Internal rogue waves in stratified flows and the dynamics of wave packets

Abstract

A theoretical study on the occurrence of internal rogue waves in density stratified flows is conducted. While internal rogue waves for long wave models have been studied in the literature, the focus here is on unexpectedly large amplitude displacements arising from the propagation of slowly varying wave packets. In the first stage of the analysis we calculate new exact solutions of the linear modal equations in a finite domain for realistic stratification profiles. These exact solutions are then used to facilitate the calculations of the second harmonic and the induced mean motion, leading to a nonlinear Schrödinger equation for the evolution of a wave packet. The dispersion and nonlinear coefficients then determine the likelihood for the occurrence of rogue waves. Several cases of buoyancy frequency (N) are investigated. For N2 profiles of hyperbolic secant form, rogue waves are unlikely to occur as the dispersion and nonlinear coefficients are of opposite signs. For N2 taking constant values, rogue waves will arise for reasonably small carrier envelope wavenumbers, in sharp contrast with the situation for a free surface, where the condition is kh > 1.363 (k = wavenumber of the carrier envelope, h = depth). Finally, a special N2 profile permits an analytical treatment for a linear shear current. Unexpectedly large amplitude waves are possible as the dispersion and nonlinear coefficients can then be of the same sign.