Trapping, percolation, and anomalous diffusion of particles in a two-dimensional random field

We analyze from first principles the advection of particles in a velocity field with Hamiltonian H(x, y)=- V₁y-- V₂x+W₁(y)-W₂(x), where W_i, i=1, 2, are random functions with stationary, independent increments. In the absence of molecular diffusion, the particle dynamics are very sensitive to the streamline topology, which depends on the mean-to-fluctuations ratio ρ=max(|-V₁|/Ū; |-V₂|/Ū), with Ū=〈|W₁′|²〉^1/2=rms fluctuations. Remarkably, the model is exactly solvable for ρ ≥1 and well suited for Monte Carlo simulations for all ρ, providing a nice setting for studying seminumerically the influence of streamline topology on large-scale transport. First, we consider the statistics of streamlines for ρ=0, deriving power laws for p_nc(L) and 〈λ(L)〉, which are, respectively, the escape probability and the length of escaping trajectories for a box of size L, L » 1. We also obtain a characterization of the "statistical topography" of the Hamiltonian H. Second, we study the large-scale transport of advected particles with ρ > 0. For 0 <ρ < 1, a fraction of particles is trapped in closed field lines and another fraction undergoes unbounded motions; while for ρ≥ 1 all particles evolve in open streamlines. The fluctuations of the free particle positions about their mean is studied in terms of the normalized variables t^-v/2[x(t)-〈x(t)〉] and t^-v/2[y(t)-〈(t)〉]. The large-scale motions are shown to be either Fickian (ν=1), or superdiffusive (ν=3/2) with a non-Gaussian coarse-grained probability, according to the direction of the mean velocity relative to the underlying lattice. These results are obtained analytically for ρ ≥ 1 and extended to the regime 0<ρ<1 by Monte Carlo simulations. Moreover, we show that the effective diffusivity blows up for resonant values of {Mathematical expression}) for which stagnation regions in the flow exist. We compare the results with existing predictions on the topology of streamlines based on percolation theory, as well as with mean-field calculations of effective diffusivities. The simulations are carried out with a CM 200 massively parallel computer with 8192 SIMD processors.