Ageostrophic baroclinic instabilities develop within the surface mixed layer of the ocean at horizontal fronts and efficiently restratify the upper ocean. In this paper a parameterization for the restratification driven by finite-amplitude baroclinic instabilities of the mixed layer is proposed in terms of an overturning streamfunction that tilts isopycnals from the vertical to the horizontal. The streamfunction is proportional to the product of the horizontal density gradient, the mixed layer depth squared, and the inertial period. Hence restratification proceeds faster at strong fronts in deep mixed layers with a weak latitude dependence. In this paper the parameterization is theoretically motivated, confirmed to perform well for a wide range of mixed layer depths, rotation rates, and vertical and horizontal stratifications. It is shown to be superior to alternative extant parameterizations of baroclinic instability for the problem of mixed layer restratification. Two companion papers discuss the numerical implementation and the climate impacts of this parameterization.

The ocean circulation is a cause and consequence of fluid scale interactions from millimeters to 10,000km. Although the wind field produces a large energy input to the ocean, all but about 10% appears to be dissipated within about 100m of the sea surface, rendering very difficult observations of the energy divergence necessary to maintain the full water column flow. Attention thus shifts to the physically different kinetic energy reservoirs of the circulation, and their maintenance, dissipation, and possible influence on the very small scales representing irreversible molecular mixing. Oceanic kinetic energy is dominated by the geostrophic eddy field, and depending upon the vertical structure (barotropic versus low-mode baroclinic), direct and inverse energy cascades are possible. The pathways towards dissipation of the dominant geostrophic eddy kinetic energy depend crucially on the direction of the cascade, but are difficult to quantify because of serious observational difficulties for wavelengths shorter than about 100-200km. At high frequencies, kinetic energy is dominated in the internal wave band by inertial motions (frequencies near the local Coriolis parameter), whose shears appear to be a major source of wave breaking and mixing in the ocean interior.

The restratification of the oceanic surface mixed layer that results from lateral gradients in the surface density field is studied. The lateral gradients are shown to be unstable to ageostrophic baroclinic instabilities and slump from the horizontal to the vertical. These instabilities, which are referred to as mixed layer instabilities (MLIs), differ from instabilities in the ocean interior because of the weak surface stratification. Spatial scales are O(1-10) km, and growth time scales are on the order of a day. Linear stability analysis and fully nonlinear simulations are used to study MLIs and their impact on mixed layer restratification. The main result is that MLIs are a leading-order process in the ML heat budget acting to constantly restratify the surface ocean. Climate and regional ocean models do not resolve the scales associated with MLIs and are likely to underestimate the rate of ML restratification and consequently suffer from a bias in sea surface temperatures and ML depths. In a forthcoming paper, the authors discuss a parameterization scheme to include the effect of MLIs in ocean models.

Significance The ocean’s role in regulating atmospheric carbon dioxide on glacial–interglacial timescales remains an unresolved issue in paleoclimatology. Many apparently independent changes in ocean physics, chemistry, and biology need to be invoked to explain the full signal. Recent understanding of the deep ocean circulation and stratification is used to demonstrate that the major changes invoked in ocean physics are dynamically linked. In particular, the expansion of permanent sea ice in the Southern Hemisphere results in a volume increase of Antarctic-origin abyssal waters and a reduction in mixing between abyssal waters of Arctic and Antarctic origin.

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