A prominence model in a simple magnetic arcade

A PROMINENCE MODEL IN A SIMPLE MAGNETIC ARCADE

V. M. BARDAKOVInstitute of Solar-Terrestrial Physics SD RAS, 664033 Irkutsk, P.O. Box 4026, Russia

(Received 12 May 1997; accepted 3 October 1997)

Abstract. This paper offers an evolution scenario for a simple magnetic arcade where the frozen-inmagnetic field decreases with the ascent of its arches together with the plasma. Uplift is producedby the movement of photospheric plasma with a frozen-in magnetic field, which is divergent withrespect to a neutral line. A decrease in magnetic field leads to the appearance in the arcade of aheight range of arches, with no high-temperature thermal equilibrium present, and to a variation ofthe nonequilibrium range with time. Uplift of the arcade is accompanied by the consecutive entryof new arches into this range. All arches entering the nonequilibrium range experience a transientprocess. Some of the earlier inquiries into the physics of this process were instrumental, in the firstplace, in identifying those arches which – through the production of an ascending plasma flow fromthe base of the arcade – are involved in the formation of a prominence (with magnetic dips appearingand evolving at the tops of these arches) and, secondly, in synthesizing a computational algorithm forthe final state of the transient process, the quasi-steady-state dynamic structure of the prominence.The arcade evolution scenario, combined with the computational algorithm, constitutes a unifiedprominence model, a model for the transition from a simple static magnetic arcade to a quasi-steadydynamic prominence structure. The model has been used in numerical calculations of parameters oftwo classes of prominences: in and outside active regions. Results of the calculations are in goodagreement with observations.

1. Introduction

In the classical view (Kippenhahn and Schluter, 1957), a prominence is stably sup-ported by the field of a magnetic arcade, with field lines that are convex downwardto the tops of the arches, the prominence location. A dynamic model that furtherdevelops this point of view (Pikelner, 1971; Priest and Smith, 1979) suggests acoronal formation of cold plasma, with a mass balance between a hot plasmaupflow along field lines from the base of the magnetic arcade and a downflowof cold prominence material across field lines. It is believed that the arcade baseis a layer above the transition region where the main upward flow of waves isdissipated and heats the plasma up to a temperatureT0 � 106 K. This layer is athermal reservoir for heating the overlying region of the magnetic arcade due toheat conduction along field lines.

Until recently, only a general scheme for the formation process of the prom-inence dynamic structure has been developed. Principal elements of the schemeare contained in the pioneering publications of Pikelner (1971) and Priest andSmith (1979). The physical principles of this scheme are as follows: destabilisingthe high-temperature thermal equilibrium in a certain height range of arches of atwo-dimensional magnetic arcade; plasma cooling at the tops of these arches dueto radiative losses; the appearance of an excess pressure gradient for hydrostatic

Solar Physics179: 327–347, 1998.c 1998Kluwer Academic Publishers. Printed in Belgium.

328 V. M. BARDAKOV

equilibrium; and the production, under its action, of a plasma upflow from thearcade base. Noteworthy also is one important point. Beginning with the paperof Pikelner (1971), in the context of the above-mentioned scheme the belief wasstrengthened that prominences are able to be produced only in magnetic arcadesthat have field lines with a downward convexity (magnetic dips) at the tops (Priestet al., 1989; Amariet al., 1991; Demoulin and Priest, 1993). Based on qualitativeconsiderations, the view was formulated as a selection rule (Demoulin and Priest,1993). This rule forbids, for example, prominences from being produced abovebipolar active regions where simple arcades have no arches with magnetic dips atthe tops. At the same time it is known from observations that filaments are fre-quently produced along a neutral line in these regions. The discrepancy betweenthe selection rule and observational evidence plus a lack of well-developed studieson the formation process casts some doubt upon the need to formulate this rule.

In recent publications (Bardakov, 1996, 1997; Bardakov and Starygin, 1997a)the unsteady regime that develops in magnetic arches after a loss of high-temperaturethermal equilibrium was treated as a common physical process. Hereinafter, weshall use the term ‘transient process’ because it is through this process that archesevolve into a new steady, not necessarily static, state. It has been found that, ina simple arcade, magnetic dips can be produced at the tops of nonequilibriumarches directly at the time of the transient process itself (Bardakov and Starygin,1997a). Further it was shown that the transient process ends in the formation ofa steady-state dynamic prominence structure (Bardakov, 1997). The previouslyobtained results are of reasonably fundamental importance, and they introduceserious modifications into existing schemes for prominence formation. Therefore,the goal of this paper is to construct – using existing results derived by investig-ating the physics of the transient process – a model for the prominence formationprocess in a simple magnetic arcade. Such a model would be a substantial furtherdevelopment of the dynamic model.

2. Conditions for the Commencement of the Prominence Formation Process

In terms of a dynamic model, a necessary condition for a prominence to be producedin a two-dimensional magnetic arcade is a loss in high-temperature thermal equi-librium in some height range of arches (Priest and Smith, 1979; Hood and Priest,1979; Bardakov, 1996). In Section 3, we shall consider the question of the loca-tion of the nonequilibrium range and conditions for its appearance. At this point,we discuss a sufficient condition for prominence formation. For nonequilibriurnarches, which originally have magnetic dips at their tops, a necessary conditionis at the same time a sufficient one. In fact, it is reasonably clear that magneticdips accumulate plasma that cools down following a loss in thermal equilibrium,based on the physical principles of a general scheme for prominence formation asmentioned in the Introduction. In this case there is even no special need to examine

A PROMINENCE MODEL IN A SIMPLE MAGNETIC ARCADE 329

the transient process because the final result (a steady-state dynamic structure of aprominence) is independent of the details of this process.

