- Email: [email protected]
Solar atmospheric responses to magnetic flux eruptions. I. A mechanism for coronal mass ejections
Chin.Astron.Astrophys.(1991)15/3,284-291 A translation of
Acta Astrophys.Sin.(1991)11/2,181-186
0 Pergamon Press plc Printed in Great Britain 0275-1062/91$10.00+.00
SOLAR A’IMOSPHERIC RESPONSES TOMAGNETIC FLUXJiMJPTIoNS. I. AMEEHANISMFOR CORONAL MASSElJErT1oNs’ IRJ You-qiu
ZRANG Hui-nan
Department of Earth and Space Sciences, University of Science and Technology of China ABSTRACT We carried out a numerical simulation of atmospheric responses to magnetic flux eruptions from the base of a helmet streamer. The results show that the material originally contained in the streamer is compressed and lifted upward by the erupting magnetic field, forming a high density bright loop. At the same time, fast magnetosonic waves appear in front of the loop and a low density dark region, below it. These results suggest that magnetic flux eruption is probably an important triggering mechanism of loop-like coronal mass ejection.
Key words:
Sun-magnetic
field-coral
mass ejection
1. INTRODUCTION Since the discovery of coronal mass ejection events in the seventies, a succession of theoretical models have appeared, analysing in detail the dynamics and generating mechanism of look-alike mass ejections. In the majority of analytical models, the mass ejection is regarded as an outwardly expanding magnetic structure carrying with it a froeen-in field and emphasis is laid on the dominant role played by the magnetic field in triggering off and accelerating the ejection [l-5]. On the other hand, in some numerical models [6-g], the high density loop is regarded as a kind of magnetic fluid wave. In these models, thermal pressure is the driving source and the initial atmosphere is in hydrostatic equilibrium (the magnetic field being a potential field or force-free) or magentostatic equilibrium and the the normal component of the field at the lower boundary is fixed and constant. Then, the effect of the magnetic field on the mass motion is one of obstruction rather than of acceleration. True, field
for a proper estimate of the physical effect on the coronal mass ejection we should await
of magnetic real-time data
1 Program supported by the National Natural Science Foundation. Received 1989 July 13; revised version 1989 September 15; passed by Referee 1990 April 4
Solar
Atmospheric
Response I
285
on the field; but a theoretical exploration of the effect is not without its value. Generally speaking, the action of a magnetic field on a plasma takes two for-: one is release of free energy through some type of instability 1101, and the other is acceleration of the plasma through eruption of magnetic flux, When the erupting field is antiparallel to the background field, there will appear neutral electric current sheet between the two, on this we have carried out analytical analyses [ll, 121 and numerical simulations [13], connecting together the current sheet formed with the bright loop structure of the ejection. Observations have shown that rany ejection events took place on top of helmet streamers and that the ejected urtter was initially located in the corona [14, 153. Based on this observational fact we carried out a numerical analysis [16] of the atmospheric responses to magnetic flux eruptions under a high density coronal arch in a closed background field (the erupting and background fields are in the same direction); the results reproduce the main observed features of loop-like ejections. For coronal streamers, an open background field with neutral sheet and wind is more reasonable. Hence, in this paper, we adopt an open field with a neutral sheet as the background field and carry out a numerical simulation of the atmospheric responses to flux eruptions at the base of the streamer. For simplicity, we again neglect the solar wind.
2. BASIC EQUATIONS AND INITIAL
BOUNDARY CONDITIONS
For axisyuetric problems in spherical coordinates (r, 8, +) we can introduce the magnetic flux function p(t, r, e) and express the field by
(11 Then, the equations
for
the two-dimensional
i!.?t+v-(pu)-0,
ideal
magnetic
fluid
are
(2)
ar
~+u.va+RvT+
(31
ar where p is the density, temperature, R, C, M, 7 gravitational constant, the permeability of the
v(vr, ve) is the velocity, T is the and p are the gas constant, the mass of the Sun, the polytropic index and vscuum.
