Anisotropic small-scale constraints on energy in rotating stratified turbulence

TL;DR

This paper investigates how energy distribution in rotating stratified turbulence is constrained by potential enstrophy, revealing anisotropic spectral decay patterns and validating predictions through numerical simulations.

Contribution

It provides the first anisotropic energy spectrum scaling predictions in rotating stratified flows based on potential enstrophy constraints, extending previous isotropic results.

Findings

Horizontal kinetic energy decays as $k_h^{-3}$ in large, nearly horizontal modes.

Potential energy decays as $k_z^{-3}$ in large, nearly vertical modes.

Numerical simulations show reasonable agreement with the predicted anisotropic scalings.

Abstract

Rapidly rotating, stably stratified three-dimensional inviscid flows conserve both energy and potential enstrophy. We show that in such flows, the forward cascade of potential enstrophy imposes anisotropic constraints on the wavenumber distribution of kinetic and potential energy. The horizontal kinetic energy is suppressed in the large, nearly horizontal wave modes, and should decay with the horizontal wavenumber as $k_h^{-3}$. The potential energy is suppressed in the large, nearly vertical wave modes, and should decay with the vertical wavenumber as $k_z^{-3}$. These results augment the only other exact prediction for the scaling of energy spectra due to constraints by potential enstrophy obtained by Charney (J. Atmos. Sci. 28, 1087 (1971)), who showed that in the quasi-geostrophic approximation for rotating stratified flows, the energy spectra must scale isotropically with total wavenumber as $k^{-3}$. We test our predicted scaling estimates using resolved numerical simulations of the Boussinesq equations in the relevant parameter regimes, and find reasonable agreement.