# Shallow water models with constant vorticity

## TL;DR

This paper extends classical shallow water models to include constant vorticity, analyzing wave breaking, stability, and instability, revealing how vorticity influences wave dynamics and modulational behavior.

## Contribution

It introduces modifications to shallow water equations for constant vorticity, proposes a full-dispersion model, and studies wave stability and breaking phenomena in this setting.

## Key findings

- Wave breaking occurs with increased elevation and higher vorticity.
- The proposed model shows stability thresholds depending on wave number and vorticity.
- Vorticity significantly affects modulational stability and surface tension effects.

## Abstract

We modify the nonlinear shallow water equations, the Korteweg-de Vries equation, and the Whitham equation, to permit constant vorticity, and examine wave breaking, or the lack thereof. By wave breaking, we mean that the solution remains bounded but its slope becomes unbounded in finite time. We show that a solution of the vorticity-modified shallow water equations breaks if it carries an increase of elevation; the breaking time decreases to zero as the size of vorticity increases. We propose a full-dispersion shallow water model, which combines the dispersion relation of water waves and the nonlinear shallow water equations in the constant vorticity setting, and which extends the Whitham equation to permit bidirectional propagation. We show that its small amplitude and periodic traveling wave is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin-Feir instability in the irrotational setting; the critical wave number grows unboundedly large with the size of vorticity. The result agrees with that from a multiple scale expansion of the physical problem. We show that vorticity considerably alters the modulational stability and instability in the presence of the effects of surface tension.