On fully nonlinear, vertically trapped wave packets in a stratified fluid on the f-plane

Abstract

The ubiquity of solitary and solitarylike internal waves in the coastal ocean has been recognized for some time. Recent theoretical studies of a strongly nonlinear, weakly nonhydrostatic set of layer-averaged model equations have predicted that rotation, for example, on the f-plane, can lead to the decay and subsequent reemergence of internal solitary waves. We reconsider this problem using high resolution numerical simulations of the rotating stratified Euler equations. We find that in certain cases the initial disturbances indeed fission into nonlinear wave packets, with the constituent waves making up the wave packet being, in themselves, nonlinear. However, for typical coastal ocean parameters this only occurs at rotation rates higher than those on Earth on the time scales we are able to simulate. We confirm, using the Dubreil–Jacotin–Long equation, that the vertical structure of the wave-induced currents is well predicted by the fully nonlinear theory of nonrotating internal solitary waves and that weakly nonlinear Korteweg–de Vries equation-based theory fails to describe this structure accurately. Subsequently, we consider flat-crested solitary waves that allow us to fix the wave amplitude while varying the horizontal wavelength. We find that as the waves’ horizontal extent nears the baroclinic Rossby radius more energy is deposited into the wave tail. However, no wave overtaking is observed, and an explanation for this fact is proposed. Finally, we discuss the effects of the horizontal component of the rotation vector and derive an exact equation for rotation modified waves near the equator. This equation demonstrates that in this situation, rotation modifies the structure of the fully nonlinear waves but does not lead to solitary wave decay.