized basin-configuration of the oceans of the world are shown in Fig. 12. Here, sources of equal intensity were taken to be at the north pole and at 60"S, 40"W. These correspond to the observed sources for deep water in the Greenland Sea and the Weddell Sea, respectively. The transports in the western boundary layers, specified as T / w ,u2, are shown to the left of each basin. There are stagnation points of transport in the western boundary layer at approximately 5"N in the Indian Ocean and 22.5"N in the Pacific Ocean. The flow in the interior of each of the basins is poleward, consistent with (10.5). An arbitrary recirculation, R, around Antarctica can be added. The deterministic part of the transport around Antarctica is shown at the bottom of the figure. Sources for both the Indian and Pacific abyssal circulations are provided via western boundary currents which receive the necessary transport from the Circumpolar Current. The abyssal circulation in Fig. 12 is taken from Kuo and Veronis (1971) and forms the basis for a study of the distribution of tracers in the ATLANTIC OCEAN
1
ANTARCTIC ClRCUYPOLAR CURRENT R + T wh... shown on tn.
..~YoI.
T
IS
0 right
INDIAN OCEAN
I
PACIFIC OCEAN
I
-1
-2
FIG.12. Particle trajectories in abyssal waters for an idealized world ocean with equal sources of water at the North Pole and at 605,40"W, including a recirculation in the Antarctic Circumpolar Current of arbitrary transport R .
Large Scale Ocean Circulation
75
deep oceans of the world. Tracer distributions calculated with the given abyssal circulation show overall agreement with observed abyssal tracer distributions in the world's oceans. However, the uniform upward velocity through the base of the thermocline is an assumption and should be deduced as part of a thermohaline circulation model. Furthermore, the effects of bottom topography should show up strongly in the circulation of abyssal waters. Attempts to include such effects are presently being made.
XI. Laboratory Simulation of Large Scale Circulation" Certain features of large scale ocean circulation can be simulated in laboratory models with containers of fluid on a rotating turntable. Qualitative features of the circulation were reproduced in the laboratory in some early studies by von Arx (1952). A more quantitative basis for laboratory simulations where sources and sinks were used to drive the fluid was given by Stommel et ul. (1958). As the theory of rotating fluids has evolved, more detailed features of laboratory models have been analyzed. Pedlosky and Greenspan (1967) presented a linear analysis of a laboratory model and Beardsley (1969) has substantiated those results with a series of careful experiments and a more comprehensive analysis. Baker and Robinson (1969) used a different laboratory model to exhibit similar features. Kuo and Veronis (1970) and Veronis and Yang (1972) have provided a firmer quantitative basis for the study of the flow driven by sources and sinks (the model of Stommel et ul.). A discussion of the latter will be given here and some recent results from laboratory experiments will be shown. A. THEBASISFOR
THE
SIMULATION
It was shown earlier for models of both wind-driven and abyssal circulation that the principal balance in the interior of the ocean is geostrophic. If the effect of wind stress is incorporated as a vertical flux below the Ekman layer, both models can be interpreted as being driven by sources or sinks at the surface. Thus, the planetary vorticity equation for a homogeneous fluid is awlaz = (cot (6/a)v, (11.1)
and a vertical integration from the bottom of a basin with uniform depth (where w = 0) to the base of the Ekman layer yields z, = (aw,/H)tan
* Jointly authored with C. C. Yang.
(6,
(11.2)
76
George Veronis
where H is the (constant) depth of the fluid and wo is the Ekman flux. Hence, an upward suction into the Ekman layer has the same effect as an upward flux through the base of the thermocline and gives rise to a poleward transport. Now consider a pie-shaped basin rotating about a vertical axis through the apex (Fig. 13). Th e interior, geostrophic equations for steady flow (in nondimensional form) are
2k
XV=
-Vp.
(11.3)
Here, the dimensional variables, v, p , and V have been scaled by means of the dimensional quantities V , P, and 1/a and the order-of-magnitude balance P = paVu has been used.
FIG.13. The source flow experiment in a pie-shaped basin of radius a, rotating with ! about the axis through the apex. angular velocity 2
T he height of the equilibrium free surface is given by (6.2). The equation for the disturbed free surface is z = h(r)
+ 5(r, 6, 9,
(11.4)
where 5 is the change in height due to a rising or sinking of the free surface because of a source or sink. Th e linear condition that the top be a free surface is at ah (11.5) w=-+uat ar
Large Scale Ocean Circulation
77
If u and w are nondimensionalized with respect to V and r with respect to (1 1.5) takes the nondimensional form
a,
w=
[ + Fru,
(11.6)
where [ = (1/V) al/at and F = Q2a/gis the Froude number. T h e incompressibility condition is as before,
v.v=o.
(11.7)
Hence, the interior flow is subject to the constraints of the Taylor-Proudman theorem and, in particular, awI/az = 0.
(11.8)
Since wI vanishes at the bottom (for an inviscid fluid), the vertical velocity at the top must also vanish. Therefore, from (11.6) uI = - [ / F r .
(11.9)
This relation is the laboratory counterpart of the meridional flow (1 1.2) which results from the planetary divergence equation (11.1). Thus, for a source-driven flow ([ > 0) the radial velocity, uI, is negative, i.e. the interior flow is toward the apex (irrespective of the location of the source). For a sink-driven flow the radial velocity is toward the rim of the basin. Hence, the apex models the northern boundary (pole) and the rim the equator. There are two points to be made in connection with (11.9). T h e first is that for source-driven flow the free surface rises locally because a column of fluid moves radially inward without changing its height. Since the equilibrium height increases with radius, such an inward flow raises the level of the fluid at any location. There must be boundary layers near the rim and near the apex since the radial flow must vanish at those extremes. For sink-driven flow a column of fluid moves radially outward thereby decreasing the level of the fluid. The second point has to do with how the change of height affects the local vorticity. Conservation of mass requires that a local increase of height of a column of fluid be accompanied by a horizontal convergence. For an inviscid fluid the angular velocity of inward-moving particles must increase in order to conserve angular momentum. T h e result is an increase in the magnitude of the vorticity. Thus, when S2 is positive (counterclockwise rotation) a source-driven flow will cause the relative vorticity to increase locally throughout the interior. Hence, the overall vorticity pattern must exhibit positive vorticity, i.e., there must be a flow toward the rim in a boundary layer on the left side looking downward. In an analogous manner it is easy to verify that a boundary current on the left side and toward the
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apex is associated with sink-driven flow. Hence, a western boundary current is generated in the pie-shaped basin when the flow is driven by a source or a sink.
B. DISCUSSION OF MORECOMPLETE SOLUTIONS A complete linear analysis of the source-sink flows is not difficult to carry out (Kuo and Veronis, 1971). However, the details require an analysis making use of Stewartson shear layers in a rotating fluid and such a treatment would take us too far afield. Consequently, the present discussion will be confined to a qualitative description of the flow and the reader is referred to the Kuo-Veronis paper for details. Figure 14 illustrates the circulation driven by a source located at the junction between the western boundary and the rim for a pie-shaped basin with a semicircular cross section. T h e fluid in the experiment is a thymal blue solution and the flow is visualized by dye streaks generated by pulsing wire electrodes in the middle layers of the fluid (Baker, 1966). This example is especially instructive because both the eastern and western boundaries lie along the diameter, the western boundary along the left half of the diameter and the eastern boundary along the right half. Hence, the eastern and western boundaries meet at the apex. T h e flow is essentially radially inward everywhere in the interior so that the apex serves as a sink for the interior flow, irrespective of where the actual source is. T h e intense western boundary layer flow is obvious in the figure. Flow along the eastern boundary is slow, consistent with the partial analysis given in the previous section. Hence, at the apex there is a divergent flow parallel to the boundary because the slower moving eastern boundary flow cannot supply the amount of fluid required by the intense western boundary layer so fluid is sucked into the vicinity of the apex from the interior. This convergent flow from the interior is evident in the figure. I n a linear flow the structure of the western boundary layer is controlled by frictional processes. Consequently Stewartson. layers of thickness Ell3 and Ell4 ( E = v/fiu2is the Ekman number introduced earlier) serve to redistribute the fluid from the source and also to satisfy the conditions of zero flow at the solid side boundaries. T h e simplest equations for the western boundary layer admit a solution which satisfies only the condition of zero normal flow at the wall. It is an exact laboratory analog of Stommel’s theoretical model for wind-driven ocean circulation. When a more complete mathematical model is formulated so that tangential velocities also vanish at the side walls, the model is the analog of the HidakaMunk lateral friction model of wind-driven ocean circulation.
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FIG. 14. T h e circulation generated by a source at the southwest corner of a p'ie-shaped basin with semicircular cross section (radius = 45 cm, R = 1.9 cm sec-l, Reyiiolds numbc:r = 1.33, mean height = 9 cm).
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Near Y = a a boundary layer must be included in order to adjust the radial flow to zero at the rim and also to bring the tangential flow to zero there. This boundary layer turns out to be thicker than that near the western boundary. T h e Ekman layer near the bottom serves no function in the lowest order equations except within the boundary layers at the sides and at the rim. Hence, the solution found in the previous section is the exact lowest order solution for the interior flow. T h e analysis for the more complete problem requires including nonlinear as well as frictional terms in the equations. The nondimensional form of the momentum equations for this steady flow is
Rv* VV + 2 k
XV=
-Vp
f
EV'V,
and the free surface condition takes the form w =[
+ Fru +F R v . V[.
Here, the velocity scale is chosen as V = Q/AE1/',where Q is the volume flux per unit time from the source and A is the area of the basin ; the length scale is a, the radius of the basin; E = v / Q a 2 ;the dimensional a[/at is set equal to V [ = Q / A so tha [ = Ell2; the [ scale is chosen as QVa/g= RFa; and the Rossby number is defined as R = V / Q a= Q/QaAE'/'. T h e specific choice [ = E l l 2 is made because for the linear problem the velocities are all O(1) as a result. When the Rossby number is E1/', inertial effects are important. T h e ratio R/E1l2is the Reynolds number, denoted here by y , so linear flow occurs for y < 1 and nonlinear effects are important for y 2 1. A study of nonlinear flows by perturbation methods is possible for y-1 and theory and experiment agree for these moderate Reynolds numbers (Veronis and Yang, 1972). A more direct simulation of the North Atlantic gyre including the Gulf Stream is obtained when the flow is driven by a sink, because the flow in the western boundary layer is toward the north (apex). T h e laboratory Gulf Stream is more intense than the flow some distance from the western boundary and, as a result, the apex serves as a point of convergent flow parallel to the boundary. Hence, the boundarycurrent must fanout at the apexto provide fluid for theinterior flow. As the strength of the sink is increased, the flow in the western boundary layer intensifies. T h e perturbation analysis for y-1 can no longer be 1 and it is used to study the nonlinear flow patterns that occur for y necessary to consider a numerical analysis for the latter cases. A numerical study has been made by Beardsley (1972) for the nonlinear flow in the sliced cylinder model of oceanic circulation. Several features emerge in our experiments with the flows driven by a sink at the southwest corner of the basin. T h e western boundary layer is
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formed near the rim and then flows to the apex before it couples again to the interior flow. Between the rim and the apex the western boundary layer flow is extremely stable. In none of the experiments, even with Reynolds numbers as high as 70, has the western boundary layer shown an indication of becoming unstable. A second feature is that as the flow becomes nonlinear a high pressure region forms in the interior near the apex. Th e western boundary jet leaves the boundary in the vicinity of the apex and much of the fluid is swept around the high pressure region and flows toward the rim on the offshore side of the western boundary layer. Hence, a recirculating gyre is generated near the western boundary. Such a feature appears also in numerical analyses of wind-driven ocean circulation (Veronis, 1966) as well as in other experimental studies (Beardsley, 1969). A photograph of the flow pattern for an experiment with y = 5 is shown in Fig. 15 and the high pressure region at the center of the recirculating gyre is clearly present just to the southwest of the apex. A third feature is that sufficiently intense flows become unstable. As noted earlier, the western boundary layer remains stable. However, as the western boundary jet leaves the vicinity of the apex and penetrates into the interior, swirls formon either side and the flow becomes transient. Figure 16 exhibits both the stability of the western boundery layer and the instability of the jet after it leaves the apex and flows into the interior. For this flow the value of the Reynolds number is 20.
C. THEFLOWDUE TO
A
SOURCE OF DENSEWATER
The experiments reported above have all been carried out with working fluids of constant density. When the density of the fluid varies, it is not possible to simulate the beta effect in the simple manner outlined above. Nevertheless, since abyssal circulation is driven by sinking of dense water in polar regions, it is of some interest to determine what the effect is when the source fluid has higher density than the ambient fluid of the basin. Such experiments are basically transient because the relative amounts of water of different densities change monotonically with time. Even so, however, we were surprised to observe the magnitude of the change from the homogeneous case to the two-fluid case. Photographs from three experiments are shown below. I n all three cases the flow is in the linear range (Reynolds number = 0.75), the rotation rate is 1.9 rad sec-l, the mean depth of the fluid initially is 8 cm, and the ~ . the first density of the fluid in the basin initially is 0.998 gm ~ r n - In experiment the density of the source water is also 0.998gmcm-3. For
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FIG.15. Same as Fig. 14 except that the flow is sink-driven and Reynolds number = 5. For the nonlinear flow shown here a high pressure region forms near the western boundary south of the apex.
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FIG.16. Same as Fig. 15 except that Reynolds number = 20. T h e jet bifurcates as it leaves the apex and forms swirls on either side. These swirls eventually separate and become self-contained eddies which are swept around the high pressure region southwest of the apex.
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the second and third experiments the densities of the source water are 1.001 and 1.075 gm ~ m - respectively. ~ , The motion was observed by two different methods. In the first, illustrated in Figs. 17-19, the source fluid is colored and the penetration of the colored fluid is shown shortly after the experiment was started and then toward the final stage of the experiment. Figures 17a and 17b show the penetration of homogeneous source fluid after 2 min and after 90 min, respectively. The ball of fluid at the apex reflects the disturbance created by the source itself. Since the transport in the bottom Ekman layer is to the left of the flow above the Ekman layer, the increased thickness of the dark band of fluid around the sides and rim of the tank in Fig. 17b shows
FIG.17. Dyed fluid with the same density, p = 0.998, as the ambient fluid is introduced through a source at the apex and makes its way around the sides of the basin. T h e penetration of the dyed fluid is shown (a) 2 min and (b) 90 min after source was turned on.
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the inward penetration associated with the Ekman layer. Note especially that the ball of fluid near the apex penetrates very slowly into the interior and even after 90 min it is confined to a sector within 13 cm of the apex. Figures 18a and 18b illustrate the change that takes place when the source fluid is slightly more dense than the fluid in the basin. Here, it is important to note that the flow pattern in Fig. 18b was photographed after a delay of only 21 min. The denser fluid was generally observed to make the circuit around the sides in a time substantially shorter than that required by the homogeneous fluid. The most striking difference between Figs. 17b and 18b is in the penetration of the ball of fluid from the source near the apex. The penetration into the interior is much greater for the dense fluid and, in fact, it works its way to the rim boundary current near the western
FIG.18. Same as Fig. 17 except source fluid has density 1.001. (a) After 2 min and (b) after 21 min.
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side. This penetrative character of the dense fluid is important in connection with the observed high concentrations of tritium in the bottom 100 m of the ocean at mid-latitudes (Rooth, 1971). Tritium is a bomb-produced radioactive tracer which has a relatively high concentration in the surface waters and throughout the depth in polar regions but its presence is barely observable in deep water at mid-latitude. The present experimental result suggests that the high concentration near the bottom is associated with the Ekman layer flux of dense water which sinks to the bottom in the Greenland Sea and then spreads southward in the Ekman layer. The source fluid in the experiment stays close to the bottom whereas in the homogeneous case it tends to distribute vertically. When the density of the source water is large, as in Figs. 19a and 19b, the circuit around the sides of the tank is even faster (there is a ten-minute
FIG.19. Same as Fig. 17 except source fluid has density 1.075. (a) After 2 min and (b) after 10 min.
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FIG.20. Experiment of Fig. 17 repeated. The flow of fluid originally in the basin is visualized by the thymol blue technique. Shows flow after 5 min.
interval between the two photographs). The penetration of the ball of fluid from the source is now erratic but very strong. Earlier photographs show that the ball extends to the rim within four minutes after the beginning of the experiment. Figure 19b shows another feature of the highdensity source flow. There is a strong flux of fluid out of the western
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boundary current about a third of the distance from the apex. This feature was observed in all of the experiments when the density of the source fluid was 1.075. The source fluid in this experiment also tends to stay close to the bottom. The second method of observation is shown in Figs. 20-22, where the thymol blue technique mentioned earlier was used to mark the fluid which is initially in the basin before the experiment was started. Figure 20 shows a flow similar to that of Fig. 14 although the source is located at the apex in the former and at the junction between the western boundary and the rim in Fig. 14. The flow here is steady and consistent with what is expected from linear theory. The ball of fluid resulting from the source near the apex is visible in Fig. 20 and a steady eddy occurs near the rim as the intense western jet impinges on the boundary.
FIG. 21. Experiment of Fig. 18 repeated. The flow of fluid originally in the basin is visualized by the thyrnol blue technique. (a) After 5 min and (b) after 7 min.
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The startling contrast introduced when the source fluid is slightly denser ( p = 1.001 gm cm-3) is shown in Fig. 21. The photographs of the fluid pattern in the homogeneous fluid were taken two minutes apart, and the disordered character of the flow is apparent in each photograph. The western boundary current is toward the rim only in the southern half of the region in Fig. 21a but even there it has reversed and exhibits eddy-like structure in Fig. 21b. Near the rim the eastward jet of homogeneous fluid reverses as the source flow works its way around the rim. The latter behavior is seen in the marked fluid near the rim on the westernmost radial marker. Later photographs show that this reversal progresses eastward with the source fluid. The chaotic behavior associated with a source of dense fluid is exhibited in Fig. 22a where just after the start of the experiment even the
FIG.22. Experiment of Fig. 19 repeated. The flow of fluid originally in the basin is visualized by the thymol blue technique. (a) After 2 min and (b) after 12 rnin.
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interior flow no longer exhibits behavior expected for the homogeneous experiment. T he boundary layer near the rim is completely reversed and there is no longer a southward western boundary current. T h e flow pattern after 12 min is shown in Fig. 22b. Th e intense flow out of the western boundary is again apparent and the interior flow is a collection of completely disordered eddies. There is not much hope of obtaining a complete analysis of the flows generated by a dense fluid source. However, some features, such as the rapid penetration of the dense source fluid via the Ekman layer and the diverging fluid from the western boundary layer, can be obtained in a model more suited for the particular purpose.
ACKNOWLEDGMENT Support from the National Science Foundation under Grant GA 25723 is gratefully acknowledged. REFERENCES BAKER, D. J. (1966). A technique for the precise measurement of small fluid velocities. J. Fluid Mech. 26, 573-575. BAKER,D. J. and ROBINSON, A. R. (1969). A laboratory model for the general ocean circulation. Phil. Trans. Roy. SOC.London Ser. A 265, 533-565. R. C. (1969). A laboratory model of the wind-driven ocean circulation. J. Fluid BEARDSLEY, Mech. 38(2), 255-271. BEARDSLEY, R. C. (1972). A numerical model of the wind-driven ocean circulation in a circular basin. J. Geophys. Fluid Dynam. (in press). BLANDFORD, R. R. (1965). Notes on the theory of the thermocline. J. Mar. Res. 23, 18-29. BLANDFORD, R. R. (1971). Boundary conditions in homogeneous ocean models. Deep-sea Res. 18, 739-751. BRYAN, K. (1963). A numerical investigation of a non-linear model of a wind-driven ocean. J . Atmos. Sci. 20, 594-606. BRYAN, K. (1969). Climate and the ocean circulation. 111. The ocean model. Mon. Weather Rev. 97, 806-827. BRYAN,K., and Cox, M. D. (1967). A numerical investigation of the oceanic general circulation. Tellus 19, 54-80. CARRIER, G. F., and ROBINSON, A. R. (1962). On the theory of the wind-driven ocean circulation. J . Fluid Mech. 12, 49-80. CARRITT, D. E., and CARPENTER, J. H. (1958). The composition of sea water and the salinity-chlorinity-density problems. Physical and chemical properties of sea water. Nut. Acad. Sci.-Nut. Res. Counc., Publ. 600, 67-86. CHARNEY, J. G. (1955a). The generation of ocean currents by wind. J . Mar. Res. 14, 477-498. CHARNEY, J. G. (1955b). The Gulf Stream as an inertial boundary layer. Proc. Nut. Acad. Sci. U S . 41, 731-740. CRAIG,H., SCHLATER, J. G., CHUNG, Y., EDMOND, J. M., KROPNICK, P. M., and WEISS, R. F. (1970). Geochemical and temperature profiles in equatorial Pacific bottom water. Trans. Amer. Geophys. Union 51, 326 (abstr.).
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DEARDORFF, J. W. (1972). Numercial investigation of neutral and unstable planetary boundary layers. J. Atmos. Sci. (in press). V. W. (1905). On the influence of the earth’s rotation on ocean currents. Ark. Mat., EKMAN, Astron. Fys. 2, 1-53. EKMAN, V . W. (1908). Die Zusammendruckbarkeit des Meerwassers. Publ. Circ. Cons. Explor. Mer 43, 1-47. ERTEL,A. (1942). Ein neuer hydrodynamischer Wirbelsatz. Meteorol. Z. 59, 277-281. FOFONOFF, N. P. (1954). Steady flow in a frictionless homogeneous ocean. J. Mar. Res. 13, 254-262. FOFONOFF, N. P. (1962a). Physical Properties of sea water. “ T h e Sea,” Vol. 1, pp. 3-30. Wiley (Interscience), New York. FOFONOFF, N. P. (196213). Dynamics of ocean currents. “ T h e Sea,” Vol. 1 , (Interscience) pp. 323-395. Wiley FOFONOFF, N. P., and FROESE, C. (1958). Program for oceanographic computations and data processing on the electronic digital computer ALWAC 111-E. PSW-1 Programs for properties of sea water. Fish. Res. Ed. Can., Manuscript Rep. Ser. No. 27 (unpublished manuscript). H. (1968). “ T h e Theory of Rotating Fluids.” Cambridge Univ. Press, GREENSPAN, London and New York. L. N. (1963). On a time-dependent motion of a rotating GREENSPAN, H. P. and HOWARD, fluid. J. Fluid Mech. 17, 385-404. HIDAKA, K. (1949). Mass transport in ocean currents and lateral mixing. J . Mar. Res. 8, 1 932-1 936. Holland, W. (1967). On the wind-driven circulation in an ocean with bottom topography. Tellus 19, 582-600. K. (1966). Ekman drift currents in the Arctic Ocean. Deep-sea Res. 13, 607HUNKINS, 620. JEFFREYS, H. (1962). “ T h e Earth.” Cambridge Univ. Press, London and New York. Knudsen, M. (1901). “ Hydrographical Tables.” Williams & Norgate, London. Kozlov, V. F. (1966). Certain exact solutions of the non-linear equation for density advection in the ocean. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okean 2, 1205-1207; see Atmos. Oceanic Phys. 2, 742-744 (1 100). Kuo, H. H. and VERONIS, G. (1970). Distribution of tracers in the deep oceans of the world. Deep-sea Res. 17, 29-46. Kuo, H. H., and VERONIS, G. (1971). The source-sink flow in a rotating system and its oceanic analogy. J. Fluid Mech. 45, 441-464. LYNN,R. J . , and REID,J. L. (1968). Characteristic and circulation of deep and abyssal waters. Deep-sea Res. 15, 577-598. MARGENAU, H. and MURPHY, G. M. (1949). “ T h e Mathematics of Physics and Chemistry.” Van Nostrand-Reinhold, Princeton, New Jersey. MORGAN, G. W. (1956). On the wind-driven ocean circulation. Tellus 8, 301-320. MUNK,W. H. (1950). On the wind-driven ocean circulation. J. Meteorol. 7 , 79-93. MUNK,W. H. (1966). Abyssal recipes. Deep-sea Res. 13, 707-730. MUNK,W. H., GROVES,G., and CARRIER, G. F. (1950). Note on the dynamics of the Gulf Stream. J . Mar. Res. 9, 218-238. NEEDLER, G. T . (1967). A model for thermohaline circulation in an ocean of finite depth. J . Mar. Res. 25, 329-342. NEEDLER, G. T. (1972). Thermocline models with arbitrary barotropic flow. Deep-sea Res. 18, 895-903. NIILER,P. A,, ROBINSON, A. R., and SPIEGEL,S. L. (1965). On thermally maintained circulation in a closed ocean basin. J. Mar. Res. 23, 222-230. PEDLOSKY, J. and GREENSPAN, H. P. (1967). A simple laboratory model for the ocean circulation. J. Fluid Mech. 27, 291-304.
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PHILLIPS,N. A. (1963). Geostrophic motion. Rew. Geophys. 1, 123-176. PHILLIPS,N. A. (1966). The equations of motion for a shallow rotating atmosphere and the “ traditional approximation.” J . Atrnos. Sci. 23, 626-628. RAMSEY,A. S. (1964). “ Newtonian Gravitation.” Cambridge Univ. Press, London and New York. REID, J. L. (1965). “ Intermediate Waters of the Pacific Ocean.” Johns Hopkins Press, Baltimore, Maryland. ROBINSON, A. R., and WELANDER, P. (1963). Thermal circulation on a rotating sphere; with application to the oceanic thermocline. J. Mar. Res. 21, 25-38. ROOTH,C. (1971). Private communication. ROOTH,C. and ~ S T L U N D ,G. (1972). Penetration of tritium into the Atlantic thermocline. Deep-sea Res. 19,481-492. STOMMEL, H. (1948). The westward intensification of wind-driven ocean currents. Trans. Amer. Geophys. Union 29, 202-206. STOMMEL, H. (1955). Lateral eddy viscosity in the Gulf Stream Systems. Deep-sea Res. 3,88-90. STOMMEL, H. (1957). A survey of ocean current theory. Deep-sea Res. 4, 149-184. STOMMEL, H. (1965). “ The Gulf Stream.” Univ. of California Press, Berkeley. STOMMEL, H., and ARONS, A. B. (1959). On the abyssal circulation of the world ocean. I. Stationary flow patterns on a sphere. Deep-sea Res. 6, 140-154. STOMMEL, H., ARONS,A. B.,and FALLER,A. J. (1958). Some examples of stationary planetary flows. Tellus 10, 179-187. SVEFUIRUP,H. U. (1947). Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific. Proc. Nut. Acad. Sci. U S . 33, 318326. SVERDRUP, H. U., JOHNSON, M. W., and FLEMING, R. H. (1942). “ T h e Oceans.” PrenticeHall, Englewood Cliffs, New Jersey. J. C., and HAMON, B. V. (1960). Some measurements of deep currents in the SWALLOW, eastern North Atlantic. Deep-sea Res. 6, 155-168. SWALLOW, J. C., and WORTHINGTON, L. V. (1961). An observation of a deep countercurrent in the western North Atlantic. Deep-sea Res. 8, 1-19. VERONIS, G. (1963). On inertially-controlled flow patterns in a ,%plane ocean. Tellus 15, 59-66. VERONIS, G. (1966). Wind-driven ocean circulation. Deep-sea Res. 17, 17-55. VERONIS,G. (1969). On theoretical models of the therrnohaline circulation. Deep-sea Res. 16, Suppl. 301-323. VERONIS, G., and YANG,C. C. (1972). Non-linear source-sink flow in a rotating pie-shaped basin. J. Fluid Mech. 51, 513-528. VONARX,W. S. (1952). A laboratory study of the wind-driven ocean circulation. Tellus 4, 311-318. WARREN, B. (1963). Topographic influences on the Gulf Stream. Tellus 15, 167-183. WEBSTER, F. (1961). The effect of meanders on the kinetic energy balance of the Gulf Stream. Tellus 13, 392-401. WELANDER, P. (1959). An advective model of the ocean thermocline. Tellus 11, 309-318. WELANDER, P. (1968). Wind-driven circulation in one- and two-layer oceans of variable depth. Tellus 20, 1-16. WELANDER, P. (1971a). Some exact solutions to the equations describing an ideal-fluid thermocline. J . Mar. Res. 29, 60-68. WELANDER, P. (1971b). The thermocline problem. Phil. Trans. Roy. SOC.London Ser. A 270.69-73.
The Wave Resistance of Ships JOHN V. WEHAUSEN Department of Naval Architecture University of California. Berkeley. California
I . Introduction . . . . . . . . . . . . . . . . . . . . . . I1 . The Measurement of Wave Resistance . . . . . . . . . . . A . Residuary Resistance and Refinements-Froude’s Method . B. Momentum Considerations . . . . . . . . . . . . . . C. Wave-Pattern Analysis . . . . . . . . . . . . . . . . 111. The Analytical Theory of Wave Resistance . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . B. The Exact Problem in an Inviscid Fluid . . . . . . . . . C . Perturbation Expansions . . . . . . . . . . . . . . . . D . Methods of Solution . . . . . . . . . . . . . . . . . E. Further Results, Variations. and Extensions . . . . . . . F‘. Numerical Methods . . . . . . . . . . . . . . . . . . G . Comparison of Theory and Experiment . . . . . . . . . H . Applications of the Theory . . . . . . . . . . . . . . I . Higher-Order Theories . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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For Georg Weinblum for his 75th birthday
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I Introduction A newcomer to the subject would probably ask the following questions . What is wave resistance? How do you measure it? How do you compute it? We shall discuss in the following pages all three questions and try to describe how they have been answered up to now . The answer to the question. “What is wave resistance?”. is not as elementary as it might seem at first glance. One observes that waves follow a moving ship. supposes that some expenditure of energy is required for 93
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their generation, and probably recognizes that the propagation of these waves is associated with the presence of a gravitational field. However, wave formation is not the only expense of energy. One also knows that water is endowed with viscosity and that any body moving through a viscous fluid experiences a resistance, partly due to tangentially acting stresses on the body, but partly also because boundary-layer growth and, of course, separation if it occurs yield a resistance resulting from integrating normal components of the stress over the body. A new question then is, “Can we clearly separate resistances due to gravity alone and to viscosity alone?” That is, is there a “gravitational resistance ” and a “ viscous resistance ”? A moment’s thought shows that this is not likely. T h e frictional resistance, i.e., the part due to tangential stress, will certainly depend upon the wave profile along the ship. On the other hand, the wave pattern itself is going to depend in some fashion upon the ship’s boundary layer and wake. T h e effects of gravity and viscosity interact in essential ways. There seems to be no neat practicable definition of gravitational or wave resistance without introducing assumptions or approximations. Since any discussion of measurement or calculation of wave resistance necessarily touches on its definition, we shall have to return to this question subsequently. Finally, one may ask, “Why make any decomposition at all?” The answer is primarily economic. One would like to be able to predict the power requirements of a full-scale ship from its designed lines. T o do this by calculation from the Navier-Stokes equations is beyond our powers. T o do it by model tests leads to dilemmas that can be resolved, at least in part, if we can find the components of resistance, especially as they depend upon the effects of gravity and viscosity. This has usually been simplified further into trying to separate the resistance into parts due to wave making and to friction. As we shall see below, the separation has been moderately successful, in fact, enough so that in this account of the subject we shall be able to avoid dealing with viscous resistance except in a superficial way. A further reason for trying to separate wave resistance and to understand its relationship with hull shape is ship design. For conventionally shaped ship hulls one cannot reduce the frictional resistance significantly by redesign of the hull. Eddy resistance can be kept small by properly designing the afterbody. Added resistance resulting from bilge vortices can also be avoided by proper design. (Wave-breaking resistance, although it shows up in certain experiments as viscous resistance, will be lumped here with wave resistance). This still leaves great freedom in designing the hull, and experience and observation have shown that optimum designs (from thestandpoint of resistance) are those that make the smallest waves. Rational
The W a v e Resistance of Ships
95
hull design requires an understanding of how a suitably defined wave resistance is connected with hull geometry. Although apparently not the first person to try to examine in a systematic and scientific way the resistance of a ship, William Froude (1810-1879) seems to have been the first to appreciate fully the differing roles played by friction and wave making in ship resistance and the significance of this difference in trying to project data from model tests to full-scale size. This is already evident in a memorandum of December 1868 from Froude to the Chief Constructor of the Navy in which he outlines the advantages of model tests and proposes construction of a model tank. In a later paper (1876) Froude discusses the components of ship resistance and their implications for ship design with such insight that this paper can still be read with profit. Froude’s analysis laid solid foundations for future tests of ship hulls and his ideas still dominate the subject. However, Froude’s method, which will be briefly described later on, has one important disadvantage, explicitly recognized by him, namely, that it did not allow one to predict by purely analytical means the resistance of a ship. T h e first significant step in this direction was taken by J. H. Michell(l863-1940) who in a paper published in 1898 derived an analytic expression for the wave resistance of a ship moving in a calm, inviscid fluid. I n much the same way that Froude’s ideas have dominated the field of ship model testing, Michell’s results have dominated much of the subsequent analytical study of wave resistance. T h e present account of the subject is divided as follows. T h e first part treats the measurement of wave resistance. This in turn is divided into a brief exposition of Froude’s method followed by a more lengthy treatment of the determination of wave resistance from wave-pattern measurements. The second part treats analytical methods for calculating wave resistance and some applications. There exist already a number of expository accounts of the separate parts of this article. For a discussion of the problem of separation of wave and viscous resistance one can hardly do better than to read Sharma’s (1964, 1965) account of this subject. For questions concerning the determination of wave resistance from direct wave measurements a paper by Eggers, Sharma, and Ward (1967) gives not only an excellent survey of the subject but also a valuable bibliography. For the analytical theory of wave resistance there are a number of expository accounts directed at different audiences. There is a fairly comprehensive treatise by Kostyukov (1959, 1968). Of the others listed below perhaps the most comprehensive are by Wigley (1949), Lunde (1951a), and Sabuncu (1962b), the last not being really very accessible because of language. T h e others are more restricted
John V . Wehausen
96
in scope or in prospective audience. See the following references: Bessho (1957b), Gadd (1968), Havelock (1926b, 1951), Inui (1954,1957), Kostyukov (1959, 1968), Lunde (1951a, 1957, 1969), Maruo (1957), Sabuncu (1962b), Wehausen (1956), Weinblum (1950, 1959, 1963), Wigley (1930b, 1935, 1949).
11. The Measurement of Wave Resistance A. RESIDUARY RESISTANCE AND REFINEMENTSFROUDE’S METHOD Determination of the wave resistance by experiment has proceeded along two rather different paths. I n the older one, to be described here, wave resistance is usually identified with a component extracted from the total resistance by a method similar to that originally devised by Froude. This component is the “residuary resistance,” defined below, or some part of it. It is easy to criticize the method and to propose modifications, and many persons have done so. Although some of the analyses that have been proposed in recent years are ingenious and throw considerable light on the components of ship resistance, I believe that it will be clear from the brief description below of Froude’s method that, in fact, the effects of gravity and viscosity are “ inextricably interwoven,” to borrow a phrase from Froude himself. The purpose of this section is then not to expound all the recent developments in this field but rather to give some perspective concerning this method of trying to assess the contribution of wave making to the resistance of a ship. Consider a ship moving steadily on a constant course in a calm sea. Let its length, say a t the waterline, be L and its velocity U , and let the density of the water be p and its kinematic viscosity v. T h e ship moves at some expense of energy. Let E be the rate of energy expenditure. We may call R = E / U the resistance of the ship. If we neglect certain other physical parameters, especially the vapor pressure of the water and surface tension, and consider our ship as one of a family of geometrically and dynamically similar ships, we may write R = F ( L , U , p, v, g). We must include g, for we know that the gravitational force plays an important role in the formation of the wave pattern behind the ship. We need not dwell upon the ways of making this dimensionless, but write down immediately
C, =
R zp U2L2
=f( U/(Lg)1‘2, UL/v)= f ( F n , Rn),
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97
where F n is the Froude number and Rn the Reynolds number. C, is evidently a function of the two variables Fn and Rn and can be represented by a surface over the (Fn , Rn)-plane that will be characteristic for this family of ship hulls. If one experiments with just one of these hulls, how much of the surface can one construct! Experiments at one place and during a restricted time interval will allow variation only in U, i.e., variation along a ray in the (Fn , &)-plane defined by
With a given hull very little variation in A is possible in practice. T h e value of g is practically constant, although one can make thought experiments with dropping elevators, merry-go-rounds, or towing tanks on the moon ; and v can be varied rather little for practically useful fluids, perhaps by a factor of two. We shall call this result the “ ship-model tester’s dilemma ” (although it is only one of several), for it shows why tests on a model cannot in principle give information about the prototype (which we assume to be much larger). Since Ap $ A,, the model experiments simply explore the wrong part of the surface C,(Fn , Rn), as shown schematically on Fig. 1. If one wishes to allow for some variation in v for the model test, one may replace the ray labeled with A, by a narrow sector. It is not customary to use three-dimensional representations. Instead, one projects the curves cut out of the surface C,(Fn, Rn) onto the (Rn, C,)-plane, and, in order to bring the curves conveniently onto one graph, uses a logarithmic scale for
FIG.1. Sketch showing parts of surface CT(Rn,Fn) determined by a model test and needed for prototype ship.
98
John V . Wehausen
0 0
30.00 30.00 24.00
6.350 6.350 7.938
+ (1
X 0
12.00 9.00 6.00 4.00
15.875 21.167 31.750 47.625
REYNOLDS NUMBER
FIG.2. Values of resistance coefficient CT obtained from tests with models of Lucy Ashton. [From Hinterthan, 1956 (Fig. 18).]
R n . Figure 2 shows a typical plot of such resistance curves for a family of models. T h e constant-A curves are labeled by the model lengths, which may not be consistent in a dimensionless plot but is hardly confusing since in practice v changes but little. T h e curves cut out of the surface C,(Fn, Rn) by constant-Fn planes are also labeled, but again not dimensionlessly. T h e prototype in this case, Lucy Ashton, is only 190 f t long. I n order to give some idea of the magnitudes of F n and Rn , we remark that usually 0.1 < F n < 0.5 and that for the usual merchant ship 0.15 < F n (0.3. T h e values of Rn depend then upon ship size. For five-foot models (the smallest usually tested because of laminar-flow difficulties) the Rn range corresponding to 0.15 < F n < 0.3 is approximately 1 x lo6 < Rn < 2 x lo6, for a 20-foot model, 7.5 x lo6 < Rn < 1.5 x lo7, and for a 500-foot ship, 1 x lo9 < Rn < 2 x lo9. T h e dilemma mentioned above was resolved after a fashion by a bold assumption of William Froude. T h e assumption as we shall state it below is not really in the form given by Froude. Froude did not know about Froude or Reynolds numbers, or about dimensional analysis. He discovered were by experimenting with geometrically similar models that, if UL kept constant, the wave patterns of models of different sizes were similar.
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However, the equation given below represents (almost) in our notation his assumption, and in any case is usually called Froude’s hypothesis.
C,(F?Z, Rn) = C,(Fn)
+ CF(R7.2).
It follows immediately from this assumption that the curves Fn = const. are all at a constant vertical distance from one another in the ( C , , Rn)-plane and that the curves A = const. are all congruent, one being obtained from the other by sliding along the curves Fn = const. Hence, in applying Froude’s assumption to model testing, one tests the model at the same Froude number as the prototype, which in any case is more practical than testing at the same Reynolds number, and then finds the total resistance coefficient of the prototype from cTp
cT?n
- cFm -k
cFp
*
What is necessary now, of course, is to know the curve C,(Rn) for the given hull form In fact, it would be sufficient to know the function CT(Fn, Rn) for any fixed value of Fn . Any such curve is known as an “ extrapolator.” One could consider Froude’s assumption as purely an empirical one to be supported by the examination of data like those shown in Fig. 2. However, there was a rationale behind Froude’s assumption. Let us decompose C , into parts obtained by integrating over the ship’s wetted surface the components in the direction of motion of the normal and tangential components of the stress vector:
CT(Fn Rn) == cnoi-m(Fn> Rn) f Ctang(Fn Rn). This decomposition is, of course, precisely defined. Now one argues that the tangential force is primarily determined by the viscosity of the water and the normal force primarily by the effect of gravity as a result of the = waves produced by the ship, or, in dimensionless variables, that Cnorm Cnor,(Fn)and C,,,, = Ctang(Rn).In a final bold step one replaces the tangential force by the tangential force on a flat plate of the same water-line length and wetted-surface area as the ship at rest. If S is this area, one defines a frictional-resistance coefficient by C,(Rn) = (flat-plate resistance)/ipUU2S. (Note that L2has been replaced by S . Other coefficients will now be similarly redefined). Th e difference C, - C, = C, is now assumed to be a function of Fn only and is called the “ residuary-resistance ’’ coefficient. The advantages and weaknesses of Froude’s assumption are nearly obvious. T he choice of C, has yielded a function of Rn that is independent of hull form. T he only problem is to measure it with adequate accuracy over a sufficiently wide range of Reynolds numbers. This problem, for years a lively topic among naval architects, and still not a dead one, will not be discussed here. The curve in Fig. 2 labeled “ Schoenherr Line ” is one of
100
John V . Wehausen
the standard formulas for C, , but there are others. T h e residuary resistance contains everything else : wave resistance, eddy resistance, curvature effects, effects of trim and sinkage, the difference between the true wetted area and the wetted area at rest, etc. T h e weakness of the assumption is, of course, to assume that C , is a function of Fn alone. A frequent refinement is to estimate an eddy or form resistance as a function of the form: C, = a bC, , where a and b are empirical coefficients depending upon the hull form. I t is not uncommon to find a = 0 and b g 0.16. T h e constants are chosen so that at the lower end of the measured total-resistance curve, where the C , curve becomes approximately parallel to C,, the curves C , and C, a bCF run together. T h e curve C , -- CF- C , is often identified with the wave resistance, although not without awareness on the part of naval architects of the shortcomings of the definition. A more sophisticated version of Froude’s method, but also one requiring more effort in application, is to define the viscous resistance as half the resistance of a double model tested at appropriate Reynolds numbers in a wind tunnel or deeply submerged in a towing tank. If this is denoted by C,(Rn), then C , - C, will still depend upon Rn as well as Fn inasmuch as the double-model experiment does not include the effect of varying wave profile and sinkage and trim upon frictional resistance, or even upon such phenomena as separation. Even at low Froude numbers when there are almost no waves, the wakes of the single and double model may be different because of the lack of turbulent interchange across the free surface with the single model. On the other hand, C, comes much closer to including all the effects of viscosity associated with a given hull form than do the methods based upon flat-plate frictional resistance. Many further refinements of Froude’s assumption are possible and, since any improvement makes more accurate the prediction of prototype performance, there are frequent proposals. Since it is not our main purpose to discuss such matters, we cite only a few recent papers concerning this topic: Lackenby (1965), Shearer and Cross (1965), Weinblum (1970), Shearer and Steele (1970). As mentioned earlier, the purpose of this somewhat lengthy but still superficial introduction to the phenomenological theory of ship resistance is to try to provide some perspective about the nature of the resistance derived from such experiments, for a theoretically computed wave resistance is often compared with them.
+
+ +
B. MOMENTUM CONSIDERATIONS In the following paragraphs we shall derive some general expressions, in terms of certain surface integrals, for the force acting upon a body in steady translational motion. The derivation will be carried out for a viscous fluid
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101
satisfying the Navier-Stokes equations and the boundary conditions appropriate to them. However, the final formulas are also correct for an inviscid fluid if one sets v = 0. It is not obvious that this will be so, for the boundary conditions appropriate to an inviscid fluid are different from those for a viscous fluid. However, one may verify the correctness of the formulas for an inviscid fluid by examining, as one proceeds through the proof, the effect of altering the boundary conditions on the body. We begin with a coordinate system fixed in the fluid, 07 directed upward (against gravity), & in the direction of motion, and 0 5 to the starboard; &% coincides with the undisturbed water surface. We recall that for an incompressible Navier-Stokes fluid the stress tensor is given by Tij
+p(ui, +
== -@ij
j
(2.1)
uj. i),
where (&, Z 2 , 2,) = (3,7, Z), the velocity components of the absolute . shall use the conmotion are (u, w , w ) = (ul, u2, u,) and ui, = a ~ , / & ~We vention that repeated indices denote summation. The velocity components must then satisfy the equations
pauilat
+
ui, i = 0, puk u i , k
= -psi - pg6i2
+
pu, k k
i=
i
3*
2i
(2.2) (2*3)
If the velocity at any point on the surface S of the body is V = (U,, U,, U3) = ( U , V , W ) ,then the boundary condition on the body is u i = Ui on S. For an inviscid fluid it is only the kinematic condition u 1. n1.= U.n. I I'
(2.4)
(2.5)
Since we shall not be considering the effects of cavitation, there is no loss in generality in taking the atmospheric pressure to be zero. We shall also neglect surface tension, for it plays practically no role for bodies of the size of concern to us. The dynamical boundary condition on the free surface is then simply Ti,
n, = -pni
+ p ( u i ,,+ u,,i)nk= 0,
i = 1, 2, 3.
(2.6)
If the fluid is inviscid, one need only set p = 0 above, to obtain the usual
p
= 0.
(2.7)
In addition, the kinematic condition must be satisfied. If we express the free surface by jj = Y(2,5,t ) , this takes the well-known form
Y$,jj, t)u - v
+ Yrw + Y t = 0.
(2.8)
John V . Wehausen
102
Consider now the fluid bounded by the wetted surface S of the body and a (possibly moving) control surface t; enclosing the body. Part of t; may consist of free surface but the combined surface S and C must bound only fluid. Figure 3 shows schematically two possibilities. The total momentum of the fluid in the volume V is Qi
pui dV,
=
and the rate of change of momentum is
If we now use the Navier-Stokes equations to replace aui/at and use the continuity equation ui,{= 0, we obtain
=
s,
bui(Uknk-Uknk) -tTiknk]dS-pgl VIsiZ,
(2.10)
where I VI is the volume of V. On any part of I; that is the free surface both terms in the integrand vanish, so that the integral over Z extends only over its submerged portions. In the integration over S the first part of the integrand vanishes because of the kinematic condition (2.5). The remainder is just the force acting on the body:
Fi = - J j T i k nk dS. S
If we let Zc stand for the submerged part of C, we finally obtain
FIG.3.
Sketches illustrating surfaces S, C, , and Z;,
(2.11)
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103
We now add a further restriction to the surface C. I t is to move together with the body. If we now assume that the motion has continued for such a long time that the mean motion of the velocity field within V no longer changes with time, then dQi/dt = 0. (Note that we are allowing turbulent motion in the wake as long as the mean motion is steady.) This now gives us a formula for the mean force acting on the body in terms of an integral over the control surface Zc on which
U, nk = Un,,
(2.13)
where U is the velocity of the body in the direction of 62. I n finding the mean value of (2.12) one must take especial care with the quadratic term pui uk nk . If we write temporarily u i = G i uit, where C i gives the mean velocity field, then
+
pui uknk = pGi zik nk
+pui’uk’nk.
We now drop the bars over the mean velocity and denote it by ui . We then find for the mean resistance R = -Fl, the only component of Fi in which we are interested here, the following expression :
R =-
1
[pul(un, - ukn k )-pn, - p u i n k
+
ZC
+
dS. (2.14) T h e quantities u i ,p , R are now all mean quantities, and -Pzdltz&‘nk is a Reynolds stress, which may be much more than the viscous stress. Although we have used only one component from (2.12), one should be aware that much of the ensuing analysis could also be carried through for side forces and sinkage, and, with a further extension of the calculations, for trim. This is an exact formula in the sense that no mathematical approximations have been made in deriving it from the original assumptions. As was mentioned earlier, this is also an exact formula for an inviscid fluid if one sets p = 0. Even if the viscosity of water is taken into account, the term with p as coefficient is usually quite small compared with the other terms. This does not mean, however, that the values of the integral will be nearly the same in either case, for the velocity uk and the pressure p satisfy different differential equations and boundary conditions in the two cases. I t will be convenient to make here a slight change in notation. Since the motion has been assumed steady in the mean in the frame of reference of the ship (taken here as Oxyz), we may write p(ul,k
uk,l)nk]
7,z, t ) = Ci(R - ut,7, 2 ) = C i ( X , y, z), P(%7,%t ) =$(a - ut,7 , 2 )= y , z),
Ui(X,
m,
Y(2,x, t ) = F(X - ut,X) = F(x, z),
(2.15)
John V . Wehausen
104 so that au, -=-u-
at
as,
au, -=-
azk
ax
asi,
etc.
ax,
With this understanding we shall drop the tildes. Formula (2.14) is not altered by this. It is convenient to make some special choices for 6,. Let us first suppose that the ship is moving down a rectangular canal and let V consist of the fluid bounded by the side walls, bottom, and planes ahead of and behind the body perpendicular to the walls and bottom. Of the plane areas bounding this volume we denote the one on the starboard wall by C,, on the port wall by X p ,on the bottom by X H ,ahead of the body by C A ,and behind by C B. Taking account of boundary conditions on the solid surfaces and of the simple form of the normal vectors (always directed out of the fluid) to C,, we find
R=-J
puz dx dy
+ J puZdx dy + J =P
ZS
-
LA
[pu(U-~)-~-pU'2+2pu,]
+J
[pu( U - U) - p
pu,, dx dx
ZH
- pU12
dydx
+2 p , ] dy dz.
(2.16)
We now modify this formula by taking X A far enough ahead of the body so that there is no disturbance of the fluid, a possibility that we assume as one of our boundary conditions. Then only p = pgy remains in the integrand and the upper limit of the y-integral is 0. We may imbed this integral in the Z A integral if we take account of the fact that the upper limit in the y-integral over Z B is Y (xB, x). We then have R=-J
p u, dx dy ZS
+J ZP
pu, dx dy
+J
pug dx dx
ZH
(2.17) where xBis the x-coordinate of the moving plane C B, and z p and xs are the x-coordinates of the walls X P and Cs, respectively. Finally we make one more change suggested by Landweber and Jin Wu
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105
(1963) and, in the form used below, by Sharma (1965). Define a “total head ” as follows : P g H h y , z ) =P
+pgy +
*PP[(U
+ +4
- y2 v2
(2.18)
and let pgH0 = &pU2.
I t follows from Euler’s integral that for irrotational motion of an inviscid fluid H = Ho everywhere. Hence H - H,, represents a kind of measure of the effect of viscosity. A straightforward manipulation allows one to put (2.17) into the form
R
=-
J
p ~ dx ,
dy
+J
pu,
dx dy
ZP
ZS
+21 p J
(- u2 + v2
+J
pug dx
dz
ZH
+ w2) dy dz
ZB
(2.19) I n the case of irrotational flow of an inviscid fluid, one is left with only the last two terms:
1 R = - p J ( - u2 v2 w 2 )dy dz
+ +
ZB
+ -21 pg J
eS
Y2(xB, Z ) dZ.
(2.20)
ZP
This is, of course, a genuine “ wave resistance,” but, as mentioned earlier, its value will not be the same as that of the corresponding terms in (2.19). T h e integrals over C s , C p ,and C H and the contribution of the term 2pu, in the integral over X B are generally negligible in comparison with the other terms. However, this is not the same as neglecting viscosity. If the body is moving in an infinitely deep fluid unbounded horizontally, one may choose as the control surface Xc a rectangular box as we did above for the canal, noting, however, that Xs, C p, and C, are no longer physical surfaces, so that the integrands in the integrals over these surfaces are more complicated :
Z’ H
[puv
+PUT - p(uY+ v,)] dx dx.
(2.21)
John V . Wehausen
106
If one lets the planes, Xs, X p , X H recede to infinity, these integrals converge to zero and one is left with the last three integrals of (2.19) with zp=-a, xs =00. If one sets p = 0, one obtains again (2.20). On the other hand, if one lets X B recede to infinity and sets p = 0, one obtains R = J puw dx dy - J puw dx dy - J puv dx dx.
(2.22)
LY
BP
&S
Whenever we have set p = 0 above, we have also supposed the Reynolds stress to vanish.
1. Separation of Wave and Viscous Resistance Since the last two terms of (2.19) coincide in appearance with (2.20), a true wave resistance in an inviscid fluid, one might be tempted to simply define these two terms as the “ wave resistance ” and
as the “viscous resistance.” However, if one applies (2.19) to a flow without a free surface and with no gravity, one still has the term 1 2
-p
”!
(-u 2 + v 2
+w”)dydz,
EB
even though one now has a true viscous resistance. This integral must evidently play a role in each component in any attempt to separate viscous from wave resistance. The first attempt to base a separation of viscous and wave resistance upon momentum considerations goes back to Tulin (1951). Since then the ideas have been further developed and refined by Jin Wu (1962), Landweber and Wu (1963), Sharma (1964, 1965), Tzou and Landweber (1968), Baba (1969a), Brard (1970a,b), and others. The main idea in these attempts is that there is a boundary-layer plus wake region (hereafter BLW) where the velocity field is rotational, but that outside this region it is irrotational. Then, from a well-known theorem concerning vector fields, we may decompose v as follows : v=v,+v,
(2.23)
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107
where
1 v --curl
S,,
~-+rr
curl v -dv, r
curl v = curl vR, curl vI= 0,
v I ~ v - v R ,
div vR= 0,
(2.24)
div vI= 0.
