Two-dimensional turbulence and dispersion in a freely decaying system

We report experimental results obtained on freely decaying two-dimensional turbulence. The flow is produced in a thin stratified layer of electrolyte, using an electromagnetic forcing. The velocity and vorticity fields are measured using a particle image velocimetry technique. The study of the temporal evolution of the system confirms in detail the scaling theory of Carnevale et al. [Phys. Rev. Lett. 66, 2735 (1991)]; the experimental value we find for the exponent characterizing the decay of the vortex density is $\xi =0.7\mathrm{}\pm \mathrm{0.1.}$ We further measure the collision time $\tau ,$ the mean free path $\lambda ,$ and the mean square displacement ${\sigma }_v^2$ of the vortices. We find the following laws: $\tau \sim t^{0.57},\hspace{0.5em}\lambda \sim t^{0.45},$ and ${\sigma }_v^2\sim t^{1.3}.$ The statistics of passive particles (albeit virtual) in the system is also studied. They move hyperdiffusively, with an exponent similar to that obtained for the vortex motion. The dispersion of the particles is controlled by Lévy flights, produced by the jets formed by the dipoles. The distribution of flight times $t_f$ is $t_f^{-2.6}.$ Further analysis of the data indicates that the vortices undergo collisions whose geometrical aspects are analogous to those of an ordinary gas, and their motion is essentially Brownian diffusion in an expanding geometry. We finally underline the close relationship between the decay of turbulence and the dispersion phenomena.