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We study the mixing and dissipation properties of the advection-diffusionequation with diffusivity $0 < \kappa \ll 1$ and advection by a class of randomvelocity fields on $\mathbb T^d$, $d=\{2,3\}$, including solutions of the 2DNavier-Stokes equations forced by sufficiently regular-in-space, non-degeneratewhite-in-time noise. We prove that the solution almost surely mixesexponentially fast uniformly in the diffusivity $\kappa$. Namely, that there isa deterministic, exponential rate (independent of $\kappa$) such that allmean-zero $H^1$ initial data decays exponentially fast in $H^{-1}$ at this ratewith probability one. This implies almost-sure enhanced dissipation in $L^2$.Specifically that there is a deterministic, uniform-in-$\kappa$, exponentialdecay in $L^2$ after time $t \gtrsim |\log \kappa|$. Both the $O(|\log\kappa|)$ time-scale and the uniform-in-$\kappa$ exponential mixing are optimalfor Lipschitz velocity fields and, to our knowledge, are the first rigorousexamples of velocity fields satisfying these properties (deterministic orstochastic). This work is also a major step in our program on scalar mixing andLagrangian chaos necessary for a rigorous proof of the Batchelor power spectrumof passive scalar turbulence.

Almost-sure enhanced dissipation and uniform-in-diffusivity exponential mixing for advection-diffusion by stochastic Navier-Stokes