Anisotropic triadic closures for shear-driven and buoyancy-driven turbulent flows

Accepted Manuscript

Anisotropic triadic closures for shear-driven and buoyancy-driven turbulent flows Claude Cambon, Vincent Mons, Benoˆıt-Joseph Grea, ´ Robert Rubinstein PII: DOI: Reference:

S0045-7930(16)30389-9 10.1016/j.compfluid.2016.12.006 CAF 3346

To appear in:

Computers and Fluids

Received date: Revised date: Accepted date:

4 March 2016 10 November 2016 8 December 2016

Please cite this article as: Claude Cambon, Vincent Mons, Benoˆıt-Joseph Grea, ´ Robert Rubinstein, Anisotropic triadic closures for shear-driven and buoyancy-driven turbulent flows, Computers and Fluids (2016), doi: 10.1016/j.compfluid.2016.12.006

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Highlights • Implementation strategies of triadic closures are reviewed and com-

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pared

• A general formalism to analyze anisotropic turbulence is proposed

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• New results concerning scaling laws of anisotropic spectra are shown

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Claude Cambon

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Anisotropic triadic closures for shear-driven and buoyancy-driven turbulent flows LMFA, UMR 5509, ECL 69134 Ecully Cedex, France

Vincent Mons

Sorbonne Universit´es, UPMC Univ. Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France

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Benoˆıt-Joseph Gr´ea

CEA, DAM, DIF, 91297 Arpajon, France

Robert Rubinstein

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Newport News, Virginia, USA

Abstract

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We show how two-point turbulence models based on triadic closures, initially developed for homogeneous isotropic turbulence, can be applied to

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anisotropic turbulent flows. Efficient numerical solution of these models is now feasible thanks to new computational capabilities. This permits explo-

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ration of their dynamics and structure at high Reynolds numbers and offers a complementary approach to direct numerical simulations. In this paper, im-

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plementation strategies, developed for shear and buoyancy driven turbulence, are reviewed and compared. A single formalism for anisotropic turbulence is proposed for both problems. Analyses of weakly anisotropic turbulence, for Email address: [email protected] (Claude Cambon )

Preprint submitted to Computers & Fluids

December 12, 2016

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scales much smaller than Corrsin’s scale (shear case) and Ozmidov’s scale (stratified case) are revisited and new scaling laws are proposed. The effect

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of initial data is analyzed. The entire spectral range, including the infrared range, is considered.

Keywords: Anisotropic turbulence, Triadic closures, Two-point turbulence model

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1. Introduction

Statistical turbulence models based on two-point closures address the dynamics of second order correlations. In the history of turbulence, they have played a central role in understanding how energy is exchanged between different scales of motion. Their successes and equally importantly,

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their failures have contributed greatly to a better understanding of cascade phenomenology. These models were first developed for the canonical case of

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homogeneous isotropic turbulence (HIT). Starting with the pioneering works of Kovaznay, Heisenberg and many

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others, a class of two-point turbulence models, referred as classical closures [16], expresses the transfer term in the Fourier representation in terms of the

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energy spectrum alone. The Leith model [37], which treats energy transfer as a nonlinear diffusion, belongs to this category. These models provide very useful information concerning the dynamics of turbulent spectra. However,

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they suffer from various limitations, for instance, imposing forward energy transfer from large to small scales only. This is due to an overly simplified treatment of the triadic interactions among the Fourier modes. In order to deal with these issues, more elaborate two-point models using

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triad interactions and two-time correlations were introduced. The Direct Interaction Approximation (DIA) proposed by Kraichnan [33] is perhaps the

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first and most characteristic example. It also introduces the important concept of the response tensor, as a kind of Green’s function linearized around a fully nonlinear basic state. The initial version of DIA for HIT was not sat-

isfactory, as shown by Kraichnan himself, because the nonlinear time-scale of the response tensor was dominated by a sweeping effect. This violated

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statistical Galilean invariance and led to a prediction E(k) ∼ k −3/2 for the energy spectrum in the inertial range. To overcome this flaw in the theory,

the Lagrangian history versions of DIA [35] were formulated. Similar but analytically more tractable theories were proposed later, such as the Lagrangian Renormalized Approximation (LRA) by Kaneda [31]. Although the two-time

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approach was essential in these models, single-time models such as the Test Field Model (TFM) [36] were also derived. All of these models recovered

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the classical Kolmogorov k −5/3 inertial range spectrum. Review of closures, extended from fluids to plasmas, can be found in [62, 61].

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The complexity of triadic closures is certainly their greatest practical drawback. Orszag’s Eddy-Damped-Quasi-Normal-Markovian (EDQNM) clo-

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sure [44] illustrates what can be called a heuristic triadic closure. Flaws identified in a purely quasinormal evaluation of fourth-order correlations were cured by adding Eddy Damping (ED) to mimic the effect of fourth-order cu-

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mulants. Markovianisation of the two-time dependence together with eddy damping then gave a more satisfactory model of the memory time of triple correlations. Whereas in DIA, the eddy damping is intimately connected to the nonlinear part of the response function, in EDQNM, the response equa-

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tion is replaced by an a priori choice of a phenomenological time scale, which is usually based on the inverse Kolmogorov straining time-scale η ∼ k 3/2 E 1/2 .

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Some variants are discussed later in this paper. Not surprisingly, the recovery of the classical k −5/3 inertial-range spectrum at high Reynolds number follows from this heuristic choice of the eddy-damping term.

It is a truism that turbulent flows encountered in nature are rarely isotropic. One can legitimately ask to what extent two-point models can be applied in

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a more realistic anisotropic context. From a theoretical viewpoint, the twopoint description of turbulence can address interactions both between different scales and different directions. On the one hand, extended versions of

classical closures have been proposed for anisotropic turbulence [17, 6, 15] by considering spherically-averaged spectra which depend only on the wavenum-

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ber modulus k but not on the directional of the vector k. On the other hand, triadic closures have also been derived for axisymmetric flows [26],

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rotating flows [12], and stably stratified flows [23] among others. The numerical solution of anisotropic triadic models can be very costly compared

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to their isotropic counterparts. This has limited the practical implementation of these models. However, recent increases in computational capabilities

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have led to renewed interest in anisotropic triadic models, for which efficient numerical solutions are now possible. The goal of this article is to derive triadic closure models for different

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cases of anisotropic turbulence, and to present detailed comparisons with DNS results and asymptotic theories of weak turbulence. More specifically, we would like to discuss and compare two different anisotropic flows: shear and buoyancy driven turbulence, which have been recently addressed in [42]

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and [10] respectively. A general closure strategy using DIA was proposed by Kraichnan [34], even for inhomogeneous flow cases (both shear and thermally-

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driven), but no tractable numerical method exists to solve the equations. Accordingly, we strictly restrict our survey in this paper to homogeneous flow cases, in which full numerical method is given.

We consider general mean flows with space-uniform velocity gradients Aij =

∂Ui ∂xj

in our approach to shear-driven turbulence, even if new results

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only address the case of pure plane shear flow, usually referred to as shear

flow. Definitions will be given or recalled at each step in order to avoid ambiguity. In this extended sense, a particular case of shear-driven turbulence is given by purely rotating flows. Such flows are dominated by interacting nonlinear inertial waves, and there is no direct energy production. Inertial wave

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turbulence theory [22, 4] can be derived from generalized EDQNM in the case of vanishing eddy damping. The Quasi-normal relationship for closing

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third-order velocity correlations becomes exact in this case [5]. Then Markovianization is consistent with derivation of equations for statistics of “slow”

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wave amplitudes, that are the envelopes of “rapid” wave phases (some key points are recalled in Appendix.) In a different presentation [58], the Quasi-

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Normal approximation follows from a random phase approximation (RPA). Other studies have shown that in order to insure realizability in the pres-

ence of waves, the eddy-damping term must be determined by evolution

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equations. An initial study [8] addresses a one-dimensional stochastic model, in which a linear term, with dispersion law ωk is added. Differences between “quasistationary” EDQNM, DIA and a new RMC (Realizable Markovian Closure) are found during transient evolution. In a second investigation [7],

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the test-field model is addressed as well, and RMC is applied to anisotropic drift-wave dynamics. A similar approach is extended by [21] to the inhomo-

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geneous context of Rossby wave turbulence over topography. The need for renormalizing the “bare” dispersion law of Rossby waves, together with a TFM-type closure, is also advocated by [27] but this is not usual in classical wave turbulence theory.

