pattern selection in long-wavelength convection e....

Physica D 41 (1990) 450-479 North-Holland

PATTERN SELECTION IN LONG-WAVELENGTH CONVECTION

E. KNOBLOCH x Department of Physics, Kyoto University, Kyoto 606, Japan

Received 13 August 1988 Revised manuscript received 24 October 1989 Communicated by F.H. Busse

The long wavelength of the first instability in Rayleigh-Brnard convection between nearly thermally insulating horizontal plates is typical of a variety of physical systems. The evolution of such an instability is described by the equation

ft = a f - / ~ v 2 f - v 4 f + ~¢V ° ]Vfl 2 v f + / 3 V • V z f v f - 3'V " f ~ f + 8 v 21WI 2,

where f is the planform function, ~ is the scaled Rayleigh number and ~ = +_ 1. The quantities a, fl, 7 represent the effects of finite Blot number, asymmetry in the boundary conditions at top and bottom, and departures from the Boussinesq approximation, respectively. The quantity 8 = fl when the original problem is self-adjoint, but 3 * /3 otherwise.

Planform selection is studied for a < 0, g = + 1 using equivariant bifurcation theory. On the square lattice both rolls and squares can be stable, depending on the parameters/3, 3, and 3. The possible secondary bifurcations located near various codimension-two singularities are analyzed. On the hexagonal lattice the primary bifurcation is always degenerate when /3 = 7 = 8 = 0. Of the six primary solution branches possible in this case the hexagon branch is stable. When fl - 8 = 3' = 0, both rolls and hexagons bifurcate supercritically, hut rolls are stable. Finally, when/3 * 6 a n d / o r 3' ~ 0 a hysteretic transition to H + or H - occurs depending on sgn(fl + "y/k~ - 8); if [/31, [71,181 << i stable H +, H - coexist at larger amplitudes, but if I/3 + "t/k2e- 81 << 1/31, h ' l , [81 = 0(1), a further hysteretic transition takes place with increasing amplitude in which the hexagons are replaced by stable rolls.

I. Introduction

Pattern formation in continuous systems has received increasing attention in recent years. Usu- ally it is assumed that the pattern-forming instability sets in at /~ =/~c with a finite wavenumber k¢. In unbounded isotropic systems such a wavenumber corresponds to a circle of marginally stable wavevectors. In addition to this orienta- tional degeneracy one finds that for/~ >/~c a whole band of wavenumbers, centered on k~, is also unstable. Both facts pose considerable difficulties for any rigorous analysis of pattern formation. A

1On leave from the Department of Physics, University of California, Berkeley, CA 94720, USA,

0167-2789/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)

tractable problem can, however, be formulated by restricting the class of solutions to those lying on a doubly periodic lattice. Guided by experiments, mainly those on Rayleigh-Brnard convection, pattern selection has been studied on both square and hexagonal lattices. The lattice spacing is typi- cally chosen to correspond to the critical wavenumber k c, thereby not only fixing the wavenumber of the resulting pattern, but also selecting a finite subset from the circle of marginally stable wavevectors. The resulting bifurcation problem is finite-dimensional: for a steady state bifurcation on a square lattice this dimension is four, while on a hexagonal lattice it is six. These dimensions are doubled when the bifurcation is a Hopf bifurcation. In these situations the rigorous techniques of

E. Knobloch / Pattern selection in long-wavelength convection 451

equivariant bifurcation theory [1] allow one to establish the relative stability of the various patterns that lie on either lattice. In the past such techniques have met with considerable success. For example, an analysis of near-degenerate steady state bifurcation on the hexagonal lattice [2] com- pleted the bifurcation diagram for the transition between hexagons and rolls in slightly non- Boussinesq Rayleigh-Brnard convection [3].

In many physical systems of interest the critical wavenumber kc either vanishes or is very small. This occurs, for example, in Rayleigh-Brnard convection between thermally insulating boundaries [4-9], Marangoni convection [10] and binary alloy solidification [11]. A recent model of Lang- muir circulation in the ocean [12] provides an additional example. Finally, recent studies of convection in binary fluid mixtures show that for sufficiently positive separation ratios k c also vanishes [13, 14]. In such circumstances the analysis based on finite k¢ formally breaks down. On the other hand one can take advantage of the long scale of the instability and show that the instability in all the above cases is described by an evolution equation of the form

f, --- v2y - v 4 f + IvYl2 v f

+ f l Y " V2 f v f - ' r V of v f + r v 2 1 v f ) 2, (1.1)

where f ( x , y, t) describes the horizontal planform of the instability, and x, y, t are appropriate slow variables. In one spatial dimension, this equation has been derived for a variety of problems [6-11]. We give an explicit derivation of the two-dimensional case for Marangoni convection [10] in the appendix.

Eq. (1.1) may be thought of as a "normal form" for this class of problems. The parameter # is the distinguished bifurcation parameter (e.g. the departure from the critical Rayleigh number), while a measures departures from an ideal situation (a = 0) in which the critical wavenumber vanishes.

In the convection problem [7, 8] this term appears owing to the finite, though small, conductivity of the boundaries (i.e. owing to finite Biot number effects). When fl ~ 0, 3 :~ 0 a n d / o r y ~ 0 the reflection symmetry f ~ - f , inherited from the reflection symmetry in the layer mid-plane, is broken. These terms are present when the boundary conditions at the top and bottom plates are not identical (fl :~ 0, 3 :~ 0), or when non-Boussinesq effects are taken into account (y :~ 0) [6, 7, 9]. The parameter ~ = fl when (1.1) is derived from a self-adjoint problem, but 8 ~ fl otherwise. The parameters a, r , y, and 8 are all important in what follows, and it is helpful to think of them, loosely speaking, as unfolding parameters. Except in ref. [11] where x = 0, the coefficient r can be scaled to be _ 1.

In this paper we study pattern selection in eq. (1.1). The linear stability of the trivial solution f = 0 is described by the growth rate

s -- a 4-/.tk 2 - k 4. (1.2)

Thus the neutral stability curve, given by s = 0, has a minimum at

k c - ( - a ) 1/2, 2 ( - a ) 1/2. (1.3)

Consequently, when a < 0, the instability has a finite (though, in the original variables, long) wavelength, while if a > 0 the wavelength remains infinite (see fig. 1). In most of this paper we

.• s=O

t~=O t~c

~ k

/.t

a<0 b ~ k

ko

Fig. 1. The neutral stability curve for (a) a = 0 and (b) a < 0.

452 E. Knobloch / Pattern selection in Iong-wat, elength cont~ection

assume that a < 0, and formulate the pattern selection problem on a doubly periodic lattice in the plane. We begin, in section 2, by constructing small-amplitude solutions in the form of rolls (R), squares (S) and hexagons (H). These solutions provide the basic data motivating much of the remainder of the paper. The discussion focuses on two cases of particular interest. The self- adjoint problem ( / 3 = 8 ) with non-Boussinesq terms (7 4: 0), hereafter case A, is typical of the Rayleigh-Brnard problem, and is studied first. The non-self-adjoint problem (/34:8) without non-Boussinesq terms (7 -- 0), hereafter case B, is typified by the Marangoni problem (see appendix), and is discussed next. Both problems are important, and give rise to distinct transition sequences among the possible patterns as # is varied. In addition, they lead to a two-parameter analysis that is much more tractable than the complete problem (1.1).

On the square lattice, discussed in section 3, we use the results of section 2 to show rigorously for

= + 1 that there are three codimension-two singularities in case A and two in case B. We analyze the bifurcation behavior in the vicinity of these singularities, and determine the conditions under which either rolls or squares can be stable. When

= - 1 , a situation that apparently does not arise in the physical examples already mentioned, a number of codimension-three singularities can also occur, but these are beyond the scope of the present paper.

The discussion of the hexagonal lattice must be carried out in two parts. In the symmetric case (/3 = 7 = 8 = 0) the results of section 2 show that the pattern forming bifurcation is degenerate in the sense that a particular non-degeneracy condition on the third-order terms does not hold. This degeneracy, implicit in refs. [5, 8], is resolved by extending the existing analysis of steady state bifurcation on the hexagonal lattice with reflection symmetry [2] to fifth order. This analysis is carried out in complete generality, without taking advantage of the special structure of (1.1). It is found that in addition to the four patterns, rolls (R),

hexagons (H), regular triangles (RT) and a pattern called patchwork quilt (PQ), that arise in the generic case, two additional patterns, termed rectangles (RA) and imaginary rectangles (IRA), can also bifurcate from the trivial state at /~ = / ~ . The relative stability of all the states is established, and it is shown that in several cases two stable states can coexist. However, neither of the two new states, RA and IRA, can ever be stable. This abstract theory, discussed in section 4, is then applied in section 5 to eq. (1.1). This application requires the computation of three fifth-order coefficients in the generic equivariant vector field. The calculations show that for ~ = + 1 the RA branch does not exist, while the remaining five branches all bifurcate supercritically. Of these only the hexagon branch is stable. The relative stability of the H and S solutions cannot, however, be established using these techniques. In section 6 we turn our attention to the non-symmetric problem (fl 4~ 0, 7 4: 0, 6 4: 0) on the hexagonal lattice. We first analyze the self-adjoint case/3 = 8 4: 0, 7 = 0 in which both hexagon branches H +, H - bifurcate supercritically, as do the rolls. We show that in this case rolls are stable. In contrast, when /3 4:8 a n d / o r 7 ~ 0, there are no stable solution branches near the primary bifurcation. Consequently we

study the case I / 3 -8 [ <<1, 171 <<1 with /3, 8 both 0(1), and show that a hysteretic transition to H + ( H - ) will occur if sgn(/3 + 7 / k ~ - 8) > 0

(< 0). To relate these results to the symmetric case we finally consider the case 1/31,171,181 << 1, and carry out a plausible symmetry-breaking analysis. We determine some of the subsequent secondary bifurcations among the various solution branches. The paper concludes in section 7 with a summary and a discussion of the results.

When the system is symmetric (/3 = 7 - - 8 = 0), eq. (1.1) admits a Lyapunov functional V [ f ] [8]. Consequently the system necessarily relaxes to time-independent solutions, the local minima of V corresponding to stable patterns. This method for studying pattern selection is described in section 5, and compared there with the one employed here.


2. Preliminary results

In this section several small-amplitude time- independent solutions of eq. (1.1) with the fun- damental wavenumber k~ given by (1.3) are calculated. In terms of a small parameter ~ << 1 we write

f = ' f o + '2 f l + '3f2 + . . . . (2.1a)

# = #¢ + c~t t + e2/~2 + . . . . (2.1b)

Eq. (1.1) can be written in the form

LU= - (t~ - b%) V 2f+ x V " I v f l a v ~

+ f l Y " v z f v f - YV " f v f + 3 v z I v / 1 2 ,

(2.2a)

where

L -= V4+/.LeV 2 - - 0¢, (2.2b)

subject to periodic boundary conditions. At each order in c the unknown quantities /~x,/22 . . . . are determined from a solvability condition, which may be written quite generally in the form

(b t - ~c) (Vfo" V f>

= K(Ivf l 2 vf0" V f ) + /3 (V 2f ~Tf0" W/)

- v ( f x 7 f o " v f ) + 3<X7fo" V IXTf12), (2.3)

where ( . . . ) - - - f o ( . . . ) with D representing the appropriate unit cell, and fo solves the leading- order problem Lfo = O.

