  We report on an instability arising when surface gravity waves propagate in a
rotating frame. The Stokes drift associated to the uniform wave field, together
with global rotation, drives a mean flow in the form of a horizontally
invariant Ekman-Stokes spiral. We show that the latter can be subject to an
instability that triggers the appearance of an additional
horizontally-structured cellular flow. We determine the instability threshold
numerically, in terms of the Rossby number Ro associated to the Stokes drift of
the waves and the Ekman number E. We confirm the numerical results through
asymptotic expansions at both large and low Ekman number. At large E the
instability reduces to that of a standard Ekman spiral driven by the
wave-induced surface stress instead of a wind stress, while at low E the
Stokes-drift profile crucially determines the shape of the unstable mode. In
both limits the instability threshold asymptotes to an Ekman-number-independent
critical Rossby number, which in both cases also corresponds to a critical
Reynolds number associated to the Lagrangian base-flow velocity profile.
Parameter values typical of ocean swell fall into the low-E unstable regime:
the corresponding "anti-Stokes" flows are unstable, with possible consequences
for particle dispersion and mixing.
