Numerical MHD Simulation of Flux-Rope Formed Ejecta
Interaction With Bimodal Solar Wind
Wang, A. H.,* Wu, S. T.*and Tan, A.**
*Center for Space Plasma and Aeronomic Research
Department of Mechanical and Aerospace Engineering
University of Alabama in Huntsville
Huntsville, AL 35899
**Department of Physics, Alabama A & M University
Normal, AL 35762
Abstract. A theoretic numerical simulation of interaction between CME ejecta and bimodal solar wind from solar
surface to 30 solar radii has been presented. A comparison with an interaction between CME ejecta with
homogeneous and bimodal solar wind is given. The results show that the bimodal solar wind changes the topology
of CME ejecta with faster propagation speed away from the equator, and fast wind will energize the CME. Also the
results of steady-state bimodal solar wind characteristics that extend to 1AU are presented.
volumetric heating, momentum addition, and thermal
conduction. The flow is calculated in a meridional
plane defined by the axis of the magnetic field. The
detailed description of the governing equations can
be found in [3].
In simple solar wind model the polytropic index
γ=1.05. Temperature and density at lower boundary
(solar surface) are uniform from equator to pole. The
plasma β=1.0 at equator and 0.2 at pole. In bimodal
solar wind model the polytropic index γ=5/3. The
temperature and density are not uniform from equator
to pole. The plasma β=1.0 at equator and 0.02 at
pole. The volumetric heat source is given by
INTRODUCTION
Observation, especially Ulysses, showed that the
solar wind consists of both fast and slow components
in the solar minimum [1,2]. In this bimodal
characteristics of solar wind there are an almost
homogeneous high-speed flow over coronal hole and
the low-speed flow over coronal streamer with the
division between them at about 70o from the pole.
The bimodal solar wind property is very important to
study the propagation of Coronal Mass Ejection
(CME) which is major large scale solar eruption. The
fast solar wind at the coronal hole will affect the
propagation speed and the topology of CME.
We combined our bimodal solar wind model [3]
and the flux-rope model [4] to investigate the
interaction between CME and bimodal solar wind.
The purpose of this study is to reveal the physical
mechanism that causes the changes of the speed and
topology of the CME during the propagation. CMEs
are also the major disturb to the Earth environment.
In order to study the disturbed solar wind parameters
at 1AU due to the CME propagation in a bimodal
solar wind environment, we have build a steady-state
bimodal solar wind model to present here. Their
effects due to propagation of CME will be given
elsewhere [5].
ρ −( r − Rs ) /( LRs )
(1)
e
ρ0
where Q0 is 5x 10-8 erg cm-3s-1 and ρ0 is the base
density. A heating term similar to this was used by
Hartle and Barnes [6]. The thermal conductive fluxes
for a Lorentz gas is given by
Q = Q0
&
q = κ ||T 5 / 2 ( B ⋅ ∇T )
&
B
B2
(2)
where κ is the collisional thermal conductivity along
the magnetic field lines as given by Spitzer [7]. The
momentum addition is given by
MHD MODELS AND RESULTS
D=
In our simple solar wind model we assume an
axisymmetric, time-dependent, MHD flow of a
single-fluid, polytropic, and fully ionized plasma. In
order to obtain bimodal solar wind model, we added
D0 a 2
(r − a ) + a
2
2
{1 − arctan[5(
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
457
θ
− 14.5)] / 4} (3)
∆θ
where D0 with the value of 5x103 dyn/g, D reaches its
maximum value at a , and θ is latitudinal angle.
The computational domain is in meridional plane
from the pole to the equator in the θ direction and
from the solar surface to 30Rs for the interaction of
flux rope and solar wind and to 1AU for quasi-steady
state Sun-Earth bimodal solar wind in the radial
direction, respectively.
The symmetric boundary conditions are used at two
side boundaries. The lower boundary is a physical
boundary and the non-reflected characteristic
boundary conditions are used. The upper boundary is
a computational boundary, and since the flow at this
boundary is supersonic and superAlfvenic the linear
extrapolation is used. The detailed description of the
boundary conditions are given by [4].
