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.

This article reports on a theoretical and numerical study of noneroding turbulent gravity currents moving down mildly inclined surfaces while depositing sediment. These flows are modeled by means of two-layer fluid systems appropriately modified to account for the presence of a sloping bottom and suspended sediment in the lower layer. A detailed scaling argument shows that when the density of the interstitial fluid is slightly greater than that of the ambient and the suspension is such that its volume fraction is of the order of the aspect ratio squared, for low aspect ratio flows a two-layer shallow-water theory is applicable. In this theory there is a decoupling of particle and flow dynamics. In contrast, however, when the densities of interstitial and ambient fluids are equal, so that it is the presence of the particles alone that drives the flow, we find that a consistent shallow-water theory is impossible no matter how small the aspect ratio or the initial volume fraction occupied by the particles. Our two-layer shallowwater formulation is employed to investigate the downstream evolution of flow and depositional characteristics for sloping bottoms. This investigation uncovers a new phenomenon in the formation of a rear compressive zone giving rise to shock formation in the post-end-wall-separation phase of the particle-bearing gravity flow. This separation of flow from the end wall in these fixed volume releases differs from what has been observed on horizontal surfaces where the flow always remains in contact with the end wall.

In this paper we study various aspects of gravity (or density) currents arising from instantaneous releases of heavy fluids in a rectangular channel with a horizontal bottom. It is shown, by means of a scaling argument, that these plane currents can be successfully modeled by a two-by-two system in conservation form together with a pair of algebraic relations. A number of numerical experiments are carried out using this "weak stratification" model to elicit information concerning the behavior of gravity currents. A weakly nonlinear analysis is employed to clarify some aspects that were uncovered by the numerical experimentation.

A theory is presented to describe the propagation and structure of coherent cold-core (mesoscale) eddies on a sloping bottom including dynamical and thermodynamical interaction with the surrounding fluid. Based on parameter values suggested by oceanographic and rotating-tank experimental data, the evolution of the baroclinic eddy is modelled with nonlinear 'intermediate lengthscale ' geostrophic dynamics which is coupled to the surrounding fluid. The process of ventilation is modelled with a simple cross-interfacial mass flux parameterization. The surrounding fluid is governed by nonlinear quasi-geostrophic dynamics including eddy-induced vortextube compression. Assuming a relatively weak ventilation rate, a multiple-scale asymptotic theory is constructed to describe the propagation of an initially isolated or coherent baroclinic eddy. Throughout the evolution the eddy is assumed to be interacting strongly with the surrounding fluid. To leading order, the eddy and surrounding fluid satisfy the Stern isolation constraint. The magnitude of the Eulerian velocity field in the surrounding fluid above the eddy is shown to be larger than the swirl velocities in the eddy interior as suggested by experimental data. Also, to leading order, the along-shelf translation speed is given by the Nof formula. The process of ventilation is shown to induce a slowly decaying upslope translation in the propagating eddy, and acts to stimulate a weak slowly decaying topographic Rossby wave field in the surrounding fluid. The important features of the theory are illustrated with a simple example calculation.

Abstract. Numerical simulations of the baroclinic dynamics of density-driven coupled fronts and eddies are described. The simulations are based on a two-layer intermediate length scale model which filters out barotropic instability and focuses on the subinertial baroclinic evolution of density-driven flows within the context of allowing finite-amplitude height variations in the lower layer. The baroclinic destabilization of a bottom-trapped coupled front on a sloping bottom is described. In the overlying fluid the instability takes the form of amplifying topographic Rossby waves. In the coupled front the perturbations to the downslope incropping are preferentially amplified compared to those on the upslope incropping. The perturbations to the downslope incropping develop into downslope propagating plumes which eventually evolve into relatively coherent along-slope propagating domes. We discuss the propagation characteristics of these domes. We also simulate the evolution of density-driven eddies or domes. The first eddy simulation we describe is for an initial eddy configuration which satisfies a zero topographic Rossby wave condition in the upper layer. We show that these traveling solutions remain remarkably coherent over a period of about 40 eddy circulation times or about 250 days for typical continental slope values. For a sufficiently large initial eddy height, upper layer fluid parcels can be transported in the along-slope direction by the baroclinic eddy. We also simulate the evolution of an initial eddy configuration which does not satisfy a zero topographic Rossby wave condition in the upper layer. A relatively intense cyclonic circulation develops in the overlying fluid over the traveling dome as does a topographic Rossby wave tail. However, even these solutions remain surprisingly coherent over many eddy circulation times.