Stochastic models for atmospheric particle dispersion

Simple stochastic differential equation models have been applied by several researchers to describe the dispersion of tracer particles in the planetary atmospheric boundary layer and to form the basis for computer simulations of particle paths. To obtain the drift coefficient, empirical vertical velocity distributions that depend on height above the ground both with respect to standard deviation and skewness are substituted into the stationary Fokker-Planck equation. The particle position distribution is taken to be uniform (the well-mixed condition) and also a given dispersion coefficient variation by height is adopted. A particular problem for simulation studies with finite time steps is the construction of a reflection rule different from the rule of perfect reflection at the boundaries such that the rule complies with the imposed skewness of the velocity distribution for particle positions close to the boundaries. Different rules have been suggested in the literature with justifications based on simulation studies. Herein the relevant stochastic differential equation model is formulated in a particular way. The formulation is based on the marginal transformation of the position dependent particle velocity into a position independent Gaussian ”velocity”. Boundary conditions are obtained from Itô’s rule of stochastic differentiation. It is observed that the model formulation based on the stationary FP equation is ambiguous. Another result is that the model directly point at a canonical rule of reflection for the approximating random walk with finite time step. This reflection rule is different from those published in the literature.