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Dispersive shock waves in coastal flows

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HY2W01 - Modulation theory and dispersive shock waves

Coastal or boundary currents are an integral part of global ocean circulation. For example, currents may respond to external forcing or intrinsic instability by expelling vortex filaments or larger eddies into the ocean, with implications for the mixing of coastal and ocean waters; and currents driven by outflows are important for the transport of freshwater, pollutants and land-derived nutrients. There is also much interest in the behaviour of ‘free’ fronts, i.e. those that are far from the coast, which can be used to model western boundary currents such as the Gulf Stream or the Kuroshio Extension. In the limit of rapid rotation the governing equations reduce to the quasi-geostrophic equations – a modified form of the two-dimensional Euler equations. For the problems considered here the vorticity of the flow is unity within the current and zero elsewhere. The unapproximated solution can thus be obtained numerically to high accuracy by applying the method of Contour Dynamics to the development of the current-ocean interface. These solutions provide comparisons for estimating the accuracy of asymptotic solutions. Alongshore variations in the flows take place over scales large compared to offshore scales and so analysis of the flows leads naturally to a long-wave equation for the current- ocean interface. Two examples will be discussed in depth. First, the development of the flow when fluid is discharged from a source on the coast to turn and form an alongshore current (Johnson et al. 2017) and, second, the Riemann problem for the subsequent development of a step change in width of a coastal flow (Jamshidi & Johnson 2020).The flux function appearing in the long-wave equation is non-convex and this leads to a wide variety of behaviours. Many of these are well-described following the method of El (2005) but some discrepancies remain. These and some other open questions will be noted.

This talk is part of the Isaac Newton Institute Seminar Series series.

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