Numerical solution of a simplified form of the diffusion equation for chemically reactive atmospheric species

Abstract

A quasi-steady state approximation (QSSA) method was developed by Hesstvedt et al. (1978, Int. J. Chem. Kinet. 10, 971–994) for atmospheric chemistry applications, and verified to give results in satisfactory agreement with the results obtained using methods with automatic error control in a wide range of atmospheric applications, given that a sufficiently short time step is selected. (Hesstvedt et al., 1978, Hov et al., 1978, Proc. WMO Symp., WMO No. 510, Geneva; Derwent and Hov, 1979, AERE-R9434, HMSO, London). In this paper modifications of the QSSA method are described to make possible the solution of a set of continuity equations for the combined effect of chemistry and vertical diffusion in the atmospheric boundary layer. It is found that it is possible to solve the chemical part of the continuity equation by using linear expressions only, and still keep the numerical error satisfactorily controlled. On this basis the agreement of the results obtained integrating the continuity equation in two ways, is demonstrated: (a) operator splitting whereby the chemical part is solved using the original QSSA version and the diffusion part using Crank-Nicholson numerical scheme and (b) solving the complete continuity equation using Crank-Nicholson scheme. The first approach is taken as the standard against which the results of the other approach are compared, and it is shown that the results fall very well in line when a short enough time step is selected. Application of Dufort-Frankel explicit scheme is also shown to give satisfactory results.