The problem of altering the eddy damping term in a generalized EDQNM

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closure in the presence of waves is outside the scope of the present paper, but a discussion will be given in the Conclusions.

Waves are no longer dominant in the shear-driven case if the mean velocity gradient includes an irrotational part, which yields direct energy production by straining. If the mean flow gradients correspond to weak ellipticity of

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mean streamlines, parametric instability of inertial waves is relevant at small additional strain rates, but this case is very special, with a thin angular band

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of instability that is very difficult to capture in a practical spectral model. This case will be not considered here.

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Turning to buoyancy-driven flows, we consider the cases of both stable and unstable stratification, and present some recent results of generalized

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EDQNM for the second case. Internal (gravity) waves are present in the first case, but even in the linear regime, they coexist with a three-dimensional non-propagating mode, in agreement with the so-called “Vortex-Wave” de-

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composition [47], which is a variant of a general toroidal-poloidal decomposition. Accordingly, stably stratified homogeneous turbulence does not reduce to internal wave turbulence at very small Froude number; instead, its nonlinear dynamics is dominated by a toroidal “strong” cascade which affects

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the toroidal, non-propagating, mode [23, 52]. The fact that this cascade is essentially direct is consistent with [40].

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Finally, we do not propose new results for turbulence with waves in this paper, but we try to clarify some aspects of the problem (The relevance of generalized EDQNM, with dedicated versions, is investigated for this pur-

pose.) Quantitative results can be found from the models using numerical codes, for both shear-driven flows dominated by production and for unstably

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stratified turbulence. An important remaining problem is the scale-by-scale

distribution of anisotropy. Characteristic scales, introduced by Corrsin [18] Hopfinger, [28] and Zeman [59] for shear-driven flows, and by Ozmidov [45] for buoyancy-driven flows, suggest that isotropy can be recovered at smallest scales. Possible universality with simple scaling laws have been proposed

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[29, 56, 30], and we will revisit these results for both shear-driven flows and buoyancy-driven flows, applying the same unified formalism for capturing

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moderate anisotropy.

The paper is organized as follows. Section 2 is devoted to shear-driven

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turbulent flows; the main issue is the possible inclusion of the mean shear in equations for triple correlations. Buoyancy-driven flows are addressed in

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section 3, with the similar issue of including the stratification frequency.

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2. Shear-driven turbulent flows In this section, the governing equations of the problem are recalled in

2.1, then the procedure for deriving triadic closures is shown in 2.2 and 2.3. Results are given in 2.4.

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2.1. Governing equations for second-order correlations Following Batchelor [3], and more specifically, Craya, we consider the

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general case of statistical homogeneity restricted to fluctuations, in which a mean flow can inject energy and anisotropy to the fluctuating flow via

spatially uniform mean velocity gradients Aij . The most general information on two-point second-order velocity correlations is given by the tensor

ˆ ij (k) obtained Rij (r) = hui (x)uj (x + r)i, and the related spectral tensor R

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by three-dimensional Fourier transform. In this context, our basic state vector for representing two-point second-order correlations is the set (E, Z), ˆ ij (k) with all its components (except which generates the spectral tensor R the helicity spectrum, see Appendix.) 1ˆ E(k, t) = R mm (k, t), 2

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and reciprocally

1ˆ ∗ ∗ Z(k, t) = R mn Nm (α)Nn (α), 2

(2)

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ˆ ij (k, t) = E(k, t) (δij − αi αj ) + < (Z(k, t)Ni (α)Nj (α)) . R

(1)

The set (N , N ∗ , α = kk ) generates an orthonormal frame for projecting the

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ˆ (k, t), and is closely related to the Crayavelocity field in Fourier space u Herring frame of reference [26]. N and its conjugate N ∗ are the helical

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modes [12, 55] (see Appendix for details.) The homogeneous fluctuating flow is subjected to an extensional mean

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flow Ui with space-uniform velocity gradients Aij =

∂Ui . ∂xj

(3)

Equations for multipoint statistics are derived from the one which governs the Fourier transform (overhat) of the fluctuating velocity field, or uˆ˙ i (k, t) + νk 2 uˆi (k, t) + Mij (α)ˆ uj (k, t) = fˆi (k, t), 9

(4)

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where with αi = ki /k,

k=

p kn kn ,

(5)

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Mij = (δin − 2αi αn ) Anj ,

generates the linear (mean-shear-induced, RDT-type) operator and fi holds for the solenoidal (divergence-free) projection of the quadratic term inherited from Navier-Stokes equations. k is the three-dimensional wavevector and ν

the kinematic viscosity. The “overdot” corresponds to the advection operator by the mean flow, or

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˙ = ∂ − Amn km ∂ . (...) ∂t ∂kn

(6)

This is equivalent to follow characteristic lines in Fourier space, directly related to mean flow trajectories in physical space, or to consider that the wavevector is time-dependent (e.g. linear RDT and nonlinear DNS [48]).

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Equations for the state vector (E, Z) are readily derived from Eq. (4) as

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˙ + 2νk 3 E + < (kZ(k, t)Sij Ni (α)Nj (α)) = kT (E) (k, t) (kE) and

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3 ˙ (kZ)+2νk Z+kE(k, t)Sij Ni (−α)Nj (−α)−2ıkZ(k, t)



1 W ·α − ΩE 2

(7)



= kT (Z) (k, t), (8)

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in which Sij is the symmetric part of the mean velocity gradient matrix (3), and Wi = ijn Anj refers to its antisymmetric part (mean vorticity vec-

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tor). The linear term related to mean vorticity in the Z-equation reflects

the stropholysis effect, identified in structure-based single-point models by [32]. Geometric anisotropic coefficients display projection along the helical modes, and ΩE is the rotation vector for passing from the fixed system of coordinates of k to the Craya-Herring one with time-dependent wave vector. 10

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Their linear coupling terms are simplified here (with respect to, e.g., [42]) by considering kE and kZ, using a multiplication by k-modulus.

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2.2. Generalized EDQNM procedure

The closure of the generalized transfer terms T (E) and T (Z) in terms of E

and Z is performed via an expression of three-point (triadic in Fourier space) third-order correlations Simn (k, p, t), with

Exact relationship is

T

1 (k, t) = km 2

Z

R3

∗ (Siim (k, p, t) + Siim (k, p, t)) d3 p,

1 (k, t) = km Ni∗ (α)Nj∗ (α) 2

Z

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(Z)

(E)

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ıhˆ ui (q, t)ˆ uj (k, t)ˆ un (p, t)i = δ(k + p + q)Sijn (k, p, t).

R3


∗ Sijm (k, p, t) + Sjim (k, p, t) d3 p.

