Seminar Ia - 1. year, 2. Cycle

Grossmann - Lohse theory forturbulent Rayleigh-Benard

convection

Rok Grah

Advisor: Tomaz Prosen

Ljubljana, March 2014

Abstract

We review the Grossmann-Lohse theory of global properties for Rayleigh and Prandtlnumber dependence on Nusselt number and Reynolds number in the turbulent Rayleigh-Benard convection. As we have heard very little about this kind of hydrodynamicinstability so far in the course of our studies, we shall derive the equations of motionusing the Boussinesq approximation. The goal of the first part is to show where dosome important dimensionless numbers in hydrodynamics come from and what do theyrepresent.

The main part is dedicated to the Grossmann-Lohse theory which is one of the firstsuccessful theories to describe the Ra and Pr dependence of Nu and Re over the wideparameter range. The theory is based on the decomposition of the energy dissipationrate into boundary layer and bulk contribution.

Contents

1 Reyleigh-Benard convection 1

2 The equations of motion 22.1 The Boussinesq equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Important dimensionless numbers . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Theories of global properties: Nu(Ra,Pr) and Re (Ra, Pr) 53.1 Grossmann-Lohse theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Conclusion 10

Introduction

The observation of thermal instability goes back a century to 1882, when James Thompson,the older brother of better known Lord Kelvin, had first described the convective instability.It was later Benard in 1900 who has discovered in the first quantitative experiments themaking of regular hexagons in convection cells in silicone oil (Figure 1). Impressed bythese results, Rayleigh in 1916 formulated a theory of convection instability of a layer offluid between horizontal planes. He showed that the instability would occur only when theadverse temperature gradient was large enough [1].

Figure 1: An example of Benard convection cells in hexagonal shape [2].

1 Reyleigh-Benard convection

Rayleigh-Benard convection is the buoyancy driven flow where the fluid is heated frombelow and cooled from above. When the temperature difference across the layers is largeenough, the stabilizing effects of viscosity and thermal conductivity are overcome by thedestabilizing buoyancy, and an overtuning instability is shown as thermal convection. Thisinstability can be distinguished from free convection. For example imagine a hot verticalplate, for which hydrostatic equilibrium is impossible [3], [1].

Reyleigh-Benard convection is a classical problem in fluid dynamics. It has played animportant role in the development of stability theory in hydrodynamics and it made thebasis in the study of pattern formation and spatial-temporal chaos.

Thermally driven flows play an important role in a variety of physical problems. Thermalconvection can be found in the atmosphere [4], in the oceans [5], process technology, in metal-production processes [6], and even in buildings [7]. It has also been seen in geophysical and

1

astrophysical context, such as convection of Earth’s mantle [8], in the Earth’s outer core[9], in stars including our Sun [10], and in generation and reversal of the Earth’s magneticfield [11]. As we see thermal convection is a very important physical phenomenon as it isfound in all branches of physics [3].

2 The equations of motion

The Reyleigh-Benard convection problem can be described by the following question: For agiven fluid in a closed container of height L heated from below and cooled from above, whatare the flow properties inside the container, and what is the heat transfer from bottom totop. Here we will assume spatially and temporally constant temperatures at the bottomand top.

In the notation of Cartesian tensors with position vectors x = xj and velocity u =uj (j = 1, 2, 3), the equations written in the standard notation where we sum over repeatedindices are as follows

∂ρ

∂t+∂(ρuj)

∂xj= 0. (1)

This is the the equation of continuity where ρ is the density of the fluid. The equation ofmotion is the Navier-Stokes equation:

ρDuiDt

= −gρδi3 +∂σij∂xj

, (2)

where D/Dt = ∂/∂t + u · ∇, the x3-axis is the upward vertical, g is the gravitationalacceleration, and the stress tensor is given by

σij = −pδij + µ

(∂ui∂xj

+∂uj∂xi− 2

3

∂ukxk

δij

)+ λ

∂ukxk

δij , (3)

where p is the pressure, µ is the coefficient of dynamic viscosity of the fluid, and λ is thecoefficient of bulk viscosity.

Next is the equation of energy, or of heat conduction which we obtain from [1], [3], [12]

ρDE

Dt= − ∂

∂xj

(k∂T

∂xj

)− p∂uj

∂xj+ Φ, (4)

where E is the internal energy per unit mass of the fluid, k is the thermal conductivity, andΦ is the rate of viscous dissipation per unit volume of fluid which is given by

Φ =1

2µ

(∂ui∂xj

+∂uj∂xi

)2

+

(λ− 2

3µ

)(∂uk∂xk

)2

(5)

For a perfect gas E = cvT , and for a liquid E = cT , where cv is the specific heat at constantvolume and c is the specific heat at constant pressure [1], [3].

