The Astrophysical Journal
, 649:1093 Y 1099, 2006 October 1
# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.
Y. Chen,
1
G. Q. Li,
2 and Y. Q. Hu
1
Recei v ed 2006 May 9; accepted 2006 June 2
ABSTRACT
Based on a flux rope catastrophe model for coronal mass ejections (CMEs), we calculate the Lorentz forces acting on the rope in equilibrium or eruption in the background field, which is taken to be either a partially open bipolar field or a closed quadrupolar field. The forces, including upward lifting and downward pulling ones, are exerted by the coronal currents inside and outside the rope, as well as the potential field, which has the same normal component distribution on the photosphere as the background field. The resultant of forces vanishes for a rope in equilibrium. For both cases with different background fields, the primary lifting force is provided by the azimuthal current inside the rope and its image below the photosphere. It is mainly balanced by the pulling force produced by the background potential field when the rope is in equilibrium. During an eruption of the rope caused by catastrophe, the two predominant forces mentioned above decrease rapidly with the ascent of the rope. In the meantime, the vertical and /or transverse current sheets and their images, which form and develop along with the rope eruption, contribute additional and significant pulling (or restoring) forces that decelerate the rope. When fast magnetic reconnection takes place across these current sheets, the restoring force provided by the sheets will be greatly reduced, which may play an important role in the dynamics of the rope. The implication of such a conclusion in the acceleration of CMEs is briefly discussed.
Subject headin g g Sun: corona — Sun: coronal mass ejections (CMEs)
1. INTRODUCTION
The catastrophic model of a coronal magnetic flux rope is one of the promising scenarios for solar active phenomena such as coronal mass ejections (CMEs), flares, and prominence eruptions
(reviewed recently by Lin et al. 2003; Hu 2005). With this scenario, the magnetic energy that has been stored in the corona is released suddenly to expel the flux rope outward with a velocity comparable to the local Alfve´n speed, as a consequence of the destabilization of the global magnetic field topology. Recently,
Hu and his coauthors carried out numerical calculations studying the catastrophe of the coronal flux rope embedded in various background fields. They found an occurrence of catastrophe in both cases of a dipolar and a partially open bipolar field (Hu et al.
2003b; Li & Hu 2003; Chen et al. 2006), and a quadrupolar field
(Zhang et al. 2005).
In the quadrupolar background field there exists a neutral point, which is deformed under conditions of high electrical conductivity into a transverse current sheet upon the emergence of a flux rope from the bottom of the central arcade. The current sheet exerts a sunward pulling (or restoring) force on the rope currents.
It was pointed out by Zhang et al. (2005) that the rope can levitate at a certain height in the corona after its catastrophic eruption.
This is very different from the result obtained by Hu et al. (2003b) and Hu & Li (2003) for bipolar background field cases: the flux rope goes to infinity instead as a result of the catastrophe. The main difference between the two situations lies in the presence of the transverse current sheet. It was suggested by Zhang et al. that this current sheet may be critical in sustaining the levitating flux rope. To provide a quantitative interpretation, it is necessary to carry out a detailed force balance analysis of the rope. In addi-
1
CAS Key Laboratory of Basic Plasma Physics, School of Earth and Space
Sciences, University of Science and Technology of China, Hefei Anhui 230026,
China; yaochen@ustc.edu.cn.
2
CAS Institute of Plasma Physics, Hefei 230031, China.
1093 tion, such an analysis can better our understanding of the physics of the flux rope catastrophe. As a matter of fact, similar analyses were used to investigate the equilibrium and catastrophic properties of the system as well as the dynamics of the eruptive flux rope in earlier flux rope models, such as the wire filament current model (e.g., van Tend & Kuperus 1978) and the thin-rope model
( Forbes & Isenberg 1991; Isenberg et al. 1993; Forbes & Priest
1995; Lin & Forbes 2000; Lin et al. 1998).
