$x$-independent modes, and an important part of the analysis in §\[sec:ED\] is to prove that very little enstrophy from the $x$-independent modes can be converted to $x$-dependent enstrophy at long times.

-   The evolving background shear flow determines the rate at which dissipation occurs locally in $y$, a fully nonlinear effect as the shear flow is determined by the solution. Controlling this effect will be the main challenge to proving Theorem \[thm:Main\] and requires a careful analysis of the interplay between the regularity and decay of the background shear flow with the regularity and decay of the $x$-dependent modes (indeed, the structure of the proof is essentially dictated by this balance; see §\[sec:Proof\]).

Notation and conventions {#sec:Notation}
------------------------

See §\[Apx:LPProduct\] for the Fourier analysis conventions we are taking. A convention we generally use is to denote the discrete $x$ (or $z$) frequencies as subscripts. By convention we always use Greek letters such as $\eta$ and $\xi$ to denote frequencies in the $y$ or $v$ direction and lowercase Latin characters commonly used as indices such as $k$ and $l$ to denote frequencies in the $x$ or $z$ direction (which are discrete). Another convention we use is to denote dyadic integers by 