A Flux Method for the Numerical Solution of the Stochastic Collection Equation: Extension to Two-Dimensional Particle Distributions

In the present paper a new method is introduced for the numerical solution of the stochastic collection equation in cloud models dealing with two-dimensional cloud microphysics. The method is based on the assumption that the probability for the collision of two cloud drops only depends on the water mass of each and not on the mass of the aerosol nuclei. With this assumption it is possible to reduce the two-dimensional solution of the stochastic collection equation to a one-dimensional approach. First, the two-dimensional particle spectrum is integrated over the aerosol mass yielding a one-dimensional drop spectrum in the water mass grid. For this intermediate drop distribution the stochastic collection equation is solved. The resulting new drop spectrum is redistributed into the two-dimensional aerosol-water grid. Numerical sensitivity studies are presented demonstrating that the flux method yields very good results. In the two-dimensional aerosol-water grid the drop distributions move from initially small aerosol and water masses toward larger values whereby the drops remain more or less concentrated along a straight line. In several calculations the numerical diffusivity of the method is investigated. The corresponding results show that in these investigations the artificial broadening of the drop distribution remains tolerably low.