(9)

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The generic EDQN closure for third-order correlations is given by Z t + 0 0 + 0 QN 0 0 Sijm (k, p, t) = G+ iu (q, t, t )Gjv (k, t, t )Gmw (p, t, t )τuvw (k, p, t )dt , (10) −∞

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in which G+ (k, t, t0 ) is the nonlinear Green’s function, which consists of the product of the Green’s function related to the linear part of Eq. (4) by an (QN )

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exponential eddy-damping term. The term τimn is the quasi-normal contribution of fourth-order correlations, mediated by nonlinearity in the equation

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for Sijm , so that it can be expressed in terms of second-order correlations as P for a normal law (Symbolically huuuui(QN ) = huuihuui). In the complete

equation for Sijm , not given here for the sake of brevity and because it will be revisited in the buoyancy-driven case, the departure from Gaussianity (IV )

is given by the contribution from fourth-order cumulants, τijm , which are 11

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globally closed as an extra-damping term η(k, t), as done by Orszag in basic EDQNM for isotropic turbulence, or (IV )

The Eddy damping term is chosen in agreement with [46] as s Z k p2 E(p, t)dp, η(k, t) = a0 0

(11)

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τijm (k, p, t) = − (η(k, t) + η(p, t) + η(q, t)) Sijm (k, p, t).

(12)

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that is more general (validated in both 3D and 2D cases) and less dependent on “ad hoc” scaling than Orszag’s original proposal.

Finally, different levels of Markovianization amount to restricting the time integral in Eq. (10) to the terms considered as “rapidly” evolving, whereas the “slowly” evolving terms, are fixed at their instantaneous value t = t0 .

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Basic principles of the most elaborate EDQNM2 and EDQNM3 versions are briefly recalled in Appendix. The simplest EDQNM1 version considers as 0

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rapid only the exponential ∼ e−η(t−t ) eddy-damping term, whereas both the contribution from the “bare” (linear) Green’s function related to the mean

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shear effect and the quasi-normal term are kept at their instantaneous value.

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2.2.1. Numerical issues and mathematical technicalities EDQNM integrals, whatever the version used, initially involve a triple

integral, even if axisymmetry is assumed. For instance, generalized transfer

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T (E) and T (Z) terms are expressed in terms of E and Z in terms of integrals R of the form R3 S(k, p)d3 p. The system of coordinates introduced by Chuck Leith [38] replaces the components of p in a polar-spherical system of coor-

dinates, say p (modulus), θp (polar angle) and φp (azimutal angle), by the 12

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(bi-polar) set p, q, φp = λ, so that the latter triple integral is expressed as  Z Z 2π ˜ p, q, λ)dλ pq dpdq. S(k, (13) k 0 ∆k Accordingly, p and q are the lengths of the sides of the vectors p and

q = −k − p forming a triangle, in the domain ∆k , at fixed k, and λ denotes the angle of rotation of the plane of the triad around k. In three-dimensional

(3D) isotropic turbulence, the transfer term is a scalar, and all directions of

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the plane of the triads are equivalent, so that the integration over λ reduces

to multiplication by 2π. This procedure can be applied to 2D isotropic turbulence by suppressing the λ-integration, and modifying the Jacobian term pq/k appropriately [38].

The numerical integration over the angle λ introduces a much larger cost

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and is cumbersome compared to the isotropic case. In fact, a fully computational approach, with three-dimensional integration as in eq. (13), for T (E)

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and T (Z) was performed only in the case of purely rotating flows [12, 4], in which the matrix Aij is purely antisymmetric and induces no energy produc-

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tion. A similar numerical method is extended to buoyancy-driven flows. The additional mathematical and numerical complexity comes from the

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incorporation of contributions from the anisotropic Green’s function in the most elaborate versions of Eq. (10). This is illustrated briefly in the Appendix, for rotating turbulence, and in the case of stably stratified turbu-

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lence.

2.2.2. Specificity of purely rotating flows This case is very special, because the Coriolis force does not directly produce energy. Accordingly only the most complicated EDQNM(2-3) versions, 13

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recalled in Appendix, and the AQNM (Asymptotic Quasi-Normal Markovian) limit of inertial wave turbulence theory [4] are entirely applicable.

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Looking at the simplest “isotropized model” for improving EDQNM1, however,the rotation rate Ω is incorporated into the eddy-damping term by setting

 Z 2 η(k, t) = a0 (2Ω) +

0

k

1/2 p E(p, t)dp , 2

(14)

in agreement with [46] and [1] without rotation. This heuristic closure may

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be partly relevant, e.g. for mimicking the damping, or rapid decay, of the velocity gradient (ui,j ) skewness Sk : it is derived from ui,j ωi ωj , with ωi the fluctuating vorticity, and rescaled to coincide with −u31,1 /(u21,1 )3/2 in isotropic turbulence. Comparisons of the fully anisotropic EDQNM2 model [13], DNS, and recent experimental data [43], are shown in Fig. 1.

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Some limitations of the model are that initial isotropy cannot be broken and all subtleties for the saturated transition from 3D to 2D structure are

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missed.

tors

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2.3. General case with production, reduction to sherically-averaged descrip-

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In all other cases of turbulence subjected to arbitrary mean velocity matrix Aij , which is not purely antisymmetric, energy production exists,

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and the need for a very complex closure of nonlinear transfer and pressureredistribution terms is less urgent. For such flows, it is expected that the EDQNM1 version is sufficient to deal with the complexity of the turbulence dynamics. In addition to the fact that the explicit effect of the mean gradient is not essential in the dynamics of triple correlations (as it is in purely ro14

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0

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10

−1

−S

10

−2

−2

−1

10

0

10 Roω

1

10

10

2

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Figure 1: Evolution of velocity derivative skewness factor Sk (absolute value, denoted as

S or −S on the figure). Top left: experimental results from [43], Sk plotted in terms of the instantaneous micro-Rossby number Roω = ω 0 /(2Ω), with ω 0 the rms value of

fluctuating vorticity. Top right: results from DNS (symbols) and simplified EDQNM (lines) for various initial Roω ; Sk plotted in terms of 1/Roω . The curve in plain line in p both top left and right figures is defined by15Sk = Sk |Roω =∞ / 1 + 2/Ro2ω , suggested by

isotropic EDQNM using Eq. (14). Bottom: evolution of Sk for the different indicated Roω

given by DNS (symbols in (a)), fully anisotropic EDQNM2 model (symbols in (b)) and simplified isotropic EDQNM (lines in (a) and (b)) using Eq. (14) with unsteady temporal variation of Sk , started with isotropic initial data; the elapsed time is non-dimensional in term of the initial turnover time based on the kinetic energy and its dissipation rate [13].

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tating turbulence, without production), two reasons prevent the use of most complicated EDQNM(2-3) versions:

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• Such versions render the structure of the EDQNM model much more complicated, via a three-fold product of tensorial Green’s functions

from eq. (10), and explicitly dependent on the type of mean shear, complicating easy further projection of angular harmonics.

• They are not correct when the (direct) linear effect of the mean shear/strain

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yields exponential growth, because the time integral of the abovementioned three-fold product diverges [14].

