eruptive instability of cylindrical prominences

E R U P T I V E I N S T A B I L I T Y OF C Y L I N D R I C A L P R O M I N E N C E S

B. V R S N A K

Hvar Observatory, 58450 Hvar, Yugoslavia

(Received 5 April, 1990; in revised form 14 June, 1990)

Abstract. The stability of prominences and the dynamics of an eruption are studied. The prominence is represented by an uniformly twisted, curved, magnetic tube, anchored at both ends in the photosphere. Several stages of the eruption are analyzed, from the pre-eruptive phase and the onset of the instability, up to the late phases of the process. Before the eruption, the prominence evolves through a series of equilibrium states, slowly ascending either due to an increase of the electric current or to mass loss. The eruption starts when the ratio of the current to the total mass attains a critical value after which no neighbouring equilibrium exists. The linearized equation of motion was used to obtain the instability threshold, which is presented in a form enabling comparison with the observations. The height at which the prominence erupts depends on the twist, and is typically comparable with the footpoint half-separation. Low-lying prominences are stable even for large twists. The importance of the external field reconnection below the filament, and the mass loss through the legs in the early phases of the eruption is stressed. The oscillations of stable prominences with periods on the Alfv6n time-scale are discussed. The results are compared with the observations.

1. Introduction

Quiescent prominences are condensations of a cold plasma embedded in the coronal magnetic arcades above the inversion lines of the photospheric magnetic field. Although they expose an intricate and brisk internal activity, their overall characteristics change slowly on a time-scale of months (Schmieder, 1988, 1990). The disappearance of quiescent prominences (also called a disparition brusque or DB-event) can be thermal or dynamical (Mouradian and Soru-Escaut, 1989). In the dynamical disparition brusque a prominence erupts upwards attaining an ascending velocity of the order of 100 km s - 1. The eruption is frequently followed by a two-ribbon flare, coronal transient or type II radio burst indicating an MHD blast wave propagating into the interplanetary space (Tandberg-Hanssen, 1974; Krager, 1979; Schmahl et al., 1986). Soon after the eruption, a new prominence usually forms at the same inversion line (Tandberg- Hanssen, 1974; Schmieder, 1988).

The eruption is sometimes preceded by preheating (Webb et al., 1976; Schmahl et aI., 1986), mass outflow through the legs (Rust, Nakagawa and Neupert, 1975; Vrgnak et al., 1987), or by the photospheric magnetic field reorganization such as emerging flux, moving pores, flux cancellation, etc. (Schmieder, 1990). The intricate internal structure of a prominence gradually simplifies, most often transforming into a bundle of helical threads twisted around the prominence axis. It is believed that these threads outline the helical/cylindrical magnetic field structure (Schmahl and Hildner, 1977; House and Berger, 1987; Moore, 1988; Vrgnak et al., 1988; Vrgnak, 1990a). A prominence steadily rises with a velocity in the order of 1-10 km s - 1 until reaching a critical height at which violent acceleration starts. In 10-20 min the ascending velocity can attain a value of

Solar Physics 129: 295-312, 1990. �9 1990 Kluwer Academic Publishers. Printed in Belgium.

296 B. VR~NAK

several hundred km s - ~. A phase of constant velocity and deceleration is frequently observed (Tandberg-Hanssen, 1974). The eruption is usually accompanied by 'detwisting' and rotational motions (Engvold, Malville and Rustad, 1974; Schmahl and Hildner, 1977; Vrgnak, 1980; House and Berger, 1987; Moore, 1988; Vrgnak, 1990a).

The eruption of prominences has been treated in a number of papers (Sakurai, 1976; Smith et al., 1977; Pneuman, 1980, 1984; Kuperus and Van Tend, 1981 ; Demoulin and Priest, 1986; Steele and Priest, 1989; Martens and Kuin, 1989; Hood and Anzer, 1987; Vrgnak, 1990b). The studies of the stability of current sheets, magnetic arcades and flux tubes link eruptive prominences to flares and coronal transients (Raadu, 1972; Anzer, 1978; Mouschovias and Poland, 1978; Van Tend, 1979; Hood and Priest, 1979, 1980, 1981; An, 1984; Hood and Anzer, 1987). Different instabilities were analyzed, and various effects and geometries have been considered (Priest, 1982a, b; 1985, 1990), providing a range of criteria for the eruption onset. However, the results were usually not in a form appropriate for comparison with the observations. Here we present a study of an eruptive instability of cylindrical prominences where the threshold and the dynamics of the eruption are given in a form which provides an observational test.

