The influence of rotation on the spectral energy transfer of homogeneous turbulence is investigated in this paper. Given the fact that linear dynamics, e.g. the inertial waves regime found in an RDT (rapid distortion theory) analysis, cannot affect a homogeneous isotropic turbulent flow, the study of nonlinear dynamics is of prime importance in the case of rotating flows. Previous theoretical (including both weakly nonlinear and EDQNM theories), experimental and DNS (direct numerical simulation) results are collected here and compared in order to give a self-consistent picture of the nonlinear effects of rotation on turbulence.The inhibition of the energy cascade, which is linked to a reduction of the dissipation rate, is shown to be related to a damping of the energy transfer due to rotation. A model for this effect is quantified by a model equation for the derivative-skewness factor, which only involves a micro-Rossby number Roω=ω′/(2Ω) – ratio of r.m.s. vorticity and background vorticity – as the relevant rotation parameter, in accordance with DNS and EDQNM results.In addition, anisotropy is shown also to develop through nonlinear interactions modified by rotation, in an intermediate range of Rossby numbers (RoL<1 and Roω>1), which is characterized by a macro-Rossby number RoL based on an integral lengthscale L and the micro-Rossby number previously defined. This anisotropy is mainly an angular drain of spectral energy which tends to concentrate energy in the wave-plane normal to the rotation axis, which is exactly both the slow and the two-dimensional manifold. In addition, a polarization of the energy distribution in this slow two-dimensional manifold enhances horizontal (normal to the rotation axis) velocity components, and underlies the anisotropic structure of the integral length-scales. Finally a generalized EDQNM (eddy damped quasi-normal Markovian) model is used to predict the underlying spectral transfer structure and all the subsequent developments of classic anisotropy indicators in physical space. The results from the model are compared to recent LES results and are shown to agree well. While the EDQNM2 model was developed to simulate ‘strong’ turbulence, it is shown that it has a strong formal analogy with recent weakly nonlinear approaches to wave turbulence.

International audienceAn asymptotic quasi-normal Markovian (AQNM) model is developed in the limit of small Rossby number Ro and high Reynolds number, i.e. for rapidly rotating turbulent flow. Based on the 'slow' amplitudes of inertial waves, the kinetic equations are close to those that would be derived from Eulerian wave-turbulence theory. However, for their derivation we start from an EDQNM statistical closure model in which the velocity field is expanded in terms of the eigenmodes of the linear wave regime. Unlike most wave-turbulence studies, our model accounts for the detailed anisotropy as the angular dependence in Fourier space. Nonlinear equations at small Rossby number are derived for the set e, Z, h - energy, polarization anisotropy, helicity - of spectral quantities which characterize second-order two-point statistics in anisotropic turbulence, and which generate every quadratic moment of inertial wave amplitudes. In the simplest symmetry consistent with the background equations, i.e. axisymmetry without mirror symmetry, e, Z and h depend on both the wavevector modulus k and its orientation θ to the rotation axis. We put the emphasis on obtaining accurate numerical simulations of a generalized Lin equation for the angular-dependent energy spectrum e(k, θ , t ), in which the energy transfer reduces to integrals over surfaces given by the triadic resonant conditions of inertial waves. Starting from a pure three-dimensional isotropic state in which e depends only on k and Z = h = 0, the spectrum develops an inertial range in the usual fashion as well as angular anisotropy. After the development phase, we observe the following features: (a) A k^−3 power law for the spherically averaged energy spectrum. However, this is the average of power laws whose exponents vary with the direction of the wavevector from k^−2 for wavevectors near the plane perpendicular to the rotation axis, to k^−4 for parallel wavevectors. (b) The spectral evolution is self-similar. This excludes the possibility of a purely two-dimensional large-time limit. (c) The energy density is very large near the perpendicular wavevector plane, but this singularity is integrable. As a result, the total energy has contributions from all directions and is not dominated by this singular contribution. (d ) The kinetic energy decays as t^−0.8 , an exponent which is about half that one without rotation

This paper investigates some irreversible mechanisms occurring in homogeneous stably stratified turbulent flows. In terms of the eigenmodes of the linear regime, the velocity-temperature field is decomposed into a vortex and two wavy components. Using an eddy-damped quasinormal Markovian (EDQNM) closure with the axisymmetry hypothesis, an analysis of the anisotropic energy transfers between the vortex kinetic energy, the wave kinetic and potential energy is made. Within the light of triadic exchanges, and by analogy of the resonance condition for three linearly interacting gravity waves, the closure model allows one to compute the detailed transfers for eight types of interactions. Results of the calculations include time evolution plots, for the isotropic closure model as well as two different types of the anisotropic closure. The pure vortical interactions are shown to be responsible for the irreversible anisotropic structure created by stable stratification, and this structure prevents the inverse cascade of two-dimensional turbulence.

