Magneto-hydrodynamic simulation of prominence formation

Chin.Astron.Astrophys. Act.Astron.Sin. 26

10 (1986) 129 - 134 (1985) 314- 320

MAGNETO-HYDRODYNAMIC

WANG Shui

Received

and

SIMULATION

WANG Ai-hua

1984 July

Pergarron

Journals.

OF PROMINENCE

University

of Science

Printed in Great Britain 0275-1062/86$10.00+.00

FORMATION

& Technology

of China

17

ABSTRACT Using the mechanism of radial ejection of cold mass from the a numerical simulation is made of the formation, solar surface, evolution and maintenance of prominences. The results show that, with the initial magnetic field taken to be that of a coronal streamer under magnetostatic equilibrium,the pile-up of the cooler and denser plasma will deform the closed magnetic field below the neutral sheet, making it flat and even concave in shape. The deformed field in turn supports the prominence material against gravity. This structure is maintained after the mass ejection ceases. It is thus demonstrated that magnetic field plays an important role in the formation and stable maintenance of prominences.

1.

INTRODUCTION

Quiet prominences are a kind of long-lasting, slowly-varying feature in the solar made of cold and dense plasma atmosphere, (characteristic temperature 5(+3)K, electron density (+ll)cmm3) floating in the hot and tenuous corona ( (+6) K, (+9) cme3) . Their size is generally taken to be 5(+3)km wide, 5(+4)km high and 1(+5)km long, and they can last several days to several months [l]. Their structure can be clearly seen in eclipse pictures in Ha in emission. From the Zeeman effect in the emission line we can measure a magnetic field (average, sight-line field) of about 5 G, [2]. From the observed Ho data we can also fairly accurately deduce the neutral line in the vertical field, the polarity, the location of maximum gradient and the general direction of the horizontal field, and the clearest indication of the neutral line is the orientation of the quiet prominence [3]. Observations have also shown that quiet prominences sometimes appear at the boundary of two weak fields of opposite polarities, [41, and that they could be regarded as located at the bottom of streamer-like field structures in the corona [S]. Theoretical studies of quiet prominences comprise generally two sets of problems. One concerns the maintenance and stability of To explain the stable these objects. maintenance of this cold and dense material in the solar atmosphere, some authors first resorted to a one-dimensional, isothermal model under magnetostatic equilibrium and proposed that the prominence is supported by

the magnetic field, that is, the cold and dense plasma sheets are suspended from bent field lines, [6 - 81. Low then discussed a non-isothermal model [9]. Lerche and Low [lo] obtained an accurate, non-linear solution for a horizontal, cylindrical and flow-carrying promenince. Because the prominence temperature is far below the background temperature, one must consider the energetics of thermal transfer, radiative loss, dissipation of mechanical hence some authors start energy etc., simultaneously with the equations of thermal equilibrium and magnetostatic equilibrium and discussed self-consistent models of the prominences, [ll - 141. The stability of these theoretical models forms another important topic [15]. The other set of important problems address the mechanism of prominence The importance of the magnetic format ion. configuration on the formation was discussed by Kuperus and Tandberg-Hanssen [16]. The majority of prominences are related to coronal streamers. They assumed that, during the time of cold contraction, the coronal field around the neutral sheet and the gas pressure do not vary while the net effect of mechanical energy dissipation and radiation loss is to cause a ten-fold increase in the density of the plasma and a ten-fold decrease in its density, temperature. After the cold contraction, tear-type instability leads to re-connection of field lines across the neutral sheet and the formation of the prominence, as generally observed [16]. Smith and Priest have also used the thermal instability in

130

WANG & WANG

the current sheet to study the question of prominence formation [17]. In this paper, we made numerical simulations of the magnetohydrodynamicsof the formation and maintenance of the quiate prominences. We investigate the entire process of formation and growth and propose a new physical mechanism.

2.

N~RICAL

SIMULATION

In a previous paper [IS], we used a combined analytical and numerical method to obtain the magnetic configuration of coronal streamers under magnetostatic equilibrium. Since the solar atmosphere is composed of fully ionized tenuous plasma, we can regard its electrical conductivity as infinite and its magnetic permeability as equal to 1. For simplifying the mathmatics, we neglected the rotation of the Sun, heat transfer, radiation loss and other energetic processes and used the usual ma~etohydrod~amic equations. In cylindrical coordinates r, 0, 4, the two-dimensional,magnetostatic equilibrium state satisfies the following equations:

(4) where R is the gas constant, g=se(ro/ri2 is the gravity and other symbols have their usual meanings. According to the method developed by Low [19], we introduce function n(r, 0) in (4) and obtain

Hence, from (1) - (31, we obtain a non-linear elliptic-type partial differential equation, 1 B’F

g-+

161

z-z9

where

P(R, e) - P(F,

R) =q Pp(F)exp

dR’ % 6(R’)@(F, R

I ! -

1

R’) ’

