Ageostrophic instabilities in a baroclinic flow over sloping topography

Abstract

The ageostrophic normal modes of a spatially uniform, vertically sheared flow along a sloping bottom are considered in two layers underneath a deep motionless third layer (two-and-half layer model). The variations of the layer thickness are assumed to be small to derive the six-order dispersion relation with constant coefficients valid for finite Froude number typical for oceanic currents. The dispersion curves for the Rossby waves and inertia-gravity waves (IGW) are investigated to identify different types of instabilities occurs if there is a pair of wave components which have almost the same Doppler-shifted frequency related to crossover of the branches when the Froude number increases. Ageostrophic instabilities due to a resonance between the IGW modes and the Rossby wave in either lower, or middle layer, are described. In both cases the growth rate and the width of the unstable wavenumber window are shown to be proportional to the square root of the corresponding gradient of the layer thickness. These powerful types of ageostrophic instability can coexist together (and with Kelvin-Helmgoltz instability) and may play an important role in mixing processes in geophysical fluids.