2.1 The Boussinesq equations

To upper equations of motion, Rayleigh in 1916 applied the Boussinesq approximation. Themain idea behind this approximation is that there are flows in which the temperature variesonly a little, and therefore the density varies only a little, but the buoyancy still drives the

2

motion. Then we can neglect the variation of density everywhere except in the buoyancy.For small differences in temperature we can therefore write

ρ = ρ0{1− α(T − T0)}, (6)

where α is the coefficient of thermal cubical expansion, ρ0 is the density of the fluid atthe temperature T0 at the bottom of the layer. We already know that for a perfect gasα = 1/T0 ≈ 3 × 10−3K−1, and for a typical liquid used in experiments α ≈ 5 × 10−4K−1.Therefore if T0 − T ≈ 10K or less, then

ρ− ρ0ρ0

= α(T0 − T )� 1, (7)

but the buoyancy g(ρ−ρ0) is still of the same order of magnitude as the inertia, accelerationor viscous stresses of the fluid and is therefore not negligible [1], [3].

For most real fluids dµ/µ dT , dk/k dT , dc/c dT ≤ α, so µ, k, and c or cv can bethreated as constants in the Boussinesq approximation. The coefficient of bulk viscosity λis neglected, as it only arises as a factor of ∂uj/∂xj , which is of order α.To sum up, one approximates all the thermodynamic variables as constants except for thepressure and temperature. Density is a non-constant variable only when multiplied with g[1].

In the continuity equation the differentials of density are of order α, so the Boussinesqapproximation gives (where we of cource sum over repeated indices)

∂uj∂xj

= 0, (8)

which is an approximation for incompressible fluid. If we threat ρ and µ as constants inevery term except for the buoyancy, the Navier-Stokes equations become

DuiDt

= − ∂xi

(p

ρ0+ gz

)− αg(T0 − T )δi3 + ν∆ui, (9)

where ∆ is the Laplacian operator, given by ∆ =∑

j ∂2/∂x2j , and ν is kinematic viscosity

[1]. The kinematic viscosity is defined as ratio of dynamic viscosity and the density of thefluid. Sometimes it is referred to as diffusivity of momentum as it is analogous to diffusivityof heat.

In the energy we can neglect the viscous dissipation Φ as the ratio of the rate of pro-duction of heat by internal friction to the rate of transfer of heat is

Φ

ρD(cT )/Dt≈ µU2L−1

ρ0cδTUL−1=

νU

cδTL, (10)

where U represents velocity scale of the flow, L a length scale, and δT = (T0 − T1) a scaleof temperature difference, where T1 is the temperature of the top layer of thickness L. Fora typical gas ν/cv ≈ 10−8s K and for a typical liquid ν/c ≈ 10−9s K. If we assume thatU/(δTL) is not very large, the upper ratio is very small for both gases and liquids. Thereforewe can neglect Φ in our equations [1], [3].

Now note that heating due to compression is given by

−p∂uj∂xj

=p

ρ= αp

DT

Dt(11)

3

where for a perfect gas, p = (cp − cv)ρT and α = 1/T . Now it follows

ρDE

Dt+ p

ujxj∼= cpρ

DT

Dt(12)

One might expect from the approximation for the incompressible fluid that heating dueto compression is negligible but this is not the case. For liquids, however, the heating isnegligible at normal pressures. The reason for the difference between liquids and gases ismainly that the heating due to compression is normally an order of magnitude less for aliquid than for a gas. The heat transfer is also proportional to the density of the fluid, anda typical fluid is 103 times more dense than a typical gas [1], [3].Now using all these approximations and the approximation for incompressible fluid we get

DT

Dt= κ∆T, (13)

where κ is thermal diffusivity with κ = k/(ρ0cp) for perfect gas and κ = k/(ρ0c) for a liquid[1].

Equations (8), (10) and (14) are called Boussinesq equations and describe the motion ofBoussinesq fluid.

2.2 Important dimensionless numbers

Now we will focus on the boundary conditions of the Boussinesq equation. Let the twohorizontal planes have equations z∗ = 0 and z∗ = L, where the temperatures are T0 andT1, respectively. Here we denote the dimensional variable with an asterisk as we shall soonintroduce dimensionless variables.

