Steady water waves in the presence of wind

TL;DR

This paper develops an existence theory for small steady water waves influenced by wind, modeled via a Kelvin--Helmholtz discontinuity, considering inviscid fluids with various vorticity profiles.

Contribution

It introduces a mathematical framework for steady water waves with wind effects, incorporating vortex discontinuities and general vorticity profiles, extending prior models.

Findings

Existence of small amplitude steady water waves with wind effects established.

Model accounts for Kelvin--Helmholtz discontinuity at the air-water interface.

Framework includes both finite and infinite depth scenarios with different vorticity profiles.

Abstract

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.