Modeling internal rogue waves in a long wave-short wave resonance framework

Abstract

A resonance between a long wave and a short wave occurs if the phase velocity of the long wave matches the group velocity of the short wave. Rogue waves modeled as special breathers (pulsating modes) can arise from these resonant interactions. This scenario is investigated for internal waves in a density stratified fluid. We examine the properties of these rogue waves, such as the polarity, amplitude and robustness, and show that these depend critically on the specific density stratification and the choice of the participating modes. Three examples, namely, a two-layered fluid, a stratified fluid with constant buoyancy frequency, and a case of variable buoyancy frequency are examined. We show that both elevation and depression rogue waves are possible, and the maximum displacements need not be confined to a fixed ratio of the background plane wave. Furthermore, there is no constraint on the signs of nonlinearity and dispersion, nor any depth requirement on the fluid. All these features contrast sharply with those of a wave packet evolving on water of finite depth governed by the nonlinear Schrödinger equation. The amplitude of these internal rogue waves generally increases when the density variation in the layered or stratified fluid is smaller. For the case of constant buoyancy frequency, critical wave numbers give rise to nonlinear evolution dynamics for “long wave-short wave resonance,” and also separate the focusing and defocusing regimes for narrow-band wave packets of the nonlinear Schrödinger equation. Numerical simulations are performed by using baseband modes as initial conditions to assess the robustness of these rogue waves in relation to the modulation instability of a background plane wave.