Xue, Bin Bin; (2017) Dynamics of beach vortices and multipolar vortex patches. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

The classical problem of point vortex equilibria has inspired many studies and the discovery of various equilibrium con gurations involving non-zero vorticity distributions. That in finitesimal vortices can be `smeared out' to a finite area vortex patch with constant vorticity to achieve equilibria has attracted large interest due to its relevance to large-scale geophysical flows and the coherent flow structures observed in two-dimensional turbulence. The present work starts with a consideration of monopolar vortex equilibria in 2D inviscid incompressible flows. By employing a Schwarz function method, an exact solution of monopolar equilibria under speci fic straining field has been found. Numerical considerations are then used to study multipolar vortex equilibria: the `m + 1' point vortex - vortex patch equilibria are numerically computed and consist of a nite area central patch surrounded by m identical point vortices arranged at the vertices of a polygon. Two distinct families of solutions have been found and their limiting states computed. Linear stability analysis is carried out to study the e ect of having a nite area central patch. Numerical routines are further modifi ed to compute the `m+1' multipolar vortex equilibria where the m point vortices from the previous con figuration are replaced with finite area satellite patches. Various properties are investigated including the limiting states and non-linear stability through time-dependent integrations. The existence of new, nite area multipolar vortex equilibria are suggested by the streamlines of multipolar vortex equilibria and have thus been found numerically here. A general numerical procedure is described and focus is put on particular multipolar vortex equilibria consisting of two nested polygonal vortex patches. There exist two distinct families of solutions each having a further two separate cases. Time-dependent integrations are carried out to study their non-linear stability and it has been shown in certain unstable solution regimes the nested polygonal vortex equilibria evolve into the `m + 1' multipolar vortex equilibria. The second topic here concerns vortex patches over an exponential bottom topography. This study is inspired by rip currents observed when water waves break while propagating alongshore or o shore. Steady translating beach vortices have been found numerically. Asymptotic approximation in the small slope limit is derived and shown to agree well with the numerical solution. Linear stability analysis indicates these structures are linearly stable and time-dependent integrations suggest these are robust structures.

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