  In this paper we derive a new formulation of the water waves equations with
vorticity that generalizes the well-known Zalkarov-Craig-Sulem formulation used
in the irrotational case. We prove the local well-posedness of this
formulation, and show that it is formally Hamiltonian. This new for- mulation
is cast in Eulerian variable, and in finite depth; we show that it can be used
to provide uniform bounds on the lifespan and on the norms of the solutions in
the singular shallow water regime. As an application to these re- sults, we
derive and provide the first rigorous justification of a shallow water model
for water waves in presence of vorticity; we show in particular that a third
equation must be added to the standard model to recover the velocity at the
surface from the averaged velocity.
