Two- and three-dimensional parametric instabilities in finite-amplitude internal gravity waves

Abstract

Abstract The linearized stability equations of a monochromatic finite-amplitude gravity wave in a Boussinesq fluid are solved for two-dimensional perturbations and three-dimensional perturbations propagating in the plane normal to the primary wave plane and the primary wave fronts. In statically stable primary waves, two-dimensional instability modes grow fastest. They can be divided into two classes: a continuous spectrum of small-scale modes and an isolated mode. In the limit of vanishing primary wave amplitudes, the small-scale modes reduce to second-order resonant wave interactions whereas the underlying wave interaction mechanism of the isolated mode remains uncertain. In statically unstable primary waves, three-dimensional instability modes grow fastest and are either fundamental or subharmonic. They reduce to resonant wave interactions of orders higher than three in the limit of zero primary wave amplitudes. The underlying interaction mechanisms of the fundamental modes also include vortical modes besides internal gravity waves. The results resemble those obtained from stability studies of free shear and boundary layer flows indicating a crucial role of parametric instability in the transition from two- to three-dimensional flows.