Reappraisal of the Kelvin–Helmholtz problem. Part 2. Interaction of the Kelvin–Helmholtz, superharmonic and Benjamin–Feir instabilities

Abstract

Several new results on the bifurcation and instability of nonlinear periodic travelling waves, at the interface between two fluids in relative motion, in a parametric neighbourhood of a Kelvin–Helmholtz unstable equilibrium are presented. The organizing centre for the analysis is a canonical Hamiltonian formulation of the Kelvin–Helmholtz problem presented in Part 1. When the density ratio of the upper and lower fluid layers exceeds a critical value, and surface tension is present, a pervasive superharmonic instability is found, and as u → u₀, where u is the velocity difference between the two layers and u₀ is the Kelvin–Helmholtz threshold, the amplitude at which the superharmonic instability occurs scales like (u₀−u)^1/2 with u < u₀. Other results presented herein include (a) new results on the structure of the superharmonic instability, (b) the discovery of isolated branches and intersecting branches of travelling waves near a critical density ratio, (c) the appearance of Benjamin–Feir instability along branches of waves near the Kelvin–Helmholtz instability threshold and (d) the interaction between the Kelvin–Helmholtz, superharmonic and Benjamin–Feir instability at low amplitude.