Influence of non-boussinesq effects on patterns in salt-finger convection

Transcription

1 Z. angew. Math. Phys. 49 (1998) /98/ $ /0 c 1998 Birkhäuser Verlag, Basel Zeitschrift für angewandte Mathematik und Physik ZAMP Influence of non-boussinesq effects on patterns in salt-finger convection Yuriko Yamamuro Renardy and Michael Renardy Abstract. The influence of non-boussinesq effects on the quadratic coefficient of the amplitude equations for patterns that are doubly periodic with respect to the hexagonal lattice is discussed. The vertical length scale is assumed to be long compared with the horizontal. Within this approximation, we consider a variety of non-boussinesq effects including the dependence of the fluid properties on temperature and salinity and the nonlinearity of the basic temperature and salinity profiles. We include some numerical calculations for the latter case. Oceanic data show a pronounced nonlinearity of the basic temperature profile. Our calculations, however, show this effect to be small, and, moreover, the dominant contribution in this case is not covered by our asymptotic thin finger analysis. The reasons for this are discussed. A nonlinear salinity profile, on the other hand, produces a much stronger effect, and our analysis seems to be applicable to this case. Mathematics Subject Classification (1991). 76E15, 58E09. Keywords. Convection, pattern formation. 1. Introduction Salt fingers are long narrow convecting cells that are set up when warm salty water lies above fresh water (Schmitt 1994a, and references therein). These structures are increasingly recognized as an important mechanism for oceanic mixing and salt transport (Williams 1975, Magnell 1976, Lambert and Sturges 1977). Recent observations by Osborn (1991) suggest that the finger instability is also important in near-surface waters. Of particular interest in his data was the finding of a distinct asymmetry in the fingers, which seemed to be best characterized as narrow, downward plumes surrounded by a broader, weaker upwelling. Schmitt (1994b) has Research supported by NSF Grants CTS , DMS and ONR Grant N J The authors are grateful to R. W. Schmitt (Woods Hole) for discussions. Part of this work was done while the authors were at the Isaac Newton Institute for Mathematical Sciences (Cambridge).

2 Vol. 49 (1998) Influence of non-boussinesq effects 225 found that the scales of Osborn s plumes are consistent with asymmetric solutions to the unbounded salt finger problem, but can provide no selection mechanism for the asymmetry. The paper of Renardy and Schmitt (1996) examines the mechanism of surface warming in terms of linearized stability, and the present paper shows the weakly nonlinear asymmetric planforms that are selected. It is important to attempt to understand such phenomena, because the near surface plumes must play a role in the structure of the diurnal warm layer observed under light wind conditions. The diurnal warm layer is often more than 0.5 o C and occasionally as high as 3.5 o C warmer than the water below (Stramma et al 1986). This temperature gradient has direct consequences for near-surface sound propagation, the computation of air-sea fluxes (which usually use an incorrect, cooler, bulk temperature), and the dynamical response of the ocean to the wind stress. As shown by Price et al (1986), the near-surface stratification set up by diurnal heating can trap the momentum imparted by the wind stress to a shallow layer which they term the diurnal jet. In this work, we are attempting to understand the most novel aspect of the observations, the asymmetric planform of the plumes. The question of planform selection is typically addressed by beginning with a linear stability analysis of a base state which satisfies the governing equations, followed by analysis of solutions to the nonlinear amplitude equations close to onsets of instability. A number of works have begun with a base state in which the solute concentration and the temperature are linear functions of depth (Baines and Gill 1969). Proctor and Holyer (1986) show that rolls are preferred over square cells for conditions modeling a salt-finger regime. This is the same as in the classical Bénard problem (Schlüter et al. 1965, Golubitsky et al. 1984, Renardy and Renardy 1992). Nagata and Thomas (1984) investigate the stability of rolls, rectangles, squares and hexagons, to perturbations restricted to have the same structure as each solution. Rather than rolls (sheets) or squares, the salt fountains observed by Osborn and modeled here are better interpreted as hexagonal patterns (Schmitt 1994b). The motivation for developing models which do not have midplane symmetry is that a priori it would have a chance to generate planforms that are hexagonal, rather than rolls (Busse 1989, Golubitsky, Swift and Knobloch 1984; Buzano and Golubitsky 1983). In this paper, we focus on mechanisms which break the midplane symmetry. Surface warming by solar radiation causes the base temperature field to depend non-uniformly on the depth. Surface evaporation can lead to a nonlinear salinity profile. Moreover, fluid properties like density and viscosity depend on temperature and salinity in a nonlinear fashion. An observed non-uniform temperature profile is shown in Fig. 1. Osborn describes his temperature measurements as comprised roughly of three regions. The top surface region of about 1m deep has a strong gradient followed by a layer of weakly stabilizing temperature stratification of 3 m deep, followed by a layer with a more marked decreasing temperature. The salt fingers are observed in the middle layer. In order to focus on the effect of this temperature profile, the salinity profile is taken as linear in depth. The salinity gradient is destabilizing,

3 226 Y. Y. Renardy and M. Renardy ZAMP while the temperature gradient is stabilizing. The competing mechanisms have the potential for setting up an instability in the middle layer, while stabilizing the top and bottom layers. The numerical results at the oceanic conditions show that the largest growth rate modes consist of the double diffusive modes in the middle layer, with little action in the top and bottom layers. These replicate the vertical structure of plumes that begin at the bottom of the top layer, extend through the middle layer, and disappear into the top of the bottom layer. In this paper, we shall consider the weakly nonlinear analysis of bifurcation from the base solution. Rather disappointingly, we find the effect of the nonlinear temperature profile to be small. We therefore believe that the pattern selection is dominated by other effects. In the limit of a thin finger analysis, which presumes that horizontal length scales are short relative to vertical ones, we can derive analytical results for the quadratic coefficient which determines whether upward or downward fingers are favored. We shall discuss a number of effects. A numerical example illustrates the effect of a nonlinear salinity profile. We begin with a review of the bifurcation analysis for problems with hexagonal symmetry. In Section 2, we discuss the reduction of the system to a set of amplitude equations for the dominant modes close to onset, using the center manifold reduction scheme. Bifurcating solutions and their stability are discussed in Section Amplitude equations We consider a generic system of nonlinear partial differential equations of the form LΦ :=AΦ+B dφ dt = N 2 (Φ, Φ), where A and B are linear operators, and N 2 represents a quadratic nonlinear term. The components of the solution vector Φ are functions of spatial coordinates (x, y, z), and we assume that the system is invariant under translations, rotations and reflections in the (x, y)-plane. We investigate solutions Φ that are doubly periodic with respect to the vectors x 1, x 2 where x 1 = W ( 3 2, 1 2, 0), x 2 = W (0, 1, 0). The lattice obtained from this double periodicity is invariant under the symmetries of the hexagon; that is, rotation by multiples of 60 degrees, reflection across the vectors a i defined by a 1 = 4π W 3 (1, 0, 0), a 2 = 4π W 3 ( 1 2, 3 2, 0), a 3 = a 1 a 2, (2.1) and reflection across axes perpendicular to the a i. We then expand Φ in a Fourier series with modes proportional to e ika 1 x+ila 2 x. In the linearized stability problem, the modes are proportional to exp(σt). This leads to an eigenvalue problem for σ, in which the results depend on the magnitude of the vector ka 1 +la 2, denoted α. We perform a bifurcation analysis for the case where there is a sixfold degeneracy of the corresponding eigenvalue σ = 0 at criticality, corresponding to the six choices of k and l for which α = a 1 =4π/(W 3), and all other modes are stable. The six eigenfunctions are denoted ζ k,k =1,2,..., 6, and are visualized as waves

