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Weak inertial-wave turbulence theory Se´bastien Galtier Institut d’Astrophysique Spatiale, CNRS–Universite´ de Paris XI, Baˆtiment 121, 91405 Orsay Cedex, France 共Received 21 October 2002; published 15 July 2003兲 A weak wave turbulence theory is established for incompressible fluids under rapid rotation using a helicity decomposition, and the kinetic equations for energy E and helicity H are derived for three-wave coupling. As expected, nonlinear interactions of inertial waves lead to two-dimensional behavior of the turbulence with a transfer of energy and helicity mainly in the direction perpendicular to the rotation axis. For such a turbulence, we find, analytically, the anisotropic spectra E⬃k⬜⫺5/2k ⫺1/2 , H⬃k⬜⫺3/2k ⫺1/2 , and we prove that the energy 储 储 cascade is to small scales. At lowest order, the wave theory does not describe the dynamics of two-dimensional 共2D兲 modes which decouples from 3D waves. DOI: 10.1103/PhysRevE.68.015301

PACS number共s兲: 47.27.⫺i, 05.20.Jj, 47.10.⫹g, 47.32.⫺y

Inertial waves are a ubiquitous feature of incompressible fluids under rapid rotation 关1兴. Although much is known about their initial excitation, still little is understood about their nonlinear interactions. The study of 共strong兲 rotating flows is of interest for a wide range of problems, ranging from engineering 共turbomachinery兲 to geophysics 共oceans, earth’s atmosphere, gaseous planets兲 and weather prediction. Rotation is often coupled with other dynamical factors, therefore it is important to isolate rotation to understand precisely its effects. The strength of the Coriolis force, measured in terms of the advection term in the Navier-Stokes 共NS兲 equations, is given by the dimensionless Rossby number R o ⫽U/(L⍀), where U is a typical velocity, L a typical length scale, and ⍀ the rotation rate. Typical values for large-scale planetary flows 关2兴 are 0.05–0.2. Several experiments have been performed on turbulent fluids under rapid rotation 关3,4兴. One of the main results observed is that the rapid rotation leads to two-dimensional behavior of an initial homogeneous isotropic turbulence. Evidence of the two-dimensional behavior is revealed through anisotropic spectra where energy is preferentially accumulated in the direction perpendicular to the rotation axis. Recently, energy spectrum E(k)⬃k ⫺2 has been experimentally observed 关4兴, instead of the Kolmogorov spectrum ⬃k ⫺5/3 for nonrotating fluids. This experimental spectrum is interpreted as the result of an inverse cascade of twodimensional 共2D兲 turbulence. Turbulent fluids under strong rotation have been widely investigated through numerical simulations 关5,6兴, closure models 关7兴, heuristic descriptions 关8兴, and studies of weakly nonlinear resonant waves 关9,10兴. The tendency toward a twodimensional behavior of the turbulence has been observed but surprisingly there is no theoretical prediction and no measure of the scaling law of anisotropic spectra. The nonlinear mechanism leading to such a state is still not well understood; neither are the different scalings for the energy spectrum obtained numerically when a forcing is applied at intermediate scale 关6兴. Important questions concern the origin of the mechanism leading to an interaction between the 2D and the 3D states, and the direction of the energy cascade 关6兴. Strong rotation introduces in the problem a small parameter, proportional to R o , from which it is possible to expand 1063-651X/2003/68共1兲/015301共4兲/$20.00

the NS equations in the framework of weak turbulence. In this paper, we present such an approach for inertial waves in incompressible rotating fluids. Weak turbulence provides a useful paradigm to understand several challenging problems of turbulence 关11,12兴; this formalism leads to wave kinetic equations 共WKE兲 that describe the evolution of kinetic energy and helicity spectra. The analysis of the WKE derived in this paper confirms the tendency toward anisotropy in such flows, and leads for the first time, to the best of our knowledge, to exact predictions in terms of anisotropic power law spectra and energy cascade direction. The Navier-Stokes equations for incompressible flows in a rotating frame read

