  We consider the (in)stability problem of the inviscid 2D Boussinesq equations
near a combination of a shear flow $v=(y,0)$ and a stratified temperature
$\theta=\alpha y$ with $\alpha>\frac{1}{4}$. We show that for any $\epsilon>0$
there exist non-trivial explicit solutions, which are initially perturbations
of size $\epsilon$, and grow to size $1$ on a time scale $\epsilon^{-2}$.
Moreover, the (simplified) linearized problem around these non-trivial states
exhibits improved upper bounds on the possible size of norm inflation for
frequencies larger and smaller than $\epsilon^{-4}$.