2. Description of Cylindrical Prominences

2.1. GEOMETRY AND STRUCTURE

We will represent a prominence with helical structure as a curved cylinder of diameter 2ro, anchored at both ends in the dense photosphere. The footpoint separation will be denoted as 2d and the height of the summit as h (Figure 1). If the shape of the cylinder axis is approximated by a circular arc, the radius of curvature Ro, the axis length l o and the height of the axis summit h can be expressed in the parametric form:

R = R o / d = 1/cos~, (1)

Z = h /d - - (1 + sin ~)/cos ~, (2)

l = Io/d = (zr + 2~)/cos ~, (3)

where the parameter ~ is the angle 'pointing' from one of the footpoints towards the center of curvature of the axis (Figure 1) and R, Z, and 1 are the parameters normalized with respect to the parameter d. Since the footpoints are anchored in the photosphere, the value of d is constant on time-scales shorter than the photospheric magnetic field evolution time-scale.

The internal structure in helical prominences can be represented as a bundle of cold material threads (Vrgnak and Ru~dj ak, 1982; Vrgnak, 1984a; 1985; Vrgnak et al., 1988), each twisted around the cylinder axis at a radial distance r and characterized by the pitch length 2 or by the pitch angle 0 (Figure 1). These parameters are related as

tg0(r) = 2rrc/2(r). (4)

E R U P T I V E I N S T A B I L I T Y O F C Y L I N D R I C A L P R O M I N E N C E S 297

i '.')... ..' ,.

. . � 9 "'. .: �9149 . . . . . . . . . ~ . . .

/2 i

Fig. 1. Sketch of the prominence with definitions of symbols used in the text.

Assuming constant pitch of the helix along the axis, the total twist of one helical thread can be expressed as

�9 (r) = 2rcN = 2 ~ l o / 2 = l o X ( r ) / r , (5)

where N ( r ) is the number of turns of the helix in a distance I o and X ( r ) = tg0(r). Since the observations indicate X ( r ) ~ r, i.e., 2(r) = const. (Vrgnak et aL, 1988) we will

assume hereafter the uniform twist configuration N ( r ) = const., i.e ~(r) = const. :

= X lo / r o = I D X , (6)

where D = d/r o and X = tg0(ro).

2.2. M A G N E T I C FIELD AND FORCES

Typical order of magnitude values for the prominence plasma density and temperature, and the magnetic induction are n = 1017 m - 3 , T = 10 4 K (Vial, 1986) and B = 10 -3 T

(Leroy, 1986), respectively. So the ratio of plasma to magnetic energy density fl = 2 # o n k T / B 2 is on the order of 0.01, i.e.,/? ~ 1, and the motion of plasma is controlled by the magnetic field. Using typical internal velocities v = 1-10 km s - 1 (Schmieder, 1988) in quiescent prominences and magnetic diffusivity r/= 109 T - 3/2 (Priest, 1982a),

one finds the magnetic Reynolds number in the tube to be R m = rov/q >> 1, meaning that the diffusion of the plasma across the magnetic field can be neglected. So it is reasonable to assume that the cold prominence material visible in He is bound to the magnetic field

298 B. VR~NAK

lines and that the helical fine structure threads outline the configuration of the magnetic field. The observed helical structure then indicates that the magnetic field in an element of the cylindrical prominence can be described in the first approximation as cylindrically or helically symmetric (House and Berger, 1987). Hereafter we will assume a cylindrically symmetric, pinched magnetic field configuration (Tandberg-Hanssen and Malville, 1974; Malville and Schindler, 1981). The complete magnetic field configuration in an element of the tube can be described as a superposition of this pinched field, a field caused by the inductive effects in the photosphere (Kuperus and Raadu, 1974) and eventually the background field (Van Tend and Kuperus, 1978). Furthermore, we will take into account a curvature of the tube axis which introduces the elements of toroidal geometry. Since observations indicate a uniform twist configuration (Vrgnak et al.,

1988) we will use as a base a simple uniform twist force-free field, with azimuthal and longitudinal components:

B4~(r' ) = B r ' X / ( r ' 2 X 2 + 1), (7)