Since vI is irrotational, it can be generated from a velocity potential vI such that grad rpl = v I . T h e function v1 is by its definition defined everywhere in the fluid, and in the region exterior to BLW. However, v R is also irrotational exterior to BLW and can be generated there by a velocity potential v R .Hence, exterior to BLW = grad(vl
+
YR).
Within BLW this representation does not hold, for v, is not irrotational there. However, v1 within BLW is a harmonic extension of cpI outside BLW, and since it seems reasonable to assume that the field v is continuous with continuous first and second derivatives, the field vRand hence vI may be presumed to have the same properties. I n particular, the function y I will be continuous everywhere in the fluid. Now we should like to assume that vRcan be extended harmonically into BLW. However, we cannot expect that this extension can be made without there being a discontinuity in v R .It will be convenient to define the extended q R so that the discontinuity is on the plane x = 0. Let us now define a potential flow vP
= grad(?l
+
P)R)
= grad Y
P>
which is defined everywhere, but with a possible discontinuity within BLW on z = 0. We further define a wake velocity field by
vw = v - vp. Although we know the boundary condition satisfied by v on the ship's wetted surface, namely v = ( U , 0,O), we cannot easily deduce the boundary behavior there of v, or vw . In particular, we cannot expect that vp.n= U n l , the boundary condition that would ordinarily be imposed for an irrotational flow in an inviscid fluid, for, to speak loosely, the function ' p p exterior to BLW does not know that there is a solid boundary at S. Presumably there does exist a stream surface S' starting at the ship's bow, perhaps not closed, upon which this condition will be satisfied. We now substitute this decomposition into (2.19), discarding the terms in p as being negligible, but retaining the Reynolds stress. T h e possible
John V. Wehausen
108
discontinuity of y p on x = 0 requires an easy modification of (2.19), for the control surface C, must now include the two sides of the surface of discontinuity in order to exclude any discontinuity within the fluid volume V . After some small manipulations one obtains the following formula:
1
I-zpS
[vw2+ww2 t 2 v ~ v w +2wPww] dy dx
ZW
1 + j p g / [ Y w z +2YPYw] dz, where C, is that part of C, within BLW. We now define a "wave resistance " by the first three integrals in (2.25):
1 R w = 2 p Jz,[-upz
+ up2 + wpz]dy dz (2.26)
Corresponding to this we define a
"
viscous resistance by "
R,=R-Rw
(2.27)
The definition of R, is sometimes simplified by discarding the last two integrals of (2.25), although it has not been really established that they are negligible. T h e definitions are precise. Whether they are useful depends to some extent upon how well they lend themselves to independent measurement. Measurement of R, will not be discussed, but various practical approximation procedures are discussed by Sharma (1964, 1965). Measurement of Rw will be discussed in the next section, but the underlying assumption
The Wave Resistance of Ships
109
of the measurement methods excludes the integral over the surface of discontinuity. An empirical proof of the usefulness of the definitions would consist in showing that the sum of the independently measured resistances adds up to the measured total resistance. We note that it is not true that, as defined above, Rw is a function of Fn alone and R , of Rn alone. Consequently we do not anticipate that measurement of these two components will necessarily be helpful in predicting full-scale resistance from model tests. We emphasize that Rw as defined here is not the same as the wave resistance that would be obtained if one were to find the velocity potential that satisfies rp, = Un, on S. As mentioned above, there presumably exists a surface S‘, containing S in its interior and coinciding with S at the bow, upon which rpPn = Un,. If S’ is not closed, but has a ‘ I wake ” trailing off to infinity, one must exclude from the integrals over C, that part occupied by the tail of the body. Th e surface S‘ is, however, irrelevant to the proposed definition of R , for S. On the other hand, it is useful to be aware of S’, for it suggests methods of making numerical experiments for estimating the effect of the wake upon wave resistance. 2. Wave-Breaking Resistance Recently Baba (196913) has pointed out that the rate of energy loss in breaking of ship-generated waves can be treated as a separate component of the total resistance. Th e occurrence of wave breaking is clearly a phenomenon in which gravity, i.e., the Froude number, plays the dominant role. On the other hand, energy lost in wave breaking will not be recorded in the “ wave-resistance ” measurements as defined above, but rather in the measurement of the “ viscous resistance.” This has been confirmed by Baba’s experiments. C. WAVE-PATTERN ANALYSIS Formulas like (2.17) and (2.19) can obviously serve as the basis for an experimental method of determining the resistance of a steadily moving body. However, the measurements would clearly be tedious, even after neglecting the terms in p, for measurements of H , u, v , w are necessary over the whole plane Z B . If one assumes irrotational flow, however, and hence (2.20), Eggers (1962, 1963) has shown that one may calculate R from surface-profile measurements alone. There are two versions of the method. I n one, transverse profiles are measured and R is derived from (2.20), in the other, longitudinal profiles are measured and R is calculated from (2.22). Both methods rely upon linearization of the boundary conditions at the free surface and hence upon measuring profiles far enough
John V. Wehausen
110
behind or to the side of the ship (or other moving body) that no serious error is introduced by this approximation. On the other hand, it puts no restriction upon the form of the ship or the cause of the disturbance, as will be done in the next section when wave resistance is calculated analytically. T he effect of the neglect of viscosity will be discussed later.
1. Transverse Profiles in a Canal We begin with the case of transverse profiles. Suppose, as we did in deriving (2.17),that the ship is moving steadily in a canal. Let its bottom be in the plane y =-h and its side walls in the planes z = &b. The undisturbed free surface is contained in y=O. As before, the coordinate system is moving with the velocity U of the ship. It will be convenient to suppose that the ( y , z)-plane intersects the ship amidships. Let x = x B be a plane sufficiently far behind the ship so that for x < x B it will be an acceptable approximation to replace the exact free-surface boundary conditions for irrotational flow, namely,
by the linearized conditions
+g Y(x,). =o, + UY,(x, z ) 0,
- Uyz(x, 0, 2)
yy(x, 0, ).
(2.29)
=
or, after eliminating Y , by yzz(x,0, z)
+ Kyy
= 0,
K
=g / U 2 .
(2.30)
In addition to Laplace’s equation, y must also satisfy the boundary conditions
(2.31) A complete set of solutions of Laplace’s equation that satisfy these three boundary conditions is given by the functions
I
cos k,x sin k,x cash pn(Y
nr + h)cos 2b ( z - b),
p,
> 0,
k n > 0,
(2.32)
111
The Wave Resistance of Ships where kn2= p n 2- (nT/2b)2, =pzp
(2.33)
+ (nrr/26)2.
Th e p,, must satisfy (2.34) and the p n pmust satisfy (2.35)
If n 2 1, there is a single solution for p, for each value of U2/gh. If n =0, there is one positive solution for po if U2/gh < 1, but none if U2/gh > 1. There are infinitely many solutions for p n p ,which we shall number with p = 0, 1, 2, . . . in order of increasing value. If n = 0, there is no solution pooif U2/gh < 1, but pooexists if U2/gh > 1. There is no steady-state solution if U2/gh = 1. The functions {cos(n~r/26)(z- 6)) form an orthogonal set of functions. However, the families (cosh p n ( y h), cos p n p ( y h), p = 0, 1, 2, . . .} do not, contrary to the behavior of such families in some similar situations. Since these form a complete set of functions for 'p in the region x 5 x B , we may express 'p as follows:
+
x (-a, cos k,x g
+& uk,,x
+
nrr + 6 , sin k,x)cosh p,(y + h)cos (z 2b
1 h
+
-
b)
nrr 2b
cnpexp k,, x cos p n p ( y h)cos - ( z - b).
(2.36) T he associated free surface is than given according to the linearized boundary condition by
Y(x,z ) = C (a, sin k , x n
+ b, cos k,x)cos nrr -( z - b ) 26
nnexp k,, x cos - ( z +b). (2.37) P 2b Th e first summation starts with n = 0 or n = 1 according as U2/gh < 1 or U2/gh> 1, respectively. I n the second summation the term corresponding to n = p = 0 is absent in the first case, but present in the second. f
1 c,, n.
112
John V . Wehausen
The coefficients a,, b, , cnPhave not as yet been determined, but they are uniquely defined by q in the region x 2 x B. Before substituting the expression for q into (2.20), the latter may be simplified a little by taking advantage of linearization in the double integral and integrating up to the surface y = 0 instead of y = Y(x, x). After a tedious but straightforward calculation one obtains the following expression for R. m 1 R = -pgb C e,-l(an2 2 n=Oorl
+ bn2)
where e0 = 4, E , = 1 for n 2 1. Note that the coefficients cnP do not enter into the determination of R, as indeed they should not inasmuch as (2.20) must be valid for any plane behind the ship and the contribution from the second summation in the expression for q must vanish as x B-+ -a.The first summation in q will be said to be the “ free-wave potential ” and the second to represent the “ local disturbance.” There now remains the problem of determining the coefficients a,,, b, needed in R. Multiply the expression for Y ( x , x ) by b-’ cos(mrr/2b)(x- b) and integrate from -b to b. This yields the equation
+ b, cos k,x + C cmPexp k,, x
a , sin k , x
P
=-Ib
mrr
Em
-b
Y(x,X)COS -(Z- b) d X 2b
= Y,(x).
(2.39)
Since x < 0, it is evident that the exponential terms will become negligible if I X I is large enough. Let us suppose that profiles have been measured at N locations x l , . . . , x N where this is the case. We then obtain N equations to determine the two unknowns a , , b, :
a , sin k m x i
+ b, cos k m x i= Y,(x,),
i = 1, . . . , N .
(2.40)
Any two of them will suffice provided sin k,(x, - x,) # 0. However, if N > 2, it is appropriate to use the method of least squares to obtain a more reliable estimate of a , and b, . T h e necessary formulas are as follows:
a, = - A - l
C Y,(x,)cos k,xi
sin k,(xi - xi),
i. j
b, = A - l
C Y,(x,)sin
k m x isin k,(xi - xi),
i. i
1 A = - C sin2 km(xi- x,). 2 i.j
(2.41)
The Wave Resistance of Ships
113
I n planning an experiment it is obviously advantageous to maximize A in order not to amplify measurement errors in Y ( x i , z). For N = 2, A = 4 sin2 k,(xl - x z ) is obviously a maximum for k , 1 x1 - x21 = 7r/2. However, since k , varies with m, an optimum spacing for one value of k , may be poor for another one. Since one usually plans to determine all the necessary values of a , and b, from one set of measurements, Y(xi,x), i= 1, . . . , N , this is a disadvantage of taking N = 2. This difficulty is ameliorated by taking a greater number of cuts and using least squares, for the function
A,(k,S)
1 N-1
1
C sin2 k,(xi - x j ) = C 2
=-
i,i
q=
1
( N - q)sin2 qk,6,
(2.42)
where 6 = xi + 1 - x i has an increasingly wider flat part about k,6 = 7712 as N increases. Figure 4 shows 2 W 2 A , ( k , 6 ) for N = 2, 4, 6 , 8, and 10.
FIG.4.
Graph of function showing effect of spacing and number of cuts upon A.
It is possible to avoid this difficulty with A if one can measure the slope Y , ( x i ,z) at the same time as Y ( x i ,z). T h e double measurement on a single profile then suffices in principle to determine the coefficients, for the equations become a, sin k,x
+ b, cos k,x = Y,(x),
a,,,k , cos k , x - b, k , sin k, x = Y,'(x),
(2.43)
114
John V . Wehausen
with solution
a,=-
1
[Y,(x)k, sin k,x
+ YA(x)cos k,x],
km
(2.44)
1
b,
=
knl
[Y,(x)k, cos k,x
-
YA(x)sin k,x].
If double measurements are made on N cuts x,, . . . , x N , a least-squares fit gives simply the average of the values obtained from each single profile. There remains the question of how far behind the ship the profiles must be measured before we can neglect the exponential terms. Since k,, < k,, and k,, < k, ,, it suffices to examine k,, , KO,, and k,, . It is not difficult to establish that
,-,
k,, > rr[@/h)2
+ (n/2b)2]1’z.
(2.45)
Hence k,, > r / h and k,, >.rr/2b. We recall that k,, does not exist if the speed is subcritical, U2/gh< 1. We assume this for the moment. If XI > # max(h, Zb), then both exp k,,x and exp k,,x are less than 0.009. This is an unsatisfactory result, for it does not take account of the size of the c,, and is overconservative for large h or b. Th e most satisfactory investigations have been numerical experiments by Kobus (1967) and Landweber and Tzou (1968), both carried out for channels of infinite depth but finite width. By introducing a mathematical disturbance approximating that caused by a ship, they are able to estimate the effect of the local disturbance upon Eggers’ method. Th e results of Landweber and Tzou are too elaborate to summarize briefly. However, the conclusion of both papers for shiplike forms is that profiles taken more than a ship length behind the ship will result in errors of less than 1 o/o for the range of practical Froude numbers. This problem has also been discussed in Eggers (1966, p. 667), where a computed curve displays for various distances behind the ship the ratio of the resistance computed from transverse-cut analysis neglecting the effect of the local disturbance to the true resistance. The Froude number was 0.36. Th e results conform to those of Landweber and Tzou. If U2/gh> 1, KO, appears. This may be very small if the flow is only slightly supercritical and is bounded above by .rr/2h. It seems likely that this exponential term may have to be taken into account if measurements are made at the usual distances behind a ship model.
I
2 . Transverse Profiles in Unbounded Fluid Several situations can be analyzed in a manner similar to that above: motion in an infinitely deep canal, motion in horizontally unbounded fluid of finite depth, and in horizontally unbounded fluid of infinite depth.
The Wave Resistance of Ships
115
We choose the last case, for it is most different from that above. However, we shall treat it in approximately the same way. Consider a region x < X, sufficiently far behind the ship that linearized free-surface boundary conditions can be used. Let a(k)= (KP)1/2,
P(k) = &[K+ ( K 2 + 4R2)112],
+ k2.
P2 = a'
(2.46)
One may show that the most general potential function satisfying the linearized free-surface condition and vanishing as y + - 03 is given by g w 1 p,(x, y , z ) = - dk -exp ~(k)y{[-F,(k)cos ax
U o
a(k)
+ [-F2(k)cos
ax
+ Gl(k)sin axlcos kz
+ Gz(k)sin axlsin kz}
+
x { [ K pcos py - ( k 2 p2)sin pyI x [F(k,p)cos kz G(R,p)sin kz]),
+
x < x, ,
(2.47)
where the F i , G i , F(R, p), G(R, p ) are general, but must satisfy certain conditions to ensure proper behavior of the integrals. T h e free surface is then given by ~ ( xz,) = Jo
m
~A{[F, sin ax + G, cos altos k z
+[F,sin ax + G, cos axlsin kz} + JmdpImdkexp(k2 +
p2)1/2x[~(~, p)cos
0
~z
+~
( kp)sin , kz].
0
(2.48) Formula (2.20) for the resistance takes the following form :
1 0 m R =p dyJ dz(-qx2 2 .--m -m
1
+ qy2+ y a 2 )+ 21 pg J m Y 2dz.
(2.49)
-m
Substitution of q into this expression for any fixed x 5 x B is simplified by the fact that the second double integral can be discarded and by use of the following Fourier-integral indentities. If
f n ( x )= Sm[/3,(k)cos kx 0
+ y,(k)sin kx] dk,
n = 1, 2,
(2.50)
John V . Wehausen
116 then
(2.51) m
Jm -m
n, m = l , 2.
fnfrndx=nJo (pnBrn+YnYrn)dk,
After simplification one eventually finds
(2.52) There now remains the problem of determining the Fi and G i , the analogs of the an and b, in the canal. We take Fourier transforms of Y(x,z):
1 *
57
j
Y(x,x)cos kz dz = Y,(x, k)
-m
+
1
7,
m
+ 1 d p ~ ( kp)exp(k2 , + W
= F,(k)sin a ( ~ ) x ~ , ( ~ ) c a(k)x os
p2)1/2x
0
(2.53)
I-,
Y(x,z)sin kz dz E Y2(x,k) = F,(k)sin
a(k)x
+ G, ( ~ ) c oa(k)x s +j
W
+G(K, p)exp(k2
+
p2)1/2x.
If profile measurements Y(xi, z) are made far enough aft that one can neglect the integrals in p, one obtains essentially the same formulas as were found for the a, and b,: N
F,(k) = - A - l
Y , ( x , , k)cos a(k)x, sin a(k)(xi- x i ) , i. i = 1 N
G,(k) = A - l
1 Y,(xj,k)sin a(k)x, sin a(k)(xi
- xj),
(2.54)
i.1=1
1
A = - 1 sin2 a(k)(xi- x j ) , 2 i. i
p = 1, 2.
The remarks about choosing 1 xi -xi 1 made for the canal carry over here without change. If one has simultaneously measured slope Yz(x,,z ) and amplitude Y ( x i ,z), then as before one may derive estimates of F , and G, for each x i and average for greater accuracy:
G,(k)= 2[ Y,(x,k)a(k)cos ax - YDz sin a x ] .
The Wave Resistance of Ships
117
In practice one will, of course, compute the F, G for a discrete set of k's sufficiently close to one another to allow calculation of R with the required accuracy. Under what conditions may the local disturbance be neglected? T h e only numerical experiments that are relevant are those of Kobus (1967) and Landweber and Tzou (1968) discussed earlier, and these were carried out for infinitely deep canals. It seems reasonable to suppose that the same results hold here. In fact, as pointed out by Sharma (1966), a numerical quadrature of (2.52) in which one takes Ak = r/2b and k, = nr/2b yields exactly (2.38) with appropriate correspondences between the a,, b, and the functions F, , G, , F,,G,. Consequently one may expect the two formulas to be valid under the same circumstances.
3. Longitudinal Profiles in Unbounded Fluid From the viewpoint of convenience the measurement of longitudinal profiles has important advantages over transverse profiles, the most important being that a wave gauge can be fixed in position while a ship or model passes by it and that measurements do not have to be made in the ship's own wake. We suppose that there exist values x, > 0 and z p < 0 such that when x 2 z , or z 5 z p it is allowable to use the linearized free-surface boundary condition. If zs' 2 z, and zp' 5 z p , then from (2.22)
-
R(S)+ R(P).
(2.56)
The conceivable contribution from a closing portion of C at, say, x = x B < 0 vanishes as x B-f - co in an unbounded fluid because the disturbance decreases as I x I -1/2, as we shall show later. We consider first only the part R"). In the region z 2 z,, y 5 0 any potential q P ) ( x , y , z) satisfying the linearized free-surface condition and vanishing as y-f - 00 can be represented as follows: (~(s)(x,y,
x) =
k2 f d k exp[--y(k)z]exp - y[-f(k)cos K
UO
kx +g(k)sin kx]
+ 5 Jwdkexp Kk2 - y{[-f,(k)cos kx +g,(k)sin kxlcos y(k)z + [-f,(K)cos kx +g,(k)sin kxlsin y(k)z} k2 +5 JwdpSadk exp[-(k2 +p2)1~2x][cospy + sin py] KP K
0
0
John V . Wehausen
118 where
k K
y(k) = - ( k 2 - KZ)l’Z,
k 2K ,
k y ( k ) = - ( K 2- kZ)l”,
O
K
< K.
The functions, although general, must, of course, give convergent integrals. We can, however, immediately put one restriction upon the functions because we know from the nature of the problem that waves must be outgoing from the ship. In particular, in the second integral only combinations like cos[kx
+ y(k)z],
sin[kx
+y(k)z]
are allowable for the starboard side. This requires that fz = -g1,
The free surface in the region z
YCs)(x,z ) =
gz =f1.
2 x, is then given by
dk 12 exp[-y(k)z][f(k)sin kx +g(k)cos kx] JoK
+ Smdkk([fl(k)sin kx +gl cos kxlcos y(k)z + [-gl sin kx + cos kxlsin y ( k ) z } K
fl
+ ~ m d p ~ m d k k e x p [ - ( k+pz)1’2x] 2 0
0
x u ( k , p)sin kx + g ( k , p)cos kx].
(2.58)
We now substitute the expression for p?(s) into the formula for W“. The calculation is again facilitated by using the theorems about Fourier integrals given in (2.51). After some simplification one obtains the following :
1 7ipgKJKmdk(k2- K 2 ) 1 ’ Z V 1 Z +gl”].
R‘S’ = -
An analogous analysis for z Combining them, we obtain
1
R = 7ipgK
< zp yields
a similar formula for
S, dk(k2- K 2 ) 1 ’ 2 [ f ~+giS)’ S)2 +
(2.59) R“).
m
fip)2
+g\p)z]. (2.60)
119
The Wave Resistance of Ships
If the ship is symmetric about its centerplane section, the usual case, of course, f i p )=f:"),g(:) =g$") and the formula collapses to
1 dk(k2 m
R = TpgK
- K2)1'2[f,2 +g12],
(2.61)
K
where.f,, gl may be from either side. Newman (1963) has given a rather elegant formula for R directly in terms of Y for a symmetric disturbance. Although his derivation seems to rely upon a special form of disturbance, the result does not. For fixed z he finds
where the kernal N ( x ) is given by ni2
N(x) =
sin28 sec 8 cos(x sec 8) d8
0
= - $(l
+x"Yo(x)
+ *77xY1(x)
+ t.rr2x2[YO(X)Hl(X)- Y1(x)ffo(x)l, where Yoand Yl are Bessel functions of the second kind and Ho and H I are Struve functions. A table of N(x) is provided. We must now determine the functions f(") and g(s'. As before, we take Fourier transforms of Y(x,x), but this time with respect to x instead of 2. This yields the following equations after neglecting the contribution from the double integral : 1
m
77
-m
-
Y(x, z)cos kx dx = Y,(k, z )
0
=i
+
>K , (2.63)
1 7-r -
m
J"
-a,
Y(x,z)sin Kx d x - Ys(k, z )
120
John V . Wehausen
Although we can determine the functions f ( k ) ,g(k) from profile measurements Y(x, xi)along a cut z = x i ,it is the functions fl, g,, that are of especial interest. The equations
1 fi(k)sin r(k)zi +gl(k)cos Y(~)z. - - Y,(k, x i ) ,
’- k
fr(k)co~r(k)zi-g,(k)sin
1
(2.64)
r(k)zi= k Ys(k,zi),
for i = 1, . . . , N can be used in the same way as the similar transversecut equations were used. However, in principle only one longitudinal cut is necessary, for with i fixed we find
1
fl(k) =
- 7; [Y,(R, zi)sin r(k)zi
+ Ys(k, xi)cos ~ ( k ) z i ] , (2.65)
1
gi(k) = k [ Yc(k zi)cos r(k)zi - Ys(k xi)sin r ( k ) ~ i ] , always, of course, for k > K. If multiple cuts have been measured, the appropriate use of the data is simply to average the results from each cut alone. We have not yet exhausted what is known about the behavior of Y(x,z), behavior that has important consequences for the behavior of Yc and Y , and hence for f,(k) and gl(k). If the disturbance causing the waves is properly restricted in extent, then far behind the disturbance, the free surface along any longitudinal cut will behave asymptotically like
Y(x, x) = [Asin Kx
+ B cos K X ] ( - - X ) - ~ / ~ .
This is shown in a special case in (3.35). As a consequence both Y, and Y , will behave like 1 K - k I for k near K. On the other hand, since in practice one deals always with only a finite length or record, this behavior will not show up in the Y , and Y , computed from the record. Two ways have been suggested for compensating for the finite length of record. In one, proposed by Newman (1963) and Sharma (1963, 1966), the tail of Y(x, z) is approximated by a function of the form
[Asin Kx
+ B cos Kx](C- K X ) - ~ ’ ~ ,
where the constants A, B, and C are determined by fitting this function to the trailing portion of Y(x, z). They give the necessary formulas for using this procedure. Th e other way of avoiding the difficulty with record length is to measure Yz(x,z ) instead of Y(x,z), for Y , decays very rapidly. Th e formulas corresponding to (2.65) are easily derived. There is some
121
The Wave Resistance of Ships
advantage in solving for y(k)f,(k) and yg,, for then the singularity at k = K is removed. For further discussion of the singularity and truncation formulas see Eggers, Sharma, and Ward (1967, pp. 123-124) and Eggers and Kajitani (1969). Since the regions x < xB and z > zs overlap in an octant of space, there must be some connection between the two forms for q~ used in the transverse and longitudinal wave profiles. This connection is provided by the integrals in the two expressions without exponential terms in either x or z. If in the first integral of (2.47) one makes a change of variable so that a becomes the independent variable, this integral becomes
5 lKWdxexp K y CC’
-
2a2-K2
{[-Fl(k)cos ax ka(az- K2)’”
+ [-F,(k)cos
ax
Comparing with the second integral of to be identical if one identifies
+ G,(k)sin ~ X]C Ok(a)z S
+ G,(k)sin axlsin k(cx)z}. P ) ( ~ ) , one
(2.66)
sees that they appear
This must be false, however, for we have seen that fis’ = -gl( s ) and gis)= fjs), whereas F,, G1, F, , G, are independent. What one must compare is 2a2- K 2 [-$(F, Ka(a2- KZ)l”
+ Gz)cos(ax +kz) + $(Gl - Fz)sin(ax + kz)] (2.68)
with -f:”)
cos(kx
+
’yz)
+g(s)sin(kx
+ yz).
Then one finds
where a
k(a) = - (az - K Z ) l J 2 . K Similarly one finds
(2.69)
122
John V . Wehausen
Hence either set of functions determines the other. If the wave field is symmetric about z = 0, then, as we have seen, f(") = f P ) and g(")=g") so that F z = Gz=O, as one can also deduce directly. In this case one longitudinal profile determines all functions. I n deriving the formulas for fi and gl we have assumed that z > zs was large enough so that the double integral could be neglected. Sharma (1966) has made a rather thorough study of this question as well as of the effect of truncation by means of numerical experiments [the results are also summarized in Eggers, Sharma, and Ward (1967)l. Briefly, he produces a disturbance by means of a known distribution of sources and sinks satisfying the linearized free-surface condition and computes the values of Y(x, x) and of F,(k),G,(K),and R. He then uses the computed values of Y in the various formulas above in order to test the effect of varying z , of using finite lengths of Y(x, z), and of applying the truncation correction mentioned above. I n the case considered the optimum value of z was about SUz/g. It may seem strange that there is an optimum value of 2. Suppose that 20,
I
I I
EXACT THEORETICAL VALUE
9
FROMY -CUT WITH
k
@ TRUNCATlON CORRECTION FROM Yz-CUT WITHOUT
v)
905 4
T
T 2
2
F
d
'
@ TRUNCATION CORRECTION FROM Y -CUT WITHOUT @TRUNCATION CORRECTION
The Wave Resistance of Ships
123
measurements terminate at x, and that L(z) is the useful length of profile at z. Then because of the nature of ship wave patterns, which will be discussed later,
xt?- L(z)N tan 70"32' =2.8. z
Hence increasing z shortens L(z), while decreasing x increases the unwanted effect of the local disturbance. Figure 5 from Sharma (1966, p. 772) shows the effect of truncation upon R , the importance of a truncation correction and the superiority of measurements of Y , instead of Y if no correction is made. Figure 6
THEORETICAL VALUE K2G ( k k O
-"tiL -20
I
I
1 3 TRANSVERSE WAVE NUMBER k / K +
1
I 4
FIG.6. Comparison of free-wave spectra derived directly from knowledge of disturbance and from theoretically computed Y. [From Eggers, Sharma, and Ward, 1967 (Fig. 6, p. 128), by permission of the Society of Naval Architects and Marine Engineers.]
John V . Wehausen
124
from Eggers, Sharma, and Ward (1967) shows the results of another numerical experiment where F(k) and G(k) are computed directly and also from the theoretical values of Y . This is a more discriminating test than a computation of R, and the agreement is impressive. For further discussion of such numerical experiments one should consult the cited papers.
4. Longitudinal Profiles in a Canal It is not possible in a canal to carry through a development analogous to that for an unbounded fluid. First of all, since all the wave energy is channeled down the canal, no pair of longitudinal cuts on each side of the ship can be long enough so that the contribution from a closing cut at x = x B < 0 can be neglected in computing R. Also, since all the energy passing through a longitudinal cut is (ideally) reflected from the walls, there is no net flux. However, as shown by Eggers (1962), it is in principle possible to evaluate the coefficients a, and b, from longitudinal profile data by means of the following formulas: mrr +2 sec (zs2b
a,
=
b,
= 2 sec - (z, - b) lim
mrr
2b
T-m
Y(x,y)sin k,x dx, 1
-
(2.72) 50
Y(x,z)cos k, x dx,
TJx0-T
where z = z, is the plane in which the profile is measured. In practice this procedure has not been successful, partly because tank walls are not perfectly reflecting, partly because it is not possible to get a long enough profile to approximate the necessary limits. Recently, however, Moran and Landweber (1971, 1972), using a modified procedure, have made both numerical and physical experiments with such profiles. T h e former show the theoretical feasibility of the procedure. Results of the latter are consistent with data obtained independently by other means. Perfect reflection is assumed. There are other ways of making use of longitudinal-profile measurements in a towing tank. In most tanks model dimensions are usually limited to a size that avoids the necessity of making wall corrections. Consequently, if a measured longitudinal profile is truncated before the first reflected waves from the tank walls affect it, that section of the profile should approximate well the profile that would be measured in an unbounded fluid. Th e truncated tail must, however, be supplied by a theoretical extrapolation, as proposed by Newman (1963) and Sharma (1963) and described in the last section. One should note that the difficulty of making a reliable
The Wave Resistance of Ships
125
extrapolation increases as the Froude number, and hence the wavelength ( s 2 r U 2 / g )increases, for one has fewer and fewer wavelengths upon which to base an extrapolation. Figure 7 shows two typical records up to the first reflection. However, the procedure seems to work satisfactorily in practice, as will be seen when we turn to experimental results.
FIG.7. Two typical records of longitudinal cuts, including the first reflected wave from the side wall.
5. Other Methods of Direct Measurement When the transverse-cut method was used to determine a,,, and b, for the waves in a canal, this led to a separate calculation for each value of m, preceded by taking a Fourier transform. These calculations required a knowledge of Y ( x i ,zj)= Y i jfor i = 1, . . . , P ; j = 1, . . ., Q. I t seems reasonable to ask why one should not bypass the Fourier transform and determine all the a,, b,, n = 0, . . . , N , in one calculation. If we may neglect the local disturbance, each measured value Y i , gives rise to an equation satisfied by the a, and b, : nr
C (a, sin k n x i + b, cos k,xi)cos (z,2b N
h=O
+
b) = Yij,
i = 1, . . ., P,
j = 1, ..., Q. (2.73)
If PQ 2 2 N 2, there are in principle enough measurements to determine the desired a , , b,. If PQ > 2 N 2, one may exploit the method of least squares to find a “best” solution. This procedure was first suggested by Hogben (1964) and further developed in Gadd and Hogben (1965) and Hogben (1970). Obviously there must be imposed further requirements on the grid of measurement points for the method to be applicable (for example, P > 2 N + 2 , Q = 1 will not work). These problems are discussed in the referenced reports. Another ingenious method due to Ward (1963, 1964) called by him the “XY-method,” is based upon measurement of the force acting on
+
126
John V. Wehausen
a long vertical cylinder situated in a position that would also be suitable for a wave probe measuring longitudinal wave profiles. The fundamental assumption underlying the method is that the force exerted on the cylinder by a plane oncoming wave is proportional to the wave amplitude, a prediction of the linearized theory of water-wave diffraction in an inviscid fluid. Ward’s experimental studies of this assumption are reported in Ward and Snyder (1968). It appears to be adequately substantiated for the intended use. The method relies upon combination of the formulas (2.20) and (2.22), the force components on the cylinder providing the data for the integral along the longitudinal cut and a separate wave gauge providing the data for an approximation to the closing transverse cut. For details one should consult the cited references. A recent modification of Ward’s idea has been given by Roy and Millard (1971).
6 . Eflect of the Wake The underlying theories for the determination of wave resistance from profile measurements all require irrotational flow. When the methods are applied practically, what is measured? If there is indeed a boundary-layerplus-wake region BLW surrounded by a region in which the flow differs only insignificantly from an irrotational one, then a longitudinal profile taken far enough to the side so that the wake region is not intersected should determine the velocity potential y p used in (2.26) to define R , in a viscous fluid. On the other hand, a determination of y p from a set of transverse profiles that intersect the wake, which may not be avoidable, may be expected to show an effect of the fact that part of the data used to determine y p is being taken in a region where the flow is not irrotational. This may be an explanation of some evident discrepancies between quantities deduced from the two sorts of measurements. The most important attempt to study the effect of the wake by means of numerical experiments is in a paper by Tatinclaux (1970). He takes as body an infinitely long vertical strut of ogival section in unbounded inviscid fluid. A wake is artificially produced by giving the form of the vorticity o in the fluid behind the midsection of the ogive. This is equivalent to assigning H,-H inasmuch as the Navier-Stokes equations for steady motion in the coordinate system Oxyx can be written in the form
pgV(Ho - H ) = W x
(V-
V) - ~ A fpv’ v
VV’
(2.74)
Let L be the length of the ogive and 2bo its beam. Then Tatinclaux makes the following special choice for o :
U
(2.75)
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127
for x aft of the ogive. For the region between the midsection and the after end x/L is deleted. For the computations h = 0.1, c = 0.25. A solution of the equations of motion for an inviscid fluid with linearized free-surface conditions is then constructed. The velocity is the gradient of a harmonic function outside the wake to which must be added a term V x A inside the wake. (Outside the wake V x A contributes VpR to the velocity.) T h e form of the surface Y(x,z) can then be calculated. For Fn = 0.25 and for sections X B at one, two, and four model lengths behind the model, Tatinclaux calculates the exact wave profile, an I ‘ asymptotic ” profile that neglects the local effect of thevorticity, and the profile that would exist if there were no wake at all. Differences are substantial even after four model lengths. For the resistance it makes no difference whether the exact or the
7. Experimental Observations T h e results of experiment that will be given here are mostly by way of illustration. In the next section, where it will be of interest to compare analytical predictions of wave-making resistance with measurement, further examples will be given. Figure 8 from Eggers, Sharma, and Ward (1967) shows C , = R,/+pU2S for a mathematically defined ship form, symmetric fore and aft, whose
John V . Wehausen
128
0.10
015
QZO
025
030
FADUDE NUMBER F, = (v/@)
035
ado
---c
FIG.8. Wave resistance for a mathematical ship form determined experimentally according to three methods. The theoretical curve is the Michell resistance (see below) with special corrections. [From Eggers, Sharma, and Ward, 1967 (Fig. 12, p. 133), by permission of the Society of Naval Architects and Marine Engineers.]
precise form need not concern us here. Results are shown for three of the methods discussed above : transverse profiles, longitudinal profiles, and Ward’s XY-method. The results are not inconsistent with one another except in the neighborhood of the Froude number 0.35 where Ward’s method and the transverse-cut method deviate substantially. I t is conjectured by Eggers, Sharma, and Ward that this is a result of separated flow at the stern and an associated more dominant wake. Figure 9 from CaliSal (1972) shows a compilation of results for a standard merchant-ship design, “ Series 60, block 60.” Some of the values were obtained by Ward using either his X Y-method or lateral-slope measurements of longitudinal cuts. Others were obtained by CaliSal from transverse cuts and from amplitudes or lateral slopes of longitudinal cuts. The residuary resistance is also included as well as a theoretically computed curve that will be discussed in the next section. Results from two transverse
129
The Wave Resistance of Ships
2
2o
c
0
A
+ A
x
FOR SERIES 6 0 . C g * 6 0
WARD’S ( 1 9 6 8 ) SLOPE METHOD WARD’S POINTS 11965)
2 STATIONS TRANSVERSE CUT L . S M TRANSVERSE CUT LATERAL SLOPE CUT LATERAL WAVE HEIGHT CUT
FIG.9. Wave resistance for a Series 60, C , = 60 hull determined from various sorts of profile measurements and from residuary resistance. [From CaliSal, 1972 (Fig. lo), by permission of the Society of Naval Architects and Marine Engineers.]
cuts are included with the least-squares value in order to show the spread of values obtainable. Once again the several methods do not seem to be inconsistent, even though at large Froude numbers the spread in values is large. As a test of consistency the wave resistance is not as sensitive as a direct comparison of the functions Fl(K), Gl(K),F,(K), G,(k) in (2.48). These functions, or some modification of them, form what is often called the “free-wave spectrum” of the motion. Figure 10 from Eggers, Sharma, K 2 ( F 2 , G12)1’2, and K2(F12 and Ward (1967) shows K 2 ( F I 2 GZ2)lI2, GZ2 FZ2 G12)’/’ plotted against k/K, as determined by both longitudinal and transverse cuts. Although the agreement is generally good, there is evidently a shifting of phases as k/K increases. Although the determination of wave resistance and of the free-wave spectrum from profile measurements is still undergoing development and refinement, there seems to be little doubt that the method is successful. I n the next section comparisons of computed wave resistance with measured wave resistance from profile measurements will be considered more valid than comparisons with residuary resistance.
+
+ +
+
+
130
John V . Wehausen
FROM LONGITUDINAL Y -CUT -
TRANSVERSE WAVE NUMBER k / K -D FIG.10. Components of free-wave spectrum determined by transverse and by longitudinal cuts. [From Eggers, Sharma, and Ward, 1967 (Fig. 13, p. 133), by permission of the Society of Naval Architects and Marine Engineers.]
Finally it should be emphasized that these presumably more accurate methods of determining wave resistance or of separating in a fairly precise and rational way viscous from wave resistance do not really help much in the problem of extrapolating model test data to full-scale behavior. On the other hand, the greater precision in allocation of causes, i.e., the effects of viscosity as against the effects of wave making, for the resistance of a particular ship hull will almost certainly lead eventually to more efficient hull designs.
8. Some Further References on Wave-Pattern Analysis Since Eggers, Sharma, and Ward (1967) have given a rather complete bibliography of the subject up to 1967, it has not seemed necessary to duplicate it. We list here only a few papers that have appeared since then
The W a v e Resistance of Ships
131
but are not mentioned in the foregoing text: Hogben (1971, 1972), Ikehata (1969), Ikehata and Nozawa (1968), Maruo and Ikehata (1969), Tanaka, Adachi, and Omata (1970), Tanaka, Yamazaki, Ienaga, Adachi, Ogura, and Omata (1969).
111. The Analytical Theory of Wave Resistance A. INTRODUCTION In the preceding section we were especially concerned with methods of measuring wave resistance without being at all concerned about the device that made the waves. It could be a displacement ship, hovercraft, hydrofoil boat, or what not. T h e goal was to find the rate of energy loss in the waves. T h e goal in this section is different. We now wish to start with the geometry of the ship, or the pressure distribution under the hovercraft, and whatever other dynamical specifications are necessary, and predict the wave resistance of the ship. We shall begin by formulating several problems ‘ I exactly,” or almost so, in order that it will become clear what sacrifices have been made when we begin to approximate later on. T h e formulation will be only “almost exact” because we shall neglect viscosity from the start. It would not be difficult to formulate the more exact problem with a viscous fluid, but since we would not really make use of it, little would be gained by doing so. T h e exact problem will turn out to be much too difficult to solve in closed form and methods of approximation must be sought. The methods that will be described here are systematic schemes that in principle allow improvement. There are, of course, valuable single-shot approximation methods, but ones that may be systematically improved would seem to be preferable. On the other hand, one must be aware that such schemes may also be illusory in some respects, as has been pointed out by Lighthill (1951) and others. T h e first such approximation method that was tied to the geometry of the ship was given by J. H. Michell (1898). T h e approximation will be discussed more fully later on, but essentially it consisted in replacing the exact free-surface boundary condition by the usual linearized one and the boundary condition on the ship’s surface by one on the centerplane section analogous to the ‘(thin-wing ” boundary condition. Although the approximation has since then been imbedded in a systematic perturbation scheme
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John V. Wehausen
by Peters and Stoker (1957), it has not really been surpassed. It has been extended to water of finite depth and to canals by Sretenskii (1936), but it is clear from Michell’s paper that he knew how to do this. Extensions to second order in the perturbation scheme have been proposed by several persons and will be discussed later. However, numerical computations are difficult and most of them incomplete in that the second-order computation is applied to only one of the linearized boundary conditions, the one on the body. This is sometimes defended as being “ physically realistic.” In fact, what is meant is that this “ improvement ” is within relatively easy computational reach whereas a more exact treatment of the freesurface condition is not. Since there is no theorem stating that such approximations lie between the Michell value and the true one, they must be considered as numerical experiments, but valuable as such. Another approximation scheme, a “ slender-body ” approximation, would seem intuitively to be better suited to ship forms than the thin-ship approximation. It was introduced more or less simultaneously by Maruo (1962), Tuck (1963a), and Vossers (1962a,b), Tuck’s being the more systematic approach to the method. Unfortunately, the first-order approximation turns out to be deducible from the Michell approximation and the second-order one never to have been brought to the point of computation (see Newman, 1970, p. 80), although one might anticipate that it would be simpler than the second-order thin-ship approximation. In recompense, the slender-body approximation has turned out to be very successful in the theory of ship motions. I n Michell’s original paper only one numerical calculation was given for a simple mathematical hull form and he expressed the hope that others would compare the result with experimental values. In an obituary notice written by his brother, A. G. M. Michell, it is stated that Michell’s paper “ was the subject of an enormous amount of work, expended not only in checking and rechecking, the theory and symbolic calculations, but also in making detailed arithmetical comparisons of the results of the theory with those of recorded tests of full-sized ships and models. Only a small fraction of this work was published, and the fact that the paper appeared without detailed arithmetical or graphical illustrations of the results may, in part, account for its having apparently remained unnoticed by those who undertake the designing of ships until about thirty years later when W. C. S. Wigley in England, and others in Germany, began to recognize its fundamental importance.” T he neglect of Michell’s paper for so long is curious. Havelock’s (1877-1968) first paper on water waves was published in 1908 and on ship wave resistance in 1909. There followed a series of papers on ship waves and wave resistance with the ship being replaced by a moving pressure
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133
distribution. These papers were of great interest in themselves, but it is not easy to associate a pressure distribution with the ship’s geometry. Only in 1923 did Havelock finally begin to exploit Michell’s results. I n 1925 (a) Havelock gave an alternative derivation based upon Green functions instead of the modified Fourier-integral theorem developed and used by Michell. Thereafter Michell plays practically no explicit role in Havelock‘s work, although Havelock proceeded to exploit Michell’s result in a series of outstanding papers on the effect of various modifications of ship lines upon the wave resistance. [It was perhaps Michell’s influence that later led Havelock (1929b) to devise a Fourier-integral theorem similar to Michell’s in both spirit and application.] As mentioned above in the quotation from A. G. M. Michell, the systematic exploitation and testing of Michell’s resistance formula started with W. C. S. Wigley (1890-1970), his first paper on this subject being published in 1926 and his next-to-last (with J. K. Lunde) in 1948. A paper in 1949 is a summing u p of his own and other’s work plus a few new results. (I regard his subsequent papers and reports as having a different aim.) In this work Wigley was for the most part engaged in making experiments and associated calculations to test the validity of Michell’s formula for hulls of more or less shiplike shape and to explore its limitations and applications. He was joined in this work in 1930 by G. Weinblum, although the latter was engaged more in exploitation than in testing of Michell’s work. These investigations of Wigley and Weinblum and many of those of Havelock required evaluating Michell’s formula, a quintuple integral, for hull shapes with some resemblance to ships. Although the hull forms were usually kept fairly simple, one must keep in mind in assessing the magnitude of their achievement that a point on a resistance curve whose computation might take 2-3 seconds, or less, for a modern computer, required then 2-3 days’ work with a desk computer. Some of their conclusions, as well as those of others overlooked in this brief history, will be mentioned later. In fact, few conclusions have been changed by high-speed computers. Havelock continued developing his Green-function approach in a series of papers culminating (but not ending) with a paper (1932~)in which he gave resistance formulas (based upon linearized theory) for arbitrary distributions of sources and sinks or of normal dipoles over a submerged surface as well as for arbitrary pressure distributions on the surface (not a new result, but a new derivation). Havelock continued to use this approach to investigate various aspects of wave resistance. T h e subsequent history of the subject will appear as individual results are mentioned in the following sections. Of particular importance is the paper of Peters and Stoker (1957) which clarified the nature of the approximations used in waveresistance theory. In addition, two important conferences on the subject
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John V . Wehausen
should not go unnoted: One in Moscow in 1936, whose proceedings contain important papers by Keldysh, Kochin, Lavrent’ev, Sedov, and others, and a more recent one in Ann Arbor in 1963 whose proceedings in three volumes contain many important papers.
B. THEEXACT PROBLEM IN AN INVISCID FLUID There are several “exact ” problems that could be formulated, but the one that seems most interesting is that of a surface ship free to trim and squat. It is important to include this possibility, for it has sometimes been mistakenly assumed that the thin-ship approximation requires that the ship be fixed in position. I n fact, as first pointed out by Peters and Stoker (1957), calculation of trim and sinkage is as much a part of the first-order theory as is the resistance. I n addition to the coordinate system Oxyx introduced in (2.15), moving with steady velocity U with the ship, we shall use another coordinate system O’x’y’z’ fixed in the ship. This coordinate system is to coincide with Oxyz when the ship is in its equilibrium position. T h e plane O‘y‘z‘ contains the midship section, and the plane O’x‘y‘ the centerplane section [a slight restriction from (2.15)]. Let the angle from Ox to O‘x’ be CY and let 0‘ have coordinates (0, e, 0) in Oxyz. Let (xG’,yG‘,0) be the location of the center of mass of the ship, and let m be its mass and V its displacement at rest, so that m = pV. Figure 11 shows these quantities schematically. The coordinate systems are related by x’ = x cos cc
+ (y - e)sin
y’ = -x sin CY
CL,
+ (y - e)cos a.
FIG.11. Sketch showing relation between coordinate system fixed in the ship and moving coordinate system Oxyz.
The Wave Resistance of Sh$s
135
Here a and e are unknowns to be determined along with the resistance of the ship. We shall suppose that the ship is propelled by an external thrust T = -R acting along a line parallel to O‘x’ and intersecting O’y’z’ at (0, d‘, 0). For a propeller-driven ship d’ < 0. (For a sailing vessel the formulation would have to be changed.) The hull itself will be described by =
*f(. y’). ’,
(34
We suppose it to be piecewise smooth. The part of the centerplane section below O’x’z’ will be denoted by So, the wetted surface by S and the projection of S on the centerplane section by S , . We shall suppose the motion to be irrotational, so that the free-surface boundary conditions are those given in (2.28). Corresponding to these, there will be kinematic and dynamic boundary conditions on the ship. The kinematic condition on the ship’s wetted surface may be written in the form
where
a
- f(x’, y’) =fi(x’, y’)cos a -fi(x’, y’)sin a,
ax
a
- f ( d ,y’) =fl(x’, y’)sin a
(3.4)
+
fi(x’, y‘)cos a, aY with the notation fi = aflax‘, fz = aflay’. In order to write out the dynamic conditions, we must first compute the force and moment acting on the ship. T h e force components resulting from the action of the water upon the ship and resolved along the axes of Oxyz will be denoted by ( F , , F y , 0 ) and along those of O‘x‘y‘z‘ by (F,’, Fg‘,0). The moment about the center of gravity is denoted by (0, 0, M G ) . If p(x, y , z ) is the pressure in the fluid.
F,’
=
y, zb,’ d S = 2
J p[., Y , f (x’,Y”fi SP
dx‘ dy’,
John V . Wehausen
136
T h e dynamic boundary conditions then consist of the equations for equilibrium of forces :
F, + T cos CL = 0, Fy f Tsin a- mg= 0, MG
(3.7)
+ T(y&- d’) = 0.
A kinematic condition must also be satisfied on the ocean bottom, taken to be a horizontal plane at y = --h:
vY(x,-h,
X)
= 0.
(3.8)
If the depth is taken infinite, we may replace this by lim y,, = 0.
(3.9)
y- --m
Finally we need a condition insuring that waves will only follow the ship : 94x3 y , z ) =
0([x2+z2]-1’2) o(1)
as x2+z2+co as x2 z2+ co
+
for x>O, for x (0.
(3.10)
Various modifications of this problem are possible, depending upon the physical situation. I n particular, in a towing tank a model can be restrained from trimming and squatting so that u = e = 0. Such modifications are not difficult to make, and this one actually simplifies the problems. Modification of the boundary conditions in order to accommodate a submerged vessel is also not difficult. Since U has been fixed in the problem formulated above, the unknown quantities are y , Y , T , a, and e . It is obvious from the boundary conditions that they are hopelessly intertwined with one another and that some approximations will be necessary. Before proceeding to the discussion of approximation, we shall formulate one more “ exact ” problem, that of a moving pressure distribution. Here the pressure above the water surface is given, but the surface profile itself is unknown. T h e kinematic condition (2.2813) remains the same, but the dynamic condition (2.28a) must be replaced by
where P(x, X) is a given function, which may be chosen to vanish outside a certain region in order to model a particular physical situation like a moving hovercraft. Although much of the complexity of the ship problem has disappeared, this is still a nonlinear problem. As in (3.5), one can com-
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137
pute the x-component of the force exerted by the pressure distribution upon the water:
s
F, = /P(x, z)n, ds = P(x, z) Y,(x, x ) dx dz.
(3.12)
C. PERTURBATION EXPANSIONS In order to obtain any solutions of the exact problems formulated above, or of others that one may formulate, some method of approximation is necessary. T he standard procedure has been by way of either regular or singular perturbation expansions. In either case one introduces a dimensionless parameter E > 0 connected with the problem in such a way that as E -+ 0 the disturbance near the free surface becomes smaller and smaller, except possibly at certain isolated points or lines. Each value of E labels one of a family of flows. Many perturbation approximations involving moving bodies have a different aim from this one. For example, in thin-wing theory in an unbounded fluid one wishes primarily to avoid the tedious computations associated with conformal mapping of the given wing section onto a circle. However, it still seems fair to say that in problems involving motion near a free surface the nonlinear boundary conditions at the free surface seem to present the greatest obstacle. Hence one chooses a perturbation parameter associated with vanishing of the surface disturbance. It will still be true, however, that some of the classical perturbation approximations also fulfill this requirement. There seem to be two main ideas behind free-surface perturbation schemes for moving bodies. One is to have the body sufficiently deeply submerged that the free-surface disturbance resulting from its motion is not very great, and, of course, must become smaller the deeper the submergence. I n the other, one introduces a family of bodies whose members can be made to approximate more and more closely to a body whose motion will not disturb the fluid. Th e most familiar example of the latter in fluid mechanics is thin-wing theory. For a moving pressure distribution one simply assumes that the imposed pressure P is " small." For moving submerged bodies the two approaches can be schematized as follows. Let L be the length in the direction of motion, d a typical length perpendicular to this direction (say a vertical or horizontal dimension, or perhaps the square root of the maximum cross-section area), and ( a , b, c ) a fixed point in the body. There are four relevant lengths: U2/g,b, d, L. In a typical deep-submergence approximation one keeps U21gLand d / L fixed and takes E = d/ I b I . In a typical thin- or slender-body approximation one keeps U2/gLand b/L fixed and chooses ~ = d l L I. n the former case,
John V. Wehausen
138
letting 1 b ( -+co entails E+ 0 and ever deeper submergence of the given body without any distortion of the body. In the latter case, letting 8 --f 0 requires changing the shape of the body so that it approaches a flat disk or a spindle, depending upon how one has chosen d. The mathematical problem is to formalize these ideas into a systematic approximation procedure. An extensive and thorough study of this problem has just been made by Ogilvie (1970), especially with regard to singular perturbation problems. Ogilvie’s paper is also complemented by one by Newman (1970) expounding recent results in the slender-body approximation. Consequently, we shall restrict ourselves here to a token presentation of the thin-ship approximation and to a statement of results for some other approximations.
1. Thin-Ship Approximation In the case of a surface ship there is, of course, no possibility of a deepsubmergence type of approximation. Consequently one is forced to fall back upon one involving the geometry of the ship. There are several ways of approaching the problem and perhaps the greatest recommendation of the one chosen here is that the resulting problem can be solved analytically. We begin by imbedding the hull form (3.2) in a family of hulls derivable from a standard hull f ( l ) ( x ‘ ,y’), i.e., we shall consider hulls of the form
z = &&f(l)(X’,y’),
&
> 0.
