On the hydrostatic approximation of the 3D Boussinesq equations of damped wave type

We study the hydrostatic approximation for the three-dimensional Boussinesq equations of damped wave type. This is a mixed degenerate system coupled by parabolic and hyperbolic equations. Compared with the purely hyperbolic hydrostatic Navier-Stokes equations, the parabolic equation for temperature will lead to an extra loss of derivatives. In the setting of Gevrey space with index 7/4, we prove the local well-posedness and the corresponding hydrostatic limit for the 3D Boussinesq equations of damped wave type.