In this study, we use the axisymmetric magnetohydrodynamic
( MHD) model in spherical geometry so as to obtain steady solutions with the flux rope in equilibrium before catastrophe and time-dependent solutions delineating the flux rope eruption after catastrophe. Starting from these solutions, we first calculate the distribution of the current density. In the region outside the rope the currents concentrate in the current sheet and flow in the azimuthal direction, whereas inside the flux rope both azimuthal and poloidal current components are present. It can be proven from the Green’s formula that the induced current on the photosphere induced by a source current in the corona can be represented by the image of this source with respect to the photosphere.
Then, the magnetic forces on the rope currents provided by these azimuthal currents (including their images) can be calculated with the use of Ampe`re’s law. On the other hand, the forces exerted by the background potential field and the azimuthal magnetic field inside the rope are evaluated directly from the Lorentz force formula. The relevant formulation is given in the Appendix.
It is the aim of this article to analyze the interplay among the different pieces of magnetic forces that play dominant roles in the equilibrium and eruptive process of the flux rope in a variety of field topologies, and in particular, to evaluate in a quantitative way the physical significance of the transverse current sheet in sustaining the levitating flux rope in the quadrupolar field. We briefly describe the theoretical model of the coronal flux rope system in x 2 and show the result of our force balance analysis in x 3. Section 4 provides a summary of this article with a brief discussion.

1094 CHEN, LI, & HU Vol. 649
Fig.
1.— Magnetic configurations of the flux rope system at a certain instant after catastrophe for ( a ) the bipolar background field case and ( b ) the quadrupolar background field case. Red lines delineate the border of the flux rope and the current sheets below and above the rope. Panels ( c ) and ( d ) show the temporal profiles of the height ( thin cur v es ) and velocity ( thick cur v es ) of the top in dashed green, the axis in solid red, and the bottom in dot-dashed blue of the flux rope.
2. THEORETICAL MODEL OF THE CORONAL FLUX
ROPE SYSTEM
For the axisymmetric magnetohydrodynamic ( MHD) system, one can introduce a magnetic flux function ( t ; r ; ) to express the magnetic field B ( B r
; B ; B
’
) as
B ¼ : < r sin
ˆ þ B
’
ˆ : ð 1 Þ
The derived ideal MHD equations can be found in Hu (2004) and are not repeated here. The initial background atmosphere is assumed to be polytropic and in static equilibrium with temperature and density given by (e.g., Low 1984) where
T ( r ) ¼ T
0 r
1
; ( r ) ¼
0 r
1 = 1 Þ
; represents the heliocentric distance in units of solar radius
ð is the polytropic index, taken to be 1.21 in this study,
R
2 Þ r and T
0 and
0 are the base temperature and density, taken to be
2 MK and 1 : 67
;
10 13 kg m 3 , respectively. It can be seen that from the coronal base to 10 R the temperature decreases by a factor of 10 and the density decreases by 4 Y 5 orders of magnitude. Such a thermodynamic structure is close to that predicted by regular solar wind models (see, e.g., Chen & Hu 2001; Hu et al. 2003a) and serves as a reasonable background atmosphere in the near-Sun region for the study of the dynamics of the flux rope system. We point out in passing that the choice of the initial atmosphere does not affect the flux rope system in equilibrium at all, since the corresponding field is kept force-free during simulations (see below). The initial background magnetic field is taken to be a partially open bipolar field or a closed quadrupolar field. The former can be obtained numerically from the dipole field
[ ( r ; ) ¼ sin
2
/ r ] by setting a lower limit c for the magnetic flux function at the equatorial plane, and the resulting bipolar field has
, an open flux that equals c c
(see Hu & Wang 2005). The parameter is set to be 0.4 in this article, i.e., 40% of the total magnetic flux of the bipolar field is open. The quadrupolar field is the same as that used in Antiochos et al. (1999), with the magnetic flux function given by
( r ; ) ¼ sin
2 r
þ
(3 þ 5 cos 2 ) sin
2
2 r 3
: ð 3 Þ r
N
¼ p ffiffiffi
3 and
N
¼ /2. The background fields will be deformed upon the appearance of a flux rope in the central arcade, but their corresponding potential fields are fixed and given by ¼ sin
2
/ r for the bipolar field case and by equation (3) for the quadrupolar field case. The unit of the magnetic flux function is taken to be
0
¼ 5 : 69
;
10
14 is B
0
¼
0
/ R 2 ¼
Wb, and the unit of the magnetic field strength
1 : 17
;
10 3 T. The numerical solution of the flux rope system can be obtained after the emergence of the rope into the closed arcade of the background field (see, e.g.,
Hu & Li 2003; Chen et al. 2006). The system obtained is then determined by the poloidal flux ( p
) and azimuthal flux (
’
) of the rope.