(3.13)
Evidently, as E --f 0 the hulls approach a flat disk, the centerplane section of the ship. Corresponding to each value of E there will be a velocity potential ~ ( xy,, z ; E ) , a free surface Y ( x , x ; E ) , and a trim a(&),sinkage e(&), and thrust T(E).Although it is possible, as shown by Ogilvie (1970), to treat the approximation by the method of inner and outer expansions, the results are the same in the first order as one obtains by assuming a regular perturbation expansion, and in the second order seem a little dubious. The problem of approximation has also been treated by Wehausen (1963) by use of Green’s theorem and Green functions and again (1969) by use of Lagrangian coordinates. Both are lengthy to expound, and we shall simply assume here, following Peters and Stoker (1957), that q(x, y , z ; &) = & I p ( X , y , z )
Y ( x , z ; &)
= &Y‘l’(X, z)
+
+
&2,(2)
+ . . ., + . . .,
&(2)Y(2)
+ + . .., e(&)= &e(l)+ 2 e ( 2 ) + . . . , T (E= ) E ~ T+ ‘ ~c ’3 F 2 )+ . . . . a(&)= &&)
&%(2)
(3.14)
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139
Starting the last expansion with 2 anticipates what will be forced upon us, and is not a real restriction. T h e underlying assumption that ~ c p ( l )and & Y ( lare ) small cannot be uniformly valid for a ship that does not have an appropriately shaped bow and stern, for there will be stagnation points on the stem and sternpost, where the velocity will be U , which is not small no matter how small E > 0 is. T h e expansions (3.14) are now substituted into the equations (2.28), (3.3), (3.7), (3.8), or (3.9), and (3.10), expanded in powers of E , further expanded in Taylor series where necessary, and all terms assembled according to powers of E . We omit the rather tedious details, but give the first-order results. The velocity potential ~ ( l must ’ satisfy the following equations :
Acp‘l) = 0, l?g)(x, 0, x)
+
Y<0 K q y = 0,
d l ) ( x ,y , I N ) = f Ufil’(x,y ) ,
&)(x, -h, 0) = 0
or
lim
(x,y )
#)=
in
so,
0.
y+-m
T h e free surface Y ( l )is determined by
U Y ( l ) ( xz, ) = - cpyyx, 0, 2). g The thrust T ( l )is given by T(1)= -2 PUS d l ’ ( x , y ,O)fil’(x,y)dxdy.
(3.16)
(3.17)
SO
The sinkage and trim are determined by the following pair of linear equations : e‘l’
+
’&)
-
2u y , O)fL”(X, y ) dx dy, A‘l’g J J M % SO
140
John V . Wehawen
where L = 21 and
If one is prepared to accept the approximation q~ = & y ( l ) Y , = &Y(l), etc., one may multiply the various equations by E or and drop the tiresome superscripts. We shall, in fact, do this. An important property of the above equations is that all the unknowns have fallen into separate equations. After q~ has been determined from (3.15), one can successively determine T , Y , a, and e. One should also note that the equations determining q~ are independent of a and e. Consequently Y and T are also independent of a and e. This is an important observation from the standpoint of comparing theory and experiment, for it tells us that theoretical predictions can be no more accurate than the difference in measured values of Y or T when a model is fixed and when it is free to trim and squat. T h e second-order theory will not be so simple because the values of a ( 1 )and e‘l) enter into the determination of q ~ ‘ ~ ) . T h e thin-ship approximation does not mean, as is sometimes claimed, that the ship is being replaced by a ‘(knife blade,” nor does the linearized free-surface condition mean that the free surface is being practically replaced by a plane, although the approximated boundary conditions may be presumed to be better satisfied, the closer these conditions are met. How thin the ship must be or how flat the free surface must be for the thin-ship approximation to be acceptable is a question that should be settled by computation, or if this is impossible, and it seems to be, by comparisons with experiment. 2. Submerged Bodies For deeply submerged” bodies one may retain all of (3.15) except the third equation, the “ thin-ship ” boundary condition. This will be replaced by the exact boundary condition cpa) = Un
on
5’.
(3.20)
Equation (3.16) remains the same and (3.17) is replaced by
T ( l )= - 2 p U
1s
@(x, y , 0) n, ds.
(3.21)
Without further specification of the exact problem there can, of course, be no analog of (3.18).
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If the approximation is of the “ thin-body ” type, say a thin hydrofoil, then again (3.15) holds, except that the thin-ship boundary condition is replaced by one appropriate to the body considered. In the case of a hydrofoil with sharp trailing edge a Kutta condition will be included for both the deep-submergence and thin-wing approximations. 3 . Slender-Body Approximations Slender-body approximations, either for surface ships or for submerged bodies, do not decompose into a sequence of boundary-value problems in the same neat fashion that regular perturbation approximations do. (This neatness is, however, misleading, as is evident from a pursuit of the thinwing approximation to the third order). As has been mentioned earlier, we shall not try to develop this method here but refer instead to Ogilvie’s paper (1970) where the subject is treated in considerable detail, including a comprehensive bibliography.
4. Moving Pressure Distributions The perturbation approximation here starts from an assumed imposed pressure in the form & P y x ,x), (3.22) the analog of (3.13) in the thin-ship approximation. One then assumes the first, second, and last expansions in (3.14). The resultant equations deter) consist of (3.15) with the second and third equations remining ~ ( lwill placed by a modification of the second one: r g ( x , 0, x )
+ K q p = (pU)-lPkl)(x,z ) .
(3.23) Here we suppose P ( l )to be defined over the whole ( x , z ) plane, but it can be zero outside a bounded area. The horizontal force exerted by the pressure distribution upon the water will then become
Fil)= - u-1J PCl)( x , x)p)$”(x, 0, z ) dx dx.
(3.24)
The free surface is given by a modification of (3.16):
U y y x , z ) = - #(x, g
1 0, z ) - - p y x , z). Pg
(3.25)
5 . Inconsistent Approximations An approximation scheme in which some of the equations of the problem are satisfied to a higher order of approximation than others is said to be inconsistent. For example, one might argue that since we are looking for
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John V. Wehausen
wave resistance, the free-surface boundary condition is more important than the body boundary condition and replace the second equation in (3.15) by one correct through the second order,
retaining, however, the other boundary conditions. Or, at the other extreme, one might argue that for normal ship speeds the linearized free-surface condition is adequate, but that the body boundary condition should be the exact one, at least on the part of the hull below y = 0. In this case one replaces the third equation of (3.15) by cpn= Un,
on
S o.
(3.27)
Neither argument is convincing and in principle each inconsistent problem is still only of first-order accuracy. There is, in fact, no guarantee that the change from the consistent first-order problem is in the direction of greater accuracy. On the other hand, it is certainly conceivable that under some circumstances one of the inconsistent problems will give results nearer to those of the consistent second-order problem than does the consistent first-order problem. Such investigations should be considered as numerical experimentation. Some interesting results of such experimentation will be discussed when we consider higher-order approximations. One particularly misleading version of an inconsistent problem is one in which a flow is generated by a distribution of sources and sinks selected to satisfy the linearized free-surface condition and from which a “ body ” is generated by tracing streamlines numerically. This is, in fact, just the second example of an inconsistent problem given above.
D. METHODS OF SOLUTION There seem to be two fundamental methods for solving the boundaryvalue problems provided by the perturbation approximations. One is the use of Fourier series or integrals, the other the use of Green functions. We shall use each method in turn to solve the first-order thin-ship problem in an infinitely deep unbounded fluid. Michell used the Fourier-integral method. Havelock introduced the method of Green functions. In addition, we shall treat the linearized moving pressure distribution by Fourier’s method and the moving deeply submerged body by the method of Green functions. As has been mentioned earlier, we shall not illustrate the slenderbody approximation but refer to Ogilvie’s (1970) exposition.
The Wave Resistance of Ships
143
1 . Fourier's Method We shall start from the representation (2.57) of a velocity potential for z > zs . I n the application made there zs had to be large enough that it was acceptable to use the linearized free-surface condition. I n the present application this presents no problem; we have already assumed this condition for the whole surface excepting the intersection of the centerplane section with y = 0. Hence (2.57) is valid for z > 0. An analogous formula holds for z < 0, but we may avoid dealing with this potential since we have assumed a symmetric ship. Using this representation and the second boundary condition in ( 3 . 1 5 ) , we find the following equation (where we have replaced the functionsf, g of (1.57) by F, G to avoid confusion with the function describing the hull) :
viS)(:x,Y, 0) = - Uf& =-
6
y)
k k2 loKdk ( K 2- k2)1'2exp - y[-F(k)cos
K
g m k +-i;;j dk-(k2K K -
kx
k2 K2)1'2exp-y[Gl(k)cos k x + F , sin kx]
K
+ + Ksin py k2P p)cos kx + G sin kx].
jomdpjomdk(k2 p2)1/2[cospy
x [-F(k,
+ G sin kx]
1
(3.28)
Here we suppose fz(x,y) = 0 if (x, y ) is not in S o . To find cp(s) we must find the various functions F(k, p), F(k), F,(k), etc. I n order to do this we introduce an analog of the Fourier transform for a function of two variables. Only the y dependence is different. Let M(x, y) be an integrable function defined for ---a < x < co,-a < y < 0. Then one can represent M as follows : m
M(x, y) =
jodp
cos py
k2 +sin py] [A(k, p)cos kx + B(k, p)sin kx] KP
+ l m d kexp k2y[C(k)cos kx +D(k)sin kx], where
x 1"dx
lo
dyM(x, y)exp ikx
-m
C(k)
2k2 + iD(k) = n-K --
s so m
-m
dx
-m
(3.29)
144
John V . Wehausen
The formula can be proved by reduction to the ordinary Fourier-integral theorem or directly in the same manner that that theorem is proved. Comparing (2.29) with (2.28),identifying M(x, y ) with - Ufz(x,y), etc., one finds easily the following:
G,(k)
U2 k + iFi(k)= -rr2 g ( k 2 - K2)l:'
(3.31)
We have now constructed the function yCS).After some slight manipulation it looks as follows:
k2 ( K 2- k2)l%]exp K (y
+ q)cos k ( x - 5)
k k(x - 5) + - (k2- K2)1'2z K
1
The corresponding function for z < 0 may be obtained from y'P'(x, y , z) = y'S'(x, y , -z).
(3.33)
The Wave Resistance of Ships
145
Along the half-plane z = 0, y < 0 one can verify that for all x,y, ~ ( ~ ) (y,x +0) , = qP"(x, y, -0) except on S o . ,.pks)(x,y,+O) = ,.p$(x, y , -0) = 0
(3.34)
On So the boundary condition of (3.15) must, of course, hold. Consequently we may think of q ( s )and v(') as being harmonic extensions of each other, and the two together as being the solution of the problem of finding v. (Matching of ,.p(s) and v(') along z = 0 is, of course, more complicated for an asymmetric ship.) In choosing the form of v(s) in (2.57) the functions F2 and G, were chosen so that the waves far from the ship would be moving in the appropriate direction according to our intuition. It may not be intuitively clear that this is equivalent to satisfying the fourth equation in (3.15). However, this turns out to be the case. The third integral in (3.32) is O( I XI -l) as I XI -+ co. The first and second integrals cancel as x+ $-aleaving a remainder that is O(x-l). However, as x-+ -co they reinforce one another and yield
I
"1 +o -
xcos K ( ( - x ) - - 4
% .:)(
(3.35)
Although (3.35) is a correct asymptotic expression, it is not useful for studying the asymptotic form of the wave pattern far behind the ship because dependence upon z has been lost. The resistance can be computed immediately from (2.59) :
+
R = 7 ~ p g K / ~ ( FGI2) ~ ~- (k2 - K2)'12dk K
4
(3.36)
m
7T
where
P + iQ(k) =
f.(x, y)exp(iks so
+ # y) dx dy.
(3.37)
This is Michell's integral for the wave resistance of a ship. Various other forms are possible. We give two of the common ones:
(3.38) where
I
+ iJ(X)= Iso f,(x, y)exp(iKx + X2Ky) dx dy ;
John V . Wehausen
146 and
I
m
M(X, y ) =
exp ~y cos ~x
1
dX. (A” - 1)1’”
Instead of using (2.59) to compute R, we could also have used (3.17). It is moderately instructive to do this, for one then sees how the contributions from the first two integrals in y drop out. T he function y can now also be used in formulas (3.16) and (3.13) to construct the free surface and to find the sinkage and trim. Although we have used Fourier’s method here only for infinitely deep unbounded fluid, it is equally applicable to fluid of finite depth and to motion in canals. In the former case the integral with respect to y is replaced by a sum and in the latter case the integral with respect to z is replaced by a sum. Finally we note that Fourier’s method has been applied in a different manner by Plesset and Wu (1960) to derive the velocity potential of a thin ship in a form they have found convenient for an asymptotic analysis of the wave pattern. Another problem in which Fourier’s method works well is the linearized problem of the moving pressure distribution with the free-surface boundary condition (3.23). Here y must be representable in the form ~ ( xy,, z ) = j m d mj “ d n exp(m2 0
0
x {[A(m,n)cos mx
+ n2)1’2y
+ B sin mxlcos nz + [C cos mx +D sin mx] sin nz}. (3.40)
Equation (3.23) then yields
1 -P,(x, x) = PU
I0
m
jodn[K(m2 + n2)1’2 m2] x {[A cos mx + B sinmx]cos n z + [C cos mx + D sin mxlsin nx}. m
dm
-
(3.41) This may now be inverted to give
(3.42)
The Wave Resistance of Sh$s
147
Substituting into the expression for y, one obtains after some simple manipulations
x cos m(x - 5)COS n(x - <)
x sin m(x - Ocos n(z - <)
x sin[k(x - 5)cos 8]cos[k(z - <)sin 81. We have not yet verified that the fourth equation in (3.15) is satisfied. The asymptotic behavior of rp is easily computed from the following theorems on Fourier integrals: Under appropriate restrictions on f (x) and for a>O /omf
).( sin R(x - a) dx = .-f(a) x-a x-a
dx=O(R-I)
+ O(R-l), (3.43) as R + m ,
where the second integral is a Cauchy principal value. We have not, in fact, as yet interpreted the integrals above. We now suppose the integral with respect to k to be a Cauchy principal value. After computing the asymptotic behavior of y,one sees that the formulas must be modified in order to satisfy the condition that the disturbance ahead of the pressure distribution should die out sufficiently rapidly. One finally obtains Havelock’s (1932~)formula:
x sin[k(x - ocos O]cos[k(z - {)sin 81
K
JJdt d{P(t,
--
TU
5) Jn12d8sec3 8 exp(Ky sec28)
x cos[K(x - Osec B]cos[K(z - <)sec28 sin 81.
(3.44)
John V . Wehausen
148
T h e wave resistance may now be computed from (3.24):
K2 R = - dx d z 7rp
u2
jj
11d t d(P(x, z)P((,5) Y ” d 0 sec3 0 cos[K(x
- t)sec 01
x cos[K(z - [)sec2 0 sin 01. (3.45) 2. Method of Green Functions As is usual in this method, it relies upon being able to construct a function of the form G(x,y , z ; 5, 77, 5) = G(P;Q ) = r - l + H(x,y , z ; t,77, 0,
+
(3.46)
+
where Y = [(x - t y ( y - q)2 ( z - 5)2]1’2 and H is harmonic in the region occupied by fluid, or in the case of the linearized problem, in the region below the equilibrium free surface bounded by the bottom and walls if any. Here we assume a horizontal bottom at y = --h, h 5 CO. Then we require the following of G :
AH = 0, G&y,
G,(x, y , z ; 4, --h,
z ; 5, 0,
5) +KG,
5) = 0
or
= 0,
lim G,
= 0.
Y-m
We now apply Green’s formula to the region bounded by the plane X F , bottom Z H ,y = --h, a circular cylinder C Rwith Oy as axis and radius R, and the two sides of S o . (In the second inconsistent problem discussed earlier one would replace So by the equilibrium wetted surface S . ) Then we have
y
= 0, the
The Wave Resistance of Ships
+ J"" dB 1' 0
dqo( 1).
149
(3.48)
-h
T he second integral may be integrated by parts to give
where the line integral is taken around the intersection of C, and ZF.As R-t 00, both this integral and the third one converge to zero (obviously a little extra care is required here if h = 00). (We note again that in the inconsistent problem in which the exact ship boundary condition without trim and squat is satisfied this line integral also contributes at the intersection of S and & ,) We finally have
v(x, y , 4 = -
U
JJft(5,
7)G(x, y , 2;
E,
7, 0) dE d7.
(3.50)
so
Evidently all that is required to complete the solution is construction of G. Green functions are known for several important physical situations. Many of these are given in Wehausen and Laitone (1960, pp. 483-490). Here we give first the one for infinite depth:
x cos[k(x - ~
C O S 191
cos[k(z - ()sin 81
dB sec' 0 exp[K(y
-4K 0
+ 7)sec' B]
x sin[K(x - Osee B]cos[K(z - [)sin 8 sec2 01, (3.51a)
+ + +
where r1 = [ ( x - 6)' ( y 7)' ( z - ()2]1/2 and the integral with respect to K must be interpreted as a Cauchy principal value. The derivation usually
John V . Wehausen
150
proceeds by way of Fourier transforms. The asymptotic value of the double integral cancels the single integral with remainder of O([c2 {2]-112) as (+-00 but doubles it as (++a. We also give the Green function for the case h < 00 since (as has been pointed out to the author by K. Eggers) it is incorrectly given in Wehausen and Laitone [equation (13.37)] :
+
x { e - , , sech kh cosh k(y
+ h)
+ h)cos[k(x - ()cos 81 x cos[k(z - {)sin 8](k cos2 8 + K ) - K } dk ~cosh k(q
-4s
n/2
[I
-
Kh sec2 8 sech2 kohl -I
80
x {sec28 e - koh sech k , h cosh ko(y+ h)
+ h)sin[k,(x - ()cos 81 x cos[k,(x - ()sin 8](k0cos2 0 + K ) - K ) d0
x cosh k o ( q
(3.51b)
where k, = ko(8) is the real positive root of
k, - K sec2 8 tanh k, h = 0, Oo= arc c o ~ ( K h ) l / ~ if
Kh < 1,
8,
< 8 < $7
B0= 0
if
Kh > 1
and r2
= [(x
The expression for sistance R = - T:
-
o2+ (y + 2h + d2+ (2
-
5) 11/2* 2
v may now be substituted into (3.17) to find the re-
11". d . JJ@d q f l ( x ,y)fe(t,v)G,(x, y , 0 ; t,
PU2 R = -7T
so
7, 0).
(3.52)
so
Only the last integral in G above contributes because of symmetry considerations. One finds again (3.39) with the function M ( x , y ) given in a slightly different form, obtained by letting h = sec 8. Substitution of y into (3.16) and (3.18) gives expressions for Y and a pair of linear equations for finding cc and e. This is a more convenient expression for analyzing Y than the one obtained from (3.32). I n particular, one may first determine the surface generated by a moving source and then by superposition find the one associated with the ship.
151
The Wave Resistance of Ships
Not all problems can be solved with more or less equal ease by Fourier’s method and by the method of Green functions. Fourier’s method will generally have required a coordinate system in which variables can be separated and then boundary conditions imposed on one of the coordinate surfaces. I n order to illustrate the greater power of the method of Green functions, let us consider the problem of the motion of a “deeply submerged” body. We recall that the body boundary condition is then (3.20). In order to find y , we begin as we did for the thin-ship problem, using Green’s formula. However, the first integral does not simplify, but remains
(3.53) T he other integrals all vanish as they did there. If we now use the boundary condition (3.20), we may rewrite Green’s formula as follows:
+ S v(Q)Gy(P;Q)dS
kv(P)
=
u JSn,(Q)G(P;Q)dS.
(3.54)
If in the left-hand integral we let P+Po on S, then by a well known theorem the integral converges to - 2T(Po)
+J dQ)G”(PO; Q) dS.
We then have the following integral equation for y(P),where P is a point of s:
If this equation can be solved for y on S by numerical methods, then y is determined everywhere by (3.54). (This is the same integral equation that is to be solved in the inconsistent surface-ship problem mentioned above, except that one must also take account of the line integral.) Instead of starting with Green’s formula (3.48), one may also assume that y can be represented by a source distribution:
dP)= J U(Q)G(P> Q) dS.
(3.56)
This yields immediately an integral equation for u:
2no(P) + a(Q)G,(P, Q) d S = Un,(P), S
This is essentially the same as (3.55).
PeS.
(3.57)
152
John V. Wehausen
Another common procedure is to try to expand y ( P ) in a series of singularities of all orders, all located at some fixed point within S. The boundary condition on S then gives an equation to be used in determining the coefficients of the series. Not all bodies can be treated by this method.
E. FURTHER RESULTS, VARIATIONS, AND EXTENSIONS I n the preceding section several problems of steady motion with a free surface were formulated and solved, but the solutions were not analyzed in any way, nor were any numerical evaluations shown. In this section we shall try to fill this gap. In addition, there are many problems that can be solved that are variations of the ones already solved. Although the results of these variations are important for the subject, the methods of solution are usually not different in nature from those already illustrated, although working out the details may require a very elaborate analysis. Consequently, we shall chiefly restrict ourselves here to giving a census of known results, a guide to recent literature and some computed results that seem to illustrate important behavior. Since many of the classical results of wave-resistance theory are already expounded in Lunde (1951a), Kostyukov (1959, 1968), and Wehausen and Laitone (1960), there seems to be no need to reproduce formulas that can easily be found in these places. Consequently, for results already available in these sources only references will be given. In a later section we shall discuss comparison between theory and measurement. Many computational results will be postponed to that section. T he ones reproduced here will be computations with no associated measurements. Although the emphasis in Section I1 and also in this one has been on three-dimensional motion and ships, most of the considerations can be carried over to two-dimensional motion. In fact, the possibility of using analytic functions of a complex variable allows an elegance of treatment not available in the three-dimensional problems. We shall not, however, discuss such problems except where the method or result throws light upon threedimensional problems. A fairly complete summary of results up to about 1959 can be found in Wehausen and Laitone (1960, Sections 20 and 218). Asymptotic analysis of wave patterns behind moving singularities or pressure points is only marginally relevant to the purposes of this article, principally in connection with light thrown upon the determination of wave resistance from wave-pattern measurements. The most relevant result for this purpose is already given in (3.25). However, at the end of this section various recent papers on this subject will be listed.
153
The Wave Resistance of Ships 1. Kochin’s H Function
Consider a deeply submerged (not necessarily symmetric) body with boundary S. We have showed in Section II1,D in (3.53) that
d P >=
1
f [YV(Q)G(P,Q) - vGVI dS(Q)
The form of H will depend upon the boundary conditions imposed upon G. In (3.51) we have given an example for h = a.For this case Kochin’s i@ function is defined as follows: S ( k , 8) =
1
{rp,, - kv[ny
S
+ i(n, cos 8 + n, sin B)]}exp kwdS,
(3.59)
where w =y
s i x cos B + i z sin 8.
(3.60)
Kochin (1936) derived the following representation in terms of % for the last integral in (3.58):
-n
iK
n12
K
n
4772
-n
- Re -[ S ( K sec2 8, O)exp(Kw sec2 B)sec2 B d8 2~ -n12
1
1
- Re -J’ d8 sec2 8
S ( K sec2 8( 1 - A), 0)
-m
x exp[Kw sec2 O ( 1 - A)]
ax x
(3.61)
-
where the integral with respect to A is to be interpreted as a Cauchy principal value. Other forms for the X function are possible. Kochin shows that
J
+
X ( k ,8) = {y,, i cos O[q,n2- y 3n,]
+ i sin 8 [ ~n,y-
‘pun.]}exp
kw dS.
(3.62) Furthermore, if cp is representable in the form
?(P)=
1
1
u(Q)G(P,$2)dS’
(3.63)
John V . Wehausen
154 then
1
%(A, 8) = - a(P)exp kw dS.
(3.64)
S
T h e real usefulness of the A? function is in the information that it contains concerning the force acting on the body. The following formulas are from Kochin (1936):
F,
2
-R
= --
&(K sec2 8, 8) I sec3 8 d8,
(3.65)
+-!pK2 47.f F 2 --
d8sec48
A?(Ksec28(1-h,8)12--
1-A h dh,
-n
p ( K sec2 8, 8) I sin 8 sec4 8 d8.
It is evident that the & function is closely related to the pair of functions P , Q in (3.37). Indeed, Havelock (1932~)introduced functions playing a very similar role. There is also a close relationship between the % function and the free surface Y(x,z). This is more or less evident from (3.61) and is explicitly stated and exploited in Eggers, Sharma, and Ward (1967). T h e definition and exploitation of the % function for other boundary conditions have been carried through by others, notably by Haskind (1945a,b), who has used it to treat two- and three-dimensional problems with finite depth and acceleration. (It is also useful in problems with oscillating boundaries but these are not being treated here.) 2. Properties of Michell’s Integral Several properties of Michell’s integral can be deduced immediately either from (3.15) or from inspection of (3.38) or (3.39): (1) R is proportional to B 2 ;( 2 ) R is independent of the direction of motion of the ship; (3) R, and the wave pattern, are the same whether the ship is free to trim and squat or fixed in position. Although the derivation of these properties has been given only for h = co and horizontally unbounded fluid, they are also consequences of thin-ship approximations in various other situations that will be considered later. It is evident that in any comparison of theory with experiment these should be the first predictions to be tested, for they require no prior numerical calculation.
155
The Wave Resistance of Ships
It is possible to derive still other information without numerical calculation by examining the behavior of Michell’s integral at high and low Froude numbers. There may be some question about the usefulness of such approximations, for the accuracy of the linearized theory decreases as the Froude number becomes either very small or very large, or put in another way, the ship must become ever thinner in these two limits to maintain the same accuracy. However, the low-Froude-number expansion has proved useful in calculation and both bring out interesting properties, one of them surprising, of the behavior of Michell’s integral. A low-Froude-number expansion seems to have been given first by Wigley (1942) and later further developed by Inui (1957) and exploited by him and still later by Bhattacharyya (1970) in calculations for certain mathematical ship forms representable by polynomials. A more elaborate study was made by Kotik (ca. 1956) in a paper that unfortunately had only a small circulation in manuscript. The first term in the Iow-Froudenumber expansion is
Formula (3.66) assumes that fz exists for all x in BL). If there are corners in the waterline, there will be further terms corresponding to these. One can find some formulas for a ship of the form f(x,y ) = X ( x )Y ( y ) in Wehausen (1956). One property of the Michell resistance is immediately clear from (3.66) and that is that there are an infinite number of maxima and minima as F,,approaches zero. Equation (3.66) shows also that the most important wave-making property of a smooth ship at slow speed is the tangent angle at stem and stern at the water surface. Michell (1898) himself gave a proof of the fact that R, --f 0 as Fn -+co. I n a discussion to Wigley (1942) Havelock stated the correct form of the asymptotic expansion at large Froude number, namely, R , R , log Fn , where R1 and R 2 are power series in F n - , starting with Fn-,.Newman (1964) has derived the first term in each series. We give here only the most important term :
+
R,
=
~-lpgL-~A$,Fn-~ log Fn + O(Fn-,),
(3.67)
where A,, is the waterplane area. The result is surprising, for one would expect the most important term in the wave resistance to be related to the displacement. In fact, if the body is completely submerged, the leading
156
John V . Wehausen
term is proportional to VzFn-2,as was first pointed out by Weinblum (1936a). One can find a somewhat different approach to the high-speed limit in Michelsen (1966). Havelock, in the discussion to Wigley’s paper cited above, points out that, if the ship form has been enlarged to take account of boundary layer and wake in such a way that there is a tail of finite cross section extending to infinity, then Rn will not converge to zero as Fn-t m but to some finite value. It. may seem like a trivial remark to note that R, 2 0 for all F,, for Michell’s integral. However, the wave resistance given by the slender-body approximation does not have this property and becomes negative at low and high Froude numbers. M. G. Krein, in work reported by Kostyukov (1959, 1968, Section 40), has shown that in fact R, > 0 for any thin ship of finite length, draft, and displacement, but also constructs a form of infinite length, but finite draft and displacement, that has R , = 0. Krein also shows an easy way to construct functions S(x, y) vanishing on the underwater profile of a hull z = if(., y) such that z = +(f S) has the same displacement and Michell wave resistance as f itself. It is not clear that f S 2 0, as it must be for a real ship. However, even this can be achieved by proper choice of 6. A study of mathematical properties of Michell’s integral has also been made by Birkhoff and Kotik (1954b). From this paper we give two further representations of R, . Let the domain of definition of f(x, y) be extended to the whole (x, y) plane by settingf= 0 outside S o . Define
+
+
where
W(x, y ) = ( 4 q - 1 ’ 2 exp(--x2/4y). Then
where M is defined in (3.39) and Y ois the Bessel function usually denoted by this letter. An obvious advantage of the first form is that all information about the hull is isolated in the function H .
The Wave Resistance of Ships
157
3. Moving Pressure Distributions Recent investigations of moving pressure distributions have all been made in order to clarify some aspect of hovercraft behavior. Although some of these contain computations for infinitely deep unbounded fluid, the computations are mostly for comparison with computations with finite depth or canal walls, or both. These results will be mentioned in subsequent sections devoted to these cases. Several others will be considered still later when we deal with comparison between theory and experiment.
4. Finite Depth Replacing the boundary condition vu+0 as y + co by yY(x, -h, y ) = 0 leads to somewhat more tedious computations than for infinite depth but to no real conceptual difficulties, although subcritical and supercritical speeds play a role. Th e same methods are applicable, and indeed it is evident from Michell’s paper that he knew how to carry through Fourier’s method for a thin ship in finitely deep water even though he gave no formulas. If one uses Green functions, formulas (3.16), (3.17), (3.18), (3.50) are still valid with a Green function for finite depth. T h e same Green function can be used in the integral equation (3.55). The Green function, first derived by Sretenskii (1937), has already been given in (3.51b). Th e wave resistance formula analogous to (3.36), also first given by Sretenskii (1937), can be found in Lunde (1951a, p. 51ff), Wehausen and Laitone (1960, p. 581), and Kostyukov (1968, Section 28). T he velocity potential and wave resistance for a pressure distribution moving over water of finite depth were first given by Havelock (1922) in a special case and in general by Lunde (1951b). They may also be found in Wehausen and Laitone (1960, p. 599). Calculations of the wave resistance for distributions of rectangular and elliptical planform have been made by Barratt (1965). Included are results for infinite depth. Yim (1971) has made calculations for planforms with parallel sides but pointed ends. Huang and Wong (1970) have calculated the surface displacements for rectangular planforms. Doctors and Sharma (1972) have calculated the resistance for rectangular planforms with the pressure distribution making a continuous transition at the edges from its maximum value to zero. Calculations are for finite and infinite depth, various beam/length ratios and various transition behaviors. Some calculations showing the effect of finite depth for a thin ship are given in Fig. 12 together with other calculations for resistance in rectangular canals. The behavior for pressure distributions is similar. Both show a marked maximum near Fh = U/(gh)’”= 1.
158
John V. Wehausen
5.0
4.O
3.O
2.0
1 .o
0.2
0.3
0.4 0 :4
FIG.12. Resistance coefficient R" = RW/8.rr-lpgB2TzL-' for BIT = 3 and the following cases: L / B = 5 , L / H = 3.75, h/T = 4: (a) 2b/B = 1 3 . 3 3 , (b) 2b/B = 6.66; L / B = 10, L/h = 5 , h/T = 6: (a) 2b/B = 20, (b) 2b/B = 10, (c) 2b/B = 6.66, (d) 2b/B = 5 . (2b = canal width, h = water depth) [From Kirsch, 1966 (Fig. 8b, p. 175) by permission of the Society of Naval Architects and Marine Engineers.]
5 . Motion in Rectangular Canals The velocity potential and wave resistance for a thin ship were first worked out by Sretenskii (1936, 1937) and by Keldysh and Sedov (1936). The results may be found in Lunde (1951a, p. 57ff) and Kostyukov (1968, Sections 11 and 29). The analogous problem for a moving pressure distribution has been solved by Newman and Poole (1962). Calculations have been given by Kirsch (1962, 1966) for a thin ship in a canal and earlier for a smaller range of variables by Voitkunskii (see Apukhtin and Voitkunskii, 1953, Chapter 7). Calculations similar to Kirsch's have also been made by Ueno and Nagamatsu (1971). The chief difference is that they try to satisfy the body boundary condition more accurately by solving an integral equation for centerplane source strength. The results differ noticeably but not significantly. Newman and Poole (1962) g'ive extensive calculations for moving pressure distributions of various shapes. Figure 12 is one of the figures from Kirsch's paper. The figure shows R* = R , / ~ T - I ~ ~ B ~plotted T ~ L against -~ Froude number for infinitely
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159
for unbounded fluid of finite depth (Sw)and deep unbounded fluid (a), for canals with several ratios of canal width 26 to ship beam B . The computations were made for a model of rectangular sections and parabolic waterlines. One should note especially the discontinuity in the resistance that = 1 when canal walls are present. Finite depth occurs at Fh= U/(gh)1’2 produces a sharp maximum in the neighborhood of F h = 1, but no discontinuity unless the walls are present. Kolberg (1963, 1966) has treated a generalization of the problem of ship motion in rectangular canals in which the canal bottom and walls are allowed to be rough, but with only “ small ’’ roughness protuberances, so that boundary conditions on the bottom and walls can be referred to smooth reference planes. Th e analysis becomes exceedingly complex in detail if not in principle. A similar investigation has been made by Biktimirov (1967). There are no computations.
6 . Motion with Acceleration T h e only kind of motion considered u p to now has been steady straightline motion that has been going on long enough so that the fluid motion is also steady in the ship coordinate system. However, it is obviouslyof interest to consider motion on curved paths or rectilinear motion started from rest and accelerated to some final velocity. In dealing with such problems one can proceed pretty much as we have up to now, formulating first exact problems and then linearized ones. We shall avoid here the interesting problem of combined oscillatory and forward motion, but shall mention some results on motion started from rest and along circular paths (see below). I n solving such problems it is useful to have a Green function for a singularity of variable strength moving on an arbitrary path. Such a Green function has been given by Haskind (1946) and Brard (1948) and may be found in Wehausen and Laitone (1960, pp. 490-495) for finite and infinite depth and for two and three dimensions. T he motion of a circular cylinder under a free surface starting from rest has been considered in two papers by Havelock (1949a, b) and also by Maruo (1957). This work and some computations are briefly described in Wehausen and Laitone (1960, pp. 610-617). T h e wave resistance of a thin ship in accelerated motion was first derived by Sretenskii (1939) and his result rederived by Havelock (1949a) in a different way, and also by Shebalov (1966). Th e theory is given by Lunde (1951a, p. 40ff, 55ff, 59ff), who also extends it to finite depth and canals. A different approach to the problem may be found in Wehausen (1963). Wehausen (1964) has used Sretenskii’s formula to investigate the asymptotic behavior as t -+00 of the
John V . Wehausen
160
wave resistance of a ship model started from rest and accelerated to constant speed. The problem is of obvious importance in model testing. Some computations for a mathematical ship form are included. Calculations of Sretenskii's formula for a constant acceleration have been carried out for a mathematical ship form by Efimov, Chernin, and Shebalov (1967). The result is then compared with the Michell wave resistance at each instantaneous Froude number. They have also compared different rates of acceleration and find that for 0.3 < Fn < 0.6 the wave resistance increases as the acceleration decreases. Shebalov (1962) considers a submerged body moving with variable velocity in a fixed horizontal direction, carries through the steps analogous to Kochin's in deriving his %' function and finds expressions for the force acting on the body in terms of the %' function. The essential step is to have available the time-dependent Green function. Havelock (1917a) found the wave resistance of a two-dimensional pressure distribution suddenly brought into being at t = 0 and moving with constant velocity. D'yachenko (1966) solves the analogous three-dimensional problem, but without requiring the impulsive start. As an example he treats a two-dimensional distribution moving with constant acceleration I
I
I
I
I
Curve I : Curve 2: curve 3:
I
I
o/g = o v/g = 0.05 u/g = 0.1
E/L = 0.5 h / L = 0.5
I
2
3
4 I
FIG. 13.
/
5
6
7
8
2Fn2
ResistancecoefficientR, = RWpg/4po2LB for rectangular pressure distribution
( p o )in acceleratedand steady motion. [From Doctors and Sharma, 1972 (Fig. 11, p. 258), by permission of the Society of Naval Architects and Marine Engineers.]
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161
from rest. Doctors and Sharma (1972) have rederived these results, including the effect of finite depth, made numerous calculations, and exploited the results in various ways. Here we reproduce in Fig. 13, one of their computed curves showing a comparison of steady wave resistance with two steady accelerations. The pressure distribution is rectangular, but the pressure does not drop discontinuously to zero. Finally we note that Warren and MacKinnon. (1968) have calculated the wave resistance for a “thin” disk moving in its own plane along an arbitrary, not necessarily horizontal plane.
7. Circular Path Havelock (1950) has computed the tangential and radial forces acting on a submerged sphere moving at constant angular velocity along a circular
path. We reproduce the resistance formula: If the radius of the sphere is a, the radius of the path r, the depth of the center d, and the angular velocity 0, then
R=
4X2pa6Q4m
gr
C n5Jn2(n2Q2r/g)exp(-2n2Q2
d/g).
(3.69)
1
He also computes the same quantities for a spheroid and in addition the moment acting on it. Sretenskii (1957a) independently calculated the force components acting on a sphere, but with a factor 8/3 instead of 4. In a note added in proof he attributes this to Havelock‘s having satisfied the boundary conditions on the sphere more accurately. Some results for a more general body have been stated by Perzhnyanko (1960). Havelock shows comparative graphs of R against Qr (both made appropriately dimensionless) for the sphere (r = d, 4d, and m) and for the spheroid. Shkurkina (1966) has considered the problem of variable angular velocity along a circular path and in particular the case when the sphere is suddenly set into motion along the path. 8. Stratified Fluids
It has been known at least since some experiments of Ekman (1906) that, if a ship is moving in a layer of fresh water over a layer of salt water, there is an extra resistance associated with waves generated at the interface. In particular, a large resistance maximum can occur at speeds well below those at which surface-wave resistance becomes important. One will find some discussion of the phenomenon with references to earlier literature in Wehausen and Laitone (1960, pp. 503-505).
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John V . Wehausen
The first ones to carry over the thin-ship theory to this situation were Sretenskii (1959) and Hudimac (1961). Essentially what is required is deriving the Green function for the problem, now complicated by an additional set of interfacial boundary conditions. Hudimac gives the Green function only if the singularity is moving in the upper fluid. Sretenskii gives it also when the singularity is in the lower (infinitely deep) fluid. They then give equivalent generalizations of Michell's integral to the case of a thin ship moving in the upper fluid. Sretenskii gives an additional formula for a " thin submarine " in the lower fluid. Sretenskii's results were extended to finite depth by Uspenskii (1959). Later Sabuncu (1961) rederived these results and completed them by finding also the Green function when the lower fluid was of finite depth, the upper one bounded by a horizontal plane (a rigid ice sheet), and both simultaneously. The associated resistance is also given. Later Sabuncu (1962a) made some calculations of the resistance for a body generated by a source and sink of equal strengths that
FIG.14. Resistance coefficient 2 5 0 R / 7 r ~ U ~ Vfor ~ ' ~'' Rankine solid " moving in upper layer of a stratified fluid. [From Sabuncu, 1962a (Fig. 4).]
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163
would have generated a Rankine ovoid of length/diameter 10.5 in an unbounded homogeneous fluid. It is not easy to state exactly what sort of body is generated, but presumably something like a Rankine ovoid. I n any case the computations show clearly the effect of the internal wave at low Froude numbers. Figure 14is reproduced from Sabuncu's report and shows clearly the low-speed " dead water " resistance. He calculated the resistance also for other relative depths of the upper fluid, with the " Rankine ovoid" always resting on the interface. He also calculated the moment acting on the body as a result of the two sorts of wave. 9. Nonuniform Current I n all considerations up to now we have assumed the ship to be moving in still water, which is equivalent, of course, to having the ship held still and a uniform stream of water flow past it. In rivers, however, the flow is known not to be uniform, so that it is of special interest to those concerned with inland waterways to study the effect of nonuniform flow upon the resistance. T he theoretical investigations have been carried out chiefly by Kolberg (1959b, 1961) and Cremer and Kolberg (1964). Since the flow is no longer irrotational, there is no longer a velocity potential and this initial hurdle must be gotten over. Let the velocity of the undisturbed stream relative to the ship be U ( y )and let V = (u, v, w ) be the disturbance velocity caused by the ship. Then Euler's equations for an inviscid fluid, linearized by neglecting second-order terms in (u, v, w ) , become uu,+ uv= -p-'pz, (3.70) u v , = -p - 'p, -g ,
uw, = -p -'p2. By using the continuity equation u,
+ v, + w , = 0,
one can eliminate
u, v, w and obtain the following partial differential equation inp'
Ap' - 2 u ' U - l ~ ;= 0.
=p
+pgy:
(3.71)
T he boundary conditions can also be reformulated in terms of p'. If there is an external pressure being applied (see 3.23), then the free-surface boundary condition becomes n
(3.72)
and the surface is given by 1 Y(x,2) = - [pyx, 092) +PI. Pg
(3.73)
John V. Wehauserz
164
Bottom and side-wall conditions (if necessary) become
p;(% -4 y ) = 0,
P;(% y , 9) = 0.
(3.74)
An equation analogous to the fourth one of (3.15) is also necessary. It is evident that one may try to carry through a program similar to that already completed for, say, the “ thin ship ” and the moving pressure distribution. T he first step for the former will be to construct a Green function for the equation and boundary conditions above satisfied by p‘. Kolberg is able to do this in the usual way for both h = 00 and h < co if he assumes
U ( y )= v exp Py,
P > 0.
(3.75)
The construction is extended to canals in Cremer and Kolberg (1964). Kolberg then shows that the linearized body boundary condition (see 3.15, Eq. 3). (3.76) w(x, y , f0) = F w x x , y ) can be satisfied by a distribution of his Green functions with strength proportional to U(y)fz(x,y ) . Th e analog of (3.17) follows immediately from (3.5):
R =2
J P’(% y, 0 ) f h y ) dx dy.
(3.77)
SQ
T he solution for a moving pressure distribution can also be carried through, and this is done by Kolberg (1961) and Cremer and Kolberg (1964). Finally, in the latter paper there is also an investigation of the asymptotic behavior of the wave pattern behind a moving singularity: They use this pattern to establish some rules of “ equivalent velocities ” allowing one to pass from nonuniform to uniform flows.
10. Viscosity EfJects Within the context of this chapter the effect of viscosity can be taken into account only by some sort of ad hoc procedure. T h e most straightforward idea of this sort is to increase the ordinates of the ship hull by the amount of the displacement thickness at each point and then to complete the afterbody by either an infinite or finite wake. Such ideas were put forward by Havelock (1926a), together with calculations, and then subjected to a more thorough and critical analysis later (1948) in a paper with many valuable insights. Havelock’s ideas were formalized somewhat by Lavrent’ev (1951) and computations based on Lavrent’ev’s formulas have been made by Wigley (1962, 1963, 1967). In comparing the wave resistances of Havelock’s modified ship forms with the originals, it is clear that the modifications have produced resistance curves conforming more closely to the
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165
behavior of observed ones than do those of the originals. Rather than reproduce some of these results, we prefer to show a recent calculation of Milgram (1969). Milgram’s “ ship ’’ has cosine waterlines and triangular sections. The “ wake ” consists of a triangular prism joined to the hull by a parabolic transition. The modified ship has the following equation :
@?(1 +y/T)cos r x / L ,
5 +L,
+ (.rrl2L)(x xo)”(*L +xo)], x 5 BL. @( 1 +y/T)(+L)(BL + x*),
B q1 +y/T)[cos z=[
xo 5 x
7
.
4
-
-*L F x <xo,
-
Computations of the Michell wave resistance were made for LIT = 10 and for xo= -0.40L and -0.45L. The wave resistance coefficient R,/ ;pU2($B)2is shown in Fig. 15. The effect of the wake evidently is more important at the smaller Froude numbers, as had been observed by Havelock.
0’40F-----l 0.36 0.32
-
0.28
-
NJ 0.24m
-
-LN0.20 -
N
3
U
-
0.08
-
NO SEPARATION SEPARATION AT 0.95 L SEPARATION AT 0.90 L
-
-
-
Fn
FIG.15. Effect of wake upon wave resistance. [From Milgram, 1969 (Fig. 2, p. 71), by permission of the Society of Naval Architects and Marine Engineers.]
Wigley (1938) follows a more empirical method in trying to correct for viscosity. He observes that the afterbody of a ship seems not to play as large a role in wave making as predicted in the Michell theory. He therefore proposes an empirical correction factor in which the effect of the afterbody is reduced by multiplication by an empirical factor depending upon the Froude number. The corrections seem generally to deform curves in the right direction, although they do not fully account for
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John V . Wehausen
discrepancies. It should be noted that this correction may be considered also as a correction to the thin-ship boundary condition. Similar but more extensive correction methods of Inui will be mentioned later. There have also been attempts to consider the effects of viscosity in a more fundamental way. The first of these was by Sretenskii (1957b) who used a linearized form of the Navier-Stokes equation (Oseen’s approximation) to treat a moving pressure distribution. Tangential forces are neglected. Formulas for resistance are derived that are not dissimilar in structure to those for an inviscid fluid. If P(x, z) represents the imposed pressure, then
(3.78) where
m2 = k2
+ la,
D = (2vm2 + ikU)2+ mg - 4vam3(m2- ikU/v)l12, and
R=-
1
k 2 P Im
jj
mk
( I 2+J 2 )dk dl,
(3.79)
-m
where m
I
+ iJ= jj exp i(kx + Zz)P(x, x) dx dx. -m
Kolberg (1959a) has extended the results to water of finite depth. Essentially the same problem is studied by Gruntfest and Nikitin (1966) except that they treat an initial-value problem in which the pressure distribution is suddenly brought into existence at t = 0. They examine the asymptotic behavior as t --f co for a pressure distribution degenerated to a singular line distribution of length L in the direction of motion. No reference is made to Sretenskii’s work and it is not clear that the two results are consistent with one another. A different sort of approach to the effect of viscosity is made by Lurye (1968). He also assumes small disturbances with respect to U and uses Oseen’s equations. However, he starts with a solution containing a singularity and wake-like flow following it. (In the neighborhood of the singularity the disturbance is, of course, not small.) T o this he adds another velocity, also small, chosen so that the sum still satisfies Oseen’s equations
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167
but also the linearized free-surface conditions for a viscous fluid. This second flow is constructed in the form of double Fourier integrals and is not very perspicuous. T. Y. Wu (1963) has followed still another path in trying to take account of viscosity in a more fundamental way. He starts from the initial ad hoc idea of increasing the ship by its boundary layer plus wake, but instead of simply taking a nominal displacement thickness and wake or using that for a flat plate in an infinite fluid, he has considered the interaction between the waves and the boundary layer. For the boundary-layer calculation he uses von Kdrmhn’s momentum-integral equation and similarity methods associated with this equation. For the exterior potential flow he uses the thin-ship potential. In the author’s words, the work is “ a very preliminary effort.” I n a paper by Webster and Huang (1970) equations for a turbulent boundary layer are solved by an approximate method. However, the emphasis here is on prediction of separation and the influence of the Froude number upon this rather than calculation of the effect of the boundary layer upon the wave resistance. Brard’s (1970a,b) investigations are more ambitious but too complex to summarize in any detail. He assumes linearized Navier-Stokes equations and linearized free-surface conditions, as did Sretenskii and Lurye. In addition, he assumes that the position of the boundary layer and wake as well as the vorticity distribution within this region are known. He wishes to satisfy the exact boundary conditions on the hull and does this by expressing the velocity as the sum of four vector fields, three of them irrotational, the fourth determined by the known vorticity distribution. A Green function appropriate to the free-surface condition is constructed and used to derive integral equations for the irrotational vectors. Although no solutions are obtained, the decomposition of the velocity is used as a basis for a decomposition of the resistance. He especially addresses himself to the problem of a rational definition of wave resistance in a viscous fluid. Intermediate between the attempts to account for viscosity by artificially adding wakes and by using the Navier-Stokes equations are papers by Tatinclaux (1970) and Beck (1971). Th e former has been discussed near the end of Section I1 and is concerned chiefly with the effect of the wake upon direct measurement of wave resistance from wave profiles. Beck couples the thin-ship approximation with a rotational wake of known vorticity distribution and position, but does not assume a viscous fluid. Th e rotational wake is eventually taken in the form of a U-shaped vortex sheet trailing aft from the ship with draft and breadth as parameters. Th e resistance is then expressed as a sum of the Michell resistance, the wave resistance of the wake itself, the interaction of the two and an
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John V . Wehausen
" equivalent viscous resistance " resulting from a head loss associated with the wake. Computations of the wake and interaction resistance were carried out for a hull with parabolic sections and waterlines, one extensively tested by Lackenby (1965) and for which measured values of the equivalent viscous resistance, the total resistance and the Michell resistance were available. Variation of the draft of the vortex sheet showed only a small effect, variation of the ratio of the wake breadth to the ship beam showed a greater effect, but still a small one compared with the total measured resistance. However, the interaction resistance shows substantial variation as a function of the attachment point. Beck uses the attachment point (for a fixed breadth and draft of the wake) as a parameter to be selected so as to make the measured total resistance agree with the sum of the four component resistances. He is able to make the selection only for the two largest Froude numbers. However, one should remember that the model of the wake is a very simplified one, and a more elaborate one would presumably allow fitting at any Froude number. Beck's procedure may seem artificial in that he must start with experiment data to determine the vortex strength in his wake model and then must again use experiment data. However, one must keep his aim in mind. He is trying to find out whether it is possible to discover a wake model that will allow one to explain the smoothing out of the humps and hollows in the Michell resistance that are observed in measurement. Once this is established, one might hope to proceed to more elaborate models, perhaps in the manner of Brard (1970a,b). We mention finally a paper by Dugan (1969) in which the plane problem of motion of a submerged cylinder normal to its generators is considered. Oseen's equations and linearized free-surface boundary conditions are used. A Green tensor is derived and the problem reduced to solution of a pair of coupled integral equations. For the special case of a flat plate of finite length moving edgewise beneath the free surface Dugan is able to derive an approximate solution (assuming among other things small Reynolds number). Although the result is still rather remote from most ship problems, it displays a resistance component due entirely to the interaction of viscosity and gravity.
11. Surface- Tension Effects Once one has found the appropriate Green function for a fluid with surface tension, one can repeat the calculations already made for thin ships, submerged bodies, etc. T h e Green functions for both unsteady and steady motion are given in Wehausen and Laitone (1960, p. 636ff). The thin-ship theory has been carried through by Webster (1966). Computations
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169
are given for ship models of lengths 2, 5, and 10 ft. For the last two the effect of surface tension is negligible, for the first substantial. I t appears, however, that 5 ft is about the lowest one should go in model tests. Other interesting things turn up. For example, certain kinds of singular behavior present in the case of gravity alone disappear when surface tension is included. 12. Slender Ships Derivations of the wave resistance that make use of the slender-body approximation may be found in papers by Maruo (1962), Joosen (1963), and Tuck (1963ab, 1964ab) each from a somewhat different point of view. The slender-body expression for the wave resistance is expressed in terms of the section-area curve (3.80) The following formula requires S’(x) to be continuous except at the ends, where it must, however, remain finite:
L/2
- S’(-+L)
+
S”(X)Yo(K(4L x)) dx
J-L/2
+ S’(@)/L’ZS”(x)Y,(K(+L-x)) dx + S’(+L)S’(-+L)Yo(Fn-2) - LI Z
2 - - JO
-T
2
-
jodY drljz(-%
s”, p,.
;
Y)fz(-%
rl)ln(irKlr
-T
drlfZ(SL Ylf,(*-G rl)ln(iyKb
+?ID
+ rl I ),
(3.81)
where In y=O.5772.. . is Euler’s constant. If the ship is such that S‘(f1;/2) = 0, then all terms after the first vanish. This would be the case with a raked stem, for example, but not a vertical one with finite entrance angle. This form for the resistance was first given by Maruo. A somewhat different form, expressed in Stieltjes integrals, is known as Vosser’s integral :
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John V . Wehausen
It is not possible, however, to obtain the last two integrals in the first form directly from this one. [In fact, (3.82) predicts infinite resistance if S’( fL/2) # 0.1 Maruo pointed out that (3.81) could be derived from Michell’s integral as a limit when the draft becomes small. As he remarks, this in itself does not mean that Michell’s integral is more accurate than Maruo’s expression. However, Wehausen (1963) has argued that Michell’s integral does indeed carry the most important information about wave resistance independently of the beam/draft ratio. In any case, the slender-body approximation has turned out to be a disappointment for wave resistance. In some sense it ought to have been better than thin-ship theory, for ships have beam/draft ratios usually between 2 and 4 and would seem offhand to be better suited to this approximation than the thin-ship one. However, for both low and high Froude numbers the resistance becomes negative and its behavior at moderate Froude numbers usually has more extreme humps and hollows than the Michell resistance. The difficulties are discussed by Kotik and Thomsen (1963), where some computed curves are also shown.
13. “Flat” Ships Inasmuch as a ship’s beam is usually two to four times its draft, one naturally asks why a counterpart to the Michell theory has not been developed in which the ship boundary condition is satisfied on the waterplane section rather than the centerplane section. There is, in fact no difficulty in formulating the problem. The difficulty is in solving it. I t leads to an integral equation with a particularly troublesome kernel and in particular to no explicit formula. In the two-dimensional case one can solve the problem, and it has been done by several persons. References and a description of one method may be found in Wehausen and Laitone (1960, pp. 587-592). The three-dimensional problem has been studied by Maruo (1967). The problem is formulated and an integral equation derived. He proceeds then to develop two approximations. First he works out a high-aspect ratio theory (an analog of the lifting-line approximation in wing theory). This can be completely worked out. Next he treats the low-aspect ratio case. This leads to another integral equation, simpler than the original one but still too difficult. By next considering a high-Froude-number limit, he is able to find some explicit results. Three-dimensional planing at high Froude numbers has also been treated by Wang and Rispin (1971). However, they have been able to solve their integral equation for aspect ratios that are O( 1). They give computations for rectangular plates with aspect ratios between 0.5 and 2.0. Comparison
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171
with experimental data of Sottorf shows fairly satisfactory agreement. Hogner in a series of papers (1932a, b, 1936, 1954) has proposed a flatship theory further elaborated into an “ interpolation formula ” designed to interpolate between thin and flat ships. The formulas must, however, be regarded as conjectured useful expressions rather than proved ones. They have not been sufficiently explored numerically so that one can judge them from this point of view.