286
HU You-qiu
and ZHANG Hui-nan
We now construct a aaguetostatic equilibriua state to siaulate approximately the coronal atreauer structure and use it as the intitisl conditions for our nuuerical calculations. Using the uethod described in (171, we start froa the potential field
we cap construct
the following
aaguetostatic
equilibriua
state: (71
4 m &&,
(8)
,
(9) where po = m(A), derivatives with
et - ART.,
Qo’ aud qo” are first respect to A and P:
and second order
--iww[-j$-(1 --+)I
specify the pressure auci density distributions in the equilibrium atmosphere, G being the teuperature, pc the density at the base of the corona and & the radius of the Sun. A necessary and sufficient condition for this equilibrium state to hold is, 1171,
Further, let S, be the potential field of the equatorial neutral sheet extending frou radius a to infinity; its flux function can be expressed as, [ 181,
and Qc is a constant. The field configuration at a= l.!iR~ is shown in Fig. la where point P is the inner edge of the equatorial neutral sheet, below the arc PC is the closed field (A ( 01, above it, the open field (AZ 0) with the neutral current sheet. The point C is located on the surface of the Sun at a colatitude of 63,38O, approximately. For a steauer
with a tail
half
opening
angle
of x, we can take
Solar
Atmospheric
Response I
201
u I
Fig.
1
2
2
4
6
6
The initial helmet streamer: (a) the magnetic configuration with a neutral point P, and (b) contours of density enhancement (po - pl)/pi
Then, the boundary of the streamer is A= Ao, the outside (A > Ao) is the isothermal atmosphere in static equilibrium of the potential field, and the inside in the magnetostatic equilibrium state, the magnetic field monotonically decreases inward to zero at the coronal arch (As 0). It can be shown that, for any set of parameter values, the inequality (11) holds identically. This shows that magnetostatic equilibrium state formed in this way is a steady state under the ideal MHD approximation. In the present calculation we took the following parameter values: a= 1.5 b, pc = 1.61 x lO=Ub, x = 5”, pc = 1.67 x lo-l3 kg/m3, 7bc = 1.5 x 106K. The corresponding distribution of the relative streamer/backgournd density enhancement is shown in Fig. lb. At the base of the coronal arch, the density is about 10 times that of the background; at the tail, about 2.5 times. There is some arbitrariness in the above choice of parameter values and the resulting density distribution may not represent faithfully the actual structure of the streamer. Fortunately this circumstance is not important for the study of the mass ejection mechanism. Actually we calculated for different sets of parameter values and we found that, as long as the density is higher in the streamer than in the background, the results are not essentially different. Consider now the boundary conditions. The region of solution is lRgIr16Rg, O~6~90° and we used a 32 x 27 non-uniform grid (32 in the r-direction), which is dense near the bottom (r= ~F&J) and the equator (6 = 90°) so as the raise the resolving power at the eruption region and the current sheet. Symmetry about the polar
266
HU You-qiu
snd ZHANG Hui-nan
axis and the equator is imposed and for the top, a non-reflecting boundary condition [19]. The botton is the perturbed boundary, the flux eruption is confined to the bottom of the streamer, that is, ec~e490”. the perturbed boundary condition is assumed to he
where a is eruption rate. Taking the velocity at the bottom to be zero is a simplifying approximation, which roughly holds when the erupting field is far greater than the background field and when the erupting rate is not large. For the region tl< &, we have fixed T= ~0, v = O.Also, the teaperature and density of the entire bottom region are to be determined by compatibility conditions derived an almost eigenvalue method f191. Finally, the set of equations (2)~(5) are numerically integrated by a multi-step, implicit algorithm 1201.
3. THE ~ICAL
~gSULTS
In the calculation below we took y= 5/3, u=O.729 X 101’Wb/s+ Fig. 2 shows the contours of relative density enhancement [p( t,r,6) - p~]/p~ at t = 6000 8 and 12000 8. Also shown is the magnetic field configuration at the initial time (chained) and at t (broken), the thick broken curve represents the boundary of the arch (A= h/2). We see that, under the action of the newly erupted magnetic field, matter in the original arch is compressed and raised to form a high density loop, leaving below a low density region, both expanding in step with the arch boundary. The high
’ IKO
W
Fig. 2 Contours of density enhsncemnet for a=0.729x IO”Wb/s at (a) t= 60008, with a maximumenhancement of IV 7.3, and (hf t= 12000 s, with a maximumenhanceuent of _ 6.5. The thick dashed curve graphs the boundary of the coronal arch.
