Multiscale models for synoptic–mesoscale interactions in the ocean

5 Multiscale analysis is used to derive two sets of coupled models, each based on the same distinguished limit, to represent the interaction of the midlatitude oceanic synoptic scale — where coherent features such as jets and rings form — and the mesoscale, defined by the internal deformation scale. The synoptic scale and mesoscale overlap at low and mid latitudes, and are hence synonymous in much of the oceanographic literature; at higher latitudes the synoptic scale can be an order of magnitude larger than the deformation scale, which motivates our asymptotic approach and our nonstandard terminology. In the first model the synoptic dynamics are described by ‘Large Amplitude Geostrophic’ (LAG) equations while the eddy dynamics are quasigeostrophic. This model has order one isopycnal variation on the synoptic scale; the synoptic dynamics respond to an eddy momentum flux while the eddy dynamics respond to the baroclinically unstable synoptic density gradient. The second model assumes small isopycnal variation on the synoptic scale, but allows for a planetary scale background density gradient that may be fixed or evolved on a slower time scale. Here the large-scale equations are just the barotropic quasigeostrophic equations, and the mesoscale is modeled by the baroclinic quasigeostrophic equations. The synoptic dynamics now respond to both eddy momentum and buoyancy fluxes, but the small-scale eddy dynamics are simply advected by the synoptic-scale flow — there is no baroclinic production term in the eddy equations. The energy budget is closed by deriving an equation for the slow evolution of the eddy energy, which ensures that energy gained or lost by the synoptic-scale flow is reflected in a corresponding loss or gain by the eddies. This latter model, aided by the eddy energy equation — a key result of this paper — provides a conceptual basis through which to understand the classic baroclinic turbulence cycle