Rotating gravity currents. Part 2. Potential vorticity theory

Abstract

An extension to the energy-conserving theory of gravity currents in rectangular rotating channels is presented, in which an upstream potential vorticity boundary condition in the current is applied. It is assumed that the fluid is inviscid; that the Boussinesq approximation applies; that the fundamental properties of momentum, energy, volume flux and potential vorticity are conserved between upstream and downstream locations; and that the flow is dissipationless. The upstream potential vorticity in the current is set through the introduction of a new parameter $\delta$, that defines the ratio of the reference depth of the current to the ambient fluid. Flow types are established as a function $\delta$ and the rotation rate, and a fourth flow geometry is identified in addition to the three previously identified for rotating gravity currents. Detailed solutions are obtained for three cases $\delta\,{=}\,$0.5, 1.0 and 1.5, where $\delta\,{<}\,1$ is relevant to currents originating from a shallow source and $\delta\,{>}\,1$ to currents where the source region is deeper than the downstream depth, for example where a deep ocean flow encounters a plateau. The governing equations and solutions for each case are derived, quantifying the flow in terms of the depth, width and front speed. Cross-stream velocity profiles are provided for both the ambient fluid and the current. These predict the evolution of a complex circulation within the current as the rotation rate is varied. The ambient fluid exhibits similar trends to those predicted by the energy-conserving theory, with the Froude number tending to $\surd 2$ at the right-hand wall at high rotation rates. The introduction of the potential vorticity boundary condition into the energy-conserving theory does not appear to have a substantial effect on the main flow parameters (such as current speed and width); however it does provide an insight into the complex dynamics of the flow within the current.