Extending High‐Order Flux Operators on Spherical Icosahedral Grids and Their Applications in the Framework of a Shallow Water Model

Abstract

This study extends a set of unstructured third/fourth‐order flux operators on spherical icosahedral grids from two perspectives. First, the fifth‐order and sixth‐order flux operators of this kind are further extended, and the nominally second‐order to sixth‐order operators are then compared based on the solid body rotation and deformational flow tests. Results show that increasing the nominal order generally leads to smaller absolute errors. Overall, the standard fifth‐order scheme generates the smallest errors in limited and unlimited tests, although it does not enhance the convergence rate. Even‐order operators show higher limiter sensitivity than the odd‐order operators. Second, a triangular version of these high‐order operators is repurposed for transporting the potential vorticity in a space‐time‐split shallow water framework. Results show that a class of nominally third‐order upwind‐biased operators generates better results than second‐order and fourth‐order counterparts. The increase of the potential enstrophy over time is suppressed owing to the damping effect. The grid‐scale noise in the vorticity is largely alleviated, and the total energy remains conserved. Moreover, models using high‐order operators show smaller numerical errors in the vorticity field because of a more accurate representation of the nonlinear Coriolis term. This improvement is especially evident in the Rossby‐Haurwitz wave test, in which the fluid is highly rotating. Overall, high‐order flux operators with higher damping coefficients, which essentially behave like the Anticipated Potential Vorticity Method, present better results.

A Layer‐Averaged Nonhydrostatic Dynamical Framework on an Unstructured Mesh for Global and Regional Atmospheric Modeling: Model Description, Baseline Evaluation, and Sensitivity Exploration

“…The full momentum equations may also be written in a 3‐D vector‐invariant form (e.g., Ullrich et al, ). In a general vertical coordinate, the originally transformed dry air density in both the horizontal and vertical pressure gradient force terms (see Zhang, , their equations and ) has been replaced by . The model constants include the gas constant for dry air ( R d = c p − c v ) and the ratio of the heat capacities, , where c p and c v are heat capacities at constant pressure and volume for dry air, respectively.…”

“…Figure compares three primal‐cell flux operators based on the correlated tracer testing strategy (Lauritzen et al, ), performed by the parallelized shallow‐water model at ~60 km. All schemes maintain the preexisting functional relation between two tracers well, and an upwind fifth‐order formulation with a full dissipation coefficient (O5; see Zhang, ) generates the smallest mixing diagnostic values. The limited scheme with the flux corrected transport limiter (Zalesak, ; also see appendix in Zhang et al, ) shows an effective restriction of the overshooting values (to machine round off), as can be observed from the scatter points and the overshooting error norm.…”

“…At day 15, while the general patterns from the four experiments still match, the PV2 runs support computational modes in both the HDC and NDC, which are less evident in the PV3 runs. Such a difference is consistent with the shallow‐water simulations in that PV3 controls the computational Rossby mode well (Zhang, ). The vorticity structures at day 30 are generally worse than those at day 15.…”

“…The horizontal flux operators are extended from Skamarock and Gassmann (2011) for both primal and dual cell applications. A minor difference from Zhang (2018) is that the triangular stencil in the polynomial reconstruction is reduced to a 10-point stencil (a triangle with its three neighbors, and the neighbors of every neighbor). Meanwhile, a flux-form semi-Lagrangian/incremental-remapping transport method (Miura & Skamarock, 2013;Skamarock & Menchaca, 2010) based on the quadratic reconstruction has also been implemented.…”

“…where r ei = P ei − P c , l ei is the length of the ith edge of this primal cell, A c is the area of this Voronoi cell, N denotes the number of neighboring edges of this node, P ei has been projected to a tangent plane that touches P c (see Zhang, 2018), and r ei /|r ei | points to the outward normal direction of this cell on the tangent plane.…”