The equations of motion for velocity, temperature, and pressure are therefore for thebasic state (that is a state where the gradient is a linear function and no velocity is observed)given by u∗ = 0, T∗ = T0 − βz∗, P∗ = p0 − gρ0(z∗ + 1

2αβz2∗) for 0 ≤ z∗ ≤ L. Let us repeat

what the parameters are; α is coefficient of thermal cubical expansion, ρ0 the density of thefluid at bottom of the layer, and β = (T0 − T1)/L is the basic temperature gradient. Weassume that an instability can only occur when T0 > T1, meaning the temperature gradientβ > 0.

If we linearize the Boussinesq equations for small perturbations u′∗, T′∗, and p′∗ with

u∗ = u′∗(x∗, t∗), T∗ = T∗(z∗) + T ′∗(x∗, t∗), p∗ = P∗(z∗) + p′∗(x∗t∗) we get

∇∗ · u′∗ = 0, (14)

∂u′∗∂t∗

= − 1

ρ0∇∗p′∗ + αgT ′∗k + ν∆∗u

′∗, (15)

∂T ′∗∂t∗

= βw′∗ + κ∆∗T′∗, (16)

where w′∗ is the velocity in z-direction, and n is the normal to the lower plate [1], [3].Now it is convenient to define dimensionless variables using L for the scale of length,

L2/κ for scales of time, and βL = T0 − T1 for scales of temperature difference;

x =x∗L, t =

κt∗L2

, (17)

u =Lu′∗κ, T =

T ′∗βL

, p =L2p′∗ρ0κ2

, (18)

4

The linearized stability equations (14-16) then become

∇ · u = 0, (19)

∂u

∂t= −∇p+ Ra Pr n + Pr ∆u, (20)

∂T

∂t= w + ∆T, (21)

where the dimensionless Rayleigh number is given by

Ra =gαβL4

κν, (22)

One should note that the Ra is positive when T0 > T1 and it represents the characteristicratio of buoyancy to the viscous forces [1].Prandtl number is given by

Pr =ν

κ(23)

Prandtl number is an intrinsic property of the fluid, not of the flow. It is a measurement ofthe ratio of the rates of molecular diffusion of momentum and heat.

When a system is imposed by Ra, the heat flux H appears from bottom to top. We canaccordingly define dimensionless heat flux called Nusselt number as

Nu =H

k∆TL−1, (24)

where ∆T is the temperature difference between top and bottom plate.Furthermore we define Reynolds number as

Re =U

νL−1, (25)

where U represents velocity scale of the flow. As the Reynolds number has already beenvastly discussed in the first cycle of faculty courses, we will skip the description of itsderivation and importance [3].

We shall stop here and will not exploit more about the solutions of the equations (27-29)as this is not the main topic of this seminar.

3 Theories of global properties: Nu(Ra,Pr) and Re (Ra, Pr)

The key question is to ask ourselves how do Nu and Re depend on Ra and Pr.Older theories predict power laws for dependences of Nu and Re on Ra and Pr

Nu ∼ RaγNuPrαNu, (26)

Re ∼ RaγRePrαRe, (27)

and are mostly obtained from approximations at certain regimes of the variables. Theseolder theories are all of limited range and they do not offer a unifying view, accounting forall data [3], [13].

We will now describe the Grossmann-Lohse theory which offers a unifying view over awide range of parameter.

5

3.1 Grossmann-Lohse theory

The main approach of the Grossmann-Lohse theory is to decompose the kinetic energydissipation rate εu and the thermal dissipation rate εT into their boundary layer (BL) andbulk contributions (see Figure 2) [12], [14], [15], [3]:

εu = εu,BL + εu,bulk, (28)

εT = εT,BL + εT,bulk (29)

Figure 2: Boundary-bulk partition. We see the sketch splitting of the (a) kinetic and (b)thermal dissipation rates on which the Grossmann-Lohse theory is based. In both figuresthe large scale convection which roll with typical velocity amplitude U is sketched. Thetypical width of the kinetic BL is λu, whereas the typical thermal BL thicknesses and theplume thicknesses are λT . Outside the BL/plume region we see background (bg) / bulkflow [3].

The exact relations assuming statistical stationarity for the left-hand sides following[13], [14] are

εu = 〈ν[∂iuj(x, t)]2〉V =

ν3

L4(Nu− 1) Ra Pr−2, (30)

εT = 〈κ[∂iT (x, t)]2〉V = κ∆T 2

L2Nu (31)

As the next description of the Grossmann-Lohse theory is a bit difficult to understand,we will divide it into seperate parts so we will easily distinguish between different importantsteps in the understanding of the theory.