A completely different type of situation occurs in simple magnetic arcades. Inthis case a new final state depends on the character of the transient process. Forarches with heights which do not substantially exceed the scale of the homogeneouscorona, it was ascertained (Bardakov and Starygin, 1997a) that in an early stageof the transient process, radiative plasma cooling and the resulting convective heatand mass transfer along field lines from the arcade base lead to the formation ofa narrow region of plasma cooling at the tops of these arches. Plasma continuesto cool down and the density continues to increase there to the end of the stagewhen the summit temperature becomes as low asTP � 104 K � T0, belowwhich radiation losses decrease abruptly. The upflow velocity at the arcade base�0 increases during the first stage and eventually reaches a maximum value of�m.Dense cold plasma distorts the magnetic field near the tops of the arches by actionof gravity. Near the end of the first stage, magnetic dips are able to form here. Thecondition for their production is (Bardakov and Starygin, 1997a)

G =4��1gR?B

B2?B

> 1 ; (1)

whereg is gravity,�1 is plasma density at the arch top at the end of the first stage ofthe transient process, andR

?B andB?B are, respectively, the radius of curvature

of a field line and the magnetic field at the arch top projected onto a plane normalto a neutral line. The inequality (1) is known as the criterion for gravity to distortthe magnetic configuration (e.g., Priest, Hood, and Anzer, 1989). In this case thecondition (1) has a particular characteristic, namely the criterion for magnetic dipformation at arch tops of a simple magnetic arcade. The density�1 is determinedfrom a consideration of the transient process.

Early in the second stage of the transient process the cold plasma region expandsslowly, because two thermal fronts that separate cold material from hot plasma moveaway from the tops (Bardakov and Starygin, 1997a). The ensuing development ofthe transient process depends on whether condition (1) is satisfied or not. If no dipsare produced and if cold material does not leave the nonequilibrium arches acrossthe magnetic field, then the thermal fronts move far away from the arch tops. Thevelocity�0 and density of cold plasma at the tops decrease in this case. The transientprocess ends in a static low-temperature thermal equilibrium. Relatively extendedcentral parts of the arches turn out to be cold (with a temperatureT � TP ) in thisstate. In accordance with hydrostatic equilibrium, plasma density in these parts isextremely low (Bardakov and Starygin, 1997a) because the scale of a homogeneousatmosphere is two orders of magnitude smaller here,TP =T0 � 10�2.

Note that if in the beginning of the second stage powerful losses of cold materialacross the magnetic field come immediately into play, then the thermal fronts haveno time to move far away from the tops, and come to a stop when plasma inflowthrough them due to upflow is balanced by a sink of cold material moving downward

330 V. M. BARDAKOV

across the magnetic field. If in this case the density of cold plasma from the thermalfront to the tops decreases only slightly, then it is conceivable that a steady-statedynamic prominence structure is produced even if no magnetic dips appear. Itis hard to tell with assurance whether such a variant of the development of thetransient process is indeed the case. Downward motion of cold material across themagnetic field is still an unresolved physical problem. The physics of this processseems to be associated with the nonlinear stage of rearrangement instability ofdense cold plasma maintained against a gravitational force by a magnetic field.At the moment it is, however, impossible to determine whether a powerful coldplasma downflow across the field can be realized in the beginning of the secondstage without the formation of magnetic dips (in this case the cold plasma densitybetween the thermal fronts begins to decrease).

If the inequality (1) is satisfied, then it is much safer to assume that nonequilib-rium arches are involved in the process of formation of the prominence dynamicstructure. In this case, even if no losses of cold material across the field are presentin the beginning of the second stage, the thermal fronts do not move far away fromthe tops of the arches due to an increase in cold plasma density at the bottom ofdeepening magnetic dips. It was shown that the maximum excursion of the thermalfronts is small compared to the lengths of arches and is attained when�m=�1 � 3,where�m is the cold plasma density at the bottom of the dips (Bardakov, 1997). Afurther density increase in the deepeningdips is accompaniednow by the movementof the thermal fronts toward the tops. With such a variant of development of thetransient process, the plasma upflow velocity�0 throughout the entire second stageremains about the same�m. Dense plasma storage ultimately leads to triggering themechanism for discharging cold material downward across the magnetic field. Inthis case the transient process would end in the formation of a steady-state dynamicstructure. The thermal fronts in this final state remain fixed in position, and plasmainflow through them is balanced by outflow of cold material (Bardakov, 1997).

From the above line of reasoning the inequality (1) will be considered a suffi-cient condition for the arch that loses high-temperature thermal equilibrium to beinvolved in the regime of prominence formation. Otherwise this arch becomes coldand empty upon reaching low-temperature thermal equilibrium.

3. A Model for Arcade Evolution Prior to the Formation Process

We consider a two-dimensional (theXY plane) simple magnetic arcade. The axisY , along which the arcade is homogeneous, corresponds to the direction along thepolarity inversion line of the photospheric magnetic field. The base of the arcade(thermal reservoir) coincides with the levelZ = 0. In the approximation of a smallplasma� (gas to magnetic pressure ratio), the arcade’s magnetostatics are describedby the equation


Figure 1. Qualitative schematic of the dynamic structure of the prominence in a simple magneticarcade. Thin lines – magnetic field lines; heavy lines – boundaries of different arcade ranges. Thinarrows – direction of frozen-in magnetic field lines; heavy arrows – direction of plasma motion.Shaded cold plasma regions: 1 – cold plasma region of prominence, 2 – rarefied plasma region ofcoronal cavity, 3 – plasma downflow.

�?A+d

dA

B2Y

2

!= 0 ; (2)

whereA(x; y) is theY -component of the vector potentialA. The magnetic field isB = rotA. The equationA(x; y) = constant defines the projection of a magneticfield line onto the planeXZ. The magnetic field componentsB in this case arerepresented as

BX = �@A

@z; BY (A) ; BZ =

@A

@z:

To ascertain the general evolution of the arcade, we shall confine ourselves to atrivial geometry. Let us assume that the field component along the arcadeBY � 0.The magnetic field in this case will be a potential one. Suppose that the initialmagnetic field distribution is produced by a currentIY = I0, which flows nearthe origin of coordinates on the superconducting surface of a cylinder of radiusr0

(Figure 1). Field lines in the planeXZ are semicircles, andB? =qB2X+B2

Z�

constant along these lines.

