A theoretical model for oceanic submesoscales under next-order effects of strain and turbulence

TL;DR

The paper develops a perturbation framework for oceanic submesoscale fronts and filaments at $Ro=O(1)$, incorporating both mesoscale strain and boundary-layer turbulence (BLT) on equal footing. Building on generalized frontogenesis models, it derives a zeroth-order geostrophic base state and a first-order cross-front streamfunction response that includes a steady particular solution and an inertial-mode–driven homogeneous part, yielding an approximate solution $\psi^1\approx(1-\cos t)\psi^1_p$. Results show that pure strain drives frontogenesis at early times, BLT introduces distinct frontogenetic or frontolytic tendencies depending on the turbulent fluxes (horizontal diffusion tends to frontolyze, vertical viscosity tends to frontogenize), and the combined forcing can either enhance, suppress, or reverse frontal sharpening in a parameter-dependent manner. The findings emphasize the need for spatially variable BLT parameterizations in mixed-layer models to accurately capture frontal evolution under mesoscale forcing, and they complement LES studies by providing analytic insight into ageostrophic responses at finite $Ro$ and finite $Ek$.

Abstract

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.

Figures (16)

• Figure 1: A submesoscale front and filament in geostrophic balance. They correspond to the zeroth order solution $b^0=B_0(X_0)$ and $v^{0} = \,B_0'(X_0)(Z-1/2)$ with $\textit{Ro}=1$, $X_0 = (x+\textit{Ro}^2 v^0)$, and $Z = z$.
• Figure 2: Front and filament profiles of (a, b) Streamfunction solutions $\psi^1_{S}$, (c, d) Velocity solutions $v^0+\varepsilon_S v^1_S$, and (e, f) First-order frontal tendency $T^1_{b,S}$. Black contours show the buoyancy $b^0+\varepsilon_S b^1_S$. The strain parameter is $\varepsilon_S=0.03$, and the corresponding time is $t=10$.
• Figure 3: Particular solution of the diffusivity streamfunction $\psi^1_{p,VD}$, $\psi^1_{p,HD}$ for front and filament. Their viscosity counterparts have the same spatial pattern with just a flip of sign.
• Figure 4: Front and filament profiles of velocity solutions $v^0+\varepsilon_i v^1_i$ with buoyancy contours $b^0+\varepsilon_i b^1_i$ for (a,b) vertical diffusivity $i=VD$, (c,d) vertical viscosity $i=VV$, (e,f) horizontal diffusivity $i=HD$, (g,h) horizontal viscosity $i=HV$. The relevant parameter is set to $\varepsilon_i =0.03$ with $\textit{Ro}=1$, and the time corresponds to $t=10$.
• Figure 5: As in figure \ref{['fig:vVH']}, but showing profiles of the first-order frontal tendency $T^1_{b,i}$.