Solar
Atmospheric
Response I
289
density loop comes form the compressionof matter both sides, and particularly from the inner side of the boundary, while the low density region is caused by the expansion of the erupting field. The two wings of the loop move slowly and get slower and slower, forming two almost stationary roots, while the top of the loop keeps a high velocity ( -.lOOkm/s) all the time. Moreover, the density enhancement in the wings becaomes more and more marked compared to the top, this is because, as the loop forms and moves, matter in the top region is slowed down all the time by gravity along the field lines. These results are in agreement with the results of 1161, basically reflecting the main observational features of mass eruptions in coronal loops [21-23). As another example, the eruption rate was raised 1.5 times (a= 1.822 x 10’ Wb/s) and the region of calculation extended to P= lob. The results are shown in Fig. 3. Apart from the density enhancement of the high density loop now increasing more strongly and fast, all other features are largely the same as in Fig. 2. The density enhancement and the velocity vary roughly linearly with the erupt ion rate.
4
5
G
7
8
910
I
2
3
4
5
6
7
8
910
Fig. 3 at (a)
Contours of density enhancement for a= 1.822 x lO”Wb/s t= 6000 s, with a maximum enhancement of _ 10, and (b) t= 12000 8, with a maximumenhancement of _ 15. The thick dashed curve graphs the boundary of the coronal arch.
There are two important differences between the present model and that of [16]. One is that, here, the erupting flux increases linearly with time (see (13)), while in [16), it increased exponentially. The consequence is that, here, the dense loop rises roughly with a uniform velocity while in [16], it had an accelerated rise. The other difference is that, here, we use an open background field with a neutral sheet, while in [161, we used a closed background field. Obviously an open field model is more reasonable, since it is only in this case that the erupted matter can follow the field lines into the interstellar space. Besides, the fast magnetosonic waves in front of the loop have different
290
HIJYou-qiu
and ZHANG Hui-nan
propagation properties in the two cases. For the closed background field, the equatorial wave propagates perpendicularly to the field lines and its wave velcoity is the greatest; as we go towards the pole, the inclination of the wave normal to the field become less and the wave velocity is lowered. For the open field, we have the opposite situation: in the neutral sheet region the Alfven velocity is less than the sonic velocity , magnetosonic waves propagate with the Alfven velocity; away from the current sheet, the Alfven velocity is greater than the sonic velocity and the angle between the wave normal and the field increases toward the pole, so the wave velocity also increases toward the pole. Hence, the front of fast magnetosonic waves assumes a concave form in the equatorial neutral sheet region (see Figs. 2 and 3). In Ref [16], two cases of the background field, the dipole and the octapole, were analysed and it was pointed out that the thickness of the top of the high density loop increased with radial distance as r”ms4 in the one case, and decreased as r-o.76 in the other. For an open background field with a neutral sheet, as Figs. 2 and 3 show, the thickness basically keeps constant at all distances. MacGueen and Cole 1241 measured the top thickness for nine typical ejection events in coronal loops and found that two kept more or less constant, two monotonically decreased with radial distance and five monotonoically increased, but more slowly than a linear increase. Previous theories all predicted a linear increase or faster, so did not agree with the above observations. According to the analyses in of the thickness at (161 and in the present paper, the variation the top is a slower than linear increase or even a monotonic decrease with distance; this is because the matter at the top is falling toward the root under gravity. This dynamical effect has been overlooked by usual analytic models. As Figs. 2 and 3 show, this effect is not obvious in the region of wave motion, so a numerical model based on the wave point of view will also fail to give the correct dependence of the thickness of the loop top on the height.
