Samuel Walsh

Stratified Steady Periodic Water Waves

This paper considers two-dimensional stratified water waves propagating under the force of gravity over an impermeable flat bed and with a free surface. We prove the existence of a global continuum of classical solutions that are periodic and traveling. These waves, moreover, can exhibit large density variation, speed and amplitude.

Travelling water waves with compactly supported vorticity

In this paper, we prove the existence of two-dimensional, travelling, capillary-gravity, water waves with compactly supported vorticity. Specifically, we consider the cases where the vorticity is a δ-function (a point vortex), or has small compact support (a vortex patch). Using a global bifurcation theoretic argument, we construct a continuum of finite-amplitude, finite-vorticity solutions for the periodic point vortex problem. For the non-periodic case, with either a vortex point or patch, we prove the existence of a continuum of small-amplitude, small-vorticity solutions.

Steady Water Waves in the Presence of Wind

In this paper we develop an existence theory for small amplitude, steady, two-dimensional water waves in the presence of wind in the air above. The presence of the wind is modeled by a Kelvin--Helmholtz type discontinuity across the air-water interface, and a corresponding jump in the circulation of the fluids there. We consider both fluids to be inviscid, with the water region being irrotational and of finite depth. The air region is considered with constant vorticity in the case of infinite depth and with a general vorticity profile in the case of a finite, lidded atmosphere.

Continuous Dependence on the Density for Stratified Steady Water Waves

There are two distinct regimes commonly used to model traveling waves in stratified water: continuous stratification, where the density is smooth throughout the fluid, and layer-wise continuous stratification, where the fluid consists of multiple immiscible strata. The former is the more physically accurate description, but the latter is frequently more amenable to analysis and computation. By the conservation of mass, the density is constant along the streamlines of the flow; the stratification can therefore be specified by prescribing the value of the density on each streamline. We call this the streamline density function.Our main result states that, for every smoothly stratified periodic traveling wave in a certain small-amplitude regime, there is an L∞ neighborhood of its streamline density function such that, for any piecewise smooth streamline density function in that neighborhood, there is a corresponding traveling wave solution. Moreover, the mapping from streamline density function to wave is Lipschitz continuous in a certain function space framework. As this neighborhood includes piecewise smooth densities with arbitrarily many jump discontinues, this theorem provides a rigorous justification for the ubiquitous practice of approximating a smoothly stratified wave by a layered one. We also discuss some applications of this result to the study of the qualitative features of such waves.

On the Wind Generation of Water Waves

AbstractIn this work, we consider the mathematical theory of wind generated water waves. This entails determining the stability properties of the family of laminar flow solutions to the two-phase interface Euler equation. We present a rigorous derivation of the linearized evolution equations about an arbitrary steady solution, and, using this, we give a complete proof of the instability criterion of Miles [16]. Our analysis is valid even in the presence of surface tension and a vortex sheet (discontinuity in the tangential velocity across the air–sea interface). We are thus able to give a unified equation connecting the Kelvin–Helmholtz and quasi-laminar models of wave generation.

Pressure Transfer Functions for Interfacial Fluids Problems

We make a consistent derivation, from the governing equations, of the pressure transfer function in the small-amplitude Stokes wave regime and the hydrostatic approximation in the small-amplitude solitary water wave regime, in the presence of a background shear flow. The results agree with the well-known formulae in the zero vorticity case, but they incorporate the effects of vorticity through solutions to the Rayleigh equation. We extend the results to permit continuous density stratification and to internal waves between two constant-density fluids. Several examples are discussed.