Extraction of Potential Energy from Geostrophic Fronts by Inertial-Symmetric Instabilities

Submesoscale oceanic density fronts are structures in geostrophic and hydrostatic balance, which are prone to inertial and/or symmetric instabilities. We argue in this article that drainage of potential energy from the geostrophic flow is a significant source of their growth. We illustrate our point with two-dimensional Boussinesq numerical simulations of oceanic density fronts on the f plane. A set of two-dimensional initial conditions covers the submesoscale portion of a three-dimensional parameter space consisting of the Richardson and Rossby numbers and a measure of stratification or latitude. Because we let the lateral density gradient decay with depth, the parameter space map is nontrivial, excluding low-Rossby, low-Richardson combinations. Dissipation and the presence of boundaries select a growing mode of inertial-symmetric instability consisting of flow cells that disturb isopycnal contours. Systematically, these isopycnal displacements correspond to a drainage of potential energy from the geostrophic fronts to the ageostrophic perturbations. In the majority of our experiments, this energy drainage is at least as important as the drainage of kinetic energy from the front. Various constraints, some physical, some numerical, make the energetics in our experiments more related to inertial rather than symmetric instabilities. Our results depend very weakly on the Richardson number and more on the Rossby number and relative stratification.