# Stokes waves with constant vorticity: II. folds, gaps and fluid bubbles

## TL;DR

This paper investigates complex wave behaviors in flows with constant vorticity, revealing folds, gaps, and fluid bubbles, and introduces efficient numerical methods to analyze these phenomena.

## Contribution

It formulates the Stokes wave problem with vorticity as a nonlinear pseudodifferential equation and explores the intricate wave structures and limiting behaviors.

## Key findings

- Folds and gaps appear in wave speed-amplitude relations for strong vorticity.
- Touching waves tend to limiting Crapper waves or circular vortex waves as vorticity increases.
- Multiple circular bubbles of fluid are observed in higher-gap wave profiles.

## Abstract

The Stokes wave problem in a constant vorticity flow is formulated, by virtue of conformal mapping techniques, as a nonlinear pseudodifferential equation, involving the periodic Hilbert transform, which becomes the Babenko equation in the irrotational flow setting. The associated linearized operator is self-adjoint, whereby the modified Babenko equation is efficiently solved by means of the Newton-Conjugate Gradient method. For strong positive vorticity, a `fold' appears 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 at the trough line, enclosing a bubble of air. More folds and gaps follow for stronger vorticity. Touching waves at the beginnings of the lowest gaps tend to the limiting Crapper wave as the vorticity strength increases indefinitely, while the profile encloses a circular bubble of fluid in rigid body rotation at the ends of the gaps. Touching waves at the beginnings of the second gaps tend to the circular vortex wave on top of the limiting Crapper wave in the infinite vorticity limit, and the circular vortex wave on top of itself at the ends of the gaps. Touching waves for higher gaps accommodate more circular bubbles of fluid.