Rayleigh-BeМЃnard convection : structures and dynamics

= w0(x, z) + p(x, y,t)dxw0(x,

z) + wx + w2 + ■ ■ ■

(3.65)

where wi = 0(Vy>), w2 = O(VVi^), etc. The equations of phase dynamics can be found as the solvability conditions of the linear systems of equations obtained at the successive steps of expansion. The phase description is obviously limited to the cases where the amplitude of the rolls, as they are deformed, remains virtually unchanged. This is what makes the phase equation different from the amplitude equation which governs a complex amplitude function and thus takes into account the variations of both the amplitude and the phase. Pomeau and Manneville [62] used the model Swift-Hohenberg equation as the starting equation for constructing the expansion (3.65). In the lowest nontrivial order of this expansion for small t, they obtained the equation of the roll-phase diffusion in the form

d«f = Dnol

(3.66)

DLdly,

where the coefficients Dy and Dx of diffusion along and across the direction of the wavevector k = {fc,0}, respectively, depend on e, fcc, and q = k — kc. Manneville and Piquemal [63, 64] performed a gradient expansion similar to that given by Eq. (3.65) for the full Boussinesq system of equations. They investigated the cases of both rigid and free boundaries but they considered only transverse, or zigzag, disturbances of the original pattern of x-rolls (i.e., perturbations with a y-directed wavevector). A consideration of large-scale drift shows that it is determined by the curvature of the rolls and directed so as to straighten the curved rolls. In the case of rigid layer boundaries, drift does not disrupt the diffusive character of the relaxation of zigzag disturbances. For this reason, Manneville and Piquemal proposed that drift be taken into account by replacing D± with an effective coefficient of transverse diffusion D'f which is greater than DL calculated for the case where drift is neglected. They found this coefficient to be 1

TO {

kc

+

R2(P)

Rc

) '

(3.67)

where ft and T 0 correspond to Eqs. (2.42) and (2.43), N(P) = 0 . 1 6 6 + 2 3 . 0 4 ^ - ' + 6.196P- 2 , and R2{P) = 10.76 - 0.073P" 1 + 0.128P" 2 . Obviously, D± is the value of De± in the limit of P —> oo. If, however, the layer boundaries are free,

3.3. 2D MODELS OF 3D

49

CONVECTION

the process is at finite P values oscillatory rather than diffusive (the oscillatory instability of rolls will be discussed in Sec. 6.3). In Refs. 63 and 64 Df is denoted as D±. The phase equation can also be derived through the procedure of the NewellWhitehead-Segel amplitude expansion [48, 279, 276]. 3.3.6. The Cross-Newell

Phase

Equation

A more general approach to the description of the phase dynamics was originally suggested by Newell and Cross [65, 66] and then substantially extended by Newell et al. [67]. The first noteworthy feature of their technique is that its applicability is not restricted to slightly supercritical regimes since the small parameter used is the inverse aspect ratio rather than a quantity characterizing the supercriticality. Moreover, the flow pattern consists of parallel straight rolls (except at singularities) only locally and can be highly diversified. The deviation of the phase from kx is not required to be small, and the width and orientation of the rolls as well as the mean-drift velocity can vary in space and time over wide limits. It is only necessary that these variations be slow. The method is similar to the nonlinear WKB technique employed by Whitham [68, 69] to study periodic wavetrains in nonlinear dispersive systems. The basic ideas of this approach can be understood most readily from the consideration of simple model equations. For this reason, we start with the simplified version of the method [65, 66] and progress to its complete form based on the Boussinesq equations [67]. The phase equation was first obtained from two models of convection: I. II.

[d, - (A - 1)](A - l)2w + (R - wrw* - vww'A)Aw 2

d,tu + ( A + \) w-

2

Rw + w w'

=0.

= 0.

(3.68) (3.69)

In each equation, w(x,y,t) is a complex scalar field, and R is the driving parameter (the analogue of the Rayleigh number). The linear part of the model equation I is identical to the linearized equation for the vertical velocity component v2, which can be obtained in the Boussinesq approximation when P — oo and the layer has free-slip boundaries and, accordingly, the vertical dependence of v2 has the form sin TTZ. The model equation II differs from the equation (3.45) of the Swift-Hohenbcrg model only in that w is now complex. Each of these model equations has a family of spatially periodic stationary solutions of the form w

= Ae'e,

0 = k • x.

(3.70)

50

3. INVESTIGATION

TOOLS

For these solutions, k = |k| and A are related with one another by the eikonal equation I. II.

R-A*(l-vk*)={k2 R-A2

+

(3.71)

1)3

,

(k2-l)\

=

(3.72)

which reflects the rotational degeneracy of the solutions, since it does not include the orientation of the vector k. When studying a real convective flow, which is more complicated than a system of straight parallel rolls and which carries the imprint of the initial and boundary conditions, it is natural to try to describe the structure of this flow by a locally periodic solution with a local wavevector k continuously and slowly varying over the horizontal plane. We introduce the slow variables X^rfx,

Y = i12y,

(3.73)

T = rjH,

where the small parameter r}2 is the ratio of the characteristic roll width (or the layer thickness) to the horizontal size of the container. In other words, rf is the inverse aspect ratio. We define the local wavevector as the gradient of the phase: k(X,Y,T) = {m,n} = Vx6 = V x 0 , (3.74) where V x = {dx,dy}, V \ = {dx,dy}, slow phase 0 by the equality

and the fast phase 0 is related to the

e(x,y,t) =

-e(x,Y\T).

(3.75)

T We seek solutions to Eqs. (3.68) and (3.69) in the form w{x, y,t) = wW(6; X , V, T) + £ ^vwM{e-

X, Y, T)

(3.76)

p

with w(o){0; X, Y, T) = f(0; A, k) = Ae,e,

(3.77)

where A and k, being no longer constants but rather functions of slow variables X, K, and T, are still related by Eqs. (3.71) and (3.72). The last assumption is in fact a particular form of the slaving principle. On substituting Eq. (3.76) into Eqs. (3.68) and (3.69), solving the resulting ordinary differential equations for u / ° ' , u ; ( 1 ) , . . . , and demanding that each w (p) (p > 1) be a 27r-periodic function of 9, we find nontrivial solvability conditions—successive approximations to the equation of the dynamics of the phase variable Q(X,Y,T). (If R —> Rc and

3.3. 2D MODELS OF 3D

51

CONVECTION

therefore A —> 0, indeterminate forms arise for w^p\ and a special analysis is needed to develop a theory uniformly valid for all A. In this limiting case, A can no longer be regarded as an algebraic function of the local wavenumber and obeys a partial differential equation.) The Cross-Newell (CN) phase equation obtained in this way proves to have a very general form that does not depend on the details of the model. In particular, as it will be seen later, an expansion based on the Boussinesq equations yields precisely this equation in the limit of P —> oo. Furthermore, the form of this phase equation is independent of whether or not the original model is variational: we note that model II is variational whereas model I is not (however, the presence of mean drift is important--see the discussion below). The Cross-Newell equation can be written as

r(k)dTe = - v x • (kfl(A)),

(3.78)

where, for the two models studied, I.

r(fc)

=

B(k)

=

r(k)

=

A2k2(\

-vk2){k2

+

l)\

(3.79)

A^k<(l-uk*)\ A A}

II.

A\

B(k) = A2 —

(3.80)

,

respectively; A2 as a function of k2 is determined by the asymptotic expansions of the eikonal equations (3.71) and (3.72) I.

R - A2(l - vk2) = R0 + v2R2 + iy4R4 +

...,

Ro = (k + l) 3 /fc 2 , 2

II.

R-A

2

2

i

= R0 + u R2 + u' R4 +

(3.81)

...,

Ro =

(k2-l)2,

(3.82)

respectively, with R2 = 0 and R4 involving terms of the form dTA/A, d\AjA, etc. Note t h a t Eq. (3.78) can be written in a different form, convenient for analysing stability: (r,

rn2AB\,^^

2mn dB .

+

„ _

( B + f i ) * 6 " °'<3-83'

To obtain a tool to describe regions where patches of roll patterns with different orientations of rolls are contiguous (grain boundaries—see §4.3.3) and the

3. INVESTIGATION

52

TOOLS

physical quantities vary in space more steeply than within these patches, the authors also derived a phase equation of a higher-order approximation. In order to include mean drift in the analysis, corrections must be introduced into the phase equation. Within the framework of the simplified theory. Cross and Newell [66] made allowances for the drift on the basis of phenomenological considerations, writing the equations of phase diffusion and mean drift in the following modified form:

dTe =

--LVx.(kJ3(«o)-u-k, r(fc) U = V x x (z,

2

AC = 7 [ V x x k ( V x - ( k / l ) ) ] - z ,

(3.84) (3.85) (3.86)

where 7 is a coupling constant. To justify this choice, they considered an appropriate modification of the general formal scheme of the derivation of the phase equation. In particular, such a derivation was carried out on the basis of the full system of Boussinesq equations for a layer with rigid boundaries, but with the additional assumption of small supercriticality. Finally, Newell et at. [67] derived the phase-diffusion and the mean-drift equation on the basis of the full Boussinesq equations for the case of rigid, isothermal layer boundaries and of arbitrary R and P. In their theory, a steadystate, spatially periodic pattern of parallel straight x-rolls is again used as the starting point. The solution of the original nonlinear equations which corresponds to such a pattern is constructed by the Galerkin method. Particularly, the velocity field is assumed to have the form

J2umne'm0g'n(zl

vx =

Y2wmne"n6gn{z),

U2 =

(3.87) (3.88)

m,n

where 8 = kx and gn are the Chandrasckhar functions (see Ref. 3, p. 635), used in solving many problems of hydrodynamic stability. They satisfy the boundary conditions gn(0) = gn(l) = 0, ^ ( 0 ) = g'n(l) = 0. form a complete orthogonal system, and are as follows:

9n(z) = {

sinh/i n (z - 0 ; : ' sinh j / i n

sin/t„ {z - | ) * *- for n even, sin j ^ n

cosh An (z - ^)L ^ cosh j An

cos A„ [z - |Lj ^ for n odd; cos ^Xn

(3.89)

3.3. 2D MODELS OF 3D

53

CONVECTION

here p.n and An are the positive roots of the equations coth \u — cot -p and tanh ^A = —tan ^A, respectively. Similar representations are used for the temperature and pressure perturbations but their vertical structure is expanded in s'mnnz and cos7i7rz, respectively. Given R and k, the coefficients of the series are obtained by numerically solving a nonlinear algebraic equations. To find modulated roll solutions, the investigators apply a technique similar to that developed for the model equations and described above. However, in addition to the solvability condition that yields the phase-diffusion equation, a second solvability condition arises now in the expansion procedure. It results from the fact that an arbitrary slowly varying 2-independent term (i.e., a function of X, V, and 7') can be added to the pressure field without any effect upon the flow dynamics at the roll-width scale but with resulting changes in the large-scale flow. This solvability condition corresponds to the requirement of mass conservation and governs the mean drift. Unlike the phase-diffusion equation obtained at order T]2, the mean-drift equation can only be found at order as high as r]4, because the second solvability condition gives at order r}2 a trivial result. Ultimately, the Newell-Passot-Souli equations for the phase 0 and the horizontal drift velocity V averaged over the vertical coordinate with a certain nontrivial weighting acquire the form

dT0 =

—JTTVX

r(fc)

■ {kB(k)} - p(k)V ■ k,

(3.90)

z • [V x x {ka(kx VxC) • z}] - V x • (k/?k-VxC) = z- V x x (-JjkVx • k.42 - ^-Vx-ktfa)

- V x • {k[Vx x kB0] • z}, (3.91)

where (3.92)

V = V x x (z, 2

k = k/k, and B, Ba, Bp, r, r Q , p, a, /?, and A are explicitly calculated functions of k. The quantity p V , being the analogue of U in Eq. (3.84), plays the role of an effective phase-transporting mean drift (note that, generally speaking, div(pV) ^ 0). The equation (3.91) would reduce to Eq. (3.86) if the horizontal velocity averaged over a 27r-long interval of the variable 9 had a parabolic (Poiseuille) z-distribution (then the terms with BQ and Bp would drop out, and a = P = const). In reality, however, this is by no means always the case.

3. INVESTIGATION

54

TOOLS

As for the practical implementation of the calculations, the differential equations that arise in the order r]2 were reduced, by means of the Galerkin expansion in the same basis as used for the original spatially periodic pattern [see Eqs. (3.87)-(3.89)], to a singular system of algebraic equations. A special, very robust method insensitive to errors was required to solve this system. As the boundary conditions on the sidewalls, the relations (3.93)

k n = V n = 0

can be used over wide limits [67], and where thermal boundary forcing is significant (see Sec. 4.2), the first condition (3.93) is replaced by

(3.94)

k x n = 0.

If P = oo (so that there is no drift) and the coordinate axis A' is directed parallel to the local wavevector k, then the phase-diffusion equation coincides in its form with the Pomeau-Manneville equation (3.66): dTQ = D||(fc)d£e + where

D

" = -4^

(fc))

Dx(k)d$Q,

'

r[k)

(3.95)

(3.96)

Now, however, k = V x © is not a small perturbation of a fixed wavevector. 3.4. T h e o r e t i c a l Approaches: C. N u m e r i c a l S i m u l a t i o n A detailed discussion of methods for numerically solving the equations of convection falls outside the scope of this book. However, it seems useful to present here some brief notes. The numerical methods currently used are diverse but can be broken down into two basic classes: finite-difference and spectral methods. The construction of finite-difference techniques starts from the development of tools to describe local, differential characteristics of the physical fields. For this reason, finitedifference methods are more universal and eventually more widespread. In particular, they can be adapted to various geometries of the problem and, where necessary, by means of introducing nonuniform computational grids, to great drops in variables across the calculated domain. On the other hand, passing from the local to the global description of the fields may cause considerable growth in errors. The spectral (Galerkin) methods represent the fields as finite

3.4. NUMERICAL

SIMULATION

55

sums of basis functions (i.e., as truncated Fourier series), are thus less "flexible" in the indicated sense, and are usually applied to geometrically simpler problems. However, where the problem admits successful use of spectral methods, they demonstrate in many cases substantial advantages over finite-difference methods. These advantages are thoroughly discussed by Orszag [70]. We point out here only one of them, which is highly significant for the practice of calculations. The Galerkin representation is in principle global, and the methods based on it do not imply that small-scale description of the fields must underlie their large-scale description, as in finite-difference methods (where small-scale description and large-scale description are linked, as a rule, through solving systems of a large number of algebraic equations). This removes accumulation of errors that arise in the local representation of the spatial structure of the flow. Therefore, for a given number of degrees of freedom (basis functions in a Galerkin technique or spatial-mesh points in a finite-difference technique), Galerkin approximations typically prove to be considerably more accurate than finite-difference approximations. It should be noted, however, that this is true provided the basis functions are properly chosen. An insight into the finite-difference methods can be obtained from monographs by Anderson et al. [71], Fletcher [72], and Berkovskii and Polevikov [73]. The basic ideas of spectral methods are examined in detail by Orszag and Gottlieb [70, 74]. Fletcher [75] described their numerous variations and particular implementations (with instructive examples) and discussed their intrinsic relations to other numerical techniques. As numerical techniques are applied, the labour content of an investigation is less dependent on the choice of boundary conditions than in analytical treatment, so that the "realistic" case of a layer with rigid boundaries is studied much more frequently (although, as can be seen from what follows, the computational algorithms based on spectral methods are still simpler if the boundaries are free). When simulating convection in an infinite layer, most investigators impose the condition that the physical fields be periodic in horizontal directions, so that the calculated domain represents one spatial period of an infinite periodic pattern. Alternatively, some researchers study the effect of sidewalls on the flow and perform calculations for a cavity of finite horizontal sizes (see, e.g., Refs. 233 and 284). By means of special efforts, the effects of any side boundaries as well as of artificial spatial periodicity can be completely eliminated [285, 268]. When Rayleigh-Benard convection is considered, the geometry of problems is simple, and Galerkin methods are thus of considerable promise for treating them. By way of example, we write down here spectral equations typical of convection problems, using the notation of Ref. 70.

3. INVESTIGATION

56

TOOLS

Let the velocity and the temperature-perturbation field associated with a flow of an incompressible fluid be approximated by the sums v(x,t)=

£

u(k,0eikx,

(3.97)

^(M)e i k x

(3.98)

||k||<||K||

fl(x,i)=

L l|k|KI|K||

Here k = {ku fc2, fc3}, x = {x!, x 2 , x 3 } , each fc0 ( a = 1,2,3) is a multiple of some basic wavenumber in the spatial coordinate .r0, and ||k|| < ||K|| means -K0 < ka < I\'a, thus specifying a finite cutoff of the spectra of v ( x . l ) and 8(x, t). Reality of these fields dictates the relationships u(k,<) = u ( - k , < ) * ,

(3.99)

d(k,t) = ti{-k,ty,

(3.100)

where the asterisk denotes complex conjugation. It can easily be found t h a t in the spectral space the Boussinesq equations (2.15)-(2.17) acquire the following form on the elimination of the pressure:

(jt + pkA Ua(k, o = RP ^

- ^ j *(k,o u,a(p,i)u7(q,t),

(3.101)

«0(p,*)*(q,«),

(3.102)

llp||.l|q||
(%+ k2)d{k,t) = uz{k,t)-ika 0 t \

I

Y.

p+q=k HPII.INKIIKII

kQua(\
(3.103)

where 8ap is the Kronecker delta, and summation over dummy Greek indices is implied. Obviously, these very equations are applicable to the simulation of Rayleigh-Benard convection in an infinite layer with stress-free boundary conditions (2.23) because the basis functions e' k * satisfy these conditions. Where no-slip boundary conditions are posed, Chandrasekhar functions (3.89) or, as suggested by Orszag [76], Chebyshev polynomials are used in most cases as basis functions. An important feature of the Galerkin technique, which raises some problems with the computational efficiency of the method, is the presence of convolution

3.4. NUMERICAL

SIMULATION

57

sums like H u Q ( p , t)us(k - p,t) in the right-hand sides of the spectral equations [see Eqs. (3.101), (3.102)]. Direct calculation of these sums created by the nonlinearity of the original equations proves to be prohibitively time-consuming unless very few harmonics are retained in the expansion (3.97), (3.98) or its analogue. However, this difficulty can be obviated applying the transform method [70]. The algorithm of fast Fourier transform [77], being highly efficient, makes it possible to reduce the calculation of the convolution sums to the multiplication of the fields as they are represented in the physical space. At each step of the computations, the spectra are transformed into spatial fields and their product is subsequently transformed again into the spectrum. As a result, the calculation of a sum is enormously speeded up. For example, if the representation (3.97), (3.98) contains terms with \ka\ values up to 15 times the basic wavenumber of the x Q -direction, the use of the transform method reduces the computer time needed to evaluate the right-hand sides of Eqs. (3.101) and (3.102) by a factor of about 10 4 . The technical details of the calculations can be found in Refs. 70 and 78. A modification of the spectral technique described here makes use of the so-called pseudospectral approximation [70]. In this approach, the terms of the equations are calculated in either the spectral or the physical space, according to whichever representation is more natural. In particular, when transcendentally nonlinear terms are present, they can be computed in the physical space much more easily than in the spectral space. The description of a particular implementation of the pseudospectral method is given, e.g., by Orszag and Kells [79]. Examples of very fruitful applications of spectral techniques to the simulation of three-dimensional Rayleigh-Benard convection appear in papers by Curry et al. [80] and McLaughlin and Orszag [81]. In the former, the stressfree boundary conditions are posed at the horizontal surfaces of the layer, and the method employed closely corresponds to Eqs. (3.97)-(3.103). The latter deals with a layer confined between rigid horizontal boundaries and utilizes a pseudospectral approximation with Chebyshev polynomials as basis functions. A considerable fraction of numerical studies of convection (including those devoted to the problem of wavenumber selection, which is of particular interest for us) is constituted by calculations within the framework of two-dimensional geometry (e.g., Refs. 231 and 232). This seems to be quite natural, since in a wide region of the parameter space convection creates quasi-two-dimensional roll patterns (see Chapters 4 and 5). Many important questions can be elucidated along this path, and two-dimensional numerical experiments are still of significance.

58

3. INVESTIGATION

TOOLS

Calculations of three-dimensional convection started in early 1970s. First they were performed for boxes with small aspect ratios [204, 286] and more recently, also for regions of large horizontal extent. For example, in Ref. 284 the box measures 11.5/i x 16/i. Nevertheless, comprehensive investigations of the dynamics of three-dimensional convection by means of numerical experiment still remain very timeconsuming. The two-dimensional models that have been developed to reproduce essential features of the spatiotemporal dynamics of three-dimensional convection (see Sec. 3.3) make it possible (within the restrictions of the models themselves) to form, using relatively economical means, an idea of the evolution of complex convection patterns. For this reason, many studies have already appeared, in which three-dimensional convection is simulated numerically on the basis of the two-dimensional equations of these models instead of the original hydrodynamic equations (Refs. 151, 152, 184, 241, 254, 261, 287 and others).

CHAPTER 4

BASIC TYPES OF CONVECTIVE-FLOW STRUCTURES

4 . 1 . T w o - D i m e n s i o n a l Rolls and T h r e e - D i m e n s i o n a l Cells It was noted in Sec. 2.5 that, as long as supercritical regimes of RayleighBenard convection are considered within the framework of the linear problem, preference cannot be given to any convection-cell planform because of the degeneracy of the solutions with respect to planforms. This degeneracy cannot be removed by the mere construction of steady solutions to the nonlinear problem. Given R and k, such a solution can be obtained for any planform function w(x) satisfying the equation Aw + k2w — 0. Thus, there arises the question of determining the planforms that would be really observed under the corresponding conditions. The stability analysis of various planforms can be applied as the first step in resolving this issue. To gain its more complete understanding, the initial-value problem must be considered with a variety of initial perturbations. The most likely (preferred) planforms could be revealed in this way. However, there are as yet no comprehensive investigations of this sort. As pointed out above, Benard [18] observed flow patterns consisting of polygonal, mainly hexagonal, cells with upflow at the centre of each cell and downflow at the periphery (Figs. 2b, 5). The most regular patterns of this type have the honeycomb symmetry. Under the influence of Benard's experiments, there emerged the opinion that steady-state laminar convection in a horizontal layer usually creates hexagonal cells (Benard cells). Subsequent investigations evidenced a radical difference in flow patterns between the cases where the top surface of the layer is free (as in Benard's studies) and where the layer is covered with a solid lid at the top. If such a lid is present and P is not too small, in a rather wide range of parameter values the well-established flow has typically the form of a collection of rolls (Figs. 2a, 6). Then, apart from usually present irregularities, or pattern defects, the velocity field of roll convection is nearly two-dimensional (see, e.g., Refs. 164, 199, and 135). We mean here the 59

60

4. TYPES OF FLOW

STRUCTURES

Fig. 5. Polygonal convection cells in a 0.81-mm-thick layer of molten spermacety (an original photograph obtained by Benard [18]). "main" convective flow that arises as a secondary state of the fluid layer after the primary, motionless state loses its stability, and assume that this flow has not experienced its own instabilities (unavoidable at sufficiently large R) and, therefore, is not complicated by tertiary flows. Moreover, we assume t h a t there are no factors that would cause the conditions in the layer to deviate appreciably from those of the standard problem (such factors will be discussed below). Benard himself already suggested that the temperature dependence of the surface tension plays an important role in the formation of the convection patterns that he observed. This was later confirmed, and according to the current interpretation, the emergence of hexagonal cells in Benard's experiments was governed by a sufficiently strong thermocapillary effect." In contrast, buoyancydriven (thermal, or thermogravitational) convection mainly produces roll patterns. Nevertheless, the incorrect assertion that Benard's hexagonal cells are normally generated by the thermogravitational mechanism under the conditions of the standard problem is still encountered in the literature. We note, however, that this mechanism does not yet rule out the possibility of the existence of stable hexagons under these conditions but in a very narrow parameter range, as evidenced by certain recent experimental observations and theoretical analyses (see §4.1.11). We assume the statement that, under the standard conditions, quasi-twodimensional rolls represent the basic form of steady-state convection as the starting point of our discussion and consider here the most important of the known factors that can make three-dimensional cells preferred. "Of course, the idealized boundary conditions (2.14) usually imposed on a "free" boundary in theoretical analysis do not make hexagons preferred, since these conditions do not take surface tension into account.

4.1. 2D ROLLS AND 3D CELLS

61

Fig. 6. A pattern of quasi-two-dimensional convection rolls observed in a layer of ethyl alcohol at R = 5.06 x 103 by V. S. Berdmkov and V. A. Markov (unpublished). Defects called dislocations are seen.

4-1.1.

Thermocapillary

Effect

Block [82] was the first to give a conclusive experimental demonstration of the role of surface tension in the formation of hexagonal cells. In particular, he managed to observe cells at R < Rc (and even R < 0), when the thermogravitational mechanism does not operate. We shall not go into detail in discussing thermocapillary convection, or Benard Marangoni convection. It is worth addressing the reader to Chapters 1, 3, and 4 of Koschmieder's book [7], which can be used as an excellent introduction to this subject. Thermocapillary convection has by now been investigated much less thoroughly than thermogravitational convection. Significant experimental results were obtained by Koschmieder and his coworkers [83-85]. An interesting comparison of the experimental and theoretical data on the planform selection under the combined action of two instabilities, the thermogravitational and the thermocapillary one, was made by Perez-Garci'a et al. [86]. It was found experimentally [85] that in the case of the two mechanisms competing, as A T grows, the characteristic size of Benard cells first decreases after the onset of convection

62

4. TYPES OF FLOW

STRUCTURES

and subsequently increases. Since typical of therrnogravitational convection is the growth of the flow scale with A T (see Sec. 6.1), it was concluded that for sufficiently large values of A T the Rayleigh mechanism predominates. Zerogravity experiments on thermocapillary convection—with the Rayleigh mechanism "switched off"—have been performed aboard spacecraft [87]. Theoretical investigations of Benard-Marangoni convection were started with the linear stability analysis by Pearson [88], who assumed the absence of gravity, and Nield [89], who considered a combination of the two mechanisms. The planforms of thermocapillary convection were studied within the framework of nonlinear problems by Scanlon and Segel [90] (whose quantitative results need to be made more accurate, as pointed out in Ref. 52), Kuznotsov and Spektor [52] (see §3.2.2), Kraska and Sani [91], and Cloot and Lebon [92]. 4-1-2. Temperature Dependence of Viscosity Back in the 1930s Graham [93] discovered, when observing polygonal convection cells in a layer of air, that under steady-state conditions the air descends in the central part of a cell and ascends in its peripheral parts. Such cells, usually observed in gases, are accordingly referred to as g-cells to distinguish them from l-cells, typical of liquids, with the opposite direction of circulation (Fig. 2b). Meteorologists call /-cells closed and g-cells open. Graham argued that the direction of circulation should depend on the sign of the derivative dv/dT which is as a rule negative for liquids and positive for gases. This hypothesis was confirmed by Tippelskirch [94] in experiments with liquid sulphur, for which du/dT < 0 at T < 153°C and dvfdT > 0 at 153°C < T < 200°C. Convection cells in these two temperature ranges were observed to be indeed of the /- and the g-type, respectively. The ascending fluid in a convection cell is always warmer than the descending fluid. Therefore, the central part of an /-cell is less viscous for liquids, and the central part of a gr-cell, for gases. We see that the realized direction of circulation is that at which viscous stresses are weaker near the axis of the cell, where they are maximum in virtue of the flow geometry. There is a number of theoretical studies on the conditions for the formation of various planforms in slightly supercritical regimes, based on the consideration of interacting sets of parallel rolls. The basic roll wavenumber is usually put equal to the critical wavenumber kc for each set, and it is assumed that the angles between the rolls of different sets are such that their wavevectors form resonant combinations, e.g., k] + k 2 + 1<3 = 0.

4.1. 2D ROLLS AND 3D CELLS

63

Palm [95] studied, using an approach of this type, convection in a fluid whose kinematic viscosity depends on temperature as v = is0 + i cos n(T - Ti),

(4.1)

where (7/1^0)2
) cos ( -kcy)

+ Z(t) cos kcy

(4.2)

[cf. Eq. (2.49)]. An analysis of the nonlinear ordinary differential equations that describe the time variation of the amplitudes Y and Z shows that the stationary state of the system in the limit of t —> 00 is characterized by the relation Y = 2Z, thus corresponding to hexagonal cells. In agreement with the law deduced from the experiments, the direction of circulation in a cell is determined by the sign of the coefficient 7. A detailed investigation [96, 97] of the set of equations for Y and Z, corrected by Palm, showed that for sufficiently large 7 values only hexagonal-cell patterns are stable near R = Rc. The conclusion that the direction of circulation is determined by the sign of 7 remains valid. In particular, hexagonal cells are possible at subcritical R values if the amplitude of the initial perturbation is sufficiently large (hard excitation). A wider set of interacting modes was considered by Segel [98], who analysed the equations of a form similar to that of Eqs. (3.25). It was found that two-dimensional rolls, being unstable immediately beyond the bifurcation point R = Rc, as R increases, become stable at a certain R value, and hexagons lose their stability at an even greater R. Subsequently, Davis and Segel [99], applying a technique of this type, assumed not only the viscosity but also other material parameters of the fluid to be temperature-dependent and allowed for the deformation of the free layer surfaces. They obtained similar results. Conditions for the existence and stability of flows with various cell planforms will be discussed below in more detail. It is obvious that the coexistence of different linearly stable stationary solutions at a certain point of the parameter space implies t h a t transitions between the corresponding forms of fluid motion are possible only under the effect of finite-amplitude perturbations if they are

64

4. TYPES OF FLOW

STRUCTURES

superimposed onto the existing flow and have an appropriate structure. Let a parameter (say. R) be varied cyclically over some range including the band where the two solutions are linearly stable. Then one state will be spontaneously replaced with the other only after its stability threshold is passed, so that transitions between the two states will demonstrate hysteresis. Thus, in the situation considered here (and in all similar situations that will be encountered repeatedly) hysteresis will accompany the transitions motionless state £ hexagonal cells in the region of finite-amplitude subcritical instability of the motionless state, as well as the transitions hexagonal cells £ rolls in the region where both these types of motion are stable. In the above-mentioned studies [95-99] the horizontal boundaries of the layer were assumed to be free. For other boundary conditions the range of Rayleigh numbers where hexagonal cells are stable was calculated by Palm tt al. [100]. Systematic studies of the stability of convective flows trace back to the paper by Schliiter, Lortz, and Busse [41] already cited above. These investigators, expanding the Boussinesq equations (with v — const) in powers of the small flow amplitude e1''2, obtained a steady-state solution to the weakly nonlinear problem in a rather general form that represents geometrically diverse flows. The stability of these flows against a wide class of infinitesimal perturbations was analysed in the linear approximation. The results of Ref. 41, when combined with those of Refs. 95-100, form a unified picture, from which the role of the temperature dependence of viscosity is clearly seen. According to Ref. 41, in the case v = const all three-dimensional flows are unstable near the onset of convection, and there exists only a class of stable two-dimensional roll flows. This result is valid irrespective of whether the layer boundaries are free or rigid. For a wide range of values of the Rayleigh number and the Prandtl number, the stability of convection rolls against perturbations of various structure will be considered in Sec. 6.3. A method similar to that developed in Ref. 41 was applied by Busse [43, 44] (see Sec. 3.2) to a fluid whose viscosity weakly depends on temperature. (Moreover, the same was in general admitted for other material properties. The effects of their temperature dependence will be discussed below, in §4.1.3.) The stability of rolls and hexagonal /- and g-cells can be characterized by a diagram obtained in this way and shown in Fig. 7. The form of this diagram is qualitatively independent of the type of boundary conditions. In the case at

4.1. 2D ROLLS

AND 3D

CELLS

65

Fig. 7. The regions of stability of rolls and hexagonal /- and (/-cells in the (<2,('/ 2 )-plane (adapted from Refs. 12, 44); Q characterizes the vertical nonuniformity of the layer and c 1 / 2 is the amplitude of the flow and (see text).

hand, the parameter Q is proportional to the (small) coefficient in the leadingorder (linear) term of the dependence v — v{T).b Fig. 8 shows the dependence of the amplitudes of these flows on 7?. Note that the parameter e1?2 is used here in the sense of Eqs. (3.1)—(3.3) rather than according to the original definition given by Eq. (2.41); the last relationship thus holds only to the lowest order. The results represented by Figs. 7 and 8 agree with the principal findings of Refs. 9 5 100. They retain qualitatively the same form where temperature variations of other parameters are admitted, Q being correspondingly modified in this case (see §4.1.3). As can be seen from Fig. 8, departures from the Boussinesq approximation make the supercritical bifurcation imperfect, so that hexagonal cells become possible below R = Rc. Busse has calculated the critical values Ra, Rr, and /?(, t h a t determine the regions of stability for different forms of convective flows. The first such region is Ra < R < Rc, where the critical Rayleigh number Rc has the sense of the bifurcation point and, in general, slightly differs from the respective quantity found in the Boussinesq approximation. In this region, except for the stable solution representing the motionless state, only hexagonalcell solutions exist, and those are stable which correspond to a certain direction of the fluid circulation in a cell (determined by the sign of Q). For Rc < R < Rr hexagons remain stable but, in addition, unstable roll-type solutions are present. For Rr < R < Rb motions of both types are stable. And finally, for R > Ri, rolls are stable and hexagons are unstable. 'This parameter is designated as P by Busse and as V in some other papers.

66

4. TYPES OF FLOW

STRUCTURES

Fig. 8. The amplitudes of rolls and hexagonal /- and ff-cells vs. the Rayleigh number R (adapted from Refs. 12, 44); e 1 / 2 and Q have the same meaning as in Fig. 7. The solid lines pertain to stable flows and the dashed lines pertain to unstable flows.

To be more precise, the coexisting stable states are not equally stable because they are characterized by different values of the specific potential (the Lyapunov functional calculated per unit area, see Sec. 6.4). This potential was calculated for different patterns, in particular, by Malomed et al. [244]. An absolutely stable state corresponds to the global minimum of the specific potential, whereas metastable states correspond to local minima (we recall t h a t for uniform patterns, e.g., those described by Eqs. (3.25), the potential is merely a function of the amplitudes of the roll sets of which any such pattern is composed). Accordingly, the conduction regime is absolutely stable for R < RT < Rc, hexagons for RT < R < RT,, and rolls for R > RT,, where the threshold values RT and RT, have the following meaning: at R - RT the conduction state and the hexagonal-cell flow have the same value of the potential, and at R - RT, the hexagonal-cell flow and the roll flow are characterized by the same potential value. Near these thresholds the state with the greater potential is not absolutely stable but metastable and can be replaced with the absolutely stable state on imposing a sufficiently strong disturbance. The G r a n g e where the absolute stability of one mode coexists with the metastability of the other is thus the region of hysteretic transitions between these modes. Such a transition occurs typically by propagation of a front separating the regions of the two states. The state that spreads is characterized by the lower potential. Experimental investigations confirmed the general features of the theoretically predicted regularities. Fluids with different behaviour of viscosity can form different convection structures under similar experimental conditions [101]. In experiments with a

4.1. 2D ROLLS AND 3D CELLS

67

Fig. 9. Hexagonal /-cells observed in an experiment with controlled initial conditions as a result of the evolution of an unstable roll pattern at R = 2460, fmax/"min = 7 [102]. silicone oil that has a slightly varying viscosity, either convection rolls (in a layer bounded by a lid at the top) or hexagons (in a layer with a free upper surface, where the thermocapillary effect can operate) were observed. If, however, the experimenters used aroclor, whose viscosity is strongly temperature-dependent, hexagons appeared also in a closed layer provided it was sufficiently thin and, correspondingly, the temperature gradient was sufficiently large. In his experiments, Richter [102] produced controlled initial disturbances corresponding to the roll geometry and studied k- and /^-dependence of the stability of the induced rolls for different values of the ratio fmax/^min of the maximum and the minimum viscosity within the layer (R was calculated with using the value U{TQ) of the viscosity at the temperature T0 = (Ti + T 2 )/2; this definition is adopted in most papers). For the working fluids employed (glycerine and polybutene oil), the quantity vmax/i/m\n reached approximately 20. It was found that for sufficiently small values of R and large values of t-Wx/t'min the rolls change in their evolution into hexagonal cells. If R is thereafter increased, the reverse transition occurs the more easily, the less regular the pattern of hexagons was. An example of hexagonal-cell patterns obtained in Richter's experiments from unstable roll patterns is shown in Fig. 9. Much larger ratios fmax/^min (up to 3400) were reached using anhydrous glycerol in experiments with uncontrolled initial conditions carried out by Stengel, Oliver, and Booker [103, 104]. A summary of the results of their study is presented in Fig. 10. It was found that the transition from hexagons to rolls as R increases is observed only if the parameter rj = \n(um^x/umm) does not exceed, roughly speaking, 2. In this case the range of Rayleigh numbers where hexagons are stable becomes wider with increasing r), in agreement with Busse's theoret-

68

4. TYPES OF FLOW

STRUCTURES

Fig. 10. Experimental observations of various convection-cell planforms in fluids with temperature-dependent viscosity (adapted from Ref. 104); t) = ln(f m a x /i/ m i„); the Rayleigh number R is calculated on the basis of the viscosity value I>(TQ). In the stability region of rolls, the regimes studied are shown with circles. Hexagons (region 6) and squares (region 4) represent patterns of stable hexagonal and square cells, respectively. Triangles (shaded transitional region) denote patterns consisting of irregular tetra-, penta-, and hexagons. Dots correspond to the motionless state. Curve 5 illustrates the dependence of the critical Rayleigh number RQ on J7 for a layer with boundaries of finite thermal conductivity, as calculated in Ref. 103; curve B represents the theoretical upper limiting values of R for the stability of hexagonal cells [43], normalized by the indicated Rc{l) values.

ical predictions [43, 44]. For larger rj values, however, as R increases, hexagons transform not into rolls but rather into a pattern of irregular tetra-, penta-, and hexagons; in this case the interval of stability of hexagons, on the contrary, shrinks with increasing 77. As R increases further, the irregular patterns of polygons in turn transform into systems of square cells if 77 > 4. It is not surprising that wide viscosity variations significantly affect the critical Rayleigh number. The theoretical section of Ref. 103 is dedicated to a linear analysis aimed at investigating the behaviour of Rc as a function of TJ for various boundary conditions. A particular version of such a dependence is shown in Fig. 10 (the curve labelled S) and corresponds to the case where the lower boundary has a finite heat conductivity typical of glass (it is a glass

4.1. 2D ROLLS AND 3D CELLS

69

bottom plate that was used in the experimental study of Ref. 104). We shall discuss variations in Rc along with the respective variations in the form of the eigenfunctions of the linear problem, in Chapter 7. In experiments with golden syrup under controlled initial conditions, White [105] investigated the stability of squares and hexagons for R < 63 000 {R being calculated as in Ref. 102) and vm!iX/vmm < 10 3 . Bounded stability regions in the (fc, i?)-plane were found for both planforms. They are similar to the so-called "Busse balloons" obtained for rolls (see Sec. 6.3). As ^ m a x / ^ m i n increases, these regions shift to higher k values. Subcritical (with respect to Rc calculated from the dependence v(T) of the type realized in the experiment) convection was observed in the form of squares rather than hexagons. The author describes different instabilities and many types of transitions between flows of various structure. Theoretical investigations of conditions for the existence of square cells started with an analysis of their stability carried out by Busse et al. [106, 107]. As in an extensive series of studies of roll stability (Sec. 6.3), the main (original) flow was calculated by the Galerkin method and the stability of this flow was analysed in the linear approximation. The boundaries of the layer were assumed to be rigid. It was found that in the case of uniform viscosity square cells are unstable [106]. For the case of linear temperature dependence of the dynamic viscosity JJ. the stability of such cells for given R depends on the ratio T — ^max/A'min [107]. The authors of the cited study put the wavenumber of the original flow to be equal to kc and restrict the analysis to the assumption that the disturbances have the same symmetry and the same wavenumber as the main flow. It was found that as r increases, squares become stable first and subsequently rolls become unstable. Jenkins [108] analysed later the possibility of the existence of square cells by considering the evolution of a flow with the planform function w(x) = A(t) cos kcx +

B(t)coskcy

(4.3)

[cf. Eq. (2.50)] for a small supercriticality, rigid boundaries, and two variants of the function n(T)—linear and exponential. The effect of finite thermal conductivity of the boundaries was also studied (we discuss it below). For a linear function n{T) the results agree qualitatively with the findings of Busse and Frick [107] but the critical values of the parameter r which correspond to transitions between planforms were found to differ from those of Ref. 107. As Jenkins claims, the disagreement is caused by the fact that the method used in Ref. 107 is applicable for only small values of r. For an exponential function /u(7'), critical values n , r i , and r 3 of this parameter, bounding the regions of different flow regimes, were found. For r < fj the flow exhibits a roll structure, for r 1 < r < r 2

4. TYPES OF FLOW

70

STRUCTURES

square cells exist in supercritical regimes, for r 2 < r < r3 subcritical convection in the form of squares is possible, and for r > r3 both rolls and squares can exist under subcritical conditions. Stuart [109] stated his opinion that the solution with the planform function (2.50) is unphysical, since it does not incorporate all characteristic features of the actually observed convection cells. However, a numerical simulation of light refraction in shadowgraph visualization performed by Jenkins [110] with automatic output of the results in the form of model "shadowgraph patterns" by using a laser printer showed that flows of the form (2.50) produce patterns very similar to the observed ones. It is interesting that such patterns do not at all resemble a checkerboard, as could be expected on the basis of the relative arrangement of warm upflows and cold downflows. The matter is t h a t shadowgraph visualization does not involve simple vertical averaging of the temperature (for this reason, the pattern can change radically its appearance as the sign of the convective velocity is changed). Instead of a checkerboard pattern, something like a negative image of square-lined paper —dark squares with light boundaries —is observed. 4.1.3.