14. ‘‘Shallow” Water When the depth of water becomes small compared with the wave length of typical waves, the equations and linearized boundary conditions that we have been using are replaced with another set of approximating equations, the so-called “ shallow-water ’’ equations. We shall not introduce the equations here [they are nonlinear and can be found in Stoker (1957, Chapter 10) or Wehausen and Laitone (1960, Chapter E)], but remark that these equations can be further linearized. These linearized shallow-water equations can be used to examine the flow about a steadily moving ship. T h e problem was attacked by Michell in his original paper on thin-ship theory (1898), and solved for a wall-sided thin ship touching the bottom, i.e., for a “thin ” vertical strut in a steady stream. T h e result was rediscovered by Joukowksi (1909) in 1903. T h e problem has been reconsidered by Tuck (1966), who does not require the ship to extend wall-sidely to the bottom. Furthermore, Tuck does not start from the linearized shallow-water equations, but from the exact equations and derives his results directly in connection with a slender-body approximation for the ship. T h e result for resistance turns out to be the same as Michell’s but he also finds expressions for vertical force and moment. T h e character of the answer depends upon whether the ship speed is sub- or supercritical, i.e., whether F h = U(gh)-1’2is (1 or > l . T h e result for resistance follows : Fh
(3.83) where S(x) is the section area as in (3.80). Th e result is at complete variance with Scott Russell’s description of the practice of certain horse-drawn tow-boat operators on the Glasgow and Ardrossan canal “ of small dimensions ’‘ who had accidently discovered that the resistance dropped dramatically if the horse could be urged over the critical speed Fh = 1. [The description and some comments may be found in a paper by Lord Kelvin (see Thomson, 1886).] The result is also at variance with the finite-depth
172
John V . Wehausen
results reported above (see, e.g., Fig. 12), which support Russell’s description. Tuck (1967) has extended his analysis to rectangular canals, but gives only sinkage and trim results. Huang and Wong (1970) have studied the surface displacement resulting from a moving rectangular pressure distribution over “ shallow ” water. 15. Wave Patterns Investigation of the wave pattern created by either a moving submerged singularity or a moving pressure point is one of the classical problems of the theory of water waves and is often called Kelvin’s ship-wave problem. We shall not carry through here the asymptotic analysis of the wave form, for it is available in many places, e.g., Stoker (1957, Chapter 8) and Wehausen and Laitone (1960, pp. 485-487). One of the most interesting recent investigations is by Ursell (1960), who has made a careful investigation of the behavior along the track of a pressure point as well as in the neighborhood of the lines bounding the wave pattern. For this latter region he has computed contour maps of the wave surface. Figure 16 shows a sample contour. Near the track of the pressure point the amplitude becomes infinite as the track is approached (according to the asymptotic analysis of the linearized problem), although this does not happen for 8, =
FIG.16. Contours of equal magnitude of the dimensionless surface elevation pU4 Y ( x , z ) / 2gP0 due to a concentrated force Po moving with velocity U . [From Ursell, 1960 (Fig. 2, p. 428).]
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173
submerged singularities or for a distributed pressure. The effect of distributing the pressure, as well as of viscosity, is the subject of a recent paper by Allen (1968). A new derivation of some of the classical results for pressure points has been given by Crapper (1964). As was explained in the introduction to this section, we shall simply list here a number of recent papers. Steady motion: Timman (1961), Warren (1961), Unsteady motion: Cherkesov (1963a, 1968), Smorodin (1965), Viscous fluid: Cherkesov (1963b), Nikitin (1965), Gruntfest (1965) (unsteady), Cumberbatch (1965), Allen (1968). Stratified fluids : Hudimac (1961), Crapper (1967), Fedosenko and Cherkesov (1970) (unsteady). There are two aspects of the problem of calculating wave patterns for ship-shaped bodies (as against pressure points or singularities). One is the computation of the wave profile alongside the ship. The thin-ship approximation .allows this computation, and comparison of the computed profile with a measured one is one of the valuable tests of the usefulness of this approximation. Examples will be given when we take up such comparisons. Unlike the resistance calculation, the profile calculation requires knowledge of q in the neighborhood of the ship. Thus in (3.32) the first and last integrals play a role, or in the representation of (3.50), the double integral of (3.51) must be computed as well as the single integral.
DISTANCE BEHIND SHIP, gx / U L
FIG.17. Contours for computed wave pattern for ship with L = 500 ft, U = 10 knots, corresponding to Fn = 0.133. In the length scale usedaship lengthis approximately 56.4 = Fn-2.[From Tuck, Collins, and Wells, 1971 (Fig. 1, p. 14), by permission of the Society of Naval Architects and Marine Engineers.]
John V . Wehausen
174
The first person to provide a table for such computation was Wigley (1949). If one assumes that the ship has infinite draft, i.e., replaces a ship by a vertical strut, the calculation of the wave profile alongside the strut simplifies considerably. The necessary formulas were worked out by Havelock (1932a) and exploited by him (1932b), Wigley (1931, 1934), and others. The wave pattern behind a ship has been investigated by Tuck, Collins, and Wells (1971) for a ship with rectangular section and parabolic waterlines. They have used the middle integral in (3.32) to compute the “freewave ’’ pattern. Figure 17 shows the pattern for Fn = 0.225. The results are given for a ship with L = 500 ft, B = 50 ft, T = 25 ft, U = 15 knots.
F. NUMERICAL METHODS Computation of Michell’s integral and of the several variations and extensions of it has become progressively easier as computing machines have increased in speed and capacity. Calculations requiring only a few seconds now were close to impossible in the 1930’s and 1940’s and still difficult in the 1950’s. As a result the, earlier computations were made only for the simplest hull forms. Much effort was spent in the past in preparing tables that could be used to facilitate the computation of Michell‘s integral for simplified ship forms, for example, Weinblum (1955), Eggers and Wetterling (1956), Guilloton (1951a). Although these were all very useful as recently as ten years ago, I think it is fair to state that the usefulness of tables has declined rapidly (except for testing computer programs) in favor of tape storage or even of computation of the entries as needed. Of course, the methods of analysis remain valid. The actual methods of computing have not changed so much, and we shall give some brief indications of these. One procedure is to represent the hull as a series in some fundamental set of functions Tn(x,y): f(x, Y) = C anTn.
(3.84)
If this is substituted into (3.38), say, the function I form in the a,, : I
+ iJ = C a, 1
SO
Tn,(x,y)exp(ihKx
+ iJ becomes a linear
+ h2Ky) dx dy
(3.85)
and when this is further substituted into the expression for R,one obtains a quadratic form in the an : (3.86)
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175
where C,,(K) is a well-defined function easily derivable from (3.38). The same procedure can be used if one starts with another form of Michell’s integral like (3.68) (but here one gets the quadratic form immediately). The functions C,,(K) can then be computed and preserved in a table, or stored on tape. Several choices have been made for the T,(x, y). Weinblum (1955, but also many papers going back to 1930) chose polynomials Tpq(x,y ) = xpyq. We shall not describe his tables in detail, but roughly p varies from 2 to 12 and q = 0, 1, or 4,while Fn varies from 0.18 to 1. Webster and Wehausen (1962) and Lin, Webster, and Wehausen (1963) have used the trigonometric polynomials sin(2m7rx/l)cos i(2p - l ) ~ y / T , (3.87) cos(2m - l)Trx/L cos +(2p- l)Try/T.
As is well known, any continuous function f ( x , y ) defined on So can be approximated uniformly by either ordinary polynomials xpyq or by trigonometric polynomials. The latter,as chosen above, have the advantage of being orthogonal, so that the coefficients apq will generally decrease as p and q increase. This is not the case for ordinary polynomials. On the other hand, if one terminates a series of trigonometric polynomials too soon, one obtains rather wiggly lines as an approximation. This blemish is more an aesthetic than a computational one. The difficulty with ordinary polynomials can be remedied by using orthogonal polynomials, say Legendre polynomials. Michelsen (1972) has worked out the procedure for Gegenbauer polynomials, which are more general than Legendre polynomials. However, no computations seem to have been made. Finally it should be noted that Michelsen (1963) and Michelsen and H. C. Kim (1967) have used polynomial representation for f together with (3.68). This list is not by any means an exhaustive one for computations done by some variant of this procedure. The other chief method of computation is essentially direct numerical integration, given the offsets of the hull at a grid of points. The first person to set this up as a systematic procedure that could be based upon a set of tables seems to have been Guilloton(1951a). His “ method ofwedges ” appears to be essentially one in which the hull is approximated within each rectangle of the grid by a surface with parabolic waterlines and straight frame lines. However, the geometric elements used to approximate the ship are semi-infinite “ wedges ” extending aft. Adding and subtracting wedges allows one to make the approximations mentioned above. Guilloton has prepared tables of functions for the necessary component wedges that allow calculation of the streamlines alongside the ship and of the resistance. Note that the obtaining of streamlines is one of the methods’ special
176
John V . Wehausen
advantages. T he method has been further discussed by Guilloton (e.g., 1956, 1965), and has been applied by Korvin-Kroukovsky and Jacobs (1954), Emerson (1967, 1971), and by Webster and Huang (1970). A somewhat similar procedure has been developed at the University of California, Berkeley, by C. C. Hsiung and W. C. Webster in which the hull is approximated within each grid rectangle by a ruled surface with straight waterlines and straight framelines. T h e geometric elements are “tent functions:” Let ( x i , y j )be a grid point and define f(i.j)(X,
y)
(3.88)
If zii is the hull offset at ( x i , y j ) ,then the approximating hull is given by (3.89) It evidently agrees with the offsets at the grid points. Equation (3.89) is now analogous to (3.84), and one repeats essentially the same steps
taken there, eventually finding
(3.90) The functions A i j k l , in a dimensionless formulation, can be stored on tape for a particular choice of grid and a selection of Froude numbers. On a CDC 6400 computer it takes about 45 sec per Froude number to compute the A i j k lfor a given value of T / L if one uses 6 waterlines and 13 stations. Once the tape is made, it takes about 9.5 sec to compute 22 points on a resistance curve with Fn varying from 0.2 to 1.0. Test computations agree well with the results of others. T h e methods can be made more accurate by either introducing a more elaborate tent function (say, the equivalent of Guilloton’s method) or by taking a finer grid.
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We have obviously not exhausted the possibilities of numerical computation, nor have we discussed those aspects of the behavior of the functions being integrated that cause difficulty. There is discussion of this point in Birkhoff and Kotik (1954a).
G. COMPARISON OF THEORY AND EXPERIMENT As has already been emphasized in Section 11, measurement of wave resistance cannot be cleanly separated from viscous resistance. There is always some interaction between the two in a real fluid. On the other hand, practically all of the analytical results have as underlying assumption an inviscid fluid. Of the two methods of measuring wave resistance we consider the direct measurement from wave patterns as conforming more closely to the analytical theory than does the residuary resistance. Most of the comparisons are for the Michell resistance in unbounded deep water, this being the case of greatest practical interest. However, there have been a few comparisons in restricted waters and more recently, in connection with hovercraft, comparisons for moving pressure distributions. 1. Michell’s Integral There are certain consequences of the thin-ship theory that hold also in restricted waters and that can serve as tests of the usefulness of the approximation without requiring computation of the integral itself. One of these consequences was evident as soon as the linearized problem had been formulated: Neither the wave resistance nor the wave pattern depend (to this order) upon whether the ship is fixed in position or free to trim and squat. (This does not mean that trim and sinkage cannot be determined by linearized theory.) As a result, if model tests are made under each condition, one cannot expect the resistance or the far-field wave spectrum, for example, to be more accurately predicted by theory than the difference between the two measurements. That this test was not recognized very early is probably a result of the fact that the first derivations of the resistance were carried out without first formulating the exact problem. Of the next two consequences the first can also be deduced directly from the formulation, and both are evident as soon as the resistance integral is derived: R , is proportional to B2 and is independent of the direction of motion of the ship. One’s immediate reaction to the second of these is to recall that viscosity has been neglected and that different boundary-layer growth in the two directions will make the prediction invalid. This is true, but it is also true that all these results also are con-
178
John V. Wehausen
sequences of the linearization approximation and would not be true in an exact theory (or even second-order theory) in an inviscid fluid. The last consequence of this nature concerns the effect of fore and aft symmetry. It states that if y =f(x, y ) gives the hull form of an arbitrary ship, then the Michell resistance of the symmetrized ship (3.91) is not greater than that of f ( x , y). The confrontation of these last predictions with the results of experiments was carried out by W. C. S. Wigley in a series of papers. Further experiments with this aim are reported in Weinblum (1932b). Although these tests of the theory could have been made without the necessity of any preliminary calculations of Michell’s integral, this is not the way in which Wigley or Weinblum chose to proceed. I n his paper first (1926) on the Michell resistance Wigley compared residuary resistance with calculation for the forms
+
z = f0.32( 1 cos7rx/S)(1 z = &1.333~0~7rx/16 z = fb(1 -J+’)cosTX/~~,
+C O S ~ T Y / ~ ) , (3.92)
b = 1.
In his second paper (1927) he varied b = SB in the last hull form above: b = i, Q, 1. The B2 dependence was approximately confirmed at high Froude numbers, but not at a lower ones, as might be anticipated from the fact that errors in identifying residuary resistance with wave resistance become relatively greater as Fn decreases. On the other hand, the two curves become increasingly more parallel over the whole range of Fn as b decreases. There have been very few further attempts to check this point. In view of the difficulty in obtaining an estimate of a true wave resistance from the residuary resistance, it would appear to be preferable to base a study of the beam dependence upon quantities derived from the free-wave spectrum or upon the wave profile alongside the ship. An investigation of beam dependence by this means has been made by Makoto (1969), using three models and four Froude numbers. Although his thinner model conforms best to the theory, the two fatter models are more consistent with each other. It is perhaps of some significance to note that the dependence of the resistance of a moving pressure distribution (see 3.46) upon the beam will not vary as the square but in some complicated manner. The effect of asymmetry was investigated by Wigley in 1930(a) and again in 1944. Furthermore, in the 1944 paper he also investigated the effect of symmetrization as in (3.90). For the asymmetric models, tests with the fuller end leading showed less resistance than those with the finer end leading, with the two coming together in the neighborhood of the first hump (counting from the right), i.e., at about F n = 0.45, and reversing
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179
their order below Fn = 0.15. Wigley attributes the lack of confirmation of the prediction of the Michell theory to viscosity rather than linearization. In fact, he writes (1930a) as follows concerning this point: “ It is probable, though not capable of definite proof that this is not a result of any other assumptions [linearization is meant] made in order to simplify the calculations, but is accurately true for any wave motion in a perfect fluid. T h e actual differences observed ...may therefore be ascribed to the effect of viscosity.’’ Although Wigley gives no reason for this conjecture, one may guess that he has based it upon the fact that the wave resistance of a moving pressure distribution is also independent of direction of motion in the linearized theory, as one may deduce immediately from (3.46). There seems to be no evidence in the exact formulation of either problem to support this conjecture. I n the 194-4 paper Wigley shows only the difference in the resistances of an asymmetric model in the two directions of motion, thus avoiding to some extent the uncertainty in comparing directly with the residuary resistance. Figure 18 is reproduced from this paper. T h e
FIG.18. Difference in resistance coefficients for asymmetric models towed in each direction. [From Wigley, 1944 (Fig. 10, p. Sl).]
John V . Wehausen
180
calculated curve is based upon Wigley’s viscosity correction mentioned earlier; the calculated difference based upon the Michell theory is, of course, zero. The resistance coefficient is Froude’s “ circular ” C, defined by
250 R -~ ?T
pUV2‘3’
a coefficient often used in the United Kingdom. In the comparison of asymmetric models with the associated symmetrized ones Wigley again showed only the difference in resistance between the two models. Figure 19 is another figure from Wigley’s paper. Here the agreement of measurement and theory can perhaps be considered satisfactory when the full end is leading in the asymmetric model, but not 0.3 v?
w
u
J 0.2 B u. Y
0
00.’
8 y o
3
v)
v)
W U
2+0.1 w a U. w
k 0
0
0 LL 0
5-0.1
3
v)
L.W Ls. OF MODELS
0 .l
0.2
0.3
0.4
0.5
0.6
F” FIG.19. Difference in resistance coefficient for an asymmetric model and the associated symmetrized model. [From Wigley, 1944 (Fig. 4, p. 47).]
The Wave Resistance of Ships
181
when the fine end is leading. On the other hand, Wigley’s viscosity correction shows up rather well in this case. Wigley (1942) apparently made only one test to compare resistances of a model free to squat and trim and one fixed in position (one should note in this paper that the labels of models 1970B and 1977 are interchanged in Fig. 10). However, this was remedied by Shearer (1951) who made such tests for models with three different prismatic coefficients C, = 0.533, 0.587, and 0.693. (We recall here that the coefficients most often used by naval architects to characterize the geometrical properties of a hull are the prismatic coefficient C, = V / L x area of midship section,
the block coefficient
C, = VILBT, and the volumetric coefficient
c,= vp3, or one related to it.) Three of Shearer’s curves are shown in Fig. 20. For all three models the resistance coefficient is higher almost everywhere for the model free to trim than for the fixed model, and the difference is for the most part rather substantial. In assessing the results of this preliminary confrontation of the linearized theory with experiment, the last one is the most discouraging, for the discrepancies cannot be easily explained away by viscosity effects, as in the symmetry tests, or by the difficulties associated with identifying the residuary resistance with the wave resistance, as in the B2 test. Moreover, as we shall see later, the trim and sinkage enter into the second-order theory, and it may be the case that even Shearer’s models, of smaller than average BIL and BIT, are still not “thin” enough for the theory to be accurate enough to be useful in design. On the other hand, as one can see from Fig. 20, the general behavior of the theoretical and experimental curves is similar; this is pretty much true in all the tests that have been made. In a further series of papers by Wigley (1926, 1931, 1932, 1934, 1939, 1940, 1942), Wigley and Lunde (18), Weinblum (1932b, 1934, 1938, 1939), and Weinblum, Schuster, Boes, and Bhattacharyya (1962) there are given the results of comparisons between experiments and theory for systematic variations of characteristics of the ship form insofar as this could be accomplished with mathematical models of the form f ( x , y ) = X(x)Y(y). I n particular, what has been varied is the prismatic coefficient, angle of entrance at the bow, and the section form. The most complete set of experiment
John V . Wehausen
182
0.10
0.20
0.M
0.40
F"
FIG.20. See facing page for caption.
0.50
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0.10
0.20
0.30
0.40
183
0.50
F"
FIG.2,O. (Cont.) Comparison of resistance coefficient for models fixed in position and free to trim with Michell resistance. [From Shearer, 1951 (Figs. 9, 10, 11, pp. 6 5 , 66), a paper reporting work done at the National Physical Laboratory.]
data together with calculations and, with the most carefully estimated residuary resistance (using double models), is in the last cited paper. Th e variation in form parameters was as follows: C, = 0.56, 0.60, 0.64, 0.68; BIT= 2, 3.33, 1.25 (for C,=0.56). Sections were U-shaped. Several graphs from this paper are reproduced in Fig. 21. The results of the earlier experiments have been summarized and discussed by Wigley (1930b, 1935, 1949) and by Weinblum (1950), so that we shall limit ourselves here to a brief statement of conclusions, some which can be confirmed by exarnining Figs. 20 and 21. I t seems appropriate to quote Wigley's own words (1935, but almost identical with those of his first paper in 1926): The calculated curve tends to be less in mean value than the experimental curves at the higher speeds. (2) The humps in the calculated curves are exaggerated in comparison with the experimental results. (3) The hollows in the calculated curve are more exaggerated than the humps, and generally appear flat in the experimental curve. (4) The humps and hollows occur rather earlier on a speed base in the calculated curves than in the experimental, generally by about 8 percent of the speed.
(1)
184
John V . Wehausen
a4
Fn
FIG.21. See facing page for caption.
as
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185
FIG.21. (Cont). Comparison of measured and calculated wave resistance for mathematical hull forms of varying prismatic coefficient. [From Weinblum, Schuster, Boes and Bhattacharyya, 1962 (Figs. 25, 28, 30, 32).]
186
John V . Wehausen
T o these one may add a fifth conclusion from the paper of Wigley and Lunde (1948) : (5)
. . . the presence of a full midsection, and therefore of a rather flat bottom, does not cause more discrepancies between calculation and fact than occur with finer midsections.
Only the first needs to be amended somewhat, for Wigley’s experiments, with one exception noted above, were always done with models free to trim. From Shearer’s experiments one sees that for Froude numbers in the neighborhood of the first hump the Michell resistance lies below the residuary resistance when the model is free to trim, but above it when the model is fixed. However, in a comparison of the Michell resistance with the residuary resistance €or Taylor’s standard series by C. C. Hsiung and Wehausen (1969), the Michell resistance lay above the free-to-trim residuary resistance for C, = 0.48. I n the experiments of Weinblum et al. (1962), with models free to trim, the calculated and measured values were very close together in all cases. To the left of the first hump the oscillations in the Michell resistance are larger, the larger the prismatic coefficient. For C , < 0.6, the Michell resistance tends to be mostly below the residuary resistance and for C , > 0.7 to lie mostly above it. For 0.6 < C , < 0.7 the residuary resistance passes through the oscillations, approximately. None of these conclusions can be
FIG.22. Comparison of Michell and residuary resistance for several hulls from Taylor’s Standard Series. [From Hsiung and Wehausen, 1969 (Fig. 3).]
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187
considered to be really firm. However, it does seem to be the case that the larger the prismatic coefficient, the smaller must be BIL to obtain satisfactory agreement. Since Shearer’s curves and those of Weinblum et al. are for a mathematical model, it seems only fair to show a comparison for a “real ” ship. Figure 22 shows this for three models from Taylor’s Standard Series and is taken from Hsiung and Wehausen (1969). T h e selected value of C, puts the theory in a better light than either of the other values chosen there for comparison, C, = 0.48 and 0.80. Figure 9 in Section I1 shows another comparison for a “ real ” ship, one of the so-called Series 60 with C , = 0.60, C, = 0.614. Here the residuary resistance lies below the Michell resistance (which was predicted above only for C, > 0.7). T h e directly measured wave resistance, which we regard as a truer measure, lies below the residuary resistance in the region 0.2 < Fn < 0.35 and then moves up to join the former. It is evident that this better measure of wave resistance has made the agreement between theory and experiment worse in this case. T h e largest collection of data comparing residuary resistance with Michell resistance for “real” ships is in a paper by Graff, Kracht, and Weinblum (1964). It does not appear to change the conclusions already reached. We now turn to two cases where the ship is indeed “thin.” The first is from an almost classic analysis by Weinblum, Kendrick, and M. A. Todd (1952) of the resistance of a plank being used in frictional-resistance experiments. Here the body is not only genuinely “thin,” but also the estimation of the viscous resistance can be very accurate. Data for the body follow: BIL = 0.0265, BIT = 0.186, C, = C, = 0.825. The body was fixed in position during the towing test. Figure 23 shows the resistance coefficient R/*pCTZS.Here the agreement between theory and experiment is remarkably good.
1.5
c)
5’1.0 x
=
L>
0.5
0.0 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fn
FIG.2 3 . Comparison of Michell and rcsiduary resistance for a [From Weinblum, Kendriek, and M.A. Todd, 1952 (Fig. 2).]
“
friction plane.”
John V. Wehausen
188
T h e second example of this sort was given by Sharma (1969). I n this case a parabolic strut with the following characteristics was tested: BIL = 0.05, BIT = 0.33, C, = C, = 0.66. However, the wave resistance was determined by direct measurement of the longitudinal wave profile as well as by subtracting an estimated viscous resistance. Figure 24 shows the resistance coefficient above plotted against y o = 1/(2Fn2) for the Michell resistance, residuary resistance and directly measured wave resistance. Again the results agree well.
X
ln N
3 P
-iN \
u
0
25
7 5
50
10 0
12 5
y o = 1/2 Fn2
FIG.24. Comparison of measured and calculated wave resistance for a thin parabolic strut. [From Sharma, 1969 (Fig. 2, p. 7 9 , by permission of the Society of Naval Architects and Marine Engineers.]
It is evident that under the most favorable conditions Michell’s integral can give very accurate predictions of wave resistance, but that these deteriorate as B / L increases. Evidence available seems to indicate that this deterioration is more rapid for large values of C , than for smaller ones. However, there does not seem to exist any systematic investigation of this topic. The preliminary investigation of Hsiung and Wehausen (1969) is handicapped by some uncertainty concerning the data from Taylor’s Standard Series, for it is known that there was considerable smoothing of experimental results. T h e excellent systematic investigation of Weinblum et al. (1962) was not designed to examine this point. If such a set of experiments were to be planned, the wave resistance should preferably be taken from wave-profile measurements and the comparison of freewave spectra included. Inui (1957) shows theoretical curves that, after having been subjected
The Wave Resistance of Ships
189
to a three-parameter system of corrections, pass almost exactly through his measured values. Even though the values of the parameters were determined after the experimental values were obtained, this suggests the possibility of determining an empirical set of corrections that could be applied to any Michell-resistance curve. Unfortunately, the prospects for this do not seem too encouraging. Inui’s good agreement was only for two mathematical models with values of C, yielding Michell curves without too violent fluctuations. Th e comparisons in Hsiung and Wehausen (1969) for C, = 0.80 do not give much ground for hoping to reconcile theory and experiment by correction factors. Emerson’s (1954) rather remarkable success in bringing the Michell resistance into line with data from Taylor’s Standard Series was for values of C , = 0.56 and 0.64, and more important, was limited to a rather narrow Froude number range 0.25 to 0.36; he used a two-parameter correction system based upon Wigley’s viscosity-correction method. We conclude by calling attention to Inui’s (1957, pp. 260-266) census of all the reported tests up to that time of mathematical hull shapes for which calculations were also available.
2. Sinkage and Trim Another test of the thin-ship approximation is provided by the formulas for sinkage and trim derived in (3.18). A comparison between computed and measured values for a single mathematical model is shown in the book of Apukhtin and Voitkunskii (1953, Fig. 54), the work being attributed to Yu. N. Popov. More recently R.W. Yeung (1972), on the basis of formulas derived by Wehausen (1969) that also exploit the smallness of the draft/ length ratio, has computed sinkage and trim for the two mathematical hull forms tested by Wigley (1942) and Shearer (1951) and for five Series 60 models with C,=0.60, 0.65, 0.70, 0.75, 0.80. Figure 25 shows the comparison for Shearer’s model with C, = 0.587. T h e agreement for the Series 60 models is not unsatisfactory, but the data stop at Fn = 0.33 just at the knuckle in the trim curve.
3 . Wave Projiles and Spectra The wave resistance is a rather stringent test of the theory because it involves the difference of large quantities. Wave profiles should be less demanding. Wigley made a number of comparisons for the wave profiles alongside a vertical strut, the only configuration for which the calculation could be carried out at that time. Many of his papers from 1931 on contain such comparisons, and the one with Lunde in 1944 comparisons for hulls of finite draft. Th e paper by Weinblum, Kendrick and M.A. Todd (1952) on the wave resistance of a thin plank also contains profile comparisons,
190
John V . Wehausen
Fn
FIG.25. Comparison of calculated and measured trim and sinkage for a mathematical hull shape. [From Yeung, 1972 (Fig. 3), by permission of the Society of Naval Architects and Marine Engineers.]
although the calculations were made for infinite draft. In most cases the theoretical curve seems to conform to the behavior of the observed curve better than is the case for the resistance. As an example we reproduce in Fig. 26 a set of curves from Shearer’s paper (1951), for he shows observed curves with model fixed and with it free to trim. Here C, = 0.587. In addition to comparing computed and measured wave profiles alongside a model, one may also make this comparison for the wave pattern behind the model. A convenient way to do this is to compare the Fourier transform of the free-wave profile as observed and as predicted by the Michell theory, i.e., to compare the functions Fl and G , defined in (2.53) ( F , = G , = 0 from symmetry about the centerplane). From the expression (3.16) for the thin-ship wave pattern, one may obtain the free-wave pattern by using only the single integral in the expression for G, but doubling it, for the double integral contributes an equal term far behind the singularity. After some easy manipulation one may write the Michell free-wave pattern as follows:
Y(x,z) =
The Wave Resistance of Ships
191
FIG.26. Comparison of calculated and observed wave profiles for Model No. 2891 of Fig. 20. [From Shearer, 1951 (Fig. 4),a paper reportingwork done at the National Physical Laboratory.]
John V. Wehausen
192
From this a straightforward calculation yields
G(r)+ iF1(r) .
.
SO
(3.94) or, expressed as functions of a, where a > K ,
Note that if the hull is symmetric fore and aft then G l r O . Sharma (1969) has shown such a comparison for the same model for which the resistance data is shown in Fig. 24, that is, for one that would seem to be genuinely thin. The variables are the same as in Fig. 10, namely, K 2 F 1 ,K2Gl,and K2(F12 + G12)1/2plotted against y/K. Results are
0
2
4
6
Tronswrse WOW lxmbsr k I K
-
8
FIG.27. Comparison of measured and calculated free-wave spectra at Fn = 0.500 and Fn = 0.316 for parabolic strut of Fig. 24. [From Sharma, 1969 (Fig. 4, 7, pp. 76, 77), by permission of the Society of Naval Architects and Marine Engineers.]
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193
given for Fn = 0.316, 0.365, 0.447, 0.500, and 0.707. Figure 27 illustrates the graphs for Fn = 0.316 and 0.500. The theoretical value for the cosine component K2G, is zero. It is interesting to note that for Fn = 0.316, the agreement for the total amplitude is fairly good even though there is considerable discrepancy in the sine and cosine components. Sharma has given similar comparisons in other papers (1963, 1968). Another comparison of this nature is in several papers by Makoto (1969), Hogben (1968, 1971), and Everest and Hogben (1970). The latter compare observed and calculated spectra for three models with parabolic waterlines and frame lines and with beam/length ratios B / L = 0.05, 0.1,0.2.Here GI=0 as above. Everest and Hogben plot essentially - &a(2a2-- K2)F,against aL. The dependence of this function upon K L = Fn-2 is not very strong in the range 0.2 < Fn < 0.55 covered in the experiments. Consequently one test of the theory is to plot all the data for different Froude numbers together to see if they, firstly, fall on one curve and, secondly, upon the curve predicted by (3.95). The agreement for BIL = 0.05 was very good on both points for Fl . For G, , however, although the data are fairly self-consistent, they cluster around a curve oscillating about zero. The self-consistency of the data is still satisfactory for the model with B/L=0.1, but they begin to deviate from the theoretical curve for Fl for larger values of aL, and the oscillations for GI become larger. For BIL = 0.2, both self-consistency of the data and agreement with the theoretical curves have deteriorated badly. Everest and Hogben use their measured spectra in still another way. They try to reconstruct the fithat in (3.95) would have given the observed spectra. This is done approximately in that they find only the strength of a line distribution of singularities at a selected mean depth (selected, one might think, to keep K - l v constant, but this seems not to be critical and 7 =-3T/8 was used). The reason for doing this is, one supposes, to gain some insight into causes of failure of the thin-ship singularity distribution as BIL increases. The observed increase in source strength ahead of the bow as BIL increases indicates, one may conjecture, an effect of nonlinearity resulting from wave steepness at the bow. However, since all analysis is based on linearized theory, it is hard to be confident of such conclusions. A weakening of sink strength near the stern of the fattest model may indicate boundary-layer growth. T o use the resulting source distribution to calculate the wave resistance would seem to be following a rather circuitous route. However, as Everest and Hogben do this, it is a further test of consistency of their results, for they use the reconstructed source distribution to compute the wave resistance in water of finite depth (h/L= 0.425 and 1.25) and then compare with measurement. The agreement seems rather satisfactory. Hogben (1971)
194
John V . Wehausen
llas measured spectra and calculated from them the equivalent line source distributions for three trawler forms, but without comparing with the thinship spectra. For these forms the spectral data do not agree very well for different speeds, but the equivalent source distributions, determined separately for different speeds, seem to agree well, except at the bow for the fullest form, if the Froude numbers are sufficiently low, less than 0.4. For Fn>O.4 there was a considerable difference in the distributions even for the finest model. It should be noted that others, e.g., Baba (1969a), have also used wave profile measurements to deduce singularity distributions that might have generated them. As has already been mentioned, the procedure seems very indirect as a method of finding the wave resistance, and its real use, if any, is in investigations of the sort conducted by Everest and Hogben. 4. Fluid of Finite Depth and Canals In comparison with the amount of testing that has taken place in order to compare theory and observation for a thin ship in deep horizontally unbounded water, there has been relatively little for the cases of motion in water of finite depth or in canals. There have been, as one would expect, many tests of the resistance of models in restricted waters, but few with the aim of comparing with theory. The main sources of information about this subject are Inui’s (1957) comprehensive report on wave-making resistance, where he both reports on some experiments and gives a number of references to Japanese investigations, and Apukhtin and Voitkunskii’s (1953) book on ship resistance, where some Russian work is reported. In both sources the comparison is essentially for a model in a rectangular canal, although the calculated resistance for motion in water of finite depth with no walls is also shown in Inui’s paper. Inui’s model is approximately one with parabolic waterlines and rectangular sections, but not exactly this, for he has used the corresponding Michell source distribution to calculate the closed streamline surface about the source distribution and has made his model in the resulting shape. As far as linearized theory is concerned, one should not distinguish between the two hull shapes, and we shall use the dimensions of the source distribution: BIL = 0.2, BIT = 4. If 2b is the breadth of the canal, as in Section 11, then tests of this model were carried out under the following conditions: 2b/B = 2, h/L = 0.4,0.3, 0.2. In Fig. 28 we reproduce his results for the shallowest water. In the neighborhood of the depth Froude number F,,= 1, it was difficult to measure stable values and some points there are marked to indicate this.
-
-*FINITE DEPTH CANAL
'TLED
0
FIG. 28. Comparison of measured and calculated wave resistance for model (B /L= 0.123, B/T = 1.46) in unbounded water of finite depth ( h / L= 0.3) and in a canal (width = 6.67h). [From Inui, 1957 (Fig. 89, p. 348).]
196
John V . Wehausen
The most significant Russian work was by Voitkunskii. His model had parabolic sections and waterlines with B / L = 0.1, B / T = 3. The form resistance was determined by a deep-water experiment and the same value used for tests in rectangular canals. Tests were made for the following cases: h/L = 0.4, 2b/L = 0.66 and 0.4; h/L = 0.1, 2b/L = 2 and 1. In both Inui’s and Voitkunskii’s tests the agreement between theory and experiment seems somewhat better than one might have anticipated on the basis of deep-water experiments. Voitkunskii’s measurements near F h = 1 follow the predicted resistance peak there remarkably well. Some experiments with moving pressure distributions in water of finite depth will be mentioned presently.
5 . Submerged Bodies Experimental investigations of the forces acting on submerged bodies moving near a free surface are rather scarce in the published literature, although one may conjecture that a large amount of data must exist in connection with the design of submarines and torpedoes. There seem to be only four papers: Weinblum, Amtsberg, and Bock (1936), Bessho (1957b, 1961), Sharma (1968), and Chey (1970). The first gives measured resistances for three different bodies of revolution at several submergence depths, but shows theoretical curves for only one submergence. The shapes of the curves are very similar for all three cases, but with the theoretical curves being displaced to higher Froude numbers, at least in the range of the experiments 0.25 5 Fn 5 0.4. Bessho, in his comprehensive survey of the theory of wave resistance of submerged bodies, reports an experiment with an ellipsoid at three different submergences. He also notes the paucity of published data, and remarks about his own experiment, “ I n general, the experiment was difficult, the resistance value was large and unsteady.” In Bessho’s results one sees again the shift of the theoretical curve toward higher Froude numbers, but his second humps are not clearly delineated as in the experiments discussed above. Sharma’s paper concerns the wave resistance of a submerged ellipsoid of rotation with a superposed ‘‘tower ” of parabolic section. Sharma’s aim is more to study the interaction between body and tower than to investigate the wave resistance of the ellipsoid. However, one can perhaps conclude from his experiments that in the case considered theory and experiment are not inconsistent. Chey’s paper will be discussed later in connection with higher-order approximations.
The Wave Resistance of Ships
197
6. Pressure Distributions Comparison of experiment with theory for moving pressure distributions seems to have occurred almost exclusively in connection with hovercraft and to have been carried out by Everest and Hogben (1967, 1969) or by Hogben (1966). (There also seems to exist a number of reports by Everest referenced in these papers.) There are very few situations in wave-resistance theory in which the agreement between linearized theory and observation has been so good. Figure 29 shows results from the 1969 paper for a rectangular pressure distribution, the wave resistance having been deduced from wave-profile measurement. (In the wave resistance coefficientP, is the mean pressure in the air cushion under the hovercraft.) There is also rather good consistency between measurement and theory for experiments in water of finite depth and for yawed pressure distributions. MEASUREMENTS : MODEL WEIGHT: 140 Ib t MODEL WEIGHT: 180 Ib
4
4.01
FROUDE NUMBER
"
4jr
FIG.29. Comparison of calculated wave resistance with measurements determined from wave-pattern analysis for a moving rectangular pressure distribution. Here LIB = 3/2. [From Everest and Hogben, 1969 (Fig. 12).]
198
John V . Wehausen
H. APPLICATIONS OF THE THEORY I n the last section we have already dealt with the application of the theory to prediction of resistance. I n this section we shall consider applications of the theory to design. This will be divided into three parts: the effect of systematic changes in form parameters, bulbous bows, and ships of minimum wave resistance. Since the predictions of wave resistance based on the Michell resistance have not been remarkable for their accuracy in the case of normal ship dimensions, one may well ask whether information about design based upon it should carry any weight. Here the general hope (perhaps wishful thinking) has been that, even though the quantitative predictions are not as precise as one might wish for, the difference, or at least its sign, between the wave-making qualities of two ships will be correctly predicted. This is, in fact, more accurately confirmed by experiments than is the absolute amount of the wave resistance. One can find a more ample discussion of the usefulness of the theory for design in Maruo (1970).
1. Effect of Systematic Form Changes Havelock was the first one to use the Michell resistance as a tool for investigating the effect of changing some aspect of the hull form. His first paper on the subject (1925a) assumed infinite draft and dealt with the effect of hollow versus full waterlines, i.e., of change of C, . He then proceeded to investigate the effect of length of parallel middle body (1925b) (again infinite draft) and of varying draft (1926a). More extensive investigations of the effect of changing prismatic coefficient, entrance angle, waterplane coefficient, etc., are contained in Wigley (1942), Weinblum (1932b), and Weinblum, Schuster, Boes, and Bhattacharyya (1962). These computations are compared with experimental measurements and have been discussed earlier. I n Fig. 30, taken from the last cited paper, R/pgV is plotted against Fn for values of C , varying between 0.52 and 0.72. Here the entrance angle is constant for all forms, T = 0.3B, B / L = 0.125. T h e hull forms are described by f(x, y ) = X ( x )Y(y), where Y ( y ) is U-shaped and fixed for the family, and X ( x )= a ,
+ u2x2+ a 4 x 4+ a,x6.
(3.96)
T he waterlines are shown dimensionlessly in Fig. 31. Inspection of Fig. 30 shows that the forms with low prismatic are superior at lower Froude numbers, but that there is a gradual reversal of order of superiority as Fn increases.
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199
FIG.30. Graph showing variation of wave-resistance coefficient R,/pgV with prismatic coefficient. [From Weinblum, Schuster, Boes, and Bhattacharyya, 1962 (Fig. 3, p. 298).]
I n Fig. 31 one of the waterlines is shown as a dashed line. This one has a slightly different representation than (3.96), namely, X ( X ) =a,
+ a& + I
a3 X I 3
+ a4x4
(3.97)
Otherwise, all the usual geometric parameters for the two models with C , = 0.56 are the same. Figure 32 shows the calculated wave resistance for the two. The predicted difference between the wave resistance of two forms is astonishingly large in the neighborhood of Fn = 0.25. Figure 33 shows the measured total-resistance coefficients R,/pgV for the two forms at three beam/draft ratios. Although the observed difference is less than the predicted one near Fn = 0.25, the sign is correct and the reversal of order near Fn=O.3 confirmed. This is one of the pieces of evidence supporting the conjecture stated above that the Michell resistance predicts differences in resistance correctly. Unfortunately, there are only few that are as discriminating as this one.
John V. Wehausen
200
1I ++ ---
MODEL FAMILY c2.4.6. . . . C>%oI\,\\. . 11 " <2,3,4, Cp,l> I
I
I
I
I
d
MODEL FAMILY <2,4,6, Cp, I>
FIG.31. Framelines and waterlines at free surface for hull shapes for which Michell resistance is shown in Figs. 30 and 32. [From Weinblum, Schuster, Boes, and Bhattacharyya, 1962 (Figs, 1 , 2, p. 298).]
The Wave Resistance of Ships
F,=U/a
201
--+
FIG.32. Comparison of Michell resistance for two hull shapes with same values for C, , B / T , and B/L. [From Weinblum, Schuster, Boes and Bhattacharyya, 1962 (Fig. 7, p. 301).]
-
.
FIG.33. Comparison of measured values of the coefficient for the two models for which computed values are shown in Fig. 32. [From Weinblum, Schuster, Boes and Bhattacharyya, 1962 (Fig. 22, p. 310).]
202
John V . Wehausen
2. Bulbous Bows Although the theory of the bulbous bow could quite properly be considered as a part of the theory of ships of minimum resistance, it seems appropriate for historical reasons and because of its special development to keep it separate. Th e effectiveness of bulbous bows was apparently discovered accidently as a consequence of the reintroduction of ram bows on warships toward the end of the nineteenth century. David W. Taylor understood qualitatively the reason for their effectiveness, and numerous systematic experiments were carried out under his direction. A brief summary of the history and further references are given by Kracht (1968). The first theoretical investigations were carried out almost simultaneously by Wigley (1936) and Weinblum (1936b), but by different approaches. Both started from thin-ship theory. Wigley modeled the bulb by putting a submerged dipole near the bow. Th e underlying idea was to place the dipole in such a position that its wave pattern would cancel most of that created by the ship’s bow. Since a submerged dipole by itself generates approximately a sphere under a free surface, the combination of the thinship singularity system and the dipole generates a bulbous-bowed ship if one fairs the sphere into the original hull. (There is obviously some difficulty here in reconciling thin-ship and deep-submergence approximations, but a little good will bridges this gap.) Wigley found that the best position for the dipole was right at the bow. Some experiments seemed to confirm this. Weinblum’s investigation remained strictly within the thin-ship framework. T o a given ship form f ( x , y ) he adds a blister af(x, y), where f b is different from zero only near the bow and is fixed. He then uses a as a parameter to formulate and solve an optimization problem. Experiments also confirmed that improvement is obtainable with an appropriately sized bulb. Although Wigley’s and Weinblum’s investigations laid the foundations for a more thorough exploitation of the possibilities of the bulbous bow, the subject lay dormant for some years until revived by Inui, Takahei, Kumano and other colleagues of theirs. Th e first publications were in 1960, but in Japanese. Inui (1962) has given a summary of these researches together with complete references to the original papers. Inui and his co-workers exploit an idea inherent in Wigley’s paper, but not really developed by him. This is to relate the wave resistance to the observed wave pattern and to deal with the latter rather than the former, because the latter responds linearly to changes in singularity strength and because wave patterns can be shifted by simply shifting singularities. Inui’s reawakening of interest in bulbous bows has had a profound effect upon hull design,
The Wave Resistance of Ships
203
but this will not be discussed here. Instead, we turn to some recent developments of Inui’s ideas. Sharma (1966, 1968) has refined Inui’s ideas and in a sense may be said to have devised a way to use linearized theory to study small deviations from a hull form to which linearized theory would not have been properly applicable. His procedure is as follows. Tests are carried out on a model without a bulb and with a bulb. Th e free-wave spectra, i.e., the functions F,(k)and G,(k) of (2.48) (F, = G, = 0 for symmetric ships), are determined for each hull at each Froude number for which tests are made. Thin-ship theory applied to either hull might give a very poor prediction of these spectra. However, if the difference between the two hulls may be considered ‘‘ small ” (or “ deeply submerged”), one may conjecture that the difference of the two spectra represents the spectrum of the bulb itself and that one may treat it according to the procedures of linear theory. Let us denote this spectrum by Fboand G b o .If one now wants to shift the free-wave pattern of the bulb forward by an amount q and multiply its amplitude b y p , one may do this by replacing Fboand Gboby
Gb(k)=p[-Fb0(k)sin ctq
+ Gbo(k)cosas].
(3.98)
If F , and Go give the spectrum for the original bulbless ship, one may now reconstruct a new total free-wave pattern
+
(3.99) G(k) = Go(k) $- Gb(k). F(k) = Fo(k) Fb(k), Associated with this wave pattern is a wave resistance R given by (2.52). There are now two parameters available, p and q, that can be used to minimize R . Let v(q,p ) = R / R o, where Rois the wave resistance of the bulbless hull, computed from the wave pattern, and R the resistance of the ship with a bulb defined by q and p. Sharma plots the contour lines 7 = const. in the (q, p) plane, and from this plot one can determine the optimum position and size of the bulb. Figure 34 shows a typical graph and is taken from a report by Sharma and Naegle (1970). This report gives the most thorough investigation of the method up to now. There remains the question of how to relate the selected values of q and p to the geometry of the new bulb. If one may start from the assumption that the original bulb was a “thin blister” plastered onto the bulbless hull, then one may presumably simply multiply the offsets by p and shift them by an amount g, taking account of the necessity of fairing this new bulb into the original ship lines. However, this is not the only possible interpretation of how to treat p. T h e problem is discussed by Sharma and Naegle and they conclude that “ it is evident then that in the actual application of
204
John V. Wehausen
FIG.34. Typical contours in Sharma’s bulb-optimization scheme. [From Sharma and Naegle, 1970 (Fig. 19, p. 48).]
this method certain elements of personal judgement cannot be avoided.” Sharma’s various applications of the method appear to be very successful. One should be aware of the fact that one of the most essential differences between Sharma’s procedure and the earlier ones of Wigley and Weinblum is that Sharma starts from experimental data for the bulbless ship and at least one ship with bulb, thereby avoiding a commitment to, say, thin-ship approximations for these two hulls. There have also been recent researches carrying forward and refining the purely analytical approaches of Wigley and Weinblum. Among these should be especially mentioned Yim (1963, 1964, 1966), Kracht (1968, 1970), and Lee (1969). The last paper also contains experiment results with computed bulb forms. Before leaving the subject of bulbous bows, one other curious aspect must be mentioned. Although bulbous bows were introduced and designed to reduce wave resistance, they turned out in both model test and practice to be very effective in reducing resistance at low Froude numbers where one usually regards the wave resistance as a negligible part of the total resistance. Various explanations have been offered, but the mystery appears to have been cleared up by Sharma in a paper by Eckert and Sharma (1970). As a result of a carefully selected series of experiments in which deeply submerged doub!e models were tested and both the wave pattern and wake
The Wave Resistance of Ships
205
were measured, he was able to show conclusively that the unanticipated saving in resistance is a result of improved flow near the bow that avoids loss of energy through wave breaking, a loss that is not measured in the wave-pattern analysis. 3. Ships of Minimum Resistance Once one has at hand an analytical formula for the wave resistance in terms of body geometry or of a pressure distribution, an obvious next step is to try to use this formula as a basis for finding a body or pressure distribution of least wave resistance. There must, of course, be some appropriate restrictions to assure a nontrivial answer. Joukowski (1909) took this step immediately for the shallow-water formula (3.83) and found that if the displacement and length are fixed, the optimum shape will satisfy S ( X )= 6 VL - '( &L2- x').
(3.100)
The first one to use Michell's formula for deep water in a similar way was Weinblum (1930a,b). However, Weinblum's minimization was (necessarily, as we shall see later) over a rather restricted family of hull shapes. A rather thorough, but formal (in the sense that existence of solutions was not considered) study of the calculus-of-variations problem was given by Pavlenko (1937), who also provided some calculated results for the case of a strut of infinite draft. T h e history of the various investigations before 1950 is well covered in Kostyukov (1959, 1968), and we shall omit further discussion here. With regard to other resistance formulas, the situation is as follows. Weinblum (1936a) considered submerged bodies of revolution, using the approximation that a body of revolution can be represented by a line distribution of sources and sinks or of dipoles that satisfy (3.47). Experiments on the resulting bodies were carried out by Weinblum, Amtsberg, and Bock (1936). Sharma (1962) has made further calculations. Maruo (1962), in the same paper in which he introduced his version of the slender-body approximation, used the formula (3.82) as a basis for finding hull forms of least wave resistance. This problem turns out to be almost identical with the problem of finding the optimum waterline for a vertical strut of infinite draft. Finally we note that Bessho (1962b) has given a brief treatment of pressure distributions of minimum wave resistance and later (1966) a more extensive treatment. Since all of the problems mentioned above are rather similar in their mathematical character, we shall limit ourselves here to minimization of the Michell resistance, which in any case has been the most studied. There
John V. Wehausen
206
are two aspects of the problem that must be discussed. One is a purely mathematical one: " Under what conditions does a minimizing hull exist? " The other is the more practical one of computing those that do exist under appropriate restrictions. First we shall formulate a problem with as few restrictions as seem to be consistent with a sensible application and then discuss some mathematical questions and further restrictions. In order to deal directly with hull offsets, we shall integrate the expression for I iJ in (3.38) once by parts with respect to x, assumef, continuous and use the fact that f ( x , y ) must vanish at stem and stern. Then
+
4 R = -pgK3 7T
P
S (Pz+ Q') (A" m
1
A4
- 1)1/2
4
+ iQ(A) = J j j ( x , y)exp(iAKx + X2Ky)dx dy,
(3.101)
Evidently R is a quadratic functional off. We should like to find f that minimizes R, with some reasonable restraints imposed upon f.One such restraint is to fix the area Soupon which f i s to be defined. Other restraints leading to practical problems can now be imposed. For example, we may fix the definition off over some region S, contained in Soand then look for an f defined over S, = So- S, that both minimizes R and has a given volume (and, of course, joins smoothly with the givenf). The restraint is a linear functional, and one finds from the usual methods of the calculus of variations that a necessary condition to be satisfied by f on S, is the integral equation
I,,f(6,
?7)N(K(x- E),
+7))d t dT
= const.
(3.103)
If there exists a solution, one must still check to see if one exists that joins smoothly to the part on S,. The case usually considered is that when S, = So, i.e., the integral restraint is
(3.104)
The Wave Resistance of ShZps
207
and one expects f to vanish on the underwater profile, although this could be varied to accommodate a ship with a given area of flat bottom. When the whole hull is allowed to vary, it has already been mentioned [see (3.91)] that the optimum form must be symmetric about the midship section. In order to avoid dealing with an integral equation in two variables one may fix the vertical volume distribution by accepting only hulls representable in the form f(x,y) = X(x)Y(y),
Y(y) given,
Y(0) = 1.
