enhanced dissipation to be equivalent to replacing the $\nu$ in front of the $\partial_{xx}$ on the RHS of with $\nu^{1/3}$; it is acting differently for each frequency at each time. Our methods will deal with this effect by designing a semi-norm to measure the solution which respects the frequency-by-frequency anisotropy and allows to gradually pay regularity for enhanced dissipation of the vorticity; see §\[sec:Proof\].

The Orr mechanism is known to interact poorly with the nonlinear term, creating a weakly nonlinear effect referred to as an *echo*. In nearly collisionless plasmas, echoes were captured experimentally in [@MalmbergWharton68] and in 2D Euler much later in [@YuDriscoll02; @YuDriscollONeil] (see also [@VMW98; @Vanneste02]). The echoes can potentially chain into a cascade and greatly amplify the regularity loss already present in the linear theory (see e.g. [@MouhotVillani11; @BM13]), which is the reason that the nonlinear results in [@CagliotiMaffei98; @HwangVelazquez09; @MouhotVillani11; @BM13; @BMM13] all require at least Gevrey class regularity [@Gevrey18] (the work of [@FaouRousset14] is the exception, as the model studied therein does not support infinite echo cascades). For , we will not need the data to be Gevrey class in a qualitative sense (which would be somewhat non-physical for a model involving a dissipative 