We seek solutions in the form of rolls (R), squares (S) and hexagons (H). The results of the calculations through third order in ~ are summarized in table 1. Of particular interest are the following two special cases: (A) fl = 8, "r ~ 0, corresponding to a self-adjoint problem with non- Boussinesq terms (e.g. the Rayleigh-Brnard problem), and (B) fl 4~ 8, ~ = 0, corresponding to a non-self-adjoint but Boussinesq problem (e.g. the Marangoni problem described in the appendix). In case A, when x - - -+1 , both the rolls (R) and squares (S) can change their directions of branching; in case B, when ~ = +1, rolls are always supercritical. The hexagon (H) branch is transcrit- ical (~L 1 5/= 0 ) when y ~ 0 a n d / o r fl 4= 6. On the other hand, in the special case fl + 3 , / k 2 - 8 = 0

the quantity #z vanishes, and the direction of branching is determined by/~2- In both cases we must distinguish between H + (rri > 0) and H - (r u < 0) since these patterns are topologically distinct, although when /~1 vanishes this distinction matters only for the secondary branching behavior.

In section 3 we use the results in table 1 in conjunction with equivariant bifurcation theory to deduce the relative stability of rolls and squares on the square lattice and to identify the presence of higher codimension bifurcations. In addition, in section 4 we shall see that on the hexagonal lattice with the extra reflection symmetry (fl - 3' = 8 = 0) the primary bifurcation is degenerate, and deduce in section 5 the existence of three additional primary branches that bifurcate a t / ~ simultaneously

Table 1 Data for small-amplitude solutions of eq. (1.1) through third order in perturbation theory.

Rolls Squares Hexagons

fo 2rRc°s(k~x)

/~1 0

t~ 2 ~[27K + 2(2//- y/k2c + 48) × ( / /+ ~,/k~ + 2a)]r~k~

2rs[cos(k x) + cos(koy)]

}[45~ + 2(2//- v/k~ + 4a) x ( / /+ ~,/k~ + 2a) -36( / /+ v/k ) ( / /+ - 28)]

2 r . ( cos (kox) + x ) ]

- ( f l + 7/k~ - 6)rHk~

×(/ / + ~,/k~ + 2a) - - 3a) ( / / + y/k + o ) ] rt,

454 E. Knobloch / Pattern selection in long-wavelength convection

with the rolls and hexagons. These results are required in turn for the discussion of the non-symmetric problem with 0 < 1/31, I~'l, I~1 << 1.

3. The square lattice

The theory describing steady state bifurcations on the square lattice has been described elsewhere [1, 15-20]. Consequently we summarize here the essential results required in order to establish the relative stability of the two competing patterns: rolls and squares.

On the square lattice the critical wavevectors may be chosen to be

k~ = k~k, k 2 = k¢33, (3.1)

where k, 33 denote orthonormal unit vectors in the horizontal plane. The most general solution to the linear problem Lfo = 0 is then

To analyze rigorously the local bifurcation structure described by (3.3a) singularity theory transformations are carried out to obtain a normal form [17, 19, 20]. We find below that depending on x and the parameters /3, 3, and 3 not only codimension-two but also codimension-three bifurcations can all occur. Since all the possible degeneracies through codimension three have been classified [19, 20], we rely below on the existing results to provide not only the defining conditions for each degeneracy in terms of partial derivatives of the functions p and q at the origin (X = N --- A = 0), but also to discover which Taylor coefficients of (3.3a) enter into the appropriate normal form. Thus the classification results tell one ex- actly which computations have to be carried out. In all the cases the singularity theory normal form and its universal unfolding are of finite order. Consequently, no arbitrary truncation is made.

3.1. The generic problem

The codimension-one normal form is given by

f s = ( zx ei,ox + z2ei~oy + c.c.l( zx ' z2 ) ~ C2 },

(3.2)

(r,) (rl) n(r,,r2, h )=(eoX +mN) r 2 + '13 _r2 ,

m * 0, q , (3.4)

where c.c. denotes the complex conjugate. The symmetry group of the square lattice, F s = D 4 × T 2, where the two-toms T 2 denotes translations in the k, 33 directions, imposes severe restrictions on the equations for the complex amplitudes z 1, z 2. In terms of the real variables (zl, z2)=(raeiea, r 2 ei¢2) , these equations take the form [16, 17]

q;1 = q;2 = 0, (3.3b)

where p and q are smooth real-valued functions of N=-r?+r22, A=32-=(r22--r?) 2 and X - /~ - #¢.

provided the following non-degeneracy conditions hold:

o, o,

q(0) * 0, pN(0) 4~ q(0) . (3.5)

The normal form coefficients are given in terms of the above Taylor coefficients:

~o = sgn(px(0) ) ,

q = sgn (q (0 ) ) ,

m =pu(O)/Iq(O)I. (3.6)

To compute these coefficients, we return to (3.3a) and restrict attention to rolls (r~ = rR, r 2 = 0) and


squares ( r 1 = r 2 = rs). W e have

px(0))~ + [pN(0) - q(0)] r 2 + d)(4) = 0, (3.7a)

px(0) h + 2pN(0 ) rs 2 + 0(4) = 0. (3.7b)

From table 1 we find

q(O) 1-219 p-S~) = ~K~V ~

,_-7 + 4 8 /3+ ,.-7 + 2 8 k~ k~

+18( ,+ , ) ( , )] ,.--_7 ,8+ - 2 8 ,.';77 k¢ k¢

p,,(o) ~[ ~ p~(O) = ~ k c - -

)( , ) ,.--_-~+48 /3+ ,.---_7 +28 k~ k~

/3 + ,.--_-_-.~ - 28 kc

q(O)px( 0 ) - P,v(O) 2.2127 _( "Y = ~Kol~-~ + 2/3-

,---d + 28 . k¢

) (3.8a)

(3.8b)

+ 48)

(3.8c)

In the following we study separately the two cases A and B, each for x = + 1. Although r can take either sign, in all the existing derivations of eq. (1.1) the coefficient x has been found to be positive. Since the trivial solution loses stability as )~ increases, Px(0) > 0, and hence (o = + 1.

6.0(X)

~/k~

a lk i . PN(0) = q(13 i~l

q(0) = 0

~, S / ~ \ \ / n

*--~\ ill-÷ 3 ,,,____

xR' s. ,

-6.0(0) -12.000

/1 1 .2

i - < fl 12.000

5.000 b s / R

_ _ / / /

/

qlu) = u - , ~ 5.000

-5.000

/ 3

I / pN(0 ) = 0

B 5.000

Fig. 2. The lines pN(0)=0, pN(0)=q(0) and q(0)=0 for (a) case A and (b) case B. The lines divide the parameter plane into (a) 4 and (b) 3 regions with topologically distinct bifurcation diagrams (N, X) as indicated in the insets. The stability of the rolls (R) and squares (S) is indicated by the sign of the eigenvalues: negative (stable), positive (unstable). Solid lines denote stable branches.

3.1.1. Case A: f l=8, 7:~0 When /3 = 8, y :~ 0 (self-adjoint problem with

non-Boussinesq terms), the solutions to eq. (1.1) depend in an essential way on the two parameters /3, 3' only. In fig. 2a we show that in this case the lines pN(0) = q(0), p~(0) = 0 and q(0) = 0 split the (/3, 7 / k 2) plane into seven disjoint regions. Owing to the symmetry (/3, T) ---' ( -/3, - T) when f ~ - f , there are only four distinct bifurcation

diagrams. These diagrams show r~ + r 2 versus and are also shown in fig. 2a, together with the regions they characterize. The stability of each solution, labelled R, S, is indicated by the signs of the corresponding eigenvalues s. The first eigenvalue always refers to stability with respect to amplitude perturbations (the sign of this eigenvalue is therefore given by - s g n # 2 ), while the second eigenvalue refers to perturbations not of


that form, i.e. those in the form of the competing planform. Thus supercritical rolls may be unstable to perturbations in the form of squares, as in region 1.

Observe that in the symmetric problem (/3 =~, = 0) squares bifurcate supercritically and are stable, in agreement with ref. [8]. With increasing 1/31, L~'I they may lose stability, however. For example, i f /3 = 0, but the non-Boussinesq effects become larger, the branch of squares becomes subcritical, and the squares unstable. Thus in regions 3 (and 4) there are no stable patterns near the primary bifurcation. Another possibility, which requires both non-symmetric boundary conditions and non-Boussinesq effects, is that the rolls ac- quire stability, as in regions 2. Note, however, that the rolls must be unstable with respect to perturbations on the hexagonal lattice whenever -/4: O, as shown in section 6.

3.1.2. Case B: fl4:& " / = 0 When /3 4: 8, "y = 0 (the non-self-adjoint

Boussinesq problem), the solution depends on the two parameters /3 and & In fig. 2b we show the (/3, 8) plane for this case. There are now only three distinct bifurcation diagrams, since the condition pN(O) ----- q(0) has no solution, and these are also shown in fig. 2b. In the symmetric problem (/3 = 8 = 0) squares are stable, but lose stability by becoming subcritical with increasing 1/31. Thus there are no stable patterns in regions 3. Rolls can be stable only if /3 4: 0, and the problem is also non-self-adjoint (/3 :# 6). As shown in section 6 the rolls are unstable to perturbations on the hexagonal lattice whenever the system is not self- adjoint, i.e. whenever/3 4: & As in case A, there is therefore no region in which rolls are stable with respect to perturbations on both the square and hexagonal lattices.

3.2. The codimension-two degeneracies

Along the three lines pN(O)= O, q(O)----O, and PN(O)----q(O) the above analysis breaks down and the appropriate codimension-two bifurcations

must be analyzed. Since these lines do not inter- sect when ~ = + 1 the codimension-three bifurcation with pN(O) = q ( 0 ) = 0 does not occur. However, other codimension-three bifurcations may occur along the lines pN(O)= 0, q (0)= 0 or pN(O) = q(0) when additional degeneracies are encountered. These degeneracies have been analyzed by Gohibitsky and Roberts [19] and Crawford and Knobloch [20]. In these papers singularity theory methods are used to classify all possible degeneracies through codimension three, and to obtain and analyze their universal unfoldings. The former assumes the existence of a distinguished bifurcation parameter h and describes the possible bifurcation diagrams for fixed values of the unfolding parameters (which do not include X). The latter paper assumes no distinguished bifurcation parameter, and all the unfolding parameters are treated on the same footing. The parameter space now includes X and the analysis determines the topologically distinct phase portraits characteriz- ing different regions of the parameter space. The sequence of bifurcations occurring as a particular physical parameter is varied, is addressed implic- itly and may not be uniquely specified. For the degeneracies encountered here this distinction is unimportant, and there are no essential differences between their universal unfoldings. In addition, if we choose /~ as the distinguished parameter, then all the non-degeneracy conditions associated with /~ are satisfied. Consequently we choose here to draw bifurcation diagrams with /~ as the bifurcation parameter.

The results of refs. [19, 20] show that the codimension-two singularity PN(O)= q(0) is fully determined provided that the non-degeneracy conditions

o,

PuN(O) + qN(O) * 0 (3.9a)

hold. The former condition is satisfied since the codimension-three singularity pN(0) = q(0) = 0 does not occur. The latter is equivalent to the requirement kt~ 4:0 at k~ = 0. Similarly, the singu-


larity p~(0) = 0 requires that

q(O) --/= O, PNN(O) =# O, (3.9b)

or equivalently that /~ # 0, #s 4= 0 at gs = 0. Fi- nally, the theory also shows that the singularity q(0) = 0 requires the conditions

p, , (o) , o,

pN(O) qa(O) --pa(O) qN(O) O. (3.9c)

Note that the second condition requires the computation of the quantity qa(0), and hence of seventh-order terms in the Taylor series expansion of (3.3), as shown already in ref. [15]. In the calculation that follows only the fifth-order terms are zomputed; the seventh-order terms are beyond the ~cope of the present paper.