⳵ t w⫺2 共 ⍀•“ 兲 u⫽ 共 w•“ 兲 u⫺ 共 u•“ 兲 w⫹ ␯ ⵜ 2 w,

共1兲

where u is the velocity field (“•u⫽0), w the vorticity (w ⫽“⫻u), ⍀⫽⍀eˆ储 ( 兩 eˆ储 兩 ⫽1), and ␯ the kinematic viscosity. We assume that the rotation is fast (R o Ⰶ1), which implies that velocity v and vorticity w are much smaller in magnitude than ⍀. We will therefore substitute in the previous equation, u→ ⑀ u and w→ ⑀ w, where ⑀ is a small parameter (0⬍ ⑀ Ⰶ1). The dispersion law in Fourier space, setting ⑀ ⫽0, leads to ⳵ t wk⫽⫺2⍀k 储 (eˆk ⫻wk)/k⫽is ␻ k wk , with ␻ (k)⫽ ␻ k ⫽2⍀k 储 /k, s⫽⫾1, wk⫽iseˆk ⫻wk , and where wave vector k⫽keˆk ⫽k⬜ ⫹k 储 eˆ储 (k⫽ 兩 k兩 , k⬜ ⫽ 兩 k⬜ 兩 , 兩 eˆk 兩 ⫽1). This corresponds to dispersive transverse circularly polarized 共helical兲 waves with s being the wave polarity. We will adopt the Eulerian formalism 关13兴 and choose a complex helicity decomposition for inertial waves whose convenience is now well recognized 关7,9,10,14 –19兴. A supplementary advantage of such a decomposition is that it renders projection operators, inherent to a description of incompressible flows, less cumbersome. The end result of such an approach is a set of integrodifferential equations for the spectral density of the invariants of Eq. 共1兲 in the inviscid case, namely, the energy E(k) and helicity H(k) spectra. The helicity decomposition hs 共 k兲 ⬅hks ⫽ 共 eˆk ⫻eˆ储 兲 ⫻eˆk ⫹is 共 eˆk ⫻eˆ储 兲 ,

共2兲

has the following properties: hks •hks ⬘ ⫽(2k⬜2 /k 2 ) ␦ ⫺s ⬘ s , is(eˆk s ⫻hks )⫽hks , k•hks ⫽0, hk⫺s ⫽h⫺k . We project the velocity on

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the basis of helical modes, uk⫽⌺ s a s (k)e is ␻ k t hks ⬅⌺ s a ks e is ␻ k t hks , where a ks is the modal amplitude in the interaction representation for which, in the linear approximation ( ⑀ ⫽0), ⳵ t a ks ⫽0; thus, the weak nonlinearities will modify only slowly in time the inertial-waves amplitude. We also have wk⫽k⌺ s sa ks e is ␻ k t hks . We introduce expressions of the fields into the NS equations written in Fourier space, and s to obtain we multiply it by vector h⫺k

⳵ t a ks ⫽ ⑀

兺

s psq

冕

p q L ⫺kpq a pp a qq e ⫺ig k,pq t ␦ k,pq d pq ,

ss s

s

s

共3兲

ss s

with interaction operator L kpqp q ⬅L I given by: L I⫽

冉 冊 isks p p 2k⬜2

s

s

s

s

关共 q•hpp 兲共 hqq •hks 兲 ⫺ 共 p• hqq 兲共 hpp •hks 兲兴 ,

and g k,pq ⫽s ␻ k ⫺s p ␻ p ⫺s q ␻ q 共whereas g kpq ⫽s ␻ k ⫹s p ␻ p ⫹s q ␻ q ), ␦ k,pq ⫽ ␦ (k⫺p⫺q), d pq ⫽dpdq. The fundamental Eq. 共3兲 contains an exponentially oscillating term essential to the asymptotic closure: weak turbulence deals with variations of spectral densities at very large time, i.e.,for nonlinear transfer times much greater than the wave period. Consequently, most of the nonlinearities will be destroyed by phase mixing and only the resonance terms will survive. The resonance condition for inertial waves corresponds to relations k⫹p⫹q⫽0 and sk 储 /k⫹s p p 储 /p⫹s q q 储 /q⫽0, which can also be written as s p p⫺sk s q q⫺s p p sk⫺s q q ⫽ ⫽ . s q␻ q s␻k s p␻ p

再 冎 E 共 k兲

H 共 k兲

⫽

␲⑀2 8

兺 ss s

p q

冕冉

s q q⫺s p p ␻k

兺

s psq

冕

p q c pp c qq e ⫺ig k,pq t ␦ k,pq d pq 共 s q q⫺s p p 兲 M ⫺kpq

ss s

s

s

共5兲

with i sin ␣ k ⫺i(s⌽ ⫹s ⌽ ⫹s ⌽ ) ss s k p p q q , M kpqp q ⫽ 共 sk⫹s p p⫹s q q 兲 e 4 k

