Environmental Stratified Flows (Topics in Environmental Fluid Mechanics)

ENVIRONMENTAL STRATIFIED FLOWS

THE KLUWER INTERNATIONAL SERIES

TOPICS IN ENVIRONMENTAL FLUID MECHANICS Series Editors Dr. Philip Chatwin, University of Sheffield, UK Dr. Gedeon Dagan, Tel Aviv University, ISRAEL Dr. John List, California Institute of Technology, USA Dr. Chiang Mei, Massachusetts Institute of Technology, USA Dr. Stuart Savage, McGill University, CANADA Topics for the series include, but are not limited to: Small-to medium scale atmospheric dynamics: turbulence, convection, dispersion, aerosols, buoyant plumes, air pollution over cities Coastal oceanography: air-sea interaction, wave climate, wave interaction with tides, current structures and coastlines, sediment transport and shoreline evolution Estuary dynamics: sediment transport, cohesive sediments, density stratification, salinity intrusion, thermal pollution, dispersion, fluid-mud dynamics, and the effects of flow on the transport of toxic wastes Physical limnology: internal seiches, sediment resuspensions, nutrient distribution, and wind-induced currents Subsurface flow and transport (the unsaturated zone and groundwater): diffusion and dispersion of solutes, fingering, macropore flow, reactive solutes, motion of organics and non-aqueous liquids, volatilization, microbial effects on organics, density effects, colloids motion and effect, and effects of field scale heterogeneity Debris flows, initiated by lava flow from volcanic eruptions; mud flows caused by mountain storms; snow avalanches, granular flows, and evolution of deserts Oil spills on the sea surface and clean-up Indoor contamination: transport of particles in enclosed space, clean room technology, effects of temperature variation Risk assessment: industrial accidents resulting in the release of toxic or flammable gasses, assessment of air and water quality New methods of data acquisition: the use of HF radar, satellites, and Earth Observation Science Stochastic models and Mass transfer Other books in the Series: Coastal and Shelf Sea Modelling, Philip P.G. dyke; ISBN: 0-7923-7995-0 Diffusion in Natural and Porous Media, Peter Grathwohl; ISBN 0-7923-8102-5

ENVIRONMENTAL STRATIFIED FLOWS

edited by

Roger Grimshaw Loughborough University United Kingdom

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

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Contents

Preface 1 Internal Solitary Waves Roger Grimshaw

vii 1

2 Internal Tide Transformation and Oceanic Internal Solitary Waves Peter Holloway, Efim Pelinovsky, Tatiana Talipova

29

3 Atmospheric Internal Solitary Waves James W. Rottman, Roger Grimshaw

61

4 Gravity Currents James W. Rottman, P. F. Linden

89

5 Stratified Flow over Topography Ronald B. Smith

119

6 Turbulence in Stratified Fluids H.J.S. Fernando

161

7 Laboratory Studies of Continuously Stratified Flows past Obstacles Don Boyer, Andjeka Srdic-Mitrovic

191

8 Elements of Instability Theory for Environmental Flows Larry G. Redekopp

223

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Preface

The dynamics of flows in density-stratified fluids has been and remains now an important topic for scientific enquiry. Such flows arise in many contexts, ranging from industrial settings to the oceanic and atmospheric environments. It is the latter topic which is the focus of this book. Both the ocean and atmosphere are characterised by the basic vertical density stratification, and this feature can affect the dynamics on all scales ranging from the micro-scale to the planetary scale. The aim of this book is to provide a “state-of-the-art” account of stratified flows as they are relevant to the ocean and atmosphere with a primary focus on meso-scale phenomena; that is, on phenomena whose time and space scales are such that the density stratification is a dominant effect, so that frictional and diffusive effects on the one hand and the effects of the earth’s rotation on the other hand can be regarded as of less importance. This in turn leads to an emphasis on internal waves. The first three chapters deal with oceanic and atmospheric internal solitary waves, now recognised to be a highly significant component of the dynamics of the coastal ocean on the one hand, and the atmospheric boundary layer on the other hand. In the first chapter Roger Grimshaw reviews current theoretical models of oceanic and atmospheric internal solitary waves, emphasising the pivotal role of model evolution equations of the Korteweg-de Vries type. Then, in the second chapter it Peter Holloway, Efim Pelinovsky and Tatiana Talipova discuss both the theory and observations of oceanic internal solitary waves, while in the third chapter Jim Rottman and Roger Grimshaw do likewise for atmospheric solitary waves. The closely related topic of gravity currents and internal bores is then reviewed in the fourth chapter by Jim Rottman and Paul Linden. Then, in chapter five Ron Smith reviews theoretical models for internal waves generated by flow over mountains. Inevitably density-stratified flows can be turbulent and this issue is addressed in chapter six by it Joe Fernando. In density-stratified flows as elswhere in fluid mechanics there is much to be learned from laboratory studies and so in chapter seven Don Boyer and It Andjelka Srdic-Mitrovic review laboratory studies of the flow of stratified fluids past obstacles. Then in chapter eight Larry Redekopp provides a comprehensive review and tutorial of the stability theory of stratified shear flows. ROGER G RIMSHAW

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Chapter 1 INTERNAL SOLITARY WAVES Roger Grimshaw Loughborough University, UK

Abstract

1.

The basic theory of internal solitary waves is developed, with the main emphasis on environmental situations, such as the many occurrences of such waves in shallow coastal seas and in the atmospheric boundary layer. Commencing with the equations of motion for an inviscid, incompressible density-stratified fluid, we describe asymptotic reductions to model long-wave equations, such as the well-known Korteweg-de Vries equation. We then describe various solitary wave solutions, and propose a variable-coefficient extended Korteweg-de Vries equations as an appropriate evolution equation to describe internal solitary waves in environmental situations, when the effects of a variable background and dissipation need to be taken into account.

INTRODUCTION

Solitary waves are finite-amplitude waves of permanent form which owe their existence to a balance between nonlinear wave-steepening processes and linear wave dispersion. Typically, they consist of a single isolated wave of elevation, or depression, depending on the background state, whose speed is an increasing function of the amplitude. They are ubiquitous, and in particular internal solitary waves are a commonly occuring feature in the stratified flows of coastal seas, fjords and lakes (see, for instance, the reviews by Apel (1980, 1995) and Ostrovsky and Stepanyants (1989), as well as Chapter 2 of this monograph), and in the atmospheric boundary layer (see, for instance, the reviews by Smith (1988) and Christie (1989), as well as Chapter 3 of this monograph). Moreover, solitary waves are notable, not only because of their widespread occurrence, but also because they can be described by certain generic model equations which are either integrable, or close to integrability. The most

2 notable example in this context is the now famous Korteweg-de Vries equation, which will figure prominently in the sequel. In this Chapter, our aim is to develop appropriate model equations to describe internal solitary waves, and indicate, albeit rather briefly, some of their more salient properties. In the next section we will demonstrate how canonical model equations can be systematically derived from the complete fluid equations of motion for an inviscid, incompressible, density-stratified, fluid, with boundary conditions appropriate to an oceanic situation. The modifications necessary to model the lower atmosphere are readily made, and will be taken up in Chapter 3 of this monograph. Our main focus is on the Korteweg-de Vries equation, but importantly, in order to account for the large amplitudes sometimes observed, we extend this model to the extended Korteweg-de Vries equation which contains both quadratic and cubic nonlinearity. We shall describe the solitary wave solutions of these equations before turning, in the third section, to the modifications necessary to incorporate the effects of a variable background environment and dissipative processes. The outcome is a variable-coefficient extended Korteweg-de Vries equation. In general this model equation needs to be solved numerically, but to give some insight into the nature of the solutions, we describe a particular class of asymptotic solutions describing a slowly-varying solitary wave. This section also contains a brief account of unsteady “undular bores”, insofar as they can be described by the Korteweg-de Vries equation. The Chapter concludes with a discussion of some outstanding issues.

2.

LONG WAVE MODELS

2.1

Governing Equations

We shall begin by considering an inviscid, incompressible fluid which is bounded above by a free surface and below by a flat rigid boundary (see Figure 1). Initially we shall suppose that the flow is two-dimensional and can be described by the spatial coordinates where is horizontal and is vertical. This configuration is appropriate for the modelling of internal solitary waves in coastal seas, and to some extent in straits, fjords or lakes provided that the effect of lateral boundaries can be ignored. The extensions to this basic model needed to incorporate these lateral effects, the effects of a horizontally variable background state, and various dissipative processes, will be described later in this chapter. The modifications needed to adapt this model to describe atmospheric solitary waves will be developed in Chapter 3. Here, in the basic state the fluid has density a corresponding pressure such that describes the basic hydrostatic

3

equilibrium, and a horizontal shear flow in the Then, in standard notation, the equations of motion relative to this basic state are

Here

are the velocity components in the directions, is the density, is the pressure and is time. is the bouyancy frequency, defined by

The boundary conditions are

and

Here, the fluid has undisturbed constant depth and is the displacement of the free surface from its undisturbed position Note that the effect of the earth’s rotation has been neglected at this stage. To describe internal solitary waves we seek solutions whose horizontal length scales are much greater than and whose time scales are much greater than We shall also assume that the waves have small amplitude. Then the dominant balance is obtained by equating to zero the terms on the left-hand side of (2.1a-d); together with the linearization of the free surface boundary conditions we then obtain the set of equations describing linear long wave theory. To proceed it is useful to use the vertical particle displacement as the primary dependent variable. It is defined by Note that it then follows that the perturbation density field is given by where we have assumed that as the density field relaxes to its basic state. The isopycnal surfaces (i.e. are then given by where is the level as In terms of the kinematic boundary condition (2.3c) becomes simply

4

Linear long wave theory is now obtained by omitting the right-hand side of equations (2.1a-d), and simultaneously linearising boundary conditions (2.3b,c). Solutions are sought in the form

while the remaining dependent variables are then given by

and

Here c is the linear long wave speed, and the modal functions denned by the boundary-value problem,

are

and

Typically, the boundary-value problem (2.7a-c) defines an infinite sequence of modes, with corresponding speeds Here, the superscript indicates waves with and respectively. We shall confine our attention to these regular modes, and consider only stable shear flows. Nevertheless, we note that there may also exist singular modes with for which an analogous theory can be developed (Maslowe and Redekopp, 1980). Note that it is useful to let denote the surface gravity waves for which scales with and then denotes the internal gravity waves for which scales with In general, the boundary-value problem (2.7a-c) is readily solved numerically. Typically, have extremal points in the interior of the fluid, and vanish near (and, of course, also at

2.2

Time Evolution

It can now be shown that, within the context of linear long wave theory, any localised initial disturbance will evolve into a set of outwardly propogating modes, each given by an expression of the form (2.5). Indeed, it can be shown that the solution of the linearised long wave equations is given asymptotically by

5

Here the amplitudes tions,

are determined in terms of the initial condi-

by the integral expressions,

where

Assuming thats the speeds of each mode are sufficiently distinct, it is sufficient for large times to consider just a single mode. Henceforth, we shall omit the indices and assume that the mode has speed amplitude A and modal function Then, as time increases, we expect the hitherto neglected nonlinear terms to have an effect, and to cause wave steepening. However, this is opposed by the terms representing linear wave dispersion, also neglected in the linear long wave theory. We expect a balance between these two effects to emerge as time increases. It is now well-known that the outcome is the Korteweg-de Vries (KdV) equation, or a related equation, for the wave amplitude. The formal derivation of the evolution equation requires the introduction of the small parameters, and respectively characterising the wave amplitude and dispersion. A KdV balance requires with a corresponding timescale of The asymptotic analysis required is well understood (e.g. Benney (1996), Lee and Beardsley (1974), Ostrovsky (1978), Maslowe and Redekopp (1980), Grimshaw (1981a), Tung et al (1981)), so we shall give only a brief outline here. We introduce the scaled variables and then let with similar expressions analgous to (2.6a-c) for the other dependent variables. At leading order, we get the linear long wave theory for the modal function and the speed defined by (2.7a-c). Note that since the modal equation is homogeneous, we are free to impose a normalization condition on A commonly used condition is that where achieves a maximum value at In this case the amplitude is uniquely defined as the amplitude of at the depth Then, at the next order, we obtain the equation for

6

Here the inhomogeneous terms and and are given by

are known in terms of

Note that the left-hand side of the equations (2.13a-c) is identical to the equations defining the modal function (i.e. (2.7a-c)), and hence can be solved only if a certain compatibility condition is satisfied. To obtain this compatibility condition, we first note that a formal solution of (2.13a) which satisfies the boundary condition (2.13b) is

where Here is a solution of the modal equation (2.7a) which is linearly independent of and so, in particular W (2.15b) is the Wronskian of these two solutions, and is a constant independent of Indeed, the expression (2.15b) can then be used to obtain explicitly in terms of The homogeneous part of the expression (2.15a) for introduces the second-order amplitude Next, we insist that the expression (2.15a) for should satisfy the boundary condition (2.13c). The result is the compatibility condition

Note that the amplitude is left undetermined at this stage. Substituting the expressions (2.14a,b) into (2.16) we obtain the required evolution equation for A, namely the KdV equation

Here, the coefficients

and

are given by

7 where

Note that here I is just with te subscript and superscript omitted. Confining attention to waves propagating to the right, so that we see that I and are always positive. Further, if we normalise the first internal modal function so that it is positive at its extremal point, then it is readily shown that for the usual situation of a near-surface pycnocline, is negative for this first internal mode. However, in general can take either sign, and in some special situations may even be zero. Explicit evaluation of the coefficients and requires knowledge of the modal function, and hence they are usually evaluated numerically. Proceeding to the next highest order will yield an equation set analogous to (2.13a-c) for whose compatibility condition then determines an evolution equation for the second-order amplitude We shall not give details here, but note that using the transformation and then combining the KdV equation (2.17) with the evolution equation for will lead to a higher-order KdV equation for A, in which the right-hand side of the KdV equation (2.17) contains terms proportional to and (see for instance, Gear and Grimshaw (1983), Lamb and Yan (1996), and Grimshaw et al (1997)). A particularly impotant special case of the higher-order KdV equation arises when the nonlinear coefficient (2.18a) in the KdV equation is close to zero. In this situation, the cubic nonlinear term in the higherorder KdV equation is the most important higher-order term. The KdV equation (2.17) may then be replaced by the extended KdV equation,

For a rescaling is needed and the optimal choice is to assume that is and then replace A with In effect the amplitude parameter is in place of The coefficient of the cubic nonlinear term is given by

where

8

and

Note here that, although the terms with coefficients or can be omitted in the asymptotic limit it is useful in practice to retain them so that this expression for remains valid even when is not small. The function is determined from the equation set (2.20b-d), which can be recognised as an inhomogeneous form of the modal equation set (2.7a-c). Indeed, it is readily seen from (2.13a-c) that

where the function also satisfies an inhomogeneous form of the modal eqution, analogous to (2.20b-d), but with the right-hand side of (2.20b) replaced with and the right-hand side of (2.20d) replaced by Of course here, we must use the compatibility condition (2.16), which is just the KdV equation (2.17), to eliminate from and But now we see that the equation set (2.20b-d) does not define uniquely, and hence (2.20a) is not unique either. Indeed we can always add a term to which has the effect of adding a term to But this is just equivalent to the transformation and it is then or readily verified that this will asymptotically transform (2.19) into itself with replaced by Thus, the lack of uniqueness in is related to a lack of uniqueness in or equivalently in The remedy is that we are free to impose an extra condition on For instance, if we suppose that at then it follows that say, where

where is the right-hand side of (2.20b), and we recall that is defined by (2.15b). The expression (2.22) is readily evaluated numerically, and is consequently recommended as a standard for the calculation of However, if an alternative condition is required, then it can readily be found by adding a term to and using the new condition to determine For instance, it is sometimes useful to require that (and also vanish at where we recall that and locates the maximum value of In this case we simply have that and then Thus, and the amplitude

9

is uniquely defined as the amplitude of (to at the depth In some atmospheric and oceanic applications, the depth is not necessarily small relative to the horizontal length scale of the solitary wave, but nevertheless the density stratification is effectively confined to a thin layer of depth which is much shorter that the horizontal length scales. In this case, a different theory is needed, and was first developed by Benjamin (1967) and Davis and Acrivos (1967). Several variants are possible, so, to be specific, we shall describe an oceanic case when and vary only in a near-surface layer of depth below which (a constant) and while the ocean bottom is now given by (i.e. TheTmodal function is again defined by (2.7a,c) but the bottom boundary condition (2.7b) is now replaced by a matching condition that as To derive the evolution equation, we again use the asymptotic expansion (2.12) but now with and restricted to the near-surface layer. This expansion is matched to an appropriate solution in the deep-fluid region where Laplace’s equation holds at leading order. The outcome is the intermediate long-wave (ILW) equation (Kubota et al (1978), Maslowe and Redekopp (1980), Grimshaw (1981a), Tung at al (1981),

where

and

Here the nonlinear coefficient is again given by (2.18a) with now replaced by while the dispersive coefficient is defined by In the limit on the integrand of (2.21b) and (2.21a) becomes the Benjamin-Ono (BO) equation. In the opposite limit (2.23a) reduces to a KdV equation. An important variant of the ILW equation (2.23a) arises when it is supposed that the deep ocean is infinitely deep and weakly stratified, with a constant buoyancy frequency Then the operator in (2.21a) is replaced by (Maslowe and Redekopp (1980), Grimshaw (1981b))

10 where Now internal gravity waves can propagate vertically in the deep fluid region, and to ensure that these waves are outgoing, a radiation condition is needed. Thus is either real and positive for or isign for As (2.24) becomes the BO equation.

2.3

Solitary Waves

Each of the evolution equations (viz. the KdV equation (2.17), the extended KdV equation (2.19) and the ILW equation (2.23a)) are exactly integrable (see, for instance, Ablowitz and Segur (1981), or Dodd et al (1982)), with the consequence that the initial-value problem with a localised initial condition is exactly solvable. But note that the variant (2.24) is not integrable. The most important implication of this integrability from the perspective of this monograph is that an arbitrary initial disturbance will evolve into a finite number (N) of solitary waves (called solitons in this context) and an oscillatory decaying tail. This, together with the robust stability properties of solitary waves, explains why internal solitary waves are so commonly observed. Note that because solitary waves typically have speeds which increase with the wave amplitude, the N waves are rank-ordered by amplitude as Also, to produce solitary waves at all, the initial disturbance should have the correct polarity (e.g. for the case of the KdV equation (2.17)). A typical solution of the KdV equation showing the generation of solitary waves is shown in Figure 2. Note that, in applications the initial condition for the evolution equation is found by first solving the linear long wave equations, and then identifying the mode of interest. Thus is given by (2.10a) in terms of the actual initial conditions (2.9). It follows from the proceeding discussion that in describing the solution of the evolution equations, the most important step is to determine the solitary wave solution. For the KdV equation (2.17) this is given by

where Note that the speed V is for the phase variable and the actual total speed is Since the dispersion coefficient is always positive for right-going waves, it follows that these solitary waves are always supercritical (V > 0), and are waves of elevation or depression according as We also see that is proportional to and hence the larger waves are not only faster, but narrower.

11

For the extended KdV equation (2.19) the corresponding solitary wave is given by (Kakutani and Yamasaki (1978), Gear and Grimshaw (1983)),

where

and

Here we recall that is in (2.19), and so A is rescaled to There are two cases to consider. If then there is a single family of solutions such that and As b increases from 0 to 1, the amplitude increases from 0 to a maximum of while the speed V also increases from 0 to a maximum of In the limiting case when the solution (2.26a) describes the so-called “thick” solitary wave, which has a flat crest of amplitude and is terminated at each end by the bore-like solutions

where In the case solitary wave solutions of (2.19) when travelling bore solution

there are no exact but instead there is the

where

Note that this solution could be derived by observing that the transformation converts the extended KdV equation (2.19) into the modified KdV equation (i.e. (2.19) with and then utilising the solution (2.27). Note here that the amplitude of the travelling bore is a free parameter, and that the speed V < 0. For the case when and there are two families of solitary waves. One is defined by has and as decreases from 0 t o – 1, the amplitude increases from while the speed V also increases from The other is defined by has and, as b increases from the amplitude increases

12

to from and are given by

Inthis case solitary waves exist if

where

On the other hand, as reduces to the algebraic form

and the solitary wave (2.24a)

For the ILW equation (2.23a), the solitary wave solution is (Joseph, 1977)

where

In the limit and algebraic BO solitary wave

so that (2.31a,b) reduces to the

where

3.

BACKGROUND ENVIRONMENTAL EFFECTS

3.1

Generalised Evolution Equations

The KdV equation (2.17) is the basic model for the situation studied in Section 2, when the flow is unidirectional, and the background state is horizontally uniform. Our purpose now is to extend this basic model to situations where there is a variable background environment. This can arise due to a variable depth or due to horizontal variability in the basic density and horizontal velocity field where Here, for simplicity, we are considering the situation when the background variability is unidirectional and in the flow direction. The scaling indicates that we are assuming that the background varies

13 on a length scale which is much greater than that of the solitary waves, but is comparable to the length scale over which the wave field evolves. The modal functions are again defined by (2.7a-c), but now depend parametrically on X , and hence so does the wave speed An asymptotic expansion analogous to (2.12) is then introduced, but the variables and in (2.11) are here replaced by

where we recall that is defined by (2.11). The amplitude can then be shown to satisfy the variable-coefficient extended KdV equation (see, for instance, Grimshaw (1981a,b), Zhou and Grimshaw (1989)),

which thus replaces (2.19). Here the coefficients are defined by (2.18a,b), and the term represents the effects of friction. The significance of the coefficient is that is a measure of the wave action flux in the X–direction, and is a conserved quantity in the absence of dissipation. Dissipation can arise from several sources, such as turbulent mixing in the fluid interior associated with local shear unstability, scattering due to bottom roughness and viscous decay due to the bottom boundary layer. Each of these can be modelled by letting

where is the Fourier transform of A (2.23c). The index determines the type of dissipation; leads to the KdV-Burgers equation since then (with while (with corresponds to linear Rayleigh damping (see Ostrovsky and Soustova (1979) for a discussion of the physical origin of these two cases, and the determination of For a laminar bottom boundary layer and

where

is the kinematic viscosity in the bottom boundary layer, and is formally required to be For a turbulent bottom boundary layer, it is customary to replace the expression (3.3) with

14 and

is an empirical drag coefficient, usually taken to be about 2.5 × and is formally required to be Next we incorporate the effects of wave diffraction, for the case when these are relatively weak. That is, relative to the dominant X– direction there is a weak tendency for the waves to spread in the transverse The appropriate evolution equation is then

Here

where

Here we have also included the effects of the earth’s rotation, represented by the local value of the Coriolis parameter since the internal Rossby radius may well be comparable with other transverse scales. Note that is required to be at least In the absence of these rotation terms and the dissipative term, equation (3.6a) is a variable-coefficient Kadomtsev-Petviashvili (KP) equation. Further, in the absence of any background variation so that (3.6a) has constant coefficients, the KP equation is an integrable equation when and is generally accepted as an appropriate two-dimensional generalisation of the KdV equation. More generally, when the background environment varies in both spatial directions, and through the basic velocity and density fields, possibly in time also, an evolution equation analogous to (3.6a) can be derived (Grimshaw, (1981a)). In this very general situation, the modal functions and the speed c depend parametrically on slow time and horizontal spatial variables. The speed is then used to determine space-time rays which in turn then determine the dominant direction for the wave propagation, so that is a time-like variable along this ray, is a phase variable describing the wave structure and Y is a co-ordinate transverse to the ray. The incorporation of the Coriolis effects is described in Grimshaw (1985). The counterpart of the generalised KdV equation (3.2), or its two-dimensional counterpart, the generalised KP equation (3.6a) can also be derived for deep fluids (e.g. Grimshaw (1981b)), thus providing the appropriate extension of the ILW equation (2.23a).

15

3.2

Deformation of Solitary Waves

In general the gKdV equation (3.2) (or its two-dimensional counterpart (3.6a)) must be solved numerically. However, to gain insight into the expected behaviour of the solitary wave solutions, it is useful to consider the asymptotic construction of the slowly-varying solitary wave solution, in which it is assumed that the background variability and the dissipative effects are sufficiently weak that a solitary wave is able to maintain its structure over long distances. For simplicity, we consider here only the case when the cubic nonlinear term in (3.2) can be ignored, and so put hereafter. In this case a multi-scale perturbation technique (see Grimshaw (1979) or Grimshaw and Mitsudera (1993)) can be used in which the leading term is

where

Here the wave amplitude and hence also are slowlyvarying functions of Their variation is most readily determined by noting that (3.2) possesses an “energy” law,

which expresses conservation of wave action flux in the absence of dissipation. Substitution of (3.7a) into (3.2) gives

Using the relations (3.7b) this is an equation for However, although the slowly-varying solitary wave conserves “energy” it cannot simultaneously conserve mass. Instead, it is accompanied by a trailing shelf of small amplitude but long length scale whose amplitude at the rear of the solitary wave is given by

When the coefficients and are known explicitly as functions of the expressions in (3.9) and (3.10) can also be readily evaluated explicitly. However, usually these coefficients, being determined inter

16

alia from the modal functions, are known only numerically, and hence and can also only be obtained numerically. In the absence of any dissipation (i.e. equation (3.9) shows that is a constant on the ray path, and hence, on using the relations (3.7b) we see that

gives an explicit formula for Further, in this case (3.10) shows that if the wave width increases (decreases) along the ray path, then the trailing shelf amplitude has the opposite (same) polarity to the solitary wave. A situation of particular interest occurs when the coefficient changes sign at some particular location, say In the oceanic environment this commonly occurs as the depth of the ocean decreases, where is typically negative in the deeper water (here we consider waves propagating to the right so that I(2.18c) > 0). In this case, since the dispersive coefficient is always positive (2.18b), it follows from (3.7b) that the solitary wave is a wave of depression when but a wave of elevation when The issue then arises as to how the solitary will behave as (i.e. as and in particular, as to whether a solitary wave of depression can be converted into one or more solitary waves of elevation as the critical point is traversed. This problem has been intensively studied (see, for instance, (Grimshaw et al (1998a) and the references therein), and the solution depends on how rapidly the coefficient changes sign. If passes through zero rapidly compared to the local width of the solitary wave, then the solitary wave is destroyed, and converted into an oscillatory wavetrain. On the other hand, if changes sufficient slowly that the formula (3.11) holds, we see that as so does in proportion to while as and as Thus, as the solitary wave amplitude decreases, the amplitude of the trailing shelf, which has the opposite polarity, grows indefinitely until a point is reached just prior to the critical point where the slowly-varying solitary wave asymptotic theory fails. A combination of this trailing shelf and the distortion of the solitary wave itself then provide the appropriate “initial” condition for one or more solitary waves of the opposite polarity to emerge as the critical point is traversed. However, these conclusions depend on the cubic nonlinear term in (3.2) being negligible in the vicinity of When this is not the case the outcome depends on the sign of at If so that solitary waves of either polarity can exist when then the solitary wave preserves its polarity (i.e. remains a wave of depression) as the critical point is traversed. On the other hand if so that no solitary wave can exist when then the solitary wave of depression may be

17

converted into one or more solitary waves of elevation, or into a lreather solution, or into an oscillatory wavetrain; for more details of this case, see Grimshaw et al (1999). Next we use equation (3.9) to determine the effects of dissipation. Here we assume that the background is uniform so that the coefficients and are all constants. Then, for the case of a laminar bottom boundary layer in (3.3) and equation (3.9) can be solved for to give

where is the initial value of the amplitude and is related to through the expression (3.7b). On the other hand, for a turbulent bottom boundary layer the expression (3.5a) should be used in (3.9) which leads to the expression

For typical oceanic parameters both these expressions give life times which are several orders of magnitude greater than the wave’s intrinsic time scale. A different kind of dissipation can occur in the deep ocean where the governing equation is (2.24). Here the decay is due to the radiation downwards of internal gravity waves. To estimate the rate of decay a slowly-varying solitary wave theory analogous to that described in Section 3.1 needs to be developed. The outcome (see Maslowe and Redekopp (1980), or Grimshaw (1981a)) is that the solitary wave may decay in a time comparable to the wave’s intrinsic time scale. This theoretical prediction may account for the relative scarcity of internal solitary waves observed in the deep ocean. The effects of the earth’s rotation is described by equation (3.6a) where represents the Coriolis parameter; more precisely is the internal Rossby radius. In the absence of any transverse dependence (i.e. and equation (3.6a) is often called the Ostrovsky equation (Ostrovsky (1978)) and has been intensively studied. Even for the case of a uniform background (i.e. the coefficients and are all constants) and no dissipation the Ostrovsky equation possesses no solitary wave solutions when (e.g. Leonov (1981), Gilman et al (1996)). Instead a localised initial condition decays with the radiation of Poincaré waves. For sufficiently small values of the decay of a solitary wave due to this radiation can be calculated explicitly using the slowly-varying solitary wave formulation (Grimshaw et al (1998a)),

18

and it is found that

where we recall that is the initial amplitude and is related to by (3.7b). The formula (3.14) predicts the extinction of the solitary wave in finite time. For typical oceanic parameters, this extinction time is comparable with the life times due to dissipation, and hence the Coriolis effect is a candidate for the eventual decay of oceanic internal waves. However, although there are no solitary wave solutions of the Ostrovsky equation, Gilman et al (1995) showed that there exist periodic solutions, which for sufficiently small have the structure of solitary-like pulses separated by long waves of parabolic shape. In a channel with vertical side walls equation (3.6a) is supplemented with lateral boundary conditions,

In a linearised framework, such side walls can support Kelvin waves whose transverse structure is proportional to However, even in the absence of background variability or dissipation, the nonlinear equation (3.6a) cannot support a steady Kelvin wave. Instead, initial disturbances with a solitary-wave structure in the and a transverse structure of the form inevitably decay due to the radiation of Poincaré waves, while the wave crest develops across-channel curvature with the smaller amplitude part trailing the larger amplitude part (Katsis and Akylas (1987), Akylas (1991), Grimshaw and Tang (1991)).

3.3

Undular Bores

The term “undular bores” is widely used in the literature in a variety of contexts and several different meanings. Here, we need to make it clear from the outset that we are primarily concerned with non-dissipative flows, in which case an undular bore is intrinsically unsteady. In general, an internal undular bore is an oscillatory transition between two basic states whose isopycnal surfaces are at different levels. A simple representation of an internal undular bore can be obtained from the solution of the constant-coefficient KdV equation (2.17) with the initial condition where we assume that and Here is the Heaviside function (i.e. if and if The solution can in principle be obtained through the inverse scattering transform.

19

However, it is more instructive to use the asymptotic method developed by Gurevich and Pitaevskii (1974a), and Whitham (1974). In this approach, the solution of (2.17) with the initial condition (3.16) is represented as the modulated periodic wave train,

where

and

Here is the Jacobian elliptic function of modulus while and are the complete elliptic integrals of the first and second kind respectively. The mean value of A over one period is while the spatial period is As and then (3.17a) becomes the KdV solitary wave (2.25a), relative to the level As where the and spatial period is this is just a sinusoidal wave train relative to the level The asymptotic method of Gurevich and Pitaevskii (1974a) and Whitham (1974) is to let the expression (3.17a) describe a modulated periodic wavetrain in which the amplitude the mean level the speed V and the wavenumber are all slowly varying functions of and The relevant asymptotic solution corresponding to the initial condition (3.16) can now be constructed in terms of the similarity variable and is given by

and

Ahead of the wavetrain where and

A = 0 and at this end, the leading wave is a solitary wave of

20

amplitude

relative to a mean level of 0. Behind the wavetrain where and at this end and the wavetrain is now sinusoidal with a wavenumber given by Further, it can be shown that on any individual crest in the wavetrain, as In this sense, the undular bore evolves into a train of solitary waves. If in the initial condition (3.16), then an “undular bore” solution analogous to that described by (3.17) and (3.18) does not exist. Instead, the asymptotic solution is a rarefraction wave,

Small oscillatory wavetrains are needed to smooth out the discontinuities in at and (for further details, see Gurevich and Pitaevskii (1974)). Let us next suppose that a dissipative term of the form (3.3), or (3.5a), is included in the KdV equation. The consequences for these “undular bore” solutions has been explored by Gurevich and Pitaevskii (1974b), Smyth (1988) and Myint and Grimshaw (1995). The outcome depends sensitively on the form of the dissipation. For instance, for the KdVBurgers equation in (3.3)) a steady-state undular bore solution is now possible (Johnson (1970)) in which the energy flux in the oscillatory wavetrain can be absorbed by dissipation (Benjamin and Lighthill (1954)). However, for other forms of dissipation, such a steady-state solution cannot be obtained, and instead the “undular bore” solution slowly decays. The generation of an undular bore requires an initial condition of the form (3.16), that is, say as A common situation where this typical initial condition can be generated occurs when a steady transcritical flow encounters a topographic obstacle. Here a flow is said to be critical if it can support a wave mode whose speed in the frame of reference of the topographic obstacle. Let us suppose that the bottom boundary of the stratified fluid is given by where and for a KdV balance as before. The speed where is the retuning parameter; defines supercritical (subcritical) flow respectively. Then it was shown by Grimshaw and Smyth (1986) that the KdV equation (2.17) is replaced

21

by the forced KdV (fKdV) equation

where Here, the coefficients and I are defined by (2.18) with Without loss of generality we shall suppose that the oncoming flow is left to right so that indeed, it is sufficient to assume that I (2.18c)< 0. It then follows that and (2.18a) is < 0(> 0) for a solitary wave of elevation (depression). The fKdV equation (3.20a) has been derived in several other physical contexts, and is a canonical model equation to describe transcritical flow interaction with an obstacle. It was first derived in the context of water waves, and indeed equation (3.20a) can describe that case by choosing the mode (see Grimshaw and Smyth (1986) and the references therein). A typical solution of (3.20c) is shown in Figure 3 for exact criticality, when and the obstacle provides a positive, and isolated, forcing term. That is is positive, and non-zero only in a vicinity of X = 0, with a maximum value of Note that for a mode 1 wave, so that (3.20b)< 0. The initial condition is A = 0 at The solution is characterised by upstream and downstream wavetrains connected by a locally steady solution over the obstacle. When the upstream wavetrain weakens, and for sufficiently large detaches from the obstacle, while the downstream wavetrain intensifies and for sufficiently large forms a stationary lee wave field. When the upstream wavetrain develops into well-separated solitary waves while the downstream wavetrain weakens and moves further downstream. For more details see Grimshaw and Smyth (1986) and Smyth (1987). The origin of the upstream and downstream wavetrains can be found in the structure of the locally steady solution over the obstacle. In the transcritical regime this is characterised by a transition from a constant state upstream of the obstacle to a constant state downstream of the obstacle, where and It is readily shown that independently of the details of the forcing term F(X). Explicit determination of and requires some knowledge of the forcing term F(X). However, in the “hydraulic” limit when the linear dispersive term in (3.20a) can be neglected, it is readily shown that This expression also serves to define the transcritical régime, which is

22

Thus upstream of the obstacle there is a transition from the zero state to A-, while downstream the transition is from to 0; each transition is effectively generated at X = 0. Both transitions are resolved by “undular bore” solutions. That in X < 0 is exactly described by (3.17) and (3.18) with replaced by and by (note that now so the condition for an “undular bore” solution becomes instead It occupies the zone

Note that this upstream wavetrain is constrained to be in X < 0, and hence is only fully realised if Combining this criterion with (3.21) and (3.22) defines the régime

where a fully developed undular bore solution can develop upstream. On the other hand, the régime

is where the upstream undular bore is only partially formed, and is attached to the obstacle. In this case the modulus of the Jacobian elliptic function varies from 1 at the leading edge to a value at the obstacle, where can be found by putting X = 0 in (3.18a) (i.e. with The transition in X > 0 can also be described by (3.17) and (3.18) where now in (3.18) we replace with with and with This “undular bore” solution occupies the zone

Here, this downstream wavetrain is constrained to lie in X > 0, and hence is only fully realised if Combining this criterion with (3.21) and (3.22) then leads to the régime (3.25), and so a fully detached downstream undular bore coincides with the case when the upstream undular bore is attached to the obstacle. On the other hand, in the régime (3.24), when the upstream undular bore is detached from the obstacle, the downstream undular bore is attached to the obstacle, with a modulus at the obstacle, where can be founding by putting X = 0 in (3.18a) (i.e. with Indeed now a stationary lee wavetrain develops just behind the obstacle (for further details, see Smyth (1987)).

23

For the case when the obstacle provides a negative, but still isolated, forcing term (i.e. is negative, and non-zero only in the vicinity of X = 0), the upstream and downstream solutions are qualitatively similar to those described above for positive forcing. However, the solution in the vicinity of the obstacle remains transient, and this causes a modulation of the “undular bore” solutions.

4.

DISCUSSION AND CONCLUSIONS

Our main theme in this Chapter has been the development of the variable-coefficient extended KdV equation (3.2), and its two-dimensional extension (3.6a), as appropriate models for the description of internal solitary waves in the oceanic and atmospheric environments. While the structure of the solitary wave solutions is well understood, and there is a developing insight into how these waves deform due to background environment variability, or dissipative processes, it is still the case that these models have yet to be fully utilised in practical situations. Some important first steps in this direction are described in Chapter 2 of this monograph. However, while the propagation properties of internal solitary waves are potentially well-described by model equations such as (3.2), it is our perception that there is still much to learn about generation mechanisms in the ocean and atmosphere. One of the important implications of the validity of KdV-type models is that, in general, there is no unique generation mechanism, and indeed, later Chapters in this monograph will describe some of the competing processes. What is clear from the KdV-type models is that initial disturbances which distort the isopycnal surfaces will generally avolve into internal solitary waves; the only exception is when the initial distortion has everywhere, an incorrect polarity. Another outstanding issue is how to describe internal solitary waves whose amplitude is too large for a KdV equation (2.17), or even an extended KdV equation (2.19), to be appropriate. In general, it would seem that numerical solutions of the full equations of motion (2.1) is the only available option (see, for instance, Tung et al (1982), Turkington et al (1991) arid Brown and Christie (1998)), although there do exist certain parameter régimes where analytical theories can be developed for finite-amplitude waves. An instance of this arises when the background stratification and flow are both nearly uniform, and this has been exploited by Benney and Ko (1978), Derzho and Grimshaw (1997), and Aigner et al (1999) amongst others. In particular, Derzho and Grimshaw (1997) have shown that certain class of large-amplitude internal solitary waves can be constructed which contain small vertex cores, a feature

24

which has been seen in some laboratory experiments, and may possibly also occur in the oceanic or atmospheric environment.

REFERENCES Ablowitz, M.J. and Segur, H. (1981). Solitons and the inverse scattering transform. SIAM, Phildelphia, 425pp. Aigner, A., Broutman, D. and Grimshaw, R. (1999). Numerical simulations of internal solitary waves with vortex cores. Fluid Dynamics Research, 25, 315-333. Akylas, T.R. (1991). On the radiation damping of a solitary wave in a rotating channel, in Mathematical Approaches in Hydrodynamics. Ed. T, Miloh, SIAM, 175-181. Apel, J.R. (1980). Satelite sensing of ocean surface dynamics. Ann. Rev. Earth Planet Sci., 8, 303-342. Apel, J.R. (1995). Linear and nonlinear internal waves in coastal and marginal seas. Oceanographic Applications of Remote Sensing, eds. M. Ikeda and F. Dobson, CRC Press, Bca Raton, Florida, 512pp. Benjamin, T.B. and Lighthill, M.J. (1954). On cnoidal waves and bores. Proc. Roy. Soc. London. Serv A 244, 448-460. Benjamin, T.B. (1967). Internal waves of permanent form in fluids of great depth. J. Fluid Mech., 29, 559-592. Benney, D.J. (1966). Long non-linear waves in fluid flows. J. Math. Phys., 45, 52-63. Benney, D.J. and Ko, D.R.S. (1978). The propagation of long large amplitude internal waves. Stud, Appl. Maths., 59, 187-199. Brown, D.J. and Christie, D.R. (1998). Fully nonlinear solitary waves in continuously stratified incompressible Boussinesq fluids. Phys. Fluids., 10, 2569-2586. Christie, D.R. (1989). Long nonlinear waves in the lower atmosphere. J. Atoms. Sci., 46, 1462-1491. Davis, R.E. and Acrivos, A. (1967). Solitary internal waves in deep water. J. Fluid Mech., 29, 593-607. Derzho, Oleg G. and Grimshaw, R. (1997). Solitary waves with a vortex core in a shallow layer of stratified fluid. Phys. Fluids., 9, 3378-3385. Dodd, R.K., Eilbeck, J.C., Gibbon, J.D. and Morris, H.C. (1982). Solitons and nonlinear waves. Academic, London, 626pp. Gear, J. and Grimshaw, R. (1993). A second-order theory for solitary waves in shallow fluids. Phys. Fluids, 26, 14-29. Gilman, O.R., Grimshaw, R. and Stepanyants, Y.A. (1996). Dynamics of internal solitary waves in rotating fluid. Dyn. Atmos. Ocean., 23, 403-411.

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Gilman, O.R., Grimshaw, R. and Stepanyants, Y.A. (1995). Approximate analytical and numerical solutions of the stationary Ostrovsky equation. Stud, Appl. Math., 95, 115.126. Grimshaw, R. (1979). Slowly varying solitary waves. I Korteweg-de Vries equation. Proc. Roy. Soc. London, A368, 359-375. Grimshaw, R. (1981a). Evolution equations for long nonlinear waves in stratified shear flows. Stud. Appl. Maths., 65, 159-188. Grimshaw, R. (1981b). Slowly varying solitary waves in deep fluids. Proc. Roy. Soc., 376A, 319-332. Grimshaw, R. (1985). Evolution equations for weakly nonlinear long internal waves in a rotating fluid. Stud. Appl. Maths., 73, 1-33. Grimshaw, R. and Smyth, N. (1986). Resonant flow of a stratified fluid over topography. J. Fluid Mech., 169, 429-464. Grimshaw, R. and Tang, S. (1990). The rotation-modified KadomtsevPetviashvilli equation. An analytical and numerical study. Stud. Appl. Math., 83, 223-248. Grimshaw, R. and Mitsudera, H. (1993). Slowly-varying solitary wave solutions of the perturbed Korteweg-de Vries equation revisited.Stud. Appl. Math., 90, 75-86. Grimshaw, R., Pelinovsky, E. and Talipova, T. (1997). The modified Korteweg-de Vries equation in the theory of the large-amplitude internal waves. Nonlinear Processes in Geophysics, 4, 237-250. Grimshaw, R., He, J.M. and Ostrovsky, L.A. (1998a). Terminal damping of a solitary wave due to radiation in rotational systems. Stud. Appl. Math., 101, 197-210. Grimshaw, R., Pelinovsky, E. and Talipova, T. (1998b). Solitary wave transformation die to a change in polarity. Stud. Appl. Maths., 101, 357-388. Grimshaw, R., Pelinovsky, E. and Talipova, T. (1999). Solitary wave transformation in a medium with sign-variable quadratic nonlinearity and cubic nonlinearity. Physica D, 132, 40-62. Gurevich A.V. and L.P. (1974a). Averaged description of waves in the Korteweg-de Vries-Burgers equation. Sov. Phys. JETP 66, 490-495. Gurevich A.V. and L.P. (1974b). Nonstationary structure of a collisionless shock wave. Sov. Phys. JETP 38, 291-297. Johnson, R.S. (1970). A nonlinear equation incorporating damping and dispersion, J. Fluid Mech 42, 49-60. Kakutani, T. and Yamasaki, N. (1978). Solitary waves on a two-layer fluid. J. Phys. Soc. Japan. 45, 674-679. Katsis, C. and Akylas, T.R. (1987). Solitary internal waves in a rotating channel, a numerical study. Phys. Fluids, 30, 297-301.

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Kubota, T., Ko, D.R.S., and Dobbs, L.S. (1978). Weakly nonlinear long internal gravity waves in stratified fluids of finite depth. AIAA J. Hydronautics, 12, 157-168. Lamb, K.G. and Yan, L. (1996). The evolution of internal wave undular bores: comparisons of a fully nonlinear numerical model with weaklynonlinear theory. J. Phys. Ocean., 99, 843-864. Lee, C.Y. and Beardsley, R.C. (1974). The generation of long nonlinear internal waves in a weakly stratified shear flow. J.Geophys, Res., 79, 453-462. Leonov, A.I. (1981). The effect of Earth rotation on the propagation of weak nonlinear surface and internal long oceanic waves. Ann. New York Acad. Sci., 373, 150-159. Maslowe, A.A. and Redekopp, L.G. (1980). Long nonlinear waves in stratified shear flows. J. Fluid Mech., 101, 321-348. Myint, S. and Grimshaw, R. (1995). The modulation of nonlinear periodic wavetrains by dissipative terms in the Korteweg-de Vries equation. Wave Motion, 22, 215-238. Ostrovsky, L.A. (1978). Nonlinear internal waves in rotating fluids. oceanology, 18, 181-191. Ostrovsky, L.A. and Soustova, I.A. (1979). The upper mixed layer of the ocean as an energy sink of internal waves. Oceanology, 19, 973-981. Ostrovsky, L.A. and Stepanyants, Yu. A. (1989). Do internal solitons exist in the ocean? Rev. Geophysics., 27, 293-310. Smith, R.K. (1988). Travelling waves and bores in the lower atmosphere: the ‘morning glory’ and related phenomenum. Earth-Sci. Rev., 25, 267290. Smyth, N. (1987). Modulation theory for resonant flow over topography. Proc. Roy. Soc., 409A, 79-97. Smyth, N. (1988). Dissipative effects on the resonant flow of a stratified fluid over topography. J. Fluid. Mech., 192, 287-312. Tung, K.K., Ko, D.R.S. and Chang, J.J. (1981). Weakly nonlinear internal waves in shear. Stud. Appl. Math., 65, 189-221. Tung, K.K., Chan, T.F. and Kubota, T. (1982). Large amplitude internal waves of permanent form. Stud. Appl. Math., 66, 1-44. Turkington, B., Eydeland, A, and Wang, S. (1991). A computational method for solitary internal waves in a continuously stratified fluid. Stud. Appl. Math., 85, 93-127. Whitham, G.B. (1974). Linear and Nonlinear Waves. Wiley-Interscience, New York, 636pp. Zhou, X. and Grimshaw, R. (1989). The effect of variable currents on internal solitary waves. Dyn. Atmos. Oceans., 14, 17-39.

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Chapter 2 INTERNAL TIDE TRANSFORMATION AND OCEANIC INTERNAL SOLITARY WAVES Peter Holloway University of New South Wales, Australia Efim Pelinovsky Tatiana Talipova Institute of Applied Physics, Russia

Abstract

1.

The transformation of internal tides, or long internal waves, as they propagate over the variable topography of the continental slope and shelf is discussed. Development of a model to describe this transformation is presented and is based on the Korteweg-de Vries equation. The model includes cubic as well as quadratic nonlinearity, dispersion, Earth’s rotation and quadratic dissipation. An initial waveform, described in terms of a vertical modal function that is periodic in time, propagates through an environment of slowly varying water depth and stratification. The role of cubic nonlinearity and the Earth’s rotation are examined in detail using idealised conditions. Observations are presented from the Australian North West Shelf of a variety of strongly nonlinear internal waves. Numerical solutions to the KdV model are also provided for observed conditions and model predictions are compared to observations.

INTRODUCTION

Internal waves are a common feature in the ocean, frequently observed on shallow continental margins as well as in deep water. The waves occur on many scales varying from inertial and tidal periods down to the buoyancy period (typically 10 minutes). The longer period waves are often observed to steepen and become nonlinear in their character, particularly when interacting with topographic features, and can generate a variety of short, nonlinear waveforms.

