Statistically self-similar mixing by 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 ${\mathbb{R}}^d$. 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 $\mathbb{E}\mathrm{\parallel }{\theta }_t{\mathrm{\parallel }}_{{\dot{H}}^{-s}}^2={\mathrm{e}}^{-{\lambda }_{d,s}t}\mathrm{\parallel }{\theta }_0{\mathrm{\parallel }}_{{\dot{H}}^{-s}}^2$ with any $s\in (0,d\mathrm{/}2)$ and $\frac{{\lambda }_{d,s}}{D_1}:=s(\frac{{\lambda }_1}{D_1}-2s)$ where ${\lambda }_1\mathrm{/}{D}_1=d$ 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 $\mathit{\text{uniformly}}$ in diffusivity.