A theoretical model for oceanic submesoscales under next-order effects of strain and turbulence
Submesoscale currents in the oceanic mixed layer, comprising fronts, eddies, and filaments, are characterized by $\textit{O}(1)$ Rossby numbers (Ro). These features, which constantly interact with background mesoscale flows and boundary layer turbulence (BLT), are critical for mediating vertical exchange between the surface and the ocean interior. Despite growing insight into their generation and evolution, the modification of initially balanced submesoscale dynamics by finite-Ro effects under the combined influence of mesoscale strain and BLT remains unresolved. In this study, we address this question through a perturbation analysis of two-dimensional, geostrophically adjusted oceanic fronts and filaments, adapting the analytical models of \citet{shakespeare_generalized_2013} and \citet{bodner_breakdown_2020}. This framework allows for a systematic exploration across a broad range of Rossby numbers Ro, Ekman numbers Ek, and strain parameters. The first-order solution under pure mesoscale strain exhibits clear frontogenesis and closely mirrors the full model dynamics during early inertial periods, despite the absence of an exponential collapse. Under BLT perturbation, the first-order solution confirms the distinct frontogenetic and frontolytic tendencies associated with eddy viscosity and diffusivity, respectively; however, no transition between these regimes is observed across the explored Ro and Ek parameter space for vertical mixing. When both strain and BLT perturbations are present, turbulent fluxes can strengthen, weaken, or even reverse strain-induced frontogenesis depending on the parameter regime. These results suggest that mixed-layer parameterizations must carefully account for the spatial variability of BLT within submesoscale currents to accurately capture frontal evolution under mesoscale strain.