# Magnetohydrodynamics Simulation in a Sphere by Yin–Yang–Zhong Grid

### Akira Kageyama

#### Abstract

For numerical simulations in a sphere, we have recently proposed a new spherical grid system called Yin– Yang–Zhong grid. The Yin–Yang–Zhong grid is composed of three components—Yin, Yang, and Zhong— that are combined to cover a spherical region with partial overlaps on their borders. Mutual interpolations are applied to sew the components together, following the overset grid methodology. We review the idea of the Yin–Yang–Zhong grid and its applications to magnetohydrodynamics (MHD) simulations in a sphere. We also present visualization methods employed to analyze the Yin–Yang–Zhong simulations.

#### References

- Chesshire, G. and Henshaw, W. D., 1990. Composite overlapping meshes for the solution of partial differential equations. J. Comput. Phys., 90(1):1-64.
- Davidson, P.A., 2001. An Introduction to Magnetohydrodynamics. Cambridge University Press, ISBN: 0521794870.
- de Moura, C.A. and Kubrusly, C.S., ed., 2012. The Courant-Friedrichs-Lewy (CFL) condition-80 Years After Its Discovery. Springer Science & Business Media, ISBN:0817683941.
- Hayashi, H. and Kageyama, A., 2016. Yin-Yang-Zhong grid: An overset grid system for a sphere. J. Comput. Phys., 305: 895-905 .
- Kageyama, A., 2005. Dissection of a sphere and Yin-Yang grids. J. Earth Simulator, 3:20-28.
- Kageyama, A., Miyagoshi, T., and Sato, T., 2008. Formation of current coils in geodynamo simulations. Nature, 454(7208):1106-9.
- Kageyama, A. and Sato, T. 2004. “Yin-Yang grid”: An overset grid in spherical geometry. Geochemistry, Geophysics, Geosystems, 5, doi:10.1029/2004GC000734.
- Kameyama, M., Kageyama, A., and Sato, T. 2008. Multigrid-based simulation code for mantle convection in spherical shell using Yin-Yang grid. Phys. Earth Planet. Inter., 171:19-32.
- Mabuchi, J., Masada, Y., and Kageyama, A., 2015. Differential rotation in magnetized and non-magnetized stars. Astrophys. J., 806(1):10.
- Masada, Y., Yamada, K., and Kageyama, A., 2013. Effects of penetrative convection on solar dynamo. Astrophys. J., 778(1):11.
- Miyagoshi, T., Kageyama, A., and Sato, T. 2010. Zonal flow formation in the earth's core. Nature, 463(7282):793- 6.
- Ortolani, S. and Schnack, D. D., 1993. Magnetohydrodynamics of plasma relaxation. World Scientific, Singapore., ISBN:981020860X
- Taylor, J. B., 1986. Relaxation and magnetic reconnection in plasmas. Reviews of Modern Physics, 58(3):741.
- Woltjer, L., 1958. A theorem on force-free magnetic fields. Proc. Natl. Acad. Sci. USA, 44(6):489-91.
- Yan, J., Song X., and Gong, G. 2016. Averaged ratio between complementary profiles for evaluating shape distortions of map projections and spherical hierarchical tessellations. Computers & Geosciences, 87:41- 55.

#### Paper Citation

#### in Harvard Style

Kageyama A. (2016). **Magnetohydrodynamics Simulation in a Sphere by Yin–Yang–Zhong Grid** . In *Proceedings of the 6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,* ISBN 978-989-758-199-1, pages 239-243. DOI: 10.5220/0005978302390243

#### in Bibtex Style

@conference{simultech16,

author={Akira Kageyama},

title={Magnetohydrodynamics Simulation in a Sphere by Yin–Yang–Zhong Grid},

booktitle={Proceedings of the 6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,},

year={2016},

pages={239-243},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005978302390243},

isbn={978-989-758-199-1},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,

TI - Magnetohydrodynamics Simulation in a Sphere by Yin–Yang–Zhong Grid

SN - 978-989-758-199-1

AU - Kageyama A.

PY - 2016

SP - 239

EP - 243

DO - 10.5220/0005978302390243