  The classic evolution equations for potential flow on the free surface of a
fluid flow are not closed because the pressure and the vertical velocity
dynamics are not specified on the free surface. Moreover, their wave dynamics
does not cause circulation of the fluid velocity on the free surface. The
equations for free-surface motion we derive here are closed and they are not
restricted to potential flow. Hence, true wave-current interaction dynamics can
occur. In particular, the Kelvin-Noether theorem demonstrates that wave
activity can induce fluid circulation and vorticity dynamics on the free
surface. The wave-current interaction equations introduced here open new vistas
for both the deterministic and stochastic analysis of nonlinear waves on free
surfaces.
