Coastal outflow currents into a buoyant layer of arbitrary depth

Abstract

The long-wave, reduced-gravity, shallow-water equations (the semi-geostrophic equations) are used to study the outflow of a river into the ocean. While previous models have studied dynamics driven by gradients in density, the focus here is on the effects of potential vorticity anomaly (PVa). The river water is taken to have the same density as a finite-depth upper layer of oceanic fluid, but the two fluids have different, uniform, potential vorticities. Under these assumptions, the governing equations reduce to two first-order, nonlinear partial differential equations which are integrated numerically for a prescribed efflux of river water and PVa. Results are found to depend strongly on the sign of the PVa, with all fluid turning downstream (in the direction of Kelvin-wave propagation) when the river water has positive PVa and anticyclonic flow upstream of the river mouth when the PVa is negative. In all cases, a nonlinear Kelvin wave propagates at finite speed ahead of the river water. Away from the river mouth, the uniformity of one set of Riemann invariants allows for similarity solutions that describe the shape of the outflow, as well as a theory that predicts properties of the Kelvin wave. A range of behaviours is observed, including flows that develop shocks and flows that continue to expand offshore. The qualitative behaviour of the outflow is strongly correlated with the value of a single dimensionless parameter that expresses the ratio of the speed of the flow driven by the Kelvin wave to that driven by image vorticity.