4 Vol. 49 (1998) Influence of non-boussinesq effects 227 propagating towards the vertices of a hexagon, as in Figure 2. The eigenfunction from the 2D linearized analysis (no y-dependence) is denoted ζ 1 = ζ(z) exp(ia 1 x), where a 1 =(α, 0, 0), x =(x, y, z). The other critical eigenmodes are then generated by rotations, i.e. ζ 2 has an (x, y)-dependence proportional to exp(ia 2 x) and ζ 3 has an (x, y)-dependence proportional to exp(ia 3 x). Moreover, ζ 4, ζ 5 and ζ 6 are the complex conjugates of ζ 1, ζ 2 and ζ 3, respectively. We invoke the Center Manifold Theorem which says that in the neighborhood of criticality, the dynamics is governed by interactions among the six critical modes. We form appropriate inner products and projections, and calculate the quadratic interaction terms, the contributions to the mean component, and represent the total solution accurate to second order. The parameter λ denotes the bifurcation parameter, the difference between the solutal Rayleigh number and its critical value, R S R SC.Forλnear 0, the eigenvalue problem L( µ)φ := AΦ µbφ =0 has a real eigenvalue µ(λ), which vanishes at criticality λ = 0, i.e. µ(0) = 0. Instability of the basic solution occurs for λ>0, i.e. µ (0) < 0 (note that the linear time-dependent problem has solutions proportional to exp( µ(λ)t). We denote by ζ k (λ), k =1,2,3, and its complex conjugates ζ k (λ), the six eigenfunctions belonging to the real-valued µ(λ), i.e., L( µ(λ))ζ k (λ) =0, k =1,.., 6. The adjoint eigenfunctions b k satisfy (b k,l( µ(λ))φ) = 0, k =1,..., 6, for every Φ, and have the same dependence on x and y as the respective eigenfunctions ζ k.all the b k are generated from b 1 by appropriate rotations. We use the normalization condition (b 1,Bζ 1 )=1. Any real-valued function Φ can be decomposed in the form Φ=Φ 1 +Ψ, Φ 1 = 3 z i ζ i + i=1 3 z i ζi, (2.2) where z i are complex time-dependent amplitude functions and Ψ represents a linear combination of eigenvectors (and possibly generalized eigenvectors) belonging to stable eigenvalues. The center manifold reduction method yields (cf. Chapter 3 of Joseph and Renardy, 1993) the following amplitude equation i=1 dz i dt + µ(λ)z i =(b i,n 2 (Φ, Φ)), i =1,..., 6, (2.3) (A + B d ( 3 dt )Ψ = N 2(Φ, Φ) 2Re (b i,n 2 (Φ, Φ))Bζ i ). (2.4) i=1

5 228 Y. Y. Renardy and M. Renardy ZAMP Quadratic interaction terms The quadratic terms in the asymptotic approximation to the center manifold for small solutions are required. Thus, Ψ=Ψ 2 + higher order terms, ( 3 Ψ 2 =2Re z i z j ψ ij + z i z j χ ij ), (2.5) i,j=1 where ψ ij = ψ ji,χ ij = χ ji. The expansion Φ = Φ 1 +Ψ is inserted into (2.3) - (2.4). By comparing quadratic terms, (A 2µ(λ)B)ψ ij = N 2 (ζ i,ζ j ) (A 2µ(λ)B)χ ij = N 2 (ζ i, ζ j ) 3 (b k,n 2 (ζ i,ζ j ))Bζ k k=1 3 (b k,n 2 (ζ i, ζ j ))Bζ k 3 ( b k,n 2 (ζ i,ζ j ))B ζ k, k=1 k=1 Many of these inner products vanish. For example, in the equation for ψ 11, N 2 (ζ 1,ζ 1 ) is proportional to exp(2iαx) and none of the b k have this x, y-dependence. Moreover, we only need to evaluate the ψ ij and χ ij at λ =0,whereµ=0,soψ 11 is calculated from Aψ 11 = N 2 (ζ 1,ζ 1 ). Amplitude equations k=1 3 ( b k,n 2 (ζ i, ζ j ))B ζ k. The expansion Φ = Φ 1 +Ψ is inserted into (2.3) - (2.4) and yields to third order dz i dt + µ(λ)z i =(b i,n 2 (Φ 1, Φ 1 )) + 2(b i,n 2 (Φ 1, Ψ 2 )). We begin by focussing on the equation for z 1, and pick out those terms in N 2 (Φ 1, Φ 1 )+2N 2 (Φ 1, Ψ 2 ) which have the same (x, y)-dependence as b 1 = b(z) exp(iαx): dz 1 dt = µ(λ)z 1 α 1 (λ) z 2 z 3 α 2 (λ) z 1 2 z 1 α 3 (λ)( z z 3 2 )z 1, (2.6) the Landau coefficients are α 1 (λ) = 2(b 1,N 2 ( ζ 2, ζ 3 )), α 2 (λ) = 2(b 1,N 2 ( ζ 1,ψ 11 )) 4(b 1,N 2 (ζ 1,χ 11 )), α 3 (λ) = 4(b 1,N 2 (ζ 1,χ 22 )) 4(b 1,N 2 (ζ 2,χ 12 )) 4(b 1,N 2 ( ζ 2,ψ 12 )). The other amplitude equations are dz 2 dt = µ(λ)z 2 α 1 (λ) z 1 z 3 α 2 (λ) z 2 2 z 2 α 3 (λ)( z z 3 2 )z 2,

