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.
Highlights
The upper ocean is a dynamically active and important region, relevant to Earth’s climate due to exchanges at the air–sea interface, but to biogeochemical processes.Turbulence acts to vertically homogenise this upper-most layer of the ocean down to typical depths of 10 to 100 m, driven by wind stresses, surface waves, heat or salinity fluxes or internal flow instabilities
Our analysis formally extends the work by Taylor & Ferrari (2009) who implicitly considered the secondary shear instability of symmetric instability (SI) modes by applying the Miles–Howard theorem
SI occurs at density fronts in the ocean and atmosphere when the potential vorticity (PV) takes the opposite sign to the Coriolis parameter, i.e. fq < 0
Summary
The upper ocean is a dynamically active and important region, relevant to Earth’s climate due to exchanges at the air–sea interface, but to biogeochemical processes. Compared with Stone (1966) and ensuing papers which studied instability of the Eady model in the inviscid, hydrostatic limit, our analysis is no longer a function only of Ri, but depends on the front strength, Γ These purely linear analyses are unable to determine the finite contribution of SI to the momentum transport, buoyancy fluxes and energetics of the flow. We combine these two stability analyses in §§ 4 and 5 to determine the finite-amplitude contributions of SI to the energetics and momentum transport, respectively. We use the framework of Tandon & Garrett (1994) to shed light on the effects of dissipation and a finite mixing time on the adjustment response and resulting inertial oscillations
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