3.3. Coefficient calculation

In order to determine the fifth-order coefficients required for the application of the classification results to eq. (1.1) we seek a solution of (2.2) in the form (2.1) with

= 22 [cos(k¢x) + ( l / a ) cos( k¢y)],

l ~ a ~ . (3.10)

Then for a = oo f is in the form of rolls, for a = 1 it is in the form of squares, and for I < a < o0 it is a cross-roll. All three patterns can be determined using modified perturbation theory as in section 2. At each order there are two solvability conditions, obtained by eliminating inhomogeneous terms in the form of cos(kcx ) and cos(key). The perturbation expansion is presumed to be asymptotic, and so requires that at each order these solvability conditions be solved for the tti corresponding to the three possible patterns. On the other hand the theory [19, 20] shows that the bifurcation is determined by a finite number of terms in the Taylor expansion of (3.3a) and that all the remaining terms in this expansion can be transformed away by appropriate near-identity coordinate changes.

Such a theory is valid on a finite neighborhood of /~ = #¢. It follows that to reconstruct the vector field (3.3a) we may combine the solvability conditions order by order without first solving them. The zeros (q, r2) -= 2(1, I /a) of the reconstructed vector field then describe all the possible steady state patterns. Thus the perturbation theory is used here as a procedure for determining the Taylor series expansion of the vector field (3.3a). To fifth order we then obtain

rl{ px(O) X + [pN(O) -- q(O)] r?

+ [pu(O) + q(O)] r~

+ [½PNN(O) +pa(O) -- qN(O)] rl 4

+ [PNN(O) -- 2pa(O)]r?r ~

+ [½PNN(O) +pa(O) + qN(O)]r;} = 0 ,

(3.11a)

r2{ px(0) h + [ PN(0) + q(0)] r 2

+ [pN(0) - q(0)] r~

+ [½PNN(O) +Pa(0) + qN(O)]r 4

+ [ pNN(0) - 2pa(0)] r~r~

+[½PNN(O) +pa(0) --qN(O)]r~} = 0 .

(3.11b)

The three non-trivial solutions are

( rR, 0) :

Px(0) X + [ PN (0) -- q(0)] r 2

+ [½PNN(O) +pa(0) -- qN(0)] r~ = 0,

(3.12)

(rs, rs): Px(0)h + 2pN(O)r 2 + 2pNN(O)r ~= 0, (3.13)

(,1, r2): q(O) + qN(O)(r 2 + r 2) = 0. (3.14)

Note that the rolls (rR, 0 ) and squares (rs, rs) bifurcate from the trivial solution (r 1, r2)= (0,0) at X --- 0. On the other hand the cross-rolls (rt, r2) bifurcate from (q, r2) = (0,0) only when q(0) = 0.


Table 2 The signs of the defining coefficients along the codimension-one surfaces pN(O) = q(0) and pu(O) = 0 for cases A and B. The degeneracy pN(O) = q(0) does not occur in case B.

Case pn(O) = q(O) pu(O) = 0

A 1 PNN ( 0 ) + pA((}) -- qN (0) < 0 PNN (0) < 0

B - PNN(O) < 0

When q(0) is small the cross-rolls appear in secondary pitchfork bifurcations off the branches of rolls and squares provided q(0)qN(O)< 0. This observation will be used again below.

We do not give the lengthy expressions for the coefficients. The essential results are summarized in table 2. Consider first case A. As I~'l increases the system first crosses the degeneracy pu(O)= 0 at which the squares become subcritical. The calculation shows that, along the line pN(O)-~ O, PNN(O) < 0 or, equivalently, tts > 0. We conclude that when the unfolding parameter ~s < 0, I~ s] << 1, there is a secondary saddle-node bifurcation in regions 3 of fig. 2a beyond which the unstable square branch gains stability. We expect the secondary bifurcation to persist elsewhere in regions 3, although it will occur at larger amplitudes. Consequently regions 3 can be associated with a finite-amplitude instability to a square pattern. When I TI increases further, another degeneracy is encountered. Here pN(O)= q(0) and the rolls become subcritical. Since along this line 1 ~PNN(O) + pa(O) -- qN(O) < 0 or, equivalently, ~t~ > 0, we conclude that in regions 4 there will be a secondary saddle-node bifurcation on the roll branch. Al- though the branch then acquires stability with respect to amplitude perturbations it will remain unstable with respect to perturbations in the form of perpendicular rolls. Since we expect the saddle-node on the square branch to persist into regions 4 we can associate these regions with the existence of two secondary saddle-node bifurcations.

In case B the degeneracy pN(O)= q(0) cannot occur, and rolls are always supercritical. When L/3[

increases for fixed 6 sufficiently small, the system first encounters the degeneracy p N ( O ) = 0 where the squares become subcritical (cf. fig. 2b). The calculation shows that along this line PNN(O)"< 0 once again and hence that /~s > 0. Consequently we associate regions 3 with the presence of a secondary saddle-node bifurcation on the square branch where the branch acquires stability, and hence with a hysteretic transition to a stable square pattern. In neither case are there codimension- three degeneracies along the lines PN(O)= q(O) and pu(O) = 0. The possible bifurcation diagrams revealed by the above analysis are summarized in fig. 3.

The q(0) = 0 degeneracy is found in both cases A and B. Abstract theory [19, 20] shows that this degeneracy is associated with the appearance of a secondary branch of cross-rolls. The stability of the branch is described by two eigenvalues. The first, proportional to pN(O), is stable since pu(O) < 0 when q (0 )= 0 in both cases. The second depends on the seventh-order term qa(0) and has not been determined. Since, however, in both cases the accompanying qN(O)> 0, we know that the cross-roll branch lies in the regions with q(0) <0 , i.e. in regions 2 in figs. 2a, 2b. The two possible bifurcation diagrams that characterize these regions are also shown in fig. 3. It follows that although the initial bifurcation in region 2 results in a pattern of rolls, these rolls will lose stability with increasing ~t, and a transition to a square pattern will take place. This transition can occur either via a stable pattern of cross-rolls, in which case it is not accompanied by hysteresis, or via an unstable cross-roll pattern, in which case it

E. Knobloch / Pattern selection in long-wacelength conuection 459

a *-~ / b ~" . . . . . . . . . . 'ib . . . . . . . . . Region 3 : - 1 < < ps < 0, Region 4:-1 < < #R < 0

u,~>0 ~,">0

/ I R - + / I R -5', / ~ _ _ s a -

Region 2: n > 0 Region 2: n < 0

4. The hexagonal lattice with reflection symmetry

In this section we begin the analysis of a steady

state bifurcation on the hexagonal lattice by con-

sidering the case fl = 3' = 8 = 0. We take the fun- damental wavevectors on the lattice to be

Then the most general solution of Lfo = 0 can be written in the form

Fig. 3. Bifurcation diagrams (N, A) near particular codimension-two degeneracies, together with their location in fig. 2. The quantity n denotes pa(0)qN(0)- pN(0)qa(0).

f # = { 21 e ikl"x + z 2 e ik2"x + z 3 e ik3"x + c.c. I

(Z 1, Z 2, Z 3) ~ C 3 ) , (4.2)

is hysteretic. In addition, recall that the small-am-

plitude rolls are unstable to perturbations on the 2 hexagonal lattice provided fl + y / k c - 8 4= O.

The results for the special case x = 0 may be

obtained by setting x = 0 in the above analysis. In this case the codimension-two surfaces shown in

figs. 2 become pairs of straight lines through the

origin. These are shown as dashed lines in figs. 2.

A w a y f rom the origin in the parameter planes the bifurcation diagrams and the transitions between

them remain well defined, and can be readily

deduced f rom those described by figs. 2. The only

exception occurs along the line 3fl + 3 , /k~ = 0 in case A, along which pN(0) - q(0) = ½puu(0) +

pa(0) - qu(0) = 0. Thus when x = 0 and fl3, < 0 the fifth-order calculation does not determine the

transition f rom region 3 to region 4 in fig. 2a.

We conclude the discussion of the square lattice

by not ing that the presence of the extra reflection

symmetry f ~ - f when fl = 3, = 8 = 0 has no effect on the bifurcation analysis. As noted else-

where [21], this is because such a reflection is equivalent to a spatial translation by half a wave-

length in both k and .13 directions. Hence no

distinct behavior arises when fl =3, = ~ = 0, and the correct branching results for this case are

obtained simply by setting fl =3, = 8 = 0 in the preceding analysis. As we shall see next, this is not so for the hexagonal lattice.

where

k 1 + k 2 + k 3 = 0 (4.3)

and x = (x, y). When fl = 7 = 8 = 0 the reflection symmetry f ~ - f enlarges the symmetry group of the problem from D 6 x T 2 to D 6 x T 2 × Z2, where

D 6 is the symmetry of the unit cell, and T 2 is

again a two-torus of translations, this time in the

directions k~ and k 3. Unlike the square lattice the

two groups D 6 X T 2, and D 6 × T 2 X Z 2 are not

isomorphic, a difference that manifests itself in distinct bifurcation diagrams.

The group D 6 x T 2 x Z 2 acts on the amplitudes

(z 1, z 2, z3) as follows [2]. The group D 6 is generated by the operat ions

c: ( z1, :2, :3) (

r120o: ( z 1, z 2, z3) (z2 , z 3, z1),

Or" (Z1, Z2, 23) (-71,-73, Z2)-

(4.4a)

(4.4b)

(4.4c)

The translations x ~ x + d give rise to the torus action

(s , t ) : ( z 1, z 2, z3)

• it ( e i s z l , e - K S + t ) z 2 , e z 3 ) , (4.4d)

where s = k x " d and t = k 3 "d. Finally, the mid-

460 E. Knobloch / Pattern selection in long-wat~elength convection

plane reflection symmetry Z 2, yields

%: ( z t , z2, z3) ~ ( - z l , - Z a , - Z 3 ) . (4.4e)

Equivariance under the above action of the group D 6 x T a x Z 2 requires that any steady state pattern (z 1, z a, z3) satisfies the equation [2]

2 1 ( l I + //113 -Jr- U?ls)

+ 2 2 ~ 3 q ( m s + u l m v + u 2 m g ) = O (4.5)

together with the two equations obtained by cyclic permutation of (z 1, z a, z3). Here the quantities l j, mj are real-valued functions of the elementary

invariants 01, 02, 03 and q2 given by

O1 = Ul + //2 4"- U3, (4.6a)

o 2 = uxu 2 + UzU 3 + u3ul, (4.6b)

O 3 = Ul/ /2U3, (4.6c)

q = zxzaz 3 + 51~2~3, (4.6d)

there are four patterns that bifurcate simultaneously at ?t = 0. Of these the branch of rolls (R) takes the form (z 1, z2, z3)= (rR, O,0 ) with r~ determined by

x = - tl,o~(O) + 13(0) ll,x(o ) r~ + (9(4), (4.8)

and the branch of hexagons takes the form

(z 1, z a, z3) = rH(1,1,1), where

311,01(0 ) + Ij(0) X = - l l ,x(0 ) r 2 + (9(4). (4.9)

Here (9(4) denotes (9(J 4, X2 2, Xa), with 2 = r R,

r H , respectively, From table 1, we deduce that when 13 = 7 =

8 = 0

)k = 2 2 9 r 2 ~ 2 (4.10) 3 x r a k c , 2t = g HKc .

where uj = zj£j ( j = 1, 2, 3), and of the bifurcation

parameter X =-- ~t -/~c- The remainder of this section is divided into

three parts. In the first we describe the generic case, and show that the non-degeneracy conditions are violated by the present problem. In the second the resulting degenerate problem is analyzed, and all possible patterns described. In the last part the stability of the patterns is determined.

