Rogue waves for resonant triads in stratified fluids with a sharply peaked buoyancy frequency profile

Abstract

A formulation for unexpectedly large displacements at the pycnocline of the upper ocean is developed. Such "internal rogue waves" can arise from triad resonance with quadratic nonlinearities. While analytical solutions of rogue waves for triad resonance have been derived in the mathematical physics literature, the applications to fluid dynamics have not been fully examined. Here interaction coefficients of the evolution equations are calculated for a sharply peaked buoyancy frequency profile. Modulation instability refers to the growth of disturbance(s) from the interplay between dispersive and nonlinear effects. Although the instability analysis is studied based on linearization, information from the growth rate spectrum can be correlated with results from the fully nonlinear, numerical robustness tests of rogue waves. Finally, pole analysis of the analytical rogue wave solutions is considered by extending the spatial variable to be complex via analytic continuation. For rogue modes with one or two peaks, locations of maximum displacement in physical space are identical, or approximately equal, to the real parts of poles at the turning points of trajectories. These numerical and analytical studies constitute a framework for understanding large scale, transient motions in the upper ocean.