1. The local dissipation rates of both kinematic and thermal contribution are modelledin the terms of U, ∆T, L, and the widths λu, λT of the kinetic and thermal boundarylayers, respectively [14]. For the thickness of the thermal boundary layer we assumeλT = L/(2Nu) and for the kinetic width λu = L/(4

√Re), same as for Blasious type

layers [12], [14], [16]. For very large Ra the laminar boundary layer will becometurbulent and λu will show stronger Re dependence. One should also note that whilethermal boundary layer occurs only at the top and the bottom plate, the kineticboundary layer occurs at all walls of the cell [14].

2. The modeling of both dissipation rates in equations (28) and (29) is obeying theBoussinesq equations. It depends on which contribution is dominant (BL or bulk),

6

the solutions for relation for Nu, Re vs Ra, Pr differ, and therefore defining differentmain regimes; see Figure 3 [14], [3]. In regime I both εu and εT are dominated bytheir boundary contributions; regime II occurs when εT is dominated by εu,BL and εuis dominated by εT,bulk; regime III comes into consideration when εT is dominated byεu,BL and εT is dominated by εT,bulk; regime IV occurs when both εT and εu are bulkdominated.

3. The BL and bulk thermal dissipation rates therefore depend on whether the kineticBL of thickness λu is within the thermal BL of thickness λT (meaning λu < λT , thatis for small Pr) or vice versa (meaning λT < λu, that is for large Pr); see Figure 4.The boundary case when λu = λT , which corresponds to Nu = 2

√Re, cuts the phase

diagram into two parts; lower with small Pr is labelled by ”l”, upper with large Prwith ”u” (see Figure 2 and Figure 3 ) [14].

Figure 3: Phase diagram in the Ra-Pr plane. The tiny regime to the right of regime Il isregime IIIl. The dashed line is λu = λT . The shaded regime for large Pr is where Re≤ 50,and in the shaded regime for low Pr we have Nu= 1. The dotted line indicates the non-normal-nonlinear onset of turbulence in the BL shear flow which we will not discuss here[12].

Figure 4: Sketch of the boundary layers with large-scale velocity U , (a) for low Pr whereλu < λT and (b) for large Pr where λu > λT [12].

4. We first consider λu < λT (small Pr, regime ”l”). Assuming large-scale velocity Ustirring the bulk, one can estimate the bulk dissipation rate εu,bulk by balancing the

7

dissipation with the large-scale convective term in the energy equation. With thesame analogy we can also get εT,bulk. It follows from [12], [14]

εu,bulk ∼U3

L∼ ν3

L4Re3, (32)

εu,BL ∼ νU2

λ2u

λuL∼ ν3

L4Re3, (33)

εT,bulk ∼U∆T 2

L∼ κ∆T 2

L2Re Pr, (34)

εT,BL ∼ κ∆T 2

L2(Re Pr)1/2 (35)

5. If we consider larger Pr (so we go from low Pr numbers in step 4 to medium sizesPr), then the kinetic boundary layer will eventually exceed the thermal one, λu > λT ,meaning the upper range ”u” (see Figure 3). The relevant velocity at the edge betweenthermal BL and the thermal bulk is now less than U , within our approximationabout UλT /λu; see Figure 4. For the transition from λu being smaller to beinglarger than λT we introduce a new function f(x) = (1 + xn)−1/n of the variablexT = λu/λT = Nu/(2

√Re) [14], [3]. The function f equals 1 in the lower range

(called ”l”, meaning small in Pr) and ≈ 1/xT in the upper range (called ”u”, meaninglarge in Pr). The relevant velocity then becomes Uf(xT ). Grossmann and Lohse [12],[14] took n = 4 to characterize the sharpness of the transition. We should note thatthis function is (within the theory) universal for all fluids. The correspondence to theexperiments shows that the choice of function is justified. Using this correction weget

εT,bulk ∼ κ∆T 2

L2Re Prf

(Nu/(2

√Re)), (36)

εT,BL ∼ κ∆T 2

L2

[Re Prf

(Nu/(2

√Re))]1/2

, (37)

while εu,bulk and εu,BL are still given by equations (32-33). Using those equationstogether with equations (30-31) we get Ra, Pr dependences of Nu, Re in the upperregime ”u”.