4.1. The generic problem

To solve eqs. (4.5) the functions l j, mj are expanded in Taylor series about z 1 = 22= 23 =

X = 0. In ref. [2] it is shown that under the non- degeneracy hypotheses

l l ,x(0 ) ~ o,

q,ol (0) + 1 3 ( 0 ) . 0,

21~,.1(o) + l ~ ( O ) , o,

3ll,o1(0) + 13(o) * 0,

13(0) * 0, ms(0 ) 4: 0, (4.7)

On comparing (4.10) with (4.8) and (4.9) we deduce at once the important result

•3(0) = O. (4.11)

Consequently, one of the non-degeneracy conditions (4.7) is violated, and the analysis of ref. [2] does not apply.

4.2. The degeneracy 13(0 ) ~- 0

In this section we analyze the degeneracy 13(0 ) ~ 0, It should be noted that within eq. (1.1) it is not possible to change 13(0 ) away from zero. Consequently, in the analysis that follows 13(0 ) remains zero and does not become an unfolding parameter. This situation is further discussed in section 7.

To solve eqs. (4.5) we write

z, { ll, ~(o) x + 11,°,(o) ol + ~ll .... 1(o) o?

+ 11,o2(0 ) o2 + uJ3,°,(O ) 01 + u~ls(O ) }

+ms(O)~a~3q = o ( 7 ) , (4.12)


with similar expressions for the remaining two equations. In writing (4.12) we have assumed that only the linear term 1~ depends on >,. This is typical, and is the case for eq. (1.1). The following solutions are readily obtained:

(i) Suppose that u 1 > 0, u 2 = u 3 = 0. Then

q,~,(O) X + q,o,(O) o,

+ [½l , , . : , ( 0 ) + 13,o,(0 ) + 13(0)]o 2 + (9(6)

= 0. (4.13)

This solution is in the form of rolls (R), and we take (z v z 2, z3) = (2,0,0) , 2 > 0, as a representative orbit element. We assume that

l l ,x(0 ) =/t= 0, 11,o1(0) ~ 0. (4.14)

(ii) Suppose that u 1 > 0, u 2 > 0, u 3 = 0. Then

l I + ull 3 + u215 = l x + u213 + u2215 = 0

or, equivalently,

( u 1 - u2)[13,o1(0 ) +15(0 ) + d7(2)]o 1 = 0 .

(4.15)

Hence u~ = u z, provided the non-degeneracy condit ion

13,o1(0 ) + 15(0 ) :~ 0 (4.16)

holds. The solution branch has the form (~, g,0), .~ > 0, and is called the patchwork quilt (PQ).

(iii) Suppose that u I > 0, u 2 > 0, u 3 > 0. Then if we multiply (4.5) by z 2 and subtract z 1 times the corresponding equation for z 2, we obtain

- u2)(13,~,(0)o x + ( u 1 + u2)15(0) Z1Z2( Ul

e3q ms(0) + 0(4)] = 0. (4.17) Z1Z 2 :

Two similar conditions also hold. There are now four possibilities. We discuss first the more obvi- ous case

u 1 = u 2 = u 3 = r 2 (4.18)

with zg = r ei'~L Then if we multiply (4.5) by ~1 and subtract z x times the complex conjugate of (4.5), we obtain

u 2 u 3 s i n ( 2 ( b ) ( m s + u l m 7 + u 2 1 m 9 ) = O , (4.19)

where ~ = ~1 "~- ~2 "[- ~3, Hence, provided the non- degeneracy condit ion

ms (0 ) ~ 0 (4.20)

holds, we have two possibilities: ~ = n~r and • = n v / 2 , n ~ Z. In the former case we have a branch of hexagons (H) whose representative we may take to be (:~, 2, :~), £ > 0; in the latter case we have regular triangles (RT) whose representative we take to be i(-9, 9, -9), .9 > 0. The four branches, R, PQ, H and RT, bifurcate in the generic case. In the present case we have an addit ional possibility, of the form

u 2 = u 3 ~ ul, (4.21)

which exists provided

(ux + 2u2)13,o1(0) + (u 1 + u2)13(0)

- 2 u 2 e - i ~ c o s ( ~ ) ms (0 ) = 0 ( 2 ) . (4.22)

F rom the imaginary part of this equat ion it now follows that • = n~r or • = n~r/2, n ~ Z. In the former case we have

[13,o1(0) + 15(0)] ul +[2/3 ,o , (0 ) + 1 5 ( 0 ) - 2 m S ( 0 ) ] u 2 = 0 , (4.23)

and we require in addit ion to (4.16) the non- degeneracy condit ion

213,o1(0 ) + 15(0 ) - 2m5(0 ) 4: 0. (4.24)

A represen ta t ive so lu t ion takes the fo rm ( ~ , ~ / a , ~ / a ) , ~ > 0 , a4: 0,1, oo, and is called a rectangle (RA). In the latter case we have

[13,o1(0)+15(0)]u1

+ [213,~(0) + 15(0)] u 2 = 0, (4.25)

462 E. Knobloch / Pattern selection in long-wat,elength cotwection

and require the non-degeneracy condition the matrix representation

213,o~(0 ) + ls(O ) * O. (4.26)

A representat ive solution takes the form i()3, f / a , -f/a), ~ > O, and a 4= 0,1, ~ . This solution is called an imaginary rectangle (IRA). Fi- nally, it can be checked that with the above non- degeneracy conditions no solution of the form u l > u 2 > u 3 > 0 can bifurcate from the trivial state. Thus there is a maximum of six solutions that can bifurcate simultaneously, though RA and IRA exist only if (4.23) and (4.25) admit real values for a. Branching equations, analogous to (4.13) for rolls, are readily obtained for each solution type. Note that in addition to the four branches with one-dimensional fixed-point subspaces whose existence is guaranteed by general nonsense [1, 2], the two new branches have two-dimensional fixed-point subspaces.

13 0 ) (4.29) C = 0 - 1 3 "

Hence, if L - (dgL, x, z = 2(1,1/a,1/a) , takes the form

p Q

the requirement (4.28) implies that R = Q = 0. Similarly, o v has the matrix form

S 0

2f = 0 , (4.31) 1

4.3. Stability thereby forcing P and S to take the form

The stability of each solution type is determined by six eigenvalues. The computation of these eigenvalues is facilitated by the techniques employed in ref. [2]. We compute below these eigenvalues for the RA branch, and simply state the results for the remaining branches.

We begin by writing eqs. (4.5) in the form

p =

S =

A B C D C E

B) E , D

A' B' B' C' D' E ' (4.32) C' E ' D'

%, ;2, ;,)

= ( gl, g2, g3, g4, g5, g6) ' (4.27)

wtiere z j = ~ j + i-fj, j = 1,2,3. We next observe that if Y,~ is the isotropy subgroup of a solution z - (zl, z2, z3), then dgz, x commutes with all 3'

~z:

"y(dg) z,x = ( d g ) z, xT. (4.28)

For RA the isotropy subgroup is generated by {c, % } given in (4.4a), (4.4c). The operation c has

Next, we consider the effect of spatial translations:

s: 2 ( 1 , 1 / a , 1 / a ) ~ 2(eiS,e-iS/a,1/a) (4.33a)

t: 2 ( 1 , 1 / a , 1 / a ) ~ g(1,e i ' /a ,e i ' /a) . (4.33b)

The null eigenvectors corresponding to these oper- ators are obtained by differentiating (4.33) with respect to s, t and setting s, t = 0. We obtain, in real coordinates, the two eigenvectors

( 0 , 0 , 0 , 1 , -- l / a , 0 ) T, ( 0 , 0 , 0 , 0 , -- 1 ,1) T. (4.34)


Hence S must take the form

A' aA' aA' ] S = C' aC' aC' •

C' aC' aC' l (4.35)

The eigenvalues of S are therefore 0 (twice) and trS = A ' + 2aC'. This leaves the 3 × 3 block P. Here one readily finds that D - E is an eigenvalue, with the remaining two given as solutions to a quadratic equation:

s 2 - Fs + G = 0, (4.36a)

where

F = A + D + E , G = A ( D + E ) - 2 B C .

(4.36b)

In the following we shall find that F = d~(2), while G = 0(6). Hence the two eigenvalues in (4.36) are given by

s = F + 0 (4 ) , s = G / F + 0(6) . (4.37)

Consequently all six eigenvalues are real. The various elements of P and S can be deter-

mined from (4.27). Using the branching equations for RA,

l I Jr- Ull 3 q- u t l 5 -~- 2msu 2 = 0 ( 6 ) ,

l 1 + U213 + u215 + 2 m s u l u 2 = d~(6), ( 4 . 3 8 a )

(4.38b)

and evaluating the elements on the RA branch, (z 1, z 2, z3) = .~(1, l / a , l / a ) , we obtain

A = = 2 u , 11 + u16 + u?ts ) , (4.39a)

3gl B = 3x 2

= 2.~1.~2 ( 0 l 2msu2) (1 + ull, + u?l ) +

(4.39b)

0g2 C = OX 1

= 2.~13~2 ( ~} l + 2msu2) ( 1 "~- U213 U215) + '

(4.39c)

Ogz D = y-~x 2

0g3 E = ~ x 2

= 2UE~-ff~2(l I + u213 + u22l,), (4.39d)

= 2u2 ~-~-~2 (11 + u3l 3 + u215) + 4msulu 2,

(4.39e)

A, = Og4 3y I = - 2msu 2, (4.39f)

3g5 D' = = - 2msulu 2. (4.39g)

0Y2

The four non-zero eigenvalues are now given by

S l = D - E

= - 2 ( u t - u 2 ) ( u t + 2u2)

× [ 1 3 , o , ( 0 ) + 1 s ( 0 ) 1 + ~ ( 6 ) , (4.40a)

s z = F = 2o111,o~(0 ) -}- 0(4) , (4.40b)

s 3 = G / F

= 2ulu2( ut - u2)[ /5(0) + 2 m s ( O ) ] / o 1,

(4.40c)

s 4 = A' + 2D' = - 2 m s ( 0 ) 02. (4.40d)

Note, finally, that the existence condition (4.23) implies that

u2[15(0) + 2rns(O)] = 01[/3,Ol (0) + 15(0) ] .

(4.41)

Hence the eigenvalues s 1 a n d s 3 always have opposite signs, and the RA branch cannot be stable. This completes the stability analysis for the rectangles.

The analysis for the remaining branches pro- ceeds in a similar way. In particular the IRA branch also cannot be stable. We have summarized all the stability results in table 3.

Several conclusions follow at once. Depending on sgn(ll, oa(O)/lx, x(O)) all branches are either supercritical or subcritical. In addition, only one of


Table 3 The eigenvalues for the six possible primary branches. All eigenvalues are real.

