Numerical and Theoretical Results Exploring the Potential-Vorticity Dynamics of Coastal Outflows

The influx of river water into the oceans brings nutrients, sediment and pollutants, as well as driving coastal currents, and as such the dynamics of river plumes is an important area of study. Experiments and field observations have shown that there are at least two modes of behaviour for river plumes. In many cases, the plume turns to the right (in the Northern hemisphere) on leaving the river mouth and follows the direction of Kelvin-wave propagation. Alternatively, a ‘bulge’ can form in the plume and a fraction of the outflow volume becomes trapped near the mouth. The present work investigates the role played by potential-vorticity gradients in determining the behaviour of river plumes, and develops a mathematical model based on a 1 $\frac{1}{2}$-layer Boussinesq system in the long-wave limit (the semi-geostrophic equations). The model depends on two physical quantities: the volume flux of river water, and the depth of the buoyant oceanic layer. Although the system is fully nonlinear, it allows theoretical predictions to be made using the method of characteristics, which are then compared with numerical experiments.

In particular, major qualitative differences occur depending on whether the (assumed uniform) potential vorticity of the plume is greater or less than the potential vorticity of the upper oceanic layer. Two mechanisms contribute: flow driven by a nonlinear Kelvin wave, and flow driven by the jump in potential vorticity, the latter of which can drive fluid in either direction. The faster Kelvin wave disturbs oceanic fluid ahead of the river water, transporting fresh water along the coast, and our model allows for the prediction of both the local fluid velocity and the speed of wave propagation in the river and oceanic fluid. Depending on the values of the physical parameters, solutions can feature coastal currents, plumes that expand offshore or propagate upstream, and plumes that separate from the coast. We will also discuss a recent extension to outflows with non-uniform potential vorticity, which can additionally lead to the formation of quasi-steady bulges.