A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid

Abstract

A conservative multi-tracer transport algorithm on the cubed-sphere based on the semi-Lagrangian approach (CSLAM) has been developed. The scheme relies on backward trajectories and the resulting upstream cells (polygons) are approximated with great-circle arcs. Biquadratic polynomial functions are used for approximating the density distribution in the cubed-sphere grid cells. The upstream surface integrals associated with the conservative semi-Lagrangian scheme are computed as line-integrals by employing the Gauss–Green theorem. The line-integrals are evaluated using a combination of exact integrals and high-order Gaussian quadrature. The upstream cell (trajectories) information and computation of weights of integrals can be reused for each additional tracer. The CSLAM scheme is extensively tested with various standard benchmark test cases of solid-body rotation and deformational flow in both Cartesian and spherical geometry, and the results are compared with those of other published schemes. The CSLAM scheme is accurate, robust, and moreover, the edges and vertices of the cubed-sphere (discontinuities) do not affect the overall accuracy of the scheme. The CSLAM scheme exhibits excellent convergence properties and has an option for enforcing monotonicity. The advantages of introducing cross-terms in the fully two-dimensional biquadratic density distribution functions are also examined in the context of Cartesian as well as the cubed-sphere grid which has six local sub-domains with discontinuous edges and corners.