BII ( r ' ) = B / ( r ' 2 X 2 + 1), (8)

respectively. Here r' = r/r o is the radial coordinate normalized by the radius of the tube and X = Bee(ro)/Bll (ro). The axial current I = 2rcXBro/#o(X 2 + 1) associated with this field induces at the photospheric boundary eddy currents which generate the magnetic field component B,,. In the model of a straight wire situated at a height h and carrying a current I, B,, is horizontal at the prominence location and causes an upward Lorentz

force

F m = #oI2/4rch (9)

per unit length (Kuperus and Raadu, 1974). For currents on the order 101~ A, (Ballester and Kleczek, 1984) this term is the order of 10 7 N m - 1. Furthermore, the curvature of the prominence axis causes a downward tension of the longitudinal field component and an upward force due to the magnetic pressure gradient of the azimuthal component (Vrgnak, 1984b). These forces can be approximated (per unit length) as

Ft = 1~oI2/2rcRX 2 ' (10)

Fk = l~oI2/4rcR , (11)

respectively (Vrgnak et aI., 1988), being of the order of 10 7 N m - 1 In the presence of a background field B o, there will be a further component of the

Lorentz force, F o = IBo , which is directed upwards in normal polarity prominences, and downwards in inverse polarity prominences (Priest, 1990). Hereafter we will neglect this force, which implies that B o < 10 4 T. Finally, the gravitational force Fg = pgr2~ (p is the mass density in the prominence, g is the acceleration of gravitation) must be taken into account since it is on the order of 2 x 10 7 N m - 1

The forces considered provide an equation of motion of the form

= A ( l / h + l /R - 2 l / R X 2) - g ( h ) , (12)

ERUPTIVE INSTABILITY OF CYLINDRICAL PROMINENCES 299

where A ( h ) -- # o I 2 / 4 z c M , and we assume that the mass of the prominence (M) is uniformly distributed along the axis so that the gravitational force per unit length can be written as Fg = Mg/ l . The parameter A is related to the 'axial' Alfv6n velocity DA = BII (#oP)- 1/2 as A = X Z v 2 / l .

2.3. B A S I C R E L A T I O N S A N D C O N S T R A I N T S

There are two important constraints imposed by the fact that the magnetic field lines are anchored in the photosphere. In the absence of diffusion, the flux of the longitudinal component of the magnetic field must be constant along the axis of the tube, and second, in the absence of magnetic reconnection within the tube, the total twist of a field line (~) must be conserved. If the radius of the tube varies along the axis and if it changes in time, some redistribution &the twist per unit length will occur along the tube (Parker, 1974; Jockers, 1978; Browning and Priest, 1983), but the overall twist (i.e., the number of turns of a helical field line) must remain constant. For the matter of simplicity we will consider in the model a uniform radius along the axis. Conservation of longitudinal flux implies

where

ro 2 ~

~[[ =f fBll(r)rd~dr=#orol/Xf(X)=const., o o

(13)

f ( X ) = X 2 / ( X 2 + 1)ln(X 2 + 1). (14)

For the uniform twist configuration, the twist conservation implies

= X lo / r o = const., (15)

which is equivalent with the conservation of the azimuthal flux in the tube when combined with Equation (13). Eliminating r o from Equations (13) and (15) one obtains

I l o / f ( X ) = const. (16)

Neglecting the dissipative effects, we close Equations (12) and (16) by demanding that the magnetic flux through the circuit (~,) does not change during the eruption, i.e., that

I L = const., (17)

where L is the self-inductivity of the circuit. Vr~nak (1990b) considered the case I = const, representing the prominence current as a part of a global solar current system, where the change of self-inductivity is negligible during the eruption if the change of the height is smaller than the solar radius. Another possibility is to treat the prominence as a current circuit closed at the solar surface (Anzer, 1978; Martens, 1986; Martens and Kuin, 1989). The self-inductivity of a rectangular current circuit can be applied for the low-lying (Z < 1) prominences (Martens, 1986) and in that case one obtains

I ( - 1 - Z + 1"1 - F2 + Z ( F 3 - F4)) = const., (18)

300 B. VR~NAK

where

and

1+7i F,. = -2 /2 ̀ + I n - , (19)

1 - 7 i

2i = sin(arctan(Y~)), (20)

II1 = 2D, Y2 = 1 /Z , Y3 = 2h/ro, Y4 = Z . (21)

After the onset of the eruption, as the height increases, the shape of a circular ring (Anzer, 1978) becomes the more appropriate representation (Figure 2(a)). Taking into account the self-inductivity of a circular wire (Batygin and Toptygin, 1964) one obtains the relation

i R ( l n ( R / r o ) _ 7) = const. (22)

instead of Equation (18). Equations (12) and (16), supplemented by (18) or (22) provide straightforward solutions in the form h(~), i.e., acceleration as a parametric function of the height (Equation (12)).