Branch Eigenvalues

R

H

PQ

RT

RA

IRA

o, sgn(/1,ol(o)), - sgn(13,ol(o ) + 15(0)) (4 × )

0, 0, sgn(/1,o~(0)), -sgn(m5(0) ), sgn(~/3,o~((/) + 15(0) - ms(O)) (2 ×

0, 0, sgn(/l,o:(0)), sgn(/3,ol(0 ) + 15(0)), - sgn(2/3.o~(0 ) + 15(0)) -sgn(2/3,,,l(0 ) +/5(0) - 2m5(0))

0, 0, sgn(q,o~(0)), sgn(ms(0)), sgn( ~/3, ,~(0) +/5(0)) (2 × )

0, 0, sgn(/1,o~(0)), sgn(ms(0) ), + sgn(13,ol(0 ) + 15(0))

0, 0, sgn(ll,oa(0)), sgn(ms(0)), +_ sgn(ls(0))

H and RT can be stable, and only one of R and PQ can be stable. The results of table 3 are conveniently summarized in the (/3,o,(0),15(0)) plane. Two cases must be distinguished, depending on the sign of ms(0 ) , as shown in figs. 4. These figures are drawn for ll, x(0 ) > 0 and Ix, o,(0 ) < 0. The case 11,o1(0 ) > 0 can be obtained from these by changing the signs of 13,o1(0), 15(0) and ms(0 ) and taking - ( + ) to indicate instability (stability). Finally, changing the sign of ll, x(0) changes the direction of increasing the bifurcation parameter and hence the notion of subcriticality and super- criticality. Figs. 4 show that the (13,o1(0), /5(0)) plane is split into 12 disjoint regions by the lines

13,o1(0) + 15(0) = o,

~13,o,(0) + 15(0) = o,

213,o,(0) + 15(0 ) = o,

32/3,o1(0 ) + 15(0 ) - m5(O ) = o,

213.~,(0 ) + 15(0 ) - 2 m 5 ( 0 ) = 0. (4.42)

Along each of these lines codimension-one bifurcations occur. Thus each region is characterized by topologically distinct bifurcation diagrams. Since the relative amplitudes of the various branches depend on the inessential quantities ll, ~1o1(0) and ll, °2(0) we have refrained from draw- ing explicit bifurcation diagrams. Instead we indi-

cate in the inset tables the numbers of stable ( - ) and unstable ( + ) eigenvalues along each solution branch. The remaining eigenvalues vanish owing to translation invariance. Observe that in each region there is at least one stable solution. These solutions are explicitly indicated in the figures. Observe also that in several regions stable solutions coexist. This occurs for rolls and hexagons, and hexagons and the pa tchwork quilt in fig. 4a, and for rolls and regular triangles, and regular triangles and the pa tchwork quilt in fig. 4b. In these regions, which pat tern is realized depends on " ini t ia l" conditions. No te also that the states RA and IRA do not exist in all the regions. Indeed it is possible to have four, five or six pr imary branches, depending on the region.

This completes our general discussion of the degenerate bifurcat ion on the hexagonal lattice. Addit ional degeneracies, such as those along any of the lines (4.42) are beyond the scope of the paper.

5. Calculation of the fifth-order coefficients

In order to apply the analysis of section 4 to eq. (1.1) it is necessary to compute the coeffi-

cients It, x(0), la, ol(O), 13,o1(0), 15(0 ) and ms(0 ). We already know that sgn(ll, x(0)) = 1 and sgn(11, o1(0)) = -K . Hence for K = + 1, all the solution branches are supercritical, while for ~ = - 1 ,

E. Knobloch / Pattern selection in long-waoelength conoection 4~

x

2(R,H)

I(H)

¢ls(O) I

7 8 9 10 11 12

R -4* -4+ -4+ -4+ -4+ 5-

H 2-2~ 4- 4- 4- 4- 4-

PQ 4 - 4- 4- 3-+ 3-+ .~-2+

RT -3+ i-3+ 3-+ 3-+ -3+ -3+

,",,t x x 2-24 2-2+ 2-2+ 2-2* .~-2"

13,o,(0)

1 2 3 4 5 6

R -4+ 5- 5- 5- 5- 5-

H 4- 4- 4- .>-2+2-2+ 2-2+

PQi2-2* -3+ -3+ -3+ 2-2+ 3-+

RT 3-+ 3-+ -3+ -3+ -3+ -3+

RA X 3-+ 3-+ '3-+ 3-+

.,,, X2-2. ;-2+2-2.XX] (a)

6(R)

x / ~,

"~% 7(pol

8(H,PQ)

,15(o) 3(R,RT)

2(R,RT)

I(RT)

6(R)

O

7 8 9 10 11 12

R -4+ -4+ -4+ -4+ -4+ 5-

H -3+ -3+ 3-+ 3-+ -3+ -3+

PQ 4- 4- 4- 3-+ 3-+ -~-2+

RT 2-2'* 4- 4- 4- 4- 4-

IRRt 2-24 2-2+ 2-2+ 2-2+ Z-2* 3-+ 3-+ 3-* ~ ~ ] 3 - +

1 2 3 4 5 6

R -4+ 5- 5- 5- 5- 5-

H 3-+ 3-+ -3+ -3+ -3+ -3+

PQ ->-2+ -3+ -3+ -3+ 2-2+ 3-+ '

RT 4- 4- 4- 2-2+ 2-2* .~-2+

RA ~'~X .~-2+ 2-2+ 2-2+ X

IRA / .~3-+ 3-+ 3-+ 3-+

13,o,(0)

7(PQ)

(b) , \ \ ° I ,,, 8(RT.PQ)

Fig. 4. The (/3,o1(0), Is(0)) plane for (a) ms(0 ) > 0 and (b) ms(0 ) < 0, showing the twelve regions in which distinct bifurcation diagrams occur. The inset tables give the number of negative (stable) and positive (unstable) eigenvalues for each of the solution branches present in each region. The stable pattern(s) in each region is (are) indicated explicitly. Note the coexistence of stable patterns in regions 2, 3, 12 and 8, 9.

466 E. Knobloch / Pattern selection in long-wavelength com,ection

they are all subcritical. In this section we compute the remaining coefficients. This task is made sig- nificantly easier with a little forethought.

We recall first that the H and RT branches are given by

x +

+)~41911,alo1(0 ) + 311,02(0) + 313,~,(0)

+Is(O ) + 2ms(O)] = g)(6), (5.1a)

x +

+)~')[911,o,,,,(0 ) + 31,,,,,(0) + 313,,,,(0 )

+15(0)] -- d~(6), (5.1b)

for (z 1, z2, Z 3 ) = 2 ( 1 , 1 , 1 ) a n d ( z 1, z 2, z3) = )3(i,i, i), respectively. Hence by computing H and RT to fifth order we determine ms(0 ).

We next observe that the RA branch (z v z z, z3) = 2(1, I / a , l / a ) , exists provided

13,o (0 ) + I s ( 0 )

+(1/aZ)[213,o,(O) +15(0) - 2 m 5 ( 0 ) ] = 0 ( 2 ) , (5.2a)

while the IRA branch, (z 1, z2, z3) =f~(i , i /a , i /a) , exists provided

6,.1(o) + l , (0)

+(1/aZ)[213,o,(0) + ls(0)] = d~(2). (5.2b)

Hence if we determine all A and areA we obtain two independent relations between /3, or(0) and 15(0 ). Together with ms(0 ) this provides sufficient information to determine all three coefficients.

The calculations are divided into five parts.

/3 = -~ = 8 = 0, we obtain

bt4( I VfoI 2 )

= -~2(V fo" v f z ) + 3x([Vfol z Vfo" V f2), (5.3)

where fo denotes a solution of Lfo = 0 of the form

fo = 22 {cos( kcx ) + c o s [ ½ k c ( d y - x)]

+ cos[½kc(fJ-v + x ) ] ) , (5.4)

and fz solves

t f 2= - - ~ t 2 v 2 f 0 - [ - I£V ° ]vf012Vfo . (5.5)

The condition that (5.5) has a solution determines the value of /z 2 in table 1. We obtain

675 1 2 ^ 4 1~4 + ~ x t c c x = 0. (5.6)

5.2. Regular triangles

The solution of Lfo = 0 in the form of regular triangles is given by

fo = 2)3{sin(kcx) + sin[½k~(v/3-y - x)]

- sin[½k~(v~y + x ) ] } . (5.7)

= 9 ~2k2 To solve (5.5) for f2 we require /~2 y c, the same value as for the hexagons. Evaluating the expression in (5.3) we obtain

243 ) 2 ^ 4 bt4 + -~4 KKcy = U. (5.8)

On comparing the fourth-order terms in (5.1) we deduce that

ms(0) = 27 2 ~xk¢ll ,x(O ) . (5.9)

5,1. Hexagons

We begin by calculating ~t 4 for the hexagon branch. F rom the solvability condition (2.3) with

In particular sgn ms(0 ) = ~. Thus, when ~ = +1 , the possibilities shown in fig. 4a apply. In this case the regular triangles cannot be stable. When x = - 1, ll, a,(0) > 0, and all solutions are unstable.

E. Knobloch / Pattern selection in

5.3. Rectangles

We next seek a solution in the form of a rectangle:

fo = 22 (cos (kcx) + (1/a)cos[½k¢(1/3y- x)]

+(1/a)cos[½ke(¢r3y+ x ) ] } , (5.10)

where a 4= 0, 1, m. Here, the form (2.3) and (5.3) of the solvability conditions is not as convenient since it involves integration over the hexagonal domain. However, it is straightforward to elimi- nate the resonant terms at each order. We find, at 0(~3), the result

2 ) 2^2 ~2 = 3,, 1 + ~ k~x . ( 5 . 1 1 )

Note that when a = 1 this reduces to the correct result for the hexagons, while in the limit a --, oo, we recover the result #2 = 322k~ for rolls. In addition, the procedure a--* 0, with 2/a fixed, yields the correct expression for the patchwork quilt.

Note, however, that the result (5.11) holds for any a. This is no longer true at 0(~5), where the solvability condition determines the value of a. Indeed, the solvability conditions obtained by eliminating resonant terms of the form cos(kcx) and then of the form cos(½kcx)cos(½kcvF3y) are different. We obtain

( 2 7 9 4 5 ) 1~4+x - ~ + - - + - - k22 ' = 0 , (5.12a) 2a = 8a 4

243 / k224 9 4__5_5 + = O. (5.12b) /Z 4 + It: ~ + 8a 2 64a 4 ] c

Solving these equations for a, we find that

a = _+1, 5a 2 + 13 = O. (5.13)

The solution a = + 1 is just the hexagon branch, and this serves as a check on the computation. The a = - 1 solution is the same solution since it is related to a = + 1 by a symmetry operation. However, since a must be real, there is no non-

long-wavelength convection 467

trivial solution for a, and hence no rectangles are possible. Nevertheless, from (5.2a) we obtain the desired relation

313,,1(0 ) + 815(0 ) + 10ms(0 ) = 0. (5.14)

5.4. Imaginary rectangles

These take the form

fo = 2 f {sin( kcx ) + ( l / a ) s in[½kc(vr j -y - x) ]

- (1 /a)s in[½kc(v /3y+x)]} (5.15)

with a e O, 1, m. Proceeding as in subsection 5.3, we obtain at O(e 3) the result

2 ]k2^ 2 #z=3~¢ 1 + ~-7/ ~Y , (5.16) l

and at O(c 5)

27 9 9 ]k2^4 0 #4 + x ~ + 2a 2 8a a) cY = , (5.17a)

9 9 243 ) k~.94 = 0. (5.17b) t~4 + x g - 8a---- 7 +

These equations determine a:

a = +__1, a z = 7. (5.18)

As before the first solution corresponds to a regular triangle. The second solution shows that imaginary rectangles do exist as a primary branch. With the result (5.18), the existence condition (5.2b) yields

913,o,(0 ) + 815(0 ) = 0. (5.19)

5.5. Results

We are now in a position to determine the necessary fifth-order coefficients. From eqs. (5.14)

468 E. Knobloch / Pattern selection in Iong-wauelength convection

and (5.19) we obtain

l,,o,(0) = ~ m , ( O ) , 1 , ( 0 ) = - ~ m , ( o ) ,

(5.20)

with ms(0 ) given by (5.9). Since ll, x ( 0 ) > 0 , sgn ms(0 ) = x. Consequently, when ~ = +1 , 13, o1(0) > 0 and 15(0) < 0. In addition, one readily shows that

~l,,.,(0) + l ,(0) = ' l rns (0 ) > 0, (5.21a)

13,,,(0 ) + 15(0 ) = - ~ m s ( 0 ) < 0, (5.21b)

- ~ m s ( 0 ) < 0 , 2/3,01(0) + ls(O ) - 2ms(O) = ,3 (5.21c)

so that the system (1.1) falls in region 11 of fig. 4a. We conclude that when ~ = + 1 there are five pr imary solution branches that bifurcate supercritically at t~ =/~c, of which the hexagon branch is stable and the remaining four are unstable.

