On mathematical aspects of exact nonlinear rotational water waves

This thesis addresses various theoretical questions regarding exact nonlinear solutions to the geophysical governing equations which are explicit in the Lagrangian framework. Such solutions are scarce, being Gerstner’s wave the only-known explicit and exact solution of the nonlinear two-dimensional periodic gravity wave problem with a non-flat free surface under constant density. This remarkable solution was extended by Pollard to an incompressible vertically-stratified fluid in a rotating system. More recently, exact and explicit solutions of the nonlinear governing equations for geophysical water waves, describing several physical scenarios, have been derived. A thorough analysis of these solutions is conducted in this thesis. From a mathematical point of view, the Lagrangian flow map defining a flow motion needs to satisfy certain conditions. It is proven in Chapters 2 and 6 that this is the case for the equatorially-trapped, non-hydrostatic internal water waves [14] and for the generalisation of Pollard’s solution [26], respectively. An advantage of some of these explicit solutions is that they are able to accommodate currents, leading to more complicated and interesting flows. This is shown in Chapter 3 where a new solution of the f-plane approximation of the geophysical equations is constructed. This solution incorporates both a constant current in the zonal direction and a transverse current in the meridional direction. The study of this solution and of other internal water waves is further developed in Chapter 4. Relevant mean flow properties are provided, establishing a relation between Eulerian and Lagrangian quantities. Furthermore, the effects of the vorticity present in these flows are compared with those of the better-known irrotational case. In addition, Chapter 5 examines the robustness of these solutions in terms of the hydrodynamic stability. This is done for the internal wave discussed in Chapter 2 and the solution of the modified β-plane approximation equations [61] by applying the short-wavelength instability method. An important aspect of the flows studied throughout this work is that they exhibit vorticity. This is further analysed in the Eulerian framework in Chapter 7 within the context of small-amplitude two-dimensional steady periodic gravity water waves propagating over a flat bed. A dispersion relation for waves with two layers of different vorticity is derived and the existence of such waves is discussed