VOLUME 73, NUMBER 11 PHYSICAL REVIEW LETTERS 12 SEPTEMBER 1994

Entropy Cascade and Energy Inverse Transfer in Two-Dimensional Convective Turbulence

Sadayoshi Toh and Eri SuzukiDepartment of Physics, Faculty of Science, Kyoto University, Kyoto 606 Ol, -Japan

(Received 1 April 1994)

Two-dimensional system modeling a central region in an experimental box on the basis of theBoussinesq approximation is examined numerically. It is assumed that the effect of mixing andboundary layers on the central region is taken into account only as a forcing to the temperaturedynamics. Neutrally stable stratification and homogeneity are also assumed. Entropy (T2) spectraare compared with ones obtained by experiments. It is shown that smaller scales are governed by theentropy cascade process and large-scale modes are induced through the energy inverse transfer.

PACS numbers: 47.27.Eq, 47.27.Te

Buoyancy drastically changes the characteristics ofturbulence. Fully developed convective turbulence wasfound by Chicago groups and called "hard turbulence"[1—4]. Temperature power spectrum contains a power-law region which is regarded as the Bolgiano-Obukhov(BO) spectra derived in 1959 [5]. Even in fully developedturbulence, a robust large-scale circulation exists. In thissense, convective turbulence includes both fluctuation andorder.

After a series of experimental works, several authorstheoretically reexamined the BO spectra [6—8]. L'vovpointed out that two conserved quantities exist in theinviscid limit based on the Boussinesq approximation.One is the total energy and the other is the entropyapproximated by T2. Strictly speaking, any function oftemperature is conserved. Thus L'vov selected T as themost meaningful physical quantity. We will follow histerminology here. He derived the BO spectra assumingthat not the kinetic energy but the entropy is cascadedto larger wave numbers and governs the system. Hesuggested that the kinetic energy is transferred to smallerscales. Brandenburg, however, showed that to obtain theBO spectra the inverse transfer of the kinetic energy isrequired in terms of a shell model on the basis of theBoussinesq approximation [9]. Then he inferred that thisinverse transfer of the kinetic energy causes the large-scale circulation. It is a fascinating problem to determinethe direction of the energy transfer and its contribution tothe production of the large-scale order.

Numerical simulations have been performed to supple-ment the experiments [10—14]. Hard turbulent states arereproduced even in two-dimensional configurations. Thissuggests that the mechanism generating hard turbulence iscoaunon to 2D and 3D systems. However, it seems to becontradictory to the corxunon sense in Navier-Stokes tur-bulence where 2D turbulence is quite different from 3Dturbulence. Roughly speaking, this discrepancy is due tothe number of conserved quantities in the inviscid limit,that is, the enstrophy is conserved only for 2D system.This extra conservation restricts the dynamics of vortic-ity. On the other hand, for the Boussinesq approximation,

BT = —u VT+ xAT+F,Bt

(3)

where v, a, g, and sc are kinematic viscosity, volumeexpansion coefficient, gravitational acceleration, and heatdiffusivity, respectively. The buoyancy acts along the

y axis and eY denotes the unit vector of this direction.In the following, we set ng as unity without loss ofgenerality. The forcing term F is effective only in lowerwave number ranges. The drag D is used to dissipate theenergy at large scales when we treat statistically steadysituations. In the inviscid limit these equations have thefollowing two conserved quantities; the total energy andthe entropy:

2 u + ~gyTdV, (4)

S = 2TdV.We describe the BO spectra in Kolmogorov's way:

S(k) = eo «"t'(ag) 'i'g(k/kid),

(5)

(6)

only the total energy and the entropy, f 2T2dV, are con-1

served in both 2D and 3D systems. Thus the 2D systemis expected to behave in a similar way to the 3D system.It should be noted that this rough argument is not sup-ported by any rigorous proof and the similarity expectedshould be examined strictly. We are now studying a simi-lar model in 3D.