Fig. 1 shows the distributions of the magnetic field,
density and velocity of the initial steady state for the
simulation of the interaction with a simple and a
bimodal solar wind, respectively. The contrast of the
density and velocity from the pole to the equator is
larger in the bimodal solar wind than that in the
simple solar wind. Also the current sheet is thinner in
the bimodal solar wind than in the simple solar wind.
Simple Solar Wind
Bimodal Solar Wind
MAGNETIC FIELD LINES
7
6
5
4
3
2
1
1
2
MAGNETIC FIELD LINES
3
4
5
6
7
7
6
5
4
3
HELIOCENTRIC DISTANCE (Rs)
2
1
1
2
3
4
5
6
7
HELIOCENTRIC DISTANCE (Rs)
109
109
1.00R0
108
108
107
Density (cm-3 )
Density (cm-3 )
1.46R0
2.13R0
3.11R0
106
4.53R0
1.00R0
107
1.46R0
106
2.13R0
3.11R0
6.62R0
105
105
10.4R0
4.53R0
6.62R0
18
36
54
Polar Angle (degrees)
72
104
0
90
10.4R0
18
700
700
600
600
6.62R0
500
4.53R0
500
10.4R0
Radial Velocity (km/s)
Radial Velocity (km/s)
104
0
6.62R0
400
4.53R0
300
3.11R0
200
0
0
90
36
54
Polar Angle (degrees)
72
90
10.4R0
3.11R0
300
2.13R0
200
1.46R0
100
1.46R0
1.00R0
18
72
400
2.13R0
100
36
54
Polar Angle (degrees)
36
54
Polar Angle (degrees)
72
0
0
90
1.00R0
18
FIGURE 1. Initial field lines, density and velocity distributions for simple and bimodal solar wind are showed.
CME-SOLAR WIND INTERACTION
evolutionary magnetic field lines, velocity and the
density enhancement for both cases, i.e. simple solar
wind and bimodal solar wind, are depicted. From this
figure the large distortion of the flux rope after
interacting with fast speed flow in the bimodal solar
wind at open field line region could be seen. The side
edges of the flux rope move faster than its center
because the fast speed solar wind in open field region
which carries the edge of the flux rope with them.
To show CME and solar wind interaction a flux-rope
emerged from the solar surface centered at the
equator. The flux rope’s radius is 0.5Rs. The center
thermal pressure is 50 times larger than its edge. The
plasma beta at the edge is 0.1 and its emerging speed
from the solar surface is 50 km/s. Fig. 2 shows the
simulation results at 12 hours. In this figure the
458
Simple Solar Wind
Bimodal Solar Wind
FIELD LINE AND VELOCITY FIELD (T= 12 HOURS)
30
25
20
15
10
5
5
10
15
FIELD LINE AND VELOCITY FIELD (T= 12 HOURS)
20
25
30
30
25
20
15
10
HELIOCENTRIC DISTANCE (Rs)
5
5
DENSITY ((D-D0)/D0)
15
20
25
30
DENSITY ((D-D0)/D0)
5
6
7
10
HELIOCENTRIC DISTANCE (Rs)
10
8
20
40
8
0
9
40
45 4
30
440
5
5
10
7
6
30
25
20
15
10
5
5
20 5
30
10
9
10
15
20
25
30
30
25
20
HELIOCENTRIC DISTANCE (Rs)
15
10
5
5
10
15
20
25
30
HELIOCENTRIC DISTANCE (Rs)
FIGURE 2. Upper panels are magnetic field and velocity at 12 hours, and lower panels are density enhancement.
R-T Diagram of Single Flux Rope’s Front and Center
BIMODAL SOLAR WIND FROM
SOLAR SURFACE TO 1AU
35
Bi-modal Solar Wind: Front
To extent the steady-state bimodal solar wind to
1AU we have to choose the following parameters of
a and D0 in momentum addition of equation (3).
These two parameters are not constants and are
function of latitude. The specific equations for them
are given as follows:
θ
− 14.5))
a(θ ) = a 0 − 0.36 arctan(5(
(4)
∆θ
and
θ
− 14.5)) / 3.5)
D0 (θ ) = D0′ (1 − arctan(5(
(5)
∆θ
where D′0 has the same value of D0 in the equation
(3).
