Proposal for an EPSRC Network in Mathematics Wave – flow interactions

The fluids encountered in environmental and industrial applications are often endowed with restoring mechanisms which lead to the propagation of waves. Compressibility and sound waves, density stratification and internal-gravity waves, differential rotation and Rossby waves are just a few examples of such restoring mechanisms and associated waves. When they propagate in a fluid, waves can interact with the fluid motion in a variety of ways. Crucially, it is a two-way interaction: fluid motion can generate waves, but waves can also force fluid motion. The generation of sound by turbulence, and the generation of along-shore currents by surface-wave breaking provide familiar examples for this. Because waves and fluid flows often have very different spatial and temporal scales, and because they involve very different physical processes (large-distance propagation vs. turbulent cascades, for example), the modelling of wave–flow interactions often poses a considerable technical and conceptual challenge. The purpose of this Network is to address these challenges by bringing together a number of UK researchers with expertise in different aspects of the problem: mathematical modelling, numerical simulations, and laboratory experiments. The group will be strengthened by the inclusion of a few associate members from overseas who, apart from bringing in their unique expertise, will ensure the international visibility of the Network. The creation of the Network appears particularly timely: research on wave–flow interactions is currently buoyant in the UK, with several groups working on different aspects of the topic, especially in the context of geophysical (stratified, rotating) fluids. Their contributions span a whole spectrum of mathematical methods, but so far the interactions have been limited. This is a reflection of a dangerous trend: research nowadays has become increasingly compartmentalised. Particular applications have become nearly the sole foci of studies, with less attention being paid to the underlying mathematical affinity of the physical processes. The Network aims at reversing this trend by facilitating collaboration between leading research groups. The complementarity of approaches among these groups is ideal for fruitful collaborations over a wide range of problems. At the practical level the main aim of the Network, then, is to foster collaborative work between its members, especially by providing opportunities for face-to-face contact. To this end, four meetings are planned, which will gather the entire Network membership. Smaller-scale workshops will also be organised, on a flexible basis, to promote hands-on collaborations. The meetings will also provide opportunities to invite researchers working on issues of wave–flow interactions in contexts not necessarily covered by the Network memberships. The interaction between quasi-Newtonian flow and gravitational waves in relativistic media, the interaction between Alfven waves and MHD flows, and the interaction of quantum vortices with waves in superfluids are examples of such contexts. The relevance of particular mathematical techniques, such as multi-scale numerical methods and stochastic models, will also be explored through the invitation of a few selected speakers. This policy will foster cross-disciplinary interactions and, if productive, enable us to extend the Network membership. Another key feature of the Network activities will be the close involvement of PhD students. Students supervised by members will attend the meetings which will purposely include introductory