On the stationary energy spectra for unidirectionally propagating internal gravity waves

Abstract

Starting from the Euler equations in the Boussinesq approximation for a stratified fluid in two dimensions we establish a kinetic equation for a statistical ensemble of weakly non-linear internal gravity waves propagating in the same horizontal direction. We find a transformation of this equation to a standard form adopted in the weak turbulence theory and then, limiting ourselves to high vertical-low horizontal wavenumber waves, get equilibrium (Rayleigh-Jeans) and non-equilibrium (Kolmogorov) stationary energy spectra, among them a k −3 vertical wavenumber spectrum.