  We show that in the limit of small Rossby number $\eps$, the primitive
equations of the ocean (OPEs) can be approximated by ``higher-order
quasi-geostrophic equations'' up to an exponential accuracy in $\eps$. This
approximation assumes well-prepared initial data and is valid for a timescale
of order one (independent of $\eps$). Our construction uses Gevrey regularity
of the OPEs and a classical method to bound errors in higher-order perturbation
theory.