(3.105)
Equation (3.102) then becomes
I n particular, if Y ( y )= Y(O), i.e., if one is considering a vertical strut, and if one lets T - t co,then
(3.107) where Yo is the Bessel function of second kind and has a logarithmic singularity at x = 0. I n fact, the function of (3.106) has also this singularity, as is shown in Kostyukov [1959, 1968, Section 40, see formula (6.60)]. The analogs to equations (3.103) and (3.104) are L12
L2
X(()N2[K(x
-
f ) ] df
= const.,
(3.108)
Li2
-Ll2
X(Odf= 4A.
where A is the waterplane area. We now turn to the problem of existence of a solution to the integral equations (3.103) or (3.108) and to related mathematical questions. First of all, we call attention to the fact that none of the restrictions imposed so far requiref(x, y ) to be positive or to be bounded. Such a restriction would not be representable as a linear functional and would complicate considerably the formal solution of the calculus-of-variations problem. Various mathematical problems associated with the integral equations (3.103) or (3.108) have been investigated by M. G. Krein (reported in Kostyukov 1959, 1968, Section 40), Karp, Kotik, and Lurye (1960), Timman and Vossers (1960), Bessho (1962a,b, 1963a,b), Kotik (1963), Maruo and Bessho (1963), and Kotik and Newman (1964). Maruo (1964) gives a good discussion of most of the significant results, so that we may restrict our-
John V . Wehausen
208
selves here to a brief restatement of some of them. Most of them are originally due to Krein, although some have been discovered independently by others. (1) If So is bounded and f ( x , y ) 2 0 and SO, then R { f }> 0. On the other hand, if Sois allowed to be unbounded, then there exist f 2 0 , $0, such that R { f } = 0. Krein gives two examples. Yim (1963) has also constructed a similar example with vertical line distributions of dipoles at the bow and stern of a thin ship. (2) Let Sobe bounded and supposef, is a minimizing function satisfying (3.104). Then Krein and Bessho (1963) have shown that there exists a 6 function S(x, y ) vanishing on the boundary of So such that f = f o satisfies (3.104) and R { f }= R{f,-,}.Since R{S}= 0, it is evident that 6 itself must be sometimes negative, so that it is not clear that f20. However, the examples show that one cannot expect a unique solution to the problem formulated in (3.103) and (3.104). This is not in itself so bad, but it indicates that numerical methods of solution might give erratic results. (3) For fixed V and bounded So there exists lim inf R { f } > O for f ( x , y ) 2 0. However, there does not exist fo 2 0 such that R{fo}= lim inf R { f } . (4) For the equations (3.106) Krein has shown that if Sois bounded and completely submerged there exists a minimizing solution in the form of a " generalized function.'' If So intersects the free surface, there exists a solution with singularities at stem and stern of the form ( x 2 - L2/4)l'2. (5) Kotik and Newman (1964) have considered the flow about bodies generated by dipole distributions on a submerged axis 1 x I 5 L/2, y = y o < 0, x = 0 such that, if p(x) is the dipole strength, jp(x) dx = const. They construct a sequence p,, such that the associated wave resistance R(p,) -+ 0 as n + CO. It is not evident from the formula for pnthat p, 2 0. Even if not, it is possible that the streamline-generated body is acceptable. These results have sometimes been interpreted to mean that there is no way to use the Michell formula to obtain ships of minimum wave resistance (even granting the other limitations of the theory), and that many of the computational approaches have been misguided. This is not strictly true. For example, in Weinblum's calculations he has usually taken hulls of the form X ( x ) Y(y),fixed Y as a simple polynomial, e g , Y = 1 - ( Y / T ) and ~, assumed
+
c a, N
X(x)=
x2,,
n=O
c afl(L/2)2" 0. =
(3.109)
With the additional condition, following from (3.104), 2n+1
4hZ02n+1 z(fi)
2
n
-T
Y(y)dy= V,
(3.110)
The Wave Resistance of Ships
209
and N fixed he now looked for the coefficients a, that would minimize R, now a quadratic form in the a,,. Solving for the a,,becomes a simple problem in finding the minimum of a quadratic form with two linear restraints, and requires solving a set of linear equations. The difficulties raised by the more general investigations have been avoided by looking for a minimum in a more restricted set of functions. Th e problem is, however, still a genuine minimization problem whose solution can give valuable informtion about optimum ship forms. T h e procedure sketched above has been used by Weinblum on numerous occasions, most recently in a paper by Weinblum, Wustrau, and Vossers (1957). It has apparently always yielded polynomials X(x) that were positive between stem and stern even though there is nothing in the formulation of the problem to require this. This outcome can be regarded as purely good fortune. I n an attempt to extend this method so as to allow also variation in depth, Webster and Wehausen (1962) used trigonometric functions of the form (3.87) to represent the hull. T h e form of the afterbody was fixed, so that only an optimum forebody was sought. However, the class of functions among which they were seeking a minimizing function was much larger than in the cases treated by Weinblum. T h e calculations produced forms swinging wildly in each section from positive to negative offsets with amplitudes many times the beam. One may conjecture that there is a relation between this behavior and the lack of uniqueness mentioned earlier. I n any case, some further restraint seemed necessary. T h e restraint was imposed by adding the frictional resistance to the Michell resistance, following a suggestion of Weinblum (1936a, 1957). Since the frictional resistance is taken as proportional to the area, this device will function as a brake upon wild behavior of the minimizing function, but cannot guarantee its positiveness. Th e paper of Webster and Wehausen uses a linearized approximation to the area that may greatly overestimate it in some cases. In a later paper by Lin, Webster, and Wehausen (1963), a more exact calculation of the area is used, and results are given for optimum hulls symmetric fore and aft as well as optimum forebodies to a given afterbody. Figure 35 shows a symmetric hull calculated to be optimum for Fn = 0.316. The calculations were made with 15 harmonics in the longitudinal direction and 4 in the vertical. The slight waviness in the waterlines is a consequence of the limited number of harmonics “trying” to represent a bulb of substantial size. T he waviness is, of course, of no significance, and in a practical appliation the lines would be made smooth. Although the procedure discussed above is apparently successful in avoiding wildly behaved forms in the minimizing procedure, it does not guarantee against negative offsets, and indeed they do occur. What would be most desirable would be to be able to solve the following problem:
210
John V. Wehausen
BODY
I
i
1
WATERLINES
OF OPTIMUM SYMMETRIC
SHIP
FIG.35. Framelines and waterlines for a hull designed by computer to be optimum Fn = 0.316. [From Caligal, Moffitt, and Wehausen, 1968 (Fig. 5, p. 21). Program prepared by W.-C. Lin.]
at
Minimize R in (3.102) subject to the restraint (3.104), and perhaps other similar restraints expressible as linear functionals in f, but also subject to inequality restraints like
This is essentially an analog in continuous functions of a problem in quadratic programming. By representing the hull as shown in (3.89), this becomes, in fact, just such a problem. If it can be solved, the frictional resistance is no longer necessary as a device to damp wild behavior.
The W a v e Resistance of Ships
211
This problem has been studied in a doctoral dissertation at the University of California, Berkeley, by C. C. Hsiung. Using the representation (3.89) he has prepared a computer program that allows him to fix the offsets at any number of grid points and for the remaining points to impose inequalities
(3.112) As an example there is shown in Fig. 36 the plans for an optimum ship derived under the following restraints: Th e body aft of the midship section is the Block 60 hull from Series 60. Also the design waterline and tangent line (outline of the flat part of the bottom) were fixed at those of the Series 60 hull. T he block coefficient of the forebody was fixed at 1.058 times that for the Series 60 hull. As inequalities the following were imposed: xij
zi.i+l - zii
yj+~-Yj
2 z:;o),
10" on the bottom row.
The last inequality was imposed to reduce the danger of slamming at the bow and need have been imposed only for the first two stations. The middle inequality was imposed to prevent flow separation. T h e factor 1.058 was selected after test calculations with several factors, and is the value yielding the lowest wave resistance. I t is evident that the minimization problem has been formulated so as to provide a hull similar to the Series 60 Block 60 but competitive with it in resistance properties. T h e ratio of the Michell resistances for the optimum hull at Fn = 0.289 (the design Froude number for the Series 60 hull) and for the Series 60 hull is 0.723. It seems remarkable that pure computation can produce forms resembling those developed from experience. On the other hand, it is also known that small changes in hull form can produce substantial changes in wave resistance so that one may properly remain unconvinced of the quality of the computed ones. In addition, the computer design has neglected such aspects of resistance as form resistance. And finally, we have already seen that the Michell resistance does not give a very good prediction of wave resistance for customary ship forms, although one hopes that it may still give useful information about optimum hulls. I t is evident that experiments are necessary.
z
N
i.000
0.750
0.500
0.250 0.075
FIG.36. Optimum forebody designed to fit Series 60 afterbody and subject to inequality restraints. [From computer program of Chi-Chao Hsiung.]
The Wave Resistance of Ships
213
Experiments on two computed hulls have been reported in Lin, Paulling, and Wehausen (1964) and in Calisal, Moffitt, and Wehausen (1968). In the first cited paper the residuary resistance compared favorably with equivalent hulls from Taylor’s Standard Series but were not appreciably better, and not nearly as low as the Michell resistance. On the other hand, the observed wave patterns seemed of remarkably small amplitude. I n the second paper the wave resistance was measured directly by transverse cuts. Figure 37 shows both residuary resistance and the directly measured wave resistance for the model of Fig. 35. It is evident that the model has indeed performed very well. The same can be said about the other two models tested, designed to be optimum at Fn = 0.267 and 0.289. Their relatively unimpressive qualities as measured by the residuary resistance are evidently a result of separation at the stern and a large form resistance. T h e model shown in Fig. 36 has not yet been built and tested. Although the design methods for bulbous bows, modified perhaps by experience, have been successfully put into practice, one cannot say the same about hulls of minimum wave resistance designed according to the methods just described. Th e reason is that the methods of calculation have
FIG.37. Residuary resistance, Michell resistance and wave resistance determined from wave-pattern measurements for the model of Fig. 35. [From Caligal, Moffitt, and Wehausen, 1968 (Fig. 26, p. 42).]
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John V . Wehausen
been too limited in their applicability until recently, and the most recent methods, such as Hsiung’s described above, have not yet been exploited and tested under sufficiently broad conditions. On the other hand, the analyses of Weinblum and others have given valuable insights into the characteristics of optimum forms that are desirable at different Froude numbers. Some further references on ships of minimum resistance : Guilloton (1966), Ishii (1968), Maruo (1963, 1969), Maruo and Ishii (1964), Shor (1963), Sretenskii (1935), Thomsen (1970).
I. HIGHER-ORDER THEORIES In Sections 111, B and C of this article several “ exact ” problems were formulated for steady motion with a free surface, and furthermore a systematic approximation scheme was described. In the subsequent developments only the first step in this scheme, “linearized” theory, has been taken. Here we shall consider some of the attempts to carry the theory beyond this first step and the results that have been obtained. We do this first for submerged bodies and then for surface ships. Finally we mention briefly some new approximation schemes. For motion of a body with a free surface the two most troublesome boundary conditions to be met are those on the free surface and on the body. It has been emphasized in the discussion of the approximation scheme that approximating one of these boundary conditions has physical implications with regard to satisfying the other one. Put simply, this says that the nature of the disturbance is closely connected with that of the disturber. On the other hand, from a purely mathematical point of view, one could devise presumably convergent schemes in which one satisfies one of the boundary conditions exactly, or nearly so, from the start, and then step by step increases the degree of approximation of the other. This procedure is sometimes called an “ inconsistent ’’ approximation scheme because the degrees of approximation of the two boundary conditions are different. Since it is usually computationally easier to satisfy more exactly the boundary condition on the body than that on the free surface, it is of practical interest to know if, in fact, this yields a more accurate approximation in the Froudenumber range of most ships than does the consistent linear approximation. Interesting studies of just this question have been made for submerged bodies where it has been possible to compare with consistent second-order approximations. For surface ships such comparisons are rather scanty, but significant. More will certainly appear in the near future.
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1. Submerged Bodies We shall start with some results of Tuck (1965) dealing with a submerged circular cylinder [an earlier paper by Bessho (1957a,b) had an unfortunate mistake at the last stage]. T h e first result indicates some of the dangers involved in using the solution of a linearized problem in the same way one would use an exact solution in order to trace streamlines. Figure 38 shows three streamlines obtained from the linearized solution for a dipole under a free surface. Whereas streamlines computed according to the linearized theory would have given physically reasonable, although approximate, streamlines, the “ exact ” streamlines are physically nonsensical.
FIG.38.
Streamlines associated with the linearized potential for two-dimensional
flow past a submerged dipole. [From Tuck, 1965 (Fig. 1, p. 405).]
Tuck states : “ I n fact it appears likely that at no finite order of approximation is a closed body generated, which may serve as a warning to those seeking to use inverse methods to calculate ship-like bodies generated by given source distribution.” One might hope that the anomalous behavior of the near-surface streamlines was a consequence of trying to represent a circle by a dipole. However, Giesing and Smith (1967) have computed streamlines when the boundary condition on the circle was exactly satisfied and still find such behavior. Tuck has also computed the resistance and lift when the body boundary condition is satisfied to second-order but only the first-order free-surface condition is satisfied, and also when both are satisfied to the second-order. Figure 39 shows the resistance for these two cases as well as for the first-order theory. I t is evident in this case that the free-surface correction has produced a larger change than has the body correction.
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John V . Wehausen
FIG.39. Wave resistance of a submerged circular cylinder accordingto three different approximations. [From Tuck, 1965 (Fig. 3, p. 412).]
Although Tuck’s computational results are of considerable interest, the circular cylinder is not a form with which one can experiment and expect conformity of measured values with those deduced from inviscid-fluid theory. Salvesen (1966, 1969) has carried out second-order calculations for a hydrofoil and has also done experiments. Salvesen’s perturbation series is based essentially upon the deep-submergence approximation, although it is expressed somewhat differently. [One will find an interesting gloss upon Salvesen’s approximations in Section 5.3 of Ogilvie (1970).] Figure 40 shows first-order theory, consistent second-order theory, and an inconsistent second-order theory, in which the second-order free-surface condition is satisfied but not the body boundary condition. For Froude numbers below 0.7, the consistent and inconsistent calculations agree very well, for larger Froude numbers the divergence is very large. Thus for Fn < 0.7 the free-surface correction is dominant, whereas for Fn > 0.7 the body boundary condition is most important. Salvesen ascribes this divergence from Tuck’s results to the necessity of satisfying a Kutta condition. Figure 41 shows measured wave resistance, obtained in two ways, and firstand second-order calculations. T h e second-order calculations are clearly
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0.3
0.5
0.7
0.9
217
1.1
Froude number, U / J ( g b ) FIG.40. Wave resistance of a submerged hydrofoil according to three approximations: theory; - inconsistent second-order theory; ---,consistent second-order theory. [From Salvesen, 1969 (Fig. 4,p. 427).]
-.-, first-order
in much better agreement with the measured values than the first-order ones. Salvesen has also included a third-order calculation of the waves from a given singularity distribution (the particular body represented is not relevant to his purpose) in order to examine the effect of Froude number on the relative magnitude of first-, second-, and third-order contributions to the wave height far downstream. As one expects, they reverse their order of importance as the Froude number becomes smaller. The wave lengths predicted by third-order theory agree better with the measured ones than do those based upon second-order. I n addition to Salvesen’s paper two others on two-dimensional motion should be mentioned. Isay and Miischner (1966) have worked out a systematic perturbation scheme for hydrofoils based upon the thin-wing approximation and include solution of the resulting equations. However, the
218
John V . Wehausen
I .7
0.6
0.5
0A
0.3
0.2
0.1
Fig. 41. Measured wave resistance compared with first- and second-order computed wave resistance: -.-, first-order theory; ---,second-order theory; -, from measured wave; ------,difference between horizontal drag at 1.25 and 4.5 ft. submergence. [From Salvesen, 1969 (Fig. 3, p. 426).]
body boundary condition does not seem to be carried to second order. Giesing and Smith (1967) also treat second-order hydrofoil theory, but they take the free-surface condition only to the first order. They have also done experiments and compare measured and predicted pressure distributions. The discrepancies are not small and it would be interesting to see their data compared with the consistent second-order theory. W. D. Kim (1969) has made calculations for a submerged sphere analogous to those of Tuck for a circular cylinder. The calculations, carried out for the center at one-diameter submergence, show very little difference between the first-order and inconsistent second-order values for either resistance or lift. Figure 42 shows the resistance curves together with that for the consistent second-order theory.
The Wave Resistance of Ships 0.051
0.0 I
I
I
I
219 I
1
/I
FIG.42. Wave resistance of a submerged sphere according to three different approximations. [From W. D. Kim, 1969 (Fig. 4, p. 36).]
As a body to be used in experimental testing of results from inviscidfluid theory, the sphere suffers from the same disadvantages as the circular cylinder. Fortunately, a satisfactory form, an %to- 1 prolate spheroid, has been investigated by Y. H. Chey (1970), who has made first-order, inconsistent and consistent second-order calculations and resistance measurements. Resistance measurements for ellipsoids of this shape had already been made earlier by Weinblum, Amtsberg, and Bock (1936) and by Bessho (1957a,b, 1961). Chey's measurements are consistent with theirs. T h e viscous resistance was determined by testing at a sufficiently deep submergence that there was no longer a free-surface effect. T h e interaction between viscous resistance and free-surface effects as the surface is approached is assumed to be small. Figure 43 shows the wave resistance as obtained by experiment and by first-order, inconsistent second-order and by secondorder theory for a submergence-to-beam ratio of 2. It is evident that the agreement between measured and second-order wave resistance is excellent. The inconsistent second-order correction has been in the correct direction but not as important as the second-order free-surface correction.
John V . Wehausen
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--.--o-
F i r s t order 2nd a d e r (body singularity) Consistent 2nd a d e r (body singularity and non-hear free-surface e f f e c t )
8
FIG.43. Comparison of measured wave resistance with wave resistance calculated according to three approximations for a submerged prolate spheroid. [From Chey, 1970 (Fig. 16, p. 44).]
Chey also made measurements at deeper submergences. At a submergenceto-beam ratio of 4 agreement between measurement and first-order theory is excellent and even with a ratio of 3 is not unsatisfactory. For submerged bodies it seems fair to conclude that consistent secondorder theory gives a satisfactory agreement with measured values when first-order theory deviates badly and that generally, but not always, the free-surface correction plays a more important role than the body correction. It should be noted that these conclusions have been drawn from data for rather large Froude numbers, 0.4 5 Fn 5 1.O, compared with typical Froude numbers for a conventional surface vessel, 0.15 5 Fn 5 0.35.
2. Surface Vessels T h e perturbation expansion introduced by Peters and Stoker (1957) in order to give a rational basis to the thin-ship approximation and given here in (3.13) and (3.14) carries with it a procedure for going beyond the Michell theory by calculating the second term in the series. I t is not absolutely
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certain that this will improve the accuracy, for little is known about the nature of the convergence of the series, except that it is not uniform. However, the results with submerged bodies have been encouraging and it is reasonable to hope that a second-order thin-ship theory will give results in better agreement with measured wave resistance than does the Michell theory. In fact, the attempts began before the results for submerged bodies had been obtained. The first serious attempt was made by Sizov (1961). This was unfortunately marred by an error in calculation of a necessary Green function. Maruo (1966) has carried through the same procedure correctly and given formulas for the second-order correction to the wave resistance. Sizov and Maruo follow a procedure that is a natural extension of the one used in deriving the first-order theory. That is, they derive a boundary-value prob, they then solve. T h e secondlem for the second-order potential T ( ~ )which and higher-order problems differ in two ways from the first-order one. First, v(2)must satisfy the following boundary condition on the plane y = 0: Qg)(X,
0, z )
+ Kc#)
=9(rp)
(3.113)
where 9 is a functional depending upon q(l).Since has in principle already been found, the right-hand side is known and this condition is like the linearized pressure-distribution condition (3.23). Second, the condition on the centerplane section, TY(X,
y , f0) = fq Y x , y ) ,
(3.114)
is similar in form to the third condition of (3.15) except that q“) depends ) also upon the first-order sinkage and trim. not only upon f and ~ ( lbut Furthermore, the region of variation of (x, y ) in (3.1 14) must extend upward to the first-order free surface. I n the solution to the problem this results in a line integral along the intersection of the plane y = 0 and the centerplane section. T he details are given clearly by Maruo. Wehausen (1963) has proceeded differently. He first uses Green’s formula to represent the value of (the “exact ”) V(X, y , z ) at an arbitrary field point inside the fluid in terms of integrals on the boundary, i.e., the free surface and the ship’s wetted surface. Th e integrals are then expressed as integrals over the projections of the ship’s wetted surface onto the plane z = 0 and of the free surface onto the plane y = 0. A line integral over the intersection of the free surface and the ships surface appears naturally as a result of certain integrations by parts. With this equation as a starting point one may now introduce the perturbation expansions (3.13) and (3.14). A choice of the Green function is now made to satisfy the linearized freesurface condition; Taylor expansions of y(x, Y , z), ~ ( xy,, ff) and similar
John V. Wehausen
222
ones of G are made so that the integrals are in terms-of cp or Ga s defined upon the plane over which the integral is evaluated. T h e substituted perturbation P ) ,cpC2), etc., expansions now allow successive determination of p)(l), dl), in terms of already computed quantities, Th e resistance may be found by integrating the pressure over the wetted hull, but it is more elegant to use (2.20), as has been done by Maruo (1966) and Eggers (1966). If one writes (2.20) in the form
1 R = 5 Pg
1 Y2(% s'
9
"P
2) d z
+ 51 p J z"sp d z
-H
dy[-cpz2(xB, y , 2)
+ V'y2 +
9)z2]
(3.115) and from Euler's integral [see (2.18) and following] evaluates
y = &YCl)+ &2yW+ . . . U
= & - cpLl'(X,
0, z)
g
+... ,
(3.116)
then one may easily derive from (3.13), (3.14), (3.15), and (3.16)
R = &lR(l)+ &3R(2)+ . . .
T he approach to the determination of wave resistance by way of Green's formula seems no more tedious in its calculation than is the formulation and solution of a succession of boundary-value problems and has the advantage that it gives directly the solution for the successive cp@). For cp") it
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223
yields immediately (3.50). For vc2)it gives the following rather compliczted expression, which we reproduce only so that we may comment upon it:
The integrals over the (4, Z;) plane give the second-order correction to the first-order approximation of the nonlinear free-surface condition, the integrals over s,,the second-order correction to the first-order approximation of the hull boundary condition. The line integrals have their origins partly in one, partly in the other. Formally at least, both boundary conditions seem to play equal roles in the corrections to the first-order theory, although numerically it is possible that one plays a more important role than the other. One should also note that the first-order trim and sinkage enter into the expression for cp('), so that any computation neglecting this fact must be considered incomplete (although one may wish to do just this while exploring numerical procedures). If one further exploits the fact that for conventional ships the draft/length ratio is small, then the most important terms in (3.11 8) turn out to be the line integrals. If one is dealing with a hull with a flat bottom, a further line integral around its boundary must also be included, as has been pointed out by Eggers (1970).
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John V . Wehausen
Still another approach to higher-order approximations has been proposed by Yim (1968) and Wehausen (1969). A kind of aesthetic blemish on the procedures discussed so far is the necessity of extending the definition of the velocity potential outside the region occupied by water in order to be able to refer values of 9 to its value on the centerplane section of the ship or the plane y = 0. Closely connected with this is the difficulty that the free-surface condition is to be satisfied on a surface one does not know and that even the hull boundary condition is to be satisfied on an unknown surface if sinkage and trim are allowed. This difficulty can be avoided if one can make the free surface and the ship’s wetted surface into coordinate surfaces. Yim does this by introducing a new coordinate 7 to be used together with x and ,z and having the property that 7 = 0 represents the free surface. Wehausen goes a step further and introduces Lagrangian coordinates (a,p, y ) with p = 0 representing the free surface and y = 0 containing the ship’s surface. Although one gains in simplicity in some respects, one pays for it by new difficulties in others. Wehausen proposes an iterative method of solution of the equations and gives formulas for both firstand second-order results, including the simplifications resulting from small draft/length ratio. None of the above authors has made any numerical calculations. The only attempt to carry through a complete second-order calculation is that of Eggers (1966, 1970). He has followed the Green’s theorem approach described above, but with various modifications designed to make the numerical computation easier or more stable. In addition, he has extended the method to accommodate a flat bottom. The only calculation so far is for a form with parabolic waterlines and rectangular framelines. Sinkage and trim are not allowed. In presenting his calculations Eggers has given separately the corrections resulting from source distributions on the side walls of the hull, those on the bottom, and those on the free surface. Thus it is possible in this case to make an assessment of the relative importance of satisfying the body boundary condition and the free-surface boundary condition. Egger’s tabulated results are shown graphically in Fig. 44-. Together with his calculations are shown measured resistance value for this model, presented as the difference between the experimentally determined wave resistance [taken from Inui (1957)] and the Michell resistance. This is, of course, the quantity that one is trying to approximate with the secondorder correction. I n examining the curves in Fig. 44, one observes that the agreement between calculated and measured values seems better for the smaller value of B/L, as one would expect. Also, the correction is less. In comparing the relative contributions of the body and the free surface, one sees that for the range 5 < y o 5 11 (0.213 5 F,,5 0.316), the free-surface contribution is
0.4
1
-I
c
4
I
0
2
4
BILz0.1, T/L =0.05 BIT72 I 6
8
10
FIG.44. Second-order corrections to the wave-resistance coefficient Cw = RwKa/ p U 2 and associated experimental values. [Prepared from Table in Eggers (1970, p. 21).]
less important than the body contribution, and for the case BIT = 0.2 is much less. Since the speed range of most commercial vessels does not exceed Fn = 0.316, this result gives support to those who believe that improved agreement with experiment will be found by solving numerically the inconsistent problem. For large Froude numbers it is evident that the free-surface correction plays an important role. However, here one should keep in mind the neglect of sinkage and trim in the calculations. If one refers to Fig. 25, one sees that these become important for Fn > 0.316, so that the applicability of Egger's calculations to ships free to trim and heave becomes doubtful in this region. T h e smallness of sinkage and trim for Fn < 0.316 lends further support to the usefulness of the inconsistent problem in this region. On the other hand, the experiments of Shearer (1951, see Fig. 20) and others show a fairly substantial difference in measured resistance between models free to trim and ones fixed, even in the region Fn < 0.3 16. Experiments in which the wave resistance is directly measured as well as further calculations would clarify the situation.
226
John V. Wehawen
A report by Eng (1969) also gives an almost complete second-order theory for a vertical strut with parabolic waterlines. The theory is somewhat deficient in that certain free-surface corrections were simplified in order to reduce computer time. In view of Egger’s results this was perhaps not serious. In any case he found fairly satisfactory agreement with values from experiments especially designed to measure wave resistance. Eng has also modified (3.117) by retaining certain terms in e4 that guarantee that R will not be negative, an event not excluded by (3.117). One should note, however, that a situation in which the term in e3 can overpower the one in ea does not seem appropriate for application of the theory. The inconsistent approximation has been the subject of a number of investigations in recent years. Here we may distinguish between at least two approaches. In each, one assumes the representation (3.56). However, in one approach, instead of solving the integral equation (3.57) with the proper Green function, say (3.51), one simplifies G for the purpose of easier numerical solution by replacing G by the Green function appropriate to a rigid surface at y = 0, e.g., in the case (3.51) by r - l + r ; l . However, in using the solution in (3.56) the complete expression for G is used. This is often called the “ zero-Froude-number approximation ” or the “ doublebody approximation.” In the other approach one attempts to solve (3.57) with the proper Green function. This means, of course, a different source density distribution for each Froude number. A modification of this approach consists in satisfying the body boundary condition to the second order and the free-surface condition to the first order. In another modification one distributes sources on the centerplane section but determines (if possible) the source strength from the exact body boundary condition. The double-body approximation was apparently first used by Inui (e.g., 1957) to construct hull shapes from given source distributions by the inverse streamline-tracing method. However, he traced the streamlines assuming a rigid surface at y = 0. This approximation has been used either in a direct or an inverse method by Breslin and Eng (1963), Pien and Moore (1963), Yokoyama (1963), Ikehata (1965), Ogiwara, Maruo, and Ikehata (1969) (here combined with a slender-body approximation), and Bhattacharyya (1970). Several of these show comparisons with experiment and it seems fair to say that the calculated curves show no better agreement than does the Michell resistance (see, e.g., the first cited paper), in fact, usually much worse. Special experiments by Gadd (1966) seem to bear this out. This approximation has also been investigated theoretically by Kotik and Morgan (1969). They point out that a velocity potential satisfying the body boundary condition and the rigid free-surface condition can be generated by infinitely many sorts of source and/or dipole distributions, and that it is not clear that all will lead to the same wave resistance. In fact, Kotik
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227
and Mangulis (1962) had already observed that a vertical circular cylinder could be generated by either a vertical line of dipoles or by a surface distribution and that the former would yield an infinite wave resistance, the latter a finite one. Eggers in his discussion of Pien (1964) has made a similar point. Kotik and Morgan restrict themselves to source and dipole distributions on the surface of a body. They show that if the body is completely submerged the wave resistance is unique no matter how the distribution of sources and dipoles on its surface is chosen. I n the case of a body intersecting the surface one must allow an additional dipole distribution over the waterplane area in order to obtain uniqueness of the wave resistance. Numerical calculations in which the complete Green function is used have been carried out by Kajitani (1965), Nakatake and Fukuchi (1967), Gadd (1969, 1970) and Kobayashi and Ikehata (1970). Of these we note that Kajitani uses an inverse (streamline-tracing) method, Nakatake and Fukuchi distribute the sources on the centerplane section, Gadd (1969) satisfies the body boundary condition only to second order, but Gadd (1970) distributes sources over the surface, as do Kobayashi and Ikehata, to satisfy this condition exactly (except for the error inherent in the numerical methods). Several of these papers give comparisons with Michell resistance, the resistance computed by the double-model approximation, and experiment. Although we shall not reproduce any results here, it is evident that in comparison with the Michell resistance agreement with experiment has been improved for Fn < 0.35, which was not the case with the doublemodel approximation. However, there also seems to be evidence of difficulties associated with the numerical calculations. These will certainly be overcome in the near future. Brard (1971, 1972) has made a thorough study of the potential-theory problem arising in the inconsistent problem and calls especial attention to the necessity of a line integral around the intersection of the ship and the plane y = 0 in the integral equation [see (3.49) and the remarks following (3.55)]. None of the above authors has included this. Finally we mention briefly a nonsystematic correction procedure of Guilloton (1964). His idea is to start from the velocity field given by the Michell theory and to map it into a new velocity field that will provide a better approximation to the “ real ” flow. Since the procedure starts with the Michell potential, the associated source distribution is on the centerplane and Guilloton’s procedure is a kind of inverse streamline-tracing method, but with the hull forced to be a streamsurface. I n some respects it is similar to satisfying the hull boundary condition to the second order. There is also a small free-surface correction. The method has been studied further by Guilloton (1965) and has been applied by Emerson (1967, 1971). I n spite of the method’s unsystematic character Emerson’s calculations
228
John V. Wehausen
show good agreement with experimental measurements for Froude numbers below those for which sinkage and trim (neglected in the theory) become substantial.
3 . Other Higher-Order Calculations I n the approximation schemes that we have been using it has been assumed that for a given velocity U the moving body was either sufficiently deeply submerged or so constituted geometrically, say thin enough, that it did not disturb the free surface much, except possibly near the stem and stern of a surface vessel. Keeping U fixed is equivalent to keeping Fn fixed. If U is quite large or quite small, it may be necessary for the body to be correspondingly quite deeply submerged or quite thin for the approximation to be useful. Even the second- and higher-order approximations treated above are subject to this. One might, however, wish to change the conditions of the approximation and have, for example, EIFn = constant as E +0, or even E = constant as while Fn -+ 0. Not all conceivable combinations will lead to useful approximation schemes, but some have been examined. T he analysis is almost always by way of matched asymptotic expansions. T he following papers treat some aspect of such problems for low Froude numbers: Ogilvie (1968), and Dagan (1971b). Some aspect of the problem for high Froude numbers is considered in the following papers : T . Y. Wu (1967), Ogilvie (1967), Dagan (1971a). T he flow in the immediate neighborhood of the bow of a surface ship will be badly represented in any approximation scheme that assumes the flow to be a small perturbation of a uniform flow. It is evident that some different method of approximation is required there. This problem has been investigated by Dagan and Tulin (1969, 1970a,b). We omit any detailed discussion of these various results, for they have been dealt with by Ogilvie (1970). We close with mention of a paper by Newman (1971) in which the usual perturbation expansion is carried to the third order in the neighborhood of the cusp line. He finds that the solution diverges on the cusp line, or put differently, that no steady third-order solution exists. If this result can be confirmed, it will have an important effect upon analytic approaches to the theory of ship wave-resistance as well as waves. ACKNOWLEDGMENTS T h e author wishes to express his gratitude to several colleagues who have read parts of a preliminary draft of this paper for their suggested improvements and for errors pointed out. In particular, his thanks go to K. Eggers, L. Landweber, J. N. Newman, and W. C. Webster.
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229
In addition, he gratefully acknowledges the support of the Fluid Dynamics Branch, Office of Naval Research, during the summer of 1971 when most of the foregoing was written. BIBLIOGRAPHY For the most part the papers listed below are those cited in the text. An exception is formed by the papers on wave-resistance theory published before 1951. For this period I have tried to give a fairly complete bibliography, although some papers will certainly have been overlooked either through inadvertance or ignorance. In addition, because few extra titles were involved, I have included, as well as I was able to determine it, a complete list of the late W. C. S. Wigley’s papers, not including reports. Papers on wave resistance published in the USSR are not covered as thoroughly as would have been desirable. Fortunately, this lack is compensated by the bibliography in Kostyukov’s (1959, 1968) book and also by Palladina’s (1957) guide to Russian literature on the theory of ships. For more recent papers no attempt at completeness has been made, and many substantial papers are not here for no other reason than that their content did not fit easily into the exposition. For example, there are many more papers on the determination of wave resistance by means of measurement of wave patterns than occur here. And this is not the only example. Fortunately, papers on wave-resistance theory, and on ship hydrodynamics in general, appear in a relatively limited number of journals or conference proceedings, in pleasant contrast to other fields. Consequently, it is not difficult to put together an almost complete bibliography on any special topic. First there are the publications of several societies of naval architects : Bulletin de l’dssociation Technique Maritime et Akonautique (Park), Jahrbuch der Schiffbautechnischen Gesellschaft (Hamburg), Journal of the Zosen Kiokai (through vol. 122; thereafter the following), Journal of the Society of Naval Architects of Japan (Tokyo), Transactions of the Royal Institution of Naval Architects (London), Transactions of the North-East Coast Institution of Engineers and Shipbuilders (Durham), Transactions of the Institution of Engineers and Shipbuilders in Scotland (Glasgow), Transactions of the Society of Naval Architects and Marine Engineers (New York), Trudy Tsentral’nogo Nauchno-Issledovatel’skogo Instituta imeni A . N . Krylova (Leningrad), and Trudy Leningradskogo Korablestroitel’nogo Instituta (Leningrad). In addition there are several journals devoted to topics in ship research: International Shipbuilding Progress, Journal of Ship Research, Schiffstechnik. Of the journals devoted to fluid dynamics in general, one will occasionally find papers in the following: Journal of Fluid Mechanics, Physics of Fluids, Prikladnayn Matematika i Mekhanika, and Izvestiya Akademii Nauk S S S R . Mekhanika Zhidkosti i Gaza. In addition to the journal literature important papers have appeared in the proceedings of the Symposia on Naval Hydrodynamics (every two years since 1956), the International Towing Tank Conferences (every three years), and the International Seminar on Theoretical Wave Resistance held in Ann Arbor in 1963. There is also a rather extensive report literature, some of it rather informal and of limited distribution, some of it intended as a permanent record. The latter is especially true of the reports of several of the large ship research laboratories. This nearly exhausts the sources of current literature, but of course not completely, as one may easily determine from the papers below. Papers have been identified in the text by author and year. If more than one paper has been published in one year, they are further distinguished in the text by letters, e.g., 1951a, 1951b, the first listed paper being 1951a, the second 1951b. The corresponding letters have also been added to the dates in the bibliographical data below. They are not, of course, properly a part of the data, but have been included for ease of reference.
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John V . Wehausen REFERENCES
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MARUO, HAJIME. A note on the higher-order theory of thin ships. Bull. Fac. Eng. Yokohama Nut. Univ. 15 (1966), 1-21. MARUO, HAJIME.High- and low-aspect ratio approximation of planing surfaces. Scht'stechnik 14 (1967), 57-64. MARUO, HAJIME. Theory and application of semi-submerged ships of minimum resistance. Jap. Shipbld. Mar. Eng. 4, no. 1, 5-15 (1969). MARUO, HAJIME.Application of the wave resistance theory to the ship form design. Korea-Japan Seminar Ship Hydrodyn., Seoul, 1970, pp. 2-1-2-22 14 figs. MARUO, H A J I M E ; BESSHO, MASATOSHI. Ships of minimum wave resistance. J . Zosen Kiokai 114 (1963), 9-23. Translated in Selec. Pap. 3, 1-18. MARUO,HAJIME;IKEHATA, MITSUHISA. Determination of wave-making resistance of a ship by the method of wave analysis. 111. (Japanese.) J . SOC. N a v . ArchitectsJap. 125
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(1969), 69-71. MORAN, D. D. ; LANDWEBER, L. A longitudinal-cut method for computing the wave resistance of a ship model in a towing tank. 16th Amer. Towing Tank Conf., Sao Paulo,
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1-24. NEWMAN, J. N. The determination of wave resistance from wave measurements along a parallel cut. Int. Sem. Theoret. Wave Resistance,'Ann Arbor, 1963, pp. 351-376; disc.
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WARD,L. W.; SNYDER, J. D . 111. Forces due to Gravity Water Waves on a Long Vertical Circular Cylinder. Webb Inst. Naval Arch., Glen Cove, New York, 1968. v -t 36 pp. WARREN,F. W. G . A stationary-phase approximation to the ship-wave pattern. J . Fluid Mech. 10 (1961), 584-592. WARREN,F. W. G.; MacKINNON, R. F. A problem of gravity wave drag at an interface. J . Fluid Mech. 34 (1968), 263-272. WEBSTER, W. C. T h e effect of surface tension on ship wave resistance. Dissertation, Univ. of Calif., Berkeley, 1966. ii 114 pp. T. T. Study of the boundary layer on ship forms. J . Ship Res. WEBSTER, W. C . ; HUANG, 14 (1970), 153-167. WEBSTER, W. C . ; WEHAUSEN, J. V. Schiffe geringsten Wellenwiderstandes mit vorgegebenem Hinterschiff. Schtffstechnik 9 (1962), 62-67; Disk. 67-68. WEHAUSEN, JOHNV. Wave resistance of thin ships. Symp. N a v . Hydrodyn., Washington, D . C . , 1956, pp. 109-133; disc. 133-137. WEHAUSEN, JOHNV. An approach to thin-ship theory. Int. Sem. Theoret. Wave Resistance, Ann Arbor, 1963, pp. 819-852; disc. 853-855. WEHAUSEN, JOHN V. Effect of the initial acceleration upon the wave resistance of ship models. J . Ship Res. 7,no. 3, 38-50 (1964). WEHAUSEN, JOHN,V. Use of Lagrangian coordinates for ship wave resistance (first- and second-order thin-ship theory). J . Ship Res. 13 (1969), 12-22. WEHAUSEN, J. V.; LAITONE, E. V. Surface waves. “Encyclopedia of Physics,” Vol. IX, pp. 446-778. Springer-Verlag, Berlin, 1960. WEINBLUM, G. Schiffe geringsten Widerstands. Proc. 3rd Int. Congr. Appl. Mech., Stockholm, 1930a, pp. 449-458. WEINBLUM, G. Anwendungen der Michellschen Widerstandstheorie. Jahrb. Schzzbautech. Ges. 31 (1930b), 389-436; Erort. 436-440. WEINBLUM, G. Uber die Berechnung des wellenbildenden Widerstandes von Schiffen, insbesondere die Hognersche Formel. 2. Angew. Math. Mech. 10 (1930c), 453466. G., Hohle oder gerade Wasserlinien? Hydromechanische Probleme des SchiffsWEINBLUM antriebs, Hamburg, 1932a, pp. 115-131, 417419. WEINBLUM, G. Schiffsform und Wellenwiderstand. Jahrb. Schiffbautech. Ges. 33 (1932b), 419451 ; Erort. 456460. WEINBLUM, G. Untersuchungen uber den Wellenwiderstand volliger Schiffsformen. Jahrb. Schiffbautech Ges. 35 (1934), 164-192. WEINBLUM, G. Widerstandsuntersuchungen an scharfen Schiffsformen. Schiffbau 36 (1935), 355-359, 408-3 14. WEINBLUM, G. Rotationskorper geringsten Wellenwiderstandes. 1ng.-Arch. 7 (1936a), 104-1 17. G.Die Theorie der Wulstschiffe. Schiffbau 37 (1936b), 55-65. WEINBLUM, WEINBLUM, G. Beitrag zur Ausbildung volligerer Schiffsformen. Schzybau 37 (1936c), 285-292. WEINBLUM, G . Wellenwiderstand auf beschranktem Wasser. Jahrb. Schzffbautech. Ges. 39 (1938). 266-289; Erort. 289-291. WEINBLUM, G. Schiffsform und Widerstand. Schzffbau 40 (1939), 27-23, 46-51, 66-70. WEINBLUM, G. Analysis of wave resistance. David W . Taylor Model Basin Rep. 710 (1950), 102 pp. WEINBLUM, G. T h e wave resistance of bodies of revolution. (Appendix I1 by J. Blum.) David W . Taylor Model Basin Rep. 758 (1951), 58 pp. WEINBLUM, G. A systematic evaluation of Michell’s integral. David W . Taylor Model Basin Rep. 886 (1955), 59 pp.
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WEINBLUM, G. Ein Verfahren zur Auswertung des Wellenwiderstandes vereinfachter Schiffsformen. Schzflstechnik 3, 278-287 (1956). WEINBLUM, G. Applications of wave resistance theory to problems of ship design. Trans. Inst. Eng. Shipbuilders Scotland 102, 119-152 (1959); disc. 153-163. WEINBLUM, G. On problems of wave resistance research. Int. Sem. Theoret. W a v e Resistance, A n n Arbor, 1963, pp. 1-44; disc. 4 5 4 9 . WEINBLUM, G. Schiffe geringsten Wellenwiderstandes. Schiflstechnik 12 (1965), 131-136. G. Uber die Unterteilung des Schiffswiderstandes. Schifl Hafen 22 (1970), WEINBLUM, 807-81 2. WEINBLUM, G. ;AMTSBERG, H. ; BOCK,W. Versuche iiber den Wellenwiderstand getauchter Rotationskorper. Schzffbau 37 (1936), 411419. Translated in David Taylor Model Basin Rep. T-234 (1950), 22 pp. G. P.; KENDRICK, J. J. ; TODD, M. A. Investigation of wave effects produced by WEINBLUM, a thin body-TMB Model 4125. David Taylor W. Model Basin Rep. 840 (1952), 14 pp. WEINBLUM, G. ; SCHUSTER, S. ; BOES,CHR.; BHATTACHARYYA, R. Untersuchungen iiber den Widerstand einer systematisch entwickelten Modellfamilie. Jahrb. Schiflbautech. Ges. 56 (1962), 296-319; Erort. 320-324. G. ; WUSTRAU, D. ; VOSSERS,G. Schiffe geringsten Widerstandes. Jahrb. WEINBLUM, Schiflbautech. Ges. 51 (1957), 175-204; Erort. 205-214. WIGLEY, W. C. S. Ship wave resistance. A comparison of mathematical theory with experimental results. I , 11. Trans. Inst. N a v . Architects 68 (1926), 124-137 (plates X, XI); disc. 137-141 ; 69 (1927), 191-196 (plate XVIII); disc. 196-210. WIGLEY,W. C. S. Ship wave resistance. Some further comparisons of mathematical theory and experiment result. Trans. Inst. N a v . Architects 72 (1930a), 216-224 (plates XXIV,XXV); disc. 224-228. WIGLEY,W. C. S. Ship wave resistance. Proc. 3rd Znt. Congr. Appl. Mech., Stockholm, 1930b, vol. 1, pp. 58-73; disc. 73. WIGLEY,W. C. S. Ship wave resistance. An examination of the speeds of maximum and minimum resistance in practice and in theory. North-East Coast Inst. Eng. Shipbuilders Trans. 47 (1931), 153-180 (plates 11-VI); disc. 181-196 (pl. VII). WIGLEY,W. C. S. A note on ship wave resistance. Hydromechanische Probleme des Schiffsantriebs, Hamburg, 1932, pp. 132-138. WIGLEV, W. C. S., A comparison of experiment and calculated wave-profiles and waveresistances for a form having parabolic waterlines. Proc. Roy. SOC.Ser. A 144 (1934), 144-159 (4 plates). W I G L E Y ,C. ~ . S. Ship wave-resistance. Progress since 1930. Trans. Inst. N a v . Architects 77 (1935), 223-236 (plates XXVI, XXVII); disc. 237-244. WIGLEY, W. C. S. The theory of the bulbous bow and its practical application. North-East Coast Inst. Eng. Shipbuilders, Trans. 52 (1936), 65-88 (plate I). WIGLEY,W. C . S. Effects of viscosity on the wave-making of ships. Trans. Inst. Engr. Shipbuilders Scotland 81 (1938), 187-208 (1 plate); disc. 208-215. WIGLEY,W. C. S . The wave resistance of ships: a comparison between calculation and measurement for a series of forms. Congr2s Znt. Zng. N a v . , Li2ge, 1939, pp. 174-190. WIGLEY, W. C. S. The analysis of ship wave resistance into components depending on features of the form. Trans. Liverpool Eng. SOC.61 (1940), 2-25; disc. 26-35. WIGLEY, W. C. S. Calculated and measured wave resistance of a series of forms defined algebraically, the prismatic coefficient and angle of entrance being varied independently. Trans. Inst. N a v . Architects 84 (1942), 52-71 ; disc. 72-74. WIGLEY,W. C. S. Comparison of calculated and measured wave resistance for a series of forms not symmetrical fore and aft. Trans. Inst. N a v . Architects 86 (1944), 41-56; disc. 57-60.
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WIGLEY, W. C. S. L’Ctat actuel des calculs de rCsistance de vagues. Bull. Ass. Tech. M a r . Aeronaut. 48 (1949), 533-564; disc. 565-587. WIGLET,W. C. S. Water forces on submerged bodies in motion. Trans. Znst. N a v . Architects 95 (1953), 268-274; disc. 274-279. WIGLEY,W. C. S. Possible developments in calculation of wave resistance of ships. Schifstechnik 3, 17-18 (1955). WIGLEY,W. C. S. The effective virtual mass of a spheroid moving near the free surface of a fluid. Actes 92me Congr. Znt. Mec. A p p l . , Bruxelles, 1957, Vol. 1, pp. 203-206= Schifstechnik 4 (1957), 65-67. WIGLEY,W. C. S. The effect of viscosity on wave resistance. Schiffstechnik 9 (1962), 69-71 ; disc 71-72. WIGLEY,W. C. S. Effects of viscosity on wave resistance. Znt. Sem. Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 1293-1310. WIGLEY, W. C. S. A note on wave resistance in a viscous fluid.Schifstechnik 14 (1967), 10. WIGLEY, W. C. S.; LUNDE,J. K. Calculated and observed wave resistances for a series of forms of fuller midsection. Trans. Znst. N a v . Architects 90 (1948), 92-104; disc. 104-110. Wu, J I N . The separation of viscous from wave-making drag of ship forms. J . Ship Res. 6, no. 1 , 26-39 (1962). Wu, T . YAO-TSU. Interaction between ship waves and boundary layer. Znt. Sem. Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 1261-1287; disc. 1288-1291. Wu, T. YAO-TSU. A singular perturbation theory for nonlinear free surface flow problems. Znt. Shipbuilding Progr. 14 (1967) 88-97. YEUNG,R. W. Sinkage and trim in first-order thin-ship theory. J . Ship Res. 16 (1972), 47-59. YIM,BOHYUN.On ships with zero and small wave resistance. Znt. Sem. Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 1031-1075 ; disc. 1076-1079. YIM,BOHYUN.Some recent developments in theory of bulbous ships. 5th S y m p . Naw. Hydrodyn., Bergen, 1964, pp. 1065-1098. YIM,BOHYUN. Analyses on bow waves and stern waves and some small-wave-ship singularity systems. 6th Symp. N a v . Hydrodyn., Washington, D.C., 1966, pp. 681-698; disc. 699-701. YIM,BOHYUN.Higher order wave theory of ships. J . Ship Res. 12 (1968), 237-245. YIM,BOHYUN. On the wave resistance of surface effect ships. J . Ship Res. 15 (1971), 22-32. YOKOYAMA, NOBUTATSUO. On the relations between a practical ship-hull form and an attempted singularity distribution. Znt. S e n . Theoret. W a v e Resistance, Ann Arbor, 1963, pp. 1111-1128. ZHUKOVSKII, N. E. See Joukowski.
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Dynamics of Quasigeostrophic Flows and Instability Theory H . L . KUO Department of Geophysical Sciences The University of Chicago. Chicago. Illinois
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . I1. Tendency Toward Geostrophic Balance in Rotating Fluids . . . . A . Adjustment of Pressure and Nondivergent Flow Fields Toward Geostrophic Balance . . . . . . . . . . . . . . . . . . . . B. Solution of the Wave Equation and the Adjustment Process . . . 111. Simplified Hydrodynamic Equations for Large Scale Quasigeostrophic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Estimates of the Magnitudes of the Thermodynamic Variables in Quasigeostrophic Flows . . . . . . . . . . . . . . . . . . B. Scaling of the Hydrodynamic Equations . . . . . . . . . . . C . Expansion of the Flow Variables in Powers of Ro and the FirstOrder Potential Vorticity Equation . . . . . . . . . . . . . D . The Boundary Conditions in Terms of $I . . . . . . . . . . . IV . Permanent-Wave Solutions of Nonlinear Potential Vorticity Equation in Spherical Coordinates . . . . . . . . . . . . . . . . . . . A . Development of the General Permanent-Wave Solution . . . . B. The Vertical. Function and the Eigenvalues . . . . . . . . . . V . Stability of Zonal Currents for Small Amplitude Quasigeostrophic Disturbances . . . . . . . . . . . . . . . . . . . . . . . . VI . General Stability Theory-Integral Relations and Necessary Conditions for Instability . . . . . . . . . . . . . . . . . . . . . A . Stability Conditions for Pure Barotropic flow . . . . . . . . . B. The Semicircle Theorem for Three-Dimensional Baroclinic Disturbances . . . . . . . . . . . . . . . . . . . . . . . VII . Stability Characteristics of Barotropic Zonal Currents and Rossby Parameter . . . . . . . . . . . . . . . . . . . . . . . . . A . Stability of the Sinus Profile U = (1 cos y)/2 . . . . . . . . B. Stability of the Bickley Jet . . . . . . . . . . . . . . . . . C . Disturbances in a Hyperbolic-Tangent Zonal Wind Profile . . . VIII . Pure Baroclinic Disturbances . . . . . . . . . . . . . . . . . A . The Constant f Model and Boussinesq Approximation . . . . . B. Approximate Solutions of Equation (8.1) for a Nonzero b . . . . C . The General Baroclinic System . . . . . . . . . . . . . . . D . Laboratory Experiments on Baroclinic Instability . . . . . . .
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248 250 252 256 257 258 259 260 263 265 265 268 272 276 277 279 281 281 283 286 291 293 297 300 305
H . L. Kuo
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IX. Finite Amplitude Unstable Disturbances . . . . . . . . . . . . A. Method of Solution . . . . . . . . . . . . . . . . . . . . B. General Equations for Wave Perturbations in a Two-Level or Two-Layer System . . . . . . . . . . . . . . . . . . . . C. Inviscid Finite Amplitude Disturbance, @ # 0, r = 0 . . . . . . D. Viscous Equilibration for /3 = 0, Y # 0 . . . . . . . . . . . . X. Instability Theory of Frontal Waves . . . . . . . . . . . . . . A. The Basic State . . . . . . . . . . . . . . . . . . . . . B. Perturbation Equations and Boundary Conditions . . . . . . . C. Frontal Wave Solution. . . . . . . . . . . . . . . . . . . D. Nonlinear Development of Frontal Wave. . . . . . . . . . . XI. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
306 307 308 310 314 316 317 318 322 327 327 328
I. Introduction T h e earth’s atmosphere is a mixture of gases and is under the influences of earth’s gravity and rotation, hence the disturbances set up in it can behave either as sound waves, surface gravity waves, internal gravity waves, inertial waves, Rossby or other kinds of vorticity waves, or some combination of these different types of motions. This is also true of the oceans. The actual nature of the disturbance depends strongly on its frequency or wave length, which are usually determined by the way the disturbance is produced. I n this respect it is useful to classify the disturbances into two different categories, namely, (1) forced motions and (2) free motions. By forced motion we mean those motions which can be attributed directly to some known forces, such as the oceanic and atmospheric gravitational and thermal tides, monsoons, mountain and sea breezes, and motions set u p by the varying topography and differential heating. On the other hand, free motions are those which cannot be attributed to any given force directly, but are the results of some intrinsic instability of the system. Within this category we have the long Rossby waves in the upper troposphere, the low level cyclone waves, and the disturbances created by convective and shear instabilities. I n this paper we shall limit our discussions to the large scale, low frequency flows only, that is, the motion systems whose aspect-ratio H / L is much smaller than unity and whose period is longer than one day, where H is the vertical scale of variation and L is the horizontal scale of variation. It can be shown that for such large scale flows the
Quasigeostrophic Flows and Instability Theory
249
vertical acceleration can be neglected in the equation of motion and the influence of the local density variation can be disregarded in the continuity equation. We then have the hydrostatic and the anelastic approximations of the hydrodynamic equations. For the oceans, the latter can be replaced by the Boussinesq approximation, which is to take the density as constant except when it is associated with the gravity where a buoyancy force is introduced due to the density difference. These approximations have the effect of eliminating the sound waves from t h e system but leaving the low frequency disturbances represented quite accurately. A very important property of the low frequency (large scale) disturbances in a rotating fluid is the tendency toward geostrophic balance, in which the Coriolis force in the horizontal plane (i.e., normal to the gravity g) is balanced, or nearly balanced, by the horizontal pressure gradient force, so that the equation of motion in the horizontal plane is approximately given by where f = 2w sin is commonly referred to as the Coriolis parameter even though it is only the vertical component, w(=7.293 x s-l) is earth's rate of rotation, V is the velocity measured in a frame fixed on the earth, k is the unit vector along the vertical, p is density, p is pressure, and V h is the horizontal grad operator. When disturbances are produced in the atmosphere and in the oceans, they may contain many components which are not in geostrophic balance. Such flows are usually of the inertial-gravity-wave type with prominent horizontal convergences and divergences, and hence are accompanied by large cross-isobar components. These flows usually act as the agents in carrying away the unbalanced field from the source and leave a predominantly geostrophically balanced field behind. This adjustment toward geostrophic balance will be discussed in Section 11. T he flow is said to be quasigeostrophic when (1.1) is satisfied approximately, but the departure from this balance is of importance for the determination of changes of the flow fields. In such flows the velocity V is predominantly rotational, i.e., the vorticity 5 = vh X is relatively large, while the horizontal divergence vh * V is small. Such flows, when established, are governed by the nonlinear quasigeostrophic potential vorticity equation, which we shall derive in Section 111. T he inviscid nonlinear potential vorticity equation permits two- and three-dimensional barotropic permanent wave solutions containing many components with arbitrary coefficients both in Cartesian and in spherical coordinates. Specific combination of these solutions can be used to represent the observed mean flow patterns in the atmosphere.
v
250
H . L. Kuo
The potential vorticity equation is linearized with respect to a basic zonal current U(y, 2) and the general stability problem is formulated in Section V, while in Section VI the general stability theory is represented by two integral relations and an extended circle theorem. The stability properties of barotropic flows are discussed in some detail in Section VII, especially with regard to the eigenvalues for the cos2(y/2d), sech2(y/d), and tanh y / d profiles. It is found that, for the same profile, an easterly current is more unstable than a westerly current under the influence of a positive Rossby parameter p, while the westerly current is more unstable under the influence of a negative /3. The linear theories of pure baroclinic disturbances without and with the influence of Rossby parameter are discussed, and the solutions related to the baroclinic disturbances in the rotating annulus experiments are reported briefly. In Section I X a general method of obtaining finite amplitude solutions of the baroclinic potential vorticity equation is formulated, and an inviscid, oscillatory solution and a viscous, equilibrium solution obtained by Pedlosky for the two-level model are presented. Finally, the instability theory of frontal cyclones is presented in Section X as a separate problem, even though it is closely related to the baroclinic wave theory for quasigeostrophic Aow. As is unavoidable in a paper like this, only a few papers on each subject have been mentioned. Additional references can be found in the papers cited, and so extensive bibliography is not included here
II. Tendency Toward Geostrophic Balance in Rotating Fluids One very important character of the motion of a rotating fluid is the tendency toward geostrophic equilibrium, in which the Coriolis force of deflection is balanced by the pressure gradient perpendicular to the direction of motion. This equilibrium is brought about by a mutual adjustment between the mass (pressure) and the momentum distributions toward the geostrophic condition whenever an imbalance exists, such as when certain momentum is suddenly imparted to part of the fluid without an accompanying pressure gradient, or when a pressure gradient is produced by extraction or addition of mass in a certain region. This process was first discussed by Rossby (1938), and later on by many others. For example Cahn (1945), Obukhov (1949), and Raethjen (1950), have examined the adjustment problem for a homogeneous rotating fluid, while Bolin (1953) and Veronis (1956) have investigated the stratified fluid problem, and Kibel’ (1955) analyzed the three-dimensional flows. In this
Quasigeostrophic Flows and Instability Theory
25 1
section we shall demonstrate this process with Obukhov's (1949) simple, vertically averaged barotropic model, by introducing m
p V dx,
(2.la)
(Y =
:),
(2.lb)
as the dependent variables, where p , is the surface pressure, p o is its mean value, H = p o / g p o is the scale height of the atmosphere, and P is the potential energy of the air column which is equal to the vertically integrated pressure, viz., (2.lc) The vertically integrated and linearized equations of motion and continuity equation are then given by ( a V / a t ) + f k x V=-gHVm,
(2.2)
anpt = -v v = -vzV. (2.3) For convenience we decompose V into its nondivergent part V , and irrotational part V , and introduce a stream function for V* and a velocity potential for V , , viz. V = V,
+ V , = k x V$ + V V .