EDQNM1 equations can be written for arbitrary anisotropy, and yield closed expressions for generalized transfer terms T (E) and T (Z) [13, 42] in

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terms of E and Z, and thereby close equations (7) and (8). Faced with the high computational cost of solving these equations for all wavevector

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directions, a second step was applied to restrict the description to spherically averaged descriptors. This purely technical step amounts to replacing fully

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anisotropic E and Z terms by the truncated expansions E(k, t) =

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and


E(k, t) (dir) 1 − 15Hmn (k, t)αm αn 2 4πk

(15)

5 E(k, t) (pol) ∗ H (k, t)Nm (α)Nn∗ (α). (16) 2 4πk 2 mn (dir) These expansions involve the non-dimensional deviatoric tensors Hmn and

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Z(k, t) =

(pol) ˆ ij (k, t) on spherical shells of radius Hmn which are given by integrating R

k =| k |, so that   Z 1 (dir) (pol) 2 ˆ ϕij (k, t) = Rij (k, t)d k = 2E(k, t) δij + Hij (k, t) + Hij (k, t) . 3 Sk (17) 16

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The three contributions, isotropic, directional anisotropy and polarization

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anisotropy derive, term-to-term from Eq. (2), rewritten as   E(k) E(k) ˆ (δij − αi αj )+< (Z(k)Ni (α)Nj (α)) . Rij (k) = (δij − αi αj )+ E(k) − 4πk 2 4πk 2 On the one hand, it is possible to extract from an arbitrary anisotropic (dir)

ˆ ij the set of spherically-averaged descriptors (E, H spectral tensor R ij

(pol)

, Hij

in which directional anisotropy and polarization anisotropy are disentangled.

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On the other hand, it is possible to reconstruct a part of the full spectral tensor by means of these descriptors using eqs. (15) and (16). Of course, only a part of the anisotropic structure is restored, because angular harmonics of degree larger than 2 are ignored (see also [51]). It is consistent to express the generalized transfer terms using the same truncated expansion, or

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and

 T (k, t)  ˜N L(dir) (k, t)αm αn 1 − 15 S mn 4πk 2

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T (E) (k, t) =

T (Z) (k, t) =

5 T (k, t) ˜N L(pol) ∗ Smn (k, t)Nm (α)Nn∗ (α). 2 2 4πk

(18)

(19)

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Finally, a closed system of equations, given in [42], is found for the set (dir)

(E, Hij

(pol)

, Hij

), in which linear terms in the left-hand-side of eq. (7) and L(dir)

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(8) give contributions T L , Sij

yield the above-mentioned T ,

L(pol)

and Sij

T ˜N L(dir) S 4πk2 ij

, whereas nonlinear contributions

and

T ˜N L(pol) S 4πk2 ij

terms. In short,

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a model using only spherically-averaged descriptors is exactly derived from the simple EDQNM1 model in k-vector space, using second-order truncated expansions just described. The resulting simplified model, say MCS (Mons, Cambon & Sagaut [42]), is flexible, versatile, and tractable. Its nonlinear part reduces to calculations 17

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similar to those of isotropic EDQNM, because integration over the direction of the plane of the triads can be performed analytically, and the orientation

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of the wave vector is eventually removed analytically as well. This model has been validated in [42] for several flow cases, including effects of irrotational

straining processes (Aij symmetric, possibly time-dependent), plane shear, and return-to-isotropy when mean flow gradients are removed. On the other

hand, the rapid distortion limit is no longer exact in the MCS model, be-

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cause the very strong anisotropy induced by RDT calls into play higher order angular harmonics.

Combination of linear and nonlinear effects seems to be rather simple in flows subjected to irrotational strain: the exponential growth induced by the linear operator is significantly damped by nonlinear effects, but without evi-

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dence of saturation, at the level of second-order dynamics. As shown in [10], such behavior is qualitatively close to what is found in unstable homogeneous

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stratification. The case of pure plane shear is radically different and therefore merits a separate investigation, which is given below. The linear operator

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induces algebraic growth, so that the nonlinear terms are responsible for eventual exponential growth. When looking at the turbulent kinetic energy,

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the transition from ‘linear’ to ‘nonlinear’ dynamics is essentially triggered by redistribution between Reynolds stress components by pressure-strain rate tensor.

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The mean plane shear is characterized by Aij = Sδi1 δj3 , with the index

1, 2 and 3 referred to streamwise, spanwise and vertical (cross-gradient) direction, respectively. Looking at the numerical results of MCS model, the evolution of single-point statistics (turbulent kinetic energy, Reynolds stress

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components) is correct even for initially high shear rates and very long nondimensional St times. In addition, the spherically averaged energy spectrum

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and its counterpart for the cross-correlation exhibit the conventional −5/3

and −7/3 slopes respectively, for an inertial range of unprecedented length. New results are presented in the next subsection. 2.4. Discussion of results

2.4.1. Weakly anisotropic scaling for spherically-averaged descriptors

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Results from [29, 56] can be translated using the present formalism as

follows. Scaling laws are proposed for weak anisotropy, when the local in√ teraction parameter, or k 2/3 ε1/3 /S  1, with S ∼ Amn Amn and ε is the dissipation rate. This condition is equivalent to choosing the wavenumber

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k to be much larger than the threshold, the inverse of Corrsin’s scale [19], (which amounts to a Zeman’s (or Hopfinger’s) scale [28, 59] for purely rotat-

ED

ing flow), or k  kS , with

kS =



S3 ε

1/2

.

(20)

PT

In this wavenumber range, the scalings laws recover the classical Kolmogorov inertial range for the energy spectrum E(k), whereas the anisotropic spherical

CE

descriptors are (dir)

AC

Hij

(k) =

1 (B − A)ε−1/3 k −2/3 Sij , 15

(pol)

Hij

2 (k) = Aε−1/3 k −2/3 Sij , (21) 5

in which A and B are constants, assumed to be universal, with a satisfactory agreement between DNS for pure plane shear flow and LRA. These scaling laws recover the −7/3 slope for the non-diagonal component of the spherically-averaged spectral tensor in the case of pure plane shear, as found 19

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in MCS as well (and in simpler models, suggested long time ago, e.g. by Lumley [41]). In addition, separate contributions from directional anisotropy and

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polarization anisotropy are shown in figs. 2, 3 and 4 for sheared turbulence at high St (St = 50) started from isotropic initial data with Saffman infrared (dir)

spectral range (k 2 ). The two contributions from ϕ13 , 2EH13

(pol)

and 2EH13 ,

are plotted in fig. 2, both in log-log coordinates— to show the typical slope —, and in log-linear ones — to show the change of sign of the directional

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component —. Given the weak values of anisotropy for wavenumbers larger than the threshold wavenumber kS , as expected, and in the infrared range, (dir)

(pol)

the ratio H13 /H13

is plotted in fig. 4. Even if a plateau is not very

clear on the latter figure, an asymptotic ratio of about −0.6 is shown, to be compared to the ratio

2 B−A 3 A

in Eq. (21).

M

Looking at the diagonal component ϕ11 , similar results are found, but a better plateau is obtained for the above-mentioned ratio, with a positive

ED

value of about 0.6. Note that the rise of a −7/3 slope, shown in fig. 3 is not predicted by Eq. (21).

PT

In order to recover additional anisotropic diagonal components in [29, 56], the temporal variation of ε may be invoked, in accordance with the inertial

AC

CE

range energy distribution of unsteady turbulence, as in [57, 50, 49]   ε(t) ˙ 2/3 −5/3 −2/3 E(k, t) = Ck ε(t) k 1 + C1 k ε(t)

(22)

with C1 a constant of order unity. Finally, for single-point quantities, obtained by integrating the above-

mentioned spectra

20

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0.2

0

10

0 −2

10

kS

−0.4

k

−6

10

−0.6

−7/3

−0.8 −8

10

−1

kS −10

10

−2

10

0

2

10

10

4

6

10

−2

10

10

k/kL

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−0.2

−4

10

0

2

10

10

4

10

6

10

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k/kL

Figure 2: Spectral contributions for the non-diagonal component ϕ13 (k) in log-log (left) (dir)

and log-linear (right) representations: total (full line), directional 2E(k)H13

0

−2

10

−6

−8

0.4 0.2

k −7/3

CE

10

0

kS

−10

−2

10

0

10

2

10

4

10

−0.2

6

10

−2

10

k/kL

0

10

2

10

4

10

6

10

k/kL

AC

10

kS

0.6

PT

10

10

0.8

k −5/3

−4

1.2 1

ED

10

(k) (dotted

(dashed line) contributions.