Temperature

Dependence of Other Material Parameters

of the Fluid

The variation of other characteristics of the fluid with temperature plays in principle the same role as the variation of the viscosity. In Busse's studies mentioned above [43, 44], departures from the Boussinesq approximation are admitted in the following form. The temperature dependence of the thermalexpansion coefficient is written as a = «o [l + ^ ( T

- To) + 0((T

- T0)2)

,

(4.4)

where To = (Ti + T 2 )/2 [cf. Eq. (3.5)]. Quite similar dependences are assumed for the kinematic viscosity, thermal conductivity, and specific heat at constant pressure (with a coefficients 72, 73, and 74 instead of 71, respectively). A stability analysis of rolls and hexagonal cells reveals the determining role of the parameter Q = YA=\ liPi (where the coefficients p, depend on the Prandtl number P if both layer boundaries are free and are constant in the considered approximation if at least one boundary is rigid). 0 The stability diagram and the plots of the fl-variation of the amplitudes of different convection modes are qualitatively the same as in the case where the viscosity alone is temperature-dependent (Figs. 7 and 8, respectively). These diagrams thus reflect the influence of the nonunic

See footnote 6 on page 65.

4.1. 2D ROLLS AND 3D CELLS

71

formity of each quantity mentioned. The non-Boussinesq effects are estimated to be negligible if 6Q2/RC < 1. A similar conclusion concerning the character of the transition from hexagons to rolls with increasing R (including the possibility of the finite-amplitude excitation of a hexagonal-cell flow in the subcritical regime) was drawn by Davis and Segel [99], who revealed this effect as a consequence of the temperature dependence of the viscosity, thermal diffusivity, and thermal-expansion coefficient. The approach employed in Ref. 99 was similar to that used in Refs. 95, 96, and 98. The experiments of Somerscales and Dougherty [111] show that the temperature dependence of the physical properties of a fluid manifests itself in the flow structure the more sharply, the thinner the layer and the greater A T The effect of the temperature dependence of the thermal-expansion coefficient on the convection pattern was observed by Dubois et al. [112] in experiments with water at temperatures close to 4°C (a varying from zero to finite values in this temperature range). Laser Doppler velocimetry shows that near the critical convection regime the spatial distribution of the vertical velocity has a form typical of a system of hexagonal cells. As A T increases, hysteretic transition to roll convection occurs. Walden and Ahlers' experiments with liquid helium [113], which also has a density maximum (at a temperature of 2.178 K), point indirectly to the same effect. Since flows of liquid helium fail to be visualized, only the behaviour of the heat flux through the layer as a function of the Rayleigh number was investigated. Two breaks in the slope of the curve of this dependence were interpreted as transitions from the motionless state to hexagons and from hexagons to rolls. In both cases the dependence exhibits hysteresis. Bodenschatz et al. [114], using shadowgraph visualization, experimentally investigated transitions between the conduction regime (motionless state), hexagonal-cell convection, and roll convection in gaseous C 0 2 at a pressure of 21.8 bar. This gas is slightly non-Boussinesq in behaviour, all the mentioned material parameters being temperature-dependent and making significant contributions to the quantity Q [115]. A cylindrical container with T = 86 was used. Subcritical and only weakly supercritical regimes were studied. As R is increased quasi-statically, a small patch of a hexagonal-cell pattern appears. It is clearly seen that the pattern is a superposition of three sets of straight parallel rolls. T h e patch grows by a six-faceted, moving front, and the process of the establishment of the final patch size takes about 750T V . On increasing R further, this size increases (so that the hexagonal pattern fills the container full at R ~ 1.02/?c), and on decreasing R, it decreases. The threshold values of t = (R — Rc)/Rc corresponding to RT and Ra were found to be t-/- = — (2.0 ± 0.1) x 1 0 - 3 and

72

4. TYPES OF FLOW

STRUCTURES

Fig. 11. Two-armed spiral observed at t = 0.15 [114]. The spiral rotates clockwise with a. period of ~ 2400rv. ca — —(2.3 ± 0 . 1 ) x 10" 3 . When finite jumps in R are made, the elimination of pattern defects and the formation of a well-ordered pattern is accomplished in ~ 15TI,. Transitions between hexagons and rolls were observed in a narrow vicinity of c = tj' ~ 0.108, while t = Ct was not reached (here £T> and £(, correspond to RT' and /?(,; we recall that R = RT' is the point where the specific potentials calculated for a uniform roll pattern and for a uniform hexagonal-cell pattern are equal). A transition occurred first at the outer wall of the container, subsequently involving regions closer to the centre. A roll pattern that formed on increasing R was ordered into a left- or right-handed spiral with the number of arms varying from run to run within a range from 0 to 13 (Fig. 11). The outer part of such a pattern comprises concentric circular rolls. Each arm of the spiral terminates in a pattern defect called dislocation (see §4.3.1 and Fig. 24). The spiral is therefore matched with the outer rings. The spiral rotates in such a direction that the resulting waves propagate out from the spiral core

4.1. 2D ROLLS AND 3D CELLS

73

(from the centre of the container). The formation of "global" spiral patterns fitting into the container geometry is indubitably an effect of a small horizontal temperature gradient near the wall, which produces sidewall forcing (a detailed discussion of this effect will be given in Sec. 4.2 and §6.5.8). Ciliberto et al. [116-118] also experimentally studied hysteretic transitions between hexagons and rolls in a fluid exhibiting slightly non-Boussinesq properties. They used circular containers with V = 18 and 20 filled with water at temperatures of about 28°C {P w 5.6-5.8). The stability thresholds found experimentally did not agree well with the theoretical values [118]. The discrepancies could result from the finiteness of the container size [118, 114]. R e m a r k . The above-studied cases of preference for three-dimensional cells are unified by a common feature: in the fluid layer there is an appreciable asymmetry of the physical conditions with respect to the midplane z — 1/2 (an up-down asymmetry). If, however, the layer is symmetric, then, as a rule, twodimensional rolls arise. This is not difficult to understand: a transition from some roll set to its mirror reflection about the midplane is equivalent [at least, to the approximation represented by only the lowest harmonic with the planform (2.49)] to a uniform translation of the entire pattern in the direction of the vector k. Three-dimensional cells do not share this property. It is thus not surprising that just rolls are typical for the cases where the top and the bottom part of the layer are indistinguishable. On the other hand, the existence of hexagonal /- and g-cells is obviously compatible with the presence of the nonuniformity of the viscosity: the direction of circulation turns out to be such that the viscosity is minimum in the region of highest strain rates—in the central part of a cell. (If hexagonal cells are nevertheless stable in a layer symmetric with respect to the midplane, Z- and g-hexagons are equally probable; we shall discuss this possibility in §4.1.11.) One may expect that three-dimensional cells will also be preferred in some other cases with up-down dissymmetry, which are considered below. J.l.J.

Asymmetry

of Boundary

Conditions

A situation with broken mirror symmetry of the layer arises when different boundary conditions are posed at the top and the bottom layer surface. However, the effect of the preference of hexagonal cells proves to be subtle. By means of weakly nonlinear analysis, Busse found in his already mentioned study [43] t h a t the asymmetry of boundary conditions is yet insufficient for such a preference at Rayleigh number values close to the critical one. Only if the terms of the fourth order in amplitude are taken into account, this preference

71

4. TYPES OF FLOW

STRUCTURES

manifests itself. If the bottom boundary is rigid and the top boundary is free, /-cells are preferred. A linear analysis of the stability of two-dimensional convection rolls, carried out for a wide range of Rayleigh numbers and based on a Galerkin representation of the original roll flow (see §6.3.1 for more detail), detected among the disturbances that could initiate the transition to hexagonal-cell convection only decaying ones [119]. 4.1-5. Curvature of the Unperturbed Temperature

Profile

Such a curvature (which may be due to internal heat sources) also facilitates the formation of hexagonal cells [120]. We note that time-dependent heating, which can result in the origin of hexagons (see below), also induces the curvature of the temperature profile that should be considered to be unperturbed under these conditions. 4-1.6. Finite Thermal Conductivity

of Horizontal

Boundaries

Square cells can arise as a result of significant departures from condition (2.12) realizable only if the thermal conductivity of the horizontal boundaries of the layer is infinite. Let the parameter £ defined as the ratio of the thermal conductivity of the plates bounding the layer to the thermal conductivity of the fluid be small. Then the stability analysis of different steady-state solutions with k = kc, performed by Busse and Riahi [121] on the basis of the expansion procedure of Ref. 41 with taking into account this parameter, leads for P = oo and infinitely thick plates to the following result. Only those solutions are physically realizable for which N = 2 [see Eq. (2.30)] and the vectors k] and k 2 make an angle lying between 60° and 120°. Square cells are distinguished among others by the property t h a t they ensure maximum heat transfer. Moreover, the most rapidly growing disturbances of roll flows (unstable under these conditions) tend to transform the rolls into a system of squares. These results, as the authors of Ref. 121 showed, should not change substantially for finite P and finite thickness of the plates. It was subsequently found by Proctor [122] that square cells are most stable in the case of finite amplitudes as well. On the basis of the fact that at small ( the horizontal flow scale is much larger than the thickness of the layer, an expansion similar to that used in the well-known "shallow-water" theories was applied. The thickness of the bounding plates was assumed to be finite.

4.1. 2D ROLLS AND 3D CELLS

75

The stability of flows with different planforms was investigated by employing a variational principle formulated in Ref. 122. An analysis of the equations describing the evolution of the planform function (4.3), which were obtained by expansion in small amplitudes (compare the above-mentioned papers [95-100, 108]), permitted Jenkins and Proctor [123] to find the critical value of the parameter £ at which a transition from rolls to squares occurs. It depends on the ratio of the thickness of the plates to the thickness of the fluid layer and on P. For very small P values (e.g., for values characteristic of mercury, P = 0.025) the critical value of £ is very small and squares are possible only if the plates are virtually perfect thermal insulators. If P is large, squares arise even if the thermal conductivity of the plates is comparable with that of the fluid. Finally, Riahi [124], using the technique of Ref. 41, performed a linear stability analysis of stationary flows with k = kc near R = Rc under the assumption of thick bounding plates and, in general, different (not small) values of the parameter £ for the bottom and the top boundary of the layer (£b and £t , respectively). It was shown t h a t rolls are unstable when squares are stable and vice versa, and all three-dimensional solutions with N > 2 (including hexagonal-cell patterns) are always unstable. The stability diagram in the space (£ t ,£b, P) is presented

in Fig. 12. Le Gal et al. [125] observed square cells near the threshold of convective instability (at e < 0.024) in an experiment with silicone oil (P — 70) with £ = 7 (glass plates). A circular container with T = 20 was used. In the range 0.024 < t < 0.057 the amplitudes of two mutually perpendicular roll sets underwent periodic oscillations in antiphase with one another; as e was increased, one set became predominant and then (at sufficiently large e) a unique steady-state roll set was established (apart from two regions near the outer wall, where the structure remained more complicated). This experiment was subsequently modified by Le Gal and Croquette [126]: plexiglass was used for the boundaries and water was used as the working fluid, so t h a t £ = 0.4 and P ss 7. A rather prolonged process of the flow-structure settling resulted in a system of square cells, which, in contrast to the preceding experiment, was observed in a wide range of t values without any signs of destabilization. The authors believe that in the first case the silicone oil behaved as a mixture and the observed features were governed by the Soret effect. T h e finite thermal conductivity of the boundaries was also taken into account to compare the theoretical results with experiment in the already mentioned work of Jenkins [108], devoted to studying the role of the temperature dependence of viscosity v{T). According to the calculations, subcritical convection in the form of squares should be observed at larger values of tWx/t'min

76

4. TYPES OF FLOW

STRUCTURES

Fig. 12. Diagram of stability of square cells and rolls in the (Ct.Cb, P)-space (adapted from Ref. 124). In each plane P = const squares are stable inside the region bounded by the heavy line and rolls are stable outside this region.

than in the experiment of White [105]. The cause of this discrepancy was not revealed in Ref. 108. 4-1.7. Deformation

of the Free Surface

Thus far it was assumed everywhere that the layer boundaries do not deform, even if they are free in the sense that tangential stresses vanish. According to Davis and Segel [99], allowance for the deformability of the free surfaces (one or both) reveals effects of the same type as in the case of departures from the Boussinesq approximation. This conclusion was drawn with neglecting surface tension, but the effect of deformation is appreciable only in very thin layers, where the effect of surface tension is also significant.

4.1. 2D ROLLS AND 3D CELLS

77

A similar result was obtained by Kuznetsov and Spektor [52] (see §3.3.1). These authors found the following criterion for deformation effects to be significant:

d\nfi (fj. is the dynamic viscosity). For most liquids this derivative is small and approaches unity near the inversion point, where d^/dT - 0. In particular, for sulphur the inversion takes place at T = 153°C. 4-1.8.

Time-Dependent

Heating

Krishnamurti [127] studied the case where the temperatures of both layer surfaces vary slowly and linearly in time in such a way that the Rayleigh number remains constant and the unperturbed temperature profile in dimensional variables has the form

T = r]t-—-z h

+

2\

l-z(z-h).

(4.5)

Here 77 = const and \ IS t n e thermal diffusivity of the fluid. On the basis of an expansion in small parameters—the perturbation amplitude and the rate of heating (cooling) 77—and of a stability analysis of rolls and hexagons (as in Ref. 41), Krishnamurti showed that the effect of non-steady-state heating is entirely similar to that of a temperature dependence of the physical properties of the fluid (considered in Refs. 43 and 44). In particular, the stability diagram has the form shown in Fig. 7, if Q stands for the parameter 77. In the case of the increase of the boundary temperatures (77 > 0) stability is possible for 9-hexagons while in the case of cooling (77 < 0) /-hexagons can be stable. Krishnamurti [128] described an experiment in which hexagons were actually observed under conditions of non-steady-state heating and the direction of circulation in the cells agreed with the theory. Similar results were obtained by Zhelnin [129] in numerical simulations of three-dimensional flows under the same conditions, although the cells did not appear precisely hexagonal. An effect of the same type can be caused by a periodic temporal modulation of the temperature difference between the bottom and the top surface of the layer. Let this difference vary as AT(1 + Scosuit), where A T is its average value. Then the expansion of the Boussinesq equations in the small flow amplitude and the small amplitude S of modulation with a subsequent stability analysis, carried out by Roppo et al. [130] and valid for S/u> < 0.06, shows that near the convection threshold (including a narrow range of subcritical values of R) hexagonal cells are stable. At a sufficiently high supercriticality the roll

78

4. TYPES OF FLOW

STRUCTURES

flow becomes stable, and at an even larger R the hexagonal-cell flow becomes unstable. The width of the range of R values in which the hexagons are stable is of order S4 and vanishes as w —> oc. Hohenberg and Swift [131] considered the set of equations governing the dynamics of the amplitudes of several lowest harmonics in a Galerkin representation of convective disturbances (with fixed wavenumbers that specify the periods in the two horizontal directions), the total number of variables being equal to 13. d Their analysis confirmed the qualitative results of Ref. 130, being at the same time not restricted to small 6 values. However, the authors concluded that hexagons cannot be observed for small values of 5, since their stability interval is very narrow in this case, and the experimental conditions cannot be controlled with the needed accuracy. In order to be able to observe hexagons, much larger modulation amplitudes than indicated in Ref. 130 are necessary. Another inference was that there is an optimal frequency range for observing hexagons, as their stability region shrinks not only for u> —> oc, but also for u> —> 0. Meyer et al. [133, 134] observed experimentally the effects of the periodic modulation of the temperature of the plate bounding the layer from below and compared them with the theoretical predictions of Refs. 130 and 131. They used water near 50.6°C and a circular container with r = 11. Unlike the abovediscussed cases of non-Boussinesq convection, where patterns are predominantly uniform and consist of either rolls or hexagons, in this experiment the two types of cells can coexist in the same pattern if R belongs to the range where either type is stable, and no hysteresis associated with the transitions between these types is observed. The authors explain this effect by the lack of a potential for a layer with thermal modulation and, therefore, the lack of preference for one or another form of convection. The boundaries of the R ranges where hexagons or rolls are stable or both types of flows coexist, as well as the direction of circulation in the hexagonal cells, agree with the theory. All this pertains, The simplest model of this kind, which describes two-dimensional (roll) convection in terms of three variables, is known as the Lorenz model. Two of these variables are the amplitude of the velocity in a roll set and the amplitude of the temperature perturbation that has the same period. The third variable is the amplitude of the temperature harmonic that is the second one in the expansion of the vertical dependence and the zeroth one in the expansion of the horizontal dependence. Therefore, it represents a horizontally uniform temperature perturbation responsible for the formation of temperature boundary layers near the horizontal boundaries of the layer. A numerical investigation of this system allowed Lorenz [132] to discover for the first time a strange attractor in its phase space and the phenomenon of dynamical chaos, thus opening a new chapter in the research of dynamical systems. Systems containing more amplitude variables, e.g., the system considered in Ref. 131, are sometimes referred to as the generalized Lorenz models.

79

4.1. 2D ROLLS AND 3D CELLS

however, to only moderate amplitudes of modulation. At high amplitudes, the patterns consist of randomly located cells at all R values studied. We also note an experiment of Ahlers et al. [59]. They investigated the behaviour of the convective heat flux through a layer of liquid helium when R is varied in time either in a stepwise fashion or linearly. The interpretation of the data of this experiment, based on the amplitude equations, showed that convection at the moment of its onset, was not of the roll type, but rather a transient regime was present; the authors tentatively associate this regime with a system of hexagonal cells. 4-1.9. The Presence of Suspended Solid

Particles

In general, suspended particles can strongly alter the mechanical properties of a fluid. We note for reference an experimental result obtained by Berdnikov and Kirdyashkin [135]. A high concentration of a polydisperse solid-particle visualizing addition in a liquid layer can cause the occurrence of polygonal cells in circumstances where only rolls are observed if the concentration is low. 4-1.10.

Tertiary

Flows

In this subsection, we consider the flows that arise as a result of the instability of a roll flow. Alternatively, they are frequently called secondary flows, this word usage implying that the original roll flow is regarded as primary. Roll convection is observed only in a certain range of Rayleigh numbers, which depends on the Prandtl number. Beyond the upper limit of this range rolls become unstable and tertiary flows develop, making the velocity field threedimensional. We shall touch upon the role of instabilities in convection dynamics several times throughout the following presentation. Here we describe some instability types that produce flow patterns of remarkable appearance, greatly affected by tertiary flows. The cross-roll instability manifests itself in the nucleation of a new system of rolls perpendicular to the original rolls. In essence, this is an instability of the boundary layers produced by the fluid circulation in the original rolls. The crossed rolls are typically somewhat narrower than the original ones. According to the experimental data of Busse and Whitehead [120] (see § 6.3.2), at moderate values of the Rayleigh number the crossed rolls finally replace the original rolls. At higher Rayleigh numbers, R > 10RC, the cross-roll instability results in the development of a so-called bimodal flow (see, e.g., a paper by Clever and Busse [136]). This is a superposition of the two roll sets. Figure 13 shows a bimodal pattern that was observed in an experiment started from random, disordered

so

4. TYPES OF FLOW

STRUCTURES

Fig. 13. Bimodalflowexperimentally observed by V. S. Berdnikov and V. A. Markov in ethyl alcohol (unpublished): slightly bent, wide original rolls superimposed with narrower crossed rolls; tf = 1.98 x 104, P sa 16. disturbances. In a bimodal flow, there are two maxima of the intensity of its tertiary component, situated between the midplane and each layer boundary. At R values as large as R ss 105 the original rolls no longer appear as the basic flow and bimodal convection assumes the form of a pattern of rectangularcells. The cross-roll instability has also a long-wave branch, with the tertiary-flow wavenumber less than fcc, which is known as the knot instability and observed at moderate P values. At sufficiently large Rayleigh numbers it results in the formation of highly concentrated rising and falling plumes (Fig. 14). Fully developed knot convection acquires the form of spoke-pattern convection in which the original roll structure is replaced with a polygonal pattern. Sheets of hot and cold fluid, erupting from the thermal boundary layers at the upper and the lower fluid-layer boundary, are stretched radially with respect to the plumes and appear like spokes (Fig. 15) [138]. The zigzag instability bends the original rolls to form a pattern of wavy, or undulating, rolls (see an experimental photograph in Fig. 16). Ultimately, depending on the parameters, either such a pattern reaches a steady state or, as Busse and Whitehead [120] wrote, "the 'zigs' of a roll cell and the - zags' of

4.1. 2D ROLLS

AND 3D

CELLS

81

Fig. 14. Patterns of knot convection, (a) A pattern calculated by Clever and Busse for R = 10 5 , P = 2.5 [137]: contours of constant vertical velocity in the planes z = 0.5 (left) and z — 0.2 (right), (b) A stage of development of the knot instability as observed by Busse and Clever [215] in methyl alcohol at R SK 5 x 104, P = 7.1.

82

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STRUCTURES

Fig. 15. Experimental photographs of spoke-pattern convection [138]. Left: R « 5 x 10 , P = 1 (methyl alcohol); right: R = 9 x 1 0 \ P = 64 (silicone oil).

Fig. 16. A stage of the development of the zigzag instability as observed by Busse and Whitehead [120] in a silicone oil at R = 3600, P =B 100. The disturbances, being randomly distributed at the initial moment, have reached different degrees of their growth in different portions of the tank. At the top left the formation of new rolls inclined at an angle of about 45° to the original rolls is seen.

83

4.1. 2D ROLLS AND 3D CELLS

Fig. 17. Roll deformation caused by the skewed varicose instability: (a) a schematic picture (solid and dashed lines correspond to roll boundaries, near which the fluid moves upward and downward, respectively); (b) a stage of the development of the instability as observed by Busse and Clever in an experimemt with water at a temperature of 50° C [215] (ft « 1.1 x 104, P = 3.7). the neighbouring cell join to form a new roll cell" In the latter case the new rolls make an angle close to 45° with the original ones. The skewed varicose instability results in a roll deformation shown schematically in Fig. 17a. An experimental photograph of a pattern produced by this type of instability is given in Fig. 17b. We shall see that the skewed varicose instability plays an important role in processes of pattern-defect nucleation and in transition to so-called phase turbulence (see Sec. 5.2). A more complete survey of different types of roll instabilities and the conditions for their development will be presented in Sec. 6.3. 4-1.11.

Hexagonal Cells in a Vertically Symmetric

Layer

Very recently Assenheimer and Steinberg [139] reported their observations of hexagonal cells in a layer without any significant up-down asymmetry and other factors complicating the picture. Since neither of two directions of fluid circulation in a cell is preferred under such conditions, botli I- and g-cells were observed simultaneously. The Prandtl number was in excess of 2. The wavelengths of the hexagons were significantly larger than those of rolls at the same Rayleigh number. The Rayleigh number was not close to its critical value, so t h a t no explicit contradiction with the known theoretical results, obtained by a weakly nonlinear analysis, was found.

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4. TYPES OF FLOW

STRUCTURES

Fig. 18. The regions in the (fc, fi)-plane where hexagonal convection cells in a vertically symmetric layer are stable [140]: (a) P = 2.5; (b) P = 16. Different signs mark off the threshols for different, instability modes. To investigate the possibility of strongly nonlinear hexagonal-cell convection in the case where no appreciable departures from the Boussinesq approximation are present and the (no-slip) boundary conditions are symmetric, Clever and Busse [140] performed a stability analysis of corrective flows of this type. They employed the technique that was many times used by Busse and his coworkers in studying the stability of convection rolls and, more recently, the stability of some three-dimensional forms of strongly supercritical convection. This technique involves the calculation of a steady initial ("unperturbed") flow by the Galerkin method and a subsequent linear analysis of the stability of this flow against diverse infinitesimal perturbations (in §6.3.1 we shall return to the discussion of some results obtained in this way). The analysis of Ref. 140 was restricted to the disturbances that do not change the initially specified horizontal periods of the pattern. It was found that stable hexagonal cells are possible for wavcnumbers much lower than the critical wavenumber kCl Prandtl numbers P > 1.2, and Rayleigh numbers R > 3000 ss 1.8/?c. Two stability diagrams for hexagons are shown in Fig. 18. For P < 10 the region of stable hexagons is a strip in the (k, 7?)-plane, which stretches from smaller R and greater k to greater R and smaller k (Fig. 18a). At higher Prandtl numbers the boundary of the stability region has a more complex form (Fig. 18b). The authors failed to determine the stability boundaries in the

4.2. PATTERNS OF QUASI-2D ROLLS

85

long-wavelength part of the diagram because of limitations posed by the numerical scheme. No stable hexagons were found for R < 3000. In view of the above-mentioned constraint on the wavelengths of perturbations, this stability analysis cannot quite definitely answer the question of stability of hexagons. However, transitions from unstable hexagons to rolls, most likely in many cases, are covered by the analysis. Busse and Clever [314] also used the same technique to analyze the stability of square-cell patterns under the standard conditions. For P = 7 and 16, square cells proved to be stable in a certain region of the (k, R)-p\ane. These cells are asymmetric with respect to the plane z = 1/2, and a reversal of circulation in each cell does not mean a simple reversal of the velocity vector everywhere in the flow region, but corresponds to the mirror reflection of the entire cellular structure about this plane. Nevertheless, hexagons and squares are not typical of convection when it develops from noise initial perturbations under conditions similar to the standard ones. Such cells may be eliminated at the early, linear stage of the process. 4.2. Patterns of Quasi-Two-Dimensional Rolls As we saw, if there are no complicating factors, roll flows represent the basic form of steady-state convection. In this section, we shall discuss the properties of such flows in more detail. Even if an experimentally observed roll pattern is highly regular and free of defects, usually, the rolls are nonetheless not quite straight, and the roll flow is not strictly two-dimensional. This occurs, at the very least, because of the fact that in reality the flow always involves only a portion of an infinite layer, and the presence of sidewalls may considerably affect the flow structure. Within the framework of the linear problem of stability of the motionless state Davis [141], using a Galerkin technique, showed that the presence of sidewalls removes the degeneracy of the eigenfunctions: in a rectangular container with rigid horizontal and vertical boundaries the critical Rayleigh number is smaller for those rolls which are parallel to the shorter side of the container. It is such rolls that were predicted on the basis of the Newell-Whitehead-Segel amplitude equation [47] and its extension that describes a superposition of sets of mutually perpendicular rolls [55]. The conclusion that such rolls are preferred was confirmed by Stork and Miiller [142] in an experiment performed at different ratios of the sides of a rectangular container and different aspect ratios. The results of this experiment agree with the findings of Ref. 141. A linear theory developed by Edwards [143] describes rather well regimes with different numbers of rolls, which were observed in Ref. 142 with different geometries of

86

4. TYPES OF FLOW

STRUCTURES

Fig. 19. Kxperimentally observed roll patterns (the boundaries of the rolls are depicted by dotted and solid lines): (a) texture in a rectangular container, R — 4fic, P = 2.5 [148] (neighbourhoods of the short container walls are not visible); (b) schematic image of a texture in a circular container, R= I \4RQ, P = 0.7 (according to a photograph from Ref. 179; dashed lines indicate the main features of the structure of the large-scale flow calculated in Ref. 185). The tendency of rolls to approach the sidewalls at a right angle is clearly seen in both figures. the container. It was found in Ref. 143 that sets of mutually perpendicular rolls can arise in a nearly square container in the neighbourhood of the instability threshold. For an infinitely long channel with free horizontal boundaries, rigid sidewalls, and a hcight-to-width ratio A, a linear problem considered by Davis-Jones [144] predicts rolls directed across the channel as the preferred mode if A < 0.1 or A > I. For intermediate values of A the overall features of the pattern are the same, but the velocity component normal to the channel walls is significant. Chana and Daniels [145] investigated in detail the structure of convection in a channel with rigid walls using a Galcrkin method. The effect of the normal orientation of rolls with respect to the long channel walls was observed experimentally, for example, by Ozoe et al. [146] in a channel as long as of T = 18. A laboratory model for an infinitely long straight channel is an annular channel between two coaxial cylindrical walls. As observed by Stork and Muller [147], if such a channel is not too wide, the rolls are directed radially, perpendicular to the walls. Extensive experimental data have shown that the indicated roll orientation is a particular case of a more general tendency: rolls tend to approach a wall at a right angle. This tendency is especially noticeable where it results in a

4.2. PATTERNS

OF QUASI-2D ROLLS

87

considerable bending of rolls and, therefore, in forming textures (see Sec. 4.3). This is the case for a roll pattern shown in Fig. 19a where the rolls in the bulk of the container make large angles with the normals to the walls. Another example, typical of round containers, is shown in Fig. 19b (this pattern, which resembles the logo of the Pan American airlines, is sometimes called the "Pan Am texture"). Pomeau and Zaleski [149], using the Swift-Hohenberg model equation, showed t h a t in the boundary layer near a sidcwall a set of rolls parallel to this wall can be unstable: a tertiary flow in the form of rolls perpendicular to the wall and to the original rolls develops (the cross-roll instability, see §4.1.10 and Sec. 6.3). This effect was observed experimentally, for example, by Croquette et al. [288] (in a cylindrical container) and Pocheau and Croquette [242] (in a rectangular container, see §6.5.1); see also Fig. 20. Cross [61] investigated the effect of a sidewall using the NWS amplitude equation with boundary conditions reproducing Eq. (3.39) to the lowest order. He found that the Lyapunov functional for the set of rolls in the boundary layer is minimum if the rolls make with the wall an angle 7r/2, within corrections of order e 1/ ' 4 . This result was later refined by Zaleski et al. [150]. They numerically calculated the Lyapunov functional of the NWS equation for the boundary layer and found the optimal (minimizing the functional) angle between the rolls and the normal to the wall to be different from zero, of order c 1 / 4 We mean here the value of this angle in the bulk of the container, i.e., outside the boundary layer, well away from the wall; at the wall itself this angle is equal to zero in accordance with the boundary conditions. For this reason, in a container whose width is greater than twice the thickness of the boundary layer and whose length is considerably larger than the width, as it can be easily imagined, the rolls must be bent like the letter S, approaching the longer sidewalls normally and making an angle with the normal in the central part [150]. Greenside et al. [151, 152] obtained such patterns of S-shaped rolls in numerical simulations of textures on the basis of the SH equation, and Le Gal [153] observed them experimentally (Fig. 20; it is remarkable that the observed pattern is virtually identical to the simulated one). The fact that the optimal angle minimizes the Lyapunov functional does not mean t h a t this is the only realizable angle (see Sec. 6.4). For some particular pattern to be realized, the overall flow geometry is ultimately important. It is determined, in particular, by the shape and horizontal dimensions of the container and by the conditions on the sidewalls. The situation is here quite analogous to the situation with the realizability of the optimal wavenumber in a pattern of two-dimensional rolls; this point will be discussed in Sec. 6.5.

88

4. TYPES OF FLOW

STRUCTURES

Fig. 20. S-shaped-roll patterns with crossed rolls near the shorter sides of the container: (a) an expenmantal photograph [153]; (b) a numerically simulated pattern [152]. For very small supercriticalities the angle made by a roll with a sidewall can differ appreciably from the right angle. Thus, in the experiment of Ref. 179 at e = 0.05 all rolls in the circular container were practically straight and parallel. It is possible that in this case neglected factors that break the ideal boundary conditions (3.39) become important. The tendency for convective rolls to approach sidewalls normally is now a well-known experimental fact. However, three important stipulations should be made here. First, all that has been said above does not pertain to those situations where the sidewall thermal regime itself dictates a certain particular character to flows in the region near the wall. Suppose, for example, that the temperature at the wall is always higher than the unperturbed temperature in the layer at the same height (this is the case wherever the heat-conducting wall has better thermal contact with the bottom layer boundary than with the top boundary or is heated from the outside). Then a steady upflow will exist near the wall, and the rolls in the boundary region will be oriented parallel to the wall. Situations of this type, where a horizontal temperature gradient at a sidewall produces static boundary (sidewall) forcing, will further be examined in §6.5.8. In some cases, they are deliberately implemented in experiments (e.g., in Refs. 242 and 158—

4.2. PATTERNS

OF QUASI-2D ROLLS

89

see §6.5.3). As Brown and Stewartson note [50], a small horizontal heat flux through imperfectly insulating sidewalls results in the replacement of the supercritical bifurcation at the critical Rayleigh number by an imperfect bifurcation of the type that is illustrated by the right-hand graph in Fig. 2e. If the wall material has a finite thermal conductivity, the forcing effect strongly depends on the factors governing the temperature distribution within the wall thickness, i.e., on the thermal contact between the wall and the plates bounding the layer at, the bottom and top, heat exchange between the wall and the ambient air, between the wall and the working fluid within the container, etc. It is thus understandable that the thermal effect of a sidewall can be greatly reduced if this wall makes good contact with both plates, is thick, and its material differs little from the fluid in thermal diffusivity. Otherwise, the mismatch between the thermal properties of the fluid and the wall can substantially affect the formation of the convection pattern even if there arise no significant static forcing in the final steady state. Heating up the walls may lead or lag behind heating up the fluid [162], which results in dynamic forcing. It is even more noticeable if the external heat supply is time-dependent. For example, in an experimental study by Meyer et al. [154] a cylindrical wall made of 5% polyacrilamide gel, whose thermal diffusivity is very close to that of the working fluid (water), had virtually no effect on the evolution of the flow. The flow structure was not correlated with the geometry of the container and was not repeated from one experimental run to another, and the rolls approached the outer wall at nearly right angles. A polyethylene wall, however, had a forcing action on the flow, the flow developed away from the walls into the chamber, and the flow structure reflected the geometry of the container (about this, see below). Second, the roll pattern can be substantially affected even by insignificant nonuniformities of heating from below and/or cooling from above. For example, in the well-known experiments of Refs. 155 and 156 an axisymmetric pattern of circular rolls in a round container arose as a result of the presence of a radial temperature gradient in the top heat exchanger, since the cooling water was pumped into the central part of the heat exchanger and carried off near its outer edge. Third, the effect of the walls can be considerably reduced if there is a zone along these walls where the liquid layer is not covered with a solid plate at the top and has a free upper surface (see Sec. 3.1 and Fig. 4). This zone plays the role of a buffer since there arise in it three-dimensional flows with a complicated structure, which can easily match arbitrarily oriented rolls arising in the bulk of the container. For this reason, in experiments with such setups the systems

90

4. TYPES OF FLOW

STRUCTURES

of parallel rolls can make different angles with the walls, and their orientation changes randomly from one run to another [135]. It is worth discussing in greater detail the question as to what are the roll patterns that arise in a round container depending on the intensity of sidewall forcing. We saw that the Pan Am texture is typical for the case of weak forcing (becoming a set of virtually straight rolls if the aspect ratio T is large), while the experiments of Ref. 154 with a "forcing" (polyethylene) wall demonstrated formation of an axisymmetric system of circular rolls. Such patterns are characteristic of cylindrical containers (see in particular Ref. 158). It is with reference to them that one of the known approaches to determining the "selected" wavenumber (see §6.5.4) has been developed, and they contain a remarkable singular point, a focus, which plays an important role in roll-wavenumber readjustment. A comprehensive experimental study of the sidewall-forcing effect upon the flow structure was carried out by Hu, Ecke, and Ahlers [157]. They used C 0 2 gas under a pressure of 25.3 bar (P = 0.93). The working volume was bounded top by a sapphire plate and had a cardboard outer cylindrical wall. The wall either was plain, with a radius-to-height ratio of T = 43, or had an annular fin (spoiler tab) at rnidheight and T = 41. Calculations showed that the finned wall less distorts the temperature field in the adjacent region than the plain wall. Correspondingly, at T = 41 forcing was weaker than at T = 43. In slightly supercritical conditions, the wall that exerts stronger forcing created an axisymmetric system of rolls (Fig. 21a). The effect of the less forcing wall was also sufficient for axisymmetric convection to arise but, however, not strong enough for circular rolls next to the wall to be stable. T h e cross-roll instability occurred, resulting in the development of a secondary flow in the form of short roll segments directed along container radii and abutting against the wall. These crossed rolls occupied an annular region of width about 4/i (Fig. 21b). It was also possible to obtain a set of almost straight rolls even if the wall with stronger forcing was used. It was needed for this that at the stage of the formation of the pattern fluid motions be already vigorous enough, not "feeling" wall forcing. This was the case when moderately supercritical conditions (e « 1) were imposed initially and some little ordered flow developed, subsequently the supercriticality being reduced, say, to e « 0.08. Alternatively, after the formation of an axisymmetric convection pattern, the container was rotated about the vertical axis rapidly enough, so that the formed pattern broke up because of (the Kiippers-Lortz) instability, and then rotation was stopped. (We can say with virtually full confidence that the authors would achieve the same result if they stirred the working fluid in its entire volume at the beginning of

4.2. PATTERNS

OF QUASI-2D ROLLS

91

Fig. 21. Roll patterns in a circular container at < ss 0.04 [157]: (a) concentric rolls formed with stronger forcing (T = 43); (b) concentric rolls formed with weaker forcing and superposed by short crossed rolls near the wall (T = 41); (c) straight rolls formed from a disordered pattern ( r = 43).

n

4. TYPES OF FLOW

STRUCTURES

Fig. 22. Eccentric annular rolls in a cylindrical container: (a) a photograph obtained in an experiment with methanol, P = 7, R/Rc = 7.3 [161]; (b) a schematic picture of the roll pattern (light dashed lines) and the profde of the large-scale flow as observed in the experiment of Ref. 159 (actually measured velocities are shown with heavy solid curves and arrows; hypothetical velocities, with a heavy dashed curve and heavy dashed arrows).

a run.) In the first case the pattern became established in several days; in the second case, this was achied in less than a day (while TV K 3.2 s and T(, RJ 1.5 h). The formed rolls were weakly curved, and in those near-wall regions where the rolls made small angles with the wall, short crossed rolls appeared (Fig. 21c). Axisymmetric roll patterns are susceptible to a particular instability, which manifests itself the more appreciably, the greater the Rayleigh number. Croquette and Pocheau [158] performed careful measurements of concentricroll patterns they observed in experiments with silicone oils (P = 70 and 14) in a round container with r — 20 and revealed a surprising feature: the patterns turned out not to be strictly axisymmetric. The umbilicus of the system of rolls was displaced from the centre of the container (by a distance of the order of 1 mm, the container radius being 40 mm). For P = 14 the displacement was larger than for P — 70, and in experiments with methanol ( P = 7) it could even be seen with the naked eye (Fig. 22a). This indicates that a strictly axisymmetric pattern, in which the radial pressure gradient suppresses the height-averaged radial flow, is unstable against a small breaking of the symmetry. The authors suggested that the pattern asymmetry is closely related to a general circulation of the double-vortex ("dipole") type (Fig. 22b).

