Effects of a temperature dependent viscosity on thermal convection in binary mixtures
aa r X i v : . [ phy s i c s . f l u - dyn ] O c t Effects of a temperature dependent viscosity on thermal convection in binary mixtures
Markus Hilt, Martin Gl¨assl, Walter Zimmermann ∗ Theoretische Physik I, Universit¨at Bayreuth, 95440 Bayreuth, GERMANY (Dated: April 2, 2018)We investigate the effect of a temperature dependent viscosity on the onset of thermal convectionin a horizontal layer of a binary fluid mixture that is heated from below. For an exponentialtemperature dependence of the viscosity, we find in binary mixtures as a function of a positiveseparation ratio ψ and beyond a certain viscosity contrast a discontinuous transition between twostationary convection modes having a different wavelength. In the range of negative values of theseparation ratio ψ , a (continuous or discontinuous) transition from an oscillatory to a stationaryonset of convection occurs beyond a certain viscosity contrast, and for large values of the viscosityratio, the oscillatory onset of convection is suppressed. I. INTRODUCTION
Thermal convection occurs in fluids or gases heatedfrom below and it is a well known, ubiquitous phe-nomenon [1, 2]. It drives many important processesin geoscience [3–6] or in the atmosphere [7, 8], and itis a central model system of nonlinear science [9, 10].Quite often, thermal convection can be described theo-retically in terms of the so-called Oberbeck-Boussinesq(OB) approximation for a single component fluid, whereconstant material parameters are assumed, except ofthe temperature-dependent density within the buoyancyterm, which is the essential driving force of convection.However, in nature, the viscosity may strongly dependon the temperature implying that models beyond the OBapproximation have to be used or convection takes placein fluid mixtures. Both degrees of freedom considerablyaffect convection, in particular near its onset. This workdiscusses the combination of both effects.For a sufficiently large viscosity contrast between thelower warmer and the upper colder region of the con-vection cell, related non-Boussinesq effects have be takeninto account, for instance, to model convection phenom-ena in the Earth’s mantle [6, 11–21]. First studies haveshown, that a linear as well as a sinusoidal temperaturedependence of the viscosity of a fluid may lead to a reduc-tion of the onset of convection compared to the case of aconstant viscosity [11, 12, 16]. In contrast, an exponen-tial temperature dependence of the viscosity can eitherlead to an enhancement or to a reduction of the threshold[15, 17], depending on the strength of the viscosity con-trast. Further, a spatially varying viscosity breaks theup-down symmetry in a convection layer causing a sub-critical convection onset to hexagonal patterns [17], andbeyond the threshold, more complex convection regimesmay be induced in fluids having a temperature dependentviscosity [22].Research on convection in binary-fluid mixtures hasa long tradition [23, 24] with numerous applications inoceanography or geoscience [23–27], nonlinear dynamics ∗ [email protected] and bifurcations [9, 28–34], and more recently also toconvection in colloidal suspensions [35–40]. In binary-fluid mixtures, the concentration field of one of the twoconstituents enters the basic equations as an additionaldynamic quantity [41, 42]: via the Soret effect ( ther-mophoresis ), a temperature gradient applied to a binaryfluid mixture in a convection cell causes a spatial depen-dence of the concentration field, which couples into thedynamical equation for the velocity field via the buoy-ancy term. The dynamics near the onset of convection inmixtures of alcohol and water as well as in He/ He mix-tures is well investigated with a good agreement betweenmeasurements and theory [9, 30]. The possibility of astationary as well as an oscillatory onset of convection inbinary fluid-mixtures, including a so-called codimension-2 bifurcation at the transition between both instabilities,caused additional attraction [9].Although the temperature dependence of the viscosityas well as the two-component character of fluids are con-sidered to be of importance for modeling many phenom-ena in planetary science [3–6, 24–27], the influence of acombination of both effects onto convection is still nearlyunexplored [43, 44]. As turbulent convection causes a ho-mogenization of concentrations and of the temperaturefield in the center of a convection cell, the impact of acombination of both effects is expected to be less sig-nificant in the turbulent regime, but to be of particularimportance at the onset of convection, which is the focusof this work.In Sec. II, we present the dynamical equations and inSec. III A, we reconsider the observation, that for a one-component fluid, in the case of a linear temperature de-pendence of the viscosity and a small viscosity contrast,one has a reduction of the onset of convection, while thereis an enhancement of the threshold for an exponentialtemperature dependence. The influence of a tempera-ture dependent viscosity on the onset of convection in abinary mixture is considered in Secs. III B and III C, bothalong the stationary branch as well as along the oscilla-tory branch, including the codimension-2 point. Moststriking, we find that the oscillatory branch can be sup-pressed by strong viscosity contrasts. In Sec. IV, theresults are summarized and discussed.