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line) and polarization

(pol) 2E(k)H13 (k)

Figure 3: Spectral contributions for the diagonal component ϕ11 (k) in log-log (left) and (dir)

log-linear (right) representations: total (full line), directional 2E(k)H11 and polarization

(pol) 2E(k)H11 (k)

(dashed line) contributions.

21

(k) (dotted line)

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2 1.5

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1 0.5 0

M

−0.5

kS

−4

10

−2

10

0

10

2

10

4

10

k/kL

CE

PT

−1.5

ED

−1

Figure 4: Ratios H (dir) /H (pol) for both non-diagonal (dashed line) and diagonal (full line)

AC

components. Log-linear plotting.

22

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with q 2 = hui ui i = (dir) −0.22, b13

=

R∞ 0

Z

0

∞

1 + 2 q } |

(dir) 2E(k, t)Hij (k, t)dk

{z

(dir)

bij

Z

0

∞

(pol)

2E(k, t)Hij {z (pol)

(k, t)dk , }

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hui uj i 1 1 bij = − δij = 2 2 q 3 q |

bij

(23)

2E(k, t)dk, the constant, asymptotic, values: b13 =

(pol) −0.01, b13

(dir)

= −0.23, b11 = 0.14, b11

= 0.09 are

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found.

(pol)

= 0.05, b13

2.4.2. Towards higher anisotropy and full spectral range

It is important to notice that the scalings from [29, 56] give no anisotropic effect of the antisymmetric part of the mean velocity gradient. Looking at purely rotating turbulence, we agree that there is a robust trend to recover

M

isotropy for scales smaller than the Hopfinger’s (or Zeman’s) scale. On the other hand, considering stronger rotation effects, or rotation acting on a

ED

larger spectral range, including larger scales, very different non-conventional anisotropic cascades are found. For instance, even the simplest “isotropized”

PT

model resulting from EDQNM1 started with isotropic initial data with correc√ tion by eq. (14) yields energy spectra as E(k) ∝ Ω2 k −3 or E(k) ∝ Ωεk −2 ,

CE

with high sensitivity to initial data and possible bifurcation from one solution to another, as shown by an asymptotic analytical study (Julian Scott,

AC

private communication). The second scaling law was proposed by Zhou [60] under phenomenological considerations. More complex anisotropic models EDQNM2-3 give an angle-dependent spectrum E, consistent with wave turbulence theory [22, 4]. The Zeman’s wavenumber tends to infinity in the latter limit (that is the opposite of the case of weak anisotropy in the previ23

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ous subsection), with vanishing Rossby number, very high Reynolds number and very high elapsed time, but EDQNM2-3 can reconcile all trends, for any

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Rossby number and almost any Reynolds number. Finally, the simplified model for shear-driven turbulence with energy pro-

duction restricted to the dynamics of spherically-averaged descriptors gives interesting results but it must be improved. In contrast with the case of

buoyancy-driven flows addressed in the next section, global trends such as

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the eventual exponential growth rate in the case of pure plane mean shear

seems almost independent of the initial data and especially of the slope of the infrared range of the initial energy spectrum. We do not think that this lack of sensitivity is due to a strong physical difference between shear-driven and buoyancy-driven flows, and we have in mind the too crude evaluation of

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dominant RDT mechanisms, competing with nonlinear ones, in the present

ED

MCS model, especially at the largest scales. 3. Buoyancy-driven flow, from stable to unstable stratification

PT

When energy production and/or anisotropy is generated by a mean buoyancy gradient, together with a fluctuating buoyancy force, strong analogies

CE

are found with the case of shear-driven flows, but: • The problem is complicated by the presence of a fluctuating buoyancy

AC

term, whether density, temperature or concentration is the stratifying agent. The buoyancy fluctuation is governed by an equation similar to the passive scalar equation in the presence of a mean scalar gradient. The buoyancy scalar, however, is active because it has a feedback on the velocity equation through the buoyancy force. In the simplest 24

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case of the Boussinesq approximation, the density can fluctuate, as the source of the buoyancy force (imbalance between gravity force and

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Archimedean force), but the velocity field remains solenoidal. • It is possible to neglect the mean velocity field so that there is no advection term in the modified Navier-Stokes equations. From this viewpoint, basic equations are simpler than the ones in the shear-driven case: the basic linear operator is self-adjoint and there is no need to

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follow characteristic lines (eq. (6)) as in eqs. (7) and (8). In addition, basic equations are fully consistent with axisymmetry with mirror symmetry, reducing the complexity of statistics compared to, e.g., turbulence subjected to pure plane (mean) shear flow.

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In this sense, stable stratification has strong analogy with rotating turbulence, and unstable stratification has analogy with irrotational straining,

ED

so that the same theoretical and numerical tools are relevant, but special adaptations are needed, as discussed below.

PT

3.1. Governing equations before closure Basic equations for velocity, buoyancy and reduced pressure fluctuations 

 ∂ ∂p ∂ui 2 − ν∇ ui (x, t) + bδi3 + = −uj , ∂t ∂xi ∂xj

CE

are

AC

and



 ∂b ∂ 2 − χ∇ b(x, t) − N 2 u3 = −uj . ∂t ∂xj

∂ui = 0, ∂xi

(24)

(25)

The equations are first given in the stable case. In the unstable case, the sign of the square of the stratification parameter N ought to be changed. 25

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The gravitational acceleration is gi = −gδi3 and the stratification frequency q ∂ρ N = − ρg0 ∂x when a vertical mean density gradient is applied, with ρ0 a 3

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constant density reference. In the following, the Schmidt number χ/ν will

be taken equal to 1.

Equations for two-point second-order correlations are derived for a state

vector in four components, by virtue of axisymmetry with mirror symmetry: the second-order spectral tensor of velocity correlations reduces to toroidal (E − Z)) and poloidal

1 2

(E + Z) energy spectra, Z being purely real, the

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1 2

spectrum of the buoyancy variance reduces to a spectrum of potential energy E (pot) , and the cross-correlation between velocity and buoyancy reduces to a

PT

ED

M

co-spectrum of poloidal buoyancy flux Ψ [23]. Their governing equations are   ∂ k⊥ 2 + 2νk E(k, t) + 2N Ψ = T (E) (k, t), (26) ∂t k   ∂ k⊥ 2 + 2νk Z(k, t) + 2N Ψ = T (Z) (k, t), (27) ∂t k   ∂ k⊥ 2 + 2νk E (pot) (k, t) − 2N Ψ = T (pot) (k, t), (28) ∂t k     k⊥ 1 ∂ 2 (pot) + 2νk Ψ(k, t) + N (E + Z) − E = T (Ψ) (k, t). (29) ∂t k 2

CE

Generalized transfer terms are gathered in the right-hand-side. In the stable case, considered here first, the total, kinetic plus potential energy (with spectrum E +E (pot) ), is conserved in the inviscid limit, so that the sum T (E) +T (pot)

AC

has zero integral over k. The nonlinear flux term T (Ψ) has a nonzero integral; it is possible, however, to disentangle in it a conservative part (with zero integral) and a part of pressure-strain rate type, as for T (Z) in the shear-driven flow case.

