A turbulent transport model: Streamline results for a class of random velocity fields in the plane

Probabilistic methods and computer simulation are used to analyze streamline properties of flows defined by a general class of random, incompressible velocity fields. Such fields are stochastically modeled by a superposition of simple shear-flow layers. The resulting flow is governed by a nonstationary, random Hamiltonian with Hurst exponent H = 0.5 and having the form H₀ = Σ^N_k=1 b_k-W_k(n_k · r) for constants b_k, where the W_k are two-sided continuous-time random walks. The statistical topography of the flow, as characterized by the level sets or streamlines of H₀, is analyzed via an associated Brownian walk space from which asymptotic results are determined. The flow consists of a hierarchy of nested, closed streamlines. For a given box of side length 2L centered at the particle's starting position, an upper bound on the particle's probability of exit, or "noncycling" probability, p_nc, is shown to have a power law dependence on the box size, p_nc(L) ∼ L^-α, for L ≫ 1 and positive constant α. We also introduce a constant, nonzero mean flow and denote its relative strength with respect to the r.m.s. fluctuations of the random field by ρ. In the case 0 < ρ < 1, the fraction p_nc(ρ) of "percolating" or noncycling particles (in an infinitely large box) satisfies the relation ρ/ρ + 1 ≤ p_nc (ρ) < 1. All particles percolate in the case ρ ≥ 1. Computer simulations for various values of N agree well with earlier work on the N = 2 case by Avellaneda, Elliott, and Apelian, thereby confirming and validating both studies. Numerical results also show the power law exponent α to be remarkably robust with respect to changes in topology, including the existence of traps, irregularly spaced modes, and the value of N. All runs yield a common value of α ≈ 0.22. Likewise, the mean length of streamlines exiting boxes of size 2L, 〈λ(L)〉, scales like L^γ with γ ≈ 1.28 for all N. These exponent values contrast with those predicted by Isichenko and Kalda yet consistently satisfy a "sum rule," α + γ = 2 - H, relating α, γ, and H, the Hurst exponent of the flow.