The long-wave vorticity dynamics of rotating buoyant outflows

This paper discusses the evolution of coastal currents by considering, relative to a rotating frame, the flow development when buoyant fluid is injected into a quiescent fluid bounded by a solid wall. The initial rapid response is determined by the Coriolis force-pressure gradient balance with a Kelvin wave propagating rapidly, at the long-wave speed, with the bounding wall to its right (for positive rotation). However fluid columns can stretch or squash on ejection from coastal outflows so that the ejected fluid gains positive or negative relative vorticity. Depending on its sign, the image in the solid wall of this vorticity can reinforce or oppose the zero potential-vorticity-anomaly (PVa) current set up by the Kelvin wave (KW). This paper presents a simple, fully nonlinear, dispersive, quasi-geostrophic model to discuss the form of coastal outflows as the relative strength of vortex to KW driving is varied. The model retains sufficient physics to capture both effects at finite amplitude and thus the essential nonlinearity of the flow, but is sufficiently simple so as to allow highly accurate numerical integration of the full problem and also explicit, fully nonlinear solutions for the evolution of a uniform PVa outflow in the hydraulic limit. Outflow evolutions are shown to depend strongly on the sign of the PVa of the expelled fluid, which determines whether the vortex and KW driving are reinforcing or opposing, and on the ratio of the internal Rossby radius to the vortex-source scale, vertical bar V-0/D-2 Pi(0)vertical bar(1/2), of the flow (where D measures the outflow depth, Pi(0) the PVa of the outflow and V-0 the volume flux of the outflow), which measures the relative strengths of the two drivers. Comparison of the explicit hydraulic solutions with the numerical integrations shows that the analytical solutions predict the flow development well with differences ascribable to dispersive Rossby waves on the current boundary and changes in the source region captured by the full equations but not present in the hydraulic solutions.