  An important feature of the planetary oceanic dynamics is that the aspect
ratio (the ratio of the depth to horizontal width) is very small. As a result,
the hydrostatic approximation (balance), derived by performing the formal small
aspect ratio limit to the Navier-Stokes equations, is considered as a
fundamental component in the primitive equations of the large-scale ocean. In
this paper, we justify rigorously the small aspect ratio limit of the
Navier-Stokes equations to the primitive equations. Specifically, we prove that
the Navier-Stokes equations, after being scaled appropriately by the small
aspect ratio parameter of the physical domain, converge strongly to the
primitive equations, globally and uniformly in time, and the convergence rate
is of the same order as the aspect ratio parameter. This result validates the
hydrostatic approximation for the large-scale oceanic dynamics. Notably, only
the weak convergence of this small aspect ratio limit was rigorously justified
before.
