Impact of diffusion on surface clustering in random hydrodynamic flows

Buoyant material clustering in a stochastic flow, which is homogeneous and isotropic in space and stationary in time, is addressed. The dynamics of buoyant material in three-dimensional hydrodynamic flows can be considered as the motion of passive tracers in a compressible two-dimensional velocity field. The latter is of interest in the present study. It is well known that the clustering of the density of passive tracers occurs in this case. We evaluate the impact of diffusion on the clustering process by using a numerical model. In general, the effect of diffusion is negligible in the very beginning of the evolution of initially uniformly distributed passive tracers. Therefore, the clustering of the density of passive tracers can emerge in accordance with the general theory. We analyze the long time clustering affected by diffusion and show that the emerged cluster structure persists in time in spite of the diffusion effect. However, the clusters split in time.