332 V. M. BARDAKOV

Conditions for upsetting the high-temperature thermal equilibrium in the archesand the height range of these arches are determined by analyzing the thermalbalance along a field line (Priest and Smith, 1979: Priest, 1985; Bardakov, 1996):

ddl�0T

5=2dTdl= �2h0F (T )� �� ; (3)

where� is the plasma density,l is a coordinate along the field line measured fromthe arch base,h0, �0, and� are dimensional constants,�0T

5=2 = � is the heatconduction coefficient along the magnetic field. The first term on the right-handside of Equation (3) describes radiation losses, and the second term representsthe heating by the residual wave flux entering the regionz > 0 (it is assumedthat the dissipation of the waves is proportional to the density). The loss functionF (T ) is often taken as (Priest, 1985)F (T ) � 1 whenT0 > T > Tp � 104 K,andF (T ) = (T=Tp)

n whenT < Tp, wheren � 1. In this paper we use theapproximation of zero radiation losses whenT < Tp assuming thatn ! 1,or F (T ) � 0 (Bardakov and Starygin, 1997a). Together with the hydrostaticequilibrium equation along a field line and the boundary conditionsT = T0,� = �0 whenl = 0, and dT=dl = 0 whenl = L, whereL is the half-length of thearch, Equation (1) describes thermal equilibrium in a symmetric arch.

Based on results derived by studying thermal equilibrium (Priest and Smith,1979; Hood and Priest, 1979; Bardakov, 1996) we now formulate the condition forthe absence of high-temperature equilibrium:

� < �C ; " < "c(�) � 1 : (4)

Here � = b0=� is the ratio of the typical size of an archb0 to the scale of ahomogeneous corona� = kT0=Mg, wherek is Boltzmann’s constant,M is the

mean mass of a plasma particle, and the typical lengthb0 = (�0T7=20 =(h0�

20))

1=2

is the half-length of such an arch where the time taken by the temperature to leveloff due to heat conduction,tT = cV �0L

2=(�0T5=20 ), is equal to the typical time

of cooling by radiation,tR = cV T0=(h0�0), wherecV is the heat capacity ofa mass unit, is the adiabatic index; the value of�C will be determined below;" = ��0=(h0�

20) is the ratio of the heating power by the residual wave flux to the

radiation loss at the arch base; the value of"C depends on the parameter�, and"! 1 when� � 1.

The first inequality in (4) is actually the condition for the production of nonequi-librium arches in the absence of dissipation of the residual wave flux when" = 0.For a very simple arcade geometry, it was shown (Bardakov, 1996) that, in determ-ining this condition, the solution of Equation (3) gives a result in agreement withinferences of a qualitative method. This method is based on analyzing the depend-ence of the value of the parameter�TR = (htT i=htRi)

1=2 on the arch lengthL,wherehtT i andhtRi are typical timestT andtR, where the average density for thearchh�i is used instead of the plasma density at the arcade base�0. The inequality


�TR > 1 separates those arches in which heat conduction has no time to level offthe temperature throughout the arch and the main central part of the arch coolsdown due to radiation losses.

Let a qualitative method be applied to the arcade examined in this paper. Ifa dimensionless coordinate is introduced along a field line� = l=L, then for ahigh-temperature arch withT � T0 the density distribution for the arch becomes

�(�) = �0 exp

0B@�L

�

�Z

0

'(�0)d�0

1CA ;

where'(�) = BZ=B? = sin(�(1� �)=2). The mean density in the arch is definedash�i = R 1

0 �(�)d�. As a result, the inequality�TR > 1 becomes

(�) = �

1Z

0

exp(��x)p1� x2

dx > � ; (5)

where� = H=� andH = 2L=� is the height of the arch. The function (�) isplotted in Figure 2. Note that max (�) = �C � 1:1 when�m � 2:85. When� ! 1 the function (�) tends to unity, and (�) = ��=2 when� � 1.Obviously, the first inequality of (4) is a condition where the criterion (5) can besatisfied. When 1< � < �C there exists a range of nonequilibrium arches withheightsHT1 < H < HT2, whereHT1 andHT2 are the roots of the equation (H=�) = �, and when� ! 1 the upper boundary of the rangeHT2 !1. When� < 1 the range is bounded only below by the heightHT1, and when� � 1 thevalue ofHT1 = 2b0=�.

The dissipation of the residual wave flux has a substantial influence upon thethermal balance of the arch. When" > 1 there is always a high-temperatureequilibrium where the temperature at the top of the archTB > T0. When" < 1such a equilibrium exists forH > HR("), whereHR(") is some height thatdepends on" and serves as a peculiar ‘high-temperature cover’ (Bardakov, 1996).If HR < HT1, then forH > HR there exists a further high-temperature thermalequilibrium, withTB < T0, however. Arches with heightsH < HR have only sucha high-temperature equlibrium. With a decrease of", the heightHR increases, andat some"C(�) it becomes larger thanHT1. At this value of" (the second inequalityin (4)) a height range appears in the arcade with no high-temperature equilibriumpresent:Hmin � H � Hmax, whereHmin = HT1, andHmax = min(HR;HT2).

Despite the uncertainty in numerical estimates of the residual wave flux penet-rating the regionz > 0 as made in the literature, a generality displayed by a decreasein intensity of this flux and the effectiveness of its dissipation with a decrease ofthe magnetic field is universally accepted. In this connection, a decrease of themagnetic field in the arcade that is responsible for a decrease of the parameter",must become the key element in the arcade evolution model.

334 V. M. BARDAKOV

Figure 2. Plot of the function (�).

