Fluctuation dynamics in randomly advected Navier-Stokes equations below critical scaling

We study randomly advected incompressible Navier-Stokes equations, where the advecting field is a mean-zero, divergence-free, space-time stationary velocity field with smooth order-one correlations.

We introduce a two-parameter family of models in which the advection is accelerated on a fast temporal scale $\varepsilon^2$ and has spatial correlation length $\delta$; the critical regime $\varepsilon = \delta$ corresponds to the natural parabolic scaling of the Navier-Stokes equation.

In the full subcritical regime $\varepsilon = o (\delta)$, we prove a law of large numbers in dimensions $d = 2, 3$: the solutions converge to a deterministic Navier--Stokes system with an enhanced diffusion coefficient given by a Green-Kubo formula.

In two space dimensions, under the slightly stronger assumption $\varepsilon = o (\delta^{1 + \iota})$ for some $\iota > 0$, we identify the leading-order fluctuations: after subtracting deterministic macroscopic corrections satisfying a nonlinear system of Navier-Stokes type, the rescaled fluctuations converge to a Gaussian field solving a linearized Navier-Stokes equation driven by multiplicative space-time white noise.