6 Vol. 49 (1998) Influence of non-boussinesq effects 229 dz 3 dt = µ(λ)z 3 α 1 (λ) z 1 z 2 α 2 (λ) z 3 2 z 3 α 3 (λ)( z z 2 2 )z 3. We use the notation dz i dt = g i(z 1,z 2,z 3,λ), i =1,2,3. (2.7) To obtain these equations, we have used the invariance under reflection across the x-axis (ζ 2 ζ 3 ), namely g 1 (z 1,z 2,z 3,λ)=g 1 (z 1,z 3,z 2,λ). The g i are related by the symmetries of the hexagon (invariance after rotation by 2π/3, by 4π/3, by π, by5π/3andbyπ/3 radians, respectively): g 2 (z 1,z 2,z 3,λ)=g 1 (z 2,z 3,z 1,λ), g 3 (z 1,z 2,z 3,λ)=g 1 (z 3,z 1,z 2,λ), ḡ 1 (z 1,z 2,z 3,λ)=g 1 ( z 1, z 2, z 3,λ), ḡ 2 (z 1,z 2,z 3,λ) = g 1 ( z 2, z 3, z 1,λ), ḡ 3 (z 1,z 2,z 3,λ)=g 1 ( z 3, z 1, z 2,λ). The original equations governing the problem are invariant under any rotation in the x y plane, translations in the x y plane, and reflections across the axes: x x,y y.onthat,we have imposed double periodicity which restricts the rotations allowed to those in multiples of 60 deg. 3. Analysis of the amplitude evolution equations Buzano and Golubitsky (1983) consider a system of the form dz dt = g, (3.1) where z IR 6, z =(z 1,z 2,z 3 ), g =(g 1,g 2,g 3 ), and use group theory to classify new solutions and determine their stability properties by considering their symmetries. They include quintic terms which we do not. Our system is a subcase of their general system. Their z i,i=1,2,3 correspond to our z 1, z 3, and z 2, respectively. Their H 1, K 1, h i,k i in their (4.1)-(4.2) are: H 1 = µ α 2 z 1 2 α 3 ( z z 3 2 ), K 1 = α 1 = k 2, (3.2) h 1 = µ α 3 ( z z z 3 2 ), h 3 = α 2 +α 3, h 5 = k 4 = k 6 =0. (3.3) Our z 1,z 2,z 3 are denoted as x 1 + iy 1,x 3 +iy 3 and x 2 iy 2 in their paper. In equilibrium, we have g =0ordz i /dt = 0 in (3.1). For steady solutions of the amplitude equations, we need to consider only the case of z 1 and z 2 real, since the general case can be reduced to this by a translation in the (x, y)-plane. Buzano and Golubitsky (1983) list the possible classes of equilibrium solutions with y 1 = y 2 =0,y 3 0 in their Theorem 4.4. The solutions with one degree of freedom (maximal symmetry) are rolls and hexagons. Not all of their solutions are applicable to our case. In our case, K 1 0andk 2 0,so that we need to examine their solutions II (rolls), III (hexagons), IV (wavy rolls,

7 230 Y. Y. Renardy and M. Renardy ZAMP transition, false hexagons; these reduce to the patchwork quilt in the case of updown symmetry) and VII (z i real and different). In Appendix B below, we prove that generically type VII solutions do not exist for the cubic case. The stability conditions for solutions of types II, III and IV with respect to perturbations with the double periodicity are given below and have been checked against those of Buzano and Golubitsky (1983). Figure 3 shows the contour plots for a perturbation quantity; e.g.,the vertical component of the velocity for the rolls, hexagons and type IV solutions. a. Rolls. For rolls, we choose z 1 =0,z 2 =0,z 3 0. This leads to z 3 2 = µ α 2. (3.4) For this solution to exist, we need µ/α 2 > 0. We examine the stability of this solution by looking at the case Im z 3 =0. The perturbations to the rolls are expressed as z 1 = z 1 exp Λt, z 2 = z 2 exp Λt, z 3 = x R + z 3 exp Λt, z 1 = z 4 exp Λt, z 2 = z 5 exp Λt, z 3 = x R + z 6 exp Λt, wherex R = µ/α 2. This yields the six eigenvalues Λ 1 =2µ, Λ 2 =0, Λ 3,4 = µ+ µα 3 +α 1 x R, Λ 5,6 = µ+ µα 3 α 1 x R. (3.5) α 2 α 2 The condition µ<0 indicates supercriticality, where the trivial solution is unstable. The zero eigenvalue reflects the fact that we have a one-parameter family of roll solutions, due to translation in the direction perpendicular to the rolls. Translation parallel to the rolls leaves the solution as is. The last two eigenvalues have multiplicity 2. For small µ, wehavex R =O( µ) and the latter terms in the last four eigenvalues dominate over the O(µ) terms; the Λ 3,4 would then have opposite sign to Λ 5,6 and the rolls are unstable. b. Hexagons. For hexagons, z 1 = z 2 = z 3 = real, say x H. This leads to This has real solutions for µ + α 1 x H +(α 2 +2α 3 )x 2 H =0. (3.6) α 2 1 4µ(α 2 +2α 3 )>0. (3.7) The perturbations to the hexagons are expressed as z 1 = x H + z 1 exp Λt, z 2 = x H + z 2 exp Λt, z 3 = x H + z 3 exp Λt, z 1 = x H + z 4 exp Λt, z 2 = x H + z 5 exp Λt, z 3 = x H + z 6 exp Λt. The first amplitude equation yields (Λ + µ) z 1 ++α 1 x H ( z 5 + z 6 )+α 2 x 2 H (2 z 1 + z 4 )+α 3 x 2 H ( z 5 + z 2 + z 6 + z 3 +2 z 1 )=0. (3.8) Using Eq. (3.6) for µ, this becomes z 1 (Λ+x 2 H α 2 x H α 1 )+( z 2 + z 3 )α 3 x 2 H + z 4α 2 x 2 H +( z 5+ z 6 )(α 1 x H +α 3 x 2 H )=0. (3.9)

8 Vol. 49 (1998) Influence of non-boussinesq effects 231 The hexagon solution is symmetric across the origin because z 1 = z 1,z 2 = z 2,z 3 = z 3,andz i is the reflection across the origin of z i. The perturbations to a symmetric, or even, solution can be split into those that are even and those that are odd, or anti-symmetric across the origin. Consider first the even perturbations, denoting the amplitude functions for perturbations that are symmetric across the origin as Z 1 = z 1 + z 4,Z 2 = z 2 + z 5,Z 3 = z 3 + z 6. The amplitude equations yield Λ+2α 2x 2 H α 1x H 2α 3 x 2 H +α 1x H 2α 3 x 2 H +α 1x H Λ+2α 2 x 2 H α 1x H 2α 3 x 2 H +α 1x H 2α 3 x 2 H +α 1x H 2α 3 x 2 H +α 1x H 2α 3 x 2 H +α 1x H Λ+2α 2 x 2 H α 1x H Z 1 Z 2 =0. Z 3 We subtract row 3 from row 2 and add column 3 to column 2. This yields (3.10) Λ 1,2 = 3α 2 x 2 H + α 1 x H µ, Λ 3 =2µ+α 1 x H. (3.11) Next, consider the odd perturbations, denoting them Y 1 = z 1 z 4,Y 2 = z 2 z 5, Y 3 = z 3 z 6.UsingEq.(3.6)forµ, the amplitude equations becomes Λ α 1 x H Λ α 1 x H Λ α 1 x H 1 Y 1 Y 2 = 0. (3.12) Y 3 We subtract column 3 from columns 1 and 2, and add rows 1 and 3. This yields Λ 4,5 =0, Λ 6 =3α 1 x H. (3.13) Each zero eigenvalue refers to the one-parameter family of base solutions obtained by translation in x or y. c. Type IV solutions: z i real, z 1 = z 2, z 1 z 3. The equilibrium equations lead to µz 1 +α 1 z 1 z 3 +α 2 z1 3 +α 3z1 3 +α 3z 1 z3 2 =0,andµz 3 +α 1 z1 2 +α 2z3 3 +2α 3z1 2z 3 =0. We multiply the first equation by z 3 /z 1 and subtract the second equation to obtain z 3 = α 1. (3.14) α 2 α 3 This is substituted into the first equation to obtain ( z1 2 = α 2 µ 1 α )( 2 1 ) (α 2 α 3 ) 2. (3.15) α 2 + α 3 For this solution to exist, we need ( α 2 µ + 1 α )( 2 1 ) (α 2 α 3 ) 2 < 0. α 2 + α 3