Let us consider a slow, divergent (with respect to the pointx = 0) movement ofthe ends of field lines at the levelz = 0 by taking into account the frozen-in plasmacondition of the magnetic field. We suppose the movement of the ends of field linesatz = 0 to be such that the disturbed current density at this level is zero everywhere,with the exception, perhaps, of the superconducting surface of the cylinder atx = r0. The slow rate of evolution provides the potentiality in the magneticfield distribution atz > 0 also. As a result, the magnetic field is determined bya currentI(t) that decreases with the time and flows on the cylinder’s surface.The conservation condition for the magnetic fluxA(x; y; t) = constant, withthe plasma traveling with the frozen-in field, specifies a relationship between theheight of the archH(t) at timet and its initial heightH0: H(t) = H0(H0=r0)

m(t),wherem(t) = I0=I(t) � 1. Hence, whenI(t) < I0, any arch together with theplasma rises up, and the necessary additional plasma enters the arch along themagnetic field from the arcade base which serves as a storage of mass. With suchan evolution of the arcade, the magnetic field at any fixed heightH decreases asB?(H; t) = B?0I(t)=I0.

For real arcades, divergent (with respect to the axisZ) movements of magneticfield lines frozen into the plasma can travel below the level of the thermal reservoirunder the action of a dense photospheric plasma. The effect of rising arches togetherwith the plasma and the concurrent effect of a decrease of the magnetic field inarches must persist in this case. However, it is much more difficult to describe themagnetic field evolution in this case. Therefore, considering the uplifting effect


of arches in real arcades with divergent movements of photospheric plasma to bean essential element of the evolution model, we shall continue our treatment of atrivial arcade.

Because of a decrease of the parameter", the height of the ‘high-temperaturecover’HR can exceedHmin. If condition (1) is satisfied for an arch of a heightHmin at this instant of time, then the arches that go up to enter the expandingnonequilibrium range�HN = HR(t) � Hmin, are immediately involved in theprocess of prominence formation. Otherwise the transient process in these archesends in a low-temperature thermal equilibrium, forming a cold, rarefied coronalcavity. This cavity is depicted qualitatively in Figure 1 as the shaded central regionof the arches occupying the upper part of the nonequilibrium range. As timeprogresses, the boundaryHR can be stabilized, and the nonequilibrium rangewill be fixed in space. In this case some arches leave the nonequilibrium range,while they rise aboveHR and are heated again. Other arches, while they risefrom below, continue to be involved in the transient process. For some of thesearches, condition (1) is realized for the first time as soon as it reaches the heightHmin. Henceforward, this arch will be termed critical. The process of prominenceformation starts directly at this instant of time, and we will consider it further inthe next section.

4. Characteristic Time for the Formation Process, and the Mass BalanceEquation in a Quasi-Steady State

As a result of the continuing rise of the arcade, the critical arch increases its heightHC(t). The range of arches�HF (t) = HC(t) � Hmin, that are involved in theprocess of prominence formation, increases. By the time of the end of the firststage of the transient process in the critical arch, in a time�t1 � tR (Bardakov andStarygin, 1997a), the fronts of the thermal cooling waves have a final height extent� �HF (�t1). In the subsequent process, the height range�HF will continue togrow, and it will be assumed that always�HF � �Xw(H; t), where�Xw(H; t)is the distance in coordinatex from the axisZ to the thermal front in any arch fromthe interval�HF .

To carry out a quantitative description, we introduce a number of simplifyingassumptions which do not affect substantially the overall behavior of the processand the characteristics of the final quasi-steady state. Magnetic field deformationin arches will be taken into account only between the thermal fronts. The currentflowing here counterbalances (at the expense of the Lorentz force) the gravitationalforce of the cold dense plasma. However, the contribution from this current sheetto the magnetic field of the regionsjxj > �Xw will be neglected. Further, basedon the smallness of�Xw=L the upflow velocity and plasma density distribution(throughout the arch) will be considered as it was very early in the second stage.Therefore, the plasma upflow velocity at the arch base and the plasma density�w

336 V. M. BARDAKOV

at the distance�Xw from the axisZ (immediately behind the thermal wave front)are taken to be, respectively,�m and�1. The approximation of zero radiation losseswhenT < Tp permits us to consider the temperature between the thermal fronts tobeT � Tp.

Because the formation process is slow compared with the transit velocity ofsonic perturbations between the thermal fronts, it is assumed that in this case mag-netogydrodynamic equilibrium always manages to be established and is describedby the equation

�rp+1c[j�B] + �g = 0 ; (6)

wherep is the plasma pressure,j is current density, andc is the speed of light.The condition�HF � �Xw allows us to writejy � �(c=4�)@ ~BZ=@x, where~BZ is the disturbedz component of the magnetic field in the regionjxj < �Xw.The undisturbedz component near the tops of the arches can be represented byBZ � �B?(x=R?B). For the chosen geometry of the arcadeR?B = H. Themagnetic field deformation is considered to be reasonably weak, so that in theprocess of deepening magnetic dips the modulus of the magnetic field near the topsof the arches can be considered constant in magnitude and equal toB?. Becausethe thickness of the emerging prominence is small, it is assumed thatBx � B? inthe region between the thermal fronts. Finally, we shall keep in mind that

j(@p=@z)=�gj � (�=�HF )�p � 1 ;

where �p = T=Tp � 1. As a result, thex and z components of the vectorequation (6) permit us to obtain a differential equation to define the distributionof cold plasma density between the thermal fronts, corresponding to an arch withheightH:

d2 ln �ds2 = �

�

�pH(G� � 1) ; (7)

wheres = x=� and� = �=�w. Boundary conditons for (7) are: d�=ds = 0 whens = 0; � = 1 whens = sw, wheresw(t) = �Xw=� is the slowly varying (withtime) dimensionless coordinate of the front of the thermal cooling wave.

