  This paper establishes the nonlinear stability of the Couette flow for the 2D
Boussinesq equations with only vertical dissipation. The Boussinesq equations
concerned here model buoyancy-driven fluids such as atmospheric and
oceanographic flows. Due to the presence of the buoyancy forcing, the energy of
the standard Boussinesq equations could grow in time. It is the enhanced
dissipation created by the linear non-self-adjoint operator $y\partial_x
-\nu\partial_{yy}$ in the perturbation equation that makes the nonlinear
stability possible. When the initial perturbation from the Couette flow $(y,
0)$ is no more than the viscosity to a suitable power (in the Sobolev space
$H^b$ with $b>\frac43$), we prove that the solution of the 2D Boussnesq system
with only vertical dissipation on $\mathbb T\times \mathbb R$ remains close to
the Couette at the same order. A special consequence of this result is the
stability of the Couette for the 2D Navier-Stokes equations with only vertical
dissipation.