30 All of these internal waves play an important role in the ocean by dissipating energy from wind and tidal sources and in contributing to ocean mixing. Nonlinear internal waves in the ocean frequently occur as solitary internal waves and observations have often been described as solitons. Apel et al. (1985) discussed observations of nonlinear internal waves in the Sulu Sea that had distinct soliton character. The waves were generated through tidal flow over a sill that generated a lee wave which subsequently evolved into a train of internal solitary waves when the tidal flow slackened and turned. This mechanism is decribed by Maxworthy (1979). However, many observations of solitary internal waves in the ocean suggest formation of the waves over the continental slope or shelf break region. These waves can propagate shoreward or into the deep ocean. Examples have been reported from many regions including the Bay of Biscay (New and Pingree, 1990, Pichon and Mare, 1990), the east coast of Canada (Sandstrom and Elliot, 1984, Gan and Ingran, 1992, Sandstrom and Oakey, 1995), the Sea of Okhotsk (Nagovitsyn et al., 1991), and the North West Shelf of Australia (e.g. Holloway, 1987). Ostrovsky and Stepanyants (1989) and Jeans (1995) provide reviews of observations of internal solitary wave observations from the ocean. Observations discussed in this chapter are from the Australian North West Shelf (NWS), e.g. Holloway, (1987), Smyth and Holloway (1988) and Holloway et al. (1997, 1998, 1999), which show long internal waves of semidiurnal tidal origin (internal tides) with wavelengths of approximately 20km evolving into a variety of nonlinear waveforms. These include shocks (bores or internal hydraulic jumps) and groups of short period waves of soliton-like form. Some of these features are apparent in Figure 1.1 where observed time series of isotherm displacements and onshore currents are shown from a series of 3 moorings, called Slope, Break and Shelf, located in 78 to 109 m water depths, and a few kilometers apart at the outer edge of the continental shelf on the NWS, see Figure 1.19. The plots show a variety of nonlinear wave forms including bores on both the leading and trailing faces of the long internal tide, as well as short period (approximately 10 minutes, close to the buoyancy period) internal solitary waves. The nonlinear features develop as the waves propagate shorewards into decreasing water depth. The phase lag of the signals between moorings is consistent with expected phase propagation speeds of Interpretation of observations of internal solitary waves in the coastal zone is usually done through weakly nonlinear theory, for which a basic equation describing the evolution and transformation of long weakly nonlinear waves is the Korteweg de Vries (KdV) equation. Typically this model provides a permanent nonlinear wave form in shallow water including the resolution of bores and solitons, and allows an offset to the nonlinearity through diffusion. KdV-type models in general form are described by Grimshaw in chapter 1. We present the “oceanographic“ KdV model (Boussinesq and rigid lid approxi-

31

mations) in this chapter that predict the evolution of internal waves as they move across the sloping topography of a continental shelf, based on the numerical solution to the modified KdV equation. The model includes observed ocean stratification, variable depth, Earth’s rotation, cubic nonlinearity, and also allows for dissipation through a quadratic friction term. Coefficients of the model are examined under simple stratification conditions to better understand their variability (in sign and value) and their impact on the solutions. In this paper we consider the evolution of initially long sinusoidal waveforms (representing internal tides) across the continental slope and shelf region in order to gain an understanding of the underlying mechanisms of internal wave evolution. In the sections which follow, we develop the model, discuss the separate and combined effects of quadratic and cubic nonlinearity, and of rotation. The impacts of shear flow on the solutions of the KdV equation are

32 also briefly considered. Finally, observations of internal wave evolution on the NWS and model-observation comparisons are presented.

2.

THE KORTEWEG-DE VRIES MODEL

The KdV model considers the evolution of an initial internal wave that propagates over variable topography in the horizontal with variable depth in the presence of a background density profile, which produces a background buoyancy frequency profile N ( z , x ) , where is the vertical coordinate. Background shear current the Earth's rotation and quadratic frictional dissipation are also included in the model equation. The basic equation of this model, including all parameters, is called the rotated extended KdV (reKdV) equation. This equation is employed to describe the nonlinear internal wave evolution assuming the waves are long (the wavelength exceeds the water depth), are small in amplitude (the amplitude is small compared with the water depth),and the Coriolis parameter is weak (low latitudes). It is obtained from a perturbation method of second order in wave amplitude, and first order in wavelength and Coriolos parameter. For internal waves in an ocean continuously stratified in density and shear flow, the KdV-type equation was derived by Lamb & Yan (1996), Pelinovsky et al. (2000). The general approach to produce the KdV type model from governing fluid dynamics equations is described by Grimshaw in chapter 1. For weak oceanic stratification the Boussinesq and rigid lid approximations can be used, and the rotated extended KdV equation has the form (Holloway et al., 1997, Talipova et al., 1999b)

where is the wave profile, which in the small amplitude, long-wave limit gives the maximum vertical isopycnal displacement in the first mode, is a horizontal coordinate and is time. All other variables and parameters are described below. The phase speed of the linear long wave is determined by the eigenvalue problem

with the normalisation The coefficients of dispersion and quadratic nonlinearity are the following integrals from the modal function at a particular depth

33

The cubic nonlinearity

is more complex and defined as:

where is the first correction to the nonlinear wave mode which is a solution of the ordinary differential equation

with boundary conditions

and the normalised condition

The parameter is the Coriolis parameter where is the angular speed of the Earth and is the latitude. Equation (2.1) is valid for an ocean of constant depth and when dispersion, nonlinearity and rotation are considered weak. Usually, for calculations of the coefficients of the KdV equation only a single vertical profile of stratification is used assuming constant depth and the density to be horizontally uniform, but the nonlinear parameter is very sensitive to variations of the vertical stratification. The shelf/slope zone is generally characterized by large bottom slope and considerable variability in stratification and shear flow, and thus using mean stratification can be inaccurate in the calculation of and Strong spatial variability in the coefficients will cause variability in the internal wave field. Account must be taken of the horizontal variability of the ocean medium and accordingly, the KdV equation must be modified, as described below. If the horizontal variability is smooth the reflection of the wave energy from the shelf can be ignored and a solution can be sought for the vertical displacement of the pycnocline in the form where is again the vertical structure of the pycnocline displacement now varies slowly and smoothly with This effect can be included in equation (2.1) as a weak additional term I (see Holloway et al., 1997) giving

34 where

and I characterizes the amplification of the linear long internal wave. The subscript “0” defines initial values at any fixed point Note that this definition of I differs slightly from that given in Chapter 1 where a different scaling is used for the horizontal coordinate. Introducing a change in variable

and with a change of coordinate

equation (2.7), under the assumption of small nonlinearity and dispersion, can be reduced to:

Dissipation of energy comes about through bottom friction and this can be represented phenomenologically as a quadratic bottom stress term in the equations of motion. The bottom stress can be defined as where is a quadratic bottom friction parameter, typically 0.0025, and is water density. Characteristic thickness can be evaluated through as in chapter 1, or through the “equivalent” depth as in Holloway et al. (1997). Including the quadratic friction term, we obtain the modified rotated extended KdV equation, referred to as the basic evolution equation (Holloway et al. (1999),

Equation (2.10) is solved numerically with a periodic boundary condition of the form which corresponds to the evolution of the periodic internal tidal wave, and with the “initial” condition where F is the periodic function with the same frequency which characterises the form of the internal tide at the fixed point In comparison to the KdV equation with constant coefficients, where there are an infinite number of conservation laws, the variable-coefficient KdV equation (2.10) has only two conserved quantities when dissipative terms and rotation are neglected. The KdV equation with constant coefficients is a fully integrable

35 system and has an exact solution for any initial conditions, while the variablecoefficient KdV equation does not have such properties and numerical methods are required for its solution. Finite difference schemes used for the solution of the KdV equation are based on Berezin (1987) and Pelinovsky et al. (1994), and used by Holloway et al. (1997, 1999).

3.

PARAMETERS OF THE EVOLUTION EQUATION

The coefficients and are calculated (without background shear flow) for observed stratification from the NWS (Figure 1.6) and assuming a linear sloping topography, representative of a continental slope and shelf region. This provides an example of how the KdV coefficients can vary spatially. Results are plotted in Figure 1.2 showing the cross shelf distribution of the coefficients. It is important to note that both nonlinear coefficients change sign, in particular the coefficient of quadratic nonlinearity is negative for deep water and positive for shallow water. This means that any solitons will have negative polarity (dip downwards) in deep water and positive polarity (point upwards) in shallow water. The coefficient of cubic nonlinearity is negative in deep water and changes sign in the coastal zone. The amplification factor I increases up to seven times and the wave amplitude (within the linear theory of the long waves) should increase at the same rate. Real amplification depends on nonlinearity, dispersion, rotation and dissipation and will be discussed below. The ratio of the quadratic nonlinear coefficient with the dispersion coefficient, effects the nature of the waveform. Strong bores form when the ratio is large, and weak bores with a more sinusoidal motion form when the ratio is small. Since the results are sensitive to changes in this coefficient has a large impact on the analysis and prediction of wave evolution. Furthermore, without the inclusion of cubic nonliearity, only one bore (either on the leading face or trailing face) could be predicted, which does not match with some observations. The quadratic bottom friction term has the effect of dampening the amplitude of the predicted wave. We consider how the density stratification impacts on the coefficients and hence the equation solutions. Density stratification influences the solution of the evolution equation (2.10) via the coefficients of the quadratic and cubic nonlinearity, dispersion, and the long wave phase speed. For a two layer fluid, when the two layers are of equal depth the quadratic nonlinear coefficient is zero and the cubic term is essential in order to predict results. An analysis of the variability of the coefficients of the KdV equation was performed for many areas of the World Ocean (Pereshkokov and Shulepov, 1984, Fennel et al., 1991, Pelinovsky et al., 1995, Holloway et al., 1997,Talipova et al., 1998, Pelinovsky et al., 1999, Ivanov et al., 1994). The linear characteristics of the internal wave field, the long wave phase speed and dispersion parameter, are only weakly

36

dependent on details of the stratification but have a strong correlation to water depth. Therefore mean characteristics of the buoyancy frequency, for example in the Levitus (1982) Atlas, can provide the linear characteristics of the internal wave field. Variability of in the coastal zone is greater (Holloway et al., 1997). Variability of has not previously been calculated for real ocean stratification and its sensitivity to details of the vertical distribution of density is poorly known. Therefore, the coefficients of equation (2.10) are calculated for several models of stratification in order to gain some insight into the nature of the coefficients, particularly the behavior of The coefficients of the eKdV equation for a two-layer fluid, with density jump between the upper layer of thickness and the lower layer of thickness are (see, for example, Djordjevic and Redekopp, 1978 or Kakutani and Yamasaki, 1978) given as

37

It can be seen from equations (3.1) that is always negative while may be either sign depending on the interface location. Note that the parameters and do not depend on the density jump and it seems that this property can be effectively used for a more general measure of the effect of these vertical profiles of density stratification with a relatively narrow pycnocline.

To check this narrow pycnocline hypothesis, consider a single peak profile of of the form

38 Numerically computed coefficients of the eKdV equation for varying and using equations (2.3) to (2.6), for H = 500 m and are shown in Figure 1.3. Results are compared with the two-layer approximation (equation 3.1) which describes all coefficients of the eKdV equation well for a narrow peak in the buoyancy frequency profile In the two-layer model it is assumed With increasing pycnocline width the calculated phase speed moves away from the two-layer approximation. The relative coefficients of the dispersion and nonlinearity, and are well described by the two-layer formulas (3.1) over a wide range of (up to 100 m).

39 To investigate the influence of the pycnocline form on the coefficients, a triangle symmetrical profile of N is used:

Again is used and the integral over depth of is the same for both profiles of N to provide meaningful comparisons. The effective width of the pycnocline is however, is different. Although the phase speeds are different and all coefficients are different due to the difference in the density jump, the relative coefficients for both the sharp and triangle pycnocline forms are similar, as illustrated in Figure 1.4. These results indicate that the two-layer approximation is accurate for calculations of the coefficients of the eKdV equation if the vertical distribution of N has a single peak distribution. Observed profiles in the deep ocean often contain two peaks corresponding to the main and seasonal pycnoclines and in the coastal zone there is a wide variety in profiles for N. As an analytical test, a three-layer model is used with the same density jump as above where the width of the upper and lower layers coincide. This example was constructed by Grimshaw et al. (1997) and Talipova et al. (1999a) giving

where is the width of the middle layer and H is the total water depth. If the middle layer is narrow, is negative (when the width of this layer tends to zero the previous two-layer result is found), and when is large, is positive. The change in sign of occurs for a critical width of the middle layer

Plots of and against are shown in Figure 1.5 along with numerical solutions of the parameters for the two peak distribution of N with an effective width for each peak of 25m. In the first case, peaks have the sharp form, and in

40

the second case a triangle form with the same depth integrated value of Numerical calculations of the relative coefficients of the eKdV equation are close to the predictions of the three-layer model. The values are only weakly dependent on the form of the peaks. The analysis suggests that the sign of depends on the structure of the vertical distribution of N. If this distribution has only one peak, is always negative and its relative magnitude can be well described by the two-layer approximation. For the three-peak distributions of N, may have either sign or be zero. In the vicinity of the “zero” point is sensitive to the details of and could have large variability. For real oceanic distributions of the buoyancy frequency the sign of cannot be predicted without the use of numerical calculations.

4.

SIMULATION RESULTS

Model simulations of internal wave evolution are performed with an initial sinusoidal wave of 12hr period, with a variety of amplitudes, and originating at a depth of 500m. Dissipation is neglected in the initial runs and where

41

rotation is included the Coriolis parameter is calculated for a latitude 20° with The model is independent of the sign of f. To analyse the influence of nonlinearity, dispersion, rotation and dissipation on wave evolution, initial simulations use simplifications of the observed bathymetry and density stratification from a section on the NWS, as illustrated in Figure 1.2. The buoyancy frequency profile (Figure 1.6) is obtained as an average of 13 measured profiles from a cross section over the slope and outer shelf on the NWS and each profile is an average from repeated measurements over a tidal cycle to remove perturbations from the internal waves. The profile shows a narrow peak with a maximum of at 30 m, and a wider peak with maximum at 120 m. The depth profile is given by two linear gradient regions, from 500 to 75 m depth representing the continental slope and a weaker gradient representing the shelf. This provides an approximation to the bathymetry of the region where the waveforms of Figure 1.1 were measured (see Figure 1.19). The coefficients of equation (2.7) are calculated from equations (2.3) to (2.6) and are plotted in Figure 1.2. Figure 1.7 provides a general view of varying model assumptions on a sinusoidal wave propagating shoreward across this simplified shelf. This is provided as background for the detailed discussion which follows. Initially bottom frictional dissipation is neglected. For small initial amplitudes, the nonlinear effects are visible only in shallow water (less than 70 m) when the wave propagates across the shelf (which has only a small slope). The initial continental slope with large slope acts to increase the wave amplitude. For these depths the coefficient of quadratic nonlinearity is positive and the coefficient of the cubic nonlinearity is negative.

42

For a small amplitude wave the wave profiles are calculated with different approximations (with/without the cubic nonlinear term, with/without rotation). The origin of the wave is held constant in these model runs as this is not a critical parameter in determining the wave evolution, because little wave steepening occurs in deep water. Results are illustrated in Figure 1.8. No deformation of the wave profile occurs until it has propagated 130 km, reaching a depth of 60 m. The wave amplitude in shallow water is small, 1 – 2m, but the form of the wave is clearly nonlinear. Note that an increase in amplitude to 1.5 m agrees with the prediction of linear theory. When and the wave transforms into a shock and the first two undulations appear. At this point is positive and the shock is on the wave front. Quadratic nonlinearity, characterized by is 0.1 and such a value of nonlinearity is enough for visible nonlinear deformation. Calculations with the eKdV equation lead to the same result (not shown), indicating the influence of cubic nonlinearity to be negligible (the ratio is approximately 0.1). The influence of rotation however is strong. The resulting wave is smoother and does not contain high

43

44 frequency waves when The same result is also obtained from the rotated extended KdV equation (reKdV). Thus, for the internal waves with initial amplitude 0.2 m in shallow water, does not influence the wave dynamics, but rotation is important.

The next run has an increased initial wave amplitude to 1 m. Figure 1.9 illustrates these wave profiles at different distances onshore with and Nonlinear effects are visible only in shallow water (depth less than 70 m) when the wave propagates over the weakly sloping shelf. For these depths is positive and thus all generated solitons have positive polarity. The effective parameter of nonlinearity is large and the soliton generation is accelerated compared with the case of This transformation of the periodic wave to the soliton group is a typical prediction of the KdV equation. Inhomogeneity (horizontal variability) influences the growth of the soliton amplitude without increasing the negative value (depressed section of the wave profile) of the wave which provides the conditions for the development of solitons. On the shelf (160 – 170 km) large solitons are over-running small solitons and this leads to a particular recurrence phenomenon, i.e. the reemergence ofthe approximate initial sinusoidal waveform, at greater distances. The exact initial waveform would re-occur if the water depth was constant. Simulations with including cubic effects but without rotation, are shown in Figure 1.10. The development of a soliton group within the eKdV

45

equation is similar to the previous account when only the quadratic nonlinear term is taken into account, but there are some differences. The value of in shallow water is approximately and thus cubic effects should be important. But is negative and this leads to a decrease in the nonlinear correction to phase speed. In particular, the “top” of the crest in Figure 1.10 occurs after 8 hours at a distance of 115 km, while the top of the crest in Figure 1.9 is at 7 hours for the same distance. Solitons develop more rapidly on the initial wave when and “cubic-quadratic” solitons are closer together than “quadratic” solitons. The influence of rotation on the evolution of a periodic wave within the rKdV equation is demonstrated in Figure 1.11. Nonlinear effects become important only for shallow water (depth less than 70 m). The first stage is qualitatively the same as the case of no rotation, the wave steepens and then the soliton group appears, but after 140 km the dynamics are different. The number of solitons does not grow (it does without rotation), they are separated in time and their amplitudes are high (up to 40 m). The decrease in the number of solitons with increasing latitude (increasing was obtained by Gerkema (1996) with the rotated-modified Boussinesq equations for a two-layer ocean. In his calculations was negative and thus the solitons had negative polarity. In our calculations similar results are obtained but with the opposite polarity for the solitons due to positive The combined effects of cubic nonlinearity and rotation (reKdV equation) are shown in Figure 1.12. Here one “thick” soliton forms with an amplitude of 15 m

46

along with several “thin” smaller solitons. The existence of “thick” solitons is a conclusion of the eKdV equation with constant coefficients and negative With this equation it was shown (Kakutani and Yamasaki, 1978, and Miles, 1981) that the soliton amplitude has a limiting value which is 10 – 20 m for the above simulations. The solution of the Cauchy problem for this equation (Miles, 1981, Slyunyaev and Pelinovsky, 1999) shows that if an initial disturbance is large enough the first soliton will be the “thick” soliton with an amplitude close to the limiting value. In the above simulations with the eKdV equation (no rotation), the soliton amplitude reaches the limiting value after propagating a large distance and the formation of the “thick” soliton should occur at an even larger distance. Rotation increases the wave amplitude (in the rKdV equation the soliton amplitude reaches 40 m, see Figure 1.11), so the soliton reaches the limit value more rapidly than in the absence of rotation, and a thick soliton forms. On the other hand, the limit soliton is realised when and, therefore, the role of the next higher-order nonlinear terms is increased, although these are not taken into account in this model. Increasing to 5 m causes nonlinear effects to be significant in deeper water. With large amplitudes, the cubic effects are important and only solutions of the eKdV equation are considered. Figure 1.13 shows the wave profiles for the case of no rotation, and still no dissipation. A wave with two shock fronts is

47

seen as well as solitons on the crest and trough of the long wave. Solitons on different backgrounds of the long wave have opposite polarities. In the illustrated simulations the wave in shallow water has an amplitude up to and A is significantly larger than Therefore, in this case, the eKdV equation produces equivalent results to the KdV equation with Such behavior is characteristic of the solutions to the KdV equation with but with the cubic nonlinear term included (Grimshaw et al., 1997, Talipova et al., 1999a). The effect of rotation is demonstrated in Figure 1.14, which shows the results of the simulation with the reKdV equation. Rotation decreases the number of solitons (compare to the wave form at in Figure 1.13, but conserves the sharp wave form with two shocks. Simulations including bottom dissipation are carried out with and see Figure 1.15. The influence of bottom friction is to remove the “thick” soliton and decrease the number of “thin” solitons generated. It also dampens the soliton amplitude significantly as well as the amplitude of the long internal wave. Finally, simulations with (Figure 1.16) and (Figure 1.17) are considered using the full nonlinear model based on equation (2.10). In both cases, due to large nonlinearity, the waves transform significantly in deep

48

49

water. But in deep water is negative and the shock forms on the back face of the wave (distance 66 km). The coefficient changes sign here and its influence is visible. When the waves approach shallow water, they transform as in the case of smaller amplitudes. The quadratic friction has a nonlinear character and rapidly dampens the wave. According to theory, after propagating large distances the nonlinear damping leads to uniformity in amplitude for waves

50

of different initial amplitudes and this effect is seen from a comparison of the results of and (Figures 1.16 and 1.17).

5.

INFLUENCE OF CURRENT SHEAR

The KdV solutions show strong dependence of wave evolution on the distribution of the nonlinear coefficient. However, the coefficients above were calculated in the absence of shear flow. In the case of the modified rKdV equation, without cubic nonlinearity but including background shear flow, it is found that for shallow water at the top of the slope, background shear can have a large influence on the modal structure and hence on the values of Using observations from the NWS, Figure 1.18 shows modal functions for two different times, ahead and behind a bore, with temperature (defining density) and velocity profiles as shown. If the background flow is neglected, the modal functions show a maximum wave amplitude below mid-depth for the depressed stratification (0200 hr) and the reverse for the raised stratification (0600 hr). Values of are positive and negative respectively as could be anticipated. However, the shear strongly distorts the modal functions and reverses the signs of hence chang-

51

ing the nature of solitary waves that are generated and occur in these regions. Holloway et al. (1997) provide further examples of the influence of shear on the value of the nonlinear coefficient. The influence of shear on has not been investigated.

6.

OBSERVATIONS FROM THE AUSTRALIAN NORTH WEST SHELF

Example observations of internal waves are presented from the NWS. The observations show the long internal-tide evolving into internal solitary waves as well as many other strongly nonlinear wave forms. Figure 1.20 (from Holloway et al., 1997) shows isotherm displacements and onshore currents at 10 m above the sea bed from the 3 closely spaced mooring locations (Slope, Break and Shelf, shown in Figure 1.19) at depths between 109 and 78 m. The isotherm displacements are characterised by steep faced waves (shocks) at the leading face and sometimes at the trailing face of the wave. Short period oscillatory

52 waves (internal solitary waves) are often seen following the shocks. The currents at Slope are very intense and show strong internal solitary waves. The onshore propagation of the waves is also seen by tracking events from Slope through to Shelf. Considerable change in the waveform is also evident as the waves propagate, particularly from Slope to Break, a distance of 5070 m. For events and the shocks strengthen between Slope and Break whereas is seen to develop into a shock on the trailing face of the wave. There is a strong similarity between the isotherm and current signals, as would be expected, except that the current also contains a signal from the semi-diurnal barotropic tide. Further examples are given by Holloway et al. (1997).

Additional observations, from Holloway et al. (1999), are presented from Break and Shelf locations, each in 78-m water depth, in Figure 1.21. Wave propagation is predominantly on shore, and so cross-shelf currents are considered with the depth-averaged value removed, providing an estimate of the baroclinic component. Isotherm displacements are also shown. The first example (panel a) shows a 12-hour period oscillation (the internal tide) in the isotherm displacements with a large number of positive internal solitary waves. This waveform is mirrored almost exactly in the currents near the seabed and also near the sea surface with a reverse in phase over depth. The waves have an amplitude of up to 25 m. A second example, from the same location but at

53

a different time, shows a similar waveform but with negative internal solitary waves forming along the crest of the internal tide (panel b). Panel c shows a 12-hour period oscillation of internal solitary waves with different polarity at different phases of the internal tide. At the start when the stratification is depressed by the 40-m-high internal tide, positive internal solitary waves are seen, followed by negative waves along the crest of the internal tide. Positive internal solitary waves then return as the isotherms move downward with the internal tide. The variations are also seen in the velocity records. A very rapid upward transition (a shock) from one level to another is seen in both displacements and currents in Figure 1.21 panel d. A packet of high-frequency waves is formed on the shock with an average frequency of (period of 7 min), a value close to the buoyancy frequency. Further examples from Holloway et al. (1999) are shown in Figure 1.22. Panel a shows an example of a square wave form. The front shock has a height of 40 m (more than half the water depth) and does not produce any short waves. The back shock occurs about 2 hours later and is accompanied by a

54

single solion and then a series of short waves. The square wave form is also distinct in the currents at both 8- and 68-m depths. The elevations plotted in panel b show a series of widely spaced internal solitary waves, with a change in polarity after the downward shock. The largest ISW is seen last and not first as could be expected with rank ordering. An approximately triangular shaped waveform is shown in panel c. After an upward jump of 40 m and a series of internal solitary waves, an approximately linear decrease in height is seen in the isotherm displacements. This is consistent with the currents at 68 m, at least over the first 8 hours. Finally an example of broad or “thick” soliton is shown in Figure 1.22 panel d. A series of three positive internal solitary waves is seen in the displacements with an amplitude of 20 m and in the currents with

55

a strength of The lead wave has a “width” of approximately 44 min with following waves of shorter period. Many of the features of the observed waves are consistent with features from solutions to the KdV models discussed. These include packets of high frequency waves and solitons, triangular waveforms and “thick” solitons.

7.

COMPARISON OF OBSERVATIONS AND MODEL PREDICTIONS

For a comparison of the KdV model with observed data, use is made of the simultaneous measurements of isotherm displacements and currents at the

56 Slope, Break and Shelf locations. Isotherm displacements, over a 12 h period, observed at the Slope location are used as the initial waveform and the KdV model is used to predict the wave transformation as the waves propagate to Break and then to Shelf locations. In these comparisons we neglect the cubic nonlinear term and the Earth’s rotation. Isotherm displacements from 13 April are used in the example presented here. Background vertical profiles of the density and shear flow fields and the corresponding coefficients of the KdV equation are obtained by averaging the observed data from Slope over the 12 hr period 0000 to 1200. The averaging for Break and Shelf is carried out for the time interval 0300 - 1500, allowing for the propagation time of internal waves. The values of at Slope, Break and Shelf are 0.00344, 0.00161 and respectively, indicating a change in polarity of solitons between Break and Shelf. The spatial variation of the coefficients between the locations are approximated by a cubic spline.

Figure 1.23 shows the initial and predicted waveforms after the initial wave has propagated 920 m and 5248 m for a bottom friction coefficient

57 although results are similar when bottom friction is included. Figure 1.24 shows a comparison of the modelled waveform at and the observed isotherm displacement time series from Break location. The greatest change in the waveform as it propagates from Slope is the strengthening of the forward shock, formed from the positive sign of and an increase in amplitude of the short-period waves. With further propagation towards Shelf, the shock increases in strength and a large number of oscillations are seen. Comparison between model and observations shows that the model correctly describes the low frequency component and leading shock, and the period of the short waves and their locations. There are, however, differences between predicted and observed amplitudes of the short-wave tail.

8.

CONCLUSIONS

The application of weakly nonlinear theory, using a modified rotated extended KdV equation including dissipation, reproduces many observed features of bores, solitary waves and short period oscillatory waves that occur as an internal tide propagates across the continental slope and shelf. By including the cubic nonlinear term, the model resolves the second bore and is able to better predict the “square” nature of the observed waves. Including rotation has the effect of reducing the number of predicted solitons evolving on the face of a shock, and in increasing their spacing. The inclusion of both cubic nonlinearity and rotation together provided the facility to predict long period solitons. Many of the observed waves from the NWS are strongly nonlinear with heights exceeding half the water depth. In these cases weakly nonlinear theory is not strictly valid and high order corrections (fourth or fifth order) may need to be considered. Lamb and Yan (1996) have compared weakly nonlinear theory

58 to a fully nonlinear model and found reasonable agreement for large amplitude waves. The temporal and spatial variability of the generalised KdV coefficients can also be important in their impact on predicted waveforms. Furthermore, because of the effect of shear flow on the quadratic nonlinear term, research suggests that the inclusion of shear flow would also have a substantial impact on the model solutions and thus work is required to further extend this modified rotated extended KdV equation.

Acknowledgments Belinda Barnes has made valuable contributions to this work. TT received support from grants RFBR 00-05-64223 and INTAS 99-1637, EP received support from INTAS grant 99-1068 and PH received support from the Australian Research Council.

References Apel, J.R., J.R. Holbrock, A.K. Liu and J.J. Tsai. (1985). The Sulu Sea internal soliton experiment, J. Phys. Oceanogr., 15, 1625-1651. Berezin Yu. A. (1987). Modelling nonlinear wave processes. VNU Science Press, Utrecht. Djordjevic, V., and Redekopp, L. (1978). The fission and disintegration of internal solitary waves moving over two-dimensional topography. J. Phys. Oceanogr., 8, 1016 - 1024. Fennel, W., Seifert, T. and Kayser, B. (1991). Rossbi radii and phase speeds in the Baltic Sea, Cont. Shelf Res., 11, 23-36. Gan, J., and R.C. Ingran. (1992). Internal hydraulics, solitons and associated mixing in a stratified sound. J. Geophys. Res., 97, 9669-9688. Gerkema, T. (1996). A Unified Model for the Generation and Fission of Internal Tides in a Rotating Ocean, J. Marine Research 54, 421 - 450. Grimshaw R., Pelinovsky E., Talipova T. (1997). The modified Korteweg - de Vries equation in the theory of large-amplitude internal waves. Nonlinear Processes in Geophysics, 4, 237 - 350. Holloway P.E. (1987). Internal Hydraulic Jumps and Solitons at a Shelf Break Region on the Australian North West Shelf, J. Geoph. Res., C92, 5405 5416. Holloway, P.E., Pelinovsky, E., Talipova, T. and Barnes, B. (1997). A Nonlinear Model of Internal Tide Transformation on the Australian North West Shelf, J. Phys. Oceanogr. 27, 871 - 896. Holloway P., Pelinovsky E., Talipova T., Barnes B. (1998). The rotated-modified extended Korteweg-de Vries equation for the description of nonlinear internal wave transformation in the ocean. Computational Techniques and Applications: CTAC97, (Proc. 8th Biennial Conf., Adelaide, Australia, Eds: B.J. Noye, M.D. Teubner, A.W.Gill). World Sci., Singapore, 297 - 304.

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Holloway P, Pelinovsky E., Talipova T. (1999). A Generalised Korteweg - de Vries Model of Internal Tide Transformation in the Coastal Zone. J. Geophys. Research, 104, N. C8, 18,333 - 18,350. Huthnance J.M. (1989). Internal tides and waves near the continental shelf edge. Geophys. Fluid Dyn.,, 48, 81-106. Ivanov V.A., Pelinovsky E.N., Talipova T.G., Troitskaya Yu.I. (1994). Statistic estimation of the non-linear long internal wave parameters in the Black Sea test area off the South Crimea. Marine Hydrophys. J., 4, 9 - 17. Jeans, D.R.G. (1995). Solitary internal waves in the ocean: A literature review completed as part of the internal wave contribution to Morena. UCES, Marine Science Labs, University of North Wales. Rep. U-95. Kakutani, T., and Yamasaki, N. (1978). Solitary waves on a two- layer fluid. J. Phys. Soc. Japan, 45, 674 - 679. Lamb, K.G. and Yan, L. (1996). The evolution of internal wave undular bores: comparisons of a fully nonlinear numerical model with weakly nonlinear theory, J. Phys. Oceanogr., 26, 2712-2734. Levitus, S. (1982). Climatological atlas of the world ocean. Environmental Research Laboratories, Geophysical Fluid Dynamics Laboratory, Princeton, NJ, US Department of Commerce, NOAA Professional Paper 13, 173pp. Maxworthy, T. (1979). A note on the internal solitary waves produced by tidal flow over a three-dimensional ridge. J. Geophys. Res.,, 84, 338-346. Miles, J.W. (1981). On internal solitary waves, Tellus, 33, 397 - 401. Nagovitsyn, A., E. Pelinovsky, and Yu. Stepanjants. (1991). Observation and analysis of solitary internal waves at the coastal zone of the Sea of Okhotsk, Sov. J. Phys. Oceanogr., 2(1), 65-70. New, A.L., and R.D. (1990). Pingree, Large-amplitude internal soliton packets in the central Bay of Biscay, Deep Sea Res., 37, 513-524. Ostrovsky, L., and Stepanyants. (1989). Yu. Do internal solitons exist in the ocean? Rev. Geophys., 27, 293-310. Pelinovsky, E., Stepanyants, Yu., and Talipova, T. (1994). Modelling of the propagation of nonlinear internal waves in horizontally inhomogeneous ocean. Izvestiya, Atmos. Oceanic Phys., 30, 79 - 85. Pelinovsky, E., Talipova, T., and Ivanov, V. (1995). Estimations of the nonlinear properties of the internal wave field off the Israel Coast. Nonlinear Processes in Geophysics, 2, 80 - 85. Pelinovsky E., Talipova T., Small J. (1999). Numerical modelling of the evolution of internal bores and generation of internal solitons at the Malin Shelf. The 1998 WHOI/IOS/ONR Internal Solitary Wave Workshop: Contributed Papers. Eds: T.Duda and D.Farmer. Technical Report WHOI-99-07, 229 236. Pelinovsky E., Poloukhina O., and Lamb K. (2000). Nonlinear internal waves in the ocean stratified in density and flow. Oceanology, 40, N. 5.

60 Pichon A., and Mare R. (1990). Internal tides over a shelf break: Analytical model and observations, J. Phys. Oceanogr., 20, 658-671. Sandstrom H. and Elliot J.A. (1984). Internal tide and solitons on the Scotian Shelf: a nutrient pump at work. J. Geophys. Res.,, 89 (C4), 6415-6426. Sandstrom, H., and N.S. Oakey. (1995). Dissipation in internal tides and solitary waves, J. Phys. Oceanogr., 25, 604-614. Slyunyaev A.V., and Pelinovsky E.N. (1999). Dynamics of large-amplitude solitons. J. Experimental and Theoretical Physics, 89, 173 - 181. Smyth N.F. and Holloway P. (1988). Hydraulic Jump and Undular Bore Formation on a Shelf Break, J. Phys. Oceanogr. 18, 947 - 962. Talipova, T., Pelinovsky, E., and Kouts, T. (1998). Kinematics characteristics of the internal wave field in the Gotland Deep in the Baltic Sea, Oceanology, 38, 33-42. Talipova, T., Pelinovsky, E., Lamb, K., Grimshaw, R., and Holloway P. (1999a). Cubic nonlinear effects of intense internal wave propagation. Doklady Earth Sciences. 364, 824-827. Talipova T., Pelinovsky E., Holloway P. (1999b). Nonlinear models of transformation of internal tides on the shelf. Ocean Subsurface Layer: Physical Processes and Remote Sensing. Nizhny Novgorod, Institute of Applied Physics, 1, 154 - 172.

Chapter 3 ATMOSPHERIC INTERNAL SOLITARY WAVES James W. Rottman Department of Mechanical & Aerospace Engineering University of California, San Diego [email protected]

Roger Grimshaw Department of Mathematical Sciences Loughborough University [email protected]

Abstract

1.

The solitary waves that have been observed in the atmosphere fall broadly into two classes: those that propagate in a fairly shallow stratified layer near the ground and those that occupy the entire troposphere. We present a survey of the observations of both types of solitary waves. The generation mechanisms differ substantially for these two types of solitary waves. Those that propagate in a shallow stratified layer are generated by small scale or mesoscale phenomena such as thunderstorm outflows, sea breezes or katabatic winds. Those solitary waves that occupy the entire troposphere are generated by much larger scale phenomena, such as some kind of geostrophic adjustment process. We also review the previous attempts that have been made to compare these observations with weakly nonlinear solitary wave theory. It appears that models with a deep passive upper layer are generally not applicable, while Korteweg-de Vries models, perhaps enhanced with higher-order nonlinearity, provide the best comparison; but it remains unclear what constitutes a suitable upper boundary condition.

INTRODUCTION

Atmospheric solitary waves are horizontally propagating nonlinear internal gravity waves that can travel over large distances with minimal change in form as the result of a balance being achieved between non-

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linearity and horizontal linear dispersion. In theoretical models, solitary waves are the long-time, far-field solution of an initial value problem in an atmosphere with a background structure that can serve as a horizontal waveguide. An initial disturbance generates a spectrum of waves of different amplitudes and wavelengths. The nonlinearity and linear dispersion of the system then organize the waves into solitary waves and radiation; the former are observed in the far field, while the latter disperses. Thus, any method of disturbing an atmosphere with suitable waveguide characteristics, will eventually generate solitary waves, provided only that there is sufficient initial disturbance of the proper polarity. This result is supported by the laboratory experiments of Maxworthy (1980), among others; indeed, Maxworthy concluded “... if a physical system is capable of supporting solitary wave motions then such motions will invariably arise from quite general excitations.” Reports of observations of atmospheric solitary waves have been increasing in the literature over the last few decades. These observations can be divided into two classes: those for which the waves are confined to the lower few kilometers of the troposphere and those that extend over the entire troposphere. Observations of the first class of waves have been reported by Tepper (1950), Abdullah (1955), Bosart and Cussen (1973), Christie, et al. (1978, 1979), Clarke, et al. (1981), Shreffler and Binkowski (1981), Smith et al. (1982), Smith and Morton (1984), Doviak and Ge (1984), Rottman and Simpson (1989), Fulton, et al. (1990), Cheung and Little (1990), Doviak, et al. (1991), Karyampudi, et al. (1993), Rees and Rottman (1994), and Mannasseh and Middleton (1995), among others. These waves have horizontal scales of 100 m to a few kilometers, phase speeds of about and are usually an amplitude-ordered series of waves of elevation. Recent reviews of the theory and observations for this class of atmospheric solitary wave are given by Smith (1988) and Christie (1989). Observations of the second class of solitary waves, which so far are not as numerous as those of the first class, have been reported by Pecnick and Young (1984), Lin and Goff (1988), Ramamurthy et al. (1990), and Rottman et al. (1992). These waves have a horizontal scale on the order of 100 km and phase speeds of 25 to and are generally (but not always) isolated waves of depression. Most theories for atmospheric solitary waves are approximate in that they assume the waves are weakly nonlinear. Maslowe and Redekopp (1980), Grimshaw (1981), and Tung, et al. (1981), among others, have derived evolution equations for weakly nonlinear solitary waves in incompressible fluids. Grimshaw (1980/1981) has derived the equivalent equations for an approximately compressible ideal gas, and Miesen, et al.

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(1990a,b) have derived the evolution equations for a fully compressible ideal gas. These theories assume that internal solitary waves propagate in a horizontal waveguide within which the background stratification and shear (which are usually assumed horizontally homogeneous everywhere) vary as a function of height (see Chapter 1 for a summary of the theoretical development). The depth of the waveguide is assumed to be much less than the horizontal scale and much greater than the vertical displacement amplitude of the wave. The lower boundary of the waveguide is usually taken as the earth’s surface, whereas the upper boundary is defined by either an infinitely deep layer of neutrally stratified fluid or a rigid lid. These two upper boundary conditions produce different wave types. The former boundary condition leads to what is sometimes referred to as the BDO (Benjamin-Davis-Ono) theory and the latter boundary condition leads to the KdV (Kortweg-de Vries) theory. The difference between these two theories is explained in greater detail in Rottman and Einaudi (1993). A variety of generation mechanisms have been proposed for these observed atmospheric solitary waves, although there have been direct observations of the generation process in only a few cases. For the lowlevel solitary waves, the proposed generation mechanisms mainly involve a gravity current, such as a thunderstorm outflow, katabatic wind, sea breeze front, or downslope windstorm, interacting with a low-level stable layer. The motion of the gravity current produces perturbations that are trapped in the low-level stable layer and eventually evolve over time into a series of solitary waves. For the tropospheric-scale waves, a frequently proposed generation mechanism is that disturbances produced by large-scale penetrative convection are trapped in the troposphere and evolve into solitary waves. As intuitively attractive as this idea seems, Uccellini and Koch (1987), in their review of 13 case studies that describe mesoscale wave disturbances in North America having significant effects on cloud and precipitation fields, concluded that the poor correlation between the presence of gravity waves and the presence or intensity of convective storm cells ruled out convective systems as a dominant mechanism for large-scale wave generation. They found that a common synoptic setting was associated with the presence of large-scale gravity waves. The waves tended to occur north of a surface frontal boundary and east or southeast of a jet streak propagating towards a downstream ridge axis in the upper troposphere. Based on these observations, Uccellini and Koch suggested that the two most probable wave generation mechanisms are shear instability and geostrophic adjustment in the exit or entrance region of a jet streak. However, as stated earlier, the details of the initial disturbance are not central to the generation

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of solitary waves; only the bulk characteristics of the disturbance, such as its polarity, magnitude and vertical distribution, have to be of the approximately correct structure in order to generate solitary waves. An absolute necessity for the existence of solitary waves is a horizontal waveguide which prevents or inhibits wave energy from propagating away in the vertical direction. For the observations discussed above, such a waveguide is bounded below by the earth’s surface and presumably above by some suitable trapping mechanism in the upper atmosphere. According to linear theory there are three fundamentally distinct mechanisms by which waves can be trapped in the atmosphere. One mechanism is for the waveguide to be bounded above by a sufficiently deep region in which linear waves of some prescribed range of wavelengths are evanescent. This property of the background state is characterized by the so-called Scorer parameter (Scorer, 1949, Chimonas and Hines, 1986, Doviak, et al., 1991), which for long waves is less than or nearly equal to zero in regions where these waves are evanescent. For long waves this type of trapping approximates an upper boundary of infinitely deep neutrally stratified fluid. Another mechanism, which in any realistic atmosphere can trap waves only partially, is a region where there is a sufficiently sharp change with height in the stability such that a significant amount of wave energy is reflected from this region (Crook, 1986, 1988). For long waves and sufficiently sharp changes in stability, this type of trapping approximates a rigid lid. The third mechanism to trap linear waves is for the waveguide to be bounded above by a reflecting critical level. A critical level is where the speed of the background flow matches the horizontal long wave speed. To be a reflecting layer, according to linear theory (Bretherton, 1966, Booker and Bretherton, 1967, Lindzen and Tung, 1976 and Lindzen and Barker, 1985), the critical level must be imbedded in a layer in which the Richardson number is less than 1/4, otherwise the wave will be absorbed at the critical level. Under some circumstances it is possible for a reflecting critical layer to over reflect the waves; that is, the waves may absorb energy from the mean flow at these levels. According to the calculations of Lindzen and Tung (1976), for long waves a reflecting or slightly over reflecting critical level approximates an upper boundary of infinitely deep neutrally stratified fluid. How these linear theories of wave trapping carry over to the case of inherently nonlinear solitary waves is not well understood. The numerical studies of Crook (1986, 1987) indicate that a layer with sufficiently small Scorer parameter can act as a reasonable trapping layer for nonlinear low-level solitary waves of elevation, although he describes some difficulties in applying the theory since an isolated solitary wave consists

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of a whole spectrum of horizontal Fourier components. Crook’s studies also indicate that a sharp temperature inversion at least partially traps some of the wave energy as well. However, it should be kept in mind that Crook’s method of generating the solitary waves by forcing a gravity current into a low-level stable layer means that the solitary waves are being constantly forced and may behave differently from a free wave. The behavior of critical levels in the presence of nonlinear waves is also not well understood. The theories for nonlinear critical levels (reviewed, for example, by Maslowe, 1986) are commonly developed only when the waves are horizontally periodic, and Bacmeister and Pierrehumbert (1988) state that it seems unlikely that the results of these theories will carry over to horizontally localized waves, such as isolated solitary waves. Maslowe (1986) remarked that more numerical simulations of stratified critical levels are needed to obtain an understanding of their behavior. For the case of solitary waves, some initial work on this subject has been done recently by Skyllingstad (1991). He numerically simulated low-level solitary waves of elevation in the presence of critical levels and found that for the limited number of cases he simulated the linear theories of critical levels reasonably predicted their wave reflecting properties. Skyllingstad’s method of generating the solitary waves is similar to that of Crook, and so raises the same objection that he simulated forced waves and not free waves. In addition, Skyllingstad concluded that there is a particular need for a morphological study of solitary waves of depression in the presence of critical levels. On the other hand, Maslowe & Redekopp (1980) and Tung et al (1981) have demonstrated that weakly nonlinear wave theories can be constructed which contain totally reflecting nonlinear critical layers. However, these critical layers are embedded in the wave structure and do not form the upper boundary condition, which distinguishes this work from that discussed above. Our purpose here is to describe the methods used to compare observations of solitary waves with weakly nonlinear solitary wave theory, as presented in Chapter 1. For this purpose, we first describe the general procedure and then we apply it to two specific observations: the first is a spectacular example of a low-level solitary wave known as the Morning Glory and the second is a well-documented large-scale solitary wave that is described by Lin & Goff (1988). In section 2 we review the general methods used for applying weakly nonlinear solitary wave theory to atmospheric observations. In section 3 we apply these methods to observations of the low-level solitary wave known locally as the Morning Glory and in section 4 to the observations of a large-scale atmospheric solitary reported by Lin & Goff.

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2.

COMPARING THEORY WITH OBSERVATIONS

The method for comparing the weakly nonlinear theory for steadily propagating solitary waves with atmospheric observations usually consists of the following five steps: 1) obtain the appropriate rawinsonde data for the observation; 2) determine the depth of the waveguide and the appropriate upper boundary condition from the sounding; 3) using the observed wave direction and wave speed, solve the linear eigenvalue problem for the linear long-wave phase speed and eigenfunction; 4) verify consistency between model prediction and data; and 5) compute the coefficients of the appropriate evolution equation. Then, with an estimate for the wave amplitude, all the wave quantities are easily computed. The best way to estimate the wave amplitude is to adjust the theory so that the maximum measured pressure or temperature at some height (usually the surface) is reproduced. The major difficulty in applying this program in the real atmosphere is determining the waveguide depth and the appropriate boundary condition at the top of the waveguide. Some of the data needed to apply the theory to actual atmospheric observations are difficult to obtain, e.g. estimating the observed amplitude of the wave. Most measurements of the waves are made at the surface and because the waves are often associated with other phenomena in the atmosphere, such as convection, it is difficult to estimate a wave amplitude from a surface pressure or temperature measurement alone. In applying the theory there are a number of assumptions that must be satisfied by the atmospheric observations for the comparison to be appropriate. First, the theory applies only to weakly nonlinear waves, so it is important to check that the ratio of the maximum vertical displacement of the wave to the depth of the waveguide is small and that the ratio of the waveguide depth to the horizontal length scale of the wave is small, at least significantly less than one. If there are indications that regions of recirculation exist in the wave, then the weakly nonlinear theory is inappropriate, and a fully nonlinear theory must be used. Further, the theory generally used in practice is two-dimensional, so it is inappropriate to use it near a source region of the waves where there are likely to be strong three-dimensional effects. In addition, it must be noted that the theory usually assumes that the background flow is horizontally and temporally homogeneous, although in Chapter 1 it is indicated how the theory can be modified to take some account of horizontal variability. One other feature to note is that the theory allows for an infinite number of modes to be trapped in the waveguide. The lowest mode (defined

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as that with the simplest vertical structure) is usually the dominant mode in the observations. The two examples we describe in the next two sections demonstrate how this program is implemented for some commonly occurring atmospheric solitary waves.

3.

