Some considerations for high-order 'incremental remap'-based transport schemes: Edges, reconstructions, and area integration

The problem of two-dimensional tracer advection on the sphere is extremely important in modeling of geophysical fluids and has been tackled using a variety of approaches. A class of popular approaches for tracer advection include 'incremental remap' or cell-integrated semi-Lagrangian-type schemes. These schemes achieve high-order accuracy without the need for multistage integration in time, are capable of large time steps, and tend to be more efficient than other high-order transport schemes when applied to a large number of tracers over a single velocity field.In this paper, the simplified flux-form implementation of the Conservative Semi-LAgrangian Multi-tracer scheme (CSLAM) is reformulated using quadratic curves to approximate the upstream flux volumes and Gaussian quadrature for integrating the edge flux. The high-order treatment of edge fluxes is motivated because of poor accuracy of the CSLAM scheme in the presence of strong nonlinear shear, such as one might observe in the midlatitudes near an atmospheric jet. Without the quadratic treatment of upstream edges, we observe at most second-order accuracy under convergence of grid resolution, which is returned to third-order accuracy under the improved treatment. A shallow-water barotropic instability also reveals clear evidence of grid imprinting without the quadratic correction. Consequently, these tests reveal a problem that might arise in tracer transport near nonlinearly sheared regions of the real atmosphere, particularly near cubed-sphere panel edges. Although CSLAM is used as the foundation for this analysis, the conclusions of this paper are applicable to the general class of incremental remap schemes.