For completeness we also note that the result (5.12a) shows that for rolls (R)

27 - 2 ^ 4 , , ( 5 . 2 2 ) [..t 4 + ~ I g K c X = U.

Substi tut ing (5.6) into (5.1a), and (5.22) into the corresponding equation for rolls now leads to the results

½11,%01(0) -'i- /3 ,o i (0) "i'- 15(0 ) 27 2 = ~xkc l l , x(O), (5.23a)

96,<,,<,1(o) + 34,:~(o) + 3t,,<,1(o) + 15(0 ) + 2ms(0 ) = 675 2 ~-Jckcll ,x(0 ) . (5.238)

Since 13,,1(0 ), 15(0) and ms(0 ) are already known, we obtain

9 2 11, <,io,(O) = ~Ickcl,, x (0), ( 5.24a)

Ii, 02(0) = - ~-xkZ~ll, x(O). (5.24b)

The resulting bifurcation diagram is shown in fig. 5. This concludes our analysis of the symmetric problem on the hexagonal lattice.

PQ (3-I+) Z/////@ H (4- l

IRA (2-2+) i~¢//i/J RT (l-3+)

/ ~ / R (1-4÷)

Fig. 5. Bifurcation diagram (%, X) for the symmetric problem (/3 = ' / = 8 = 0) on the hexagonal lattice. Of the five branches the hexagons are stable.

5.6. The Lyapunov functional

In the remainder of this section we introduce and discuss the procedure employed in ref. [8] to study pat tern selection. The procedure relies on the structure of eq. (1.1) to observe that when /3 = "y = 8 = 0 and ~ = + 1, the quant i ty

V I i ] ~- (~ lv f l4 + ½Iv 2fl 2 - ½~lv f l 2 - ½0zf 2)

(5.25)

is a Lyapunov functional for (1.1). Specifically, one can show, provided certain surface terms vanish, that

d V / d t = - ( [Of/~tl 2) (5.26)

and that V is bounded from below. To show the latter we distinguish the two cases a < 0 and a > 0. When a < 0, we have

v i i ] >__ -~ ( I v / i ' ) - -~( iv / I 2) >__ ¼ ( ivf l2) 2 - ~#t(Ivfl 2)

1 2 (5.27) > - ~ .

When a > 0, we need to use an inequality of the form

( I v f i 2 ) / ( f 2) >__ K

to obtain

(5.28)

v t f ] >_ - ~ ( ~ + ~ / K ) 2 (5.29)

Consequent ly the system (1.1) evolves towards the local minima of the functional V. Indeed, the Eu le r -Lagrange equat ion for V is precisely the t ime-independent form of (1.1). Thus steady solu-


tions of (1.1) correspond to stationary points of V, local minima corresponding to stable solutions.

To apply this observation to small amplitude pattern selection, we substitute the expansions (2.1) into (5.25). The result can be simplified using the O(c 3) and 0(c ~) solvability conditions

t~(iVlol ~) = (IV/ol'), (5.30a)

~t4( IV/ol 2) + t~2(vfo" vA)

= 3 (IVfol2 Vfo • vf2) , (5.308)

and the relations

(foS4o) = ( Iv Void)- ~o(iVfol ~) - , ~ ( # )

-~0,

(loLl2) -~ ( f o L f 4 ) = 0

to obtain

(5.31a)

(5.31b)

vtsi = -

"l'- f 6 [ - ~ilf4( iVloi ~) _ 1.2(Vlo" yah

+ 0(~8). (5.32)

(21, 2'2) "~- (.~, 0), and squares, (z x, z2) = (2, 2). We obtain

rolls: I 4 2 [ 2 v \ 2 v ls l=-~, . , t~ ] +0(,'),

(5.34a)

squares: 1 4 2 { 2 ~ 2 v[:l=-~,.~lU ) +0(, ' )

(5.34b)

Since the latter value is smaller, this computation suggests that on the square lattice squares will be selected. Indeed, an explicit calculation [8] shows that rolls are unstable, although a similar one to show that (5.34b) corresponds to a minimum has not been made.

The surface terms also vanish when the integrations are carried out over a hexagonal domain, provided we restrict attention to planforms that are periodic on the hexagonal lattice it generates. In this case we find, for rolls (zl, z2, z3) = 2(1, 0, 0), and hexagons, (zl, z2, z3) = 2(1,1,1), the results

rolls: V [ f ] = - - -

(5.35a)

In deriving the result (5.32) we have likewise discarded all surface terms. To justify this procedure it is necessary to choose carefully the domain D over which the integrations implied by the angular brackets are to be carried out. Since neither f nor its gradients vanish at infinity in a homogeneously forced system, the surface terms vanish only for planforms that are doubly periodic in the plane, with D chosen to be the unit cell. Thus for the square lattice

D = { ( x , y ) 1 0 N x N 2 ~ / k c, O N y N 2 ~ / k e }

(5.33)

and the evaluation of (5.32) is restricted to rolls,

hexagons: v [ [ l = - y ~ - t ~ + 0 ( , ' )

(5.35b)

The fact that the two values are equal underscores the presence of the degeneracy studied in section 4. This degeneracy can be resolved by computing the 0(c 6) terms. Note, however, that the general theory shows that there are up to six possible solutions for which this would have to be done, and that since stable states can coexist, the second variation around each steady state would also have to be calculated to determine the stability properties. Given the general theory this is now an unnecessary task.

470 E. Knobloch / Pattern selection m long-wavelength convection

In ref, [8] Proctor points out that there is one additional case for which the surface terms vanish. This case arises when the angular brackets are taken to denote an average over the layer:

( . . . ) = lim l fL f;;(...)dxdy. (5.36)

In this instance the surface terms are (9(L -1) and vanish as L --+ o~, provided the spatial derivatives remain bounded. This procedure has the advantage that it is now possible to compare V[f] for rolls, squares and hexagons. For the former one recovers the results (5.34). For the latter we have

1 a 2[2~r) 2 hexagons: V[f] = - a E >21~- 7 +(9(c6) .

(5.37)

Consequently V[f] for squares is smaller than that for both rolls and hexagons, a conclusion that suggests that hexagons are in fact unstable to squares. This calculation does not exclude the possibility that the hexagons correspond to a sub- sidiary minimum of V[f] and hence can coexist with the squares, or indeed that the squares do not yield the global minimum of V[f].

Although the method described in the present paper cannot be used to determine the relative

stability between squares and hexagons, it has the advantage that it does not rely on the structure of the basic equation, but only on its symmetry properties. Consequently, it can be applied, as here, to problems for which no Lyapunov functional exists. On the other hand the variational approach

can be extended to larger amplitudes by seeking extrema of the fully non-linear functional V[f].

6. Hexagonal lattice without reflection symmetry

In this section we discuss pattern selection in eq. (1.1) on a hexagonal lattice when fl ~ 0, -/vs 0, 8 4= 0. In this case the symmetry of the problem is D 6 X T 2, and it forces the amplitude equations to

take the form [2, 18, 22]

(hi+ ulh 3+ U~hs)Zl

+ ( p 2 + U l P 4 + u 2 p 6 ) z 2 z 3 = O , (6.1)

with the corresponding equations for z 2 and z 3 obtained by cyclic permutation. Here the hi, pj are real-valued function of X and the invariants

o x, 0 2 , 0 3 and q. At small amplitudes eqs. (6.1) become

[hl,h(O ) ~k + hi,a1(O ) 01 -b//3(0 )//i] Zl

+p2(O) Z2Z3 + (9( Z 4, ~kZ 2) = O. (6.2)

In the generic case all the coefficients are (9(1) when /3, ~/, 8 are (9(1). There are then three solutions to (6.2). These correspond to rolls:

(zx, z 2, z3) = 2(1,0, 0), hexagons H+: (z 1, z> z3) = 2(1,1,1), 2 > 0, and hexagons H - : (z> z 2, z3) = 2(1,1,1), 2 < 0. Since (6.2) contains a quadratic term, pz(O)-4: O, all of these solutions are locally unstable [23]. This is a familiar result in non- Boussinesq Rayleigh-B4nard convection [3]. In contrast, when p z ( 0 ) = 0, i.e.

fl + 7 / k ~ - 8 = 0 (6.3)

(cf. table 1), there are four primary branches, rolls, hexagons (H +, H - ) and rectangles (RA) [2]. This bifurcation problem is described by the normal form

zl(X + a% + eu, + dot)

+ ~ 2 5 3 ( b % + u l + cq) =0 (6.4)

together with two additional equations obtained by a cyclic permutation of z t, z z and z 3. The analysis of ref. [2] shows that if e > 0 and a + e < 0 roils are stable regardless of the various higher- order terms. This is so in the present case.


R(5- ) ( 5 - / R (1+3-) RA

i / / / / / . . . . H ~ (2_ 9+) ( 4 _ ).,,,,,If, f,,,,,,~ - ~ , /

~ - ~ ~ ~1+3-) , . / i s _ ~_ . . . . . .

(a) (b)

H ÷ H e

' ~ / /

( .:2 . . . . ,-_ (c ) (d)

Fig. 6. Bifurcation diagrams (ol, ~) for the non-symmetric problem on the hexagonal lattice: (a) the self-adjoint case fl + "t/k2c - 8 = 0, and (b) the nearly self-adjoint case I/~ + y / k ~ - 81 << IBI, lYl, 181 = O(1) l The diagrams are drawn for ~ + g < 0 and apply to both case A and case B. (c,d) show additional bifurcations involving the equal amplitude solutions under the conditions ff + g < 0,

> 0, 3b + 1 > 0, and (c) ~'> 0, (d) ? < 0. Solid lines indicate branches stable with respect to equal-amplitude perturbations only. For e < 0 (c,d) are interchanged as are the labels H ÷, H - .

To show this we use table 1 and the condition (6.3) to show that in both case A and case B,

2 2 2 #~ = (3K + 4/3 )rRk ~, (6.5a) 2 2 #~ = (9x + 13/32)rnk¢. (6.5b)

Thus when x = + 1 both rolls and hexagons bifurcate supercritically (#2 > 0). From the normal form (6.4) we have, however,

X= - ( a+ e)o l + O(ot z) (6.6a)

for rolls, and

X= - ( a + ½e)o 1 + 0 (o~ ) (6.6b)

for hexagons. It follows therefore that

a + e = - (3K + 4/32)k z, (6.7a)

a + gel _- _ (3x + ~/32) k~.2 (6.7b)

1RZb'2 > < 0 (for +1) Since e = i t ~ ,-e 0, and a + e K= we conclude that in both cases A and B rolls are stable with respect to perturbations on the hexagonal lattice (see fig. 6a).

