is mean-zero and satisfies $\omega_{in}^{\nu} = \omega^\nu_{S,in} + \omega^\nu_{R,in}$ with $$\begin{aligned}
{\left\lVert |\nabla|^{-1} \omega^\nu_{in} \right\rVert}_{2}   + {\left\lVert \omega^\nu_{S,in} \right\rVert}_{{\mathcal{G}}^{\lambda;s}} + e^{K_0\nu^{-\frac{3(1+\delta)s}{2(1-s)}}}  {\left\lVert \omega^\nu_{R,in} \right\rVert}_{2} = \epsilon \leq \epsilon_0, \label{ineq:IDassump}\end{aligned}$$ then for all $\nu$ sufficiently small (independent of $\epsilon$ and $\epsilon_0$) the solution $\omega^{\nu}(t)$ with initial data $\omega_{in}^\nu$ satisfies the following properties (with all implicit constants independent of $\nu$, $t$ and $\epsilon$),

-   the uniform (in $t$ and $\nu$) estimate $$\label{main-omega} 
    {\left\lVert \omega^\nu(t,x + ty + \Phi(t,y),y) \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} \lesssim \epsilon$$ where $\Phi(t,y)$ is given explicitly by $$\begin{aligned}
    \Phi(t,y) = \int_0^t e^{\nu (t-\tau)\partial_{yy}} \left(\frac{1}{2\pi}\int_{\mathbb T}U^x(\tau,x,y) dx\right) d\tau;  \label{def:phi}\end{aligned}$$

-   inviscid damping on inviscid time-scales: for $t \lesssim \nu^{-1/3}$,

    \[main-omega-inv\] $$\begin{aligned}
    {\left\lVert \frac{d}{dt}\left[\omega^\nu(t,x + ty + \Phi(t,y),y)\right] \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} & \lesssim \frac{\epsilon^2}{{\left\langle t \right\rangle}^2} + \epsilon \nu {\left\langle t \right\rangle}^2, \label{ineq:scattering} \\
    {\left\lVert P_{\neq 0}U^x(t) \right\rVert}_2 + {\left\langle t \right\rangle}{\left\lVert U^y(t) \right\rVert}_{2} & \lesssim \frac{\epsilon}{{\left\langle t \right\rangle}}, \label{ineq:ID1} \\ 
    {\left\lVert \frac{d}{dt}P_{0}U^x(t) \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} & \lesssim \frac{\epsilon^2}{{\left\langle t \right\rangle}^3} + \epsilon \nu \label{ineq:P0Ux};\end{aligned}$$

-   enhanced dissipation for $x$-dependent modes on the fast viscous time-scale: for $t \gtrsim \nu^{-1/3}$,

    $$\begin{aligned}
    {\left\lVert P_{\neq 0}\omega^\nu(t,x + ty + \Phi(t,y),y) \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^\alpha}, \label{ineq:thmED} \\ 
    {\left\lVert P_{\neq 0}U^x(t) \right\rVert}_2 + {\left\langle t \right\rangle}{\left\lVert U^y(t) \right\rVert}_{2} & 