THE F L A R E P R O C E S S IN A T O P O L O G I C A L L Y N O N - E Q U I -

L I B R I U M M A G N E T O H Y D R O D Y N A M I C A L S Y S T E M

V. M. B A R D A K O V

Siberian Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation (SiblZMIR), USSR Academy of Sciences, Irkutsk 33, P.O. Box 4, 664033 U.S.S.R.

(Received 19 August, 1985; in revised form 10 January, 1986)

Abstract. The existing models for solar flares fail to treat in an appropriate manner the energy release mechanisms based on a step-like transformation of the magnetic field energy from magnetic field to energy of the hydrodynamical motion of the medium and dissipation of these motions through shock waves into heat.

In considering an example of the relaxation process in a topologicaily non-equilibrium magnetohydro- dynamical syste m resulting from the merging of two magnetic loops that possess balanced longitudinal currents, this paper suggests one such energy release mechanism. Due to a certain degree of universality for different topologically non-equilibrium systems, a variety of characteristics of the relaxation process obtained may form the basis for constructing a model of solar flares based on a step-like transformation of the magnetic field energy in topologically non-equilibrium magnetohydrodynamical systems.

I. Introduction

That a direct energy source for a solar flare is provided by the magnetic field of active regions is rpesently the most attractive idea. The main property of energy release mechanisms in the majority of solar flare models (e.g., Sweet, 1958; Gold and Hoyle, 1960; Petschek, 1964; Sturrock, 1968; Syrovatskii, 1976; Priest, 1976; Spicer, 1977) is that most of the magnetic energy is immediately converted into both thermal energy of the medium and the energy of accelerated particles (these kinds of energy will be collectively referred to as the thermal form of energy throughout the remainder of this paper). The process of transformation of the magnetic energy into the thermal form is frequently called magnetic field annihilation.

Solar flare models insufficiently reflect another kind of transformation of the magnetic field energy - conversion of the magnetic energy first into the energy of the hydrodynamical motion of a plasma and then shock-wave driven transformation of the energy of the hydrodynamical motion into thermal forms of energy. It is this step-like conversion of the magnetic energy which is a distinctive feature of the approach to be taken later in this paper to the problem of magnetic energy release in the solar flare.

For the magnetic energy to be transformed in a stepwise fashion requires firstly the presence of a mechanism which would rapidly transform an appreciable fraction of the magnetic energy in a local region of the upper chromosphere or the corona into the energy of hydrodynamical motions of the plasma, with the velocity comparable with the Alfv6n velocity C A in that region. This mechanism is obvious, provided some fragment of the active region (let us call it the magnetohydrodynamical system) is in a state which is not magnetohydrodynamical equilibrium or is in unstable equilibrium. The magneto-

Solar Physics 106 (1986) 147-164. �9 1986 by D. Reidel Publishing Company

148 v. M. BARDAKOV

hydrodynamical system (MHDS) rapidly leaves this state and when the state of stable equilibrium is far from the initial one, a substantial fraction of the MHDS in a period of time ZA g L/CA (L is the characteristic size of the MHDS) becomes involved in hydrodynamical motions with characteristic velocity CA. These motions decay through dissipative mechanisms and, most effectively, due to shock waves (Wentzel, 1964), thereby ensuring a step-like transformation of the magnetic energy and the attainment of the MHDS of a new equilibrium state with a lower magnetic energy.

So simple a scheme of magnetic energy release involves a fundamental question of how the MHDS, with external boundary conditions varying slowly, can reach such a non-equilibrium state or an unstable equilibrium state which possesses a far greater magnetic energy than does the near-lying stable equilibrium.