Equations (18) and (22) do not involve the effects of reconnection below the filament (Martens and Kuin, 1989). Such reconnection can cause a two-ribbon flare (Priest, 1982b), or in less energetic events a flare-like plage brightening (Ru2djak et al., 1987, 1989). In the presence of reconnection, the flux through the circuit is changing at the rate

~t'n = Bcvce , (23)

where Bc is the coronal magnetic field inflowing with a velocity v c into the current sheet below the filament (Figure 2(b)) and e is the length of the current sheet normal to the

I c

I , I I

Ir

b

Fig. 2. (a) The electric current system represented as a rectangular circuit for low-lying prominences (Martens, 1986) or as a circular ring for high-lying prominences (Anzer, 1978). (b) The reconnection of the external magnetic field below the filament (indicated by the shaded circle) is represented by an X-type

neutral point. Arrows denote inflow and outflow.


plane of Figure 2(b). For the low-lying prominences e g 2d and for the circular shape

~ 2R. Reconnection below the filament implies

I / I + 12/L = ~n/gin (24)

instead of Equation (17). The existence of the electric current associated with the X-type

neutral line below the prominence causes a downward Lorentz force on the prominence

(Martens and Kuin, 1989). This force can be approximated as F r = I lolIc/a , where

I c ~ 2Boa~# o is the total current in the sheet. The parameter a is the dimension of the sheet, approximating also the distance of the X-type neutral line from the prominence summit. Taking B c < 10 -4 T and I = 10 l~ A one finds F r < 10 6 N m - i and Fr will be

neglected hereafter. The upward, outflowing jet from the current sheet (Sato and Hayashi, 1979) exerts a dynamical pressure on the prominence, causing an upward force

F a = poV~ro = roB2/l~o (25)

per unit length, where Po is the density of the outflowing plasma and v o = Bc(#oPo ) 1/2 (Somov, 1986; Vr~nak, 1989). Using the values Bc < 10- 4 Tand r o = 10 4 km one finds

F a ~ 105 Nm 1 which can be neglected.

Since the parameter A in Equation (12) depends also on the mass of the prominence, a mass loss through the footpoints, which is frequently observed, must be taken into

account. The simplest presentation is

A M / M o = - m A Z , (26)

where m is a dimensionless parameter describing the rate at which mass is outflowing

and M o is the initial mass. Such a presentation is appropriate only for small A~., i.e., the

onset of the instability. Another expression which can be used, one obtains by assuming

a constant velocity of the mass outfow (rout):

A M Gut

MoAh vlo (27)

where we u sedM o = pr2~clo, A M / A t = prg ~Vou t and AM~At = v A M / A h , while v = Ah/At .

3. Stability of Prominences and Eruption Onset

The equilibrium height (ho) of the prominence is determined by Equation (12) for prescribed values of the parameters X and A. For a small displacement Ah = h - ho

Equation (12) can be written as

h = A o f 2 o - g o + A h ( ~ ( A f 2 - g ) ) , 0

(28)

where f2 = (l/h + l /R - 21/RX2), and the subscript '0' denotes the value of a respective function at h o . Since in the equilibrium Aof2 o - go = 0, this is the equation of motion

302 B. VR~NAK

for a harmonic oscillator:

A ~ = t, o t Oh)o or, Oh)o - ~ o Ah = - <4Ah.

