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

Abstract

A theory is developed describing the sub–inertial baroclinic dynamics of bottom–intensified density–driven flows within a continuously stratified fluid of finite depth with variable bottom topography. The evolution of the density–driven current is modelled as a geostrophically balanced homogeneous flow, 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–geostrophic dynamics describing a balance between the production of relative vorticity, the vortex–tube stretching/compression associated with a deforming gravity current height, and the rectifying influence of a background topographic vorticity gradient. The model is derived as a systematic asymptotic reduction of the two-fluid system in which the upper fluid is described by the Boussinesq adiabatic equations for a continuously stratified fluid and a lower 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 variational principle for arbitrary steady solutions and stability conditions in the sense of Liapunov. The general linear stability problem associated with parallel shear flow solutions is examined in some detail. Necessary conditions for instability are derived. The instability is convective in the sense that it proceeds by extracting 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 height profile. In the overlying fluid, the unstable normal modes correspond to amplifying bottom–intensified topographic Rossby waves.