On the baroclinic instability of axisymmetric rotating gravity currents with bottom slope

The baroclinic stability characteristics of axisymmetric gravity currents i n a rotating system with a sloping bottom are determined. Laboratory studie s have shown that a relatively dense fluid released under an ambient fluid in a rotating system will quickly respond to Coriolis effects and settle to a state of geostrophic balance. Here we employ a subinertial two-layer mod el derived from the shallow-water equations to study the stability characte ristics of such a current after the stage at which geostrophy is attained. In the model, the dynamics of the lower layer are geostrophic to leading or der, but not quasi-geostrophic, since the height deflections of that layer are not small with respect to its scale height. The upper-layer dynamics ar e quasi-geostrophic, with the Eulerian velocity field principally driven by baroclinic stretching and a background topographic vorticity gradient. Necessary conditions for instability, a semicircle-like theorem for unstabl e modes, bounds on the growth rate and phase velocity, and a sufficient con dition for the existence of a high-wavenumber cutoff are presented. The lin ear stability equations are solved exactly for the case where the gravity c urrent initially corresponds to an annulus flow with parabolic height profi le with two incroppings, i.e. a coupled front. The dispersion relation for such a current is solved numerically, and the characteristics of the unstab le modes are described. A distinguishing feature of the spatial structure o f the perturbations is that the perturbations to the downslope incropping a re preferentially amplified compared to the upslope incropping. Predictions of the model are compared with recent laboratory data, and good agreement is seen in the parameter regime for which the model is valid. Direct numeri cal simulations of the full model are employed to investigate the nonlinear regime. In the initial stage, the numerical simulations agree closely with the linear stability characteristics. As the instability develops into the finite-amplitude regime, the perturbations to the downslope incropping con tinue to preferentially amplify and eventually evolve into downslope propag ating plumes. These finally reach the deepest part of the topography, at wh ich point no more potential energy can be released.