9 232 Y. Y. Renardy and M. Renardy ZAMP The switch-over occurs at µ = α 2 1 α 2/(α 2 α 3 ) 2 which is the intersection point between the line (3.14) and the roll solution parabola (3.4). For x 3 /x 1 > 2, = 2, and between 0 and 2, Buzano and Golubitsky (1983) name the solution wavy rolls, transition and false hexagons, respectively. The false hexagon pattern appears to be slightly elongated in one direction, compared with the true hexagon solution. In the symmetric case when z 3 =0,α 1 =0,the type IV solution reduces to the patchwork quilt, which is not stable (Golubitsky, Swift and Knobloch 1984). When µ =0,z 1 and z 3 do not reduce to 0, except when α 1 = 0 which is the symmetric case of the patchwork quilt. Thus, the type IV solutions do not bifurcate out of the origin. As in the hexagon case, the base solution is even, and we decouple the perturbationsintoevenandoddones. Wesetz 1 =x 1 + z 1 exp Λt, z 2 = x 1 + z 2 exp Λt, z 3 = x 3 + z 3 exp Λt, z 1 = x 1 + z 4 exp Λt, z 2 = x 1 + z 5 exp Λt, z 3 = x 3 + z 6 exp Λt. The amplitude equations yield (Λ + µ) z 1 + α 1 (x 3 z 6 + x 1 z 5 ) + α 2 x 2 1 ( z 4 +2 z 1 )+α 3 (x 1 x 3 ( z 5 + z 2 )+x 2 1 ( z 6 + z 3 )+(x 2 1 +x2 3 ) z 1)=0. The even perturbations satisfy Λ+2α 2x 2 1 α 1x 3 α 1 x 3 +2x 2 1 α 3 α 1 x 1 +2α 3 x 1 x 3 α 1 x 3 +2α 3 x 2 1 Λ+2α 2 x 2 1 α 1x 3 α 1 x 1 +2x 1 x 3 α 3 α 1 x 1 +2α 3 x 3 x 1 α 1 x 1 +2α 3 x 1 x 3 Λ+µ+3α 2 x α 3x 2 1 Z 1 Z 2 =0. Z 3 (3.16) Row 2 is subtracted from row 1, column 2 is subtracted from column 1; this yields and Λ 1 = 2α 2 x α 1x 3 +2α 3 x 2 1, (3.17) Λ 2 +Λ(2α 2 x 2 3 α 1x 2 1 x 3 +2α 2 x α 3x 2 1 ) +2x 2 1 (2α 2x 2 3 α 1x 2 1 )(α 2 + α 3 ) 2x 2 1 x (α 1 +2α 3 x 3 ) 2 =0. 3 (3.18) The odd perturbations satisfy Λ α 1 x 3 x 3 x 1 x 3 Λ α 1 x 3 x Λ α 1 x 1 x 1 x 3 Y 1 Y 2 = 0. (3.19) Y 3 Column 2 is subtracted from column 1, row 2 is added to row 1; this yields Λ 4,5 =0, Λ 6 =2x 3 α 1 + α 1x 2 1. (3.20) x 3 Each zero eigenvalue refers to the one-parameter family of solutions obtained by translation in x or y.

10 Vol. 49 (1998) Influence of non-boussinesq effects Non-Boussinesq effects In this section, we study the effect of various non-boussinesq effects in salt finger convection. The quantity of interest will be the coefficient α 1 in the preceding analysis; it is this coefficient which determines whether stable hexagons will have x H > 0orx H <0 (see equation (3.6)), and hence whether downward or upward fingers are favored. We state our equations with a general dependence of the fluid properties on temperature θ and on salinity S. Specific heat is denoted by c(θ, S), thermal conductivity by k(θ, S), salt diffusivity by κ(θ, S), and viscosity by µ(θ, S). The density is assumed to be a constant ρ 0 except in the gravity terms where it depends on the temperature and salinity and is denoted by ρ(θ, S). The equations are: θ +(u )θ= [k(θ, S) θ] Ṡ +(u )u= [κ(θ, S) S], 1 c(θ, S), (4.1) ρ 0 ( u +(u )u)=div(µ(θ, S)[ u +( u) T ]) p ρ(θ, S)ge z, (4.2) divu =0. We now make a thin-finger approximation based on the assumption that solutions vary much more rapidly in the horizontal direction than in the vertical direction. This is based on the observation that saltfingers have a length on the order of meters, but a width and spacing on the order of centimeters. The approximation we make in the equations is that in the diffusion terms of the temperature and salinity equations only horizontal derivatives are taken into account. As far as the viscosity term is concerned, the most important contribution is horizontal diffusion of momentum in the equation for the vertical velocity (the dominant component); here the viscosity term is kept in full. For the vertical diffusion of vertical momentum and horizontal diffusion of horizontal momentum, we approximate the viscosity by a constant. Vertical diffusion of horizontal momentum is neglected. In this case, the only boundary condition needed is w =0atz=0,L. The equations become: θ +(u )θ= 1 c(θ, S) H [k(θ, S) H θ], Ṡ +(u )S= H [κ(θ, S) H S], ρ 0 ( u H +(u )u H )=µ 0 H u H H p, ρ 0 (ẇ+(w )w)= H (µ(θ, S) H w)+µ 0 2 w z 2 p z ρ(θ, S)g, divu =0,

11 234 Y. Y. Renardy and M. Renardy ZAMP where H denotes the horizontal gradient, and u =(u H,w). The base state is p T = T 0 (z), S = S 0 (z), u = 0, z = ρ(t 0,S 0 )g. For the study of salt finger convection, we are interested in the case where both temperature and salinity increase towards the surface, i.e. T 0 (z) > 0, S 0 (z) > 0. The linearized equations for disturbances proportional to exp(iαx) are θ = wt 0 (z) α2 k(t 0,S 0 ) c(t 0,S 0 ) θ, S = ws 0 (z) α2 κ(t 0,S 0 ) S, ρ u = µ 0 α 2 u iαp, ρẇ = µ(t 0,S 0 )α 2 2 w w µ 0 z 2 p z ρ θ(t 0,S 0 )g θ ρ s (T 0,S 0 )g S, 0=iαu + w z. Denote an eigenfunction by Then the adjoint eigenfunction is b 1 =( ρ θ(t 0,S 0 )g T 0 ζ 1 =(θ 1,S 1,u 1,0,w 1,p 1 ) exp(iαx). θ 1, ρ s(t 0,S 0 )g S 0 S 1,u 1,0,w 1,p 1 ) exp(iαx). We examine the quadratic terms in the governing equations. They are θ = (u ) θ c θ(t 0,S 0 ) c 2 (T 0,S 0 ) k(t 0,S 0 ) θ H θ c s(t 0,S 0 ) c 2 (T 0,S 0 ) k(t 0,S 0 ) S H θ+ k θ (T 0,S 0 ) c(t 0,S 0 ) H ( θ θ) + k s(t 0,S 0 ) c(t 0,S 0 ) H ( S θ), S = (u ) S+κ θ (T 0,S 0 ) H ( θ S) +κ s (T 0,S 0 ) H ( S S), ρ 0 u H = ρ 0 (u )u H, ρ 0 ẇ= ρ 0 (u )w+µ θ (T 0,S 0 ) H ( θ H w)