OBSERVATIONS OF A LOW-LEVEL SOLITARY WAVE

Our first case study is an example of a low-level atmospheric solitary wave. This particular solitary wave is quite easy to observe because it is made visible as a spectacular propagating roll cloud that goes by the local name of the “Morning Glory.” The Morning Glory is a regularly occurring (in the spring) atmospheric feature seen near the southern coast of the Gulf of Carpenteria in northern Australia. The roll cloud is often very smooth in appearance, several hundred meters in diameter and 100-1000 km long, traveling at a speed of at 500 m above the ground. A picture of one such cloud is shown in Figure 3.1. Quite often the Morning Glory is an amplitude-ordered series of solitary waves that form a series of roll clouds. In fact, a second roll cloud can be seen in Figure 3.1. Observations have shown that there are three distinct types of Morning Glory waves. The most common type propagates from the northeast and generally appears in the early morning over the southern coast of the Gulf of Carpenteria . The second type propagates from the south and tends to appear over the this region at any time of day except late in the day. Finally, the last type propagates from the southeast and tends to appear over the Gulf of Carpenteria in the early morning. The particular example we will describe in detail here is one of the waves that propagates from the northeast. It occurred on the morning of 11 October 1981. The observations of this event are described in Smith & Morton (1984). An analysis of the wave properties of this event are described in Noonan & Smith (1985) and Rottman & Einaudi (1993). A map of the area in which the observed morning glory occurred showing the horizontal length and direction of propagation is given in Figure 3.2. Smith and Morton (1984) report that the leading roll cloud of the morning glory on 11 October 1981 passed over the Macaroni station at 0123 and over the Burketown station at 0710 (all times for this event are Australian Eastern Standard Time, EST). The cloud traveled at an estimated speed of from the direction of 75° from north.

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3.1

SYNOPTIC OVERVIEW

The synoptic conditions favorable for the formation of Morning Glories are nearly identical for all three types of waves, and it is possible for all three types to occur simultaneously over the southern Gulf of Carpentaria. One such observation has been reported by Reeder, et al. (1995). The favorable synoptic conditions are a significant pressure ridge over the east coast of the Cape York Peninsula, the absence of storm activity over the Burketown region and a well-developed sea-breeze regime over the southeastern Gulf area on the preceding day. These conditions are made even more favorable by the presence of an inland heat trough and an advancing frontal trough system south of the Gulf of Carpentaria.

3.2

EVIDENCE OF A SOLITARY WAVE

The evidence for Morning Glory solitary waves is sometimes quite simple, since the waves are often marked by a distinctive roll cloud. However, on many occasions the waves exist without the accompanying roll cloud. In these cases the waves can be detected by microbarographs on the surface. The measured streamlines and microbarograph time series for the Morning Glory of 11 October 1981 is shown in Figure 3.3, which shows four distinct waves represented by pressure elevations at the surface.

3.3

GENERATION

The northeasterly morning glory disturbances originate during the previous evening when a sea breeze front from the east and another from the west collide over the highlands of the Cape York Peninsula. This collision produces a bore that propagates along the low-level stable layer of moist marine air. The bore evolves into a series of amplitudeordered solitary waves that propagate at night towards the southwest over the Gulf of Carpentaria, arriving near dawn over Burketown. The cloud formation associated with northeasterly Morning Glory waves dissipates fairly rapidly as the disturbance moves inland into drier air over northern Queensland. Even after the cloud dissolves, however, the disturbance continues to propagate inland, often for distances of several hundred kilometers. The origin of southerly Morning Glory waves is not as well understood. There is evidence that some southerly disturbances originate over the interior of the Australian continent due to the interaction of a mid-latitude cold front with a developing nocturnal radiation inversion, as described in one case by Smith, et al (1995). Even less is known about the origin or properties of Morning Glories which arrive

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at Burketown from the southeast. Their existence were first reported only recently by Christie (1992). It is speculated that their generation is associated with the mountain range that runs along the east coast of Australia. Either katabatic flows down this mountain range or outflows from thunderstorms which often develop over these mountains are possible generation mechanisms for these waves.

3.4

TRAPPING

The 0600 sounding at Burketown for the temperature and wind speed in the observed direction of propagation of the wave are plotted in Figure 3.4. The actual measurements are plotted as solid circles and the smooth analytic function fit used in our numerical calculations is plotted as a solid line. A plot of the buoyancy frequency and the Scorer parameter in which c is the wave speed and is the wind profile) computed from a smoothed fit to the data is shown in Figure 3.5. The sounding shows a very strongly stratified layer next to the ground (about 500 m deep) and a layer of weak stratification from 0.5 to 2.0 km. Above 2.0 km the buoyancy frequency is slightly unstable. No data was collected above 4.0 km, but based on the observations of Clarke (1983) and Smith and Morton (1984), among others, it is now generally accepted that a strong subsidence inversion is almost always present at a height of about 4 km over the southern Gulf of Carpentaria during the spring and summer months. The weak or neutral stratification up to 4 km is generally attributed to convective mixing during the warm afternoon, and the deep layer of strong stability near the ground at Burketown is attributed by Clarke (1983) to preconditioning by an intense sea-breeze circulation on the previous day. It is clear from the plot of the wind speed in Figure 3.4 that no critical layer exists, as the maximum background speed is less than and the wave is traveling nearly twice that fast. Smith and Morton (1984) argue that it is unlikely at this time of year and at these latitudes that a critical level would exist above 4 km. Thus it seems plausible that this is a suitable situation for the application of the BDO formulation. One is at first tempted to define the waveguide as the first 500 m or so above the ground, as this is the roughly the height at which becomes negative in each sounding. However, calculations for the Burketown sounding using this depth produce far too small a value for the wave speed. Apparently the region of negative is not deep enough to trap all the wave energy. The reduction of N to zero above about 2.0 km in the Burketown sounding, on the other hand, is more promising.

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3.5

COMPARISON WITH WEAKLY NONLINEAR THEORY

Rottman & Einaudi (1993) chose the amplitude of the wave to be that which gave an approximate match to the surface pressure amplitude of the first undulation at each station. The results indicate that an amplitude of 1100 m gives the best comparison for the Burketown sounding. The agreement for both the horizontal scale and amplitude of the surface pressure and the streamlines is satisfactory, even though the comparison is made difficult by the fact that the observed waves are still part of an evolving family of solitary waves. Nevertheless, the results show that the horizontal scale of the waves is very close to the theoretically determined values. Incidentally, Rottman & Einaudi repeated these calculations for a fully compressible fluid and, as expected, found no significant differences in the results. These results are in contradiction to those of Noonan and Smith (1985), even though they used the same waveguide depth. The discrepancy is due mostly to their use of a much smaller wave amplitude. They used an amplitude of 400 or 600 m, based on pibal measurements of the streamline structure. These measurements, however, covered a maximum height of 1 km, whereas the largest amplitude displacements should be near the top of the waveguide, which in this case is 2 km. Noonan and Smith (1985) also tried matching the surface pressure but found that to be impossible. The difference between the two methods is that one used the consistent weakly nonlinear approximation for the pressure, whereas the other used the fully nonlinear result. Even though the use of the fully nonlinear pressure formula is inconsistent with the other weakly nonlinear approximations, the difference in results indicates that the wave must be strongly nonlinear. Further evidence of the strong nonlinearity of these low-level waves is seen in the streamline plots. Although the theoretical streamlines at l km altitude match qualitatively the measured streamlines, there is some evidence in the observations that the amplitude of the streamlines reaches a constant at height of about 1.0 km. The amplitude at this altitude is about 600 m. This would indicate that the effective waveguide depth is 1.0 km and the maximum amplitude is about 600 m. However, streamlines are very difficult to accurately deduce from the coarsely measured data, particularly at the upper levels. We conclude that the comparison with weakly nonlinear theory is quite difficult to justify in these two cases, mainly because of the strong nonlinearity of the waves.

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4.

OBSERVATIONS OF A LARGE-SCALE SOLITARY WAVE

The observation reported by Lin & Goff (1988) is an example of a large-scale solitary wave of depression for which there is some dispute over what atmospheric feature serves as the upper boundary of the wave guide for this wave. Lin & Goff concluded that the sharp increase in stability at the tropopause served as the upper boundary of the waveguide for this wave. In contrast, Rottman and Einaudi (1993) proposed that the upper boundary was a critical level imbedded in a low Richardson number layer located just below the tropopause.

4.1

SYNOPTIC OVERVIEW

The three-hourly surface sectional maps for the period 0000 - 1200 UTC 6 March 1969 showing the sea level isobars and National Meteorological Center (NMC) analyzed frontal positions are shown in Figure 3.6. We have modified these maps by drawing in the positions of squall lines determined from the radar summaries shown in Figure 3.7. The synoptic situation during the evening hours of 6 March was dominated in the southern United States by a surface low pressure region associated with a cyclone located over east-central Texas. A cold front extended southward from this cyclone and a stationary front extended eastward along the coastline. An inverted surface pressure trough exists to the north of the cyclone center. Three hours later a squall line, probably associated with the northerly moving warm, moist air behind the stationary front, formed in this inverted trough. The squall line extended from the cyclone center towards the northeast terminating in central Louisiana. At 0600 the southern end of the squall line has extended towards the south along the entire length of the cold front and the northern part of the squall line has moved into southern Mississippi; the whole squall line is moving towards the east at a speed of At later times the cyclone intensified and moved slowly towards the east along the coast reaching the Mississippi-Alabama border at about 1200. The squall line also continued to travel towards the east, staying in line with the cold front, but weakened substantially by 1200. At 0600 a detailed analysis (not shown here) indicates that a mesolow (a mesoscale low-pressure center) appeared over south-central Mississippi, near the northern limit of the squall line, and began to propagate towards the northeast. By 1200 there is evidence in the more detailed surface analysis of Lin and Goff (1988) that this mesolow reached the Ohio-Pennsylvania region by 1200. The radar summaries in Figure 3.7 show the convective systems that developed over the south-

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central United States on 6 March 1969. The squall line that formed in eastern Texas at 0300 is seen to have intensified quite rapidly so that by 0600 (when the mesolow began to propagate towards the northeast) its cloud tops have reached heights of more than 10 km, penetrating the tropopause. Also, as the mesolow propagated towards the northeast the radar summaries show that thunderstorms were associated with it. The 250 mb analyses for 0000 and 1200 UTC 6 March 1969 showing geopotential heights and isotachs are presented in Figure 3.8. These analyses show a deep upper level trough over the central United States and a jet streak to the east of this trough. Both the trough and the jet intensify and propagate towards the east over this 12 hour period. It appears that the mesolow that detaches from the squall line at 0600 propagates along the direction of the jet at subsequent times. The jet maximum, which exceeds , occurs over the mid-eastern states, so there is a so-called jet entrance region (the flow into the jet maximum) ahead of the trough axis, as is required for the generation of gravity waves by geostrophic adjustment (Uccellini and Koch, 1987). This entrance region is located over central Texas at 0000 and over the LouisianaMississippi border by 1200 on 6 March. That is, the entrance region is located in the area where the mesolow formed at 0600.

4.2

EVIDENCE OF A SOLITARY WAVE

The primary evidence of a mesoscale wave of depression is the presence of a “V-shape” in many National Weather Service (NWS) surface barograms from the eastern US. Figure 3.9a, taken from Lin and Goff (1988), shows the time series of surface pressure for a selection of stations along the path traveled by the mesolow that propagated to the northeast. All of these time series have a V-shaped pressure drop, with an amplitude of 3 - 6 mb and a period of about 2 hours. The pressure depression intensifies as the wave propagates away from Mississippi and Alabama, reaches a maximum near the West Virginia-Pennsylvania region, and steadily decreases thereafter. During the nearly 12 hours of its existence the general V shape of the wave is retained, although its amplitude changes. Figure 3.9b is a plot of the hourly isochrones, constructed by Lin and Goff (1988), connecting the time of passage of the pressure dip through the NWS surface barometer network. This plot shows clearly that the wave propagates approximately parallel to the direction of the upperlevel jet shown in Figure 3.8 and just to the north of the jet maximum. Lin and Goff (1988) estimated that this wave of depression propagated with an average speed of and had a horizontal length scale of

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approximately 185 km. Note that as the wave propagated away from its source it appears to spread laterally. An isentropic cross section, based on the 1200 UTC 06 March 1969 rawinsondes, is shown in Figure 3.10. The position of this cross section is indicated in Figure 3.9b as a heavy dashed line. At this time the wave had just passed over Dayton (DAY) and appears to be directly over Huntington (HTS). The tropopause can be identified in this plot at a height of approximately 9 km where there is a sharp change in the potential temperature gradient. In general, the atmosphere along the wave path is stably stratified, with a mean buoyancy frequency of about , below the tropopause and strongly stratified, with a mean buoyancy frequency of about above the tropopause. It is clear from this plot that the wave of depression occupies the lowest 8 or 9 km of the atmosphere and there is little evidence of a wave above

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this level. Lin and Goff estimate the maximum vertical displacement of these isentropes to be about 500m, although the coarse resolution of a plot of this kind makes an accurate estimate of the wave amplitude difficult. Note that there possibly is a phase reversal between 6 and 8 km; that is, the sign of the displacements of the isentropes from their unperturbed levels changes sign in the region above Huntington. Also, note that there is some evidence of other smaller disturbances ahead of the primary wave of depression.

4.3

GENERATION

From Figure 3.9 it appears that the wave was generated some time around 0600 in central Mississippi at the same time as the squall line was located in that area. Lin and Goff (1988) propose that the strong convection associated with the squall was responsible for triggering the observed solitary wave. Indeed, the radar summaries shown in Figure 3.7 indicate that the echo tops associated with the squall line in this region

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were at an altitude of 10 km, which seems sufficiently high for the convection to trigger a wave occupying most of the troposphere. Lin and Goff presented a rather detailed linear treatment, based on the theory of Smith and Lin (1982) and Lin and Smith (1986), to describe the generation of a wave of depression by the release of latent heat in an idealized sounding of the atmosphere near the squall line. In particular they centered their heating at 550 m above a mid-tropospheric temperature inversion, located at an altitude of 5.5 km, to produce a dispersive wave of depression propagating away from the source region. As mentioned in the introduction, Uccellini and Koch (1987) have argued that thunderstorms do not contain enough energy to generate such large scale waves as observed in this case. They suggest that some larger scale generation mechanism must be at work. In the present case, the conditions discussed by Uccellini and Koch are relevant. As shown in the wind speed cross section in Figure 3.11 strong shear exists in the upper troposphere and the Richardson number is sufficiently small to make the flow just below the tropopause susceptible to shear instability. Equally important, Figure 3.8 shows that the wave source region is near a jet streak entrance zone located on the downwind side of an upper level trough. The wave is observed primarily north of the jet streak and is a wave of depression, which would be consistent with geostrophic adjustment in this case. Thus all three mechanisms, deep convection, shear instability and geostrophic adjustment, could have played a role in generating the observed wave, and it would be difficult to distinguish which is the primary forcing mechanism in this particular case. It may be that such a dramatic wave was a fortuitous result of all three mechanisms coming together at the same place and at the same time.

4.4

TRAPPING

The disturbance we are considering here is a long wavelength internal gravity wave that propagates horizontally over a large distance. As described in the introduction, such a wave requires trapping to prevent the vertical propagation of energy. The type of trapping, evanescence or reflection, determines the type of solitary wave theory, BDO or KdV, that is appropriate. A plot of wind speed in the direction of wave propagation 235°, based on the 1200 UTC 06 March 1969 Dayton rawinsonde, is shown in Figure 3.11. In general, the vertical structure of the wind speed is monotonically (nearly linearly) increasing with height to a wind speed maximum at an altitude of about 10 km and then monotonically decreasing

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at higher altitudes. The wind speed maximum is located about 1 km above the tropopause. Lin and Goff decide that in the present case the tropopause, located at an altitude of 9 km, acts as a lid on the troposphere trapping, at least partially, the wave energy associated with the solitary wave. With this assumption, the KdV theory is the appropriate solitary wave theory. For the weakly nonlinear KdV theory to be strictly valid the ratio of the buoyancy frequency above the tropopause to that below it has to be infinite. The increase in buoyancy frequency at the tropopause in this case is about a factor of three, which according to Lindzen and Tung (1976) is not large enough to make a good reflector. Instead, they conclude that a critical layer imbedded in a low Richardson number region acts as a good reflector of linear internal gravity waves. The recent numerical simulations of Skyllingstad (1991) indicate that this last conclusion may also hold for nonlinear solitary waves. Lin and Goff (1988) state that in the present case the wave “is not a critical wave phenomenon since at no altitude does the wave phase speed equal the wind component in the direction of wave propagation”. The wind speed sounding, shown in Figure 3.11, contradicts this statement, since two critical levels clearly exist at altitudes of 8 and 12 km. In addition, Figure 3.12 reveals that in the region of the critical level at 8 km the buoyancy frequency is very small (even slightly unstable), and consequently the Scorer parameter is small, making this lower critical level a candidate as a reflecting upper boundary according to the requirements prescribed by Lindzen and Tung (1976) and Lindzen and Barker (1985). Rottman and Einaudi (1993) found that the best agreement with internal solitary wave theory was obtained when the critical layer at 8 km was treated as a rigid lid.

4.5

COMPARISON WITH WEAKLY NONLINEAR THEORY

Rottman & Einaudi (1993) obtained numerical solutions of the modal equation and computed the integrals defining the coefficients of the appropriate Korteweg-de Vries evolution equation (see Chapter 1 for details), in order to determine the properties of the solitary wave of depression that would propagate in a waveguide bounded below by the earth’s surface and above by the critical level at 8 km (actually the lower edge of the low region at 7.6 km was used as the upper bound of the waveguide). Smooth functional fits to the actual temperature and wind profiles were used to represent the background state. Rottman and Einaudi calculated the wave properties with and without the shallow-

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convection (or incompressible) approximation. The fully compressible calculations produced slightly better agreement with the observations: The computed phase speed is about which is very close to the observed speed of and also obtained reasonable agreement with the horizontal scale of the observed wave. Despite the favorable results obtained by Rottman & Einaudi, there remain doubts as to whether it is appropriate to use a critical layer as a rigid upper boundary in a weakly nonlinear approximation. In particular, within the error of the measurements, their results are not significantly better than those obtained by Lin and Goff.

5.

SUMMARY AND DISCUSSION

We have given a review of observations of atmospheric solitary waves, which can be divided into two groups: those that occupy a shallow layer near the earth’s surface and those that occupy the entire troposphere. The generation mechanisms appear to be quite different for these two classes of waves. The waves that occupy the lower part of the troposphere are generated by mesoscale processes such as gravity currents and downslope winds. The waves that occupy the whole troposphere are apparently generated by synoptic scale features such as large-scale convective systems and geostrophic adjustment. For these waves to propagate the long horizontal distances that are observed, there must exist some feature in the atmosphere that serves to prevent the wave energy from propagating away in the vertical direction. It appears that these trapping mechanisms are either deep layers of the atmosphere with very low values of the buoyancy frequency, or critical layers. However, in many cases it is difficult to determine precisely what the trapping mechanism is. Finally, we have reviewed attempts to use the simplest forms of weakly nonlinear solitary wave theory to predict the character of the observed solitary waves. In general this is found to be a quite difficult task. In particular, the theory requires an understanding of what mechanism is trapping the waves. As stated above, this is difficult to determine in practice. Further, it is still unclear just what upper boundary condition should be used that accurately corresponds to the various possible trapping mechanisms. For these reasons, comparisons with theory have so far been problematic, although there have been some successes. However, these comparisons so far have been restricted to the simplest forms of the theory; the more realistic forms of the theory described in Chapter 1 have yet to be compared in detail with atmospheric observations. Also,

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the observed waves are often strongly nonlinear, and there is clearly a need to develop theories which can take account of this.

Acknowledgments We thank Julie Noonan for providing us with the sounding data for the morning glory observation of 11 October 1981 and Yuh-Lang Lin for providing us with some details of the observations of 6 March 1969. Franco Einaudi, Chaing Chen, Rich Fulton, George Huffman, Mike McCumber and William C. Skillman aided us in the interpretation of the synoptic data.

References Abdullah, A.J., 1955: The atmospheric solitary wave. Bull. Amer. Meteor. Soc., 10, 511–518. Bacmeister, J.T., 1987: Nonlinearity in transient, two-dimensional flow over topography. Ph.D. thesis, Princeton University, 187 pp. Bacmeister, J.T., and R.T. Pierrehumbert, 1988: On the high-drag states of nonlinear stratified flow over an obstacle. J. Atmos. Sci., 45, 63–80. Bosart, L.F., and J.P. Cussen, 1973: Gravity wave phenomena accompanying east coast cyclogenesis. Mon. Wea. Rev., 101, 445–454. Booker, J.R., and F.P. Bretherton, 1967: The critical layer for internal gravity waves in a shear flow, J. Fluid Mech., 27, 513–539. Bretherton, F.P., 1966: The propagation of groups of internal gravity waves in shear flow. Quart. J. Roy. Met. Soc., 92, 466–480. Cheung, T.K. and C.G. Little, 1990: Meteorological tower, microbarograph array, and sodar observations of solitary-like waves in the nocturnal boundary layer. J. Atmos. Sci., 47, 2516–2536. Chimonas, G., and C.O. Hines, 1986: Doppler ducting of atmospheric gravity waves. J. Geophys. Res., 91, 1219–1230. Christie, D.R., 1989: Long nonlinear waves in the lower atmosphere. J. Atmos. Sci., 46, 1462–1491. Christie, D.R., 1992: The morning glory of the Gulf of Carpentaria: A paradigm for nonlinear waves in the lower atmosphere. Aust. Meteor. Mag., 41, 21–60. Christie, D.R., K.J. Muirhead, and A.L. Hales, 1978: On solitary waves in the atmosphere. J. Atmos. Sci., 35, 805–825. Christie, D.R., K.J. Muirhead, and A.L. Hales, 1979: Intrusive density flows in the lower troposphere: a source of atmospheric solitons. J. Geophys. Res., 84, 4959–4970. Clarke, R.H., R.K. Smith, and D.G. Reid, 1981: The Morning Glory of the Gulf of Carpenteria: an atmospheric undular bore. Mon. Wea. Rev., 109, 1726–1750.

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Crook, N.A., 1986: The effect of ambient stratification and moisture on the motion of atmospheric undular bores. J. Atmos. Sci., 43, 171–181. Crook, N.A., 1988: Trapping of low-level internal gravity waves. J. Atmos. Sci., 45, 1533–1541. Doviak, R.J., S.S. Chen, and D.R. Christie, 1991: A thunderstorm-generated solitary wave observation compared with theory for nonlinear waves in a sheared atmosphere. J. Atmos. Sci., 48, 87–111. Doviak, R.J., and R.S. Ge, 1984: An atmospheric solitary gust observed with a Doppler radar, a tall tower and a surface network. J. Atmos. Sci., 41, 2559–2573. Fulton, R., D.S. Zrnic, and R. Doviak, 1990: Initiation of a solitary wave family in the demise of a nocturnal thunderstorm density current. J. Atmos. Sci., 47, 319–337. Noonan, J.A. and R.K. Smith, 1985: Linear and weakly nonlinear internal wave theories applied to “Morning Glory” waves. Geophys. Astrophys. Fluid Dyn., 33, 123–143. Grimshaw, R., 1980/1981: Solitary waves in a compressible fluid. Pageoph., 119, 780–797. Grimshaw, R.,1981: Evolution equations for long nonlinear internal waves in stratified shear flows. Studies Appl. Math., 65, 159–188. Karyampudi, V.M., S.E. Koch, C. Chen, J.W. Rottman, and M.L. Kaplan, 1993: The influence of the Rocky Mountains in the 13-14 April 1986 severe weather outbreak. Part II: Generation of an undular bore and its role in triggering a squall line. Mon. Wea. Rev., 123, 1423– 1446. Lin, Y.-L., and R.C. Goff, 1988: A study of a mesoscale solitary wave in the atmosphere originating near a region of deep convection. J. Atmos. Sci., 45, 194–205. Lin, Y.-L., R.B. Smith, 1986: Transient dynamics of airflow near a local heat source. J. Atmos. Sci., 43, 40–49. Lindzen, R.S. and J.W. Barker, 1985: Instability and wave over-reflection in stratified shear flow. J. Fluid Mech., 151, 189–217. Lindzen, R.S., and K.-K. Tung, 1976: Banded convective activity and ducted gravity waves. Mon. Wea. Rev., 104, 1602–1617. Mannasseh, R. and J.H. Middleton, 1995: Boundary-layer oscillations from thunderstorms at Sydney airport. Mon. Wea. Rev., 123, 1166– 1177. Maslowe, S.A., 1986: Critical layers in shear flows. Ann. Rev. Fluid Mech., 18, 405–432. Maslowe, S.A., and L.G. Redekopp, 1980: Long nonlinear waves in stratified shear flows. J. Fluid Mech., 101, 321–348.

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Maxworthy, T., 1980: On the formation of nonlinear internal waves from the gravitational collapse of mixed regions in two and three dimensions. J. Fluid Mech., 96, 47–64. Miesen, R.H.M., L.P.J. Kamp, and F.W. Sluijter, 1990a: Long solitary waves in compressible shallow fluids. Phys. Fluids, A2, 359–370. Miesen, R.H.M., L.P.J. Kamp, and F.W. Sluijter, 1990b: Solitary waves in compressible deep fluids. Phys. Fluids, A2, 1401–1411. Pecnick, M.J., and J.A. Young, 1984: Mechanics of a strong subsynoptic gravity wave deduced from satellite and surface observations. J. Atmos. Sci., 41, 1850–1862. Ramamurthy, M.K., B.P. Collins, R.M, Rauber, and P.C. Kennedy, 1990: Evidence of very- large-amplitude solitary waves in the atmosphere. Nature, 348, 314–317. Rees, J.M., and J.W. Rottman, 1994: Analysis of solitary disturbances over an Antarctic ice shelf. Boundary-Layer Met., 69, 285–310. Rottman, J.W. and F. Einaudi, 1993: Solitary waves in the atmosphere. J. Atmos. Sci., 50, 2116–2136. Reeder, M.J., D.R. Christie, R.K. Smith and R. Grimshaw, 1995: Interacting “Morning Glories” over northern Australia. Bull. Amer. Met. Soc. 76, 1165–1171. Rottman, J.W., F. Einaudi, S.E. Koch, and W.L. Clark, 1992: A case study of penetrative convection and gravity waves over the PROFS Mesonetwork on 23 July 1983. Meteor. Atmos. Phys., 47, 205–227. Rottman, J.W., and J.E. Simpson, 1989: The formation of internal bores in the atmosphere: a laboratory model. Quart. J. Roy. Meteor. Soc., 115, 941–963. Scorer, R.S., 1949: Theory of waves in the lee of mountains. Quart. J. Roy. Meteor. Soc., 75, 41–56. Shreffler, J.H. and F.S. Binkowski, 1981: Observations of pressure jump lines in the Midwest, 10-12 August 1976. Mon. Weather Rev., 109, 1713–1725. Skyllingstad, E.D., 1991: Critical layer effects on atmospheric solitary and Cnoidal waves. J. Atmos. Sci., 48, 1613–1624. Smith, R.B., and Y.-L. Lin, 1982: The addition of heat to a stratified airstream with application to the dynamics of orographic rain. Quart. J. Roy. Meteor. Soc., 108, 353–378. Smith, R.K., 1988: Traveling waves and bores in the lower atmosphere: the “Morning Glory” and related phenomena. Earth-Sci. Rev., 25, 267–290. Smith, R.K., N.A. Crook, and G. Roff, 1982: The Morning Glory: An extraordinary atmospheric undular bore. Quart. J. Roy. Meteor. Soc., 108, 937–956.

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Smith, R.K., and B.R. Morton, 1984: An observational study of northeasterly ‘morning glory’ wind surges. Aust. Met. Mag., 32, 155–175. Smith, R.K., M.J. Reeder, N.J. Tapper and D.R. Christie, 1995: Central Australian cold fronts. Mon. Wea. Rev., 123, 19–38. Tepper, M., 1950: A proposed mechanism of squall lines – the pressure jump line. J. Meteor., 7, 21–29. Tung, K.-K., D.R.S. Ko, and J.J. Chang, 1981: Weakly nonlinear internal waves in shear. Studies Appl. Math., 65, 189–221. Uccellini, L.W., and S.E. Koch, 1987: The synoptic setting and possible energy sources for mesoscale wave disturbances. Mon. Weather Rev., 115, 721–729.

Chapter 4 GRAVITY CURRENTS James W. Rottman and P. F. Linden Department of Mechanical & Aerospace Engineering University of California, San Diego [email protected], [email protected]

Abstract

1.

A review is given of the basic theory for gravity currents, as well as the closely related internal two-layer bore. These theories are then compared with laboratory experiments and numerical simulations. It is found that the simple theory of gravity currents works very well for a wide range of density differences. However, the theory for the internal two-layer bore is successful only for Boussinesq fluids and even then only for a certain class of bores. The main difficulty with the hydraulic theory for a two-layer bore is determining how the necessary dissipation is distributed between the layers.

INTRODUCTION

Gravity currents are horizontal flows of fluid of one density into a surrounding fluid of another density. The driving force is the buoyancy force that results in a gravitational field from the density difference between the two fluids. A gravity current can be thought of as a limiting case of a two-layer bore, a propagating abrupt change in the interface depth in a two-layer fluid. A gravity current results when the depth of the layer upstream vanishes. Gravity currents and internal bores are common phenomena in geophysical flows. They occur whenever there are fluids of different densities in the presence of a horizontal boundary, either a rigid surface, a density interface or a free surface. There are many examples of gravity currents in nature. One example of an atmospheric gravity current, known as a haboob, is shown in Figure 4.1. An extensive collection of naturally occurring gravity currents can be found in Simpson (1997). Possibly the most studied natural gravity current is the sea breeze, which occurs when cool, moist sea air flows in over the land. In some parts of the world this is an almost daily occurrence, and the sea breeze is a major influence on the local climate. Sea air is carried inland, sometimes over

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100 km, and the sea breeze can also transport pollution from coastal regions significant distances inland. Gravity currents also occur in industrial flows and are often associated with the accidental release of flammable or toxic gases into the atmosphere. These gases, such as natural gas or chlorine, are usually stored at low temperatures as liquids under pressure. On release to the atmosphere, the liquid vaporizes producing a cloud of cold dense gas. This cloud moves and is diluted under the influences of its own (negative) buoyancy and the wind. When the buoyancy forces are dominant the flow is a gravity current. Gravity currents occur in buildings, when air masses of different densities come into contact. This can occur when a door between two rooms is opened. Air from the cooler room flows through the lower part of the doorway and

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across the floor of the warmer room as a gravity current. Gravity currents may also occur when hot gases above a fire spread across the ceiling. These flows are particularly important in relatively enclosed spaces such as transportation or mining tunnels. As seen from Figure 4.2 a gravity current is characterized by a sharp front, immediately behind which is a raised head. The presence of the front is the defining property of the gravity current that distinguishes it from a more diffuse gravity-driven flow. Immediately behind the smooth head region shear instabilities occur. These cause the gravity current to mix with the ambient fluid, and cause it to gradually change its properties. In this chapter we will derive the basic theory describing the steady propagation of a two-dimensional gravity current along a free-slip solid boundary in a channel of finite depth. The assumptions involved in this basic theory will be explored as to their plausibility and comparisons will be made with laboratory studies of lock-exchange gravity currents. The next section describes the existing theories for two-layer bores. Currently there is no definitive theory for two-layer bores. All existing theories are flawed, although some work quite well over a limited subset of the entire parameter space. We will describe the

92 main existing theories and compare their results with laboratory experiments and numerical simulations.

2. 2.1

BOUNDARY GRAVITY CURRENTS THE BUOYANCY ACCELERATION

Take the case of fluid with two different densities and with say separated by a vertical barrier. Since the fluid is at rest the difference in pressure on either side of the barrier at the bottom of the fluid is

where H is the depth of the fluid and is the acceleration due to gravity. Since the pressure difference is the net force per unit area, which is the product of mass times acceleration per unit area, we get

where is the acceleration and is the density of the fluid being accelerated. Comparison of (4.1)and (4.2) shows that the acceleration

The quantity is known as the reduced gravity. Note that there is some ambiguity in the choice of in the denominator of (4.3), depending on which side of the barrier is under consideration.

2.2

SCALING LAWS

Before developing a formal theory, we first examine some consequences of dimensional analysis. Consider a finite volume V of dense fluid (density released from rest on a horizontal boundary in a stationary, infinitely deep ambient fluid of density Suppose the fluid is confined in a channel of unit width, so the dense fluid flows along the channel and the properties of the flow are independent of the across-channel coordinate. For simplicity we consider the channel to have a vertical wall at so that the current travels in the positive direction only, as sketched in Figure 4.3. Flow is generated by the buoyancy force and the associated acceleration is given by defined by (4.3). We suppose that initially the flow accelerates to a speed large enough that viscous forces are unimportant, and the volume V is large enough to be effectively infinite. Then the other parameter determining

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the flow is the initial depth H of the dense fluid. Dimensional analysis shows that the velocity U of the advancing current is given by

where F is a dimensionless constant, which has the form of a Froude number, and is a dimensionless time. The function describes the acceleration from rest. Clearly when and observations show that tends to a constant as becomes large compared to the acceleration time After this time the current travels with a constant speed, characterized by a constant Froude number F. The relation (4.4) may be interpreted as a balance between the buoyancy force driving the current and the inertia of the surrounding ambient fluid, or between the potential energy of the dense fluid and the kinetic energy of the resulting flow. At later times, the finite volume V of the dense fluid will influence the motion. For the flow in the channel the relevant parameter is the area of the release, which is the volume per unit width. This introduces a further dimensionless variable where is the initial length of the released fluid as shown in Figure 4.3, and (4.4) becomes

94 The variable is the dimensionless time associated with the finite volume of the initial release. When is small, the current propagates as though the initial volume was infinite, and has a constant speed given by (4.4). When becomes large the effects of the finite initial volume become important. Then it seems reasonable to assume that the front travels with a constant Froude number, but now based on the local depth at the front rather than the initial depth. We represent the current by a length and depth Conservation of buoyancy is expressed as

where is the initial value of the reduced gravity and is a shape constant, which would be unity if the current retained a rectangular shape. A constant local Froude number F implies that

Using (4.6) and integrating gives

Equation (4.8) shows that at these later times the length of the current increases as so that the velocity of the front decreases with time. It is worth noting that this result does not assume conservation of volume of the current. The current can mix with the ambient fluid, thereby increasing its volume and decreasing its density. However, the total buoyancy is conserved, as expressed by (4.6). As the current decelerates, frictional effects become important and the final stage of the motion is determined by viscosity. A further dimensionless time now enters the problem and the front speed may be written as

At large times we expect the dependence on and to be unimportant. The horizontal pressure gradient driving the current will be balanced by viscous stresses so that

95 where is a dimensionless shape constant. Using conservation of buoyancy (4.6) we obtain the following differential equation for the length of the current

To proceed further it is necessary to assume that volume is conserved. This is likely to be a good assumption in the viscous phase and is written as

where is the area of the current at the start of the viscous phase. Substituting for from (4.12) and solving the resulting differential equation gives

where is the length of the current at the start of the viscous phase. Experimental results confirming these scalings are given in Huppert & Simpson (1980), Huppert (1982) and Rottman & Simpson (1983).

2.3

CONSERVATION LAWS AND BENJAMIN’S THEORY

The theory of inviscid gravity currents is based on the seminal paper by Benjamin (1968). In order to examine the simplest possible case he considered the flow of a cavity along the roof of a rectangular channel into a region of homogeneous fluid initially at rest. The channel is wide compared to its depth, so that the flow may be considered as two-dimensional. Benjamin (1968) assumed that the motion of the cavity is steady, so that the front of the cavity is at rest in a frame of reference moving with the (constant) velocity of the front. In this frame of reference, the foremost point of the cavity is a stagnation point. The point of considering a cavity is that it provides the simplification that the pressure is constant everywhere within it, since there is no flow in the cavity. For the case of two fluids, flow within the current relative to the head, such as a recirculating flow from the rear of the current, will produce pressure variations along the interface. Therefore, the application of Benjamin’s model to that case requires that any internal flows within the current are negligible. Here we first derive the theoretical results for both a heavy current and a light current propagating into a stationary fluid. Following Benjamin we assume that the current has a constant velocity U and work in a frame of reference in which the front of the current is at rest. We also assume that far behind the front the flow

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is horizontal so that the pressure is hydrostatic. We will derive the results for the heavy and light current separately, with no restriction on the density ratio. We denote the density ratio by

Then limit being

2.4

with

corresponding to a cavity and the Boussinesq

HEAVY CURRENT

In this case a current of density propagates into a fluid of lesser density as shown in Figure 4.4. Denote the depth of the channel by H, the depth of the current by and suppose that the fluid velocity above the current far behind the front where the interface is flat is Continuity implies that

Since there are no external forces acting on the flow, the net flux of momentum into a control volume including the front is zero. We consider the control volume consisting of the two vertical planes downstream at A and upstream at B of the front and the top and bottom boundaries of the channel in between, as indicated in Figure 4.4. Conservation of the horizontal component of the momentum flux may then be written as

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The pressure distributions at the two locations A and B may be determined since the flow is assumed hydrostatic. We define the pressure at the stagnation point O to be Application of Bernoulli’s equation along the streamline OB, gives the pressure above B as

Since the velocity within the current is zero, application of Bernoulli’s equation along AO gives

Substitution of (4.17) and (4.18) into the momentum balance (4.16) and use of the continuity equation (4.15) gives

where

If there is no dissipation in the flow we may apply Bernoulli’s equation along another streamline to determine The choice of either the upper boundary of the channel or the interface between the two fluids gives the same result

Equating the two expressions for the current speed U, gives two solutions for the current depth

Hence an energy-conserving current occupies one-half the depth of the channel and travels with a non-dimensional speed

98

Before considering further properties of these solutions we discuss the case of a light current.

2.5

LIGHT CURRENT

The nomenclature (shown in Figure 4.5) and the derivation is essentially the same in this case. Again writing the pressure at the stagnation point O as and applying Bernoulli along the upper boundary, the pressures at the planes B and A are

and

The horizontal momentum balance is slightly different in this case and is

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Substituting for the pressure from (4.24) and (4.25) and using continuity

we obtain

where

For an energy conserving flow we may apply Bernoulli’s equation along either the lower boundary of the channel or along the interface between the two fluids. In both cases we obtain

and equating the speed with that in (4.28) gives again the two solutions

The energy-conserving current occupies one-half the depth of the channel and has velocity

2.6

COMPARISON BETWEEN THE LIGHT AND HEAVY CURRENTS.

The similarity between the light and heavy currents is seen by writing (4.29) in terms of the depth of the current. In this case

which is the same as the form of f for the heavy current (4.20). Thus the dimensionless speeds differ by the factors involving the density ratio in (4.19) and (4.28).

100 These two factors are for the light current and for the heavy current. Since a heavy current travels faster than a light current of the same depth for all density ratios. For Boussinesq currents, and we may write with In this case the difference in speeds is In the Boussinesq limit the heavy and light currents are identical. In this case the dimensionless speed of the current depends on the depth of the current and is given by

The left hand side of (4.34) is the square of the Froude number of the current, based on the channel depth and the reduced gravity The function is plotted in Figure 4.6 as a function of The volume flux carried by the current is also plotted in this figure. We see that the Froude number increases with reaching a maximum at and takes the value 1/2 when The volume flux Q increases monotonically with up to the energy conserving limit at which As discussed above energy conserving currents either have zero depth at the front or they occupy one half the full depth Benjamin (1968) argued that currents occupying less than half the depth were dissipative and that external energy is required for currents greater than half the depth. As we will see later, full depth lock release currents are observed to occupy half the depth, and so we will consider the properties of this flow further. In this case the Froude number of the current based on the channel depth, which in the frame of reference of the front of the current corresponds to the Froude number of the oncoming upstream flow, is 1/2. Thus the on-coming flow is subcritical. Behind the head the Froude number of the ambient stream flowing over the current, and therefore the Froude number of the flow within the current in the rest frame is Thus as the flow passes over the front of the current from sections B to A in Figure 4.4, it passes from subcritical to supercritical, and the front itself can be considered as a point of hydraulic control. In keeping with the concept ofa bore, the Froude number at the front is not well defined. The flow is not hydrostatic there, and there may be dissipation - as in the case of a bore. We will examine the relationship to bores in § 3. A consequence of this idea of a hydraulically controlled head is that its speed, in circumstances where Benjamin’s (1968) theory does not strictly apply, may be calculated by assuming that the current travels at a constant Froude number. This provides a rationale for the use of a constant Froude number in the buoyancy-inertia regime discussed in § 2.2. This idea is commonly used

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in integral models. However, since it is difficult to define the current depth unambiguously, there is some debate about the appropriate numerical value for F. Huppert & Simpson (1980) have suggested 1.12, when the depth of the current some distance behind the head is used in the definition of the Froude number.

2.7

NON-BOUSSINESQ EFFECTS.

When there is a large difference between the densities of the two layers there are significant differences in speeds between the heavy and the light currents. In the limit of vanishing upper layer density, and the velocity of the heavy layer increases without limit provided remains finite in this limit. Since, observations show that, e.g. water from behind a dam travels at finite speed, clearly the above theory must be modified in this limit. This issue is addressed by considering the dam break problem.

2.8

THE DAM BREAK

This is the flow produced by the release of liquid from the instantaneous removal of a dam. It represents the limit of the non-Boussinesq flow discussed in § 2.7, where the lighter density is a gas and so is negligible. The problem

102 was solved by St. Venant (see for example, Whitham (1974)) in the context of shallow water theory. In the dam break problem it is assumed that the flow satisfies the assumptions of shallow water theory, in particular, that vertical accelerations are small compared to This assumption is unlikely to hold precisely during the initial phase just after the dam breaks, but is reasonable thereafter. So, while the solution is not strictly valid at the start of the flow, the errors introduced will be small. In shallow water theory horizontal velocities are assumed independent of depth and, for flow in a channel, the problem reduces to the along channel momentum equation and the continuity equation, respectively,

and

These equations may be written in terms of characteristics. First nondimensionalize the variables using the initial depth as the length scale and as the timescale. Then (4.35) and (4.36) reduce to the following equation for the nondimensional velocity and depth

The fluid is initially at rest

and the depth

On any positive characteristic, the forward Riemann invariant

In the region covered by these characteristics the motion is given by the straight negative characteristic

Hence,

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Thus the free surface is a parabola between the front and the undisturbed dam level at The depth at the lock position is and the speed of the front is 2, which corresponds to a Froude number based on the original lock depth of 2. The speed at the lock is and so the Froude number based on the local depth is unity at that location. Thus we see that the front propagates at a speed so that the Froude number is 2. This implies that the limit of (4.20) as is

Given the form of

this implies that

showing that the depth of the front goes smoothly to zero in the limit as

2.9

EXPERIMENTS

Results of two typical full-depth lock exchange experiments are shown in Figures 4.7 and 4.8. These flows were started by lifting a vertical barrier (the lock gate) at the mid point of the tank that separated stationary fluid of different densities. Both the lower and upper horizontal boundaries are rigid horizontal planes. These figures show a series of shadowgraph images of the resulting flow for two density ratios: a Boussinesq case and a non-Boussinesq case In both plots time is non-dimensionalized by based on the reduced gravity and the depth. For the Boussinesq case, Figure 4.7, the speeds of the light and heavy currents are constant and almost the same. The flow is symmetrical about the centerline, with the leading part of each current occupying about one half of the depth. Although there are some small scale instabilities the overall shape of the interface between the two counterflowing layers is stable, and is at mid-depth at the lock-gate position. The Froude number calculated from this flow, F = 0.48 is close to the theoretical value 0.5 for an energy conserving current. For the non-Boussinesq case, Figure 4.8, the speeds are again constant but now the heavier current travels significantly faster than the light current. The light current travels at about the same non-dimensional speed as the Boussinesq current shown in Figure 4.7. The symmetry of the Boussinesq case is lost, but the depths of the leading parts of the two currents are again close to the half depth of the fluid. The depth at the lock-gate position is close to mid-depth.

104

105

106 In addition to the different speeds in the non-Boussinesq case there is another significant asymmetry shown in Figure 4.8. This is the formation of a region behind the heavy current front where there is a significant decrease in the depth of the dense layer. Associated with this is the evidence of turbulence and mixing. It is clear that the flow violates volume conservation if the two layers have the same depths but different speeds. In the non-Boussinesq case (Figure 4.8), the volume flux carried to the right by the heavy layer is greater than that carried to the left by the light layer. Hence the depth of the dense layer must decrease, as observed, to conserve volume. Thus the experiments suggest that both the heavy and light fronts are moving independently at constant speeds with depths close to half the channel depth. There is an adjustment in the lower layer behind the heavy front to account for the fact that the supply to the lower layer is less than that carried by the front. This suggests, and the experimental measurements of the front speeds verify, that the flow may be described by Benjamin’s (1968) energy conserving theory for steady gravity currents. Further confirmation of validity of the energy conserving theory is the comparison shown between the theoretical shape as calculated by Benjamin (1968) and the observed currents shown in Figures 4.9 and 4.10. The theoretical shape, which is strictly valid only near the front, has been extended by a straight horizontal line at mid depth to join the two fronts. The agreement with the observed currents is excellent, and leaves little doubt that these currents are the half depth energy conserving currents predicted by the theory in § 2.3. The only significant deviation is the narrowing of the lower layer from the lock towards the heavy front. There is then an abrupt increase in depth to the half depth shape closer to the front. Associated with this depth change is evidence of strong mixing. A similar feature is shown in Figure 4.8. It is our contention that this narrowing ofthe lower layer is a result of volume conservation. Since the heavy front travels faster than the light front, ifthe upper and lower layers had equal depths there would be a net flow towards the light side of the lock. In a closed channel this is impossible, and so it necessary for the lower, faster layer to be thinner than the slower upper layer, since the volume flux is a monotonic function of as shown in Figure 4.6. However, the heavy front remains at half depth and so it is necessary for the lower layer to increase again near the front. It does this abruptly as a hydraulic jump or bore.

3.

TWO-LAYER INTERNAL BORES

As stated in the introduction, two-layer bores can be thought of as a generalized form of gravity currents. A two-layer internal bore is a propagating abrupt change in depth of the horizontal interface between two fluids of different densities and a gravity current results when the upstream depth of this interface vanishes. In this section we will restrict attention for simplicity to an internal

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bore propagating at constant speed into two fluid layers at rest, although in the concluding section we will make some comments about how the results are modified if there is shear in the upstream flow. In the absence of shear this flow is the same as that of a stationary hydraulic jump. Shadowgraphs of an internal bore propagating along an interface between two fluids of different densities is shown in Figure 4.11. In this figure the three basic types of bore structure are shown for three different bore strengths (the ratio of the downstream to the upstream depth of the interface): for weak bore strengths the bore has a smooth undular character, for intermediate strengths it is undular but with some turbulent mixing occurring behind the first crest, and for strong bores the mixing completely dominates the motion obliterating any undulations. In this latter case, comparing it with the image in Figure 4.2, the bore has the appearance of a gravity current. We will use integral conservation of mass and momentum arguments, similar to those used in the previous section for gravity currents, for determining the relationship between the speed of the bore and its strength.