4.2. PATTERNS

OF QUASI-2D ROLLS

93

In order to test this hypothesis, a special experiment was performed [159]. A photochromic compound, benzothiazolinic spiropyran, which becomes coloured after ultraviolet radiation, was dissolved in the working fluid (methanol, P = 7). The experimenters marked a portion of the volume of the fluid, irradiating a selected diameter in the system of annular rolls, and were then able to observe the effects of the large-scale flow. If R is gradually increased, the asymmetry of the pattern becomes strongest before the next central rolls is annihilated and disappears after the annihilation. For this reason, at almost the same R value both a symmetric and an asymmetric pattern can be observed. It was noticed that the diameter marker is not deformed in an axisymmetric pattern (although blurred by diffusion) but is systematically transported by the mean flow in an asymmetric pattern if directed normally to the eccentric displacement of the focus. This experiment thus demonstrates the existence of large-scale flow which results, according to Refs. 53 and 54, from roll curvature. The structure and direction of this flow is determined by the roll-curvature field and the position of the sidewalls. In an asymmetrically deformed system of circular rolls fitting a cylindrical container, the drift flow is directed so as to further compress the rolls. If this compression overcomes phase diffusion which tends to restore the roll widths, the deformation increases further. The patterns thus turns out to be unstable with respect to a small displacement of the umbilicus. This phenomenon, which came to be known as the focus instability, was discovered and the mean-drift field was calculated by Newell [16] using the extension (3.84)(3.86) of the Cross-Newell equation and then by Newell et al. [67] on the basis of the full set of the phase diffusion-mean drift equations (3.90)-(3.92). This was done by linearizing the equations with respect to the small displacement of the focus (umbilicus). Another experiment in which the eccentricity of the umbilicus was observed (and the wavenumbers of axisymmetric convection were studied) was performed by Steinberg, Ahlers, and Cannell [162] with V = 7.5 and P = 6.1 (water at 25°C). For £ > 0.16 a system of circular rolls, having been generated at the beginning of an experimental run by wall forcing, then remained stable virtually without such forcing (which is undoubtedly a consequence of the small aspect ratio). For c < 0.16 a transition to a lower degree of ordering (to a complex texture) occurred over times much longer than TV The eccentricity of the umbilicus was noticeable starting with e ss 2.5 and changed with t qualitatively in the same way as in Ref. 159. As shown by Hu et al. [157] in the already mentioned experiments at P = 0.93, that the outer wall that exerts stronger forcing on convection also more

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4. TYPES OF FLOW

STRUCTURES

strongly counteracts the focus instability. The description of the regularities of roll-pattern behaviour for different Rayleigh numbers given in Ref. 157 refers mainly to the case of the less forcing wall (T = 41). For e < 0.08 the umbilicus of the roll system was shifted off the container centre and the amount of the shift increased with e. The establishment of the pattern occurred without any peculiar phenomena. In the interval 0.08 < e < 0.10 the process of the pattern settling went through the stage at which the umbilicus emitted radially travelling waves. As this took place, from time to time a new circular roll nucleated in the umbilicus, this being followed by an abrupt shift of the umbilicus, a perturbation travelled from the centre to the periphery of the container (where the outermost roll disappeared), and the umbilicus relaxed slowly to a new position. After several (from 3 to 12) such cycles a steady-state regime was achieved, when the umbilicus had returned to within h/'i of the geometric centre of the container. For c > 0.10 such a return does not take place. The umbilicus, emitting waves, moves toward the wall, eventually collides with it, and disappears. Such evolution results in a pattern of more or less straight rolls, in which defects are present (Fig. 23). In other words, the driving force of convection is so strong that sidewall forcing cannot resist the general tendency towards the formation of a straight-roll pattern. In a subsequent paper [163] the same authors reported that foci can also persist at the wall, continually nucleating rolls in the same fashion as the focus of an unstable concentric-roll pattern does and moving erratically along the wall. The frequency of the nucleation of rolls by a focus increases when other pattern defects (see Sec. 4.3) move closer to this focus. In a pattern of circular rolls, the roll-nucleation frequency increases when the focus approaches the sidewall. If the umbilicus is shifted off the container centre, there exists, as we saw, a large-scale flow. In the wall region, near the point situated in the direction of the umbilicus displacement from the centre, there is a source producing new crossed (i.e., radial) rolls [157]. They are carried off this source in both directions along the wall toward its opposite position where a sink is present—the rolls disappear. In the experiments with stronger sidewall forcing (T = 43) the behaviour of roll patterns differed qualitatively from the described one, mainly in that there was no interval of t values in which, after the stage of emitting waves, the position of the umbilicus settled down near the centre. Some features of pattern behaviour reported in Ref. 157 for the case of greater Rayleigh numbers, where a steady-state regime is not achieved, will be noted in Sec. 5.3.

4.2. PATTERNS

OF QUASI-2D ROLLS

95

Fig. 23. Destruction of concentric-roll pattern observed in a layer of CO2 gas at t = 0.102, P = 0.93, r = 41 [157]. Elapsed times (measured from an arbitrary origin) are: (a) 0, (b) 5600, (c) 7800, (d) 8520, (e) 9240, (f) 9840r v .

4. TYPES OF FLOW

96

STRUCTURES

4.3. C o n v e c t i o n T e x t u r e s . R o l l - P a t t e r n D e f e c t s We listed in Sec. 4.1 the "inherent" factors of thrce-dtmensionalizing a convection flow, which are normally present, if at all, everywhere in the layer of a convecting fluid. The content of Sec. 4.2 shows that, even without such factors, a convective flow is in general not two-dimensional. The influence of the container sidewalls can result in a rather strong bending of rolls. Moreover, if convection has spontaneously developed from noise perturbations then the regularity of the pattern is '•spoiled" to a greater or lesser degree by structural defects of different kinds. In many cases defects are typical of transient regimes and eventually disappear. Equilibrium states with defects are also possible; however, the structure of equilibrium patterns with defects is largely determined by the presence of sidewalls. Relatively ordered roll patterns in which the direction of rolls varies in space slowly, even if over wide limits, are termed textures. Also complex textures are commonly observed, in which several ordered fragments, or textures in the indicated sense, can be isolated. We shall see that the presence of defects imparts to the system additional "degrees of freedom". Readjustment of the roll wavenumber in the process of seeking the optimum value occurs most easily in the presence of pattern defects. Many observed defects of convection patterns closely resemble defects of crystal lattices. For this reason, the terminology used to describe them is borrowed from the physics of crystals. Typical defects of roll patterns are shown schematically in Fig. 24. 4-3.1.

Dislocations

A dislocation (Fig. 24a) is a defect arising at a point where an "extra" pair of rolls, "wedged" into a regular roll pattern, terminates (the rolls being somewhat bent near the dislocation). Both stationary and moving dislocations may be observed. We saw examples of patterns with dislocations in Figs. 6, 11 (where the spiral arms are fitted by dislocations with the concentric annular rolls in the outer part of the container), and in the upper right portion of Fig. 23f. In most cases dislocations move in a direction parallel to the rolls (climb), though sometimes movement in a perpendicular direction (glide) can also be observed, accompanied by topological changes near a dislocation. We shall see in §6.5.3 that the velocity of the dislocation climb is determined by the background wavenumber of the pattern and the climb is one of possible scenarios

4.3. CONVECTION

TEXTURES.

PATTERN

DEFECTS

97

Fig. 24. Defects of roll patterns (lines represent, roll boundaries): (a) dislocation, (b) disclinations (focus singularities are shown at the top), (c) grain boundary.

of pattern evolution (sometimes called the "selection mechanisms") which make the wavenumber closer to the optimal one. 4-3.2.

Disclinalions

Disclinations are defects corresponding to singularities in the field of directors (vectors without arrows) obtained from the local wavevectors of the pattern, these vectors being defined as the local phase gradients in accordance with Eq. (3.74) [1]. Typical disclinations are shown in Fig. 24b. A relevant characteristic of a disclination is the angle through which the director twists as its midpoint circumscribes the disclination in the counterclockwise direction on some contour. (In the wavevector field itself, not only point singularities may be associated with disclinations but also discontinuities along some lines; the wavevector changes its direction to the opposite one on crossing such a line. In particular, the disclinations shown in Fig. 24b at the bottom possess this property.) Among disclinations, the focus singularity is considered especially frequently in studies on the wavenumber selection. This disclination arises at the centre of an axisymmetric system of circular rolls or in fragments of such a system (possibly deformed)—target patterns, in particular, at a container sidewall. We have already discussed axisymmetric patterns and the behaviour of focus singularities in Sec. 4.2 (see, e.g., Figs. 21a, b; 23) and shall return to this discussion in §6.5.4. Note that in the process of the readjustment of a pattern containing foci, rolls usually appear or disappear just at these points. We note that a dislocation can be regarded as the superposition of two disclinations, namely, those sketched in the bottom part of Fig. 24b.

4. TYPES OF FLOW

m 4-3.3. Grain

STRUCTURES

Boundaries

A defect of a very characteristic type is a grain boundary, i.e., a line along which two ordered patches (or textures) of rolls of different orientation come together, composing a more complex pattern (Fig. 24c). e In rectangular containers where straight rolls are parallel to the shorter sides of the container, grain boundaries, being usually observed near these sides, delimit the bulk pattern from crossed rolls that meet the shorter sidewall at a right angle. Similarly, grain boundaries can arise near the outer wall of a circular container, where the cross-roll instability produces short radial (normal to the wall) rolls. If the bulk of the container is filled with nearly straight parallel rolls, grain boundaries are usually present in two regions where these rolls are parallel to the wall (Fig. 21c). If the rolls of the main system are concentric rings and form an axisymmetric pattern, they may join the crossed rolls along a circular grain boundary (Fig. 21b). Another example of a pattern with grain boundaries in a circular container can be seen in Fig. 23f, in the regions where the target pattern and the pattern of almost straight lines join. As it will become clear in §6.5.1, motion of grain boundaries can ensure very efficient roll-wavenumber readjustment over wide limits.

"The terms domain wall and domain boundary are also used to denote this type of defects.

CHAPTER 5

CONVECTION REGIMES

5.1. Regime Diagram Convection manifests itself in diverse forms: cells may have different configurations, forming more or less ordered patterns, and the flow may either reach a steady state, or undergo oscillations (also of varying degree of ordering), or be completely turbulent. To a first approximation, the regime of convection in a horizontal layer under standard conditions is determined by the Rayleigh number R and the Prandtl number P, and transitions between regimes can be described by the diagram presented in Fig. 25. This diagram summarizes the experimental data of Krishnamurti [164-166] and of a number of other investigators. It was first constructed by Krishnamurti and later modified by Busse [12, 13]. The lines delimiting the regions of different regimes are drawn, to a certain extent, arbitrarily, since the results of different experiments do not always agree with one another perfectly and, moreover, there may be difficulties with determining the R value that corresponds to a transition, especially in the cases where variations in R result in hysteretic transitions. As will become evident from what follows, the question of the transition from steady-state to time-dependent convection is especially subtle. The region where the fluid is in a stable motionless state lies below line / (R — Re). The region of steady-state roll convection lies above this line (as we know, roll convection is conventionally referred to as two-dimensional convection—see Sees. 4.2 and 4.3). For high values of P this region extends to R values of the order of l3Rc « 2 x 104. Nearly two-dimensional but, in general, not quite stationary convection can be observed right up to Rayleigh numbers of about several tens times Re (see, e.g., Ref. 167). Curve / / is the threshold above which the cross-roll instability (see § 4.1.10 and Sec. 6.3) results in steadystate bimodal convection. It is interesting that for very large P values (such as 8.6 x 103) artificially induced bimodal convection (and square-cell convection,

99

100

5. CONVECTION REGIMES

Fig. 25. Diagram of convection regimes (according to Refs. 164-166 and other experimental studies); adapted from Ref. 12.

which can be initiated quite similarly) can be stable even at R as (2 to 8) x 105 but unstable at R as 105 [168]. The transition to a bimodal flow is only observed at sufficiently large values of P. For small P values, however, time-dependent convection sets in immediately (curve / / / ) . This transition is associated with the onset of the oscillatory instability which manifests itself in wavelike bends travelling along the rolls. [It is now appreciated, however, that such a simple scenario is a theoretical prediction based on idealized models (see Sec. 6.3) rather than an experimental finding. It will be clear from what follows that for small P values it is impossible to determine universal threshold values of R for the regime changes in diverse experimental setups.] At a certain R value bimodal convection also becomes time-dependent. According to Krishnamurti, in the region P > 50 this value is almost constant, about 5.5 x 104, whereas subsequent experiments of Whitehead et al. [221, 168, 236] gave R values increasing with P (see Fig. 25). The splitting of curve III into two branches reflects the fact that the onset of oscillations strongly depends on the presence of nonuniformities in the pattern. Nonuniformities can give rise to oscillations in isolated patches at relatively small R values (branch Ilia). In contrast, highly uniform bimodal-convection patterns (which can be created artificially in experiments with controlled initial conditions by means of inducing

5.2. PHASE

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101

two-dimensional rolls at R exceeding the cross-roll-instability threshold) demonstrate the onset of time dependence at much larger R values (branch Mb) [221]. For convection at R ~ 10 5 -10 7 and moderate values of P there characteristically exist sharply time-dependent, localized thermals on the background of relatively steady large-scale cells (time-dependent knot convection [221, 137]; see also §6.3.1). Curve IV corresponds to the emergence of higher-order harmonics in the spectrum of oscillations (and, like for other transitions, to a j u m p of the derivative of the convective heat flux with respect to R). The above remark concerning the peculiarities of the exchange of regimes at small P is also valid for this transition. The region of completely turbulent convection is situated above curve V. The onset of bimodal convection as a result of the development of the crossroll instability, as well as the conditions for the manifestation of other instabilities, were investigated theoretically, through the linear stability analysis of two-dimensional roll flows, by Busse and his coworkers in a scries of works. Certain results of these studies will be discussed in §6.3.1. Here we note that the stability of bimodal flows was also analysed in a similar way [106, 136]. In particular, Clever and Busse [136] found that two types of oscillatory regimes are possible for such flows. One of them, called the wavy oscillatory bimodal convection, is characterized by waving of rising and descending sheets of fluid which are produced by the crossed rolls (i.e., the upflows and downflows oscillate in the direction of the basic rolls). In contrast, as the symmetric oscillatory bimodal convection takes place, the1 regions of upflow and downflow periodically expand and contract (being opposite in phase). The realization of the first or the second regime depends on R, P, and the wavenumbers of the original and crossed rolls. The description of these two types of oscillatory flows obtained numerically agrees well with the experimental observations reported by Busse and Whitehead [221]. A detailed numerical investigation of knot convection and its instabilities was undertaken by Clever and Busse [137]. Vel'tishchev and Zhelnin have reproduced the diagram of regimes in its qualitative features by means of numerical simulation of three-dimensional convection

[169]. 5.2. P h a s e T u r b u l e n c e An important property of convection, which has been actively studied for almost two decades and which cannot be reflected by diagrams similar to that given in Fig. 25, is that the attainability of a steady state for small and moderate

102

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REGIMES

supercriticalities and small P values depends on the aspect ratio of the container and even on its shape. Ahlers and Behringer [170, 171] (see also Ref. 14 for a review) performed experiments with normal (not superfluid) liquid 4 He in circular containers with different aspect ratios. They studied the behaviour of the temperature difference A T as a function of time for the heat flux through the layer kept constant at a given level. For T = 57 and P « 3-4, nonperiodic oscillations with a very wide spectrum peaked at zero frequency were observed even at very low R values, virtually from the very onset of convection (the Rayleigh number R is meant here to be calculated on the basis of the average A T value). At the same time, for T fa 2-5 and P ss 0.8 there exists an interval of R values, of width up to several Rc, in which convection settles down to a steady state; for higher R values either steady convection is immediately replaced by irregular oscillations (at greater T values) or they are preceded by periodic or quasiperiodic oscillations (at smaller T values). Similar experiments, limited to small values of T, were carried out by Libchaber and Maurer [172] and Ahlers and Walden [173]. Such investigations were also performed for mercury by Fauve et al. [174]. Further experiments with liquid helium, carried out in circular containers of different V (2.4 < V < 22) for 0.5 < P < 0.7 by Behringer et al. [175], demonstrated steady states even at large values of T, although for V > 15 this was the case in a very narrow range of R: Rc < R < \.09RC. In addition, for T > 4 the regime immediately following the steady-state regime is always periodic (although, generally speaking, far from harmonic oscillations). It is possible that this regime was not achieved in experiments with V — 57 because of a very long time required for its establishment. It is noteworthy that Motsay et al. [176] found in experiments with liquid helium that in rectangular containers of horizontal sizes \3Ah x 5.95/i and 18.2/i x 8.12/t time dependence ensues at appreciably higher Rayleigh numbers, 3.39/? c and 2.53/? c , respectively. Low-frequency noise detected in the behaviour of AT(t) at small supercriticalities came to be associated with a special type of turbulence. One of the names given to it is turbulence near the threshold of the convective instability or turbulence near the onset of convection. This phenomenon is also referred to as persistent dynamics. While cryogenic experiments do not permit flow visualization, experiments with other fluids, in which the flows can be visualized, showed that this turbulence looks like slow travel of rolls and roll-pattern defects. Since in many studies such processes are described in terms of space-time variations of the phase of a roll system, this phenomenon is often called weak, or low-frequency (spatial) turbulence, or phase turbulence.

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103

Gollub et al. undertook an experimental study [148] precisely in order to investigate the behaviour of the spatial structure of convection both in the case where the pattern reaches a steady state and where phase turbulence persists. Water near 70°C was used as a working fluid. For it P = 2.5, which falls in the range of values for liquid helium. A rectangular container of horizontal sizes 20/i x 30/i was employed. For R < 5RC the evolution of the pattern to a steady state was found to appear like gradual elimination of defects and transition to a relatively simple texture of smoothly curved rolls, which approach the sidewalls at a right angle (Fig. 19a). This process may last for hundreds of hours, which is four orders of magnitude longer than r v and an order of magnitude longer than Th. Nevertheless, a steady state is not always achieved in such times. (We note that for such processes, according to an estimate made by Cross and Newell [66], the establishment times are > [Yh.) Starting with R « 5/? c , much faster processes take place, a steady state is not reached, and a continuous recording of the flow velocity at a fixed point yields a picture of wide-band noise with a main spectral maximum at zero frequency. Defects arise, move, interact, and disappear in the spatial pattern. Roll necks become a characteristic feature of the pattern. For R > 9RC another peak occurs in the spectrum near 0.05 Hz, being related to roll oscillations. Various regimes of convection were also studied experimentally by Heutmaker and Gollub [177]. The same working fluid and a circular container with r = 14 were used. Automatic processing of the shadowgraph images made it possible to investigate in detail the field k(x) of local wavevectors and the distribution function f(k) of the wavenumbers. Three types of regimes were revealed depending on the reduced Rayleigh number t. If t < 0.2, nonperiodic motion associated with restructuring of defects is observed at least for a time of 507V For 0.2 < £ < 3.5 steady-state complex textures form after sufficient time has elapsed. At c > 3.5 the flow is again time-dependent. From time to time new rolls nucleate in the foci at the outer wall; rolls in the central part of the container, being compressed, pinch off to form a pair of dislocations, which results in the elimination of one roll pair; then the dislocations climb to the wall and in turn disappear; etc. (Fig. 26). It is interesting that when settling does not occur, i.e., in the first and the third case, the distribution f(k) overshoots the limits of the wavenumber band in which, theoretically (see §6.3.1), uniform spatially periodic roll patterns should be stable (Fig. 27). As for regimes with very small e values, Heutmaker and Gollub do not rule out the possibility that time dependence is due to random external influences and imperfect maintenance of experimental conditions. In a square container with a side length equal to the diameter of the circular container they observed flow settling within a time of lOOTh- On the other hand, settling was not observed at t = 0.141 and P = 5.7

104

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REGIMES

Fig. 26. Time-dependent pattern observed at t = 3.84, P = 2.5 [177]. (a)-(f) New rolls appear at the sidewall foci (see arrows), (g)-(l) A pair of dislocations forms, eliminating one roll pair; the entire sequence then repeats nonperiodically.

5.2. PHASE

TURBULENCE

105

Fig. 26 (continued)

106

5. CONVECTION

REGIMES

Fig. 27. Distribution functions of local wavenumbers for different t values as obtained in experiments at P = 2.5 (adapted from Ref. 177). The wavenumber ranges in which uniform patterns of straight parallel rolls are stable according to linear thery are marked by arrows and the thresholds for different instability types are indicated: ZZ, the zigzag instability; K, the knot instability; CR, the cross-roll instability; SV, the skewed varicose instability (see §6.3.1).

in an experiment that used a circular container with T = 15 and lasted for a time of the order of 200T|, (about a month) [178]. Experiments of Pocheau et. al. with argon at room temperature (P ss 0.7) in a round container with T = 7.66 [179-181] (see Ref. 161 in addition) also demonstrated topological changes in the roll pattern and, moreover, revealed t h a t the regime sequence is very complicated. For e < 0.126 steady-state patterns of slightly curved rolls with two focus singularities at the outer wall are observed (Pan Am textures like those shown in Figs. 19b and 28a). At larger e values the flow is time-dependent (Fig. 28). Sometimes in the central region of the pattern, which is a Pan Am texture (Fig. 28a), a pinching of a roll pair arises. This pinching results in the formation of two dislocations (Fig. 28b). They move

5.2. PHASE

TURBULENCE

107

Fig. 28. Cyclic dynamics of a pattern experimentally observed at c = 0.14, P — 0.69, and T = 7.66 [179]: (a) a Pan Am texture with two foci at the outer wall (marked with arrows); (b) nucleation of a pair of dislocations; (c) the climb of the dislocations toward the opposite parts of the wall; and (d) their glide toward the foci, which results in their disappearance and the recovery of state (a). apart, climbing to the wall (Fig. 28c), then glide to the foci (Fig. 28d), and disappear. New rolls are generated at the foci, and the pattern is restored. If 0.126 < t < 0.175, the process is periodic. In this t range five scenarios of the pattern evolution can be observed, depending on the value of t. All these scenarios mainly follow the above-described scheme. For 0.175 < c < 0.346 topologically diverse patterns arise. Their behaviour is chaotic, i.e., phase turbulence is observed. In steady-state regimes the local wavenumbers fill a certain range of values, achieving their maximum at the centre of the container and their minimum, near the wall. The width of the range increases with t. It is interesting that the upper bound of this range reaches the theoretical instability threshold found for a uniform roll pattern (see § 6.3.1) just near that t value at which the steady-state regimes are superseded by time-dependent regimes. The results of Refs. 179-181 thus share common features with the results of Ref. 177.

108

.5. CONVECTION

REGIMES

Within the range 0.346 < t < 1 there are another four transitions [181]: as e is increased, chaotic evolution changes again into a steady-state pattern, after which a periodic process is observed; it is again followed by a steady-state regime and then, once again, by a chaotic process. The settling time reached 500x1, in these experiments. Motsay et al. [176], as they used the larger of the two rectangular containers in their experiments with liquid helium (see above), also observed a "fine structure" in the e-distribution of steady-state, periodic, and nonperiodic regimes. In experiments with air, Leith showed [182] that if the ratio of the horizontal dimensions of a rectangular container ranges from 0.5 to 1, then phase turbulence associated with dislocation gliding is possible. This process arises at R values approximately corresponding to the threshold of the skewed varicose instability (see §6.3.1), which plays an important role in the disordering of patterns. It is natural to conjecture that intricate and clearly nonvariational dynamics observed in many cases at small and moderate P values is related to the presence of large-scale flow (see Sec. 3.3). Manneville [183] attempted to numerically simulate phase turbulence in a rectangular container of horizontal sizes 15.9/i x 11.5/i on the basis of Eqs. (3.40)-(3.42), which take into account the z-independent mean drift (this corresponds to a layer with free horizontal boundaries). The domain 0 < e < 0.5, 0 < 1/P < 1 of the parameter space was systematically explored. Steady states were achieved in the calculations. However, for sufficiently small P(= 1.6), a settling process of a very long duration was observed. Such regimes can be treated as a transient stage on the route toward turbulent regimes. The mean flow creates a local compression in the roll pattern, and necks appear; they give rise to dislocations, which in turn climb; and roll deformation takes place. This sequence of events resembles the experimentally observed one and can recur many times. An important feature of the process is, in Manneville's opinion, the lack of compatibility between the mean flow (which carries the rolls and whose structure depends on the overall geometry of the flow, in particular, on the shape and sizes of the flow region) and the roll curvature (which determines the speed of diffusive phase transfer). This results in "dynamic frustration" of the roll pattern, which implies topological changes, and, ultimately, makes the flow dynamics n o n r e l a t i o n a l . The results of numerical simulation of flows in a circular container carried out by Greenside, Cross, and Coughran [184] on the basis of the model (3.47), (3.49)-(3.51), (3.55), (3.56) with a - b = 0 and c = 1 showed close similarity to the behaviour of defects observed experimentally [179-181]. Chaotic regimes were revealed for both circular and rectangular regions. Pocheau [185] constructed an explicit analytical solution to the Cross- Newell equations (3.84)-(3.86) for the phase field and the mean flow in a circular region,

5.2. PHASE

TURBULENCE

109

using expansion in a small parameter related to the roll curvature. In this way, he reproduced the steady-state pattern of bent rolls for P = 0.7 and the effect t h a t this pattern loses its stability on reaching that very supercriticality at which the greatest local wavenumber passes outside the stability band for straight rolls. This occurs when the mean flow, directed toward the foci near the wall (where the roll curvature is maximum) and, further, from the foci toward the centre of the container (Fig. 19b), becomes sufficiently intense. In this case the rolls are appreciably compressed in the central region. It was later shown by Daviaud and Pocheau [186] both experimentally and theoretically (by solving the CN equations) that the pattern can be stabilized by means of making the sidewall of the container permeable to the mean flow. If a circular container is surrounded by an annular region where convection is suppressed in one way or another and where the mean flow can nevertheless easily penetrate, then this flow, being distributed over a larger area, will no longer produce "dangerous" roll compression at the centre of the container. In the experimental part of the study convection was suppressed in the outer region by placing there a thin horizontal annular plate, which divided the layer into two convectively stable sublayers. As a result, steady states remained attainable until t as high as 1.2, and phase turbulence was first observed only at t = 1.5. We give now a brief c o n c l u s i o n to this section. As it became clear from the aforesaid, the role of mean drift is at the heart of the phenomenon of phase turbulence. This phenomenon is fundamentally related to the interplay between the phase diffusion and the mean flow. Moreover, there is little doubt that the process of seeking the optimum spatial scale (the optimum wavenumber) of convection rolls, which will be discussed in detail in Chapter 6, is essentially involved. It should be added now that the mean drift is an agent of coupling between the processes that occur in different portions of the pattern and, therefore, the optimization of the flow structure at one site may require its deoptimization at another site. Daviaud and Pocheau's study [186] elucidates another key point of our comprehension of phase turbulence. We see that this phenomenon is crucially dependent on the presence of sidewalls, which confine the mean flow to a certain region. T h e less the aspect ratio of this region (within certain limits), the greater the velocity gradients and the more complicated the roll mean drift dynamics. The sidewalls prevent defects and "pieces of nonequilibrity" in the pattern—patches without conformity between the roll configuration and the mean drift—from leaving the region." The effect of sidewalls is even more important if boundary forcing is present. For these reasons, the features of "Obviously, our usage of the word nonequilibnly is in no way related to the thermodynamic meaning of this term.

110

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REGIMES

pattern dynamics in which mean-drift effects are important seem to have little in common with convection phenomena in large-scale natural objects such as atmospheres and oceans. To approximate the dynamics of convection in natural systems resembling an infinite layer, when carrying out numerical or laboratory modelling, one has to eliminate sidewall effects as completely as possible. 5.3. Spiral-Defect Chaos There is also a type of convection regimes with persistent time dependence, first observed by Morris el al. [187] (see also Ref. 188 for a description of further experiments), which has come to be known as spiral-defect chaos. The authors performed experiments with C 0 2 gas under a pressure of 32.7 bar (P = 0.96) in a cylindrical container with T = 78, whose outer wall was made of porous filter paper and produced therefore very weak forcing. If e < 0.050, the flow settles down to a steady-state pattern of straight rolls (Fig. 29a). As c is increased, the rolls show a progressive tendency to approach the sidewall at a right angle. As a result, focus singularities appear at the wall, and grain boundaries arise that separate particular texture patches (Fig. 29b). Such a state is reported (say, for t « 0.1) to be time-dependent, the motion of defects being its characteristic feature. At ( « 0.4, convection rolls begin to form rotating spirals in the interior of the container (Fig. 29c), and when c > 0.5, numerous interacting rotating spirals and other defects are observed. With increasing e, the container gradually becomes filled with them—spiral-defect chaos develops (Fig. 29d). To quote Ref. 187, "individual spirals typically rotated several times while translating a distance comparable to their diameter before being destroyed or suffering a change in the number of arms". Most of them are single-armed although two- and three-armed spirals, and patches of concentric rolls (targets) are also present. The correlation length of the pattern decreases strongly with increasing t. In contrast to what was observed in Ref. 114 (see §4.1.3), the spirals are unrelated to the lateral boundary, whence the authors conclude that their formation is part of the chaotic dynamics. The authors emphasize that the mean wavenumber of rolls in such patterns, demonstrating a decrease with increasing e (which is a normally observed behaviour—see Sec. 6.1), remains within the range found theoretically to be the region of stability of spatially periodic straight-roll patterns (within the "Busse balloon", §6.3.1). However, the distribution of local A; values extends beyond the stability limits, although to a lesser degree than in the experiments of Ref. 177. As for the two-dimensional distribution of wavevectors k, in the regimes of fully developed spiral-defect chaos it forms an annular band centred on k = 0, the pattern being thus statistically isotropic.

5.3. SPIRAL-DEFECT

CHAOS

111

Fig. 29. Patterns observed at P = 0.96, T = 78 [187]: (a) e = 0.040, (b) e = 0.116, (c) i = 0.465, (d) c = 0.721.

112

5. CONVECTION

REGIMES

Qualitatively similar transitions to spiral-defect chaos were observed by Hu, Ecke and Ahlers [157, 163]. A further investigation by these authors [189] was specially devoted to this phenomenon and described an experiment on C 0 2 gas at 32 bar (with P = 0.98) in a circular container with T = 40. It was found that a distinct transition to spiral-defect chaos takes place at c w 0.55. At this point, the e-dependence of the spatially averaged roll curvature exhibits a change in the slope of its plot, thus marking the transition. At lower c values this quantity remains virtually constant. Centre defects resembling the disclination shown in Fig. 24b, on the lower left, are reported to be a characteristic feature of chaotic patterns. For 0.55 < t < 0.8 spirals and centres appear and disappear, their number and the area covered by them fluctuate noticeably, the dynamics being thus intermittent. In such regimes the pattern acquires from time to time a fairly ordered appearance typical for lower e values and subsequently becomes little ordered, as at greater t. For c > 0.8 spiral and centre defects are always present. These experiments were subsequently extended to the range 0.05 < t < 5 for the same P value [190]. The effect of sidewall forcing was reduced by a special design of the paper wall, which was made in the form of two coaxial cylinders connected together by an annulus placed horizontally, and had thus an H-shaped cross section. A certain amount of the gas that filled the working volume was also present in between the two wall cylinders. The horizontal piece of the H inhibited convection in this region. As the thermal conductivity of paper is ten times greater than that of the gas and the paper interlayer separating two gas volumes was thin, the gas portion in between the cylinders played the role of a sidewall well matched with the working fluid in its thermal properties. For t < 0.09 stationary straight-roll patterns like that shown in Fig. 21c are observed. The size of the crossed-roll patches bracketing the main straightroll region decreases with increasing t. Above t m 0.09 the skewed-varicose instability (see Fig. 17 and §6.3.1) manifests itself, and cyclic changes occur in the pattern. Their scenario is rather similar to that observed at smaller V values in patterns with two sidewall foci [179-181] (Fig. 28): the instability results in the pinching-off of rolls in the central region of the tank and in the nucleation of defects. The defects approach the wall and then cither disappear immediately or move along the wall and disappear on reaching the crossed-roll patches. As t is increased further, the originally straight rolls near the crossed rolls bend progressively, and the tendency for the rolls to end perpendicularly to the sidewall increases. When 0.12 < e < 0.20, sidewall foci occasionally appear as a result of defects merging with the crossed rolls. For e > 0.20, foci are always present, and their number increases with increasing t. At c ss 0.20, there are typically two foci, whereas at higher e, up to 0.3, either two or three foci are

5.3. SPIRAL-DEFECT

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113

observed, depending on the prehistory of the pattern. At t as large as about 0.50, four to seven foci are present. In these regimes the foci nucleate new rolls while moving irregularly along the sidewall. At t a 0.65, an abrupt increase in the roll-nucleation frequency is observed. Over the t range from 0.55 ± 0.04 to 0.8 a continuous transition to spiral-defect chaos takes place in much the same way as described in Ref. 189. Further increases in t are not marked with qualitative changes in the pattern dynamics until the global appearance of the oscillatory instability at t « 3 (see §6.3.1). Liu and Ahlers [191] studied the effect of the Prandtl number on spiraldefect chaos, using various pure gases (Ar, CO2, and SF 6 ) as well as binary gas mixtures (He-SF 6 , H e - C 0 2 , and Ne-Ar). The P value was thus varied from 0.30 to 0.69 for the mixtures and from 0.69 to 1.00 for the pure gases. The onset of spiral-defect chaos was found to move to smaller e values with decreasing P. This is consistent with the current assessment of the role of mean drift flow in the generation of spiral defects (see below). Note that in the experiments with gas mixtures the Soret effect is substantial. The critical Rayleigh numbers for both the onset of convection and for the onset of spiral-defect chaos are rather sensitive to the Soret effect. However, the critical value of the reduced Rayleigh number t at which the chaotic state arises appears to be independent of the Soret effect. Assenheimer and Steinberg [192, 193) observed also patterns of another type, similar to spiral patterns in certain respects and comprising many targets — patches of concentric rolls with foci in the bulk of the container (Fig. 30). They performed experiments with SF6 near the gas-liquid critical point (T. = 318.7 K, p. — 37.8 bar, and p, = 0.73 g/cm 3 ) in a circular container with T = 115. The advantage of using such regimes is that one can easily control the value of the Prandtl number P. continuously varying it within a very wide range (from about unity to virtually infinity), and also the value of the parameter Q characterizing non-Boussinesq effects. These quantities are functions of the departure of the mean temperature in the layer from the critical temperature (because the thermodynamic and kinetic properties of the fluid depend on this quantity), and, in addition, Q varies with the layer thickness h and the temperature difference across the layer A T . In different regions of the parameter space the investigators observed hexagonal cells, rolls, targets, and spirals. Both targets and spirals arise at Q < I. T h e combination 6Q2/RC is very small in this case. However, the possibility of the occurrence of hexagonal cells at Q ~ 1 [116, 114], a value satisfying the formal criterion 6Q2/Rc
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Fig. 30. Target pattern observed at 3.5, being observed at least until P = 30. Transitions between spirals and targets occur in a certain range of P values, above a certain R value, without hysteresis. Spirals are, as a rule, one-handed. Stationary and rotating spirals coexist. They may rotate in both directions with respect to the winding direction. It is rather obvious that in the case of spiral-defect chaos the dynamics is nonvariational, just as in the case of phase turbulence. Xi et al. [194-196] carried out finite-difference numerical experiments aimed at investigating the roles of two known origins of nonvariational behaviour—the mean flow and the departures from the Boussinesq conditions—in the formation of spiral patterns. Their calculations were based on the set of equations (3.52)-(3.54), generalizing the SH equation, and the boundary conditions (3.55), (3.56). In order to ensure on the quantitative level the best fit of the model to the experimentally observed regimes, the authors derived from the governing equations a set of three-mode amplitude equations of the form (3.25). By comparison with the experimental data of Bodenschatz et al. [114], they found the proportionality factor relating e that appears in the equations to the reduced Rayleigh number and other parameters of the equations. The sidewall forcing was simulated by putting / ( x ) equal to a nonzero value at the meshpoints closest to the boundary. With the mean flow (and the vertical vorticity component) ignored [194] and without sidewall forcing, the simulated process of the nucleation of a hexagonal-cell pattern was found to occur nearly in the same way as observed by Bodenschatz et al. [114] (§4.1.3), i.e., by the propagation of six fronts from

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the central region. After c is increased to above the corresponding threshold value, hexagonal cells merge to form rolls, and the area occupied by rolls grows progressively. This transition (in a circular region with T = 60) also resembles that described in Ref. 114; however, spirals do not appear. Sidewall forcing results in that the rolls form a concentric-ring pattern or, when the computations are started from randomly distributed disturbances, a stable, one-armed spiral ending at a dislocation. In contrast to experimentally observed spirals, it does not rotate. Both forms of the quadratic term in the model equation [w2 and wAw instead of w2 in our Eq. (3.52)] resulted in virtually the same behaviour of patterns. The presentation in the paper refers to the case of wAw. The mean flow was included in the model (along with sidewall forcing) in Ref. 195. The quadratic term was chosen to be proportional to w2. The transition from hexagons to rolls led to the formation of a set of concentric circular rolls, which subsequently transformed into a rotating spiral closely resembling the spirals observed by Bodenschatz et al. (see Fig. 11 in §4.1.3). This process took a time of about 12TH, which agrees with the experimental observations. The number of arms (equal to the number of dislocations matching the spiral with the outer circular rolls) decreased with increasing e, as in the experiment. It is remarkable that non-Boussinesq effects play a crucial role: at 32 = 0 the spiral does not appear, even if the mean flow is present. The rotation of the spiral is apparently related to the mean flow. Finally, Xi et al. [196] succeeded in simulating spiral-defect chaos. Calculations were carried out for circular containers. When the mean flow was taken into account, for T = 32 and P = 1 the simulated patterns in their final state comprised many locally rotating spirals, which filled the entire container. At T = 16 chaotic states were not achieved; rather a globally ordered pattern with a single two-armed spiral was obtained. Likewise, spiral-defect chaos did not develop at P = 6. In general, large aspect ratios and small Prandtl numbers were found to play a decisive role in the formation of such states. In contrast to what was said regarding single-spiral patterns, non-Boussinesq effects seem to be unnecessary for the occurrence of chaotic states with many spirals, because they can be obtained for g2 = 0. Spiral-defect chaos was also reproduced by Decker, Pesch and Weber [197] in numerical experiments based on the Boussinesq equations. Calculations were carried out for a square (in plan) region with periodic boundary conditions in the horizontal directions (T = 25 or 50) and in some cases for a circular region (T = 25). The horizontal layer boundaries were assumed to be rigid. Calculated patterns turned out to be very similar in their visual appearance to those observed experimentally by Morris et al. [187]; moreover, very good agreement with the results of this work (even quantitative) was achieved in the time scale

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of the changes of the patterns, in the distribution of local wavenurnbers, as well as in the behaviour of the mean wavenumber and of the correlation length as functions of t. Being puzzled with a dramatic difference between the observed chaotic, timedependent pattern of spirals and a well-ordered, steady-state picture of straight parallel rolls theoretically predicted for the conditions under study, Morris et al. [187] suggested that "the attractor basin of straight rolls apparently does not overlap with the initial conditions and boundary conditions accessible to the experiment". This statement can however hardly be adopted for the following reason based on the quite justified assumption that the numerical model of Decker et al. [197] is adequate. When the initial conditions chosen for the simulation of the flow impose a preferred direction, a spatially periodic straightroll pattern was found to have just a fairly wide basin of attraction: "Even an initial superposition of noise with a strength of 80% of the roll amplitude is not sufficient to prevent the eventual recovery of the stable roll-pattern attractor at t = 0.7". In addition, we know that situations where initial conditions lead to straight-roll patterns are fairly typical in experimental studies. How can the apparent contradiction between the theory and experimental observations be resolved? First, it is quite plausible1 that in fact there is no such a contradiction. Spiral-defect chaos could be merely a transient regime that precedes reaching a steady state, although it may last very long. In the experiments of Morris et al. [187] the characteristic time of observation was about 2rh, which is much less than rYh. We saw, however, that in a number of cases where a very complex dynamics was observed, the time of flow settling was many times greater than

rv h . Second, such a phenomenon as spiral-defect chaos, which always arises after a certain critical R value is exceeded, could be merely the result of the roll instability of a particular type not investigated theoretically. In this case the stability region in the (k, P, #)-space would be additionally restricted as compared with the results of the available theoretical analyses (see §6.3.1). Finally, as the experiments of Assenhcimer and Steinberg [192, 193] suggest, one should not rule out the possibility that non-Boussinesq effects favour the formation of spiral-defect chaos (although, according to Decker et al. [197], they seem not to be necessary). Let us s u m m a r i z e some important features of the exchange of convection regimes discussed in this chapter. For small and moderate P values the character of convection in finite-size containers is not determined in a universal way

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by the numbers Ft and P only. The corresponding (left-hand) portion of the diagram of regimes (Fig. 25) illustrates only the roughest regularities in the regime sequence. A detailed description must be very specific, pertaining to a container of the given dimensions and shape, with the given boundary conditions at the sidewalls. It seems plausible that as the influence of the sidewalls of the container is reduced and the aspect ratio is simultaneously increased (the conditions of an infinite layer being thus approached) the structure of convection patterns that spontaneously develop from weak noise perturbations should become more predictable on the basis of the R and P values. At the same time, the stability of finite-amplitude convection strongly depends on the overall geometry of the flow—this will become clear from the discussion of the laws governing the behaviour of the wavenumbers of convection rolls (Chapter 6).

CHAPTER 6

SELECTION OF THE WAVENUMBERS OF CONVECTION ROLLS

Even if the formulation of a problem concerning supercritical convection or the conditions of a planned experiment enable one to predict the planform of convection cells, the question of the scale of the realized flow (which can be characterized by the width of a roll, the radius of a polygonal cell, etc.) remains nevertheless open. This question arises from the fact that, given a supercritical R value, steady-state solutions exist for any wavenumber k specifying their horizontal period provided this wavenumber lies within a certain finite-width range. In the literature this subject is referred to as selection of wavenumbers (or wavelengths). We discuss here this problem for the simplest case where rolls are the main form of the convective flows (this is how matter stands under the standard conditions). Even in this variant the problem has by no means been resolved, although over the past one and a half decade many studies have been devoted to it. As a rule, roll convection, spontaneously set in, demonstrates some distribution f{k) of local wavenumber values, which is peaked at a certain k value (Refs. 135, 177, and others), thus evidencing the existence of a distinguished optimal scale (see Fig. 27). It is natural to associate the expression "selection of scales" precisely with such a distinction. We shall call the preferred, or optimal, wavenumber the most likely wavenumber kp of a well-established convection-roll pattern that has developed in an infinite layer from weak random (noise) initial perturbations. We consider this quantity to be an inherent characteristic of roll convection, determined by the regime parameters—the Rayleigh and the Prandtl number -as well as by the boundary conditions on the horizontal layer surfaces. For real experimental conditions t h a t differ little from the above-indicated idealized conditions (and, in particular, do not involve any significant influence of sidewalls), the most likely k value will hereinafter be referred to as the observed preferred wavenumber.

