Wave turbulence in rapidly rotating flows

Abstract

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

Inertial Wave Turbulence Driven by Elliptical Instability

“…Indeed, each inertial wave excited by elliptical instability of the base flow can be itself unstable to a triadic resonance with another pair of inertial waves [22]: these secondary instabilities have been observed both numerically [17,23] and experimentally [24]. Whether these multiple resonances asymptotically lead to a wave turbulence regime [25,26], similar to the recently observed regimes with flexural waves in plates [27], gravity-capillary waves [28], and internal waves [29], remains to be seen. Barker and Lithwick [19], in their local model of the elliptical instability, nonetheless showed that strong geostrophic flows emerge during the saturation and disrupt the inertial wave resonances, leading instead to growth and decay cycles.…”

Kinematic simulation for stably stratified and rotating turbulence

“…This underlying anisotropic aspect is often ignored. It is worth mentioning here some attempts at filling the gap between pure isotropic 3D and pure 2D in rotating turbulence: an asymptotic wave-turbulence approach by Bellet et al (2006) and the DNS by Liechtenstein et al (2006).…”

Hysteretic transitions between quasi-two-dimensional flow and three-dimensional flow in forced rotating turbulence

“…The Taylor-Proudman theorem has succeeded in explaining the cylindrical flow in laboratory experiments and field observations in terms of the Taylor column. However, the theorem cannot describe transitions between the Q2D and 3D flows, because energy is exchanged between the Q2D mode and the 3D mode by nonlinear mechanisms [1]. The energy transfers to the Q2D modes were demonstrated by an instability analysis [2] and weak-turbulence theory in the large-rotation limit [3].…”