II. BASIC EQUATIONS AND HEATCONDUCTING STATE
Compared to the common basic equations for con-vection in binary fluid mixtures in Boussinesq approx-imation [31, 33], we replace the constant viscosity bya temperature dependent kinematic viscosity of a fluid ν = ν ∞ exp(¯ γ/T ), whereby we assume that both com-ponents of the mixture have the same temperature de-pendence [45, 46]. With the mean temperature in theconvection cell, T , and a Taylor expansion of the ex-ponent around T up to the leading order, the viscositytakes the following form ν = ν e − γ ( T − T ) , (1)where γ = ¯ γ/T and ν = ν ( T = T ) = ν ∞ exp(¯ γ/T ).In a binary mixture, a temperature dependent viscosityimplies via D ∼ /ν also a temperature dependent ther-mal diffusion constant D : D = D e γ ( T − T ) . (2)We would like to stress that this relation does not holdin general, but is appropriate, when the dependence ofthe viscosity on the temperature is roughly identical forboth components or when the concentration of the sec-ond component is very small, such that the viscosity ofthe mixture is almost exclusively determined by the firstcomponent. In Sec. III, we will restrict our analysis tothese two cases.The basic transport equations for an incompressiblebinary fluid mixture involve a dynamical equation for thetemperature field T ( r , t ), the mass fraction of the secondcomponent N ( r , t ) and the fluid velocity v ( r , t ): ∇ · v = 0 , (3a)( ∂ t + v · ∇ ) T = χ ∆ T , (3b)( ∂ t + v · ∇ ) N = ∇ · (cid:18) D (cid:16) ∇ N + k T T ∇ T (cid:17)(cid:19) , (3c)( ∂ t + v · ∇ ) v = − ρ ∇ p + ∇ · S − ρρ g ˆ e z . (3d)Herein, S = ν (cid:0) ∇ v + ( ∇ v ) T (cid:1) (4)describes the stress tensor, χ denotes the thermal diffu-sivity of the mixture, k T is the dimensionless thermal-diffusion ratio, that couples the temperature gradient tothe particle flux and is related to the Soret coefficient S T via k T /T = N (1 − N ) S T , and p ( r , t ) denotes the pressurefield. As in the common Boussinesq approximation, weassume that χ and k T /T ∼ N (1 − N ) S T are constantsand the dependence of the density ρ on T and N is takeninto account only within the buoyancy term, where weassume a linearized equation of state of the form ρ = ρ [1 − α ( T − T ) + β ( N − N )] . (5) The Eqs. (3) are completed by no-slip boundary con-ditions. For a fluid that in the z-direction is confinedbetween two impermeable, parallel plates at a distance d that are held at constant temperatures and extend in-finitely in the x-y-plane, the following set of boundaryconditions results at z = ± d/ T = T ∓ δT , (6a)0 = ∂ z N + k T T ∂ z T , (6b)0 = v x = v y = v z = ∂ z v z . (6c)In the absence of convection (i.e., for v = 0), the time-independent and with respect to the x-y-plane transla-tional symmetric heat-conducting state is given by T cond ( z ) = T − δT zd , (7a) N cond ( z ) = N − δN zd , with δN = − k T T δT. (7b)For the further analysis, it is convenient to separate thisbasic heat conducting state from convective contribu-tions setting T ( r , t ) = T cond ( z ) + T ( r , t ) and N ( r , t ) = N cond ( z ) + N ( r , t ). Making use of the rotational sym-metry in the fluid layer, we can restrict our analysis tothe x-z-plane and introduce a scalar velocity potential F ( x, z, t ) via v x = − ∂ x F, v z = ∂ z ∂ x F, (8)with the help of which Eq. (3a) is fulfilled by construc-tion. Rescaling distances by d , times by the vertical dif-fusion time d /χ , the temperature field T by χν /αgd ,the concentration field N by − k T χν /T αgd and thevelocity potential F by χd , all material and geometryparameters are regrouped in 5 dimensionless parameters:the Rayleigh number R , the Prandtl number P , the Lewisnumber L , and the separation ratio Ψ P = ν ξ , L = D ξ , R = αgd χν δT, Ψ = βk T αT (9)are well known from common molecular binary-fluid mix-tures and the fifth dimensionless quantityΓ = χν αgd γ (10)characterizes the viscosity contrast ¯ ν between the viscos-ity at the upper and the lower boundary via¯ ν = ν ( z = +1 / ν ( z = − /
2) = e Γ R . (11)In the following, we will discuss our results mainly independence on Ψ and Γ, whereas P and L are fixed to P = 10 and L = 0 .
