Fronts, or regions with large horizontal density gradients, are common and important features of the upper ocean. On sufficiently large scales, the hydrostatic pressure gradient associated with the horizontal density gradient is often nearly balanced by a geostrophic (or thermal wind) vertically-sheared along-front flow. However, fronts in the ocean and atmosphere co-exist with small-scale turbulence which disrupts this balance. Analytic models of fronts have often neglected the effects of small-scale turbulence; however, more recent work has employed various simple parametrisations to describe the mixing properties of this turbulence and hence its effect on the frontal evolution.

Here, we use idealised models to examine the influence of this small-scale turbulence on the evolution of ocean fronts in the case of strong mixing. First we consider the evolution of an initially balanced density front subject to an imposed viscosity and diffusivity as a simple analogue for small-scale turbulence. At late times, the dominant balance is found to be the quasi-steady turbulent thermal wind balance with time-evolution due to an advection-diffusion balance in the buoyancy equation. In the absence of surface forcing this advection-diffusion equation admits similarity solutions describing a spreading front where the spreading results from shear dispersion associated with the cross-front flow and vertical diffusion of density. In response to shear dispersion, the front evolves towards a density profile that is nearly linear in the cross-front coordinate. At the edges of the frontal zone, the density field develops large curvature and these regions are associated with narrow bands of intense vertical velocity. The presence of surface wind stress and heat flux modifies the leading order velocity and buoyancy fields which, through the advection-diffusion balance, can lead to frontal gradients sharpening or reaching an equilibrium depending on the wind stress direction and forcing magnitude. These predictions are tested using numerical simulations and found to be valid for a wide range of parameters.

Secondly we examine baroclinic instability in the presence of small-scale turbulence using a simple model for vertical mixing of momentum and buoyancy. The governing equations for buoyancy and vorticity are found to exhibit a normal mode linear instability which is studied using an analytical stability analysis and numerical simulations. The instability is found to be similar to the classical Eady instability however vertical mixing reduces the growth rate of the most unstable mode. Additionally, vertical mixing causes the wavenumber vector associated with the most unstable mode to point at an angle to the cross front direction with the angle being determined by the relaxation timescale. The most unstable modes grow by converting potential energy associated with the basic state into kinetic energy of the growing perturbations. However, unlike the inviscid Eady problem, the dominant energy balance is between the buoyancy flux and the energy dissipated by the modeled vertical mixing. We test our analytical predictions for the angle and growth rate of the most unstable mode using numerical simulations and generally find good agreement. In the absence of horizontal mixing, the most unstable mode has an infinite wavenumber. We discuss mechanisms providing a finite wavenumber cutoff including horizontal mixing and stratification.