Dynamics of two-dimensional turbulence

Abstract

Mixing of Lagrangian particles by a vortex singularity is shown to correspond to a continuous curdling process with the Lipschitz–Hölder exponent equal to (4)/(3) , demonstrating that Richardson’s l4/3 scaling is derivable from the two-dimensional kinematics of the vortex singularity. Scaling relationships describing the Lagrangian geometry of two-dimensional turbulence are linked to the rate of enhanced diffusion. The value of the two-dimensional intermittency exponent μ= (2)/(3) , derived from the author’s Lagrangian structure function theory [Phys. Fluids A 1, 1836 (1989)], when used in the Lévy flight theory of enhanced diffusion developed by Shlesinger et al. [Phys. Rev. Lett. 58, 1100 (1987)] predicts 〈R2〉∝t2.6 for the time rate of separation of labeled pairs of vortices in two dimensions. This is found to be in good agreement with the rate of dispersion measured in numerical simulations of inviscid two-dimensional turbulence. Kinetic theory arguments, supplemented by the previously derived Lagrangian structure relationships, yield an asymptotic P∝s−5/3 probability distribution function for path lengths s between vortex pair collisions. This result is consistent with path length distributions obtained from the two-dimensional simulations.