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-dime nsional 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 assumptio n it is possible to reduce the two-dimensional solution of the stochastic c ollection equation to a one-dimensional approach. First, the two-dimensiona l particle spectrum is integrated over the aerosol mass yielding a one-dime nsional drop spectrum in the water mass grid. For this intermediate drop di stribution 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 me thod yields very good results. In the two-dimensional aerosol-water grid th e drop distributions move from initially small aerosol and water masses tow ard larger values whereby the drops remain more or less concentrated along a straight line. In several calculations the numerical diffusivity of the m ethod is investigated. The corresponding results show that in these investi gations the artificial broadening of the drop distribution remains tolerabl y low.