To obtain force-free solutions of the flux rope system in equilibrium, we keep the temperature and density fixed to their initial values given by equation (1). This is just a mathematical way of constructing a force-free field solution by relaxation, as described in Hu et al. (2003a, 2003b). To locate the catastrophic point we keep
’ fixed and increase p slightly until this point is met. Physically speaking, the increase of the poloidal flux can be achieved by a twist of the roots of a long three-dimensional flux rope anchored to the photosphere. To properly describe the dynamics of the flux rope that is erupting upward after catastrophe, we solve the full set of the MHD equations to find out the timedependent solutions without resetting the values of the temperature and density. The azimuthal flux of the rope
’ is taken to be

No. 2, 2006 FORCE BALANCE ANALYSIS OF A FLUX ROPE 1095
Fig.
2.— Dependence of magnetic forces on the rope per radian on the poloidal flux of the rope azimuthal flux of the rope is fixed at
’
¼ 0 : 18.
p for equilibrium solutions for the bipolar background field case. The
0.18 and 0.05 in the bipolar and quadrupolar background field cases, respectively. Both the total magnetic energy of the system and the height of the flux rope axis increase monotonically with increasing p
, as expected from previous calculations. When p reaches a certain critical value, the catastrophe sets in, causing the eruption of the flux rope. Figure 1 illustrates the magnetic field configurations at a certain instant after catastrophe for the bipolar field case ( Fig. 1 a ) and the quadrupolar field case ( Fig. 1 b ) along with the corresponding temporal profiles of the height and velocity of the top, axis, and bottom of the flux rope ( Figs. 1 c and 1 d , respectively). Starting from the obtained magnetic field, one can calculate the distribution of current density. In the bipolar field case the catastrophe takes place when
(
’ p changes from 0.33 to 0.34
¼ 0 : 18). The flux rope starts to break away from the photosphere at about t 20 minutes. The velocity of the flux rope reaches its maximum in about 1 hr and becomes almost constant thereafter. The maximum speed is about 1300 km s 1 for the rope top and 800 km s
1 for the rope bottom. This means that the rope undergoes a rapid expansion during its eruption. In the quadrupolar field case, the catastrophe sets in when to 0.44 (
’ p changes from 0.43
¼ 0 : 05). The flux rope starts to leave the photosphere at t 55 minutes and is accelerated to about 900 Y 1100 km s
1 within 10 minutes. Then, the velocity of the flux rope decreases with bumps to zero, and eventually, the rope stays at a height of
4 Y 6 R in equilibrium. To shed a good light on the flux rope dynamics, we calculate the magnetic forces acting on the flux rope in equilibrium or eruption in x 3.
