QNSE theory of turbulence anisotropization and onset of the inverse energy cascade by solid body rotation

Abstract

Under the action of solid body rotation, homogeneous neutrally stratified turbulence undergoes anisotropization and onset of the inverse energy cascade. These processes are investigated using a quasi-normal scale elimination (QNSE) theory in which successive coarsening of a flow domain yields scale-dependent eddy viscosity and diffusivity. The effect of rotation increases with increasing scale and manifests in anisotropization of the eddy viscosities, eddy diffusivities and kinetic energy spectra. Not only the vertical (in the direction of the vector of rotation $\unicode[STIX]{x1D734}$) and horizontal eddy viscosities and eddy diffusivities become different but, reflecting both directional and componental anisotropization, there emerge four different eddy viscosities. Three of them decrease relative to the eddy viscosity in non-rotating flows while one increases; the horizontal ‘isotropic’ viscosity decreases at the fastest rate. This behaviour is indicative of the increasing redirection of the energy flux to larger scales, the phenomenon that can be associated with the energy backscatter or inverse energy cascade. On scales comparable to the Woods’s scale which is the rotational analogue of the Ozmidov length scale in stably stratified flows, the horizontal viscosity rapidly decreases, and in order to keep it positive, a weak rotation limit is invoked. Within that limit, an analytical theory of the transition from the Kolmogorov to a rotation-dominated turbulence regime is developed. It is shown that the dispersion relation of linear inertial waves is unaffected by turbulence while all one-dimensional energy spectra undergo steepening from the Kolmogorov $-5/3$ to the $-3$ slope.