  In this work, we obtain the hydrostatic approximation by taking the small
aspect ratio limit to the Navier-Stokes equations. The aspect ratio (the ratio
of the depth to horizontal width) is a geometrical constraint in general large
scale geophysical motions that the vertical scale is significantly smaller than
horizontal. We use the versatile relative entropy inequality to prove
rigorously the limit from the compressible Navier-Stokes equations to the
compressible Primitive Equations. This is the first work to use relative
entropy inequality for proving hydrostatic approximation and derive the
compressible Primitive Equations.
