  We investigate the hydrostatic approximation for inviscid stratified fluids,
described by the two-dimensional Euler-Boussinesq equations in a periodic
channel. Through a perturbative analysis of the hydrostatic homogeneous
setting, we exhibit a stratified steady state violating the Miles-Howard
criterion and generating a growing mode, both for the linearized hydrostatic
and non-hydrostatic equations. By leveraging long-wave nonlinear instability
for the original Euler-Boussinesq system, we demonstrate the breakdown of the
hydrostatic limit around such unstable profiles. Finally, we establish the
generic nonlinear ill-posedness of the limiting hydrostatic system in Sobolev
spaces.