There possibly exists only one way that such an unstable equilbrium state may be attained, namely a long evolution of the MHDS in the metastable equilibrium state. Here, apart from a temporal equilibrium state in which the MHDS finds itself, there also exist in the system stable equilibrium states with a lower magnetic energy. In addition, the difference in magnetic energy between the metastable equilibrium and the lower-lying equilibrium states grows with time due to a slow variation of external boundary conditions. At a certain moment of time, the metastable equilibrium is transformed into an unstable equilibrium which is followed by the above-mentioned fast relaxation process (Sweet, 1964; Barnes and Sturrock, 1972).

As regards the attainment of a desired non-equilibrium MHD S, there exists a method which has not been discussed previously within the context of conversion of a significant portion of magnetic energy into the energy of hydrodynamical motions. The magnetic field topology of the MHDS is able to alter due to reconnection processes during the course of a slow evolution with varying external conditions. This may give rise (but not always, of course) to a configuration of the magnetic field which, in principle, has no equilibrium state, the so-called topologically non-equilibrium MHDS (Parker, 1982, 1983). The appearance of such a magnetic field topology provides a trigger mechanism for subsequent energy release in the MHDS. Indeed, within the topologically non- equilibrium MHDS there appear fast motions of the medium, whose energy is dissipated due to a system of shocks arising in the MHDS; moreover, the MHDS gives rise to such a process of magnetic field reconnection which tends to eliminate the topological non-equilibrium conditions of the system. Both of these processes constitute the content of the relaxation process that brings the MHDS into a stable equilibrium state and ensures the flare energy release in the MHDS.

When the fraction of the magnetic energy that is directly transformed into the thermal form in processes of reconnection of a topologically non-equilibrium MHDS is small comapred with the energy transformed into the thermal form through shock waves from the energy of the hydrodynamical motion of the medium arising in a non-equilibrium system, it is then possible to speak of a step-like transformation of the magnetic field energy as the principal form of magnetic energy conversion during a relaxation process.

The main characteristic of the process in the topologically non-equilibrium MHDS is the time taken by the system to relax to a topologically equilibrium state.

THE FLARE PROCESS IN A SYSTEM 149

It is clear that the relaxation time depends upon the particular topologically non- equilibrium MHDS. Although such MHDS has attracted interest due to Parker's papers (Parker, 1982, 1983), there are at present no sufficiently simple, and rigorously mathematically substantiated examples of MHDS which are topologically non- equilibrium systems. It is our hope that in the future an analysis will be made of possible topologically non-equilibrium MHDS which are capable of being produced in an active region of the Sun.

In this paper, however, we will try to examine a relaxation process in the topologically non-equilibrium MHDS that has been reported in the literature (Vainshtein, 1984). This implies a magnetic configuration produced by merging of two magnetic cylinders which have shielded longitudinal currents.

Let there exist in an active region two chains of sunspots of opposite polarity which in the upper chromosphere and the corona produce an arch-like structure of the magnetic field. Such structures with an inversion polarity line of the magnetic field are rather common in active regions.

Assume that there also exist smaller-scale features in the form of magnetic loops with a shielded longitudinal current, i.e., having such a density distribution of the longitudinal current that its net current is zero. The loops are distinguished in the overall arch-like structure of the toroidal magnetic field Bt by the presence of a poloidal component of the magnetic field, Bp which is there precisely because the shielded longitudinal current flows in local channels. The relaity of such features and how they can be produced will not be discussed in the present work.

Unless the loops make contact with each other, they evolve independently. With general activation of the entire arch-like structure, the loops 'floating' in this structure are able to contact with each other and the oppositely directed poloidal components of the magnetic field will start reconnecting due to finite conductivity. The MHDS which arises is similar (except that the poloidal component of the field only exists in a limited volume in this case) to a double twisted flux tube (Gold and Hoyle, 1960; Sweet, 1964). We assume that along the length of contact the loops are homogeneous along the toroidal field and B t ~> Bp. This will allow us to restrict our attention to a two- dimensional problem in a plane perpendicular to the toroidal magnetic field and to exclude entirely from consideration the toroidal field Bt, by assuming the plasma motion to be incompressible (Kadomtsev, 1975; Vainshtein, 1984). Figure 1 shows such a two-dimensional MHDS before the loops make contact and after initial reconnection of the poloidal magnetic fields.

