Title: Shallow water models with constant vorticity,
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.