3. FORCE BALANCE ANALYSIS OF THE CORONAL
MAGNETIC FLUX ROPE
Using the method presented in the Appendix, we calculate the magnetic forces on the rope provided by different current systems
(including the corresponding images) and the background potential field. As mentioned above in x 2, the background potential field is the dipole field for the bipolar field case and is given by equation (3) for the quadrupolar field case. The forces are calculated for the rope per radian in azimuthal width in units of f
0
¼ 5 : 34
;
10
17
¼ B 2
0
R 2 /
0
N. For simplicity, the forces exerted by a current and its image are added up, and the sum is regarded as the force associated with this current. From the symmetries of the system, the resulting force on the rope is radial along the equatorial plane and vanishes for the flux rope in equilibrium. Using this equilibrium condition, we can check the accuracy of our numerical calculations. Meanwhile, a comparison among the component forces enables us to determine their relative importance. In the following discussion the magnetic forces on the rope are divided into three terms. The first term is attributed to the currents inside the rope. From the symmetry of the system, we need only to consider the force among poloidal currents and that among azimuthal currents (see the Appendix), and they are denoted by f
Rp and f
R ’
, respectively. We emphasize that the self-interaction of the azimuthal current inside the rope by itself results in an outward radial force on the rope. This force comes from the curvature of the rope surrounding the Sun, which is called the toroidal or ‘‘hoop’’ force by
Chen (1989) and Krall & Chen (2000) and the rope curvature force by Lin et al. (1998). Note that in the Cartesian models (e.g., van
Tend & Kuperus 1978) this self-force is trivially zero by the symmetry of an infinitely long straight current. The second term is the force produced by the background potential field, denoted by f p
. The third term is associated with the current in the current sheets, denoted by f c 1 for the current in the equatorial current sheet above the rope in the bipolar background field case, by f c 2 for the current in the newly formed vertical current sheet below the erupted rope in both cases, and by f c 3 for the transverse current sheet above the flux rope in the quadrupolar background field case. While the rope is erupting, various types of MHD waves are excited and carry additional body currents in the background atmosphere.
The corresponding magnetic forces are negligible, as shown by our calculations, and thus they are not discussed in the following analysis.
3.1.
Force Balance Analysis for the Bipolar
Back g round Field Case
For the partially open bipolar field case, a series of equilibrium solutions can be obtained for point ( p
’ acterized by a specific value of
¼ p
0 : 18, and each of them is charthat is below the catastrophic
< 0 : 34). The forces acting on the rope include f
R ’
, f
Rp
, f p
, and f c 1
. Their variations with p are shown in Figure 2, with positive values denoting upward lifting forces and negative values representing downward pulling forces. Among the component forces, f
R ’ and f p make the main contributions to the lifting and pulling forces, respectively, and they are plotted in Figure 2 a .
The other two components f
Rp f ¼ f
R ’
þ f p
þ f
Rp
þ f c 1 and f c 1 and the resultant of forces are shown in Figure 2 b . For ease of comparison, we show f p in Figure 2 a (also in the other figures to be presented). For all equilibrium solutions in the range of 0 < p
0 : 33, f is less than 0.003, implying that the force-free condition is well fulfilled. The lifting force f
R ’ is mainly counterbalanced by the pulling force f p
. Both forces are larger than f
Rp and f c 1 in amplitude by about 1 order of magnitude. In addition, and f p decrease monotonically with increasing p f
R ’
, whereas f
Rp and f c 1 are almost independent of p
. Therefore, the lifting force

1096 CHEN, LI, & HU Vol. 649
Fig.
3.— Temporal profiles of magnetic forces on the eruptive flux rope per radian for the bipolar background field case. The poloidal and azimuthal fluxes of the rope are taken to be p
¼ 0 : 34 and
’
¼ 0 : 18, respectively.
f
R ’ provided by the azimuthal component of the rope current and the pulling force f p provided by the background potential field are the dominant magnetic forces that maintain the flux rope in equilibrium.
When p increases from 0.33 to 0.34, a catastrophe takes place and results in an eruption of the flux rope. The associated magnetic field configuration and the temporal profiles of the height and velocity of the top, axis, and bottom of the flux rope have been shown in Figures 1 a and 1 c , respectively. Figure 3 a gives the magnetic forces acting on the erupting flux rope as functions of time. In comparison with the equilibrium case, there appears one more force f c 2 acting on the eruptive rope, which is provided by the current in the newly formed current sheet below the rope.
