Dispersion of passive tracers in a velocity field with non-δ-correlated noise

The diffusive properties in velocity fields whose small scales are parameterized by non-δ-correlated noise is investigated using multiscale technique. The analytical expression of the eddy diffusivity tensor is found for a two-dimensional (2D) steady shear flow and it is an increasing function of the characteristic noise decorrelation time τ. In order to study a generic flow v, a small-τ expansion is performed and the first correction O(τ) to the effective diffusion coefficients is evaluated. This is done using two different approaches and it results that at the order τ the problem with a colored noise is equivalent to the δ-correlated case provided by a renormalization of the velocity field v↦ṽ depending on τ. Two examples of 2D closed-streamlines velocity field are considered and in both the cases an enhancement of the diffusion is found.