The Cubic Vortical Whitham Equation

TL;DR

This paper introduces the cubic-vortical Whitham equation, a model for wave motion in sheared currents that extends the classical Whitham equation by including vorticity and cubic nonlinearity, analyzing solution stability.

Contribution

It generalizes the Whitham equation to include constant vorticity and cubic nonlinear terms, enabling larger amplitude solutions and stability analysis.

Findings

Large-amplitude solutions are unstable regardless of wavelength.

Vorticity increases the amplitude of solutions.

Small-amplitude solutions are modulationally unstable if kh > 1.252 with zero vorticity.

Abstract

The cubic-vortical Whitham equation is a model for wave motion on a vertically sheared current of constant vorticity in a shallow inviscid fluid. It generalizes the classical Whitham equation by allowing constant vorticity and by adding a cubic nonlinear term. The inclusion of this extra nonlinear term allows the equation to admit periodic, traveling-wave solutions with larger amplitude than the Whitham equation. Increasing vorticity leads to solutions with larger amplitude as well. The stability of these solutions is examined numerically. All moderate- and large-amplitude solutions, regardless of wavelength, are found to be unstable. A formula for a stability cutoff as a function of vorticity and wavelength for small-amplitude solutions is presented. In the case with zero vorticity, small-amplitude solutions are unstable with respect to the modulational instability if kh > 1.252, where k is the wavenumber and h is the mean fluid depth.