An Asymptotic Model for the Propagation of Oceanic Internal Tides through Quasi-Geostrophic Flow

We derive a linearized, asymptotic model for the slow evolution of hydrostatic internal waves in quasi-geostrophic flow, and thus applicable to the propagation of oceanic internal tides through mesoscale eddies and currents. The derivation uses a multiple-time-scale asymptotic method to isolate the evolution of a single-frequency wave field over intervals longer than a wave period, and exploits an asymptotic ‘method of reconstitution’ to yield a description of both resonant and near-resonant wave-flow interactions. The resulting approximate description is reduced relative to the linearized hydrostatic Boussinesq equations while more general than asymptotic, spectral space models that rely on a restrictive resonant interaction assumption. A numerical comparison with the linearized Boussinesq equations defines the range of model validity and illustrates the manners of model failure when the mesoscale flow is strong and the wave frequency is close to inertial.