(71

and the non-dimensional quantities are defined by R 1;(R,

rJf&,

R, -

@> - A(r,

mw

-

~wm,

r,/H,, G(R, e) -

@)/Am

@(R,

8) -

GIF(R,

T(r,

elf

WTd,

in which Hi, A~, B , pO, TV are constants, and po is an arbitrary function which we can take to have the form, P181,

P,(F)a

Co-t- C,F -I- GF’,

191

with Cot CIr C2 constants. Solving (6) numerically, after having set G=G, we obtain the base state of the field of a streamer under magnetostatic equilibrium. (see Fig. 3(a)). Different values of CO, Cl, ~2 will give different configurations.

131

Formation of Prominence

To discuss further the process of formation of prominence under the coronal streamer field, we use the following set of MUD equations in the two-dimensional (a/a+=0) case:

(101 (111

(12)

(13) (14) (15) (16)

LIT ---~~~(r'TY,)-_~((TYg6ine)

+_T_B(r~~)+ 3r'6r '

at

T

P-PR,T.

This set of equations, (10)- (19), can be solved numerically using the Euler implicit scheme [20]. In our calculations, the iteration accuracy was taken to be IP??- P?,,I < 10-3 P?.,

(19)

and the range of calculation was lre(r(6re, oO< 8590". At the pole (8 = 0") and on the equator (6 = 900), symmetrical boundary conditions were used. The physical boundary conditions at the bottom of the calculated region and the calculational boundary conditions at the top were determined using the method of projective characteristicsand the condition of no reflection, [20]. In the case of Vz 0, 5 parameters at the bottom can be assigned arbitrarily and the other 3 determined by the consistency equations. We can then use the equations of projective characteristicsto determine 5 parameters at the top and the condition of no reflection at the boundary to determine the other 3. We used the following values in calculating the base state: coronal temperature 2(+6)K, and at the equator on the solar surface, plasma density 3(-16) 9/cm3, and gas to magnetic pressure ratio 8*=0.1. The constants in (8) were taken to be Ho= Ire, A0 =Bore2A0, Af3=4.5'; pa and Ba were given the values at the equator on the surface, determined from (18) and the 13*value. In the calculation, the steplength in time was taken to be At=2Os and the steplength in

radius was taken to be (Ar)i=riAB. After obtaining the stable base-state, we assumed that at t=O, cold matter is ejected upwards in the region 83.2S"&8i 90.00" on the solar surface, the velocity of ejection increasing linearly to the equator. It also increases in time reaching its maximum value of 200 km/s (at e=90°) at t=200s. This velocity is kept up until t=4500swhen the prominence is completely formed. At this point, the velocity falls to zero and the ejection terminates. Also, we assumed that, during the ejection, the density in the bottom part of the ejection region is twice the base state value, and the temperature is half. The results of the present calculation can be expected to simulate the processes of formation and maintenance of prominences generated by radial ejections of cold matter from thq solar surface.

3.

RESULTS OF CALCULATION

Part of the calculated results is shown in Figs. l- 3. Figs. l(a) and l(b) show the variation of the relative density over the equatorial latitudes, near the bottom of the neutral sheet (r= 1.46 re), at two instants of time t= 4500s and t= 6000 s. We see that, following the ejection of cold matter from the surface, a high-density region is gradually built up above the equator. It is formed by the piling-up of the ejected material and the falling material. This

132

WANG & WANG

01,.,,.20 polar distance (b) polar distance

(a)

Fig. 1

IL Ml

~--L-_-1.._-.

-/II

..i_

Distribution of the relative density over the equatorial region, 65"~0~115", at r= 1.46ro, (a) t=4500s, (b) t=6000s.

-L.

KI1

!I11

/____&_I 1wt

polar distance

Fig. 2

I111

12(1

”Iill

’

70

“-x-*

120

HO

polar distnace

Distribution of the relative tmeperature over the equatorial region, 65'~0~115~, r=1.46 ro, (a) t=4500s, (b) t=6000s.

(d)

Fig. 3

Evolution of magnetic configuration during the formation of prominence. (a) t=Os, (b) t=3500s, (c) t =4500 s, (d) t =6000 s. 6 times enlargement in (b)- (d) and 65'~8t115".

