Near-inertial parametric subharmonic instability

Abstract

New analytic estimates of the rate at which parametric subharmonic instability (PSI) transfers energy to high-vertical-wavenumber near-inertial oscillations are presented. These results are obtained by a heuristic argument which provides insight into the physical mechanism of PSI, and also by a systematic application of the method of multiple time scales to the Boussinesq equations linearized about a ‘pump wave’ whose frequency is close to twice the inertial frequency. The multiple-scale approach yields an amplitude equation describing how the 2f0-pump energizes a vertical continuum of near-inertial oscillations. The amplitude equation is solved using two models for the 2f0-pump: (i) an infinite plane internal wave in a medium with uniform buoyancy frequency; (ii) a vertical mode one internal tidal wavetrain in a realistically stratified and bounded ocean. In case (i) analytic expressions for the growth rate of PSI are obtained and validated by a successful comparison with numerical solutions of the full Boussinesq equations. In case (ii), numerical solutions of the amplitude equation indicate that the near-inertial disturbances generated by PSI are concentrated below the base of the mixed layer where the velocity of the pump wave train is largest. Based on these examples we conclude that the e-folding time of PSI in oceanic conditions is of the order of ten days or less.