A Wavelet Monte Carlo Method for Turbulent Diffusion with Many Spatial Scales

Abstract

A new algorithm is developed here for Monte Carlo simulation of turbulent diffusion with random velocity fields having a power law spatial spectrum, arbitrarily many statistical scales, long range correlations, and infrared divergence. This algorithm uses an expansion of the moving average representation of a Gaussian random field via a specific orthonormal basis which exploits the scaling behavior of the random field and yields a compact representation of the field despite infrared divergence. The orthonormal basis involves a family of wavelets due to Alpert and Rokhlin and the vanishing of a large number of moments for this basis (order 3 or 4) is crucial for compact representation of the statistical velocity field. The authors also develop a rigorous practical a priori energy criterion for truncation of the wavelet expansion; furthermore, for moment cancellations with orders 3 or 4 and a fixed energy tolerance, the number of Gaussian random basis elements needed by the method is sublinear in the exponent m, where 1 < |x| < 2m denotes the range of active velocity scales. The algorithm developed here has remarkably low variance as regards sampling errors. For example, for the infrared divergent velocity spectrum corresponding to the Kolmogoroff spectrum in an exactly solvable model, the velocity structure function is simulated accurately for separation distances over more than five decades with only 792 random basis elements and 100 ensemble realizations; with this same singular spectrum in the model problem, the pair dispersion statistic for the passive scalar can be determined within errors of 3% throughout over five decades of pair separation distance with only 792 random elements, 1000 realizations, and a few hours calculation on a workstation. All of the computational algorithms and results are presented for a nontrivial exactly solvable model and are described within a mathematical framework of error analysis where analytic, sampling, and discretization errors are treated separately.