  Two methods to treat wave breaking in the framework of the Hamiltonian
formulation of free-surface potential flow are presented, tested, and
validated. The first is an extension of Kennedy et al (2000)'s eddy-viscosity
approach originally developed for Boussinesq-type wave models. In this
approach, an extra term, constructed to conserve the horizontal momentum for
waves propagating over a flat bottom, is added in the dynamic free-surface
condition. In the second method, a pressure distribution is introduced at the
free surface that dissipates wave energy by analogy to a hydraulic jump
(Guignard and Grilli, 2001). The modified Hamiltonian systems are implemented
using the Hamiltonian Coupled-Mode Theory, in which the velocity potential is
represented by a rapidly convergent vertical series expansion. Wave energy
dissipation and conservation of horizontal momentum are verified numerically.
Comparisons with experimental measurements are presented for the propagation of
a breaking dispersive shock wave following a dam break, and then incident
regular waves breaking on a mildly sloping beach and over a submerged bar.
