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

Abstract

AbstractThis 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.