Stability analysis from fourth order evolution equation for small but finite amplitude interfacial waves in the presence of a basic current shear

Abstract

A fourth-order nonlinear evolution equation is derived for a wave propagating at the interface of two superposed fluids of infinite depths in the presence of a basic current shear. On the basis of this equation a stability analysis is made for a uniform wave train. Discussions are given for both an air-water interface and a Boussinesq approximation. Significant deviations are noticed from the results obtained from the third-order evolution equation, which is the nonlinear Schrodinger equation. In the Boussinesq approximation, it has been possible to compare the present results with the exact numerical analysis of Pullin and Grimshaw [12], and they are found to agree rather favourably.

The current-modified nonlinear Schrödinger equation

“…Another form of current that is of interest has shear in the vertical. That type of current has been discussed by Pullin & Grimshaw (1986) and, for the particular case of Dysthe's (1979) MNLS equation, by Dhar & Das (1994).…”

Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline

“…Considering the importance of the fourth order evolution equation, which was first pointed out by Dysthe [8] and later elaborated by Janssen [14] and considered by many authors ([2, [5][6][7]11,13,21]) in studying stability of water waves, two coupled nonlinear evolution equations correct to fourth order in wave steepness are obtained for a surface gravity wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave packet.…”

Non local Models for Envelope Waves in a Stratified Fluid

“…Therefore, we introduce the asymptotic expansion, This system has the form of the Davey-Stewartson equations [3] describing gravity-capillary wave packets on the surface of homogeneous fluid. On the other hand, the system (4.3a)-(4.3b) with parameters presented in Appendix A follows also from the general equations for quasi-harmonic interfacial waves [6][7][8][9] in the shallow-deep limit of the fluid depths. Note that for arbitrary fluid depths the coupling of mean flow n and amplitude 'I' of the interfacial waves is given by a fourth-order differential equation that is different from the Davey-Stewartson system.…”

“…For the last 30 years the problem of self-modulation of small-amplitude nonlinear waves has been investigated intensively both for surface water waves [1][2][3][4] and for internal waves in a density-stratified fluid [5][6][7][8][9]. It was shown that the cubic nonlinear Schrodinger (NLS) equation is a universal model for a description of wave propagation when there is no resonance between the main quasi-harmonic wave and the second or zero (mean flow) harmonics induced by nonlinear effects.…”