As will be shown later, ifHmin is not too large, then the parameterG(HC(t))initially increases as the critical arch rises. In this caseG(H) > 1 for all archeswith Hmin < H < HC(t). The dimensionless density� varies from� = �m > 1whens = 0 to� = 1 whens = sw(H; t), where�m is the dimensionless density atthe bottom of the magnetic dip whenx = 0. For simplicity’s sake, unity as againstG� on the right-hand side of Equation (7) may be neglected. The solution of sucha simplified equation gives a distribution of�(s):

s = ��1=2p

1p�m(t)

ln

p�m(t) +

p�m(t)� �(s)p�(s)

!; (8)


where� = (2H=(�G))1=2. The position of the thermal front is found from (8) tobe

�Xw(t) = ��1=2p W (�m) ; (9)

whereW (�m) = (1=p�m) ln(

p�m +

p�m � 1):

Using (8) it is easy to express the mass of cold plasma�p(t) =R�XW0 � dx at

the top of any arch in terms of the dimensionless density�m:

�p(t) = ��w�1=2p �

q�m(t)� 1 : (10)

On the other hand, a growth in mass�p(t) is ensured by the plasma upflow, andd�p=dt = �0�m. Whence�p(t) = �0�mt if time is reckoned from the beginningof the second stage of the transient process in any arch. Using (10) we find atime dependence of�m(t):

p�m � 1 = (�0=�w)�mt=(��

1=2p ). When�m > 3,

the semi-thickness�Xw begins to decrease because the functionW (�m) has amaximum when�m � 3. The stage of cold plasma storage in deepening magneticdips would conclude at a certain�m = �A, when cold plasma downflow across themagnetic field comes into play. If�A is not too large, then the time of the storagestage may be estimated as

�A � ��w

�0�1=2p

�

�m: (11)

Cold plasma downflow across the magnetic field is currently a central physicalproblem which demands special investigation. In this paper we take advantageof experimental results, according to which cold material inside the prominencemoves downward along a host of fine-structured vertical filaments with velocities� 1 km s�1 (see, for example, Priest, 1985). We now introduce the plasma downflowvelocity �L averaged over the entire prominence. It is reasonable to suggest thatthe mean velocity is lower than the observed velocity in vertical filaments. It istherefore assumed that�L � 0:5 km s�1.

If by the timet � �A the range of arches involved in the formation process hasreached some extent in height�HF (�A); then the time taken to reach the finalquasi-steady state with a mass balance can be estimated as�L � �HF (�A)=�L.A total formation time is� �A + �L. The time�A is the ‘internal’ time of theformation process as opposed to the time�L which depends, in terms of�HF (�A),on the divergence rate of field lines at the base of the arcade due to external effects.It is the ‘internal’ time�A that is taken here as the characteristic time of prominenceformation.

The state in which a mass balance is realized is a quasi-steady state since theheight of the critical archHC(t) varies slowly with the time because of variationsin the base of the arcade. The mass balance equation in a quasi-steady state will

338 V. M. BARDAKOV

permit the�m(H)-distribution in heights of arches to be determined in the range�HF = HC(t) � Hmin. In this case relationship (9) may be treated as a heightdependence of the prominence semi-thickness in a quasi-steady state�Xwq(H).Expression (10) will define the mass of cold plasma at the arch top�pq(H; �(H))as a function ofH and�m(H). The mass balance equation itself is

�0�m(H) = �Ld

dH�pq(H; �m(H)) : (12)

WhenH > HC(t) we have�pq = 0, and whenH = HC the upflow velocity�m undergoes an abrupt change from zero to�m(HC) 6= 0, hence the value of�pq(HC ; �m(HC)) = 0, which serves as a boundary condition for (12). Thiscondition immediately shows that for a dynamic structure of the prominence in aquasi-steady state�Xwq(Hc) = 0. Qualitatively, the dynamic structure is depictedin Figure 1. The figure also shows the range of arches whenH < Hmin, where coldplasma downflow proceeds along the magnetic field. The density and temperaturedistribution in these arches calls for special investigation, and this issue does notcome within the province of this paper.

To solve Equation (12), one has to know an explicit dependence of�m(H) and�w(H) on the heightH. Such dependences were obtained in the approximation

of a constant heat conduction coefficient�C = �0T5=20 (Bardakov and Starygin,

1997a). In the next section, we shall determine the functions�m(H) and�w(H)by taking into account the real temperature dependence� = �0T

5=2 and considerdifferent prominence parameters in a quasi-steady state.

5. Plasma Upflow Characteristics

We have already introduced the simplifying assumption that in any arch withH < HC after the first stage of the transient process the upflow velocity andplasma density (throughout the arch) distribution may be taken to be such as itwould be if this arch were initially in the second stage of the transient process.This remains also valid in a quasi-steady state. Provided that the flow velocityis small compared with the velocity of sound, and this is indeed the case, thedensity distribution throughout the arch can be considered in the approximation ofhydrostatic equilibrium:

�(�) = (�0=�(�))exp(��(�)) ; where �(�) = D

�Z

0

['(�0)=�(�0)] d�0 ;

D = L=� and �(�) = T (�)=T0 is dimensionless temperature. A constancy ofplasma flow along the arch gives for the dimensionless velocityV (�) = �(�)=�T

V (�) = Vm�(�)exp(�(�)) ; (13)


where�T = L=tT , andVm = �m=�T . The energy equation, in view of the smallnessof the kinetic energy of plasma motion compared with thermal energy, is writtenas

Vmd�d�=

dd�K(�)

d�d��

q2

�2F (�)exp(�2�)� �DVm'(�) ; (14)

whereK(�) = �5=2 is the dimensionless heat conduction coefficient,q = L=b0,and � = ( � 1)= . Equation (14) was considered forK � 1 in an earlierstudy (Bardakov and Starygin, 1997a), and forK(�) = �5=2 only for archeswith homogeneous pressure whenD � 0 (Bardakov and Starygin, 1997b). In theapproximationF (�) � 0 when� < �p, Equation (14) has boundary conditions(Bardakov and Starygin, 1997a)

�(� = 0) = 1 ; �(� = 1) = �p ; �0(� = 1) = 0 : (15)

The third boundary condition permits the velocityVm at the base of the magneticarch to be determined.

