Sub-inertial dynamics of density-driven flows in a continuously stratifiedfluid on a sloping bottom. I. Model derivation and stability characteristics

A theory is developed describing the sub-inertial baroclinic dynamics of bo ttom-intensified density-driven flows within a continuously stratified flui d of finite depth with variable bottom topography. The evolution of the den sity-driven current is modelled as a geostrophically balanced homogeneous f low, which allows for finite-amplitude dynamic thickness variations and for which the pressure fields in each layer are strongly coupled together. The evolution of the overlying fluid is governed by baroclinic quasi-geostroph ic dynamics describing a balance between the production of relative vortici ty, the vortex-tube stretching/compression associated with a deforming grav ity current height, and the rectifying influence of a background topographi c vorticity gradient. The model is derived as a systematic asymptotic reduc tion of the two-fluid system in which the upper fluid is described by the B oussinesq adiabatic equations for a continuously stratified fluid and a low er homogeneous layer described by shallow-water theory appropriate for an f -plane with variable bottom topography. The model is shown to possess a non -canonical Hamiltonian formulation. This structure is exploited to give a v ariational principle for arbitrary steady solutions and stability condition s in the sense of Liapunov. The general linear stability problem associated with parallel shear flow so lutions is examined in some detail. Necessary conditions for instability ar e derived. The instability is convective in the sense that it proceeds by e xtracting the available gravitational potential energy associated with the lower-layer water mass sliding down the sloping bottom. For the normal-mode instability problem, a semicircle theorem is derived. The linear stability characteristics are illustrated by solving the normal-mode equations for a simple linearly varying lower-layer weight profile. In the overlying: flui d, the unstable normal modes correspond to amplifying bottom-intensified to pographic Rossby waves.