01, respectively.Finally, by introducing a rescaled temperature devia-tion θ = ( R/δT ) T , a rescaled concentration deviation˜ N = − ( T R/k T δT ) N as well as a rescaled velocity po-tential f = 1 / ( χd ) F and using the combined function˜ c = ˜ N − θ instead of ˜ N , we obtain( ∂ t − ∆) θ + R∂ x f = − (cid:0) ∂ z ∂ x f ∂ x − ∂ x f ∂ z (cid:1) θ , (12a) ∂ t c − L ∇ · (cid:16) e Γ( − Rz + θ ) ∇ c (cid:17) + ∆ θ = − (cid:0) ∂ z ∂ x f ∂ x − ∂ x f ∂ z (cid:1) c , (12b) ∂ t ∆ ∂ x f − P ∆ (cid:16) e − Γ( − Rz + θ ) ∆ ∂ x f (cid:17) + P Ψ ∂ x c + P (1 + Ψ) ∂ x θ + 2 P h(cid:16) ∂ z e − Γ( − Rz + θ ) (cid:17) ∂ x + (cid:16) ∂ x e − Γ( − Rz + θ ) (cid:17) ∂ z i ∂ x f − P (cid:16) ∂ x ∂ z e − Γ( − Rz + θ ) (cid:17) ∂ x ∂ z f = − (cid:0) ∂ z ∂ x f ∂ x − ∂ x f ∂ z (cid:1) ∂ x f , (12c)together with the no-slip, impermeable boundary condi-tions θ = ∂ z c = ∂ x f = ∂ z ∂ x f = 0 at z = ± / , (13)where, for simplicity, all tildes have been suppressed. III. ONSET OF CONVECTION
The parameters at the onset of convection are deter-mined by a linear stability analysis of the basic, noncon-vective state θ = c = f = 0, as for instance described inmore detail in Ref. [38].For this purpose, the linearized equations ∂ t θ = ∆ θ − R∂ x f (14a) ∂ t c = − ∆ θ + L ∇ · (cid:0) e − Γ Rz ∇ c (cid:1) (14b)1 P ∂ t ∆ ∂ x f = − Ψ ∂ x c − (1 + Ψ) ∂ x θ + ∆ (cid:0) e Γ Rz ∆ ∂ x f (cid:1) − R e Γ Rz ∂ x ∂ x f (14c)are solved by a Fourier ansatz along the horizontal di-rection: ( θ, c, f ) = (cid:0) ¯ θ ( z ) , ¯ c ( z ) , ¯ f ( z ) (cid:1) exp( i k x + σ t ). The z -dependence of the fields ¯ θ ( z ), ¯ c ( z ), ¯ f ( z ) are expandedwith respect to orthogonal polynomials that fulfill theboundary conditions in Eq. (13). By a projection ofthe linear equations onto these polynomials (Galerkin-Method, see, e.g., Refs. [47–49]), the dynamical equa-tions are transformed into an eigenvalue problem. Bythe condition Re( σ ) = 0, the neutral curve R ( k ) forthe Rayleigh number is determined, whose minimum( R c , k c ) at the critical Rayleigh number R c and the crit-ical wavenumber k c determines the onset of convection.With ω c = Im( σ ), we denote the frequency at the thresh-old of the oscillatory onset of convection. R c (a)3.03.54.04.55.00.000 0.002 0.004 0.006 0.008 0.010 k c Γ (b) exponentiallinearexponentiallinear FIG. 1. (color online) (a) The critical Rayleigh number R c and (b) the critical wavenumber k c for a one-component fluidas a function of Γ. The solid line marks an exponentialtemperature-dependence of the viscosity, the dotted line rep-resents a linear one. The dotted line ends at Γ ≈ . × − (black point), where the viscosity becomes negative at the up-per boundary. A. Simple fluids ( ψ = 0 ) At first, let us concentrate on the effect of an expo-nentially temperature-dependent viscosity on the onsetof convection for one-component fluids ( ψ = 0).For this case, the critical Rayleigh number R c and thecorresponding critical wavenumber k c are shown in Fig. 1as a function of Γ (solid lines). Both quantities reveala non-monotonic