To examine the characteristics of fully developed con-vective turbulence, we propose a model situation wherethe central region is separated from boundary and mixinglayers. The effect of the two layers on the central regionis taken in only through the forcing to the temperaturedynamics. We also assume neutrally stable stratificationand homogeneity. These assumptions are supported bythe 3D experiments and our 2D simulations. We used thefollowing two-dimensional Boussinesq approximation:

BQ

at—= -u Vu —Vp + veau + ogTe + D,

V'Q=0,

0031-9007/94/73 (1 1)/1501 (4)$06.001994 The American Physical Society

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VOLUME 73, NUMBER 11 PH YS ICAL REVIEW LETTERS 12 SEpTEMmR 1994

E(k) = a~ v"'(ag) "6'(k/kd)

where ez is the entropy dissipation rate, k@d =ay (ag)'~, and kd = v ~ a~ (ag)'~ .We expect

that the functions g and g are universal, though theyshould be dependent on the Prandtl number Pr = v/~.In the inertial range, S(k) and E(k) take the power laws

S(k) = Cga (ag) ~ k

E(k) = C a (ag) k

(8)

where Cq and C~ are some constants and also expectedto be universal. These two forms are first derived byBolgiano and Obukhov for stably stratified situations.The time dependent entropy dissipation rate is governedby the following equation:

ey(t) = — = 2aR(t) = k S(k)dk. (10)dt 0

We solve Eqs. (1), (2), and (3) numerically usingthe pseudospectral method in a doubly periodical box[0, 2n. ] X [0, 2m. ]. Aliasing terms are removed in termsof the 1/2-shifted grids, thus the effective modes are(8/9)N2. The time marching is performed by the fourthorder Runge-Kutta method. We calculated statisticalquantities for relatively long temporal periods after tran-sient stages started with randomly phased initial condi-tions. In this paper we treat the case that Prandtl numberis unity. While we employed two types of forcing, ran-

dom forcing and fixed forcing, the results do not dependon the form of the forcing.

First, we examine the cases for normal viscosity (NV)(see Table I). In Fig. 1 spectra scaled by (6) and (7) areshown. The scaled entropy spectrum obtained by theexperiments is also plotted there. The coincidence ofthose spectra is clear. It should be noted that the entropyspectrum of the 3D experiments is the frequency one. Wehave no experimental energy spectra to be compared withours [15].

The entropy flux function H& and the energy fluxfunction II& are defined as follows:

TABLE I. Data of simulations. The forms of forcingare Ff = f cos(2x) cos(2y) (fixed) and F„=f P4 ~k~,X exp[iti„(t,k) + ik . x] (random) where f is a constant and

0„is random phase. The drag is effective for 0 ( (k( ( 3 andits form is D = dh 'u.

No. Modes Forcing; fNV1NV2NV3NV4NVSHV1

128225622562256251222562

8 x 10-'3 x 10--'

3 x 1Q--'

3 x 10--'

8 x 1Q-'

5 x 10-"

Fixed; 0.5Fixed; 0.3Fi.xed; 0.3

Random; 1.0Fixed; 0.2Fixed; 1

0

2

00.5

79.873.791.0

113.9

10

process. The flux function of the kinetic energy takesnegative value at lower wave numbers than the dissipa-tion range. This means that the energy is transferred tolower scales on average.

To see the inertial range more clearly, we introducehyperviscosity (HV) terms v, b, T and v, A u instead ofthe normal viscosity terms in Eqs. (1) and (2). In Fig. 3both the entropy and the energy spectra are shown. Thepowers of the both spectra are —1.19 and —2.47 for the

entropy and the energy, respectively. Those powers weredetermined by the least squares fitting for 5 ~ k ~ 80.The energy spectrum is steeper and the entropy one isless steep than the BO spectra by about a 10% in power.We do not clearly know the reason for the discrepancy ofthe powers between the BO spectra and ours. We onlysuggest the following [16]: (1) Speciality of 2D systemlike 2D Navier-Stokes turbulence, (2) use of hyperviscos-ity. The two-dimensional spectra of the entropy and the

energy are shown in Fig. 4. The spectra gain isotropy atlarger wave numbers including inertial range. Hence it isreasonable to deal with one-dimensional spectra averagedover shells. The flux functions are plotted in Fig. 5.