3.1

THE BORE SPEED

A schematic diagram of a steady two-layer bore in a channel is shown in Figure 4.12. In this figure, we are in a reference frame in which the bore is at rest, such that both layers upstream of the bore have uniform fluid speed U. The

108

109 density is assumed constant in each layer. This sketch serves to define much of the nomenclature to be used in this discussion. To determine the relationship between the speed U and the bore strength we assume that mass is conserved within each layer and that overall momentum is conserved. Mass conservation in each layer implies

Since there is no external horizontal force on this system, the overall horizontal momentum must be conserved with a control volume containing the bore front. Consider the control volume consisting of the two vertical planes downstream at A and upstream at B and the top and bottom boundaries of the channel. The integral theorem of horizontal momentum conservation can be written as

The energy dissipation rate D in our control volume is equal to the difference in the energy flux through both ends and positions A and B, with the difference taken such that the dissipation rate is positive if energy decreases within the control volume

In these equations, the two-layer stratification is represented by

Far upstream and downstream of the bore turbulence is assumed to be sufficiently weak so that the flow is horizontal and uniform with depth within each layer so that upstream U is independent of and downstream

Also as a consequence of this assumption, the pressure is hydrostatic far upstream and downstream of the bore. Therefore, the vertical distributions of

110 pressure at locations A and B are

and

where and are the values of the pressure at the top of the channel at locations A and B, respectively. Using (4.45), (4.51) and (4.52) in (4.46) we obtain

where is the reduced gravity. This equation gives a relation between the bore speed U and the bore strength given that we know the pressure difference along the top of the channel. For one-layer flow with a free surface, and so (4.53) reduces to

and here

The dissipation rate for this flow is from (4.47)

Therefore, for free-surface bores mass and momentum conservation completely determine the bore speed as a function of the bore strength. The dissipation equation tells us that energy is lost in traversing the jump if otherwise energy is gained as the fluid traversed the jump which is physically impossible. In two-layers the application of the principles of conservation of mass in each layer and the overall conservation of momentum through the jump does not produce a closed problem. Using these conservation relations does not allow us to determine the pressure difference across the bore at the top of the channel. The reason for this is that it is unclear how the necessary energy dissipation should be distributed between the two layers. There have been three distinct attempts to resolve this ambiguity within the context of hydraulic theory. The first attempt to describe the relationship between the strength and the speed of a two-layer bore was made by Yih & Guha (1955). This theory assumes

111

that the pressure remains hydrostatic through the jump and results in an overall energy loss through the jump although one of the layers actually experiences an energy gain. The hydrostatic assumption would appear to be valid for jumps of small amplitude and, indeed, numerical simulations and laboratory experiments have confirmed that this theory is valid for small amplitude bores propagating into two-layer fluids at rest. The second attempt at resolving the ambiguity in the two-layer bore theory was made by Chu & Baddour (1977) and independently by Wood & Simpson (1984). They proposed that a better approximation for a two-layer bore would be to abandon the assumption that the pressure is hydrostatic through the bore and to replace it with an assumption about energy conservation in each fluid layer through the bore. Specifically, they assumed that the energy must be conserved in the contracting layer and that energy must be lost in the expanding layer. This idea is based on earlier results of hydraulic flow in a one-layer fluid with boundary layer separation. The hope was that this new theory would apply to larger amplitude bores than the earlier theory of Yih & Guha (1955). But it turns out that the results of the new theory and the old theory were almost indistinguishable and in particular the new theory was not any better at predicting large amplitude bores. The third attempt at resolving this issue was made by Klemp et al. (1996). They proposed a variation of the second theory for two-layer bores. They sug-

112 gested that the opposite assumption of energy conservation in the expanding layer and energy dissipation in the contracting layer is consistent with the Benjamin (1968) theory of gravity currents in the limit of very large amplitude bores. The theory derived with this assumption appears to give better agreement with both numerical simulations and experiments over the whole range of bore amplitudes, at least for bores propagating into a fluid at rest. However, Klemp et al. (1996) made comparisons mainly for flows in which one layer was significantly deeper than the other. Klemp et al. (1996) gave physical arguments for why their theory should be approximately correct for bores propagating into a fluid at rest. The same arguments suggest that this theory will not be accurate for bores that are propagating into a shear flow, as would be the case for hydraulic jumps that form in the lee of the hill in two-layer flows over hills. In this case, the upstream vorticity field makes the internal jump behave more like the hydraulic jump in a free surface flow. Such jumps are characterized by forward breaking waves and intensive mixing through the entire active layer. Since the first and second theories give nearly the same result, we will focus here on the details of the second and third theories. Both theories make the additional assumption that the densities ofthe two layers are sufficiently similar that the Boussinesq approximation can be used. For the Wood & Simpson theory we can use Bernoulli’s equation to compute the pressure difference at the top of the channel

where Substituting this result into (4.53) we get the following expression for the bore speed

The dissipation rate for the bore in this theory is

For the theory of Klemp et al. (1996) the pressure difference is given by

Substituting this result into (4.53) we get the following expression for the bore speed

113 The dissipation rate for the bore in this theory is

In the limit as

expression (4.60) reduces to

which is the linear long-wave speed for a two-layer fluid. And in the limit as expression (4.60) becomes

which is the expression for the gravity current speed derived in the previous section. A plot of the bore speed as computed from both theories as a function of bore strength is shown in Figure 4.13 for several values of the parameter Note that the two theories agree very well for small bore strengths but for larger bore strengths the Wood & Simpson theory predicts substantially larger speeds. This difference between the theories gets more pronounced as decreases. In fact, the Wood & Simpson theory is singular in the limit

3.2

MAXIMUM BORE STRENGTH

The strength of free-surface bores is theoretically unlimited, but the strength of two-layer bores is bounded. Baines (1995) argues that the largest amplitude internal bore occurs when the dissipation rate through the bore vanishes. For larger bore strengths the dissipation rate changes sign and the fluid must gain energy when traversing the bore, which is physically unacceptable. This of course must be true, but there is another more fundamental restriction on twolayer bore strength within the hydraulic approximation. That restriction is that the bore cannot travel faster than the long interfacial waves propagating behind it. Otherwise, there is no way for the fluid to keep up with the bore. In this section, we will investigate the consequences of requiring that the bore speed be less than or equal to the speed of the long interfacial waves that follow it and compare the restriction on the bore strength this imposes with that imposed by the requirement that energy is dissipated in the bore. In a reference frame in which the bore is propagating to the right with speed U into fluid that is at rest, the characteristic speeds on the interface behind the bore are given by

114

in which

and in this case is the speed of the fluid in the lower layer downstream of the bore, which from the previous section (see (4.19) and (4.20)) can be shown to be

Of course, is the speed of the right propagating wave and is that of the left propagating wave. We seek the value of for specified for which which means that the bore is moving at or greater than the speed of the long waves. Figure 4.15 shows the results of this calculation for both the Wood & Simpson theory and the Klemp et al. theory for the bore speed.

3.3

NUMERICAL SIMULATIONS AND EXPERIMENTS

A number of laboratory experiments and numerical simulations have been performed to test the validity of the theories developed in the previous sections. The most common method of generating a two-layer bore in the laboratory is to tow a hill along the bottom of a tank filled with a two-layer fluid. For the appropriate values of the hill height and interface depth, the motion of the hill

115

will generated a bore propagating along the interface upstream of the hill. This is how the experiments of Wood & Simpson (1983) were performed. Here we will present some results of a two-dimensional numerical simulation of the uniform flow of a two-layer fluid over a Gaussian-shaped hill that were performed by Cummins (1995). The mean amplitudes and speeds of upstream propagating bores were measured for three values of the parameter 0.11,0.16, and 0.23. In this case we are interested only in the intrinsic properties of the bores and not in their relationship to the height and width of the hill. Since the bores were in general undular, their mean amplitude was determined by applying a strong spectral filter to the mean density contour to eliminate the undulations. The speed of the bore was determined by plotting the position of the interface at half the amplitude of the bore as a function of time. The results of these measurements are plotted in Figure 4.13, along with the results obtained from the bore theories outlined in the previous sections. First note that the theory of Klemp et al. (1996) clearly compares best with the numerical simulations. Indeed, for the case of and perhaps even the agreement is all one could hope for in a theory. The other theory, while valid for small amplitude bores, significantly overestimates the speeds for larger amplitude bores, whereas the theory of Klemp et al. is valid over the full amplitude range. However, for values of there is a small but clear overprediction by Klemp et al.’s theory. This indicates that their

116 assumption that no energy is dissipated in the lower layer is probably strictly valid only for small

4.

DISCUSSION AND CONCLUSIONS

We have reviewed the classical theory of two-dimensional gravity currents and found that the theory first put forward by Benjamin (1968), when appropriately applied provides a reasonable approximation for the relation between the front speed and the depth of the current, despite its many simplifying assumptions. We have also reviewed the classical theory of internal two-layer bores, of which gravity currents may be thought of as a limiting case. The simple hydraulic theory for two-layer bores is less secure. It appears that a reasonable theory exists for Boussinesq fluids when the upstream undisturbed depth of the lower fluid layer is small compared with the total depth of the two-fluid system. However, if the fluids deviate from Boussinesq or the upstream lower layer is larger than about 10% to 20% of the total depth of the tank, then all theories become invalid for sufficiently large amplitude bores. The reason for this is that it is indeterminate in which layer the required dissipation occurs.

References Baines, P. G. 1995 Topographic effects in stratified flows. Cambridge University Press. Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248. Chu, V.H. & Baddour, R.E. 1977 Surges, waves and mixing in two-layer density stratified flow. Proc. 17th Congr. Intl. Assn. Hydraul. Res., Vol. 1, pp. 303– 310. Cummins, P.F. 1995 Numerical simulations of upstream bores and solitons in a two-layer flow past an obstacle. J. Phys. Ocean. 25, 1504–1515. Huppert, H.E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58. Huppert, H.E. & Simpson, J. E. 1980 The slumping of gravity currents. J. Fluid Mech. 99, 785–799. Klemp, J.B., Rotunno, R. & Skamarock, W.C. 1996 On the propagation of internal bores. J. Fluid Mech. 331, 81–106. Rottman, J.W. & Simpson, J. E. 1983 Gravity currents produced by instantaneous releases of heavy fluids in a rectangular channel. J. Fluid Mech. 135, 95–110. Simpson, J.E. 1997 Gravity Currents: In the Environment and the Laboratory. Cambridge University Press, pp. 258.

117

Whitham, G.B. 1974 Linear and Nonlinear Waves. John Wiley and Sons. Wood, I.R. & Simpson, J.E. 1984 Jumps in layered miscible fluids. J. Fluid Mech. 140, 329–342. Yih, C.S. & Guha, C.R. 1955 Hydraulic jump in a fluid system of two layers. Tellus 7, 358–366.

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CHAPTER 5 STRATIFIED FLOW OVER TOPOGRAPHY

Ronald B. Smith Yale University, New Haven, Connecticut, USA

1. INTRODUCTION The analytical study of stratified airflow over hills began with G. Lyra in Germany (1943). Lyra was recruited by L. Prandtl to investigate the pioneering 1933 wave-assisted 7000-meter glider ascent by J. Kuettner. Kuettner published his observations and interpretations of mountain waves and wave clouds in 1939. P. Queney, then at the University of Chicago, joined the effort shortly thereafter (1947). Their fundamental theoretical contributions showed how airflow over mountains could generate steady gravity waves; waves whose restoring force arises from the background gravitational stability of the atmosphere. They established a consistent smallamplitude theory of mountain waves. In 1949, R. Scorer discovered that if the wind speed increased or the stability decreased with height, that a gravity wave "resonant cavity" could be formed between the jet stream aloft and the earth's solid surface. In this situation, trapped periodic "lee waves" would be generated by the hills and extend downstream for a considerable distance. During the period 1940 to 1955, the mathematical and physical issues of energy radiation, causality and the appropriate upper boundary condition for solution uniqueness were hotly debated and finally resolved (Eliassen and Palm, 1954). In the 1950’s, R. Long showed that under special circumstances, finite amplitude disturbances could be treated analytically (1953, 1955). All of these remarkable early contributions were summarized in a technical note of the World Meteorological Organization (Queney et al., 1960). Since that time, research on the subject has expanded rapidly, encouraged by practical applications to aviation safety, severe wind damage and atmospheric mixing. Mountain waves have also been studied to understand their contribution to the momentum balance of the atmosphere. Corresponding phenomena in the stratified ocean have also been studied. There is no doubt that the beauty of wave clouds and the intrinsic elegance of the mathematical analyses have also stimulated this work. In the development of lee wave theory, a great debt is owed to Rayleigh, Kelvin and other mathematical physicists from the previous century who

120 showed that acoustic, electromagnetic and surface gravity waves could be treated analytically. In spite of this solid foundation, the effort to understand lee waves has been challenging, due to the dispersive and anisotropic nature of internal gravity waves. The study of complex linear and nonlinear wave dynamics has been aided by rapid advances in the numerical simulation of wave-like flows (Durran, 1998, Doyle et al., 2000). Extensive reviews of mountain wave dynamics have been given by Smith (1979, 1989a), Durran (1990), and Wurtele et al. (1996). Lee wave theory has also been discussed in textbooks on atmospheric dynamics (Gill, 1982), stratified flow (Turner, 1973), mesoscale meteorology (Atkinson, 1981 and Durran, 1986a) and atmospheric waves (Gossard and Hooke, 1975). The most complete treatment of this field is the monograph by P. Baines (1995). Baines describes layered flow, 2-dimensional flow, upstream blocking and laboratory experiments among other subjects. In this Chapter, we present the basic theory of mountain waves, with an emphasis on newer developments. We use a new more flexible linear theory model to illustrate the various physical attributes of lee waves. We also discuss recent work on non-linear dissipative lee wave dynamics. Special attention is placed on the subject of potential vorticity generation, an issue that has arisen in the last decade.

2. INTERNAL GRAVITY WAVES AND GROUP VELOCITY The basic properties of internal gravity waves have been discussed elsewhere (e.g. Turner, 1973, Gossard and Hooke, 1975, Gill, 1982) and in other chapters of this book. Here we give a brief review of the subject as a foundation for mountain wave theory. The linearized Boussinesq equations for waves in a stagnant stably stratified fluid are:

121 governing the five fluctuating fields u(x,y,z,t), v(x,y,z,t), w(x,y,z,t), p(x,y,z,t) and The "z" coordinate is directed upward, opposite to the gravity vector. Subscripts indicate partial derivatives. The Boussinesq formulation neglects density variations in the inertial terms and the kinematic divergence of the velocity field associated with compressibility. Density variations play a role only through the action of gravity. The buoyancy effect is proportional to the value of the buoyancy frequency N defined by

The quantity is a reference density. The quantity is the ambient vertical gradient in density. A similar set of equations to (1) can be derived for a compressible atmosphere using scaled variables. In this case, the buoyancy frequency is written

where the potential temperature is given by If the coefficients in (1) are constant, it has plane-wave solutions with each dependent variable written in the complex exponential form. For example, the vertical velocity is written

In (2), k, l and m are the three components of the wavenumber vector,

whose magnitude is

The plane wave expressions satisfy (1) if the frequency dispersion relation

satisfies the

The three components of the group velocity vector can be computed by taking partial derivatives of (5) according to

122

Physically, the group velocity represents the propagation of wave energy through the fluid by the action of oscillating piston-like fluid motions correlated with pressure anomalies. In simple terms, the part of the fluid where the wave is, does work on the part of the fluid where the wave will be, to propagate the wave energy. From these expressions (6), three important characteristics of internal gravity waves can be seen: These waves are dispersive and anisotropic. The wave speed depends on the wavenumber vector, in particular on its orientation relative to the vertical direction. The frequency of the gravity wave is always less than the buoyancy frequency N. Disturbances with higher frequency do not propagate. The group velocity vector is perpendicular to the wavenumber vector.

These results can be applied to the problem of stationary mountain waves by adding a mean flow (U) to the formulation. A positive mean flow advects the waves to the right, adding a “Doppler” frequency component Uk. We consider waves that propagate to the left relative to the fluid. For simplicity, we reduce the problem to two dimensions (x,z) by setting so that (5) becomes.

Using (6,7) the group velocity vector is

For the wave to be stationary, the rightward advection and leftward phase propagation must cancel so that From (7), this condition establishes a relationship between and

123 This relation can be substituted into (8) to determine the group velocity vector in fixed earth-relative coordinates

where

and

The slope of the ray path is the ratio of the two group velocity components

as illustrated in Figure 1 (Bretherton, 1966; Bretherton and Garrett, 1968; Lighthill, 1978).

In the hydrostatic limit, k is small compared to N/U and the slope (12) increases towards infinity. Thus, long waves are found directly above the terrain that generated them. The essential lesson from (12) is that all steady gravity waves will be found either downstream or directly overhead from their source. The nature of the gravity wave dispersion relation (5) is that steady waves will never be found upstream. The term "lee wave" is consistent with

124

3. LINEAR THEORY OF MOUNTAIN WAVES The equations of linear mountain wave theory, with the Boussinesq approximation, are:

where x and y are the horizontal coordinates and z defines the vertical coordinate; parallel to the gravity vector. The functions u(x,y,z), v(x,y,z), w(x,y,z), p(x,y,z) and are the perturbation velocity component, pressure and density fields. U(z), V(z), are the background environmental wind and density profiles. Subscripts indicate partial derivatives. The derivation of (13) will not be given here, but is found in the references. In (13), the non-linear advection of momentum and density are neglected under the assumption that the disturbance amplitude is small.. The time derivative terms are dropped under the assumption of steady state flow. The steady state assumption is an essential part of mountain wave theory, justified by the steadiness of the incoming flow and the fixed geometry of the terrain. All the coefficients in (13) are independent of x and y, suggesting that a Fourier transform method might provide a compact solution. As seen below, the Fourier method has the additional advantage of identifying the up- and down-going wave solutions. This identification is necessary for applying the upper boundary condition. Combining (13) into a single equation for w (x, y, z), and performing a Fourier Transform from physical space (x, y) to Fourier space (k, l) according to

we obtain a single equation for the transformed vertical velocity;

125 where

is the intrinsic frequency

The intrinsic frequency is the frequency felt by a parcel of fluid moving through the stationary wave field. In stationary waves, it plays the role of the temporal frequency seen in (2,5). This transformed equation (15) governs mountain waves in three dimensions. It was first analyzed by Scorer (1956), Wurtele (1957), Crapper (1962) and Sawyer (1962). In two-dimensional flow, we set and (15,16) become Scorer’s equation.

It is often convenient to use vertical displacement variable, defined by

as the dependent

In Fourier space, (18) is

The governing differential equation for

combines (15, 16, 19)

Equations (15) and (20) differ slightly, due to the somewhat different effect of vertical shear on vertical velocity and displacement. Note the similar formulation in Chapter 1. The properties of the solution of (20) depend on the sign of the bracket in the last term When this coefficient is positive, the solutions are approximately trigonometric in form indicating vertical propagation. When negative, the solutions are approximately exponential. This behavior is consistent with the idea from Section 2 that when the frequency, in this case the intrinsic frequency, is greater than the buoyancy frequency, the wave can no longer propagate. An interesting limiting case in mountain wave dynamics is the hydrostatic limit. When the vertical acceleration in equation (13c) is neglected, the transformed equation (20) becomes

126 The coefficient of the last term is now positive definite, so vertical propagation is guaranteed. Another interesting situation is when the properties of the atmosphere (i.e. U(z), V(z), N(z)) vary slowly in the vertical. In this case we can write the solution to (17) as

according to Bretherton (1966). When U(z) and N(z) are constant, a(z) is constant and increases linearly. With slowly varying U and N, the “fast” terms in (17) are

so that the phase function

is given by

For practical purposes, the term situation.

can usually be neglected in this

The “slow” terms in (17) give, neglecting the small

term,

so that

In the physics literature, the quantity in (26) is referred to as an “adiabatic invariant”. As the wave propagates into layers of increasing N(z) or decreasing U(z), will increase (23) and the amplitude of the perturbation vertical velocity will decrease (26). According to (19) however, the amplitude of the vertical displacements will increase as U(z) decreases, as the parcels spend a longer time in the updraft and downdraft regions. In this scenario, upward changes in the wave field and basic state can promote the role of non-linearity (see Section 5). In the case when the wind speed decreases to zero at a so-called critical level, the intrinsic frequency in 16) approaches zero and equations (15,17,20,21) become singular. Analysis of the singularity by Booker and

127 Bretherton (1967) showed that this could lead to nearly complete wave absorption. In most cases however, non-linear processes will occur near the critical level (Clark and Peltier, 1984, Winters and D’Asaro, 1994, Dörnbrack et al., 1995, Grubisic and Smolarkiewicz, 1997). A closely related discussion can be found in Chapter 8. When the wind speed increases and N(z) decreases aloft, will decrease until it becomes zero (23). The asymptotic method described by (2226) then breaks down. Beyond this point, the wave structure is evanescent and wave energy will be reflected downward. Mathematical methods for solving (15) and (20) in sheared mean flows have been presented by Klemp and Lilly (1975), Wurtele et al. (1987), Smith (1989a), Grubisic and Smolarkiewicz (1997) and several others. For the present purpose, we return to Sawyer's three-layer formulation. In this approach, the atmospheric profile of velocity and static stability is approximated by three layers with constant properties (U, V, N). The interfaces between layers are at specified heights and If wind turning is neglected, the three wind speed values, three stability values and two interface heights amount to eight control parameters. This number of parameters is sufficient to illustrate several ways that wave structure depends on the mean flow. Additional parameters enter the problem through the mountain shape specification. Three layer models have also been discussed by other authors such as Marthinsen (1980). Our three-layer three-dimensional formulation reduces easily to the two-layer two-dimensional formulation of Scorer and to the one-layer twodimensional formulation of Lyra and Queney. Thus we can trace the full history of linear mountain wave theory with our model. Our formulation does not include wind turning with height. The turning of the wind gives rise to complex distributed critical layers. Work has just begun on this problem (Broad 1995, Shutts and Gadian 2000) Within each layer (i =1,2,3) of constant wind and stability, and N and are constant so the solution to (20) is

where the vertical wavenumber

is given by

In (27), and are the amplitude coefficients for the up and downgoing wave respectively, provided that a consistent sign for is given by The upgoing wave is characterized by an upwind phase tilt, an upward energy transport and a downward flux of horizontal momentum. When the magnitude of the intrinsic frequency is much smaller than the buoyancy frequency (i.e. the vertical wavenumber is nearly

128

upward energy transport and a downward flux of horizontal momentum. When the magnitude of the intrinsic frequency is much smaller than the buoyancy frequency (i.e. the vertical wavenumber is nearly independent of especially for The wave is hydrostatic and nondispersive. When is close to N, the wave is dispersive due to nonhydrostatic effects. When the intrinsic frequency is greater than the buoyancy frequency (i.e. the vertical wavenumber in (28) is imaginary and the solutions (27) are exponential rather than trigonometric. In this case, nonhydrostatic effects are dominant and we describe the wave as “evanescent”. Across the interfaces between the layers, continuity of mass and pressure require

and

assuming that there is no jump in density across the interface. These jump conditions can be derived directly from (20) if desired, by integrating across the interface between layers, and assuming that and are finite there. The upper boundary condition requires decay in the upper layer if is imaginary. If is real, a radiation condition is applied in the top layer by setting the coefficient of the down-going wave equal to zero (i.e. The linearized lower boundary condition is

which, in Fourier space, is written

In the mountain wave examples discussed in this Chapter, we use an ideal Gaussian hill shape given by

as used by Smith and Grønås (1993). In (33), "a" and "b" are the minor and major axes of the elliptical mountain planform shape. The Fourier transform (14) of (33) is

129 To give the model more flexibility, we introduce a reflection coefficient (q) at the lower boundary to represent partial absorption of down-going waves (Smith et al., 2000). Equation (32) is modified to become

Written in this form, the upgoing wave amplitude is the sum of the wave generated by the terrain and the reflected, and phase reversed, down-going wave Dissipation of the down-going wave by boundary layer turbulence or by critical layer absorption at the lower boundary can be parameterized by setting 0
at one or several altitudes using an inverse Fast Fourier Transform (FFT). In (36), is given by (27) with A and B computed from (26,28,29,35), as shown in the Appendix. For the purposes of this Chapter, evaluation of (36) is carried out on a 1024 by 1024 grid with a grid cell size of one kilometer. The hill (33) is centered in the domain (i.e. at 512, 512). In the FFT technique, solutions are forced to be periodic. Thus waves that reach the downstream boundary will enter the domain on the upstream boundary. Our large domain includes a significant buffer region to allow waves to decay before they reach the downstream boundary. Nevertheless, the parameters chosen for each case must allow some decay mechanism to operate if well-behaved solutions are sought (see Section 4.4). For display, we select an interior region with upper left corner at (412,412) and the lower right corner at (712,612). This subdomain thus has a size of 300 by 200 kilometers. Within the subdomain, the hill is located at a point (100,100) from the lower left corner.

4. A CATALOG OF LINEAR THEORY SOLUTIONS A three-layer three-dimensional formulation allows us to reproduce, in a consistent way, most of the aspects of linear lee wave theory discussed by

130

4. A CATALOG OF LINEAR THEORY SOLUTIONS A three-layer three-dimensional formulation allows us to reproduce, in a consistent way, most of the aspects of linear lee wave theory discussed by previous authors. To demonstrate lee wave properties, we have carried out twelve wave field computations with different wind and stability profiles and different mountain shapes (Table 1). All solutions are computed for westerly flow (i.e. U>0 and V=0). The hill has a height of 1000m. The wave fields are shown in planview in Figure 2. The physics of each case is described below, with reference to figure 2 and the relevant literature.

Blank cells in Table 1 indicate that the solution is independent of this value. For example, the interface altitudes have no meaning if the layers have identical properties. The hill rotation has no meaning if the hill is axisymmetric. The value of is irrelevant if there are no layer contrasts to reflect waves downward. The value “z” is the height of the displayed wave field. While we have chosen to use dimensional quantities in our discussion, the non-dimensional parameter Na/U is given in Table 1 for each case. This parameter is a measure of the degree of hydrostatic balance in the flow. When the parameter is as large as 10, the flow is nearly hydrostatic in all respects. For smaller values, non-hydrostatic effects, such as dispersion, evanescent behavior and wave trapping occur. The nondimensional parameter is also given in Table 1, when relevant. This parameter provides an estimate of the phase shift across layer #1 for First mode trapped lee waves require a phase shift between and (i.e. between 1.55 and 3.1 radians). Second mode lee waves require a phase shift between and

131

4.1.

Vertically propagating waves; hydrostatic

To illustrate a hydrostatic field of mountain waves, we choose a mountain width “a” such that the parameter Na/U >>1. Due to the rapid decay of the wavenumber spectrum forced by broad smooth hills (33,34), little energy is put into waves with The energetic waves then satisfy k<
This parabolic wave energy zone is evident in Figure 2a. The second example of the hydrostatic limit is flow over a ridge. In this case, the forcing is anisotropic. The dominant waves have k>>l. Equation (28) becomes

The group velocity for all the wave components making up the disturbance is directed vertically (12). Thus wave energy is found only in the region above the hill. At the level sampled (z=5000m), the wave has been phase shifted by about 3/4 of a wavelength so the parcels first fall and then rise as they cross the ridge. An x-z cross section through this wave field would be similar to Queney's celebrated diagram for 2-D hydrostatic waves (see also Section 5.1).

132

133

134

Other hydrostatic wave patterns are possible. Contrasts between layers allow partial downward reflections not seen in Figures 2a,b. The reflected waves will reflect again from the solid lower boundary (Klemp and Lilly, 1975, Blumen, 1985). A layered representation of the atmospheric profile, such as our three-layer model, may exaggerate these reflections. The hydrostatic assumption simplifies the derivation of closed form mountain wave solutions such as (37). The drag caused by the pressure difference between the windward and leeward slopes can also be given in closed form. For an axisymmetric Gaussian h i l l (i.e 33 with a=b) with uniform wind and stability, the drag is

For a long ridge (33 with b>>a) the drag per unit length is

where

is the local ridge height. If the ridge height varies slowly along its

length, (39b) can be integrated along the ridge to obtain the total drag. If the ridge is skewed with respect to the wind direction, U in (39b) should be the wind component perpendicular to the ridge.

4.2. Vertically propagating waves; non-hydrostatic When the parameter Na/U is near to unity, the wave field will contain both hydrostatic and non-hydrostatic components. The non-hydrostatic components have a group velocity vector with a more downstream orientation. In an elevated horizontal plane therefore, we will see these shorter components downstream of the hydrostatic components. Two examples of this pattern are shown. In both cases we use a uniform wind and stability so that wave components do not change their propagation characteristics as they enter the next layer aloft. Figure 2c shows a 3-D field generated by an axisymmetric Gaussian hill with a=b=2km. Figure 2d shows the wave field from a long ridge. An x-z cross-section through Figure 2d

135 would be similar to Queney's figure for 2-D dispersing non-hydrostatic waves.

4.3. Trapped lee waves : Diverging and transverse As shown by Scorer (1949), if the lower layer is slower and/or more stable than the upper layers, waves propagating upward through the lower layer may become evanescent aloft. This will result in the downward reflection of the wave. If the down-coming wave reflects from the earth's solid surface, a resonant cavity will form and a trapped lee wave may exist. The wavelength of the trapped stationary wave will be that which allows the wave's phase propagation upstream to balance the downstream advection, so that the wave is steady. The group velocity is directed downstream when expressed in fixed earth coordinates. As shown by Sawyer (1962), Gjevik and Marthinsson(1978), Marthinsson (1980), Simard and Peltier (1982) and Sharman and Wurtele(1983), trapped lee waves are of two types, diverging and transverse. Diverging waves splay outward from the downstream centerline while the transverse waves are nearly perpendicular to the flow direction. The resonant condition associated with lee waves can be derived from (27-30) with a homogeneous condition at the lower boundary

For a two-layer profile in three dimensions, the result is compactly described by the transcendental condition

where

and

The positive sign conditions in (41) and (42) require that the Scorer Parameter, N/|U|, must decrease aloft. This requirement is called the Scorer Condition. The sets of (k,l) pairs that satisfy (40-42) form families of curves in (k,l) space (Fig 3). These branches lie between the reference lines given by the Scorer Parameter values in the two layers, and The branch

136 with the highest wavenumbers is the fundamental mode, with the simplest vertical structure. If a solution branch crosses the l=0 axis, there exist transverse lee waves. If no branch crosses the l=0 axis, only diverging lee waves exist. In the original 2-D treatment of lee waves (Scorer, 1949) only transverse waves were considered. In fact, diverging waves may be more common as they encompass a wider parameter range. They require a trapping mechanism, but they do not have to stand steady against the full incoming flow. Their oblique orientation requires them to stand against only a reduced component of the flow speed. The appearance of a lee wave field depends sensitively on the existence of transverse modes and the forcing of the two lee wave types by the terrain. Here, the forcing ellipse in (k,1) space (34) is a central concern (see Fig 3). If the hill is circular, it will force diverging and transverse waves alike. In the absence of a transverse lee wave mode, diverging waves will still exist. For a ridge oriented across the flow, the forcing is concentrated into waves with small l. Diverging waves are not forced. If no transverse mode exists, no lee wave will be found.

137 In Figure 2e, we show a wave field for a simple two-layer atmosphere with a strong enough decrease in Scorer parameter so that both wave types exist. The mountain is circular so both diverging and transverse waves are forced. The diverging waves are easy to identify in the figure. The transverse waves along the centerline are more difficult to see, because of their longer wavelength (i.e. about 25km), but they are substantial. In Fig 2f, we force the same atmosphere with a long ridge. The transverse waves are now strong. They exhibit a slow decay due to dispersion laterally. As the forcing ellipse (34) is narrow, no train of diverging waves is seen. In figure 2g, we return to the small circular hill, but we increase the flow speed so that only diverging waves exist. No waves are seen on the centerline. A Gaussian ridge would generate no lee waves at all with this mean flow.

4.4. Lee wave decay In the inviscid linear mountain wave problem, there exist three mechanisms of lee wave decay: lateral dispersion, leakage aloft and absorption at the lower boundary. Decay by lateral dispersion was seen in Figures 2e and 2f. No leakage is possible in these cases as the upper layer is infinitely deep and the value of q is taken so large that it only has a decaying influence in the outer domain (not shown). In Figure 2h, we reduce the depth of layer #2 to 3000m so that lee wave energy can leak through and resume propagating in layer #3. The energy lost in this way results in the downstream decay of wave amplitude. The value of q=1 is chosen in this case so that no absorption occurs at the lower boundary. The wave decays rapidly; only about four crests can be seen before the wave disappears. In Figure 2i, we return to the deep upper layer but reduce the reflection coefficient at the lower boundary to q=0.5. Every time a downward reflected ray hits the lower boundary, it loses a portion of its energy. The wave amplitude decays rapidly downstream. Only about five wave crests can be seen. The rate of downstream decay depends on the value of q, the depth of the trapping layer and the ray path angle. A shallow layer and steep rays will cause more rapid decay as the waves impinge on the bottom more frequently. While Figures 2h and 2i look very similar, a detailed analysis of the wave field would reveal significant differences. In the case with leakage aloft, the pressure (p) and vertical velocity (w) are positively correlated in the wave field giving an upward propagation of energy. Likewise, the u and w oscillations are in negative correlation giving a downward flux of horizontal momentum. Mountain drag is carried upward, just as in the vertically propagating examples. With absorption at the lower boundary, both these phase relationships reverse. Energy in the wave train moves downward and the wave drag is returned to the lower boundary from whence it came.

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A striking kinematic difference is the tilt of the lee wave structure. Non-decaying trapped lee waves have no vertical tilt of the crest and trough phase lines. Leakage establishes a slight upstream tilt. Low level absorption establishes a downstream tilt. In Figure 2j, we show a ridge flow with the reflection coefficient q=0. The Scorer Condition is well met, but the downward reflected waves are completely absorbed at the lower boundary. No resonant cavity exists and no trapped lee waves are seen. The wave field includes only dispersing vertically propagating waves (like Figure 2d) and a set of waves reflected downward from the evanescent layer aloft. These two wave trains interfere to give a weak and irregular train. According to Smith et al. (2000), total absorption of the down-going wave will occur when there is a stagnant layer near the surface of the earth.

4.5.

Second lee wave mode

To illustrate a second transverse lee wave mode, we deepen the stable layer from 2 to 4 km and decrease the wind speed from 10 to 8 m/s. The nondimensional number rises to 6. The function for the second mode has a node in the first layer. Using a long ridge we can compare the pure single mode in Fig 2e with double mode in Fig 2k. The short wave in Figure 2k (i.e. is the fundamental mode. A second mode has a longer wavelength. It beats against the fundamental mode giving an irregular appearance to the lee wave. As the higher order mode generally has a smaller propagation speed than the fundamental mode, it must compensate by having a longer wavelength. Like the first mode, it must have a sufficient speed to stand steady against the mean flow.

4.6.

Left-right asymmetry

Finally, we consider an example without left-right symmetry. We choose a simple two-layer configuration, similar to Fig. 2k, with two transverse modes. The ridge is rotated clockwise by 45 degrees (Figure 21). For reasons to be explained, we increase the wind speed slightly from 8 to 10 m/s. The new wave field has two wave trains, a longer wavelength train with a ray path angle of about 30 degrees north of east and a shorter wavelength train with a ray angle about 10 degrees north of east. In the proximity of the ridge, the flow can be considered to be twodimensional; independent of distance along the ridge. In this local 2-D problem, the incoming flow speed perpendicular to the ridge is reduced by a

139 factor cosine(45)=0.707 below the actual speed 10m/s. This speed reduction allows a second transverse lee wave mode to exist, as in Figure 2k. The far wave field in Figure 21 is particularly interesting. Both lee wave families are found in the northeast quadrant of the diagram, indicating that they have a northward component of group velocity. This was anticipated because in the x-y plane the group velocity is normal to the wave crests. Thus the lee wave mode with NE-SW oriented crests has a NW-ward oriented group velocity. The westward component of group velocity is overcome by the mean flow. Its northward component is unopposed by the mean flow and thus the wave train propagates into the NE quadrant. The longer waves have a larger northward component of group velocity than the short waves. The shorter waves barely show their northward component. Because of their different group velocity orientations, the waves separate nicely downstream so that we see each one without interference from the other. From this example, one can imagine what would happen to a wave field behind a N-S oriented ridge as the wind slowly turned from westerly to southwesterly. Initially of course, the wave field would be located east of the ridge. As the wind turned, the wave field would rotate counterclockwise faster than the wind. When the wind reached SW, the waves would be found, not NE, but NNE of the ridge.

4.7.

Applications of Linear Theory

To conclude this Section, we note that linear theory is more than just an idealized model of mountain waves. There are a growing number of observational studies in which linear theory compares well with direct measurements of the atmosphere. Examples include lee waves over western England (Vosper and Mobbs, 1996) and over Mt. Blanc (Smith et al. 2000). The types of patterns shown in Figure 2 are common in satellite images of clouds in the atmosphere. Nevertheless, there is evidence that under certain conditions, non-linearity and dissipation play a role in stratified flow over topography. Examples of nonlinear flow are found over large mountains, e.g downslope winds over the Front Range (Lilly and Zipser, 1972) and waves over the Pyrenees (Bougeault et al., 1997), and over smaller mountains, e.g. lee waves over the Appalachians (Smith, 1976) and the Adriatic Bora (Smith,1987 ). We review these aspects in the next Section.

5. NONLINEAR AND DISSIPATIVE EFFECTS The study of nonlinear effects in mountain waves began with R.R. Long's laboratory experiments and his mathematical formulation of a finite amplitude wave equation; the so-called Long's Equation. Long's Equation was elegantly

140

used by Huppert and Miles (1969) to predict the onset of wave breaking. Long (1955) and Houghton and Isaacson (1968) considered one and two layer hydraulic formulations. Dissipative effects are also important, sometimes forced by non-linearity. In the Section below, we summarize current knowledge of non-linear and dissipative phenomena such as flow splitting, gravity wave breaking, severe downslope winds, hydraulic jumps, rotors and turbulent boundary layers. Some of these subjects have been reviewed in Smith (1989a), Durran (1990), Baines (1995) and Wurtele et al.(1996).

5.1. Flow splitting and gravity wave breaking One of the most important predictions of mountain wave theory is the onset of flow splitting and gravity wave breaking. Flow splitting is defined as the horizontal splitting of the incoming flow so that it passes around rather than over the mountain peak. Streamline splitting requires that the low-level flow first be decelerated to a stagnation point. Gravity wave breaking, in a uniform background state, begins by the steepening of the wave front and decelerating the flow, leading to overturning. Work on this problem has mostly been confined to the hydrostatic l i m i t where the parameter Na/U is large. In this case, the non-linearity parameter Nh/U plays a dominant role, along with parameters describing the mountain planform shape. We define H= Nh/U as the non-dimensional mountain height. The mountain width plays no role, so intuitive ideas about mountain steepness and splitting must be discarded. The mechanism of flow deceleration is the same for both flow splitting and wave breaking. In the regions of upward parcel displacement, a positive density anomoly is created by the ascent of denser or potentially cooler air (Figure 4). According to the hydrostatic law, areas of high pressure will exist at the base of these dense fluid anomalies. According to Bernoulli's Law

as parcels approach a high pressure region, the speed decreases due to the adverse pressure gradient (Smith and Grubisic, 1993; Vosper and Mobbs, 1997). The height term in (43), once thought to be dominant, plays little role (Smith, 1988, 1990). As the non-dimensional mountain height (H) increases, the strength of the high pressure regions increases at two special locations in the flow; on the windward mountain slope (point B) and at a point directly above the hill at an altitude of approximately (point A). The relative magnitude of these two deceleration points determines whether flow splitting or gravity wave breaking occurs first (Smith, 1989b; Stein,

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1992; Smith and Grønås, 1993; Baines and Smith, 1993; Olafsson and Bougeault, 1996). For a long ridge, or in strictly two dimensional flow (i.e. x,z), the deceleration at point A is stronger than at point B. Thus wave breaking occurs first, starting approximately when H=0.85. For an isolated hill with circular contours, the two points (A and B) are similar in their deceleration potential. Splitting and wave breaking begin approximately when H=1.2. In 3-D flow, the lateral dispersion of waves aloft weakens the density anomalies, so a larger hill is required to stagnate the flow. Once flow splitting begins, the wake region takes on a complex vortical structure which has been investigated in the laboratory (Brighton, 1978, Snyder et al., 1985, Gheusi et al., 2000) and with numerical simulation (Rotunno and Smolarkiewicz, 1991; Miranda and James, 1992). The mechanism of vorticity generation will be discussed in Section 6. This relatively simple picture for splitting and wave breaking can be modified considerably when the ambient atmospheric profile has vertical structure or a turbulent boundary layer. For example, strong shear or a shallow stable layer aloft may promote wave breaking by a Kelvin-Helmholtz mechanism without requiring deceleration and overturning (See Chapter 8).

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5.2. Severe downslope winds In 1977, Clark and Peltier showed in a numerical simulation that when mountain wave breaking begins in a two-dimensional setting, the entire flow field would transform itself into a new configuration, quite different than the pre-breaking wave field. This new configuration includes spilling or plunging flow down the lee slope, leading to the name "severe downslope wind". This flow also has a magnified mountain drag, turbulence in large region above the lee slope, weaker waves in the stratosphere and some unsteadiness. Peltier and Clark (1985) showed that a similar severe downslope wind structure can be found with smaller mountain heights (i.e. H<0.85) if there is a wind reversal or critical layer at certain altitudes. A self-consistent theory of downslope winds was put forward by Smith (1985), using Long's Equation for finite amplitude disturbances and an assumption that the low level flow would decouple from any waves which might exist aloft. This model predicts the shape of the isentropes and isotachs, the location of the turbulent flow aloft, the mountain drag, the depth of blocking and the special critical layer heights that can trigger severe winds with small mountains. The theory has been tested in numerical studies (Durran and Klemp, 1986, 1987; Bacmeister and Pierrehumbert,1988; Crook et al., 1990; Miranda and James, 1992) and limitations have been noted.

5.3.

Hydraulic jumps

According to atmospheric observations and numerical simulations, at the downstream limit of the severe downslope wind, the flow speed drops quickly and the streamlines zoom skyward (e.g. Doyle et al., 2000). Such structures have often been equated with the phenomena of the hydraulic jump. This flow structure has been seen both in flows with and without layered profiles upstream. Yet, the hydraulic jump is primarily thought to be layered fluid phenomena. Possibly, when layering is absent upstream, the severe wind descent can create a layered flow where none existed. The occurrence of a hydraulic jump is related to the fact that a layered flow, with a given flux of mass and momentum, can exist in two so-called conjugate states. Typically, one of these states is subcritical and the other is supercritical (i.e. slower and faster than the long wave speed). When a supercritical flow decelerates, non-linear steepening tendencies act to create a jump, which almost discontinuously converts the fast flow to its slower subcritical conjugate state. An essential property of a jump is that energy is dissipated and the value of the Bernoulli function (43) decreases. This can have consequences for vorticity generation and for the flow pattern further downstream. Baines (1995) gives a thorough treatment of the dynamics of

143 layered flows, including jump dynamics. The discussion of gravity currents in Chapter 4 is also relevant to this issue. When the jumps are weak, they can take on a wave-like form, similar to the Morning Glory phenomenon discussed in Chapters 1 and 3. The lee-side hydraulic jump then becomes a mechanism for creating finite amplitude lee waves (Rottman et al, 1996, Nance and Durran, 1998). Recent work, not included in Baines (1995), is the resolution of the existence and uniqueness problem for jumps in two active fluid layers (Yih and Guha, 1955; Mehotra and Kelly, 1973). New analysis has provided a closed form expression for the Bernoulli loss in each layer resulting from energy dissipation, (Jiang and Smith, 2000). A surprising result was that even when both layers are active in the jump, Bernoulli loss is concentrated in one of the layers. An equal sharing of dissipation seems to be impossible.

5.4. Rotors The concept of a rotor was put forward in the early papers on mountain waves (i.e. Kuettner, 1939, Queney et al., 1960). According to common usage, the term refers to a compact low level vortex with horizontal axis, downstream of a mountain ridge. The vortex axis lies normal to the mean flow and the sense of rotation is clockwise, if the mean flow is from left to right. The vortex lies underneath, and is causally related to, the first crest of a trapped lee wave. To be defined as a rotor, the vortex must be strong enough to cause reverse flow at the ground. In extreme cases, the reverse flow can be intense and damaging. It can be easily distinguished from a severe downslope flow by its opposite flow direction. There may be some confusion between a rotor and a hydraulic jump. Both exhibit strong deceleration and upward jumping streamlines. Perhaps rotors may be identified by the reversal of flow near the ground, the reattachment of the flow or the existence of a trapped lee wave. Numerical models have been shown to capture rotor-like structures, but little theoretical or numerical work has been done to understand the rotor or to more clearly define its character. Derzho and Grimshaw (1997) used Long’s Equation to establish a connection between waves and rotors.

5.5. Turbulent Boundary layers The action of a turbulent boundary layer at the earth's surface has been shown to decrease the amplitude of topographically generated gravity waves (Richard et al, 1989; Olafsson and Bougeault, 1997). Boundary layer waves and turbulence can interact (Carruthers and Hunt, 1990). The separation of the boundary layer by adverse wave-generated pressure gradients may play a

144 role in rotor formation. A stagnant boundary layer may also absorb waves which have been reflected downward from evanescent layers aloft (Smith et al. 2000). Welch et al. (2000) discusses the blocking and stagnation of boundary layer air in complex terrain.

5.6. The onset of turbulence in breaking gravity waves The onset of turbulence in breaking gravity waves is a difficult problem involving multiple space and time scales. In spite of frequent aircraft encounters with wave-induced turbulence, the time sequence of turbulence evolution has never been observed. Most numerical models do not resolve smaller scales of motion and so they tell us little about turbulence cascade of energy. Many models simply parameterize the diffusive and dissipative effects of turbulence with an assumed eddy viscosity and diffusivity. These transport coefficients are usually assumed to be strong functions of the resolved Richardson Number or shear magnitude to replicate some properties of shear instability. Other models include a turbulent closure scheme involving a prognostic equation for turbulent kinetic energy (Mellor and Yamada, 1974,1982; Zilitinkevich and Laikhman, 1965, Schumann 1977; Duynkerke, 1988, Trini Castelli and Anfossi, 1997, Xu and Taylor, 1997). These schemes however do not distinguish different scales of turbulence, so they tell us nothing about turbulent cascades. Also, with their current state of development, there is no evidence that their mixing length formulations are appropriate for gravity wave breaking (private communication: Dr. Branko Grisogono). They have been tested only in cases of developed boundary layer turbulence. The advance of high-speed computers has offered another way forward in this problem. Several authors have recently reported progress using large and/or nested grid arrays that are capable of resolving the first decade of turbulent granulation (Bacmeister, and Schoeberl, 1989, Andreasson et al., 1994, Afanaseyev and Peltier, 1998, Fritts and Isler, 1994, Winters and d’Asaro, 1994; Dörnbrack et al. 1995; Fritts et al. 1996, Schmid and Dörnbrack, 1996, Scinocca 1996, Gheusi et al. 2000). They have been able to identify a sequence of instabilities as the gravity wave steepens. Eventually, the sequence leads to a "three-dimensionalization" of the flow, followed by full cascading turbulence. The resulting turbulent fluxes of heat and momentum can generate macroscopic potential vorticity, as discussed below.

