Some considerations for high‐order ‘incremental remap’‐based transport schemes: edges, reconstructions, and area integration

Abstract

SUMMARYThe 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. Copyright © 2012 John Wiley & Sons, Ltd.

On Mass Conservation in High-Order High-Resolution Rigorous Remapping Schemes on the Sphere

“…For fully two-dimensional polynomial reconstructions of degree 2 (h 5 2) choices of c (0,0) are given in Ullrich et al (2009Ullrich et al ( , 2012. For fully two-dimensional polynomial reconstructions of degree 2 (h 5 2) choices of c (0,0) are given in Ullrich et al (2009Ullrich et al ( , 2012.…”

“…Note that in the original formulation of CSLAM, mass conservation relied on the analytical integration along line segments coinciding with grid lines, which was possible on the gnomonic cubed-sphere grid (Ullrich et al 2009). This limited the application of CSLAM to FIG.…”

“…Perhaps its first application to atmospheric transport in Cartesian geometry was by Ran ci c (1992). Rezoning or remapping on the sphere has also received considerable attention in the atmospheric sciences because of its applications in the conservative coupling of components in global climate system models (Jones 1999;Lauritzen and Nair 2008;Ullrich et al 2009) and conservative semi-Lagrangian tracer transport on global domains (e.g., Lauritzen et al 2010). Mass conservation in rigorous remapping schemes is more stringent compared to flux-based discretizations (e.g., Lauritzen et al 2011).…”

A high‐order fully explicit flux‐form semi‐Lagrangian shallow‐water model

“…The explicit time‐stepping procedure described in this paper is applicable to any conservative semi‐Lagrangian transport scheme, including the family of semi‐Lagrangian integrated‐mass schemes , the Semi‐Lagrangian Inherently Conserving and Efficient scheme , and certain dimension‐split semi‐Lagrangian formulations . However, the flux‐form implementation of CSLAM is particularly well suited for implementation as a dynamical core on the cubed‐sphere grid because it does not use dimension splitting (which is potentially problematic near cubed‐sphere corner points) and can be modified to guarantee formal third‐order accuracy . In particular, the use of quadratic upstream edges is important for avoiding errors in the numerically computed divergence, which can pollute the solution.…”

“…However, the flux‐form implementation of CSLAM is particularly well suited for implementation as a dynamical core on the cubed‐sphere grid because it does not use dimension splitting (which is potentially problematic near cubed‐sphere corner points) and can be modified to guarantee formal third‐order accuracy . In particular, the use of quadratic upstream edges is important for avoiding errors in the numerically computed divergence, which can pollute the solution.…”