So the criterion for the instability (coo 2 < 0) can be expressed in the form

. ~ ~ 1 7 6

(29)

(3o)

2 C l = t g ~ o + - , (36)

n + 2~ o

in (Xg + 1) - X 2

C2 = (X [ + 1)ln(X 2 + 1) (37)

where

The gravitational term (0g/O~) o can be written as

( O g ~ _ 544d R~ (31) \0U o (1 - sin r (R, + ho) 3 '

where R s is the solar radius and g = 272 ms-2 at h = 0. Using Equations (1), (2), and (3), the function ~ can be transformed into the form

~? = (re + 2r - 2IX 2 + 1/(1 + sin ~)) (32)

and one obtains

(~?~) =2(1 -2 /X~ ~~

cos o + ( r c + 2 ~ ~ o \ 0 ~ J o ( l+s in~o)2J

where (0X/0~)o depends on the function I(h). Neglecting the reconnection, Equation

( 2 4 ) gives

L {OI'~ + I (~?L'~ o!,,~) ~ o\ OUo = O. (34)

Near the onset of the eruption the prominence is still at low heights, and we can adopt the rectangular circuit approximation for the self-inductivity (Equation (18)) to obtain

(OX) Xo(C1Lo-C3) (35)

o- 2(C~o ~ o ( 1 + ZOO)) '


and

Zo Alkl C 3 - - (F 4 - F3 - i) - A2k2C 1 + - -

cos {o 1 + sin ~o +

4ZgD~ ~ 2Zo 2 + A4k4 2Z~176 ~-+~oo) - A 3 k 3 - - c o s ~o (38 )

Here F 3 and F 4 are defined by Equations (19), while

0F,. 2 A~ - - (39)

07; 77(11 7;)2

The parameters 7i are defined by Equation (20) and

k~ - ~7i _ cos (arctg(Y~)) , (40)

o5 1-

where Y~ are defined by Equations (21). The remaining term (c3A/0f)o can be written as

(41) 1 ~ M oHo)

where (M/O~)o can be expressed for the rectangular circuit as

(42) (=) ) ~ - c ' '

and the term (OM/a~)o can be approximated by Equation (26) or (27). The results obtained for various approximations are shown in Figure 3, for a range

of the values of parameters appropriate for prominences. We present the border between

the stable and unstable region by the functions Xorit, (r t and r The stable region is to the right of a particular curve. The onsets of the eruptions of prominences should generally be expected in the area filled with vertical branches of the curves presented. The full lines correspond to the rectangular circuit approximation, where the complete Equation (29) was used and the reconnection was neglected. We want to note that the results for different values of the parameters A and d are identical if the ratio A/d is the same. The mass loss was approximated by Equation (26) for several values of the parameter m. Furthermore, the approximation I = const, and M = const. (Vrgnak, 1990b) for the circularly shaped axis is shown (dotted line), as well as the result for A ~ 12/M = const. (broken line). Finally, the previous result by Vrgnak et al. (1988) is also presented in Figure 3(b), where gravitation, mass loss and the change of the current were neglected and where an elliptical form for the prominence axis was assumed. Let us stress that although we used different approximations providing different behaviour of I(h), and substituted a wide range of values for the parameters, the form

304 B. VRgNAK

X l

7 !

6.

5

Z

3'

2-

I

/,"it

r

15~

I0~.

51I

..... ~,,; ......... ii ...........................

1 2

2 b

i ...f...

1 2

6~ ~4

k "

Z

1 2 - Z

Fig. 3. Cri t ical values of the pa rame te r s X (a), ~/D (b), and 4~ (c) as a function of Z . Full l ines represen t the rec tangular circuit approx imat ion : (1)A = 20 m s - : , R = ~, m = 0.3; (2)A = 20 m s - 2, R = ~,1 m = 0.3; (3)A = 10 ms -2 , R = ~, m = 0; (4)A - 20 ms - z , R ~-�89 m - 0. Broken lines represen t the circular axis approx imat ion : (5)A = 20 m s - z = const. , R = �89 (6)A = 20 m s - a, R = �89 m = 0.3. The dot ted line represents the I = const, approx imat ion for the circular axis (A --- 20 ms - 2) and the b roken-do t t ed l ine in (b)

resul ts from Vr~nak et al. (1988). In all cases we t o o k d = 50000 kin, the results depending only on the rat io A/d. The observa t ions of three prominences are indica ted by 6ors.


of the results is basically the same. For the comparison we included in Figure 3 values of the parameters X, �9 and 4~/D in three eruptive prominences, measured soon after the onset of the eruption and presented by Vr~nak et al. (1988) and Vrgnak (1990a). These values are based on the 'effective' twist (Vrgnak, 1990a) which is obtained neglecting the changes of the parameter X along the axis due to the width variation.

