The Alber equation has been proposed as a model for stochastic ocean waves,
and it is associated with a nonlinear "eigenvalue relation" which controls the
possible linear instability of given wave spectra. We call this condition the
"Penrose condition" after a similar one appearing in plasma physics, and we
show that it can be easily understood by adapting tools developed in plasma
physics. Our main result is linear Landau damping: we prove that if a spectrum
is stable in the sense of the Penrose condition, then any perturbations of it
vanish in time. This is stronger than what the well-known formal linear
stability analysis indicates, which would only be slow growth of perturbations,
and not decay of perturbations. This is the first quantification of a mechanism
that can explain the observed robustness of stationary and homogeneous spectra
in the ocean. Finally, numerical investigation indicates that typical real-life
spectra are stable, while if they become appreciably more narrow they would
become unstable, further supporting the plausibility of Landau damping as a
real-life phenomenon taking place in the ocean.