119

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We shall identify the process of flow readjustment to the optimal scale as the selection of the preferred wavenumber, the factors forcing the flow to readjust as selective factors, and the problem of theoretical prediction of kp as the selection problem. Under particular artificial conditions, where the flow structure is in some special way predetermined at the initial moment, and/or sidewalls are present, such readjustment may be hampered. We shall say in what follows that in these cases contraselective factors operate, which endow the flow with stability against the action of natural selective factors. The selection process stops at one or another stage of the flow evolution from the initial state, and the characteristic scale of the well-established flow does not correspond to the preferred wavenumber. Especially pronounced contraselective properties are characteristic of such simple (highly ordered) patterns that have in the steady-state regime a distribution f(k) resembling ^-function. Virtually, only a unique wavenumber is observed in this case. We shall call it the final (meaning the ultimate result of the temporal evolution), or realized, wavenumber. The distinction of the realized wavenumber seems to be quite unambiguous. And it is therefore tempting to speak about selection keeping in mind precisely such situations in which the pattern arrives at a single realized wavenumber. It is to do in this manner that is the common practice in the literature. In any such case one or another ''selection mechanism" is said to operate, and the realized wavenumber is called the "selected" one. In general, this wavenumber is not optimal: given R and P , different "mechanisms" may produce different realized wavenumbers. Moreover, by means of special efforts (e.g., by imposing controlled initial conditions in the experiment), flows with different "man-made" wavenumbers can be realized. And it would thus be highly incongruous to attribute the word selection to such cases! Generally, it makes little sense to call selection (or selection mechanism) any act of the realization of a flow with a definite wavenumber, the "distinction" of this wavenumber being possibly restricted to this case only. Meanwhile, since much investigators' attention has been concentrated on the "selection mechanisms", the question of the optimal scale has fallen into the background. The disagreement between the final wavenumbers achieved in different cases is considered by many researchers as a reason for the assertion that not only is there no universal principle of wavenumber selection but there is not any unique (for given R and P) distinguished spatial scale at all. The term "wavenumber selection" has thus lost its meaning to a considerable degree. We shall see that understanding the final state of a convective flow as a result of the combined action of selective and contraselective factors enables one to

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reduce the available facts to a unified, consistent picture of phenomena. Among other things, it becomes possible to formulate on this basis such requirements for a calculated model that the optimal scale (which emerges under natural conditions) turns out to be predictable with using this model. 6 . 1 . W a v e n u m b e r s in E x p e r i m e n t s with R a n d o m Initial D i s t u r b a n c e s As already noted, at sufficiently large P values steady-state roll convection can be observed up to R m \0RC. Not-quite-stationary roll patterns can also occur at larger Rayleigh numbers (for example, in the experiments of Gollub and McCarriar [167], even at P as small as 2.5 the flow was mainly two-dimensional up to R ftJ 40fl c ). In a fairly wide range of parameter values, the flows that exhibit phase turbulence are quasi-steady-state with respect to the characteristic turnover time of the fluid in a roll. Defects break two-dimensionality only locally. The larger the value of R, the greater the natural spread in local wavenumber values, which is measured by the width of the peak of the distribution f(k) (see Fig. 27). Numerous experiments revealed a general regularity in the behaviour of the observed preferred wavenumber. 0 Such a kp decreases with increasing R (see, in particular, Refs. 198, 164, 199, 156, 200, 135, 167, 201, 202, 179, 180, 177, and others), and this is also true for the experiments in which annular axisymmetric rolls are observed [203, 156] (see also a review by Koschmieder [9] and his book [7]). An example of such behaviour of kp is presented in Fig. 31. Experiments performed with different fluids under similar conditions show that the /^-variation of kp is typically the weaker, the greater P (see Fig. 32, which represents Koschmieder's summary of the /^-dependences of the preferred roll wavelength A = 2ir/kp measured for different P values [7]). For P > 10 the dependence kp{R) is hysteretic [164, 199]. In Ref. 167, kp (for P ='2.5) was reported to be almost constant within the range 6 < t < 40 (at larger values of R two-dimensionality was lost). In some cases, however, results contradicting the general line of the behaviour of the mean wavenumber (k) as a function of R were obtained. In particular, in the experiments of Hu et al. [190] (described in Sec. 5.3) an increase in (A:) with R was observed for t < 0.09 and P = 0.98, which was replaced with a decrease after the onset of time dependence due to nucleation of defects. It is worthwhile to note in passing that in general, the presence of defects can "According to our definition, it corresponds to the position of the maximum of f[k). In many cases, however, the average k value, {k), is measured experimentally instead of the most likely k value, kp We do not distinguish between kp and (k), which are close to each other, and attribute the term observed preferred wavenumber to both.

122

6. WAVENUMBERS OF CONVECTION ROLLS

Fig. 31. Average wavenumbers of steady rolls for different c values as observed at P = 2.5 (adapted from Ref. 177). Thresholds are shown for the skewed varicose (SV), cross-roll (CR), zigzag (ZZ), and Eckhaus (E) instabilities of rolls (see §6.3.1).

by itself contribute to the short-wavelength part of the wavenumber spectrum, since, as a rule, defects introduce fine details into a pattern. In the example mentioned, short crossed rolls may be such fine details, although they allow the straight rolls that fill the bulk of the container to easily change their width, thus ensuring the decrease of the straight-roll wavenumber with increasing R (this possibility will be discussed in §6.5.1). If so, the formally calculated average wavenumber may substantially disagree with the most likely wavenumber and may not have a clear physical meaning. In such cases, a somewhat arbitrary procedure of selecting "representative" (free of defects) fragments of the pattern may turn out to be useful in the determination of kp. For a long time, attempts to theoretically describe the behaviour of the wavenumbers of convection in an infinite layer have resulted in a contradiction with experiment: k increased with R. This is also the case for the k value at which the heat transfer through the layer is maximum (and which should be realized according to the so-called Malkus principle—see below), as well as for the k values that correspond to the extrema of some other characteristics—see Sec. 6.2. Lipps and Somerville [204] revealed the same effect in two-dimensional numerical simulations and had to use a three-dimensional model to achieve reasonable agreement with experiment. Therefore, these investigators concluded that the preferred wavenumber cannot be reached in purely two-dimensional processes. In their opinion, the formation of a steady-state two-dimensional flow involves a three-dimensional transient process, which affects the final wavenumber.

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Fig. 32. Summary of wavelengths A = 2ir/kp of steady convection, measured in experiments with different fluids, as a function of the Rayleigh number [7]: (l) silicone oil, P as 3000 [198]; (2) silicone oil, P as 870 [164]; (3) silicone oil, P ss 950 [156]; (4) air, P a 0.7 [201]; (5) water, P as 7 [199]; (6) water, P as 7 [200]; (7) air, P as 0.7 [199]. It was also suggested that the decrease of kp with increasing R is an effect of accessory factors such as the presence of sidewalls (for a bounded volume the condition of maximum heat transfer gives a qualitatively correct prediction of the behaviour of k [205]; see § 6.5.8 for further discussion of the role of sidewalls) or the finiteness of the heat conductivity of the plate bounding the layer top (in this case, according to Ref. 206, the critical wavenumber fcc is reduced, and it is by this effect that a reduction in A:p was accounted for; however, experiments with plates of very good conductivity show nevertheless a decrease in kp with increasing R [203]). The question of the factors affecting the wavenumbers of two-dimensional rolls will be discussed in detail in subsequent sections. We only note here that the decrease of kp with increasing R can be obtained in a purely two-dimensional numerical experiment simulating the conditions of an infinite layer with perfectly conducting horizontal boundaries if the requirement that the flow be spatially periodic, usual for the practice of numerical simulations, is lifted [266, 268] (see §6.5.6).

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6. WAVENUMBERS OF CONVECTION ROLLS

6.2. Searches for Universal Selection Criteria The very idea of the existence of preferred forms and scales of convective flows appeared long ago. Searches for a general principle that could make it possible to single out such forms and scales were apparently started with a work by Malkus [207]. It makes sense to summarize the investigations aimed to implement this idea, which has been repeatedly and undeservedly criticized for a long time. The Malkus principle was suggested on an intuitive basis. According to this principle, those flows should be realized that maximize the convective heat transfer. Later, Malkus and Veronis [40] made an attempt to relate this principle to the stability of steady-state solutions to the equations of convection. However, many cases have been found in which predictions made on the basis of the Malkus principle disagree with experiment or accurate theoretical analyses of stability. First of all, for two-dimensional roll flows the wavenumber that maximizes the convective heat flux through the layer increases with increasing R (see, for example, Busse's studies [211, 13]), while an opposite behaviour is observed experimentally for the mean wavenumber. (True, Davis noted [205] that in a bounded container, in contrast to infinite layers, the wavenumber selected according to the Malkus principle should nevertheless decrease with increasing R. However, we shall see in Sec. 6.5 that the decrease in A; with increasing R is not necessarily an effect of sidewalls.) The fact that the realizability of a flow is not directly related to the condition of maximum heat transfer is evidenced by Foster's numerical experiments [208]. Further, the experimentally observed motion of dislocations results in such wavenumber changes that reduce the heat transfer [256]. Finally, for a nonlinear temperature dependence p(T) there exists an interval of Rayleigh numbers where rolls are unstable although they transfer more heat than stable hexagons [12, 98]. Busse [44] formulated an extremum principle according to which, at small supercriticalities, among steady solutions with different planforms and a fixed k = kc, those solutions are stable that minimize a certain functional. Under some conditions this principle is equivalent to the Malkus principle as well as to the requirement that the kinetic energy of convection be maximum. This approach, however, does not reveal a preferred wavenumber. In general, the kinetic-energy maximum, as also the heat-transfer maximum, does not agree with the observed kp values [13]. This is also true for the maximum of the reduction of potential energy in a layer of convecting fluid as compared with the static value [13], for the maximum of the growth rate of disturbances as found from linear theory [31], and for some other characteristics of convection [209]. Catton [209] concluded that the ^-dependence of the deter-

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125

minant of the second variation of the mean-square time derivative of the mean temperature gives a much better result. The determinant can be obtained by means of solving the so-called mean-field equations. We note that the solution procedure is not doubtless. The thermodynamic approach based on applying the principle of maximum production of entropy [210] makes it possible to investigate stability in various particular situations. The stability criterion obtained in this way does not, by itself, permit one to single out a unique preferred wavenumber within the interval of stability (see Sec. 6.3). However, the stability functional 0 has a single maximum in this interval, which shifts toward smaller k values as R increases [209]. But this shift is much sharper than the experimentally observed shift of fcp. From general considerations, Catton [209] proposed a selection criterion in the form of the following condition: d2 . . QJpl\k=kp{R) = 0.

(6.1)

When combined with an equation for (p, this condition determines the dependence kp(R), provided that the initial conditions specifying this dependence, i.e., the values of kp and dkp/dR at R — R^, are given. Clearly, kp(Rc) = kc. Catton uses experimental data to determine k'p(Rc). The dependences obtained in this manner agree satisfactorily with those specific experiments from which the corresponding values of k'p(Rc) were found. Possibilities to determine the preferred wavenumber using the Lyapunov functional for the equations that describe convection will be discussed in Sec. 6.4 and further. 6.3. Stability of Two-Dimensional Roll Flows As noted in the introductory chapter, a widely used approach to solving the problem of readability (and, therefore, the problem of selection) involves investigation of steady-state flows. The greatest attention has been focused on the question of stability of two-dimensional spatially periodic roll patterns. 6.3.1. Theoretical Results This direction of research traces back to a study by Schliiter, Lortz, and Busse [41] (see Sees. 3.2 and 4.1) based on expanding the unperturbed flow in small amplitudes (small reduced Rayleigh numbers) and analysing, in a linear approximation, the stability of the obtained solutions against small perturbations. These investigators showed that all three-dimensional flows are unstable

6. WAVENUMBERS

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OF CONVECTION

ROLLS

provided the standard formulation of the problem is adopted. 6 As for twodimensional flows, it was found that irrespectively of the P value and boundary conditions on the horizontal surfaces, rolls with k < kc are unstable and their stability takes place in a certain interval kc < k < fci(e). The subsequently published stability diagrams (see Figs. 33 and 34 below) agree with the inference concerning the region k < kc only for P = oo. The cause for this disagreement became clear after the elucidation of the role of the vertical vorticity component at finite P values by Siggia and Zippelius [53, 54] (see below). In Ref. 41 this component was eliminated by assumptions about the flow structure introduced there, whereas in later works it was taken into account. A detailed analysis of the linear stability of roll flows for a wide range of values of the Rayleigh and the Prandtl number was performed mainly by Busse and his coworkers [211-220, 137, 106, 119]. In these studies, a steady-state spatially periodic flow regarded as the initial (unperturbed) one is calculated on the basis of the complete nonlinear equations (2.15)-(2.17) by the Galerkin method and has the form vz = ^2 Amnfn[z)

cos mkx

(6.2)

m,n

(with corresponding expressions for vx and 6).^ Here fn[z) (n = 1,2, ...,,/V) are the functions of a complete orthonormal system which satisfy the boundary conditions for vz; m = 1,2, ...,N — n, where the truncation parameter N is chosen such that the solution changes by a negligible amount as N is replaced by N + 2. If the layer is bounded by two rigid boundaries, the Chandrasekhar functions gn defined by Eq. (3.89) stand for fn. In the case of two stress-free boundaries / „ = sm nz. Solution (6.2) is symmetric with respect to x ~ 0. In addition, convection rolls that are really observed are symmetric with respect to the roll axis:

v„(x,z) = -vj--x,

1 - z)

(6.3)

(the same relation is valid for vx and 9). Obviously, such a symmetry implies that Amn assume nonzero values for only even m + n. Let three-dimensional infinitesimal perturbations of the form v'2 =

£(£

m n

cos mkx + Cmn sin rnkx)fn(z)

e'^+W+"*

(6.4)

,m,n 6

Recall, however, §4.1.11. To be more precise, in the studies under discussion the velocity field v is decomposed into the poloidal and the toroidal component:

c

v = V x (V x id>) + V x zV, where and xjj are scalar functions.

6.3. STABILITY

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(with corresponding expressions for v'x, v', and 0') be superposed on this twodimensional flow. Then the linearization of the original equations with respect to the perturbations leads to an eigenvalue problem for the growth rate a. A detailed description of the procedure of analysis is given in Ref. 213. In further investigation, Busse and coworkers also obtained nonlinear solutions representing tertiary flows that can result from the development of certain types of roll instability. Moreover, they analysed the instabilities of the tertiary flows in the linear approximation. In Sec. 5.1 we already mentioned certain results obtained in this way and concerning the instability of bimodal flows [106, 136] (note that first experimental observations of such instabilities were made long before these studies [221]). Nonlinear solutions for tertiary flows of another considered type are of the form vz = ^2[Aimn m.n

cos lky(y-ct)

+ A,mn sin lky(y-ct)}

I

*X

\

I COSTTlKxX J

fn(z) (6.5)

(with similar expressions for other variables) and describe waves travelling along the rolls. The coefficients of this representation are calculated by the Galerkin technique. The equation for the determination of the phase speed c can also be obtained in a straightforward way. Some characteristic instabilities of two-dimensional, spatially periodic roll flows have already been mentioned in §4.1.10. They play an important role in convection-pattern dynamics and, particularly, in the processes of wavenumber adjustment. It is reasonable to give here a summary of their properties with an indication of the P- and fc-ranges where the instabilities can manifest themselves (we are now talking about a layer with rigid boundaries). We denote as am and bm, respectively, the values of a and 6 which maximize Re a. The disturbances with a ^ 0 are either symmetric ( C m n — 0) or antisymmetric (Bmn — 0) with respect to the plane x = 0. For oscillatory instabilities, the imaginary part Im 2, and k < kc. Results in sinusoidal curving of the rolls. 2. Cross-roll instability (CR): a = 0, |6 m | > kc, Imcr = 0, m + n is odd, Cmn = 0, P > 1.1, and k > kc. Forms a system of rolls perpendicular to the initial ones. 3. Knot instability (K): a = 0, |6 m | < kc, I m a = 0, m + n is odd, Cmn = 0, 1.1 < p < 10, and k > kc. This is a branch of the CR instability for which the growth rate a as a function of 6 has two maxima. The K instability corresponds to the maximum that is located at a relatively small b.

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ROLLS

4. Skewed varicose instability (SV): \am\ < 1, |6mj < 1, ajb is finite, Im<7 = 0, m + n is even, P < 30, and k * kc. Roll deformation caused by this instability was illustrated in §4.1.10 by Fig. 17. 5. Eckhaus instability (E): \am\ < 1, b — 0, m + n is even, Imtr = 0, Cmn = 0, P < 1, and fc > fcc. This is the only instability that does not break the two-dimensional geometry of the flow; it leads to x-alternating compressions and expansions of groups of rolls. 6. (Even) oscillatory instability (EO): a = 0, |6 m | « 2, Imcr ~ ft*, m + n is even, Bmn = 0, P < 2.5, and fc < kc. Leads to the development of sinusoidal wavy disturbances that travel along the rolls. 7. (Oscillatory) two-blob instability ( B 0 2 ) : a = 0, |6 m | » 3.1, Imcr ~ 2ft*, m + n is odd, C m „ = 0, 2 < P < 8, and k < kc. In the cross section of a roll there arise two spots of elevated temperature, which are located in diametrically opposite parts of the section, and two spots of reduced temperature, which are located similarly, between the "hot" spots (see below). 8. (Oscillatory) one-blob instability (BE1): a = 0, |6 m | ss 4, Imcr ~ ft*. m + n is even, Cmn = 0, 7 < P < 12, and k < kc. Unlike the case of the two-blob instability, in the cross section of a roll there arise one spot hotter and one spot colder than the ambient fluid. They are located at diametrically opposite parts of the section. Some results of the investigation of stability for the case of two rigid boundaries are summarized in Figs. 33 and 34. The former shows for each of four chosen P values the curves in the (k, P)-plane which correspond to the thresholds ( R e a = 0) of different modes of instability; the region of stable roll flows (where Re a < 0 for all types of instability) is bounded by these curves and sometimes called the "Busse balloon". The latter, constructed by Busse and Clever [222], represents a composite three-dimensional stability diagram in the (k, P, P)-space ("Busse windsock"), which interpolates seven two-dimensional diagrams plotted in particular sections P = const. In addition, the results for small Prandtl numbers [220] are presented below in Fig. 36. The criterion of the onset of the Eckhaus instability [223, 224] has in the limit of t —> 0 a rather universal form, not only independent of P but, moreover, applicable to two-dimensional perturbation fields described by equations of a very wide class if the critical value of the corresponding control parameter is slightly exceeded (in particular, in fluid mechanics Taylor vortices represent a class of structures very similar to convection rolls in behaviour). In this limiting case the threshold curve for the Eckhaus instability is given by the equation

e - 3$(fc - kc)2 = 0.

(6.6)

6.3. STABILITY

OF 2D ROLL FLOWS

129

Fig. 33. Regions of stability of infinite, spatially periodic roll patterns in the (k, H)-plane ("Busse balloons") for P = oo [211] (filled with dots), P = 1 [215] (horizontal hatching), P = 0.71 [214] (oblique hatching), and P = 0.1 [214] (vertical hatching). No-slip conditions are assumed for the horizontal surfaces of the layer. The boundaries of the regions are formed by segments of the threshold curves for different types of instability, which are labelled by the letters denoting the instability type and by the P value. The dash-dotted line at the bottom (Ri) is the neutral curve for the onset of convection.

According to Eq. (2.40), the neutral curve of the primary instability of the unperturbed system (motionless fluid layer) corresponds to the equation e - e0(k - kc)2 = 0.

(6.7)

The wavenumber bandwidth for spatially periodic roll patterns stable with respect to the Eckhaus perturbation mode is therefore \/3 times as narrow as the wavenumber bandwidth for the patterns of this type that can exist at all. The types of instability most typical at large P values and significant at moderate P values are the zigzag instability, which results in an efficient decrease of the characteristic flow scale and thus sets the long-wavelenghth boundary of the stability region, and the cross-roll instability, which can either increase or

130

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Fig. 34. Region of stability of spatially periodic roll patterns in the (fc, P, R)-space ("Busse windsock"); adapted from Ref. 222. By convention, the stability diagram calculated for P = oo is placed in the plane P = 300. No-slip boundary conditions are assumed for the horizontal surfaces of the layer. The portions of the surface bounding the stability region which correspond to particular types of instability are labelled by the letters denoting these types. decrease the characteristic scale [211, 215]. As we noted in §4.1.10, at moderate values of R the development of the CR instability replaces the original rolls with new rolls directed normally to them [120]. If, however, R > 10/? c , crossed rolls develop mainly in the thermal boundary layers created by the basic flow and have smaller horizontal dimensions. Ultimately a three-dimensional flow in the form of a superposition of the original and crossed rolls (bimodal convection) is established (Fig. 13, §4.1.10; see, in particular, Ref. 136). As already noted, the stability of steady-state bimodal convection against infinitesimal perturbations was investigated in Refs. 106 and 136. If the development of the zigzag instability does not reach the stage of the transition to a new set of straight rolls turned with respect to the original rolls by an angle of about 45° (see §4.1.10), the pattern settles down to a steady state and assumes the form of a collection of wavy (undulating) rolls. Busse and Auer [225] analysed possible tertiary instabilities of such patterns. Under the assumption of negligibly weak mean drift (i.e., of sufficiently large P values) the Newell-Whitehead-Segel amplitude equation was used for the analysis. The

6.3. STABILITY

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131

Fig. 35. Steady domain pattern of undulating rolls as obtained by numerical simulation for stress-free boundaries at ( = 0.101, P - 12 [225]. Lines of constant vertical velocity in the midplane of the layer are shown. investigators found that the set of undulating rolls obtained by superposing a wavy disturbance with a wavenumber p onto the original straight rolls with a wavenumber k = kc + q can exist provided q < -p2£o/4kc and is stable if, in addition, q > -(f./3-p 4 £ u ; /3A^) 1 / 2 The evolution of an unstable undulating-roll flow was simulated numerically by means of calculating the interaction of 9, 15, or 25 modes. The most remarkable feature of the development of the instability is the formation of domain patterns. In such a pattern (Fig. 35), domains of nearly two-dimensional rolls (whose wavenumber is somewhat larger than that of the original rolls) alternate with domains of undulating rolls (which have a wavenumber smaller than the original one; however, the effective characteristic scale of a flow of this type is reduced by undulations). Accordingly, this type of instability received the name domain instability. At intermediate P values, the knot instability (Fig. 14, §4.1.10) can come into play [214, 215]. Clever and Busse [137] investigated in detail, by means of numerical experiment, the behaviour of a well-established pattern of rolls with a tertiary flow produced by the K instability and the proper oscillatory instabilities of such a superposition. Steady knot convection is stable in only a small region of the parameter space. If R is sufficiently large, the development of the K mode results in the formation of concentrated updrafts and downdrafts. When this mode is fully developed, it forms a characteristic spoke pattern (Fig. 15, 84.1.10; see Ref. 138) in which large polygonal cells rather than rolls are clearly

132

6. WAVENUMBERS OF CONVECTION ROLLS

seen. At Rayleigh numbers close to the onset of this type of convection the pattern is almost stationary, while at high Rayleigh numbers (of order 105-106) it fluctuates in time and behaves chaotically. Spoke-pattern convection is the predominant form of high-/? convection at moderate Prandtl numbers [13]. It should be kept in mind that the term spoke-pattern convection was originally used to designate oscillatory knot convection (i.e., for a type of patterns in which the original roll structure is still apparent) [221], while the knot instability of rolls was called the collective instability (although it would be more appropriate to attribute this term to the instability of bimodal convection). At moderate P values the boundary of the stability region also has a small segment (at the top, near the intersection of the neutral curves for the CR and K modes) that corresponds to the onset of the one-blob (for larger P values) and/or two-blob (for smaller P values) instability [216]. In particular, both modes are present at P — 7, and they are not shown in Fig. 33 only because the corresponding diagram was not given in the original paper (see, however, Fig. 34). Flows that evolve from the oscillatory blob instabilities can assume the forms of standing or travelling oscillatory blob convection. The structure of tertiary flows due to the two-blob instability was analysed in detail by Clever and Busse [226] on the basis of nonlinear solutions of the form (6.5). This structure and the flow dynamics are largely determined by the process of periodic eruption of thermal blobs from the "cold" (top) and the "hot" (bottom) boundary layer. Blob formation is possible at sufficiently high temperature gradients within the boundary layers as they become convectively unstable. While at high Prandtl numbers their instability can produce steady tertiary motions (crossed rolls), at moderate Prandtl numbers the effects of advection described by the term P _ 1 ( v - V ) v are fairly strong, and the interplay between blob formation and advection results in oscillations. A linear stability analysis of the solutions representing blob convection was also performed and showed that standing oscillations are always unstable. A distinctive feature of travelling blob convection is a mean flow directed along the rolls. The longitudinal velocity component averaged in the longitudinal direction has a complex distribution over the cross section of a roll, being sign-alternating. Integration of this component over the cross section yields a resulting stream in the direction of wave propagation. At moderate and small values of P the short-wavelength boundary of the stability region is determined by the skewed varicose instability [214,215], which has been observed experimentally [215]. It increases the characteristic flow scale. Finally, for P < 1 one boundary of the stability region is the neutral curve for the even oscillatory mode [213]. We note that an analytical investigation of roll stability in the limit of P -> 0 for a layer with free boundaries [212] determined the threshold value of the Rayleigh number for the EO instability to

6.3. STABILITY

OF 2D ROLL FLOWS

133

Fig. 36. The region of stability of two-dimensional convection rolls for P = 0.01 (solid lines) and P < 0.003 (dashed lines) [220]. No-slip conditions are assumed for the horizontal surfaces of the layer. The stability boundaries are labelled by the letters designating the types of instability. The solid line below is the neutral curve for the onset of convection. be R; = Rc(l + 0.31P 2 ) (under the assumption that k = kc). This means that the interval of R values where rolls are stable vanishes in this limit. In contrast, if the boundaries are rigid, the region of stability remains finite [220] (Fig. 36). T h e reason for this difference lies in the fact that the EO mode is related to the appearance of a nonzero value of the vertical vorticity component, absent in an unperturbed roll flow. In the case of free boundaries, in the limit of b —> 0, an undamped perturbation with a vertical vorticity uniform throughout the layer can exist (which corresponds to the rotation of the layer as a whole). The critical Rayleigh number for oscillations corresponds to such a perturbation. For rigid boundaries, a vertical vorticity constant in z is forbidden by the boundary conditions, and the critical value of R is reached at a finite 6. Clever and Busse [217] studied also the nonlinear properties of oscillatory convection in a layer with no-slip boundaries. Infinitesimal perturbations were assumed to have the same structure and the same wavenumbers as the unperturbed flow (6.5), and the strongest growing disturbance was selected by means of a linear analysis out of four classes that differ in their symmetry with respect to the unperturbed travelling waves. Subsequently the evolution of this disturbance to a finite amplitude was studied by numerical integration, the coefficients Aimn, Aimn, etc. being considered time-dependent.

134

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OF CONVECTION

ROLLS

Fig. 37. A stage of the evolution of an asymmetric travelling wave according to a numerical experiment for R = 104, P = 0.71 [217]. Lines of constant vertical velocity in the midplane of the layer are shown. No-slip conditions are assumed for the horizontal surfaces of the layer. In particular, it was shown that the onset of oscillations reduces the heat transport in comparison with the case of steady straight rolls. Another remarkable feature of travel ling-wave convection, first noted by Lipps [227] and McLaughlin and Orszag [81] and discussed in more detail by Clever and Busse [217], is the property that symmetric travelling waves of the form (6.5), losing their stability at a sufficiently high Ilaylcigh number, become asymmetric (Fig. 37). This is because the disturbance that starts growing first has a symmetry different from that of the unperturbed solution. This analysis was extended to the case of very low Prandtl numbers in Ref. 220. The interaction of the horizontally averaged flow with the mean pressure gradient that depends on the presence of sidewalls was taken into account. While the properties of travelling-wave convection near its onset does not show any marked P-dependence, the tertiary transition to asymmetric waves experiences a qualitative change at P w 0.02. Two types of instability of symmetric travelling waves differ in their behaviour in the frame of reference moving with the phase speed c of these waves. At very low Prandtl numbers the symmetry-breaking perturbation grows monotonically near the transition. At higher Prandtl numbers the monotonic instability is preceded by the oscillatory instability (which is characterized by a finite imaginary part of the growth rate).

6.3. STABILITY

OF 2D ROLL FLOWS

135

In the case of stress-free boundaries, the ranges of the parameter values in which stable roll flows are possible are, under otherwise identical conditions, considerably narrower than in the case of no-slip boundaries. An investigation of convection in a layer with free boundaries performed by Busse and Bolton analytically for small supercriticalities [218] and numerically for their wide range [219] showed that for P < 1 the wavenumber bandwidth of stable rolls is very small (for example, at P = 0.71 its width does not exceed 0.0065 for any value of R). The long-wave boundary of the stability region is determined in this case by the oscillatory skewed varicose instability (which is not revealed in the case of rigid boundaries) while the short-wave boundary is determined by the standard (monotonic) SV instability. For P < Pc = 0.543 stable flows are impossible at all. We note (and it will be important in what follows) that the interval of wavenumbers where rolls are stable against the Eckhaus instability, is nevertheless fairly wide for any value of P. For P — oo the boundaries of the Busse balloon, which are determined in this case by the CR instability, were found by Schnaubelt and Busse [228]. For the case of asymmetric boundary conditions (a rigid lower and a stressfree upper boundary) the stability of rolls was considered by Kvernvold [229] and Clever and Busse [119]. The results of these analyses do not disagree markedly with those pertaining to the case of two rigid boundaries. In particular, the oscillatory skewed varicose instability, which is typical for a layer of a lowPrandtl-number fluid with two free boundaries, does not manifest itself. In contrast to what vvas found for the case of two rigid boundaries, the oscillatory one- and two-blob instabilities do not determine the position of any segment of the stability-region boundary at any P, being always preceded by monotonic instabilities. Two important c o n c l u d i n g r e m a r k s can be made on the basis of the results described here. First, if the case of two free boundaries and P < Pc is excluded, then in a certain range of Rayleigh numbers there always exists an interval of wavenumbers which corresponds to stable two-dimensional roll flows. For not-too-small P values, this interval is rather wide, and its shrinkage with decreasing P is always due to three-dimensional instabilities. Thus, all flows with wavenumbers lying in the stability range appear to be equally realizable. It will become clear from what follows that this is not the case. Second, if the flow should readjust since its k lies outside the stability region, the readjustment is in most cases governed by three-dimensional processes. This raised the widespread belief (supported by numerical experiments [204]) t h a t two-dimensional deformations are not an efficient means to change k (in

136

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ROLLS

particular, to ensure the experimentally observed decrease in A: with increasing R). We shall demonstrate below the fallibility of this point of view. Now we note some results of the investigation of roll stability by other methods. Greenside and Cross [230] undertook such a study on the basis of model equations in order to choose a model that would best reproduce the stability properties that were found by solving the Boussinesq equations. The authors studied two classes of models: a generalization (3.47) of the Swift Hohenberg equation and the Herzberg-Sivashinsky equation (3.48). Since the vertical vorticity component plays a fundamental role in the development of some instabilities (in particular, the SV instability), the researchers included in the equation a term that is governed by this component and describes mean drift. Moreover, in a number of cases they introduced a special filtering procedure that suppresses the short-wave instabilities (e.g., the CR instability). The best qualitative agreement with the rigorous theory was obtained for the model (3.48) at d = 3 with the drift and filtering added. In the language of the theory of phase dynamics [62, 64, 48, 65-67] (see §§3.3.5 and 3.3.6) the Eckhaus instability corresponds to the case D\\ < 0 and the zigzag instability in the absence of mean drift, to the case of D±. < 0. Obviously, mean drift affects the stability of rolls [53, 54, 218, 219, 67]. As drift tends to straighten curved rolls, it strongly suppresses the ZZ instability at small values of P (in which case this instability takes place if De± < 0). Conversely, drift has a destabilizing effect on the skewed varicose mode. An analysis based on Eqs. (3.90)-(3.92) shows [67] that, as the SV instability develops, the mean drift caused by the roll deformation amplifies the deformation, thus giving rise to positive feedback. As a result, pinches are produced, which can ultimately lead to reconnection of constant-phase lines and creation of pairs of dislocations. Hence, the SV instability plays an important role in the origin of phase turbulence (see Sec. 5.2). The long-wave boundaries of the Busse balloon, which are the thresholds for the E, ZZ, and SV instabilities, are reproduced very well on the basis of the above-mentioned equations [67], and it turns out to be possible to obtain explicit expressions for the growth rates and the form of the most "dangerous" disturbances. Numerical simulations of convective flows by no means always allow one to represent the results in terms of the stability theory although agreement with this theory is, as a rule, present. We note only that in two-dimensional calculations with the periodicity conditions on the side boundaries of the calculated domain and either free [231] or rigid [232] horizontal boundaries the wavenumber band of stable rolls was found to be very wide (just as in the case of the

6.3. STABILITY

OF 2D ROLL FLOWS

137

Eckhaus mode), filling most part of the linear-instability range for the motionless fluid. If, however, convection in a box with rigid sidewalls is simulated, the stability band shrinks [233] (see §6.5.8). Of the three-dimensional numerical experiments aimed at investigating the stability of rolls, it is worth mentioning a study by Travis et at. [234] (P = oo , free boundaries, small aspect ratios of t h e calculated domain) where the results are compared with those of Ilefs. 219 and 228. 6.3.2. Experimental

Results

In parallel with theoretical investigation, stability of two-dimensional flows was studied experimentally, by observing artificially created rolls of a given width. It is interesting that in the first study of this series, performed almost simultaneously with Busse's calculations [211], Chen and Whitehead [235] revealed certain features of the phenomenon which were not mentioned in subsequent discussion for a long time. Apparently, they could receive little attention because the researchers had failed to directly associate these features with theoretically predicted instabilities. Chen and Whitehead, using the technique of controlled initial conditions (see Sec. 3.1), investigated the behaviour of rolls with different initial widths. Circular containers with T = 10-16, fluids with P ~ 103 (silicone oils), and shadowgraph visualization were used. It was found that in the studied range of Rayleigh numbers Rc < R < 2.5RC, the region where rolls are stable against three-dimensional perturbations is clearly delimited in the (k, /?)-plane, being bounded by the threshold of the cross-roll instability on the short-wave side and by the threshold of the zigzag instability on the long-wave side. In these qualitative features the results agree with Busse's findings [211]. But within the stability region the rolls by no means always remained unchanged. In general, their width varied, approaching a certain optimal value close to l.l/i. The process of such roll readjustment is mainly two-dimensional and possible because of the fact that at the sidewalls, near the focus singularities (see Fig. 19b), new rolls appear or old rolls disappear. The characteristic readjustment time amounts to several tens of hours, which is a few hundred times longer than r v and comparable with r h . T h e range of roll widths (or wavenumbers) where significant changes of rolls are not observed at all is much narrower than the range where the rolls expand or contract, being nevertheless stable against three-dimensional perturbations. Conversely, in the regions of three-dimensional instability the development of three-dimensional modes is accompanied by variations in the roll width.

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ROLLS

The ability of rolls to undergo two-dimensional expansions or contractions thus appears to be a more general property than their three-dimensional instabilities and a more universal mechanism for achieving the optimal scale. Later experiments with controlled initial conditions were aimed at directly testing the theoretical predictions of Busse and Clever (see §6.3.1). Rectangular containers were used. Busse and Whitehead [120] used a silicone oil with P ~ 102 in their experiments for Rc < R < 6 x 104 « 35/? t and Whitehead and Chan [236], a silicone oil with P = 16 and water at 70°C ( P = 2.74), also for a wide range of Rayleigh numbers covering the region of the existence of stable flows. Depending on R and the artificially imposed k value, the investigators observed either stable regimes (without explicit restructuring of the roll pattern) or development of instabilities of one or another type, which could be identified in most cases with theoretically predicted modes. In Ref. 120 the distribution of points representing the experimental regimes in the (k, /?)-plane showed satisfactory agreement with the calculated stability diagram [211] for P = oo (see Fig. 33). In Ref. 236 the regions of stability were appreciably deformed and displaced toward smaller k values as compared with those calculated for the corresponding values of P (especially if P = 2.74). Whitehead and Chan attribute this discrepancy to the finiteness of the thermal conductivity of the top and the bottom boundary of the layer. Nevertheless, the characteristic sizes of the stability region do not differ much from those obtained theoretically even in the latter case, so that the theory can be considered to be qualitatively confirmed. An important circumstance should, however, be kept in m i n d / In Refs. 120 and 236, in contrast to Ref. 235, the conclusion that the rolls are stable was made after a relatively short timespan elapsed since the forcing lamp was switched off and free evolution of the flow started. This time interval was only two or, at most, several times as large as r v , i.e., vanishingly small compared with Tj,. This means that in those cases where the authors recognized the regime to be stable, one could not in general rule out the possibility of slow quasi-twodimensional readjustment of the rolls in their width with a tendency toward a certain optimal final wavenumber—a process observed in Ref. 235. We shall see below (in §6.5.7) that under certain conditions this process can proceed rapidly. Busse and Clever [215] verified the conditions for the onset of the knot and the skewed varicose instability, found in the same work, by an experiment with controlled initial conditions. Whitehead et al. [221, 236, 168] studied also the stability of an artificially created bimodal flow. Further experimental investigations of roll stability in a container with a moderate V value were I am grateful to E. L. Koschmieder, who called my attention to this point

6.4. LYAPUNOV

FUNCTIONAL

AND

SELECTION

139

performed by Kolodner et al. [237] and demonstrated qualitative agreement with the theory. Experimental observations of the SV, the CR, and the EO instability in an argon layer (P = 0.7, which is typical for gases) were discussed by Croquette [160] in comparison with the theoretical stability diagram. The reaction of a roll pattern to artificially induced Eckhaus and zigzag disturbances was studied experimentally and compared with the theory of phase dynamics by Croquette and Schosseler [238]. 6 . 4 . L y a p u n o v Functional and Selection The fact that the dynamics of convecting fluid is in general nonvariational can be seen from the mere possibility of low-frequency turbulence associated with the motion of defects (Sec. 5.2). In particular, the monotonic decrease of the potential is incompatible with the cyclic evolution of patterns, observed in many cases. However, in certain situations potential dynamics can nevertheless be expected. For a finite-size box this happens, for example, if P —> oo or if the flow is two-dimensional and the conditions of applicability of the Newell-WhiteheadSegel equation are satisfied. For spatially periodic flows in an infinite region the integral representing the Lyapunov functional diverges. Sometimes a potential calculated for an integer number of spatial periods and recalculated per unit length in the direction x of the wavevector (and, of course, per unit length in the (/-direction) is considered. We shall term it the specific potential.* In an infinite region, to an arbitrarily small change in the wavenumber there corresponds a finite variation in the periodic function (representing the flow velocity). It is therefore obvious that a radical distinction exists between the variation in the functional of a finite system due to a small variation in the velocity field and the variation in the specific potential of an infinite system due to a small change in the wavenumber. According to the calculation of Pomeau and Manneville [62] for the Swift Hohenberg model (3.45), the specific potential has a minimum at the long-wave boundary of the wavenumber band corresponding to stable spatially periodic patterns. As the supercriticality t increases, the wavenumber kp at which this minimum is reached (and which the authors called the optimal wavenumber) slightly decreases, its departure from the critical wavenumber kc being proportional to c2. It is clear from what has been said that the presence of such a minimum does not mean that a roll pattern must necessarily be readjusted from a nonoptimal "Some authors call it the Lyapunov functional density.

