Submesoscale fronts with large horizontal density 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 — a form of convective–inertial instability which occurs when the potential vorticity is of opposite sign to the Coriolis parameter. Symmetric instability is characterised by growing slantwise convection cells nearly aligned with isopycnals and which encourage vertical transport of important biogeochemical tracers in addition to geostrophic momentum. This momentum transport destabilises the balanced thermal wind and can prompt geostrophic adjustment and re-stratification which often leaves remnant inertial oscillations. This thesis sets out to model the impacts of symmetric instability on the structure, evolution, and equilibration of the broad range of submesoscale fronts.

We develop a theory for the linear growth and weakly-nonlinear saturation of symmetric instability in the Eady model. This idealised front configuration is motivated by nearly-uniform symmetrically-unstable frontal zones observed during the SUNRISE field campaign in the Gulf of Mexico. Both the fraction of the balanced thermal wind mixed down by symmetric instability and the primary source of energy are strongly dependent on the front strength, defined as the ratio of the horizontal buoyancy gradient to the square of the Coriolis frequency. Strong fronts with steep isopycnals develop a flavour of symmetric instability we call 'slantwise inertial instability' by extracting kinetic energy from the background flow and rapidly mixing down the thermal wind profile. In contrast, weak fronts extract more potential energy from the background density profile, which results in 'slantwise convection.' Using a set of direct numerical simulations, we further investigate the non-linear consequences of these flavours of symmetric instability together with the effect of lateral constraints imposed on the growing modes within finite-width fronts by varying the balanced Rossby number. While weak fronts develop narrow frontlets and excite small-amplitude vertically-sheared inertial oscillations, stronger fronts generate large inertial oscillations and produce bore-like gravity currents that propagate along the top and bottom boundaries. Furthermore, fronts with a super-critical balanced Rossby number equilibrate to a nearly self-similar profile dependent only on the deformation radius, but for small enough Rossby number, self-similar frontlets form and sub-divide the frontal region. We describe each of these mechanisms and energy pathways as the front evolves towards the final adjusted state. These emergent front properties are incorporated into a scaling model framework to describe the ultimate state of the equilibrated front, and broader implications of these results are discussed in the context of current parameterisations of symmetric instability affecting the upper ocean mixed layer and ultimately climate & earth system models.