Azimuthal averaging–reconstruction filtering techniques for finite-difference general circulation models in spherical geometry

Abstract

When solving hydrodynamic equations in spherical or cylindrical geometry using explicit finite-difference schemes, a major difficulty is that the time step is greatly restricted by the clustering of azimuthal cells near the pole due to the Courant–Friedrichs–Lewy condition. This paper adapts the azimuthal averaging–reconstruction (ring average) technique to finite-difference schemes in order to mitigate the time step constraint in spherical and cylindrical coordinates. The finite-difference ring average technique averages physical quantities based on an effective grid and then reconstructs the solution back to the original grid in a piecewise, monotonic way. The algorithm is implemented in a community upper-atmospheric model, the Thermosphere–Ionosphere Electrodynamics General Circulation Model (TIEGCM), with a horizontal resolution up to 0.625∘×0.625∘ in geographic longitude–latitude coordinates, which enables the capability of resolving critical mesoscale structures within the TIEGCM. Numerical experiments have shown that the ring average technique introduces minimal artifacts in the polar region of general circulation model (GCM) solutions, which is a significant improvement compared to commonly used low-pass filtering techniques such as the fast Fourier transform method. Since the finite-difference adaption of the ring average technique is a post-solver type of algorithm, which requires no changes to the original computational grid and numerical algorithms, it has also been implemented in much more complicated models with extended physical–chemical modules such as the Coupled Magnetosphere–Ionosphere–Thermosphere (CMIT) model and the Whole Atmosphere Community Climate Model with thermosphere and ionosphere eXtension (WACCM-X). The implementation of ring average techniques in both models enables CMIT and WACCM-X to perform global simulations with a much higher resolution than that used in the community versions. The new technique is not only a significant improvement in space weather modeling capability, but it can also be adapted to more general finite-difference solvers for hyperbolic equations in spherical and polar geometries.