Stochastic trajectory models for turbulent diffusion: Monte Carlo process versus Markov chains

Abstract

Abstract Turbulent diffusion of passive scalars and particles is often simulated with either a Monte Carlo process or a Markov chain. Knowledge of the velocity correlation generated by either of these stochastic trajectory models is essential to their application. The velocity correlation for Monte Carlo process and Markov chain was studied analytically and numerically. A general relationship was developed between the Lagrangian velocity correlation and the probability density function for the time steps in a Monte Carlo process. The velocity correlation was found to be independent of the fluid velocity probability density function, but to be related to the time-step probability density function. For a Monte Carlo process with a constant time step, the velocity correlation is a triangle function; and the integral time scale is equal to one-half of the time-step length. When the time step was chosen randomly with an exponential pdf distribution, the resulting velocity correlation was an exponential function. Other time-step probability density functions, such as a uniform distribution and a half-Gaussian distribution, were also tested. A Markov chain, which presumes one-step memory, has a piecewise linear velocity correlation function with a finite time step. For a Markov chain with a short time step, only an exponential velocity correlation function can be realized. Thus, a Monte Carlo process with random time steps is more versatile than a Markov chain. Direct numerical calculation of the velocity correlation verified the analytical results. A new model which combines the ideas of the Monte Carlo process and the Markov chain was developed. By examining the long-time mean square dispersion, we found an exact solution for the Lagrangian integral time scale of the new model in terms of the intercorrelation parameter and the mean and the variance of the time steps. Using this new model, we can generate any velocity correlation, including one with a negative tail. Two approximate solutions that give upper and lower bounds for the Lagrangian velocity correlation are proposed.