4. CONCLUSION In this paper numerical simulation was carried out of the atmospheric responses to magnetic flux eruptions at the base coronal streamers and the results of the calculation largely reproduced the main observed features of mass eruptions in coronal loops. This shows that flux eruption may be an important mechanism in generating such events. Under this mechanism, the high density loop and the low density region below it are plasma structures moving with the magnetic field rather than waves. The results of the calculation also revealed the density structure within the loop and the relevant dynamical effects and the wave phenomenon in front of the loop. Adopting an open background field with a neutral sheet allows the erupted material to follow the field into the interstellar space, and this is advantageous for further study of the interplanetary effects of coronal mass ejections. To check the effectiveness of the magnetic flux eruption we should secure the relevant magnetic data during the regular observations of coronal ejections. Lastly, a more reasonable numerical analysis should take
Solar
Atmospheric
291
Response I
into account the effect of the solar wind in the construction more realistic coronal streamer model.
of a
REFERENCES
[II 121
Pnc-,
[31
Low.
B. C,
141
Low,
B. C,
I51
Hu. W. R,
t61
Dryer.
173 [aI
Wu, Wu.
[91
Sccinolflon,R. S., Awon.
[lOI
Pncuman. G,
IllI
Cl21 1131
Houubo~i~s,
M.
and P&ad.
A. I., Anrophys.
65( 1980).
1.
220(1978),
Munro. P. H., and Fisher. R. R., A~wopLys. Aurophyr. A~rroplyr. and
J,
281(1984),
Space Sri,
Maxwell.
A.,
675.
369. I,
254(1982).
Solar Phyr,
92(1983).
Anrophyr.
Awophyhyr, 94( 1984).
335.
392. J.
373. 251(1979),
945.
S. T., Dryer, hf.. Nakagawa, Y. and Han. S. hi., Aswophya. J, S. T., Wang. S., Dryer, M. et al. Solar Phys, 85(1983), 3.51. 115(1982).
218(1978),
324.
39.
387.
HU You-qiu, Scientia Sinica, Ser. A, No. 10 (1985) 928. HU You-qiu and WANGran-ping, Chin.Astron.Astrophys. 11/l (1987) 83-86 = Acts Astropbys.Sin. 6/4 (1986) 312-316. HU You-qiu and JIN Shu-ping, Chin.Astron.Astrophys. 11/2 (1987) 171-178 = Chin.J.Space Sci. 7/l (1987) l-9.
[I41
Illing,
Cl51
Hu~+usen,
R. M. E. and Hundhauseo,
Ejections, Cl61
Ch. T.
G., Solar Phys.,
A. J., Long-Term Prep&&
[I71
Hu.
[ 181
f-w
[I91
Hu. Y. Q. =d
PI
Hun Y.
Y. Q,
91(1986),
A. J., A Possible Mechanism
10 J. Gcop+.
Awophyr.
I,
B. C., AwopLyr. Q,
Rrs,
10951.
of aronrl
bfarr
1987.
Hu. Y. Q, and Hundhouren, m be rubmiued
A. J. J. Gtophys.
Variations in the Charrcteristiu
J.
JS1(19118), SlO(l986),
WY 6. T., J. Compur.
A Multistep Implicit
Mwnetohydrodynamic
for Coronal
402. 953.
Php., 65(1984),
33.
Scheme for Tiime_&pendcnl
Flows, J. Compur. Phyr.,
Tw~imenrion~f
1989(in press).
[?I1
Jackson,
R. V. and
WI
Mx.Queco,
R. M.
v31
Sime. D. G., MacQuccn. R. M. and Hundhattsm, A. J.. J. Geophyr.
t241
MacQuecn, R. hf. aud Cole. D., Arrrophys.
Hindlcr.
Mar: Ejections,
RCS, 1969.
E., Solar Php,
60(1978),
and Fisher, R. R.. Solar Phy#. I,
155.
89(1983),89.
299(1985),
526.
Rer,
90(1985),
563.
Acta Astrophys.Sin.(1991)11/2,181-186
0 Pergamon Press plc Printed in Great Britain 0275-1062/91$10.00+.00
SOLAR A’IMOSPHERIC RESPONSES TOMAGNETIC FLUXJiMJPTIoNS. I. AMEEHANISMFOR CORONAL MASSElJErT1oNs’ IRJ You-qiu
ZRANG Hui-nan
Department of Earth and Space Sciences, University of Science and Technology of China ABSTRACT We carried out a numerical simulation of atmospheric responses to magnetic flux eruptions from the base of a helmet streamer. The results show that the material originally contained in the streamer is compressed and lifted upward by the erupting magnetic field, forming a high density bright loop. At the same time, fast magnetosonic waves appear in front of the loop and a low density dark region, below it. These results suggest that magnetic flux eruption is probably an important triggering mechanism of loop-like coronal mass ejection.