26

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In the linear inviscid limit, for the stable case, at vanishing right-hand sides, one recovers the conservation of toroidal energy ( 12 (E − Z)) whereas

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the dynamics of gravity waves only affects the poloidal mode, with spectral energy ( 12 (E + Z)), whereas poloidal and potential energy spectra are affected by periodic oscillations with opposite phases, with amplitude given by

their initial imbalance, and call into play the dispersion frequency of gravity waves σs = N

(30)

p 1 − α32 , using the notations from the shear-driven case. In

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with k⊥ /k =

k⊥ , k

the basic equations, the unstable case is obtained by changing the sign of the

factor N 2 in eq. (25). Examples of evaluation of N in geophysical flows are obtained by changing the sign of the mean gradient. A very recent applica-

M

tion of USHT (Unstably Stratified Homogeneous Turbulence) addressed the dynamics of the mixing zone induced by Rayleigh-Taylor instability, when

ED

turbulence is sufficiently developed to permit a quasi-homogeneous approach, with a given value of N [10]. As in the stable case, N is still called the

PT

“stratification frequency”, even if temporal oscillations are replaced by an exponential growth. The buoyancy variance, divided by N 2 , is again called

CE

potential energy, even if this terminology is less obviously applicable than in the stable case. In the linear inviscid limit, it is found that both poloidal and

AC

potential energy spectra exhibit the same exponential growth exp(2N t kk⊥ ). Even if the total energy is not conserved, this is due to linear “production” effects, so that the integral over k of the sum T (E) + T (pot) is still zero.

27

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3.2. Third-order statistics and EDQNM closure As for the shear-driven flow case, the generalized transfer terms T (E) ,

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T (Z) , T (pot) , T (Ψ) are not directly expressed in terms of the state-vector (E, Z, E (pot) , Ψ), but the triadic closure is performed at the level of threepoint third-order correlations. 3.2.1. Stable case

As for shear-driven eq. (10), the general EDQN equation for three-point

M

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third-order correlations is written as      k⊥ 0 p⊥ 00 q⊥ 2 2 2 +s +s (t − t0 ) Sss0 s00 (k, p, t0 ) Sss0 s00 (k, p, t) = exp − ν(k + p + q ) − ıN s k p q # " Z 0   Z t t p q k ⊥ ⊥ ⊥ (QN ) + s0 + s00 (t − t0 ) τss0 s00 (k, p, t0 )dt0 . + exp − µkpq (t00 )dt00 + ıN s k p q t0 t0 (31)

ED

For mathematical convenience, a diagonal form of the RDT Green’s function is used, projecting the correlation tensor on the eigenmodes of the linear regime, mixing toroidal (“vortex”) mode and gravity wave modes.

PT

Accordingly the signs, s, s0 , s00 take not only the value ±1, for waves (gravity waves here), as in purely rotating turbulence, but also the value zero, for

CE

the toroidal mode. Another difference is that the resonant operator in the previous equation involves the dispersion frequency of gravity waves (30). In

AC

this case, EDQNM2, in which the temporal integral is only applied to the “resonant” N -operator, which reflects the three-fold product of inviscid RDT

Green’s function as in eq. (10), is much more suitable than the EDQNM1 version, in which this term is neglected.

28

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3.2.2. From stably to unstably stratified homogeneous turbulence

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The EDQN equation is now      k⊥ 2 2 2 0 p⊥ 00 q⊥ Sss0 s00 (k, p, t) = exp − ν(k + p + q ) − N s +s +s (t − t0 ) Sss0 s00 (k, p, t0 ) k p q " Z 0 #   Z t t k p q ⊥ ⊥ ⊥ (QN ) µkpq (t00 )dt00 − N s + exp − + s0 + s00 (t − t0 ) τss0 s00 (k, p, t0 )dt0 . k p q t0 t0 (32)

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The essential difference with the stable case is now the presence of real terms,

instead of purely imaginary ones, that account for the explicit effect of N . Accordingly, as noted in [14], the EDQNM2 version cannot be applied as in the stable case, because the temporal integral of the term with N factor no longer converges. Consequently, a strict EDQNM1 version, which amounts to

M

setting N = 0 in the above integral, was first checked when the full numerical code used in the stable case was adapted to the unstable one.

ED

The related model and numerical code adapted to USHT gave satisfactory results compared to high resolution DNS, but the accumulated energy transfers appeared somewhat overestimated compared to their DNS counter-

PT

part. Much more accurate EDQNM1/DNS agreement was finally obtained in [10] by reintroducing N in a global quasi-isotropic way. This procedure is

AC

CE

consistent with a heuristic closure of the EDQN equation for USHT,   k⊥ (IV ) 0 p⊥ 00 q⊥ −τss0 s00 (k, p, t) − N s +s +s Sss0 s00 (k, p, t) = k p q
= η N (k, t) + η N (p, t) + η N (q, t) Sss0 s00 (k, p, t), s, s0 , s00 = 0, ±1,

with

N

η (k, t) = a0

Z

k 2

p E(p, t)dp

0

29

1/2

+ a1 N.

(33)

(34)

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We do not attempt to justify the additional term in eq. (34) that represents a global (isotropized) estimate of the explicit linear anisotropic N -term in the EDQN equations, or the contribution from fourth-order cumulants

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(IV )

τss0 s00 . However, it is shown that a reasonable fitting of the constant a1 gives good overall results [10].

Interestingly, the analogy with similar adjustments in wave turbulence, as in eq. (14) for purely rotating turbulence, is not really convincing. It is cer-

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tain that the effect of interacting dispersive waves results in a global damping of triple correlations. On the other hand, we cannot explain why the explicit

linear operator related to N yields amplification of second-order correlations, but seems to induce global damping of triple correlations. Two perspectives can help lead us to a rational explanation and perhaps to improved modelling

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as well:

i) As done long time ago in rotating turbulence (e.g. fig. 11 in [12]),

ED

it is possible to calculate the impact of the transient term, the first line in eq. (32), which gives the rapid distortion of triple correlations from the

PT

initial value problem. For consistency, initial isotropy can be assumed at t = t0 . This transient contribution was not calculated in almost all related

CE

EDQNM versions, because either t0 was taken at its far past value or the initial contribution of third-order correlations was neglected, in order to isolate the secular term (second line in eq. (32). The optional calculation of

AC

the transient term, which is present in EDQNM2 codes for both rotation and stable stratification, can be reactivated in the USHT code, derived from the EDQNM2 SSHT one. ii) The problem of the lack of convergence of the temporal integral for

30

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a three-fold product of Green’s functions with possible exponential growth, can be treated using a temporal integral from past to future, with non-causal

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Green’s function1 3.3. Discussion

3.3.1. Scalings in terms of spherically-averaged descriptors

Spherically averaged descriptors, extracted from DNS and from the improved EDQNM model for USHT, then subsequently from the model at much

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higher Reynolds numbers, were used to derive new scaling laws and to check the relevance of the Ozmidov scale as a threshold for recovering isotropy.

Regarding velocity correlations, axisymmetry with mirror symmetry yields simplification of eqs. (15,16) as  
(dir) E(k, t) 15 2 E(k, t) = 1− 3 cos θ − 1 H33 (k, t) 4πk 2 2

M

(35)

15 E(k, t) 2 (pol) sin θH33 (k, t), 2 4 4πk

ED

and

Z(k, t) =

(36)

PT

from Hij = 21 (3δi3 δj3 − δij ) H33 , with θ the polar angle of k versus the vertical direction, referred to index 3. The same truncated expansion is applied to

CE

E (pot) as

AC

E

(pot)

E (pot) (k, t) (k, t) = 4πk 2

 
(pot) 15 2 1− 3 cos θ − 1 H33 (k, t) . 2 (pot)

Note that polarization has no meaning for a scalar, so that H33

(37)

only char-

acterizes directional anisotropy. Finally, a similar expansion applied to the 1

from Julian Scott, unpublished.