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6. WAVENUMBERS

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ROLLS

to the optimal wavenumber, and does not contradict the fact that flows with different wavenumbers lying within a certain finite-width band are stable. A completely similar situation arises in determining the "optimal" value of the angle a made by the rolls and the normal to a sidewall [150] (see Sec. 4.2). Although the potential is minimum at a = aopl / 0, a linear stability analysis for rolls shows that at a = 0 they are stable. Since a semi-infinite region is considered, for an arbitrarily small change in a the variation of a function describing the roll structure (e.g., vz in the plane z = 1/2) turns out to be finite at sufficiently large distances from the wall. Within corrections of higher order in e, the optimal wavenumber ky in the SH model corresponds to the wavenumber kzz at which the transverse-phasediffusion coefficient DL vanishes (see §§3.3.4, 3.3.5) or, in other words, to the threshold of the zigzag instability [62]. This result has also been obtained for the general case of potential dynamics [239]. In a number of cases that will be studied in the subsequent sections (although not in all of them) the realized wavenumber coincides with kp provided the dynamics is potential. Cross [61] made an attempt to use the Lyapunov functional to investigate the relative stability of different (not too complicated) textures. The behaviour of the Lyapunov functional calculated according to the formula (3.63) of the SH model for experimentally observed evolving textures was studied by Heutmaker et al. [240, 177]. More precisely, the calculations were performed using the formulae obtained in Ref. 61 for the contributions made by different components of a texture to the functional. Although in this experiment the Prandtl number was comparatively small (P = 2.5), in the range of supercriticalities 0 < e < 3 the functional either decreased monotonically or exhibited weak short-duration increases, related primarily to the nucleation of new defects, against the background of an overall tendency for a decrease. For larger t values the nonrelaxational character of the evolution became more obvious. For very small e values the temporal variation of the functional is very slow and difficult to study. In the well-established regimes the values of the Lyapunov functional have a spread of about 25%; this reflects the fact that possible steady-state textures are not unique in the case of sufficiently large aspect ratios. 6.5. "Selection M e c h a n i s m s " Now we consider certain particular cases in which the flow geometry and the scenario of flow evolution result in the fact that a definite realized wavenumber is singled out. As was already mentioned, such situations are sometimes referred

6.5. "SELECTION

MECHANISMS"

111

Fig. 38. Roll pattern with two grain boundaries (adapted from Ref. 241).

to as particular "selection mechanisms''; different mechanisms may disagree in their final wavenumbers. We shall see that the closeness between the final (realized) wavenumber and the preferred one is directly related to the possibility of roll-width readjustment. Such readjustment, i.e., the contraction or widening of rolls without topological changes and appreciable breaking of two-dimensionality, will hereinafter be called relaxation. A concluding discussion of the attainability of the optimal scale, based on the consideration of diverse situations from a unified point of view, will be given in §6.5.9. 6.5.1. Grain-Boundary

Motion

Let a finite set of x-rolls be sandwiched between two sets of transverse y-rolls, joining them along two grain boundaries (Fig. 38). Motion of these boundaries ensures efficient relaxation of the central x-rolls: if their wavenumber is not op­ timal, the y-to\\s can easily change their length and do not offer any significant resistance to t h e readjustment of the x-rolls. Although such a flow pattern is on the whole three-dimensional, the readjustment of the x-rolls is a substan­ tially two-dimensional process and the sets of y-rolls play the role of movable sidewalls changing their position to fit the width of relaxing x-rolls. As a result of relaxation, the central rolls arrive at a certain final wavenumber kgh, which depends on R (the notation is derived from the term grain boundary). Tesauro and Cross [241] studied this process by solving model equations, the potential Swift-Hohenberg equation (3.45) and the nonpotential equation (3.48) with d = 3. The equations were numerically integrated over time, the

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OF CONVECTION

ROLLS

conditions of periodicity in x and y being imposed on the boundaries of the calculated domain (Fig. 38). In addition, the technique of amplitude equations, which were obtained from the original model equations, was applied in order to find the steady states. These states coincide, within a good accuracy, with the states to which temporal evolution leads. It was found for both model equations that to each e there correspond unique values of the wavenumbers of the x- and y-rolls, kx = ksx and ky = fcj, for which a steady state is possible. If initially kx > fc*, the x-rolls expand (and correspondingly, the y-rolls become shorter), as a result of which a steady state with kx = ksx is ultimately reached. For an initial wavenumber A;,, > ky new xrolls appear, i.e., the area occupied by nonoptimal y-rolls decreases. If initially kx > ksx and ky > ksy, a combination of both processes occurs. In all cases the final value ksx = fcgb of kx is independent of both ky and the initial value of kx. For the potential model of Eq. (3.45) fcgb = ksx = k\ = kF - kzz, where kF is the wavenumber value that minimizes the specific Lyapunov functional and kzz is the threshold wavenumber for the zigzag instability. For the nonpotential model of Eq. (3.48) with d = 3 it was found that ksx / ksy , both values differing from kzz obtained by Greenside and Cross [230]. An experimental investigation of the behaviour of a roll system confined between two grain boundaries was carried out by Pocheau and Croquette [158, 242]. The experiment was performed with a silicone oil, which has P = 70, under controlled initial conditions. The initially induced x-rolls did not completely fill the container, and near the sidewalls parallel to them two systems of short yrolls arose owing to the cross-roll instability (see Sec. 4.2). They were delimited from the main set by two grain boundaries. The flow was allowed to settle down at each value of R. which was changed in small steps. Study of the sequence of equilibrium states gave a pronounced unambiguous (nonhysteretic) dependence of the wavenumber /cgb of the main roll set on R (bow ties in Fig. 39). As Fig. 39 shows, in most part of the range of R values this wavenumber exhibits a systematic, although not too large, departure from kzz- At the same time, fcgb is virtually identical to the wavenumber fcj (dotted squares and circles) at which dislocations were found (in the same work) to be in equilibrium (see §6.5.3). Nepomnyashchy et al. [243, 244] assumed in their analysis of a set of cou­ pled amplitude equations (3.24) that there is a single grain boundary x = 0 separating two semi-infinite patterns, both of which are stable. In particular, for two systems of rolls whose wavevectors k; (/ = 1,2) make arbitrary angles 6i with the x-axis (neither 0/ being close to TT/2), it was found that a steady state is possible only if ki — kc for both systems. For a uniform steady system

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MECHANISMS"

143

Fig. 39. Roll wavenumbers in experiments with grain boundaries, dislocations, and axisymmetric patterns of annular rolls for P = 70 (a composite diagram based on the data of Refs. 158 and 242). Solid curve at the bottom: neutral curve of stability of the motionless state. Bow ties: mean wavenumbers fcgb of the steady-state central set of rolls in the experiment with grain boundaries. Dotted circles and squares: wavenumbers £<j that correspond to the steady state of a dislocation as obtained in two ways—by direct measurements in the pattern and by extrapolating the climb velocity to zero, respectively (see §6.5.3). Solid circles: wavenumbers fca of axisymmetric convection (jumps in &a(() are due to changes in the number of annular rolls). Heavy solid straight line: dependence &zz(0 according to Eq (3.67) [63, 64]. Light solid line (close to the heavy line): the same dependence as erroneously shown in Refs 158 and 242. Dashed line: dependence i a ( 0 according to Eq. (6.14) [64] (see §6.5.4).

144

6. WAVENUMBERS

OF CONVECTION

ROLLS

of straight rolls with a wavenumber k = kc + q, according to Eq. (3.22),

w=izili, and the specific potential for the Newell-Whitehead-Segel equation (3.22), equal in this case to the integrand in Eq. (3.58), is minimum for q = 0. Therefore, the relation fcgb = kp- is also valid in this case. The above-mentioned authors also obtained stationarity conditions for a grain boundary between a hexagonal-cell pattern and a roll pattern in weakly supercritical regimes as well as between a hexagonal-cell pattern and a motion­ less fluid in weakly subcritical regimes (in both cases, the equilibrium is possible only if either pattern is in a metastable state). We note that the technique of amplitude equations is in general of limited applicability to studying real grain boundaries. Sharp spatial transitions are characteristic of defects of this type, which contradicts the idea that the ampli­ tude varies slowly. 6.5.2. Spatial Ramp of

Parameters

Imagine that the temperature difference A T between the surfaces of the layer and/or its thickness h vary in the x-direction, so that the local Rayleigh number is also some function R(x). It is said in such cases that there exists a (spatial) ramp of the Rayleigh number and of the parameters determining it. We shall be interested in such ramps that at a certain point x = xc the function R{x) passes through the critical value Rc. To be specific, we assume that everywhere dR/dx < 0, so that the conditions are supercritical in the region x < xc and subcritical in the region x > xc. We also assume that to the left of a certain point x = Xi < xc the supercritical region is uniform: h = const, A T = const. Let a two-dimensional flow of the x-roll type exist in the supercritical region. Then, if the ramp slope |d/?/dx| is everywhere sufficiently small, the roll am­ plitude should be expected to gradually decrease in the positive x-direction, in which R decreases passing to the subcritical region. Such a ramp should act as a "soft sidewall"; in particular, it should exhibit little resistance to roll relaxation. (Generally, large-scale fluid circulation should occur in a system with a ramp, involving a region somewhat more extended than the ramp region. If, however, the x-variation of the temperatures of the layer surfaces is fitted with the shape of these surfaces in such a way that the unperturbed isotherms are horizontal planes everywhere in the layer, circulation does not occur.) The question arises of whether or not, if a slow ramp is present, a regime will be established at which the wavenumber kr in the uniform part of the supercritical region coin-

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145

cides with the preferred wavenumber kp. At first glance, the results of existing theoretical investigations by no means answer this question positively. We shall see in what follows that they nevertheless do not contradict the concept of the inherent optimal scale. Kramer et al. [245] (see also Ref. 246) studied a rather general problem on the basis of a system of reaction-diffusion equations n

&"•• = 5 Z £ , i J ( c r i , . . . , Q m ) A u J + / , ( u i , . . . , w n ; a 1 , . . . , Q m ) ,

i=

l,2,...,n,

j=l

(6.8) for which a certain function i of the parameters at is an analogue of the Rayleigh number. Let the parameters Q/ be independent of spatial coordinates. Then the transition from subcritical conditions (under which all perturbations of some initial stationary state are damped) to supercritical conditions (under which the development of the perturbations can result in the formation of spatially periodic patterns) corresponds to a certain critical value of the function c. The authors assumed that the parameters Q; are slowly varied with the z-coordinate. As a result, there occurs a spatial transition from subcritical con­ ditions to supercritical ones. The slow coordinate X and the slow time T were introduced using a small parameter that is a measure of the variation rate of Q;, and an expansion of the equations, similar to that applied for the derivation of the Cross-Newell equation (see §3.3.6), was performed. The obtained phasediffusion equation includes in general the drift of the pattern as a whole, which results from the nonuniformity of the conditions. For a steady state this equa­ tion yields a first-order differential equation that unambiguously determines the distribution k(X) of the local wavenumber if k is given at some point. In order to choose such a unique dependence, the authors made a step that has become standard for problems of this type, namely, they put k = kc at the critical point. (The justification of this assumption is a key element and will be discussed be­ low.) It was found that all ramps that can be transformed into one another by a transformation of the spatial variable result in the same dependence of k on at(X). In potential systems all ramps lead to the same value k = kr in the uniform supercritical region, namely, to the value kp that minimizes the specific potential of the uniform pattern. The relation between the pattern wavenumbers in systems with slow ramps of one parameter t and the wavenumbers minimizing the specific Lyapunov functional was demonstrated by Pomeau and Zaleski [247]. For a steady-state original equation of a very general form, a differential equation was derived that relates c[X) and the local wavenumber k(X) of an "adiabatic" solution (a solu­ tion slowly modulated in A') to the original equation. The explicit ^-dependence

146

6. WAVENUMBERS

OF CONVECTION

ROLLS

of k obtained for a certain family of amplitude-type equations is identical to the e-dependence of fcf in the case of a potential system. (The set of reactiondiffusion equations discussed in Ref. 245 was next studied as an example; the amplitude equation obtained for this set was shown in Refs. 246 and 248 to be incorrect.) An important point is that, according to Poineau and Zaleski's con­ siderations [247], a transition from a gently sloping ramp to a steep ramp should manifest itself in a transition from a single wavenumber to a finite wavenumber band. In nonpotential systems kT depends on the form of the ramp. Kramer and Riecke [249] studied convection of a fluid with P = oo assuming that the top layer surface is an isothermal horizontal plane and the height and temperature of the bottom boundary depend on the slow coordinate X. An expansion of the Boussinesq equations, which followed the scheme proposed in Ref. 245, resulted in an equation for k(X), which simplifies greatly for small supercriticalities e and free boundaries and becomes in some cases solvable an­ alytically. In general, the system of rolls undergoes drift. If there is no drift (which does not necessarily mean the absence of a large-scale flow, e.g., a flow with zero mean flux), the local k value can be expressed directly in terms of the local t value. The curves in the (fc,e)-plane which represent such dependences and pass through the critical point (kc, 0) can, depending on the ramp structure, have slopes of very different characteristic magnitude and even of different sign. And an impression arises that the preferred wavenumber does not manifest itself in any way. The statement of the problem considered by Buell and Catton [250] differs from the preceding one in that the top and the bottom layer boundary are assumed to be rigid and the considered values of the Prandtl number range from 0.025 to infinity. The layer thickness is constant, and a ramp is present only in the distribution of A T . The equation for the roll phase in a steady state (where a large-scale flow is nevertheless present but phase advection is compensated by phase diffusion) relates the local k value with the local R value. The dependence k(R) was found by numerical integration for a wide range of Rayleigh numbers (Fig. 40). The Prandtl number strongly affects the form of the integral curve drawn from the point (fcc, Rc). In particular, for P < 0.7 the k value increases with increasing R, which is radically different from the behaviour of the observed kp in a uniform layer. Experimental studies of ramped convection are not extensive. Moreover, in some works, although a ramp is present, it is very steep and cannot play the role of a "soft" side boundary (see Ref. 251 for an example of such situations). A smooth ramp was reproduced in experiments by Rehberg et al. [252], but

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147

Fig. 40. The local wavenumber k as a function of the local Rayleigh number R in a layer with a ramped temperature difference AT [250]. The curves are labelled with the values of the Prandtl number. these authors did not investigate the behaviour of the local wavenumber as a function of the coordinate x for given distributions R(x). 6.5.3. Motion and Equilibrium

of Dislocations

Efficient adjustment of a roll pattern to the optimal wavenumber value is offered by the mechanism of dislocation climb. It is climb that has been dealt with in the great bulk of the investigations of dislocation motion. Some results concerning dislocation glide will be noticed for the reader's reference although they are not related directly to wavenumber selection.

148

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OF CONVECTION

ROLLS

Siggia and Zippelius [253] were probably the first to theoretically study motion of dislocations. They investigated this problem both analytically (on the basis of the Newell-Whitehead-Segel equation) and numerically, by integrating the full Boussinesq system of equations as well as the NWS equation. A pair of dislocations was considered which arise at the endpoints of a segment of an '"extra" pair of rolls, "wedged" into a roll pattern. Such a configuration can be obtained if the pattern shown in Fig. 41 is supplemented by its mirror image with respect to the top boundary (and the boundary layer situated near this boundary is ignored). Since the authors used the amplitude equation without taking into account the vertical component of the vorticity, the results of their analysis are valid only in the limit of P —> oo. The analytic calculation was performed under the assumption that an unper­ turbed (dislocation-free) roll pattern, filling an infinite layer, has a wavenumber k — kc + Sk, Sk -C 1 (only the case 8k > 0 was studied because for Sk < 0 and P 3> 1 the initial roll pattern is unstable under slightly supercritical condi­ tions). It was also assumed that Sk*1'1 is much less than the distance between the dislocations (the length of the wedged-in segment of the roll pair). For such a choice of the wavenumber, the dislocations approach each other, i.e., the wedged-in segment is shortened. This means that the rolls that were com­ pressed as a result of the insertion of the segment tend to expand. For the climb velocity V of the dislocations, an expression was obtained whose structure does not depend on the type of boundary conditions on the layer surfaces: f2 8k3'2 (6.9) V = 1 47 ^° /2 \f2kl To where f0 and r 0 are given by Eqs. (2.42)-(2.44). Thus, in the approximation used by the authors, the wavenumber of the background pattern in which the introduced dislocation turns out to be stationary (we denote this wavenumber ad hoc as k^\ the meaning of k^ will be somewhat different in other problems) is equal to kc (i.e., for the NWS equation, to kp also). The numerical experiments were carried out by a pseudospectral method with x- and y-periodic conditions at the boundaries of the calculated domain. Assume that the size of this domain in the x-direction is equal to Lx, so that the unperturbed pattern of n roll pairs has accordingly the wavenumber 2nn/ Lx. Then in the horizontal strip confined between the y-coordinates of the two dislocations (i.e., the top portion of Fig. 41 and its mirror image), where the roll pattern is perturbed by the presence of the extra roll pair, the wavenumber is equal to 27r(n -f- \)/ Lx. The authors were mainly concerned with determining the dislocation ve­ locity. They found that its values obtained on the basis of the full equations

6.5. "SELECTION

MECHANISMS"

149

Fig. 41. Roll pattern with a dislocation (according to Ref. 254).

and from the amplitude equation do not always agree well with one another, even for P = oo. In many cases the evolution of the pattern is complicated by instabilities and the results are difficult to analyse. If the pattern remains unchanged in the main for a sufficiently long time, then the dislocation velocity settles down quite rapidly and subsequently varies very little. The relation be­ tween the velocity and the wavenumbers was not studied systematically. It was noted that as n and Lx are increased with the wavenumber of the unperturbed pattern being fixed and equal to A;c (or, in other words, the pattern becomes progressively less perturbed on average), the dislocation velocity approaches zero. A different result is hardly to be expected since in this limit the average wavenumber of the pattern approaches the unperturbed value kc. The force determining the dislocation climb in a crystal structure is called the Peach-Kohler (PK) force [255]. If the additional layer that is wedged into the structure and terminates at a dislocation is squeezed by the adjacent layers, the PK force strives to expel this layer and eliminate the dislocation. If, however, the pressure of the adjacent layers is negative, the PK force pushes the additional layer in deeper. In the theory of convection patterns an analogue of this force is considered. In the case of potential dynamics the change in the Lyapunov functional associated with the displacement (climb) of a dislocation over some distance is interpreted as the work performed by the PK force. We calculate this force following Tesauro and Cross [254]. We assume that the main parameter of a roll pattern with a dislocation is the wavenumber {k) obtained by averaging the local wavenumbers over the region under consideration. Let the dislocation be at a point (0, yj) within a strip —L<x
6. WAVENUMBERS

150

OF CONVECTION

ROLLS

the Lyapunov functional Fi of the pattern in the region at hand. The quan­ tity SFL is determined only by the fact that in a rectangle of area 2L5yd rolls with the wavenumber k+ = k\ +(X> are replaced by rolls with the wavenumber k- = &!„_>_«>• Thus, dF

6FL=-[F(,k+)-F(k.)]2L5yd

= - —

dfc

=

■ 2L(k+ - k.) 6yd k={k)

'2?r<jj/d'

~ dk aK

(6.10)

k=(k)

where F(k) is the specific potential for a uniform pattern characterized by a wavenumber k, the finite difference is replaced by a differential under the as­ sumption that L is large, and (k) = (k+ + fc_)/2. Hence, the PK force is equal to f

_

d

^ _ dy d

dF

(6.11)

9n

dfc

k={k)

Obviously, the dislocation is stationary if /p« = 0, and the corresponding wavenumber (k) = kd is equal to the wavenumber kp minimizing F\ in ad­ dition, it corresponds to the threshold of the zigzag instability [239]:

kd = kF = kzz-

(6.12)

Pomeau et al. [239] analysed the relationship between the dislocation ve­ locity and the wavenumber of the roll pattern for two problems: the problem of convection in a layer bounded by plates with poor heat conductivity (this problem admits a variational formulation--see §§ 3.3.3 and 3.3.4) and the prob­ lem of convection in a layer filled with a porous material. As was done in the analytical part of Ref. 253, Pomeau et al. studied a pattern that passes at large distances from the dislocation into a regular roll pattern with a wavenumber k. The technique of amplitude equations was applied. For a steady state the solvability condition of the equation for the complex amplitude determines the kd value; otherwise, for a given k, it determines the climb velocity V In the first problem it was found that V oc [k -

fcd)3/2,

(6.13)

with kd — kF — kzz- A relationship between this velocity and the phasediffusion coefficients D± and D\\ was also obtained. The second problem, which is nonvariational in the approximation studied, leads to a kd value that differs from kzz-

6.5. "SELECTION

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151

T h e glide of dislocations, i.e., their motion in the direction perpendicular to the rolls, was also studied in Ref. 239. It was found that if the dynamics is variational, glide is impossible in a system of uniformly curved rolls. In this case the potential of the system would not vary if the dislocation glided and there would be nothing to compensate the energy losses due to viscosity. Tesauro and Cross [254] studied the behaviour of dislocations both analyt­ ically and numerically on the basis of model equations, the potential SwiftHohenberg equation (3.45) and the nonpotential equation (3.48) with d — 3. A modification of these models in the form (3.49)-(3.51) was also used to take into account large-scale drift. In potential models the dislocation velocity is proportional to ((k) — k^)3?2 [cf. Eq. (6.9) obtained in Ref. 253]. In nonpotential models k^ turns out to differ from the other distinguished wavenumbers, kzz and k a , and for small (k) — fcj the climb velocity increases linearly with this quantity. The basic regularities of the motion of dislocations do not thus depend on t h e details of the statement of the problem and can easily be interpreted from the standpoint of the idea of the preferred wavenumber. Climb tends to make the roll-pattern wavenumber approach the preferred wavenumber kp, whereas a dislocation is in equilibrium if the mean wavenumber (k) of the pattern is equal to kp and the rolls tend neither to expand nor to compress. For potential systems kp = kp. Experiments specially designed to study dislocations were apparently first undertaken by Busse and Whitehead [120, 256]. (The first paper contains only qualitative results, the word dislocation is not yet used, and the process of expelling a dislocation is called the pinching mechanism. The authors regard it as a possible mechanism that could change the pattern wavenumbers.) The technique of controlled initial conditions was applied. In this way, a chain of dislocations was produced on the line of contact between two sets of parallel rolls, joining at their ends, the wavenumbers of the sets being in the ratio 2 : 3. The values of the wavenumbers turned out to be such that the narrower rolls were expelled by the wider rolls. The speed of the dislocation climb increased nearly linearly with increasing R and decreased with increasing P [256]. Pocheau and Croquette performed a detailed experimental study of the be­ haviour of an isolated dislocation [158, 242]. The dislocation was produced at the stage of imprinting the initial flow pattern by illuminating the layer through an appropriately shaped mask. Precautions were taken to prevent the develop­ ment of the cross-roll instability near the sidewalls of the tank, in contrast to what was done in the grain-boundary experiment described in the same work. Thin copper wires glued to the bottom of the container near the sidewalls, being

6. WAVENUMBERS

152

OF CONVECTION

ROLLS

in thermal contact with the bottom, created upflows, which fixed the position of the extreme rolls. A silicone oil with P = 70 was used. The motion of the dislocation was, as a rule, almost steady; only near a wall it sometimes slowed down and even stopped (this effect is referred to by the authors as the trapping of the dislocation by the wall). The basic results of measurements of the optimal wavenumber fa that was defined to correspond to the stationary position of a dislocation not trapped by the wall (i.e., in the bulk of the container) are presented in Fig. 39. In the cases where a steady state of the pattern was possible to be achieved by adjusting the R value, fa was determined as the arithmetic mean of the two wavenumbers present in the pattern, unperturbed and perturbed. These fa values are indicated in the figure by dotted circles. Another procedure for determining fa was based on applying the ((A;) - fa)3/2 law [see Eqs. (6.9) and (6.13)] to the climb velocities measured for different k values and a fixed R (here k also means the half-sum of the two values). The corresponding fa values are shown in the figure by dotted squares. It is evident that the measured values of fa are close to the fab values obtained in the same study, throughout the entire range of supercriticalities where data on both quantities are available, i.e., right up to c ss 4. Such a coincidence strongly suggests that in the case of dislocation motion, as in the case of grain-boundary motion, the geometry of the flow does not have any appreciable contraselective effect, and the pattern thus arrives at a wavenumber very close to the preferred one. The role of dislocation climb in the readjustment of the wavenumbers of roll patterns has also been investigated by Leith in experiments with air [182]; it was also noted that glide can favour the onset of phase turbulence. Finally, Whitehead [257, 258] performed some interesting studies of the be­ haviour of dislocations in bimodal convection. Such dislocations can also exhibit glide [258] and play an important role in the transition to chaotic motion [257]. 6.5.4- Axisymmetric

Flows

Pomeau and Manneville [259] considered a steady-state axisymmetric set of annular rolls in an infinite layer at large distances r from the symmetry axis. A general procedure of expanding the equations in 1/r yields, to the first order in this parameter, a system of equations whose solvability condition determines a unique roll wavenumber fa. The authors concluded that this wavenumber is always equal to the threshold value faz for the zigzag instability and ensures that the condition D± = 0 is satisfied. It was later found that this criterion is applicable only in certain cases.

6.5. "SELECTION

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153

Manneville and Piquemal [64] introduced an important refinement into this question. As said in §3.3.5, they obtained (63, 64] for a layer with rigid bound­ aries the effective value D'f of Dt [see Eq. (3.67)] taking into account the large-scale drift flow, which has a Poiseuille type vertical profile and tends to straighten roll zigzags. In a pattern of concentric annular rolls such a flow is impossible since it must be directed radially toward the symmetry axis and result in the accumulation of substance at the centre. Therefore, in an axisymmetric pattern the radial flow acquires, owing to pressure redistribution, a different vertical profile—with two nodes and zero total flux. If this flow is taken into account, then for large r values and sufficiently small supercriticalities the roll-pattern wavenumber A:a will be determined by the relationship [64] fca - kc _ kc

N'{P) R~R2(P)

Rc

Rc '

(6.14)

where N'(P) = 0.L66 + 1 . 4 2 6 P " 1 - 1 . 2 2 0 P " 2 , and R2[P) has the same form as in Eq. (3.67). The comparison of Eqs. (6.14) and (3.67) shows that fca corresponds to the condition De± = 0 only in the limit of P —► oo, when D^ff = Dj.; if, however, P is finite, then for k = ka it turns out that D'f > 0. When analysing the modified CN equation (3.S4) [together with Eqs. (3.85), (3.86)], Cross and Newel] [66] wrote in relation to this fact that allowance for the mean drift has no effect on the wavenumber of an axisymmetric pattern (since drift does not arise in this geometry), but changes the value of kzz, stabilizing the transverse perturbations of the rolls. In the case of free layer boundaries, as already noted, diffusive relaxation of zigzag disturbances does not occur, the zigzag mode of instability is replaced by the oscillatory mode, and the coefficient De± does not exist for finite values of P As Manneville and Piquemal [64] showed using the same technique, in this situation the wavenumber of axisymmetric convection is determined by the relationship fca - kc _ 1 / 5 \ R - Rc (6.15) kc \6P\P I Rc The further development of this work was based on lifting the assumption t h a t the supercriticality is small. To find the /ca value that ensures the solvability of the equation obtained in the first order in 1/r, Buell and Catton [260] applied a combined numerical method. They used a Galerkin representation for the horizontal dependence of the variables describing the flow, and a finite-difference approximation, for the vertical dependence. The authors investigated a wide range of values of R and P T h e results obtained in Refs. 64 and 260 have a characteristic feature that contrasts sharply with the experimentally known behaviour of the wavenumbers.

154

6. WAVENUMBERS

OF CONVECTION

ROLLS

For P < 0.784 according to Eq. (6.14), as well as for, roughly speaking, P < 0.7 according to calculations of Ref. 260, the wavenumber fca increases with increasing R (let us recall the fact, mentioned in Sec. 6.1, that it is for small P values that the decrease of (k) with increasing R is most clearly evidenced by experiments with uncontrolled initial conditions). It should be noted here that, in general, the above-presented results of the analysis of wavenumber "selection" in an axisymmetric roll pattern should not necessarily demonstrate good agreement with experiment. This analysis is re­ lated to rolls that have large radii in plan, and at the same time it does not assume that the container has an outer boundary. For this reason, under real ex­ perimental conditions, the radius of this boundary (of a cylindrical wall) should in turn be much greater than the radii of the rolls studied, in order that the effect of the boundary be insignificant. But this means that it will hardly be possible to maintain such an axisymmetric pattern, and the rolls will no longer form concentric rings. There should arise either a common set of almost straight and almost parallel rolls or a complex texture in which patches containing such rolls dominate. The obtained wavenumbers will therefore be of fairly academic interest. On the other hand, as shown by Newell et al. [67], the phase-diffusion equa­ tion (3.90) makes it possible to obtain the exact value of the wavenumber of a steady-state axisymmetric pattern for any value of P without assuming roll radii to be large. Because of the absence of a mean drift in such a pattern, rkB = const and in order that this relation hold right up to the centre of sym­ metry, the constant in its right-hand side must be zero to the leading order. This means that fca must have a value /eg such that B(ks) = 0 (cf. expression (3.96) for D±; we note that in Ref. 67 the coefficient D\_ does not include drift effects, and Eqs. (3.96) are always correct, but fczz = kg only for P = oo). The mean roll wavenumbers measured by Heutmaker and Gollub [177] in complex textures (for P = 2.5 and three R values, see Fig. 27) and by Steinberg et al. [162] (for P — 6.1, see below) are in good agreement with the fcg values calculated for the corresponding values of P. (These experimental data are also close to the A:a values found by Buell and Cation [260] for large P; as we know, there is no agreement between the experiments and the fca values obtained in Ref. [260] for small P.) Newell et al. [67] showed that for k < ks (k > kg) a focus of the roll pattern acts like a source (sink) of rolls. It was already mentioned in Sec. 6.1 that the experimentally measured wavenumbers of annular rolls decrease with increasing R, sharing this prop­ erty with the wavenumbers of rolls in different patterns. As R increases, at certain stages of the evolution an "extra" annular roll disappears at the centre

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MECHANISMS"

155

of the pattern [155, 156, 158]. And as R is decreased, it is at the centre that new rolls originate [158]. Precisely in the same way roll-pattern foci act if they are situated at the sidewalls of the container [235, 179, 180, 181, 177, 163]. The focus singularity thus plays a role important for the processes of wavenumber readjustment, and this role is adequately reflected by the finding of Ref. 67. Croquette and Pocheau [158] compared their experimental values of /ca for T = 20 and P = 70 and 14 (silicone oils) with the theoretical values of A:a and fczz obtained from Eqs. (6.14) and (3.67), respectively (both the experimental values and the theoretical dependences fca(e) and kzz{t) are shown in Fig. 39 for P = 70; in the case of P = 14 the line kzz{t) is inclined to the left even more strongly.) A significant discrepancy with the computed data was found. At P = 70, from a certain R value on, the experimental ka value was appreciably less than both theoretical figures (and, from the standpoint of the theory, it should fall into the region of instability). At P — 14 this wavenumber A;a was found to lie between the two theoretical values. The change in the number of rolls in the pattern that occurs as a result of the creation or annihilation of a roll at the focus is a discrete process accompanying a continuous variation of R. It is understandable that, as R alternately increases and decreases, the changes of the wavenumber exhibit hysteresis. But the nonuniqueness that results from the hysteresis is considerably less significant than the discrepancy between the measured value of ka and the calculated values of fca and kzzThe authors associated this discrepancy with the eccentric deformation of the pattern of circular rolls and with the mean drift flow of the double-vortex ("dipole") type (see Fig. 22 in Sec. 4.2), which is what makes Eq. (6.14) inap­ plicable. In an experimental investigation by Steinberg et al. [162] (T = 7.5, P = 6.1), as already noted, the deformation due to the focus instability was also observed. The measured wavenumbers were found to be close to those obtained by Koschmieder and Pallas [156] for P sa 500-900 but differed sharply from fca calculated for P — 6.1 according to Eq. (6.14). However, they are in good agreement with the kg values found for the same P value in Ref. 67. Hysteresis was observed in the variation of (k) with c. It is interesting that, according to Ref. 67, the focus instability is in partic­ ular possible in a certain region of the (A.'g, /?)-plane that lies within the Busse balloon. To s u m m a r i z e what has been said in §§6.5.1-6.5.4, we note that the con­ sidered situations demonstrate a noteworthy common feature. If the dynamics

156

6. WAVENUMBERS

OF CONVECTION

ROLLS

is variational, the calculation of the realized wavenumbcrs yields Kgij — kx — k^ — /c a — kf

— k'L'lt

(6.16)

where ky is the wavenumber minimizing the specific potential of the system and fczz is the threshold wavenumber value for the onset of the zigzag instability, determined by the condition DL = 0. It is therefore reasonable to interpret the wavenumber kp as the preferred wavenumber. In nonpotential systems these realized wavenumbers are in general different (Fig. 42). It should be noted, however, that this statement was based on particu­ lar model equations [261] and its validity for the natural Navier- Stokes equations is not clarified. It is therefore especially interesting to note that the realized wavenumbers coincide, even under the conditions of nonpotential dynamics, in some differing situations that share nevertheless a common feature—the lack of significant contraselective factors and, hence, the possibility for the preferred wavenumber to be attained. One such example is very good agreement between fcgb and fc<j, up to relatively large R values, in l'ocheau and Croquette's experi­ ments [158, 242). We shall return to this point in the concluding discussion of the selection problem (in §6.5.9). In the following subsection the formation of patterns by the propagation of a convection front will be discussed. And, although this process is frequently referred to as a "selection mechanism", we shall see that it can produce patterns with different final wavenumbers in different cases. Moreover, even if only vari­ ational dynamics is considered, the wavenumber of the pattern formed behind a front may differ from the wavenumbers realized by another "mechanisms" An interpretation for these facts will be given in §6.5.6. In particular, it will emerge that the process of roll relaxation, which is of fundamental importance for the selection of the unique preferred wavenumber, in some cases drops out of the consideration merely because of particular features of the statement of the mathematical problem. 6.5.5. Convection-Front

Propagation

The propagation of a convection front, which separates an already formed roll pattern from a region of motionless fluid, is considered one of "selection mechanisms'', because under certain conditions the wavenumber of the pattern that arises behind the front can be predicted/ ■■This problem is considered in the literature in the contexts of various pattern-forming sys­ tems. It is reasonable to prefer in our discussion the terminology related to convection.

157

6.5. "SELECTION MECHANISMS"

Fig. 42. Disagreement between the realized wavenumbers predicted for different "selection mechanisms" on the basis of the model equation (3 48) with d - 3 [261]. Reduced wavenum­ bers q = k/kc (abscissas) are plotted against the control parameter ( (ordinates). Heavy lines represent the results of analytical calculations [230]: qt corresponds to the wavenumber /:a of an axisymmetnc pattern ("focus selection"); qj., to the midpoint of the band of wavenumbers possible in the presence of lateral boundaries (see below § 6.5.8) and to the wavenumber kr calculated for a ramped system ("one-dimensional selection"). Solid circles correspond to the wavenumbersfcdof patterns with zero dislocation-climb velocity; crosses, to the final wavenumbers qgb of rolls that relax in the presence of grain boundaries (both from numerical calculations). Dashed lines are drawn through the numerical points and the linear asymptotes calculated analytically for small t. Light lines: neutral curves for the instability of the unperturbed state (N) and for the Eckhaus instability of spatially periodic patterns of straight parallel rolls (E).

We first consider, following Dee et al. [262, 263], the simplest version of the Newell-Whitehead-Segel equation dTA = d2xA + A-A3

(6.17)

with a real function A, which is a particular form of the nonlinear diffusion equation. We are interested in such solutions of this equation that have the form A{X,T) v

'

= Ae(X-cT), '

x

lim AC(X) = 1, X->-oo

lim AC(X) = 0,

(6.18)

A-»+oo

where c is the constant speed of the front. They satisfy the equation Ai'c = -cA'c -Ac + A3e.

(6.19)

158

6. WAVENUMBERS

OF CONVECTION

ROLLS

Obviously, Eq. (6.19) can be interpreted as the equation of motion of a material point of unit mass in the potential field $(AC) = A2J2 - A*/4, if Ac is regarded as the coordinate of the point; A', as the time; and c, as the coefficient of friction. According to Eqs. (6.18), the particle leaves the point of the potential maximum (Ac = 1) at zero velocity and moves toward the final position—to the point of the potential minimum (Ac = 0). It is obvious that solutions to this problem exist for any c > 0. The larger c, the more slowly the particle moves, i.e., the wider is the front. For the problem (6.17) there exists a class of initial conditions that lead to selection of a definite speed of the front. This was shown by Aronson and Weinberger [264] in relation to problems of population genetics. Specifically, all initial states of the system which are described by functions A(X,0) that are confined to the strip 0 < A < 1, do not vanish everywhere, and decrease with X at least as fast as e ~ x , generate fronts moving (in the limit of T —> oo) with the speed c — 2. This speed is the minimum at which A(X,T) remains nonnegative every­ where (in the language of the above-mentioned analogy, c — 2 is the lowest value of the friction coefficient at which a particle, having rolled into the potential well, does not pass the point of the potential minimum with a finite velocity). The speed c = 2 is also remarkable in another aspect. It corresponds to the marginal stability of the system in the frame of reference moving with a speed c. The term marginal stability is used here in the following special sense. Let us consider small localized perturbations superimposed onto the unperturbed, uniform unstable state of the system. It is said that the marginal stability is achieved at the speed c = c" if it is possible to choose a perturbation of this sort which neither grows nor decays at some fixed point of the frame moving at the speed c", even if a growing disturbance is generated that moves away from this point. This is the case if the velocity of the frame of reference is such that the observed point always remains on the leading edge of the disturbance (Fig. 43). If Eq. (6.17) for real A is regarded as precisely an amplitude equation (for example, the NWS equation or the amplitude equation for the Swift Hohenberg model, which has the same form), then, according to Eq. (3.16), the roll pattern has everywhere the wavenumber k = kc. Patterns with k ^ kc can be described using complex amplitude functions whose phase depends on X. The amplitude equation for complex A has a class of solutions with a propagating front and a ''winding'' phase. However, Ban-Jacob et al. [263] argued that if the initial perturbation is strictly localized (vanishes outside some finite region of the Xaxis), then it is highly unlikely that it would evolve into a complex function A(X, T) with a finite (even small) value of k — kc behind the front. Accordingly, the front speed should be c = 2. We shall see in §6.5.6 that this statement,

6.5. "SELECTION

MECHANISMS"

159

Fig. 43. Schematic picture of the evolution of a perturbation in (a) the laboratory frame of reference; (b) a frame of reference moving at velocity c > c'; and (c) the frame of reference moving at velocity c = c' (the case of marginal stability). After Ref. 263. being based on an idealized statement of the problem [see Eqs. (6.18)], may really pertain only to patterns formed at large distances from the zone of initial perturbation, long after the process of front propagation was initiated. The NWS equation written in terms of the usual physical variables, i.e., in the form (3.22), has steady-state phase-winding solutions

\

9

J

to

(6.20)

each of them corresponding to a uniform pattern with a wavenumber k = kc + q— const. As already noted in §6.5.1, for such patterns the specific potential, equal to the integrand in Eq. (3.58), reaches its minimum at q = 0, being an even function of q. In other words, kp = kc. It is thus seen that, if the dynamics of the process is governed by the NWS equation and the regime predicted in Ref. 263 has been established, we have for the wavenumber k{ of the pattern formed behind the front (6.21) k( = kF, which is similar to Eq. (6.16). A different conclusion can be drawn from the consideration of the SH model equation (3.45) [262, 263]. The correspondence of the "natural" front speed (i.e., the speed acquired by fronts in most cases if initial perturbations are varied) to the condition of marginal stability has not been proved for the SH equation and is assumed in the mentioned papers as a hypothesis in order to single out a unique regime with the "natural" speed. The values c = c* and kt = k" obtained in this manner differ from those found from the amplitude equation,

160

6. WAVENUMBERS

OF CONVECTION

ROLLS

c = 2 and A:f = kQ, by 0(ca) corrections, where Q > 0. The wavenumber A:* increases with t. Further, this process was studied by numerical simulation on the basis of the SH equation with a localized initial disturbance [262, 263]. The values of c and k were found to agree well with the values that follow from the marginal-stability hypothesis. The quantity k" differs appreciably from the wavenumber value kp minimiz­ ing the specific Lyapunov functional for the SH model. Moreover, in contrast to km, the quantity kj> shows a slight decrease with increasing e. Thus, in this case (with variational dynamics) fcf yt

kF.

On the basis of the amplitude equation, Dee [265] investigated the propaga­ tion of a front in the case where the initial (unstable) state is already perturbed and corresponds to a periodic pattern with some wavenumber lying outside the stability band. A new stable pattern with a different wavenumber forms be­ hind the front. Numerical simulation with initial conditions corresponding to an abrupt stepwise spatial transition from a stable to an unstable state at some point x — x0 showed that the front speed agrees with the marginal-stability hypothesis. In the context of our discussion, it is interesting that the final wavenumber generated behind the front depends on the wavenumber of the ini­ tial pattern ahead of the front. Moreover, there exist ranges of final k values that cannot be reached for any initial k value. Numerical simulation of convection-front propagation in an infinite layer, based on the full two-dimensional Boussinesq equations, was done by Getling using a spectral technique [266-268]. At a later time, Liicke et al. [269, 270] performed finite-difference calculations for a long rectangular cavity. The main qualitative difference between the results of these studies is that in the first case roll relaxation was revealed, in contrast to the second case. The same dissimilarity is also seen between the experimental study of Berdnikov, Getling and Markov [271] and that of Fineberg and Steinberg [272]. Possible causes for these disagreements will be discussed below (in §6.5.9). It is worthwhile to consider here the studies that were specially aimed at checking the predictions made on the basis of simple theoretical models. Liicke et al. [269, 270] studied convection-front propagation in a cavity with rigid boundaries and the aspect ratio T = 25 within the framework of a twodimensional problem, solving the Boussinesq equations by a finite-difference method. It was assumed that P = 1. The process was initiated by introducing a short-time heating of a vertical wall. Calculations were used for the range of reduced Rayleigh numbers 0.01 < t < 0.2. The wavenumber of the pat-

6.5. "SELECTION

MECHANISMS"

161

tern behind the front was found to increase with t and to agree well with the wavenumber fcmax = fcc(l + 0.245e) for which, according to linear theory, the growth rate of a perturbation of the motionless state is maximum [31]. The front-speed values were determined to be close to those predicted in Refs. 262 and 263. We note that the front speed V obtained by recalculating the speed c to the standard physical time and length units, which do not depend on the supercriticality e, is

V=c^e1'2.

(6.22)

TO

As seen from Eq. (6.22), the front traverses a fixed distance in a time 0{t~1^2). According to Eq. (2.40), the development of convection from small disturbances takes a time 0(t~l). It is therefore obvious that one can rely on experimen­ tal observability of convection-front propagation in an unstable motionless fluid only at very small supercriticalities. Otherwise, spontaneous nucleation of con­ vection patterns will occur before the front has advanced appreciably. Fineberg and Steinberg [272] conducted a laboratory experiment with water at 30.2°C (P = 5.373) for the range 4 x 10" 4 < e < 2.5 x l f r 1 . A tank was used whose length is equal to 27.3/i and whose width is about four times smaller. At the beginning of each run the fluid layer was carried from a subcritical into a supercritical state by an increase in heating from below, and simultaneously an additional heating of one of the short sidewalls was also introduced. A front began to propagate away from this wall, leaving behind it a roll pattern. The pattern wavenumber was observed to increase with t. The values of the front speed in the studied e-range were found to agree to a high accuracy with Eq. (6.22) at c = 2, if £o and r 0 are calculated for a layer with rigid boundaries.

