Nonlinear Energy Transfer from Semidiurnal Barotropic Motion to Near-Inertial Baroclinic Motion

A theoretical investigation into the subharmonic parametric instability of internal gravity waves to progressive surface waves in a continuously stratified medium is presented. A perturbation and multiple scales analysis is utilized and is carried out to the third order. Recent parallel efforts for a two-layer model have shown that surface waves can resonate, exponentially in time, two oblique internal wave trains whose frequencies are nearly subharmonic to that of the surface wave. This analysis is generalized in the current paper to allow for continuous variation of density with depth, as well as planetary rotation. Specific attention is focused on the energy transfer from the semidiurnal barotropic tide to near-inertial baroclinic motion. At low latitudes, linear analysis demonstrates that such a resonant triad indeed exists, suggesting that this mechanism may play a role in the development of the inertial peak seen in many internal wave spectra. At the second order, evolution equations are obtained that demonstrate that the internal waves initially grow exponentially in time. Finally, when the analysis is carried to the third order, it is shown that an equilibrium exists. Nonlinear detuning of the internal wave frequencies results in the internal wave amplitudes reaching a steady-state value, which is readily calculated.