(2.4) On applying the operators V x and V . to (2.2) we then obtain the following vorticity and divergence equations, respectively : V 2 [ h+fvl= 0, V2[vt-f* +@TI = 0. Combining (2.5) with (2.3) we find the potential vorticity equation V2$bt-fTr, = 0.
(2.51 (2.6) (2.7)
This equation shows that the potential vorticity Q, given by = v2* -f T ,
(2.7a)
is independent of t and hence is a function of x , y only. We can use this equation to determine the final distributions of and m from their initial distributions.
H . L. Kuo
25 2
For wave perturbations we can remove the operator V2 from (2.5) and (2.6) and obtain (2.5a) (2.6a) On applying a/at to (2.6a) and substituting $t and rt from (2.5a) and (2.3) we then obtain the following wave equation in p): Co2=gH.
f f 2 y - Co2V2p)= 0,
(2.8) We assume that the initial values y o , $o, and ro are given. Equation (2.6a) then furnishes the initial y t , so that (2.8) can be solved with the initial conditions p)tt
yo = el(% y ) , yto
=f$o
(2.8~)
- gHr0 = ez(x,y ) .
(2.8b)
Note that if p) is decomposed into its Fourier wave components exp i(k,x k 2 y - ut), we then find that u is given by
+
u2
=f2 +
(K,2
+
K22)C02.
(2.8~)
This relation shows that these waves are dispersive and hence an initially localized distribution of the unbalanced wave energy represented by ez(x,y ) will be carried away by these surface waves and spread out over the whole domain, thereby also altering the mass distribution. Thus, the wave system represented by (2.8) and (2.8a,b) actually furnishes the mechanism through which the adjustment toward geostrophic equilibrium is established, and the rate at which the equilibrium is reached is also determined by this system of equations in the present model.
A. ADJUSTMENT OF PRESSURE AND NONDIVERGENT FLOWFIELDS TOWARD GEOSTROPHIC BALANCE T o obtain the solutions of (2.7) and (2.8) as an initial value problem, we first decompose I+$r,and p into a steady field and an unsteady or wave field, viz.,
$=*+*‘,
(2.9) where 4, 7j, p) are functions of x, y only and $’, r’,and y’ are functions of x,y , and t. From (2.5a) and (2.6a) we then find 7T=++r’,
p=q+p)’,
(2.10)
(2.11)
Quasigeostrophic Flows and Instability Theory
25 3
Therefore the F field has no steady part so that the steady flow is nondivergent, while ii and $ satisfy the geostrophic relation. Further, since the potential vorticity is independent of t, Q must be given by the steady field and also equal to its initial value. Thus we have
n = V2$- f 77 = no =
v2*o-fro
(2.12)
>
Q' = VZ*' -fT' = 0.
(2.13)
Substituting ii from (2.11) in (2.12) we then obtain the following equation for $: (2.14) where h = (gH)li2/f is called the radius of deformation or the radius of influence of the system. For any given initial distribution of Q, the solution of this equation can be obtained from the following general formula *(Xi
y) = -2T
SSn,(E,
'I)Ko(P) dt
4,
(2.15)
where Ko(p)is the zeroth-order Bessel function of imaginary argument, also known as Kelvin's function, whose asymptotic expressions are
+ log(2/p)
K,(p) =-0.5772
=(42p)
112
e
--P 7
and p is given by P2 = [(x - ElZ
+ (Y
for p < 1 for p 9 1
(2.15a)
'I)21/h2. (2.15b) When the disturbance depends only on one space coordinate, the solutions of the steady and unsteady systems (2.14) and (2.7) and (2.7a,b) can be obtained rather easily. For example, with an initial unbalanced vortex motion given by -
(2.16a) T o = To = 0,
we find that the final solutions of the potential vorticity equation (2.14) are
(2.16b)
H . L. Kuo
254
Here R is a horizontal scale length and AIR is the strength of the flow, and
5 = r/R,
r2 = x2
+y2,
p = R2/h2.
These solutions show that, when p is small, the difference between 6 and coo is insignificant, while for relatively large values of p, the change of z, is appreciable. For example, the curves in Fig. l a represent the function 5, v o , and ii for AIR = 5 m s - l and p = 0.0516, which corresponds to R = 500 km and A = 2200 km. It is remarkable that in this example z, changes so little while the change of pressure is so drastic, amounting to a reduction (A < 0) of 20 mb at the center. This situation is characteristic of the disturbances whose horizontal dimension R is small compared with A. For large values of AIR, the change of z, becomes significant, as can be seen from the values of 6 / v 0 in Table I, which are based on h = 2700 km.
2.0
l 1.2 . b y ;
0.0
m s'
0.6 0.4
0.4
0
it@.!
0.2
-
c
Hours
I I I I I
0 500 1000 1500 2000 km r
0 500 lo00 1500 km
r
FIG.1. Adjustment between pressure and velocity distributions in rotating fluid. (a) Adjustment of pressure to given initial velocity field; (b) diminution of unbalanced pressure field; (c) -pressure change at the center of an unbalanced vortex.
TABLE I
VALUES OF U/v0
500 3000
5000
I
1000
2000
3000
4000
5000
0.99 0.75 0.52
0.99 0.74
0.99 0.71 0.47
0.98 0.64 0.39
0.98 0.51 0.27
0.51
Quasigeostrophic Flows and Instability Theory
255
If, on the other hand, the initial disturbance is in the form of an unbalanced pressure, such as the sudden elevation of the free surface or localized distribution of potential energy, we shall find that most of it will be carried away by the gravity waves and only a very small fraction will be left behind to be balanced by a wind system, and the more so the smaller the initial horizontal scale of the perturbation. This case is demonstrated by the following example.
Initial jield:
IIIo=o, n=o= A [ 2
yo=(),
+ p - (4+ p)x + x2]e-.?.
(2.17a)
Final steady jield:
Afr
= - [x - 2]e-",
2
=
ARZf
( I - x)e-*,
(2.17b)
ii = Ap(1 - x)e-'.
where x = r2/2R2. Thus, when p < 1 the final pressure perturbation ii is very small compared with its initial value n o ,as is illustrated in Fig. lb. Thus we conclude that when p is small, the pressure always adapts to the velocity. Since p is proportional t o p , we expect pressure adjustment to take place on larger scales at low latitude. T h e physical reason for the behavior discussed above is that when an imbalance beween the pressure gradient and Coriolis force is present in a region, adjustment of mass distribution can be accomplished rapidly near the edge of the region but only very slowly far away from the boundary. Therefore unbalanced pressure gradient in a small region can readily be obliterated by a mass flow in the direction of the pressure gradient or Coriolis force, whereas far away from the boundary the velocity has to adjust to the pressure gradient. Another point worth mentioning is that large scale variations can only be established gradually and hence the pressure and velocity distributions have ample time to adjust to each other, therefore we do not expect strong imbalances to occur over a large area. On the other hand, large velocity or pressure gradient concentrations can easily develop over small regions. According to the adjustment theory discussed above, only the small scale velocity concentration can persist while the unbalanced pressure gradient will soon be obliterated. Thus, the adjustment mechanism seems to be responsible for the streakiness of the velocity distributions in the atmosphere and in the oceans.
H . L. Kuo
256
B. SOLUTION OF THE WAVEEQUATION AND THE ADJUSTMENT PROCESS As has been pointed out already, the adjustment of the pressure and the velocity distributions toward geostrophic equilibrium is actually accomplished through the divergent flows. For the problem under consideration, these flows are governed by the dispersive wave equations (2.8) and (2.8a,b). T h e solution of this system can be obtained more readily by using
4 x 9 Y,
5, t ) = COS(f5/Co)d~,y , t )
as the dependent variable, so that (2.8) is transformed into the following simple three-dimensional wave equation Utt = V,%.
(2.18)
The initial conditions (2.8a,b) are then given by
u(x, Y , 5, 0) = 4% y)cos(f5/co),
(2.18a)
4%y, 5, 0) = 4% y)cos(f5/co).
(2.18b)
The solution of this system can readily be obtained by Cauchy’s method and the function ~ ( xy , t ) is simply given by u(x,y, 0, t). Thus we have
where the argument offi and 5 ct, and
f2
are x
+ p cos 6, y + p sin 6, the limit for
p is p
.q = ( C 2 t 2
-p
y .
T h e pressure variation can be obtained from (2.19) by first applying V2 to it and then integrating over t. For the initial disturbance represented by (2.16a), the change of the pressure at the center with time is represented in Fig. lc. It is seen that the geostrophic value of the pressure at the center is established within three to four hours. Many works on the adjustment problem, such as those of Cahn (1945), Bolin (1953), and Veronis (1956), are centered on the solution of the dispersive wave equation. However, from the point of view of the large scale flows, it seems that the solution (2.15) is more important and interesting. I t is evident that in a stratified medium, the adjustment process works essentially within individual layers bounded by isentropic surfaces, even though adjacent layers influence each other to a certain degree. T h e equation that governs the adjusted state is the general potential vorticity equation
QuasigeostrophicFlows and Instability Theory
257
( 3 . 1 3 ~ )to be derived in the next section, while the mass adjustment will be accomplished both by surface waves and by the internal gravity waves.
111. Simplified Hydrodynamic Equations for Large Scale QuasigeostrophicFlow We consider that the motion system under consideration is characterized by a horizontal scale length L, a vertical scale length D, a velocity U, and a time scale T . When the ratio LID is much larger than unity, the motion will be referred to as of large scale. For the large scale motions in a stably stratified and rotating fluid, the following parameters are of paramount importance in determining the nature: of the motion, (i) (a) T h e thermal Rossby number
gD vh8 gD Ah8 = -ROT= fo2L 8, fo2L2 0, ' (b) The mechanical Rossby number
(ii) The planetary Richardson number
gD A,0, &=--gD2 38, - -fO2L2t7,az -fO2L2 0, ' (iii) 'The planetary Froude number P=-*
fo2L2 gD
(iv) The Ekman number
E = vifo D2, where fo is an appropriate value of the Coriolis parameter for the region under consideration, 0 is the potential temperature and 0, is its normal value, which is taken as a function of z only, 0, is a vertical average of 8, for the level considered, A,0 is the horizontal difference of 0 within the horizontal distance L , and A20, is the vertical difference of 8, within the vertical distance D.The motions are considered as the results of the horizontal temperature gradient AhO, so that U may be identified with the thermal wind gAhO/ffls. We then have ROT= Ro.
25 8
H . L. Kuo
Because of the existence of the mean stratification in the earth's atmosphere, the vertical scale length D of the large scale motions is of the order of the depth of the troposphere, which is about lo4 m, while the typical , horizontal wave length) is of the order of horizontal scale L( ~ ! x / 4A= lo6 m, essentially 1/10 of the distance from the equator to the pole. Thus, if we use the equator to pole difference of 8 [ %SO "C] as the representative horizontal temperature gradient, we shall have V,8 m 5.0 x "C m-l. On using fo = 10-4s-1 we then find
ROTM 0.15. On the other hand, the normal vertical stratification of 8, gives aO,/az= 6.0 x "C/m, and hence S, M 2.0. The planetary Froude number p depends only on fo , D, and L and is of the order of 0.1 for the values of fo, L, and D used above. Thus there is an important class of motions for which R, and p are much smaller than 1 while S, is of the order of unity. I n what follows we shall derive the simplified versions of the hydrodynamic equations that are adequate for this class of motion.
A. ESTIMATES OF THE MAGNITUDES OF THE THERMODYNAMIC VARIABLES IN QUASIGEOSTROPHIC FLOWS
As will be shown later in this section, the large scale motion of a stably stratified fluid is mainly geostrophic. That is to say, the horizontal velocity V and the pressure perturbation p' =p -ps(z)are related approximately by the geostrophic wind equation
fopsk x V= 0 , ~ ' . (3.1) On replacing V by U and Vhp' by p'/L for order of magnitude consideration we then find
[ P I = [fopsLu1= LfoLU/RT,lp, = [(pQ'gH)ROlp,,
(3.la)
where H = RTJg is the equivalent depth of a homogeneous atmosphere with total pressure p , and uniform density p, , and is of the order of 7.5 km, which is nearly equal to D.Thus we have [p'lp,] = E = p R o M 0.015. Since, according to the equation of state, p'/ps and T'/Tsare of the same order of magnitude as p ' / p s , we have
x/xs = EX*
(34
for all the thermodynamic variables, where X ' represents either p', p', T', or 8' and 2,represents the vertical average of the corresponding basic state thermodynamic variable, and X * represents the nondimensionalized perturbation of these quantities. In this form X * is of order unity.
Quasigeostrophic Flows and Instability Theory
259
B. SCALING OF THE HYDRODYNAMIC EQUATIONS The nonsteady quasigeostrophic flows usually propagate with speeds comparable to the mean current velocity U and hence the quarter period of the time variation is of order TI = L/U. I n addition, the stable stratification of the atmosphere renders w' to be of order UR, D/L instead of UDIL. Therefore the appropriate scalings of this class of motion are
x' = Lx, u' = uu,
pi +AmpsQ,
Y' = LY, v'
=
uv,
= Epsp*,
Z' = Dz,
t' = tL/U,
W' = wUDR,/L,
el = &eSs,
(3-3)
f'=E,
where the primed quantities are dimensional and the unprimed quantities are dimensionless, and fi is an appropriate mean value off' for the area under consideration. Substituting these transformations in the horizontal and the vertical equations of motion, the continuity and the heat equations we then obtain the following nondimensionalized equations
dV Ra - +fk x V = -V@ dt
+ EV3'V,
(3-4)
where
and V' = V, is the dimensionless horizontal velocity, u, = -( l / p s ) (a p , / a z ) , Q = Q'/CpTsp,Q' being the rate of accession of heat per unit mass, V = V, is the horizontal gradient, and V32 is the three-dimensional Laplacian. We mention that in the scalings in (3.3) both the dependent variables and their space and time derivatives are assumed to be of order unity. Thus, the terms on the right-hand side of (3.5) can be neglected whenever the ratio D2/L2is much smaller than unity, and hence for such large scale motions the pressure distribution is hydrostatic even in the disturbed state. Further, the viscous term in (3.4) can also be neglected except close to the surface where it becomes important in creating the Ekman flow.
H . L. Kuo
260
Further, since E is much smaller than R, for almost all large flows under consideration, the last term in (3.6) can also be neglected. This approximation to the continuity equation is called the anelastic approximation. Thus for E = p R < R, (3.4)-(3.7) can be simplified to the following: d V +fk x
V = -V@,
(3.4a)
dt
a@
- - a,@
ax
+p"
a@
-s =
= - +s,@
ax
apsw = 0, v v+-Ro -
(3.5a) (3.6a)
*
ps
0,
ax
(3.7a) where
1
ae,
e,
ax
s, = - -= ps,.
With R, set to unity, these equations are called the primitive equations in meteorology and they represent the proper simplification of the original equations (3.4)-(3.6), involving only the neglecting of the viscous dissipation, the vertical acceleration in ( 3 4 , and the local density change in the continuity equation (3.6a). The results of the hydrostatic and the anelastic approximation are that high frequency internal gravity waves and acoustic waves are excluded from the system, but otherwise the system is still able to carry the relatively low frequency internal gravity waves as well as inertial waves and vorticity waves.
C. EXPANSION OF THE FLOW VARIABLES IN POWERS OF R, AND THE FIRST-ORDER POTENTIAL VORTICITY EQUATION For a Rossby number R, of the order of 0.15 or smaller, the series expansions of the flow variables in powers of R, can be expected to converge rapidly so that the first few terms should give sufficiently accurate results for many problems. We shall therefore develop the dynamic equations from a formal expansion in R, by setting
X=
C ROmXm,
(3.8a)
m=O
w = w1+ R o w ,
+
* * *
,
(3.8b)
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261
where X stands for any one of the variables u,cu, @, or s. Further we write
f =J’K = (1 +ROPY),
(3.8~)
where h=pS,R;l is of order 1, s,= S z p = h R o , P=/3’L2/LT, which is taken as of order unity, where p‘( =dJ’/dy‘) is the dimensional Rossby parameter. Substituting these expansions in (3.4a)-(3.7a) and equating to zero the coefficient of Rooand Ro, we then obtain the following systems of zeroth- and first-order equations :
k x Vo=-V@o,
(3.9a)
a(Do/az= so,
(3.9b)
v.vo=o;
(3.9c)
8% - s, - m0, --
(3.10b)
ax
(3.10~) as0
-
dt
+ Vo V S ,+ Szwl= Q1. *
(3.10d)
This set of equations is essentially the same as those obtained by Charney and Stern (1962) and by Pedlosky (1964a). The higher order hydrostatic and continuity relations are of the same forms as (3.10a) and (3.10b) while the higher order equations of motion and heat equation are given by the following for m 2 1
(3.11a) (3.11b) Thus the zeroth-order equations (3.9a-c) signify that V o , @, , and so are connected by the geostrophic and the hydrostatic relations, and that V , is nondivergent to the extent that a constant fo is used in the geostrophic relation. Notice that these relations do not contain the time rate of change explicitly and hence they are not able to reveal the time changes of the flow variables, even though time variations may be implied implicitly.
262
H . L. Kuo
T o obtain the time changes we must make use of the higher order equations (3.10a-d) and (3.11a,b). Since (3.10a) contains Vl and Q1in addition to aVo/at,it is difficult to use them directly even though (3.4a) and (3.7a) form a closed system. However, by applying V x to (3.10a) and making use of (3.10~)we can eliminate O1and Vl from this equation and obtain
where
Here we have included a first-order frictional term even though this term is small. Note that the geostrophic motions possess vorticity because the @-field forms closed contours. Eliminating w1 between (3.12) and (3.10d) and including the first-order dissipative terms for generality, we obtain the following first-order nonlinear quasigeostrophic potential vorticity equation in terms of a single dependent variable Q 0 : (3.13) where q is the potential relative vorticity, given by (3.13a) The dimensional form of q is
(3.13b) where t,h =p’/fopsis the dimensional stream function. Equation (3.13) can also be obtained from Ertel’s potential vorticity equation by the application of the geostrophic relation (see Ertel, 1942; Kuo, 1972). For a homogeneous incompressible rotating fluid of variable depth H( =Ho + h), the quasigeostrophic potential vorticity equation takes the form
d (f3) = 0. dt
H
(3.14)
Quasigeostrophic Flows and Instability Theory When f and H , are functions of y only, and h written as
263
< H , , this equation can be ay ax
=0,
(3.14a)
where
a*
The second part of represents the influence of the variation of the undisturbed depth on the flow. It can often be used in laboratory experiment to simulate the Rossby parameter. Because of the change of depth, it is no longer possible for the particles to move along the contours of constant H and hence steady geostrophic flow is replaced by wave motions, just as under a variable f.
D. THEBOUNDARY CONDITIONS IN TERMS OF 4 For flows in a region bounded by rigid side boundaries, the kinematic condition to be satisfied is the vanishing of the normal velocity. Therefore we have = 0. (3.15a)
vo,
On the other hand, the conditions for w at the bottom surface and aloft must be expressed in terms of so through (3.10d) and then in terms of Q, through (3.10b). For example, when the surface is not horizontal but has a topography given by h(x, y ) , we shall have W, =
V, * Vh
T h e condition at an upper level can be specified in terms of Qo or psQo in a similar manner. It is rather remarkable that the complete set of the first-order hydrodynamic and thermodynamic flow fields in nonsteady quasigeostrophic flow can be determined by the solution of the single potential vorticity equation together with the associated boundary conditions alone. We note that the internal gravity waves have been excluded from the present system by the condition of geostrophy, as is evidenced by the fact that (3.13) is of first order in t and hence it cannot represent gravity waves. One shortcoming of the formal expansion used above is the approximation off by the mean value fo in both the geostrophic relation and in the
H . L. Kuo
264
coefficient of the divergence term in the vorticity equation (3.12), resulting in the appearance of f o 2 in the expression (3.13b) for q. Even though the approximation o f f by f o is justifiable in middle and higher latitudes, it can hardly be valid in lower latitudes. Further, the neglectingof the variable part off is based on the assumption that L < ~ a / 2and , hence is not valid for the very long waves. Therefore, for some problems it is desirable to relax the ordering consistency in the R,-expansion since higher order equations are not being used anyway. Thus, we may replace fo2 by f 2 in (3.13b) so that the potential vorticity is given by (3.13~) Since the hydrostatic relation is valid for all the large scale flows, it is often convenient to employ p as the vertical coordinate and use w = dp/dt as the measure of the vertical velocity. T h e horizontal pressure gradient is then measured by the gradient of the geopotential @ =gz of the isobaric surface. T h e primitive equations in dimensional forms are then given by
dV
-++fk dt
x
v=
-V@,
(3.4b) (3.5b)
am v . v+--0,
aP
(3.6b) (3.7b)
where
r = -T-ae= -(yd),
e
aP gP yd and y being the dry adiabatic lapse rate and the actual lapse rate, respectively, and Q the rate of accession of heat. It is seen that the continuity equation (3.6b) is formally like that for a homogeneous and incompressible fluid, even though inhomogeneity and compressibility are included implicitly. T h e quasigeostrophic potential vorticity in pressure coordinate is given by (3.13~") where S = - R r l p and ah,
= @/f.
Quasigeostrophic Flows and Instability Theory
265
On multiplying (3.10a) scalarly by ps V oand (3.10d) by psso7integrating over the entire volume T , and making use of (3.10b,c) and (3.15a), we then obtain the following equations for the changes of the kinetic energy K = 4j r p s V o V , dr and the available potential energy E =g/2 Jz pssgdr/s2: (3.16a)
-aE_ - _ 2t
jrgPsWls0 + D*
(3.16b)
9
where D, is the rate of viscous dissipation of K and D , is the rate of generation or destruction of E by diabatic heating. These equations are of the same forms as that given by the primitive equations (3.4a)-(3.7a). It is seen that the sum of K and E is conserved for inviscid and adiabatic changes and hence the quasigeostrophic system of equations represents an energetically consistent system.
IV. Permanent-Wave Solutions of the Nonlinear Potential Vorticity Equation in Spherical Coordinates
A. DEVELOPMENT OF THE GENERAL PERMANENT-WAVE SOLUTION For inviscid and adiabatic flows the three-dimensional quasigeostrophic potential vorticity equation (3.13) in spherical coordinates reduces to
a%,
+
*A
411 - *11 q A
+ 2Q*A
=0
9
(4.1)
where a is the radius of the earth, the subscripts denote partial differentiations, and h is the longitude, q(= sin @) is the sine of the latitude, and q is the relative potential vorticity, which is given by
(4.la) where S = -p-la log e/ap, and Vs2 is the horizontal Laplacian operator in spherical coordinates, viz., (4.lb)
H . L. Kuo
266
In this section we shall seek permanent-wave solutions of (4.1), namely, solutions of the type #(A, 77, P, 4 = *(A - at, 7, PI?
(4.2) where CL is the constant angular phase-velocity of the perturbation. For such disturbances (4.1) can be written as
'FAG, -yP, G, where"" and G are given by
= 0,
(4.3)
+a 2 q , G = 2(!2 + a ) + ~ A2Y. Y =$
(4.3a) (4.3b)
The first integral of (4.3) is
G = F(Y), (4.4) where F is an arbitrary function ofY. In this investigation we shall restrict ourselves to the case where F ( Y ) is a linear function of Y, that is, G = -(p/a2)Y, (4.4a) where p is a constant which can be chosen to fit the specific situation. In this case $ is given by
A'$
P +$ = -(ZQ a2
+p~)q.
(4.5)
We write the solution of this equation as
where Pz(v) is the associated Legendre function and Niis the vertical amplitude function, given, respectively, by
where n j is a positive integer, including zero, 5 =plp, ,p , being the pressure at the surface, and S* and 1," are given by
T a log 8, S"=----..--
TC
RT
ac
+
'
ljz = -[nj(nj 1)- p]. fo2aZ
(4.8a) (4.8b)
QuasigeostrophicFlows and Instability Theory
267
Since the solution (4.6) satisfies the finiteness condition forI/,I over the entire globe, we only need to impose boundary conditions at the top (5 = 0) and at the ground surface, which we shall assume to be flat and rigid, so that the condition at the bottom surface is the vanishing of the vertical velocity. According to the hydrostatic and the quasigeostrophic approximations, we have w = - +apw - = p ap
at
ax
;( ). --
Substituting this expression in the adiabatic heat equation (3.10d) and setting w to zero we then obtain the following relation for 5 = 1 : a2(*ct
+ + s*t)
*A
*tv - *v
$4, = 0,
(4.9)
where s = -8 log t9,/8<. This relation is satisfied if t,hc= -s+
for
5=
1.
(4.9a)
Therefore we shall take this relation as the lower boundary condition. T h e function N j must satisfy the same relation as I/ at 5 = 1, viz.,
dNjldl;= -sNj.
(4.10)
It may be remarked that it is permissible to neglect the right-hand side term in (4.10) since s# is usually much smaller than $c. That is to say, (4.10) can be replaced by dN,ld(
= 0.
(4.10a)
However, this simplification corresponds to a zero temperature change on the surface and hence is more restrictive than the condition (4.10). T h e condition (4.9a) or (4.9) can also be applied at the tropopause level provided the sudden increase of the static stability at that level is so large to render w to become very small there. However, a better approximation to the real atmospheric condition can usually be achieved by using two different solutions, one for the troposphere and one for the stratosphere, corresponding to two different stability factors in these two layers. Thus the constants lj or n j in (4.8b) are the eigenvalues of the system (4.8), (4.10) and the upper boundary condition, and the summation over j in (4.6) is a summation over all eigenvalues. We point out that the last term of the solution (4.6) represents the solid rotation part of the horizontal velocity, viz.,
H . L. Kuo
268
Therefore the angular phase velocity of the disturbance is given by Haurwitz's (1940) formula cc = w - (2/p)(Q
+ 6).
B. THEVERTICAL FUNCTION AND
THE
(4.11)
EIGENVALUES
When S* is equal to a positive constant in the whole domain 0 5 5 5 1, the solutions of (4.8) and the boundary condition (4.10a) are given by
N j ( 5 )= cos j d ,
(4.12)
where j is an integer, including zero, and n j is given by (4.12a) The solution for j = 0 corresponds to the pure barotropic solution obtained by Ertel (1943) and Neamtan (1946), viz.,
c A,, p,", (7)sin m(h "0
$0
=
m=O
%(no
+ 1)
= p,
-
at
+
Em),
(4.13) (4.134
while the solutions for j 2 1 correspond to internal modes. For example, for S* = 1/3, T = 290 K, and p = 156 we have no = 12,
(4.144
n, = 3.
(4.14b)
Thus, for the large p and the constant S" under consideration, an internal mode j = 1, n, = 3 and a barotropic mode j = 0, no = 12 exist simultaneously. For this case the summation over j in (4.6) covers the first two eigenvalues. T h e results obtained above can readily be modified to correspond more closely to the real situation in the earth's atmosphere, where the stability factor S varies appreciably with height, especially across the tropopause and in the stratopause. As can be seen from Fig. 2, the annual mean value of the stability factor S,( = - T a log 0/ap=pS/R)remains fairly constant in the troposphere and goes through a large and an almost discontinuous increase across the tropopause, and then increases continuously within the stratosphere in inverse proportion to Cb, with b lying between 1 and 2. Because of the variation of the pressure of the tropopause, the discontinuous nature of the change of S , across the tropopause is more evident at
Quasigeostrophic Flows and Instability Theory
269
r (“C/km) FIG.2. Variation of the global average static stability factor atmosphere with height.
I?(
= S, = p S / R ) of
the
individual stations and for shorter time intervals than the annual mean value of S, illustrated in Fig. 2 for the whole United States. T o take into account the dynamic influence of such discontinuous variations of S, it is necessary to divide the atmosphere into a number of layers, each with a different but continuous distribution of S. For simplicity, we take the atmosphere as composed of a troposphere, with a nearly constant S, , and an isothermal stratosphere, extending from 5 = tt to 5 = 0. T h e stability factors S for these two layers can be represented by (4.15) where Fz = 1 is for the troposphere and k = 2 is for the stratosphere. T h e values C, = 100 K, ccl = 0.5 fit the tropospheric I? closely, while C, = 50 K, a2= 0 may be taken as representative of the isothermal stratosphere. For the S distribution given by (4.15) the substitutions Nk(5)
+ cck)-1’2Rk(5),
transform (4.8) into where bk2 is given by 6,’
= (Rck/f2a2)Ap- $,
A p = p - n(n
+ 1).
(4.17a)
H . L. Kuo
270
The condition (4.10) at the bottom
+
(5 = 1) is
(dR,/dq) hR, = 0
at q = 0,
(4.18)
+
where h = 0.5 a log 6,lag w 112. Instead of (4.10), we require R , = C1I2N,to remain finite at 5 = 0, which implies finiteness of the kinetic energy density at 5 = 0. Thus, for positive bj2 the vertical function N can be written as
Nl(5)= A({ +a)-ll2(sin b l q - - bl cos 6,q) h
N2(5)= t-ll2(A,sin b,q
+ B, cos b,q)
for q1 iq I a. (4.1%)
By making use of the continuity relations N , = N , and N ; = N; at the tropopause we can express A, and B, in terms of A, so that only A remains as an arbitrary constant. It can be seen that for small Ap, Nvaries only slightly in the troposphere while for large Ap, the distribution of N is like that of the ordinary internal gravity waves. We point out that when only the finiteness of R, is required at 5 = 0, all integral values of n which yield positive A p are eigenvalues. On the other hand, if either N = 0 or a radiation condition is applied at a fixed height, then only certain values of A p represent permissible eigenvalues. Hence, in this latter case the number of eigensolutions is reduced. Besides the internal-wave type solutions represented by (4.19a,b), there exists another type of solutions for (4.17) and (4.18), namely, the solutions which are continuous at the tropopause but with discontinuous derivatives. Since the tropopause is a surface of discontinuity in the stability factor S, (or N 2 )and apparently also of T , especially when individual situations are considered, we expect the disturbances to possess discontinuities in temperature and in vertical shear of the horizontal wind across the tropopause. Assuming that such atmospheric disturbances are quasipermanent and are governed by the potential vorticity equation, we may then represent them by the solutions of (4.17) with negative values of biz. That is to say, we take them as given by
(4.20a)
Quasigeostrophic Flows and Instability Theory
27 1
where
This vertical function is illustrated in Fig. 3 for various values of Ap. It is seen that they resemble the commonly observed vertical distribution of the large scale flows in the atmosphere.
FIG.3.
Variation of the vertical function R with height.
We point out that even though these permanent-wave solutions satisfy the nonlinear potential vorticity equation, their components are not interacting with each other, including the zonal flow. Hence these components can be treated separately within each family characterized by the individual value of p or n. Solutions corresponding to a variable f = 2w sin y in the coefficient of z,bzz of the potential vorticity equation have been discussed by the author (Kuo, 1959). T he horizontal functions F(q) are then of the Hough function type instead of the Legendre functions. T h e horizontal streamline patterns given by these solutions are illustrated in Fig. 4.
H . L. Kuo
272
FIG.4.
Horizontal streamline patterns for (a) m = 3 and (b) m = 6 .
V. Stability of Zonal Currents for Small Amplitude Quasigeostrophic Disturbances Observations show that the large scale flow field in the atmosphere is usually dominated by two distinctly different kinds of systems, namely, the short waves and the long waves, with wave lengths of the order of 1500 km and 5000 km, respectively. T h e shorter waves are distinctly low-level phenomena, characterized by higher temperature in the low pressure region and low temperature in the high pressure region, and they are usually associated with the surface fronts. On the other hand, the long waves usually have their maximum intensities at the tropopause level, and they are characterized by colder low pressure centers and warmer high pressure centers.
Quasigeostrophic Flows and Instability Theory
273
Further, the low level, short waves often occur in the region to the east side of the upper troughs, suggesting that the surface front is intimately connected with the upper flow pattern. This relation is schematically illustrated in Fig. 5. The occurrence and development of these large scale, quasigeostrophic disturbances in the region of the jet stream in middle latitude indicates that these disturbances owe their existence to the instability of the basic flow, which is characterized by the existence of both vertical and horizontal shears. Hence the instability theory occupies a central position in meteorological fluid mechanics. T he vertical shear of the mean zonal current can be attributed directly to the mean meridional temperature gradient, which is always present in the atmosphere on account of the differential heating by solar radiation. On the other hand, the large horizontal shear of U can only be explained by the specific circulation that governs it, which is usually difficult to describe in terms of a simple mechanism, except along the intertropical convergence zone, where the large U , can be attributed directly to the
FIG.5. Schematic prevailing flow pattern at the 500 mb level and surface fronts (after Palmen, 1951).
H . L. Kuo
274
horizontal convergence and the deflection by the Coriolis force. Since these vertical and horizontal shears represent utilizable potential and kinetic energies for perturbations, we expect to find certain types of disturbances developing under a given zonal wind system. For simplicity, we consider that the flow is composed of a zonal current U which is independent of x and can also be taken as independent of the time t within the time scale of the perturbations, and a small perturbation, represented by the stream function #(x, y, z, t). For such small disturbances, the potential vorticity equation (3.13) can be linearized to the following:
where p stands for the undisturbed density ps and (5.la)
a aY
- (Po
+f)=B-
f 2 uyy--
gsz
[Um- o z u,]= Q o y
(5.lb)
where = df/dy is the Rossby parameter. According to (3.14a), fi may also arise from the variation of the depth H with y. In pressure coordinate we have (5.la') (5.1b') where S = -(ae/ap)/pe. For the stability problem we take Q* as zero and consider as represented by a Fourier series or a Fourier integral in x and t, which we shall represent by the real part of the complex expression
+'
f(x, y, %, t , =
@k(y,z)exp[ik(x- ct)],
(54
+
where c = c, ici . Here the real part c, represents the phase velocity and Rc, represents the amplification rate of the wave component R. For convenience we introduce the transformation
$k(y, %)
= @D,(y,z ) e - u 2 z / 2 = ( p / p O O ) 1 ' 2 @ k ,
(5.2a)
where poo is the surface value of p. It is readily seen that V,,$k . V,t,hk represents the kinetic energy of the horizontal motion per unit volume.
QuasigeostrophicFlows and Instability Theory
275
Substituting (5.2) and (5.2a) in (5.1) we then find that the amplitude function i,bk satisfies the following equation :
where
i,b
is written for
$k
and the operator L is given by (5.3a)
It may be noted that (5.3) contains the Orr-Sommerfeld equation for parallel flow as a special case, namely, for nonrotating or purely horizontal and nondivergent motions. For atmospheric and oceanic flows the direct influence of the viscous dissipation are of secondary importance, especially away from the boundaries. We may therefore consider the interior flows as inviscid and adiabatic, so that (5.3) reduces to
(U-c)L(*)
+--+=o. aQo
(5.4)
aY We assume $I or a$/ay vanishes along the parallels y
+=O
or
dr,h/dy=O
=y1 and y
at y = y 1 , y 2 .
=y z , viz.
(5.5a)
We assume also that the vertical velocity w vanishes at the earth’s surface, and at z = z, , so that the adiabatic heat equation (3.10d) reduces to
( U - c ) - a* az
U,$=O
at z=O, z 2 .
(5.5b)
Because of the large increase of the static stability across the tropopause, the vertical velocity can be effectively reduced to relatively small values at this level. Hence z , can be taken at the tropopause level. However, it should be pointed out that, even though the influence of viscous dissipation is small in the interior, the influence of surface friction is of some importance, especially in creating the Ekman flow and the boundary layer suction wb at the top of the Ekman layer zb, which, according to the Ekman boundary layer solution, is given by the Charney and Eliassen (1949) formula wb
<,,
= s
where = v2$bis the relative vorticity at z b , and 6 = ( ~ / 2 f ) ~is” the boundary layer thickness. At the edge of the top boundary layer, the suction velocity wt is proportional to -C t . Thus, the influence of the boundary
276
H . L. Kuo
layer friction on the quasigeostrophic disturbances is included by the application of the boundary conditions
I n pressure coordinate the perturbation equation and the boundary conditions are
(5.4")
T h e system represented by (5.4) and the boundary conditions (5.5a,b,c) govern all inviscid and adiabatic small amplitude quasigeostrophic disturbances. In accordance with the common usage among dynamic meteorologists and geophysical fluid dynamists, we shall refer to the flow as barotropic when the vertical shear of the mean current, U , , is zero, and as purely baroclinic when the horizontal shear U, is zero. We note that all the parameters that occur in (5.3) and (5.5a)b) are real. Thus when c becomes complex, it will occur in conjugate pairs, and hence instability is implied whenever we have ci # 0.
VI. General Stability Theory-Integral Relations and Necessary Conditions for Instability On multiplying (5.4) by $* = z,hr -- i$, and integrating over y and z from y l to y 2 and from 0 to x z and making use of the appropriate boundary conditions (5.5a,b,c), we then obtain an integral relation. Separating this equation into its real and imaginary parts we then obtain
Y2
Quasigeostrophic Flows and Instability Theory
277
where A, X = X ( z z )- X ( z o )and
(6.lb) (6.1~) Other forms of integral relations are given by Pedlosky (1964a) and by Miles (1964b). Here AW, and A H j are the jumps of W a n d H across the singular points or points of discontinuities m and j of (5.4), respectively. Notice that A is proportional to the sum of the kinetic energy and the potential energy of the perturbation, W represents the horizontal Reynolds stress and measures the northward transport of momentum, while H measures the northward transport of heat by the disturbance. These relations can be used to obtain some general and useful conclusions about the nature of the solution. For example, for continuous solutions without jumps in W a n d H , unstable modes can exist only when the coefficient of ci in (6.2) vanishes. This can be achieved in different ways under different situations. For clarity, let us consider some special cases first.
A. STABILITY CONDITIONS FOR PUREBAROTROPIC FLOW The stability of a barotropic zonal current characterized by U , = 0, under the influence of differential rotation or variable depth in rotating fluid as represented by the Rossby parameter p, was first studied by Kuo (1949). For such barotropic flows the most unstable disturbance corresponds to pure horizontal and nondivergent flow, that is, the flows whose 4 are independent of z. Thus (6.1) and (6.2) reduce to
where Qoy = /3 - Uyyis the gradient of the basic state absolute vorticity. The relation (6.3) shows that for unstable disturbances (c, # 0) to exist in the barotropic system, the gradient of the absolute vorticity, Qoy , must
278
H . L. Kuo
change sign within the range (yl, y2). That is to say, for inviscid unstable barotropic disturbances to exist, the basic velocity distribution must be such that U,, is able to over-balance /3 to make /3- U,, change its sign. This is KUO’S(1949)extension of Rayleigh’s theorem of instability to include the influence of /3 on the stability of the zonal flow. Thus, a positive (negative) /3 has a stabilizing (destabilizing) influence on the wave motions in the westerlies and a destabilizing (stabilizing) influence on wave motions in the easterlies. This last mentioned effect will be demonstrated in detail in Section VII. Equation (6.4) also shows that ( U - c,) and Qoymust be positively correlated within (yl,y 2 )for free disturbance to exist, whereas (6.3) together with (6.4) show that, for unstable disturbances to exist, the product UQoy must be positive at least in part of (yl,y 2 )in addition to the change of sign of Qoy . Further, this relation requires I 4 1 to be large in the region of positive UQoyand small in the region of negative UQoy. This condition has often been stated incorrectly to imply that the existence of unstable disturbances requires UQoyto be positive at every point. When Qoy= /3 - u” changes its sign across the critical point yk within the range (yl,yz) ,then a regular neutral mode with c = U k = U(yk), ct = ak > 0, J!,I = +k(y)exists provided the mean value P of the function
exceeds a certain minimum value p c . In other words, there must be a critical neutral solution for the unstable solution to exist. In case there exists a second critical point y k 2 with u k 2 # uk1)pk > Pc , then another regular neutral disturbance exists and is given by c2 = uk, = U(yk2)’a = a k 2 , 4 = 4 k 2 . Under these circumstances there exists a band of unstable disturbances with akl and ctk2 as their wave number limits, with c, lying between uk1 and uk2. In case there is only one critical point, or all the critical points have the same uk, such as the two critical points of a symmetric jet-type profile, but Rossby-Haurwitz type retrograding neutral solutions are in existence, then the lower limit of the wave number of the unstable modes corresponds to c = U,,, and is given by (see Kuo, 1949)
where
Quasigeostrophic Flows and Instability Theory
279
When no such regular retrograding neutral modes exist, then a,, is equal to zero, so that the longest wavelength of the unstable modes is infinity. Thus for symmetric channel flow the following two conditions appear to be sufficient for the existence of unstable disturbance :
Qoy= /3 - U" changes sign in ( y l ,y z ) , P = mean value of Pk> P, .
(6.7a) (6.7b)
For antisymmetric profiles extending to infinity, such as the U = tanh by profile, the condition (6.7a) alone appears to be sufficient for the existence of unstable disturbances. We pointed out that the stability characteristics of an easterly current differ from that of the corresponding westerly current on account of the influence of p. I n fact, the stability characteristics of an easterly current under the influence of /3 are exactly the same as that of the westerly current under the influence of -/3, as can readily be seen by a change of the direction of the x-coordinate. Hence we shall use a negative /3 to characterize the flow properties of an easterly current.
B. THESEMICIRCLE THEOREM FOR THREE-DIMENSIONAL BAROCLINIC DISTURBANCES
It has been shown by Howard (1961) that for the two-dimensional inviscid disturbances in a nonratating system, the upper bounds of c, and ci are given by the half of the difference between the maximum value (U,) and the minimum value (Urn)of the basic current. A similar result can also be derived for the three-dimensional quasigeostrophic and baroclinic disturbances under the influence of p. For this purpose we shall at first nondimensionalize the potential vorticity equation (5.49 in pressure coordinate by setting u'= u,
u, a=kL,
c' =
u, c, y' = LT, p = (fo L / J S)5 , / 3 = p f L 2 / U M , $'=LU,$,
where the primed quantities are dimensional. For simplicity, we assume that the stability factor S is constant. In terms of these dimensionless variables, (5.4) becomes
Th e boundary conditions for $ are
H . L. Kuo
280
Here we limit ourselves to the unstable disturbances only, so that we have ci# 0 and hence c # U. We transform (6.8) further by setting
$ = ( U - c)F(r, 5).
(6.10)
Equation (7.9) and the boundary conditions (5.5a,b) then become d
( U - c)' F=O F,=O
"1
-
d5
- a'(
U - c)'F
+p( U - c)F = 0. (6.11)
at 7 l = % , % , at 5=51,52.
(6.11a) (6.1 lb)
For the unstable disturbances both c and F are complex. Thus on multiplying (6.11) by the conjugate complex F" of F and integrating over 7 and 5 from T~ to q 2 and from to C2, respectively, and making use of the conditions (6.11a,b) we then find
J"JU- c)'Q dA = /3
J" ( U -
c)l
A
+
FI ' d A ,
(6.12)
+
where Q ='a I F I ' 1 F,, I ' 1 F, 1 ', which is positive definite, A = (yz -q1)(12 - 5,). This equation can be separated into its real part and its imaginary part, which are given by
J [( U - c,)'
-
ci']31Q d A = p
J(U
- c,)
I F I ' dA,
(6.12a)
1
ci[J(U- cr)Q d A - P- I FI d A ] = 0. (6.12b) 2 Since ci differs from zero for the unstable modes under consideration, the quantity in the bracket must vanish. On substituting (6.12b) in (6.12a) we find
s,
[U'-(C,' + c i 2 ) ] Q d A = p J " U I F I ' d A .
(6.13)
A
Now we have
( U - Urn)(U- U M ) = U 2 - ( U m + U M ) U + U m U M 1 0 ,
10- UI I A U ,
(6.14a) (6.14b)
where UM and U , are the maximum and the minimum of U and (Urn UM)/2, AU = ( U , - Urn)/2.Hence (6.13) yields
+
(c,-
o=
Quasigeostrophic Flows and Instability Theory
281
where P2 and Q are the area averages of I FI and Q. Thus c, - 0 and ci are bounded by the square root of the right-hand side and hence are bounded by the absolute value of ( U , - Urn+pF2/Q]/2.Equation (6.15) becomes identical with the relation derived by Pedlosky (1964a) when F2/Qis taken as equal to (k2 kI2)- l. Another limit of c,2 has been obtained by Miles (1964b) for the baroclinic system, viz.
+
+
4ci2< { ( p / ~ s ) ~ U " ~K( T a ~ / Y-Tl } )m a x .
(6.16)
VII. Stability Characteristics of Barotropic Zonal Currents and Rossby Parameter For the nondivergent perturbations in a barotropic zonal current (5.4) reduces to
T he appropriate boundary conditions for
$=O
t,h
are
at y1,yz.
(7.la)
For convenience we nondimensionalize this system by setting
Y"
u=- u m a x
Y'd' $=-
U"
P ,
,
c=-,
C"
umax
b = - ,Pd2
(7 4
~=kd, umax duma, where the starred quantities are dimensional, Urn,, is the maximum value of the basic current and d is a measure of the horizontal scale of U. Expressed in terms of these dimensionless variables, (7.1) becomes
We take b positive for westerly current and negative for easterly current, corresponding to a positive and a negative Urn,, , respectively.
A. STABILITY OF THE SINUS PROFILE U = (1
+ cos y ) / 2
T he solutions corresponding to this sinusoidal current have been discussed by Kuo (1949). It is readily seen that, for this basic current the condition for instability is simply b < 1/2. When this condition is satisfied,
H . L. Kuo
282
the flow is unstable and the upper transition from stability to instability is given by the neutral solution = cos y/2 = U1l2,
= J3/2,
=
= 112 - b.
(7.4) For 0 5 b 5 1/2, the lower transition from the unstable waves to the stable modified Rossby-Haurwitz type waves is given by the neutral solution $k
ffk
= cos2"y/2) =
$0
ao2=
UA,
u k
c, = Umin(=O),
[4h = 1
1 - ha,
ck
+ (9 - 16b)1'2].
(7.5)
This neutral solution does not exist for the easterly current ( b < 0). Besides these symmetric neutral solutions, there exists an antisymmetric neutral solution given by y, aka= 0, c k 2 = 1/2 - b. (7.6) Since this velocity profile is symmetric about y = 0 and since the boundaries are symmetrically located, the most unstable solution of (7.3) is also symmetric about y = 0. Therefore the appropriate boundary conditions are $k2
= sin
$'(O) = 0, $(r)= 0. (7.7) The solutions for this profile have been obtained by an iterative method, both for positive and for negative b values, which is to integrate (7.3) for a given a and a given c numerically, and then to correct c by reducing the discrepancy at the boundary to zero. The dimensionless phase velocity c, and growth rate S = a c i so obtained are illustrated in Figs. 6a and 6b,
umaxTh umax~rlx I
l
l
,
,
0.6
c,
Cr/
00
812
2
,
( 0 )
4
6
8
10
I
1214
L/ D
0.8
a
FIG.6 . Dimensionless eigenvalues for barotropic disturbances in a sinus profile. (a) c,; (b) growth rate a t i .
Quasigeostrophic Flows and Instability Theory
283
respectively. Of particular interest is the result that the easterly current (i.e., b < 0 ) is made more unstable by the /3 effect within the range 0 > b 2 -0.25 (approximately), while the westerly current is made more stable. In addition, the most unstable wave length is slightly longer in the Easterlies than in the Westerlies under the influence of a positive b. Numerical results have also been obtained by Yanai and Nitta (1968, 1969), who also found that when the horizontal shear exceeds certain critical values, the neutral, Rossby-Haurwitz type waves change into singular wave solutions of the continuum type, with some discontinuities in $’, while regular neutral solutions cease to exist. We point out that for this velocity profile the actual half width is d’ = Td. Thus, for d’= lo6 m, Urn,, = 10 m s-l, fl= 2.29 x 10-l1 m - l s-l, b is of the order of 0.23. For this case, the most unstable disturbance in the easterly current corresponds to a = 0.5. Hence the most unstable wave length is L = k d = 4000 km. Before we discuss the other stability problems, it appears worthwhile to mention briefly the destabilization of the easterly sinus profile U=-siny, O < ~ < T (7.8) by the influence of /3. Since the inflection points of this profile are either on the boundaries or outside the range of y, this velocity distribution is stable when /3 is absent. However, with b(=fld2/Um,,) < 1, the absolute vorticity gradient QOy = b - sin y changes its sign in (0, T) and hence the flow is made unstable by b. The same result applies also to a westerly current when a negative b is present.
B. STABILITY OF THE BICKLEY JET For the jet with the profile U = U+/Umax=sech2y,
-co
(7.9)
(7.3) becomes (7.10) where S = sech y, b = i3d2/Um,,. Here it is assumed that the fluid extends to f00 and the boundary conditions are
+=O
at y = + m .
(7.11)
However, for the symmetric modes we only need to calculate II, for the half space 0 < y = co and the conditions for 4 are $’(O)
= 0,
*(
00)= 0.
(7.12)
H . L. Kuo
284
T h e condition for instability for this profile is
-2 5 b 5 2/3.
(7.13)
A symmetrical critical solution for this problem has been found by Lipps (1962) and it is given by = 2[1
& (1 - 3b/2)ll2],
b = (a,2,/6)(4 - a&).
$,
c, = as2/6,
= sech2y.
