The occurrence of parametric instabilities in finite-amplitude internal gravity waves

Abstract

The parametric instability of a plane internal gravity wave is considered. When the two-dimensional equations of vorticity and mass conservation are linearized in the disturbance quantities, partial differential equations with periodic coefficients result. Substitution of a perturbation of the form dictated by Floquet theory into these equations yields compatibility conditions which, when evaluated numerically, give the curves of neutral stability and constant disturbance growth rate. These results reveal that, for an internal wave of even infinitesimal amplitude, disturbance waves can begin to grow in amplitude. Moreover, these parametric instabilities are shown to reduce to the classical case of the nonlinear resonant interaction in the limit of vanishingly small basic-state amplitude. The fact that these unstable disturbances can exist for an internal wave of any amplitude suggests that this phenomenon may be an important mechanism for extracting energy from an internal gravity wave.