Magnetohydrodynamics Simulation in a Sphere

by Yin–Yang–Zhong Grid

Akira Kageyama

Department of Computational Science, Kobe University, Rokkodai 1-1, Kobe, Japan

Keywords:

Magnetohydrodynamics, Computational Fluid Dynamics, Sphere, Visualization.

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.

1 INTRODUCTION

Magnetohydrodynamics (MHD) is a theory for elec-

trically conducting ﬂuid ﬂows (Davidson, 2001).

Computer simulations of MHD in spheres are im-

portant in astro- and planetary physics because many

stars and planets have electrically conducting ﬂuids in

their bodies.

One of the most popularly used methods to dis-

cretize the basic equations of MHD, i.e., MHD equa-

tions, in the spherical geometry is the spectral meth-

ods in which physical variables are expanded by or-

thonormal functions deﬁned by the spherical harmon-

ics. The time development of a set of mode ampli-

tudes is numerically integrated. In this kind of spec-

tral approach, it is common that nonlinear terms ap-

pearing in the equations are calculated in the real

space as products, to avoid the costly computations

of convolutions in the spectral space. This approach,

called the pseudo-spectral approach, requires trans-

formations of variables between the real space and the

spectral space every time step. There is, however, no

de facto standard of “fast” algorithm for the spher-

ical harmonics transformations for massively paral-

lel computers. It means that the computational speed

of the spherical harmonics expansion method does

not linearly scale as functions of the maximum mode

number, i.e., spatial resolution, and the processor

number used in the parallel computation.

On the other hand, the grid-based approaches, that

are exempliﬁed by the ﬁnite difference method and

the ﬁnite volume method, are relatively easy to at-

Figure 1: Yin–Yang–Zhong grid. Three component grids,

Yin, Yang, and Zhong, are combined to cover a full sphere

including the origin. The overset grid method is used to

stitch up the three component grids together.

tain the linear scaling in massively parallel computa-

tions. However, it is impossible to discretize a sphere

with a structured grid system without a coordinate

singularity. Take the spherical polar coordinate sys-

tem (r, ϑ, ϕ), for example, where r, ϑ, and ϕ are the

radius, colatitude, and longitude. It has two types of

coordinates singularities: One is at the poles (ϑ = 0

and π) and the other is at the origin (r = 0).

The coordinate singularity itself is not a serious

problem because one can always apply L’Hˆopital’s

rule to convert an equation on a coordinate singular-

ity into a non-singular form. The challenge resides

around a coordinate singularity, rather than on it. In

a structured grid system, a coordinate singularity in-

evitably leads to a nearby concentration of grid points

Kageyama, A.

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

DOI: 10.5220/0005978302390243

In Proceedings of the 6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2016), pages 239-243

ISBN: 978-989-758-199-1

239

which degrades computational efﬁciency, especially

when one uses an explicit scheme for temporal inte-

gration. The Courant–Friedrichs–Lewy (CFL) condi-

tion (de Moura, 2012) imposes an impractically se-

vere limit on the time step due to the small grid spac-

ings. Even if an implicit time integration scheme is

used, the grid concentration implies unphysical high

resolution of the numerical accuracy around there.

We proposed a grid system, Yin–Yang grid, to

avoid the coordinate singularities on the poles of

the spherical polar coordinates (Kageyama and Sato,

2004; Kageyama, 2005). The Yin–Yang grid is a kind

of the overset grid (Chesshire and Henshaw, 1990)

that is applied to the spherical geometry. It has two

congruent grid elements—Yin and Yang—that are

combined to cover a two-dimensional spherical sur-

face or a three-dimensionalspherical shell volume be-

tween two concentric spheres.

We have applied the Yin–Yang grid to geody-

namo simulations (Kageyama et al., 2008; Miyagoshi

et al., 2010), solar dynamo simulations (Masada et al.,

2013; Mabuchi et al., 2015), and mantle convection

simulations (Kameyama et al., 2008). The Yin–Yang

grid is also used in other ﬁelds and by other groups,

from geophysics to astrophysics, from climate mod-

els to image proccessings. The spherical tessellation

problem (Yan et al., 2016) would be one of the most

promising applications in future in which the Yin–

Yang grid is potentially useful.

While the Yin–Yang grid system avoids the coor-

dinate singularities at the poles (ϑ = 0 and π), an-

other singularity at the origin (r = 0) is laid aside.

Yin–Yang simulations have, therefore, a “cavity” at

the center of the sphere, unless some symmetries are

assumed on the solutions at r = 0.