dependence on Γ, similar to the re-sults reported in Refs. [15, 50]: while for small Γ, R c rises compared to the case of a constant viscosity, thethreshold is reduced in the limit of large Γ. This con-trasts to related studies [16], where a linear temperature-dependence of the viscosity has been assumed and whichpredict a monotonic decrease of the threshold with risingviscosity contrast. However, we can reproduce that re-sult by a linear approximation of the exponential termsin Eqs. (12), which is also shown in Fig. 1 (dotted lines)and which clearly demonstrates the importance of termshigher than the leading linear order. The velocity poten-tial f at the onset of convection is shown in Fig. 2 for twovalues of Γ. The stronger the viscosity varies in space,the more the center of the convection rolls is shifted to-wards the lower boundary and the more the fluid motionis suppressed near the upper boundary, where a highlyviscous layer forms. -0.50.00.5 -1.0 0.0 1.0 2.0 z (a) -0.50.00.5 -1.0 0.0 1.0 2.0 z x (c) (b) η/η (d) FIG. 2. (color online) Contour lines of the velocity potential f at the onset of convection for (a) Γ = 0 .
002 and (c) Γ = 0 . η ( z ) /η . The critical values are k c = 3 .
09 and R c = 1997 in (a, b) and k c = 4 .
59 and R c =1464 in (c, d). B. Binary mixtures with positive Soret effect( ψ > ) In the range of a positive Soret effect, i. e. ψ > R c and k c as functions of ψ for two rep-resentative finite values of Γ as well as for the limitingcase Γ = 0. For moderate values of Γ (dashed lines), R c and k c are higher than for Γ = 0 (dotted lines) and theirbehavior as functions of ψ , in particular the shift of k c towards zero for rising values of ψ , is pretty similar tothat for Γ = 0. However, for higher values of Γ (solidlines) and small ψ , the threshold is reduced compared toΓ = 0, which is similar to the case of a simple fluid asshown in Fig. 1. In addition, for large Γ, the decay of R c as a function of ψ becomes much weaker and, mostimportantly, at a certain value of ψ , the threshold dis-continuously jumps down to much lower values, whichare comparable to those for Γ = 0.To understand this discontinuous behavior, Fig. 4shows the neutral curves R ( k ) for two values of ψ , whichare to the right or left of the jump, respectively, andwhich are marked by circles in Fig. 3. For the largervalue of ψ [cf. Fig. 4(b)], an additional region of station-ary instability forms in the ( R, k )-plane with a minimumat lower Rayleigh numbers, which explains the disconti-nuity shown in Fig. 3.As the viscosity contrast [cf. Eq. (11)] at the onsetof convection is given by the product of Γ and R c , thejump in the critical Rayleigh number leads for ψ close tothe discontinuity to a strong change in the viscosity con- R c (a) k c ψ (b) Γ = 0 . . . FIG. 3. (color online) Critical values (a) R c ( ψ ) and (b) k c ( ψ )for Γ = 0 . . . ψ , for which neutral curves are shown in Fig. 4. R k (a) k (b) FIG. 4. (color online) Neutral curves corresponding to thecircles in Fig. 3 with (a) ψ = 0 .
02 and (b) ψ = 0 .