Key words:
Sun-magnetic
field-coral
mass ejection
1. INTRODUCTION Since the discovery of coronal mass ejection events in the seventies, a succession of theoretical models have appeared, analysing in detail the dynamics and generating mechanism of look-alike mass ejections. In the majority of analytical models, the mass ejection is regarded as an outwardly expanding magnetic structure carrying with it a froeen-in field and emphasis is laid on the dominant role played by the magnetic field in triggering off and accelerating the ejection [l-5]. On the other hand, in some numerical models [6-g], the high density loop is regarded as a kind of magnetic fluid wave. In these models, thermal pressure is the driving source and the initial atmosphere is in hydrostatic equilibrium (the magnetic field being a potential field or force-free) or magentostatic equilibrium and the the normal component of the field at the lower boundary is fixed and constant. Then, the effect of the magnetic field on the mass motion is one of obstruction rather than of acceleration. True, field
for a proper estimate of the physical effect on the coronal mass ejection we should await
of magnetic real-time data
1 Program supported by the National Natural Science Foundation. Received 1989 July 13; revised version 1989 September 15; passed by Referee 1990 April 4
Solar
Atmospheric
Response I
285
on the field; but a theoretical exploration of the effect is not without its value. Generally speaking, the action of a magnetic field on a plasma takes two for-: one is release of free energy through some type of instability 1101, and the other is acceleration of the plasma through eruption of magnetic flux, When the erupting field is antiparallel to the background field, there will appear neutral electric current sheet between the two, on this we have carried out analytical analyses [ll, 121 and numerical simulations [13], connecting together the current sheet formed with the bright loop structure of the ejection. Observations have shown that rany ejection events took place on top of helmet streamers and that the ejected urtter was initially located in the corona [14, 153. Based on this observational fact we carried out a numerical analysis [16] of the atmospheric responses to magnetic flux eruptions under a high density coronal arch in a closed background field (the erupting and background fields are in the same direction); the results reproduce the main observed features of loop-like ejections. For coronal streamers, an open background field with neutral sheet and wind is more reasonable. Hence, in this paper, we adopt an open field with a neutral sheet as the background field and carry out a numerical simulation of the atmospheric responses to flux eruptions at the base of the streamer. For simplicity, we again neglect the solar wind.
2. BASIC EQUATIONS AND INITIAL
BOUNDARY CONDITIONS
For axisyuetric problems in spherical coordinates (r, 8, +) we can introduce the magnetic flux function p(t, r, e) and express the field by
(11 Then, the equations
for
the two-dimensional
i!.?t+v-(pu)-0,
ideal
magnetic
fluid
are
(2)
ar
~+u.va+RvT+
(31
ar where p is the density, temperature, R, C, M, 7 gravitational constant, the permeability of the
v(vr, ve) is the velocity, T is the and p are the gas constant, the mass of the Sun, the polytropic index and vscuum.
286
HU You-qiu
and ZHANG Hui-nan
We now construct a aaguetostatic equilibriua state to siaulate approximately the coronal atreauer structure and use it as the intitisl conditions for our nuuerical calculations. Using the uethod described in (171, we start froa the potential field
we cap construct
the following
aaguetostatic
equilibriua
state: (71
4 m &&,
(8)
,
(9) where po = m(A), derivatives with
et - ART.,
Qo’ aud qo” are first respect to A and P:
and second order
--iww[-j$-(1 --+)I
specify the pressure auci density distributions in the equilibrium atmosphere, G being the teuperature, pc the density at the base of the corona and & the radius of the Sun. A necessary and sufficient condition for this equilibrium state to hold is, 1171,
Further, let S, be the potential field of the equatorial neutral sheet extending frou radius a to infinity; its flux function can be expressed as, [ 181,
and Qc is a constant. The field configuration at a= l.!iR~ is shown in Fig. la where point P is the inner edge of the equatorial neutral sheet, below the arc PC is the closed field (A ( 01, above it, the open field (AZ 0) with the neutral current sheet. The point C is located on the surface of the Sun at a colatitude of 63,38O, approximately. For a steauer
with a tail
half
opening
angle
of x, we can take
Solar
Atmospheric
Response I
201
u I
Fig.
