  We show that, under reasonably mild hypotheses, the solution of the
forced--dissipative rotating primitive equations of the ocean loses most of its
fast, inertia--gravity, component in the small Rossby number limit as
$t\to\infty$. At leading order, the solution approaches what is known as
"geostrophic balance" even under ageostrophic, slowly time-dependent forcing.
Higher-order results can be obtained if one further assumes that the forcing is
time-independent and sufficiently smooth. If the forcing lies in some Gevrey
space, the solution will be exponentially close to a finite-dimensional "slow
manifold" after some time.
