Parametric subharmonic instability of internal gravity wave beams

Internal gravity wave beams are time-harmonic plane waves with general spatial profile that arise in continuously stratified fluids owing to the anisotropy of this wave motion. In the last decade, these wave disturbances have been at the forefront of research, both from a fundamental perspective and in connection with various geophysical flow processes. Oceanic internal wave beams, in particular, form the backbone of the internal tide, generated by the interaction of the barotropic tide with sea-floor topography. The internal tide breakdown and its role in deep-ocean mixing have attracted considerable attention. In this context, it is of interest to understand mechanisms by which internal wave beams become unstable and eventually breakdown, thereby contributing to mixing. A possible instability mechanism is via resonant triad interactions that amplify short-scale perturbations with frequency equal to one half of that of the underlying wave. For spatially and temporally monochromatic internal waves, this so-called parametric subharmonic instability (PSI) has been studied extensively and indeed can lead to breakdown. By contrast, the focus here is on understanding how wave beams with locally confined spatial profile, such as those in the field, may differ, in regard to PSI, from monochromatic plane waves. To this end, an asymptotic analysis is made of the interaction of a small-amplitude wave beam with short-scale subharmonic wavepackets in a nearly inviscid stratified Boussinesq fluid. A novel system of coupled evolution equations that govern this nonlinear interaction is derived and analyzed. For beams with general localized profile, unlike monochromatic wavetrains, it is found that triad interactions are not strong enough to bring about instability in the limited time that subharmonic perturbations overlap with the beam. On the other hand, for quasi-monochromatic wave beams whose profile comprises a sinusoidal carrier modulated by a locally confined envelope, PSI is possible if the beam is wide enough. In this instance, a stability criterion is proposed which, under given flow conditions, provides the minimum number of carrier wavelengths a beam of small amplitude must comprise for instability to arise. Furthermore, the effect of the Earth’s rotation on PSI of internal wave beams is in-