  The Boussinesq system for buoyancy driven fluids couples the momentum
equation forced by the buoyancy with the convection-diffusion equation for the
temperature. One fundamental issue on the Boussinesq system is the stability
problem on perturbations near the hydrostatic balance. This problem can be
extremely difficult when the system lacks full dissipation. This paper solves
the stability problem for a two-dimensional Boussinesq system with only
vertical dissipation and horizontal thermal diffusion. We establish the
stability for the nonlinear system and derive precise large-time behavior for
the linearized system. The results presented in this paper reveal a remarkable
phenomenon for buoyancy driven fluids. That is, the temperature actually
smooths and stabilizes the fluids. If the temperature were not present, the
fluid is governed by the 2D Navier-Stokes with only vertical dissipation and
its stability remains open. It is the coupling and interaction between the
temperature and the velocity in the Boussinesq system that makes the stability
problem studied here possible. Mathematically the system can be reduced to
degenerate and damped wave equations that fuel the stabilization.