025 forΓ = 0 . trast at the threshold. This finally leads to very differentvelocity fields at the onset of convection for values of ψ that are to the right or to the left of the jump, which isillustrated by the velocity potential in Figs. 5 (a) and (c),respectively. For the smaller value of ψ , R c is higher [cf.Fig. 4(a)], leading to a stronger viscosity contrast [cf.Fig. 5(b)] and therefore to a pronounced shift of the flowfield towards the lower boundary [cf. Fig. 5(a)]. In con-trast, for the larger value of ψ , the threshold R c is smaller[cf. Fig. 4(b)], the viscosity contrast is much weaker [cf.Fig. 5(d)], and thus, there is only a slight shift of theconvection rolls [cf. Fig. 5(c)]. Further, the different lat-eral extension of the role structure shown in Figs. 5(a) -0.50.00.5 -1.0 0.0 1.0 2.0 z (a) -0.50.00.5 -1.0 0.0 1.0 2.0 z x (c) (b) η/η (d) FIG. 5. (color online) Contour lines of the velocity potential f at the onset of convection corresponding to (a) Fig. 4(a)( ψ = 0 . . k c ∼ = 4 . R c ∼ = 1376) and (c) Fig.4(b) ( ψ = 0 . . k c ∼ = 1 . R c ∼ = 426). (b) and(d) depict the corresponding decay of the viscosity. and (c) reflects the jump in k c [cf. Fig. 3(b)]. C. Binary mixtures with negative Soret effect( ψ < ) The most interesting effect of a strongly temperature-dependent viscosity occurs in the range of a negativeSoret effect, i. e., for ψ <
0, where with increasing valuesof Γ, the divergence of the stationary instability (in thecase of a constant viscosity [31–34]) vanishes. Further,beyond a certain L-dependent value of Γ, the onset ofconvection is no longer oscillatory for all ψ <
0, as itis known in the case of a constant viscosity. Instead, atstrongly negative values of ψ , the oscillatory instability isreplaced by a stationary one. Depending on the strengthof the exponential temperature-dependence of the viscos-ity, the transition from a Hopf-bifurcation to a stationaryinstability with decreasing ψ can show a discontinous ora continous threshold behavior.
1. Discontinous transition from an oscillatory to astationary instability
For moderate values of Γ, the transition between bothtypes of instabilities is characterized by a discontinousjump in R c , k c , and ω c , as exemplarily illustrated inFig. 6 for Γ = 0 . R ( k ) for different ψ are shown in Fig. 7, where red linesindicate those parts of the neutral curves where the fre-quency ω is finite, while blue lines represent a station-ary instability with ω = 0. For small | ψ | , the minima R c k c k c ω c ψ k c FIG. 6. (color online) (a) Critical Rayleigh number R c , (b)critical wavenumber k c , and (c) critical frequency ω c as func-tions of ψ < . of the neutral curves (green line) belong to an oscilla-tory instability. However, with increasing | ψ | , this regiontransforms into an oscillatory island, which finally dis-appears, while the stationary branch of the curve, whichshows a minimum at larger values of k , remains. In con-sequence, for even larger | ψ | , convection sets in stationaryat a higher threshold and a considerably increased criticalwavenumber. These changes are directly reflected in thevelocity potential at the onset of convection, as shownin Fig. 8 (a,c): while Fig. 8(a) shows travelling waves inthe regime of the oscillatory instability, Fig. 8(c) displaysstationary convection rolls with a much smaller lateralwidth (due to the jump in k c ), which are also much moreshifted to the lower boundary [due the higher thresholdand, hence, the more pronounced viscosity contrast, cf.Figs. 8(b,d)].
2. Continous transition from an oscillatory to a stationaryinstability
For larger values of Γ, the transition between the Hopfand the stationary bifurcation is still characterized byjumps in the critical wavenumber and the critical fre-quency, but does no longer show a discontinuity in thethreshold, as illustrated in Fig. 9 for Γ = 0 . R k ψ = − . ψ = − . ψ = − . ψ = − . ψ = − . FIG. 7. (color online) Neutral curves for Γ = 0 .
003 anddifferent ψ < c ∼ = − .