6. THE GENERATION OF POTENTIAL VORTICITY (PV) The Ertel Potential Vorticity (Ertel, 1942) for compressible flow

145

becomes

for Boussinesq flow. Potential vorticity satisfies parcel conservation

in the absence of heat and momentum diffusion. With the Coriolis term neglected, PV provides a useful diagnostic quantity for gravity wave studies and a possible tool to link gravity wave breaking with downstream phenomena such as wakes, eddies, jets and convection (Smith, 1989a,c). Its usefulness arises from the fact that PV remains zero, even in the most nonlinear of mountain waves, until wave breaking begins. In laminar waves, the baroclinically generated vorticity vectors lie parallel to density (or isentropic) surfaces and thus the dot product of vorticity and density (or potential temperature) gradient vanishes. Only turbulent transport of heat or momentum or mixing to destroy parcel identity can significantly alter PV in a dry atmosphere (Haynes and McIntyre, 1987, 1990, McIntyre and Norton, 1990). Danielsen’s (1990) argument that PV is conserved even in mixing flow is probably incorrect. Once generated, potential vorticity’s conservation property (45) allows it to be carried downstream. PV “banners” downstream contain fluid parcels which have passed through the dissipative regions over the mountains. Existing theorems concerning potential vorticity inversion indicate that when balanced flow is reestablished downstream, the PV field can be used to compute the velocity and density anomalies (Blumen, 1972, Hoskins et al., 1985, Raymond, 1993). The plausibility of PV generation by breaking gravity waves has been established theoretically by detailed investigation of the shallow water system by Schär and Smith (1993a,b), Grubisic et al (1995), Smith and Smith (1995) and Pan and Smith (1999). Samelson (1992), Tjernström and Grisogono (2000) and Jiang and Smith (2000) have examined the shallow water solutions in cases with small or no PV generation. In the shallow water system, the potential vorticity is given by

where is the vertical vorticity and H is the layer depth. In this framework, Bernoulli losses in hydraulic jumps induce lateral gradients in the Bernoulli function (43) which are related to vorticity according to

146

in steady flow. In (47), “n” is the direction perpendicular to the streamline. Thus, complex flows with jumps of varying strength will have patterns of potential vorticity downstream. If the generation is weak, these anomalies will advect directly downstream in PV-banners (Figure 5). If the generation is stronger, the PV can wrap itself into stable leeside eddies or alternating drifting vorticies. The merging of two airstreams with differing histories can also generate PV. A number of field programs have given some support to the potential vorticity generation hypothesis. Smith and Grubisic (1993) showed that large stationary eddies in Hawaii's wake may contain vorticity generated in shallow jumps on the flanks of the island. Smith et al (1997) showed that the stable 300km long wake behind the mountainous island of St. Vincent, can be attributed to wave breaking over the lee slopes (Figure 5). Radar-observed wakes behind the Aleutians have been explained by wave breaking and Bernoulli loss (Pan and Smith, 1998). The recent Mesoscale Alpine Programme, from September to November 1999, verified the predicted PV banners arising from features along the complex crestline of the Alpine massif, particularly in the Rhone Valley and near the Gotthard and Brenner passes (Bougeault, 2000). Numerical models have shown that they can capture PV generation in continuously stratified flows (Thorpe et al. (1993), Schär and Durran (1995), Smith et al (1995), Aebischer and Schär (1998). The diagnostic evaluation of how PV is generated in numerical models has been more difficult. Schär (1995) provided a theoretical framework. Rotunno et al.(1999) used a numerical model to examine how the conventional viscous terms act to generate PV in dissipating gravity waves. Examples of the importance of orographic PV generation to downstream weather are appearing in the literature. Hawaii's lee eddy returns volcanic gases to the lee shore, and by weakening the influence of the trade winds there, allows a diurnal sea-land breeze cycle to operate (Smith and Grubisic, 1993). An eddy from the Palmer Divide near Denver controls the spread of urban smog (Crook et al., 1990). Bernoulli loss in breaking waves and PV banners control downstream “gap winds” in the Aleutians (Pan and Smith, 1998). Eddy formation by the mountains of Taiwan is so strong that it can deflect approaching typhoons (Smith and Smith, 1995). South of the Alps, a strong PV eddy can interact with the larger scale baroclinity to trigger cyclogenesis (Aebischer and Schär, 1998). PV generation may also provide a framework for analyzing the influence of wave drag on the general circulation.

147

148 We conclude this Section by outlining the mechanisms that can generate PV. The question of which mechanism dominates is still unanswered. The analysis of PV generation in a continuously stratified fluid begins with the basic vorticity equations for a Boussinesq fluid. The vector vorticity is influenced by advection, stretching and tilting, and is created by baroclinic and viscous terms according to

where is a viscous force. The potential vorticity (44) is unaffected by stretching, tilting or baroclinic terms and is created and destroyed by viscous and heating terms according to

where H is the rate of heating per unit mass. According to Haynes and McIntyre (1987), the generation of PV in a compressible flow can be written in flux form according to

where

is the non-advective flux of P. The first term on the right hand side of (49) or (51) represents viscous torques acting with a component perpendicular to a The second term acts if there is a gradient in the heating rate H along the vorticity vector Another view of PV dynamics is given by expressing the total flux of PV in steady flow as

after Schär (1993).

In ideal steady flow, with

the

vector lies parallel to the flow direction and the PV flux is purely advective (i.e. In the presence of heating or viscous force may have an additional component representing a dissipative non-advective

149 flux. For example, in a wave breaking region, is directed upstream while is directed upward. According to (52) the PV flux vector is directed laterally, resulting in a pair of positive and negative PV banners or eddies (Figure 5). Constraints on the generation of vorticity by viscous stresses can be analyzed using the stress tensor for an isotropic Newtonian fluid.

where is the total stress tensor, “p” is the pressure (i.e. usually the thermodynamic pressure), is the rate-of-strain tensor

and is the rate of volume change (Batchelor, 1967). The Newtonian model can be formally extended if the viscosity (in 53) is allowed to be a function of any rotation-invariant scalar state variable such as the local instantaneous strain rate or Richardson number, or an inherited quantity like turbulent kinetic energy (TKE). This flexibility for encompasses most of the turbulence parameterization schemes proposed in the last 35 years; for example Zilitinkevich and Laikhtman (1965), Lilly (1962), Smagorinsky (1963), Mellor and Yamada (1974, 1982), Schumann (1977), Duynkerke (1988), Dörnbrack (1996), Ying and Canuto (1996), Xu and Taylor (1997), Castelli and Anfassi (1997), Afanaseyev and Peltier (1998). Using (53,54) and

the torque in (49) is

In (56), the first two terms diffuse vorticity. The third term can create vorticity when none was present before. We write it schematically as

indicating that each term in is a product of a first spatial derivative of velocity and a second spatial derivative of viscosity. For convenience, we introduce the term “internal boundary” to represent all subregions in the

150 domain where

. An internal boundary is a gradient zone between

regions with different viscosity or different turbulent eddy viscosity. Within it is possible for vorticity to be created by viscous stresses where there was none before. If is oriented with a component perpendicular to a its action will generate potential vorticity. To illuminate the generation of PV by mountains, we consider the simplest prototype problem; uniform wind and stability approaching a hill in the absence of background rotation. In this case, and PV = 0 for each incoming fluid parcel. Using (56,57), four distinct PV generation pathways can be identified. Pathway #1 involves the direct creation of vorticity with a cross isentrope component. This can only occur by the action of in (56), within internal boundary regions Pathway #2 and #3 are twostep processes. First, vorticity is created by the baroclinic mechanism (in 48) as part of the gravity wave propagation mechanism, so while PV = 0. Then, through the action of heating (Pathway #2;

or vorticity

diffusion (Pathway #3; the vorticity is “converted” to potential vorticity. The last possibility, Pathway #4, is that vorticity diffuses in from the lower boundary of the domain. Rotunno, et al. (1999) have investigated some of these Pathways numerically by using various sets of assumptions about the form of dissipation. To remove P4, they used a free-slip lower boundary and argued, from scale analysis, that curvature effects at the boundary (Batchelor, 1967) are insignificant. P1 and P3 can be eliminated by setting viscosity equal zero, but allowing the diffusion of heat. This assumption isolates P2; the thermal reorientation of the so that In another simulation, constant non-zero values of viscosity and thermal diffusivity were used, allowing Pathways P2 and P3. In both simulations, conditions were set to give steady flow with weak wave breaking. They concluded that PV generation in 3-D mountain waves occurs primarily by Pathways 2 and 3; i.e. the “conversion” of baroclinic vorticity to potential vorticity. It remains to be learned whether Pathway #1 needs to be considered in fully turbulent wave breaking. P4 will be important in real flows near a no-slip lower boundary.

ACKNOWLEDGEMENTS The author thanks Dr. Steven Skubis for helping to develop the linear model used in this chapter. Qingfang Jiang, Jie Zhang and Larry Bonneau

151 helped with the figures. Sigrid R-P Smith did the formatting. Through their papers, reviews, lectures and private discussions, scientists working on the subject of mountain waves have helped the author to understand the subject. Due to space limitations, it was not possible to include a description of all the significant recent work. Part of this chapter was prepared while the author was a visitor at the University of Stockholm. Support for the author's research has come primarily from the NSF Division of Atmospheric Sciences, especially the Mesoscale Meteorology section.

APPENDIX 1 The up and down-going wave amplitude coefficients of the 3-layer model, and are computed from the transform of the terrain h(k,l) and the parameters that define each layer The expressions are:

where

where

All these quantities are complex. Note that when the atmospheric conditions are uniform with height, and the down-going wave amplitudes (Equations from Smith et al., 2000)

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Chapter 6 TURBULENCE IN STRATIFIED FLUIDS H.J.S. Fernando Environmental Fluid Dynamics Program Department of Mechanical and Aerospace Engineering Arizona State University, Tempe, AZ 85287-9809

Abstract

Turbulence and stratification are common features of environmental flows, and their interactions lead to a myriad of flow phenomena of dynamical interest. These flow phenomena are described based on a dynamical framework provided by the equations of motion. Various broad categories of stratified turbulent flows are identified and their characteristics are briefly reviewed.

1. INTRODUCTION Environmental flows are usually turbulent and coexist with background stable stratification. Some examples of environmental stratified turbulent flows are the atmospheric (ABL) and upper-ocean (OBL) boundary layers, oceanic thermocline and atmospheric inversion layers. The night time ABL is strongly stably stratified due to cooling of the ground and turbulence is generated in the stratified region by the ground-based shear flow. The nocturnal surface stratification so generated is destroyed during the daytime by the thermal turbulence generated due to heating of the ground, and the interaction between stratification and turbulence is confined to the boundary between the near-surface turbulent layer and the overlying stratified layer (i.e., the entrainment interface). The OBL is also subjected to similar diurnal forcing, with stable stratification developing during the day and convective mixing at night. Stabilizing buoyancy forces profoundly influence the nature of turbulent flows so as to bestow unique physical effects, for example, the appearance of jet-like velocity profiles, formation of (density) structure and retardation of vertical diffusion of contaminants. Stable stratification also supports internal gravity waves, cohabitation of which with turbulence in stratified media produce complex flows having both "wave" and "turbulence" like properties. Yet these flows are unfathomable in the frameworks of waves and turbulence studies. Inhibition of vertical turbulent diffusion in stratified

162 fluids causes vertical length scale in ocean pycnocline and nocturnal ABL to be limited to a few tens of meters. Thus, stratified turbulence cannot be resolved in predictive numerical models and needs to be parameterized. One of the main aims of stratified turbulence studies, therefore, is to develop state-of-the-science parameterizations for environmental models with the hope of improving forecasts of atmospheric and oceanic circulation patterns. It has been demonstrated that existing forecasting models perform poorly under stably stratified conditions and that model predictions are quite sensitive to the parameterizations used to represent sub-grid scale stratified turbulence (Gregg 1987; Fernando & Hunt 1996). Turbulence in stratified flows can be characterized by the nature of forcing and stratification. Fernando & Hunt (1996) identified, in broad sense, four types of flow. These are: (i) decaying homogeneous turbulence within uniform stratification; (ii) decaying inhomogeneous turbulence in a region of uniform stratification; (iii) forced turbulence in stably stratified layers; and (iv) homogeneous turbulence in uniform mean shear and uniform stratification. Given the diversity and complexity of the types of turbulent flows plausible, the above classification cannot be treated as unique, and a given turbulent flow can be in the realm of more than one category. Such a classification, however, is helpful to identify a given flow based on the dominant physical processes present, and hence, in broad sense, to prescribe appropriate parameterizations. In addition, within a given broad category, different flows may show widely different behavior, a striking example being the nocturnal ABL. At high cooling rates (conditions with clear skies and weak winds) the turbulence therein is intermittent, non-stationary and patchy with pronounced "wave"-like motions (known as the very stable boundary layer). On the other hand, at low cooling rates (significant winds/cloud cover), the turbulence is more-or-less continuous, though weak, and obeys classical similarity laws. Therefore, even within a given category, the parameterizations of turbulence can be subjected to constraints imposed by the dynamical mechanisms of turbulence generation and mixing rates (buoyancy fluxes) associated therewith. The aim of this paper is to provide a brief overview of the nature of turbulence is stratified fluids. Several past articles have provided comprehensive reviews on different facets of this subject, which include Sherman et al. (1978), Lin & Pao (1979), Muller (1984), Gregg (1987), Hopfinger (1987), Fernando (1991, 2000) and Riley & LeLong (2000). Therefore, this communication mainly concerns the recent work, paying attention to physical insights gained using laboratory and numerical experiments as well as theoretical analyses.

163

2. GENERAL ASPECTS OF STRATIFIED TURBULENCE Gibson (1991) defines "active turbulence" as random motions where inertial-vortex forces are balanced by pressure gradient forces and where the body (e.g. buoyancy, Coriolis and electromagnetic) and viscous forces do not play a role to the leading order. Such flows encompass a range of motion scales, from large forcing scales represented by the integral length scale L to the smallest Kolmogorov scale where the motions are still random, yet viscous effects are sufficient to cause substantial kinetic energy dissipation. Here is the turbulent kinetic energy dissipation rate and the kinematic viscosity. When the body forces are of the same order as the inertial forces at any of the scales, then the turbulence is said to be "fossilized" at those scales. In general, the buoyancy influence is first felt at larger length scales of turbulence, say when where is a threshold scale. When then a range of length scales exits in which turbulence is active, and this state is called the “ active-fossil” turbulence. When no active three-dimensional turbulence can be present, and the flow belongs to the “ fully fossilized” state (or simply the "fossil" turbulence state). The hydrodynamic state of a given turbulent flow is dependent on a variety of factors, including the characteristics of stratification, the length ( L ) and r.m.s. velocity

scales of turbulence

and prevailing forcing mechanisms; here the primed quantities denote turbulent fluctuations and is the velocity Some general features of stratified turbulence can be illustrated in terms of the governing equations as follows. Consider a motion field with typical horizontal and vertical velocity and lengthscale of and respectively. The scales for pressure

and buoyancy

and time scales

are defined

similarly, where is the time scale of buoyancy fluctuations. The normalized form of continuity equation becomes

where the non-dimensional

variables are defined as

with being the vertical coordinate anti-parallel to gravity. The superscript appended to the equation number indicates that the variables are normalized, but (*) has been dropped from the variables. When the motion field is three dimensional,

164 The momentum equations, to the Boussinesq approximation, can be written as

where

is the Coriolis parameter,

a reference density,

the buoyancy, which can be written in terms of its mean and perturbation values as the density and gravitational acceleration. The buoyancy conservation becomes

the

where is the (constant) background buoyancy frequency and is the molecular diffusivity of the stratifying agent. Several familiar nondimensional numbers appear in (2.2-2.4), namely the Reynolds number the Schmidt number and the bulk Richardson number Assume that the turbulence is imposed on a stratified fluid at time For high Reynolds number “active” turbulence to exist, the inertial terms should be of the leading order, and hence the initial time evolution of the flow is expected to follow

165 This initial growth occurs until the buoyancy forces affect the growth when the conditions or are satisfied. Therefore, the vertical scale of turbulence at the onset of fossilization is given by This lengthscale can also be derived by a simple vertical energy balance, wherein fluid parcel having a vertical velocity

is the lengthscale to which a rises against the buoyancy forces.

Also it is evident that the two length scales

and

can be used to describe stratified turbulence. The first lengthscale characterizes the vertical scale of evolving turbulence (which is suitable even for the case of passive scalar mixing) and the second signifies the onset of buoyancy effects. They are called the Ellison and the buoyancy length scales, respectively. The buoyancy influence sets in at a time scale of

typically

(DeSilva and

Fernando, 1998). The Coriolis forces become important when Since the time scale ratio is large for oceans (~10) or and the atmosphere stratification effects are expected to come into play first during the evolution of environmental turbulence. At the onset of fossilization, the vertical scale can be expressed in a different form using the well-known parameterization the turbulent kinetic energy dissipation as is the well-known Ozmidov length scale. Since

for where can be

measured in oceans with a higher degree of reliability than it is common practice to employ in oceanographic turbulence studies (however, see the limitations described in Section 3.4). Another problem encountered in geophysical measurements is the inadequacy of sampling. For example, in oceanic measurements, a vertical profile at a given space-time location can be obtained during a single cast; owing to the ship drift combined with nonstationarity and inhomogeneity of turbulence, successive vertical casts cannot be averaged to obtain extensive statistics. Therefore, each vertical profile ought to be used to extract as much information as possible, for example, as is done in the method of Thorpe (1977). In this method, an instantaneous density profile obtained through a turbulent region containing density inversions (Figure 1a,b) is rearranged so as to obtain a monotonically varying stable density profile. The associated (Thorpe) displacements of fluid parcels are used to construct the so-called Thorpe length scale

The

166

maximum of all is called the maximum Thorpe displacement which is a measure of the physical vertical size of a turbulent region. Although the dynamical limit for the vertical length scale of stratified turbulence is there are no constraints on the growth of horizontal scales in the flow, which may follow

causing the

flow to assume a state with characterized by large-scale anisotropy of the flow. This anisotropy can be felt even at smaller dissipation scales, when the buoyancy scales

become on the same

order as the Kolmogorov scale or when the parameter drops below a critical value (Gibson 1980; Gargett et al. 1984). As becomes smaller, turbulence tends to confine to horizontal sheets or thin layers.

As discussed, the fast (internal wave) time scale signifies the onset of fossilization. Active turbulence is associated with no effective internal wave propagation (as the excitation frequencies are larger than N), but once the fossilization begins the internal waves can propagate out, carrying energy away from the generating region. At large times, the layered structure develops and the dynamics of flow evolution changes substantially. When the leading order terms in (2.3) can be several orders of magnitude smaller than (2.2), and hence the flow tends to be rich in the evolution of horizontal motions (and hence the vertical vorticity) compared to vertical motions. (Because of strong vertical shear, however, the

167 horizontal vorticity is intense and dominates the flow). In motions that follow, term has a magnitude at most on the order as far as the condition

is satisfied.

The buoyancy contribution

also has the order Therefore, internal wave dynamics is expected to be of less importance during the long-time evolution. Owing to the leading order of non-linear effects, the slow evolution described by (2.2) is characterized by vorticity dynamics, as has been clearly seen in laboratory experiments on decaying turbulence in turbulent jets and wakes (Figure 2) and in continuously forced stratified turbulent flows (Figure 3). The vertical and horizontal vorticity equations (excluding Coriolis effects) become

which show how the buoyancy effects becomes inconsequential at large times, following the generation of sheet-like (layered) structure. Prior to the viscous terms becoming important, the inertial effects govern vortex dynamics. It is also of interest to note that the potential vorticity of stratified (incompressible) turbulence

can be approximately represented by the vertical vorticity mode, given that is vertical when isopycnals are approximately horizontal for Therefore, stratified turbulence can be signified by two principal components,

168

169

the "fast" propagating internal wave modes and "slow" non-propagating potential vorticity mode (Riley et al 1981: Lilly 1983). Although stratified turbulence is sometimes referred to as twodimensional because of its sheet-like appearance, it is far from being twodimensional. The sheets posses rich vertical structure, and the vertical and horizontal components are strongly coupled in (2.2) by the leading order nonlinear term (note that and hence coupling introduced though the pressure term is weak. Owing to the layering developed during its evolution, the final period of decay of stratified turbulence is much different from that of three-dimensional unstratified turbulence (e.g., Batchelor 1953) as will be described later. When fluctuations decrease to sufficiently small values, however, linear analysis is valid (Pearson & Linden 1983; Hanazaki & Hunt 1996) and the final period of decay can be analyzed using the techniques employed by Batchelor (1953). Once fossilization begins, (which is the same as given that note that is invalid in this case) decreases and may continue to increase. The demise of turbulence is expected when viscous forces become influential in any component of the velocity field, which can occur in one of the following ways. First, may reduce to the size of the Kolmogorov scale conjectured by Gibson (1980). Using

as it is possible to evaluate

the corresponding lengthscale as Second, the viscous influence can set in when the motions determined by (2.2) and (2.3) have an aspect ratio of the two scales becomes

or

Note that the ratio

Since it appears that active turbulence becomes extinct and degenerates into viscous shearing motions concurrent with the shrinking of overturning eddy sizes to the Kolmogorov scale. Note, however, that the assumption used in the above derivation should be viewed with circumspection. As the overturning eddy size decreases, the separation between integral and dissipation scales also decreases, and hence an inertial sub-range, in which may not exist. The above parameterization, however, has been used in developing the fossil turbulence theory of Gibson (1980), which shows that turbulence goes extinct and fully fossilization

170 occurs

when

or when the

non-dimensional

parameter

exceeds a critical value Laboratory experiments show that may take values between 10 - 30 (Ivey & Nokes 1989; Stillinger et al. 1983). A part of this variability can be attributed to the different definitions used to identify the demise of turbulence. Once the active turbulence decays, the flow assumes layered structure, and the dissipation of kinetic energy occurs due to viscous friction between laminar-like motions of different layers. Needless to say that the turbulent kinetic energy dissipation where is the rate of strain tensor, should be strongly anisotropic at this time, mainly contributed by the vertical gradients of horizontal velocities, of (2.2). Measurements in decaying stratified turbulence indeed show that the in-plane dissipation, for a plane defined by the direction normal to it, is mainly contributed by the horizontal components and Fincham et al. (1996) found that about 90% of the total dissipation is due to these two components, which is principally contributed by the vertical gradients of horizontal velocities. (Figure 4a,b). Once stratified turbulence evolves into horizontal layers dominated by viscous dissipation, usual parameterizations such as are not valid. Because is mostly contributed by the vertical shear terms, such as it is possible to parameterize the dissipation as A simple model based on the following equations can describe the evolution of horizontal layered structure,

and

The first expression describes the decay of dominant velocity component and the second represents the evolution of horizontal scale. At large times these have solutions of the form

171

and

where and are the respective values of and at the onset of the viscous decay of layers. From Figure 4a, it is possible to estimate or Therefore, based on (2.11a), the velocity decay law can be expressed as This is in reasonable agreement with the high Reynolds number experiments shown in Figure (4b). An important aside of the above discussion is the evolution of scalar fluctuations. In three-dimensional non-stratified turbulence, the scalar inhomogeneities smear off at scales

(for

or

172 (for but because of the unique structure of stratified turbulence, the scalar dissipation may occur by a different mechanism wherein the diffusion between layered structure plays a key role. This should be contrasted from the three-dimensional vortex stretching and amplification that occur in unstratified turbulence (Batchelor 1959). According to (2.4), scalar inhomogeneities may smear off due to strong vertical gradients when water

Therefore, for Sc ~ O(1) (e.g. thermal stratification in or air the onset of viscous dissipation between the

layers is concurrent with the onset of scalar dissipation over the same vertical scales. When Sc>1 (e.g. salt stratification, however, required for the diffusive smearing is much smaller and hence scalar inhomogeneities prevail for a longer time. As far as active turbulence prevails, the buoyancy flux in the flow is non-zero, with suitably correlated buoyancy and vertical velocity fluctuations. With complete fossilization of turbulence upon reaching the character of the flow changes, “layered” structure is formed and the buoyancy flux due to active mixing ceases. Buoyant fluid parcels having a characteristic displacement scale now find themselves in an environment of viscous influence and turbulent eddying motions with which they were being advected are becoming extinct. Consequently, these fluid parcels drift to their equilibrium density levels under own buoyancy forces (which is also known as the restratification). In so doing, they are subjected to viscous drag, with their vertical velocity being determined by the balance where

is the characteristic size of fluid parcels,

Therefore, and the time scale at which the fluid parcels return to their equilibrium density level is given by or Note that the restratification is associated with buoyancy transport in counter (up) gradient direction (leading to positive buoyancy fluxes), a phenomenon that has been noted in previous stratified-flow studies (Stillinger et al. 1983). Based on above discussions, this counter-gradient buoyancy flux is expected to diminish with decreasing Sc and ought to be vanishingly small when Interestingly enough, only meager counter gradient fluxes have been observed in heat stratified experiments, in comparison to those observed with salt stratification (Lienhardt & van Atta 1990). At times, unstable buoyancy fluxes associated with restratification are strong enough to regenerate turbulence, and the turbulence so spawned from a fossilized state is known as “zombie” turbulence (Gerz & Yamazaki 1993).

173 Finally, it is of interest to discuss the decay of fluctuations in unforced stably stratified turbulence. As pointed out by Batchelor & Townsend (1948), the later (final) decay of initially isotropic turbulence occurs while maintaining a balance between unsteady inertial and viscous forces, with non-linear inertia terms being inconsequential in view of weak fluctuations. Under these conditions, (2.2)-(2.4) simply become linear internal wave equations with molecular viscosity and diffusivity terms included. The vorticity equations (2.6)-(2.7) imply that both the “potential vorticity” and “internal wave” modes should decay due to viscous friction. The horizontal gradients of buoyancy (c.f. 2.7) make only a small contribution to the evolution of horizontal vorticity. As was shown by Pearson & Linden (1983), the linearized final decay equations accept oscillatory and decaying solutions. For example, the Fourier amplitude of the vertical velocity can be written as

with

where number

and

is the inclination to the horizontal of the resultant wave is the horizontal wave number. Oscillatory, but decaying,

solutions are possible when For only the decaying modes are possible without oscillations, and a limiting case of this is the situation wherein the dissipation dominates the buoyancy, for example, as a result of or The wave number is near vertical in the latter case, corresponding to horizontal layering, and the former alludes to scales smaller than those of fossil turbulence. Thence, for (2.13) becomes (Pearson and Linden 1983)

and the slowest decaying mode has a frequency the minimum of which becomes

corresponding to

Note that the slowest decaying mode satisfies corresponding to motions in horizontal layers.

174 It should be noted that the analysis of Pearson & Linden (1983) considers only a special class of solutions involving the dynamics of individual wave numbers, and it does not take into account the integral effects of all wave numbers spanning the space. A complete analysis requires consideration of all possible wave numbers and frequencies satisfying linearized governing equations subject to initial conditions that specify velocity and density fluctuations at the onset of the final decay period. Because of the ill-defined nature of transition to the final decay period, it is difficult to specify such conditions. Studies reported hitherto, therefore, have considered the decay of turbulence from an initial fully three-dimensional state (specified by appropriate spectra) to the final state entirely based on linearized equations (Hanazaki & Hunt 1996). Pearson & Linden’s (1983) work is a special case of such a treatment. Hanazaki & Hunt (1996) find that the slowest decaying mode is not given by (2.14), instead the velocity and density fluctuations should obey the decay laws

if the low wave number end of the spectra of initial velocity and potential energy fluctuations are given by the Saffman form and,

if the Loitsiansky form is assumed. The decay of vertical wave numbers or the horizontal motions (though they are not the slowest decaying) were in agreement with (2.14a,b). When considering the integral effects of all wave numbers, the two constraints and are not necessary in arriving at (2.14a,b). 3. EXAMPLES OF STRATIFIED TURBULENT FLOWS Some of the most conspicuous examples of stratified turbulent flows are found in nature. They often are associated with aesthetic atmospheric flow phenomena visible to the naked eye and intriguing oceanic flow structures detectable using suitable instrumentation (Figure 5). These visual effects are largely associated with refractive index fluctuations akin to density fluctuations in stratified flows, and in laboratory experiments these refractive index variations can be captured by shadowgraph flow visualization (Figures2b, 6). Traditionally laboratory experiments have played an important role in studying basic fluid mechanics of stratified turbulent flows, which lineage to the work of Rouse & Dodu (1955) who used a two-layer fluid stirred by an oscillating grid to investigate turbulent mixing across density

175 interfaces. More recently, following Riley et al. (1981), direct numerical simulations have also contributed much to our understanding of stratified turbulence. Various related studies will be briefly reviewed below in the framework of the classification provided by Fernando & Hunt (1996) outlined in Section 1.

176

3.1 Decaying Homogeneous Turbulence within Uniform Stratification This is perhaps the simplest type of stratified turbulent flow possible, which has been a building block of understanding the basic features of such flows. Most laboratory studies in this context have been performed using water (Stillinger et al. 1983; Itsweire et al. 1986; Huq & Britter 1995) or wind tunnels (Lienhard & van Atta 1990; Yoon & Warhaft 1990). These experiments illustrate how turbulence behind a grid in a uniformly stratified flow initially evolves as if there is no stratification, is arrested by the buoyancy forces (onset of fossilization) after several buoyancy periods and finally decays to a state of internal wave motions (fossil turbulence). As turbulence evolves, interesting flow phenomena appear, such as the alternation of energy storage in potential and kinetic energy modes, the formation of layering and restratification in large scales, thus providing support for the dynamical framework discussed in Section 2. Figure 6 shows the evolution of turbulence when a part of a tank containing a quiescent stratified fluid is induced with turbulence by a towing grid. Note how initial

177 small-scale turbulence evolves into horizontal layers. The evolution of turbulence in these experiments is sensitive to the initial conditions, since the nature of turbulence produced and thus the subsequent evolution depend on the turbulence generation mechanism near the grid. Therefore, parameters such as the mesh/bar sizes and the configuration of the grid become important in laboratory experiments (Itsweire et al. 1986). In addition, turbulent fluctuations present in the flow upstream of the grid can interact with shear layers generated at the grid bars so as to modify the nature of grid-generated turbulence. For example, in the absence of upstream turbulent fluctuations, such as in the case of towing grids, turbulence generated at the grid bars is governed by the shear layers only. If the turbulence so produced is not sufficiently strong (i.e., at low and moderate grid Reynolds numbers), the signatures of shear layers persist downstream as a series of density layers. Experiments of Liu (1995) and Rehmann & Koseff (2000) illustrate the formation of such artificial layers in grid turbulence, which is a direct result of the initial conditions used. As such, comparison between different experiments should be done with caution, paying attention to the details of initial conditions. Uniform flow past a grid, towing grids and free falling grids (Dickey & Mellor 1980) have been used hitherto for such studies. In all of the above flow configurations, the degeneration of turbulence to internal wave motions appears to be rapid, as evident from the successful prediction of flow evolution by linear theory at times (Hanazaki & Hunt 1996).

3.2 Decay of Inhomogeneous Turbulence in a Region of Uniform Stratification This is prototypical of natural flows that are characterized by spatially patchy and evolving turbulence. A simple laboratory example is a turbulent patch in a linearly stratified fluid, either generated mechanically by an external source (Fernando 1988) or by an internal instability mechanism such as breaking internal waves (Ivey & Nokes 1989). If the source acts only for a short time period, say by instantaneous imparting of turbulent kinetic energy into a stratified fluid (Figure 2a), then the patch grows while redistributing energy within the patch. In addition, the turbulence decays simultaneously. The combined effect is a rapid reduction of turbulence intensity within the patch, and hence the maximum patch size is achieved, and turbulence fossilizes rather quickly. The growth of a patch in a stratified fluid by an impulsive source can be modeled using simple theoretical arguments. Assume that the turbulent event produces a patch of size with a characteristic velocity The growth of the patch size can be written as (Townsend 1976)

178

where

are constants and

is a characteristic r.m.s.

velocity. The decay of turbulence in the patch can be written as

and the rate of change of kinetic energy within the patch due to combined dissipation and growth becomes

where is the patch volume and is a shape factor. Here the buoyancy effects have been neglected, since (3.2) and (3.3) deals with the initial growth of the patch. It is possible to rewrite (3.1)-(3.3) as

where and at As the patch size grows, the scale at which the viscous effects become important, say increases influence

and

the

scale

decreases. Using

normalizing length scale, and non-dimensionalize (3.4) and the other relevant scales as

of

buoyancy being the

it is possible to

179

where

and

at

for a patch starting with

Here is the normalized lengthscale of fully fossilized turbulence. Figure 7a shows the growth of the turbulent patch for and for typical values of and Note how various scales grow relative to the patch size and the onset of buoyancy effects before the viscous influence comes into play. Also shown is the case of where the onset of buoyancy and viscous effects occur simultaneously. In this case, the patch becomes completely fossilized without going through an active-fossil state. Figure 7b shows the case of and In this case also the patch degenerates into a completely fossil state at

The above results point to the importance of initial Reynolds

number and in the evolution of stratified turbulence. Several laboratory studies have been performed to investigate the evolution of impulsive turbulent sources in stratified fluids, for example, by squirting a blob of fluid as a jet pulse into a stratified fluid as shown in Figure 2a (Gibson 1987; Flor et al. 1994; Fonseka et al. 1998). The results of these experiments, in broad sense, are in agreement with the evolution scenario described by the above model. Another interesting case is the turbulent forcing that sustains for a finite period of time, as in most oceanic and atmospheric situations. In order to mimic such events, experiments have been conducted by oscillating a horizontal grid in a linearly stratified fluid for a finite period of time. The turbulent patch achieves its maximum integral scale typically at (DeSilva & Fernando 1992). In the experiments, the forcing was stopped abruptly so as to allow turbulent eddies to collapse in response to rapidly decaying The flow then evolves to a state with conducive for the formation of layered structure. The initial Reynolds number used was on the order of 80, but the instantaneous Reynolds number rapidly decreases with time upon the removal of source, achieving whence the flow is dominated by horizontal layering and viscous dissipation of motions therein. Also, with the onset of viscous friction, the molecular diffusion is expected to come into play immediately if Sc~1. Conversely, for Sc>>1, scalar inhomogeneities are expected to last for a longer period.

180

Figure 8 shows density profiles obtain at in two experiments carried out with salt and heat, the grid forcing being removed at Except for the solute used to obtain stratification, the experimental conditions were identical. It is clear that the fine structure (or the wiggliness) of the two experiments are vastly different, even after a short time of the removal of forcing, indicating how the molecular diffusion comes into play via the formation of horizontal layering even when some active turbulence may be still present. This observation is consistent with the framework of stratified turbulence presented in Section 2. Figure 9 shows the decay of density

181 fluctuations obtained during the above experimental program, for both heat and salt stratified cases. The data represent the decay since the removal of forcing at Note that turbulence in this case is non-stationary, and hence the statistics shown represent the average of eighty identical experiments, with and identical grid conditions. The r.m.s. buoyancy fluctuation has been normalized by the averaged buoyancy frequency N at the time of the forcing removal and the characteristic r.m.s. velocity of turbulence corresponding to the edge of the turbulent region at The results clearly show a vast deference of decay rates from the outset, in view of the differences in the fine structure of two cases from very early stages of decay. In addition, the r.m.s. salinity fluctuation show a distinct increase of intensity starting at Nt ~ 7, consistent with the restratification phenomena discussed in Section 2. The temperature fluctuations, however, did not show this intensification, possibly due to molecular diffusive effects that smear off inhomogeneities from the early stages of decay.

3.3 Forced Turbulence in Stable Layers The atmospheric (ABL) and oceanic (OBL) boundary layers are examples of forced stably stratified layers. At night, air near the ground is radiatively cooled, leading to the formation of a stable nocturnal layer (NBL) or a surface inversion. The vertical wind shear near the ground generates turbulence (characterized by a friction velocity the diffusion of which is opposed by stable stratification. Two competing processes -- the development of turbulence and stable stratification -- lead to an equilibrium turbulent

182

boundary layer of thickness on the order

where

is the

Monin-Obukhov lengthscale, the von Karman constant and the stabilizing buoyancy flux. Nevertheless, it has been frequently noted that scaling is unsuitable for very stable NBLs, where stable stratification suppresses turbulence to an extent that continuous turbulence with the characteristic velocity scale is untenable. Turbulence therein tends to be intermittent with sporadic local breakdown of stratification (Mahrt 1998). Because of its practical importance in predicting surface-level pollution built up in the evening hours and the paucity of fundamental understanding of its nature, very stable NBL is an area of extensive current research. An analogous situation arises in the upper ocean during the daytime heating of the surface, where heating creates stable stratification whereas wind stirring generates turbulence, leading to the formation of a shallow thermocline (~25-50m depth) with the characteristic scale (Kitaigoroskii 1960). On the other hand, in the atmosphere during the day or in oceans at night, the buoyancy flux is destabilizing (heating from below in the ABL and vice versa in oceans), and the combined effects of winds and buoyancy flux destroy the stable stratification. At different regions of the boundary layer, disparate turbulence generating mechanisms dominate; in the ABL, at heights less than about the shear production dominates whereas, above convective turbulence with velocity scale where H is the boundary-layer depth, is dominant. Therefore, in the modeling of daytime ABL or night time OBL, the additive effect of these two mechanisms is parameterized by expressing the r.m.s. velocity of the turbulent layer as either (Andre et al. 1978) or (Deardorff 1983), where and are constants. Forced turbulent patches in stratified fluids have been studied using laboratory experiments. Sustained energy sources that provide energy directly to turbulence [e.g. an oscillating grid in the experiments of DeSilva & Fernando (1992, 1998) or jet injection as in Gibson (1987)] or continuously forced internal waves that grow and break intermittently (McEwan 1983; Figure 3) have been used. In the former case, turbulent patch is arrested at a physical size

where

is the background stratification, at a

dimensionless time of The buoyancy frequency within the patch N is much less than typically (DeSilva & Fernando 1992) and hence the limiting patch size based on the local buoyancy frequency becomes

(Also see the experiments

described in Section 3.1). Although the turbulent patch achieves a quasi

183

equilibrium state at this height, it can grow further by a different mechanism, in that the entrainment interface separating the turbulent patch and surroundings can degenerate into secondary turbulent patches by breaking of waves (Fernando 1988; Lozovatsky & Fernando 2000). The merger of secondary patches with the primary patch causes the growth of the turbulent region. If an isolated turbulent patch is not confined in lateral directions, horizontal pressure gradients are set up and intrusions that propagate out of the turbulent patch are formed. The dynamics of the patch growth in this case is much different (DeSilva & Fernando 1998).

3.4 Homogeneous Turbulence in Uniform Mean Shear and Uniform Stratification Homogeneous stratified shear flows have often been used as a prototype of understanding complex stratified turbulent flows. Consider the evolution of turbulence in a uniform shear flow with a velocity gradient and a buoyancy frequency N. The turbulence imbedded in this flow is specified by its kinetic energy and dissipation Depending on the governing parameters, turbulence in such flows can either decay or evolve into an active mixing state, and criteria that demarcates such disparate flow regimes are of utmost interest in geophysical studies. The above governing variables give rise to the following length scales:

184

where the shear length scale

represents the lower bound of eddy sizes that

are deformed by the mean shear, plays the role of the buoyancy scale, which is the dynamical bound imposed by stratification on the vertical displacement of fluid parcels, and is the integral scale of embedded turbulence. Note the following length scale ratios:

where

is the gradient Richardson number. The shear number

S represents whether the turbulence is affected by the mean shear or not, irrespective of stratification. When S >> 1 or alternatively where is a critical shear number, turbulence interacts with (or is effectively strained by) and extracts energy from the mean shear, thus allowing the turbulent intensity to grow in time. It should be noted, however, that for the non-linear turbulence cascade to exist, the condition or where is the turbulent Reynolds number, should be satisfied. Therefore, one might expect that the turbulence embedded in the flow to extract energy from the mean shear when where is a critical value of This criterion for the growth is in general agreement with past numerical and laboratory studies (Harris et al 1977; Rohr et al. 1988a,b; Piccirillo & Van Atta 1997; Jacobitz & Sarkar 1999). The stratification can also interact with turbulence and mean shear, depending on the relative magnitudes of and For example, when then possible. Here

and S<<1; thus,

and

are

is a critical gradient Richardson number where the

stratified shear flow interacts with turbulence, either by feeding energy from the mean flow or by breaking down the flow via instabilities (note that for the two different processes need not be the same). Under these conditions, the shear is expected to interact with stratification, thus producing instabilities (for example, Kelvin-Helmholtz billowing; Strang and Fernando 2000) and intermittent (patchy) turbulence, but turbulence already existing in the flow is not significantly affected by shear and is expected to decay Under

185 such conditions, the buoyancy length scale can be parameterized as and hence the commonly used Ozmidov scale for stratified turbulent studies has limited utility. When and S << 1, and hence and are possible. In this case, neither the instabilities nor shearturbulence interaction may produce turbulence, and the existing turbulence in the flow is expected to decay. The length scale where the buoyancy influence comes into play is now given by

(note that

is permissible in this case). Similarly, when and

are possible, and hence stratified turbulence can be

sustained due to extraction of energy from the mean flow (rather than due to the growth and breakdown of shear instabilities), with the buoyancy scale of turbulence being the Ozmidov scale Finally, perhaps the most interesting case arises when

or specifically when

and Such stratified flows may sustain substantial turbulence and mixing activity due to the combined effects of shear instabilities and production of turbulence by the mean shear-Reynolds stress interaction.

4. CONCLUSIONS Some important dynamical aspects of stratified turbulent flows were expounded in this paper. Using the governing equations of motion, interesting features of such flows were elicited and evidence for their existence was presented by alluding to past laboratory, numerical, and field observations. Due to space limitations, only a selected amount of such works has been referenced, and the reader is referred to other extensive reviews of the subject mentioned in Section 1 that covers specific research areas. The emphasis of this paper was limited to describing fundamental processes of stratified turbulence and classification of stratified turbulent flows. In geophysical scales, stratified turbulence can also be influenced by background (Earth’s) rotation, which requires a separate treatment (Lindborg 1998). In the development of forecasting models for environmental and engineering applications, many of the processes intrinsic to stratified turbulence need to be parameterized as they are not resolved or under-resolved. A host of parameterizations is currently available and used with varying degrees of success, some examples being Mellor & Yamada (1974), Gibson (1980), Ivey & Imberger (1991), and Pacanowski & Philander (1981). Major improvements for these models are, however, desired for improvements in

186

predictions, which require further understanding and development of mathematical descriptions of such flows.

ACKNOWLEDGEMENTS Stratified turbulent work at Arizona State University is funded by the Environmental Meteorology Program of the Department of Energy and the Fluid Mechanics/Atmospheric Sciences Programs of the National Science Foundation. The author wishes to thank Dawn Lervick, Frank Yu and Destry Lucas for their help in experiments and calculations and Jennifer McCulley for her editorial work on the manuscript.

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187 Fernando, H. J. S. Turbulent mixing in stratified fluids. Ann. Rev. Fluid Mech. 1991; 23:455493. Fernando, H. J. S. Aspects of stratified turbulence. Kluwer Academic Publishing 2000 (in Press).

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188 Ivey, G. N., Imberger, J. On the nature of turbulence in a stratified fluid. 1: The energetics of mixing. J. Phys. Oceanogr. 1991; 21:650-658. Jacobitz, F. G., Sarkar, S. On the shear number effects in stratified shear flow. Theor. Comp. Fluid Dyn. 1999; 13:171-188. Kitaigorodskii, S. A. On the computation of the thickness of the wind mixing layer in the ocean. Bull. Acad. Sci. U.S.S.R. Geophys. Ser. 1960; 3:284-287. Lienhard, J. K., van Atta, C. W. The decay of turbulence in thermally stratified flow. J. Fluid Mech. 1990; 210:57-112. Lilly, D. K. Stratified turbulence and mesoscale variability of the atmosphere. Amer. Meteor. Soc. 1983; 40:749-761. Lin, J.-T., Pao, Y. H. Wakes in stratified fluids, Ann. Rev. Fluid Mech. 1979; 11:317. Lindborg, E. Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? J. Fluid Mech. 1998; 388:259-288. Liu, H.-T. Energetics of grid turbulence in a stably stratified fluid. J. Fluid Mech. 1995; 296:127-157. Liu, Y. N., Maxworthy, T., Spedding, G. R. Collapse of a turbulent front in a stratified fluid, 1. Nominally two-dimensional evolution in a narrow tank. J. Geophys. Res. 1987; 92(5):54275433. Lozovatsky, I. E., Fernando, H. J. S. Turbulent mixing on a shallow shelf, Submitted to J. Phys. Oceanogr. 2000. Mahrt, L. Stratified atmospheric boundary layers and breakdown of models. Theor. Comp. Fluid Dynamics 1998; 11:263-279. McEwan, A. D. The kinematics of stratified mixing through internal wave breaking. J. Fluid Mech. 1983; 128:47-57. Mellor, G. L., Yamada, T. A hierarchy of turbulence closure models for planetary boundary layers. J. Atmos. Sci. 1974; 31:1791-1806. Müller, P. Small-scale vortical motions. Proceedings of the Internal Gravity Waves and SmallScale Turbulence Hawaiian Winter Workshop, (ed. P. Muller, and R. Pujalet). Hawaii Institute of Geophysics, Honolulu. 1984. Pacanowski, R. C., Philander, S. G. H. Parameterization of vertical mixing in numerical models of tropical oceans. J. Phys. Oceanogr. 1981; 11:1442-1451. Pearson, H. J., Linden, P. F. The final stage of decay of turbulence in stably stratified fluid. J. Fluid Mech. 1983; 134:195-203. Piccirillo, P.S., van Atta, C.W. The evolution of a uniformly sheared thermally stratified turbulent flow. J. Fluid Mech. 1997; 334:61-86. Rehmann, C. R., Koseff, J. R. A unified scaling theory for the mixing efficiency of decaying stratified turbulence. Submitted to J. Fluid Mech., 2000.

189 Riley, J. J., Lelong, M. P. Fluid motions in the presence of strong stable stratification. Ann. Rev. Fluid Mech. 2000; 32:613-657. Riley, J. J., Metcalfe, R. W., Weissman, M. W. Direct numerical simulations of homogeneous turbulence in density-stratified fluids. In: Nonlinear Properties of Internal Waves (ed: B.J. West). AIP Conf. Proc. 1981; 76:79-112. Rohr, J. J., Itsweire, C., Helland, K. N., van Atta, C. W. Growth and decay of turbulence in a stably stratified shear flow. J. Fluid Mech. 1988a; 195:77-111. Rohr, J. J., Itsweire, C., Helland, K. N., van Atta, C. W. An investigation of growth of turbulence in a uniform-mean-shear flow. J. Fluid Mech. 1988b; 188:1-33. Rouse, H., Dodu, J. Turbulent diffusion across a density discontinuity. Houille Blanche 1955; 10:522-532. Sherman, F. S., Imberger, J., Corcos, G. M. Turbulence and mixing in stably stratified waters. Ann. Rev. Fluid Mech. 1978; 10:267-288. Stillinger, D. C., Helland, K. N., van Atta, C. W. Experiments on the transition of homogeneous turbulence to internal waves in a stratified fluid. J. Fluid Mech. 1983; 131:91122. Strang, E. J., Fernando, H. J. S. Entrainment and mixing in stratified shear flows. J. Fluid Mech., Submitted, 2000. Thorpe, S.A. Turbulence and mixing in a Scottish loch. Phil. Trans. Roy. Soc. London Ser. A. 1977; 286:125-181. Townsend, A. A. Structure of Turbulent Shear Flow. Cambridge University Press, 1976. Yoon, K., Warhaft, Z. The evolution of grid generated turbulence under conditions of stable thermal stratification. J. Fluid Mech. 1990; 215:601-638.

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Chapter 7 LABORATORY STUDIES OF CONTINUOUSLY STRATIFIED FLOWS PAST OBSTACLES

Don L. Boyer and Andjelka Srdic-Mitrovic Arizona State University, Tempe, AZ, 85287, USA

Key words:

stratification, rotation, obstacles, laboratory experiments

Abstract:

Owing to the societal importance placed on weather prediction, pollution transport and dispersion, and the commercial and recreational use of our atmospheric and oceanic environment, there continues to be a substantial interest in conducting research in the broad field of environmental fluid mechanics. This area of inquiry encompasses the role of background stratification and rotation on fluid motions and includes the effects of turbulence, complex bathymetry, time-dependent forcing (e.g., tides), phase changes in the fluid media (e.g., precipitation) and a myriad of other complex factors affecting the fluid system. The present communication is concerned with the effects of stratification on the motion of fluids past a variety of topographic shapes. In particular the focus is on a review of selected laboratory studies that have been carried out to better understand the effects of mountain ranges in the ocean or atmosphere on the motion fields in their respective environments and the motion of vehicles (e.g. submarines) through stratified waters. The general conclusion is that, while much has been learned from past laboratory studies, the rapid development of new data acquisition methodologies will allow laboratory studies to play an even greater role in the future in obtaining a deeper understanding of environmental flows. In particular laboratory studies have the potential for being a key factor in the development of improved numerical models of these phenomena, leading eventually to improved prediction. It is also concluded that increased efforts should be made in the laboratory studies to (i) conduct in-depth comparisons between the detailed data sets now available from laboratory studies and appropriate numerical and theoretical models, (ii) investigate flows past complex terrain for which the Rossby number is of order unity or smaller (rotation is important), (iii) consider experiments with more realistic buoyancy frequency and background velocity distributions, and (iv) investigate the vertical motion of obstacles through stratified fluids.

