Resonant interactions of water waves with vorticity

This research project examines an important aspect of the water wave motion: resonant interaction between two or more waves that combine to build a new one. The phenomenon of resonance is of high relevance in coastal navigation, where the need to know whether particularities of the flow beneath, like the presence of underlying currents, can be detected from the examination of the free surface, emerges very naturally. To model the occurrence of non-uniform currents (noted above) and to describe wave-current interactions, one needs to allow for swirling motions in the fluid, captured by the vorticity function. The scarcity of rigorous mathematical studies on resonances for water waves with vorticity is a result of the enormous difficulties raised by allowing the presence of swirling motions. Although former studies on three and four-wave resonances considered stratification, they remained until recently largely confined to the setting of irrotational flows of infinite depth and did not address the possibility of accommodating underlying non-uniform currents. This project will investigate novel aspects pertaining to the more realistic assumption of finite depth and to the inclusion of a (piecewise) constant vorticity, which allows for consideration of wave-current interactions and flow reversal. The difficulties caused by vorticity are circumvented by the availability of a variational formulation that permits the writing of the fluid motion in terms of the ``wave variables``. This formulation was achieved by the PI together with collaborators and by the PI alone for water flows of finite depth and allowing for a piecewise constant vorticity. By means of this variational formulation we will also attack the important issue of the development of instabilities, first in the relatively simpler case of constant vorticity and thereafter we will treat the case with piecewise constant vorticity. We will also address these topics in the setting of flows opposite to the propagation direction of surface waves. Other ingredients that enter the successful study of resonances are the dispersion relations, of which the PI has quite a good understanding evidentiated by several publications on this subject. The PI will consider in this research project the resonance problem by taking into account the nature of restoring forces acting upon the fluid, the stratification and the distribution of vorticity. Namely, we will consider stratified flows driven by capillarity, capillarity and gravity and gravity alone. Moreover, for a better understanding of the dynamics of water waves, we plan to carry out several numerical investigations.

The main focus of the project was to investigate resonant interactions for water waves arising as the free surface of rotational water flows. The resonant interaction is the process by which two or more waves combine to build a new wave. It is an event of paramount importance given the significant energy transfer among the wave trains that occurs in the course of resonance. Along these lines, three-wave resonances might set in motion \emph{rogue waves}--oceanic phenomena believed to have been the cause for a number maritime disasters. The investigation of resonances was until relatively recently largely confined to the case of infinite depth, irrotational flow, that is, a flow in which fluid elements endure no net rotation with respect to chosen coordinate axes from one instant to other. Clearly, the previous described scenario fails to encompass many reasonable physical occurrences like complex vertical structures specific to subsurface currents, ubiquitous wave-current interactions in oceans and seas or flow reversal. Mindful of the previous aspects we have proved the emergence of three-wave resonances for capillary-gravity water waves over \emph{rotational} water flows with piecewise constant distribution of the vorticity, a setting that describes bilinear shear currents. Instrumental in this endeavour (and a by-product of the project) was the \emph{dispersion relation}--a formula that provides the relative wave speed in terms of the physical parameters of the flow. While the dispersion relation is, in general, a convoluted algebraic equation, we were able to obtain explicit plain formulas in the situation when the ratio ``thickness of the near-surface vortical layer/wavelength of the surface wave" is sufficiently large. Once three -wave resonance established we investigated the dynamics of wave packets participating in the resonance. More precisely, utilizing a variational formulation of Hamiltonian type of the rotational water wave problem, we derived evolution equations for the envelopes of the three waves entering the resonance process. Other upshots of the project concern the derivation of explicit and exact solutions describing (from the perspective of the rotating Earth) water flows that exhibit an azimuthal propagation direction and accommodate a general continuous density that varies with depth and latitude. These solutions satisfy the full nonlinear governing equations expressed in spherical coordinates and refer to the large-scale equatorial ocean dynamics of a fluid body with a free surface. Within the same category of geophysical water flows, we have obtained characterizations of three-dimensional water flows displaying a constant vorticity vector. The novel analytical techniques (from partial differential equations, dynamical systems, calculus of variations) used to overcome the difficulties raised by the project have the potential to be of relevance in approaching new challenges arising in fluid dynamics.