Relative dispersion in two-dimensional turbulence

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Abstract
In this paper we study the statistical laws of relative dispersion in two-dimensional turbulence by deriving an exact equation governing its evolution in time, then evaluating the magnitude of its various terms in numerical experiments, which allows us to check the validity of the classical dispersion laws: the equivalent to the Richardson-Obukhov t3 law in the energy cascade range, and the Kraichnan-Lin exponential law in the enstrophy cascade range. We examine theoretically and experimentally the conditions of validity of both laws. It is found that the t3 law is obtained in the energy inertial range provided the separation scale of the particles is smaller by an order of magnitude than the injection scale. When the t3 law is reached, the relative acceleration correlations are observed to have reached a statistical quasistationary stage: this would tend to justify in the energy inertial range of two-dimensional turbulence a working hypothesis formulated by Lin & Reid (1963); also, the necessity of starting from very small initial separations to get the t3 law may be explained by the time necessary for relative acceleration correlations to reach the statistical quasi-stationary regime. On the other hand, the Kraichnan-Lin exponential law is, strictly speaking, never observed; it is in fact reduced to a very short transient stage when the relative dispersion characteristic time reaches its minimum value, as predicted by Batchelor.