Fractal trajectories and anomalous diffusion for chaotic particle motions in 2D turbulence

We consider a simple, nonlinear Hamiltonian model for particle motions in two-dimensional turbulence in which the stream function power spectrum has the form k^-γ ({γ} constant). We show that fluid particle trajectories, (1) while differentiable and chaotic at small scale, may be fractal space curves at larger scale, (2) undergo anomalous transport, and (3) have frequency spectra {~}f^-α in fully stochastic limit, where we obtain {α}={α}({γ}), a scaling law. Anomalous diffusion is shown to arise partly as a result of the fractal nature of the particle trajectories.