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.
In order to determine the steady-state subcritical gravity-capillary waves that are produced by potential flow past a wave-making body, it is typically necessary to impose a radiation condition that allows for capillary waves upstream, but disallows those corresponding to gravity. However, this radiation condition is not known a priori and consequently, the computation of accurate numerical solutions to the steady-state problem remains problematic. Although the physical model might be modified (e.g. with viscosity), recovery of the original problem is not always possible.In this work, we discuss the above radiation problem, and show how, in the low-speed limit, the steady gravity-capillary waves can be resolved using a Sommerfeld-type boundary condition applied to an asymptotically reduced set of water-wave equations. These results allow us to validate the specialized classes of low-speed waves theoretically predicted by Trinh & Chapman (2013b) using methods in exponential asymptotics [J. Fluid Mech. 724,. The issues of numerically solving the full set of gravity-capillary equations for a potential flow are discussed, and the sensitivity to errors in the boundary conditions is clearly demonstrated.
Experiments and field observations have shown that there are at least two modes of behavior 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. This paper discusses how bulge formation can be affected by the vorticity profile at the river mouth. Due to the image effect, regions of cyclonic vorticity tend to propagate rightwards, whereas regions of anticyclonic vorticity propagate leftward upon exit from the source. If an outflow consists of regions of cyclonic vorticity to the left of regions of anticyclonic vorticity, the two image effects are in competition. We explore this phenomenon using a quasi-geostrophic model with piecewise-constant potential vorticity, which allows the vorticity profile at the source to be set as part of the problem. We present analytic solutions valid in the source region and at the head of the plume and show that all of the outflow travels rightwards if and only if the region of cyclonic vorticity is dominant. The initial-value problem for the model is integrated numerically using the method of contour dynamics, and the full parameter space is explored. We find that if the cyclonic and anticyclonic contributions cancel, as in the experiments of Avicola and Huq (2003), then steady solutions are unstable and a bulge can form downstream of the river mouth.
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