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Surface and Interfacial Waves over Currents and Point-Vortices

Banff International Research Station for Mathematical Innovation and Discovery

The computation of surface and interfacial waves is a central problem in fluid mechanics. While much has been done, the effect of vorticity on surface and internal wave propagation is still poorly understood. To address this, we first look at shallow-water propagation in density stratified fluids with piecewise linear shear profiles. We show that by allowing for jumps in the shear across the interface, strong nonlinear responses can be generated resulting in phenomena like dispersive shock waves. Thus depth varying currents could play a larger role in interface dynamics than is currently understood. Second, we study the problem of collections of irrotational point vortices underneath a free fluid surface. We present a derivation of a model and numerical scheme which allows for arbitrary numbers of vortices in a shallow-water limit. While we are able to recreate much of the classical results for how surface waves form over two counter-propagating vortices, we go beyond this case and look at an example involving four vortices. Again, our approach allows for any number of vortices to be present, and this lets us provide some hint as to how underwater eddies might generate free surface waves.

Mathematics; Fluid mechanics; Partial differential equations; Fluid dynamics

video/mp4

Author affiliation: SDSU

2017-05-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61521

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/