The subsequent temporal evolution of the MHDS depends on the way in which the loops are brought into contact. One way involves the collision of two loops moving toward each other with a certain relative velocity. In this case, back-reconnection and inelastic reflection of the loops may occur provided they have a certain relative velocity. Another way implies that the loops are slowly pressed onto eahc other by external forces. A fraction of the poloidal magnetic flux reconnects (Figure l(b)), thereby leading to dissipation of the extra magnetic energy of the poloidal field which is produced within the system under the action of external forces. Once the action of the external forces

150 V. M. BARDAKOV

(a) (b)

Y a:

Fig. 1.

(c)

Different states of the MHDS. (a)Non-interacting loops, (b)a topologically non-equilibrium MHDS, (c)final state of the system.

ceases, a total energy of the system (the energy of the poloidal field plus the energy of possible hydrodynamical motions) may not exceed the energy of the poloidal field of the two original loops (Figure l(a)). It is clear that in this case the reflection process is not possible because the system merely lacks the energy excess in order for it to return to its original state (Figure l(a)). The first, quite natural, possibility for a subsequent evolution involves the arrival of the M H D S at a state of stable equilibrium under the conditions of a new topology of the magnetic field, i.e., with the partly reconnected poloidal ftux which has been roduced by the action of the external force. This possibility is easy to realize provided the M H D S with any given reconnected flux (Figure l(b)) involves a sufficiently large number of equilibrium states or at least one such state.

On the other hand, when there are no equilibrium states at all in the M H D S with any given reconnected flux, we are dealing with an example of a topologically non- equilibrium MHDS. In this case, the M H D S evolves through non-stationary processes with a subsequent reconnection of the poloidal flux to its complete reconnection and to the merging of the two loops into one which arrives at a stable, axisymmetric equilibrium state (Figure l(c)). If it is assumed that the two original loops were totally identical, then, using the approximation of an incompressible medium and conservation of the poloidal magnetic flux during the reconnection process (Kadomtsev, 1975) it is easy to determine the energy of the poloidal magnetic flux during M H D S following the merging process. This energy is found to he precisely a factor of two smaller than that of the poloidal field of the two original loops. Thus, half the magnetic energy of the poloidal field is released during the relaxation process of a topologically non-equilibrium M H D S which implies the merging of two loops.

Earlier implicit suggestions that a two-dimensional M H D S with the magnetic topology depicted in Figure l(b) is topologically non-equilibrium are contained in

THE FLARE PROCESS IN A SYSTEM 151

Parker's paper (Parker, 1983). The physical considerations adduced in that paper and more recently in Vainstein's paper (Vainshtein, 1984) appear reasonable. The mathe- matical proofs, however, contained in these two papers should be recognized as incorrect. Mathematically, the problem is a very complicated one and, therefore, the question of a rigorous proof of topological non-equilibrium conditions of the required MHDS remains open.

For discussion, let us formulate the following postulate. A two-dimensional MHDS with a limited region occupied by a magnetic field and

comprising three topologically distinct flux systems, divided by a separatrix (i.e., the MHDS involves two zero points of the magnetic field of the type of O-points and one zero point of the type of an X-point or two X-points with a current sheet that lies on the separatrix) is devoid of the state of magnetohydrodynamical equilibrium.

Possibly, this postulate is a consequence of a more general theorem such as one which Parker (1983) and Vainshtein (1984) have attempted to prove and which states that an equilibrium state of a two-dimensional MHDS with a limited region occupied by a magnetic field can only be the axisymmetric state. In the following, we will be using the postulate formulated.