5, wherethe transition from an oscillatory to a stationary instabilitytakes place. At that point, the minimum of the neutral curvesshows a discontinous jump in k c and R c (points). -0.50.00.5 -1.0 0.0 1.0 2.0 z (a) -0.50.00.5 -1.0 0.0 1.0 2.0 z x (c) (b) η/η [10 ] (d) FIG. 8. (color online) Contour lines of the velocity potential f at the onset of convection corresponding to Fig. 7 for (a) ψ = − . k c ∼ = 4 . R c ∼ = 3864 and (c) ψ = − . k c ∼ =11 . R c ∼ = 5060. (b) and (d) depict the corresponding decayof the viscosity. of forming an oscillatory island, that, for rising | ψ | , iseventually disappearing, here, as displayed in Fig. 10, forrising | ψ | , the minimum of the oscillatory branch of theneutral curves moves higher and higher. At a certainvalue of ψ , the minima of the oscillatory and stationarybranches are of equal height and with further increasing | ψ | , the minimum of the stationary branch is finally lowerand determines the onset of convection. The changes ofthe velocity potential near this new codimension-2-point R c (a) k c (b) ω c ψ (c) FIG. 9. (color online) (a) Critical Rayleigh number R c , (b)critical wavenumber k c , and (c) critical frequency ω c as func-tions of ψ < . are similar to those depicted in Fig. 8. The stationarybranch of the critical Rayleigh number, shown in Fig 9(a)(blue line) in the range of ψ < . ψ > | ψ | . When further in-creasing Γ, this trend continues, i.e., for rising strength ofthe exponential temperature dependence of the viscosity,the region in the parameter range ψ <
0, where convec-tion sets in via a Hopf bifurction, becomes smaller andsmaller.
IV. SUMMARY AND CONCLUSIONS
The parameters at the onset of convection are deter-mined in a binary fluid mixture where the viscosity de-pends exponentially on the temperature.As explicitely shown for a single component fluid, thecritical values at the onset of convection behave as a func-tion of the viscosity difference between the lower, warmerand the upper, coulder boundary differently for a lineartemperature dependent and an exponentially tempera-ture dependent viscosity, respectively.In the range of a positive separation ratio ψ , we find, asa function of ψ , for larger values of the viscosity contrast R k ψ = − . ψ = − . ψ ≈ − . ψ = − . FIG. 10. (color online) Neutral curves for Γ = 0 .
004 anddifferent ψ < ψ c ∼ = − . k c (points). a discontinuous change of the critical Rayleigh numberas well as of the critical wavelength of the convectionrolls, in contrast to their continuous behavior in the rangeof a constant viscosity and small values of the viscositycontrast.The strongest qualitative influence of an exponentially dependent viscosity at the onset of convection we findin the range of negative values of the separation ratio ψ . In molecular binary mixtures, for ψ <
0, below theonset of concevtion, the minor and heavier component ofthe fluid mixture is, via the Soret effect, enriched near thelower and warmer boundary. In geophysical applications,where also double diffusive models are applied, the Soreteffect does not play a very strong role, but due to grav-itation, the heavier minor component of the mixture issimilarly accumulated in the lower warmer range of theconvection layer. For molecular binary fluids, such aswater-alcohol mixtures, it is common that in closed con-vection cells, one has an oscillatory onset of convectionin the range ψ <
0. However, beyond the threshold, theconcentration gradient is quickly reduced by the convec-tive motion, which soon leads to a stationary convectionpattern again [30]. In the case of an exponentially tem-perature dependent viscosity of the binary mixture, wefind in the range ψ < ψ <
0, in which the onsetof convection is still oscillatory, shrinks with increasingviscosity contrasts. According to this result for closedconvetion cells, we expect also in model systems, whereone has nonvanishing currents of the minor componentthrough the lower boundary [26, 44] and that are of im-portance for geohysical situations, a stationary onset ofconvection.