31

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flux term yields F3 (k, t) = − sin θΨ(k, t) =

3 2 F sin θE3 (k, t), 2

(38)

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in which F3 denotes the vertical component of the buoyancy flux, in agree-

ment with a general (non-axisymmetric) scalar flux vector Fi used in [9] as well.

It is important to notice that the preceding equations which truncate at

the first nontrivial degree of angular harmonics, are not used in the model

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and numerical code by [10, 11]. Angular discretization in the computational approach allows expansions of much higher degree. These expansions are completely consistent with the analysis of [30], in spite of different notations and terminology.

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In addition, the spherically-averaged descriptors used here can be extracted from any axisymmetric spectral distribution, independently of its

ED

degree of angular variability. Similarly, the additional spherically-averaged

Z

PT

descriptors of potential energy and flux are obtained by Z E (pot) (k, t)d2 k = E (pot) (k, t),

CE

2 (pot) sin2 θE (pot) (k, t)d2 k = E (pot) (k, t) + 2E (pot) (k, t)H33 (k, t), 3 Sk

AC

and

(dir)

Instead of H33

(39)

Sk

Z

Sk

F3 (k, t)d2 k = E3F (k, t).

(40)

(41)

(pol)

and H33 , an angular parameter can characterize the scale-

by-scale directional axisymmetry in axisymmetric turbulence as proposed in [11] for instance sin2 γu (k, t) =

2 (dir) + 2H33 (k, t), 3

sin2 γb (k, t) = 32

2 (pot) + 2H33 (k, t). 3

(42)

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Of course, similar indicators could be referred as cos2 γ (e.g. in Quasi-Static MHD [20]) or 3 cos2 γ − 1 (first relevant Legendre polynomial). For instance, the 3D isotropic case, the 2D limit (relevant for USHT) and the 1D limit

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(dir)

(relevant for SSHT) yield 2/3, 1, 0 for sin2 γ and 0, 1/6, −1/3 for H33 , reciprocally.

As for shear-driven flows, the scalings from [30], revisited in [53] and [11]

can be translated into our formalism. These scalings are proposed in an

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inertial wavenumber range which correspond to k  kO , where kO is the threshold wavenumber related to the Ozmidov scale, or  3 1/2 N . kO = 2π ε

(43)

pol dir iso and R33 , R33 in [25] correspond to (2/3)E(k, t), The indicators referred to as R33 (pol)

= E(k, t)(sin2 γu (k, t)−2/3), 2E(k, t)H33 , respectively. From

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(dir)

2E(k, t)H33

ED

the appendix (eqs. A1) in [10], and using Z Z

2 8 2 2 2 sin θk d k = 4πk , sin4 θk d2 k = 4πk 2 3 15 Sk Sk

is found

PT


5 2 N Ce Ck + Co εB ε−1 k −3 , 16
4 (dir) 2E(k)H33 (k) = N 2 Ce Ck + Co εB ε−1 k −3 , 45

CE

E(k) = Ck ε2/3 k −5/3 −

AC

and

(pol)

2E(k)H33 (k) =


8 2 N Ce Ck + Co εB ε−1 k −3 . 15

(44) (45)

(46)

For the spectrum of scalar variance, is found E

(pot)

(k) = Co εB ε

−1/3 −5/3

k

− Cp N

2

C k + C o εB ε

33

−1




k

−3



 5 25 −1 − εB ε , 3 12 (47)

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(pot)

2E (pot) (k)H33 (k) = using sin2 θ =

2 3


8 Cp N 2 k −3 Ck + Co εB ε−1 , 45

(48)

− 13 (3 cos2 θ − 1). Finally, the flux term amounts to

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4 E3F (k) = Cf N Ck ε1/3 + Co εB ε−2/3 k −5/3 9

(49)

Under this form, the proportionality of the various anisotropic indicators (dir)

(pol)

(pot)

H33 , H33 , and H33

is immediately displayed with simple coefficients.

It is also clear that there is no reason based on tensorial structure to dis-

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card terms proportional to N k −7/3 in eqs. (45,46,47) a priori rather than

discarding terms proportional to N 2 k −3 . The scaling in N k −7/3 is in closer accordance with results from [11] at the highest Reynolds number. We have also to keep in mind that the study of [30] implicitly addresses the case of

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stable stratification only. As a final remark on the N k −7/3 scaling, such a term can be added in Eq. (44) via unsteadyness, as for Eq. (22). This term

ED

is not significant from our new numerical results, given as follows for the anisotropic descriptors only.

The different contributions to ϕ13 are plotted on Fig. 5, left, similarly as (dir)

(pol)

PT

in Fig. 2, with the ratio H33 /H33

given on right. The agreement between

DNS and (USHT) EDQNM is very good, but the turbulent Reynolds number

CE

(Re ≈ 5000) allowed by DNS (20483 ) from [25] is not sufficiently high to yield very clear and long-range spectral slopes. In addition, the change of sign of

AC

the polarization term at larger k in DNS is not understood, and DNS is not

very useful to explore this range. Consequently, results are extended to a very high Reynolds number (Re = 3 × 105 ) using (USHT) EDQNM only

in Fig. 6 with set up described in [24]. The k −7/3 slope is ascertained for

both contributions from directional anisotropy and polarization anisotropy, 34

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0

10

k −7/3 −5

10

kO 0

2

10

k/kL

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10

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k −5/3

Figure 5: Left: spectral contributions for the component ϕ33 (k) in log-log representation: (dir)

total (full line), directional 2E(k)H33

(pol)

(k) (dotted line) and polarization 2E(k)H33

(dashed line) contributions; EDQNM (black), DNS (green). Right: ratio

(k)

(dir) (pol) H33 /H33

in

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log-linear representation; EDQNM (black), DNS (green).

for wavenumbers much larger than the Ozmidov’s threshold kO , even if a

ED

small k −3 zone could be identified for wavenumbers just after the energy peak.

PT

3.3.2. Role of initial conditions on the late time dynamics A lesson learned from the decay of HIT [2, 39] is that the initial distribu-

CE

tion of energy at large scales influences the late time dynamics of turbulent flows. This characteristic is also observed in USHT, and is expressed by

AC

a formula relating the growth rate exponent to the initial slope of infrared spectra [54]. These predictions are difficult to check with DNS as the large scales become confined by the finite size of the computational domain. However, anisotropic two-point models based on triadic closures can reach the late time self-similar regimes. Accurate measurements of the growth rate 35

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0

10

−5

10

kO k −7/3

−10 −2

0

10

10

2

k/kL

4

10

10

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10

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k −5/3

Figure 6: Same as for the preceding figure. EDQNM only at high Reynolds number Re = 3 × 105 .

exponents confirm the theory [11]. It is important to stress that this kind of

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study can be generalized to anisotropic flows, as in the case of shear driven

ED

turbulence.

4. Conclusion and perspectives

PT

In this article, the relevance of two-point turbulent models based on triadic closures is investigated for homogeneous anisotropic turbulence, with a

CE

focus on generalized EDQNM. Applications are illustrated here by shear and buoyancy driven turbulence. The generic form of the closure relationship for third-order three-point

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correlations is given by Eq. (10) for shear-driven turbulence, with two building blocks: A three-fold product of nonlinear tensorial Green’s functions, and the quasi-normal expression of the contributions from fourth-order correlations. Anisotropy can affect both terms, thus the transfer terms mediated

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by third-order correlations. Anyway, anisotropy is reflected in the equations governing second-order correlations, via linear terms inherited from RDT.

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Letting aside the anisotropic form of the quasi-normal term, built from products of anisotropic second-order correlations, which is a purely technical cal-

culation, the choice or the derivation of the nonlinear Green’s function is the essential problem of closure.