12 Vol. 49 (1998) Influence of non-boussinesq effects 235 +µ s (T 0,S 0 ) H ( S H w) ρ θθ (T 0,S 0 )g θ 2 2 ρ sθ (T 0,S 0 )g θ S ρ ss (T 0,S 0 )g S 2 2. The rotated eigenfunctions appear in the expression for the Landau coefficient α 1. These are ζ 2 =(θ 1,S 1, u 1 3 2, 2 u 1,w 1,p 1 ) exp(iαx/2 i 3αy/2), ζ 3 =(θ 1,S 1, u 1 3 2, 2 u 1,w 1,p 1 ) exp(iαx/2+i 3αy/2). The relevant term is (b 1,N 2 ( ζ 2, ζ 3 )). This inner product leads to the integral over 0 to L with respect to z of T 1 + T 2 + T 3, as follows. ρ θ (T 0,S 0 )g T 1 = θ 1 T 0 [( u 1 2 )(iα 2 )θ θ 1 1 w 1 z u 1iαθ 1 ] ρ s (T 0,S 0 )g +S 1 S 0 [( u 1 2 )(iα 2 )S S 1 1 w 1 z u 1iαS 1 ] w 1 g[ρ θθ (T 0,S 0 ) θ ρ θs(t 0,S 0 )θ 1 S 1 +ρ ss (T 0,S 0 ) S2 1 2 ]. The term T 1 incorporates effects of the nonlinear profile in the base state and nonlinear dependence of the density ρ on θ and S. Upon integration by parts and with the use of incompressibility, the integral of T 1 equals the integral of T 1 = θ2 1 2 w 1gρ θ (T 0,S 0 ) z ( 1 T 0 )+ S2 1 2 w 1gρ s (T 0,S 0 ) z ( 1 S 0 ) + θ2 ( 1 gs ) 2 w 1ρ θs (T 0,S 0 ) 0 T 0 + S2 ( 1 gt ) 2 w 1ρ θs (T 0,S 0 ) 0 S 0 θ 1 S 1 w 1 ρ θs (T 0,S 0 )g. The terms with the same sign as w 1 favor upward fingers. In the first term, ρ θ < 0, so that upward fingers are favored if T 0 > 0, and downward fingers are favored if T 0 < 0. In the second term, density increases with salinity, ρ s > 0, so that upward fingers are favored if S 0 < 0, and downward if S 0 > 0. The latter case, with salinity gradient increasing toward the ocean surface, appears more realistic. In the nonlinear density dependence, ρ θθ and ρ ss cancel out. Since S 0 /T 0 is positive, the final contributions have a sign equal to that of ρ θs: S ρ θs (T 0,S 0 )gw 0 T 1 2T 0 (θ 1 S 0 1 S 0 ) 2.

13 236 Y. Y. Renardy and M. Renardy ZAMP base salinity base temperature z*=l surface region I S=S * * 0 * = * 0 > * 1 region II z*= 0 region III S=S * * 1 * * = 1 Figure 1. A sketch showing profiles of base salinity and temperature. The quantity ρ θs is negative (Landolt-Börnstein 1986) since an increase in salinity lowers the anomalous temperature, and hence increases thermal expansion. The next term is from the nonlinear thermal properties. We form the integral from 0 to L with respect to z of ρ θ (T 0,S 0 )g T 2 = θ 1 T 0 ( c θ(t 0,S 0 ) c 2 (T 0,S 0 ) k(t 0,S 0 )θ1 2 α2 + c s(t 0,S 0 ) c 2 (T 0,S 0 ) k(t 0,S 0 )S 1 α 2 θ 1 k θ(t 0,S 0 ) c(t 0,S 0 ) = θ3 1 ρ θgα 2 c [ k ] 2T 0 c 2 θ 2 1 α2 2 θ θ2 1 S 1ρ θ gα 2 c st 0 k s(t 0,S 0 ) S 1 θ 1 α 2 ) c(t 0,S 0 ) 2 [ k ] c 2 Note that the signs of θ 1 and S 1 are opposite to that of w 1, ρ θ < 0, and T 0 > 0. Downward fingers are favored if [k/c 2 ] θ > 0or[k/c 2 ] s > 0. We have k θ > 0and c θ < 0, so [k/c 2 ] θ > 0. Moreover, k s is negative, but very small, while c s is negative, so that [k/c 2 ] s becomes positive (see Landolt-Börnstein 1986). The term from the nonlinear diffusion coefficient is the integral of T 3 ρ s (T 0,S 0 )g = S 1 S 0 [ κ θ (T 0,S 0 ) S 1θ 1 α 2 κ s (T 0,S 0 ) S2 1 α2 2 2 ]. Diffusivity increases with temperature, κ θ > 0, and this favors upward fingers. No information on κ s is given in Landolt-Börnstein, but it seems likely that it is negative, also favoring upward fingers. The term from viscosity is [ w1 θ 1 α 2 ] [ w1 S 1 α 2 ] w 1 µ θ + w 1 µ s. 2 2 Viscosity decreases with temperature, which favors downward fingers, and increases with salinity, favoring upward fingers (Landolt-Börnstein 1986). s.

14 Vol. 49 (1998) Influence of non-boussinesq effects Figure 2. The six eigenfunctions illustrated as wave vectors. 5. Non-uniform temperature and salinity profiles Renardy and Schmitt (1996) give a detailed analysis of the linear stability problem for a temperature profile consisting of three layers, motivated by observations of Osborn (1991). The profile is sketched in Figure 1. Regions III, II and I denote 0 <z <d 1,d 1 <z <d 2 and d 2 <z <L, respectively. The temperature at the ocean surface z = L is θ0,atz =d 2 it is θ 3,atz =d 1 it is θ 2,at the bottom z = 0 it is θ1,andθ 0 >θ 3 >θ 2 >θ 1. At the ocean surface, the solute concentration is S0, and at the lower boundary, it is S 1 ; S 0 >S 1. At temperature θ0, the fluid has thermal coefficient of volume expansion ˆα, solute coefficient of volume expansion ˆβ, thermal diffusivity κ T, solute diffusivity κ S, viscosity µ, densityρ 0, and kinematic viscosity ν = µ/ρ 0. Dimensionless variables (without asterisks) are as follows: (x, y, z) =(x,y,z )/L, t = κ T t /L 2, v = v L/κ T, θ =(θ θ0 )/(θ 1 θ 0 ), S =(S S0 )/(S 1 S 0 ), p = p L 2 /(ρ 0 κ 2 T ). We define a thermal Rayleigh number R T = g ˆα(θ 1 θ 0 )L3 /(κ T ν), a salinity Rayleigh number R S = g ˆβ(S 1 S 0 )L3 /(κ T ν), a Prandtl number P = ν/κ T, a Lewis number τ = κ S /κ T,andG=gL 3 /κ 2 T. Regions III, II and I occupy 0 <z<d 1,d 1 <z<d 2,andd 2 <z<1, respectively. The governing equations are the transport equations for the temperature and solute concentration, the Navier-Stokes equation with the Oberbeck-Boussinesq approximation and incompressibility. At z = d 1,d 2, the velocity, shear stress, normal stress, temperature, heat flux, salinity and salinity flux are continuous. At the top surface z =1,S=0,θ= 0, and at the bottom surface z =0,S=1,θ =1. At both the top and bottom, the vertical component of velocity and the shear stresses vanish. In the context of the oceanic values relevant to the observations on asymmetric salt fingers (Osborn 1991), the thermal diffusivity is κ T 0.001, the salt diffusivity is κ s =10 5,ρ= 1, and the kinematic viscosity is ν 0.01 in cgs units. The Prandtl number varies slightly with temperature; we set P = 8, τ= The

