On the baroclinic instability of cold-core coupled density fronts on a sloping continental shelf

Abstract

A theory is presented to describe the linear baroclinic instability of coupled density fronts on a sloping continental shelf. The new baroclinic model equations used to study the instability process correspond to an ‘intermediate lengthscale’ dynamical balance. Specifically, the frontal dynamics, while geostrophic, is not quasigeostrophic because frontal height deflections are not small in comparison with the frontal scale height. The evolution of the frontal height is strongly coupled to the geostrophic pressure in the surrounding slope water through the hydrostatic balance which expresses the continuity of the dynamic pressures across the frontal interface. The deeper surrounding slope water evolves quasi-geostrophically and is coupled to the front by baroclinic vortex-tube stretching/compression associated with the perturbed density front (allowing the release of mean frontal potential energy) and the topographic vorticity gradient associated with the sloping bottom. It is shown that the baroclinic stability characteristics are principally determined by a so-called non-dimensional interaction parameter (denoted μ) which physically measures the ratio of the destabilizing baroclinic vortex-tube stretching/compression to the stabilizing topographic vorticity gradient. For a given along-front mode wavenumber it is shown that a minimum μ is required for instability. Several other general stability results are presented: necessary conditions for instability, growth rate and phase speed bounds, the existence of a high wavenumber cutoff, and a semicircle theorem for the unstable modes. The linear stability equations are solved exactly for a parabolic coupled density front and a detailed description of the spatial and temporal characteristics of the instabilities is given. For physically realistic parameter values the instabilities are manifested as amplifying topographic Rossby waves in the slope water, and on the density front the unstable perturbations take the form of amplifying anticyclones which have maximum amplitude on the offshore side.