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

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 k 2 < N 2 /2 U 2 , where N is the Brunt—Väisälä frequency and 2 U is the change in velocity across the discontinuity. For N 2 /2 U 2 < k 2 < N 2 / U 2 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 /2 ½ U . 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.