Coupled triads in the dynamics of internal waves: Case study using a linearly stratified fluid

Abstract

In a linearly stratified fluid, two sets of resonant triads with one member in common, hence coupled, can arise for a wide range of wave numbers where all five waves travel in the same direction. The nonlinear dynamics is investigated by deriving the evolution equations of slowly varying wave packets. The dependence of the interaction coefficients on the channel depth and the buoyancy frequency are explicitly demonstrated. Coupling may produce instabilities with growth rates bigger than those displayed by either triad in isolation. Similar properties are expected for layered fluids with shear currents. By drawing on a recent link between modulation instability and the occurrence of rogue waves, surprisingly large displacements in the interior of the fluid are thus more likely. For applications to oceanography, the interaction coefficients of the evolution equations will affect the magnitude of the actual amplitude of an internal rogue wave. Computations indicate that displacements much larger than those of surface rogue waves are possible. From a more theoretical perspective of wave resonance through the Madelung representation, special ranges of dynamical phase angles are identified for the existence of time invariants. Finally, numerical simulations of a simplified system are performed to highlight the spontaneous generation of modes due to coupling.