We now consider the mathematical meaning of the postulate. Within the frames of the ideal magnetohydrodynamics, the equations of two-dimensional incompressible plasma motion are (Parker, 1982):

dv 7AAA Po 7p , divv = 0, (1)

dt 4re

where A are components of the vector potential of the magnetic field along a toroidal field, p is the generalized pressure including, apart from the customary gas dynamics pressure of the medium, the magnetic pressure B~/8r~, and Po is the density of the incompressible medium which, for the sake of simplicity, we will assume constant throughout the cross section of the loops in order to exclude from consideration the convective transfer of the density. The poloidal magnetic field lines are the lines of equal value A and the fines of equal poloidal magnetic flux. The magnetic field of the two- dimensional MHDS is represented thus:

Bp = [VA X et], (2)

where e t is the unit vector along the toroidal magnetic field. From (1) follows the equilibrium equation:

AA + 4top(A) = 0, (3)

where p(A) is an arbitrary function of A. On a closed curve that determines the localization of magnetic field, B/,, we impose a boundary condition 7.4 = 0 or, equiva- lently, Bp = 0.

The mathematical meaning of the postulate implies the absence of such solutions of the boundary problem (3) for any boundary and any function p(A) that have the magnetic field topology indicated in the formulation of the postulate.

152 v. M. BARDAKOV

Now, let us make some remarks concerning the method of analyzing the relaxation process. It is a complex problem to treat the dynamics of a topologically non-equilibrium MHDS using accurate magnetohydrodynamics equations with proper allowance for dissipative processes and with due regard for the deviation from an incompressible medium, in order to give proper consideration to dissipative processes in shock waves.

We, therefore, intend to separate the general problem into individual, rather idealized, physical elements (fragments) in the overall picture of the process. Each fragment will be described qualitatively. A reasonable relation of all the fragments will provide us with a complete physical picture of the relaxation process. The result will be a general temporal evolution of the MHD S, starting from reconnection of a small fraction of the poloidal flux under the action of the external force and will be presented in Section 4.

2. On Non-Physical Equilibrium and Maximal Free-Magnetic Energy

Let at some moment of time the MHDS depicted in Figure l(b) be at rest, i.e., v = 0. Assume further that at this moment of time the magnetic energy Wb of the MHDS is equal to that of two original loops depicted in Figure l(a), i.e., Wb = 2Wo. Finally, let, for simplicity, the longitudinal current in the original loops be distributed in such a way that the magnitude of the poloidal field does not depend on the radial coordinate. Then, throughout the cross section of the loops Bpo = B o = const., and Wo = (B2/8z)rcL2o, where L o is the radius of the original loops.

Since the MHDS is not in equilibrium, it will be set in motion at subsequent moments of time. In this, a current sheet is able to form near the X-point and the magnetic field will continue reconnecting due to finite conductivity. Let us examine another idealized situation. We impose a prohibition on reconnection assuming that conductivity a = oo. But let us introduce into our consideration dissipative processes which lead to a damping of hydrodynamical motions of the medium. For the moment, we are not interested in what these processes actually are.

We then ask the question: How does the MHDS evolve in such a situation? As long as the magnetic configuration of the MHDS satisfies the postulate, the

magnetic energy will go into the excitation of the hydrodynamical motions which decay due to dissipative processes. Therefore, the final state of the MHDS must have a magnetic configuration which does not satisfy the conditions of the postulate (the existence of one X-point and of two O-points). Furthermore, the magnetic energy in the final state, Wf, must necessarily be smaller than W b. A reduction in the length of the reconnection lines of force leads to a decrease in magnetic energy of the portion of the MHDS with the reconnected magnetic flux. A minimum magnetic energy for this part of the MHD S is achieved when the reconnected lines of force become concentric circles.

The only possible finite state for the MHDS with a fixed reconnected flux which does not satisfy the conditions of the postulate, is shown in Figure 2. In this state, the X-point and the two O-points have merged into one. This merging should be viewed as a reduction in distance between the X-point and the O-point to infinitesimal with respect to the asymptotic in time. Also, as the points come closer together, the magnetic flux

THE FLARE PROCESS IN A SYSTEM

Fig. 2. Non-physical asymptotic equilibrium.

