On echo chains in the linearized Boussinesq equations around traveling waves

We consider the 2D Boussinesq equations with viscous but without thermal dissipation and observe that in any neighborhood of Couette flow and hydrostatic balance (with respect to local norms) there are time-dependent traveling wave solutions of the form $\omega=-1+f(t)\cos(x-ty)$, $\theta=\alpha y + g(t)\sin(x-ty)$. As our main result we show that the linearized equations around these waves for $\alpha=0$ exhibit echo chains and norm inflation despite viscous dissipation of the velocity. Furthermore, we construct initial data in a critical Gevrey 3 class, for which temperature and vorticity diverge to infinity in Sobolev regularity as $t\rightarrow \infty$ but for which the velocity still converges.