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 $t\to \mathrm{\infty }$. 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.