Ils(k) = — g g bkI p+qik, 'T'(k')ui(p)T(q),)k'))k p,q

1 iklk'11m(k) = — g g~k, v+q, (1 —~i,.)

)k'()k p,q

X u, (k')u~(p)u„(q), (12)

10

10

1010 10 10

k/ked, k/kd

where subscripts I, m, n (= 1,2) denote the x or Y direc-tion, tilde and asterisk stand for Fourier component and

complex conjugate. In Fig. 2 the flux functions of therun NV3 are shown. The entropy flux function has aflat region which indicates the existence of the cascade

FIG. 1. Normalized 1D spectra for NV. The energy spectraare multiplied by 100. Rigid lines indicate k '/' and k "/ .Norma1ized entropy spectrum of Fig. 2 in Ref. [2] is traced.Error bars represent the thickness of the line in Fig. 2.Coordinate is shifted arbitrarily to overlap with ours. x: NV1,0: NV2, : NV3, 6: NV4, +: NV5.

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VOLUME 73, NUMBER 11 PHYSICAL REVIEW LETTERS 12 SEPTEMBER 1994

0.10 10:

0.05 10

0.00

-0.05

10

10

10

-0.10 (,10I

10k

I

10 10 pI

10k

10

FIG. 2. Flushes for NV. 6: II~, 0: II~. II~ is multipliedby 4.

FIG. 4. Fluxes for HV. h: Iis, 0: )II~(. II~ is negativeat lower wave numbers like IIE in Fig. 2.

L'vov estimated the power of the energy flux function as4/5. This scaling fits our result well, though the directionof the energy transfer is opposite. The inverse transfer ofthe kinetic energy is predicted by Brandenburg [9). How-ever, his shell model disagrees with the assumption of theneutral stratification. An improved model also supportsthe inverse transfer of the kinetic energy. The details ofthe results obtained by the shell model will be reportedin forthcoming papers [17]. Anyway, this inverse transferseems to be one of the mechanisms for maintaining thelarge scale circulation.

The drag is employed to keep the system statisticallysteady. The inertial and dissipative ranges are not affectedby the drag. On the other hand, larger scale structuresare formed and depend on the largest scale saturated bythe drag. A snapshot of the temperature field is shownin Fig. 6. The temperature field locally repeats to be com-pressed and stretched by relatively larger-scale velocityfield.

We also examined probability distribution func-tions (PDF) of temperature and its derivatives (seeFig. 7). In contrast with the experimental results, the

10:

PDF of temperature takes Gaussian form rather thanexponential-like. On the other hand, the PDFs ofderivatives are exponential-like. The skewness factorsof PDFs of T and BT/Bx are null. However, that ofBT/By takes a finite value. This asymmetry of PDFis due to the large-scale anisotropy induced by buoy-ancy. In fact, PDFs reconstructed with low-pass-filteredFourier components retain symmetry. Still in con-vective turbulence a finite skewness is not necessarilyrequired to induce cascade or explosion because theinduction term of R cannot be expressed by the skew-ness factors unlike the enstrophy in 3D Navier-Stokesturbulence [18]. From the direct simulation with a 2Dbox, we concluded that the exponential-like form of PDFis brought about by the intermittent visit of hot or coldplumes to the central region [7]. Moreover, the indepen-dence of the form of the PDFs from Rayleigh numbersuggests that the characteristics of turbulence in the cen-tral region such as power spectra is not so strongly con-trolled by the plumes. In this paper therefore we took intoaccount a role of plumes on the central region only as theforcing to the temperature dynamics. The separation ofthe turbulence in the central region from the dynamics

10

10

(a)

- 10G

(b)

- 100

10

10

'

-Oky -0 ky

10 () 10k

10

0I

100 0I

100

FIG. 3. 1D spectra for HV. 6: S(k), 0: E(k). Rigid linesdenote k " and k . obtained by the least squares fitting for5~k~80.

k, kx

FIG. 5. 2D spectra for HV. (a) in[S(k, kY)], (b) 1n[E(k„kY)].Isoline increment is 1 and minimum value is —8.

