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 ? = ay with a > 1/4. We show that for any ? > 0 there exist non-trivial explicit solutions, which are initially perturbations of size ?, and grow to size 1 on a time scale ?(-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 ?(-4).