Resonances and coupling of waves in rotational flows

The resonant interaction between two or more water waves that combine to build a new one is a process of paramount importance for the evolution of waves given the significant energy transfer that occurs during the unfolding of the interaction. Another feature displayed by many water flows is the presence of internal waves due to density stratification which is quite pronounced in the tropical part of the ocean. Often, a coupling effect between surface and internal waves arises and at times might trigger concealed internal waves of big amplitude than can imperil ship maneuvering. Mindful of the previous aspects we direct the proposed project towards an interdisciplinary study of the resonances and of the coupling between surface and internal water waves. We will also aim at understanding the characteristics of the flow beneath like pressure, velocity field and the interactions of the surface and internal waves with the underlying currents in rotational flows. While previous studies have dealt with the coupling between internal and surface waves, they remained confined to the irrotational setting, thus being unable to seize the vertical structure specific to currents in oceans and seas. Moreover, they did not address the geophysical effects (arising from the Earth`s rotation) either. Our proposed project aims at filling these gaps by applying a mixture of tools from partial differential equations, variational calculus and fluid mechanics. A successful treatment of the previous mentioned ocean flow features hinges upon finding a concise formulation of the water wave problem that highlights its structural properties. Of high relevance here is the Hamiltonian formulation that has the advantage that allows the reformulation of the water wave problem in terms of the wave variables. Acquiring a Hamiltonian formulation that encompasses the mentioned aspects will be one of the key contributions of the project. It will also allow for the development of structure preserving approximations of the nonlinear governing equations and also the derivation of model equations that are amenable to an in-depth study. While the proposed research will need to handle challenging aspects of mathematical analysis and fluid mechanics its implications are broader and have potential relevance for the field of ocean engineering. Indeed, the accurate prediction of internal and surface waves of large amplitude represents an imperative need for routine navigation activities. The implementation of the project requires a truly interdisciplinary approach that is envisaged to be a resolute blend of novel analytical results guided by numerical results and experimental data. Moreover, the project has the potential to facilitate the two-way knowledge transfer between fundamental mathematical research and field/experimental data.

The general objective of the project was to study surface and internal waves and their potential coupling in water flows that exhibit depth-depedent currents, (discontinuous) density, and Coriolis effects stemming from the Earth's rotation. To exemplify the previous aspects we would like to mention the pronounced density stratification in the equatorial Pacific within a band of 150 km on each side of the Equator where changes in temperature and salinity result in density fluctuations which yield a vertical layering of the flow whose hallmark is the presence of an interface, called thermocline or pycnocline. Furthermore, the Coriolis forces together with the westward winds generate an underlying current field that displays flow-reversal. Towards the achievement of the previously formulated goal, we utilized a mathematical approach that consisted of an in-depth analysis of a system of nonlinear partial differential equations satisfied by the velocity field and the pressure (representing conservation of momentum), of an equation stating the conservation of mass and of some nonlinear boundary conditions, stating the impermeability of the surface wave, of the interface (playing the role of an internal wave) and of the fluid's bottom. The more specific tasks concerned the development of structure-preserving approximations to the fully nonlinear governing equations and the derivation of model equations (amenable) to an in-depth study which included numerical simulations of the derived model equations. To mitigate the difficulties displayed by the mentioned nonlinear system we have resorted to equivalent reformulations of the governing equations. These reformulations are able to write the nonlinear equations only in terms of the wave variables: the free surface and the internal wave. We were able to derive in the long-wave approximation a KdV-type equation with variable coefficients. Moreover, emphasis was put on the derivation of explicit/exact solutions for discontinuously stratified geophysical water flows that manifest an azimuthal propagation direction. An important feature of these solutions is that they are presented in spherical coordinates: that is, no approximations by way of simplifying the Earth's geometry were made. Another key aspect relating to the derived exact solutions was to also achieve qualitative/quantitative results about the behavior of the pressure function in relation to the free surface and the interface of the two-layered fluid domains taken into account. The mentioned features render this analysis suitable for representing the gross dynamics of the Equatorial Undercurrent (EUC) and of the Antarctic Circumpolar Current (ACC)-the world's longest and strongest current. Along the lines of exact solutions, we have also derived a family of radial explicit solutions in Eulerian coordinates for a fluid domain with a free surface and of finite depth and exhibiting vertical structure. The implications of constant vorticity on the dimension reduction of three-dimensional flows were also studied.