Moreover, the resultant of forces f ¼ f
R ’
þ f p
þ f
Rp
þ f c 1
þ f c 2 may deviate from zero, since the rope is in a dynamic state. Figure 3 b is a local enlargement of Figure 3 a to illustrate clearly the contributions of f
Rp
, f c 1
, and f ure 3 we can see that both f
R ’ c 2 and the variation of and f p f . From Figdecrease quickly with the eruption of the flux rope. At about t 20 minutes, f changes from nearly zero to an apparent positive value. This corresponds to the instant at which the flux rope starts to erupt. At t ¼ 40 minutes f reaches its maximum value of 0.035 and then decreases slowly.
At about t ¼ 80 minutes, f p
0 : 04, so f
R ’
¼ 0 : 01, f c 2
¼ 0 : 02, and f
R ’
¼ is always the dominant lifting force acting on the flux rope. The pulling force produced by the newly formed vertical current sheet, f c 2
, appears and increases in magnitude after the eruption of the rope, and it becomes larger than f p in magnitude and serves as another main pulling force after t 70 minutes.
3.2.
Force Balance Analysis for the Quadrupolar
Back g round Field Case
In this subsection, we continue our analysis for the quadrupolar background field case. This case differs from the bipolar field case in two aspects. First, there exists a neutral point in the quadrupolar potential field, and the field changes its direction across this point. Consequently, f p may change its role from a pulling force to a lifting one during the rope eruption. Second, the neutral point turns into a transverse current sheet upon the emergence of a flux rope. The sheet develops when the rope expands and erupts. No equatorial current sheet exists above the rope in this case. Correspondingly, f c 1 is replaced by f c 3 contributed by the transverse current sheet. A catastrophe takes place when
0.43 to 0.44 for a fixed
R ’ p changes from
¼ 0 : 05. In the solutions with 0 < p
< 0 : 43 the flux rope remains attached to the photosphere.
f
The force components f
R ’
Rp
, f c 3
, and f ¼ f
R ’
þ f p and f
þ f
Rp p
þ are plotted in Figure 4 f c 3 a , while are shown in Figure 4 b . As seen from Figure 4, f
R ’ is the only lifting force and f p dominant pulling force. Both f
R ’ and f p is still the are larger in amplitude by more than 1 order of magnitude than f
Rp and f c 3
. The resultant of forces f deviates from zero, ranging from 0.01 to 0.01, and the numerical error is slightly larger than that in the bipolar field case, but still acceptable. The main error source comes from the determination of the location and current density of the transverse current sheet. The component forces f
R ’ monotonically with increasing p and f p decrease
, whereas there is only a slight declining trend for j f
Rp j and j f c 3 j with increasing p
.
Fig.
4.—Same as Fig. 2, but for the quadrupolar background field case (
’
¼ 0 : 05).

No. 2, 2006 FORCE BALANCE ANALYSIS OF A FLUX ROPE 1097
Fig.
5.—Same as Fig. 3, but for the quadrupolar background field case ( p
¼ 0 : 44,
’
¼ 0 : 05).
