On the wind generation of water waves

TL;DR

This paper develops a rigorous mathematical framework to analyze the stability of wind-generated water waves, unifying different models and extending previous instability criteria to more complex conditions.

Contribution

It provides a comprehensive derivation of linearized evolution equations and proves the instability criterion of Miles, including effects of surface tension and vortex sheets.

Findings

Unified equation connecting Kelvin--Helmholtz and quasi-laminar models

Complete proof of Miles' instability criterion

Analysis valid with surface tension and vortex sheet effects

Abstract

In 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. 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.