Golubitsky et al. [2] show how this result is related to the earlier result that as soon as (6.3) is violated there are no stable solutions near the

primary bifurcation. Specifically, if P2(0)=~ 0 but is small, then the bifurcation problem is described by the normal form

z,(X + ao, + +

+ go1 + .1 + aq) = 0, (6.8)

where d - a = 0(c), etc., and c 0c-P2(0). Pro- vided fl = 0(1) the results (6.7) show that the non-degeneracy conditions

a + ~ ¢ 0 , 2a+ ~#=0, 3 ~ + g:/: 0, g :~0 (6.9)

are satisfied. The additional non-degeneracy conditions 3/, + 1 =~ 0 and E=~ 0 have not been veri- fied but are not critical for the small-amplitude behavior. The bifurcation diagram that obtains in the relevant case ~ + g < 0 is shown in fig. 6b for c(3b + 1 )< 0, and shows the branching behavior expected for both case A and B. Observe that there is a hysteretic transition to H + ( H - ) as ~, increases, depending on sgn(B + 7/k~ - 8) > 0 ( < 0). The branch H + ( H - ) acquires stability at a secondary saddle-node bifurcation, but loses it again at larger amplitude when it interacts with a branch of rectangles (KA). The rolls bifurcate


supercritically but are initially unstable. They ac- quire stability at a secondary bifurcation to RA

and are the stable patterns at larger amplitude. Thus at larger amplitude the results agree with those for p2(0 )= 0, while at small amplitude they agree with those for P2(0)4= 0. Note that the RA branch is never stable, and that there is an interval

of X in which stable H + ( H - ) and R coexist. The resulting bifurcation diagram is familiar from the study of weakly non-Boussinesq convection between conducting boundaries [2, 3].

When e(3/~ + 1) > 0 an additional secondary bifurcation takes place at large amplitude. This is a bifurcation to a branch of triangles (T) and allows the H + and H - branches to exchange stability [2]. However, in the present case ( a + Y<0) the hexagons lose stability already at a smaller amplitude when the secondary bifurcation to the RA branch takes place. Thus none of the bifurcations associated with the T branch produce stable states. Consequently we do not calculate here the coefficients D and 6, and restrict ourselves to reproduc-

ing the two possible bifurcation diagrams in figs. 6c, 6d. In these figures only the equal-amplitude solutions of the form (z, z, z), z ~ C, are shown, and stability is indicated with respect to perturbations of this form only, i.e. with respect to a subset of the possible perturbations on the hexagonal lattice.

It remains to relate the preceding results to those of the symmetric problem/3 = 3' = 8 = 0. To this end we again consider the case where p2(0) is small, but now assume that all of/3, 3', 6 are small:

0 < 1/31, 13' I, 181 << 1. This case poses a delicate problem in symmetry breaking, and a complete

theory remains unavailable [2]. If we follow the analysis of ref. [2] we conclude that the simplest possible system that can describe the present situation is of the form

2 [ l l , x (0 ) ~k -t-/1,or(0) o I + l/1,otot(0 ) o?

+ 11,02(0 ) 02 + ]3,o1(0) 12101 + 15(0 )/,/? + caq]

+ ~z~3[q + c2o~ + c3u~ + m,(O) q] + 0(7) = O, (6.10)

where 0(7) denotes (~(z 7,/3z 6, 3'z 6, (~z 6 . . . . ). Here

the coefficients q , c 2, c3, and c 4 depend on the symmetry-breaking parameters 13, 7, 8 and vanish when /3 = 3' = 8 = 0. The necessity of including all four terms was demonstrated in ref. [2] for the

case h3(0, 13 = 3' = (~ = 0) ¢ 0, although it was found that the resulting bifurcation diagrams de- pended in essential ways only on the coefficients

c a and 3c2+ C 3. Note that, when /3 =3' = 8 = 0 (6.10) reduces to the symmetric case (4.5).

At small amplitude eq. (6.10) has only a roll solution given by

/,,x(O) X + 11,,:,~(0) 2 2 + (9(4) = O, (6.11a)

and the two hexagon solutions given by

+ 3t1,o,(0) 2 + 0(4) =0.

(6.11b)

Note that the rolls are unaffected by the absence of reflection symmetry, while the hexagons are not. The hexagons undergo a secondary saddle- node bifurcation when d ~ / d 2 = 0. Then

3/1,o~(0 ) .f + c x = d~(3). (6.12)

From table 1, we have that for )~ = 1 la, o~(O ) < 0, ca = (/3 + 7 / k ~ - 8)k~. Consequently, the sad- d le-node bifurcation occurs on the H + branch when / 3 + 7 / k 2 - 8 > 0 and on the H - branch

when/3 + 3'/k~ - 8 < 0, The saddle-node bifurcation stabilizes the branch. Thus for/3 + 3"/k 2 - 8 >0 (/3+3"/k 2 - 8 < 0 ) we expect a hysteretic transition from the trivial solution to H + ( H - ) .

In section 5 we had shown that when 13 =3 ' = 8 = 0 and ~ = 1, the hexagon branch is stable. This branch is either H + or H - , the two solutions being related by reflection symmetry. We expect the symmetry-breaking effects to be confined to small amplitudes, and conclude that both the H + and H - branches must have identical stability characteristics at larger amplitudes. In particular,

E. Knobloeh / Pattern selection in long-wavelength convection 473

when r - 1, both must be stable, a l though they will differ slightly in amplitude. To see how the second b ranch becomes stabilized we restrict eq. (6.10) to the equal ampli tude invariant subspace:

(z l, z 2, z3) = (z, z, z). Then

z { ll, x (0)X + 31i,o,(O)u + [~lx,o,o,(O )

+ 31,,o,(0) + 3/3,o,(0 ) + I s (0) ] u 2 + c4q}

+ ~ 2 [ q + (3c2 + C3 ) U + m s ( 0 ) q]

+ O ( V ) = 0, (6.13)

where u = Izl 2, q = z 3 + £3. One can readily show

that nei ther the zu 2 nor the zq term affect qualita-

tively the bifurcat ion diagrams. I f we set these terms to zero, the resulting system is the singularity theory normal form describing the symmet ry

breaking D 3 X Z2--~ D 3, and is completely analyzed in ref. [2]. The appropr ia te results for x = 1

a

T #

? f

are summar ized in fig. 7. The figure shows that the plane spanned by c 1 and 3c 2 + c 3 generically splits into four regions character ized by topologically distinct bifurcat ion diagrams. We see that, as hy- pothesized above, bo th H + and H - branches are stable at large ampli tudes, one gaining stabili ty at

the secondary s a d d l e - n o d e bifurcat ion and the other at a secondary bifurcat ion to an unstable triangle branch.

In the present p rob lem the coefficients Cl, c2,

c 3, and c 4 all depend on the symmet ry-break ing parameters r , y, & If we determine this depen- dence then the four possibilities depicted in fig. 7

may be translated into the (fl, 7) p lane (case A), or the (fl, 6) p lane (case B).

We sketch below the details of this calculat ion since they are not wi thout interest. We will look for solutions of the form

z = ~ + i g , I.~1 <<1,

3

, : ~ 2C~. 0

3

(6.14)

H- H*

H • ' t

H* H-

: YS I) b ¢ - 7 r -

H* -

S Fig. 7. Analysis of symmetry breaking within the equal amplitude invariant subspace for sgn(ll,o~(0) ) = - 1 and sgn(ms(0)) = 1. The

1 (cl, B) ptane (B-~ - c 2 -~c3) splits into 4 regions with topologically distinct bifurcation diagrams. The hexagon and triangle branches are labelled by H e and T, respectively. Solid lines indicate stable solutions within the equal amplitude subspace.


corresponding to triangles (I)31 > 0) and hexagons ()3 = 0). The restriction I)3[ << 1 is purely for com- putational convenience. With this assumption (6.13) becomes

ll,X(0) X()~ + i)3)

= - c12 ( 2 - 2i)3) - 311,o~(0 ) 9~2( 2 + i)3)

- 24(2c4+ 3c 2+ c3) - 223 i ) ( c 4 - 3c2- c3)

+ 0 ( 2 ' , )32, 2)32, 22)32). (6.15)

Hence the coefficients ca, 3c 2 + c 3 can be determined from the real and imaginary parts of (6.15). Explicitly, we note that the solution (6.14) corresponds to the solution

If we also identify Jk =/~1 + ~t2 + . . . . we see that (6.18) is precisely of the form of the second-order terms in (6.15). We conclude that

q v ) -8 (6.19)

Using this procedure one can construct the Taylor expansion (6.15) to any order. For the low-order terms in the normal form we do not require the details of the contact transformation eliminating the higher-order terms. We find

6,o~(0) /1,x(0 ) = -3~¢k~ + 0(2) (6.20)

)Co = 2 2 [ c o s ( k ~ x ) + 2cos(½k~x)cos(½k~v~y)]

- 2 )3 [sin( kcx ) - 2 sin( ½k~x )cos( ½kcv~-y )]

(6.16)

of t fo = 0. We now carry out a perturbation calculation starting with (6.16), and keeping only terms independent of )3 and linear in ) . In addition we only work to leading order in (/3, -/, 8).

There are two solvability conditions for the next-order equation Lfl = 0, given by

btl ) ~ + k /[~ + ~-2c2 -- 8 3~2 ----- 0 ,

/~xP- 2k~(fl + Y ) ,-5-8 22=0. k¢

(6.17a)

(6.17b)

Of course these equations are incompatible unless )3 = 0. This comes as no surprise since we know that there are no T solutions branching from the trivial solution. However, our purpose is to derive an equation of the form (6.15). To do this we combine (6.17) into the form

/ ~ l ( : ~ + i ) 3 ) = - k 2 /3+~--~2-8 ~ ( 2 - 2 i ) 3 ) .

(6.18)

and

3c 2 + c 3 - - = 27 - - 2 [ ~ q_ "Y ) +o(3),

(6.21a)

c4 3. , . 2 [ v 11,~(0) = - ~ c 7-~ J Ikc-38 +0(3) (6.21b)

Here O(n) denotes homogeneous terms of order n in (/3, ,/, 8).

The analysis summarized in fig. 7 shows that the bifurcation diagrams depend fundamentally on the parameters c 1 and 3c 2 + c 3 but not on c 4. The (c 1, 3c 2 + c3) plane splits into four regions characterized by distinct bifurcation diagrams. Two of these regions are large in the sense that their boundaries subtend a finite angle at the origin in the (c 1, 3c 2 + c3) plane, and hence the corresponding bifurcations are more "likely" to be realized. In contrast, regions 2 and 4 subtend a vanishingly small angle at the origin (the two boundaries are tangent there) and the corresponding bifurcation diagrams are "unlikely". Note, however, that they must occur during a passage from region 1 to region 3 or vice versa.

In addition to the secondary bifurcations just discussed, other bifurcations involving the four


other patterns present in the symmetric problem must be expected to occur, but they cannot alter the ultimate stability of the larger amplitude H ÷, H - branches.

Note finally that we may identify (6.21a) as a leading-order contribution to the quantity 3/~ + 1 in (6.8). Consequently with (f l + y / k 2 + 8)( f l +

y / k 2 - 8) > 0 with I t + y / k 2 - 81 << 1/31, Ivl, 181 << 1, the bifurcation diagram 6b applies and the triangle branch is absent. The secondary bifurcations involving the triangle branch occur only when ( f l + r / k 2 + 8)(f l + y / k 2 - 8) < O.

7. Discussion and summary

In this paper we have studied pattern selection in systems described by eq. (1.1). This equation arises in a very natural way when the first instability of a translationally invariant state has a very long wavelength, and consequently finds a number of applications [4-14]. We have focused attention on doubly periodic patterns, since such patterns are commonly observed in pattern-forming systems. To formulate a tractable problem we studied eq. (1.1) on square and hexagonal lattices. A third possible doubly periodic lattice, the rhombic lattice, was not considered. On the square lattice an existing and rigorous theory enabled us to conclude that, depending on the parameters/3, y, and 8, both rolls and squares can be stable. These parameters represent asymmetry in the boundary conditions (/3, 8 #: 0), non-Boussinesq effects (y :~ 0) and the effects of non-self-adjointness of the problem from which (1.1) is derived (fl :~ 8). Consequently these parameters can, at least in principle, be varied independently. We considered two cases in detail, a self-adjoint problem with non-Boussinesq terms (case A) and a Boussinesq non-self-adjoint problem (case B). In all cases squares are stable in the symmetric problem (/3 = y = 8 = 0); stable rolls require non-zero fl and y (case A), or non-zero fl and /3 - 8 (case B). For K = + 1 the analysis revealed the existence of several codimension-two singularities the study of

which revealed a variety of secondary saddle-node bifurcations. In all cases considered these bifurcations stabilized an otherwise unstable pattern with respect to amplitude perturbations. Additional possibilities exist when x = - 1 .