Starting from the equilibrium solution with p
¼ 0 : 43, we increase p to 0.44 so as to cause a catastrophe. The associated magnetic field configuration and the temporal profiles of the top, axis, and bottom of the height and velocity of the flux rope have been shown in Figures 1 b and 1 d . In Figure 5 a we show the variation of magnetic forces with time. Figure 5 b is a local enlargement of Figure 5 a so as to illustrate clearly the contributions of f
Rp
, f c 2
, and f c 3 and the variation of f . It can be seen from Figure 5 that both f
R ’ and f p decrease slowly with the expansion of the flux rope at first, then abruptly once the rope starts to leave the photosphere at t when the rope goes beyond r
N
¼ p ffiffiffi
3 f p changes its direction
, and thus turns into a lifting force. During the rope eruption, the transverse current sheet increases rapidly in both scale and current density. As a result, the corresponding pulling force f c 3 jumps in amplitude from 0.1 to a maximum of 0.25 and becomes comparable with the main lifting force f
R ’
. It is apparent that such a quick increase of f c 3 resultant of forces f ¼ f
R ’
þ f p
þ f
Rp
þ f c 3
þ f c 2 makes the change from positive to negative, which accounts for the rapid deceleration of the eruptive flux rope and the subsequent formation of a new equilibrium state with the flux rope levitating in the corona.
4. CONCLUSIONS AND DISCUSSION
To investigate the physics underlying the equilibrium and catastrophe of the coronal flux rope system, we calculate the magnetic forces acting on the rope currents. Two cases are treated with different background fields, one bipolar and the other quadrupolar.
We show that the dominant lifting force is provided by the azimuthal current inside the rope. It is mainly balanced by the pulling force produced by the background potential field when the rope is in equilibrium. For the eruptive flux rope after catastrophe, the above two forces decrease rapidly with the ascent of the rope. On the other hand, the restoring force provided by the newly formed current sheets rises and becomes another main pulling force decelerating the flux rope. A vertical current sheet appears below the eruptive rope in both cases, but the restoring force provided by it is too small to prevent the rope from escaping. An additional restoring force comes from the transverse current above the eruptive rope in the quadrupolar field case, and it accounts for the rapid deceleration of the flux rope and the formation of a new equilibrium with the flux rope levitating in the corona after catastrophe.
According to our study, the restoring force produced by the newly formed current sheet(s) constitutes a major component of the pulling force no matter what the configuration of the background field is (bipolar or quadrupolar). This conclusion is helpful in understanding the acceleration caused by magnetic reconnections in CMEs. In the frame of ideal MHD, the flux rope is thrown up by the unbalanced magnetic force after catastrophe.
However, electric current sheets are bound to form and develop, and provide restoring forces so as to decelerate and even stop the rope eruption. On the other hand, the newly formed current sheets offer proper sites for fast magnetic reconnection. The reconnection heats and accelerates the plasmas in the reconnection site, which contributes to the acceleration of the flux rope. Based on our study, this is only one aspect of the acceleration caused by reconnection. Another aspect, perhaps more important, is the attenuation and suppression of the current within the sheets, which leads to a remarkable reduction of the related pulling force and thus a larger acceleration of the eruptive rope (see also Low &
Zhang 2002). Taking the quadrupolar background field case as an example, we have obtained a new equilibrium with the flux rope levitating in the corona after the catastrophe if magnetic reconnection is prohibited. It is obvious that the flux rope will inevitably erupt once magnetic reconnection takes place across any one or both of the current sheets, as demonstrated by Zhang et al.
(2006). We point out in passing that a similar magnetic configuration with a transverse current sheet above the rope may exist in a bipolar field without invoking a quadrupolar field, as was illustrated qualitatively by Low & Zhang (2002). It is interesting to examine how the magnetic free energy released by reconnection is distributed to thermal and kinetic energies of the plasma and to identify the roles played by Ohm’s dissipation and work by the
Lorentz force.
Incidentally, the impact of the solar wind on the equilibrium and eruption of the flux rope was not taken into account in the current study. Such an effect is largely negligible in finding flux rope equilibrium solutions and locating the catastrophic point (e.g.,
Sun & Hu 2005). Nevertheless, the solar wind must have a significant influence on the dynamics of the eruptive flux rope. This effect should be taken into account especially when one applies the flux rope catastrophe model to interpreting CMEs. The calculations considering both the effect of the solar wind and magnetic reconnection will be presented in another paper.
This work was supported by the National Natural Science
Foundation of China ( NNSFC 40404013 and 10233050) and the
Ministry of Science and Technology of China (2006 CB806304).

