The evolution and fate of a submesoscale frontal instability -- simulation and spectral analysis*

We analyze submesoscale-resolving simulations of frontal instabilities inspired by recent observations in the Northeastern Pacific. The initial state is a finite-width front in thermal-wind balance, i.e., a horizontal buoyancy gradient $\bar{b}_x$ balanced by the vertical shear $\overline{V}_{\!z}$ of the geostrophic flow. The choice of the base flow profiles and frontal width are motivated by recent saildrone measurements off the northern California coast as part of NASA's Submesoscale Ocean Dynamics Experiment (S-MODE). To probe this configuration for instabilities, we superimpose upon the background flow field a white noise density perturbation. The simulations have a horizontal resolution of 100 m and a vertical resolution of 2 m and are performed using flow\_solve, a spectral code that solves the non-hydrostatic, primitive equations under the Bousinnesq approximation. Turbulence closure is achieved through eigth order hyperviscous and hyperdiffusive operators that act to dissipate energy and buoyancy variance only at the smallest resolved scales, leaving the larger scales untouched. Over $\mathcal{O}(1)$ days a mixed layer baroclinic instability manifests, with $\mathcal{O}(1)$ Rossby number and horizontal scale comparable to a mixed layer deformation radius scale $\mathcal{O}(Nh_{ml}/f)$ ($\mathcal{O}(1)$ km). This instability acts to partially restratify the front by slumping isopycnals. By day 10, secondary barotropic instabilities are observed along the flanks of the baroclinically unstable mode. By day 15, there is a forward energy flux towards the smallest resolved scales, as confirmed by a $k^{-2}$ slope of horizontal kinetic energy spectra over the entire submesoscale range. A spectral proper orthogonal decomposition (SPOD) of the vorticity field on different 2D planes provides insight into the modal structure of the instabilities.