6.5.6. Convection-Front

Propagation

Combined with Roll

Relaxation

Now we pass to the studies that reveal roll relaxation behind the front. On the basis of the full set of Boussinesq equations, Getling [266-268, 273] performed numerical simulations of the evolution of two-dimensional roll flows in an infinite layer with free horizontal boundaries. In contrast to traditional approaches, the condition of spatial periodicity was not imposed. The depen­ dences of the velocity and temperature on the horizontal coordinate x were represented by Fourier integrals and the vertical dependence of each variable, by several Fourier harmonics. Initial conditions of different types were considered.

162

6. WAVENUMBERS

OF CONVECTION

ROLLS

Fig. 44. Development of convection in an infinite layer from initial perturbations of type I at R = 103 = l.5Rc as calculated in Ref. 268 (lateral boundaries of frames are not the boundaries of the calculated domain). Contours of the stream function ip (streamlines) and of the temperature T are shown, (a) Initial perturbation, (b) Flow at the moment t = 10 for P = 0.1. (c) Flow at the moment ( = 0.6 for P = 10. Increment of T is 0.1 in all cases. Increment of ip: (a) 0.02 and 2 for P = 0.1 and 10, respectively; (b) 0.5; (c) 0.5.

6.5. "SELECTION

MECHANISMS"

163

Fig. 45. Time variation of the extremum stream-function values I^KI in individual rolls and of roll wavenumbers kx for several central rolls of a pattern that develops from a single-roll-type initial perturbation (a variant of the type-I initial conditions [273]) and is symmetric with respect to the plane x = 0 (K is the number of a roll counted from the central roll K = 0); R= 10 3 = l.bRc. P = 1 Type I corresponds to a perturbation of the velocity and/or temperature field with narrow localization in x [as usually, immobility of fluid and a linear (conductive) temperature distribution Ts[z) are taken as the unperturbed state]. An example of such an initial state is shown in Fig. 44a. In the course of evolution, convection involves a progressively wider region, two convection fronts propagating in both directions of the rr-axis. Already formed rolls attain a well-established state (see middle rolls of the patterns in Figs. 44b and 44c). The process of the establishment of the velocity amplitude and local wavenumber 7r/A determined for individual rolls (A being the roll width) is illustrated by Fig. 45. The limiting value to which the wavenumber of an individual roll tends will be called here the calculated preferred wavenumber (calculated kp). This wavenumber is determined by the regime parameters R and P We note for the subsequent discussion that in the case shown in Fig. 45 fcp = 2.37-2.38, whereas the wavenumber maximizing the growth rate A! of the lowest linear convective-perturbation mode [see Eq. (2.40)] is A;max = 2.46. The difference fcmax — kp is thus far beyond the uncertainty in the determination of A:p. Figs. 44b and 44c reflect an important property of the calculated kp revealed in Ref. 268: at P = 0.1 it decreases rapidly with growing R while at P = 1 and

164

6. WAVENUMBERS

OF CONVECTION

ROLLS

Fig. 46. Development of convection in an infinite layer from initial perturbations of type II at R = 103 = 1.5tfc, P = 0.1, k0 = 2.22 as calculated in Ref. 268 (lateral boundaries of frames are not the boundaries of the calculated domain). Contours of the stream function i/> (streamlines) and of the temperature T are shown. Increment of i/> is 0.5; increment of T, 0.1. (a) Initial perturbation of temperature (initial velocity is zero), (b) Flow at the moment i = 3. (c) Flow at the moment t = 12.

10 it does not vary almost at all. This agrees qualitatively with the regularity in the behaviour of the observed kp known from numerous experiments. Initial conditions of type II have a quite different form and involve only a temperature perturbation (Figs. 46a and 47a) such t h a t it generates very early in the flow evolution a set of z-rolls of a given width, which occupies a region of finite extent in x (Figs. 46b and 47b). Then, simultaneously with the propagation of two convection fronts, there occurs readjustment (relaxation) of the rolls that have already arisen. And it turns out that the wavenumbers of these rolls, no matter what they were initially, tend to approach the same calculated kp t h a t is obtained in numerical experiments with localized initial conditions of type I. Figures 46c and 47c illustrate roll readjustment of this kind. An important feature of such a process is that the wider the region

6.5. "SELECTION

MECHANISMS"

165

Fig. 47. Same as in Fig. 46 but for R = 103 = 1.5flc, P = 0.1, k0 = 1,4. Increment of T is 0.1 in all cases, (a) Initial perturbation of temperature (initial velocity is zero), (b) Flow at the moment t — 1.5; increment of ip is 0.25. (c) Flow at the moment t = 13.5; increment of tf> is 0.5. involved by the initial disturbance (i.e., the more rolls were created at the early stage), the slower the subsequent readjustment proceeds. We consider here two-dimensional flows. It makes sense to compare the obtained results with only those data of roll-stability investigations which char­ acterize the stability against disturbances not breaking two-dimensionality. For the case of an infinite layer these studies, as it was seen, reveal a very wide wavenumber band of stability (with respect to the Eckhaus mode). It covers most part of the range of admissible wavenumbers, i.e., of the range within which convection can at all develop owing to instability of the motionless fluid. In the above-described numerical experiments with the initial conditions of type II the local roll wavenumbers are within the range of stability against Eckhaus perturbations from the very beginning of evolution, or almost so; nevertheless, readjustment does occur. Thus, different wavenumbers within the stability band of infinite, uniform spatially periodic roll patterns are not equally realizable. In the evolution of

166

6. WAVENUMBERS

OF CONVECTION

ROLLS

finite fragments of such patterns the selective factor clearly manifests itself: lo­ cal roll wavenumbers tend to the preferred value. On the other hand, the more rolls a fragment contains, the slower they readjust. This is quite understandable because changing the width of the rolls situated in the central part of the frag­ ment requires moving the outer rolls. The rolls hinder one another to readjust, and roll interaction thus produces a contraselective factor. The mechanism of two-dimensional roll readjustment was revealed in these numerical experiments owing to lifting a very strict limitation on the flow struc­ ture, spatial periodicity. As a result, for sufficiently small P , the effect of the decrease of the preferred wavenumber with increasing R was also revealed as an inherent property of convection. This effect, being well known from exper­ iment, was explained before only by invoking accessory factors (see Sec. 6.1), such as three-dimensional transient processes preceding the establishment of a two-dimensional flow. A combined action of the selective and contraselective factors can be very clearly seen in the behaviour of roll patterns that evolve from initial conditions of type III[274]. They correspond to introducing two temperature perturbations localized near x = ±x0 (Fig. 48). At the early stage of the evolution, two roll pairs arise, and in each of them the central upflow corresponds in position to the maximum of the initial temperature perturbation. Subsequently, in the case of x0 — 2.5 presented in the figure, between the induced pairs another, central pair appears with an initial roll width determined by the position of more powerful neighbouring rolls. This width is less than the optimal one. Therefore, the rolls of the central pair, as their energy increases, force their outer neighbours apart. In other words, the tendency for the central-roll width growing toward the optimal value (the selective factor) overcomes the counteraction of outer rolls (the contraselective factor). The limiting (for t —> oo) value of the wavenumber of several central rolls is in excellent agreement with the calculated kp as found for the initial conditions of type I at the same R and P. If, for some given XQ, the central rolls turn out to be very narrow, they develop slowly. As a result, during their development, many rolls have time to form in peripheral regions. They exert a strong resistance to the expansion of the central rolls. Because of this, the width of the central rolls is virtually "frozen". Effects of this kind will be discussed in detail somewhat later. Finally, if the width of the central rolls exceeds the optimal one, these rolls contract and "draw" the outer rolls into the central region. Numerical experiments with initial conditions of type III show especially emphatically that a mature convection roll behaves like an elastic object. It ex­ erts pressure on other, adjacent rolls, can in turn be compressed as the pressure

6.5. "SELECTION

MECHANISMS"

167

Fig. 48. Development of convection in an infinite layer from initial perturbations of type III at R = 10 3 = 1.5/JC> P = 1. *o = 2.5 [274] (lateral boundaries of frames are not the boundaries of the calculated domain). Contours of the temperature T are shown for the initial moment and contours of the stream function ill (streamlines), for subsequent moments. Increment of 4> is 0.5; increment of T, 0.1.

6. WAVENUMBERS

168

OF CONVECTION

ROLLS

of the neighbouring rolls increases and, vice versa, can expand as this pressure decreases. We note that roll contraction or expansion is necessarily accompa­ nied by a flux of the fluid through the roll boundaries, because the fluid is assumed to be incompressible in all calculations described here. The motion of the boundary between two rolls is the motion of the separatrix between two streamline families that are associated with these two rolls. Such a separatrix corresponds to a local maximum in the x-distribution of the pressure at a given level z = const. The elastic properties of rolls as interacting individual objects generally play a substantial role in convection dynamics, including selection processes. And the elastic roll interaction, which can impede the attainment of the optimal width by one or another roll, operates independently of therrnogravitational phenomena. Getling [274] calculated flows geometrically quite similar to roll convection in a plane layer but with the Archimedean force "switched off" (Ft = 0, 0 — 0). It was found that a set of several rolls (vortices) expands, no matter whether the initial width of each roll was slightly greater than the layer thickness or exceeded it by a factor of 1.5-3.5. It can be said that such a roll system is always "under pressure". Therefore, in the case of convection, as rolls tend to the optimal scale, their elastic interaction inhibits their readjustment. An exception may be provided by only such special cases in which a shift of the interface between two particular rolls makes both closer to the optimal size. Getling [273] showed that roll relaxation accompanying front propagation can also be revealed by numerically solving the Newell-Whitehead-Segel am­ plitude equation. Such an approach enables one to carry out calculations for long times and long paths traversed by fronts. However, the NWS equation predicts according to Eq. (6.21) that k{ = kp = kp; for this reason, to study roll relaxation in this way, one has to artificially introduce a difference between the initial pattern wavenumber fc'0' and the preferred wavenumber kp. In Ref. 273 an implicit finite-difference method was used. An initial per­ turbation was introduced in the form of a set of x-rolls with some prescribed wavenumber, which had a finite width but was much more extended in x than in the above-described calculations with the initial conditions of type II: r

A=\

T

-g(t-ZW)

1/2

( t a n h ^ U + tanh^-p)e'"*

at t = 0,

(6.23)

where x\ > 0, S > 0, and q are fixed parameters; obviously, A;'0' = kc + q. Such a perturbation gives rise to two fronts propagating in opposite directions, the solution being symmetric with respect to x = 0. When the fronts are sufficiently far apart, each of them does not feel the presence of the other.

6.5. "SELECTION

MECHANISMS"

169

Fig. 49. Variation of the front-propagation speed c as calculated on the basis of the NWS equation for t = 0.5, P=l,g = Q,x1 = 20, and different <5 [273]. As said in §6.5.5, both the "natural" front speed c = 2 and the correspond­ ing wavenumber fcr of the pattern behind the front can be deduced from the marginal-stability condition. The initial condition (6.23) satisfies the require­ ments listed on page 158, thus necessarily leading to the marginal stability, only if e1/25/£o < 2. And it is not difficult to construct, for example, an exact solution to the NWS equation

-nr {'-'"■"[^(f "^')]} 1

describing the propagation of two fronts with the speed c = 3 / v 2 « 2.12. Thus, the question arises: How general is the marginal-stability criterion? Calculations for patterns with real amplitude functions (q = 0, i.e., A;'0' = kc) and different values of the initial front width 5 indicated that at early stages of the process the speed of the fronts can lie within a wide range, depending on S, and undergo also wide variations with time (Fig. 49). As should be expected, wider fronts propagate with greater speeds. But at later times the front, changing its profile, tends to a definite shape, and c —> 2 in all cases. An

170

6. WAVENUMBERS

OF CONVECTION

ROLLS

Fig. 50. Time variation of the wavenumber of the central roll of the pattern [K — 0) as calculated on the basis of the NWS equation [273] for P = 1 and different values of q (or /fc<0)). Solid lines: ( = 0.5; dashed lines: t - 0.125, dash-dotted lines: i = 0.01; heavy lines: S — 10; light lines: 6 — 2 (in all these runs xi = 8). Dotted solid lines: ( = 0.5, 6 = 2, x\ = 4 . attempt to numerically reproduce the analytical solution (6.24) by putting 5 = 2£ u (2/e) 1/ ' 2 showed only that at first c varies slowly, remaining for a time close to 3/-\/2. Later, front deformations cause the speed to tend to the "attracting" value c = 2. The marginally stable (in the above sense) solution with c = 2 thus appears to be the only real solution stable in the usual sense. However, for the speed settling down to its final value, times much longer than the vertical-thermal-diffusion time r v are needed. This means that the process of roll readjustment (relaxation), taking place over this period, can in general substantially affect the final wavenumber. Such an influence is clearly demonstrated by calculations with q ^ 0 (A:'0' ^ kc), i.e., with complex amplitude functions (Fig. 50). First of all, these calculations reveal quite distinctly the effect already men­ tioned above: the rate of roll relaxation strongly depends on the width of the region occupied by the roll pattern (this width being initially equal to 2xi) and thus decreases as the fronts move apart. But the initial front width has the most dramatic effect on the relaxation of rolls. The comparison of the curves corresponding to 5 = 2 and <S = 10 shows that, both situations being otherwise identical, in the second case the wavenumber is frozen almost at all. This can be understood in the following way. At S = 10, because of a very high initial front speed, the conditions approach very fast those of infinite spatially peri­ odic patterns. In periodic patterns, smooth and uniform two-dimensional roll

6.5. "SELECTION

MECHANISMS"

171

relaxation (i.e., equal and simultaneous expansion or contraction of all rolls) is inhibited since a small roll-width change cannot be obtained as a small flow perturbation. This ensures a more or less wide wavenumber band of stable pat­ terns. The fast creation of a widely extended regular roll pattern thus gives rise to a fairly strong contraselective factor. A separate consideration is needed for the question of the dependence of the relaxation rate upon the Rayleigh number (or the parameter t). As noted above, the characteristic time of spontaneous development of a roll pattern caused by weak priming disturbances is O^'1), whereas the time required for the front to traverse a fixed path is 0{t~l/2). Further, as shown by Daniels [279], the time of the lateral positioning of rolls (in a long box) is 0(e~2). On comparing the three mentioned time scales, one may naively suppose that the relaxation of rolls should become unobservable in the limit of t -> 0. However, it is seen from Fig. 50 that the parameter c has a nontrivial effect on roll relaxation. In con­ trast to what could be expected, as t decreases, roll relaxation becomes faster rather than slower. Like before, this can be explained in terms of the behaviour of the front speed, by its decrease with decreasing t. Furthermore, the greater e, the stronger the roll elasticity and, therefore, the more significant the con­ traselective properties of the pattern should be. Hence, in patterns that develop from localized initial perturbations there operates a different mechanism of roll relaxation than in patterns closely packing finite-size boxes. In the former case thermal driving does not crucially affect the relaxation, the elastic properties of rolls and the width of the region occupied by them being of prime importance. The wavenumber-freezing effect observed at large 8 remains highly pro­ nounced at small e. Now we point out certain details of roll dynamics revealed in Ref. 273, which are useful for constructing a general picture. First, as new rolls are formed, their original wavenumber is determined by the linear growth of priming perturbations. Hence, if "white noise" is present, there arise rolls with the wavenumber k = kmAX at which the growth rate is maximum (this was also found by Liicke et al. [269, 270]). If, however, a priming perturbation is characterized by some particular wavenumber A:'0', this very wavenumber will also describe newly formed rolls. Second, a roll relaxes the slower, the farther it is from the front at the moment under consideration or, in other words, the greater is the "mass" of the neighbouring rolls, which counteract the readjustment of the given roll. Third, the wavenumber of a roll varies, in general, nonmonotonically because of the competition between the proper tendency of the roll to relax and the pressure (either positive or negative) of other evolving rolls.

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Generally, the readjustment of a roll to the optimal wavenumber can be in­ hibited to a higher or lower degree by the contraselective factor—the interaction with the mass of other rolls. 6.5.7. Relaxation

of Rolls in Contact with a Disordered Flow

The experiment of Berdnikov, Getling, and Markov [271] was aimed at repro­ ducing the essential features of the roll-relaxation process revealed in numerical simulations [267, 268]. This turned out to be possible although the initial con­ ditions in the experiment and in the calculations were essentially different. In this experiment, performed with ethyl alcohol (at P — 15.8), the be­ haviour of the rolls initially induced in the central part of a rectangular con­ tainer was studied. Thin parallel wires stretched horizontally across the tank were arranged a given spacing apart (the set of these wires will be referred to as the initiator), and an electric current was passed through them for several seconds at the beginning of each run. As soon as the induced rolls became clearly visible, the current was switched off. In contrast to the experiments of Pocheau and Croquette [158, 242], these rolls occupied a relatively small portion of the container. Thus, it was possible for a much less ordered roll flow to spontaneously develop on both sides of the induced rolls even before the current was switched off. Since along the perimeter of the tank there was a gap where the top surface of the fluid layer was free (see Fig. 4), these spontaneously growing rolls in the side parts of the working domain could be directed arbitrarily. They exerted some action on the induced rolls, gradually destroying them. For this reason, the induced rolls could never settle: only a globally ordered roll pattern could reach a steady state under such conditions at a later stage of evolution—generally speaking, after the induced rolls had been demolished. It was reasonable to follow the variations in the wavenumber of the induced rolls only while they remained ordered. Nevertheless, even this relatively short time (several minutes as a maximum; cf. r v ~ 3 min) turned out to be sufficient for very significant readjustment (relaxation) of the induced rolls to occur. As a rule, the initial conditions of a run were chosen in such a way that the point in the (k, /?)-plane representing the induced rolls at the initial moment was within the theoretical domain of stability for spatially periodic roll patterns in an infinite layer (the Busse balloon). In Fig. 51 illustrating typical experimental results, the Busse balloon is given for P = 7; however, as Fig. 34 suggests, it differs little from the stability domain for P = 15.8 in position and sizes. Each arrow shows the variation of the mean wavenumber of the induced rolls over the period of observation in an individual run. There were some cases in

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Fig. 51. Experimentally observed behaviour of the induced rolls that develop in contact with a disordered flow (P = 15.8) [271]. Arrows: time variation of the mean wavenumber in a run. For each of two runs shown by arrows with ^ff = 0 curve computed for P = 15.8 according to Eq. (3.67) valid, however, for only small supercriticalities. Dotted line: mean wavenumber of steady rolls as a function of R as obtained in experiments with random initial conditions at the same P [135]; hatched band represents width in individual wavenumber values (at the 0.7 level in their distribution).

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which an increase in the wavenumber was followed by a decrease. This effect is indicated by the turn of the arrow to the opposite direction, and the turning point corresponds to the wavenumber maximum; note also the caption. The main regularity is that the average wavenumber of the induced rolls, irrespective of its initial value, approaches that very kp value which was observed in experiments with random initial disturbances [135] (performed with the same apparatus and the same working fluid). The fact that the mean wavenumber goes away from kp at the initial stage of some runs with a large number of closely spaced initiator wires evidently results from a considerable amount of heat released by the wires in these cases. The heat release should produce a vertical distribution of temperature such that convection does not initially develop in the whole layer but involves only a minor interval of heights. The roll width accordingly decreases. Later on, the extra heat diffuses, and further evolution proceeds in the standard way. (In a few individual runs the anomalous course of evolution resulted in the wavenumber passing beyond the stability band—see the lowest arrow at the right of the diagram.) The relaxation process is mainly two-dimensional. In the {k, /{J-plane, the points that represent the regimes of roll convec­ tion, as they approach the curve k — kp(R), move in the long-wave region of the diagram away from the threshold curve Dc± = 0 for the zigzag instability (probably, in some cases after crossing this curve). Hence, the threshold of this instability does not correspond to the preferred wavenumber; moreover, it is seen from the results of the "uncontrolled" experiments that there is no such a correspondence even at small c values. 9 The value P = 15.8 thus seems to be too small for the dynamics to be variational. No effect of sidewalls on the behaviour of the induced rolls was observed. In particular, the readjustment process took place in time intervals short compared with the times T|, and i\,f P of the transmission of influences along the layer by thermal conductivity and viscosity, respectively. Sidewall effects could be additionally inhibited because of the above-mentioned buffer role of the space near the walls where the fluid surface is free. Thus, the preferred wavenumber manifests itself as an inherent characteristic of convection for fixed values of 11 and P 9

We note that earlier experiments with random initial disturbances showed this disagreement even for very high Prandtl numbers (P ~ 102-104) starting from f a; O.T>. In the already mentioned Fig. 9 of Ref. 164, a considerable gap is seen between the threshold of the ZZ instability (at long wavelengths) and the mean wavenumbers of the well-established flow. Moreover, as the Prandtl number is increased, the departure of the mean wavenumber from the threshold curve for P = oo increases!

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It is interesting that, despite the presence of disordered flows on both sides of the induced (ordered) rolls, these rolls evolve approximately in the same manner as in the simulations described in §6.5.6 according to Refs. 267 and 268, where the fluid outside the zone of initial disturbances was assumed to be completely motionless. And, just as in these simulations and in Pocheau and Croquette's experiments [158, 242], three-dimensional processes are not necessary for the rolls to achieve the optimal wavenumber, which shows a usual decrease as the Rayleigh number increases. 6.5.8. The Effect of Sidewalls The situation that will now be considered is, strictly speaking, not a "selec­ tion mechanism" because it does not single out a unique wavenumber but only narrows the band of possible wavenumber values. Instabilities apart, a steady-state spatially periodic solution describing weak­ ly nonlinear convection in an infinite layer can have any wavenumber within a band of an 0(e 1 / ' 2 ) width. For the solutions stable against the Eckhaus pertur­ bation mode, the bandwidth has the same order of magnitude: near R = Rc it is •v/3 times as narrow as the band of the wavenumbers admissible for steady-state solutions (this has been shown for a wide class of problems—see §6.3.1). A different situation with the wavenumber band arises if a roll pattern is bounded by one or two sidewalls. Cross et al. [275], using amplitude equations, investigated steady-state regimes of two-dimensional convection in a box with stress-free horizontal bound­ aries and a large aspect ratio at small supercriticalities t = (R— Rc)/lSn2 (here Rc is taken for an infinite layer). The study was aimed at analysing the pos­ sibility of flows with different wavenumbers k (or different q — k — kc). In an infinite layer such flows can be represented by the "phase-winding'' solutions /!(*) = (l-Q2)1/2eiOX,

|Q|<1,

(6.25)

of the Newell-Whitehead-Segel equation (3.21) for the steady-state case. Ac­ cording to Eq. (3.19), this means that —e 1/2 < q < e 1 / 2 . If, however, the region is bounded in the x-direction (even if it is semi-infinite), then the phase-winding solutions are impossible under the sidewall condition A = 0 [see Eq. (3.38)] but are possible in the presence of boundary forcing, when a nonzero near-wall A value of the order of a small quantity A is prescribed. It was found that far from the wall, where A = 0 ( 1 ) , the band of admissible q values has a width of order Ae'/ 2 . In a more realistic case (with sidewalls of finite thickness and finite thermal conductivity thermally insulated on the outside) it follows from the amplitude

176

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equation (obtained by Cross et al. to the next order in e1/2 as compared with the NWS equation) that A = e1''2. This gives for a semi-infinite region (with one wall) a wavenumber band <j_ < q < q+ of an 0(c) width, and for a finite region — L<x< L, a discrete set of admissible q values which lie in the same band (the number of these values being of the order of eL/n). The quantities <j_ and q+ are proportional to t, and c/_ < 0 always, whereas the sign of q+ depends on P and on the thermal conductivity of the walls. Thus, the insertion of sidewalls reduces the bandwidth of the wavenumbers of possible steady-state solutions from 0 ( e 1 / 2 ) to 0 ( e ) . Instabilities can make this band even narrower. Similar results were obtained by Pomeau and Zaleski [149], who dealt with model equations—the potential Swift-Hohenberg equation (3.45) and the nonpotential Eq. (3.46)—with the conditions w = dxw = 0 on the boundaries of the region. We note that the band c/_ < q < q+ does not depend on the size 2L of the system. Cross et al. [275] explained the impossibility of a limiting process of passing directly to the case of an infinite layer by the fact that the time of the propagation of wall influence along the layer increases without band as L —> oo. However, if we accept this explanation, it remains unclear why the admissible values of q are limited to the same band in the cases of a semi-infinite and of a finite region. Meanwhile, the explanation of the effect of sidewalls turns out to be quite convincing when based on the concept of the preferred scale of convection. In contrast to the case of an infinite layer, in a bounded domain the adjustment of rolls to the optimal wavenumber does not necessarily require dramatic changes in the velocity field. As Cross et al. themselves point out [275], in the bulk of the container, smooth readjustment within certain limits can be ensured by the boundary layers near the sidewalls. And even if it is impossible to approach the optimum through such smooth adjustment of rolls without changing their number, the process of roll creation or disappearance also occurs most easily in the boundary layers, where the flow velocities are reduced (as shown by nu­ merical integration of the phase-diffusion equation [276]). It is clear that if the boundary layers set up some resistance to roll readjustment, this readjustment can be of threshold nature, occurring only if the nonoptimality k - kp is suffi­ ciently large. In this case, there will exist a band of admissible wavenumbers more narrow than for patterns filling the entire infinite layer. All that has been said here should also hold true for the case of a semi-infinite pattern with one wall: if its initial wavenumber is outside the mentioned band, then the pattern will readjust, and steady-state regimes are impossible.

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The role of boundary layers is evident, in particular, from the results ob­ tained by Zaleski [277], whose statement of the problem is similar to that stud­ ied in Ref. 275 but assumes (for more general boundary conditions) that the boundary-forcing parameter A is not small and can attain 0 ( 1 ) values [in the notation of Ref. 277, 0 ( c l / 2 ) ] . In the case of strong forcing, the wavenumber band is wider and its bounds can reach the thresholds of the Fxkhaus instability. From the standpoint of the approach presented here, it is interesting that numerical simulations of the evolution of disturbances performed by Pomeau and Manneville [278] on the basis of Eqs. (3.45) and (3.46) for a long region (into which 50-80 rolls fit) and for long time intervals demonstrate in all cases t h a t a single wavenumber is generated. For the potential model (3.45) it min­ imizes the specific potential and, in addition, coincides with the threshold of the zigzag instability determined by the condition D± = 0 (see §3.3.5). For the nonpotential model (3.46) the realized wavenumber falls in the region where D x < 0 (recall however that Df > DL for finite P). As mentioned above, the evolution of convective flows involves different char­ acteristic times. Daniels [279], extending the work performed in Ref. 275, in­ vestigated the stability of phase-winding solutions for a box — L < x < L and separated stable and unstable solutions among a discrete set of admissible so­ lutions with g_ < q < q+. He found that the establishment of the final k value takes an 0{t~2) time, which is much longer than the 0(e~l) time in which the amplitude is established. For sufficiently small P values, the steady-state k shows an appreciable decrease with increasing c. i.e., qualitative agreement with experiment is achieved although the model is two-dimensional. Cross et al. [280] supplemented the results of Refs. 275 and 279 by numerical calculations of the evolution of rolls on the basis of the NWS equation. As in Ref. 279, it was found that stable regimes are not unique. This, however, does not rule out (as in Ref. 278, where the model equations were considered) the possibility that a unique wavenumber can be obtained using an amplitude equation of a higher-order approximation than the NWS equation. In the experiments with controlled initial conditions [120, 236, 215] described in §6.3.2, good agreement with Ref. 275 could be expected but there were no such agreement in reality. In addition to the fact that, the duration of the experiment might be insufficient, it is not inconceivable that the discrepancy was associated with strong sidewall forcing: in these experiments, the thermal boundary conditions could give rise to a stable upflow at the sidewalls, thus fixing the position of the extreme rolls. There is however, an experimental study by Bensimon [281], who observed the narrowing of the wavenumber band due to the presence of sidewalls, al­ though with a significant role of the thermocapillary effect. The experiments

178

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were performed in an annular channel, which imitated an infinite layer, and a radial partition could be inserted into the channel to pass to the case of a finite container. This effect was also revealed by Arter et al. [233] in two-dimensional nu­ merical experiments based on the full Boussinesq equations. The number of rolls produced by the initial temperature perturbation at the beginning of a computational run was varied from 12 to 20, whereas the aspect ratio of the box was 16. Flow restructuring usually began near the sidewalls in the manner typical of the development of the Eckhaus instability. Over times < 12r v , the wavenumber of convection was found to fall within the stability band predicted in Ref. 216. Further evolution took times several times longer and resulted in the narrowing of the band of k values by a factor of three. But the wavenumber remained nevertheless nonunique. Thus, in both laboratory and numerical experiments only fast processes can be identified with the instabilities found theoretically for an infinite layer. 6.5.9. The Preferred Wavenumber Conclusion)

and Realized Wavenumbers

(Discussion

and

We saw that, generally speaking, there exists disagreement between the wavenumbers realized as a result of the action of different "selection mecha­ nisms". This largely disengaged the interest of investigators in the idea of the universal selection criterion. However, it will be shown here that the discrep­ ancy between the final wavenumbers in no way contradicts the existence of the optimal wavenumber and can be naturally explained if the role of contraselective factors inhibiting the relaxation of rolls is taken into account. These factors, as already noted, are closely related to the degree of ordering of the flow. We shall give here a descriptive, purely qualitative discussion of the results presented in this section, based on a unified approach. First of all, we compare various situations in which the flows are two-dimensional and convection rolls have different ability to readjust (or relax). We shall see that as this ability increases, the tendency to single out the preferred wavenumber manifests itself more and more clearly. On the basis of this consideration, seeming contradiction between the results related to different cases—in particular, to various "selection mechanisms"—will be removed. In the context of our discussion, the processes of two-dimensional roll read­ justment are most important. Thus, let us first mentally "switch off" the mech­ anisms of three-dimensional deformation for simplicity. A uniform, spatially periodic pattern of straight parallel rolls which fills the entire infinite layer is the least readjustable flow. As already noted, smooth

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adjustment of the wavenumber of such a pattern is impossible. An arbitrarily small, equal change of the widths of all rolls involves finite changes in the veloc­ ity field because the expansion or contraction of all rolls by any fixed quantity entails as large roll displacements in the layer as is wished. Hence, any "global" change of the wavenumber of the pattern must involve the birth or disappear­ ance of rolls. These processes can result from the development of the Eckhaus instability. As this takes place, in some roll groups, the rolls approach the opti­ mal width; in others, they even more depart from this optimum and ultimately either split into smaller rolls or merge into larger ones. The groups of expanding and contracting rolls alternate in space. But the deviation of the roll-pattern wavenumber from the preferred (optimal) value must be very great even for the initial emergence of the E instability. Therefore, a uniform roll pattern, when created artificially, will conserve any given wavenumber if it falls in a very wide stability range. This is a manifestation of a strong contraselective factor—the occurrence of spatial periodicity (high ordering) in the entire infinite region. [If a widely accepted tendency is followed formally (see the beginning of this chapter), each of these wavenumbers can be called the "selected" wavenumber, although such a usage of this term is obviously a piece of nonsense.] Quite similar conditions are offered by two-dimensional numerical models with the periodicity conditions imposed on the lateral boundaries of the calculated domain, the spatial period of the flow being determined by the horizontal size of this domain [231, 232]. In this case, only wavenumber quantization arises additionally, since an integer (even) number of rolls must necessarily fit in a spatial period. In agreement with the results obtained by linear stability analysis for uniform patterns, two-dimensional roll flows formed in rectangular boxes exhibit stability within a very wide wavenumber range. Now we turn to semi-infinite patterns formed behind a convection front mov­ ing with a well-established speed. Note that the marginal-stability hypothesis predicts an increase in the pattern wavenumber with increasing supercriticality, and both the numerical experiments (except those of Refs. 266 and 268) and the laboratory experiments demonstrate this effect. The wavenumber "selected" by a front disagrees with the values realized in other "mechanisms" as well as with the minimum of the specific potential (in the Swift-Hohenberg model). This fact is cited by a number of authors as an illustration of the lack of any universal selection criterion. The thesis on the disagreement between the wavenumbers "selected" (i.e., realized) in different situations can even be reinforced. As we saw, propagating fronts are capable to create patterns with different wavenumbers. The matter is t h a t roll readjustment takes place only in a bounded region behind the front and by the moment when the front has passed a sufficiently great distance from some

180

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chosen point, the local wavenumber in the vicinity of this point settles down. The faster the front moves, the faster the pattern approaches the conditions of "close packing" of rolls in a bounded container, the more rapidly roll relaxation slows down, and the farther the final wavenumber remains in general from the preferred one. But the front speed itself can vary over a wide range before its steady value is achieved. Obviously, the less steeply the amplitude of the initial disturbance drops in the x-direction, the more rapidly the front will move initially. The value of the realized wavenumber can be controlled by imposing weak priming disturbances with one or another period in a more or less wide region ahead of the front. Apparently, in a situation close to realistic ones, immediately behind the front a wavenumber will emerge from a weak noise which is close to A;max that corresponds to the maximum of the linear-growth rate. High-speed fronts can freeze this wavenumber. All this means that, in general, the wavenumber realized in a semi-infinite pattern with a propagating front does not characterize the intrinsic properties of the convection mechanism and bears an imprint of accessory (in fact, accidental) circumstances. This fact can also be illustrated by the results of the above-cited Ref. 265. In this work, the cases were studied in which a periodic pattern with a wavenum­ ber lying outside the stability range already exists ahead of the front. It has been shown that the wavenumber of the stable pattern formed behind the front depends on the initial wavenumber ahead of the front. The fact, surprising at first glance, that readjustment proceeds more rapidly at lower supercriticalities, shows that the degree of supercriticality affects the relaxation rate mainly through the front speed. In the numerical experiments of Ref. 269 based on the full Roussinesq equa­ tions and in the laboratory experiments of Ref. 272, the relaxation of rolls could remain unnoticed because of small lengths of the working domains and, correspondingly, short times of observation. In the numerical experiments of Refs. 262 and 263 based on the SH equation, the i-distribution of local wavenumbers k has a minimum kmm at the point x = 0 of the initiation of the process; this km;„ is much closer to the wavenumber kf minimizing the specific potential than the wavenumber k" of the rolls formed later (k' > kf). This fact could be a reflection of the relaxation process in which k decreases, approach­ ing kp = kf. (True, the authors claim that, in contrast to our suggestion, k increases near x = 0. This is possible at a later time, when many rolls with k — k" have already been formed: they also tend to expand, thus compressing the rolls near x — 0.) All this means that the known results of the studies of front propagation in no way contradict the possibility of the existence of the preferred wavenumber.

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As noted in § 6.5.8, passing from an infinite uniform pattern to a semi-infinite pattern bounded by a sidewall or to a finite pattern bounded by two sidewalk separated by a distance 2L results in narrowing the band of the wavenumbers admissible in steady-state regimes [275]. This finding was interpreted by some authors in the sense that the sidewalls themselves create a selective factor. However, as we saw, there are no reasons for such a belief. The sidewalls merely reduce the contraselective effect as against the case of an infinite pattern. The role of sidewalls outlined here (in fact, the role of boundary layers) can be seen most clearly if the sidewalls themselves do not exert forcing on the flow. Now assume that sidewall forcing is present: for example, the wall is heated, and there exists a steady upflow near it. Then the boundary layers will be less pronounced (or even completely lacking), and the boundaries of the rolls adjacent to the walls will be more or less rigidly fixed in space. The stronger sidewall forcing, the less the velocity is reduced in the boundary layers and the less the flow differs from a fragment of a uniform pattern. Accordingly, the wider the wavenumber range proves to be [275, 277]. If sidewall forcing is strong, the boundaries of this range can reach the thresholds of the Eckhaus instability as found for uniform patterns. This means that, when varying the strength of boundary forcing, one can also vary the magnitude of the contraselective effect and obtain different intermediate degrees of inhibiting roll readjustment to the optimal wavenumber. The ability of rolls to readjust can also vary over a wide range in finite pat­ tern fragments that arise between two convection fronts moving apart. This ability depends, as we saw, upon the width of the fragment, the front speed (consequently, upon the front steepness and the Rayleigh number) and the sep­ aration of a given roll from the fronts. Let us turn to the results of §6.5.6 and compare the calculations based on the Newell Whitehead-Segel equation with those based on the full Boussinesq equations and simulating flows localized much more narrowly. We can trace a gradual transition from very weakly readjustable extended structures, in which the attainability of the preferred wavenumber is generally not obvious, to disturbances localized so narrowly that the rolls arising first virtually reach this wavenumber at the initial stage of the process. Various situations at hand constitute a wide set of intermediate cases that are between the two extremes—the case of a uniform, strictly periodic roll pattern in the entire infinite layer (a (5-function in the spectrum) and the case of a localized initial disturbance (a ^-function in the physical space). Therefore, to calculate the optimal wavenumber in practice, one can make use of a numerical experiment with an initial disturbance localized as narrowly as possible, observing the development of the first formed rolls. Their fast

182

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ROLLS

relaxation will result not only from their closeness to the fronts but also from the low front speed: later, the fronts become in general more gentle, and their propagation speeds up. Roll relaxation with achieving the preferred wavenumber is clearly demon­ strated by some patterns which are three-dimensional on the whole but in which two-dimensional roll readjustment can easily occur. As already shown in §6.5.1, such a property is exhibited by a finite set of parallel rolls to which two systems of crossed rolls are adjacent on both sides, thus forming two grain boundaries (Fig. 38). The crossed rolls do not exert pressure on the main rolls, easily change their length and do not considerably resist the readjustment of the main rolls. A similar situation arises in the experiments described in § 6.5.7. Despite the presence of fairly disordered flows on both sides of the ordered (induced) rolls, the latter relax in much the same way as the rolls that develop from type-II initial perturbations in the numerical simulations assuming full rest of the fluid outside the initially perturbed zone (§6.5.6) or as the main roll set in the grainboundary experiment of Pocheau and Croquette [158, 242]. (These authors studied quasi-steady states without imposing large departures of the main-rollset wavenumber from the optimal wavenumber; on the whole, the flow was more ordered in their experiment. The investigators did not make a comparison with the case of spontaneous development of convection.) A pattern with a dislocation (Fig. 41) is also three-dimensional. But it is quite obvious that the process of expelling or drawing-in an "extra" roll pair is directly related to the tendency of rolls to change their width arid, correspond­ ingly, with the elevated or lowered pressure in the vicinity of the dislocation. Somewhat later in this subsection, when discussing axisymmetric convection, we shall compare the final wavenumbers obtained in certain experiments with grain boundaries, dislocations, and annular-roll patterns to the characteristic theoretical values. An important remark should be made here. On the one hand, as we saw in §§6.5.1, 6.5.6, and 6.5.7, the local wavenumber of a roll can be changed in the course of relaxation even if it falls from the very beginning in the stability domain found for uniform patterns. On the other hand, it follows from calcu­ lations for a circular domain based on model equations [184] that at certain T values steady states are possible in which the rolls passing through the central part of the domain (see Fig. 19b) have in this region local wavenumbers well beyond the threshold of the skewed varicose instability. We see again that, in general, the stability criteria obtained for uniform straight-roll patterns in an

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infinite layer cannot be applied locally to an arbitrary fragment of a convection pattern on the basis of only the local wavenumber for this fragment, with the general flow geometry ignored. In particular, it is important that mean drift, disregarded in the stability analysis of uniform patterns, significantly affects the stability criteria. A special examination is needed for the situations in which a ramp of pa­ rameters determining the Rayleigh number is present. As we saw, the solutions of the phase equations obtained from the Boussinesq equations for ramped fluid layers [249, 250] show that in certain cases steady-state distributions of the roll-pattern phase are possible. The corresponding distribution of the local wavenumber as a function of the local Rayleigh number may have different ap­ pearance, depending on the ramp structure [249] and on the Prandtl number [250]. It is not necessarily consistent with the observed dependence kp(R) for a uniform layer. In general, any manifestation of the preferred wavenumber is not in evidence therewith. Let us consider the physical interpretation of these results. The theoretical conclusions regarding the value of the realized wavenumber were drawn on the basis of the key assumption that locally k = kc where R = Rc. If one considers only the possibility of the existence of steady-state convective flows with one or another k value at a given R in the uniform region, then this step seems to be doubtful from the logical standpoint. The values of Rc and kc were found for spatially periodic flows in a uniform layer, where the interaction of convection rolls does not generate a mean energy flux along the layer. Generally, in ramped systems this is not the case. More energetic rolls located in the region with a higher supercriticality transfer their energy to less energetic rolls that occur under less supercritical conditions. Moreover, this energy flux cannot become zero at the point where the regime is critical, and it will unavoidably penetrate into the subcritical region. This means that the conditions near the x value at which R — Rc differ from the conditions in a uniform layer at R = Rc, and rolls will not necessarily have there the wavenumber k = kc. The mere fact of the existence of solutions that describe roll sets extending into the subcritical region and do not satisfy the requirement that k = kc at R = Rc, is far and away selfexplanatory. There arises here an effect similar to the well-known phenomenon of penetrative convection (see Sec. 7.2): if the unperturbed temperature profile Ts(z) is not linear, and in a certain range of heights z convective stability takes place (dTs/dz > 0), convective motions penetrate into this stable region. If a r a m p is present, penetration into the region of stability should also occur but in the horizontal rather than vertical direction. And since there exists not a

184

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ROLLS

unique solution but a family of such solutions reflecting the effect of horizontal penetration, various distributions k(R) can in principle be created, at least artificially. Similarly to the fact that the criterion of the stability of already existing rolls cannot be applied locally, it is also beyond reason to apply locally the criterion of the stability of motionless fluid and, moreover, to assume that local conditions also dictate the local wavenumber value. The situation turns out to be different if we address ourselves to the question of roll-flow stability. We note that the threshold curve of the Eckhaus instability, which lies above the neutral curve of the convective instability of a motionless fluid, is tangent to this neutral curve at the point (kc, Rc) of the (fc, fl)-plane (see Fig. 33). Therefore, any curve representing a dependence k{R), unless it passes through the point (kc, Rc), has necessarily a portion in the supercritical region outside the domain of stability with respect to E perturbations. Hence, if it is nevertheless granted that the E instability occurs in the ramped-layer region that corresponds to this portion of the k{R) curve (which means that the stability criterion obtained for a uniform pattern is applied to the ramped system locallyl), then it can be expected that this instability will really rule out the possibility of steady-state regimes with k(R) curves not passing through the

point (fcc, Rc). However, in this case we must note another fact. To speak in terms of the phase-diffusion theory, in the point where the k(R) curve intersects the neutral curve of the E instability, the coefficient D\\ vanishes. In the vicinity of this point, a set of rolls should be in a state close to the neutral equilib­ rium with respect to variations in the phase distribution which do not break two-dimensionality (i.e., to variations in the roll wavenumber). Thus, the E in­ stability limit in the spatial structure of the ramp should act as a soft sidewall and ensure the readjustment of a roll pattern localized in the region of higher supercriticality (where D\\ > 0). But in the region of the E instability (Dy < 0) a dramatic growth of disturbances can occur, and a unified phase description of a pattern that extends from the uniform supercritical region through the ramp region to the subcritical region proves to be impossible. For these reasons, the results of Refs. 249 and 250, based on the phaseequation technique, in no way indicate that in a ramped system steady-state patterns "selected" by the requirement that k = kc at R = Rc must be es­ tablished. The discrepancy between the wavenumbers calculated for systems with different ramps as well as the disagreement of these wavenumbers with the optimum wavenumber cannot serve as an argument against the very existence of this optimum. Note one more regularity characteristic of ramped systems. Subcritical con­ ditions suppress convection. And the steeper the transition to subcritical condi-

6.5. "SELECTION

MECHANISMS"

185

tions due to the ramp, the more similar the effect of this transition on the flow to the effect of an ordinary rigid sidewall, and the stronger the contraselective factor. It is clear that as the ramp steepens, the unique wavenumber observed in the supercritical region should transform to a progressively wider band, just as indicated in Ref. 247. It is also worthwhile to note certain subtle points in the interpretation of theoretical results concerning axisymmetric systems of annular rolls. The maintenance of an axisymmetric pattern in an experiment requires in general either a radial temperature gradient on at least one horizontal surface or sufficiently strong sidewall forcing. Otherwise, there arises a roll system like that shown in Fig. 19b or even a more complex texture comprising individual ordered fragments. Therefore, if an axisymmetric pattern is observed, its wavenumber may differ from the preferred one. Also, this wavenumber should not necessarily coincide with the fca value found for large distances r from the symmetry axis on the basis of the expansion of the original equations in 1/r [64, 260]—the greater the container radius, the more strong forcing is needed to maintain annular rolls. An interesting illustration of the relation between the theoretical A:a and the measured kB and kp under the conditions of nonpotential dynamics is offered by the results of already mentioned Croquette and Pocheau's experiments with axisymmetric convection [158], dislocations, and grain boundaries [158, 242] (Fig. 39). Although a rather large P = 70 is dealt with, a considerable dis­ agreement between the theoretical /ca and kzz indicates that the dynamics is nonvariational. However, the measured /ca values differ not only from kzz but even more from the theoretical fca. On the other hand, a remarkable feature is seen in the behaviour of other presented quantities. The measured wavenumbers fcgb and fed are in reasonably good agreement with kzz up to c ~ 1.5. The authors regard this agreement to be good in the entire investigated range of Rayleigh numbers, extending to this range the linear law of the variation of kzz with e which was obtained for small t [63, 64]. It seems, however, that there are no reasons sufficient for this conclusion (and, correspondingly, for the inference about the variational character of the dynamics for all c), especially if we observe that the fczz(e) l i n e >s drawn in Refs. 158 and 242 inaccurately (even being not a straight line—see Fig. 39), and its correction results in an increased departure of fcgb and kd from the theoretical kzz- But it is all the more interesting that these fcgb and k<\ demonstrate excellent agreement between each other right up to t a 4. This confirms the existence of a single preferred wavenumber, irre­ spective of the potentiality of the system. The wavenumbers of axisymmetric flows can be inconsistent with this kp in view of the considerations presented above.

