Ageostrophic, anticyclonic instability of a geostrophic, barotropic boundary current

Abstract

After having previously demonstrated the occurrence of ageostrophic, anticyclonic instability (AAI) for several other steady flows, in this paper we further test the hypothesis that rotating, stably stratified, shear flows are generally subject to AAI at finite Rossby (Ro) and Froude (Fr) numbers by calculating the inviscid, incompressible normal modes of a geostrophic, barotropic boundary current that vanishes toward the interior of the domain. We focus on a background current profile with no inflection point and nonvanishing absolute vorticity to exclude other known instability types. The hypothesis is again confirmed. The instability occurs for finite Ro only in an anticyclonic background flow. Its modal growth rate steeply decreases (approximately inverse exponentially, ∼e−c/Ro) as Ro→0. The downstream phase velocity of the unstable mode lies within the speed range for the background current, so the eigenmode has a near-critical layer within the zone of strong background shear. Its unstable eigenmode has an aspect ratio of downstream and vertical wave numbers, l/m, on the order of the ratio of the Coriolis and Brünt-Väisällä frequencies (as is typical for rotating, stratified flows). The maximum growth rate is independent of Fr for small Fr (the usual geophysical regime), and it decreases with large Fr, disappearing when there is no stratification. The eigenmode has a weakly decaying, oscillatory, interior far-field structure similar to an inertia-gravity wave. This supports the interpretation that the modal instability arises through the coalescence (i.e., resonance) of two stable normal-mode branches, one associated with the background shear flow and the other an inertia-gravity or Kelvin wave mode.