  In this article we consider the stability and damping problem for the 2D
Boussinesq equations with partial dissipation near a two parameter family of
stationary solutions which includes Couette flow and hydrostatic balance.
  In the first part we show that for the linearized problem in an infinite
periodic channel the evolution is asymptotically stable if any diffusion
coefficient is non-zero. In particular, this imposes weaker conditions than for
example vertical diffusion. Furthermore, we study the interaction of shear
flow, hydrostatic balance and partial dissipation.
  In a second part we adapt the methods used by Bedrossian, Vicol and Wang in
the Navier-Stokes problem and combine them with cancellation properties of the
Boussinesq equations to establish small data stability and enhanced dissipation
results for the nonlinear Boussinesq problem with full dissipation.
