Parametric subharmonic instability of inertial shear at ocean fronts

The two-dimensional stability of vertically sheared inertial oscillations at ocean fronts is explored through a linear stability analysis and nonlinear simulations. Baroclinic effects reduce the minimum frequency of inertia-gravity waves to an extent determined by the balanced Richardson number ${{Ri}}$ of the front. Below a critical value of ${{Ri}}$, which depends on the strength of the inertial shear, the inertial oscillations become unstable to parametric subharmonic instability (PSI) resulting in growing perturbations that oscillate at half the inertial frequency $f$. Since the critical value is always greater than 1, PSI can occur at fronts stable to symmetric instability. Although modest in weak inertial shear, growth rates exceeding $f/2$ can be achieved for inertial shear greater than or equal to the thermal wind shear. Our formulation allows for non-hydrostatic perturbations and can be applied to initially unstratified geostrophic adjustment problems. We find that PSI will almost totally damp the transient oscillations that arise during geostrophic adjustment. The perturbations gain energy at the expense of the inertial oscillations through ageostrophic shear production. The perturbations then themselves become unstable to secondary Kelvin–Helmholtz instabilities creating a pathway by which the inertial oscillations can be dissipated rapidly. In contrast to symmetric and baroclinic instabilities that draw on a front's kinetic or potential energy, PSI acts to increase the energy stored in the balanced front as the convergence and divergence of the eddy-momentum fluxes set up a secondary circulation in the sense to stand up the front.