Compared with more elaborate (in principle) two-time triadic closures,

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the nonlinear Green’s function corresponds to the Kraichnan’s response function. On the one hand, the generic equation for two-time second-order correlations involves a single response tensor, but a three-fold product is recovered, as in Eq. (10) when “converting” two-time into single-time statistics. The

other two factors appear in line with the use of a fluctuation-dissipation

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theorem that again calls into play the response tensor. How the “exact” linear operator coexists with the specific nonlinearity of the response tensor, or

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generic Green’s function, for the closure of third-order correlations? Classical wave turbulence theory, discussed here for purely rotating flows, gives an ex-

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treme case in which the nonlinear part is unimportant, so that the three-fold product of “bare” linear Green’s functions amounts to the resonant operator.

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In the other theories surveyed in our introduction, such as RMC (Realizable Markovian Closures), the capture of transient wave effects needs to keep the Eddy Damping in altering its evolution equation. Without wave effects, and

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with energy production, the opposite situation is found, with a completely dominant part of the response tensor given by the Eddy Damping term. Note again that generalized EDQNM, with different levels of Markovianization or some specific unsteadiness of the Eddy Damping, can match all these situ-

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ations in a tractable way. In contrast, elaborate two-time theories cannot even recover classical wave turbulence theory (anisotropic case, e.g. purely

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rotating flows), if started with isotropic zeroth-order response function. To answer an important question suggested by the RMC approach, we

find in practice that a “quasisteady” and isotropic Eddy-Damping term is sufficient to model unsteady anisotropic flows. We have numerous examples

in decaying turbulence (e.g. Fig. 1), and even in turbulence subjected to

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time-dependent straining processes [42]. Particularly in USHT, generalized

EDQNM, compared to DNS, reproduces well the dramatic increase of energy, anisotropy, and Reynolds number, that result from the exponential amplification at constant N [10]. Promising applications in progress take into account time-dependent N (t). Its temporal evolution can be induced by a feedback

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from fluctuating to mean flow, permitted by weak inhomogeneity. Beyond the theoretical interest, two-point triadic models can also provide

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useful informations concerning the dynamics and structure of complex turbulent flows as the numerical integration can be done explicitly. These predic-

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tions can now be tested against experiments or direct numerical simulations and fully exploited in order to develop new simplified closures for engineer-

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ing purpose. Two different numerical methods, using truncated expansions in terms of spherically-averaged descriptors (shear case) or not (USHT), allow to extend the numerical procedure of Leith [38] for HIT to anisotropic

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cases. Our numerical results here give a better picture of how the turbulence cascade is modified in an anisotropic flow and isotropy is restored at small scales. In addition, the unified formalism is of interest, as summarized in Table 1.

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.

Spherically averaged descriptors Isotropy

Directional anisotropy

E(k, t)

E(k, t)

Z(k, t)

0

E (pot) (k, t)

E (pot) (k, t)

Fi (k, t)

0

(dir)

Hij

(k, t)

0 (pot)

Hij

Polarization anisotropy

(k, t)

0

0

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k-vectors descriptors

(pol)

Hij

(k, t)

0

(F ) Ei (k, t)

Table 1: Description of anisotropy at the velocity, scalar and scalar flux levels, using k-

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vectors descriptors and their corresponding spherically-averaged contributions. Adapted from [9]

The first two lines display the general description of two-point second-order velocity correlations, from full angle-dependence to the first-order spherically-

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averaged descriptors. This is extended to scalar spectrum and scalar-velocity flux co-spectrum in the last two lines, and finally the axisymmetric descrip-

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tion of buoyancy-driven flow is recovered by restricting the angular dependence of basic spectra (in k-vector) to the polar angle only, and by restricting the five independent components of the tensors Hij to the component H33

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only. In this table, a general vector flux, Fi , holds for the spectrum of hui bi; it is restricted to F3 = − sin θΨ in SSHT and USHT here.

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Beyond this formal aspect, cross-fertilization between the two domains,

around the common theme of anisotropic closures, is well advancing. A more

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complex model, as in USHT, is suggested for shear-driven flows with energy production, especially for a better capture of terms inherited from RDT. In turn, a much simpler model could be derived for USHT in terms of nonlinear interactions only restricted to spherically-averaged descriptors. This

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could allow to evaluate quantitatively the impact of higher order harmonics for the transfer terms, probably responsible for long-term transients and

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departure from the simplest scaling laws discussed here. About simpler models, one can explore to which extent results from triadic closures in terms of spherically-averaged descriptors can be obtained by (apparently) local mod-

els (in wavenumber space) in which “diadic” or “pair interactions” dominate,

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following derivation of new anisotropic Leith’s models. Acknowledgments

CC and BJG wish to acknowledge Y. Kaneda and O. Soulard for interesting discussions concerning the different scaling laws of anisotropic spectra.

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Appendix. Helical modes and rotating turbulence

ˆ ij is not a general rank-3 Independently of closure, the spectral tensor R

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complex matrix, but has a number of special properties, including the fact ˆ ij kj = 0, obtained from that it is Hermitian, positive-definite, and satisfies R

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ˆ ij ∝ hˆ its definition as R u∗i uˆj i and the incompressibility condition kj uˆj = 0. Taken together, these properties mean that, instead of the 18 real degrees of

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ˆ ij has but freedom of a general complex tensor (9 for Rij in physical space), R 4. Indeed, these 4 components are identified using a spherical polar coordi-

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nate system in k-space [12, 52] (e.g. the Craya-Herring frame of reference and its variants of helical modes, recalled below). where N (α) = e(2) (α) − ıe(1) (α),

N ∗ (α) = N (−α) = e(2) (α) + ıe(1) (α) (50)

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are the helical modes [12, 55], defined from the local Craya-Herring frame of reference [26] (e(1) , e(2) , α) attached to a polar-spherical system of coordi-

e(1) =

α×n , |α×n|

e(2) = α × e(1) ,

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nates for the wave vector, with polar axis n, or α=

k . |k|

(51)

ˆ ij is generated by four scalar (or pseudo-scalar) spectra but Z In fact, R

is complex-valued (equivalent to two real terms, with its modulus only be-

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ing frame-invariant,) and only the spectrum of helicity, which generates the ˆ ij (e.g. [12]) is ignored purely antisymmetric, purely imaginary, part of R here, because it cannot be created by mean flow gradients, and must be initialized or forced, often in an artificial way.

Helical modes allow a rigorous separation between amplitudes and phases

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in rotating turbulence. The velocity field is expressed as ˆ (k, t) = u

X

as (k, t)e2ısΩ·αt N (sα).

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s=±1

In the purely linear limit, the as , s = ±1 are constant, with their values re-

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lated to initial data. In the “weak” approximation of inertial wave turbulence theory, it is possible to use two-time asymptotics considering “slow” ampli-

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tudes, or as (k, t) and “rapid” phases σt = 2Ω.·αt, with  a small parameter of the order of magnitude of the Rossby number. Even if less conventional in

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classical wave turbulence theory, it is perhaps better to consider amplitudes with regular time as (k, t) and rapid phases σ t , as suggested by the approach

of Lifshitz and Hameiri (details in [52]). It is then possible to transfer the EDQNM machinery from velocity Fourier

ˆ (k, t) to “slow amplitudes” as (k, t). Contribution of “rapid” phases modes u 41

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averages out, so that they only survive in the resonance operator, similar to the one in Eq. (31) for s, s0 , s00 = ±1. The most elaborate EDQNM3 model

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[4] rigorously matches classical wave turbulence theory in the asymptotic limit of vanishing Rossby number, where the eddy damping is unimportant

(or σ(α)  η(k)), and allows to treat all Rossby numbers, recovering the other limit of conventional EDQNM for HIT at large Rossby number, when

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