  We investigate the long-time properties of the two-dimensional inviscid
Boussinesq equations near a stably stratified Couette flow, for an initial
Gevrey perturbation of size $\varepsilon$. Under the classical Miles-Howard
stability condition on the Richardson number, we prove that the system
experiences a shear-buoyancy instability: the density variation and velocity
undergo an $O(t^{-1/2})$ inviscid damping while the vorticity and density
gradient grow as $O(t^{1/2})$. The result holds at least until the natural,
nonlinear timescale $t \approx \varepsilon^{-2}$. Notice that the density
behaves very differently from a passive scalar, as can be seen from the
inviscid damping and slower gradient growth. The proof relies on several
ingredients: (A) a suitable symmetrization that makes the linear terms amenable
to energy methods and takes into account the classical Miles-Howard spectral
stability condition; (B) a variation of the Fourier time-dependent energy
method introduced for the inviscid, homogeneous Couette flow problem developed
on a toy model adapted to the Boussinesq equations, i.e. tracking the potential
nonlinear echo chains in the symmetrized variables despite the vorticity
growth.