1098 CHEN, LI, & HU Vol. 649
APPENDIX
FORCE BALANCE ANALYSIS OF THE AXISYMMETRIC FLUX ROPE
A1. TWO DIMENSIONAL PROBLEM OF THE MAGNETIC FIELD IN SPHERICAL GEOMETRY
In spherical coordinates ( r ; ; ’ ), when the current density has only an azimuthal component J
’
( r ; ), the magnetic vector A ¼ A
’ satisfies the Poisson equation (Lin et al. 1998)
9 2
( A
’ cos ’ ) ¼ J
’ cos ’; ð A1 Þ which can be solved by means of the Green’s function method with a given boundary condition A
’
(1 ; ) or equivalently (1 ; ) ¼
A
’
(1 ; ) sin , where is the magnetic flux function. The solution is
A
’
( r ; ) cos ’ ¼
4
þ
4
1
Z Z Z
Z
0
2
G ( r ; r
0
) J
’
( r
0
;
0
) cos ’
0 r
0 2 sin d ’
0
Z
0
A
’
(1 ;
0
) cos ’
0
@ G ( r ; r
0
)
@ r 0
0 dr
0 d r 0 ¼ 1
0 sin d ’
0
0 d
0
; ð A2 Þ where
G ( r ; r
0
) ¼
( r 2 þ r 0 2
1
2 rr 0 cos )
1 = 2
1
( r 2 r 0 2 þ 1 2 rr 0 cos )
1 = 2
; ð A3 Þ cos ¼ cos cos
0
þ sin sin
0 cos ( ’ ’
0
) ; ð A4 Þ
@ G ( r ; r
0
)
@ r 0 r
0 ¼ 1
¼
( r 2 r
2
1
þ 1 2 r cos )
3 = 2
: ð A5 Þ
It can be proven that the right-hand side of equation (A2) is also proportional to cos ’ , which can then be canceled from both sides of this equation. Without loss of generality, we set ’ ¼ 0 and substitute equations (A3) and (A5) into equation (A2) to give
A
’
( r ; ) ¼
4
þ r
Z Z Z
2
4
1 1
1
Z
0
2
( r 2 þ r
Z d ’
0
0 2
0
2 rr 0 cos )
1 = 2
A
’
(1 ;
0
) sin
0
( r 2 r cos ’
0
0 2 þ 1 2 rr 0 cos )
1 = 2 d
0
:
( r 2 þ 1 2 r cos )
3 = 2
J
’
( r
0
;
0
) r
0 2 cos ’
0 sin
0 dr
0 d
0 d ’
0
ð A6 Þ
The second term on the right-hand side of the above equation corresponds to the potential field with the given magnetic flux distribution at the lower boundary, and the first term represents the magnetic field produced by the source current J
’ seen that the image is located at r
00 ¼ 1/ r
0
,
00 ¼ 0 with a current density of J
’
( r
0
;
0
)/ r
0
.
( r
0
;
0
) and its image. It can be
A2. FORCES BY THE AZIMUTHAL COMPONENT OF CORONAL CURRENTS
The magnetic forces exerted by a given azimuthal current and its image on the rope currents can be directly calculated from Ampe`re’s law, which is written as
Z Z
F ¼
4 dV dV
0
J
< ½ J
0 < ð r r
0
Þ = j r r
0 j
3
; ð A7 Þ where J ( r ; ) is the rope current, located at r ( r ; ; ’ ), and J
0
( r
0
;
0
) is the acting current, located at r
0
( r
0
;
0
; ’
0
).