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1.

INTRODUCTION

The study of the fluid motion past obstacles having a variety of shapes is one of the principal topics considered in the study of classical fluid mechanics. If the fluid is homogeneous and incompressible and the fluid medium is "infinite in extent”, the dynamics depends on a single parameter, the Reynolds number; for an airfoil, for example, the primary interest is on determining such integrated effects as the lift and drag. Flows past obstacles in environmental flows (i.e., those occurring in the Earth’s atmosphere or oceans or the atmospheres of other planets), are vastly more complicated and thus more difficult to fully understand. Factors leading to this complexity include the background stratification and rotation of the medium, the extent and complexity of the topography of the regions considered, the difficulty of properly specifying the initial and boundary conditions, the inherent timevarying nature of the flows, the complex physics of natural processes and the fact that they occur simultaneously, as well as the need to understand how to parameterize the effects of small-scale processes into numerical models for the large-scale flow. A deep understanding of environmental flows as they interact with complex terrain is important if one is to (i) obtain improved weather prediction for local regions, (ii) reliably predict the transport and dispersion of pollutants over complex terrain, and (iii) provide predictions which are sufficiently precise to maximise the safety of aircraft traversing mountainous regions. The effects of stratification are critical to the occurrence of many environmental phenomena. It is well known, for example, that stratified flows past long mountain ranges may be blocked and thus may lead to serious air pollution problems; e.g., the vicinities of Los Angeles, California, and Phoenix, Arizona, USA. Stratification effects are also critical to the occurrence and prediction of severe downslope winds (e.g., as found in the vicinity of Boulder, Colorado, USA). Down-slope or katabatic winds, as caused by the radiative cooling of the Earth’s surface, are characteristic of many regions; in these flows radiative cooling leads to the fluid parcels near the surface becoming heavier relative to their surroundings and thus to their flowing downhill. These so-called drainage flows then interact with the local orography. For relatively small horizontal length scales, as exemplified by the “Valley of the Sun” in Phoenix, Arizona, USA, the effects of the Earth’s rotation may be important but are not dominant (i.e., the Rossby number is estimated to be of order 1). Such drainage flows are important in determining regions for which severe pollution episodes might occur owing to the fact that these surface flows gather pollutants as they advect downslope and then pool in certain low-

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lying areas. The problem is exacerbated by the fact that the pollution concentration in these pools is strongest during the early morning hours when the work traffic is adding to the pollution levels (see Chen et al. 1999). When such drainage motions occur along topographic features having extensive horizontal length scales such as the Antarctic continent, the Earth’s rotation is a dominant factor in the dynamics. In the vicinity of the seaward side of drainage basins, such katabatic flows lead to some of the strongest winds found on the Earth (Schwerdtfeger, 1984). On the recreational side, the standing lee waves found, for example, on the eastern slopes of the Sierra Nevada Mountains prove to be an excellent natural environment for glider pilots. The von Karman vortex streets found in the wakes of islands with substantial mountains also owe their presence to the fact that the atmosphere is stratified; e.g., Madeira (Chopra and Hubert, 1965) and Cheju (Tsuchiya, 1969). Background stratification also has an important influence on the nature of the ocean circulation in the vicinity of seamounts and seamount chains although the Earth’s rotation is also an important consideration in these problems; see Roden (1987). One concludes that the effects of stratification are important in a wide range of environmental flows. Additionally, one also concludes that the Earth’s rotation is important in most environmental flows of interest. To begin a study on the topic of orographic effects on stratified flows, the reader is referred to the excellent monograph by Baines (1995). Long (1972), Maxworthy and Browand (1974), Lin and Pao (1979), Baines and Davies (1980), Hopfinger (1987), Riley and Lelong (2000), and Boyer and Davies (2000) have published review articles on various aspects of the problem. Here the focus is on laboratory experiments and, in particular, what has been learned in the past and the perceived directions for the future in light of recent developments in data acquisition methodologies. Laboratory experimentation has been used extensively in the study of the flow of stratified and/or rotating fluids past obstacles with a variety of shapes and no attempt will be made to present an in depth literature review. The nature of the background stratification can vary greatly and thus, in order to focus the review, we will consider primarily fluids having linearly undisturbed density profiles; i.e., constant Brunt-Väisälä or buoyancy frequencies. Other characteristic stratification profiles are also important, but are not be considered further. These include (i) flows in open channels such as rivers and canals, for which the layers are water and air, respectively, (ii) two-layer for which the density difference between the layers is small as approximated by the upper mixed layer and the deeper ocean, and (iii) naturally occurring,

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continuous nonlinear profiles that are characteristic of the atmosphere and oceans. For the present purposes then, we consider the flow configuration given in Figure 1; i.e., we wish to investigate the motion and density fields resulting from a linearly-stratified, vertically-sheared fluid impinging on an isolated obstacle fixed on a horizontal plane surface; we take the characteristic fluid speed to be U and the characteristic horizontal and vertical length scales of the obstacle to be D and h, respectively. It is noted that numerous laboratory experiments have been conducted by placing a symmetric obstacle in the central region of a water tunnel test section; e.g., placing a long circular cylinder with its axis horizontal in a uniform test section flow. For sufficiently large Reynolds numbers, the upper half of such a flow will approximate that which would be obtained by placing a half cylinder on the horizontal surface as in Figure 1;here we assume that boundary layer effects and large scale eddy shedding phenomenon are negligible. In order to relate the laboratory configuration to possible applications consider a uniform, steady background current with a topography that may or may not be complex. In so doing we will not address such important factors as (i) the influence of unsteady background flows (e.g., tides), (ii) the effects of horizontal shear, (iii) the presence of turbulence in free stream, and (iv) the beta-effect.

195 The motion characteristics resulting from this physical system will depend not only on U, D and h as defined above, but also on the buoyancy or Brunt-Väisälä frequency , the kinematic viscosity ν and possibly the background rotation, as characterized by the Coriolis parameter, f. The flow characteristics can thus be expressed in the following dimensional form

Employing dimensional analysis, one finds that the following dimensionless parameters characterize the flow:

internal Froude number,

Reynolds number,

Rossby number, and

topography aspect ratio. In many studies of rotating and stratified fluids, one introduces the Burger number Bu defined by

but, as noted, Bu is not independent of the above.

196 Tables 1 and 2 provide typical values for a submarine mountain such as Fieberling Guyot (Erickson, 1991), a typical mountain in the atmosphere, a submarine and a representative laboratory experiment. As one notes, the Reynolds numbers in all of these representative cases are significantly greater than one. Thus applying Reynolds number similarity, it is assumed that the Reynolds number is unimportant in determining the various flow patterns. This Reynolds number similarity assumption leads naturally to the hypothesis that the various flows to be discussed can all be represented on an internal Froude number against Rossby number regime diagram. It is further reasonable to hypothesize, and laboratory experiments support, that the effects of rotation and stratification become negligible as the Rossby and internal Froude numbers become greater then approximately ten, respectively. As will be discussed below, the long-time behaviour of stratified wake flows will depend on the fluid stratification even though F >> 1 because as the flow dissipates and

Figure 2 depicts such a regime diagram showing the location for each of the application cases and the laboratory example.

The atmospheric and oceanic cases indicate regimes for which rotation effects are of practical interest. As one notes, the oceanic and laboratory examples are clearly ones for which background rotation must be considered. The atmospheric example is one for which rotation plays a lesser role but with the Rossby number being of order one, may still be a factor. The present discussions address the lower right hand region of the plot whereby the Froude number is small but the Rossby number is large.

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2.

LABORATORY FACILITIES AND EXPERIMENTAL TECHNIQUES

The initial step in setting up the experiments is of course to stratify the working fluid. The most common and the easiest way to achieve a linear stratification is to use the two-tank method described by Oster and Yamamoto (1963). It is possible to use a variation of this method, using computer-controlled mixing from two tanks of specified, but different densities, to establish a predetermined nonlinear vertical density profile; see Hill (2000). The principal laboratory approaches to developing controlled background motions of continuously stratified fluids relative to the topographic feature to be studied are either to force the fluid past the obstacle that is fixed in the laboratory frame or to tow the obstacle through the fluid. The former, of course, is the wind tunnel approach used in aeronautics whereby a test section flow is established and the obstacle under consideration placed in the test section. This approach is more difficult for stratified fluids because the fluid forcing used may tend to destroy the density stratification. Odell and Kovasznay (1971), however, developed a unique approach in which the concept was to drive each layer of the stratified fluid separately around a closed-circuit stratified fluid channel. This was accomplished by forcing the fluid by viscous effects through a “pump” consisting of two stacks of

198 overlapping, counter-rotating disks. This method forced the stratified liquid to move through the plates with very little mixing and then traverse a closed loop channel, including the test section, before again passing through the “pump’. The Odell-Kovasznay method is not as useful if used in conjunction with background rotation since the boundary layer transports along the channel walls will tend to destroy the stratification. Stratified wind tunnels have been developed to produce a layered freestream stratification (Scotti and Corcos, 1972) and to obtain a continuous stratification by forcing the upstream flow through a heat exchanger (Hewett et al., 1971). Huq and Britter (1995a, 1995b) reported on a stratified water channel facility in which fluid was removed at given levels at a downstream location by a series of pipes and then returned to the same vertical positions at an upstream location. This method is somewhat akin to the OdellKovaszanay approach but is not as effective in maintaining a given density and velocity profile. Perhaps the easiest approach to obtaining a background flow relative to the topography is by towing the obstacle through a stratified fluid, otherwise at rest relative to the laboratory. For geophysical applications of flows over and around mountains in which the model mountain is towed along the channel floor, one shortcoming of this approach is that the lower surface of the channel upstream and downstream of the obstacle is moving toward or away from an observer fixed to the model mountain. Should viscous effects not be of primary concern, this may not be a significant problem. Otherwise, consideration should be given to placing the moving obstacle on a long horizontal tray so as to properly simulate the boundary layers near the mountain. One disadvantage of the tow-tank approach is that the facility has a finite length and thus fluid blocking effects may propagate to the upstream end of the tank and reflect back into the “free stream,” thus contaminating the flow in the vicinity of obstacle. In the application of laboratory results to specific physical systems one must consider both “finite depth” and “infinite depth” fluids as exemplified by flows over mountains in the ocean and atmosphere, respectively. All laboratory experiments, of course, use finite depth fluids. As such, upward propagating internal waves generated by the topographic feature will be reflected back into the flow region being investigated and thus may not adequately simulate the infinite depth case. Baines and Hoinka (1985) developed a tow-tank facility with a vertical-dividing wall oriented along the tank but not reaching the free surface of the fluid; see Figure 3. The facility also included a flat plate along the channel axis that was oriented at 45° to the horizontal. The authors demonstrated on theoretical grounds and by laboratory observations that this configuration reflected the upward propagating internal waves away from the section of the tow tank for which

199 observations were made, thus approximating the radiation condition in an infinitely deep fluid. In principal, the Baines and Hoinka (1985) method should work for rotating fluids as well. For the most part, measurements obtained in the past for this class of problems have been of a qualitative nature and have employed such observation techniques as dye tracers (e.g., Baker, 1956), and neutrally buoyant particles (see below), hydrogen bubbles (Schraub et al., 1965) and shadowgraph and Schlieren (see Merzkirch 1974) techniques. While these techniques provide good qualitative information on the flow fields being investigated, it is submitted that past work in this area, while helpful in leading to a better understanding of the various flows in question, does not give the level and quality of data to clearly compare the observations with theories and numerical analyses. Density profiles are normally measured by securing liquid samples from various locations and then using a refractometer to deduce the density. Alternatively, conductivity probes (Head 1983) can be used directly to determine instantaneous density profiles. When used in arrays, these probes can also provide a measure of the developing density field. It is the thesis of this brief review that laboratory work in the future should be more directed toward obtaining higher quality data from these laboratory experiments so that numerical and analytical modelers can have available benchmarks for their model development.

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In recent years, flow diagnostics have benefited greatly from rapid developments in the area of image processing and computer hardware. Each day it seems, the equipment available increases in quality making the prospects for obtaining ever-improving data in the years ahead. These enable development of a variety of diagnostic tools based on digital image processing. Some of the methods that are currently being rapidly incorporated into laboratory research programs concerned with fundamental fluid mechanics are those of particle imaging velocimetry (PIV), particle tracking velocimetry (PTV), laser induced fluorescence (LIF), 3D imaging, synthetic Schlieren, and velocimetry techniques based on conserved scalar field measurements. Both PIV and PTV are particle-based velocimetry techniques, and require that the background fluid be seeded with neutrally buoyant particles, which act as flow tracers. The flow is then illuminated by a light source such as a laser beam, typically shaped into a thin light sheet. In the PTV methods the concept of determining the velocity field is quite simple; i.e., by recording particle motions, one can determine the velocity by measuring the mean particle displacements over specified periods of time. Thus, the PTV method measures the velocity at the instantaneous position of the tracer particle (fluid particle), i.e., the Lagrangian velocity. In order to obtain the Eulerian velocity field, the Lagrangian velocity data are extrapolated on a regular grid. In the PIV method, the velocity is obtained for the probe volume defined by the size of the interrogation cell. First, images are segmented into an array of interrogation cells, such that each of them contains a number of particles. Then, the average displacement of the particles within the cell is obtained as the one that maximizes the crosscorrelation function. The PIV approach is excellent when fine features of the Eulerian flow field are to be resolved with high temporal resolution, and if the focus is on the motion of the particles themselves (two-phase flow), the PTV method is indispensable due to its inherent Lagrangian nature. These two techniques are the oldest and the most widely used full-field velocity measurement methods. An enormous number of publications are devoted to the subject, here, only a few are mentioned as a starting point for one who would like to explore the field (Dalziel 1993; Fincham and Spedding 1997, Raffel et al. 1998). Another group of flow diagnostic methods is based on conserved scalar field measurements. In this class of methods, fluorescent dye is used to visualise the fluid motion instead of particles. The intensity of the fluorescence is monitored in space and time and related to dye concentration. Inversion of the time- and space-evolving concentration field reveals the underling velocity field. An excellent review of the methods built around

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this concept is given by Dahm et al. (1996). This technique can be adjusted for the four-dimensional (space and time) measurement of the concentration or density field as well as the measurement of the velocity field and its derivatives. The best known method from this group is laser-induced fluorescence (LIF); it is suited for full field concentration (density) measurements. Typically, fluorescent dye is added to the fluid in one tank before stratification is produced, and as a result, the concentration of the fluorescent dye will vary in the vertical direction as well as the fluid density. The fluorescent dye should be chosen such that its diffusivity matches the diffusivity of the salt. The wavelength of the laser is chosen to excite fluorescence of the dye and the laser sheet is used to illuminate the region of interest. A flow is recorded by a CCD camera employing the appropriate filter. Images of the initial undisturbed stratification are used to relate intensity of fluorescence and dye concentration. One of the difficulties, involved with using the described optical methods in stratified fluids is that the refractive index varies in the vertical direction. Consequently a horizontally directed light beam will not remain horizontal as it passes through the fluid. Also, images recorded in vertical plane will be distorted. Consider, for example, a light sheet directed horizontally through a linearly stratified fluid. The estimate of the vertical location of the laser beam, as it propagates through a stratified fluid whose buoyancy frequency is N = 1, is shown in figure 4 as a solid line (see Voropayev et al. 1983). The bending of the light beam becomes significant at a distance larger then 50 cm. This estimate is valid for a perfectly horizontal beam, which is usually difficult to produce, and even a slight displacement from the horizontal will produce significant bending of the beam in the vertical plane (see figure 4, curve marked by circles). During experimentation with stratified fluids, mixing of the fluid will contribute to further local changes of initial refractive index. The illuminated plane will thus constantly change its position, making measurements at the known position impossible. The method typically used to eliminate this effect is the matching of refractive indices of the solutions in two tanks before stratification is produced, by adding alcohol into the solution with lower density. If measurements are to be made in the vertical plane, a correction for the change of the refractive index may be introduced through non-linear mapping between the image space and the laboratory coordinate frame.

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Finally, one of the above methods does not require matching of refractive index, on the contrary, it relies on the analyses of the distorted mask patterns produced by imaging through the stratified flow. This method is called synthetic Schlieren and may be used for qualitative and quantitative studies of the density field in two-dimensional stratified flows (see Dalziel 2000). The analysis involves the comparison of the initial image of the mask pattern obtained by imaging through an undisturbed stratified fluid and its image obtained at a certain instant when the flow is initiated, and solution of the inverse problem to obtain refractive index field (density field) which produced that distortion.

3.

SOME BASIC LABORATORY OBSERVATIONS

3.1

Two-Dimensional Topographies

In an elegant laboratory/theoretical study Long (1954,1955) investigated the motion field resulting from the flow of a two-layer- and a linearlystratified fluid over a long ridge of constant cross-section; the analysis later become known as Long’s model. One of the fundamental assumptions made

203 by Long was that there was no upstream influence caused by the topographic feature; in particular he assumed that is a constant independent of height far upstream, where is the mean density and U the velocity. While no quantitative assessment was made in comparing the theory and experiments, the qualitative agreement was considered very good. Browand and Winant (1972), using a tow tank, conducted experiments for flow past a right circular cylinder at small internal Froude and Reynolds numbers; i.e., and Their experiments demonstrated nicely a strong blocked region far upstream of the cylinder and brought into question, from a laboratory perspective, the assumption by Long of no up-stream influence. Wei et al. (1975), in a similar set of experiments, but at substantially larger Froude and Reynolds numbers and considered the flow past a right circular cylinder and a vertical plate extending across the full width of the tank. They showed that even at these much higher Reynolds numbers there were unattenuated columnar disturbance modes upstream of the obstacle. It should also be noted that Martin and Long (1968) showed that upstream disturbances even occur when the disturbance is caused be the boundary layer formed by a flat plate translating through a stratified fluid. Baines (1977, 1979) conducted an extensive series of laboratory experiments on the flow past two-dimensional obstacles focusing on examining the validity of Long’s model and examining the degree to which the laboratory results were in consonance with linear theory. Honji (1984) presented some beautiful flow visualization photographs of the wakes of stratified fluids passing a right circular cylinder. Baines and Hoinka (1985), henceforth noted as BH, conducted a systematic study of five topographic features with a range of cross-sections using the simulated “infinite extent” tow-tank facility described above. Their results demonstrated that for sufficiently small F (large NH/U) for each of the topographic features studied, upstream blocking occurs. This physically means that the kinetic energy of the particles near the lower surface is too low for the particles to advect over the ridge. Furthermore, because the ridge is infinitely long (i.e., blocked by the lateral walls of the channel, these slow moving particles cannot “go around the ridge”. As such the flow is blocked and care must be taken in interpreting the data since this blocked flow can reach the “upstream end of the channel”, reflect and interfere with the upstream flow. The resulting flow thus cannot represent the motion field in an infinite medium. The conditions for which blocking was found for the various topographies are given in Figure 5. In particular, BH found that for the low-level fluid upstream of the topography

204 is blocked and the value of F is only slightly affected by the shape of the topography.

In the range the velocity of the approach flow near the bottom surface increases from near zero to approximately U. The lee waves steepen progressively as F decreases until at which “rotors” or semi-stagnant regions are embedded in the lee wave field. As F decreases further, the wave amplitude above the obstacle decreases. Finally in the range the flow consists of lee waves of wave numbers of order N/U with the upstream disturbances to the flow being negligible. Boyer and Tao (1987) showed that the time history of the start up to a uniform flow can be an important factor in determining the final steady state motion field for the flow over long ridges. Boyer et al. (1989), henceforth noted as BDFZ, investigated the flow past a right circular cylinder using a tow tank which included a solid horizontal floor and a free surface; the cylinder was towed through the center of the tank with the cylinder axis being horizontal and normal to the channel axis. This study employed both shadowgraphs and particle tracers and showed clearly that the nature of the flow field, for the range of parameters studied, depended not only on the internal Froude number F (defined in terms of the cylinder diameter), but also the Reynolds number Re. The BDFZ study identified ten rather distinct flow regimes for the experiments in the parameter ranges and ;

205 see Figure 6. Figures 7a-d are shadowgraphs for four of these regimes: viz. the (a) attached flow, (b) single centerline structure, (c) isolated mixed regions and (d) turbulent symmetric wake regimes. Figure 7a shows clearly the upstream wake, including the long blocked region of fluid. One also notes that the flow in the lee does not detach from the cylinder. Figure 7b shows the single centerline structure regime in which the downstream profile has a weak jet in the vicinity of the streamwise centerline. Figure 7c shows the rather remarkable flow associated with large regions of recirculating fluid downstream of the cylinder. Similar phenomena called “stationary humps” were observed by Baines and Hoinka (1985). Figure 7d shows a typical turbulent symmetric wake in a stratified fluid. In particular, in homogeneous flows, the wake width is considered to be a monotonic increasing function of the downstream coordinate. As Figure 7d demonstrates, a turbulent wake in stratified flows is found to decrease in width, a situation called wake collapse. At early times, wakes in stratified fluids increase at the same rate, as do those in homogeneous flows, other parameters being held fixed. Mixing, however, causes an increase in potential energy in the wake. This mixed fluid then seeks an equilibrium level that corresponds to a collapse in the vertical. These collapsed wakes then lead to vortex street type eddy fields of thin vortices generally known as “pancakes.”

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208 Mitkin et al. (1998), using shadowgraphs and acoustic echo sounding methods conducted experiments similar those of BDFZ. While using different regime definitions, they demonstrated that the Froude and Reynolds numbers values for their experiments in the so-called “multiple centerline structures” and “turbulent – vortical structures” regimes were in consonance with the Boyer et al. (1989) regime diagram. Mitkin et al. (1998) also provided data on the time-dependent characteristics of the spiraling vortices in the turbulent-vortical structure regime. Similarly, Chashechkin and Mitkin (1998), using shadowgraph methods, conducted experiments which replicated the single centerline structure and multiple centerline structures regimes of BDFZ. Chashechkin (1999) showed how modified Schlieren techniques can be used to obtain photographs with a high spatial and temporal resolution; he presents, as an example, a photograph of the single centerline structures regime. Using the same experimental facility, Mitkin and Chashechkin (1999) focus on the low Reynolds, low Froude numbers region of the BDFZ regime diagram. They show that instabilities owning to the diffusion of salt occurs in the thin density boundary layer associated with the slow flow of the cylinder through the stratified fluid. In summary, the experiments of Chashechkin and his colleagues support strongly that the BDFZ regime diagram gives a good indication of the location of the various flow regimes as a function of the Froude and Rossby numbers. One weakness of these laboratory measurements is the fact that these data are all essentially of a qualitative nature. As discussed below, it is important that recognition be given that these are gross qualitative features of the synoptic flow field. What is needed is better quantitative data, including some sense of the errors involved in the measurements. This will be discussed further below.

3.2

Three-dimensional Obstacles

The case of three-dimensional topographic features is more important from the point of view of applications and more complex with regard to theory, numerics, and laboratory experimentation. Because the flow in such cases can go around as well as over the topography, the obstacle provides less of a barrier to the flow than does its two-dimensional counterpart. Thus, the upstream influence of the topography at small F is substantially weaker for the three-dimensional case then for its two dimensional counterpart. Baines (1979) conducted laboratory experiments on the flow past a long ridge of constant cross-section, with the exception of a gap along one end of the ridge. The flow was realized by towing the topography through a towtank having a rectangular cross-section. Blocked flow was observed for the

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two-dimensional case (i.e., no gap) for For all cases with a gap and for all of the fluid below a certain height flowed around the barrier and through the gap. One of the earliest reported laboratory studies of stratified flow past an isolated topography was that of Brighton (1978). Using principally dye tracers, he observed the general characteristics of the flow past a body of revolution noting the general nature of the flow to include flow over and around the topography as well as the existence of pairs of cowhorn eddies in the lee of the obstacle. Hunt and Snyder (1980) conducted tow-tank experiments with a bell-shaped hill, a linearly stratified fluid and a uniform approach flow to test Drazin’s (1961) theory for low Froude number flow over three-dimensional obstacles; these experiments were conducted in the Froude number range and the Reynolds number range Good agreement with Drazin’s theory was found for Greenslade (1992) and Hunt et al. (1997) considered theoretically steady stably stratified flow with a buoyancy frequency profile N(z) and mean velocity profile U(z) past an isolated three-dimensional mountain under the conditions of small Froude number F (i.e., F << 1) and large Reynolds number Re (i.e., Re >> 1); see Figure 1. Mathematical solutions for this class of problems have not been achieved and numerical solutions, while obtained, have not been adequately tested against either field data or laboratory experiments. Greenslade divided the flow fields obtained into three principal regions and, at least qualitatively, these are in consonance with the observations of Brighton (1978) and Hunt and Snyder (1980); see Figure 8. In the middle region, which extends over the greater portion of the mountain height, the flow goes around the mountain. In the top layer, the flow rises and goes over the mountain, while in the bottom layer, the flow must go around the mountain. Hunt et al. (1997) give an approximate analysis for this complex problem, including matching at the interfaces. It is fair to say, however, that no detailed comparisons of the velocity or density fields have been made between the theory and any laboratory models. One qualitative observation consistent between laboratory observations and the theory (middle layer) is the laboratory finding by Sysoeva and Chashechkin (1988) that the wake in the lee of a sphere at low Froude number near the obstacle has an approximately rectangular cross-section. Another qualitative laboratory observation supporting the theory is that shed vortices tend to leave the obstacle at an angle, but their axes quickly become vertical as they advect downstream; Hunt and Fernando (1996). One quantifiable observable, investigated in most of the threedimensional topography laboratory studies, is that of the dividing streamline

210 height introduced by Sheppard (1956) and inherent in the Greenslade model. He postulated that a fluid parcel can rise over the hill only if it has sufficient kinetic energy to overcome the potential energy required to raise the parcel from its upstream level to the top of the hill. The concept is simply then, that streamlines below advect around, while those higher, pass over the topography. For a uniform approaching flow and linear density gradient, Sheppard’s formula leads to which is consistent with Drazin’s prediction, but is only valid as The results of Hunt and Snyder’s (1980) experiments supported this simple formula for the dividing streamline height. Castro et al. (1983) utilizing principally flow visualization techniques conducted an extensive series of experiments on stratified flow over and around triangular ridges with various aspect ratios. They concluded that does not depend on the body shape and is in general agreement with the Sheppard and Drazin relation for F << 1. Snyder et al. (1985) conducted a wide range of experiments with a variety of hill shapes and orientations to the currents and with different upstream density and velocity profiles. They concluded that for symmetric hills and small upstream vertical shear the Sheppard estimate of is a good one. For asymmetric geometries and strong upstream vertical shear, however, the estimate is the lower limit.

The classic obstacle shape of interest for three-dimensional flows is the sphere. Lin et al. (1992) have conducted an extensive series of laboratory experiments aimed at delineating the range of flow phenomena (flow regimes) observed for a wide range of Froude and Reynolds numbers; in particular, the study included the ranges and

211 Eight (8) distinct flow regimes were identified and the region of their location on a Froude against Reynolds numbers diagram delineated. These regimes ranged from “steady two-dimensional attached vortices” at low F and low Re to “turbulent wakes” at high F, Re combinations. At low Froude numbers, the two-dimensional attached vortices regime is characterized by a symmetric pair of leeside vortices whose vertical extent encompasses only the height of the sphere and whose horizontal flow pattern is roughly independent of depth; this owes to the tendency of stratified fluids to stay in horizontal planes. At somewhat larger F, Re combinations, these attached vortices shed from the sphere and, owing to the stratified nature of the fluid, tend to develop strongly stable horizontal flows which include the shedding of vertically-oriented vortices having roughly the same height as the sphere diameter: see Figure 9. Vortex lines cannot end within the fluid and thus weak vortex lines, which diffuse rapidly by viscous effects, connect these strong vertically oriented structures. The vortex pattern throughout the level of the sphere diameter is qualitatively similar to a Karman vortex street formed by a circular cylinder translating uniformly through a homogeneous fluid.

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Chashechkin et al. (1995) employed a laser scanning refractometer (LSR) to investigate the density distribution in the vicinity of a sphere being towed in a horizontal direction through a linearly stratified fluid. It was found that the vertical gradient ahead of the body (but in the disturbed region) had a decreased gradient relative to the undisturbed fluid, while this gradient was greater than in the undisturbed in the wake of the sphere. Thin high densitygradient layers were observed along the periphery of the wake. Hopfinger et al. (1991) studied the internal wave field produced by spheres being towed at constant Reynolds number, Re = 3,000 (turbulent regime), and a range of Froude numbers from 0.5 to 12.5. Two distinct wave fields were recognised: lee waves, generated by the body prevailed when F < 4 , and waves generated by the large-scale coherent structures of the wake F > 4. Spedding (1997) focused on the dynamics and structure of the late wakes of towed spheres. At that stage, wakes are characterised by high mean wake defect velocities (in comparison to non-stratified wake) and similar decay rates of energy and enstropy. An important class of problems with great practical significance is that of the motion of self-propelled undersea vehicles. It would appear that both rotation and stratification effects for a submarine wake would be unimportant and that is indeed the case for small characteristic times based on the advection time D/U. The submarine however is a source of turbulence in the wake which decays leading to a characteristic speed which decreases and a characteristic length-scale, which increases. These lead to representative Froude and Rossby numbers, which approach O(1) . This in turn implies that at large times both stratification and rotation become important in spite of their unimportance at early times; the flow field itself evolves into so-called pancake eddies which are a topic of considerable interest owing to the possibility of being detected by remote means. Gilreath and Brandt (1985) have studied the wave fields produced by the motion of self-propelled and towed elliptical bodies and their wakes. Voropayev et al. (2000) focused on the flow produced by a moving continuously acting jet in a stratified fluid (a situation akin to the accelerated motion of a self-propelled body). These flows are characterized by turbulent wakes, which eventually collapse and excite internal waves. One important aspect of this problem is the nature of the wave field produced in this process, and recognising the wave signature that may be used to detect the presence and the action of the self-propelled body. While the studies to date on stratified flow over three-dimensional topography have clearly improved our understanding of this complex topic, it is fair to say that extensive comparisons between the field properties

213 (velocity, density and pressure), laboratory observations, theories, and numerical analyses are yet to be done.

4.

A COMMENTARY ON THE FUTURE

The increasing demands society is placing on the research community for relevance will be a driving force in setting the future directions for research in all areas of science including the ones being addressed presently. This trend will play a significant role in determining the future directions of laboratory experimentation. Beginning with Long’s seminal experiments (1954, 1955), laboratory observations have been a key component in developing our knowledge base concerning stratified flow over and around topographic features having a variety of shapes. It seems fair, however, to note that for the most part these experiments (i) have not been focussed on direct applications (e.g., to specific atmospheric or oceanic flows), (ii) have been highly idealized and thus, in many respects, not good models of natural flows and (iii) have, for the most part, been qualitative in nature. Furthermore, it seems clear that the improvement of predictive capabilities for atmospheric and oceanic flows generally, and these involving topography, in particular, will depend on the continuing development and improvement of numerical models. Our view is that the increasing pressures for research relevance and the recognition of the key role numerical models must play in providing information of use to the researcher’s customer, the public, is that the interplay between laboratory experiments and numerical model development will accelerate in the years ahead. This is further supported by the need for reliable and understandable data for the development of numerical models and the recognition of the difficulty of obtaining such data from the natural environment. This opens the possibility that laboratory experiments may find a role in providing data sets in support of numerical model development and testing. Needless to say, laboratory data cannot fully substitute for in situ observations, in the development of numerical models; reliable field data will continue to be essential in determining “at the end of the day”, the degree to which the numerical model can predict the state of the natural environment. The following are a few selected areas for which laboratory experiments could play key roles, with the support of numerical model development and testing (Section 3.1) being an overarching theme of the laboratory studies.

214

4.1

In-Depth Comparisons Between Laboratory Observations and Numerical Models

Most of the comparisons to date between laboratory observations and the results of numerical models concerned with the interaction of stratified flows with topography have been qualitative in nature. The experimental techniques for full field measurements, discussed above, provide the capability of obtaining data sets for the velocity and density fields that have sufficient spatial and temporal resolution to be excellent tests for numerical models. These techniques also allow for repeating experiments at seemingly the same parameter values and thus one can readily assess the errors inherent in the laboratory experiments. Hanazaki (1988), in his excellent paper on the numerical analysis of the flow past a sphere at a Reynolds number of 200 and a Froude number range of 0.25 < F < 200, presented detailed data on the velocity fields and isopycnal surfaces at selected locations. Owing to the fact that data simply were not available to test these results, the comparisons made were not with the velocity fields themselves, but with such integrated parameters as the drag. While such comparisons are helpful, it is more desirable to have comparisons, which test the flow field predictions, because the integrated values may mask errors in the details of the model output. It should also be noted that the development of techniques to measure the drag on obstacles moving through rotating and/or stratified flows has moved very slowly since the pioneering results of Mason (1977) and Lofquist and Purtell (1984). That is, to the authors’ knowledge, capabilities do not presently exist for measuring the drag (and lift) forces on an obstacle of arbitrary shape immersed in a stratified (and/or rotating) flow. This area of laboratory inquiry is ripe for new ideas on how to make these measurements. Note that Mason (1977) and Lofquist and Purtell (1984) worked with spheres. This geometry allowed for the relatively easier measure of the drag force. The need is to be able to determine, accurately, the drag (and lift) on obstacles having complex bathymetry. The comparisons between laboratory observations and numerical models must first address the question of how the comparisons are to be made. Hanazaki (1988), for example, presented detailed velocity fields for selected regions of the flow and for various parameters. Assuming comparable velocity fields are available from laboratory experiments, how should these fields be compared? Two methods were recently employed for a similar class of problems; i.e., the motion resulting from (i) a barotropic, oscillatory forcing of a rotating, stratified fluid in the vicinity of a submarine canyon (Perenne, Haidvogel and Boyer, 2000), and (ii) an impulsive, upwelling or downwelling favorable barotropic current past a canyon (Perenne, et al,

215 2000). The former, or oscillatory flow study, used as a comparison a normalized kinetic energy difference, NKE, averaged per grid point, between the numerical model and the laboratory experiments. This proved to be a robust measure but suffered from the fact that the domain size chosen was a factor in determining the precise value of the NKE. The method used in the second study was to develop laboratory- against numerical-model velocity-component scatter diagrams of each of the horizontal velocity components in a flow domain of interest. These nicely demonstrated modelmodel differences, but also suffered from being domain size dependent. In summary, while integral methods such as determining the drag on an obstacle provide a good overall comparison between the lab and the numerical model, the need is for more critical comparisons relative to the overall flow fields of the systems involved. Even a determination of some objective measures to make comparisons would appear to be a fruitful line of inquiry. Recognizing that the laboratory observations in Brighton (1978) and Hunt and Snyder (1980) were mainly qualitative in nature and the fact that an understanding of the motion past such symmetric isolated topographies are fundamental to the subject, it seems in order that detailed quantitative measurements be made concerning this problem so that (i) the motion fields can be delineated for a range of governing parameters of interest and that theoretical and numerical models of this system be compared with the data obtained. The overall purpose is to fully understand this basic flow and to determine the degree to which current models can capture the essence of the resulting flows. It is submitted that (i) the rapidly developing flow diagnostic techniques for 2D and 3D full-field measurements of flow properties, (ii) the need to have improved benchmarks for numerical models of environmental flows, and (ii) the high societal importance placed on improved prediction models for weather, pollution and climate, suggest that laboratory experimental programs, focused on topographic effects in stratified flows, be directed toward obtaining more quantitative data than in the past. Furthermore, it is important that the laboratory work be directed toward problems having some societal significance, because the basic physics of the phenomena involved are at this point relatively well understood.

4.2

More Realistic Background Conditions

Etling (2000), in a recent article on gravity waves and vortices in the wake of large islands, points out that most theoretical, numerical, and laboratory studies of these phenomena neglect such aspects as the vertical

216

variation in the velocity and stratification of the approaching flow (not noted are such effects as horizontal shear and unsteadiness in the free stream) and that such variations may lead to motion fields which are totally different than the results from flows with constant free stream speeds and buoyancy frequencies. The matter of investigating flows with vertical variations in the free stream density and velocity distributions thus seem rich areas for further investigation. Hill (2000) demonstrated how specified, stable, nonlinear density profiles can be established in a laboratory tank. The system involves a modification of the commonly used two-tank method in that solutions of the unsteady conservation equations for mass and salt are employed to determine the flow rates necessary for establishing a specified density profile. Hill, using computer-controlled peristaltic pumps, shows the viability of his approach by producing hyperbolic tangent and double-tanh density profiles of quite good accuracy While a vertical shear can be established in a non-rotating, stratified fluid by driving the surface of a clindrical test cell by a rotating horizontal circular disc (this leads to a linear shear for a linearly stratified fluid), the matter of the establishment of specified vertical velocity profiles in, say, a long straight test section is desirable; such techniques are currently not avaliable.

4.3

Order Unity Rossby Number Flows; Complex Topography

While most laboratory studies of topographic effects in stratified flows involve single topographic features, it is clear that most bathymetric features in the atmosphere and oceans cannot be considered as isolated. In the following we consider the flow over either an isolated topography or envelope of a group of hills of characteristic width D and height h. For the complex topography case (i.e., group of hills), the characteristic widths, heights and spacing of the individual hills forming the envelope will also be parameters of the problem. If one considers typical background currents, U, buoyancy frequencies, N, and Coriolis parameters, f, for the oceans and atmosphere (see Table 1), and assumes mountain heights of 200 m and 1,000 m, respectively, one can write the pertinent dynamical parameters in terms of the characteristic width of the topography, D; i.e. F = 0.1, 0.25 and for the ocean and atmosphere, respectively. Now Coriolis effects surely come into play when the Rossby number is of order one or smaller, thus implying that background rotation is important for mountains of the order of 1 km and 50 km or greater for the ocean or atmosphere, repectively. Because our interest is typically to understand the

217 motion fields on these scales, problems involving both stratification and rotation are fundmental. Essentially no laboratory work has been done on such mesoscale problems in which Ro ~ O(1) and F> 1. This fundamental finding that there is a mechanism for having disturbances in stratified flows inducing motions on much larger length scales is important and should be carefully tested in the laboratory. This upscale energy transfer mechanism was first discussed by Merkine (1975).

4.4

Vertical motion of a body

Most of the investigations on the problem of obstacles moving through stratified fluids have been focused on their motion in the horizontal direction. Obstacles exhibiting motion in vertical direction will produce internal wave fields if the characteristic frequency of the motion is smaller than the buoyancy frequency; this problem is less explored especially the interaction of the obstacle and flow field generated by its vertical movement. Many authors have studied the wave field produced by either the steady vertical motion of an obstacle or stagnant oscillating source; there is abundant literature on this topic. Mowbray and Rarity (1967) considered experimentally internal waves produced in linearly stratified fluids by the steady vertical motion of a sphere. The Toepler-Schlieren system has been employed to obtain a measurement of the phase configuration and to demonstrate agreement between linear theory and experimental results. Liu et al. (1989) considered the interactions of the internal wave field produced by an oscillatory disturbance in a linearly stratified fluid and horizontal

218 shear. It has been demonstrated that the St. Andrew’s cross-wave is modified by a horizontal shear above the level of the source. Although the study of the vertical motion of an obstacle in linearly stratified fluids with constant speed or oscillating with constant frequency is a good way to approach the problem of interactions between obstacle and stratification, it usually cannot be applied successfully in natural and engineering situations. The typical situation for natural or engineering flows would be the motion of unpropelled bodies under the action of gravity. A typical example would be settling of particulate matter (PM) in the atmosphere and in the oceans, or the rising of plumes. Unpropelled bodies moving vertically under the influence of gravity change their speeds accordingly to changes of the buoyancy force and other forces resulting from the effects of stratification. Such motions may produce non-monochromatic wave fields, thus making the physics of the problem even more complex. Nikiforovich and Dudchak (1992) studied the dynamics of a thermally conducting sphere settling through a thermally stratified fluid. The energy radiated by internal waves can exert additional drag on the body, as was shown by Warren (1960) in a study dealing with the uniform vertical motion of a body in linearly stratified fluid. Larsen (1969) has treated finiteamplitude oscillations of a sphere at its neutrally buoyant level theoretically. Another aspect of the problem is related to the ‘drift’ and its possible interaction with stratification. Darwin (1953), who considered the motion of a solid body through an ideal, homogeneous, incompressible fluid, delving into actual trajectories of fluid particles, has introduced the concept of ‘drift’. It is defined as the deformation of a material surface initially perpendicular to the direction of motion (the isopycnal lines in the case of vertical motion through stratified fluid). Interest on this topic has been resuscitated recently and for the first time studied in the context of stratification by Eames and Hunt (1997). The authors considered theoretically and numerically the motion of a body in a weak density gradient without buoyancy effects. The modification of the drift by stratification and effects that this modification has on a settling sphere has been demonstrated in an experimental study of Srdic-Mitrovic et al. (1999). It has been demonstrated that a sphere settling under gravity through a density interface of miscible fluids, experiences additional drag forces due to the buoyancy force acting of the fluid carried by the sphere from the upper layer; the effect has been found significant for moderate Reynolds numbers. Recently the motion of the descending sphere in linearly stratified fluid has been studied numerically by Hanazaki and Torres (2000) who demonstrated increased drag on a sphere, at low Froude numbers. To our knowledge, this effect has never been examined experimentally for the vertical motion of a body in infinite linearly stratified fluids.

219 To conclude the effect of stratification on an obstacle moving in the vertical direction is an area of research not very well understood, and deserves further investigation. Acknowledgements: The authors wish to thank the National Science Foundation (Grant OCE96-17639) and the Office of Naval Research (Grants N00014-1-9543 and N00014-99-1-0344) for support of their work.

REFERENCES: Baines, P.G., 1977: Upstream influence and Long's model in stratified flows. J. Fluid Mech. 82, 147-159. Baines, P.G., 1979: Observations of stratified flow over two-dimensional obstacles in fluid of finite depth. Tellus, 31, 351-371. Baines, P.G., 1979: Upstream blocking and airflow over mountains. Ann. Rev. Fluid Mech., 19, 75-97. Baines, P. G., 1995: Topographic Effects in Stratified Flows. Cambridge, U.K. Cambridge Univ. Press, 482 pp. Baines, P. G., and Davis, P. A. 1980: “Laboratory studies of topographic effects in rotating and/or stratified fluids”. In Orographic Effects in Planetary Flows, GARP Publ. Ser. 23, ed. R. Hide, and P. W. White, pp. 235-293. Geneva: World Meteorol. Org. Baines, P. G., and Hoinka, K. P. 1985: Stratified flow over two-dimensional topography in fluid of infinite depth: a laboratory simulation. J. Atmos. Sci., 43, 1614-1630. Baker, D. J., 1956: A technique for the precise measurement of small fluid velocities. J. Fluid Mech 26, 573-575. Boyer, D.L., P.A. Davies, H.J.S. Fernando and X. Zhang, 1989: Linearly stratified flow past a horizontal circular cylinder. Phil. Trans. Roy. Soc., Lond., 328, 501-528. Boyer, D.L. and P. A. Davies, 2000: Laboratory Studies of Orographic Effects in Rotating and Stratified Flows. Ann. Rev. Fluid Mech., 32, 165-201. Boyer, D.L. and Tao, L., 1987: Acceleration effects for linearly stratified flow over long ridges. Meteorol. Atmos. Phys. 37, 271-296. Brighton, P.W.M., 1978: Strongly stratified flow over three-dimensional obstacles. Q.J.R. Met. Soc., 104, 289-307. Browand, F.K. and C.D. Winant, 1972: Blocking ahead of a cylinder moving in a stratified fluid: An experiment. Geophys. Fluid Dyn, 4, 29-53. Castro, I.P., 1987: A note on lee wave structures in stratified flow over three-dimensional obstacles. Telleus 39A, 72-81. Castro, I.P., W.H. Snyder and G.L. Marsh , 1983: Stratified flow over three-dimensional ridges. J. Fluid Mech. 135, 261-282. Chashechkin, Y.D., 1999: Schlieren visualization of a stratified flow around a cylinder. J. of Visualiza., 1, 345-354. Chashechkin, Y.D., E.V. Gumennik and E.Y. Sysoeva, 1995: Transformation of a density field by a three-dimensional body moving in a continuously stratified fluid. J. App. Mech. and Tech. Phys., 36, 19-29.

220 Chashechkin, Y.D. and Mitkin, V. V., 1998: High-gradient interfaces in continuously stratified fluid in the field of two-dimensional attached internal waves. Doklady-Physics, 43, 636-640. Chen, R-r., Berman, N.S., Boyer, D.L. and Fernando, H.J.S., “Physical Model of Nocturnal Drainage Flow in Complex Terrain,” In Contributions to Atmospheric Physics, 72(3), 219242, 1999. Chopra, K. P. and Hubert, L. F., 1965: Mesoscale Eddies in Wake of Islands. J. Atm. Sci. 22, 652-657. Dahm, W. J. A., Lester, K. S. and Tacina, K. M., 1996: Four-Dimensional Measurements of Vector Fields in Turbulent Flows. AIAA Fluid Dynamics Conference, New Orleans, LA. Dalziel, S. B., 1993: “Decay of rotating turbulence: some particle tracking experiments”. In F.T.H. Nieuwstadt Ed., Flow Visualisation and Image Analysis. Kluwer, Dordrecht. Dalziel, S. B., 2000: “Synthetic Schlieren measurements of internal waves generated by oscillating a square cylinder”. In Proc. of the Fifth International Symposium on Stratified Flows, Vancouver, Canada. Darwin, C. 1953: Note on hydrodynamics. Proc. Camb. Phil. Phil. Soc. 49, 342-354. Drazin, P. G., 1961: On the steady flow of the fluid of variable density past an obstacle. Tellus 13, 239-251. Eames, I. and J. C. R. Hunt, 1997: Inviscid flow around bodies moving in weak density gradients without buoyancy effects. J. Fluid Mech., 353, 331-355. Etling, D., 2000: “Gravity waves and vortices in the wake of large islands”. In Proc. of the Fifth International Symposium on Stratified Flows, Vancouver, Canada. Eriksen, C., 1991: Observations of amplitude flows atop a large seamount. J. Geophys. Res., 96(C8), 15,227-15,236. Fincham, A. M. and Spedding, G. R., 1997: Low cost, high resolution DPIV for measurement of turbulent fluid flow. Exp. Fluids, 23, 449-462. Gilreath, H. E. and Brandt, A. 1985: Experiments on the generation of internal waves in a stratified fluid. AIAA J. 23, 693-700. Greenslade, M. D., 1992: Strongly stratified airflow over and around mountains. Ph.D. thesis, University of Leeds Hanazaki, H. 1988: A numerical study of three-dimensional stratified flow past a sphere. J. Fluid Mech., 192, 393-419. Hanazaki, H. and Torres, C. R. 2000: “Jet and internal waves generated y descending sphere in a stratified fluid”. In Proc. of the Fifth International Symposium on Stratified Flows, Vancouver, Canada. Head, M. 1983: The use of miniature four-electrode conductivity probes for high resolution measurements of turbulent density or temperature variation in salt-stratified water flows, Ph.D. Thesis, University of San Diego. Hewett, T.A., J.A. Fay and D.P. Hoult, 1971: Laboratory experiments on the motion of a buoyant plume in a stratified flow. Atmos. Environ, 5, 767-789. Hill, D. F., 2000: “Making waves... in nonlinearly stratified fluids”. In Proc. of the Fifth International Symposium on Stratified Flows, Vancouver, Canada. Honji, H. and K. Masafumi, 1984: Wakes of a circular cylinder in stratified fluids. Reps.Res. Inst. for App. Mech., Vol XXXI, No. 98, 89-95. Hopfinger, E. J. Flor, J. B. and Chomaz, J. M. and Bonneton P. 1991: Internal waves generated by a moving sphere and its wake in a stratified fluid. Exp. Fluids, 11(4) 255261. Hopfinger, E. J., 1987: Turbulence in stratified fluids – A Review. J. Geophys. Res., 92:C(5), 5287-5303.