1
2
2
4
6
6
The initial helmet streamer: (a) the magnetic configuration with a neutral point P, and (b) contours of density enhancement (po - pl)/pi
Then, the boundary of the streamer is A= Ao, the outside (A > Ao) is the isothermal atmosphere in static equilibrium of the potential field, and the inside in the magnetostatic equilibrium state, the magnetic field monotonically decreases inward to zero at the coronal arch (As 0). It can be shown that, for any set of parameter values, the inequality (11) holds identically. This shows that magnetostatic equilibrium state formed in this way is a steady state under the ideal MHD approximation. In the present calculation we took the following parameter values: a= 1.5 b, pc = 1.61 x lO=Ub, x = 5”, pc = 1.67 x lo-l3 kg/m3, 7bc = 1.5 x 106K. The corresponding distribution of the relative streamer/backgournd density enhancement is shown in Fig. lb. At the base of the coronal arch, the density is about 10 times that of the background; at the tail, about 2.5 times. There is some arbitrariness in the above choice of parameter values and the resulting density distribution may not represent faithfully the actual structure of the streamer. Fortunately this circumstance is not important for the study of the mass ejection mechanism. Actually we calculated for different sets of parameter values and we found that, as long as the density is higher in the streamer than in the background, the results are not essentially different. Consider now the boundary conditions. The region of solution is lRgIr16Rg, O~6~90° and we used a 32 x 27 non-uniform grid (32 in the r-direction), which is dense near the bottom (r= ~F&J) and the equator (6 = 90°) so as the raise the resolving power at the eruption region and the current sheet. Symmetry about the polar
266
HU You-qiu
snd ZHANG Hui-nan
axis and the equator is imposed and for the top, a non-reflecting boundary condition [19]. The botton is the perturbed boundary, the flux eruption is confined to the bottom of the streamer, that is, ec~e490”. the perturbed boundary condition is assumed to he
where a is eruption rate. Taking the velocity at the bottom to be zero is a simplifying approximation, which roughly holds when the erupting field is far greater than the background field and when the erupting rate is not large. For the region tl< &, we have fixed T= ~0, v = O.Also, the teaperature and density of the entire bottom region are to be determined by compatibility conditions derived an almost eigenvalue method f191. Finally, the set of equations (2)~(5) are numerically integrated by a multi-step, implicit algorithm 1201.
3. THE ~ICAL
~gSULTS
In the calculation below we took y= 5/3, u=O.729 X 101’Wb/s+ Fig. 2 shows the contours of relative density enhancement [p( t,r,6) - p~]/p~ at t = 6000 8 and 12000 8. Also shown is the magnetic field configuration at the initial time (chained) and at t (broken), the thick broken curve represents the boundary of the arch (A= h/2). We see that, under the action of the newly erupted magnetic field, matter in the original arch is compressed and raised to form a high density loop, leaving below a low density region, both expanding in step with the arch boundary. The high
’ IKO
W
Fig. 2 Contours of density enhsncemnet for a=0.729x IO”Wb/s at (a) t= 60008, with a maximumenhancement of IV 7.3, and (hf t= 12000 s, with a maximumenhanceuent of _ 6.5. The thick dashed curve graphs the boundary of the coronal arch.
