Stability of the slow manifold in the primitive equations

TL;DR

This paper demonstrates that solutions to the forced-dissipative rotating primitive equations of the ocean tend to a slow manifold, exhibiting geostrophic balance and losing fast inertia-gravity components over time, especially under smooth, time-independent forcing.

Contribution

It establishes the stability and exponential approximation of solutions to a finite-dimensional slow manifold in the primitive equations under certain regularity conditions.

Findings

Solutions approach geostrophic balance as time progresses.

Fast inertia-gravity components diminish in the small Rossby number limit.

Solutions become exponentially close to a finite-dimensional slow manifold with smooth forcing.

Abstract

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.