On the stability of barotropic prograde and retrograde jets along a bottom slope

Abstract

Barotropic slope currents following isobaths are common features of shelf seas in nonequatorial regions. They are usually remotely forced and stabilized locally by the bottom slope. The stability of these barotropic currents is investigated to determine how well they can be modeled numerically as stationary flows. Both prograde jets, with their cyclonic flank in the deeper water, and retrograde jets, with their cyclonic flank in shallower water, are considered. A necessary condition for instability is at least one extremum in the distribution of the potential vorticity across the jet, which is equivalent to the planetary beta stability problem, and corresponds to the well‐known Rayleigh inflection‐point theorem for the stability problem of a nongeophysical flow. A criterion for instability that involves the bottom slope parameter as a stabilizing factor and the velocity shear as a destabilizing factor has been derived. This criterion suggests that even mild slopes are capable of stabilizing both prograde and retrograde jets. For the case of a cosine jet along an exponential topography, neutral solutions are found to be governed by the Mathieu equation. Because of our use of exponential topography our equation contains more terms than the instability problem for planetary beta on a flat bottom, but the difference tends to vanish when the Rossby number is small. Our results also imply that a given bottom slope stabilizes more wave modes for a retrograde jet, which flows in a direction opposite to topographic Rossby waves.