On stability estimates for the inviscid Boussinesq equations

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 $\epsilon^{-4}$.