Resonant Long–Short Wave Interactions in an Unbounded Rotating Stratified Fluid

Abstract

A theoretical study is made of finite‐amplitude modulated internal wavetrains and the attendant nonlinear interaction with the mean flow induced by the modulations, in an unbounded uniformly stratified Boussinesq fluid rotating around the vertical axis. When the rotation is relatively weak, in particular, ‘flat’ wavetrains, that feature stronger vertical than horizontal modulations, are resonantly coupled with the mean flow in a manner analogous to the resonant long–short wave interaction between gravity and capillary waves on the surface of deep water. A coupled set of evolution equations for the vertical wavenumber, the wave amplitude, and the mean flow is derived under resonant conditions, and is used to examine the propagation of locally confined wavetrains with initially uniform wavenumber and no pre‐existing mean flow. The resonant interaction causes radiation of energy away from a flat wavetrain by means of the induced mean flow which forms a trailing wake; this furnishes a possible mechanism for generating low‐frequency inertial–gravity waves in the atmosphere, as suggested by field observations. Moreover, owing to refraction by the mean flow, a finite‐amplitude wavetrain may experience rapid wavenumber variations in certain locations, consistent with prior numerical simulations. Eventually, in these regions, the wavenumber tends to become multi‐valued, suggesting the formation of caustics.