A numerical study of the frequency and the energetics of nonlinear internal gravity waves

Abstract

Internal gravity wave motions in a two‐dimensional vertical plane of fluid having constant mean Brunt‐Vaisala frequency are numerically simulated. The waves are integrated to statistical equilibrium under forcing and dissipation with forcing applied to low wave number modes. The model is used to study the variation of the internal wave energy dissipation rate with the Richardson number Ri, the effect of nonlinear wave interaction on the wave frequency, the wave number spectrum of the buoyancy flux, and the balance of the spectral energy flux. The energy dissipation rate is shown by our simulations to vary strongly near Ri ≈ 1 and to increase proportionally to Ri−1. The rms frequency fluctuation is found to increase approximately linearly with Ri−1/2 and with the wave number magnitude. The simulations show that the buoyancy flux varies nonuniformly with wave numbers, kinetic to potential energy conversion occurs mostly at low wave numbers, and an opposite conversion occurs in the rest of the wave number space. The mean spectral energy balance in the region where negative conversion occurs is marked by positive (gain) potential energy transfer and negative (loss) kinetic energy transfer in the intermediate wave number range and positive transfer for both kinetic and potential energy in the high wave number range; the latter is balanced by dissipation. The spectra of kinetic and potential energy are presented. The similarity in the slope of the kinetic energy spectra between the stratified turbulence from our simulation and the two‐dimensional unstratified, homogeneous turbulence is noted.