Dispersion of passive tracers in closed basins: beyond the diffusion coefficient

We investigate the spreading of passive tracers in closed basins. If the characteristic length scale of the Eulerian velocities is not very small compared with the size of the basin the usual diffusion coefficient does not give any relevant information about the mechanism of spreading. We introduce a finite size characteristic time $\tau(\delta)$ which describes the diffusive process at scale $\delta$. When $\delta$ is small compared with the typical length of the velocity field one has $\tau(\delta) \sim \lambda^{-1}$, where $\lambda$ is the maximum Lyapunov exponent of the Lagrangian motion. At large $\delta$ the behavior of $\tau(\delta)$ depends on the details of the system, in particular the presence of boundaries, and in this limit we have found a universal behavior for a large class of system under rather general hypothesis. The method of working at fixed scale $\delta$ makes more physical sense than the traditional way of looking at the relative diffusion at fixed delay times. This technique is displayed in a series of numerical experiments in simple flows.