The heating length L in equation (1) is redefined as
follows:
Bi-Modal Solar Wind: Center
30
HELIOCENTRIC DISTANCE (Rs)
Simple Solar Wind: Front
Simple Solar Wind: Center
25
20
15
10
5
0
0
2
4
6
8
TIME (HOURS)
10
12
14
FIGURE 3. Distance vs time of flux rope front and center
Further, Fig. 3 show the distance-time curve for the
edge of the flux-rope. It is clearly indicated that the
solar wind has energized the propagation of the fluxrope. As expected this effect will modify the CME
induced shock strength which related to particle
acceleration.
L=10
L=12.5
459
r ≤ 20Rs
r > 20Rs
Some steady-state results are shown in Fig. 4 and
Fig. 5.
SUMMARIES AND CONCLUSIONS
Quasi-Steady State Sun-Earth Structure
Theoretic numerical simulations for interaction of
CME with simple solar wind or bimodal solar wind
have been presented up to inner heliosphere in this
study. Also the steady-state bimodal solar wind
results are showed up to 1AU. The purpose is to lay
the ground to study the disturbed solar wind
properties due to the CME propagating to the Earth’s
environment. We have the following conclusions:
1. Fast speed solar wind will energize the
CME.
2. The shape and speed of the flux rope will
change during the propagation because the
interaction with bimodal solar wind.
3. The forms of heating and momentum
addition functions are important to obtain
realistic Sun-Earth bimodal solar wind.
Density
108
107
106
105
104
103
Equator
102
101
100
0
Pole
20
40
60
80
100
120
140
Heliocentric Distance
160
180
200
220
160
180
200
220
800
700
Pole
Velocity
600
500
400
Equator
300
200
100
0
0
20
40
60
80
100
120
140
Heliocentric Distance
FIGURE 4. Densities and velocities vs heliocentric
distance at the pole and the equator are showed.
ACKNOWLEDGMENTS
This research work performed by AHW and STW
is supported by NSF grant (ATM0070385) and
AFOSR grant (F49620-00-0-0304). AHW and AT
also are supported by NASA grant (NAG5-10202).
Density & Velocity at 1AU
80
Density (cm-3 )
70
60
50
40
REFERENCES
30
20
10
0
0
18
36
54
Polar Angle (degrees)
72
1.
90
Radial Velocity (km/s)
800
2.
700
600
500
400
300
3.
200
100
0
0
18
36
54
Polar Angle (degrees)
72
90
4.
FIGURE 5. Densities and velocities vs latitude at
1AU are showed.
5.
By examining Figs. 4 and 5, the solar wind
characteristics of this study are : at the solar surface n
(at equator) = 108/cm3 and n (at pole) = 2x107/cm3; vr
(at equator) ≅ 0 and vr (at pole) = 25km/s. At 1AU, n
(at equator) = 73/cm3 and n (at pole) = 12/cm3; vr (at
equator) = 420km/s and vr (at pole) = 790km/s.
Comparing with the average observational data the
number density at equator and pole in our bimodal
solar wind model are 2-3 times higher, but the
velocity is matched well.
6.
7.
460
Phillips, J. L., et al., Geophys. Res. Lett., 22,
3301 (1995).
Grall, R. R., Cole, M. T. Klinglesmith, A. R.
Breen, P. J. S. Williams, J. Markkanen, and
R. Esser, Nature, 379, 429 (1996).
Wang, A. H., Wu, S. T., Suess, S. T.,
Poletto, G., J. Geophys. Res., 103, 1913
(1998).
Wu, S. T. and Guo, W. P., Coronal Mass
Ejections, in Geophysical Monograph 99
(AGU Press, Washington DC), edited by N.
Crooker, J. Joslyn, and J. Feynman, 1997,
pp. 83-89.
Wang, A. H. and Wu, S. T., Solar Physics
(to be submitted).
Hartle, R. E. and Barnes, A., J. Geophys.
Res., 75, 6915 (1970).
Spitzer, L., Physics of Fully Ionized Gases.
2nd rev., edited by John Wiley, New York,
1962.