## Abstract

Considered is a pointwise energy diagnostic scheme for a multilayer, primitive equation, nonisopycnic ocean model. Both conservative as well as nonconservative energy exchange terms are considered. Moreover, the scheme is worked out for both the finite depth as well as the reduced gravity versions of the model. The work is motivated by the need to discern the various instability processes responsible for the observed and modeled mesoscale flow structures commonly found in oceanic frontal regions, for example, upwelling areas, and regions separating coastal and adjacent deep ocean currents. As is common the mathematical form of the conservative energy exchange terms are ambiguous. A careful analysis is therefore effectuated to interpret them in terms of known physical processes. The analysis reveals that four basic instability processes are supported. One is the barotropic or horizontal shear instability, which extracts its energy from the horizontal shear in the mean current. The remaining three are the vertical shear instability, the frontal instability, and the conventional baroclinic instability and are, thus, different forms of baroclinic instability. The first, the vertical shear instability, obtains its energy from the velocity difference between adjacent layers (the model’s rendition of a vertical shear). The second, the frontal instability, elicits the potential energy stored in the lateral layer density gradients, while the third, the conventional baroclinic instability, gets its energy from the lateral gradient in the layer thicknesses (the model’s rendition of a vertical density gradient). It is also further shown that the bottom topography contributes to the conservative energy exchange by releasing potential energy when the integrated mass transport in a water column is directed downslope. Moreover, the analysis reveals that the traditional reduced gravity models, that is, models employing uniform layer densities, only support horizontal and vertical shear instabilities. Finally, it is shown that the entrainment process always leads to a loss of kinetic energy and that some of this lost energy may, under certain circumstances, be retrieved as potential energy.

* Additional affiliation: Institute of Geophysics, University of Oslo, Oslo, Norway.

*Corresponding author address:* Lars Petter Røed, Norwegian Meteorological Institute, P.O. Box 43 Blindern, N-0313 Oslo, Norway.

Email: larspetter.roed@dnmi.no