[@BM13] and *concentrate only on the steps that are specific to Navier-Stokes.* The proof of Theorem \[thm:Main\] is outlined in §\[sec:Proof\], however let us quickly highlight the main ideas and the main challenges.

-   The coordinate system used in [@BM13] needs to be adapted to account for the viscous dissipation in the background shear flow (see below). In the Lagrangian view, we are accounting for the stochastic wandering of particle trajectories in the $y$ variable. In the Eulerian view, we are accounting for the momentum transport induced by viscosity. Note that this is physically relevant even for short times, as the viscous dissipation always dominates for high enough frequencies. Like the choice of coordinates in [@BM13], the choice used here is determined essentially uniquely by the analytic properties we require.

-   As can be seen from , the enhanced dissipation effect is highly anisotropic in time and frequency: it is only active for $x$-dependent frequencies and slows down near the critical time $t \sim \eta/k$. To correctly capture the enhanced dissipation we use a special semi-norm that is well adapted to the anisotropy (defined in and analyzed in §\[sec:Anu\]). The anisotropy also creates a huge imbalance between $x$-dependent modes and 