Transition to self-similarity of diffusion of tracer in turbulent patch

A mixing of a passive tracer inside a turbulent patch generated by a locali zed short-time perturbation is studied numerically and analytically. Two ki nds of an initial distribution of a tracer are considered: two-layer and co ntinuous with constant gradient. For the turbulent patch shaped as a layer, it is shown that, regardless of details of initial distributions of a turb ulent energy and dissipation, a tracer concentration evolves to self-simila r regimes as time elapses. Analytical self-similar solutions to turbulent d iffusion equations are found for three symmetric shapes of a turbulent patc h: layer, cylinder, and sphere. Distributions of the concentration inside a patch are found to be substantially nonuniform, with a typical ratio of a concentration gradient in the middle of a patch to its initial value of abo ut 0.5.