(7.14a) (7.14b)
+
T h e solution corresponding to the sign in (7.15a) represents the upper transition curve for the unstable disturbances in either a westerly ( b > 0) or an easterly ( b < 0) current, while the solution with the - sign in (7.15b) exists only for b > 0. This latter solution represents the lower transition for stability to instability in the westerlies. Another symmetric solution, given by
c=l, J! C ,
(9
&“=[3-
+4 ~ 2 1 ,
= (sech ~ ) “ ‘ / ~ ( t a n y)2-a2!3 h
(7.15)
(7.16)
exists in the Easterlies ( b < 0). This solution represents a submode of the stable modified Rossby wave and is not a stability boundary for small -b. However, for b < -1, this solution does closely represent the lower transition from instability to stability. Another neutral solution is given by
2 = 2[1 - (3b/2)I1I2- 1,
+ 3)/6,
c = (2
$ = sechy . tanh y. (7.17)
This is an antisymmetric solution and it exists in the b < 1/2 region but it does not represent transition from stability to instability. T h e solutions of the system (7.11) and (7.12) have been obtained by an iterative method; the dimensionless eigenvalues c, and 6 = aci so obtained are represented in Figs. 7a and 7b. The numbers attached to the curves in Fig. 7a are the values of b. It is seen that the phase velocity of the unstable disturbance is always within the range of the basic current and for positive b its value decreases as the wavelength increases, while for negative b, c, decreases to a minimum and then increases with the wavelength. Here again, the most interesting results are the fact that the easterly jet is made more unstable by the b influence within the range b 5 0.84, while the westerly jet is made more stable by b. In contrast to the case of the sinus profile, here the most unstable wavelength is reduced by the b effect both for the Westerlies and for the Easterlies, and this reduction is very promi-
Quasigeostrophic Flows and Instability Theory
1.5
2.0
285
2.5
b FIG.7.
Eigenvalues for U = sech2y. (a) c,; (b)
OLC,
nent. Notice that the growth rates in Fig. 7b are of the same order of magnitude as that in Fig. 6b. For an easterly current with a halfwidth = lo6m = 1.76 d, U,, = 10 m s-l, we find b = 0.74. The most unstable disturbance for this b corresponds to cc =: 2n-d/L = 1.28, therefore the most unstable wavelength is L = 2790 km.
H. L. Kuo
286
C. DISTURBANCES IN A HYPERBOLIC-TANGENT ZONALWIND PROFILE The intertropical convergence zone (ITCZ) is a region in the tropics roughly parallel to the equator, containing deep, intense cumulus convection. The convectively active part of the zone is usually very narrow, of the order of 100 km. Its latitudinal position varies from 3" or so to 20" or more, with the average of about 10" throughout the year. But what is most astonishing is the fact that it is almost never found at the equator. Satellite pictures of the ITCZ usually reveal it not as a truly parallel zone, but often in a disturbed state with wavelike disturbances superposed on it. Wind observations also show the regular appearances of wave disturbances and vortices along the ITCZ, and many of these disturbances develop into hurricanes and typhoons. The mean wind in the intertropical convergence zone is usually characterized by a nearly uniform current on one side and a different uniform current on the other side, with a rapid change of direction across a relatively narrow zone of transition. Such zonal velocity distributions are illustrated in Fig. 8a by the mean zonal wind profiles observed over the Pacific and the Atlantic, given by Yanai (1961) and by Riehl(1969), respectively. Since the ITCZ is a zone of active convection and upward motion, such a zonal wind distribution can surely be expected theoretically because the convergence toward the ITCZ will definitely create such a wind system under the influence of the Coriolis deflection, provided the ITCZ is not located at the equator. These observed mean wind distributions in the intertropical convergence zone can be represented analytically by a hyperbolic tangent profile, viz.
where U* is the dimensional mean zonal current, 0 is its mean value, and 2U, is the total shear. It can be seen that, with d - 150 km, the profile (7.18) can fit the observed Atlantic profile in Fig. 8 quite well. The dimensionless gradient of the absolute vorticity of this wind distribution is given by Qo,=b-
U,,=b-2~(1-~'),
(7.18a)
where x = tanh 77. Thus, for this mean wind (7.3) becomes
4m-
b - 2 4 1 - 9) [.2+
Z+c
1*=o.
(7.19)
287
Quastgeostrophic Flows and Instability Theory -20
-30 0 '
- I Om/s
' I
I
IOOmb
300 500 700
050
20'N.
I
U 8SOmb
Majuro
EQ
I
1
(b)
A*€
he+
Torowo
I
lo"
Ae+
20'
. I Om/s
0
F I G . 8. Observed mean zonal wind profiles and structure of perturbations in the tropics. (a) Vertical distributions; (b) horizontal distributions over the Pacific; (c) mean velocity profile over the Atlantic.
H . L. Kuo
288
The boundary conditions for a,b is
+-to
as
(7.19a)
7 p - f ~ .
The velocity profile (7.18) and the vorticity gradient (7.18a) are represented in Fig. 9. From (7.18a) we find that the necessary condition for instability is
I bl < b, = 4 x 3 -3"
(7.20)
= 0.7698.
From Fig. 9 we see that, when (7.20) is satisfied, QOnchanges its sign at two values of 7,viz., 7 = r)cl and 7 = vC2.Thus two neutral solutions exist when 161 < b, . The phase speeds of these neutral disturbances are equal to the current velocities at vCland 7 c 2 ,which are given by the roots of the following cubic equation 2zC3- 2xc
+ b = 0.
Hence zCjis given by
+
cos[(% 2Trj)/3],
z,j = ($)1/'
j
= 1, 2,
3;
% = ~ 0 ~ - ~ ( - 2 2 7 ~ ' ~ b / 4 )7, ~ / 2 5 % < 7 ~ .
(7.21)
For b = 0 we have zC1= - 1, zc2= 0, zc3= 1; hence all three values are allowed. However, for b > 0 the magnitude of zC1becomes larger than 1 and hence is outside the velocity range and must be excluded, while the other two roots zCzand zcg lie within the range 0 5 zC25 3 -11' and zC35 1 and hence each one gives rise to a neutral solution. These 3-lI2 solutions are given by (7.22).
<
I
U
FIG.9(a). page.
P
Caption on facing
Quasigeostrophic Flows and Instability Theory
289
a
P
-180
I80
0
FIG.9. (continued) Eigenvaluesfor U = -tanhy. (a) c,; (b) ac,; ( c ) total streamline pattern forb = 0.3, a = 0.5.
#,( I7) = ( 1 - 2 ) ( l + = c j ) / 2 ai=(1-2zj)1/2,
cI .- - - z
'(1 + z ) ( l - z c i ) / z 7
c 1. 7
j = 2 , 3.
(7.22)
Lipps (1962) obtained this solution for b > 0, with the sign of c, reversed. It can be shown that the solution (#3, a 3 , c3) represents the short-wave cutoff of the amplifying waves, while the solution ( # 2 , m Z , cz) represents a transition from amplifying to neutral perturbations only for b > 0.17. T h e problem represented by (7.19) and (7.19a) for b = 0, which corresponds to the shear instability problem in a nonrotating system, has been
290
H. L. Kuo
investigated by Michalke (1964), Betchov and Criminale (1967), and many other hydrodynamicists, and the eigenvalues for the rotating systems with b > 0 have been calculated by Lipps (1962) for the nearly neutral solutions by a perturbation method. In this work the exact solutions of the equation system (7.19) and (7.19a) have been obtained by an iterative Runge-Kutta method, and the values of c, and the dimensionless growth rate 6 = acI so obtained are represented in Figs. 9a and b, respectively, as functions of a and b for the whole unstable range b 5 b, . From Fig. 9a we see that the dimensionless relative phase velocity c, is negative for all a and b, which means that the waves propagate toward west relative to the mean current 0.Further, the absolute value of c, increases with b for a fixed a and decreases with a for a fixed b in such a way that the value of (-c,) is less than 1 in the whole (b, a) domain of the unstable disturbances, but has a maximum value of about 0.9 near the lower neutral solution (Icl2, d 2 , c2) at a = 0.35. For b = 0, the maximum ci (= 1) occurs at a = 0, but it shifts toward a larger a as b increases. For a fixed a, ci decreases with increasing b, showing the stabilizing influence of b. One interesting and unexpected result of this investigation is that it showed that for b 5 0.175, the whole region 0 5 a 5 a s is unstable and hence the lower neutral solution is not a transition from stability to instability, but an isolated solution. This result is in agreement with the conclusion reached by Howard and Drazin (1964) that the disturbances close to a = 0 are unstable for sufficiently small b. The dimensionless growth rate 6 = aci is shown in Fig. 9b as a function of b and a. It is seen that 6 has a maximum at a = a, and a, increases gradually with increasing b, from a, = 0.4449 at b = 0 to a, = 0.817 at b = 0.7698, while the maximum value of 6 decreases from 0.188 to zero. Thus increasing b causes the most favored disturbance to shift to a larger a. However, if the increase in b is due to an increase in the value of the halfwidth d, then the wave length of the most favored disturbance is increased. From the observed wind distribution in the tropics we find that U , is of the order of 7 m s - l and d is of the order of 250 km, corresponding to b e 0.2. The most unstable wave length is about 3000 km and the e-folding time of the most unstable disturbance is a little over two days. The westward relative phase velocities (c," = 0 U , c,) of the unstable disturbances are mostly under 2.0 m s - l , in agreement with the available observations. The amplitude and the phase of the stream function of the most unstable disturbance with b = 0.3, a = 0.5 are represented in Fig. 9c. It is seen that for b > 0 the maximum amplitude occurs on the equator side of the shear zone. The average net momentum transfer function W , defined by
+
Qm&eostrophic Flows and Instability Theory
29 1
has also been calculated. It is found that this transfer reduces the maximum shear in the shear zone and hence it intensifies the perturbation.
VIII. Pure Baroclinic Disturbances Even though in reality the basic current U depends both on y and z, for the investigation of the baroclinic influence of the system we shall consider U as independent of y so that all the parameters in the perturbation equation (5.4) and the boundary condition (5.5a) are functions of x only. Then the function t,b can be taken as given by the product of a vertical function ~ ( zand ) a harmonic function sin kly, so that (5.4) and (5.5a) become
+
( U - ~)[(d’v/dz’)- a2v] bv = 0, ( u - c)[(dv/dx) where q~ is the amplitude of p’p;lI2, a2= (N2/f2)(K2 +k12)
+ r2,
+ Fvl=
r = a,/2
u z
v,
(8.1)
(8.4
and
+
b = (N2/f2))B 2 r U 2 - U,, , (8.la)
N 2 =gs,. When the influence of the Ekman-layer friction is taken into consideration, the condition (8.2) is replaced by
( U - c)yZ= Uz[l F ib”]p
at
xb
, xt,
(827
where a’ = k2
+ k12.
For this system the integral relation (7.3) reduces to
This equation shows that, for b > 0 ( < O ) , instability requires the quantity U Z v 2I /U - cI to have a larger (smaller) value at the bottom than at the top, while for b = 0 this quantity must be the same at the two boundaries. Since the overall baroclinic effect in the troposphere can be represented by the mean vertical shear in this layer, for the investigation of the baroclinic instability we may take U , as constant so that U becomes a linear function of z and b independent of z , viz.,
U = Uo
+ U,z,
b =p N ” f - 2
+ 2 r U , = constant,
where U , is the basic current at the bottom. This model corresponds to a basic state whose entropy S o = c, log Oo increases linearly with x and toward the equator with gradient f UJg.
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With this distribution of the basic state variables it is convenient to introduce a dimensionless independent variable 5, a dimensionless relative phase velocity A, and a parameter Y, defined by
5 = U( U - c ) / U ,= p[(X/h)+ A], p=ah,
U"=hU,,
Y
h=H/2,
= bh2/2pU",
(8.4)
A=(U,-C)/U",
where H is the total depth. When H is infinitely large, we take h as an arbitrary length scale unrelated to H . Denoting the differentiation with 5 by a prime we then find that (8.1) and (8.2) become [(V" - p)
+2YV
~ [ ~ ' + ( r / u ) ~ ] - y = Oat [ = 5 , = p A
(8.la)
= 0,
and
5=ct=p(2+A). (8.2b)
When the Boussinesq approximation is used, 1 N/3h2 2ah f U"'
r / a term
is given by
a2=k2+kI2
y=---
and the become
Y
disappears in (8.2a), so that the boundary conditions 5V'-F=O
at
5=L,5t.
(8.2aQ)
Thus, in the Boussinesq approximation r becomes the sole parameter which appears explicitly. On the other hand, when the r / u term is retained in (8.2a), the eigenvalues then depend upon this quantity explicitly. Since Flu is large for the long waves and since its upper limit is 1/2, it has an appreciable influence on the eigenvalue of the long waves. In addition, the contribution of 2 r U , to b and hence to Y is also significant. However, the basic mechanism of the baroclinic system can be demonstrated adequately by the Boussinesq approximation with l7 = 0. This version of the baroclinic model was first formulated by Charney (1947), who has also obtained the formal solutions of the system (8.la) and (8.2a) and discussed the general characters of the solutions but did not obtain the unstable eigenvalues. Later on, the author (Kuo, 1952) proved that the phase velocities of all possible continuous neutral disturbances are less than U,, and that all disturbances with critical layers (cr = U at some level) are unstable, and also obtained the eigenvalues of the disturbances in the most unstable range 0 S Y5 1. The fact that the system is unstable for disturbances of all wavelengths except those corresponding to a positive integer r has been proved by Burger (1962), while Green (1960), Hiroto (1968), and Garcia and Norscini (1970) have obtained the eigenvalues for a finite depth by different numerical calculations. The analytic properties of (8.la) have also been analyzed in more detail by Miles (1964a,b). In
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293
addition, the baroclinic influence has also been investigated by the use of simpler versions of this model, such as Eady's (1949) constant f and Boussinesq model (b = 0) and the two layer model. Since the results from these simple models are more easily grasped than those of the model represented by (8.1) and (8.2), we shall present these results first.
A. THECONSTANT f MODELAND BOUSSINESQ APPROXIMATION Whenfis taken as constant (/3 = 0) and u2(= -d log pol&) is set to zero, we have Y = I'= 0, so that (8.la) reduces to T" - ql = 0.
(8.5) This model is equivalent to Eady's model of the baroclinic problem, here we expressed it in terms of q,while Eady (1949) expressed it in terms of the amplitude w of wl, which, for this model, satisfies the following equation
w" - (215) w'
+ w = 0.
(8.5a)
Note that (8.5) has no singularity, while (8.5a) requires w' = 0 at 5 = 0. The solution of (8.5) can be written as q~ = Aer
+ Be-'.
(8.6) On shifting the origin of z to z = h and applying the boundary conditions (8.2a) we obtain the two following homogeneous equations for A and B: {,(Ae'j - Be- 'j ) - Ae'j + Be-'j, j = 1, 2, (8.7)
+
where C1 = p ( A - 1/2), C2 = p ( A 1/2), p = ah, A = (O- c)/U*,h = H/2, U* =:hU,, and 0 is the value of U at the mid-level z = h. The existence of nontrivial solutions demands the vanishing of the determinant of the coefficients of A and B in (8.7), which gives h2 = K ( p )f1
+p2- 2p coth 2p,
(8-8) where h = u( 0 - c ) / U z .This equation shows that the stability is defined by (8.8a) and the critical value of p is p, = 1.1997. Since p is proportional to the wavenumber k, this criterion shows that short waves are stable while long waves are unstable. Further, since p is proportional to N/f, for the same stability factor N , the critical wavelength increases toward lower latitude. Thus, for a normal stability factor N = 1.2 x 10-2s-1 and H = lo4m, we find that the critical wavelength L, is 3140 km at 45 deg. lat. and I,,+ 12,000 km at 10 deg. lat. Thus only very long waves can be baroclinically unstable at low latitudes.
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Since the right-hand side of (8.8) is always real, this equation also implies
A,&
= 0,
(8.8b)
where A, and A, are the real part and the imaginary part of A, respectively. According to this relation, we can divide the disturbances into three different regimes, viz., (i) the stable or neutral short wave regime with A, # 0, A, = 0; (ii) T he transitional regime with A? = A, = 0; (iii) T he unstable or self-excited long wave regime with A, = 0, A, # 0. (i) T he stable regime-these disturbances are characterized by c = c, = 0 fK ( p ) ,
c, = 0.
(8.8c)
Hence, for each wavelength L < L, , there always exist two neutral disturbances which move in opposite directions relative to the mean current velocity 0. For example, for p = 2pc we have c, - 0 = f0.5833 U". (ii) T he transitional disturbance-coalescence of moving disturbances and linear growth. As L increases to L, ,the value of K decreases and the difference between the speed of these two waves diminishes to zero, hence these waves coalesce into a single unit. When such a wave is created by given initial values of and v t ,it will grow linearly with time, as can be shown by the following consideration. Let us assume that q is composed of two neutral waves with the same wavenumber k but slightly different phase velocities c and c 6:
+
= A cos k ( x - ct)
+ B cos k [ -~ + 8)t], (C
where A and B are given by the initial values of
~ ( 0= ) pocos kx,
and p, viz.,
p,(O) = kpl sin kt.
Solving for A and B and setting them in p=yo
(8.9)
we then find
cos 5- (Fl - CPO) {cos f[1 - cos k8t] - sin 5 sin kSt), (8.9a) S
where 5 = k(x - ct). Thus, when S is finite, the two waves move away from each other. On the other hand, when S approaches zero, as for the transitional disturbance L = L, , the above expression reduces to q~ = y o cos 8
+ kt(y, - cpo)sin 5.
(8.9b)
Thus, a sin &wave is created through the coalescence of the two waves and the amplitude of this wave grows linearly with time. (iii) T h e amplifying disturbances with L > L, . For L > L, , (8.8b) gives
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c,= u, Cl # 0. (8.10a) Thus, all these disturbances move with the mean current velocity 0and their amplitudes grow exponentially with t . Th e disturbance that grows fastest is given by p = pm= 0.8031 = ha,,,,
ci
= 0.386U*,
ym = k,,,ci
(8.10b)
= 0.31 U , f / N .
Th e variations of the quantities c, , ci and y = kc, with p are illustrated in Fig. 10. I n addition, Eady (1949) has also shown that, when the system is composed of three layers with different stability factor N , it will be more (less) unstable if N is larger (smaller) in the middle layer. 0.0 0.6 = 0.4 -
--0.6 -0.4 --0.2
0 0.5
FIG.10. Eigenvalues as functions of wavelength for the baroclinic disturbances in Eady’s model.
1. The Structure of the Disturbances T he amplitudes and phases of the functions p, w , and q for the unstable disturbance are symmetric about z = h, as illustrated in Figs. 1l a and b. It is seen that both p and w have their maxima at the midlevel z = h. Further, the troughs and the ridges of the pressure- and v-fields incline westward and upward, while the axis of the temperature field inclines eastward with height. Th e low pressure region is warm at the surface but is cold at the top. We mention that the integral requirement (7.3) is satisfied by these unstable modes on account of the symmetry of p. For the short stable wave with c, = 0+ K112, has its maximum at the top, while for the other wave moving with c, = U - P I 2 , p has its maximum at the bottom, as is illustrated in Fig. 11b for the disturbance p = 2pc. Thus for these short stable disturbances the p distribution is asymmetric. On the other hand, w is nearly symmetric about z = h for these disturbances and has a zero gradient at the critical level where c, = U .
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0.251
A Amplitude
Phase
FIG.11. (a) Variations of the amplitudes of @ and w and phases of T' and p or @ of the most unstable disturbance with height. (b) Variations of CP and w of the neutral disturbance with height.
2. Infruence of Surface Friction on the Baroclinic Waves As has been mentioned before, the influence of surface friction on the wave disturbances is through the appearance of the boundary layer suction w,, in the boundary condition (5.5b"). When this influence is taken into consideration in the present problem, the boundary condition (8.2a") is replaced by 5q' - (1 'F ib*)q = 0 at 5 = C b , Ct , (8.2a"") where
T he equation which corresponds to (8.8) is pA
= ib
coth 2p
[b2(1 - coth2 2p) - K ( p ) ] 1 ' 2 .
(8.8")
From this relation we find that instability occurs only when U , exceeds a critical value, given by
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297
Under the normal atmospheric conditions in middle latitudes we have gs, = s-', f = lo-* s-l, h = 5000 m. Using an eddy viscosity coefficient v = 5 m2 s - l and p = 0.83, which corresponds to the most unstable mode for inviscid flow, we find U,, = 0.83 x 10 - 3 s -l. Thus the influence of the surface friction is quite appreciable for the baroclinic disturbances in the atmosphere. We point out that, with friction, both of the two c-solutions yield exponentially damped solutions when either U , is below U,, or when ci is larger than the stability limit a,, while in the unstable region one mode grows while the other is damped.
B. APPROXIMATE SOLUTIONS OF EQ. (8.1) FOR A NONZERO b When r differs from zero, the solution of (%la) becomes much more complicated than that given by (8.6). In order to gain some insight on the influence of the r-term of this equation on the baroclinic solution, we shall at first obtain two approximate solutions by two different methods of approximations.
1. The Two Levels Approximation The solution of (8.lj can be obtained numerically by converting it into a finite difference equation. The simplest and yet quite appropriate and revealing solution is given by a two interior point approximation, which is in common usage in meteorological research. For convenience, we use the pressure as the vertical coordinate and use the linearized versions of the vorticity equation (3.12) and the heat equation (3.10d) instead of the potential vorticity equation (8.1). For the wave perturbations under consideration, these two equations may be written as (8.11)
+
(8.12) cp;t W X - U*& = --sw', where y ( = f l ) is the geopotential of the isobaric surface, u2 = k2 kI2, w f = dp/dt is a measure of the vertical velocity, S = --pol a log O,/ap represents the static stability and the subscripts denote partial differentiations. For simplicity, we shall take S as constant. In this two-level approximation y p is represented at the midlevel 3 while y is represented at the levels 2 and 4 halfway between p , and p , and p , and p , =p , , where p , and p b are the pressures at the top and bottom of the layer under consideration, as indicated in Fig. 11. On setting y',
+
w f = (p, w ) e i k ( x - c t )
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298
in (8.11) and (8.12) and applying these equations at the levels 2, 4, and 3 respectively, we then find
where U* = &ApUp,Ap = (p5-p1)/2,and on the right side of (8.1la) the sign is for j = 2 and the - sign is for j = 4.For convenience we introduce
+
Fj = a(v2
+
v* = Hv4 - v2)
v4)
Further, we set 4= p),, 0= 0,. On taking the sum and the difference of the two equations of (8.11a) we then obtain
(0- c +?*)a + u*p= 0,
where P*
=
PI.".
(8.14)
Eliminating 0 , between (8.12) and (8.15) we obtain
+ p2)(0 -
[(a2
C)
- 839"
+ (a2- p2)U*q = 0,
(8.16)
where
On equating the ratio v*/q given by (8.14) to that given by (8.16) we then obtain the following expression for c:
u-
'*
(2a2+ p 2 ) 1 [P2p4 - 41P2a4(p4 - a4)]1'2,(8.17) 2a2(a2 pz) 2(a2 Thus the marginal curve is given by c=
+
+
(8.18) This result was first obtained by Thompson (1953), while a similar result, based on a physical model composed of two layers of homogeneous incompressible fluid, was obtained by Phillips (1951). It is seen that when a2 > p 2 , or when U* is below the critical value
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(8.17) gives two differentreal values of c, whereas when u
Uc*, the two solutions for c are complex conjugate with the same c, . The solution with ci > 0 yields exponentially growing mode while its conjugate is exponentially damped. The absolute minimum value of U,* occurs at u2 = p2/2 and is given by (8.18b) We mention that in this case the marginal disturbance is also the result of the coalescence of two neutral disturbances moving with two different velocities and hence it tends to grow linearly with t , as has been demonstrated by (8.9b). Urnin = PIP2.
2. Truncated Power Series Solution A different approximate solution, also based on (8.11) and (8.12)' has been obtained by the author (Kuo, 1953) by a different method. In this method, we make use of the w-equation obtained by eliminating yt from (8.11) and (8.12), viz.,
f" 2
wpp
- sw
(-
B =f a2
),
-2upv
where w = y z / f . It is assumed further that the vertical variation of be represented by
=p(p-Pb)M,
(8.19) w
can
(8.20)
which satisfies the requirement w = 0 at p = 0 and p = p , . Here M is taken as a slowly varying function of p and is found by substituting this expression in (8.19), which gives
(8.20a) where 5' =PIP,, h = ~ ~ S ( A p )f ~2 = 1 2p - 2 . Substituting this in (8.20) and using this w in (8.12) and rearranging we then obtain the following equation for v:
where
The solution of (8.21) is given by
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300
where 5 = 5' - 1/2 = p / p o - 1/2, and v o is the value of v at the midlevel 5' = 0. Expanding this solution in powers of 5 we then obtain -V = I + VO
[^-
A-1 5 A+lS,
A l+hA+l
(p' - 1) + 26,
where /3'=/3/a2Um and
61 --6--=---
u,
If + -?;
*
*
, (8.22a)
A
0-c
urn urn
1
+2'.
The eigenvalue c is determined by substituting v(=yz/f) in (8.11) and integrating from 5 = -1/2 to 5 = 1/2 and making use of the conditions w = 0 at the two ends. When v is truncated at the 5 term we then find c
-u---
r-
(T=-
2A+1 P * o , 2(A 1) 2
+
0 1 A + l [p2-1)+-
B2
1.
(8.2213)
a*U2m
These results are similar to those in (8.17). On the other hand, when the t2term is included, it is found that the short waves are also unstable, just as in the exact solution of @.la). Further, this solution also reveals that the maximum amplitude of the short waves ( L 5 2000 km in middle latitude) occurs at low level, while that of the long waves occurs at the top or the tropopause level, in agreement with the observed distributions. This feature of the amplitude distribution has also been revealed by more recent calculations from the continuous baroclinic wave equation (8.1) (Hiroto, 1968; Simons, 1969). C. THEGENERAL BAROCLINIC SYSTEM
Equation (8.la) is a confluent hypergeometric equation and it can be brought into the standard form by the transformations d 5 ) = rle - "2@(rl), rl = 25. (8.23) Substituting in (8.la) and (8.2a) and using the prime to denote differentiation with 7 we then find v@" (2 - ?$D' - (1 - r)@ = 0, (8.24) q 2 [ W - y@] = 0 at = rl,, = 2pA and = qt = 2p(2 A), (8.25) where y = i(1- Flu) and p and A are as defined in (8.4). The general solution of (8.24) is
+
+
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301
Q = A@,+BQ,,
(8.26)
where A and B are two arbitrary constants and Qland Q 2 are two linearly independent solutions (8.24). When r is not a positive integer, it is convenient to choose these solutions as
1--r =1+-7+ 1!2
(l-~)(Z-r) 212.3 q2+
*
-,
(8.26a)
(8.26b) where (x)m= x(x
+ 1) - - (x + m-
1)
and +(x) is the logarithmic derivative of the gamma function r(x). For large q, these solutions are given by the following asymptotic expressions :
Q2=r(l-r)qr-l
Y( 1 - r)( 1 - r)(2 - r )
q2
+
***I.
(8.2613")
Thus Qlbehaves as q - r - l e n while Q2 behaves as q r - l for large q. On the other hand, when Y is a positive integer, Ql terminates and reduces to the Laguerre polynomial of (Y - 1)th degree, while Q., as given by (8.26b) ceases to be valid. The proper form of Q2 for this case can either be obtained by taking the limit of r(l - r)Q2 or by using Q, and integrating (%la) by the Wronskian method. It can readily be shown that in this case Q 2 is of the form Q 2= q - * - l e n
+al log q + *
(8.2613"")
and hence it behaves as q-*-'en for large 7. Thus, when r approaches an integer, Qland Q., change their behaviors. For negative real -q(= -q), log q must be taken as given by the principal value for a continuous solution, viz., log -q = loglqJ - 772..
(8.27)
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Hence Q 2 is complex for all negative 7, which can only occur when c is complex. This result shows that for a real c, vb must either be equal to or greater than zero, therefore the upper limit of the phase velocity c, of the neutral disturbances is the surface current velocity U , . Substituting (8.26a,b) in (8.25) we then obtain
+
$[A(@b - # l j ) B(@hj- y@Zj)] = 0, (8.28a,b) where j = 1, 2 refers to the bottom (ql = qb) and the top (qz = vt), respectively. When 7 1 b differs from zero, the existence of nontrivial solutions requires the vanishing of the determinant of the coefficients of A and B. For convenience we write this relation as (8.29) where Q k represents the solution which'behaves as ~ - , - l e "for large T and 0,represents the other solution. The term on the right is of order e-nt and hence it can be set to zero when rltr, which is the real part of T ~ , is sufficiently large. This equation determines the eigenvalue c in terms of Y, y, and qt - T b , and when ( T ~ Tb) is large and when r / a is neglected in y , c becomes a function of r alone. On the other hand, when T b = 0, (8.28b) then merely determines the ratio BIA. The solution then represents a neutral disturbance with c = U , . 1. Injnitely Deep Layer When H is very large, the right-hand side of (8.29) vanishes so that the solution is given by the not exponentially increasing function O j alone. This model has been used by Charney (1947) and by the author (Kuo, 1952). It also yields accurate results for systems of finite depths whenever qtr is sufficiently large. a. Neutral Solutions for Integer Y. Since for this case Q2 contains q-,-'eq, we must set B = 0 in (8.26) so that @ is given by the terminating solution O1,i.e., the Laguerre polynomial. The lower boundary condition (8.28a) is then (8.30) Tb2(@);b - Yalb) = 0.
+
Since Q1 is a polynomial of (Y - 1) degree, there are altogether (Y 1) roots, two of them correspond to qb = 0, i.e., with c, equal to the surface current velocity U , , while the other (r - 1) roots correspond to c, < U , . For example, for r = 2 we have one additional disturbance with c, = U , -2U*[p, while for r = 3 we have two additional disturbances with c,= u, - ( 5 & u*/2p. From the terminated form of Q1we see that @ has ( r - 1) nodal points
Jr)
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3 03
and hence the structure of the disturbance becomes complicated when r is large. b. The Unstable Solutions for r Not an Integer. When r differs from an integer, the solution for the present problem is given by the function Q2 of (8.26b) and (8.26b*). This solution contains log q and hence is complex when q, , which denotes the real part of q, changes its sign within the range of z. I t has been shown by the author (Kuo, 1952) that, for the range 0 5 r 5 1 - E , Trb is negative, hence the solutions for this whole range of r are unstable. T he instability of the other ranges m + E _< r < m 1 - E for m 2 1 has been shown by Burger (1962). T he eigenvalues of the unstable solutions can be calculated by an iterative method in the following manner. First we write the boundary condition (8.25) in terms of z,h = vQ2 in the form
+
(#k/z,hb)
(8.31)
- ('/Tb) - Y = E,
with the exact solution corresponding to E = 0. Next we consider the disturbance with r = m - Sr with a small 6r and hence small q i b . We can then assume that c, differs little from that of the neutral disturbance for r = m while the magnitude of c, is much smaller than c,. From (8.26b) we find that the main imaginary part of
E + E'Av
+ E"(Aq)'/2 = 0.
(8.32)
The process is repeated until E(T#))is sufficiently close to zero. Usually, a linear approximation is more convenient than the quadratic equation. A convenient first estimate of $,l)(r) is the known eigenvalue for a neighboring r. Thus we can start the calculation with r = m fSr and use the eigenvalue for r = m (integer) as the first approximation, and then use @(r + Ar) = Tb(r) for any other r when ~ ~ (is rknown. ) T h e eigenvalues have been obtained in this manner by the author in the previous study (Kuo, 1952) for 0 5 r 5 1. Additional values for 1 < r 5 5 have been calculated and those for 1 < r 5 3 are represented in Fig. 12 together with the improved results for 0 5 r 5 1. We point out again that for this problem the eigenvalue c is a function of the parameter r( = b/2aUz= /3N/2fkU,) alone, and its real and imaginary parts are given by the following expressions in terms of qbrand 7b1: c,-
u,= --2a
uzr]br
c,=
--. U z q b i 2a
(8.33)
Since a = (A2 + k12)1'2N/f, the growth rate kci is directly proportional to -?),,Iuzwhen k, is small in comparison with k . T he results in Fig. 12 show that all the disturbances with Y f m are amplified and that the growth rate has a maximum within each interval m 5 r 5 m + 1. Notice that the value of the maximum kc, diminishes only
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304
r FIG.12. Eigenvalues -qbr and -ybl of the baroclinic disturbances in Charney's model as functions of r for an infinitely deep layer.
slowly with increasing m, contrary to the expectations held before. Since these maxima are well separated, we expect to find the corresponding disturbances excited separately. That is to say, we expect the baroclinic atmosphere to have high energy concentrations at the scales of the prominent maxima at r = 0.55, 1.7, 2.7, etc. T he amplitude $ and phase E of the stream function a,h for the amplifying disturbances r = 0.5 and r = 1.7 are illustrated in Fig. 13. It is seen that for r = 0 . 5 , the amplitude at the surface is fairly large and the upper trough lags behind the surface trough by about l l O o , while for r = 1.7 the amplitude is quite small at lower levels and very large at higher levels.
t 77,
-I
'0
I
2
3
4
5
0
2
4
6
8
lOl'#l
FIG.13. Variations of the amplitude 141 and the phase E of the stream function with height. (a) r = 0.5; (b) r = 1.7.
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305
Further, for the latter case the phase difference between the lower and the higher parts of the system is close to 180", and the upper pressure trough coincides with the temperature trough, indicating that these disturbances are akin to the upper-air systems analyzed by Palmen (195 1). I n view of the fact that the boundary layer suction gives rise to prominent damping to the disturbances, we do not expect the disturbances with r > 5 to be truly self-excited. 2. Bounded Layer T he solutions for a finite layer have been studied by Green (1960), Hiroto (1968), and Garcia and Norscini (1970) and are being investigated again by the author in order to clarify many important questions. T h e detailed solutions of this problem will be presented elsewhere and here I shall mention only the fact that when the condition (8.2813) is imposed at z = H , the eigenvalue c is then given by (8.29) and hence it becomes a function of both aH and r. When aH is much larger than 1, the results will be almost the same as that for H = 00. However, when aH is not very large, the results depart appreciably from those for an infinitely deep layer. For example, the disturbances with r equal to an integer are also unstable in a finite layer because for this case the function Q2 must be included and the term Ql log 7 is complex when qr is negative. Therefore for this case there is no purely neutral wavelength even though neutral solutions still exist for Y > 1. D. LABORATORY EXPERIMENTS ON BAROCLINIC INSTABILITY Laboratory experiments on baroclinic quasigeostrophic flows have been carried out by the groups headed by Fultz et al. (1964), Hide (1958), Pfeffer and Chiang (1967) and others in their laboratories during the last two decades and their results have helped scientists gain much deeper understandings on the mechanisms that govern this type of flows. Further, the successful predictions and interpretations of the transitions from one regime of motion into another regime in Fultz's well-known diagram by the theoretical investigations of Davies (1956, 1959), Kuo (1956, 1957), Lorenz (1962), Barcilon (1964), and many others, as illustrated in Fig. 14, which is taken from a paper by Kaiser (1970), have also demonstrated the correctness of the basic concept on the nature of these motions. In addition to the Rossby number R, = U*/f L , the static stability factor represented by the global Richardson number R,= -gAp/4pQ2H plays a decisive role in determining the transition from one regime into another, even though the basic heating mechanism may only be horizontal. In fact, the development of the disturbances automatically creates a more stable stratification. The importance of viscosity for the transition from lower
H . L. Kuo
306
'Expwimentol 0-31
'
0
I
I
lo2
03
lo4
lo5
I
lo6
I
lo7
I
IO*
lo9
Tg FIG.14. Theoretical and observed transitions from symmetrical regime to wave regimes in baroclinic annulus experiments (after Kaiser, 1970).
symmetry to the wave regime is indicated by the closeness of the results obtained by Kuo (1956) and by Barcilon (1964) to the observed results and also to each other. However, these theoretical values of Roc are still somewhat higher than the observed critical Rossby number, especially at higher rotation rates. It is the opinion of the author that this discrepancy is due to the neglecting of the centrifugal acceleration in the theoretical model. A more detailed comparison between the different theoretical results and observations has been given by Kaiser (1970). T h e importance of the side boundary layers in these rotating annulus experiments has been explained successfully by Williams (1967a,b, 1968). Obviously, the very interesting vacillation phenomena observed in these annulus experiments can only be explained by the nonlinear solutions such as those discussed in the next section.
IX. Finite Amplitude Unstable Disturbances T h e solutions of the linearized vorticity equation given above predict that, when the basic flow is unstable and when only small amplitude random perturbations are present at the initial moment, the flow field will become dominated by the disturbance with the highest or nearly highest growth rate. However, it is evident that the exponential growth cannot proceed forever since the total energy of the whole system is limited. I n fact, a consideration of the law of conservation of energy suggests that, when the disturbance is interacting with the mean flow and when no dissipative force is acting, the disturbance and the mean flow will undergo a coupled oscillation, with the energy flow in one direction during one
Quasigeostrophic Flows and Instability Theory
307
phase of change and in the reverse direction during another phase. On the other hand, under the influence of dissipation a steady wave motion may be established.
A. METHOD OF SOLUTION I n order to determine the behavior of the finite amplitude disturbance, we must go back to the nonlinear vorticity equation
+ UP=+
$yqz) = -J,
(9.1) where q is the relative potential vorticity and Qoy is the basic state absolute vorticity gradient, which are given by (5.la,b)or (5.1ar,b'), and J(Jacobian) represents the nonlinear transport. For convenience we also use the x-average of this equation, viz., qt
- ($zqy-
Qov$s=
Subtracting (9.2) from (9.1) we then obtain
+ uq,' +
J' = -J
=-
+J,
(9.3) where $' = $ - $, q' = q - g are the departures of $ and q from their x-averages. T he boundary conditions for $' are those in (5.5a,bx), while the condition for is obtained from the x-average of the equation for ut and is given by qtr
Qou
$zr
I,&
0 at y = 0, y z . (9.4) For the unstable disturbance with x-wave number k and x-phase velocity c under consideration, $ can be taken as represented by the sum of the fundamental mode $ ( y , z , t)sin k(x - ct) and its higher harmonics, viz. $ty=
m
$(x,
y , z, t ) = 1 $ n=O
4z,~t)sin~nk(x - ct)
m
=
c
n= -m
$(y, x, t)exp[nk(x- ct)i].
(9.5)
Substituting these expressions of $ and q in (9.1) we find the following systems of equations :
308
H . L. Kuo
T he boundary conditions for $,, are the same as those for 9. These two spectral equations are equivalent to (9.3) and (9.2). T he finite amplitude solution of this system of nonlinear equations can be obtained in the following manner. We first write +,(y, z ) as the product of a time varying amplitude A( T )and a phase function, and then expand A( T ) in powers of the expansion parameter T , which we take as a convenient function of the difference (A) between the actual value of the stability parameter and its critical value, i.e., we set +L(y, Z, t, T )= A,(T)+,(y, z ) e i m k ( L - c t ) ,
(9.8a)
and choose
T = X(A)t cc ci t,
(9.8~)
where ci is the imaginary part of c given by the stability theory. Unlike the linear theory, here we use c to represent only the real phase velocity, leaving the change of the amplitude to the function A(T). The exponent r in (9.8b) represents the order of ,which is determined by the cascading process represented by the potential vorticity equation in creating 4; from the fundamental disturbance #:. Th e method outlined above can be applied to any type of finite amplitude wave disturbance resulting from instability, but in what follows I shall discuss only some results obtained from the two-level model.
EQUATIONS FOR WAVE PERTURBATIONS IN A TWO-LEVEL B. GENERAL OR A TWO-LAYER SYSTEM T he behavior of the finite amplitude disturbances under slightly unstable conditions in the two-layer model has been investigated by this method by Pedlosky (1970, 1971, 1972). This model is equivalent to the two-level and Boussinesq approximation of the continuous model discussed in Section VII1,B. Here I shall present only the essential results obtained by Pedlosky concerning the behavior of these small but finite amplitude baroclinic waves with a somewhat different derivation. As mentioned before, steady wave motion can evolve in the unstable system only when viscous dissipation is present and the simplest way to include this effect is through the introduction of the Ekman suction velocity w, in the heat equation at the boundaries. According to the Ekman theory of the planetary boundary layer flow, wb is equal to -8 * c2 at the top
Quasigeostrophic Flows and Instability Theory
309
c4
and equal to 6 at the bottom, where 6 = (v/2f is the Ekman layer thickness. The potential vorticity equation for the levels 2 and 4 can be obtained from (9.1) by setting $PP2
= ($P3 - $PJAP = ($4 - $2)/(AP)2- *Pl/AP,
=( A 5
$PP4
- $P3)/AP = - ($4
(9.9)
+$P5/AP.
- $2)/(AP)z
Substituting in (9.1) and making use of the heat equation at levels 1 and 5 to eliminate d$,,/dt and d&,/dt we then find
(9.10) where 92
= v2*2
+ F$*,
Q oy 2 =
** = $4
94 = v2*4- F$*,
B + FA u,
Qoy4
=
,
B - FA u,
AU = U , - U , ,
F =f”/S(Ap)2= p2/2,
-*2
(9.10a)
r =f 6/h.
It is seen that (9.9) is of exactly the same form as that for the two-layer model used by Pedlosky. Before we proceed to the nonlinear solutions, let us at first obtain c for the linear stability problem by considering the perturbation
$j= y j e t k ( x - C t ) sin my.
(9.11)
The linearized version of (9.10) then gives [( u, - c ) ( a 2 [(u4- c)(a2
+F ) +F )
-
6,
- Be
-
FAU]yz = F( u2 - c ) ~ 4
+ F A u l ~ 4 F( ==
u 4-c
)~z,
where
+ iru2/k,
/3* = /I
a2 = k 2
+ m2.
(9.12)
On equating the two ratios y 4 / y 2 and solving for c from the resulting equation we find c=
u- -
(aZ++)
B*
+ 2F) CL.
(a2
1
* 2a2(a2+ 2F) [4F2peZ
- ( AU)2a4(4F2 -
(9.13) This result agrees with (8.17) for r = O since we have 2 F = p 2 , A U = -2U”. Notice that with r > 0, the stable waves are damped.
H . L. Kuo
310
C. INVISCID FINITE AMPLITUDE DISTURBANCE, p # 0, r = 0
It has been shown in Section VII1,B that, for u2
2FP u, = a2(4F2 - a4)1’2
(9.13a)
and the minimum value of U , is /3/2F, which occurs at Suppose AU is slightly supercritical, i.e.,
A U = U2where A -g U, . For
a2
U4=
a2 = F.
U,+A,
(9.14)
< 2F we then have
Cl =
&
JZFP
+ 2F) (&)
112
(9.15)
a2(a2
’
Since instability occurs only for positive A, we shall limit A to positive values. The relation (9.8) indicates that for this case the slow time scale T can be chosen as
T = r)t, of
(9.16)
r ) = All2.
We now expand the amplitude viof the fundamental mode.& in powers r ) so that a,h; may be written as
#i(x, y, t,
T )= (r)#)
+
+ - )eik(z-ct)sin my, * *
j
= 2,
4.
(9.17)
According to the definitions of v, and T given above, we have (9.18) According to the linear stability theory, there is a phase difference between a,b2 and $4 for A > 0, hence we also expect to find a second-order zonal mean created by the nonlinear transport in (9.2). Therefore we set
$,(y, T ) = r)2&2)
+ r)3&3) +
According to (9.14) and (9.10a) we have
*
.
(9.19)
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311
Substituting these expansions in (9.5) and equating to zero the coefficients of 7 and 7 2 to zero we find the following system of equations:
+ K 2vL1 = 0, K3yL1’+ K4vL1)= 0,
K1vL1)
(9.2 1a)
)
(9.2 1b)
K,vL2)+ K2 vk2’= - k q‘(1) z
(9.22a)
9
K3v(22)+K4vL2)=-K441), i 2 Klv(23)+ K 2 v 4( 3 ) = _ _ 42 (2) k
K , vL3)+ K , ~ $ 3 = ) -
2
q,C 2 )
(9.22b)
+ qL1’+ FvP’,
(9.23a)
+ qL1’- Fcpi”,
(9.2313)
where q y ) = dq(*)/dTand
+
+
+
K1= ( U , U, - C ) ( L Y ~ F ) - /3 - FUc , K2 = - F( U4 U, - c), K , = - F( U , - c), K , = ( U4- c ) ( d F ) - /3 F U , , (9.24a) F ( r ) - ( F ct2)vr’. 4“’ = Fyr’ - (a2 F)v$’, 44(r) - ~2 (9.24b)
+
+
+
+
Eqs. (9.21a,b) simply yield the relations for the marginal solution obtained already (9.25a)
---_K3 -
F
(9.25b) * K, Notice that the left-hand sides of the inhomogeneous equations (9.22a,b) and (9.23a,b) are the same as in (9.21a,b), therefore their right-side member must bear definite relations to have a solution. Dividing (9.22a,b) by K , and K , , expressing q$l)in terms of vj through (9.24b) and making use of the relations in (9.2513) we then find u2
+F
-
(p - FU,)/( U , - C )
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312
where A=&). It can readily be shown that the two coefficients of A are equal, so that these two relations are identical. Hence we have (9.27) This relation shows that a phase different between i,h2 and i,h4 exists when A differs from zero. For convenience, we set q$) to zero so that we have
i,h2 = Re(qA ~ , b= ~ Re[yqA - iq2 -
+ q3yL3)+
(B+FUc)
k F ( U g + UC-c)"
* *
.)eik(t-Ct) sin my,
A+,,3p)i3)+
(9.28a)
...]eik(z-ct)sinmy. (9.28b)
The mean stream functions $2") and I&$) can now be determined. T o simplify the real forms of i,h2 and i,h4 we set
A = Reie,
t, = 5 + 0.
f = k(x - ct),
(9.29)
We then find
i,h& = [qR cos i,hk = {(y
c1+ q3yL3)+
* *
. ]sin my,
+ C17 d)qR cos t1+ Clq2R sin tl+ .
-
(*L 4 2 ) Y
= - (*L 4 ; ) Y = -
* *
(9.3Oa) }sin my, (9.30b)
kmFCl
2 RR sin 2my,
(9.30~)
where
(B + FUC) c, = kF(U4+ U,-C)~'
(9.30d)
Thus, the solution of (9.2) and (9.2a) gives
1-
sinh(2F)lI2Cy- (y2/2)] m c o ~ h ( F / 2 ) ~ ' ~ y , (F/2)1/2
(9.31)
where Ro is the initial value of R and
c -- (FkmC, +2m2)
*
(9.31a)
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313
The corresponding change of the vertical shear is given by
Similarly, when (9.23a) is divided by K , and (9.23b) divided by K 4 , we find that the left sides of these two equations are the same and therefore their right-hand members must be equal. On making use of the results obtained above in the various functions involved and equating these righthand members one finds the following equations for R and 8:
R = Co2R- N oR(Rz- Ro2) + L2/R3,
(9.33a)
R28= L,
(9.33b)
where L is a constant of integration and co2
No
2k2/YF2 + 2F)2UC’
(9.35c)
= a4(a2
k2mC2 8(F + 2m2)(a2 2F) ((2F-
+
8m2 (2m2 F ) tanh(F/2)1’2y2 x (F/2)l
[
+
a”) a’+
] +2m2(2a2- F)].
The first integral of (9.35) is given by
1
- R2-
2
1
+
+N
(COz N oRO2)R2 - R4 = 2E, 4
(9.34)
where E stands for the initial amount of the total energy. Just as in the truncated nonlinear two-level baroclinic wave problem discussed by Lorenz (1963), R(T) is given by the elliptic function R( T )= R,,, dn[kR,,,(N/2)1/2(
T - TO)].
(9.34a)
The most interesting result revealed by this solution is that the disturbance and the basic flow change together rhythmically, keeping the total energy constant at every moment. The amplitude of the wave oscillates between the maximum R,,, and the minimum Rmin,which depends on the initial amplitude R,. The variation is such that the wave extracts energy from the basic state when R is increasing, but as R grows beyond its equilibrium value R e , the environment becomes increasingly more stable and finally the direction of the energy transfer reverses and the amplitude of the disturbance diminishes.
H . L. Kuo
3 14
D. VISCOUS EQUILIBRATION FOR / I = 0, r # 0 In nondimensional form r is the ratio between the square root of the Ekman number and the Rossby number. Here we take r = O(1). For this case we find from (9.13) that the critical vertical shear UC=AUmi,is given by
U, =
2ra k(2F - a2)1/2*
(9.35a)
We consider again that I AUI is slightly above U, such that
AU= Uc+A, ci =
A < U,
(A>O),
(2F - a2)l” aA. 2(a2 F )
+
(9.35b)
For sufficiently small A, the behavior of the disturbance is again determined by its interaction with the mean flow. However, (9.35b) indicates that for this case the slow time scale should be
T = At = q2t,
(9.36)
7 = All2.
Using the same expansions for & and qj as given by (9.17) and (9.21) we then find from (9.10) that the 7- to q3-order equations are given by
K,q$)
+ K2q$) = G$),
(9.37a) s=
+
1, 2, 3 (9.3713)
K3cp(a) K49)(4S) = Gf),
where K, , K 2 ,K3, K , are given by (9.24a) except j3 is replaced by ira2/k and G$) and Gc,“)are given by GL1)= G(1)= 0 (9.38a) (32)
(33)
=J63)
(33)
where
JP)
=J&2),
+
Gi2) =Ji2);
+ $’&I)
=Ji3) - F&)
(9.3813)
2
-- pL1)
k
- -z pi 1 ) k
’
(9.38~)
’
are the s-order contributions from the nonlinear transport
J’ of (9.1”) or (9.4). The two homogeneous equations for s = 1 yield the phase velocity c = U‘I +(U$).
(9.39)
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315
Using this relation in the coefficients K , in (9.24a) we find
uc (a' Kl = 2
-F ) -
ira2 k '
K , = -K,
UC K --(F - a') 4 2
= - FUC
-
2 '
ira2 - -, k
(9.40a)
Notice that the amplitude ratio y is complex in this solution and that the upper wave is lagging. Therefore there is an energy transfer from the mean flow to the perturbation for this marginal solution, which is needed for the wave to be maintained against the viscous dissipation. This spatial structure of the wave accounts for the +order contribution of the nonlinear vorticity transfer J'. Thus, the equations (9.38a,b) for s = 2 give (9.41) When J is taken as J' =J -J, this equation is satisfied identically. The solution of the 'q order part of (9.5) gives
R2ak(2F' - -$p-
$W-
8rm
Similarly, the equations for s = 3 require the equality of 1/K2 times the right side of (9.37a) and l / K 4 times the right side of (9.37b). The result is the following first-order equation for the amplitude R of z,&): d dT
-R' = 2R2[kCoi- k2NR2],
(9.43)
where R 2 is the square of the amplitude and coi =
(2F - a2))"a , 2(a2 F )
+
N=
F %(a2
+F ) [4m2(a2- F ) + 3a2(2F-
a')].
(9.43a) The solution is therefore of the same form as that given by Stuart and Watson (1960), viz. (9.44) Thus, the amplitude of the wave approaches asymptotically a steady value
R' = Co,/kN.
(9.44a)
H . L. Kuo
316
There is also a linear phase change
e=
( k / 2 ) ~ (A= Re 1. (9.44b) This is simply a reflection of the fact that the phase velocity of the wave is equal to the mean current velocity 0 = U , ( U , A)/2, which differs from that of the marginal wave speed given by (9.39). T he cases with smaller Y , e.g., Y
+
+
X. Instability Theory of Frontal Waves It has been shown by the continuous model in Section VIII that the normal atmosphere is baroclinically unstable for disturbances of all wave lengths and that the long waves ( L > 4000 km) have their maximum intensities at upper levels while the shorter waves ( L < 2000 km) are limited to the lower part of the troposphere, therefore we may identify these shorter baroclinic disturbances with the cyclone waves and attribute the origin of the cyclones to the general baroclinic instability. However, surface extratropical cyclones usually form on a frontal surface, wherein a major part of the temperature or density contrasts between neighboring air masses is concentrated into a narrow transition layer which, on the scale of the large scale flow, amounts essentially to a surface of discontinuity in temperature or density. Thus, treating the cylone wave as a disturbance on the front will definitely bring the theoretical result closer to reality. T he frontal cyclone theory goes back to the Bjerknes-Solberg (1922) cyclone model, which depicts the extratropical cyclone as an unstable wave which develops from a small perturbation on a quasistationary front characterized by a cyclonic shear. A mathematical model was first formulated by Solberg (1928; cf. also V. Bjerknes et al., 1933, Chapter 14; J. Bjerknes and Godske, 1936), with two planes parallel to the frontal surface serving as boundaries for mathematical expediency. A physically sound and mathematically tractable model was later on formulated by Kotschin (1932), who also obtained a neutral solution of the system. A significant advance on the instability theory of frontal waves has been made by Eliasen (1960), who obtained solutions of Kotschin’s equations
Quasigeostrophic Flows and Instability Theory
317
for a range of values of the important parameters relevant to cyclone waves and demonstrated that the flow pattern given by the unstable solutions are very similar to the observed flow pattern in developing cyclones. The Kotschin equations have also been integrated numerically by Orlanski (1968), who also covered other ranges of values of the parameters and showed that the Margules type front is unstable for inviscid disturbances of all wave lengths except a number of isolated neutral disturbances, just as in Charney's continuous model of the baroclinic problem.
A. THEBASICSTATE The basic state is characterized by a balanced stationary front which separates two homogeneous fluids moving with constant velocities U, and U , in the x-direction in a rotating system, as illustrated in Fig. 15. The pressure distribution of this state satisfies the hydrostatic and the geostrophic relations (10.la) (10.lb) where j = 1 refers to the layer below and j the front. From (10.la) we find
Pdz, y ) =pz"(y)+gp@ P1@,
- x)
Y ) =Pz(h0)+gp1(ho - .) =Pl"(y) + g p l ( H - x)
=2
refers to the layer above
0I ho I 2I H, 0I
2
I ho < H ,
0I zI H,
X
FIG.15.
