# Stokes waves in a constant vorticity flow

## TL;DR

This paper formulates and analyzes Stokes waves in a constant vorticity flow using a conformal mapping approach, revealing complex wave behaviors including folds, gaps, and limiting wave profiles as vorticity varies.

## Contribution

It introduces a modified Babenko equation for Stokes waves with constant vorticity and explores the wave behavior and bifurcations as vorticity strength changes.

## Key findings

- Development of a fold in wave speed versus amplitude for strong vorticity
- Existence of gaps bounded by touching waves with self-contact profiles
- Limiting wave profiles include Crapper waves and rigid body rotation disks

## Abstract

The Stokes wave problem in a constant vorticity flow is formulated via a conformal mapping as a modified Babenko equation. The associated linearized operator is self-adjoint, whereby efficiently solved by the Newton-conjugate gradient method. For strong positive vorticity, a fold develops in the wave speed versus amplitude plane, and a gap as the vorticity strength increases, bounded by two touching waves, whose profile contacts with itself, enclosing a bubble of air. More folds and gaps follow as the vorticity strength increases further. Touching waves at the beginnings of the lowest gaps tend to the limiting Crapper wave as the vorticity strength increases indefinitely, while a fluid disk in rigid body rotation at the ends of the gaps. Touching waves at the boundaries of higher gaps contain more fluid disks.