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

The ring average technique is adapted to solve the issue of clustered grid cells in polar and spherical coordinates with a finite-difference method.

The ring average technique is applied to develop a

The high-resolution TIEGCM shows good capability in resolving mesoscale structures in the ionosphere–thermosphere (I–T) system.

Mesoscale structures with a typical horizontal size of 100–500 km have gained more and more attention in research on the dynamics of the upper-atmospheric system. A number of studies have been carried out to investigate these structures, including the formation and evolution of polar cap patches and tongues of ionization

Spherical or cylindrical coordinates are commonly used in solving geophysical problems, including the modeling of the upper-atmospheric systems. As a workhorse for space weather research, a number of general circulation models (GCMs) for the coupled ionosphere–thermosphere (I–T) system have been developed based on spherical coordinates using finite-difference schemes (e.g.,

The major difficulty in increasing longitudinal resolution in spherical-geometry-based GCMs is that the explicit time stepping is constrained by the clustering azimuthal cells near the pole due to the Courant–Friedrichs–Lewy (CFL) condition

Recently,

This paper is organized as follows: in Sect.

An example of standard polar grids with a horizontal resolution of

The

The finite-difference adaption of the ring average algorithm is based on a similar averaging–reconstruction process over a predefined “effective” azimuthal grid as used in the finite-volume version of the algorithm. Figure

Schematic of grid cells within effective chunks.

We use the following example of solving the linear advection equation to illustrate the averaging–reconstruction process within each chunk. Consider the following linear advection equation of an incompressible fluid in the azimuthal direction as an example:

In the reconstruction step, the above algorithm uses the piecewise linear method (PLM) to reconstruct solutions within each chunk for the next time step of the GCM calculations, resulting in second-order accuracy. To achieve higher accuracy in the reconstruction step, a piecewise parabolic reconstruction method (PPM)

The algorithm shown in Fig.

The ring average algorithm with both the PLM and PPM.

The procedures in PPM are the same with PLM except for Step 3.

Step 0 in Fig.

The distribution of density at three simulation snapshots (

In this section, in order to illustrate the implementation of the ring average algorithm in a finite-difference code, we solve the two-dimensional (2D) linear advection equation in the polar geometry as an example. The code used in the 2D linear advection solver is a main subroutine used in the ring average module for GCMs. This two-dimensional advection test in polar geometry is also useful to demonstrate the effectiveness of the finite-difference ring average technique in handling a strong, narrow shear flow near the pole. A fourth-order central-finite-difference scheme is used to solve the following mass continuity equation under the incompressible assumption:

Figure

As shown in Fig.

We use the NCAR TIEGCM to demonstrate the effectiveness of the ring average technique in resolving mesoscale upper-atmospheric structures. TIEGCM is a physics-based 3D global model that solves the coupled equations of momentum, energy, and continuity for neutral and ion species in the upper-atmospheric I–T system using a fourth-order and centered finite-difference scheme to evolve the advection terms on each pressure surface with a staggered vertical grid

The thermospheric energy equation is

The zonal momentum equation is expressed as

The thermospheric major species in the TIEGCM include

The ions of the ionosphere in the TIEGCM include

By assuming a thermal quasi-steady state, the electron energy equation is

The main ring average algorithm in the TIEGCM.

For the electrodynamics, i.e., the “neutral wind dynamo process”, TIEGCM assumes steady-state electrodynamics with a divergence-free current density

The ionospheric convection pattern for computing the plasma advection velocity

In this study, the ring average technique is implemented in the TIEGCM v2.0 to solve the issue of clustering grid cells near the poles in the development of a high-resolution version of the TIEGCM. This technique is applied as a post-processing treatment of the fluid variables including oxygen ion density

The simulated polar maps of electron densities using

On the basis of the ring average technique, a new high-resolution version of TIEGCM with a horizontal longitude–latitude resolution as high as

The basic ring average settings of variables (column 1) and the corresponding reconstruction method (column 2), Fourier reduction (column 3), and sub-cycling (column 4) in the TIEGCM.

