Shallow water models with constant vorticity

We modify the nonlinear shallow water equations, the Korteweg-de Vries equation, and the Whitham equation, to permit constant vorticity, and discuss 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 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 find that a small amplitude, 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 of a Stokes wave in the irrotational setting; but the critical wave number grows unboundedly large with the size of vorticity. Moreover, constant vorticity qualitatively changes modulational stability and instability in the presence of the effects of surface tension.