Figure 4 shows the dependence of the equilibrium height on A. Results are presented for different values of the parameter X' -- X ( Z = 1) (related to �9 as X' = ro~/2drO, d and R = ro/d. We also show the results for I = const. (Vrgnak, 1990b). Figure 4 provides a scenario for the instability onset in terms of the slow evolution of the parameter A, caused either by increasing current or by mass leakage in the footpoints. As the value of the parameter A increases, the prominence slowly rises. The curves characterized by X > Xtcrit (corresponding to �9 > I~crit ) have a 'knee' and when the parameter A attains a critical value Acrit a t /crit, no neighbouring equilibrium exists and the prominence abruptly leaps from Zcrit to the upper equilibrium position (the eruption is denoted by the thick arrow). Larger prominences (larger d) erupt more violently and for smaller

~ c r i t �9

Z

/ /

. / ~X'=Z,

Z.O 50 60 A (ms-')

Fig. 4. The dependence of the equilibrium height as a function of the parameter A for the rectangular circuit approximation. Thin lines represent d = 50 000 km, R = 0.5 and various X'. Stable prominences are characterized by X' < 2.5. The broken line represents X' = 3, d = 105 kin, R = 0.5; the dotted-broken line represents X' = 3, d = 50000 km, R = 0.2; and the dotted line represents X' = 4, d = 50000 kin, I = const. For one case we have indicated the lower equilibrium position at which the prominence becomes unstable (the

circle) and erupts (the arrow) to the upper equilibrium position (the cross).

306 B. VR~NA~

4. Prominence Oscillations

If coo 2 > 0, the prominence is stable and various eigenmodes of oscillations can develop when the prominence is displaced from the equilibrium position (Malville and Schindler, 1981; Vrgnak, 1984b, Vr~nak et al., 1989, 1990). The best known examples are winking filaments (Ramsey and Smith, 1966; Kleczek and Kuperus, 1969).

Other examples are oscillations caused by photospheric oscillations, observed as Doppler shifts of the prominence spectral lines (Balthasar, Stellmacher, and Wiehr, 1988, and the references therein). Photospheric 'waves' shake prominence footpoints and cause forced oscillations if the frequency is in resonance with some eigenmode frequency of a freely oscillating prominence.

The frequency of free oscillations in the lowest mode given by Equation (27) can be expressed as

(O o = (VA/10) X/(TT + 2~o)/Z o x /2 - X 2 + O(X o, d, ~o) , (43)

where v A = B II/#o P. The form of the function 0 depends on the form of the function I(h). The approximation I = const, and M = const, gives

0 = Xo 2 l~ - 2Z~176 2 + (lo/d) sin ~-o goloho (44)

2Zoho c2 (Ro + ho)vA

In Figure 5 we present the period T of the oscillations as a function of X for different Z and d/A, where we used the form of I(h) for the rectangular circuit without reconnection (Equations (35), (42)) and without mass loss. For small values o fX (X < 0.5) the period is approximately proportional to X and only weakly depends on the normalized height Z and R --- ro/d. In this range of the values of the parameter X, the isoperiod lines should be approximately parallel to the abscissa (Z) in the X ( Z ) graph presented in Figure 3(a). For X > 1 and Z < 0.5 the period depends only weakly on X and R = ro/d, and isoperiod lines in the X ( Z ) graph should be approximately parallel to the ordinate.

An important role in prominence oscillations is played by a 'frictional' term which can be due either to coronal viscosity (Hyder, 1966) or internal viscosity, or due to emission of sound waves (Kleczek and Kuperus, 1969) or MHD waves. This term might be especially important in the late phases of the eruption, when the prominence approaches the upper equilibrium position (Vrgnak et al., 1989, 1990; Vr~nak, 1990a). If the damping term b is larger than the natural frequency of the oscillations at the upper equilibrium position, oscillations do not occur. Instead, the prominence monotonically approaches an equilibrium. However, higher eigenmodes can appear, leading to a rather complex relaxation of the prominence axis (Vrgnak et al., 1989, 1990).

5. Process of Eruption

The dynamics of the eruption depends crucially on the form of the function I(h), and can be significantly influenced by the mass loss. Anzer (1978) treated a circular ring


T [mi l l

90

60.