ACKNOWLEDGMENT
We are grateful to Georg Freund for instructive dis-cussions about how to implement the Galerkin methodin an efficient way. [1] M. Lappa,
Thermal Convection: Patterns, Evolution andStability (Wiley, New York, 2010).[2] P. Ball,
The Self-Made Tapestry: Pattern Formation inNature (Oxford Univ. Press, Oxford, 1998).[3] D. L. Turcotte and G. Schubert,
Geodynamics (Cam-bridge Univ. Press, Cambridge, 2002).[4]
Mantle Dynamics , edited by D. Bercovici (Elsevier, Am-sterdam, 2009).[5] F. H. Busse, in
Convection: Plate Tectonics and GlobalDynamics , edited by W. R. Peltier (Gordon and Breach,Montreux, 1989), pp. 23–95.[6] A. Davaille and A. Limare, in
Mantle Convection , editedby D. Bercovici (Elsevier, ADDRESS, 2009), p. 89.[7] R. A. Houze,
Cloud Dynamics (Academic Press, NewYork, 1994).[8] B. Stevens, Annu. Rev. Earth Planet Sci. , 605 (2005).[9] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. ,851 (1993).[10] M. C. Cross and H. Greenside, Pattern Formation andDynamics in Nonequilibrium Systems (Cambridge Univ. Press, Cambridge, 2009).[11] E. Palm, J. Fluid Mech. , 183 (1960).[12] O. Jensen, Acta Polytech. Scand. , 1 (1963).[13] K. E. Torrance and D. L. Turcotte, J. Fluid Mech. ,451 (1970).[14] J. R. Booker and K. C. Stengel, J. Fluid Mech. , 289(1978).[15] K. C. Stengel, D. S. Oliver, and J. R. Booker, J. FluidMech. , 411 (1982).[16] F. H. Busse and H. Frick, J. Fluid Mech. , 451 (1985).[17] D. B. White, J. Fluid Mech. , 247 (1988).[18] F. M. Richter, H.-C. Nataf, and S. F. Daly, J. Fluid Mech. , 173 (1983).[19] U. R. Christensen and H. Harder, Geophys. J. Int. ,213 (1991).[20] S. Balachandar, D. A. Yuen, D. M. Reuteler, and G. S.Lauer, Science , 1150 (1995).[21] P. Tackley, J. Geophys. Res. , 3311 (1996).[22] S. Androvandi et al. , Phys. Earth Planet. Inter. , 132(2011). [23] J. S. Turner, Buoyancy Effects in Fluids (CambridgeUniv. Press, Cambridge, 1973).[24] H. E. Huppert and J. S. Turner, J. Fluid Mech. , 299(1981).[25] E. Giannandrea and U. R. Christensen, Phys. EarthPlanet. Inter. , 139 (1993).[26] A. Manglik, J. Wicht, and U. R. Christensen, EarthPlant. Sci. Lett. , 619 (2010).[27] T. Tr¨umper, M. Breuer, and U. Hansen, Phys. EarthPlant. Int. , 53 (2012).[28] H. R. Brand, P. C. Hohenberg, and V. Steinberg, Phys.Rev. A , 2548 (1984).[29] W. Hort, S. Linz, and M. L¨ucke, Phys. Rev. A , 3737(1992).[30] M. L¨ucke et al. , in Evolution of Spontaneous Structuresin Dissipative Continuous Systems , edited by F. H. Busseand S. C. M¨uller (Springer, Berlin, 1998).[31] M. C. Cross and K. Kim, Phys. Rev. A , 3909 (1988).[32] E. Knobloch and D. R. Moore, Phys. Rev. A , 860(1988).[33] W. Sch¨opf and W. Zimmermann, Europhys. Lett. , 41(1989).[34] W. Sch¨opf and W. Zimmermann, Phys. Rev. E , 1739(1993).[35] R. Cerbino, S. Mazzoni, A. Vailati, and M. Giglio, Phys.Rev. Lett. , 064501 (2005).[36] B. Huke, H. Pleiner, and M. L¨ucke, Phys. Rev. E ,046315 (2008). [37] G. Donzelli, R. Cerbino, and A. Vailati, Phys. Rev. Lett. , 104503 (2009).[38] M. Gl¨assl, M. Hilt, and W. Zimmermann, Eur. Phys. J.E , 265 (2010).[39] F. Winkel et al. , New J. Phys. , 053003 (2010).[40] M. Gl¨assl, M. Hilt, and W. Zimmermann, Phys. Rev. E , 046315 (2011).[41] J. K. Platten and L. C. Legros, Convection in Liquids (Springer, Berlin, 1984).[42] L. D. Landau and E. M. Lifshitz,
Course of TheoreticalPhysics, Vol. 6Fluid Mechanics (Butterworth, Boston,1987).[43] J. Tanny and V. A. Gotlib, Int. J. Heat Mass Transfer. , 1683 (1995).[44] A. Mambole, G. Labrosse, E. Tric, and L. Fleitout, Stud.Geophys. Geod. , 519 (2004).[45] C. V. Raman, Nature , 532 (1923).[46] R. H. Eweell and H. Eyring, J. Chem. Phys. , 726(1937).[47] R. M. Cleber and F. H. Busse, J. Fluid Mech. , 625(1974).[48] C. Canuto, M. Hassaini, A. Quarteroni, and T. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag,Berlin, 1987).[49] W. Pesch, Chaos , 348 (1996).[50] M. Kameyama, H. Ichikawa, and A. Miyauchi, Theor.Comput. Fluid Dyn.27