The influence of front strength on the development and equilibration of symmetric instability. Part 1. Growth and saturation

Abstract

Submesoscale fronts with large horizontal buoyancy gradients and $O(1)$ Rossby numbers are common in the upper ocean. These fronts are associated with large vertical transport and are hotspots for biological activity. Submesoscale fronts are susceptible to symmetric instability (SI) – a form of stratified inertial instability which can occur when the potential vorticity is of the opposite sign to the Coriolis parameter. Here, we use a weakly nonlinear stability analysis to study SI in an idealised frontal zone with a uniform horizontal buoyancy gradient in thermal wind balance. We find that the structure and energetics of SI strongly depend on the front strength, defined as the ratio of the horizontal buoyancy gradient to the square of the Coriolis frequency. Vertically bounded non-hydrostatic SI modes can grow by extracting potential or kinetic energy from the balanced front and the relative importance of these energy reservoirs depends on the front strength and vertical stratification. We describe two limiting behaviours as ‘slantwise convection’ and ‘slantwise inertial instability’ where the largest energy source is the buoyancy flux and geostrophic shear production, respectively. The growing linear SI modes eventually break down through a secondary shear instability, and in the process transport considerable geostrophic momentum. The resulting breakdown of thermal wind balance generates vertically sheared inertial oscillations and we estimate the amplitude of these oscillations from the stability analysis. We finally discuss broader implications of these results in the context of current parameterisations of SI.