The ring average setup for different TIEGCM horizontal resolutions (column 1) associated with the number of longitude grids (column 2) and the number of averaging chunks in each azimuthal ring near the pole (column 3).

Comparisons of horizontal resolution in a geographic latitude–longitude grid (column 1), vertical resolution (column 2), time step (column 3),

To show the capability of the new high-resolution TIEGCM based on the ring average technique in resolving mesoscale I–T structures, we have simulated the ionospheric and thermospheric variations during the 17 March 2013 major geomagnetic storm as an example. Figure

The polar maps of electron densities at pressure surface 2 (near the

Figure

Polar maps of the

Benefiting from the ring average technique, the newly developed high-resolution TIEGCM has been applied to explore the mesoscale variations in the I–T system during space weather events. For instance, based on the

Simulating the mesoscale structures also requires a more realistic input from the upper boundary, corresponding to the electric field and auroral precipitation from the magnetosphere, and the bottom boundary, corresponding to the upward propagation of tides and waves from the lower atmosphere, of the I–T system. In the TIEGCM, these inputs are directly adopted from two empirical models, the Weimer model and the Global Scale Wave Model (GSWM), which might not necessarily represent the complexity of the actual physical processes from the boundaries. To obtain a more physical upper boundary condition, the CMIT has been developed

Furthermore, the ring average technique has also been applied in the WACCM-X, which can provide a relatively more realistic bottom boundary for the I–T simulation. The WACCM-X is a whole-atmosphere chemistry–climate general circulation model spanning the range of altitude from the Earth's surface to the upper thermosphere to simulate the entire atmosphere and ionosphere

Polar map of the electron density in the Southern Hemisphere at 14:00 UT on 17 March 2013 from the WACCM-X 1

In summary, a post-processing technique with an averaging–reconstruction (ring average) algorithm is developed to solve the problem of clustering of azimuthal cells in a spherical coordinate with the finite-difference method. The ring average technique is applied based on a reduced effective polar grid by first averaging quantities within azimuthal effective “chunks” and then reconstructing them within each chunk. The ring average technique shows the advantages of inexpensive computational cost, easy implementation, time step relaxation, and maintenance of the mesoscale structures without introducing artifacts, which allows for the development of high-resolution GCMs to resolve mesoscale structures. We have developed a new version of the TIEGCM, which has a horizontal resolution of

The source codes of TIEGCM 2.1 are provided through the GitHub repository (

The simulation data used in this study are available from the Zenodo archive (

TD developed the model code, performed the simulations, and wrote the paper. BZ, JL, and KAS proposed the original idea and revised the paper. WW and AB helped to develop the code and edit the paper. HLL and KP contributed to coupling the high-resolution TIEGCM to WACCM-X and CMIT, respectively.

The authors declare that they have no conflict of interest.

We are grateful for support from the ISSI/ISSI-BJ workshop “Multi-Scale Magnetosphere–Ionosphere–Thermosphere Interaction”. We would also like to acknowledge the Supercomputing Center of the University of Science and Technology of China for the numerical simulations in this study.

This work was supported by the B-type Strategic Priority Program of the Chinese Academy of Sciences (XDB41000000), the National Natural Science Foundation of China (41831070, 41974181), the pre-research project on Civil Aerospace Technologies no. D020105 funded by China's National Space Administration, and the Open Research Project of Large Research Infrastructures of CAS – “Study on the interaction between the low- and mid-latitude atmosphere and ionosphere based on the Chinese Meridian Project”. Tong Dang was supported by the National Natural Science Foundation of China (41904138), the National Postdoctoral Program for Innovative Talents (BX20180286), the China Postdoctoral Science Foundation (2018M642525), and the Fundamental Research Funds for the Central Universities. Binzheng Zhang was supported by the RGC General Research Fund (17300719, 17308520) and the Excellent Young Scientists Fund (Hong Kong and Macau) of the National Natural Science Foundation of China (41922060). Han-li Liu's work was supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under cooperative agreement no. 1852977. The National Center for Atmospheric Research is sponsored by the National Science Foundation.

This paper was edited by Josef Koller and reviewed by two anonymous referees.