221 Hunt, J.C.R., Y. Feng, P.F. Linden, M.D. Greenslade, 1997: Low-Froude-number stable flows past mountains. Il Nuovo Cimento. 20 C, 261-272. Hunt, J.C.R. and W.H. Snyder, 1980: Experiments on stably and neutrally stratified flow over a model three-dimensional hill. J. Fluid Mech., 96, 671-704. Hunt, J.C.R. and H. J. S. Fernando, 1996: Separated flow round bluff obstacles at low Froude number; vortex shedding and estimates of drag. IMA Stably Stratified Flow Conference, Dundee, UK. Huq, P. and Britter, R. E., 1995: Mixing due to grid-generated turbulence of a two layer scalar profile. J. Fluid Mech., 285, 17-40. Huq, P. and Britter, R. E., 1995: Evolution and mixing in a 2-layer stably stratified fluid. J. Fluid Mech., 285, 41-67. Larsen, L. H., 1969: Oscillations of a neutrally buoyant sphere in a stratified fluid. Deep Sea Res. 16, 587-603. Lin, Q., W.R. Lindberg, D.L. Boyer and H.J.S. Fernando, 1992: Stratified flow pas a sphere. J. Fluid Mech., 240, 315-354. Lin, J. T. and Pao, Y. H., 1979: Wakes in Stratified Fluids. Ann. Rev. Fluid Mech., 11, 317338. Liu, R., Nicolaou, D., and T. N. Stevenson, 1989: Waves from an oscillatory disturbance in a stratified shear flow. J. Fluid Mech. 219, 609-619. Lofquist, K. and Purtell, P. 1984: Drag on a sphere moving horizontally through a stratified liquid. J. Fluid Mech., 148, 271-284. Long, R.R., 1954: Some aspects of the flow of stratified fluids, II. Experiments with two-fluid systems. Tellus, 6, 97-114. Long, R.R., 1955: Some aspects of the flow of stratified fluids, III. Continuous density gradients. Tellus, 7, 341-357. Long, R.R., 1972: Finite amplitude disturbances in the flow of inviscid rotating and stratified fluids over obstacles. Ann. Rev. Fluid Mech., 4, 69-92. Mason, P. J. 1977: Forces on spheres moving horizontally in rotating stratified fluid. Geophys. Astrophys. Fluid Dyn. 8, 137-154. Martin, S. and R.R. Long, 1968: The slow motion of a flat plate in a viscous stratified fluid. J. Fluid Mech. 31, 669-688. Maxworthy, and T. Browand, F. K., 1974: Experiments in rotating and stratified flows: oceanographic applications. Ann. Rev. Fluid Mech., 6, 273-305. Merkine, L. O., 1975: Steady finite-amplitude baroclinic flow over long topography in a rotating stratified atmosphere. J. Atmos Sci., 32(10), 1881-1893. Merzkirch W. 1974: Flow visualisation. Academic Press, New York.. Mitkin, V.V. and Y.D. Chashechkin, 1998: Recurrence and reconnection effect in an attached two-dimensional internal wave field. Fluid Dyn., 33, 753-760. Mitkin, V.V., V.E. Prokhoov and Y.D. Chashechkin , 1998: Investigation of the variability of the structure of a stratified wake behind a horizontal cylinder using optical and acoustic methods. Fluid Dyn. 33, 303-312. Mowbray, D. E. and B. S. H. Rarity, 1966: The internal wave pattern produced by a sphere moving vertically in a density stratified liquid. J. Fluid Mech., 30, 489-495. Newley, T. M. J., Pearson, H. J. and Hunt, J. C. R. 1991: Stably stratified rotating flow through a group of obstacles. Geophys, Astrophys. Fluid Dyn., 58, 147-171. Nikiforovich, Y. I. And Dudchak, K. P., 1992: Fall of a rigid, thermally-conducting sphere through thermally stratified fluid. Fluid Mech. Res., 21, 7-14. Odell, G. and Kovasznay, L. S. G. 1971: A new type of water channel with density stratification. J. Fluid Mech., 50, 535-543.

222 Oster, G. and Yammamoto, M. 1963: Density gradient techniques. Chem Rev., 63, 257-268. Pérenne, N. Lavelle, J.W., Smith IV, D.C. and Boyer, D.L., Impulsively-Started Flow in a Submarine Canyon: Comparison of Results from Laboratory and Numerical Models, Appl. Math. Modeling, under review. Pérenne, N, Haidvogel, D. and Boyer, D.L., Laboratory-Numerical Comparisons of Flow over a Coastal Canyon, J. Atmos. Tech., in press. Raffel, M., Willert, C. and Kompenhans, J., 1998: Particle Image Velocimetry. Springer. Riley, J. J. and Lelong, M. P., 2000: Fluid motions in the presence of strong stable stratification. Ann. Rev. Fluid Mech., 32: 613-657. Roden, G. I., 1987: “Effects of seamounts and seamount chains on ocean circulation and thermohaline structure”. In Seamounts, Islands and Atolls, ed. B. Keating, P. Frye, R. Batiza, G. Boehlert, pp. 335-54. Washington, DC: Am. Geophys. Union. Scotti, R.S. and G.M. Corcos, 1972: An experiment on the stability of small disturbances in a stratified free shear layer. J. Fluid Mech., 52, 499-528. Schwerdtfeger W., 1984: Weather and Climate of the Antarctic. Amsterdam: Elsevier, 261 pp. Schraub, F.A., S.J. Kline, J. Henry, P.W. Runstadler and A. Littell, 1965: Use of hydrogen bubbles for quantitative determination of time-dependent velocity fields. Trans. ASME J. Basic Eng., D, 87, 429-444. Sheppard, P. A. 1956: Airflow over mountains. Q. J. R. Met. Soc., 83, 528-529. Snyder, W.H., R.S. Thompson, R.S. Eskridge, R.E. Lawson, I.P. Castro, J.T. Lee, J.C.R. Hunt and Y. Ogawa, 1985: The structure of strongly stratified flow over hills: dividingstreamline concept. J. Fluid Mech., 152, 249-288. Spedding, G. R., 1997: The evolution of initially turbulent bluff-body wakes at high internal Froude number. J. Fluid Mech.,337,283-301. Srdic-Mitrovic, A. N., Mohamed, N. A. and H. J. S. Fernando, 1999: Gravitational settling of particles through density interfaces. J. Fluid Mech., 381, 175-198. Sysoeva, E. Y. and Chashechkin, Y. D. 1988: Spatial structure of a wake behind a sphere in a stratified liquid. J. Appl. Mech. Tech. Phys., 5, 655-660. Tsuchiya, K., 1969: The Clouds with the Shape of Karman Vortex Street in the wake of Cheju Island, Korea. J. Meteor. Soc. Japan, 47, 457-464. Voropayev, S.I., Gavrilin, B. L., and Zhur, V. V., 1983: The limitations due to light refraction on the holographic-interferometry of stratified fluids. Okeanologiya, 23(2) 348-350. Voropayev, S. I., Smirnov, S. A., Filippov, I. A. and Boyer, D. L., 2000: “Large eddies and vortex streets behind moving jets in stratified fluid”. In Proc. of the Fifth International Symposium on Stratified Flows, Vancouver, Canada. Warren, F. M. G., 1960: Wave resistance to vertical motion in stratified fluid. J. Fluid Mech. 7, 209-229. Wei, S.N., T.W. Kao and H.P Pao, 1975 Experimental study of upstream influence in the twodimensional flow of a stratified fluid over an obstacle. Geophys. Fluid Dyn. 6, 315-336.

Chapter 8 ELEMENTS OF INSTABILITY THEORY FOR ENVIRONMENTAL FLOWS Larry G. Redekopp University of Southern California

Abstract

1.

An introduction to instability theory is provided, together with application of the theory to a number of idealized flows. Essential issues pertaining to the distinction between, and the application of, spatial and temporal approaches are given particular attention. The selection of specific examples is motivated by both their relevance to environmental contexts and their pedagogical value. Emphasis is placed primarily on the instability of density stratified shear flows, but the chapter closes with a brief consideration of the overturning (Rayleigh-Taylor) instability of statically-unstable interfaces when diffusion of the stratifying agent is suppressed.

INTRODUCTION

Among the abundance of hydrodynamic flows arising in nature or generated in technological applications, almost all are either benefited by or plagued by some type of instability. Instabilities underlie the appearance of many intriguing and novel flow patterns, the selection of prominent space and time scales inherent to most unsteady dynamics, the enhancement of mixing and transport of fluid properties and particulates, and the rise in the drag force on a body as its speed relative to its environs increases. As a consequence, studies related to flow instability have wide-ranging relevance, encompass complex physics, and frequently involve the application of non-trivial mathematics. Spurred by our incessant natural curiosity about flow physics in general, and motivated by a need to understand, predict and control many flow processes, instability theory has been an active and fertile area of

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research for well over a century. In fact, questions concerning hydrodynamic instability issues provided the impetus for many developments in mathematics, and played a major role in the emergence of the discipline of hydrodynamics. The accomplishments to date in this subject area are both multitudinal and profound. However, when progress is measured relative to the ultimate goal of attaining predictive tools for specifying (reasonably) precise conditions for the onset of, and the ensuing characteristics of, the complex dynamics often termed loosely as ‘turbulence’, the accomplishments are at best only modest. Certainly the state of either our physical understanding or our predictive capability is more primitive in some flow contexts than in others. Not unexpectedly, the most significant advances have occurred where a long standing, synergistic interplay has been possible between laboratory and theoretical investigations. Nevertheless, progress in the theory of hydrodynamic instability has been largely limited to the specification of conditions under which the amplification of infinitesimal disturbances is expected in a well-defined base state, and the physical mechanisms whereby an initial transfer of energy occurs from the base state to the disturbance field. The present article deals almost exclusively with these issues. When approaching a subject having an extensive history, and where the physical and mathematical foundations are well established, one expects that excellent monographs already exist setting forth both the theoretical essentials and a discussion of results pertaining to various applications. This is particularly true in the case of hydrodynamic instability. In the opinion of this author, the primary references providing the important foundations for the work described here are: Chandrasekhar (1961); Craik (1985); Drazin & Howard (1966); Drazin & Reid (1981); and Huerre & Rossi (1998). It is not the intention to duplicate here foundational material contained in the primary references. The present chapter should be viewed as being supplementary to these definitive contributions, and covering only a narrow focus of limited extent into aspects of the instability of environmental flows. It is certainly worth noting in regard to this particular focus that Turner (1973) presents a discussion of instability and mixing issues in environmental flows that is far broader in scope and more comprehensive in detail than is conceived or attempted here. The instability of environmental flows is addressed exclusively in the context of highly-idealized models. The models are proposed as extractions of either isolated or embedded flow features appearing on different scales in gravity currents, exchange or wedge flows, intrusive flows, wave and wind generated shear flows, etc. Some of these flows are described, analyzed and discussed in other chapters in this volume. The appli-

225 cability of specific models to some of the wide range of environmental flows encountered in nature will only be alluded to in passing at various points in the presentation. A closer description of the relevance of different models, and the comparison of observed and predicted dynamics in specific contexts, must be reserved for a different forum. Nevertheless, the monograph by Turner (1973) already provides an excellent resource toward this goal. The presentation that follows is necessarily limited in space and scope and will allow only discussion of a restricted set of instability issues. As a consequence, we consider exclusively linear instability theory, and its application primarily to stratified shear flows. Furthermore, the presentation focuses on elemental considerations with a view toward providing a pedagogical introduction to the subject, using the context of environmental flows as illustrative examples of fundamental topics. Some familiarity with the introductory material in the primary references noted above is helpful, but for the most part, the text is intended to stand alone. Also, only sporadic (and selective!) pointers to a few of the more relevant references are included, either for historical context or to guide the reader to recent pertinent research. Connections of the limited topics discussed here with the extensive literature is not attempted.

2.

PRELIMINARIES

2.1

Basic Concepts

In general, instability considerations require the definition of a base state plus the prescription of a class of disturbances against which the stability of the base state is examined. The structural features of the base state are defined in terms of a set of control parameters P. The disturbance is primarily characterized by some energy norm relative to a measure of the energy of the base state. The disturbance can also be classified in terms of its spectral content, but the term ‘stable’ usually implies stability in the presence of a broad spectrum of disturbances. Then, given the base state specified in terms of the control parameter(s) P and the (essentially unrestricted) class of disturbances specified by an energy level a primary objective of instability analyses is to determine those domains (if any) in the space where the prescribed base flow is stable. Because of analytical and computational limitations, this ultimate goal is necessarily compromised in most applications. The compromises usually made in practice fall into either of two classes. First, an ‘energy’ formulation of the fully-nonlinear, primitive equations for the disturbance field is employed to establish necessary conditions which ensure

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the global stability of the flow. The approximation procedures are almost always extremely conservative, yielding a lower-bound estimate for the unconditional stability of the base state irrespective of the disturbance energy level Since the estimated bound is frequently considerably below the true global stability bound and since the analysis generally fails to provide much insight to the nature of the unstable dynamics when the control parameter is marginally greater than the results of this approach are frequently of quite limited practical value. The relation between these different values of the control parameter is shown schematically in Figure 8.1.

A second and more widely pursued compromise employs the linearized equations of motion, linearized about the postulated base state, and seeks sufficient conditions for instability. The pivotal question being addressed in linear instability analysis is: Under what (parameter) conditions is the postulated base state unstable with respect to all possible infinitesimal disturbances? That is, one seeks sufficient conditions under which a vanishingly small disturbance, either spontaneous or noisesustained, will cause the base flow to tend toward an altered state, either steady or unsteady. The outcome of this approach is a critical value of the control parameter beyond which an infinitesimal disturbance will always be amplified. It also provides insight to the spectral content and the structural shape of those disturbances which are most susceptible to amplification. What this linear approach cannot provide is insight into the nature of ensuing dynamics for That is, whether sta.ble, finite amplitude states exist; what hierarchical mechanisms of instability might enter; etc. These latter issues comprise some of the questions addressed in the active field of weakly nonlinear instability studies.

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Linear instability analyses examine the tendency for a prescribed base flow state to be altered either spontaneously or by the injection (or superposition) of infinitesimal disturbances. The base flow is generally supposed to be quasi-parallel in which inhomogeneous distributions (profiles) of vorticity, density stratification, etc. vary rapidly in a cross-flow (or wave guide) direction. In idealized models employed for instability analyses (and frequently in reality) a cross-flow, modal or wave-guide direction is available across which gradients of the profiles of the base state inhomogeneity are steep in comparison to their gradients in the orthogonal directions, directions which define the propagation space for disturbances. There are, of course, many practical situations where the steep regions are embedded within an ambient medium which supports disturbances having a propagation component in a direction normal to these steep regions. In this case no strict wave guide exists. However, even in these cases the origin of the energy release in the instability process is still located in the steep gradient regions. With the distinction between the modal and propagation spaces in hand, a separable problem can be defined in terms of a (complex) modal amplitude function and a phase function The disturbance stream function

for example, can then be expressed as1

The spatial gradient of the phase function defines the wave vector in the propagation space and is the frequency of propagating disturbances. In this case, and in the presentation which follows, the coordinate direction is aligned with the cross-flow (wave-guide) direction, and the propagation space is

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To provide a physical context for this discussion we consider the vertically-sheared, horizontal flow of an inviscid density stratified fluid as illustrated in Figure 8.2. The linearized equation for the disturbance field for this flow, within the confines of the Boussinesq approximation, is

In writing this equation we suppose that the base state consists of a parallel, uni-directional shear flow directed along the and that the density-stratification (represented by the Brunt-Väisälä profile varies only in the cross-flow direction, which is aligned anti-parallel to the gravitational body force. denotes the three-space-dimensional Laplacian and is the horizontal Laplacian (i.e., in the plane of the propagation directions). It is important to note that the presence of velocity shear breaks the reflectional symmetry of the underlying system. This fact points immediately to the important distinction between downstream and upstream development of disturbances in shear flow instability, a distinction that will be elaborated further in what follows. Since space considerations limit the scope of the presentation here, we choose to restrict our focus to the instability of the base state flow to plane wave disturbances in the reduced propagation space It is true of course that instability may set in through other classes of disturbances. For example, it may occur via oblique or three-dimensional disturbances, although a Squire’s theorem stating that the most susceptible disturbances are in fact two-dimensional can be proved for a number of shear flows. Instability may also arise through non-normal (continuous spectrum) disturbances exhibiting algebraic growth. Neither the analytical framework for nor specific examples of such classes of unstable disturbances will be discussed. Hence, the phase function for the strictly-uniform 2 base flow is taken in the form where are the frequency-wave number pair for a plane wave disturbance (single Fourier mode). The modal amplitude function is then defined by the Taylor-Goldstein equation

subject to appropriate boundary conditions at the upper and lower boundaries of the wave guide. The latter equation is written in terms of the phase speed and can be interpreted as an eigenvalue problem with discrete eigenvalue depending on the wave number as

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parameter (plus other control parameter(s) P specifying the base state). The solution of (8.3) yields a dispersion relation

defining the properties of the solution in the propagation space, and an eigenfunction defining the physical structure of the solution in the wave guide. The analytical properties of the dispersion relation are at the heart of instability analyses, and deserve further discussion. Although these analytical properties are mathematical, involving complex analysis, they have profound effects in physical application and interpretation. One of the primary purposes of this presentation is to point out some of these issues in a few elementary models of environmental flows. Once a phase function is assumed, the dispersion relation allows multiple mathematical interpretations of the nature of the spatio-temporal structure of the disturbance field, interpretations which have important physical consequences. First, one can postulate that the wave number is real and the frequency is complex. In this case we suppose that a disturbance with the spatial structure given by a single Fourier mode is imposed along the whole line of the propagation space, and the temporal development of this periodic disturbance is sought. That is,

and one seeks, in particular, the band of wave numbers (and phase speed for which temporal growth occurs The disturbance structure for temporal instability is illustrated in Figure 8.3a. Alternatively, one can consider the signaling problem whereby a temporally periodic excitation of real frequency is injected at a fixed point say) whereupon one asks the following question: Does this excitation grow spatially as it propagates away from the source location? In this case the phase function representation must be interpreted as

That is, the wave number is complex and one seeks the spatial growth rate (assuming development for and the wavelength of disturbances of prescribed frequency The disturbance structure in this case is illustrated in Figure 8.3b. Now, the alert reader will immediately note the hypothesis that disturbances “propagate away from the source” and that the assumed development of the disturbance is in “the downstream direction”. The question as to whether the disturbance propagates away from the

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source is definitely not peripheral, and points to the fundamental role of group velocity in instability considerations (just as it does in the case of neutral waves). This issue will be addressed shortly. However, before moving on to such issues we note that the existence of unstable disturbances for implies, for the same spatial branch of the dispersion relation, decaying disturbances for (and vice versa for Furthermore, causality considerations for the thought experiment consisting of a prescribed frequency injected at a fixed spatial position requires that unstable (damped) disturbances grow (decay) as distance from the source increases. This points to the important issue of the existence of different spatial branches for and upstream (downstream) growth. The issues of group velocity and distinct spatial branches are implicit in the dispersion relation (8.4). However, the thought experiment implied by the hypothesized disturbance structure (8.1), namely a single Fourier disturbance encompassing the entire propagation space is not the most convenient one to elucidate these issues. Rather, it is more convenient to consider a localized injection of disturbances with a broad spectrum at a fixed position and time; that is, to construct the Green’s function for the initial-boundary value problem for the linearized disturbance field for a given base state. The spatio-temporal approach to instability will be illustrated in the context of a model problem in the next section, but some aspects can be revealed here by consideration of the group velocity

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and by reference to the wave packet structures illustrated in Figure 8.4.

First of all, the group velocity has physical relevance only if it is purely real. Hence, the meaningful values of are those where (8.7) is evaluated at complex values of (and generally, therefore, of also) for which the temporal development of the disturbance is either amplified, neutral, or damped in a frame moving with the (real) speed Now, if all unstable disturbances originating from a localized injection source (i.e., all disturbances that experience temporal amplification in their respective frame moving with have non-zero group speeds, there will be no in situ growth at the injection source. That is, all unstable disturbances propagate away from the source and no residual amplified disturbances will remain at any fixed spatial position as after the source is turned off. Unstable systems fulfilling these conditions are termed convectively unstable. If the source contains a sustained oscilla-

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tory excitation, the system will behave as a spatial amplifier in which the linear response is characterized by the spatial amplification rate. The wave guide in this case behaves as a spatial amplifier and the unstable dynamics (response) will be quite susceptible to ambient noise propagating into or advecting through the wave guide. Furthermore, the behavior of the disturbance field in the wave guide upstream or downstream of the source will require explicit distinction between the upstream and downstream spatial branches of the dispersion relation in order to satisfy causality requirements (i.e., the requirement that all disturbances originate from the source and none are propagating toward the source from plus or minus infinity). Group velocity considerations clearly permit another possible scenario. If any amplified disturbance exists with a vanishing group velocity, then the injection of energy at a localized source will lead to in situ unbounded growth (on a linear basis) as Furthermore, if there are neighboring (complex) wave numbers with group velocities greater than and less than zero, amplification of disturbances originating from the source will progressively invade both the domains upstream and downstream of the source until the flow in the entire wave guide is affected. Unstable systems fulfilling these conditions are termed absolutely unstable, and there is an unambiguous temporal amplification at every fixed spatial position as Clearly, temporal instability analyses are relevant to absolutely unstable systems, and absolutely unstable systems exhibit spontaneous, intrinsic dynamics because the in situ temporal growth will rapidly overwhelm any injected control disturbance.

2.2

An Illustrative Example

The instability issues discussed in the previous section are perhaps best clarified by considering a concrete, simple example. The example we select is the linear version of the familiar Ginzburg-Landau equation which we express in the form

This equation for the complex amplitude function can be derived as part of a rational approximation of the dynamics of marginal disturbances in dissipative systems. As such, the coefficient of the diffusive/dispersive term is normally complex For causality we require (i.e., the diffusion coefficient is positive) to avoid unbounded growth of disturbances as the wave length decreases toward zero. Also, the equation is invariant under the transformation so we consider only upstream and downstream growth relative to the system with We assume further that

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the advection coefficient U and the growth rate (alt., the criticality or control parameter) are both real. In the derivation of (8.8) in the context of a flow in a wave guide, the cross-stream modal structure of the disturbance field has been eliminated and the equation describes the evolution solely in the propagation space The derivation of amplitude equations of this sort will be illustrated in several specific cases later. Equation (8.8) is a convenient model in that it captures the minimum structure necessary to reveal the instability issues of interest. That is, it possesses a single temporal mode (the equation is first-order in time) which is dispersive (the equation is second order in space so the dispersion ‘tensor’ is non-zero); it has two spatial instability branches of the single temporal mode; and it has an explicit control parameter measuring the degree of super-criticality of the system. The dispersion relation conjugate to the linear, constant coefficient PDE is

where

The diffusive/dispersive coefficient b is related to the (complex) curvature of the dispersion relation and the two forms given in (8.9) are used interchangeably in expressions that follow. The complex parameter pair define the conditions for vanishing of the group velocity

and are referred to as the absolute frequency and wave number. The temporal growth rate for this wave system is showing that the maximum temporal growth rate is irrespective of the convection parameter U, and that the system is unstable in a limited band of wave numbers centered around for However, examination of the expression for shows that the nature of the instability is such that the system at fixed value of the control parameter is absolutely unstable for low values of U and convectively unstable for high values of U In what follows the convective-absolute velocity scale which has physical meaning when is used as a substitute measure for the control parameter.

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For the purpose of illustrating the spatial branches, and the spatial growth rate when the instability is of convective type, we present the relation for the spatial branches in the special case when Equation (8.9) can then be inverted and expressed in the form

This expression can be used directly to compute the spatial growth rate for when the flow is convectively unstable and the frequency is purely real, provided the proper distinction is made between upstream and downstream branches. What should be noted especially in (8.11) is the fact that the spatial growth rate, in contrast with the temporal growth, will vary with the frame of reference (i.e., U) as well as the criticality parameter (alt., the value of This is a fundamental quality of convectively unstable flows and points out the fallacy of using temporal growth rates to describe instability characteristics in such flows. There is an additional aspect regarding convectively unstable flows not revealed here in the simplified context of the amplitude equation (8.8). Namely, the nature of the modal structure functions (cf., as defined in equation (8.3), for example) can be quite different for absolutely and convectively unstable conditions, differences which are often manifest through significant changes in the distribution and peak levels of such quantities as the kinetic energy density, buoyancy flux, and Reynolds stresses across the wave guide. As noted following equation (8.11), proper specification of the spatial branches of the dispersion relation is needed to compute spatial growth rates that are relevant for This issue is readily clarified through considerations underlying the construction of the causal Green’s function for the linear instability problem. To this end we consider the forced version of (8.8) written as

where represents a source function which is compact in space and is non-zero only for times greater than zero By use of Fourier methods, the solution can be expressed in the integral form

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The double transform of is denoted by the function is the dispersion relation

and

The integration contour F in the is nominally along the real axis and, to ensure causality (i.e., that all disturbances emanate from the source), the contour L must be positioned above all singularities in the (so that the solution for is A = 0). These contours are illustrated in Figure 8.5, and the images of the contours are shown in respective planes for a particular set of parameters.

The disposition of the spatial branches can be ascertained by following the images of the L contour. When the L contour is raised sufficiently high in the upper half of the the images will be separated into the (upper, lower) halves of the respectively. The branch(es) in the upper half plane applies for and the branch(es) in the lower half plane applies for Now, as the level of the contour L is lowered toward the real axis in the the branches deform and may cross the real axis of the If so, the path of the contour F must necessarily be deformed to respect the upstream/downstream applicability of the branches. For example, if descends into the lower half of the the down-

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stream branch will acquire a positive spatial growth rate for a range of frequencies. It should be apparent to the reader that the separate branches (images of the L contour) may collide (pinch) at some point as the height of L is lowered. At this point the deformation of the contour F between the two branches becomes pinned. This pinch point corresponds to a pair where the group velocity vanishes, and is a branch point of the dispersion relation. If the motion is damped along the limiting ray and the instability is of convective type. Spatial growth rates for real values of can be calculated in this case. However, if the contour L cannot be lowered down to the real axis in the and no spatial growth rate calculation is possible. The motion along the limiting ray is amplified and the instability in this case is of absolute type. The advantage of the simple model defined by (8.12) is that the Green’s function can be evaluated exactly. Supposing that the solution is

where

It is clear that the growth rate is positive and the flow is unstable when Also, when U decreases toward the ray which remains fixed at the source location for all time is critical, with incipient temporal growth. This condition marks the onset of absolute instability, which ensues for On the other hand, all rays with proceed downstream from the source when in which case the flow is convectively unstable. The temporal amplification rate vanishes as at any fixed in this latter case, even though the system is unstable This points convincingly to the failure of temporal instability theory when the system is convectively unstable.

2.3

Shear Flow Fundamentals

Before analyzing some prototypical flows that reveal important elements pertaining to the instability of stratified shear flows, we con-

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sider several celebrated results implicit to the eigenvalue problem defined by the Taylor-Goldstein equation (8.3) where the continuously differentiate functions and are defined in and the disturbance field satisfies homogeneous boundary conditions First, consider the limiting case where the density stratification vanishes The Taylor-Goldstein equation then collapses to the Rayleigh equation. Multiplying (8.3) in this case by the complex conjugate of the eigenfunction, integrating across the wave guide, and considering temporally unstable disturbances where is real and c is complex (with one obtains from the imaginary part

This expression is the basis for the Rayleigh inflection point criterion. Clearly, there must be at least one point where passes through zero, being negative and positive on opposite sides of this point, in order for the integral to vanish. Hence, it is shown that a point of inflection must be present in the mean velocity profile in order for an unstratified flow to be unstable. More precisely, as stated by Fjortoft’s theorem (cf., Drazin & Reid 1981 or Huerre & Rossi 1998), the base state must have a point where the vorticity has a local maxima within the wave guide. Next, consider temporal instability again and take (8.3) with The eigenvalue problem can be recast in terms of an auxiliary function through the transformation Multiplying the resulting equation by the complex conjugate of H and integrating across the wave guide dimension, and assuming homogeneous boundary conditions at leads to a quadratic functional (cf., Howard 1961), of which the imaginary part is

It is clearly contradictory to suppose that both everywhere across the depth of the wave guide and that the flow is unstable This consequence leads to the elegant and widely known Miles-Howard theorem. That is, everywhere in a stratified shear flow is a sufficient condition for the linear stability of that flow. This result is most often stated in terms of the gradient Richardson number with the flow being stable for

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The role of the Richardson number in determining the stability of stratified shear flows is revealed further by considering the real part of the quadratic functional. After a few manipulations one can derive the following upper bound for the temporal growth rate:

Increasing values of the Richardson number in the range implies a reduction in the growth rate of unstable waves. The Miles-Howard theorem is usually pointed to as establishing the stabilizing influence of density stratification in a shear flow. However, one needs to give careful consideration to the relation between the (vertical) length scales for the B-V profile and for the vorticity profile in establishing the relevance of the theorem in a given flow (cf., Figure 8.2). If and the scale encompasses the entire range of the scale then the Richardson number tends to infinity at the edges of the vorticity layer. In such cases the presence of a stable density stratification can act to suppress vertical motions everywhere across the shear flow and the Miles-Howard theorem is certainly relevant. By contrast, the situation is quite different if Now the gradient Richardson number will vanish at the extremities of the vorticity layer and the theorem is not applicable. Even if over a substantial portion of the wave guide, there will be no stabilizing influence to suppress vorticity waves at the edges. The same effect arises if the flow is asymmetric and the two lengths scales only partially overlap. There is another possible scenario that deserves mention in the case when for which the Miles-Howard theorem is not strictly applicable. The theorem was established under the condition where the disturbance field vanishes at both boundaries at When there exists the possibility of vertically-propagating internal waves outside the vorticity layer, waves which are not ducted along the vorticity layer. In this case radiation conditions must be imposed and the strict basis for the theorem is compromised. In flows where both the shear and the density gradient develop through diffusive processes from velocity and density “discontinuities” (jumps) of similar vertical scale, the ratio of lengths will scale with some power (1/2 for laminar flows) of the Prandtl or Schmidt number. For example, for a thermally stratified shear flow in air is only marginally smaller than and the criterion is likely to be relevant. On the other hand, for a salinity stratified shear flow in water is expected to be considerably larger than and the criterion is not applicable. Environmental shear flows fre-

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quently develop under different larger-scale force fields and one should always examine the length scales of local profiles before any observed dynamics, or lack thereof, are attributed to shear instability on the basis of the criterion. It is also true that there are other linear processes, secondary instabilities, as well as nonlinear mechanisms, that can lead to instability in stratified shear flows, but the limited scope here precludes discussion of these important, related issues.

3.

THE INSTABILITY OF STRATIFIED SHEAR FLOWS

Instability analyses of stratified shear flows are concerned with the interaction between stable density stratification (i.e., a statically-stable vertical density gradient with horizontal isopycnal surfaces) and vertical gradients in horizontal vorticity (i.e., horizontal shear flows). The existence of a stable density stratification permits propagating internal waves, and the existence of a gradient in mean vorticity permits propagating vorticity waves. The primary interaction between these fields arises from the fact that propagating gravity waves distort the isopycnal surfaces and inevitably give rise to a baroclinic generation of (primarily) horizontal vorticity, a disturbance vorticity field that adds to, or subtracts from, the local vorticity concentrations induced by vorticity waves propagating in the mean vorticity field. In the same way, propagating vorticity waves that distort the horizontal isovorticity surfaces are influenced by the ‘stiffness’ imposed on the flow by the stable stratification. There are further secondary interactions between, for example, the baroclinically-generated vorticity derived from the disturbed isopycnal field and the disturbance vorticity induced by distortions of the mean vorticity gradient. However, as the focus here is directed exclusively to the primary, linear instability characteristics in stratified shear flows, these other (interesting and important) secondary and nonlinear instability issues are left for another forum. The abbreviated presentation that follows examines only idealized flow models designed to elucidate the interaction between the two basic fields: (stable) density stratification and (stable or unstable) vorticity stratification. The elementary flow models are chosen so that analytical expressions for the eigenvalue relation can be obtained. This requires the use of ‘layer’ models of the base state, where gradients of the density or vorticity field are confined to ‘interfaces’ (i.e., delta functions). In this sense the prototypical flow models studied here are at best ‘artificial’ representations of more realistic flows with continuously differentiable profiles of base state quantities like the density field and the vorticity

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field. Nevertheless, these simplified models are known to often yield semi-quantitative results useful for forming an intuitive understanding of physical processes and for guiding or interpreting some laboratory or numerical studies. The goals of the presentation are several. First, the simple models are analyzed to reveal important physical mechanisms that provide a foundation for understanding basic processes operative in many environmental flows. Second, the prototypical model flows discussed here provide a convenient vehicle for exposing and illustrating many of the important elements in linear instability theory discussed above, and to point out the relevance of these concepts in physical contexts. In regard to important elements of instability theory, the prototypical models are selected to emphasize the following points. a) Inviscid instability derives from a resonance where the phase speeds of individual waves coalesce.3 b) The group velocity of unstable disturbances is of fundamental importance, determining the type of instability (convective or absolute) existing in any region of parameter space.

c) The type of instability occurring in any situation is essential to: i) determining how its growth in space-time is to be characterized. ii) determining whether the flow exhibits intrinsic, spontaneous dynamics or if it acts essentially as a spatial amplifier.

As stated above, the discussion is intended to give insight to the mechanism of and condition for instability. However, the determination as to whether a flow is unstable is only the first important step in the application of even linear theory. There are further issues which are vital to the interpretation and application of linear theory to any specific context. These issues are illustrated to different degrees in the discussion of the various flow models analyzed below. The analytical foundations for deriving the dispersion relations examined below are presented in Appendix A.

3.1

Class-1 Prototypical Flows

The prototypical problem revealing the destabilizing effect of velocity shear and its interaction with inhomoeneous density fields is the classical Kelvin-Helmholtz (KH) problem. This flow model was originally posited as one that could provide an estimate for the critical wind speed

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to initiate wind-generated capillary-gravity waves. In the presentation here the base flow state is generalized slightly to allow uniform vorticity in the bulk flow to couple with the vortex sheet existing along the interface separating two immiscible, homogeneous fluids of different densities. The flow model and definition of symbols are given in Figure 8.6. The dispersion relation for small amplitude disturbances superposed on the steady base state is

The parameter is the coefficient of interfacial tension for the two immiscible fluids forming the interface. The dispersion relation is written for the “interface-centered” reference frame shown in Figure 8.6 where the mean velocity immediately above and below the interface is equal in magnitude and oppositely directed.

The generalized KH model is presented here to reveal the character of wave propagation in several limits. Also, it serves to make a distinction between, in the terminology used here, first-order and second-order vortex sheets. A first-order vortex sheet is an ‘interface’ supporting a discontinuous jump in mean velocity and, therefore, possessing a non-zero circulation per unit length (i.e., vortex sheet strength) around a limiting circuit that encompasses the interface. A second-order vortex sheet is an ‘interface’ supporting a discontinuous jump in mean vorticity (i.e., a j u m p in the gradient of mean velocity), but across which the mean velocity is continuous and the circulation per unit length vanishes. In this

242 terminology, a first-order vortex sheet is comprised of the coalescence of two second-order vortex sheets.4 To reveal some of the general aspects of instability theory discussed earlier we consider a normalized form of (8.20) in the deep fluid limit All variables are normalized by the intrinsic length and time scales and where is the reduced gravity across the interface 5 and is the interface parameter (i.e., the interfacial tension divided by the average density of the two fluids). The scaled dispersion relation in this limit involves four parameters: measuring the velocity difference across the interface relative to the reference velocity the dimensionless bulk shear rates and the Atwood number measuring the density anomaly. The dimensionless dispersion relation can then be written in the form

where C is the scaled phase speed in the “interface-centered” frame and is the scaled wave number. Several limiting cases of (8.21) will be considered. Case KH – 1 Consider initially the familiar case of a first-order vortex sheet in the Boussinesq limit in the absence of any bulk vorticity In this case the phase speeds are given by

The only remaining parameter is which forms an effective control parameter for the onset of instability. As long as as there are two real solutions for C describing left- and right-going waves relative to the centered reference frame. However, for the two waves coalesce at the wave numbers where

and instability occurs for in the wave number band That is, there is a threshold velocity shear beyond which the vortex sheet is unstable, with the criticality condition These parameter conditions are shown in Figure 8.7 in reference to the

243 neutral curve defined by the bracket expression in (8.22). The reader should note that the instability depends only on This symmetry of the threshold condition is preserved even in the non-Boussinesq case, for which the phase speed relation (8.22) is

but will be altered by the presence of ambient vorticity. The Doppler shift on the left-hand side of the latter equation accounts for the difference between the “interface-centered” reference frame and the “massaveraged” reference frame.

It is important to note that instability occurs through the coalescence of two waves, that is, by a resonance. This is illustrated by the section of the neutral curve included in Figure 8.7. Resonance is always the mechanism whereby inviscid (in this case Hamiltonian) systems become unstable. This point will be emphasized further in later sections, but it is exhibited in Figure 8.7 for the dispersion relation given in (8.22) to stress the point here. For Hamiltonian systems the phase speed has square root (branch point) singularities at and The low wave number cut off comes about from the stabilizing effect of gravity on long waves, and the cut off at derives from the stabilizing of short waves by surface tension. The relation (8.23) defines the instability boundary in space, but it does not provide any clue as to whether the temporal growth

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rates computed from (8.22) for are relevant in any real flow. Such a determination, as discussed previously, requires a spatio-temporal instability analysis of the base-state flow in an arbitrary reference frame. For this purpose the reference frame is transformed to one where the velocity in the “interface-centered” frame is advected with a uniform speed to the right (see Figure 8.6). The dispersion relation in this new frame, here termed the laboratory frame, is obtained from (8.20) by applying the transformation (Doppler shift) It is common in studies of shear layers to define a velocity ratio R as

which measures the strength of the interfacial shear relative to (twice) the mean velocity in the laboratory frame. Then, letting denote the dimensionless frequency, the dispersion relation for wave motions relative to the laboratory frame is

This equation defines a functional relation between the complex frequency and the wave number K, and involves the two parameters and . Of course, one could alternatively use the parameter in place of (i.e., Equation (8.26) can be used to identify the boundary in parameter space separating the region of local absolute instability from that of local convective instability for the present flow model. This is accomplished by computing the wave number corresponding to the pinching of distinct upstream and downstream spatial branches of the dispersion relation; that is, computing the (complex) pairs satisfying simultaneously the dispersion relation (8.26) and the vanishing of the group velocity Results of the calculation of parameter pairs satisfying (8.26) and vanishing of the group velocity for fixed (level curves) are shown in Figure 8.8, where varies continually along these curves. The dashed curve for defines the boundary demarcing respective regions of convective and absolute instability. For example, at a supercritical value of absolute instability occurs for (alt., Clearly, higher values of the velocity ratio (i.e., smaller values of are required to realize the pinching of spatial branches for higher values of It is shown that absolute instability occurs in this flow even when both streams are co-flowing. As the mean velocity in the laboratory frame increases, however, the instability changes to convective type.

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To offer an alternate perspective of the onset of instability at finite, non-zero wave number when the critical value of interfacial shear is attained, and to provide an example of the derivation and use of amplitude equations describing a reduced dynamics, we consider the evolution of disturbances in the immediate vicinity of the critical conditions For this purpose we introduce the criticality parameter and the perturbed wave number defined by

where is an order parameter such that is of unit order. Substituting into (8.24) and retaining only the leading order terms yields the dispersion

where is the frequency measured relative to the critical value The dispersion relation (8.28) corresponds to the Klein-Gordon equation

where the multiple-scale variables interface displacement is represented as

are used and the

246 The local instability characteristics represented by (8.28) in the Boussinesq limit are illustrated in the insert in Figure 8.7 enlarging the region of the neutral curve in the immediate vicinity of the critical shear The weakly nonlinear extension of (8.29) has been developed by Drazin (1970) and Weissman (1979). The amplitude equation (8.29) can be used to examine the nature of the instability near the critical shear for onset of interfacial wave growth. For this purpose we write (8.29) in laboratory coordinates, obtaining

The absolute/convective instability characteristics described by this KleinGordon equation can be obtained analytically (cf., Huerre 1987), whereupon one finds that the instability for is of absolute type provided and of convective type when the same quantity is larger than unity. One obtains, therefore, the result that the onset of Kelvin-Helmholtz instability is of absolute type when the critical shear is marginally greater than and the mean convective speed of the contiguous layers of fluid is less than Within the confines of this highly-idealized model one then finds that wind blowing over stationary water is absolutely unstable once the critical interfacial shear is exceeded.6 As a consequence, the effect of fetch should be of minor importance since the interface is an unstable oscillator for this flow condition and not a spatial amplifier. Of course, the air-side boundary layer thickness grows with fetch, and so also the water-side boundary layer to a lesser extent, which introduces a pair of extrinsic length scales into dynamics. The effect of these length scales can sometimes be quite subtle and may well alter the nature of the instability, rendering it of convective type. The effect of extrinsic length scales on the instability characteristics of shear layers with stable density stratification is considered in subsequent sections. Case K H – 2 We consider briefly the limit case of (8.21) when In this case the interface is a second-order vortex sheet (a discontinuity in vorticity only, not in velocity) and one readily finds that the flow is neutrally stable for all and The presence of a jump in the ambient vorticity across an isolated interface affects only the propagation speeds of neutral capillary-gravity waves on the interface. Furthermore, if the density anomaly across the interface vanishes and the effect of interfacial tension is neglected there is a single dispersive neutral mode whose propagation speed is proportional to the difference

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in vorticity between the two layers

The interaction between this type of vorticity wave and internal gravity waves will be examined in subsequent sections.

3.2

Class-2 Prototypical Flows

In the previous section we examined the instability of a flow possessing only an isolated interface supporting a discontinuous change in mean flow properties. It was shown that instability occurs in the presence of stable stratification only if a jump in velocity of sufficient strength is present. An isolated discontinuous change in vorticity alone (i.e., a second-order vortex sheet) supports an entirely neutral propagating wave, or a pair of neutral propagating waves if density stratification is present. In this section the instability of prototypical flows possessing a finite layer sandwiched between two ambient layers with differing vorticities and/or densities is examined. That is, we consider flows with two distinct interfaces which contain at least one extrinsic length scale - the distance between the two interfaces. Case TGH (Taylor-Goldstein-Holmboe Flow) The simplest stratified shear flow one can consider containing an extrinsic length scale consists of an isolated second-order vortex sheet separated a distance from an isolated density interface. The flow model, termed the TGH model after Taylor, Goldstein and Holmboe, is sketched in Figure 8.9. The coordinate system is fixed in the undisturbed position of the density interface, and the vorticity interface is situated a distance above the density interface and moves with a speed relative to it. As is already clear from (8.32) above, an isolated vorticity interface supports only a neutral wave propagating in only a single direction. An isolated density interface, even when embedded in a flow with uniform vorticity, supports gravity waves propagating to the right and to the left. The question of interest pertaining to this prototypical flow with three temporal modes is: What are the conditions, if any, for resonance (i.e., phase speed coalescence) between the vorticity wave and one of the gravity waves when the interfaces are in proximity to each other? Representing the displacement of the (upper) vorticity interface by a disturbance of the type and the (lower) density interface by the pressure (normal stress) matching condition applied at the two interfaces yields the following system of equations for the independent wave amplitudes , and

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Introducing dimensionless quantities based on the length scale and velocity scale the elements in the coefficient matrix are given by the relations

In writing these expressions we have invoked the Boussinesq approximation in which case and is the reduced gravity for the density interface. Also, a Richardson number parameter is defined, and specifies the difference between the scaled vorticity in the upper ambient layer from that in the intermediate layer, measured in terms of the scale The individual elements in the coefficient matrix, when equated to zero, give the dispersion relation for waves on the respective interface when the opposite interface is replaced by a non-deformable, stress-free boundary. The terms represent the coupling between waves propagating along the two interfaces. It is evident that these coupling terms vanish as that is, as the wave length becomes short with respect to the separation distance between the interfaces. In this latter case, the expressions for the provide the dispersion relations for waves on the separate interfaces

249 when they are isolated in an unbounded domain. These general aspects pertaining to the elements in the coefficient matrix apply to all models discussed in subsequent sections. To obtain a preliminary estimate regarding the potential for instability in this TGH flow model one can examine the conditions for phase speed coalescence in the limiting case Wave resonance in this limit of ‘interface independence’ occurs along the locus in the (J, K) plane given by

Dotted curves corresponding to this approximate coalescence condition, together with contours of constant temporal growth rate for the complete eigenvalue problem defined by the vanishing of the determinant of the coefficient matrix (8.33), are shown for two choices of the ambient vorticity parameter in Figure 8.10. The instability derives from a coalescence of the phase speeds of the vorticity wave on the upper interface with one of the gravity waves on the lower interface. When the phase speed of the vorticity wave is decreased from the value as K decreases (i.e., it moves upstream relative to the streamline speed At the same time, the speed of the downstream propagating (C > 0) gravity wave increases as K decreases. When these phase speeds coalesce, the two waves become ‘locked’, moving in unison with a single speed and in a phase configuration favorable for mutual reinforcement. The reinforcement or growth of the locked waves can be interpreted either from the local concentrations of vorticity or from the work done by the disturbance pressure fields. The phase relation between the two waves in the ‘locked’ state can be computed using either of the two equations in (8.33). For example, one can compute a relation of the form where the phase depends on the imaginary part of the phase speed The reader is referred to Baines (1995) for further discussion and illustration of these issues. It should be noted in this example that the presence of an extrinsic length scale confines the region of instability to lower values of the wave number, even in the absence of any high wave number cut off imposed by a stabilizing effect like surface tension. This is a general consequence of the presence of a non-zero extrinsic length scale in instability problems.

