Linear inviscid damping for stably stratified Boussinesq flows

We study the linear asymptotic stability of stably stratified monotone shear flows for the Boussinesq equations in the periodic channel. By means of the limiting absorption principle, we obtain a precise description of the inviscid damping experienced by the perturbed velocity field and density, with time-decay rates that depend on the local Richardson number $\mathcal{J}(y)$ and split into four stratification regimes (non-stratified, weak, mild, and strong) reflecting qualitative changes in the structure of the Green’s function at the critical thresholds $\mathcal{J}(y)=0$ and $\mathcal{J}(y)=\frac{1}{4}$. The velocity and density decay estimates are later used to prove quantitative sub-linear growth of the vorticity and gradient of density. As a byproduct of our analysis, we show that, under mild hypotheses on the underlying shear-type equilibrium, the spectrum of the linearised Boussinesq operator is purely continuous.