Numerical solution of the reaction–advection–diffusion equation on the sphere

A finite volume algorithm for the solution of the reaction–advection–diffusion equation on the sphere is derived and evaluated using analytical solutions. The proposed approach is based on the principle of semidiscretization. The convective and diffusive fluxes are approximated first, and then the resulting set of the ordinary differential equations (ODEs) is solved using the appropriate time stepping algorithm. In the first part of the paper, solutions to both the linear advection and the advection–diffusion problems for a single conservative scalar are discussed. The monotonicity of the scheme is achieved with the explicit adaptive dissipation. The development as well as the selected applications of the method are illustrated using a finite volume mesh constructed on the basis of geodesic icosahedral grid, which, in the past 40 years, has been frequently applied in different models of geophysical fluid dynamics. The performance of the solver is assessed using a suite of standard tests based on solid body rotation for different initial conditions. After analysis of the advection–diffusion problem, the extension of the method for the equations with reactive terms is presented. The performance of the solver is assessed by comparing the results to the analytical solution of the linearized reaction–diffusion system. In the final part of the paper, the application of the solver for studies of nonlinear reactions on the sphere is illustrated. The main intended application of the proposed method includes the simulation of transport of chemical constituents in the Earth’s atmosphere as well as the forecasting of moisture and cloud water fields in numerical weather prediction and climate models.