Wave turbulence in rapidly rotating flows

Abstract

An 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 $\theta$ to the rotation axis. We put the emphasis on obtaining accurate numerical simulations of a generalized Lin equation for the angular-dependent energy spectrum $e(k, \theta, t)$, 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:

1. 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.

2. The spectral evolution is self-similar. This excludes the possibility of a purely two-dimensional large-time limit.

3. 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.

4. The kinetic energy decays as $t^{-0.8}$, an exponent which is about half that one without rotation.