Formation

high-density region is already conspicuous at t=4500 s (Fig. l(a)), the density reaching a maximum at the equator and falls rapidly on Even after the ejection had both sides. stopped, (Fig. l(b)), this high-density region was kept up. Fig. 2(a) and 2(b) shows this distribution of the relative reciprocal temperature at the same two instants. We see that a lowtemperature region is formed over the equator and is maintained, its position matching that of the high-density region. Again, the temperature reaches a minimum at the equator and rapidly increases on both sides and, again, it was kept up even after the ejection had stopped. Thus, the formation and growth of this cold and dense structure can simulate the formation and maintenance of prominences. Figs. 3(a) - (d) sketch the evolution of the magnetic field configuration during the formation of the prominence. We can clearly see that, under the action of the cold and dense matter, the field lines over the equator undergo a remarkable transformation. As the cold matter piles up, the field lines first spread sideways, become flattened in shape and slowly become pressed downwards, form a hollow near the surface and assume a concave shape. This result is in excellent agreement with Low’s sheet model [9] of quiet prominence obtained from magnetostatic equilibrium. We can also see that, not only does the cold dense matter correspond to the location where the field lines are concave, it is also situated in the closed region of the streamer field, which supports a neutral sheet above. This picture agrees very well with the results deduced from observations The results of calculations also [4, 51. show that the pressure in the cold dense region is not very different from the pressure in the surrounding regions and it mainly depends on the bent field lines, i.e., action from the magnetic field, that provides the necessary support. After the cessation of the radial ejection and the completion of the prominence, this magnetic configuration keeps on being stably maintained (Fig. 3(d)). Thus the importance of magnetic field on the formation and persistence of prominence is amply demonstrated.

4.

DISCUSSION

In this paper, of magnetostatic

we started from the equations equilibrium and obtained a

of

Prominence

133

magnetic configuration that includes a closed region, a neutral sheet and an open region, the first two constituting a streamer structure. Following the continuous accumulation of cold dense matter, the field in the closed region is radically altered; it is flattened into a concave shape, to support the cold dense matter. The results of our simulation agree well with the observational data and the results of theoretical analysis by other authors; they also demonstrate the very important role played by magnetic field in the formation and maintenance of prominences. To simplify the mathmatics, a number of assumptions and approximations were made, especially the following two items should be improved upon in future studies. First, in our calculations we neglected heat transfer, radiation, and other energy processes and our equations did not include the equation of energy equilibrium. In the actual solar atmosphere, such processes as heat transfer, radiation, heating of the heat source (e.g. due to wave dissipation) are all very important. In discussing questions of equilibrium, we should consider the mutual consistency between the thermal and magnetostatic equilibria, [14]. In discussing the dynamical processes, we should use the MHD equations that include the equation of total energy. However, from the discussion by Kuperus and Tanderg-Hanssen of 1161,it appears that a consideration energetics may further assist the formation of prominence. Second, in this paper, we assumed that during the ejection of cold matter, the density at the bottom of the ejecting region is twice the density in the base state, and the temperature, one-half. There is a large degree of arbitrariness in this assumption and it is the main reason for the substantial differences between our calculated density and temperature and the observed values. If we regard the ejected matter to originate in the photosphere or the lower part of the chromosphere, then the density can be as high as 3(-7) g/cm3 and the temperature as low as 6(+3)K. However, it is very difficult to take directly these values in the numerical simulation, in any case it will not alter our basic conclusion that radial ejection of cold matter below the coronal streamer field configuraton can lead to the formation and steady maintenance of prominence.

WANG & WANG

134

REFERENCES III 121 131 I41 I51

I61 171 181 [91 [ 101 [Ill II21 [I31 1141 1151 [ 161 [171

P81 PI WI

‘I’nndber~IImssen, E., Solar I’:ominmce. D. Reidel Pub. Corn.. (1974). Rust. D. /lp. J. 150 (1966), 313. Gibson, E. G., The Quiet Sun, hASA, (1973). Murk, S., Solar Phyr., 31 (19731, 3. Saito, K. and Tandberg-Hanssm, E. Solar Phyr.. 31(1973l. 105. Dungey. J. W., hf. N. R. A. S.. 113(1953). 180. Kiprxnhslm. R. and Schlurw, A., Zs. f. Ap.. 43 (1957). 46. Ikowvn. A.. Ap. I., 128 (195R). 646. law, B. C.. Ap. I.. 1% (197jk, 211. Lack I. and Low, B. C.. SOLI Nlyr., 66 (1980), 285. kche, I. and Low, B. C. S&r Phyr.. 53(1977), 385. Milne, A. M, Priest. E.P.and Roberts, B, Ap. I.. 232 (1979). 304. Low, B. C. and Wu. S.T.. Ap.I., 248 (1981), 335. 1:/k, ItYJlIilw. 27(l’J82), 74. Freidbcrg, Rev. Moden Phy~, 54 (1982). 801. Kupem,M. andTmdberg-Hansen, E..Solar Phys.. 2(1967), 39. Smith, E. A. and Priest, E. K.. Solar Phyr., 53 (1977). 25. WANG Shui & WANG Ai-hua, Kexue Tonqbao 29 (1984) 1360. Low, B.C., Ap. J. 197 (1975) 251. WANG Shui, HU You-qiu & I'JU Shi-can, Scientia Sinica Ser.

A

(1982)