186

6. WAVENUMBERS

OF CONVECTION

ROLLS

Now assume that an axisymmetric pattern, having arisen in some way, per­ sists for one reason or another without appreciable forcing (as was the case, e.g., for t > 0.16 in the above-mentioned experiments of Steinberg et al. [162] at T = 7.5 and P = 6.1). The presence of a focus singularity in the centre of such a pattern facilitates roll formation or disappearance, i.e., reduces the contraselective factor. It can be expected that in this case the system will "output" a wavenumber close to the preferred one. In fact, the wavenumbers measured in the indicated cases [162] as well as in complex textures observed by Heutmaker and Gollub [177] proved to be very close to the wavenumber ks of a steadystate axisymmetric pattern as obtained for the corresponding R and P from the phase-diffusion equation (3.90) without the assumption that 1/r is small [67]. But the data of Ref. 162 radically differ from the /ca values calculated according to Ref. 64. Thus, in the absence of forcing, annular rolls arrive on their estab­ lishment at the preferred wavenumber value (possibly, within a geometric factor reflecting roll curvature). Now we pass to cases in which three-dimensional deformations of rolls play a considerable role. In particular, such deformations arise as a result of the development of in­ stabilities other than the Eckhaus instability (see Sec. 6.3) in infinite uniform two-dimensional roll patterns. As seen from Fig. 33, the possibility of changing the characteristic scale of a roll flow without its radical breaking (inevitable if only two-dimensional processes are admitted) results in an appreciable shrink­ age of the stability domain. Because these instabilities are more "dangerous" than the E instability, their growth was for a long time believed to be the basic (or even the only) mechanism of changing the flow scale [204, 120, 214, 215]. Although an infinite uniform roll pattern can most easily readjust through threedimensional deformations, even in this case the stability band can nevertheless be fairly wide (and, what is most important, is of finite width). Therefore, this highly ordered system strongly resists scale changes. In experiments with controlled initial conditions [120, 236, 237, 215], the development of three-dimensional disturbances is obviously not forbidden. The results of these investigations of roll stability turned out to be in qualitative agreement with the theoretical predictions for infinite patterns (made with tak­ ing into account three-dimensional instability modes). Most likely, this means that the conditions of these experiments ensured rather strong sidewall forcing, so that boundary layers with reduced flow velocities did not develop near the sidewalls, and these walls played largely the same contraselective role as the spatial periodicity of flows in an infinite layer. However, as already noted in §6.3.2, in this case we cannot rule out the possibility of very slow processes that result ultimately in the readjustment of

6.5. "SELECTION

MECHANISMS''

187

roll wavenumber even if this wavenumber was initially within the theoretical stability band. Let us recall in this context the already mentioned numerical simulations [233] of flows in a cavity, which illustrate the role of sidewalls. It turns out that in both laboratory and numerical experiments, only fast pro­ cesses (of duration only several times longer than r v ) can be identified with the instabilities revealed theoretically for an infinite layer. Very high freedom in readjustment to the optimal wavenumber is charac­ teristic of flows that develop from random (noise) initial disturbances and are observed in uncontrolled experiments. Under such conditions, there arise many defects in the process of roll-pattern formation. They move, and most of them disappear sooner or later at sidewalls or by annihilating one another. These processes favour the "output" of the preferred scale. Ultimately, an equilibrium or almost-equilibrium state is established, in which there are few or no defects (except boundary layers—see below), and local wavenumbers are clustered near the preferred wavenumber. In general, very long times, > Tr h , may be needed for the full establishment of a pattern [66]. To all appearances, under certain conditions (small P , small e, large I\ and, likely, predominantly circular containers) steady states cannot be achieved at all. In this case, vanishing defects are constantly replaced by new ones, which also disappear in turn. Phase turbulence is observed, and its possibility is evidently closely related to the equilibrium conditions for textures. The more complex the texture, the more stringent should be the conditions for its steady state. Different portions of the texture may be close to the opti­ mum state to varying degrees and may have different degrees of stability; some fragments may approach the optimum at the expense of the deoptimization of others, and steady states may turn out to be unattainable. Nevertheless, the peak of the distribution of local wavenumbers will probably correspond to the optimal scale even in this case. A comparison of the experiment described in §6.5.7 with an uncontrolled experiment performed under the same other conditions [135] gives grounds to emphasize once again that the existence of a scale optimum is not correlated with the variational nature of the dynamics. An obvious disagreement between the preferred wavenumber and the threshold value for the zigzag instability (seen in Fig. 51) indicates that the evolution is nonvariational in the studied range of parameter values. Nevertheless, in the two sharply different situations— with induced and with spontaneously arisen flows - t h e same preferred scale manifests itself. An important general regularity is evident from the above-presented ma­ terial. It is natural to regard a uniform spatially-periodic pattern of straight parallel rolls filling an infinite layer as a flow of the highest degree of ordering. It

188

6. WAVENUMBERS

OF CONVECTION

ROLLS

has a wide stability range in the wavenumber axis, and the preferred wavenumber of rolls cannot be revealed by examining patterns of this type. A finite fragment of such a roll pattern is in essence a flow ordered somewhat less. And it becomes progressively less ordered as the domain of flow localization shrinks, because the characteristic scale of the pattern nonuniformity decreases. As this takes place, the rolls readjust better and better, and the optimal scale of roll convection manifests itself more and more clearly. Boundary layers near sidewalls can be considered a special sort of roll-pattern defects. Their presence, as we saw, also reduces the contraselective factor. Likewise, other defects—grain boundaries, foci, and dislocations—impart additional degrees of freedom to the pattern, making it easier to seek the optimal wavenumber. In other words, de­ fects provide "margins" for pattern readjustability. This means that the less ordered the pattern, the greater its freedom and the closer the final wavenumber (at least, the final wavenumber averaged over the spatial pattern) to the pre­ ferred one. The contraselective factor thus depends on the overall geometry of the flow and on the degree of its ordering. Therefore, as concerns systems of two-dimensional rolls, disagreement be­ tween the final wavenumbers output by various "selection mechanisms" in no way means that these wavenumbers are equally realizable. Flows with all these wavenumbers are stable (or at least do not exhibit instabilities of the known types). But, whereas each of these wavenumbers can be obtained by introducing artificial initial conditions, not any of them can be realized if the flow evolution starts from typical, natural initial conditions. There is no fact contradicting the idea of the existence of the preferred wavenumber, or the inherent optimal scale. As shown above, this wavenumber can be predicted if the problem is properly stated. In the cases of relaxational dynamics, the optimal scale corresponds to the minimum of the specific potential. We found it reasonable to mean by the scale selection (or wavenumber se­ lection) the process of flow evolution in which the characteristic flow scale ap­ proaches the optimal value. This optimum is, however, not always attainable. The final (realized) state is a result of the combined action of the selective and contraselective factors and depends on the overall flow geometry, which is in turn determined by the initial and boundary conditions. At the same time, the optimal scale clearly manifests itself in natural situations. Such a relation between the optimal state and realized states can be il­ lustrated by the following analogy. Assume that there is a force field with a potential ip(x) like that shown in Fig. 52. If a motionless particle is placed at some local potential minimum, it will stay there indefinitely long and none of the minima will be distinguished among others in any way. If, when placing a particle at some point, one imparts an initial velocity to it, and friction operates,

6.5. "SELECTION

MECHANISMS"

189

Fig. 52. For explanation see the text. then one or another steady final state will eventually be realized, depending on the initial coordinate, initial velocity, and friction coefficient. The particle will take up one or another local minimum of
190

6. WAVENUMBERS

OF CONVECTION

ROLLS

nonlinear effect. If initial disturbances are weak, then, for a time, their devel­ opment obeys the linearized equations, according to which the wavenumber of the most rapidly growing mode increases with R rather than decreasing. But after reaching a considerable amplitude, the calculated flow will not be able to reduce its wavenumber if the computational model does not ensure its sufficient readjustability. Apparently, this is the case in the two-dimensional version of the model of Ref. 204. To conclude this discussion, it is worth emphasizing once again that closer inspection is needed for slow processes. It cannot be ruled out that in many cases where the optimum seems to be unattainable, contraselective factors can nevertheless be overcome within sufficiently long periods of time. As concerns essentially three-dimensional (cellular) convective flows, there are good reasons to expect that for such flows the general regularities of seeking the optimal scale and of the occurrence of contraselective factors are similar to those revealed for two-dimensional flows.

CHAPTER 7

PECULIARITIES OF STRATIFICATION AND VERTICAL STRUCTURE OF CONVECTION

In the standard Rayleigh-Benard problem, convection cells fill the whole layer thickness, and their horizontal size is comparable with the vertical size. Meanwhile, in astrophysics and geophysics different situations are frequently encountered. Convection cells may be confined to a certain portion of the layer depth. In particular, small-scale (both in plan and in vertical extent) cells may be localized in a thin surface sublayer, without extending over the entire depth of the convection zone. Moreover, in some cases such small-scale motions coexist with usual large-scale motions, whose characteristic vertical scale is determined by the layer thickness. It is of importance to know the conditions under which convective motions that do not involve the entire layer can be generated exclusively owing to in­ ternal factors present in this layer. Such factors may be related to the vertical distribution of parameters, to their dependence on the state of the substance, etc. In this chapter we consider the effect of some of these factors on the char­ acter of convection within the framework of simple models. 7 . 1 . Effects of S t r o n g T e m p e r a t u r e D e p e n d e n c e of V i s c o s i t y In §4.1.2, when discussing convection in a fluid with temperature-dependent viscosity, we were mainly interested in the effect of this dependence on the planform of convection cells. As we saw, even a weak nonuniformity in the viscosity distribution can result in the replacement of two-dimensional rolls with three-dimensional, polygonal cells. It is not surprising that the temperature dependence of viscosity can also affect the vertical flow structure. However, to cause considerable changes in the vertical distribution of convection velocity, this dependence must be sufficiently strong. 191

7. VERTICAL STRUCTURE

192

OF

CONVECTION

We already mentioned in §4.1.2 (and illustrated by Fig. 10 therein) the re­ sults of the linear stability analysis performed by Stengel et al. [103] for a layer with large viscosity variations. Some details and extensions of these results were also discussed by Richter et al. [289]. An iterative numerical scheme was used to find the eigenvalues Ri(k) and eigenfunctions fi(z) of the linear stability problem (the Rayleigh number R being defined in terms of the kinematic vis­ cosity v\. that corresponds to a temperature equal to the mean To = (2~i + T2)/2 of the two boundary temperatures; for the nondimensional temperature, we put T\ = 1, T2 = 0). Different forms of the temperature variation of viscosity were investigated: (7.1) 0 < 7 < 1 v = i/i[l - 7 C O S T T ( 1 - T ) ] , 2

—

~

(a so-called Palm-Jenssen fluid first considered by Palm [95]); (7.2)

■2

(an "exponential" fluid); and some empirical functions v = aeF

(7.3)

representing the behaviour of the real fluids used in the experiments (glycerol [103]; L-100 polybutene oil and golden syrup [289]). Here 7] = In

■

(7.4)

For the last two liquids, F = be~Ttc was assumed in Eq. (7.3). Generally, for dependences (7.2) and (7.3), Rc as a function of r\ is nearly constant for low 77, then increases, reaches a maximum in the vicinity of 77 = ln(3 x 10 3 ) w 8, and decreases. It is such a regularity that can be seen in Fig. 10. The decrease of Rc in the region of sufficiently large viscosity variations 77 is related to the fact that a transition takes place in this region from motions involving the whole layer thickness to those localized in a certain low-viscosity sublayer (Fig. 53). This transition is accompanied by the growth of the critical wavenumber. A simple description of the effect of the confinement of convection to a lowviscosity sublayer can be given in terms of the local Rayleigh number calculated for a sublayer of depth z:

_

agATz3

(7.5)

2A

where vi is the viscosity at the midheight of the sublayer. For an exponential fluid, the critical eigenfunction assumes a form typical of sublayer convection if

7.1. STRONG

TEMPERATURE

DEPENDENCE

OF

VISCOSITY

193

Fig. 53. The vertical-velocity eigenfunctions /i(z) (arbitrarily normalized) of the critical perturbations in the linear stability problem for exponential fluids with different rj values (adapted from Ref. 103). Both layer boundaries are assumed to be rigid. the viscosity variation is so large that i?ioc is for a certain sublayer greater than R for the entire layer. In this case, the relative viscosity variation across such a sublayer turns out to be exactly e 8 ss 3 x 10 3 , and the sublayer thickness equals 8/7?. For the onset of sublayer convection, r? must thus be markedly greater t h a n 8. The experiments of Ref. 103 were aimed at testing the dependence RC{TJ) found theoretically. In Ref. 289 the flow structure was experimentally studied by measuring the horizontally averaged temperature. To this end, the investigators measured the electrical resistance of platinum wires stretched in a horizontal direction across the working volume. Large viscosity variations resulted in an asymmetry between the top (cold) and the bottom (hot) boundary layer. The experiments basically confirm the findings of the linear theory. The behaviour of Rc as a function of 77 agrees with the theoretical predictions. For viscosity variations greater than about 100, most of the viscosity drop occurs within a stagnant, viscous uppermost sublayer, or "lid". Convection takes place in a low-viscosity sublayer underlying the lid. These experiments also demon­ strate the possibility of finite-amplitude instability at subcritical Rayleigh num­ bers, already discussed in Sec. 4.1 in the context of weak non-Boussinesq effects.

194

7. VERTICAL STRUCTURE

OF

CONVECTION

Supercritical convection regimes were studied by Ogawa et al. [290]. They performed numerical simulations of steady three-dimensional convective flows of an infinite-Prandtl-number fluid for the case in which the temperature de­ pendence of dynamic viscosity has the form -ET

(7.6)

where E is the parameter determining the viscosity ratio r = eE between the top and the bottom layer boundary. The Rayleigh number was calculated on the basis of the viscosity value for the top boundary and denoted as Rt. Horizontal layer boundaries were assumed to be stress-free, and the conditions of reflection symmetry were imposed at the vertical sidewalls of the calculated domain. As a rule, the horizontal dimensions of the domain were l.7hx0.5h. The investigators found it possible to determine the critical value of r at which a transition takes place between the usual, whole-layer convection mode and the regime in which a stagnant zone of viscous fluid is formed in the uppermost coldest part of the layer, the latter situation being referred to as the stagnant-lid convection mode. Near the critical point, both regimes are possible, and the transition is hysteretic. The regions in the (Rt, r)-plane in which various convection regimes take place are shown in Fig. 54. First, this diagram represents the transition from two-dimensional roll convection to three-dimensional bimodal convection, which occurs as the Rayleigh number, being increased, passes through the instability threshold for rolls. Second, the transition from whole-layer to stagnant-lid con­ vection at sufficiently large r values is reflected. Stagnant-lid convection was found to be three-dimensional in all cases. There is little reason for analysing the structure of the simulated flows in detail, since the calculations were restricted to specific, fairly small aspect ratios of the calculated domain. However, the vertical distribution of the horizontally averaged convective velocity is an important feature. Two examples of such distributions are shown in Fig. 55 along with the distributions of the averaged perturbed temperature and averaged perturbed viscosity. The r values differ in these cases by a factor of 10. The distinction between the regimes of whole-layer and stagnant-lid convection is seen quite clearly. 7.2. P e n e t r a t i v e C o n v e c t i o n In principle, it is not difficult to imagine a different class of situations in which the region of the development of convection is restricted to a certain interval of heights. This is possible where a convectively unstable layer is con­ tiguous in the vertical direction with a region of stable stratification (where the

7.2. PENETRATIVE

CONVECTION

195

Fig. 54. Convection-regime diagram in the [Rtl r)-plane [290]. Open circles: two-dimensional whole-layer convection (WL-2D rolls); small solid circles: three-dimensional whole-layer con­ vection (WL-3D); large solid circles: three-dimensional stagnant-lid convection (SL-3D); solid square: both forms of convection are possible; solid line: critical Rayleigh number for the on­ set of convection as found from the linear stability analysis [103]; dashed line: the boundary separating the regimes of two-dimensional and three-dimensional flows; double solid line: the boundary between the regimes of whole-layer and stagnant-lid convection; cross: the point of the transition between whole-layer and stagnant-lid convection in the critical regimes accord­ ing to by the linear analysis [103]; short vertical arrow: the point of the transition between two-dimensional and three-dimensional flows at r = 1 according to the stability analysis of supercritical convection [228].

196

7. VERTICAL STRUCTURE

OF

CONVECTION

Fig. 55. Vertical structure of convective flows [290] for two values of r: horizontally averaged temperature Tav (solid line), horizontally averaged (root-mean-square) velocity uav (dashed line), and horizontally averaged dynamic viscosity 77av as a function of height z. (a) r = 3.2 x 104, fit = 102 (stagnant-lid convection); (b) r = 3.2 x 103, Rt = 103 (whole-layer convection). density grows from top to bottom). Such a "locking" sublayer decelerates and can stop the motion of fluid blobs that enter this sublayer from the unstable region (penetrative convection). In some cases, penetrative convection, affecting stable layers through viscosity, can produce there "countercells" with the oppo­ site direction of circulation and with a horizontal period enforced by the basic convective flow. This section does not present a comprehensive survey of the studies on pen­ etrative convection. The principal features of this phenomenon, revealed in early investigations, are quite pronounced and understandable, and subsequent analysis involved merely some variations in the statement of the problem and a more detailed description of different possibilities. Probably, Gribov and Gurevich [291] were the first to study penetrative convection. They considered a convectively unstable fluid layer bounded above by a semi-infinite region where the fluid is stable and bounded below either by another such region or by a rigid surface. The unperturbed temperature profile was assumed to be piecewise linear, with a constant (negative) value of the temperature gradient dT 0 /dz in the unstable layer and another constant (positive) value in each stable region. In this case, the density of heat sources or sinks can be represented by one or two ^-functions, each of them being peaked at a height where the unstable layer contacts a stable region. The solution of the problem can therefore be obtained by matching the analytical solutions found

7.2. PENETRATIVE

197

CONVECTION

separately for each of the regions of constant temperature gradient (it is this technique t h a t will be used in §7.3.1). The authors obtained the critical Rayleigh number Rc and the critical wavenumber fcc for the limiting cases of small and large temperature gradients in the stable regions (in the definition of R and k, the thickness of the unstable layer and the temperature difference across this layer were used). It was found t h a t in the first case the depths of the region of convective circulation grows indefinitely as the gradient in the single stable region or the gradients in both stable regions tend to zero. Accordingly, an unlimited growth in the critical convection-cell size is observed, and Rc approaches zero. In the second case, the depth of flow penetration into a stable region tends to zero as the gradient in this region becomes infinite. Thus, the conditions approach those of the standard problem with stress-free boundaries. Although the qualitative features of the behaviour of the solutions were subsequently confirmed by other investigators, certain numerical values given in Ref. 291 were found to be in error and corrected [292, 293]. Variations of this problem were considered by Rintel [294], Ogura and Kondo [295], and Whitehead and Chen [292]. In parti cular, Ogura and Kondo analysed various cases in which the upper, stable region has either finite or infinite depth. They studied in detail how the critical parameters and the flow structure depend on the depth ratio of the stable and the unstable region and on the ratio of the temperature gradients in these regions. It was found that at moderate values of the gradient ratio, as the stable-layer thickness is increased, the eigenfunction fi{z) representing the profile of the vertical velocity component shows an increase in the number of its nodes. This means that in sufficiently thick layers, above each convection cell produced by the instability of the lower sublayer, one or more additional cells arise, which have the same horizontal wavenumber as the primary cell. The additional cells, usually referred to as countercells, are created by the primary cells because of viscous forces, which entrain the overlying fluid. Therefore, the direction of the fluid circulation alternates in a series of vertically arranged cells. A different situation in which penetrative convection can take place was first considered by Veronis [296] and then repeatedly discussed in the literature. It arises in fluid layers with the quadratic law of thermal expansion P - Po = -poa(T

- T0)2.

(7.7)

This corresponds to the behaviour of the water density near a temperature of 4°C at which this quantity attains its maximum value po- Let the boundary temperatures be chosen in such a way that the density is peaked at a certain

198

7. VERTICAL STRUCTURE

OF

CONVECTION

height within the layer. In this case a convectively stable sublayer overlies an unstable sublayer. It is remarkable that in the linear approximation the convective-stability problem is in this case mathematically equivalent to the problem of stability for the Taylor-Couette flow between counter-rotating coaxial cylinders [297] (see also Refs. 3, 7). The ratio A of the total layer thickness to the unstable-sublayer thickness is an important parameter of the problem. The critical Rayleigh num­ ber and the critical wavenumber calculated for the unstable sublayer were found to be less than in the standard problem. This is because the replacement of an impermeable fluid-layer surface with a permeable interface, at which this layer contacts a contiguous stably stratified layer of the same fluid, implies that the boundary conditions become less restrictive. Accordingly, both the vertical and the horizontal cell size become larger. As A is increased, the number of the nodes of the eigenfunction fi{z) increases, thus indicating the formation of countercells. In addition to the linear analysis, Veronis obtained finite-amplitude solu­ tions of the problem by means of expansion in small amplitudes and revealed a possibility for subcritical instability. Numerical simulations of flows under the conditions of the Veronis problem largely confirmed the analytical results (see, in particular, Refs. 298-300). These conditions were also reproduced in laboratory experiments by Townsend [301] and Furumoto and Rooth (unpublished; see Ref. 296). In these experiments a water layer bounded below by an ice surface was dealt with. The temperature of the upper layer surface was above 4°C. A case similar to that discussed by Veronis was also investigated by Sparrow et al. [302]. They assumed the profile of the unperturbed temperature to be parabolic, which corresponds to a uniform distribution of heat sources through­ out the layer. The reduction of the critical Rayleigh number caused by the curvature of the profile was also revealed. Whitehead and Chen [292] considered, along with piecewise-linear profiles of the unperturbed temperature, certain parabolic profiles and certain profiles produced by an exponential heat-source distribution. The main features of the solutions are in agreement with other similar studies. If the transitions between the stable and unstable regions are gradual, the reduction of the critical Rayleigh number due to the penetration of motions into the stable region can be masked because the curvature of the temperature profile in the unstable region tends to increase the critical Rayleigh number. Other examples of this effect of raising Rc will be given in the next section. Whitehead and Chen performed also an experimental investigation of pen­ etrative convection. Complicated temperature profiles in the fluid layer were produced by irradiating it with light, which was absorbed by the fluid (a mineral

7.3. SMALL-SCALE

MOTIONS IN AN UNSTABLE

199

LAYER

oil) in the topmost sublayer. Reasonable agreement between the predicted and measured values of the critical Rayleigh number was noted. In supercritical regimes, the flow consisted of jets that plunged downward into the interior of the fluid layer. Thus, we see that the phenomenon of "locking'' convection in a restricted height interval and partial penetration of motions into adjacent stable regions are typical of many situations with the unperturbed temperature gradient chang­ ing its sign. As a rule, this leads to an increase in the critical cell size and has a destabilizing effect. However, if the temperature gradient varies with height within the unstable region, some stabilization is possible. If a stable region is sufficiently thick, forced motions in the form of countercells arise in this region. 7 . 3 . S m a l l - S c a l e M o t i o n s in a Globally U n s t a b l e Layer In the preceding section we considered motions of reduced scale which can develop in a stable sublayer and are in essence forced. Meanwhile, there exist far less obvious and less trivial ways to obtain the effect of the reduction of the flow scale. We shall discuss here within the framework of the linear approximation two situations in which the unperturbed temperature gradient is unstable every­ where in the layer but experiences a dramatic change at a certain height. This has a profound effect on the vertical structure of the flow. 7.3.1. A Layer with a Piecewise-Linear

Unperturbed Temperature

Profile

Our first problem concerns a fluid layer to which the Boussinesq approxima­ tion is applicable [303]. In contrast to the conditions of the standard problem (as specified in Chapter 2), we assume here that concentrated heat sources op­ erate at a certain height in the layer, their density being independent of the state of the substance. In this case the unperturbed temperature gradient 0(z) is no longer constant. Eqs. (2.15)—(2.17) linearized with respect to infinitesimal perturbations remain unchanged as compared with the standard situation. Eq. (2.31) assumes now the form ( D a _ k2 - A)(D 2 - k 2 -

^ A ) ( D 2 - k2)f

=

-Rr(z)k2f.

(7.8)

Here the Rayleigh number R is calculated, as before, on the basis of the fixed t e m p e r a t u r e difference A T between the bottom and the top layer boundary, and r(z) is the nondimensional temperature gradient, r(z)=/3(z)—.

AT=h

I Jo

0(z)dz,

I r{z)dz = 1. Jo

(7.9)

7. VERTICAL STRUCTURE OF CONVECTION

200

We assume T0(z) to be a piecewise-linear function such that r(z)

1 - hi < z < 1 (region 1),

rr « r(l)

)

0 < z < 1 - hi (region 2),

(7.10)

where the quantities hi < 1 (the nondimensional thickness of region 1), r(1) (/ = 1,2), and the ratio 7 of the temperature gradients are positive constants [303]. This means that the unperturbed temperature gradient makes a jump from one constant value to another at z = 1 — h%. We denote (7.11)

flW = i?r«; then, according to Eqs. (7.9), Rm =

2*

i + (7-i)V

(7.12) 7

and Eq. (7.8) reduces to two equations with constant coefficients. The functions obtained by means of solving these equations should be matched at the interface 2 = 1 — hi. To this end, we require that the function f(z) and its first five derivatives be continuous. These matching conditions determine the eigenvalues R* of R and the corresponding eigenfunctions fn{z) for given P, 7, hi, A, and k. The results that will be discussed below correspond to the case where the bottom layer boundary is rigid and the top boundary is stress-free. This combi­ nation of boundary conditions emphasizes the effect in which we are interested, viz., the tendency for convective flows to concentrate within a relatively thin surface sublayer. First of all, we note that the layer can be either stabilized or destabilized by the presence of the jump in the unperturbed temperature gradient, depending on the parameters of the problem. For hx < 0.4 such a peculiarity of the thermal stratification exerts, on the whole, a stabilizing effect: as 7 is increased from the standard value 7 = 1 , the critical Rayleigh number Rc = minflj(fc) = i?J(/cc) increases, ultimately reaching a certain limiting value. If hi > 0.4 and 7 > 1, the fluid layer is destabilized: Rc decreases as hi is increased to hi « 0.6 and then grows, approaching the standard (Rayleigh) value as hi —>■ 1. The eigenfunction fx (z) represents the profile of the vertical velocity com­ ponent vz(z) for the neutral disturbance. It was shown that for 7 ^> 1, hi -C 1, and large k values these normal modes correspond to convection cells that have not only small horizontal scales but also small vertical scales. The flows are

7.3. SMALL-SCALE

MOTIONS IN AN UNSTABLE

LAYER

201

Fig. 56. Contours of the growth rates of the lowest harmonic of disturbances at hi = 0.01, 7 = 106, P = 1 as calculated in Ref. 303 (the second harmonic becomes unstable at higher R values). Dashed lines: the wavenumber k, at which R*(k) is a maximum, as a function of R for P = 1, 10, and 100; the dots (for P = 10, 100) are labelled with the A values. The short vertical arrow points to the minimum of Rl(k) (at k = kc). thus localized within a relatively thin sublayer near z = 1. Nevertheless, at the critical Rayleigh number only large-scale cells can exist. However, the preference of small-scale flows becomes quite obvious if we turn to considering the spectrum of growing disturbances for R exceeding Rc. In a certain region of the parameter space, it is small cells that prove to exhibit the fastest growth. If P, hi, and 7 are specified, the spectra of growth rates A for different R values can be represented by a family of the contours of constant A in the (k, i?)-plane. Each contour is given by the equation R = R^{k), where R^k) is the n t h eigenvalue of the problem, corresponding to certain A and k. Two most remarkable examples of such families are shown in Figs. 56 and 57. Equation (7.8) is invariant with respect to the replacement of A by A' and P by P' according to the formulae P = 1/P' and A = \'P' = X'/P' but the last of boundary conditions (2.32) does not possess this invariance. For a perturbation with a given k, the more substantial the role of the temperature profile relative to the role of the boundary condition at 2 = 0, the closer the coincidence in eigenvalues and eigenfunctions between the analogous cases (A,P) and (A', P'). The case illustrated by Fig. 56 is typical of very thin surface sublayers (hi 1). The contours A = const are of almost the same form at P — 1,10, and 100; they virtually coincide in analogous

202

7. VERTICAL STRUCTURE

OF

CONVECTION

Fig. 57. Contours of the growth rates of four lowest harmonics of disturbances at hi = 0.01, 7 = 10 4 , P = 10; the analogues (P = 0.1) are labelled by A. Solid lines: n = 1; dashed lines: n = 2; dash-dotted lines: n = 3; dotted lines: n = 4. Short vertical arrows point to the minimum of i?J(fc) and two minima of R^1,5(&).

cases ( P = 10 and 0.1, P = 100 and 0.01). Each dependence R^(k) has a mini­ m u m at a certain k = km. Obviously, if we fix R = R*(k,), the harmonic with k — k„ proves to grow most fast out of the entire spectrum. It is seen t h a t this maximum of A is sharply displaced to greater k values as R is increased. If P = 1 (and this effect is most pronounced), the value fc» w 20 corresponds to R = 10RC (whereas kc w 3). At these regime parameters, a normal-mode disturbance with a wavenumber k = fc„ differs dramatically from the critical disturbance (k = fcc, R = Rc) in that fn(z) forms a rather narrow peak in the region of small 1 — z (see Fig. 58a). In other words, this mode represents convection cells that occupy less than a quarter of the layer thickness. It may appear that at such a large 7 the entire drop of the unperturbed tem­ perature across the layer actually takes place in region 1 (very thin as compared with the whole layer), while in the remainder of the layer dT0/dz is negligible, so t h a t virtually 7 = 00. However, at R ~ 107 the local Rayleigh number cal-

7.3. SMALL-SCALE MOTIONS IN AN UNSTABLE LAYER

203

Fig. 58. Lowest eigenfunctions fi(z) representing the profiles of the vertical velocity compo­ nent of disturbances with wavenumbers k = kt at R = min R^(kt); hi = 0.01. The vertical dashed straight line marks off the position z = 1 — hi of the interface between regions 1 and 2. (a) 7 = 106, P = 1, different A values (and, respectively, R values), (b) 7 = 104, P = 10, A = 61.5, k.i = 5.9, k,2 = 13.2, R = 8.13 x 10s. ciliated for region 2 is R\JC = i?' 2 '(l — hi)4 « R/ihi ~ 10 3 , whereas for region 1 it is R^l = R^h* n Rh\ ~ 10. Reg ion 2 thus proves to be locally much more unstable than region 1. The distribution of A over the (k, ii)-plane does not vary with 7 almost at all for 7 > 10 8 , i.e., starting from 7 values such that "loc/^loc

X

-

T h e intermediate case 7 = 104 is of particular interest. If P = 10 (Fig. 57), in a certain range of A values the functions Ri{k) have two minima. For A = 61.5 these minima are at the same height, R = 8.13 x 106 » 80i? c ; they are marked off by short vertical arrows in the figure. In this case, the highest growth rate is

204

7. VERTICAL STRUCTURE

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CONVECTION

exhibited by two disturbances with different wavenumbers, fc»i and /c, 2 . Their profiles / i ( z ) are shown in Fig. 58b. It should be noted, however, that the range of 7 values in which the described transition from large cells to small, near-surface, cells takes place (and, in certain particular cases, the disturbances of both scales can grow equally fast) is as narrow as a few tenths of a decimal order of magnitude. For hx < 0.01, all principal results hold true but the effect of the fast growth of near-surface disturbances is pronounced more sharply and at smaller R/Rc (although at larger 7). Hence, we see that, if the action of heat sources results in a dramatic change in the unperturbed temperature gradient at a certain height near a layer bound­ ary, a nontrivial effect of the excitation of small-scale motions in a surface sub­ layer is possible. In certain cases, the maximum growth rate selects two different scales of disturbances, and these scales differ the more widely, the more widely deviates P from unity. The size of the small cells is not directly determined by any characteristic scale of the problem; moreover, it strongly varies with R/Rc. 7.3.2. A Layer with Radiative Energy

Transfer

The effect of the development of small-scale surface flows may be even more pronounced if radiative energy transfer occurs in the layer. The problem on convection under such conditions was considered by Goody [304]. He determined the values of the critical Rayleigh number Rc by a variational technique and found that the trial functions imitating boundary-layer flows yield in some cases smaller Rc values than the functions representing whole-layer flows. However, Goody himself cast some doubt on t h e physical realizability of small-scale flows because the technique of trial functions he used made it possible neither to completely describe flows of various spatial scales and the conditions for their development nor to accurately determine the verti­ cal velocity profiles of these flows. Further investigations were concentrated on the effects of the spectral selectivity of the emission and absorption of radiation, and the possibility of small-scale flows was not touched upon. We present here the results of a more detailed study [305] of convection in a layer with radiative transfer within the framework of the model of Ref. 304. If internal heat sources are present, the steady-state form of the dimen­ sional equation of heat transport (2.3) linearized with respect to infinitesimal perturbations of variables is rji

-0vz = — + XA0, s

(7.13)

7.3. SMALL-SCALE

MOTIONS IN AN UNSTABLE

205

LAYER

where (3 = —dT0/dz, H' is the perturbation of the rate H of radiative heating per unit volume (in other words, H is the volumetric density of heat sources), s = pcp is the heat capacity per unit volume, and x is the thermal diffusivity. The dependence 0 = /3(z) can be found by means of solving the equation (7.14)

s

We make use of the equation of radiative transfer (see, e.g., Ref. 306) (7.15)

^-1*-/(»!.

where K is the absorption coefficient per unit volume, 1(1) is the radiation in­ tensity in the direction 1, d/ is the displacement in the direction 1, and B is the Planck intensity of black-body radiation (nB = a T 4 , where a is the StefanBoltzmann constant). In the limiting cases of an opaque and a transparent layer, the equation set (2.2), (2.4), (7.13) for a steady regime is reducible to the following equations for the vertical velocity component (represented by the function / ) , which are the analogues of the steady-state version of Eq. (2.31): (D 2 -k2)3(l 2

2 2

2

2

(D - k ) (T> -k ~

for«2/i2»fc2,

+ X)f=-Rr{z)k2f 2 2

3K h X)

2

= ~Rr{z)k f

2

2

2

for K /I < k .

(7.16) (7.17)

Here, the nondimensional temperature gradient r(z) (equal to unity on average) and the Rayleigh number R are defined as in § 7.3.1 [see (7.9)], and the quantity x

_ 16ar03 3«x 5

(7.18)

is treated as a constant under the assumption \(3\h 0 and the other for all vectors with lz < 0. Then the gradient normalized to its mean value proves to be" r(z) = L cosh A

(z--)+M,

(7.19)

"In Ref. 304, the expressions tor L and M (see below) were erroneously written witnout dividing the factor \/3 + 3X by 2. As one can infer from good agreement between the results of Refs. 305 (where the needed corrections were made) and 304, the calculations in Ref. 304 were carried out by the correct formulae, and the mistake crept only into the text of the paper.

7. VERTICAL STRUCTURE

206

OF

CONVECTION

Fig. 59. A family of neutral curves for the case of a transparent layer [305]; A = 102

where A2 = 3 K 2 / I 2 ( 1 + X ) ,

(7.20)

. „(2X . LA v/3 + 3X . , A , A\ L = X ^ T s m h - + — ~ s , n h - + cosh-j ..

L (V3 + 3X . , A

,A\

' ,

(7.21)

(7.22)

For A > 1, the function r(z) is nearly constant in the bulk of the layer but forms narrow peaks at the layer boundaries, thus representing thin thermal boundary layers. If, in addition, the layer is opaque (A 2 3> X), the bulk value of r(z) is close to unity and the total drop of the unperturbed temperature across the layer is mainly due to its almost linear growth in the downward di­ rection throughout the bulk region. In this case, the situation closely resembles the conditions of the standard Rayleigh-Benard problem. This is not surpris­ ing because radiative energy transport in an opaque medium can be described in terms of an effective, radiative heat conductivity. Moreover, the governing equation for an opaque layer (7.16) has the standard form, with R/(l + \) a s the effective Rayleigh number. If, however, X > A 2 > 1, i.e., the layer is transparent, the temperature gradient r(z) is virtually zero in the bulk region and the boundary peaks are extremely high, accounting for most of the total temperature variation.