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VOLUME 73, NUMBER 11 PHYSICAL REVIEW LETTERS 12 SEpTEMsER 1994

FIG. 6. A snapshot of temperature field for HV. Temperatureincreases as brightness. The hottest region is white and thecoldest region is black.

of the entropy and the energy decomposed into severalscales by 2D wavelet. The correlation function of theentropy between adjacent scales has a sharp peak. Thispeak is scaled as k / which is the time scale in theinertial range (results are not shown here and will bereported in forthcoming papers). That is, on average theentropy of some scale is transferred to the next scale withthe period t„proportional to k„/5. Moreover, we caninfer that the entropy is transferred to infinitesimally smallscales in a finite time T = g t„in the inviscid limit.

We believe that further research on convective turbu-lence will bring fruitful results.

of plumes and boundary layers is one of the main pointsof our research.

Pumir and Siggia suggested the existence of a finitetime singularity for the 2D-inviscid Boussinesq equationswhich correspond to axisymmetric 3D Euler equations[19]. If the scaling (6) and (7) is valid, eB and I~ shouldbe independent. In the inviscid limit, then R defined inEq. (10) should explode. This situation is quite similar to3D Navier-Stokes turbulence. Nevertheless, the existenceof singularity and its relation with the cascading processare not clear so far [20]. We examined temporal evolution

010

10:

10:

10 -8 -6 -4 -2 0 2 4 6 8

FIG. 7. Histograms of T, BT/Bx, and BT/By for HV normal-ized by their mean values and standard deviations. The flat-nesses are 3.8, 6.4, and 6.4, respectively. The skewness factorfor BT/By is 0.24 and others are null within numerical error.

[1] E.D. Siggia, Annu. Rev. Fluid Mech. 26, 137 (1994).[2] X.Z. Wu, L. Kaddanoff, A. Libchaber, and M. Sano,

Phys. Rev. Lett. 64, 2140 (1990).[3] B. Castaing et al. , J. Fluid Mech. 204, 1 (1989).[4] S.C. Ching et al. , Physica (Amsterdam) SSD, 414 (1992).[5] A. S. Monin and A. M. Yaglom, Statistical Fluid

Mechanics: Mechanics of Turbulence, Vol 2(MIT. Press,Cambridge, 1975), p. 387.

[6] I. Procaccia and R. Zeitak, Phys. Rev. Lett. 62, 2128(1989).

[7] V. S. L'vov, Phys. Rev. Lett. 67, 687 (1991).[8] V. Yakhot, Phys. Rev. Lett. 69, 769 (1992).[9] A. Brandenburg, Phys. Rev. Lett. 69, 605 (1992).

[10] R. M. Kerr (to be published).[11] L. Sirovich, S. Balanchandar, and M. R. Maxey, Phys.

Fluids A 1, 1911 (1989).[12] S. Toh and E. Suzuki, Unstable and Turbulent Motion of

Fluid (World Scientific, Singapore, 1993), p. 272.[13] J. Werne, Phys. Rev. E 48, 1020 (1993).[14] E.E. DeLuca, J. Werne, and R. Rosner, Phys. Rev. Lett.

64, 2370 (1990).[15] P. Tong and Y. Shen, Phys. Rev. Lett. 69, 2066 (1992).[16] This discrepancy has been suggested to be due to finite

size correction to the scaling exponents. [V.S. L'vov andI. Procaccia, Phys. Rev. E 49, 4044 (1944).]

[17] E. Suzuki and S. Toh (to be published).[18] M. Lesieur, Turbulence in Fluids (Kluwer Academic

Publishers, Dordrecht, The Netherlands, 1990), p. 145.[19] A. Pumir and E.D. Siggia, Phys. Rev. Lett. 6$, 1511

(1992).[20] E. Weinan and C.W. Shu, Phys. Fluids 6, 49 (1994).

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