Solar
Atmospheric
Response I
289
density loop comes form the compressionof matter both sides, and particularly from the inner side of the boundary, while the low density region is caused by the expansion of the erupting field. The two wings of the loop move slowly and get slower and slower, forming two almost stationary roots, while the top of the loop keeps a high velocity ( -.lOOkm/s) all the time. Moreover, the density enhancement in the wings becaomes more and more marked compared to the top, this is because, as the loop forms and moves, matter in the top region is slowed down all the time by gravity along the field lines. These results are in agreement with the results of 1161, basically reflecting the main observational features of mass eruptions in coronal loops [21-23). As another example, the eruption rate was raised 1.5 times (a= 1.822 x 10’ Wb/s) and the region of calculation extended to P= lob. The results are shown in Fig. 3. Apart from the density enhancement of the high density loop now increasing more strongly and fast, all other features are largely the same as in Fig. 2. The density enhancement and the velocity vary roughly linearly with the erupt ion rate.
4
5
G
7
8
910
I
2
3
4
5
6
7
8
910
Fig. 3 at (a)
Contours of density enhancement for a= 1.822 x lO”Wb/s t= 6000 s, with a maximum enhancement of _ 10, and (b) t= 12000 8, with a maximumenhancement of _ 15. The thick dashed curve graphs the boundary of the coronal arch.
There are two important differences between the present model and that of [16]. One is that, here, the erupting flux increases linearly with time (see (13)), while in [16), it increased exponentially. The consequence is that, here, the dense loop rises roughly with a uniform velocity while in [16], it had an accelerated rise. The other difference is that, here, we use an open background field with a neutral sheet, while in [161, we used a closed background field. Obviously an open field model is more reasonable, since it is only in this case that the erupted matter can follow the field lines into the interstellar space. Besides, the fast magnetosonic waves in front of the loop have different
290
HIJYou-qiu
and ZHANG Hui-nan
propagation properties in the two cases. For the closed background field, the equatorial wave propagates perpendicularly to the field lines and its wave velcoity is the greatest; as we go towards the pole, the inclination of the wave normal to the field become less and the wave velocity is lowered. For the open field, we have the opposite situation: in the neutral sheet region the Alfven velocity is less than the sonic velocity , magnetosonic waves propagate with the Alfven velocity; away from the current sheet, the Alfven velocity is greater than the sonic velocity and the angle between the wave normal and the field increases toward the pole, so the wave velocity also increases toward the pole. Hence, the front of fast magnetosonic waves assumes a concave form in the equatorial neutral sheet region (see Figs. 2 and 3). In Ref [16], two cases of the background field, the dipole and the octapole, were analysed and it was pointed out that the thickness of the top of the high density loop increased with radial distance as r”ms4 in the one case, and decreased as r-o.76 in the other. For an open background field with a neutral sheet, as Figs. 2 and 3 show, the thickness basically keeps constant at all distances. MacGueen and Cole 1241 measured the top thickness for nine typical ejection events in coronal loops and found that two kept more or less constant, two monotonically decreased with radial distance and five monotonoically increased, but more slowly than a linear increase. Previous theories all predicted a linear increase or faster, so did not agree with the above observations. According to the analyses in of the thickness at (161 and in the present paper, the variation the top is a slower than linear increase or even a monotonic decrease with distance; this is because the matter at the top is falling toward the root under gravity. This dynamical effect has been overlooked by usual analytic models. As Figs. 2 and 3 show, this effect is not obvious in the region of wave motion, so a numerical model based on the wave point of view will also fail to give the correct dependence of the thickness of the loop top on the height.
4. CONCLUSION In this paper numerical simulation was carried out of the atmospheric responses to magnetic flux eruptions at the base coronal streamers and the results of the calculation largely reproduced the main observed features of mass eruptions in coronal loops. This shows that flux eruption may be an important mechanism in generating such events. Under this mechanism, the high density loop and the low density region below it are plasma structures moving with the magnetic field rather than waves. The results of the calculation also revealed the density structure within the loop and the relevant dynamical effects and the wave phenomenon in front of the loop. Adopting an open background field with a neutral sheet allows the erupted material to follow the field into the interstellar space, and this is advantageous for further study of the interplanetary effects of coronal mass ejections. To check the effectiveness of the magnetic flux eruption we should secure the relevant magnetic data during the regular observations of coronal ejections. Lastly, a more reasonable numerical analysis should take
Solar
Atmospheric
291
Response I
into account the effect of the solar wind in the construction more realistic coronal streamer model.
of a
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