Existence, Stability, and Qualitative Theory of Traveling Water Waves

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Water waves are a fundamental subject of study in atmosphere and ocean science. They have been investigated for hundreds of years, yet many of their most basic features remain poorly understood mathematically. For instance, only very recently has significant progress been achieved in developing a rigorous existence theory for steady rotational water waves, that is, waves with vorticity in the bulk of the fluid. These types of waves are ubiquitous in nature: vorticity is generated by incoming currents, the wind blowing over the ocean, or the density stratification due to salinity. A deeper understanding of the mathematics underlying these phenomena would have many scientific and practical applications. For example, traveling waves in density-stratified water (internal waves) are a generic feature of coastal flows in the ocean; they are known to play a major role in driving the circulation of ocean. Vorticity in air is likewise strongly connected to the process of wind generation of water waves, a topic of great importance to ocean and climate modeling as well as ocean engineering. The investigator studies traveling waves in density-stratified water, traveling waves with localized vorticity, and the stability of several important classes of steady waves. He also studies the question of whether stratified waves depend continuously on their density distribution. This question has serious practical implications for how well layered-fluid models can reliably represent flows in the ocean. The specific goals of this project are to (i) expand the existence theory for steady stratified waves and rotational traveling waves; (ii) develop new tools for studying the qualitative properties of internal waves; and (iii) ascertain the stability/instability of several important classes of steady waves. The investigators and colleagues improve on the current existence theory for large-amplitude internal waves by allowing for a general velocity profile in the far field. Concurrently, they study the question of whether stratified waves depend continuously on their density distribution. Confirmation or denial of this fact has serious practical implications: if it is not true, then there is an unbridgeable gap between continuously stratified fluids and the layered-fluid systems that are commonly used to model them. Conversely, if the dependence is smooth, then the modulus of continuity gives bounds on the error arising from the layered approximation. The investigator studies traveling waves with localized vorticity. In earlier work, he and collaborators proved the existence of classes of traveling wave with compactly supported vorticity (including point vortices and vortex patches), but completely unexplored is the intermediate regime where the vorticity is localized but not compact. This important regime is attacked using techniques from bifurcation theory and semi-linear elliptic theory on unbounded domains. Using newly developed tools for determining stability/instability of bound states for non-canonical Hamiltonian systems, they also study the orbital stability of traveling waves with point vortices. Lastly, the problem of wind generation of water waves is considered. This entails studying the stability of small-amplitude steady waves in a two-fluid Euler system, which sheds light on the role of the air-sea interface in the transfer of energy from wind to water.