The frontal surface model.
y
y I 0, (10.2a)
a
0I y I
2 D,
(10.2b)
H. L. Kuo
318
where H is the total depth, D is the width of the frontal belt and p*(y) is the pressure on the top boundary. From these relations we find aj2
ap2*
aY
aY
dH dY
-= -+gp2 -=
u,,
-fpz
( 10.3a)
Therefore the slope of the front is given by
-dh0 _ -tancr dY
f fP --[pzU2-p1U1]=-(U2O -gAP gAP
Ul),
(10.4)
where Ap=pl-p2. We assume that the stratification is stable so that Ap > 0. We also take Ap as much smaller than either p1 or p 2 , so that p1 = p 2 = p can be used when they occur individually.
B. PERTURBATION EQUATIONS AND BOUNDARY CONDITIONS We assume that every flow variable is composed of an undisturbed part and a small departure, viz.,
Vj = (U,
+ u’)i + vj’j + wj’k,
p , = p j +p,’,
h = ho(yj
+ c‘,
j = 1, 2. (10.5)
The pertubation pressure p i is also taken as hydrostatic, hence we have PAX,
Pl(X,
y , z 2 , t)=P2+p2’(x,y, zz, t)=p”+gp,(H--z,)
y , z, 4 =I1 + P l ’ ( X , y , z1,t ) =P+ +gpz(H-h) +gpl(h-zz,),
hIz25H (10.6a)
0 i z 1 i h iH. (10-6b)
The linearized equations of motion and the continuity equation are
;( + uj $i.
1
- fv; = - -piz, P
1 +fUj’ = - -piy, P u;$+ vjy + w;, = 0. Vj‘
(10.7a) (10.7b) (10.8)
Quasigeostrophic Flows and Instability Theory
319
From the continuity of pressure across the interface we find
Pl'
-P 2 '
=g(Pl - P 2 ) 5 ' ( X , y ,
t),
a p j p Z=o
(10.9a) (10.9b)
so that the perturbation pressure is independent of x. From the equations of motion we also conclude that the horizontal velocities uj' and vj' are also independent of height within each individual layer, viz.,
(10.9~) Therefore the continuity equation gives w j as a linear function of x , viz., j
au
avji
(ax
ay)
-L+- ( x - x o j )
w.'--
(10.9d)
where xol = 0 and xO2= H . In addition, we have at the interface x = h ( 10.10)
Combining this equation with (10.9~)for z = h , we then obtain ah
(10.11)
where d1= ho i
d2 = - ( H -
ho).
(10.1 la)
The equations (10.71, (10.9a), and (10.11) constitute a closed system for the variables p j ' , u j ' , vj', and ['. As in other stability problems, we take the perturbations as represented by the product of a wave factor and their amplitude, viz., (u' v'
p' 5') = ( 0 9 , v*, p , [)ei',
(10.12)
where 6 = (kx + w t ) is the phase of the wave disturbance and w is the frequency. It is understood that only the real parts of these complex expressions are to be taken to represent the real variables. Substituting this representation in (10.7a,b) and solving for ujDand vj* in terms of p j we then obtain (10.13a) uj8 = (kwj*Pj -fPjv)lpFj > vj* wj#=
= i(kfPj - wj*Pupju)lPF~9
w
+k U j ,
Fj =f
-my2.
(10.13b) (10.13~)
H . L. Kuo
320
Substitutions of (10.12) in (10.9a) and (10.11) result in the following: Pl
-
Pz =d P 1 -
(10.14)
P2)L
+ v jdho - + d,(iku,* + vry) = 0.
(10.15) dY Inserting u,", v,*, and 5 from (10.13) and (10.14) in (10.15) we then find the two following equations for p , and p , : iw,*<
U
(10.16)
[(I +r))Pl'l' - [x"1 +7)+;-=i]Pl=Bl(pl-P2),
where the prime denotes differentiation with respect to the dimensionless independent variable 7 and the other parameters and 7 are defined by u=--
f D - Richardson number, 2u*
4pu*2 k U"
P = 7= Rossby number,
B,= (T - [I - P"(T - I)'], 2
U
B, = 2 [1 - P'(T
f
(10.18)
I)'],
v = 2Y - - 1. 2
Equations (10.16) and (10.17) are valid within the frontal belt - 1 5 7 5 1. Outside this belt w' vanishes at all levels and therefore (10.8) reduces to ikUj*
+ vy* = 0.
(10.19)
Substituting uj* and vj* from (10.13) in this equation we then find
p ; - P p , = 0.
(10.20)
The conditions that must be satisfied by the solutions are (i) finiteness of all flow variables for all 7 ; (ii) continuities of pressure and normal velocity across the interface. Thus the solutions of (10.20) must be of the forms
pI1I(r))= C,e - A n
piv(r))= C,eAn.
(10.20a)
Since the regions I1 and IV (see Fig. 15) are occupied by the same lighter fluid, both p , and p,' must be continuous across r ) = -1. Similarly, p , and
Quasigeostrophic Flows and Instability Theory
321
p,' must be continuous across 7 = 1. That is to say, p , and p , must satisfy the following relations (a)
P,I=P{~I,
(b)
dp,I - dp:I1 -- -- --hPII' 1 d., 4
- -$,I
at 7 = 1, (10.21)
In addition, it appears that p , and p , must satisfy additional relations at the points where the front cuts the bottom and the top boundaries, namely, at 7 = - 1, z = 0 and 7 = 1, x 5 H because w vanishes at these points. Thus, on setting w to zero in (10.10) we obtain
where the subscript e denotes values at the two end points of the front. Substituting zlj from (10.13b) and 5 from (10.14) in this relation we then obtain (10.23) On making use of the relations (10.21b,d) we then obtain the following relations between p , and p , :
1
7-1
[ P - - ](P7 -i = 1)2
[ P - x ] P z
at 7 = 1.
(10.24b)
We now rewrite the conditions (10.21b,d) as
dP1 -= d7
-Ap1
at
7= 1
(10.24~) (10.24d)
The four conditions (10.24a-d) constitute the four proper boundary conditions for the solutions of (10.16) and (10.17). The conditions (10.24a,b) have not been properly applied to the frontal wave problem in the previous investigations and their influences on the solutions are not yet known.
H . L. Kuo
322
C . FRONTAL WAVESOLUTION It is evident that many very different phenomena are covered by this model when very different values are chosen for the various parameters. For example, by letting Ap approach zero, the frontal surface becomes vertical and the problem becomes identical with Rayleigh’s problem of shear-layer instability, whose eigenvalue is T = +i. However, since our primary interest in this model is on the frontal-wave type disturbance, we shall restrict our attention to the range of the values of the parameters relevant to this problem, for example, with
5 5 ~ 5 1 0 , /3-0.15,
h=0(1).
1. Marginal Solution with 7 = 0 Notice that (10.16), (10.17), and (10.24a,b) remain unaltered if 7 and 7 and -7, pl(7), and p2(7)are interchanged simultaneously. Hence, if T is an eigenvalue and p1(7)and p z ( q ) are the eigenfunctions, then - 7 , ~ ~ ( - 7and ) p,( -7) represent another solution. Thus the eigenvalues occur in pairs (7,-7) in this problem. When the system is unstable for the disturbance in question, the conjugates of 7 and -T are also eigenvalues. Therefore there are two growing waves with 7 = &7r and two decaying waves with 7 = T~ h i . Let us consider at first the neutral solutions with 7 = 0. For these neutral disturbances (10.16) and (10.17) reduce to -7,
+
+
[(I
+ 7I)Pl’l’
-
h2(1
+7)Pl + APl + BP2
[(I - 7)P2’1’- h2(1- 7 ) P z
+ + BPl 4
2
= 0,
(10.25)
= 0,
(10.26)
where
A
U
0
B = - (1 - p2). 2
= - (1 -+/3’)),
2
(10.27)
These equations take simpler forms when expressed in terms of the variables
U(7)= (1
+ 7)(Pl’ +
B(1 - 71P2 ,
(10.28a)
+ B(1 +7)Pl.
(10.28b)
4
1
) -
V(7) = (1 - 7)(P2’ - 4
2
)
Th e new equations are
u’- A U + B V = 0, V’+ AV- BU=O.
(10.29) ( 10.30)
Quas@eostrophicFlows and Instability Theory
323
For /I# 0, 1, the general solutions of these two equations can be written as
+
U(7)= CIBeAq C,(A - A)ecAn,
( 10.3la)
+ C , Be-“”.
(10.31b)
V(q)= Cl(A - A)eAq
On applying the boundary conditions (10.24c,d) to these solutions we find
Cla2/3(1 - p)2= C2a”@(1- /3)z
= 0.
(10.32)
Thus, when a2/3(1- /3)2 differs from zero, we must have C, = C , = 0, i.e., both U and V vanish identically. Hence (10.28a,b) reduce to (1
+d ( P l ’ + 4 5 )
= B(1 - T)PZ 9
+
(1 - d(p2’ - Ap,) = --BU dp1. The boundary conditions consistent with (10.32) are
PI’(1) =p,( 1) =p2’(-1) =p2(- 1) = 0.
(10.33) (10.34) (10.35)
Eliminating p , from these equations we obtain (1 - q y p ;
+ 2p,‘ + [2A
-
A,( 1 - 72)]p,= 0.
(10.36)
It can readily be shown that this equation remains valid as A+ 0. In this case the solution is given by
where P,(q) is the Legendre function of the first kind and
n(n + 1) = 0.
(10.37a)
The boundary conditions (10.35) can be satisfied by these solutions only when n is a positive integer, i.e., when u = 2,6, . . . . Thus we expect to find that the solutions of (10.36) to approach these solutions for small p. In fact, Kotschin (1932) has expanded a neutral solution near u = 2 in power series and found the following relation between u and B: (5
=2
214 + 2-3 p z + +* . 875 p4
*.
( 10.38)
The relation between these neutral modes and the unstable modes are still not clearly known. They are not transitions from stability to instability as Kotschin assumed. 2. Unstable Solutions In seeking the solutions of (10.16) and (10.17), it is convenient to shift the origin of q to y = 0 by using 7 = 2y/D. Eliminating p , from these
H . L. Kuo
324
equations we then obtain the following fourth-order equation in p , :
where U
A, = --tB,, 7-1
u
U
A , = - -+ B,, 7+1
A2
B, = -- - ( T - l),, 2 20 (10.39a)
M==4h2 +A,M2 = A,A,
Al, - BIB,
M I == M - 2X2,
+ (1 + 2A,)X2.
The points 7 = 0 and q = 2 are regular singular points of this equation. The boundary conditions are
p2'=Ap2,
pz"=h2p2
at
+
w
, - 112 - P(. =(T+I)[B(T-I)~-T--
1131
v=O, 11P 2
9
(10.40a) (10.40b)
The general solution of (10.40) can be written as 4
PAT)=iC KjRi(q), =1
(10.41)
where R j ( j = 1,2, 3,4) are four fundamental solutions of (10.40). These solutions can be written in the form m
Rk(q)= C a',"'qm + pk,
a , # 0.
(10.42)
/33=0.
(10.42a)
m=O
The indicia1 equation then gives pl=p4=2,
/32=17
The recurrence formula for the coefficients a',")is given by i = O jja::j
= 0,
(10.43)
Quasigeostrophic Flows and Instability Theory where a?; fo=
=0
b + &(p
325
for all positive integer v and
+m
- 2),
+
+ m-2)[(p + m-
f z == -$(P
+ 3)M + Mi],
fl = -&b m],- (1 Al)[p + m - 112, (10.43)
where p = pk and [m+s],= (m+s)(m + s - 1) ... (m+ s - r + 1). Since the p k in (10.42a) differ by an integer, one of the solutions with p = 2 involves log r] and the other three solutions are regular. Thus these solutions can be taken as
(10.44) m
+
R,=Ca',4'r]m+2 R1(r])log7. m=O
We take ahk)= 1 for all four solutions, while the other coefficients can be obtained from (10.43). Since R, gives rise to an infinite p;(O), it must be excluded from the solution, so that we have pz(7) = KiRi
+ Kz Rz + K3 R3
(10.45)
*
This solution is convergent up to r] = 2 - E ( E > 0), but it breaks down at the other singular point r] = 2. This difficulty was avoided by Eliasen (1960) who placed a wall at r] = q N < 2, which is justifiable because the real front usually does not reach the tropopause. Then (10.45) represents the proper solution of the problem. With this solution the conditions (10.40a) give K , = hK3. Using this K , in p , and equating the two ratios of K 3 / K lgiven by the two conditions v1(vN)= v2(rlN) = 0, we obtain the equation for the eigenvalue T in terms of the other parameters. This equation is usually of rather complicated form and the root can only be extracted numerically. T h e growth rate kci and T~ obtained by Eliasen from this model are plotted in Fig. 16a and 16b against LID( =n-/A), while Fig. 17 illustrates the pressure distribution at the x = 0.1H level for the most unstable disturbance for 0 = 5.0, U , = -2U1, U , - Ul = 20 m s - l , and with an amplitude exp(kc,t) =
c0
H . L. Kuo
326
oL--(a)
I
Tr
0.4
0.2 OO O
A
I
2
2
L
'0
x
2
~
46
D
FIG.16. Eigenvalues of the frontal surface disturbances. (a) T* as a function of A; (b) variation of the growth rate act with the wavelength for different values of the Richardson number.
0.1H. It is seen that this pressure distribution is very similar to that in actual well-developed cyclones. Th e most unstable wave length is about 2000 km when D = 1000 km. We point out that the symmetry property for T discussed in Section X,C,1 is absent in this model with a wall at v N , hence all the phase speeds are larger than 0. T he problem represented by (10.16) and (10.17) for the whole domain -1 < 7 < 1 was treated by Orlanski numerically together with the boundary conditions (10.24c,d) and the requirements that p , and p , are regular
0.0
0.2
0.4
0.6
I .o
0.8
I .2
x/L
FIG.17.
Isolines of ( p + p ' ) / g A p H at the height z
I ho' I ekeit= H/lO. Heavy line indicates position of interface.
= HjlO
for
U, = -Ul,
Quasigeostrophic Flows and Instability Theory
327
at q = f l . As is to be expected, different regimes of motion prevail for different ranges of 0 and 8. However, most of Orlanski’s calculations are for u < 3 and hence may not be really representative of the frontal wave disturbances. T h e wave structure given by Orlanski for 0 = 5 is very similar to that in Fig. 17, but is for a much longer wave length. The exact relation between the neutral solutions with T = 0 discussed in Section X,C,1 and the different regimes of motions are still not very clear. It appears that they do not represent transition from stability to instability but rather from one unstable regime to another, as indicated by Orlanski’s results.
D. NONLINEAR DEVELOPMENT OF FRONTAL WAVE T h e nonlinear equations of the frontal wave model are extremely difficult to integrate because every point of the intersection of the frontal surface with the ground is in essence a singular point. Kasahara et al. (1965) integrated a simplified version of this model, i.e., with the upper layer at rest, in a quasi-Lagrangian scheme and obtained very realistic results with the cold front advancing faster than the retreating of the warm front.
XI. Concluding Remarks Aside from the adjustment process discussed in Section I1 and the permanent-wave solutions given in Section IV, our discussions in this paper are confined mainly to the instability aspect of the quasigeostrophic flows. Since such flows are characteristic of all large scale, low frequency motions in a stably stratified and rotating fluid, they can also occur as forced motions, for example, motions resulting from the influences of large-scale topography and non-uniform heating. T h e quasigeostrophic potential vorticity equation has also been used as the tool for numerical forecasting, especially during the early stage of the development of this branch of theoretical meteorology. Here we shall leave these subjects to other reviews. Essentially nongeostrophic stability problems have also been left out of this paper. ACKNOWLEDGMENT This work was supported by the National Science Foundation under Grant No. GA 25161.
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HOWARD, L. N. (1961). Note on a paper of John Miles, J . Fluid Mech. 10, 509-512. HOWARD, L. N., and DRAZIN, P. G. (1964). On instability of parallel flow of inviscid fluid in a rotating system with variable coriolis parameter. J . Math. Phys. (Cambridge, MUSS.)43, 83-99. KAISER,J. A. C. (1970). Rotating deep annulus convection. Part 2. Wave instab vertical stratification and associated theories. Tellus 22, 275-287. KASAHARA, A,, ISAACSON, E., and STOKER, J. J. (1965). Numerical studies of frontal motion in the atmosphere. I. Tellus 17, 261-276. KIBEL’,A. I. (1955). 0 prisposoblenii dvizheniya vozdukha k geostroficheskomy. Dok Akad. Nauk SSSR 104, 6@63. KOTSCHIN, N. (1932). Uber die Stabilitat von Marguleschen Diskontinuitatsflachen. Beitr. Phys. Frei. Atmos. 18, 129-164. Kuo, H. L. (1949). Dynamic instability of two-dimensional non-divergent flow in a barotropic atmosphere. J . Meteorol. 6, 105-122. Kuo, H. L. (1952). Three dimensional disturbances in a baroclinic zonal current. J . Meteorol. 9, 260-278. Kuo, H. L. (1953). Stability properties and structure of disturbances in a baroclinic atmosphere. J . Atmos. Sci. 10, 235-243. Kuo, H. L. (1956). Energy-releasing processes and stability of thermally driven motions in a rotating fluid. J . Atmos. Sci. 13, 82-101. Kuo, H. L. (1957). Further studies of thermally driven motions in a rotating fluid. J . Meteorol. 14, 553-558. Kuo, H. L. (1959). Finite amplitude three-dimensional harmonic waves on the spherical earth. J . Meteorol. 16, 524-534. Kuo, H. L. (1972). On a generalized potential vorticity equation for quasigeostrophic flows. Geofis. Pure Appl. 96, 171-175. LIPPS, F. B. (1962). The barotropic stability of the mean winds in the atmosphere. J . Fluid Mech. 12, 3 9 7 4 0 7 . LORENZ, E. (1962). Simplifieddynamic equations applied to the rotating basin experiments. J . Atmos. Sci. 19, 39-51. LORENZ, E. (1963). The mechanics of vacillation. J . Atmos. Sci. 20, 448-464, MICHALKE, A. (1964). On the inviscid instability of the hyperbolic-tangent velocity profile. J . Fluid Mech. 19, 543-556. MILES,J. W. (1964a). Baroclinic instability of the zonal wind. Parts I1 and 111. J . Atmos. Sci. 21, 550-556 and 603-609. MILES, J. W. (1964b). Baroclinic instability of the zonal wind. Rev. Geophys. 2, 155-176. NEAMTAN, S. M. (1946). The motion of harmonic waves in the atmosphere. J . Meteorol. 3, 53-56. OBUKHOV, A. K. (1949). On the question of geostrophic wind. Bull. Acad. Sci. USSR, Geogr.-Geophys. Ser. 13, 281-306. ORLANSKI, I. (1968). Instability of frontal waves. J . Atmos. Sci. 25, 178-200. PALMEN, E. (1951). The aerology of extratropical disturbances. In “ Compendium of Meteorology,” pp. 599-620. Am. Meteorol. SOC.,Boston, Massachusetts. PEDLOSKY, J. (1964a). The stability of currents in the atmosphere and the oceans. Part I. J . Atmos. Sci. 21, 201-219. PEDLOSKY, J. (1964b). Part 11. J . Atmos. Sci. 21, 342-353. PEDLOSKY, J. (1970). Finite amplitude baroclinic waves. J . Atmos. Sci. 27, 15-30, PEDLOSKY, J. (1971). Finite amplitude baroclinic waves with small dissipation. J . Atmos. S C ~28, . 587-597. PEDLOSKY, J. (1972). Limit cycles and unstable baroclinic waves. J. Atmos. Sci. 29, 53-63.
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PFEFFER, R. L., and CHIANG, Y . (1967). Two kinds of vacillations in rotating laboratory experiments. Mon. Weather Rev. 94, 75-82. PHILLIPS,N. A. (1951). A simple three-dimensional model for the study of large-scale extratropical flow patterns. J . Meteorol. 8, 381-394. PHILLIPS, N. A. (1963). Geostrophic motion. Rev. Geophys. 1, 123-176. ~ H J E N P. , (1950). uber gegenseitige adaptation der Druck und Stromfelder. Arch. Meteorol., Geophys. Bioklimutol., Ser. A 2, 207-222. RIEHL,H. (1969). Some aspects of cumulus convection in relation to tropical weather disturbances. Bull. Amer. Meteorol. SOC.50, 585-595. ROSSBY,C. G. (1938). On the mutual adjustment of pressure and velocity distributions in certain simple current systems. 11. J . M a r . Res. 1, 239-263. SIMONS,T. J. (1969). “ Baroclinic Instability and Atmospheric Development,” Atmos. Sci. Pap. No. 150. Department of Atmospheric Sciences, Colorado State University, Fort Collins. SIMONS,T. J. (1972). The nonlinear dynamics of cyclone waves. J . Atmos. Sci. 29, 509-5 12. SOLBERG, H. (1928). Integration der atmospheric Storungs gleichungen. Geofys. Publ. 5 , No. 9. SONG,R. T. (1970). A numerical study of the three-dimensional structure and energetics of unstable disturbances in zonal currents. J . Atmos. Sci. 28, 549-586. STUART, J. T., and WATSON,J. (1960). On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows, Parts I and 11. J . Fluid Mech. 9, 353-370 and 371-389. THOMPSON, P. D. (1953). On the theory of large-scale disturbances in a two-dimensional baroclinic equivalent of the atmosphere. Quart. J . Roy. Meteorol. SOC.79, 51-69. VERONIS, G. (1956). Partition of energy between geostrophic and nongeostrophic oceanic motions. Deep-sea Res. 3, 157-177. WILLIAMS, G. P. (1967a). Thermal convection in a rotating fluid cumulus. Part I. The basic axisymmetric flow. J . Atmos. Sci. 24, 144-161. WILLIAMS, G. P. (196713). Part 11. Classes of axisymmetric fl0w.J. Atmos. Sci. 24, 162-174. WILLIAMS, G. P. (1968). Part 111. Suppression of the frictional constraint on lateral boundaries. J . Atmos. Sci. 25, 1034-1045. YANAI,M. (1961). Dynamical aspects of typhoon formation. J . Meteorol. SOC.Jup. 39, 282-309. YANAI,M., and NITTA,T. (1968). Finite difference approximations for the barotropic stability problem. J . Meteorol. SOC. Jap. 46, 389-403. YANAI,M., and NITTA, T. (1969). A note on the barotropic instability of the tropical easterly current. J . Meteorol. SOC.Jap. 47, 127-130.
Author Index Numbers in italics refer to the pages on which the complete references are listed.
A
Carpenter,, J. H., 19, 90 Carrier, G. F., 49, 53, 90, 91 Carritt, D. E., 19, 90 Charney, J. G., 38, 49, 53, 90, 261, 275, 292, 302, 328 Cherkesov, L. V., 173, 231, 232 Chernin, K. E., 160, 231 Chey, Y. H., 196, 219, 220, 231 Chiang, Y., 305, 330 Chung, Y., 71, 90 Collins, J. I., 173, 174, 242 Cox, M. D., 50, 90 Craig, H., 71, 90 Crapper, G. D., 173, 231 Cremer, H., 163, 164, 231 Criminale, W., 290, 328 Cross, J. J., 100, 241 Cumberbatch, E., 173, 231
Adachi, H., 131, 242 Allen, R. F., 153, 173, 230 Amtsberg, H., 196, 205, 219, 244 Apukhtin, P. A., 158, 189, 194, 230 Arons, A. B., 73, 75, 92
B Baba, E., 106, 194, 230 Baker, D . J., 75, 78, 90 Barcilon, V., 305, 306, 328 Barratt, M. J., 157, 230 Barrillon, E. G., 230 Beardsley, R. C., 75, 80, 90 Beck, R. F., 167, 230 Bessho, M., 96, 196, 205, 207, 208, 219, 230, 239 Betchov, R., 290, 328 Bhattacharyya, R., 155, 181, 185, 186, 188, 198, 199, 200, 201, 226, 230, 244 Biktimirov, Yu, K., 159, 230 Birkhoff, G., 156, 177, 230 Bjerknes, J., 316, 328 Blandford, R. R., 55, 60, 61, 90 Bock, W., 196, 205, 219, 244 Boes, C., 181, 185, 186, 188, 198, 199, 200, 201, 244 Bolin, B., 250, 256, 328 Brard, R., 106, 159, 167, 168, 227, 231 Breslin, J. P., 226, 231 Brown, J. A , , Jr., 328 Bryan, K., 32, 41, 49, 50, 53, 90 Burger, A. P., 292, 302, 328
C Cahn, A., 250, 256, 328 Cali~al,S., 127, 128, 129, 210, 213, 231
D Dagan, G., 228, 231 Davies, T. V., 305, 328 Deardorff, J. W., 41, 71, 91 Doctors, L. J., 157, 160, 161, 231 Drazin, P. G., 290, 329 Dugan, J. P., 168, 231 D’yachenko, V. K., 160, 231
E Eady, E. T., 293, 295, 328 Eckert, E., 204, 231 Edmond, J. M., 71, 90 Efimov, Yu. N., 160, 231 Eggers, K. W. H., 95, 109, 114, 121, 122, 123, 124, 127, 128, 129, 130, 154, 174, 222, 223, 224, 225, 232
331
Author Index
332
Ekman, V. W., 24, 37, 40, 91, 161, 232 Eliasen, E., 316, 325, 328 Eliassen, A., 275, 328 Emerson, A., 176, 189, 227, 232 Eng, K., 226, 232 Ertel, A , , 16, 91 Ertel, H., 262, 268, 328 Everest, J. T., 193, 197, 232
Hogben, N., 125, 131, 193, 197, 232, 233, 235, 236 Hogner, E., 171, 236 Holland, W., 56, 91 Howard, L. N., 39, 91, 279, 290, 329 Hsiung, C . - C . , 186, 187, 188, 189, 236 Huang, T. T., 157, 167, 172, 176, 236, 243 Hudimac, A. A , , 162, 173,236 Hunkins, K., 40, 91
F I Faller, A. J., 75, 92 Fedosenko, V. S., 173, 232 Fleming, R. H., 18, 19, 21, 92 Fofonoff, N. P., 19, 20, 24, 32, 49, 53, 60, 91 Froese, C . , 20, 24, 91 Froude, W., 95, 232 Fukuchi, N., 227, 239 Fultz, D., 305, 328
G Gadd, G . E., 96, 125, 227, 232, 233 Garcia, R. V., 292, 305, 328 Gertler, M., 233 Giesing, J. P., 215, 218, 233 Godske, C. L., 316, 328 Graff, W., 187, 233 Green, J. S . A , , 292, 305, 328 Greenspan, H. P., 39, 75, 91, 92 Groves, G., 5 3 , 91 Gruntfest, R. A , , 166, 173, 233 Guilloton, R., 174, 175, 176, 214, 227, 233
H Hamon, B. V., 30, 92 Haskind, M. D., 154, 159, 233, 234 Haurwitz, B., 268, 328 Havelock, T. H., 96, 133, 147, 154, 157, 159, 160, 161, 164, 174, 198, 234, 235 Hidaka, K., 51, 91 Hide, R., 305, 328 Hinterthan, W. B., 98, 235 Hiroto, I., 292, 300, 305, 328
Ienaga, I., 131, 242 Ikehata, M., 131, 226, 227, 236, 237, 239, 240 Inui, T., 96, 155, 188, 189, 194, 195, 202, 224, 226, 236 Isaacson, E., 327, 329 Isay, W. H., 217, 236 Ishii, M., 214, 236, 239
J Jacobs, W. R., 176, 237 Jeffreys, H., 6, 91 Johnson, M. W., 18, 19, 21, 92 Joosen, W. P. A , , 169, 236 Joukowski, N. E., 171, 205, 236
K Kaiser, J. A. C., 305, 306, 329 Kajitani, H., 121, 227, 232, 236 Karp, S., 207, 236 Kasahara, A , , 327, 329 Keldysh, M. V., 158, 237 Kendrick, J . J., 187, 189, 244 Kibel', A. I., 250, 329 Kim, H. C . , 175, 239 Kim, W. D., 218, 219, 237 Kirsch, M., 158, 237 Knudsen, M., 24, 91 Kobayashi, M., 227, 237 Kobus, H . E., 114, 117, 237 Kochin, h '. E., 153, 154, 2.37 Kolberg, F., 159, 163, 164, 166, 231, 237 Korvin-Kroukovsky, B. V., 176,230, 237
Author Index Kostyukov, A. A., 95, 96, 152, 156, 157, 158, 205, 207, 229, 237 Kotik, J., 155, 156, 170, 177, 207, 208, 226, 230, 236, 237 Kotschin, N., 316, 323, 329 Kozlov, V. F., 33, 59, 91 Kracht, A., 187, 202, 204, 233, 238 Kropnick, P. M., 71, 90 Kuo, H. H., 74, 75, 78, 91 Kuo, H. L., 262, 271, 277, 278, 281, 292, 299, 302, 303, 306, 329
L Lackenby, H., 100, 168, 238 Laitone, E. V., 149, 152, 157, 159, 161, 168, 170, 171, 172, 243 Landweber, L., 104, 106, 114, 117, 124, 238, 239, 242 Lavrent’ev, V. M., 164, 238 Lee, A. Y. C., 204, 238 Lighthill, M. J., 131, 238 Lin, W., 175, 209, 213, 238 Lipps, F. B., 284, 287, 289, 290, 329 Lorenz, E., 305, 313, 329 Lunde, J. K., 95, 96, 152, 157, 158, 159, 181, 186, 238, 245 Lurye, J., 207, 236 Lurye, J. R., 166, 238 Lynn, R. J., 25, 26, 91
M MacKinnon, R. F., 161, 243 Makoto, O., 178, 193, 238 Mangulis, V., 226, 237 Margenau, H., 7, 91 Maruo, H., 96, 131, 132, 159, 169, 170, 198, 205, 207, 214, 221, 222, 226, 238, 239, 240 Michalke, A., 290, 329 Michell, J. H., 95, 131, 155, 171, 239 Michelsen, F. C . , 156, 175, 239 Miles, J. W., 277, 281, 293, 329 Milgram, J. H., 165, 239 Moffitt, F. H., 210, 213, 231 Moran, D. D., 124, 239 Morgan, G. W., 49, 53, 91
33:
Morgan, R., 226, 237 Muschner, W., 217, 236 Munk, W. H., 32, 41, 42. 51. 53, 91 Murphy, G. M., 7, 91
N Naegle, J. N., 203, 204, 241 Nagamatsu, T., 158, 242 Nakatake, K., 227, 239 Neamtan, S. M., 268, 329 Needler, G. T., 33, 58, 60, 61, 65, 66, 67, 68, 91 Newman, D. J., 207, 208, 237 Newman, J. N., 119, 120, 124, 132, 138, 155, 158, 228, 239 Niiler, P. A.. 50, 91 Nikitin, A. K., 166, 173, 23’, 239 Nitta, T., 283, 330 Norscini, R., 292, 305, 328 Nozawa, K., 131, 236 0
Obukhov, A. K., 250, 251, 329 Ogilvie, T. F., 138, 141, 142, 216, 228, 240 Ogiaara, S., 226, 240 Ogura, M., 131, 242 Omata, S., 131, 242 Orlanski, I., 317, 329 Ostlund, H. G., 71, 92
P Palladina, 0. M., 229, 240 Palmen, E., 273, 305, 329 Paulling, J. R., 213, 238 Pavlenko, G. E., 205, 240 Pedlosky, J., 75, 92, 261, 277, 281, 308, 316, 329 Perzhnyanko, E. A., 161, 240 Peters, A. S., 132, 133, 134, 138, 220, 240 Pfeffer, R. L., 305, 330 Phillips, N. A., 15, 17, 30, 33, 91, 92, 298, 330 Poole, F. A. P., 158, 239
Author Index
334 R
Raethjen, P., 250, 330 Ramsey, A. S., 5, 92 Reid, J. L., 25, 26, 91, 92 Riehl, H., 286, 330 Rispin, P., 170, 242 Robinson, A. R., 33, 49, 50, 53, 58, 90, 91, 92 Rooth, C., 71, 86, 92 Rossby, C. G., 250, 330
T Tanaka, H., 131, 242 Tatinclaux, J.-C., 126, 167, 242 Thompson, P. D., 298, 330 Thomsen, C1.-P., 214, 242 Thomsen, P., 170, 237 Thomson, Sir W., 171, 242 Timrnan, R., 173, 207, 242 Todd, M. A,, 187,189, 244 Tuck, E. O., 132, 169, 171, 172, 173, 174, 215, 216, 242 Tulin, M. P., 106, 228, 231, 242 Tzou, K. T . S., 106, 114, 117, 238, 242
S
Sabuncu, T., 95, 96, 162, 240 Salvesen, N., 216, 217, 218, 240 Schuster, S., 181, 185, 186, 188, 198, 199, 200, 201, 244 Sclater, J. G., 71, 90 Sedov, L. I., 158, 237 Sharma, S. D., 95, 105, 106, 108, 117, 120, 121, 122, 123, 124, 127, 128, 129, 130, 154, 157, 160, 161, 188, 192, 193, 196, 203, 204, 205, 231, 232, 240, 241 Shearer, J. R., 100, 183, 189, 190, 191, 225, 241 Shebalov, A. N., 159, 160, 231, 241 Shkurkina, Z. M., 161, 241 Shor, S. W. W., 214, 241 Simons, T. J., 300, 330 Sizov, V. G., 221, 241 Smith, A. M. O., 215, 218, 233 Smorodin, A. I., 173, 241 Snyder, J. D., III., 126, 243 Solberg, H., 316, 328, 330 Song, R. T., 330 Spiegel, S. L., 50, 91 Sretenskii, L. N., 132, 157, 158, 159, 161, 162, 166, 214, 241 Steele, B. N., 100, 241 Stern, M., 261, 328 Stoker, J. J . , 132, 133, 134, 138, 171, 172, 220, 240, 242, 327, 329 Stornrnel, H., 3, 32, 41, 43, 53, 73, 75, 92 Stuart, J. T., 315, 330 Sverdrup, H. U., 18, 19, 21, 92 Swallow, J. C., 30, 53, 92
U Ueno, K., 158, 242 Ursell, F., 172, 242 Uspenskii, P. N., 162, 242
V Veronis, G., 33, 49, 53, 54, 55, 67, 68, 71, 74, 75, 78, 80, 81, 91, 92, 250, 256, 330 von Arx, W. S., 75, 92 Voitkunskii, Ya. I., 158, 189, 194, 230 Vossers, G., 132, 207, 209, 242, 244
W Wang, D. P., 170, 242 Ward, L. W., 95, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 154, 232, 242, 243 Warren, B., 56, 92 Warren, F. W. G., 161, 173, 243 Watson, J., 315, 330 Webster, F., 42, 92 Webster, W. C . , 167, 168, 175, 176, 209, 238, 243 Wehausen, J. V., 96, 138, 149, 152, 155, 157, 159, 161, 168, 170, 171, 172, 175, 186, 187, 188, 189, 209, 210, 213, 221, 224, 231, 236, 238, 243
.
Author Index Weinblum, G . , 96, 100, 156, 174, 175, 178, 181, 183, 185, 186, 187, 188, 189, 196, 198, 199, 200, 201, 202, 205, 209, 219, 233, 243, 244 Weiss, R. F., 71, 90 Welander, P., 33, 50, 58, 60, 61, 64, 68, 70, 92 Wells, W. H., 173, 174, 242 Wetterling, W., 174, 232 Wigley, W. C. S., 95, 96, 155, 164, 165, 174, 178, 179, 180, 181, 183, 186, 189, 198, 202, 244, 245 Williams, G. P., 306, 330 Wong, K. K., 157, 172, 236
335
Worthington, L. V., 53, 92 Wu, J., 104, 106, 238, 245 Wu, T. Y., 167, 228, 245 Wustrau, D., 209, 244
Y Yamazaki, Y., 131, 242 Yanai, M., 283, 286, 330 Yang, C. C., 75, 80, 92 Yeung, R. W., 189, 190, 245 Yim, B., 157, 204, 208, 224, 245 Yokoyama, N., 226, 245
Subject Index A
Boundary-layer-plus-wake region, in wave resistance, 106-107 Boussinesq equations or approximation in baroclinic disturbances, 291-296, 308 in large-scale ocean circulation studies, 18-27 momentum conservation and, 21-23 and motions of intermediate scale, 31-32 scaling and, 28-33 small-scale motions and, 30-31 use and limitations of, 23-27 Boussinesq fluid, temperature-salinity relations and, 56 Bulb-optimization scheme, 204 Bulbous bows, wave resistance and, 202-205
Abyssal circulation, 72-75 Adiabatic hydrostatic field, 19-20 Angular momentum, Coriolis acceleration and, 17 Antarctica, circulation around, 74 Asymmetric models, wave resistance differences in, 179-180 Atmosphere stratification in, 258 vorticity waves in, 248
B Baroclinic disturbances bounded layer in, 304-306 eigenvalues of, 303 pure, 290-306 semicircle theorem in, 279-281 structure of, 295-296 truncated power series solution in, 298-300 Baroclinic system, general, 300-306 Baroclinic waves, surface friction and, 296 Barotropic flow, stability conditions for, 277-281 Barotropic mode diffusion in, 69 ideal-fluid model and, 67 in thermohaline circulation, 65-68 Barotropic zonal currents, stability characteristics of, 281-290 Beam-length ratios, in wave resistance, 193 Bickley jet, stability of, 283-285 Bjerknes-Solberg cyclone model, 316 Block coefficient, in wave resistance, 181
L
Canal finite-depth fluids and, 194-196 longitudinal profiles and, 124-125 rectangular, 158-159 transverse profiles in, 110-1 14 Circumpolar Current, 74 Conservation of momentum, equations for, 7 Coriolis force or acceleration, 14-18, 249 angular momentum of, 17-18 components of, 15 pressure gradient and, 255 Current systems, modeling of, 42-56 Cyclone extratropical, 316 frontal, 250 instability of, 250 336
337
Subject Index D Dead-water resistance, 163 Deformation radius, geostrophic balance and, 253 Dense fluid, chaotic behavior of, 90 Dense water Ekman layer and, 84-85 as source of flow, 81-90 Diffusion thermohaline circulation and, 68-71 vertical, 69-70 Downwelling region, convergence in, 70
E Earth atmosphere of, 248, 258 ellipticity of, 2-5, 12 as oblate spheroid, 6-7 Eddy resistance, wave resistance and, 94 Ekman boundary layer solution, 275 Ekman flow, 275 Ekman fluxes, in thermal wind relation, 51 Ekman layers, 2, 48-49, 52, 57-58, 65, 75, 80, 84-86, 90, 275 depth of, 38, 41 frictional dissipation and, 3 6 4 0 transport in, 40, 47 Ekman number, 78 Ekman pumping or suction, 39-41, 4 7 4 9 , 58, 65 Ellipticity, of earth, 2-5 general expression for, 11 gravitational bulge and, 6 Equation of state, pressure-fluctuation term in, 22
F Finite-amplitude disturbances, in geostrophic flows, 306-316 “ Flat” ships, wave resistance for, 170-171 Flow, quasigeostrophic, see Quasigeostrophic flow Flow variables, potential vorticity equation and, 260-263
Fluid Boussinesq system and, 23-27 conservation of momentum for, 21 finite-depth, 194-1 96 geostrophic balance in, 249 low-frequency disturbances in, 249 multicomponent, 19 rotating, 254 single-component, 23-24 stratified, 161-162 transformation of motion equations for, 6-14 unbounded, 114-124 vorticity of, 15 Fluid mass, equilibrium figure for selfgravitating and rotating form of, 3-6 Fourier method, vs. Green’s functions, 151 Framelines, wave resistance and, 210 Free-boundary surface, irrotational flow and, 110 Free-wave potential, transverse profiles and, 112 Free-wave spectra disturbance and, 123 measured vs. calculated, 192 wave resistance and, 130 Frontal cyclone theory, 316 Frontal dissipation, Ekman layers and, 36-42 Frontal waves basic state in, 317-318 eigenvalues for, 326 instability theory of, 316-326 nonlinear development of, 327 perturbation equations and boundary conditions in, 318-321 solutions for, 322-327 Froude number, 77, 98, 155, 178 wave-breaking resistance and, 109 wave resistance and, 97, 196, 225
G Gegenbauer polynomials, 175 Geostrophic balance, pressure and nondivergent flow fields in, 252-255 Geostrophic flow, 33-36 vorticity of, 260
338
Subject Index
Geostrophic-hydrostatic flow, 57 equations for, 63 Gravitational bulge, ellipticity and, 5-6 Gravitational resistance, wave resistance and, 94 Gravity potential, 10 Greenland Sea, 74, 86 Green’s functions finite depth and, 157-158 in motion with acceleration, 159-160 thin-ship theory and, 164 in wave-resistance solutions, 148-152, 221-222, 226 Gulf Stream, 30, 42, 48, 52-54, 56, 80 Gyres in North Atlantic, 80 wind-driven, 54, 56 wind stresses and, 42, 48
H Height, vorticity and, 77 Hidaka-Munk lateral friction model, 78 Hull shape Michell resistance and, 201 wave resistance and, 94, 200-201, 210-214 Hydrodynamic equations quasigeostrophic flow and, 257-265 scaling of, 259-260 Hydrostatic equation, Stommel’s model of, 50 Hyperbolic-tangent zonal wind profile, disturbance in, 285-290 Hyperboloids of one sheet, 8
I Ideal fluid model, barotropic mode and, 67 Ideal-fluid thermocline, 63-65 Indian Ocean, 74 Instability theory, quasigeostrophic flows and, 247-327 see also Quasigeostrophic flows Intertropical convergence zone, defined, 285-286 Inviscid finite-amplitude disturbance, 310-313
Inviscid fluid exact problem in, 134-137 formula for, 102-103 irrotational flow of, 105 wave resistance and momentum in, 100-101, 219 Irrotational flow free-surface boundary and, 110 of inviscid fluid, 105 velocity potential and, 163 Isopycnals for ideal-fluid thermocline, 64 for thermohaline circulation, 61
K Kochin’s function Green’s function and, 160 in wave-resistance solutions, 153-1 54 Kutta condition, hydrofoil and, 141
L Laguerre polynomial, 301 Laplace equation, transverse canal profiles and, 110-111 Large-scale ocean circulation, 1-90 abyssal circulation and, 72-75 adiabatic hydrostatic field and, 19-20 Boussinesq equations and, 28-33 dense-water source flow and, 81-90 geostrophic flow and, 33-36 hydrostatic and geographic flow in, 29 laboratory simulation of, 75-90 modeling of current systems in, 42-56 simulation basis in, 75-78 thermodynamic simplifications in, 18-27 thermohaline circulation and, 56-72 Legendre polynomial, 175 Longitudinal profiles in canals, 124-125 in unbounded fluid. 117-124
M Mass, conservation of, 36 Meridonal transport, equation for, 47
Subject Index Michell’s integral, 170, 174-175, 177-189 mathematical properties of, 154-156 for wave resistance, 145 Michell potential, 227 Michell wave resistance, 165, 168, 186-187, 213, 224, 227 equation for, 178 hull shapes and, 201 Momentum, conservation of, 7, 21 Motion equations for oblate spheroidal coordinates, 9-10 transformation of, for fluids, 6-14 Moving pressure distributions, wave resistance and, 141, 157
N Navier-Stokes equations, 94, 101-102, 126 viscosity and, 166 wake and, 167 Newtonian gravitational potential, 4 Nondivergent flow field, geostrophic balance and, 252-255 Nonlinear potential vorticity equations, solutions of, 265-272 North Atlantic gyre, 80
0 Oblate spheroidal coordinates earth and, 7-8 motion equations for, 9-10 Ocean circulation, 1-90 see also Large-scale ocean circulation ; Seawater dense-water source flow and, 81-90 Ekman layers in, 4&42 frictional dissipation and, 36-42 inertial effects in, 53-56 large-scale motion in, 32-33 modeling of current systems and, 42-56 quasigeostrophic balance in, 34 small-scale motions in, 30-31 wind-driven, 31, 43-49, 53-56, 78 Oceanic atmospheric gravitational tides, 248 Oceanic thermal tides, 248 Oceanographic studies, spherical coordinates in, 14
339 P
Pacific Ocean, stagnation points in, 74 Perturbation expansion methods, 137-142 in wind-driven ocean circulation studies, 54 Pie-shaped basin, steady flow and, 76-79 Planetary vorticity, 46 change of, 44 Potential vorticity, conservation of, 15-1 6 Potential vorticity equation, flow variables and, 260-263 Pressure distribution in rotating fluid, 254 wave resistance and, 197-198 Pressure equation, in therrnohaline circulation, 57-58 Pressure field, geostrophic balance and, 252-2 55 Pressure fluctuation time, in state equation, 22 Pressure gradient, Coriolis force and, 2.55 Prismatic coefficient, in wave resistance, 181, 184-185 Prolate spheroid, wave resistance for, 220
Q Quasigeostrophic flows baroclinic disturbances and, 290-306 boundary conditions in, 263-265 conditions for, 35 constant f model, 292-296 defined, 249 finite-amplitude instable disturbance in, 306-316 general baroclinic system and, 300-306 general stability theory and, 276-281 hydrodynamic equations and, 257-265 instability of frontal waves and, 316-326 instability theory and, 247-327 permanent-wave solutions in, 265-272 stability conditions for, 277-281 thermodynamic variables in, 258 vertical function and eigenvalues of, 268-272 viscous equilibration in, 314-316 wave equation and, 256-257 zonal-current stabilities in, 272-276 Quasigeostrophic potential vorticity equation, 262-263
340
Subject Index R
Rankine ovoid, dead-water resistance and, 163 Rectangular canals, motion in, 158 Residuary resistance, 213 Resistance coefficient, in wave-resistance computations, 182-183 Reynolds number, 81-83, 98-99 Reynolds stress, 107-108 Rossby-Haurwitz waves, 283 Rossby number, 28, 80, 260 Rossby parameter, 250, 274, 281-290 Rossby vorticity waves, 248, 284 Rotating fluid, pressure and velocity distributions in, 254 Rotating system, linear frictionless flow in. 33
S
Salinity conservation of, 22-23 variations in, 24-25 Schoenherr Line, 99-100 Seawater density of, 24-25, 27 isopycnals for, 25 potential density for, 26 properties of, 24-25 stability and instability factors in, 25 temperature-salinity relations in, 25-27, 56 Self-gravitating fluid mass, equilibrium figure for, 3-6 Semicircle theorem, in baroclinic disturbances, 279-281 Shallow-water wave resistance, 171-172 Ship boundary layer and wake of, 94 frictional resistance of, 94 wake of, 94 (see also Wake; Wake resistance) wave resistance of, 93-229 (see also Wave resistance) Ship-model tester’s dilemma, 97 Similarity solutions, in thermohaline circulations, 57-62, 66 Sinkage and trim, wave resistance and, 189
Sink-driven flow, dense water and, 82-83 Sinus profile, stability and, 281-282 Slender-body wave resistance, 169-170 Source flow dense-water, 81-90 in pie-shaped basin, 75-79 Source-sink flows, models of, 75-77 Spherical coordinates, in oceanographic studies, 14 Spherical mass, potential due to, 5-6 Stability theory, in geostrophic flow, 276-281 Steady flow, in pie-shaped basin, 76 Stewartson layers, 78 Stieltjes integrals, 169 Stommel model, 43-49 extensions of, 44, 49-53 Stratified fluids, wave resistance and, 161-162 Submerged bodies, wave resistance of, 196, 215-220 Surface vessels, wave resistance for, 220-228 Sverdrup transport balance, 44, 47 Systematic form changes, wave resistance and, 198-201
T Thermodynamic variables, in quasigeostrophic flows, 258 Thermodynamics, first law of, 22 Thermal tides, oceanic, 248 Thermal wind equations, 35, 51 Thermohaline circulation, 3, 56-72 barotropic mode and, 65-68 boundary conditions in, 58, 71 diffusion and, 68-71 ideal-fluid thermocline and, 63-65 isopycnals for, 61 models of, 71-72 pressure equation in, 57-58 similarity solutions in, 57-62 thermocline and, 72 Thin-ship theory or approximation, 140 for stratified fluids, 162, 164 wake and, 167 wave resistance and, 138, 173-174, 177-178, 187-188
Subject Index Transverse profiles free surface and, 111 in unbounded fluid, 114-117 Two-level system, wave perturbations in, 308-309
U Unbounded fluid longitudinal profiles in, 117-124 transverse profiles in, 114-1 17 Unstable disturbances, in quasigeostrophic flows, 306-31 6
V Velocity distribution, in rotating fluid, 254 Velocity potential, irrotational flow and, 163 Vertical function height and, 271 in quasigeostrophic flow, 268-272 Vertical shear, in quasigeostrophic flow, 313 Vertical stratification, in atmosphere, 258 Viscosity Navier-Stokes equation and, 166 vorticity distribution and, 167 Viscous resistance equivalent, 168 wave resistance and, 94, 106-109 Volumetric coefficient, in wave resistance, 181 Vorticity change of, 44 of geostrophic motion, 262 height and, 77 in large-scale oceanic flows, 15 planetary, 44-46 relative, 77 Vorticity equation, 249-250 Vorticity waves, 248 Vosser’s integral, 169
W Wake rotational, 167-168 viscous resistance and, 168 vorticity distribution and, 167 wave resistance and, 126-127, 165-166
341
Wave-breaking resistance, 94, 109 Wave equation, in quasigeostrophic flows, 256-257 Wave-making resistance, experimental observations in, 127-128 Wave pattern, wave resistance and, 172 Wave-pattern analysis, 109-131 and canal transverse profiles, 110-1 14 Wave perturbations, general equations for, 308-309 Wave profiles calculated vs. theoretical, 191 in wave resistance calculations, 189-194 Wave resistance, 93-229 analytical theory in, 131-229 application of theory in, 198-214 approximating hull in, 176 in asymmetric models, 179-180 beam/length ratios in, 193 Bessel function and, 156 boundary-layer-plus-wake region in, 106-107 bulbous bows and, 202-205 circular path and, 161 computer design and, 211 coordinate system in, 101 “ dead-water ” resistance and, 163 deep water formulation for, 205 defined, 93-94, 108 early papers on, 132-134 eddy resistance and, 94 exact formulation in, 132, 214-215 experimental observations in, 127-1 30 finite depth and, 157-158 finite-depth fluids and canals in, 194-1 96 first- and second-order theory in, 216-217, 223 for “flat” ships, 170-171 Fourier’s method in, 143-148 framelines and waterlines for, 210 free-wave potential and, 112 free-wave spectrum and, 130 frictional resistance and, 94 Froude method in, 96-100 Froude number in, 196, 225 gravity vs. viscosity in, 94 Green’s functions in, 148-152, 157, 221-227 Havelock formula for, 147 higher-order theories of, 214-228
342
Subject Index
hull approximation and, 176-177 hull shape and, 94, 200-201, 211-213 inconsistent approximations and, 141-142 Kochin’s function and, 153-154 measured vs. calculated, 184-185, 195, 218, 220 measurement of, 96-131 method of solution in, 142-152 method of wedges in, 175-176 Michell’s integration for, 145, 154-156, 170, 174-175, 177-189, 205 Michell resistance and, 178, 186-187, 213, 224, 227 Momentum considerations in, 100-109 motion with acceleration in, 159-161 moving pressure distributions and, 141, 157 Navier-Stokes equations and, 101 numerical methods in, 174-177 perturbation expansions in, 137-142 potential flow and, 107 pressure distributions and, 197 prismatic, block, and volumetric coefficients in, 181, 184-185 in rectangular canals, 158-159 residuary resistance and, 96-100, 213 resistance coefficient in, 182-183 Reynolds number and, 99-100 in shallow water, 171-172 in ships of minimum resistance, 205-214 sinkage and trim in, 189 slamming bow and, 211 slender-body approximations and, 141 for slender ships, 169-170 stratified fluids and, 161-163 submerged bodies and, 140-141, 196, 215-220 for submerged prolate spheroid, 220 of submerged sphere, 219 surface-tension effects and, 168-169 for surface vessels, 220-228
systematic form changes and, 198-201 theory vs. experiment in, 177-197 thin-ship approximations and, 138-140, 162, 164, 173, 177-178, 187-188 thin-wing boundary condition and, 131-132 third-order theory in, 217 transverse-cut method in, 125 trim and sinkage in, 223 underwater profile and, 207 viscosity effects and, 164-168 vs. viscous resistance, 106-109 wake and, 126-127, 165-167 waterline waviness and, 209 wave patterns and, 172-174 wave profiles and spectra in, 189-194 wave source and, 131 wedges method in, 175-176 Weinblum formulation for, 207-209 XY-method in, 125-126 zero Froude number approximation in, 226 Weddell Sea, 74 “ Wedges ” method, in wave resistance calculation, 175-1 76 Wind-driven ocean circulation, 78 inertial effects in, 53-56 Wind-stress curl, 47 vanishing of, 56 Wronskian method, 301
X
XY-method, in wave-resistance measurement, 127-128
z Zero Froude number approximation, 226 Zonal currents, stability of in quasigeostrophic flow, 272-276
