Nonlinear aspects of an internal gravity wave co-existing with an unstable mode associated with a Helmholtz velocity profile

Abstract

Recently Lindzen (1974) has proposed a model of a shear-layer instability which allows unstable modes to co-exist with radiating internal gravity waves. The model is an infinite, continuously stratified, Boussinesq fluid, with a simple jump discontinuity in the velocity profile. Linear stability theory shows that the model is stable for wavenumbers k such that k2 < N2/2U2, where N is the Brunt—Väisälä frequency and 2U is the change in velocity across the discontinuity. For N2/2U2 < k2 < N2/U2 an unstable mode may co-exist with an internal gravity wave. This paper examines the weakly nonlinear aspects of this model for wavenumbers k close to the critical wavenumber N/2½U. An equation governing the evolution of the amplitude of the interfacial displacement is derived. It is shown that the interface may support a stable finite amplitude internal gravity wave.