We have recently proposed an overset grid sys-

tem, Yin–Yang–Zhong grid, for the spatial discretiza-

tion of a full sphere, or a ball, including the ori-

gin (Hayashi and Kageyama, 2016). The Yin–Yang–

Zhong grid has three components; Yin, Yang, and

Zhong (see Figure 1). The new component grid

(Zhong) is a set of cuboid blocks based on the Carte-

sian grid. (“Zhong” stands for “center” in Chinese

language.) The Zhong grid component is placed to

cover the “cavity” of the Yin–Yang grid. The three

component grids cover the full sphere with partial

overlaps on their borders. The boundaries are sewed

together by mutual interpolations, following the gen-

eral overset grid methodology (Chesshire and Hen-

shaw, 1990). We performed a couple of valida-

tion tests of the Yin–Yang–Zhong grid (Hayashi and

Kageyama, 2016). For example, we compared damp-

ing rates of various eigenfunctions of the diffusion

equation in a sphere with analytical solutions.

The Yin–Yang–Zhong grid is a straightforward

extension of the Yin–Yang grid, by just adding a new

component grid (Zhong) at the center. Therefore, it is

relatively easy to modify an existing Yin–Yang code

into a Yin–Yang–Zhong code.

In the following, we summarize our recent appli-

cations of the Yin–Yang–Zhong grid for MHD simu-

lations in a sphere. We then brieﬂy review visualiza-

tion methods that we have developed for those simu-

lations.

2 SIMULATIONS OF MHD IN A

SPHERE

2.1 MHD Relaxation in a Sphere

MHD relaxation is a fundamental process in MHD

physics. When an MHD ﬂuid with a magnetic ﬁeld

is placed in a vessel (with no initial ﬂow), the MHD

system shifts spontaneously toward another state if

the initial state is unstable. After a short period of

transition, the system calms itself down to a quasi-

equilibrium state. This process is called MHD relax-

ation (Ortolani and Schnack, 1993). Various plasma

experiments show surprisingly good agreements with

a relaxation theory proposed by Woltjer (Woltjer,

1958) and Taylor (Taylor, 1986). Although plasma

instabilities, and therefore ﬂows, play essential roles

in the Woltjer-Taylor theory, the ﬂow velocity is as-

sumed to be absent in the relaxed state in the theory.

Figure 2: Streamline visualization of the ﬂow of a quasi-

stationary state of an MHD relaxation simulation in a

sphere. The color denotes the velocity amplitude (blue to

red for slow to fast). The simulation is performed using the

Yin–Yang–Zhong grid.

We have performed an MHD simulation inside a

sphere using the Yin–Yang–Zhong grid to investigate

the MHD relaxation processes that has a ﬂow in the

relaxed state. Figure 2 shows streamlines in a relaxed

state obtained by the simulation. The quasi-stationary,

relaxed state has both the magnetic ﬁeld and ﬂow ﬁeld

SIMULTECH 2016 - 6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

240

with the same levels of energy; this is a solution be-

yond the Woltjer-Taylor theory.

2.2 MHD Convection in a Thin Shell

We have also performed an MHD simulation of ther-

mal convection in a thin spherical shell layer with the

Yin–Yang–Zhong grid. The layer is between two con-

centric spheres of radii r = 0.9 and r = 1.0, whose

temperatures are kept hot and cold, respectively. A

central gravity toward the center is assumed. The pur-

pose of this simulation is to investigate the pattern for-

mation of the MHD convection and the MHD dynamo

effect by the ﬂow. (The MHD dynamo is an energy

conversion process from the ﬂow’s kinetic energy into

the magnetic energy through the electromagnetic in-

duction effect.)

Figure 3: A ﬂow pattern of thermal convection of an MHD

ﬂuid in a thin spherical shell. It shows the radial component

of the ﬂow at the average radius of the shell. Magnetic ﬁeld

is generated by this convection ﬂow and the magnetic ﬁeld

diffuses into the inner conductive sphere under the convec-

tion layer.

The MHD convectionexhibits a roll-like pattern in

the spherical shell as shown in Figure 3. The Zhong

grid component is critically important in this simu-

lation because the dynamo-generated magnetic ﬁeld

diffuses into the inner sphere of r ≤ 0.9, in which we

solve the diffusion equation for the magnetic ﬁeld on

the Zhong grid.

Magnetic ﬁeld is generated by the MHD dynamo

action by the ﬂows in the convection rolls. Drawing

magnetic ﬁeld lines, we have found that they wind

around the rolls and the magnetic energy is converted

from the ﬂow’s kinetic energy through the work done

by the ﬂow against the ﬁeld line tension force in the

windings. Generated magnetic energy is concentrated

in dislocations of the columns, i.e., the Y–shaped

forks of the rolls in Fig. 3. The ﬁeld lines are an-

chored to the inner core.

3 VISUALIZATIONS OF MHD IN

A SPHERE

As in other simulations, data visualization is a crucial

step in analyzing the Yin–Yang–Zhong simulations.

Visualization methods are, in general, divided into

two categories, i.e., post-process visualization and co-

process visualization. A post-process visualization is

applied to numerical data that are saved to a disk drive

system after a simulation job. A co-process visualiza-

tion is, on the other hand, applied while a simulation

is running. The output data of the co-process visual-

ization is a set of images.

3.1 Post-process Visualization on

Supercomputer

We use Armada as a post-process visualization tool.