A general conclusion about a steady-state upflow can be drawn by consideringan integral energy balance in the arch and integrating (14) over the entire length ofthe arch:

Vm(1� �P )� �0(� = 0) =2�D�

Vm +

1Z

0

q2 exp(�2�)�2 d� : (16)

Here the left-hand side represents the energy flux into the arch caused by theconvective transfer and heat conduction, and the right-hand side describes theexpenditure of energy to overcome the gravitational field by the upflow and energylosses by radiation from the arch. It is evident from (16) that whenD > DK =

�=(2�) the convective energy flux into the arch is less than gravitational losses ifVm > 0. But such a situation is energetically disadvantageous, andVm < 0 mustcorrespond to arches withD > DK . Thus, it can be immediately expected thatupflow exists only for arches with heightsH < HK = �=� � 2:5� ( = 5

3).For the boundary-value problem (14) with boundary conditions (15), it is

impossible to find an exact analytical solution. Still, it is possible to constructan approximate solution in different regions for�. In each of these regions, twoterms of Equation (14) are competitive. A matching between the regions is donebased on the continuity of�(�) and�0(�). The second derivative�00(�), however,undergoes abrupt changes on the boundaries of these regions. A similar methodof approximate solution of the boundary-value problem (14) and (15) was used inearlier studies (Bardakov and Starygin, 1997a, b). This method provided a goodagreement with results of a numerical investigation of the transient process. Wenow use this method to find the functions�m(H) and�w(H).

340 V. M. BARDAKOV

Let one-half of the arch be divided into three regions. In the first region,�1 � � � 1 because the derivative�0 and the function'(�) are small, onlythe two first terms of the right-hand side of (14) are competitive. Assume that inthis region�(�) changes little and that�(�) � �1 = �(�1). The solution (14) inthis region, in view of�0(1) = 0, then gives

d�d�

= � 2p3

q exp(��1)

�5=2(�3=2� �

3=2P )1=2 : (17)

Suppose that�(�1) = �1 � �p. In this case solution (17) and the conditiondefining the position of the point�1 from the comparison of the convective termjVm�0(�1)j with radiation losses, permit us to write two relationships:

�1 =

p3

2q exp(��1)

Vm

!4

; (18)

(1� �1)exp(��1) =2p

311q

�11=41 : (19)

In the second region,�2 � � < �1, energy losses by gravity are still smallbecause of the smallness of'(�) andD. At the same time it is easy to see that inthis region the gradient of convective heat flux and the power of radiation lossesare competitive. Let us also assume that here the quantity�(�) changes little, thatis, j��j = j�1 � �2j � �1, and�(�) � �1 in the second region. The solution(14) in this region then becomes

�3 = �32 �

3q2 exp(�2�1)

Vm(� � �2) ; (20)

where�2 = �(�2).In the third region, 0� � � �2, the convective energy flux goes to overcome

the gravitational field. The solution (14) in this region, in view of�(0) = 1, is

�(�) = 1� D

DKsin��

2�

�: (21)

Note that, with a smallD, the third region is totally lacking, that is losses goingto overcome the gravitational field do not manifest themselves. In fact, this is thecase whereD � 0, a consideration of which (Bardakov and Starygin, 1997b) gave

Vm = 3q2 ; �m =32�H

tR; �w =

�0

�p: (22)

The third region manifests itself when then parameterD exceeds a certain value ofDB . This value will be determined in what follows.

A calculation of�1 using (21) gives


�1 =1�

ln�

1�2

�= � ln(�5=2

2 ) : (23)

At the point� = �2 where gravitational losses are equal to radiation losses, wehave

q2�

32 = �DVm sin

��

2(1� �2)

�: (24)

A matching of the function�(�), which in the second region is specified by formula(20), on the boundary� = �1 gives the relationship

�31 = �

32 �

3q2�52

Vm(�1� �2) : (25)

We obtain from (21) on the boundary� = �2

�2 = 1�D

DKsin��

2�2

�: (26)

In view of (23), five relationships (18), (19), (24)–(26) define five parameters:�1,�2, �1, �2, Vm.

We obtain from (18), (19), and (25) this relationship:

�32 +

111

p3

2q�

5=22

Vm

!12

� 3q2�52

Vm(1� �2) ;

from which, provided thatp

32

q

Vm�

5=22 < 1 ; (27)

it follows that

Vm = 3q2�

22(1� �2) : (28)

We obtain from (24) and (26) the equation for defining�2:

D

DK= f(�2) =

�3�2(1� �2) sin

��

2(1� �2)

�+ cos

��

2(1� �2)

��1

: (29)

The functionf(�2) increases monotonically in the region 0� �2 � 1 from2=(3�) when�2 = 0 to unity when�2 = 1. Therefore, the solution of Equation(29) exists only for 2=(3�) < D=DK < 1, and henceDB = 2DK=(3�). It iseasy to ascertain that condition (27), under which Equation (29) is obtained, isalways satisfied. The inequality�1 � �p that, if satisfied, was used to obtain the

342 V. M. BARDAKOV

relationships (18) and (19), is not satisfied asD approachesDK . It can be shownthat the relationships (18) and (19) will be altered in this case, but the expression(28) and Equation (29) undergo no changes. A simple analytical formula can beobtained whenQ(H) = 1�H=HK � 1. The parameter��=�1 in this case isestimated as 1=(5 ln(1=Q)). It is indeed small, as expected, even for not too largeQ�1. The solution of (29) then becomes: 1� �2 � (2=�) (DK=(3D))1=2Q1=2. Thevalue of�2 � Q, and forVm and�m we have

Vm =2p

3�

q2Q5=2�DK

D

�1=2

; �m =p

3H

tR

�HK

H

�1=2

Q5=2 : (30)

In the upper limit of validity, these formulas (30) can be used right up toQ = 1 � DB=DK � 0:8. For cold plasma density behind the thermal wavefront we get

�w = �0Q5=2=�p : (31)

Formula (31) does not differ from the case of a constant heat conduction coefficient.Yet, we will not consider here the reason for the substantial difference for thevelocityVm, since it was discussed in detailed in an earlier study (Bardakov andStarygin, 1997b).