7.3. SMALL-SCALE

MOTIONS IN AN UNSTABLE

LAYER

207

Fig. 60. Profiles of the eigenfunction /i(z) for the critical regimes with coexisting large-scale and small-scale motions [305]. Light lines: k = kci\ heavy lines: k — kc2; solid lines: X = 107, A = 102; broken lines: X = 109, A = 102 5 This problem, stated in Ref. 304, was solved numerically in Ref. 305. For a given wavenumber k, the employed finite-difference technique yields the lowest eigenvalue Ri and the corresponding eigenfunction fi(z) of the problem. The form of the neutral curve R = Ri(k) can therefore be analysed. In the case of an opaque layer, this form is qualitatively the same as in the standard problem, only flows in the form of large cells being possible in the critical regime. However, the situation is radically different if the layer is transparent. For sufficiently high A values, within a certain range of X values, the neutral curve shows two local minima of Ri(k) at certain wavenumbers kci and kc2. Let the respective minimum values be Rci and Rc2. The critical Rayleigh number for the onset of convection is therefore Rc = m'm(Rcl, Rc2). A typical illustration for such a case (A = 102) is given in Fig. 59. It can be seen that there exists a X value close to 10 7 at which Rc\ = Rc2, so that motions of two types are possible in the critical regime. Their vertical structure is shown in Fig. 60 by the profiles of the eigenfunction fi(z) for k = kcX = 3.2 and k = kc2 = 26. It is evident that the short-wave solution represents small convection cells localized in surface layers, whereas the long-wave solution describes usual large cells occupying the entire layer thickness.

7. VERTICAL STRUCTURE

208

OF

CONVECTION

As A is increased further, the short-wave minimum of Ri(k) progressively moves away from the long-wave minimum, and the vertical scale of the small cells accordingly decreases. In particular, if A = 10 2 5 , the value of kc2 lies within the range 75-80, whereas fccl remains close to 3.2; the minimum values Rci and Rc2 of Ri{k) become equal at X « 10 9 . For this case, the profiles of fi(z) are also shown in Fig. 60. A comparison between the structure of the thermal boundary layer (not illustrated in the figure) and that of the small-scale flow shows that this flow involves a range of heights substantially greater (in the last case, by a factor of about 15) than the thickness of the boundary layer. As we shall see in the next section, the effect of the two-scale structure of convective motions revealed here on the basis of two simple models is of particular interest in astro- and geophysical contexts. 7.4. A s t r o - and Geophysical A p p l i c a t i o n s It is quite plausible that the above-studied effects should manifest themselves under the conditions of a complex fluid-parameter stratification, such as the conditions encountered in the solar convection zone or in the Earth's mantle. We discuss here on the qualitative level certain points concerning these objects. 7.4-1- Solar Convection

Zone

Understanding the dynamics of the convection zone is a key to understanding the mechanisms of solar activity. All active phenomena are related to magnetic fields, and all solar magnetic fields eventually depend on the motion of the convection-zone plasma. On the global scale, convection is the driving mecha­ nism of the solar hydromagnetic dynamo. To describe in general terms the stratification of the convection zone, vari­ ous mixing-length theories were proposed at different times. They are a crude investigation tool, and many optional assumptions underlie such theories. On the other hand, they deal with the actual solar values of physical quantities. The calculations based on different guesses as to the magnitude of the mixing length yield markedly different models of the convection zone, which neverthe­ less agree with one another in their basic qualitative features. The most widely known model calculated by Vitense [307] gave a vertical distribution of the spe­ cific entropy S/kgN of the form illustrated by Fig. 61 (here S is the entropy per unit volume, &B is the Boltzmann constant, and N is the concentration of atoms). More precisely, this quantity is plotted in the figure as a function of the gas pressure ps; for its two values the corresponding depths beneath the photospheric surface are indicated. It is seen that in the depth range, roughly, from

7.4. ASTRO- AND GEOPHYSICAL

APPLICATIONS

209

Fig. 61. Structure of the solar convection zone, after Vitense (adapted from Ref. 307). 1000 to 65000 km (in other versions of the model of Ref. 307, to 165000 km) the substance is well mixed, thus being almost isentropic. Below the level of 65000 km the distribution of entropy (and temperature) is convectively stable; therefore, this depth approximately corresponds to the base of the convection zone. Above 1000 km, there exists a highly unstable layer where the entropy increases downward despite mixing (the zone of partial ionization of hydrogen— see below). This layer underlies a region of stable stratification, and motions penetrate into this stable region from below (i.e., penetrative convection takes place). Subsequent calculations [308, 309] that took into account not only the ionization of hydrogen but also the ionization of helium, resulted in greater values of the convection-zone thickness, up to 200000 km. Models of this kind can be used only tentatively. To draw more definite inferences about the structure and dynamics of the convection zone, a detailed, "deterministic" description of flows must be developed. One of crucial points is here the splitting of the scale spectrum of struc­ tures, or typical elements of the convection-velocity field. At least, four types of structures t h a t can be interpreted as convection cells are observed on the Sun: granules, mesogranules, supergranules, and giant structures. Their char­ acteristic scales are respectively 10 3 km, a few thousands of kilometers, 3 x 104 km, and 3 x 10 5 km. 6 We know that the vertical and the horizontal scale of a convection cell are normally of the same order of magnitude. As compared with the convection-zone thickness estimated to be about 2 x 10 5 km, all the structures mentioned, except giant structures, are small-scale cells and should 6

In the subsequent discussion, we imply that these scales correspond to the principal energycontaining peaks in the spatial spectrum of the velocity field and that motions of smaller scales can be regarded as fluctuations responsible for a certain effective eddy viscosity.

210

7. VERTICAL STRUCTURE

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CONVECTION

be localized in a relatively thin surface layer. However, it turned out to be a challenging task to find the conditions under which a possibility exists for the development of convective flows not involving the entire thickness of a globally unstable layer. Because of this, for a long time, the ideas concerning the lo­ calization of the cells of different types were based on only phenomenological considerations such as taking account of the fact that a cell of a small size in plan cannot extend vertically to a depth much greater than this size. The problems considered in the last two sections demonstrate that the scalespectrum splitting can be caused even solely by peculiarities of the stratification of the layer in which convection occurs. Apart from internal heat sources, a sharp change in thermal conductivity near a certain height can result in such an effect. In the case of compressible gas, the distribution of the specific heat is an important factor because the local isentropic temperature gradient depends on this quantity, while in the equations of convection the actual unperturbed temperature gradient is replaced by its excess over the isentropic gradient (see Sec. 2.1). In the solar convection zone, the superisentropic gradient varies over a wide range, depending on the degree of ionization of hydrogen and helium. The specific heat reaches its maximum values and the isentropic gradient reaches its minimum values in the regions where the ionization degree is not too low and not too high. Precisely in such conditions, because of low transparence, the radiative heat conductivity proves to be a minimum, so that the conductive temperature gradient becomes a maximum. The distribution of the specific entropy over the solar convection zone, as described by the known models (see Fig. 61), corresponds to the conditions of fully developed convection and intense mixing of the substance. This dis­ tribution is highly smoothed as compared with the unperturbed distribution, which could be established in a motionless atmosphere. Nevertheless, near the level p g w 6 dyn/cm 2 , a rather dramatic change in the specific-entropy gradient and, consequently, in the superisentropic temperature gradient takes place. This seems to be favourable for the splitting of the scale spectrum of solar convection. Note in addition that large-scale convective circulation involving the entire zone, from top to bottom, should produce by itself a thermal boundary layer at the top of the zone, which favours the development of small-scale convection. Let us recall bimodal convection (§4.1.10) in this context. It is seen from the considered problems and from the present discussion that solar convection should appear as a superposition of convection cells of different sizes rather than forming a number of layers with cells of a particular size localized in a particular layer (this suggestion is frequently encountered in the astrophysical literature).

7.4. ASTRO- AND GEOPHYSICAL

APPLICATIONS

211

T h e obtained results show how arbitrary are the usual assumptions of the mixing-length theory, such as that the mixing length is equal to or, say, twice as large as the local density scale height. If convection cells exist, it is their vertical scale t h a t should be taken as the mixing length. This quantity is determined by certain peculiarities in the distribution of different parameters, being not directly correlated with the density scale height. 7.4.2. The Earth's

Mantle

Thermal convection in the Earth's mantle, if it really occurs, should be an important mechanism governing the dynamics of the lithosphere. The spatial structure and characteristic scales of convection are of fundamental geodynamic importance. However, even the mere possibility of convection still remains a controversial subject. Among all physical quantities upon which this possibility depends, the vis­ cosity of the mantle substance is known with the least certainty. At different times, various viscosity estimates were proposed, their spread being as large as about 14 orders of magnitude. According to some data, the drop of the vis­ cosity value across the mantle thickness attains six orders of magnitude. L Most of best justified estimates for the lower mantle [310, 311] are in the vicinity of the value 10 22 P , no significant nonuniformity being revealed in these layers. For the asthenospheric viscosity minimum in the upper mantle, the estimates (0.4 to 1.0) xlO 2 1 P were obtained [311]. This means that, in any case, the vertical nonuniformity in viscosity should not be ignored. Moreover, if the above-mentioned dramatic variation of the viscosity really occurs, it should oc­ cur within the upper mantle, thus being especially sharp. Let us estimate the Rayleigh number for the mantle. We assume \i ~ 10 22 P, g ~ 10 3 c m / s 2 , and, according to Ref. 312, a ~ 1 0 - 5 K _ 1 ; we also choose the following values least favourable for convection: x ~ 10 _ 1 cm 2 /s (out of the range l O ^ - l O - 1 c m 2 / s ) , h ~ 2.5 x 108 cm (out of the range 2500-2800 km), A T ~ 10 3 K, and p ~ 3 g / c m 3 (out of the range 3-5 g/cm 3 ). Then we obtain

i ? ~ 5 x 105.

The critical Rayleigh number that determines the possibility of mantle con­ vection can substantially differ from the Rc value found for the standard problem because the physical conditions in the mantle deviate widely from the conditions of this problem. c

As a rule, the dynamic viscosity /i is estimated. Likely, the mantle is relatively unnorm in density, so that all considerations concerning the spatial distribution of the dynamic viscosity are directly applicable to the kinematic viscosity.

212

7. VERTICAL STRUCTURE

OF

CONVECTION

First of all, owing to phase transitions and radioactive decay, internal heat sources affecting the temperature distribution should operate within the mantle thickness. For this reason, Rc can be either increased or decreased [302-304]. However, as it is clear from the results discussed in two preceding sections, only powerful sources can cause a considerable change in Rc, since very large variations in the heat flux and in the unperturbed temperature gradient are necessary for such a decrease. To all appearances, there is no so powerful heat release in the mantle [312]. Likewise, the critical Rayleigh number is affected by viscosity variations. If the upper, low-viscosity layer were much thinner than the mantle as a whole, the situation would be similar to the case of a thin thermal boundary layer because the temperature gradient and the reciprocal of the viscosity are two equivalent factors in the expression for the Rayleigh number. In reality, however, the thickness of the upper mantle is rather large, 500-600 km. Comprehensive theoretical and experimental studies of the convective instability of a fluid layer with a strong temperature dependence of viscosity but without a low-viscosity boundary layer [103, 105, 289] showed that even for iViax/^min ~ 103—104 the Rc value is less than two times as large as the standard value calculated for u = const. As fmax/^min increases further, Rc, after reaching a maximum, approaches the standard value again (see, e.g., curve S in Fig. 10). Finally, if global lithospheric movements result from convective flows in the mantle, the upper boundary of the layer of the convecting mantle material should not be considered as a rigid "lid". Therefore, Rc should be less than in the case of a no-slip boundary. Our estimate for R is thus several hundred times as large as Rc. Despite the uncertainty in the available viscosity estimates, it can be stated t h a t not only is thermal convection very likely to occur in the mantle, but it should be highly supercritical. Now we discuss the question of the spatial localization of convective flows. The fact that the upper mantle is far less viscous than the lower mantle and the lack of observations of earthquakes that would occur below a depth of about 650 km led some investigators to the suggestion that convection takes place only in the upper mantle. Alternatively, the belief was stated that convection occurs independently in the upper and in the lower mantle. We shall see t h a t there are no reasons sufficient for such a conjecture if the above values of the physical characteristics of the mantle material are valid. First, we note that A T and h for the lower mantle are relatively not much smaller than for the entire mantle, the characteristic values of other parameters being the same as for the mantle as a whole. Therefore, the Rayleigh number

7.4. ASTRO- AND GEOPHYSICAL APPLICATIONS

213

for the lower mantle is about half as large as that for the entire mantle. This means that the lower mantle should be in the state of convective motion. On the other hand, convection can form a flow system in the lower mantle independent of the flows in the upper mantle only if upflows that issue from the lower mantle are then decelerated in the upper mantle. This is possible if the rising material enters the region in which the viscosity is much greater than in the lower mantle rather than being smaller. The deceleration can also be due to a stable temperature gradient in the upper region. If neither of these situations is the case, both the upper and the lower part of the mantle should be involved in a common system of convective circulation. Furthermore, the upper and the lower mantle do not differ greatly in density, and the viscosity of the mantle substance is determined by its thermodynamic state rather than external factors independent of convective motion. Changes in viscosity are therefore "inherent" transitions, which accompany the motion of the material and which are in fact a secondary effect of this motion. All this also favours the existence of a circulation system common to the upper and the lower mantle. In this case, the characteristic horizontal scale 2A of convection cells (or the spatial period of the flow) should be of the order of 5-6 thousand kilometers or somewhat larger, which agrees with the basic scale of lithospheric movements. If convection is assumed to be localized in the upper mantle, there is no such agreement. Since the mantle is vertically nonuniform, the polygonal planform of con­ vection cells seems to be a plausible conjecture. The possibility of small-scale convection that develops from the thermal boundary layer produced by the basic circulation is a delicate subject. There are no data sufficient to resolve this problem with certainty. As we already noted, it is unlikely that either a thermal boundary layer with a very sharp temperature-gradient jump is present in the mantle or a similar situation takes place with viscosity nonuniformities, which play the same role as temperaturegradient nonuniformities do. However, the existing viscosity differences between the upper and the lower mantle can nevertheless reinforce the instability to which the basic convective flow should be subject at large Rayleigh numbers. This instability can produce small-scale motions superimposed onto the basic circulation. The planform of small convection cells should be determined by the distribution of parameters and by the velocity field in the layer where these cells are localized. In particular, if rolls are the preferred cell type, they should be aligned with streamlines of the basic, large-scale flow. The characteristic scale 2A of such surface convection can be estimated in the following way. On the basis of the observed speeds of lithospheric move-

214

7. VERTICAL STRUCTURE

OF

CONVECTION

ments, we assume the horizontal velocity u of the basic convective flow in the upper mantle to amount to several centimeters per year. As A ~ ft, the ver­ tical velocity in a zone of the mantle-material upwelling is of the same order as u. Then, from the balance of the advection and the diffusion term in the heat-conduction equation, we find the thickness of the boundary layer to be of the order of \/u ~ 10 km, and the scale of the surface flows should range from several tens to a few hundreds of kilometers. Such flows can be responsible for the spatial modulation of the heat flux near oceanic ridges [313] and can manifest themselves in some structures of the Earth's crust such as transform faults.

CHAPTER 8

CONCLUSION

We have considered a number of important aspects of the self-organization of convective flows in a horizontal fluid layer heated from below. Now, after a detailed examination, let us try to give a general outline of the fundamental features of the convection-pattern dynamics as they can be understood on the basis of the presented information. Although the principal regularities in the behaviour of convective flows have been revealed for patterns of quasi-two-dimensional cells (rolls), it seems to be highly plausible that they also hold for patterns of three-dimensional cells. We thus discuss here both two- and three-dimensional flows without distinguishing between them. The considered material clearly demonstrates that the realizability of a flow is not identical to its stability. The realization of various stable states is not equally probable, so that the class of stable states can in general be much wider than the class of states that are realized under the natural conditions. On the one hand, we see that the tendency toward the formation of a welldefined spatial structure of the convective flow and the emergence of its welldefined spatial scale can as a rule be revealed with certainty from experimental observations. On the other hand, the overall geometry of the flow (which essentially depends on the initial and boundary conditions) can in general counteract changes in the shape and size of convection cells, thus inhibiting the mentioned tendency to a certain degree. As a result, in many typical situations, the optimal shape and size can be detected in the flow pattern only as averaged characteristics. In addition, these optimal shape and size sought by the flow are themselves highly sensitive to various factors related to the properties of the fluid and to the character of the external influences. In view of the aforesaid, the prediction of t h e geometry and characteristic scale of the flow is a rather delicate matter. As concerns roll flows (two-dimensional or even more complex but demonstrating quasi-two-dimensional relaxation), the problem of determining their

215

216

8.

CONCLUSION

scales (wavenumbers) appears to be substantially more advanced in its compre­ hension than the extremely complicated general problem. It is known that the wavenumber band in which rolls are stable is in many cases wide whereas the normally observed wavenumbers occupy a minor portion of this band, their distribution being peaked at a certain optimal wavenumber. Particular "selection mechanisms" result in particular final states of the pattern with final wavenumbers that generally differ from the optimal wavenumber. We introduced definitions that make a clear-cut distinction between the preferred (optimal) wavenumber and particular realized (final) wavenumbers of roll pat­ terns and used the notions of the selective and contraselective factors. The preferred wavenumber manifests itself as an inherent characteristic of roll con­ vection, while the contraselective factors are governed by the degree of ordering of the pattern (if the word ordering is used in a certain generalized sense clar­ ified in § 6.5.9). In particular, the presence of pattern defects (in the broad sense) facilitates the readjustment of roll wavenumber to the optimal value. The approach we used allowed us to systematize various situations differing in the strength of the contraselective factors, to explain from a unified point of view the observed differences between the wavenumbers realized in various cases, and to find an approach to the evaluation of the optimal wavenumber. It became clear that the realized wavenumbers do not necessarily coincide with the optimal wavenumber because the evolution of the flow toward the optimal state can stop at one or another stage. An important conclusion relates to the mathematical statement of problems aimed at determining the possible roll wavenumbers. The condition of spatial periodicity, which is traditionally used by theorists, imparts additional stability to the flow simulated mathematically and can therefore prevent the pattern from reaching the optimal scale. If roll textures are dealt with, mean drift introduces significant extra com­ plications. We saw that this flow component is responsible for the phenomena of persistent dynamics observed in certain parameter ranges—phase turbulence and spiral-defect chaos. Persistent time dependence appears to be a state of a convection pattern which results from the impossibility to simultaneously achieve throughout the pattern both the optimal roll wavenumber and the bal­ ance between the phase diffusion and the mean drift (in general, the optimal wavenumber value does not necessarily imply such a balance). Mean drift is an agent coupling different portions of the pattern, and some of them may approach the optimum at the expense of deoptimization of others. As a result, steady states prove to be unattainable. The occurrence of such phenomena strongly depends on the boundary conditions at the sidewalls of the container, first of all, on the behaviour of the mean drift near the sidewalls. The more restrictive

8.

CONCLUSION

217

the boundary conditions and the less the aspect ratio of the container (within certain limits)—i.e., the greater the drift-velocity gradients, the more complex can in general be the dynamics of the pattern. We regard here the boundary conditions as restrictive if they resist the penetration of pattern defects and "nonequilibrium" patches (in which the rolls are not optimal and the phase diffusion is not balanced by the mean drift) through the boundary. It appears to be plausible that as the sidewall boundary conditions become less restrictive and the aspect ratio of the container is simultaneously increased, the conditions of an infinite layer being thus approached, the structure of convec­ tion patterns that spontaneously develop from weak noise perturbations should become more predictable on the basis of the R and P values. The features of persistent time dependence (in the above-specified sense) that are characteristic of moderate aspect ratios and essentially depend on mean-drift effects seem not to be related to convection patterns in large-scale natural objects such as atmospheres, oceans, and stellar convection zones. If a natural system resembles an infinite layer, it can be successfully simulated in a numerical or laboratory experiment provided sidewall effects are eliminated as far as possible. In thick fluid layers with a complicated vertical structure, the scales of convective motions can vary over a wide range, depending on the particularities of the stratification of the layer. The considered approaches make it possible to predict in many cases the essential features of the behaviour of convective flows. The development of a technique for the quantitative characterization of the selective and contraselective factors as well as a more complete understanding of such points as slow relaxation processes (possible when the flow is stable against the investigated perturbation modes) and the equilibrium of complex textures could substan­ tially contribute to resolving the fascinating problem of the structure and dy­ namics of convection.

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298. S. Musman, Penetrative convection, J. Fluid Mech. 31(2), 343-360 (1968). 299. A. J. Faller and R. Keylor, Numerical study of penetrative convective instabilities, J. Geophys. Res. 75(3), 521-530 (1970). 300. D. R. Moore and N. O. Weiss, Nonlinear penetrative convection, J. Fluid Mech. 61(3), 553-581 (1973). 301. A. A. Townsend, Natural convection in water over an ice surface, Quart. J. Roy. Meteorol. Soc. 90(385), 248-259 (1964). 302. E. M. Sparrow, R. J. Goldstein, and V. K. Jonsson, Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile, J. Fluid Mech. 18(4), 513-528 (1964). 303. A. V. Getling, Concentration of convective motions near the boundary of a horizontal fluid layer with a height-nonuniform, unstable temperature gradient, Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza no. 5, 45-52 (1975) [English translation: Fluid Dyn.}. 304. R. M. Goody, The influence of radiative transfer on cellular convection, J. Fluid Mech. 1(4), 424-435 (1956). 305. A. V. Getling, On the scales of convective flows in a horizontal layer with radiative energy transfer, Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 16(5), 529-532 (1980) [English translation: Izv. Acad. Sci. USSR, Atmos. Oceanic Phys.]. 306. V. Kourganoff, Basic Methods in Transfer Problems (Oxford Univ. Press, Oxford, 1952). 307. E. Vitense, Die Wasserstoffkonvektionzone der Sonne, Zs. Astrophys. 32(3), 135-164 (1953). 308. K. H. Bohm, The outer solar convection zone, In Aerodynamic Phenomena in Stellar Atmospheres, IAU Symp. No. 28 (Academic, London, 1967), pp. 366-384. 309. H. S. Spruit, A model of the solar convection zone, Solar. Phys. 34(2), 277-290 (1974). 310. R. J. O'Connell, Pleistocene glaciation and the viscosity of the lower man­ tle, Geophys. J. Roy. Astron. Soc. 23(3), 299-327 (1971). 311. L. M. Cathles, The Viscosity of the Earth's Mantle (Princeton Univ. Press, Princeton, N. J., 1975). 312. V. N. Zharkov, Vnutrennee stroenie Zemli i planet [Internal Structure of the Earth and Planets] (Nauka, Moscow, 1983). 313. K. Horai and S. Uyeda, Terrestrial heat flow in volcanic areas, In The Earth's Crust and Upper Mantle, ed. P. J. Hart, Geophysical monograph no. 13 (Am. Geophys. Union, Washington, D.C., 1969). 314. F. H. Busse and R. M. Clever, Asymmetric squares as an attracting set in Rayleigh-Benard convection, Phys. Rev. Lett. 81(2), 341-344 (1998).

SUBJECT INDEX

Bold numbers indicate where entries are defined most strictly

adiabatic approximation, 33, 35, 36 subcritical (backward), 2 1 , 22 adjustment of rolls (wavenumbers), see supercritical (forward, normal), 21-23, wavenumber adjustment 32 air, 29, 62, 108, 123, 152 transcritical, 21, 22 amplitude, 31, 41, 45, 73, 78, 150, 159, bimodal convection, see flow, convective, 169, 177 bimodal amplitude equation, 5, 31, 35-47, 79, 142, boundary conditions 144, 146, 158-160, 175 on horizontal boundaries, 119, 126, 148, Newell-Whitehead-Segel, 36, 37-40, 201 asymmetric, 73, 135 45-47, 85, 87, 130, 139, 144, 148, finite-thermal-conductivity, 14, 27, 156-159, 168-170, 175, 177, 181 28, 41, 44, 68, 69, 74, 75, 123, 138, argon, 106, 113 150 aroclor, 67 infinite-thermal-conductivity aspect ratio, 16, 33, 41, 49, 50, 58, 85, 90, (isothermal), 13, 27, 28, 52, 123, 93, 102, 109, 115, 117, 160, 175, 192, 205 178, 194, 217 rigid (no-slip), 13-15, 17-20, 39-41, atmosphere, 110, 217 48, 52, 57, 59, 69, 84, 85, 115, 126, 130, 133-136, 146, 153, 161, 193, basin of attraction (in the space of states), 200 3, 24, 30, 116 (stress-)free (free-slip), 13-15, 17-20, Benard cells, 9, 59; see cells, hexagonal 38, 39, 41, 48, 49, 57, 59, 64, 126, Benard-Marangoni convection, see 135-137, 146, 153, 161, 175, 194, convection 197, 200, 205 bifurcation, 5, 20, 2 1 , 22, 63 on sidewalls (lateral boundaries), 175, imperfect supercritical, 22, 23, 32, 65, 216, 217 89 finite-thermal-conductivity, 41, 89 Landau-Hopf (Hopf), 2 1 , 22 for amplitude function, 41, 87, 175 pitchfork, 21-23 239

240

SUBJECT

INDEX

periodic, 23, 55, 115, 123, 136, 142, rectangular, 27, 85-88, 102, 103, 108, 161, 179, 216 115, 138, 160, 161, 171, 172, 176, refection-symmetry, 194 177, 180 rigid (no-slip) 42, 45, 46, 54, 85, 87, contraselective factors, 6, 120, 152, 156, 137, 185 166, 171, 172, 178, 181, 185-190, thermal-insulation, 41, 46 216, 217 thermally forcing, see sidewall, control (driving, stress) parameter, 15, forcing effect 43, 157 boundary forcing, see sidewall, forcing convection, 1; see also flow, convective effect Benard-Marangoni, 1, 61, 62 Boussinesq approximation (equations), 5, oscillatory, 99, 101-103; see also 9, 10-13, 21, 23, 30, 32, 42, 47, 49, instability of roll flows, 51, 52, 65, 115, 136, 146, 148, 160, oscillatory 161, 178, 180, 181, 183, 199 blob, 132 box, see container knot, 132 Busse balloon, 69, 110, 128, 129, 135, penetrative, 183, 194, 196-198 136, 155, 172 Rayleigh-Benard, 1-3, 5, 30 Busse windsock, 128, 130 stagnant-lid, 194-196 thermal (thermogravitational, carbon dioxide, 29, 90, 95, 110, 112, 113 buoyancy-driven), 9, 60, 61 cavity, see container thermocapillary (surface-tensioncells, 3, 5, 23, 24, 33, 59, 70, 73, 99, 202, driven), 9, 60-62 215 time-dependent, 100-105, 107, 108 hexagonal, 9, 25, 30, 59, 60, 63-75, travelling-wave, 133, 134 77-79, 83-85, 114, 115, 124, 144 turbulent, 99, 101 closed, 62 vertical structure of, see flow, <7-type, 25, 62, 64-66, 73, 77, 83 convective, vertical structure l-type, 25, 62, 64-67, 73, 74, 77, 83 whole-layer, 194-196, 204 open, 62 convection zone polygonal, 59, 60, 62, 68, 79, 119, 131, solar, 4, 7, 208-210 191, 213 stellar, 217 rectangular, 26, 80 countercells, 196, 197-199 square, 25, 30, 38, 68, 69, 74-76, 85, critical regime, 19 100, 124 Cross-Newell equation, see phase centre defect, 112 equation chamber, see container Chandrasekhar functions, 52, 56, 126 defects of patterns, 3, 6, 33, 83, 94, 96-98, channel, 29, 86, 178 102, 103, 121, 122, 139, 188, 216; coherence length, 19 see also centre defect, disclination, container (box, cavity, chamber, tank), dislocation, focus (singularity), 16, 24, 33, 87, 89, 110, 124, 139, grain boundary, spiral-defect 142, 216 chaos, target circular (cylindrical), 16, 27, 71, 75, 78, deformation of free surface, 76, 77 86-94, 102, 103, 106-110, 112, 113, director, 97 115, 137, 154, 182, 185, 187 disclination, 97

SUBJECT

INDEX

dislocation, 30, 96, 97, 103-108, 115, 136, 143, 147-152, 182, 185, 188 trapping of, 152 climb of, 96, 103, 107, 108, 143, 147152, 157 glide of, 96, 107, 147, 151, 152 domain instability, 131 domain pattern, 131 drift (large-scale flow), 14, 39, 40, 44, 45, 47, 48, 51-53, 86, 92-94, 108110, 115, 136, 146, 151, 153-155, 183, 216, 217 driving parameter, see control parameter dynamic frustration, 108 dynamical chaos, 78 Earth's mantle, 7, 208, 211-214 envelope function, 36 ethyl alcohol, 29, 61, 80, 172 exchange of stabilities principle of, 18 excitation finite-amplitude (hard), 23, 71 soft, 23 "exponential" fluid, 192, 193 extremum principle, 124

241 vertical structure of, 4, 7, 24, 191, 199, 200 wavy oscillatory bimodal, 101 focus singularity {and umbilicus), 92-94, 97, 103-107, 109, 110, 112, 113, 137, 154, 155, 157, 186, 188 free energy, see potential front (propagation), 66, 156-161, 163, 164, 168, 179-182 speed of, 157-161, 169-171, 179-182 Galerkin method, 42, 52, 54, 69, 74, 78, 84-86, 127, 128, 153 of numerical simulation, see spectral technique gas mixtures, binary, 113 giant structures, 209 glycerine (glycerol), 29, 67, 192 golden syrup, 69, 192 grain boundary, 51, 97, 98, 141-144, 152, 157, 182, 185, 188 granules, 209

heat exchanger, 27-29, 89 heat flux, 3, 4, 32, 71, 74, 79, 101, 102, 122-124, 134 helium finite-difference technique (of numerical gaseous, 14 simulation), 54, 55 liquid, 29, 71, 79, 102, 103, 108 flow, convective Herzberg-Sivashinsky equation, 44, 47, bimodal, 30, 79, 80, 99-101, 130, 138, 136 152, 194 Hopf bifurcation, see bifurcation disordered, 91, 172-175, 182 hysteresis, 64, 66, 78, 155, 194 horizontal scale of, 3, 191, 200, 209 large-scale, 7, 191, 204, 207, 210, 213; initial conditions (in experiment) see also drift controlled, 5, 29, 67, 69, 137, 138, 142, quaternary, 21 151, 172, 173, 177, 186 secondary, 21, 79, 90 uncontrolled (random), 121, 173, 174, small-scale, 7, 191, 204, 207, 210, 213 187 symmetric oscillatory bimodal, 101 initiator, 172 tertiary, 21, 79, 87, 127, 130, 131, 132, instability 134 convective, 18, 19, 60, 79, 85, 86 vertical scale of, 191, 200, 208, 209 finite-amplitude, 23

242 of roll flows, see also stability of roll flows collective, 132 cross-roll, 79, 87, 88, 90, 91, 98, 99, 101, 106, 112, 122, 127, 129, 130, 135, 137, 142, 151, 182 Eckhaus, 128, 129, 135, 136, 139, 157, 165, 175, 177, 178, 179, 181, 184, 186 focus, 92, 93, 94, 155 knot, 80, 81, 106, 127, 131, 138 one-blob, 128, 132, 135 oscillatory, 49, 100, 113, 128, 132134,153 oscillatory skewed varicose, 135 two-blob, 128, 132, 135 skewed varicose, 83, 106, 108, 122, 128, 132, 135, 136, 138, 139, 182 zigzag, 80, 82, 106, 122, 127, 129, 130, 136, 137, 139, 142, 150, 152, 153, 156, 174, 177, 187 of travelling-wave convection, 134 subcritical, 23, 32, 64, 65, 193, 198 kinetic energy, 124 Kuppers-Lortz instability, 90 knot pattern, 101; see also instability of roll flows, knot large-scale flow, see drift laser Doppler velocimetry, 29, 71 "lid", viscous, stagnant, 193, 194 linear problem, 10, 16, 31, 192, 193 Lorenz model, 78 generalized, 78 low-frequency turbulence, see phase turbulence Lyapunov functional, 45-46, 87, 125, 139, 140, 142, 145, 149, 160 density of, see potential, specific Malkus principle, 122, 124 Manneville equations, see microscopic equations

SUBJECT

INDEX

marginal stability, 158-160, 169, 179 maximum production of entropy, principle, 125 mercury, 14, 75, 102 mesogranules, 209 metastability, 66, 144 methyl alcohol, 29, 82, 92, 93 microscopic (Manneville's) equations, 41, 42-43, 47 Milne-Eddington approximation, 205 mineral oil, 198, 199 minimum principle, 32 mixing-length, 208 theory, 208, 211 model equations, 43-45, 49, 156, 157, 182 neutral curve, 18, 19, 36, 129, 132, 133, 143, 157, 173, 184, 206, 207 neutral regime, see critical regime Newell-Passot-Souli equations, see phase equation Newell-Whitehead-Segel (amplitude) equation, see amplitude equation non-Boussinesq effects, 32, 37, 38, 45, 63, 65, 70, 71, 73, 76, 78, 84, 113-116, 193 nonvariational dynamics, see variational and nonvariational dynamics numerical simulation, 6, 33, 44, 54-58, 87, 88, 101, 108, 110, 114, 116, 122-124, 134-137, 141, 148, 151, 157, 160-161, 164-166, 177, 179181, 189, 194, 198, 217 ocean, 110, 217 optical interferometry, 29 order parameter, 43, 45 ordering of flow, 3, 6, 33, 93, 96, 98, 99, 108, 112, 115, 116, 120, 172, 178, 179, 182, 187, 188, 216 Palm-Jenssen fluid, 192 pattern-forming systems, 2, 3

SUBJECT

INDEX

pattern of convection, 3, 99 axisymmetric, see rolls, circular disordered, see flow, convective, disordered roll, see rolls, 2D or quasi-2D S-shaped, see rolls, S-shaped target, see target Peach-Kohler force, 149, 150 persistent dynamics, see phase turbulence and spiral-defect chaos perturbation of static quantities 11, 125-127 technique, 30-32, 40, 125 phase diffusion, 48, 52, 54, 93, 108, 109, 153, 154, 176, 184, 186, 216, 217 coefficients of, 48, 54, 136, 140, 153, 154, 173, 177, 184 phase equation, 5, 31, 47, 49, 51-54, 145, 176, 183, 184 Cross-Newell, 49, 5 1 , 52, 93, 109, 145, 153 Newell-Passot-Souli, 52, 53, 93 Pomeau-Manneville, 47, 48, 54 phase turbulence, 6, 83, 101, 102, 103, 107-109, 114, 121, 139, 152, 187, 216, 217 pinching mechanism, 151 planform, 3, 24-26, 30, 32, 35, 59, 61-63, 68, 69, 119, 191, 213 function, 24, 59, 63, 69, 70, 75, 85 preferred, 24, 59 polybutene oil, 67, 192 Pomeau-Manneville equation, see phase equation potential, 46, 47, 78, 140 specific, 66, 139, 142, 145, 150, 156, 159, 177, 179, 180, 188 potential dynamics, see variational dynamics Prandtl number, 14, 119 pseudospectral approximation, 57 radiative energy transfer, 204 equation of, 205

243 radiative heat conductivity, 206 ramp of parameters, spatial, 144-147, 183-185 Rayleigh-Benard convection, see convection Rayleigh-Benard problem, 13, 14 Rayleigh number, 14, 20, 30, 68, 119, 145, 199, 200, 205, 211 critical, 19, 21, 30, 63, 68, 85, 192, 195, 197-201, 204, 211, 212 local, 144, 147, 182, 192, 203 reduced, 19, 114 reaction-diffusion equation, 145, 146 readjustment of rolls (wavenumbers), see wavenumber adjustment readability, 2, 3, 24, 125, 165, 188, 215 regimes of convection, 6, 99 diagram of, 99-101 nonlinear, 20 supercritical, 19, 21, 59 relaxation of rolls, 141, 144, 156, 160, 161, 164, 168, 170-172, 174, 178182 relaxation time, 19 relaxational dynamics, see variational dynamics rolls circular (annular, concentric), 38, 8998, 115, 121, 143, 152-155, 157, 182, 185, 186 eccentric (asymmetric), 92, 93, 155 crossed, see instability of roll flows, cross-roll, and flow, convective, bimodal spiral, 72, 110, 114, 115 S-shaped, 87, 88 2D or quasi-2D, 9, 24, 30, 38, 40-42, 46-50, 59-61, 63-68, 71, 73-79, 85-94, 96-101, 106, 109, 110, 112116, 119, 121-146, 148-152, 154, 157, 161, 163-166, 168-184, 186, 187, 194, 215, 216 undulating 130, 131

244 scale of flow; see also wavenumber optimal (preferred), 3, 6, 109, 120, 124, 137, 138, 141, 145, 168, 176, 178, 186-188, 215-217 realized (final), 3, 6, 120 Schwarzschild criterion, 12 selection, 6 of wavenumbers, 6, 19, 24, 57, 90, 97, 119, 120, 139, 147, 154, 156, 168, 178, 188 "selection mechanisms", 6, 97, 120, 140, 156, 157, 175, 178, 179, 187, 216 selective factors, 6, 120, 166, 181, 188, 216, 217 shadowgraph visualization, 29, 70, 71, 137 sidewall, 16, 41, 45, 54, 85, 87-91, 94, 97, 103, 107, 109, 110, 113, 117, 120, 123, 124, 134, 137, 140, 142, 151, 152, 154, 157, 160, 161, 174-179, 181, 185-188, 217 forcing effect, 45, 54, 88-91, 93, 94, 110, 112, 115, 177, 181, 185-186 dynamic, 89 static, 88 permeable, 109 siliconeoil, 14, 29, 67, 75, 82, 92, 123, 137, 138, 142, 155 Sivashinsky equation, see HerzbergSivashinsky equation slaving, 33, 35, 42 principle, 5, 34, 36, 50 solid particles, suspended, 79 Soret effect, 75, 113 specific potential, see potential, specific spectral (Galerkin) technique (of numerical simulation), 54-57 spiral-defect chaos, 6, 110-116, 216, 217 spirals, see rolls, spiral spoke pattern, 80, 82, 131, 132 stability, 2, 3, 21-24, 29, 30, 32, 60, 6264, 66, 68-70, 74, 75, 84, 85, 125, 126, 192, 215 of bimodal flows, 101, 127, 132, 138 of hexagonal-cell flows, 64-66, 68, 69, 73, 77, 78, 84

SUBJECT

INDEX

of knot convection, 101, 131 of roll flows, 6, 64-66, 68, 69, 70, 73, 74, 76, 77, 79, 83, 84, 101, 107, 110, 116, 127-133, 135-140, 165, 178, 182, 188 of square-cell flows, 69, 74-76, 85 of undulating-roll flows, 130, 131 standard conditions (statement of problem), 5, 14, 32, 119, 126, 197199, 206, 207 state optimum (preferred), 30, 188, 190 primary, 21, 23 static quantities, see unperturbed quantities strange attractor, 78 stratification, 191 stress parameter, see control parameter sulphur hexafluoride, 113 supercriticality, relative, see Rayleigh number, reduced supergranules, 209 surface tension, 9, 61, 76 Swift-Hohenberg equation (model), 43, 44, 46, 48, 49, 87, 114, 136, 140141, 151, 158, 159, 176, 179, 180 tank, see container target pattern, 97, 110, 113, 114 Taylor-Couette flow, 2, 198 Taylor vortices, 2, 128 temperature gradient isentropic (adiabatic), 12, 210 superisentropic (superadiabatic), 12, 210 unperturbed, 210, 212 temperature profile, unperturbed, 205, 206, 210 curvature of, 74, 77, 183, 198 piecewise-linear, 196, 198-200 texture, 33, 42, 86, 96, 98, 103, 140, 187, 216 complex, 33, 93, 96, 103, 154, 185-187, 217 "Pan Am", 87, 90, 106, 107

SUBJECT

INDEX

thermal conductivity of boundaries, 69 thermocapillary effect, 28, 39, 60, 61, 177; see also convection, thermocapillary time-dependent heating, 77, 78 turbulence near onset (threshold) of convection, see phase turbulence umbilicus, see focus unperturbed quantities, 10 up-down dissymmetry, 6, 37-39, 73, 83 variational and nonvariational dynamics, 46, 47, 108, 114, 139-142, 146, 150, 151, 156, 160, 174, 176, 185, 187, 188 viscosity temperature-dependent, 38, 39, 62-71, 75, 191-194, 212 vorticity vertical component of, 15, 16, 39, 47, 114, 126, 133, 136, 148

245

164-166, 168, 170, 174, 176, 178182, 184, 186, 188, 216 critical, 19, 30, 36, 63, 123, 192, 197, 198 freezing effect, 166, 170, 171, 180 local, 51, 103, 106, 107, 109, 110, 116, 119-121, 145-147, 149, 165, 180, 182, 187 mean, 110, 116, 121, 122, 124, 149, 151, 154, 155, 173, 174 preferred (optimal), 119, 120, 122-125, 137-140, 145, 146, 151, 152, 156, 166, 168, 172, 174, 177, 178-182, 185-189, 216 calculated, 163, 164 observed, 119, 121, 124 realized (final), 120, 122-124, 141, 154-157, 160, 170, 176, 178-180, 182, 183, 187, 188, 216 wavevector local, 33, 36, 50, 51, 54, 97, 103 water, 29, 75, 78, 83, 93, 103, 123, 138, 161, 198 waves in roll patterns, 94 weak turbulence, see phase turbulence

wall, see sidewall wavelength, see wavenumber wavenumber adjustment (readjustment), 90, 96-98, zigzag perturbations, 48, 153; see also 127, 135, 137-141, 147, 152, 155, instability of roll flows, zigzag