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As implied by the two cases shown in Figure 8.10, and as expected from earlier discussion regarding an isolated density interface in a uniform vorticity field, the instability in this flow vanishes when the vorticity becomes everywhere uniform The instability is confined to positive, non-zero values of the Richardson number because an isolated second-order vortex sheet is neutrally stable at all wave numbers. Also, the flow is neutrally stable when the two interfaces coincide so that any instability can exist only for finite, non-zero It is important to note also that instability in this TGH flow model occurs only for that is, when the vorticity in the ambient region is less than the vorticity in

251 the region surrounding the density interface. When the phase speed of the vorticity wave increases as K decreases and a coalescence with the downstream propagating gravity wave along the density interface is precluded. As seen from (8.32), the sign of the vorticity change across a second-order vortex sheet determines the direction of the phase velocity and can, therefore, influence the possible coalescence of phase speeds and initiation of instability. Case GTG (Generalized Taylor-Goldstein Flow) The general prototypical flow model analyzed in remaining parts of this section is depicted in Figure 8.11. The flow shown in this figure is a generalized version of that commonly known as the Taylor-Goldstein model (in which case the ambient layers are of unbounded extent and have vanishing vorticity). As shown in Figure 8.11 the density in successive layers is given by and Also, in order to exploit certain ‘symmetric’ limits to be discussed later, the origin of the coordinate system is positioned at the mid-depth level in the central layer having thickness (the important extrinsic length scale). The reference frame is chosen so that the velocities of the upper and lower interfaces of the central layer are equal, but oppositely directed, with magnitude The vorticities in the ambient layers are referenced to the value in the central layer, being given by where the are dimensionless constants. Hence, the vorticity is uniform across all three layers when and the flow is uniform with vanishing vorticities in the ambient layers when

Representing the independent displacements of the two interfaces in the same way as in the preceding TGH model flow, application of the

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pressure matching condition at the two interfaces yields a system identical to (8.33) where the dimensional elements in the coefficient matrix are

The quantities denote the reduced gravities for the respective interfaces. Slight and straightforward alterations of this system for different reference frames will be employed later when discussing different limiting cases of this flow. Case GTG-1 The first example of the GTG model to be considered is that of a flow with perfect anti-symmetry in the Boussinesq limit Scaling all variables using the length scale and time scale the elements of the coefficient matrix have the dimensionless forms

where and The control parameters in this flow model are the ambient vorticity parameter and the (bulk) Richardson number Consider first the possibility of phase speed coalescence under the conditions of ‘isolated’ interfaces for which The approximate condition for resonance (which occurs for as expected from symmetry) is

Suppose, for sake of discussion, the limit of unbounded flow Since it is clear that relation (8.38) has a solution for real K at

253 J = 0 only if that is, provided the vorticity in the central layer exceeds the vorticities in the ambient layers. This is an expected result based on Fjortoft’s theorem (cf., Drazin & Reid 1981) which states that neutral disturbances which are contiguous to unstable ones must have a critical level (i.e., a level where here) where the local vorticity is a maximum, not a minimum. As increases from negative values through zero, there is no possibility of phase speed coalescence when J = 0, indicating that the unstratified flow is necessarily neutrally stable when the vorticity in the central layer is a minimum. On the other hand, it is apparent from the approximate resonance condition (8.38) that instability is possible even for provided J is sufficiently large. Hence, we encounter the curious (and quite remarkable!) result that stable density stratification can cause a stable vorticity distribution to become unstable (cf., Howard & Maslowe 1973). And further, the more stable the vorticity distribution (i.e., larger the flow can only become unstable at ever higher values of the density anomaly across the interface.

Domains in the (J, K) plane where the model flow is unstable are shown in Figure 8.12 for selected values of the ambient vorticity parameter and depth parameter D. Included between the instability boundaries is a dotted line which tracks the simplified resonance condition (8.38) for each choice of The boundaries for when the flow is unbounded coincide with those computed originally by Taylor (1931) and Goldstein (1931). Also the variation of the temporal growth rate with K for the limiting value J = 0 in this case was determined first by Rayleigh (1894). The domain of instability for corresponds to a flow with neighboring density interfaces in an entirely

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uniform vorticity field. It is clear that density stratification can render this neutrally-stable flow unstable, and similarly for when the central layer has a local minimum of vorticity. The stabilizing effect of confining boundaries is shown in a separate panel in Figure 8.12. It is seen that instability is possible only at ever increasing values of the Richardson number as the depth D of the ambient layers decreases. As the critical value of J tends to infinity, whereupon the limit of stable Couette flow of a homogeneous fluid is attained as the density interfaces coincide with the confining boundaries. Inspection of the result shown in Figure 8.12 reveals that the intersection with the J-axis of the limiting boundaries for the unstable band for any and D encompasses a unit span of the Richardson number. This property emerges straight away from the longwave limit of the term independent of C in the full dispersion relation. Performing this calculation shows that the band of Richardson numbers for which the limit is contiguous with unstable disturbances is bracketed according to the relation

Decreasing the depth and increasing the vorticity of the ambient layers have an additive influence in moving the unstable band to higher values of Richardson number. The lower bound in (8.39) contains the known result that the Rayleigh shear layer (J = 0) is stabilized by the presence of boundaries when D = 1, a case that is shown in Figure 8.12. We point out that the fact that the instability boundary curves intersect the J –axis with a finite slope when and with a horizontal tangent when D is finite drives from the different limit properties of the term K coth K D in (8.38) as As evident from the lower bound in (8.39), the critical Richardson number for onset of instability is given by which prevails at K = 0. The long wave dynamics of instability near this onset condition can be captured in terms of an asymptotic analysis in which case a small parameter is defined by the departure of the control parameter J from its critical value Since the scope is restricted here to linear instability, and since the linear part of any amplitude equation describing asymptotic dynamics is given straightway by the asymptotic limit applied to the full dispersion relation, the desired evolution equation can be obtained directly from the expressions in (8.37). Constructing the longwave expansion of the dispersion relation

255

assuming finite depth D, gives the truncated expression

where

This relation implies that the appropriate slow space and time scales for the dynamics near onset are Then, the PDE conjugate to the approximate dispersion relation (8.40) is the linear Boussinesq equation for an amplitude function A(X, T)

The scaled criticality parameter can be related to a scaled amplitude parameter and a weakly nonlinear analysis can be carried forward to obtain the leading nonlinear term entering (8.41). An analysis of this type was performed by Hickernell (1983) for the symmetric case of the flow model studied here. The temporal growth rate for the marginal case represented by (8.40) is given by

which has a branch point at K = 0 and at the cut off wave number The maximum temporal growth rate occurring at the wave number is

However, as already noted several times, the applicability of this temporal analysis is predicated on the flow being absolutely unstable. To establish the conditions for absolute instability we must investigate the characteristics of the dispersion relation (8.40) in the Doppler-shifted frame locating the value of as a function of the criticality for which the absolute growth rate is incipiently positive. Toward this end one can investigate the instability characteristics of the scaled system

256

Analysis of this system shows that the instability for is absolutely unstable for and convectively unstable for Transforming this scaled result to match that for the original dispersion relation (8.40) yields the critical reference frame velocity (or velocity ratio

It should be noted, of course, that this predicted value of is valid only in the limit for the flow model considered here, but is a general result for the linear Boussinesq equation (8.44). Case TG At this point we present some instability characteristics of the shear layer profile originally studied by Taylor (1931) and Goldstein (1931). The dispersion relation for the classical TG model is given by the matrix elements (8.37) with (vanishing vorticity in the ambient layers) and (unbounded flow). The instability boundaries for this flow are included in Figure 8.12. The first issue addressed here is the condition under which the instability of this classical shear layer model is of either absolute or convective type. To calculate this transition boundary, the reference frame is necessarily shifted from the ‘centered’ or ‘symmetric’ frame used in writing the relations (8.37) to a laboratory frame. This change in the frame of reference is accomplished via the transformation where is the velocity ratio introduced in (8.25). Then, locating the critical condition in the plane where an unstable wave with the complex pair has a vanishing group velocity and a vanishing absolute growth rate leads to the boundary curve (solid line) shown in Figure 8.13. It is seen that the transition velocity ratio moves from the value for onset of absolute instability in the Rayleigh shear layer (cf., Huerre & Rossi, 1998), to higher values of as the stabilizing effect of density stratification increases. That is, the amount of reverse flow on the low-speed side of the shear flow must increase with in order to experience instability of absolute type. Included in Figure 8.13 is the boundary curve (dashed line) demarcing the transition from convective to absolute instability for a family of continuous profiles examined by Lin & Pierrehumbert (1990). They chose the profiles defined by

and obtained the dispersion relation by numerical solution of the TaylorGoldstein equation (8.3). These profiles form a one-parameter family in

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terms of the minimum value of the gradient Richardson number This is an example of a flow where the length scale exceeds that of As a result, the local Richardson number increases without bound at the extremities of the mixing layer, imposing a stabilizing effect of density stratification everywhere across the flow for A temporal instability analysis of this continuously stratified flow yields the neutral curve where It is readily seen that the flow is stable for in consistency with the Miles-Howard theorem, as

258 expected when As the critical Richardson number is approached, the velocity ratio R tends to infinity. This implies that the flow is convectively unstable in all possible reference frames as J approaches the limit for stability of the flow. Representative spatial growth rates are shown in Figure 8.14 for the unbounded TG model. The fact that different growth rate characteristics arise when boundary walls are placed either on the high speed or low speed sides is evidence that the eigenfunctions are skewed to either side, and that the instabilities for these cases have different phase and group speeds. Case UML (Upper Mixed Layer Flow) A special case of the GTG flow model is useful for exploring the effect of wind-generated shear in the upper mixed layer (UML) of a lake on the possible generation of high frequency internal waves in the metalimnion. The modified form of the GTG model shown in Figure 8.15 is employed where the mixed layer depth is and the uniform vorticity in the mixed layer is The metalimnion depth and the weak shear velocity allowed across the metalimnion are both referenced to corresponding scales for the mixed layer. For the present discussion the vorticity in the hypolimnion is identically zero and the hypolimnion depth is unbounded. The remaining flow parameters, in addition to H and are the bulk Richardson numbers and (where

The presence of a free surface with a large density change across it is represented simply as an impermeable boundary in this model. Inclusion of a deformable free surface supporting gravity waves would increase

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the number of temporal modes from four to six. This would allow the presence of another mode of instability, but likely with growth rates proportional to the small air-to-water density ratio. This mode is neglected here, together with any consideration of any harmonic or triad resonance conditions between the surface and internal wave modes. Including a free surface on the upper boundary also brings in the possibility of Langmuir circulations. The existence of coherent circulatory motions forming in the mixed layer and having counter-rotating cells aligned with the wind is well established, and the vertical transport associated with these motions is quite important. The underlying instability giving rise to these ordered motions is wave-induced (i.e., dependent on the presence of a surface wave field), and beyond the scope of this presentation. Interested readers are directed to the review by Leibovich (1983). The linear wave characteristics for this UML model are given by the system (8.33) where the elements in the coefficient matrix are

The quantity specifies the ratio of the vorticity in the metalimnion to that in the mixed layer.8 Of particular interest with this flow model are the conditions for which an instability exists, and the frequency and scale of waves selected by the instability. It has been shown already that a single second-order vortex sheet is never unstable, either with or without a finite density change across it. Hence, there can be no instability in this UML flow as or when On the other hand, a second-order vortex sheet in proximity to a density interface, as examined in the TGH model at the beginning of this sub-section, can exhibit an instability. However, it is not clear whether the appearance of instability in this simplified flow model depends on the presence of a non-zero shear across the metalimnion having finite, non-zero thickness. Only sample temporal instability results are presented here to expose some of the basic features of unstable solutions of the system defined by (8.47). It is found that instability arises only when a non-zero shear is present across the metalimnion, and that the strength of this instability is strongly influenced by the depth H of the metalimnion and the symmetry of the density anomalies at its boundaries.

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These aspects are illustrated in Figure 8.16 where contours of constant temporal growth rate about the peak value (denoted by an asterisk) for three configurations are shown. The growth rate contours have not been extended to the instability boundary to avoid excessive overlap of unstable regions near the origin. The two lower groups of contours in the figure correspond to the same density and geometric conditions, but with increasing shear across the metalimnion. It is found that the island of instability shrinks to zero as for fixed parameters H and It is probably worth noting here that linear instability characteristics such as those shown in Figure 8.16 have practical relevance only when the mixed-layer shear is sustained for time periods long enough to realize measurable growth of unstable waves. This loss of instability as is entirely consistent with the results of the TGH model presented earlier. It was shown there that the instability vanishes when the vorticity in the top layer exceeds the vorticity in the central and lower layers. Now, the vorticity is assumed to be maximum in the mixed layer (bounded upper layer) and an instability can therefore exist only if there is also a vorticity jump across the lower interface (i.e, base of the metalimnion). The top set of contours in Figure 8.16 correspond to a flow with the same values for H and as used to compute the central island, but the symmetry of the density anomalies across the boundaries of the met-

261 alimnion is broken. Specifically, the density anomaly across the upper interface is chosen to be five times larger than that across the lower interface. This type of asymmetry is seen to enhance the peak growth rate at any fixed value of J, and to extend the domain of instability to higher values of J. On the other hand, the strength of the instability is weakened if the density anomaly across the upper interface is diminished relative to that across the lower interface. In most natural environments the density changes (continuous gradients) are generally larger across the upper boundary and smaller across the lower boundary, a configuration which is more favorable for wind generated internal waves in the metalimnion. Case SBL (Separated Boundary Layer Flow) The GTG model can be adapted to another form relevant to environmental flows. It consists of a representation of the separated boundary layer structure appearing in the footprint of a finite-amplitude wave propagating upstream relative to an ambient flow. The simplified model of this flow is shown in Figure 8.17 and consists of two free interfaces separating regions of different vorticity in close proximity to a boundary. Although the linear instability of a nominal boundary layer flow with mean vorticity decaying monotonically with distance from the wall depends on viscous effects, a separated boundary layer is unstable on an inviscid basis because of the presence of an inflectional velocity profile. Hence, this simplified model can be expected to provide some guidance concerning the instability characteristics of separated boundary layers.

To analyze the instability of the base state shown in Figure 8.17, the boundary layer thickness and the flow speed at the edge of the

262 boundary layer are used as reference scales. Relative to these scales, a reversed flow of peak strength at a height exists within the boundary layer. The nominal boundary layer vorticity is modilied by the wave-boundary layer interaction creating contiguous layers of disparate vorticity. When the amplitude of the wave is sufficiently large, the vorticity generated at the wall by the wave-induced flow can become strong enough to create a near-wall layer with vorticity of opposite sign and a reversed flow with Although one can, in general, incorporate the effect of density stratification by including a stable density j u m p across both the edge of the boundary layer and the vorticity interface within the boundary layer, the discussion here ignores an internal density change across the interface at The structure of a separated boundary layer involves a region of recirculating flow in which the fluid is likely to be well mixed. Consequently, the most prominent effect of stable density stratification enters at the edge of the boundary layer, and is denned in this model by the bulk Richardson number The parameters entering the analysis then are and where measures the vorticity in the ambient flow relative to that in the nominal or undisturbed boundary layer. The instability characteristics for this SBL model are given by the system (8.33) where the matrix elements are

General instability characteristics for this model flow can be readily calculated from these relations. However, one of the most relevant issues pertains to the potential onset of absolute instability, and therewith global instability (cf., Huerre & Rossi 1998), in the wave induced boundary layer. This statement is based on the conjecture advanced elsewhere that onset of global instability in benthic layers might be the root cause of dynamics giving rise to pronounced resuspension events (cf., Bogucki & Redekopp 1999; Wang & Redekopp 2000). With this motivation, the boundary in space demarcing regions of absolute and convective instability for (no vorticity in the ambient flow) and several values of the Richardson number J are shown in Figure 8.18. Several points about the data shown in this figure deserve comment. First, results are shown for three values of the Richardson number when the critical growth rate at transition is This value of was

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deliberately chosen to overestimate the condition for onset of absolute instability on an inviscid basis, compensating somewhat for viscous effects coming from the influence of the bounding wall. The difference between the transition boundary calculated for and for a fixed value of is shown. The change is quite substantial. It is believed that the curves for should yield an upper bound for the effect of finite Reynolds number. Second, the critical backflow needed to initiate absolute instability decreases quite rapidly as the depth of the separation increases. The constraining effect of the bottom boundary requires a marked increase in the critical value of as the separation bubble becomes shallow. Third, the effect of a stable density ‘gradient’ at the edge of the boundary layer is quite surprising. One notes that increasing the Richardson number decreases the critical value of the backflow velocity at fixed depth This effect is not anticipated based on the ‘nominal’ understanding that a stable density stratification (in this case where an internal layer of maximum vorticity exists) will decrease the susceptibility of the flow to onset of instability. However, the convective-absolute instability transition depends principally on the group velocity of unstable disturbances, not their growth rate, and the group velocity may vary more strongly with the stratification than the magnitude of, say, temporal growth rates. It is seen that the effect of density stratification is more pronounced for shallow separations, indicating a stronger effect of the bounding wall on the group velocity associated with pinching of spatial branches as the stratification increases.

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3.3

Class-3 Prototypical Flows

We consider in this section another class of physically relevant flow models. They comprise an extension of the Class-2 flows discussed in the previous section to include three free interfaces and at least two extrinsic length scales. The particular models examined here expose some crucial length scale effects, effects which underlie some important environmental applications. Case GH (Generalized Holmboe Flow) The encompassing flow model in the class examined here is shown in Figure 8.19. It is termed the generalized Holmboe (GH) model in that it incorporates several straightforward generalizations of a mixing layer flow proposed by Holmboe (1962) to expose the effect of a disparity between the length scale of the vorticity field and that of the density gradient field. The disparity in scales was extreme in the original Holmboe model, with the ratio but yet retaining a finite density j u m p across the mixing layer. In the coordinate system used here, the origin is placed in the central density interface, with density jump and the remaining interfaces are displaced a distance and away. The velocity of the upper interface is relative to the central interface, and the velocity of the lower interface is oppositely directed with speed Following the GTG model discussed in the previous section, the excess vorticities in the ambient layers are now represented by the parameters which measure the vorticities in the ambient layers relative to their values in neighboring layers of the primary shear flow. As such, the vorticities in these ambient layers are equal to that in the adjoining layers when and the ambient vorticities vanish when

265

In developing the eigenvalue relation for infinitesimal disturbances in this flow, the interface displacements are represented by descending in order from the highest to the lowest interface. There are now three pressure matching conditions across the three internal interfaces, leading to the third-order system

The dimensional elements in the coefficient matrix for the GH model have the following definitions in dimensional form (cf., Figure 8.19):

The effect of interfacial tension along the central interface is included (cf., the term involving in to facilitate application of this system to a model for wind-generated capillary-gravity waves. To this end, the Boussinesq approximation has not been invoked for either interface and the relations for the matrix elements are valid for arbitrary density changes across these interfaces. The system therefore possesses six temporal modes consisting of left- and right- running waves along each of the three interfaces. In the discussion which follows, however, only two restricted cases of this general system are considered, each possessing just four temporal modes. The linear instability characteristics of restricted cases with four temporal modes have been presented for various ‘symmetric’ and ‘asymmetric’ configurations, together with companion experimental studies of flows that closely mimic these idealized configurations (cf., Lawrence et al 1991; Pouliquen et al 1994). The work by Pouliquen et al (1994) included the effect of interfacial tension acting along the density interface, as does that by Caponi et al (1992) who consider a model for windgenerated capillary-gravity waves.

266 Case SH (Symmetric Holmboe Flow) The only flow in this class for which we consider specific instability characteristics is the symmetric Holmboe (SH) model. Following Holmboe, the density anomalies across the upper and lower interfaces are neglected and the flow has perfect anti-symmetry across the central interface (i.e., and Furthermore, the effect of surface tension is neglected along the central interface and the density change across this interface is assumed to be small Then, the relevant length and time scales are and and the members of the coefficient matrix assume the following dimensionless forms:

The parameters in this formulation have their familiar definitions It should be evident by now that the eigenvalue problem defined by these relations can be expected to possess two modes of instability: one arising (essentially) from a phase speed coalescence between and and another from the coalescence between (alt., by symmetry) and The latter resonance condition should be similar to that described previously in Case TGH, and the former by the limit of Case TG (i.e., the familiar Rayleigh shear layer). That is, there is in addition to the Kelvin-Helmholtz mode a second mode termed the Holmboe mode. The modifications to the stability diagram shown in Figure 8.10 for the TGH model when and in the present case are well-known (cf., Holmboe 1962; Lawrence et al 1991; Baines & Mitsudera 1994; Baines, 1995). The primary alteration to Figure 8.10 arising from the addition of another second-order vortex sheet positioned symmetrically with respect to the density interface is the appearance of a stationary mode of instability. This stationary mode is the continuation of the Rayleigh mode for to non-zero values of the Richardson number. The peak growth rate for this mode occurs at and diminishes gradually as J increases, vanishing at a value For larger values of J the temporal growth rates closely approximate those for the TGH model given in Figure 8.10. The peak dimensionless growth rate for this SH model is 0.142 while that for the TGH model is 0.137. Furthermore, the Richardson number where the peak growth

267 rate occurs is only slightly lower for the SH model for SH and for TGH). Hence, the Holmboe mode of instability is qualitatively and semi-quantitatively captured by the coupling between a neutral wave on an isolated second-order vortex sheet and a gravity wave on an isolated density interface in a uniform vorticity field. When the symmetry of the flow is broken either through or (or both) differing from unity, the speed of neutral waves on the two vorticity interfaces differ (in magnitude) and two different wave coalescences appear. The consequence is the presence of two modes of instability at all values of the Richardson number J. The splitting of the symmetry of the vorticity interfaces preserves the Holmboe mode, but extends the Rayleigh mode to high(er) values of J in a manner resembling the Taylor-Goldstein mode (cf., Lawrence et al 1991). Further studies pertinent to this flow model include those by Koop & Browand (1979), Smyth & Peltier (1989), Lawrence et al (1998), and Pawlak & Armi (1998). Since the Holmboe model is a useful idealization of a class of environmental shear flows where the vorticity layer and the density layers have disparate scales (i.e., in Figure 2), it is of considerable relevance to to identify the conditions for which the instability is of absolute or convective type. The transition boundaries in space defining regions where the instability changes type are depicted in Figure 8.20 for the ‘symmetric’ conditions The velocity ratio the value for incipient absolute instability in the Rayleigh limit forms a boundary curve for all values of J. Interestingly, the strength of the density interface in the center of the shear layer does not affect the position of this boundary. However, and in contrast to the other flow models discussed herein, there is a reversal in the type of instability as R increases (i.e., as the backflow on the low-speed side increases). As such, absolute instability persists in this flow only for a diminishing range of velocity ratios as J increases. Absolute instability exists in the range at high values of the Richardson number. 9 The reappearence of convective instability at higher values of the velocity ratio can be understood by reference to Figure 8.4a and the following argument. In this symmetric Holmboe flow there exists a wave coalescence between the vorticity wave on the upper interface and the downstream-propagating gravity wave on the density interface. Relative to the centered reference frame used in Figure 8.19, this coalescence occurs for a positive phase speed and gives rise to a downstreampropagating packet as shown in Figure 8.4a. Correspondingly, and by symmetry, a coalescence also exists between the vorticity wave on the lower interface and the upstream-propagating gravity wave on the density interface, giving rise to an upstream-propagating packet. Now, when

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one shifts to an arbitrary laboratory frame the downstream/upstream symmetry is broken and the two packets both move in the downstream direction as increases. In fact, the window of absolute instability shown in Figure 8.20 captures precisely that range of R for which the two limiting rays of the upstream packet shown in Figure 8.4a are rotated clockwise through the ray as increases. Hence, the convectively unstable domain for corresponds to reference frames where the two packets move in opposite directions, and the convectively unstable domain at high values of (R > 2 for corresponds to reference frames where both packets move in the same (downstream) direction. In even more complicated flow profiles admitting multiple wave resonances one can expect multiple domains of absolute instability.

4.

INSTABILITY OF STATICALLY-UNSTABLE INTERFACES

The density differences across the interfaces in all the flow models considered in the previous section were statically stable. Any instability appearing in these model flows derives from the presence of an unstable distribution of vorticity or the interaction between neutral gravity waves and a stable vorticity distribution. In this section we examine the tendency for overturning motions when the density anomaly across an interface is statically unstable. The onset of overturning, or convective, instability will be examined in the absence of any diffusive transport of the scalar property giving rise to the variation in density. As such, we

269 examine only the instability of immiscible interfaces in the non-diffusive limit of either an inviscid or a viscous fluid. The subject of thermal convection, the onset of unstable overturning motion in a layer of a viscous and diffusive fluid, is reviewed by Kelly (1994), including the important environmental effect of velocity shear in the convecting layer.

4.1

Inviscid Overturning Instability

The eigenvalue relation for the inviscid instability problem is contained in the dispersion relation (8.20) for the generalized Kelvin-Helmholtz model. In the present case we take the statically unstable density distribution and examine instability in the limit of the Boussinesq approximation Then, taking so that the interface is at most a second order vortex sheet, and defining the reduced gravity and the surface tension parameter the eigenvalue relation for an isolated interface in an unbounded domain is

In the absence of any change in the mean vorticity across the interface a stationary instability arises for Surface tension is effective in stabilizing the short waves with wave numbers beyond the cut-off wave number On the other hand, when surface tension is neglected and a finite change in mean vorticity exists across the interface, the differential in vorticity stabilizes the long waves in the band and the instability is propagating (non-stationary). It seems clear then that a statically-unstable, second-order vortex sheet can be stabilized when exceeds that is, differential shear can inhibit overturning instability provided surface tension effects are strong enough to stabilize the high wave number part of the spectrum. In fact, employing the intrinsic scales introduced prior to equation (8.21), the instability boundary in space as given by the vanishing of the discriminant in (8.52) is

One can show straightway that instability is suppressed when

270

4.2

Viscous Overturning Instability

The onset of overturning instability in the presence of a staticallyunstable interface will obviously be affected by the finite viscosities of the contiguous fluids. Furthermore, many environmental applications involve thin interfaces of miscible fluids and the stabilizing effect of interfacial tension is not available. Yet, it is of considerable practical interest to have at least some semi-quantitative instability characteristics in hand for statically-unstable interfaces in a viscous fluid. Now, a quite extensive literature exists exploring a range of issues and applications pertaining to this general problem area, commonly referred to by the rubric of Rayleigh-Taylor instability. Some generalized models for the instability of statically-unstable layers formed by internal wave motions in a continuously-stratified, viscous and diffusive fluid, have been studied by Thorpe (1994ab). The approach taken here follows that employed in the preceding section; namely, the linear instability of a few simple interfacial models is examined to reveal general characteristics of the instability mechanism and the role of some key parameters. The formulation of the linear instability problem, and the requisite matching conditions across an interface when the effect of viscosity is accounted for, are set forth in Appendix B. The results presented there are used without any further elaboration of details necessary to obtaining the eigenvalue relations for particular models discussed below. Case The first case we consider is that of an isolated interface in an unbounded domain. An incompressible fluid with density and viscosity overlays another fluid with density and viscosity As shown in Appendix B, the eigenvalue problem can be cast in terms of a 4 × 4 determinant. Since the density differences may be small in many environmental applications, a reduced gravity is defined for the statically unstable interface with the density of the heavy, overturning fluid used as a reference. Also, as there is no extrinsic length scale for the problem, we define intrinsic length and time scales based on and the kinematic viscosity of the overturning fluid:10

Employing these scales the problem is reduced to three parameters: the density ratio the viscosity ratio and the Bond number based on the intrinsic length scale Making the Boussinesq approximation and ignoring viscosity stratification leaves an eigenvalue problem for the dimensionless growth

271

rate in terms of the dimensionless wave number K and the Bond number B as parameter. Some representative growth rate curves for this limiting case are shown in Figure 8.21. Both the peak growth rate and the bandwidth of the instability decrease, as expected, with increasing surface tension. There is no high wave number cut-off when the Bond number vanishes, but any finite amount of interfacial tension limits the instability to the range where is the cut-off value.

We point out that the lubrication approximation discussed briefly in Appendix B is quite reasonable for dimensionless wave numbers of order unity. The approximation ignores the growth rate relative to in the terms of the form However, when the instability occurs at low wave numbers, the computed growth rate can be significantly overestimated when using the approximation In fact, increasing the viscosity ratio generally shifts the instability to lower wave numbers tending to further jeopardize the validity of the lubrication approximation. On the other hand, increasing viscosity ratios also decrease the growth rate, so the applicability of this approximation must be assessed carefully in some cases. Case The next case we consider is that of a statically unstable interface near a stress free boundary. In particular, we suppose a (thin) layer of depth of heavier fluid overlays a lighter fluid of unbounded extent. This configuration introduces the effect of an extrinsic length scale or in nondimensional form. Although the author prefers to introduce

272 nondimensional variables through the use of intrinsic scales, one can alternately use the geometric scale and a related diffusive time scale In this case the Rayleigh number (alt., Grashof number) appears explicitly, but it is readily seen that the ratio of these different scales depends directly on the Rayleigh number:

Following the analysis procedure outlined in Appendix B one obtains an eigenvalue relation defined in terms of a 6 × 6 determinant. The objective here is to reveal the effect of the parameter on the RayleighTaylor instability of a thin layer. For this purpose the eigenvalue relation is evaluated for the Boussinesq limit with uniform viscosity and in the absence of surface tension effects. The maximum growth rate as a function of is displayed for this limiting case with a confining stress-free upper boundary in Figure 8.22. It is seen that the depth of the fluid layer has a significant stabilizing effect for For the peak growth rate asymptotes to the value 0.2825 corresponding to B = 0 in Figure 8.21, and the overturning instability of a statically-unstable interface is unaffected by the presence of a confining stress-free boundary. Similar calculations have been made for a no-slip boundary in place of the stress-free condition used to obtain the results in Figure 8.22. The peak growth rates are reduced quite dramatically by the presence of a no-slip boundary for low values of but the difference between the noslip and the stress-free cases falls to less than five percent when Case It was pointed out in the discussion of Case RT-2 that the instability characteristics of a heavy layer are dependent on the thickness of the layer when In some cases the heavier layer may be positioned between an otherwise stable layer so that a finite stable density difference exists across this heavy layer. Hence, we consider here the instability of a heavy layer with density and thickness sandwiched between an upper, ambient layer with density and a lower layer with density As such, there is another stability parameter, say, measuring the stability of the ambient layers in relation to the destabilizing density anomaly across the lower interface separating the heavy fluid and the lighter ambient which it overlies. The parameter is defined as

273

When the ambient state is statically stable and the (intruding) heavy layer is statically unstable relative to the lower layer. The goal is to assess the role of when the central, heavy layer is thin. The analytical problem requires the matching of the field solutions in three layers across the two free interfaces. As there are four matching conditions at each interface, the eigenvalue problem can be represented in terms of the vanishing of an 8 × 8 determinant. There are clearly a number of parameters in this configuration, so only results for a restricted limit case will be included here. The viscosity is supposed to be uniform and surface tension effects are neglected. Furthermore, the effect of density ratios entering the terms and are ignored (see definition of following B.2 in Appendix B). In this special case (and covering a somewhat inconsistent range) the only parameters entering the eigenvalue relation are and Carying forward a sequence of calculations where the maximum growth rate is computed over a spectrum of wave numbers for each parameter setting, the results shown in Figure 8.23 are obtained. As is increased over the range for fixed the growth rates are reduced by the presence of the ambient stable density difference. This model provides a zeroth-order assessment, for example, of the instability of thin layers, often comprised of biological or detrital material, embedded in an ambient fluid. Such layers may be created by the organizing effect of hydrodynamic motions associated with internal waves acting on distributed particulate matter in the fluid. They may also be formed by intrusive layers which subsequently overturn across

274

statically-unstable ‘interfaces’ when the stabilizing effect of shear weakens (cf., statements following eqn. 8.52),

5.

CLOSING REMARKS

The range and depth of topics discussed herein confronts only a small fraction of the host of physical processes operative in environmental flows. Only two principal mechanisms of instability have been addressed: shear or vorticity-driven instability and gravitational or overturning instability. Although these mechanisms are operative in a wide range of naturally-occurring contexts, either as primary or secondary instabilities, there are a variety of other instability mechanisms that select the dominant dynamical characteristics in many contexts. For example, and to name just a few, baroclinic instability, centrifugal instability, convection from distributed or point sources, double-diffusive convection, thermo-baric instability, etc. Nevertheless, it is hoped that this chapter illustrates features important both to the approach and to the analysis of linear instability issues in general, and that the specific choice of topics, point of view, and results serve as a useful supplement to material already obtained in available references.

275

Appendix A. Layer Matching Conditions for Inviscid Flows Interface matching conditions are required to connect the solution for the field variables in one layer where the base state is uniform (i.e., uniform density and vorticity) with the corresponding solution in an adjoining layer. Assuming that the fluids on both sides of the interface are immiscible, and that there is no material transport across the interface by, for example, evaporative phase change, there are two conditions that must be applied for an inviscid flow. First, there is the continuity of the normal velocity across the interface. Now, the normal component of the fluid velocity at an interface can be related to the motion of the interface by the kinematic condition which ensures that the interface remains a material surface (alt., that the fluid immediately adjacent to the interface must have a component of velocity normal to the interface which is identically equal to the velocity of the interface). Assuming that the undisturbed interface is a horizontal plane where the points normal to the plane and anti-parallel to gravity, and that the disturbed interface is at the position the interface can be defined by the level surface

The kinematic constraint

is then given by the equivalent, but alternate, relations

where

is the unit vector normal to the interface and is the velocity vector in the right handed Cartesian coordinate system with These relations are inherently nonlinear as they are applied at the disturbed position of the interface. Since the discussion here focuses exclusively on linear wave motion, the linearized kinematic condition is given simply by

where U is the base state velocity of the fluid. This relation is applied at the equilibrium interface positions The second requirement is that the difference between the normal stresses acting on the two sides of a curved interface must be balanced by the interfacial tension acting along the surface. Denoting the hydrodynamic pressure by the coefficient of interfacial tension by T, and the Gaussian curvature of the interface by this condition can be

276

represented as

Supposing that the pressure is represented as the sum of the hydrostatically-balanced value of the base state (where and the perturbation to this state, and assuming that the distortion to the interface is small, the pressure immediately above the interface is, after use of a Taylor series approximation for the continuous functions and

The linearized version of the pressure matching condition can then be written as Assuming a Fourier representation for the dependent variables as

the linearized Euler equations for base state flows having uniform density and vorticity can be reduced to the second-order field equation

for the vertical velocity modal function. Solutions for must be constructed for each layer within which the base state is prescribed to have a unique density and/or vorticity. The perturbation pressure in each layer is then related to the modal function by

where is the uniformly-sheared horizontal velocity of the base state within the same layer. With these relations the formulation of the linearized instability problem is complete. As an example illustrating the application of these results, consider the GTG flow model sketched in Figure 8.11. Solving the field equation (A.7) in each of the three layers, and using the boundary conditions that W = 0 at each of the external confining boundaries, the linearized

277

kinematic relation (A.2) can be used to represent the modal function in terms of the amplitudes and of the independent interface displacements. The results for the domains and respectively, are:

Use of the pressure relation in (A.8), and substitution into the linearized pressure matching condition (A.5) for each of the free interfaces, yields the system represented by (8.33) with the individual matrix elements defined in (8.36).

Appendix B. Layer Matching Conditions for Viscous Fluids The matching conditions across an interface separating two immiscible, viscous, nondiffusive, incompressible fluids are summarized here. It is assumed that the interface is a horizontal plane at level in its equilibrium position. The discussion is restricted to consider entirely stationary base states (i.e, no relative motion across the interface and no mean vorticity). The matching conditions for planar motion under these restrictions are as follows: i) continuity of normal velocity ii) continuity of tangential velocity iii) continuity of tangential stress and iv) continuity of normal stress

As already noted in (A.2), the distortion of the interface is related to the velocity field via the kinematic constraint Considering the Navier-Stokes equations for an incompressible, constant density and viscosity fluid linearized about a stationary base state, the field equation for the modal amplitude function for the vertical velocity is where we have assumed modal separations of the dependent variables in the form and It is convenient to write the time dependence in terms of the growth rate

278

since the onset of instability occurs via a stationary bifurcation (i.e., often referred to as ‘exchange of stabilities’). It is worth pointing out that a number of studies discussed in the literature invoke the ‘lubrication’ or ‘creeping-flow’ approximation in which the (linear) inertial terms are neglected. In this case the growth rate in (B.2) and As a consequence, the characteristic equation for (B.2) contains repeated roots and the fundamental solutions are Time dependence and onset of instability then enters solely through the kinematic condition (A.2) in this limit. The set of matching conditions can be written entirely in terms of the modal function for the linearized problem. The conditions to be applied at the equilibrium level are, in respective order,

The kinematic condition (A.2) has been used to eliminate the interface displacement amplitude from the last relation. The remaining conditions needed to completely specify the solution for a given flow configuration are those applicable at confining boundaries. For example, at an impermeable, no-slip boundary the requisite conditions are Alternately, at an impermeable, stress-free boundary we have the conditions As an illustration of the use of these conditions, consider the simplest situation of an isolated interface at separating two fluids of unbounded extent (cf., Case RT-1 in Section 4.2). Denoting the upper fluid with subscript ‘1’ and the lower fluid with subscript ‘2’, there are four solutions of (B.2) satisfying the constraint of bounded motion at Substitution of these four solutions with arbitrary amplitudes into the four matching conditions yields a homogeneous system of four equations. The eigenvalue problem is, therefore, given by the vanishing of the 4 × 4 determinant

The rows of the determinant correspond, respectively, to the four matching conditions in (B.3), and the columns correspond to the ordered set

279 of four independent solutions of (B.2). For instability one seeks the largest eigenvalue (s) for a given set of the physical parameters. The reader should note that the eigenvalue appears explicitly in the last row and implicitly in the expressions and If the lubrication approximation is employed, the terms involving must be modified and the eigenvalue only enters via its explicit appearance in the last row.

6.

ACKNOWLEDGEMENTS

The author is especially grateful to B.-ji Wang for his generous help in preparing many of the figures. Thanks are also due to F. Chandler, A. C. H. Hu and A. Riaz for their contributions to the graphical results for several models. Partial support during preparation of this material was provided by the Office of Naval Research under grant N00014-95-1-0041.

Notes 1. The reader is reminded that whenever a complex representation is emplayed for a physical variable, as in (8.1) or elsewhere in this chapter, the true field is computed by considering the real part of the right hand side. 2. The theoretical framework presented here can be extended to consider, for example, flows which are slightly inhomogeneous in the propagation direction by defining a slow space scale and writing the phase function as Space limitations preclude discussion of these extensions here. Interested readers can consult Huerre & Rossi (1998). 3. In the linear, primary instability discussed here this merging of phase speeds occurs at the same wave number and, therefore, frequency. Other cases certainly exist where, for example, the phase speed of a wave (with wave number merges with an harmonic (with wave number 2 of the same (or different) mode, but space prevents discussion of such cases. 4. It is useful to conceptualize the usual KH model as consisting of two second-order vortex sheets separated by a distance (an upper one with and a lower one with in the limit where is the wave length of a disturbance. That is, two second-order vortex sheets appear as a first-order vortex sheet in the long wave limit, where ‘long’ wave is relative to the separation distance between the second-order sheets. 5. For the present discussion we assume so the density stratification is statically stable. A statically unstable interface is considered in Section 4. 6. Approximating the critical conditions for large density ratios yields the relations:

7. Temporal instability analyses of an extended set (two-parameter family) of these profiles has been carried forward by Hazel (1972) in which the ratio of these two length scales has been varied.

280 8. If the hypolimnion depth is taken to be finite, the factor unity in the square bracket term in is replaced by coth where is the depth of the hypolimnion relative to that of the mixed layer. 9. During preparation of this chapter the author became aware of identical calculations for this flow model by Ortiz et al (2000). The transition boundary for low values of where the Holmboe and Rayleigh modes overlap was computed by Ortiz et al, and the data shown in Figure 8.20 was graciously provided by these authors. They also present details pertaining to spatial instability of the SH flow. 10. Chandrasekhar (1961) defines intrinsic scales in terms of directly, not in terms of a reduced gravity. Use of the reduced gravity defined here is especially convenient when considering the Boussinesq approximation in the limit of small density differences as it then incorporates the entire effect of the density ratio.

References Baines, P.G. Topographic Effects in Stratified Flows. Cambridge University Press, 1995. Baines, P.G. & H. Mitsudera. On the mechanism of shear flow instabilities. J. Fluid Mech., 1994; 276: 327-342. Bogucki, D. & L.G. Redekopp. A mechanism for sediment resuspension by internal solitary waves. Geophys. Res. Lett., 1999; 26:1317-1320. Caponi, E.A., M.Z. Caponi, P.G. Saffman & H.C. Yuen. A simple model for the effect of water shear on the generation of waves by wind. Proc. R. Soc. Lond., 1992; A438:95-101. Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press, 1961. Craik, A.D.D. Wave Interactions and Fluid Flows. Cambridge University Press, 1985. Drazin, P.G. & L.N. Howard. Hydrodynamic stability of parallel flow of inviscid fluid. Adv. Appl. Mech., 1966; 9:1-89. Drazin, P.G. Kelvin-Helmholtz instability of finite amplitude. J. Fluid Mech., 1970; 42:321-336. Drazin, P.G. & W.H. Reid. Hydrodynamic Stability, Cambridge University Press, 1981. Goldstein, S. On the stability of superposed streams of fluid of different densities. Proc. R. Soc. Lond., 1931; A132:524-548. Hazel, P. Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech., 1972; 51:39-61 Hickernell, F.J. The evolution of large-horizontal-scale disturbances in marginally stable, inviscid, shear flows. I. Derivation of amplitude evolution equations. Stud. Appl. Math., 1983; 69:1-21. Holmboe, J. On the behaviour of symmetric waves in stratified shear layers. Geophys. Publ., 1962; 24:67-113. Howard, L.N. Note on a paper of John W. Miles. J. Fluid Mech., 1961; 10:509-512. Howard, L.N. & S.A. Maslowe. Stability of stratified shear flow. Boundary-layer Meteor., 1973; 4:511-523. Huerre, P. Spatio-temporal instabilities in closed and open flows. In Instabilities and Nonequilibrium Structures (E. Tirapagui & D. Villoroel, eds.). D. Reidell Publ. Co., Dordrecht, 1987. Huerre, P. & M. Rossi. Hydrodynamic instabilities in open flows. In Hydrodynamics and Nonlinear Instabilities (C. Godréche & P. Manneville, eds.). Cambridge University Press, 1998.

281 Kelly, R.E. The onset and development of thermal convection in fully developed shear flows. Adv. Appl. Mech., 1994; 31:35-112. Koop, C.G. & F.K. Browand. Instability and turbulence in a stratified fluid with shear. J. Fluid Mech., 1979; 93:135-159. Lawrence, G.A., F.K. Browand & L.G. Redekopp. The stability of a sheared density interface. Phys. Fluids A., 1991; 3:2360-2370. Lawrence, G.A., S.P. Haigh & Z. Zhu. In search of Holmboe’s instability. In Physical Processes in Lakes and Oceans (J. Imberger, ed.) AGU, Wash. D.C., 1998; 54:295304. Leibovich, S. The form and dynamics of Langmuir circulations. Ann. Rev. Fluid. Mech., 1983; 15:391-427. Lin, S.J. & R.T. Pierrhumbert. Absolute and convective instability of inviscid stratified shear flows. In Stratified Flows (Proc. 3rd International Symp. Strat. Flows, 1987, E.J. List & G.H. Jirka, eds.), Am. Soc. Civil Engineers., New York, 1990. Ortiz, S., T. Loiseleux & J.M. Chomaz. Absolute and convective instability in shear flows with an interface. Fifth International Symposium on Stratified Flows (G.A. Lawrence, R. Pieters & N. Yonemitsu, eds.) Univ. British Columbia, Vancouver, Canada. Also, Spatial Holmboe instability, submitted to Phys. Fluids, 2000. Pawlak, G. & L. Armi. Vortex dynamics in a spatially accelerating shear layer. J. Fluid Mech., 1998; 376:1-35. Poliquen, O., J.M. Chomaz & P. Huerre. Propagating Holmboe waves at the interface between two immiscible fluids. J. Fluid Mech., 1994; 266:277-302. Rayleigh, Lord. The Theory of Sound. (2nd ed.), Macmillan, London; p.393, 1894. Smyth, W.D. & W.R. Peltier. The transition between Kelvin-Helmholtz and Holmboe instability: An investigation of the overreflection hypothesis. J. Atmos. Sci., 1989; 46:3698-3720. Taylor, G.I. Effect of variation in density on the stability of superposed streams of fluids. Proc. R. Soc. Lond., 1931; A132:499-523. (Reprinted in Scientific Papers, IV:147-162). Thorpe, S.A. The stability of statically unstable layers. 260:315-331.

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Thorpe, S.A. Statically unstable layers produced by overturning internal gravity waves. J. Fluid Mech., 1994b; 260:333-350. Turner, J.S. Buoyancy Effects in Fluids. Cambridge University Press, 1973. Wang, B.-ji & L.G. Redekopp. Long internal waves in shear flows: Topographic resonance and wave-induced global instability. To appear in Dyn. Atm. Oceans, 2000. Weissman, M.A. Nonlinear wave packets in the Kelvin-Helmholtz instability. Phil. Trans. R. Soc. Lond., 1979; A290:639-685.

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Index A amplitude, see finite-amplitude waves atmosphere, internal solitary wave, 1-2, 9, 23, 61-88 internal waves, 119-160 mountain waves, 119-160 B Benjamin-Ono equation, 9, 12, 63 bores, undular, 2, 18-22, 30, 41-60, 67-73, 89 -91, 106-114 turbulent, 89-91, 106-114 blocking, 120, 142-144, 192, 199, 204, 209 211 boundary layer, 11-12, 129, 143, 161, 181, 249 Boussinesq, approximation, 30-32, 95, 99-100, 102, 120, 164, 228 equations, 252 Brunt-Vaisala frequency, see buoyancy frequency buoyancy frequency, 3, 33, 121, 164, 193-195 Burgers number, 197

C cnoidal waves, 19 compressibility, 62-63 continental shelf, 30-31, 42-50 continuously stratified fluid Coriolis frequency, 3, 14, 18, 42-50, 58, 164 -165 critical layer, 64-65, 84, 126 current shear, 51, 124-129, 224, 236-238

D dispersion relation, 122-124 downslope winds, 142, 192 drag, 134-135, 138, 142, 192, 216 E equations of motion for a stratified fluid, 2-3, 120

F finite-amplitude waves, 1-2, 23, 58, 139-141,143, 203 fossil turbulence, 163-172 free surface, 3, 270-276 frequency, see buoyancy frequency wave, 123, 126 friction, 2, 13-17, 36-37, 94, 139 Froude number, 93-94, 100, 195, 203-204, 208, 212

G Ginsburg-Landau equation, 232 gravity current, 67-73, 89-118, 224 group velocity, 122-123 H Holmboe instability, 247-250 hydraulic, jumps, 106-107, 112, 122, 142 theory, 19-20 hydrostatic, 131-134 I incompressible, 2-3 internal tide, 29-60 internal waves, see solitary waves, or mountain waves instability, see stability absolutely unstable, 232 convectively unstable, 231 temporal instability, 229 spatial instability, 230 inviscid, 2-3 K Kadomtsev-Petviashvili equation, 15,17-18 Kelvin- Helmholtz instability, 239-246 Kolmogorov length scale, 163, 166, 169 Korteweg-de Vries equation, 2, 5-7, 23, 30, 63, 66, 83 extended, 7, 11, 30-34 variable-coefficient, 12-17, 33-34

L laboratory experiments, 191-222 lee waves, 32, 119-160

M mixed layer, 32, 161, 262 mixing, 91, 106-107, 111, 198-199, 202, 206 Miles-Howard theorem, 239 modes, long wave, 4, 34, 66 equation, 228 momentum equations, 3 Morning Glory, 67-73 mountain waves, 119-160 N nonlinear, cubic, 7, 33, 43-50 quadratic, 6, 33, 43-50

O observations, atmosphere, 61-88 ocean, 29-60 ocean, internal solitary waves, 1-2, 9, 23, 28-60 internal tide, 29-60 Ozmidov length scale, 165 P Potential temperature, 121 Potential vorticity, 120, 144-150, 167 Pycnocline, 38-41 R Radiation condition, 129 Rayleigh instability theorem, 237 Reynolds number, 209 Richardson number, 237 Rossby number, 197-198, 209 Rotation of the earth, 3, 14, 18, 42-50, 58, 192-193, 208

S Scorer parameter, 64, 135 separation, 143, 264 solitary wave, atmospheric, 1-2, 9, 23, 61-88 internal, 1-26 oceanic, 1-2, 9, 23, 29-60 slowly-varying, 2, 15-17 speed, linear long wave, 4-5, 30-32, 66 solitary wave, 9-10 stability of stratified shear flows, 4, 63, 223-281 subcritical, 21-22, 142 supercitical, 21-22, 142

T Taylor-Goldstein equation, 229 thermocline, 38-41, 161 Thorpe length scale, 165 troposphere, 62, 73-82 turbulence, 4, 106, 109, 161-190, 205, 224 two-layer fluid, 38-39 U undular bore, 2, 18-22, 30, 41-60, 67-73 W waves, see internal waves, mountain waves and solitary waves wind, 119-160, 269