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We analyze the anisotropy of turbulence in an electrically conducting fluid in the presence of a uniform magnetic field, for low magnetic Reynolds number, using the quasi-static approximation. In the linear limit, the kinetic energy of velocity components normal to the magnetic field decays faster than the kinetic energy of component along the magnetic field [Moffatt, JFM 28, 1967]. However, numerous numerical studies predict a different behavior, wherein the final state is characterized by dominant horizontal energy. We investigate the corresponding nonlinear phenomenon using Direct Numerical Simulations. The initial temporal evolution of the decaying flow indicates that the turbulence is very similar to the so-called "two-and-a-halfdimensional" flow [Montgomery & Turner, Phys. Fluids 25(2), 1982] and we offer an explanation for the dominance of horizontal kinetic energy.

Stably stratified freely decaying homogeneous turbulence is investigated by means of direct numerical simulations (DNS) and a two-point closure statistical model of the EDQNM type; a careful comparison with laboratory experiments is also made. Several aspects of anisotropy in the flow are studied, both at large and small scales. DNS and EDQNM approaches give very similar results up to the finest indicators of the flow, namely anisotropic spectra of velocity fields. Hence the statistical model predicts the structure of the flow at all scales. Large-scale anisotropy appears in the Reynolds stress components and in the directional integral length scales. The well-known collapse of vertical turbulent motion, which yields the organization of the flow into quasi-horizontal vertically decorrelated vortex structures, is retrieved and quantified. Thus, the thickness of the vortex structures is shown to be set by their Froude number being of order one, in agreement with a previous dimensional analysis for an inviscid flow. Smallscale anisotropy is quantified from the components of the velocity and temperature gradients, whereby models for the dissipation rate of kinetic energy and available potential energy are discussed. The mixing properties of the flow are also investigated: the counter-gradient heat flux that exists at small scales appears to inhibit mixing when diffusivity is low enough and the Cox number varies linearly with the parameter ǫ/νN 2. All results agree very well with laboratory experiments on stably stratified grid turbulence, though the initial condition of our computations is different from the flow just behind the grid. This suggests a relative independence of decaying stably stratified turbulence of initial conditions. * (k) •û(k),

In this work, three different approaches are used for evaluating some Lagrangian properties of homogeneous turbulence containing anisotropy due to the application of a stable stratification and a solid-body rotation. The two external frequencies are the magnitude of the system vorticity 2Ω,chosen vertical here, and the Brunt-Väisälä frequency N,w h i c hg i ves the strength of the vertical stratification. Analytical results are derived using linear theory for the Eulerian velocity correlations (single-point, twotime) in the vertical and the horizontal directions, and Lagrangian ones are assumed to be equivalent, in agreement with an additional Corrsin assumption used by Kaneda (2000). They are compared with results from the kinematic simulation model (KS) by Nicolleau & Vassilicos (2000), which also incorporates the wave-vortex dynamics inherited from linear theory, and directly yields Lagrangian correlations as well as Eulerian ones. Finally, results from direct numerical simulations (DNS) are obtained and compared for the rotation-dominant case B =2Ω/N = 10, the stratificationdominant case B =1/10, the non-dispersive case B =1, and pure stratification B =0 and pure rotation N =0. The last situation is shown to be singular with respect to the mixed stratified/rotating ones. We address the question of the validity of Corrsin's simplified hypothesis, which states the equivalence between Eulerian and Lagrangian correlations. Vertical correlations are found to follow this postulate, but not the horizontal ones. Consequences for the vertical and horizontal one-particle dispersion are examined. In the analytical model, the squared excursion lengths are calculated by time integrating the Lagrangian (equal to the Eulerian) two-time correlations, according to Taylor's procedure. These quantities are directly computed from fluctuating trajectories by both KS and DNS. In the case of pure rotation, the analytical procedure allows us to relate Brownian t-asymptotic laws of dispersion in both the horizontal and vertical directions to the angular phase-mixing properties of the inertial waves. If stratification is present, the inertia-gravity wave dynamics, which affects the vertical motion, yields a suppressed vertical diffusivity, but not a suppressed horizontal diffusivity, since part of the horizontal velocity field escapes wavy motion.

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