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

On echo chains in the linearized Boussinesq equations around traveling waves