6. Geometrical and Physical Parameters of the Prominence in the FinalQuasi-Steady State

We now apply our developed model to two classes of prominence: prominencesemerging above a bipolar active region, and quiescent long-lived prominencesexterior to active regions. For these two classes of prominence, the initial difference,from which ensue (in accordance with the model in hand) all other differences, liesin different physical conditions of the arcade base (thermal reservoir). It is known(see, for example, Gibson, 1973) that the pressure above the transition region foractive regions is an order of magnitude higher than for quiet regions, and thetemperature is higher by about a factor of two. Let us assume, for estimates to bemade later in the text, that in the active regionT0 � 2� 106 K andn0 = �0=M �5� 109 cm�3, and outside the active regionT0 � 106 K andn0 � 109 cm�3.

The parameter� is expressed in terms ofn0 andT0 as

� =

��0

h0

�1=2 T3=40 g

kn0:

(32)

With the provision (Priest, 1985) that�0 = 10�6 erg s�1 cm�1 K�7=2 andh0 =4 � 1025 erg cm3 s�1 g�2 we obtain for the active region and outside of it,


respectively,� � 0:3 and� � 1:0. This difference in the parameter� immediatelyspecifies a difference in the height of the lower boundary of the prominence,Hmin,defined by the equation (Hmin=�) = �: i.e.,Hmin � (2=�)�� = 2� 109 cm forthe active region andHmin � 7�109 cm� 1:5� for the quiet region. Temperaturein the thermal reservoir for the active region is twice as high; therefore, the scale of ahomogeneous corona for the active region was taken as� = 1010 cm, while for thequiet region it was� � 5�109 cm. SinceDB=DK = HB=HK � 0:2, for the activeregion and outside of it, respectively, we haveHmin=HK � 8� 10�2 < DB=DKandHmin=HK � 0:6 > DB=DK . Arches that are involved in the process ofprominence formation in the active region can therefore be regarded as arches withhomogeneous pressure using formulas (22) for the velocity�m and density�w. Foridentical arches exterior to the active region, formulas (30) and (31) will be used.

Consider the characteristic time of the formation process. For active region,formulas (11) and (22) give

�A =2p

23�

��

Hmin

�1=2 tR

G1=2�1=2p

� 9tR � 3:6� 104 s :

Here,G � 1, �p � 0:5� 10�2, and

tR = kT0=(h0M2n0) � 104(T0=106) (109=n0) � 4� 103 s

are used. Outside of active regions, from (11), (30), and (31) we obtain

�A =

�2

3�

�1=2 tR

G1=2�1=2p

� 5tR � 5� 104 s ;

whereG � 1, �p � 10�2, andtR � 104 s are taken. It is evident that the charac-teristic time of formation is almost the same for the two classes of prominences.

The mean magnetic field in the prominence is bounded above by the field of thecritical arch at the time of its appearance at the heightHmin. Based on (1) one cantherefore write the condition

hB?i < BC =p

4��wgHmin : (33)

For active regions, (33) givesBC � 25 G, and for quiet regionsBC � 5 G.Finally, we consider the solution of Equation (12) for the two classes of promi-

nence considered. It is assumed thatG(H) = Gm(!=!m)3(1�!)5=2=(1�!m)5=2

and � = (2Hmin=�)1=2!m(1 � !m)

5=4=(!(1 � !)5=4), where! = H=HK ,!m = Hmin=HK , andGm = G(Hmin). For active regions, in the approximation ofH,Hmin � HK , the solution (12) becomes

p�m � 1 =

3�4Hmin

VLtR

�Hmin

2�

�1=2

G1=2m �1=2

p

"�HC

Hmin

�2

��

H

Hmin

�2#: (34)

344 V. M. BARDAKOV

Let the altitude extent of the prominence be in the range fromHmin toHC = 2Hmin.With r0 � Hmin, the value ofGm virtually does not differ fromG = 1. Hence,with H = Hmin, the value of�m � 4 andnm = n0�m=�p � 2� 1012 cm�3.

Outside the active region, let! and!m be close to unity. In this caseG decreaseswhenH > Hmin, and in a quasi-steady state the upper boundary of the prominenceis not the archHC(t) if G(HC) < 1 holds for it, but an arch of a heightHh, forwhichG(Hh) = 1. Assume thatHh � 1010 cm. ThenGm � 1:5 if it is assumedthatHmin � 7� 109 cm. The solution of (12) gives

p�m � 1=

27

�32

�1=2 HK

VLtR

�HK

�

�1=2

G1=2m �1=2

p

(1� !)7=2� (1� !)7=2

(1� !m)5=4(1� !)5=4:

(35)

WhenH = Hmin, the value of�m � 2,Q(Hmin) � 0:4, and

nm = n0�mQ5=2(Hmin=�p � 2� 1010 cm�3 :

The dependenceof the prominence thickness on the height is defined by formula(9) where� is dependent on the heightH. Maximum values of the prominencethickness occur whenH = Hmin. For active regions and outside of them, we have,respectively,dm = 2�Xwq(Hmin � 5� 108 cm anddm � 9� 108 cm. Upflowvelocities at the arcade’s base, inferred by formulas (22) and (30) forH = Hmin,are 25� 105 cm s�1 for active regions and 2� 105 cm s�1 for quiet regions.

For clarity the calculated geometrical and physical parameters in the final quasi-steady state are performed in Table I. The initial parameters for the model areperformed here also.

Thus, except for the characteristic time of the formation process, all the otherparameters, inferred in terms of the model, differ markedly when the two classes ofprominence are intercompared. Note that, as follows from the model, an increase inthe height and thickness of prominences, and also a decrease of the magnetic fieldand cold plasma density inside prominences at the transition from active to quietregions, corresponds also to similar trends displayed by observations (Tandberg-Hanssen, 1974; Hirayama, 1978; Leroy, 1988).

