Stability of the slow manifold in the primitive equations

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.