On the hexagonal lattice we were forced to distinguish between the cases fl = y = 8 = 0 and fl 4: 0, "t 4: 0, 8 4: 0. In the former the additional reflection symmetry f ~ - f exerts a profound in- fluence on the bifurcation behavior. We found, in section 4, that here the initial pattern-forming bifurcation is degenerate, with an essential cubic term vanishing identically. We were able to give a complete discussion of the generic behavior with this degeneracy. In particular we found that up to six primary branches can now bifurcate simultaneously from the trivial state, and were able to establish their stability properties. We found that when the branches bifurcate supercritically, at least one and at most two branches are stable. The new branches, called rectangles and imaginary rectangles, cannot be stable, however. On the other hand, in contrast to the non-degenerate problem, the solution called a patchwork quilt could now be stable. Our results are summarized in figs. 4. Ap- plying this general theory to the specific problem (1.1) we found that the coefficients are such that the hexagon branch is supercritical and stable, with the remaining branches all unstable. The appearance of stable hexagons here is unusual, and suggests that it may be possible to observe such states in physical systems. When the symmetry is broken (fl 4: 0, "y 4= 0, 8 4= 0) there are no stable branches near the first bifurcation, unless /3+ 2 y / k ¢ - 8 = 0. We showed that in this special case of the four primary branches the rolls are stable. To relate this result to the symmetric problem we considered two additional situations: I t +

Y~ k 2 - 81 << 1/31, I~'1,181 = 0(1) and 1/31, I~'l, 181 << 1. In the former we found that a secondary saddle-node bifurcation stabilizes H + ( H - ) depending on the sgn(/3 + y / k 2 - 8 ) , but that the hexagons lose stability again at larger amplitude where a hysteretic transition to stable rolls takes place. This result is familiar from the theory of

476 E. Knobloch / Pattern selection in long-wavelength colwection

weakly non-Boussinesq convection between conducting boundaries i.e., for convection in which the instability wavelength is of order of the layer thickness. In contrast, in the latter case we found that both hexagon branches are stable at larger amplitudes: the hysteretic transition to rolls no longer takes place, and the second hexagon branch gains stability through a bifurcation to a triangle branch. We have been unable to map out the intermediate regime in which the rectangle branch moves off to infinity and the triangle branch enters the picture.

We have argued that at the present time techniques are not available that would establish the relative stability between hexagons and squares. Consequently experiments, both numerical and otherwise, have an important role to play in set- tling the question of whether hexagons or squares are selected in these systems. In fact simulations with (1.1) should be considerably simpler, being two-dimensional, than the corresponding problem in three-dimensional convection. In addition, such simulations could tackle the question of non-periodic patterns and the nature of the defects admit- ted by eq. (1.1), both important issues that are beyond the scope of the present paper.

As an immediate application of the results pre- sented here we note from the appendix that the Marangoni problem is described by eq. (1.1) with

7 v (7.1) /3 ~- 9 6 ' ~/ = 0 , ~ = 16"

It follows from fig. 2b that squares are stable on the square lattice; on the hexagonal lattice no pattern is stable near the primary bifurcation but we expect a secondary bifurcation to stabilize H ÷ and consequently a hysteretic transition to hexagons in which the flow rises in the center and descends along the periphery.

The presence of the degeneracy in the symmetric problem on the hexagonal lattice raises important issues. It is commonly assumed that such degeneracies which cannot be removed by varying

the system parameters render the basic equation suspect as a model of the physical world [24]. And yet such systems are encountered not infrequently. A case in point is provided by the Hopf bifurcation in binary fluid mixtures heated from below [25]. In the present case it might be argued that the reflection symmetry can never be exact, and indeed /3 ~ 0, 7 4: 0, 8 ~ 0 appears to largely re- solve the degeneracy. But even within the symmetric problem there are 0(42 ) terms, where the original wavenumber is 0 ( ( ) , that have been omit- ted from (1.1). In an asymptotic derivation of (1.1) to drop such terms is in fact the only consistent mathematical procedure, since the procedure guar- antees a good approximation only in the limit k $0. More important is the possibility that the equations from which (1.1) is derived are them- selves inconsistent when k = 0(4). For example, the Boussinesq approximation used in convection theory, and hence implicit in (1.1) in many of the applications cited in section 1, breaks down in the limit k ~0 unless additional small parameters are appropriately tuned (cf. [26]). But if the Boussinesq approximation is itself viewed as the first term in an asymptotic expansion in these parameters of the full fluid equations, then no difficulties with the derivation of (1.1) are encountered. In this case the smallness of both types of correction terms makes (1.1) the appropriate equation to study since the effect of any new and degeneracy-breaking terms will be confined to asymptotically small amplitudes. In physical applications, however, neither asymptotic limit can be taken, and whether (1.1) applies to a particular situation has to be determined by the physical parameters of the problem. Thus the effect of higher-order corrections to (1.1) is not without interest.

Eq. (1.1) is incomplete, moreover, when the basic state is not only translationally invariant, but also invariant under Galilean transformations. This occurs, for example, in a convection layer between stress-free boundaries at top and bottom. Under these conditions the requirement of Galilean invariance couples eq. (1.1) to a velocity


field u [27]:

Of 3--[ + u . v f = a f - I . t v 2 f - v a f + r, V • Ivf l 2 v f

+fix7 " v 2 f v f - v v " f r y

+ ~V 2lvfl 2, (7.2a)

0u O--i- + u . xTu = g [ f , v f . . . . ]. (7.2b)

Appendix. Derivation of eq. (1.1) for Marangoni convection

In this appendix we show how to derive eq. (1.1) for a specific system. The system we choose had been studied before [10] in one dimension, but the generalization to two dimensions was carried out incorrectly. We consider Marangoni convection described in the steady state by the dimen- sionless equations

These equations are equivariant under translations

x - o x + d , f ~ f , g ~ g (7.3a)

and the Galilean transformation,

x - - * x + v t , f ~ f , u - - * u - v . (7.3b)

In eq. (7.2a) it is helpful to think of f = 0(1), f l = 0 ( 1 ) , V=O(~2), ~ = 0 ( 1 ) , r = 0 ( ~ ) , /.t ~--- 0 ( ~ 2 ) , O/Ot = 0 ( ~ 4 ) , and a = tV(~4). Conse-

quently for the coupling to be consistent with the asymptotic perturbation theory, u = 0(~3). Thus g = 0(~7). In addition we expect [27] that (7.2) has time-independent solutions of the form ( f , u) = ( f , 0), imposing an additional constraint on the vector field g. Such coupling reveals the possibility of secondary instabilities to drifting modes (cf. ref. [28]), although in the present case the drifts would require seventh-order perturbation theory for their discovery and description.

Acknowledgements

The author is grateful to Professor Y. Kuramoto for his kind hospitality in Kyoto, where most of this work was done and to the Japan Society for the Promotion of Science for financial support. Very helpful correspondence with Dr. M.R.E. Proctor is also acknowledged.

V4w = 0, (A. la)

xT~u = -Wxz, (A. lb)

V 2 o = -Wyz, (A. lc)

V 20 Jc" W = U" V O , (A. ld)

V ' u = 0, (A. le)

where u -= (u, v, w) denotes the velocity field relative to (x, y, z) coordinates, and 0 the tempera- ture departure from the conduction solution. The quantity x7 2 denotes 32 + 32. Ignoring for the moment the effects of finite Blot number, the boundary conditions are

u = o = w = 0 z = 0 on z = 0, (A.2a)

W = W z z - M a v 2 0 = O z=O o n z = l , (A.Zb)

and correspond to surface tension driven convection in a thermally insulated layer with a rigid no-slip lower boundary (z = 0) and a free upper surface (z = 1). Here Ma is the Marangoni number.

With the boundary conditions (A.2) the dimen- sionless wavelength of the initial instability is long, of order c-1, say, where 0 < c << 1. Consequently, we introduce the slow scales

X = ~x, Y = ~y, (A.3)

478 I£. Knobloch / Pattern selection in long-wavelength convection

and let

( u, v, w,O) = ( {U,{V, ezw, o ) , (A.4)

where U = U(X, Y), etc. Eqs. (A.1) become

D 4 W + 2{2 D2vI~W+ { 4 v ~ W = 0, (A.5a)

v ~U = - Wx~, (A.5b)

VnZV = - Wry, (A.5c)

D 2 0 + { 2 V 2 0 + {2W

= e2UOx + {2VO v + {2W DO, (A.5d)

Ux+ Vr+ D W = 0 , (A.5e)

with D - 3/3z. We define e by setting

Ma -= Mac(1 + {2). (A.6)

Using (A.le) the boundary conditions then become

W = D W = D O = 0 o n z = 0 (A.7a)

W = D 2 W - MAC(1 + {2) Vn20 = DO = 0

on z = 1. (A.7b)

Proceeding as in ref. [10], we expand

the solvability condition:

M a c = 48. (A.11)

We now obtain

@2 v 2 f ( - 1 2 Z4 = + -

+ Ivnf [2(4z 3 - 3 z 4 ) + f z ( X , Y ) , (a .12a)

U 2 = V 2 f x ( 1 5 2 z 2 - 8z 3 + 6z 4)

+ 12{ IvHfJ 2 + f + f 2 } x( 2 z - 3z2),

(a .12b)

V 2 = v z f v ( ~ z 2 - 8z 3 + 6z 4)

+ 12( IVn/ [ 2 + f + f 2 } v( 2 z - 3z2),

(A.12c)

W2 = v 4 f ( 4 3 _ - 3z + 2z 4 ~Z 5)

- 12(vZIVHfl 2 + v ~ f + v z f 2 ) ( z 2 - z3).

(A.12d)

Finally, at (.0(£ 4) the solvability condition for D204 yields the required equation for the planform function f (X, Y):

(A.13)

v ~ f + ~ v ~ f - 48 2 VH ° IVHfl V H f

+ l v I I . V H 2 f V H f + 3 2 2 VHIvHfl =0"

@ = 69o(X, Y) +,202(X, Y) + . . .

and similarly for the other obtain

O o = / ( X, Y), (A.9a)

~Ma c U o= ' f x (2Z 3z2), (A.9b)

Vo-= ¼Mac f v ( 2 z - 3 i f ) , (A.9c)

~M a ~vH f ( z3). (A.9d) W0 1 2 z 2 -

At 0({2), we obtain

D202 = - v ~ f + x 2 z 2 zMacVHf( --z 3)

+ ~Ma c IvHfl2(2z -- 3z2). (m.10)

(A.8)

fields. At (-o(c °) we

The critical Marangoni number is obtained from

In one dimension this equation reduces to

13 ! P! t f " + ~,f . . . . . ~ ( f ' 3 ) ' + ~6( f f ) = 0 , (A.141

as derived in ref. [10]. When the effects of finite Biot number at z = 0

are taken into account a term a f is added to the left-hand side of eq. (A.13) [10]. This completes the derivation of eq. (1.1) for this problem. Note that f l ¢ 6, a fact that can be traced to the non- self-adjointness of the boundary conditions (A.2). In contrast the planform equation for infinite Prandtl number Rayleigh-B6nard convection between nearly thermally insulating boundaries one of which is free-slip and the other no-slip also takes the form (1.1) but with fl = 6, a consequence of the self-adjointness of the boundary conditions [291.


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