By the symmetry of the system, we need only to consider forces between azimuthal currents and between poloidal currents. A poloidal current creates only azimuthal fields that do not act on an azimuthal current, of course. An azimuthal current creates only poloidal fields, and their force on a poloidal current is in the azimuthal direction, and should be zero because of axisymmetry. Let us first consider the force between azimuthal currents. From the north-south symmetry, it can be known that the total force of any azimuthal current on the flux rope is along the radial direction in the equator. Without loss of generality, we set ’ ¼ 0 in equation (A7) and calculate the force on the rope per radian, and the result reads dF d ’
¼
4
Z r
2 sin dr d
Z r
0 2 sin
0 dr
0 d
0 d ’
0
J
’
( r ; ) J
0
’
( r
0
;
0
)( r
0 sin
( r 2 þ r 0 2
0 r sin
2 rr 0 cos )
3 = 2 cos ’
0
)
; ð A8 Þ

No. 2, 2006 FORCE BALANCE ANALYSIS OF A FLUX ROPE 1099 where the integration over ’
0
Z
2 can be rewritten in terms of the complete elliptic integrals as d ’
0
4 E
0
ð r 2 þ r 0 2 2 rr 0 cos Þ
3 = 2
¼
ð 1 k 2 Þ ½ r 2 þ r 0 2 2 rr 0 cos ð þ
Z
2 cos ’
0 d ’
0
0
ð r 2 þ r 0 2 2 rr 0 cos Þ
3 = 2
½ ð k 2 Þ E ð 1 k 2 Þ K
¼ k 2 ð 1 k 2 Þ ½ r 2 þ r 0 2 2 rr 0 cos ð þ
0 Þ
3 = 2
;
0 Þ
3 = 2
; where
0
K ( k ) ¼
R
0
/2
) . Substituting 1/ r
0
1 k 2 for r
0 sin
2
’ and J
0
’
1/2
/ r
0 d ’ and E ( k ) ¼ for J
0
’
R
0
/2
1 k 2 sin
2
’
1/2 d ’ with k 2 ¼ 4 rr
0 sin sin
0
/ ½ r 2 þ r
0 2 2 rr
0 cos ( þ in equation (A8), we get the expression for the contribution of the corresponding image current. Note that the self-interaction of the azimuthal current inside the rope has a net contribution to the rope, which comes from the curvature of the rope surrounding the Sun, as mentioned and calculated by Chen (1989), Krall & Chen (2000), and Lin et al. (1998).
The force between this current and its image is also present. From our calculations, the two forces are comparable, and both of them are upward in direction.
A3. FORCES BY THE POLOIDAL COMPONENT OF CORONAL CURRENTS
For the numerical examples treated in this study, the poloidal current, referred to as J
Rp
, exists only inside the rope, and so does the associated azimuthal magnetic field B
’
. Thus, there is no induced current of J
Rp on the photosphere, and no corresponding image current appears. In the following discussion we derive the magnetic force provided by J
Rp on the rope per radian. Consider a segment of the rope,
S
1
’ in azimuthal width, with a volume V enclosed by a cylinder and two bases, denoted by S
1
( S
2
) is perpendicular (parallel) to ˆ . The force exerted by J
Rp on the rope segment is and S
2
. Note that the outward normal of
Z
V
J
R p
<
B
’ dV ¼
1
¼
1
Z
I
V
(
: <
B
’
)
<
B
’ dV ¼
S
1
þ S
2
B
’
B
’
1
Z
V
: =
B
’
B
’
1
2
B
2
’
I
= dS ¼
2
1
I
S
2
B
2
’ n dS ;
1
2
B
2
’
I dV
ð A9 Þ where I is the unit matrix and n is the normal of each of the two bases. It can be seen from equation (A9) that an outward tension acts on downward pulling force, given by ð 1/2 Þ
R R
B 2
’
B
2
’
/2 . Through some manipulations the resultant force is found to be a r dr d ’ , so the contribution made by the poloidal current inside the rope reads f
R p
¼ dF
R p d ’
¼
2
1
Z Z
B
2
’ r dr d : ð A10 Þ
Finally, the magnetic force on the azimuthal current inside the rope exerted by the background potential field can be directly calculated with the Lorentz formula.
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