30 �84

2 11 / / /

' 3

I1/// / ~ / ,9

//t l / i

,',,' / / 3

/ / / ]

[/I/II t I I

•

Fig. 5. Pe r iod o f osc i l la t ion as a func t ion o f the p a r a m e t e r X for A = 20 m s 2 a n d R = 0.2. The full lines

r e p r e s e n t d = 5 0 0 0 0 k m a n d the b r o k e n lines d = l0 s kin: (1) Z = 1 ; (2) Z = 0.75; (3) Z = 0.6.

current without reconnection (i.e., assuming constant magnetic flux through the circuit). The results were applied to coronal transients, and the main feature was an approximately constant velocity in the late phases of the eruption. Steele and Priest (1989) considered the equation of motion of the prominence and coronal transient, with the

reconnection below the filament included, and obtained a similar result. Martens and Kuin (1989) also treated a current system involving the reconnection below the filament and took into account dissipative effects, which relate the filament eruption directly to two ribbon flares. Vrgnak (1990b) treated I = const, and the form of )z (Z) revealed equilibrium solutions for stable low-lying prominences and high coronal loops, as well as metastable and unstable equilibria which are related to the eruptions. These results were qualitatively the same as the ones determined by Equations (12), (16), and (17). In Figure 6 we present functions h (Z), X(Z), and ro(Z ) obtained for a prominence with rectangular axis (full lines) and circular axis (broken), where we have neglected reconnection and we let the mass loss compensate for the changes of the axial current (A = const.). We also show the case with the mass loss determined by Equation (26) to illustrate the influence &the form of the mass loss used. The values of the parameters used were chosen to reproduce the observations of the prominence of September 19, 1979 (Vrgnak etal., 1988) as much as possible. The fit is the best for the A = const. approximation. Obviously significant differences and variations can occur depending on

308 B. VR~NAK

1.5

1.0

0.5

X

3

i2

(ms -21

60

50

c0

30

20

10

0

-10

2 3 ~ 5 Z

: . . \

�9 ' / ' . . \ ,; 2 " X

/ " \ ' , , \ \ D -

i 2 i i S Z

Fig. 6. The parameters X and R (a) and the acceleration (b) as a function of height in the early stages of the eruption. (a)The bold line represents the parameter X(Z) which does not differ significantly for the circular and rectangular axis approximation. The thin and the broken line represent R(Z) for circular (rout = 30 ms - 2, X' = 3.5, R = 0.5, d = 50000 km, A = 42 ms - 2) and rectangular (A = 42 ms - 2 = const.) approximations, respectively. (b) The thin line represents the circular axis approximation, and the broken line the rectangular one for the same values of parameters as in (a). The broken-dotted line represents the rectangular axis, A = const, approximation for R = 2.5 and it illustrates that 'thinner' prominences erupt more violently. Dotted lines represent the observed values in (a) and (b) for the prominence of

September 19, 1979.

the form of the mass loss. However , the m a i n physical difference is governed by the

na tu re of the current . I f the current is a par t of a global Solar cur ren t sys tem (I ~ const . ) ,

r o decreases dur ing the e rup t ion (Vrgnak, 1990b) which is somet imes observed (Vrgnak,

1990a). O n the other h a n d if the cur ren t represents a local system, r o increases, which

is thought to be a more c o m m o n behaviour a nd is i l lustrated by the obse rva t ions

p resen ted in Figure 6.

Figures 4 and 6 rep roduce several m a i n character}stics of the dynam ica l disparition brusque. The e rup t ion is p receded by a slow rising mot ion , related to an increas ing value

of the pa rame te r A. The pi tch angle of the field l ines decreases dur ing the e rup t ion while

the width can increase (Ru~djak and Vrgnak, 1981) or decrease (Vrgnak, 1990a)

depend ing on the evolu t ion of the electric cur ren t I(h). The end of the process can be


Fig. 7.

Ires -2)

60

20

I 2 3 z, 5 Z

The acceleration in the late phases of the eruption as a function of Z for different values of the outflowing velocity (km s - ~ ) in the circular axis approximation.

described qualitatively (Vrgnak, 1990b and Vrgnak et al., 1990) by incorporating the coronal 'friction' (Hyder, 1966) which can stop the prominence at the upper equilibrium position, where eventually damped oscillations can appear (Vr~nak et al., 1989, 1990). In Figure 7 we present the results for late phases of the eruption, when a circular axis approximation is appropriate. Equation (27) was used for mass outflow and different values of the outflow velocities were considered. In this approximation the acceleration ceases and the ascending velocity becomes constant.