6. Let us now see the theory for very large Prandtl numbers. In step 4 we have smallPr numbers, in step 5 medium sized Pr numbers and now the extension to very largePrandtl numbers. ⇒ As Rac (critical Ra number when instability occurs) is indepen-dent of Pr, the wind still exists for Ra> Rac = 1708 [14]. But for large scale convectionRe slows down with increasing Pr. It eventually becomes laminar throughout the cell[14]. λu in the equation λu = L/(4

√Re) does no longer increase with decreasing Re,

but will close up to a value of order L. We shall call the corresponding number Recand λT = L/(4

√Rec) in the regime of very large Pr. As the transition for Re after

Rec is not smooth, we use the crossover function g(x) = x(1 + xn)−1/n of the variablexL = λu(Re)/λu(Rec) =

√Rec/Re, and here also n = 4 [14], [3]. Below the transition

(xL small) function g increases linearly g(xL) = xL, and equals 1 at very large regimesof Pr with Re ≤ Rec. All this put together means that we have to replace λu withg(xL)λu(Rec) [14].

8

In our case equation (32) still holds even at very large Pr regime as εu,bulk doesnot contribute much to εu due to large extension of the kinetic boundary layers.Nevertheless the equations (33-35) have to be generalized

εu,BL ∼ν3

L4

Re2

g(√

Rec/Re), (38)

εT,bulk ∼ κ∆T 2

L2Re Prf

[Nu

2√

Recg

(√RecRe

)], (39)

where equation (39) simplifies for large enough Ra or very large Pr (meaning large fargument or large g argument, respectively) to

εT,bulk ∼ κ∆T 2

L2

Pr Re

Nu(40)

For example how to obtain power laws describing heat flux and velocity, one insertsthe equations (38) and (39) into the right side of equations (38) and (39) in the regimecalled III∞ for large enough Ra and Pr [12], [14]

Nu ∼ Ra1/3Pr0, Re ∼ Ra2/3Pr−1. (41)

The generalized εT,BL (equation 37) beyond Il (relevant for medium Ra) stays thesame as its derivation did not involve λu and function f = 1.

7. Putting all the generalized equations in equations (30) and (31), Grossmann and Lohse[14], [3] obtained the results of the Grossmann-Lohse theory

Nu Ra Pr−2 = c1Re2

g(√

Rec/Re) + c2 Re3, (42)

Nu = c3 Re1/2 Pr1/2

{f

[Nu

2√

Recg

(√RecRe

)]}1/2

+ c4 Pr Ref

[Nu

2√

Recg

(√RecRe

)] (43)

Some dimensionless prefactors were added to complete the modelling of the dissipationrates. They were adopted by nonlinear fit from a set of experimental data points forNu(Ra,Pr) and the results for aspect ratio Γ = 1 were c1 = 120, c2 = 74, c3 =0.89, c4 = 0.048, and Rec = 0.28 [17], [14]. We should note that ci and Rec slightlydepend on the aspect ratio and are not universal.

All limiting, pure scaling regimes derived from equations (42) and (43) can be seen inTable 1 where the approximations for typical sizes of variables (Pr and Ra) at each regimeswere made and typical sizes of arguments in functions f and g were used. The phase diagramhas been drawn in Figure 3. One has to note that even though lines indicate transitionshave been made, the crossovers are nothing but sharp.

The success of the theory is of course in the correspondence to the actual experimentaldata. We can see on Figure 5 data together with GL prediction for Re in regime I. Theagreement of the slope is very good. For more information see Figure 5.

9

Regime Dominance of BLs Nu Re

Il εu,BLεT,BL λu < λT 0.22Ra1/4Pr1/8 0.063Ra1/2Pr−3/4

Iu λu > λT 0.31Ra1/4Pr−1/12 0.073Ra1/2Pr−5/6

IIl εu,bulkεT,BL λu < λT 0.37Ra1/5Pr1/5 0.17Ra2/5Pr−3/5

IIu λu > λT 0.51Ra1/5 0.19Ra2/5Pr−2/3

IIIu εu,BLεT,bulk λu > λT 0.018Ra3/7Pr−1/7 0.023Ra4/7Pr−6/7

III∞ λu = L/(4√

Rec) > λT 0.027Ra1/3 0.015Ra2/3Pr−1

IVl εu,bulkεT,bulk λu < λT 0.0012Ra1/2Pr1/2 0.025Ra1/2Pr−1/2

IVu λu > λT 0.050Ra1/3 0.088Ra4/9Pr−2/3

Table 1: The pure power laws for Nu and Re. The prefactors ci are based on the valuesgiven below equation (43) [14].