Armada was originally developed by N. Ohno for

Yin–Yang simulation data. We have recently im-

proved this program so that it can visualize Yin–

Yang–Zhong data, too. Armada is a software ren-

dering program that is parallelized with MPI and

OpenMP. Visualization methods implemented in Ar-

mada are volume rendering, contour colors on cross

sections, vector glyphs, and stream tubes. Since it

does not need GPU (Graphics Processing Unit), we

can execute Armada on general supercomputers. Fig-

ure 4 shows sample snapshots of the visualization by

this software.

Figure 4: Post-process visualizations by Armada. Armada

is a parallel visualization program with volume rendering

(left), isosurface (middle), vector arrow glyphs (right), and

other visualization methods.

3.2 Co-process Visualization using

ParaView

In the post-process visualizations, we need to save

three-dimensional numerical data for the visualiza-

tion. The required storage size and the network band-

width degrade the usefulness of the post-process vi-

sualization. As a result, another approach to the visu-

alization, i.e., co-process visualization, is getting at-

tentions of simulation researchers these days.

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

241

ParaView

is one of the most popularly used gen-

eral purpose visualization programs. Although it is

basically for post-process visualizations, ParaView

can also be used for co-processvisualizations by mak-

ing use of a special library called Catalyst

. We use

ParaView/Catalyst as a co-process visualization tool

to analyze Yin–Yang–Zhong simulations. Shown in

Figure 5 are sample snapshots of movies obtained

by this approach. This is a powerful approach for

the visualization of large scale parallel simulations

since ParaView has a rich set of advanced visualiza-

tion methods.

Figure 5: Snapshots of visualization movies taken by

co-process visualizations of an MHD simulation by Par-

aView/Catalyst. (a) Allocations of MPI process in the sim-

ulation & visualization. (b) Vorticity amplitude in the Yin

grid component. (c) Vorticity amplitude in the equatorial

plane. (d) Vorticity amplitude in a sphere.

3.3 Co-process Visualization by Vector

Graphics Format

We have also developed our original co-process

visualization tool that is much simpler than Par-

aView/Catalyst. The tool, insitu2d, is implemented

as a Fortran90 module. It visualizes only two-

dimensional cross sections (the equatorial plane and

meridian planes) of Yin–Yang–Zhong grid simula-

tions. Consequently, it enables us to perform a quick

rendering without damaging the simulation speed.

The output images of insitu2d are stored in EPS

(Encapsulated PostScript) format. We can magnify

the images keeping sharp outlines thanks to the vector

graphics format of EPS. Sample ﬁgures of insitu2d

visualizations are shown in Figure 6.

http://www.paraview.org

http://www.paraview.org/in-situ/

Figure 6: Co-process visualization of MHD simulations by

our original tool insitu2d. (a) A meridional cross section

of the MHD relaxation simulation presented in Section 2.1.

(b) An equatorial cross section of the MHD simulation pre-

sented in Section 2.2.

4 CONCLUSIONS

We have recently proposed a new overset grid system,

Yin–Yang–Zhong grid, for numerical simulations in

a sphere (Hayashi and Kageyama, 2016). The Yin–

Yang–Zhongis an extension of the Yin–Yang grid that

is for the spherical shell geometry between two con-

centric spheres. In many cases, Yin–Yang simulations

have a cavity at the center of the sphere because of the

coordinate singularity at the origin r = 0. The Zhong

component grid is placed to cover the cavity region.

Three component grids (Yin, Yang, and Zhong) are

combined to cover a full sphere with partial overlap

between them on the borders that are stitched by mu-

tual interpolations based on the standard overset grid

method. The Yin–Yang–Zhonggrid enables us to per-

form a simulation in a full sphere without any care

about the severe CFL conditions caused by concen-

trated points.

We presented in this paper two MHD simulations

as application examples of the Yin–Yang–Zhonggrid:

One is MHD relaxation simulation, and the other is

MHD convection simulation in a sphere. For three-

dimensional visualizations of those simulations, we

use Armada for post-process visualizations and Par-

SIMULTECH 2016 - 6th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

242

aView/Catalyst for co-process visualizations. For

two-dimensional, co-process visualizations, we have

developed a simple and concise library, insitu2d.

The combination of the Yin–Yang–Zhong grid

and the specially designed visualization tools for the

grid system will be useful for various simulations in

the sphere.

ACKNOWLEDGEMENTS

This work was carried out with graduate students,

Kohei Yamamoto (§ 2.1), Takuya Furuzono (§ 2.2),

Arata Suga (§ 3.2), and Takashi Shimizu (§ 3.3).

Simulations reported in this paper were performed

on Earth Simulator (NEC SX-ACE) at JAMSTEC,

Plasma Simulator (Fujitsu FX100) at NIFS, and π–

computer (Fujitsu FX10) at Kobe University. This

work was supported by Grant-in-Aid for Scientiﬁc

Research (KAKENHI) 23340128.

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