共6兲

where ␣ k refers to the angle between p and q in triangle k ⫹p⫹q⫽0. The local decomposition allows one to concentrate concisely complex information in a unique exponential function which will help simplify notably the derivation of the WKE. We define energy density tensor e s ⬘ (k⬘ ) for homogeneous turbulence, such that 具 c s (k)c s ⬘ (k⬘ ) 典 ⬅e s ⬘ (k⬘ ) ␦ (k ⫹k⬘ ) ␦ ss ⬘ , for which we shall write a ‘‘closure’’ equation. The presence of delta function ␦ ss ⬘ means that the correlations between opposite polarities have no long-time influence in the weak turbulence regime. The derivation of the WKE is a technical and lengthy but classical calculation of weak turbulence 共see, e.g., 关12,13兴兲, which will be presented elsewhere. In this paper attention is rather focused on the main properties of the WKE, which to our knowledge are given for the first time here:

⳵ t e s 共 k兲 ⫽

4␲⑀2 s␻k

兺

s psq

冕

共 s qq

ss s

⫺s p p 兲 2 兩 M kpqp q 兩 2 ␦ 共 g kpq 兲 ␦ kpq 关 s ␻ k e s p 共 p兲 e s q 共 q兲

共4兲

⫹s p ␻ p e s 共 k兲 e s q 共 q兲 ⫹s q ␻ q e s 共 k兲 e s p 共 p兲兴 d pq . 共7兲

These relations will help simplify the WKE and give a proof of the conservation of their ideal invariants as well. We now take an orthonormal vector basis local to each ˆ 2 (p)⫽nˆ,O ˆ 3 (p) ˆ 1 (p)⫽nˆ⫻eˆp ,O triad 关9,18,19兴 as follows: O ⫽⫺eˆp where eˆp⫽p/ 兩 p兩 and nˆ⫽(k⫻p)/ 兩 k⫻p兩 ⫽(p⫻q)/ 兩 p ⫻q兩 ⫽(q⫻k)/ 兩 q⫻k兩 . Vector nˆ is normal to any vector of triad (k, p, q) and it changes sign by interchanging p and q but not by cyclic permutation. One introduces vectors s ˆ 1 (p)⫹is p O ˆ 2 (p) and we define rotation ⌶ s p (p)⬅⌶ pp ⫽O angle ⌽ p such that cos ⌽p⫽nˆ•(eˆp⫻eˆ储 ) and sin ⌽p⫽nˆ• 关 (eˆp ⫻eˆ储 )⫻eˆp]. It leads to relation hks ⫽(k⬜ /k)⌶ ks e ⫺is⌽ k . With c s (k)⬅c ks ⫽(k⬜ /k)a ks , we have

⳵t

⳵ t c ks ⫽ ⑀

冊

Equation 共7兲 it is the first main result of this paper; it describes statistical properties of inertial wave turbulence. Mass s trix M kpqp q is taken as a modulus, which means that the complex information concentrated in the exponential function does not enter into account in the dynamics. Note that the resonance condition appears as delta function ␦ (g kpq ). As expected, the WKE conserve in detail 共for each triadic interaction兲 ideal invariants, i.e., energy E(t)⬅ 兰 ⌺ s e s (k)dk and helicity H(t)⬅ 兰 ⌺ s ske s (k)dk. After simple manipulations, in particular to introduce spectra E(k) and H(k), we obtain the general expression of the WKE at the level of three-wave interactions:

2

共 sk⫹s p p⫹s q q 兲 2

冉 冊 sin ␣ k k

2

␦ 共 g kpq 兲 ␦ kpq s ␻ k s p ␻ p

再 冎 XE

XH

d pq

共8兲

with

再 冎再 XE

XH

⫽

E 共 q兲关 E 共 k兲 ⫺E 共 p兲兴 ⫹ 共 H 共 q兲 /s q q 兲关 H 共 k兲 /sk⫺H 共 p兲 /s p p 兴 sk 兵 E 共 q兲关 H 共 k兲 /sk⫺H 共 p兲 /s p p 兴 ⫹ 关 H 共 q兲 /s q q 兲共 E 共 k兲 ⫺E 共 p兲兴 其 015301-2

冎

.

