Mixing by Statistically Self-similar Gaussian Random Fields

We study the passive transport of a scalar field by a spatially smooth but white-in-time incompressible Gaussian random velocity field on ℝ^d . If the velocity field u is homogeneous, isotropic, and statistically self-similar, we derive an exact formula which captures non-diffusive mixing. For zero diffusivity, the formula takes the shape of 𝔼 ‖θ _t ‖ _Ḣ^-s^2 = e^-λ _d,s t‖θ _0 ‖ _Ḣ^-s^2 with any s∈ (0,d/2) and λ _d,s/D_1:= s(λ _1/D_1-2s) where λ _1/D_1 = d is the top Lyapunov exponent associated to the random Lagrangian flow generated by u and D_1 is small-scale shear rate of the velocity. Moreover, the mixing is shown to hold uniformly in diffusivity.