15 238 Y. Y. Renardy and M. Renardy ZAMP rolls hexagons wavy roll transition false hexagon Figure 3. Plot in the x y plane of the patterns: rolls, hexagons and type IV solutions. coefficient of thermal expansion ˆα forseawateris /deg.c. The linear salinity and temperature profiles considered previously are obtained as equilibrium solutions of the diffusion equation. In reality, however, temperature and salinity of the ocean are influenced by a number of factors which fluctuate with time, and equilibrium is never reached. In addition, diffusion is a slow process, and even profiles which are not steady-state solutions will not change appreciably over a period of hours if they are used as an initial condition. We shall then make the approximation of regarding these profiles as stationary and doing a linear stability analysis around them. If the timescale for the growth of the instability is short relative to the timescale over which the base profile is varying, such an assumption is justified. The situation is analogous to the familiar case of boundary layer profiles in shear flows, which also do not really exist as steady solutions of the Navier-Stokes equations. We note also that, within the thin finger approximation considered in the preceding chapter, any temperature and salinity depending only on depth is an equilibrium solution to the equations.

16 Vol. 49 (1998) Influence of non-boussinesq effects z_ rolls µ hexagons type IV hexagons rolls z_ µ Figure 4. Bifurcation diagram for the three-region model for surface warming, d 1 =0.385,d 2 =0.846,θ 2 = 0.833,θ 3 =0.769, R T = 20540,R S = 191.7,α =10,R ρ = 15. Stable branches are shown by continuous lines, unstable ones by dash. a. Base solution. The velocity field for the base solution is zero. The salinity field is destabilizing: S =1 z. (5.1) The sharp temperature gradient in region I is due to warming from solar radiation and we model this by introducing a source term which is a function of z, intothe right hand side of the heat equation. This is a forcing term which maintains the nonlinear base temperature profile: linear in region II and quadratic in regions I and III. We denote by [f ] at z = d i the jump in f defined by f(z = d i ) f(z = d i +). The base temperature profile satisfying the continuity conditions on the temperature (θ = θ 2 at z = d 1, θ = θ 3 at z = d 2 ) and the heat flux ([ θ/ z] = 0

17 240 Y. Y. Renardy and M. Renardy ZAMP z_ rolls hexagons type IV µ rolls z_ µ Figure 5. Bifurcation diagram for the three-region model for surface warming, d 1 =0.385,d 2 =0.846,θ 2 = 0.833,θ 3 =0.769, R T = ,R S = ,α=50,R ρ = 15. Region at the origin is magnified. Stable branches are shown by continuous lines, unstable ones by dash. at z = d 1,d 2 )is 1+A 1 z 2 +A 2 z for 0 <z<d 1 θ= A 3 (z d 1 )+θ 2 for d 1 <z<d 2 A 4 (z 1) 2 + A 5 (z 1) for d 2 <z<1, (5.2) where A i are given in Renardy and Schmitt (1996). Their Fig. 9 shows the basic dimensional temperature profile θ = θ(θ1 θ 0 )+θ 0, and this compares with the temperature profile in Fig. 5 of Osborn (1991). The data show gradients in regions II and III to be close to linear, with almost a discontinuity in slope. b. Perturbation equations. The equations for the perturbations u, w, θ, p, S are v P v + p +(R S P S R θ T P θ)e z = (v )v,divv=0, θ+w z θ = (v ) θ, S+w S z τ S = (v ) S, where the base temperature gradient θ/ z is found from (5.2). In stating the equations, we have also allowed a nonlinear salinity

18 Vol. 49 (1998) Influence of non-boussinesq effects w(x=0,z) z Figure 6. Contour plot of the vertical component of the velocity for the onset mode at α =50forthe data of figure 5, d 1 = 0.385,d 2 = 0.846,θ 2 = 0.833,θ 3 = 0.769, R T = ,R S = ,α=50,R ρ = 15. Upper plot: horizontal axis extends one wavelength,0 <x<2π/α, vertical axis extends over 0 <z<1. Lower graph shows the vertical component of the velocity taken from the upper plot at x =0,vsz. profile S/ z, which we shall consider later. If (5.1) applies, then S/ z = 1, Renardy and Schmitt (1996) use the following boundary conditions: θ =0at z= 0,1, S = 0 at z = 1, S = 0 at z = 0. Stress free conditions w = 0, u/ z + w/ x =0, v/ z + w/ y = 0 are used at z =0,1, since the salt finger phenomena occur in a layer within the ocean, sandwiched between fluid layers. At z = d 1,d 2, we have the continuity conditions for velocity shear stresses, normal stress, temperature, heat flux, salinity and salinity flux: [v] z=d i = 0, [ u/ z ] z=d i =0, [ v/ z] z=d i =0, [ p] z=d i =0, [ θ] z=d i =0, [ θ/ z] z=d i =0, [ S ] z=d i =0, [ S/ z] z=d i =0. The set of unknowns, v, p, Sand θ, is denoted by Φ. The equations, boundary conditions, and continuity requirements at z = d 1,d 2 for these variables is conveniently expressed in the schematic form LΦ =N 2 (Φ, Φ), where the opera-

19 242 Y. Y. Renardy and M. Renardy ZAMP tor L incorporates the linear terms, N 2 stands for the nonlinear terms, which are quadratic, and these are defined in Appendix A. The real linear operator L can bewrittenintheforma+ d dtb. We also introduce the notation L(σ) =A+σB. 6. Numerical results We did a number of numerical calculation with the linear salinity profile (5.1) and the three-layer temperature profile (5.2). The specific values which we determine from a rough match to Osborn s data are d 1 =0.38, d 2 =0.85, θ 2 =0.83, θ 3 = The region where the instability develops is the central region II, where we define the local density ratio to be R ρ = R T II /R S II in terms of Rayleigh numbers defined over region II, i.e., R T II /R T =(θ 2 θ 3 )(d 2 d 1 ) 3,R S II /R S =(d 2 d 1 ) 4. For our parameter values, R ρ =0.14(R T /R S ). Table 1 of Schmitt (1994b) gives typical values in the salt-fingering regime to be between 10 and 20, based on the classical double diffusion problem with linear profiles for temperature and salinity, and for our calculations, we picked R ρ = 15. Table 1 Neutral stability conditions where σ = 0 for the surface warming model of Renardy and Schmitt (1996) with the three-region non-uniform temperature profile. The density ratio for the middle region is R ρ =15=0.14(R T /R S ). We list the critical wavenumber α, the Landau coefficients α 1,α 2,α 3 and the last column denotes whether the onset plumes are upward or downward at the center. R T α α 1 α 2 α down down up For the linear stability problem, we consider normal mode solutions proportional to exp(iαx + σt). We fix α and look for critical situations where σ =0. In all the calculations, we fixed P =8,τ=0.01, R ρ = 15, and for given α, wethen adjust R T and R S (the ratio of these is fixed by R ρ until criticality is reached. Once we have a critical situation, then we calculate the nonlinear coefficients α 1, α 2 and α 3. Table 1 shows a number of data sets for varying wavenumbers α. The trend is that for the lower wavenumbers, the plumes are downward at the center, and for the larger wavenumbers, the plumes have upward motion at the center. The Landau coefficient α 1 is small compared with the other coefficients, and becomes even smaller as the wavenumber increases. The bifurcation diagram for the second and third values of wavenumber are shown in figures 4 and 5, respectively. The velocity component of the eigenfunction is shown in figure 6 for the third set. For the larger Rayleigh numbers and wavenumbers, the instability is increasingly confined to the middle region, as evident from figure 6. The vertical axis in the