共9兲

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WEAK INERTIAL-WAVE TURBULENCE THEORY

Several properties are observed. First, we see that an initial state with zero helicity will not generate any helicity at any scale. Second, we observe that there is no coupling between helicity states associated with wave vectors p and q, when they are collinear (sin ␣k⫽0). Third, there is no coupling between helicity states associated with these vectors whenever magnitudes p and q are equal if their associated polarities s p and s q are also equal. This property holds for both energy and kinetic helicity; it is a standard property of helical waves 关9,15,18,19兴. Fourth, a strong helical perturbation, localized initially in a narrow band of wave numbers, will lead therefore to a weak transfer of energy and helicity 关15,20,21兴. Because anisotropic turbulence prevails in several numerical simulations 关5,6兴, it is of interest to investigate the local interaction limit 共equilateral triadic wave coupling k ⬇p⬇q) of Eqs. 共8,9兲 in order to understand precisely the primary dynamics leading to anisotropic turbulence. Then the resonance condition 共4兲 reads (s p ⫺s)/s q q 储 ⬇(s q ⫺s p )/sk 储 ⬇(s⫺s q )/s p p 储 . From equations 共8,9兲, we see that only interactions between two waves (p and q) with opposite

⳵t

再 冎 Ek

Hk

⫽

⍀ 2⑀ 2 4

兺

ss p s q

冕

冉

sk 储 s p p 储 s q q⬜ ⫺s p p⬜ ␻k k⬜2 p⬜2 q⬜2

⫹s q q⬜ 兲 2 sin ␪ ␦ 共 g kpq 兲 ␦ k 储 p 储 q 储

再

冊

polarities (s⫽s p ⫽⫺s q or s⫽⫺s p ⫽s q ) will contribute significantly to the nonlinear dynamics. It implies that either q 储 ⬇0 or p 储 ⬇0, which means that only a small transfer is allowed along ⍀. In other words, local nonlinear interactions lead to anisotropic turbulence where small scales are preferentially generated perpendicular to the external rotation vector. Note the similarity with electrically conducting magneto hydro dynamic fluids for which the presence of a strong uniform magnetic field prevents any energy transfer along it 关22兴. The previous reasoning allows us to consider the anisotropic limit, i.e., the k⬜ Ⰷk 储 limit. This means that, for example, we assume that the turbulence is rather generated initially by a source in a limited band of 共large兲 scales. Local interactions will therefore dominate and they will lead essentially to anisotropic turbulence, i.e., structures elongated along the rotation axis like the vortices observed experimentally in 关4兴. At leading order in k 储 /k⬜ , and for an inertial wave turbulence that is axially symmetric with respect to the rotation vector, the simplified WKE read

2

共 sk⬜ ⫹s p p⬜

E q 共 p⬜ E k ⫺k⬜ E p 兲 ⫹ 共 p⬜ sH k /k⬜ ⫺k⬜ s p H p /p⬜ 兲 s q H q /q⬜ sk⬜ 关 E q 共 p⬜ sH k /k⬜ ⫺k⬜ s p H p /p⬜ 兲 ⫹ 共 p⬜ E k ⫺k⬜ E p 兲 s q H q /q⬜ 兴

冎

d p⬜ dq⬜ dp 储 dq 储 , 共10兲

with ␪ the angle between k⬜ and p⬜ in triangle k⬜ ⫹p⬜ ⫹q⬜ ⫽0, ␻ k ⫽2⍀k 储 /k⬜ and E k ⫽E(k⬜ ,k 储 ) ⫽2 ␲ k⬜ E(k⬜ ,k 储 ), H k ⫽H(k⬜ ,k 储 )⫽2 ␲ k⬜ H(k⬜ ,k 储 ). Exact solutions of Eqs. 共10兲 as power laws can be found by applying the Kuznetsov-Zakharov conformal transformation 关11兴 which is a 2D generalization of the Zakharov transformation. The most interesting solutions are those for which the flux is finite 共instead of being null, as in the thermodynamic solutions兲. These exact solutions, the KuznetsovZakharov-Kolmogorov 共KZK兲 spectra, read , H 共 k⬜ ,k 储 兲 ⬃k⬜⫺3/2k ⫺1/2 . 共11兲 E 共 k⬜ ,k 储 兲 ⬃k⬜⫺5/2k ⫺1/2 储 储 The anisotropic limit is the only case where the theory works well in the sense that the collision integral does not suffer from infrared divergences which require corrections to the spectra, e.g., logarithmic 关23兴. Although it is possible to derive the exact expressions of the Kolmogorov constants appearing in front of spectra 共11兲, it is difficult to give them precise values since they depend on the cutoffs introduced by the anisotropic limit 关24兴. However, the sign of the energy transfer can be computed for a reasonable range of cutoffs; it is found positive, hence a direct energy cascade. These latter two exact results, which cannot be found by simple heuristic

