"The proposal concerns the mathematical study of interactions of vortices, waves and mixing processes, in idealised models relevant to experiments and geophysical systems. Vortices, regions of rotating fluid, are common in the oceans and atmosphere, for example Gulf Stream rings and hurricanes, and in astrophysical systems, for example Jupiter's Great Red Spot. Such systems also support many kinds of waves, including inertial, Rossby and gravity, and the interactions of vortices and waves provide a rich variety of problems for mathematical and numerical study.Recent work of the group in Exeter has focused on vortex dynamics - purely two-dimensional fluid flow - in the absence of waves, and has quantified aspects of the stability of vortices, thresholds for the formation of structures such as tripoles, and development of vorticity staircases. There has been particular emphasis on linking the formation of structures to the stability of vortex profiles using the technique of tracking Landau poles.The aim of the proposal is to extend these studies by including surface gravity waves, using the so-called shallow water equations. The presence of such waves allows vortices to be destabilised by radiating away wave energy, and also allows vortices to be forced by streams of incoming waves. The research is focused on developing an understanding of stability, mixing and structure formation with these additional physical effects, which although idealised using a shallow water system, are an important aspect of many applications. The research programme involves a combination of asymptotic analysis and large-scale numerical studies, and is linked to recent experimental studies of Cobelli, Maurel, Pagneux and Petitjeans, in Paris. The proposed fellowship builds on the expertise of the researcher (in theory, experiment and numerics) and opens up opportunities for him to develop his career, by pursuing these new research directions in collaboration with the Exeter group."