20 Vol. 49 (1998) Influence of non-boussinesq effects 243 bifurcation diagrams is z 3. The horizontal axis is chosen to be µ, andthebase solution is unstable for µ > 0. The parabola (3.6) of the hexagon solution passes through the origin. On the same plot, we show the two other solutions: parabola (3.4) for the rolls which has the vertex at the origin, and the half-line (3.14) for the type IV solution which exists to the right of the intersect with the roll solution. Note that in the case of up-down symmetry, z 3 =0forthetype IV solution. There are two points where stability can change on the hexagon branch: at the vertex µ = α 2 1 /(4(α 2 +2α 3 )),x H = z 3 = α 1 /[2(α 2 +2α 3 )] and the intersection with the type IV solution at z 3 given by (3.14). The computation of the eigenvalues Λ i of Section 3 shows that the lower hexagon branch is stable between the vertex and the intersect with the type IV solution. The roll solution becomes stable after intersection with the type IV solution. The stable branches are shown by continuous lines, the unstable ones by dash. The bifurcation to the hexagonal solution is transcritical. There are two types of hexagonal solutions, the l(liquid)-hexagons (upflow at the center) and g(gas)-hexagons (downflow at the center), according to the nomenclature of Busse (1978). In his work, the problem of heating from below a layer of fluid with temperature-dependent viscosity was addressed, and the type of hexagon excited depended on whether the layer was a liquid or gas; the viscosity of a liquid decreases with temperature and that of gas increases with temperature. Figure 4 shows that x H < 0 along the stable hexagon branch, and the velocity contour plot was checked, that the sign is positive at the center of the hexagons as in figure 6. Thus, we have g-hexagons. Horizontally, the hexagon solution is a superposition of three cosine solutions with the center of the hexagon at the origin, where the three are all 1. In figure 5, the hexagon branch intersects the type IV solution on the upper branch and we have z 3 = x H > 0. As shown in figure 6, the eigenfunction describes upward motion; thus the weakly nonlinear solution is upflow at the center with downflow around. The roll solution has stable branches as shown in bold. In all cases α 1 is very small. On the other hand, α 2 and α 3 become quite large as α increases. The latter feature is due to a flattening out of the neutral stability curve (as a function of α) in that case. As a result, second order harmonic interaction terms become near-resonant. Of course, this questions the single mode onset paradigm for studying the formation of observed patterns. Our main interest, however, is in the coefficient α 1, which determines whether hexagons with upward or downward motion can be a preferred pattern. Our analysis from Section 4 leads us to expect that the top and bottom layers have opposite effects on α 1. To separate their influences, we looked at a two-layer arrangement, where the temperature gradient increases towards the top, but remains uniform up to the bottom. Specifically, we modified the three-layer arrangement in such a way that the temperature gradient is uniform across the bottom two layers, but increases in the top layer. The values are θ0 =18.5o C, θ3 =16.5o C, θ2 = 16.3 o C, θ1 =15.9o C, the layer depths are 1m, 3m, 2.5m from the top. The density ratio R ρ = 0.18R T /R S. Table 2 shows the Landau coefficients. The α 1

21 244 Y. Y. Renardy and M. Renardy ZAMP becomes smaller for the higher wavenumbers compared with α 2 and α 3.Themotion is downward at the center of the hexagons. The bifurcation diagrams are similar to figure 4. The hexagons bifurcate subcritically. The density ratios are R ρ =15,20, 15, respectively, for the three cases. For the third data set, the contributions to the coefficient α 1 from the various equations are shown in Table 3. The contributions from the momentum equation (the reduced equation for the vertical component of the velocity w) is small, as are the contributions from the heat transport equation. The dominant terms come from the salinity transport equation in the lower layer. The eigenmode is shaped as in figure 6, with convection confined in the middle layer. Similar results are found for the first data set in Table 2, the case of α = 10. Table 2 Neutral stability conditions for the model with sharp temperature gradient in top 1m, and moderate gradient for the lower 5.5m. Base salinity profile is linear. θ 2 =0.9296,θ 3 =0.845,d 1 =0.385,d 2 =0.846,R ρ =0.183R T /R s. The last column indicates whether the flow is upward or downward at the center of the hexagons. R T R s α α 1 α 2 α down down down Table 3 Contributions to Landau coefficient α 1 forthethirddatasetoftable2. θ 2 = ,θ 3 =0.845,d 1 =0.385,d 2 =0.846,R ρ =15,R T = ,α= 70. layer momentum equation heat equation salinity equation top -0.5E-7-0.7E middle bottom We still find α 1 to be small. Moreover, the sign of α 1 is opposite from that expected by the analysis in Section 4. While the result of Section 4 would lead us to believe that an increase in temperature gradient at the top favors upward fingers, we actually find that downward fingers are favored. Obviously, the thin-finger approximation does not apply, and we believe that this is due to the presence of another small parameter in the problem, namely the Lewis number. In Section 4, the contribution to α 1 in the case of a nonlinear temperature gradient came from the temperature equation. Our numerical results indicate that the contribution to α 1 coming from the temperature equation does indeed have a sign consistent with the result of Section 4, but the dominant contribution to α 1 comes from the salinity equation. Specifically, the bottom layer yields a contribution favoring downward fingers, while the upper layers yield a contribution favoring upward

