  We study the horizontal dispersion of passive tracer particles on the free
surface of gravity waves in deep water. For random linear waves with the
JONSWAP spectrum, the Lagrangian particle trajectories are computed using an
exact nonlinear model known as the John--Sclavounos equation. We show that the
single-particle dispersion exhibits an unusual super-diffusive behavior. In
particular, for large times $t$, the variance of the tracer $\langle
|X(t)|^2\rangle$ increases as a quadratic function of time, i.e., $\langle
|X(t)|^2\rangle\sim t^2$. This dispersion is markedly faster than Taylor's
single-particle dispersion theory which predicts that the variance of passive
tracers grows linearly with time for large $t$. Our results imply that the wave
motion significantly enhances the dispersion of fluid particles. We show that
this super-diffusive behavior is a result of the long-term correlation of the
Lagrangian velocities of fluid parcels on the free surface.
