The Evolution of Nonlinear Wave Trains in Stratified Shear Flows

Abstract

The propagation of an internal wave train in a stratified shear flow is investigated for a Boussinesq fluid in a horizontal channel. Linear effects are primarily reflected in the dispersion relation for the various modes. The phenomenon of Eckart resonance occurs for more realistic stratification profiles. The evolution of nonlinear internal wave packets is studied through a systematic perturbation analysis. A nonlinear Schrodinger equation for the envelope of the internal wave train is derived. Depending on the relative sign of the dispersive and nonlinear terms, a wave train may disperse or form an envelope soliton. The analysis demonstrates the existence of two types of critical layers: one the ordinary critical point where ū=c, while the other occurs where ū=cg. In order to calculate the coefficients of the nonlinear Schrodinger equation a numerical code has been developed which computes the second‐harmonic and induced mean motions. The existence of these envelope solitons and their dependence on environmental conditions are discussed.