\lesssim \frac{\epsilon}{{\left\langle t \right\rangle}{\left\langle \nu t^3 \right\rangle}^\alpha}; \label{ineq:ID2}.\end{aligned}$$

-   and standard viscous decay for $x$-independent modes on the slow viscous time-scale: for $t \gtrsim \nu^{-1}$, $$\begin{aligned}
    {\left\lVert P_{0}\omega^\nu(t) \right\rVert}_2 \lesssim \frac{\epsilon}{{\left\langle \nu t \right\rangle}^{1/4}}.  \label{ineq:thmSD}\end{aligned}$$

It is useful to compare Theorem \[thm:Main\] with the linear behavior in Proposition \[prop:LinBehave\]. The qualitative behavior predicted by the nonlinear theorem is essentially the same except for two details: the algebraic rate of the enhanced dissipation and the correction to the shear flow characteristics $\Phi(t,y)$. The correction $\Phi$ is to account for the fact that the perturbation induces a small shear, $<U^x(t)>$, which varies in time and decays at a non-integrable rate on the very long $O(\textbf{Re})$ time scale (in particular, in the $\nu = 0$ case the flow never returns to Couette flow). Therefore, gradients would grow without bound in if we did not account for this. The algebraic rate of decay arises since, due to un-mixing, we pay regularity in return for enhanced dissipation. The current proof pays $3\alpha$ derivatives; with some additional technical effort it seems we could trade Gevrey-2 regularity (or Gevrey-$\frac{1}{\beta}$ for any $\beta < s$) and likely get a decay like $\exp\left[-(\nu t^3)^{1/6} \right]$. However, getting the decay 