Chaotic dynamics of super‐diffusion revisited

Abstract

[1] Super-diffusive mixing in geophysics occurs in atmospheric turbulence, near surface currents in the oceans, and fracture flow in the subsurface to name a few examples. Models of super-diffusion have been around since L. F. Richardson's pioneering work in the 1920's. Here we construct a family of super-diffusive stochastic processes Xα(t), 0 < α, with independent, nonstationary increments, but a priori defined mean-square displacement given by tα+1. The case α = 2 corresponds to Richardson super-diffusion. The family of processes is a nonstationary extension of Brownian motion and hence completely characterized by its first two moments. The Fokker-Planck equation for Xα(t) is classical diffusive with time dependent diffusion coefficient given by tα/2. In contrast to what is found in fractional Brownian motion, where the fractal dimension depends on the Hurst exponent, here the fractal dimension and hence complexity of the process is the same for all exponents α as that of classical Brownian motion. An analytical expression is developed for the finite-size Lyapunov exponent and numerical examples presented.