The energy cascade in a strongly stratified fluid

Abstract

A cascade hypothesis for a strongly stratified fluid is developed on the basis of the Boussinesq equations. According to this hypothesis, kinetic and potential energy are transferred from large to small scales in a highly anisotropic turbulent cascade. A relation for the ratio, $ l_{v}/l_{h} $, between the vertical and horizontal length scale is derived, showing how this ratio decreases with increased stratification. Similarity expressions are formulated for the horizontal and vertical spectra of kinetic and potential energy. A series of box simulations of the Boussinesq equations are carried out and a good agreement between the proposed hypothesis and the simulations is seen. The simulations with strongest stratification give horizontal kinetic and potential energy spectra of the form $ E_{K_{h}} \,{=}\, C_{1} \epsilon_{K}^{2/3} k_{h}^{-5/3} $ and $ E_{P_{h}} \,{=}\, C_{2} \epsilon_{P} k_{h}^{-5/3}/\epsilon_{K}^{1/3} $, where $ k_{h} $ is the horizontal wavenumber, $ \epsilon_{K} $ and $ \epsilon_{P} $ are the dissipation of kinetic and potential energy, respectively, and $ C_{1} $ and $ C_{2} $ are two constants. Within the given numerical accuracy, it is found that these two constants have the same value: $ C_{1} \approx C_{2} \,{=}\, 0.51 \pm 0.02 $.