6. Discussion and Conclusions

Let us repeat and discuss the assumptions and the approximations used, to establish the limits of the model.

(a) The forces were considered only at the prominence summit and we used a very

simplified geometry. (b) We assumed a specific basic internal magnetic field configuration, force-free in

the radial direction. (c) The inductive effect of the photospheric boundary was estimated by a straight

wire approximation. (d) We have used several forms for the self-inductivity (Equations (18), (22))

depending on the geometry appropriate for different regimes of the eruption. (e) Dissipative processes were neglected. (f) The internal structure was treated in cylindrical symmetry, while helical symmetry

might be the more realistic one (House and Berger, 1987), which would lead to a 'solenoidal' description of the prominence.

(g) The reconnection below the prominence was neglected. However, a stronger

310 B. VR~NAK

coronal magnetic field would ensure a more important role for reconnection and then the equation of motion must necessarily be completed by the term F r (Section 2.2).

The electric current flowing along the prominence can be partly related to the 'global' solar current system and partly can be generated by a local dynamo. The first component would not change significantly during the eruption, unless the height becomes comparable to the radius of the Sun. Another component is closed at the solar surface, and interferes significantly with the eruption. The relative contributions of these two components determine the dynamics of the process. The basic difference which can be observed is the change of the prominence width: in the first case the stretching dominates and the width decreases, while in the second one the width increases. We believe that both possibilities are illustrated by the prominences of August 16, 1988 (Vrgnak, 1990b) and September 19, 1979 (Vrgnak et al., 1988), respectively.

Although simple, the model relates the threshold for the instability with the prominence internal structure, provides a scenario for the eruption onset and relates kinematics to the evolution of the internal structure. There are several possible causes for the eruption onset in the increasing A scenario (Figure 4). The most simple one is a mass loss. Another one is the increase of the electric current caused by, e.g., slow reconnection below the filament driven by the -converging motions in the photosphere. The axial current can also increase due to the emerging flux process (increasing ~n through the circuit), or the twisting motions in the footpoints. On the other hand, resistive instabilities, usually having lower thresholds (Priest, 1985) can possibly also trigger the eruption. The released heat can increase the prominence width and conse- quently the parameter ~/D, or can initiate mass loss and so increase the parameter A. An increasing value of the parameter A corresponds to a slow rising motion of the prominence due to the increse of the equilibrium height. When the prominence reaches the point where no neighbouring equilibrium exists, it erupts. During the eruption the width of the prominence increases or decreases depending on the development of the electric current. However, the pitch angle of the field lines decreases in both cases if the redistribution of the twist due to the width variation along the axis is negligible.

The stability criteria depend on the geometry and the form of the mass loss function, but not significantly. Low-lying prominences are stable to the instability described even for large values of the parameters X or ~/D or ~, if Z is below the critical value (Figure 3). The Tayler limit (Tayler, 1957) and the Kruskal-Schafranov limit can be surpassed before the eruption. Some resistive instability such as cylindrical tearing can be triggered in this regime, and although it develops on a slower time scale (Priest, 1985), it can change a process significantly. The result of such an instability is a fragmentation of the tube into helical X-type and O-type neutral lines (Waddell et aI., 1978) which can be related to the appearance of helical threads near the onset of the eruption.

The equation of motion provides two sets of equilibrium solutions, one for the low-lying prominences and the second one for the coronal loops, with a possibility for eruptive transition from the first to the second form. Unlike the stability criteria, the course of the eruption can vary significantly depending on the details of the processes involved (mass loss, reconnection, etc.). The prominence can stop at the upper equilib-


rium position, or can oscillate there, or can enter the regime of constant velocity in the late phases of the eruption. All of these cases are known observationally.

Finally, let us stress that most often eruptive prominences are helically/cylindrically symmetric, although there are also very complex events which can not be easily explained by the model.

Acknowledgements

I would like to thank Drs U. Anzer, E. R. Priest, and T. Forbes for helpful discussions during IA U Colloquium 117 'Dynamics of Quiescent Prominences', held at Hvar in 1989.

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eruptive instability of cylindrical prominences

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