7. Conclusions

Let us summarize the key points, with special emphasis on the merits of theimproved dynamic model of prominences, a model for the transition from a statichigh-temperature state of a simple magnetic arcade to a quasi-steady dynamicstructure of the prominence. The key features of the model are:

(1) A decrease of the magnetic field in the arcade at the expense of slow,divergent (with respect to a neutral line) movements of photospheric plasma with


Table IParameters of the prominence in the final quasi-steady state

In active Outside ofregions active regions

Input Velocity of cold plasma downflow 0.5 0.5parameters across the magnetic field inside

of prominence,�L, km s�1

Temperature at the arcade’s base, 2:0� 106 106

T0, KPlasma density at the arcade’s base, 5� 109 109

n0, cm�3

Calculated Temperature of prominence, 104 104

parameters Tp, KLower boundary of prominence, 2� 109 7� 109

Hmin, cmMaximum value of the prominence 5� 108 9� 108

thickness,dm, cmThe mean magnetic field in the 25 5prominence,hB?i, GPlasma density inside 2� 1012 2� 1010

prominence,nm, cm�3

Upflow velocities at the arcade 25 2base,�m, km s�1

Characteristic time of formation, 3:6� 104 5� 104

�A, s

a frozen-in magnetic field. A decrease of the field leads to the following con-sequences: the appearance of a nonequilibrium range where arches lack a high-temperature thermal equilibrium; the process in which arches rise up to enter thenonequilibrium range; and the establishment of conditions for the formation ofmagnetic dips.

(2) The production of magnetic dips at the tops of nonequilibrium arches duringthe transient process that evolves after these arches have entered the nonequilibriumrange. This leads to a subsequent deepening of these dips, to accumulation of colddense plasma in them, and to triggering the mechanism for cold material downflowacross the magnetic field.

(3) The formation, during the transient process, of two thermal fronts nearthe tops of the nonequilibrium arches to form a sharp boundary between coldprominence material and hot coronal plasma. The small thickness and stationarityof these fronts is ensured by the competition of the steepening effect of convectiveheat transfer by a plasma upflow with the spreading effect due to heat conduction.

346 V. M. BARDAKOV

The formation of steady-state thermal fronts is a property inherent in a dynamicmodel.

The merits of the proposed model include:In the first place, a high degree of the model’s self-consistency. The main

prominence parameters are inferred in terms of the model if density and temperaturevalues are specified at the arcade base.

Secondly, the model gives answers immediately to three questions of the prom-inence problem: (1) the maintenance of the prominence – a stable location of thecold dense plasma in the magnetic dips that are produced at the tops of archesduring the transient process; (2) the mechanism for a sharp boundary betweencold prominence material and the hot coronal plasma – stationary thermal coolingwaves where a balance between spreading and steepening is ensured due to heatconduction and convective heat transfer in the upflow; and (3) the mechanism forcompensating for mass loss caused by a constant cold material downflow – theplasma upflow from the arcade base occurring in nonequilibrium arches during thetransient process.

Thirdly, a consistency of different consequences of the model with observation-al results. This consistency may be itemized as follows. First we must point outthat the divergent (with respect to a neutral line) slow movement of photosphericplasma with a frozen-in magnetic field, as the reason for prominence formation, isconsistent with observational evidence of a decrease in longitudinal photosphericmagnetic field gradient near a neutral line prior to filament formation (Maksimovand Ermakova, 1985, 1987; Schmiederet al., 1991). Further, numerical estimatesof different geometrical and physical parameters of prominences as made by modelformulas (Section 6) are in good agreement with measurements (Hirayama, 1989;Jensen and Wiik, 1989). Trends in the variation of parameters are also in agreementwhen comparing prominences within and outside active regions. Finally, the modelassumes the appearance of a cold rarefied region in the upper part of the nonequi-librium range, to which there corresponds the coronal cavity observed above aprominence.

Acknowledgements

I wish to thank V. P. Maksimov for helpful discussions. I am indebted to V. G. Mik-halkovsky for his assistance in preparing the English version of the manuscript.

References

Amari, T., Demoulin, P., Browning, P.et al.: 1991,Astron. Astrophys.241, 604.Bardakov, V. M.: 1996,Astron. Letters22, 273.Bardakov, V. M.: 1997,Pisma v Astron. Zh., in press.Bardakov, V. M. and Starygin, A. P.: 1997a,Pisma v Astron. Zh.23, 307.Bardakov, V. M. and Starygin, A. P.: 1997b,Astron. Zh., in press.


Demoulin, P. and Priest, E. R.: 1993,Solar Phys.144, 283.Gibson, E. G.: 1973,The Quiet Sun, NASA, Washington.Hirayama, T.: 1978, in E. Jensen, P. Maltby, and F. Q. Orrall (eds.), ‘Physics of Solar Prominences’,

IAU Colloq.44, 4.Hirayama, T.: 1989, in V. Ruzdjak and E. Tandberg-Hanssen (eds.),Dynamics of Quiescent Promin-

ences, Springer-Verlag, Berlin, p. 156.Hood, W. and Priest, E. R.: 1979,Astron. Astrophys.77, 773.Jensen, E. and Wiik, J. E.: 1989, in V. Ruzdjak and E. Tandberg-Hanssen (eds.),Dynamics of Quiescent

Prominences, Springer-Verlag, Berlin, p. 298.Kippenhahn, P. and Schluter, A.: 1957,Z. Astrophys.43, 36.Leroy, J. L.: 1988, in E. Priest (ed.),Dynamics and Structure of Quiescent Solar Prominences, Kluwer

Academic Publishers, Dordrecht, Holland, p. 770.Maksimov, V. P. and Ermakova, L. V.: 1985,Soviet Astron.29, 323.Maksimov, V. P. and Ermakova, L. V.: 1987,Soviet Astron.31, 458.Pikelner, S. B.: 1971,Solar Phys.17, 44.Priest, E. R.: 1985,Solar Magnetohydrodynamics, MIR, Moscow.Priest, E. R. and Smith, E. A.: 1979,Solar Phys.64, 267.Priest, E. R., Hood, A. W., and Anzer, A.: 1989,Astrophys. J.344, 1010.Tandberg-Hanssen, E.: 1974,Solar Prominences, D. Reidel Publ. Co., Dordrecht, Holland.Schmieder, B., Van Driel-Gesztelyi, I., Henoux, J. C.et al.: 1991,Astron. Astrophys.244, 533.

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A prominence model in a simple magnetic arcade