descriptions, are particularly significant since they suggest experimental measurements to compare with. On the other hand, the KZK spectra can be obtained phenomenologically. Dimensional analysis leads to relation ¯⑀ ⬃u 2 / ␶ tr ⬃E(k⬜ ,k 储 )k⬜ k 储 / ␶ tr , where ¯⑀ is the mean rate of energy dissipation per unit of mass and ␶ tr is the transfer time whose form is given by the WKE. We have ␶ tr 2 ⬃ ␶ NL / ␶ ⍀ , with ␶ NL the nonlinear characteristic time and ␶ ⍀ the inertial-wave period. Anisotropic turbulence leads to scaling ␶ NL ⬃ᐉ⬜ /u⬃(k⬜ u) ⫺1 , and ␶ ⍀ ⬃k⬜ /⍀k 储 . Finally, . Note that we obtain spectrum E(k⬜ ,k 储 )⬃(¯⑀ ⍀) 1/2k⬜⫺5/2k ⫺1/2 储 if we forget the anisotropic hypothesis and assume k⬜ ⬇k 储 ⬇k, we recover the earlier prediction E(k)⬃(¯⑀ ⍀) 1/2k ⫺2 for the 1D isotropic energy spectrum 关8兴, which is moreover a solution of isotropic direct interaction approximation equations 关25兴. The primary dynamics leading to anisotropic turbulence may stop at some point since nonlocal interactions develop as well. According to direct numerical simulations 共DNS兲, anisotropy is generated and preserved. It seems therefore that the possible balance between local and nonlocal interactions does not lead to an isotropization of the turbulence. Analysis of resonance condition 共4兲 shows indeed that strongly non-

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local interactions lead also to anisotropic turbulence 关10兴. Therefore, our theoretical analysis confirms that there is a global nonlinear tendency to develop and maintain anisotropy. Inertial-wave turbulence theory is an asymptotic theory based on a time scale separation ␶ tr Ⰷ ␶ ⍀ . 共For strong turbulence, we have ␶ tr ⬃ ␶ NL ⬃ ␶ ⍀ .) The consequence is that the theory is not valid in the entire k space. Evaluation of the ⫺3/2 关26兴. The transfer time gives 共see above兲 ␶ tr ⬃ ⑀ ⫺2 ⍀k 1/2 储 k⬜ time scale separation condition 共with ⍀⬃1) leads to k 储 Ⰷ ⑀ 4/3k⬜5/3 . The previous relation combined with the anisotropic assumption defines the domain of validity in k space of our theory at the level of three-wave interactions. We have a nonuniform validity of the WKE which means, in particular, that the theory is not valid for too small values of k 储 or too large values of k⬜ . It is important to note that inside the

prohibited region, other interactions like higher-order processes 共four-wave interactions, . . . ) have to be taken into account 共see, e.g., 关12,27兴兲. In particular, Eq. 共10兲 shows that the nonlinear transfer, for energy and helicity, decreases 共linearly兲 with k 储 . For forbidden value k 储 ⫽0 the transfer is exactly null. As previously mentioned 关6,10兴, the 2D 共geostrophic or slow兲 modes decouple from the 3D inertial waves. Such decoupling underlies the validity of quasigeostrophic models, e.g., for the atmosphere or the oceans 关28兴. Forced DNS 关6兴 show the generation of slow modes. From the present work, we see that weak turbulence at the lowest order cannot describe such an observation; however higherorder processes could play a significant role 关6兴.

