The spatio-temporal evolution of the probability density function for a passive scalar advected by a rapidly varying random wind

Abstract: We explore the evolution of the Probability Density Function (PDF) for an initially deterministic passive scalar diffusing in the presence of a rapidly fluctuating wind field. For spatially Gaussian intial data, we calculate the exact pdf by two methods, one direct, one indirect appealing to statistical moments. These exact solutions provide an extremely important class of test problems for more general situations where only moments, are partial moment information is available.

For general initial data, exact PDF's are not available, though exact statistical moments are still accessible from the stochastic solution of the problem. Using a proper normalization for this problem, the measure can be proved to be supported upon the compact interval [0,1], and we develop several orthogonal polynomial reconstruction methods for re-summing the measure from its moments. We are most interested in the location of the singularities of the probability density function. We further document the convergence of monte-carlo algorithms to faithfully construct the probability distribution, and in turn use both orthogonal polynomial reconstruction and monte-carlo simulations to study problems where only partial statistical information is available. A fairly complete picture of the spatio-temporal evolution of the probability measure will be presented.