On Figure 6 one can see the functional dependence of Ra over a larger area on Nu. Theagreement with the GL theory is again very good, for larger Ra some deviations occur.The theory is much better for data that is not corrected for finite plate conductivity as ourapproximation did not include it. See Figure 6.

Figure 5: Data for Raynolds numbers as a function of Ra and Pr from [18]. (a) showsRe/Pr−3/4 vs Ra. The expected slope (in regime Il) is 1/2, the linear regression fit (solidline) gives 0.492 ± 0.002. (b) shows Re/Ra1/2 vs Pr. The extended slope is −3/4 forPr ≤ 2 (regime Il) and −5/6 for Pr≥ 2 (regime Iu). The linear regression fit (dashedline) gives −0.77± 0.01. The agreement of the prefactor is also excellent. According to thetheory, following table 1 we get log10(Re/Ra1/2) = −1.432−(3/4) log10Pr (solid line, hardlydistinguishable from the dashed one); the fit value for the prefactor from linear regressionis −1.413± 0.005 [12].

4 Conclusion

We have seen a global theory for Ra and Pr dependence of Nu and Re. We should rememberwhat is the main idea behind the theory and how do we take into account different sizes ofboundary layers and their transitions into other regimes.The main result of GL theory involves 6 dimensionless prefactors which are slightly de-

10

Figure 6: Nusselt number versus Rayleigh number. Reduced Nu for Γ = 1, obtained usingwater (Pr≈ 4.4) and copper plates, as a function of Ra. Open simbols represent uncorrecteddata. Solid symbolds, after correction for the finite plate conductivity. Circles [19], squares[20]. The upwards and downwards triangles are upper and lower bounds on the actual Nu atlarge Ra. The diamond originate from an estimate. The solid line is the Grossmann-Lohseprediction [3].

pendent on the aspect ratio and are therefore not entirely universal. The results of thetheory are very good, the prediction of the theory is in agreement with data obtained fromsimulations and experiments. We can conclude that this field which has been discussed forcenturies will still continue to grow rapidly in the future.

References

[1] P. Drazin, and W. H. Reid, Hydrodynamic stability (Cambridge University Press, Cam-bridge, 1981)

[2] http://openlandscapes.zalf.de/openlandscapeswiki glossaries/Ecosystem%20Theories%20-%20Thermodynamics.aspx (5.3.2014)

[3] G. Ahlers, S. Grossmann, and D. Lohse, Rev. Mod. Phys 81, 503 (2009).[4] D. L. Hartmann, L. A. Moy, and Q. Fu, J. Clim. 14, 4495 (2001).[5] J. Marshall, and F. Scottm, Rev. Geophys. 37, 1 (1999).[6] A. D. Brent, V. R. Voller, and K. J. Reid, Numer. Heat Transfer 13, 297 (1988).[7] G. R. Hunt, and P. F. Linden, Build. Environ 34, 707 (1999).[8] J. Marshall, and F. Schott, Rev. Geophys. 37, 1 (1999).[9] P. Cardin, and P. Olson, Phys. Earth Planet. Inter. 82, 235 (1994).

[10] F. Cattaneo, T. Emonet, and N. Weiss, Astrophys. J. 588, 1183 (2003).[11] G. A. Glatzmaier, and P. H. Roberts, Nature (London) 377, 203 (1995).[12] S. Grossmann, and D. Lohse, J. Fluid Mech. 407, 27 (2000).[13] B. I. Shraiman, and E. D. Siggia, Phys. Rev. A 42, 3650 (1990).[14] S. Grossmann, and D. Lohse, Phys. Rev. Lett. 86, 3316 (2001).[15] S. Grossmann, and D. Lohse, Phys. Rev. E 66, 016305 (2002).[16] L. D. Landau, and E. M. Lifshitz, Fluid Mechanics (Pergamom Press, Oxford, 1987)[17] G. Ahlers, and X. Xu, Phys. Rev. Lett. 86, 3320 (2001).[18] X. Chavanne et al, Phys. Rev. Lett 79, 3648 (1997).[19] D. Funfschiling, E. Brown, A. Nikolaenko, and G. Ahlers, J. Fluid. Mech. 536, 145

(2005).[20] C. Sun, L. Y. Ren, H. Song, and K.Q. Xia, J. Fluid. Mech. 542, 165 (2005).

11