关1兴 H.P. Greenspan, The Theory of Rotating Fluids 共Cambridge University Press, Cambridge, England, 1968兲. 关2兴 J.H. Shirley and R.W. Fairbridge, Encyclopedia of Planetary Sciences 共Kluwer Academic, Dordrecht, 1997兲. 关3兴 E.J. Hopfinger, F.K. Browand, and Y. Gagne, J. Fluid Mech. 125, 505 共1982兲; L. Jacquin, O. Leuchter, C. Cambon, and J. Mathieu, ibid. 220, 1 共1990兲; S.V. Veeravalli, Ann. Res. Briefs, NASA-Stanford Center for Turbulence Research 共1991兲. 关4兴 C.N. Baroud, B.B. Plapp, Z.-S. She, and H.L. Swinney, Phys. Rev. Lett. 88, 114501 共2002兲. 关5兴 J. Bardina, J.M. Ferziger, and R.S. Rogallo, J. Fluid Mech. 154, 321 共1985兲; N.N. Mansour, C. Cambon, and C.G. Speziale, in Studies in Turbulence, edited by T.B. Gatski et al. 共Springer-Verlag, Berlin, 1992兲, p. 59; P. Bartello, O. Metais, and M. Lesieur, J. Fluid Mech. 273, 1 共1994兲; M. Hossain, Phys. Lett. 6, 1077 共1994兲. 关6兴 L.M. Smith and F. Waleffe, Phys. Fluids 11, 1608 共1999兲. 关7兴 C. Cambon and L. Jacquin, J. Fluid Mech. 202, 295 共1989兲; C. Cambon, N.N. Mansour, and F.S. Godeferd, ibid. 337, 303 共1997兲. 关8兴 O. Zeman, Phys. Fluids 6, 3221 共1994兲; Y. Zhou, ibid. 7, 2092 共1995兲. 关9兴 F. Waleffe, Phys. Fluids A 4, 350 共1992兲. 关10兴 F. Waleffe, Phys. Fluids A 5, 677 共1993兲. 关11兴 V.E. Zakharov, V. L’vov, and G.E. Falkovich, Kolmogorov Spectra of Turbulence 共Springer-Verlag, Berlin, 1992兲. 关12兴 A.C. Newell, S.V. Nazarenko, and L. Biven, Physica D 152153, 520 共2001兲. 关13兴 J. Benney and A.C. Newell, Stud. Appl. Math. 48, 29 共1969兲. 关14兴 S. Chandrasekhar and P.C. Kendall, Astrophys. J. 126, 457 共1957兲; A. Craya, Contribution a` l’Analyse de la Turbulence

associe´e a` des Vitesses Moyennes 共P.S.T. Ministe`re de l’Air, France, 1958兲, Vol. 345. R.H. Kraichnan, J. Fluid Mech. 59, 745 共1973兲. J. Herring, Phys. Fluids 17, 859 共1974兲. M. Lesieur, Turbulence in Fluids, 2nd ed. 共Kluwer Academic, Dordrecht, 1990兲. L. Turner, J. Fluid Mech. 408, 205 共2000兲. S. Galtier and A. Bhattacharjee, Phys. Plasmas 共to be published兲. Note that these properties can be seen on the fundamental Eq. 共5兲 as well. A strong similarity exists with weak whistler 共helical兲 wave turbulence for which the present formalism is well adapted; it leads to very similar WKE whose properties are therefore common in many points 关19兴. S. Galtier, S.V. Nazarenko, A.C. Newell, and A. Pouquet, J. Plasma Phys. 63, 447 共2000兲. R. Rubinstein, ICASE-NASA Report No. 99-7, 1999 共unpublished兲. The same difficulty arises for internal gravity waves. P. Caillol and V. Zeitlin, Dyn. Atmos. Oceans 32, 81 共2000兲. See R. Rubinstein and Y. Zhou, ICASE-NASA Report No. 97-63, 1997 共unpublished兲. This isotropic scaling is in agreement with Ref. 关6兴 where no distinction is made between wave numbers k⬜ and k 储 . Note that we have now included small parameter ⑀ . S.V. Nazarenko, A.C. Newell, and S. Galtier, Physica D 152153, 646 共2001兲. O.M. Phillips, J. Fluid Mech. 34, 407 共1968兲; P. Bartello, J. Atmos. Sci. 52, 4410 共1995兲. Note that, contrary to magnetohydrodynamics 关22兴, the slow modes do not drive the dynamics.

I gratefully acknowledge A. Pouquet for useful correspondence, as well as C. Baroud, C. Cambon, G. Carnevale, J. Herring, and S. Nazarenko for their comments.

关15兴 关16兴 关17兴 关18兴 关19兴 关20兴 关21兴

关22兴 关23兴 关24兴 关25兴

关26兴 关27兴 关28兴

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