  The primitive equations are derived from the $3D$-Navier-Stokes equations by
the hydrostatic approximation. Formally, assuming an $\varepsilon$-thin domain
and anisotropic viscosities with vertical viscosity
$\nu_z=\mathcal{O}(\varepsilon^\gamma)$ where $\gamma=2$, one obtains the
primitive equations with full viscosity as $\varepsilon\to 0$. Here, we take
two more limit equations into consideration: For $\gamma<2$ the
$2D$-Navier-Stokes equations are obtained. For $\gamma>2$ the primitive
equations with only horizontal viscosity $-\Delta_H$ as $\varepsilon\to 0$.
Thus, there are three possible limits of the hydrostatic approximation
depending on the assumption on the vertical viscosity. The latter convergence
has been proven recently by Li, Titi, and Yuan using energy estimates. Here, we
consider more generally $\nu_z=\varepsilon^2 \delta$ and show how maximal
regularity methods and quadratic inequalities can be an efficient approach to
the same end for $\varepsilon,\delta\to 0$. The flexibility of our methods is
also illustrated by the convergence for $\delta\to \infty$ and $\varepsilon\to
0$ to the $2D$-Navier-Stokes equations.