22 Vol. 49 (1998) Influence of non-boussinesq effects 245 fingers. This is because a downward motion would increase the salinity gradient at the bottom, further enhancing the instability, but decrease the salinity gradient at the top, suppressing the instability. In the classical double diffusion problem with uniform temperature and salinity gradients these two effects cancel each other, but here there is an overall contribution favoring downward motion which remains. Evidently, this contribution is not included in the thin finger analysis presented above. Since the overall effect is small and may well depend on factors such as boundary conditions which must be regarded as highly uncertain in oceanographic applications, we are inclined to take our results with a grain of salt (or maybe we should say a finger of salt). Table 4 Landau coefficients for base salinity profile S =1 z 2, linear temperature profile. Last column shows direction of flow at center of plumes. R T R s α α 1 α 2 α down down Overall, we conclude that, even though the nonlinear temperature profile is quite a prominent feature in the data, it is probably not the dominant influence on pattern selection, since the coefficient α 1 turns out to be rather small. We also looked at the case of a linear temperature profile and nonlinear salinity profile. To model this, the base temperature profile varies linearly from θ 1 =15.9o Cat the bottom to θ 0 =18.5o Cat the top. The coefficients in Eqn (5.2) are A 1 =0,A 2 = 1,A 3 = 1,A 4 =0,A 5 = 1. The depth coefficients are d 1 =0.3,d 2 =0.7. The base salinity profile is S =1 z 2, with salinity gradient increasing toward the surface. The density ratio is defined R ρ = R T /R s. The pattern selection results are shown in Table 4. The eigenmode consists of cells floating in the upper half of the region. The bifurcation diagrams are similar to figure 4. In this model, α 1 is substantially larger, and the sign is as expected from the analysis in Section Conclusion We have investigated the effect of nonlinear fluid properties and nonuniform temperature and salinity profiles on pattern formation in doubly diffusive convection. In particular, we focus on the breaking of up-down symmetry and on the resulting quadratic term in the amplitude equations which could lead to a preference for hexagonal patterns with downflow at the center of the hexagons; such patterns would be of interest because downward fingers are observed in the ocean. For the limit of a thin-finger analysis, we were able to derive closed form expressions and assess a variety of effects. We present numerical calculations for the cases of a nonuniform temperature and salinity profile. We conclude that the overall effect of a nonuniform temper-

23 246 Y. Y. Renardy and M. Renardy ZAMP ature profile is likely to be small. The results for the nonuniform salinity profile are in qualitative agreement with the thin finger analysis. Appendix A The discretized form of the solution vector Φ denotes (v 1, θ 1, p 1, S 1, v 2, θ 2, p 2, S 2, v 3, θ 3, p 3, S 3 ), v =(u, v, w), where subscripts refer to the regions I,II and III. This satisfies the vector form equation LΦ =N 2 (Φ, Φ) where θ+w θ z θ in region I u P u + p x in region I v P v + p y in region I ẇ P w + p z + R SP S R T P θ in region I S + w S z τ S in region I θ + w θ z θ in region II u P u + p x in region II LΦ = v P v + p y in region II. ẇ P w + p z + R SP S R T P θ in region II S + w S z τ S in region II θ + w θ z θ in region III u P u + p x in region III v P v + p y in region III ẇ P w + p z + R SP S R T P θ in region III S + w S z τ S in region III The operator L is defined to incorporate the boundary conditions at z =0,1the continuity requirements for v i, S i, θ i and their first derivatives, the continuity of p i at z = d 1,d 2, and the incompressibility condition. The real linear operator L is also written in the form A + d dt B where

24 Vol. 49 (1998) Influence of non-boussinesq effects 247 w θ z θ in region I P u + p x in region I θ 1 P v + p y in region I u P w + p z + R SP S R T P θ 1 in region I v w τ S 1 in region I w w θ z θ 1 in region II S 1 P u + p θ x in region II 2 AΦ = P v + p u y in region II 2 P w + p z + R SP S R T P θ, BΦ= v 2. in region II w w τ S 2 in region II S w θ z θ 2 in region III θ 3 P u + p x in region III u 3 P v + p v 3 y in region III P w + p z + R SP S R T P θ w 3 in region III S 3 w τ S in region III The corresponding nonlinear terms in the equations are written (v 1 ) θ 1 H 1 (v 1 )u 1 H 2 (v 1 )v 1 H 3 (v 1 )w 1 H 4 (v 1 ) S 1 H 5 (v 2 ) θ 2 H 6 (v 2 )u 2 H 7 N 2 (Φ, Φ) = (v 2 )v 2 = H 8. (v 2 )w 2 H 9 (v 2 ) S 2 H 10 (v 3 ) θ 3 H 11 (v 3 )u 3 H 12 (v 3 )v 3 H 13 (v 3 )w 3 H 14 (v 3 ) S 3 H 15 The operators A, B, N 2 also incorporate the boundary and continuity conditions and incompressibility. The definition of N 2 is extended in a symmetric fashion to the case when the arguments are different: N 2 (Φ, Ψ) := 1 ( ) N 2 (Φ + Ψ, Φ+Ψ) N 2 (Φ Ψ, Φ Ψ). 4

25 248 Y. Y. Renardy and M. Renardy ZAMP This represents the average of the two permutations of the quadratic expression. For example, the first component H 1 of N 2 (ζ 1,ζ 2 )is (1/2)[(v ζ1 2 ) θ ζ2 +(v ζ2 1 ) θ ζ1 ]. These are referred to in section 5. Appendix B: Nonexistence of type VII solutions The Type VII solutions of Buzano and Golubitsky (1983) are solutions of (2.7) with all three amplitudes z i real, nonzero and different from each other. In this appendix, we show that, except in a degenerate case, the cubic amplitude equations have no such solutions. We write z i = x i to emphasize it is real. The equilibrium amplitude equations are µx 1 + α 1 x 2 x 3 + α 2 x α 3(x x2 3 )x 1 =0, (B1) µx 2 + α 1 x 3 x 1 + α 2 x α 3(x x2 1 )x 2 =0, (B2) µx 3 + α 1 x 1 x 2 + α 2 x α 3(x x2 2 )x 3 =0. (B3) We subtract (B2) from (B1) and divide by x 1 x 2. This yields µ α 1 x 3 + α 2 (x x 1x 2 + x 2 2 )+α 3(x 2 3 x 1x 2 )=0. (B4) In an analogous fashion, we obtain µ α 1 x 1 + α 2 (x x 2x 3 + x 2 3 )+α 3(x 2 1 x 2x 3 )=0, (B5) µ α 1 x 2 + α 2 (x x 3x 1 + x 2 1 )+α 3(x 2 2 x 3x 1 )=0. (B6) We now subtract (B5) from (B4) and divide by x 1 x 3. This yields α 1 +(α 2 α 3 )(x 1 + x 2 + x 3 )=0. (B7) We next solve (B4) for µ and insert into (B3). We find α 1 (x 1 x 2 + x 2 3 )+(α 2 α 3 )(x 3 3 x2 1 x 3 x 2 2 x 3 x 1 x 2 x 3 )=0. (B8) Now we solve (B7) for α 1 and insert the result into (B8). After some simple algebra, the result is (α 3 α 2 )(x 1 + x 2 )(x 1 + x 3 )(x 2 + x 3 )=0. (B9) Unless α 3 = α 2, two of the amplitudes must be equal in modulus, so the solution is not Type VII. If α 3 = α 2, then it follows from (B7) that α 1 =0. Inthiscase, (B1)-(B3) simplify to µx 1 + α 2 (x x2 2 + x2 3 )x 1 =0, (B10) µx 2 + α 2 (x x2 2 + x2 3 )x 2 =0, (B11) µx 3 + α 2 (x x2 2 + x2 3 )x 3 =0. (B12) We see that the entire sphere x x2 2 + x2 3 = µ/α 2 (B13) consists of solutions. Obviously this includes solutions of Type VII.