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 Rd. 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‖θt‖H˙-s2=e-λd,st‖θ0‖H˙-s2 with any s∈(0,d/2) and λd,sD1:=s(λ1D1-2s) where λ1/D1=d is the top Lyapunov exponent associated to the random Lagrangian flow generated by u and D1 is small-scale shear rate of the velocity. Moreover, the mixing is shown to hold uniformly in diffusivity.