and a common theme in [@MouhotVillani11; @BM13; @BMM13]. Note further that this implies that in general, if one does not pull back by the shear flow we have ${\left\lVert \omega(t) \right\rVert}_{H^\sigma} \approx {\left\langle t \right\rangle}^\sigma$ for $\nu t^3 \ll 1$. Thirdly, notice that the $x$-dependent contribution to the vorticity decays rapidly in after $\nu t^3 \gg 1$ whereas the $x$-independent contribution takes much longer to decay. Finally, the reason that $c < 1/3$ in the above theorem is due to the fact that the enhanced dissipation is anisotropic in time and frequency, as can be seen in the degeneracy of the exponentials in . Like the loss of regularity, this is also due to a transient un-mixing effect.

In our work we study the long time, inviscid limit of ‘small’ solutions to and seek to find the appropriate nonlinear analogue of Proposition \[prop:LinBehave\]. The primary motivation of our work is to study the enhanced dissipation by mixing exhibited in . A second, deeply related, motivation for our work is to affirm the physical relevance of the inviscid ($\nu = 0$) work completed by two of the authors in [@BM13], where inviscid damping is exhibited in the Euler equations.

The enhanced 