Via a sequence of approximations of the Lagrangian in Hamilton's principle
for dispersive nonlinear gravity waves we derive a hierarchy of Hamiltonian
models for describing wave-current interaction (WCI) in nonlinear dispersive
wave dynamics on free surfaces. A subclass of these WCI Hamiltonians admits
\emph{emergent singular solutions} for certain initial conditions. These
singular solutions are identified with a singular momentum map for left action
of the diffeomorphisms on a semidirect-product Lie algebra. This
semidirect-product Lie algebra comprises vector fields representing horizontal
current velocity acting on scalar functions representing wave elevation. We use
computational simulations to demonstrate the dynamical interactions of the
emergent wavefront trains which are admitted by this special subclass of
Hamiltonians for a variety of initial conditions. In particular, we
investigate: (1) A variety of localised initial current configurations in still
water whose subsequent propagation generates surface-elevation dynamics on an
initially flat surface; and (2) The release of initially confined
configurations of surface elevation in still water that generate dynamically
interacting fronts of localised currents and wave trains. The results of these
simulations show intricate wave-current interaction patterns whose structures
are similar to those seen, for example, in Synthetic Aperture Radar (SAR)
images taken from the space shuttle.