is, the mode depends on $x$), the mode is strongly damped by the viscosity by the time $t \gtrsim \nu^{-1/3}$, whereas for $k = 0$ the evolution is that of the 1D heat equation with a slower $t \gtrsim \nu^{-1}$ dissipative time-scale; hence the $k \neq 0$ modes experience *enhanced dissipation*. For shorter times, the behavior of essentially matches that of the inviscid problem, studied in detail by Orr in [@Orr07]. One of Orr’s key observations was the *inviscid damping* predicted by , that is, even for $\nu = 0$, shows that if the vorticity is sufficiently regular, then the velocity field $U = {\nabla}^\perp \psi$ will return to a shear flow at an algebraic rate. Both inviscid damping and enhanced dissipation are manifestations of the same *vorticity mixing* induced by the background shear flow, which gives rise to a linear-in-time (for each $k \neq 0$) transfer of enstrophy to high frequencies (as seen in ). Together, the effects can be summarized by the following.

\[prop:LinBehave\] Let $\omega(t)$ be a solution to with initial data $\omega_{in} \in H^\sigma$ for any $\sigma \geq 3$ such that $\int \omega_{in} dx dy = 0$ and $\int {\left\verty \omega_{in}\right\vert} dy dx < \infty$. Then 