153

between the X-point and the O-point remains unchanged because we have imposed a prohibition on subsequent reconnection. Conservation of flux yields the relation BoL ,.~ BI' , where l' is the distance between the X- and O-points, and L is the radius in the original loops, out of which the reconnection of flux was organized. When l' ---, 0, there appears an infinitely large magnetic field; however, the scale l in a direction perpendicular to the line connecting the two O-points is limited for this increased field. As will be apparent in the next section, the force from the magnetic pressure on the scale l, i.e., BZl, must not exceed B~L. Then l ~ l '2/L and the magnetic energy in the region of the increased field W~ ..~ (B2/8rt)ll ' ~ (B2o/8rOL2(I' /L) --* 0 when l' /L ~ O.

As a hypothesis which must be a consequence of the postulate formulated previously, we can make the following statement. The magnetic field configuration below the separatrix in a final asymptotic state is determined in a unique way. Moreover, the magnetic energy below the separatrix is nearly twice the magnetic energy of a portion of the unperturbed, axisymmetric original loop which was left unreconnected.

In the case under consideration, Bpo = Bo - const., and the magnetic energy enclosed below the separatrix is 2c~(L) (B~/8rc)rcL 2 where e(L) > 1. Hence, the total magnetic energy W z = (BZ/8rc)rt[L~ + (2c~ - 1)L2]. We will call the difference between Wb = 2Wo and Wf the maximal free magnetic energy W(L) for a given reconnected flux:

# (L) B~ rcLo 2 1 - ( 2 c ~ - 1) . (4) 8re

Note that when AL = L o - L ~ 0, c~ --, 1. It is also possible that the coefficient ~ is a universal one and equals unity for any reconnected flux. Then, the final state of the MHDS with a fixed reconnected flux possesses an absolute minimum of magnetic energy.

The state of the MHDS which we have considered is a non-physical equilibrium state. Long before this asymptotic state is reached, the finite conductivity effect should manifest itself and lead to a subsequent reconnection of magnetic flux. The following

154 v . M . BARDAKOV

point is essential. When the magnetic Reynolds number R o = CA(Bo)/Lo/~?o >> 1 (rlo = c2/4rca is the magnetic viscosity which we will, throughout the text, consider a constant) the MHDS can then move towards the non-hysical equilibrium state until finite conductivity starts to operate. How close the MHDS approaches the asymptotic state depends on the ratio of the damping rate of the hydrodynamical motions to the reconnection rate due to finite conductivity within a current sheet that forms in the vicinity of the X-point. It, in turn, appears that the rate of reconnection increases as the MHDS approaches the non-physical equilibrium state and the duration of the relaxation process (the time taken by the loops to merge) is reduced substantially. These issues will all be considered in the next section.

3. Acceleration of the Reconnection Process in the Presence of Dissipation of the Energy of the Hydrodynamical Motions

In order to describe to some extent the damping of the hydrodynamical motions requires an understanding of the basic characteristics of the possible motions of the medium in our MHDS. Since from the outset we have abandoned the idea of analyzing the accurate equations of magnetohydrodynamics, it is necessary to rely on reasonable physical conceptions. Let us consider an MHDS that is at rest at an initial moment of time, as described at the beginning of Section 2. Assume further, however, that the fraction of the reconnected flux is comparable with the unrec0nnected flux. In this case,

Wo. The principal physical content of non-equilibrium of such an MHDS is a simple

physical phenomenon, the attraction of two parallel currents. Indeed, a current of equal magnitude and direction flows in a direction perpendicular to the plane of Figure l(b) in each of the two magnetic islands. Based on this physical conception it is possible to take up a discussion of the hydrodynamical motions in the MHDS.

To begin with, we examine a totally idealized case when a prohibition is imposed on a subsequent reconnection of the magnetic flux and there is no damping of the hydrodynamical motions in the system. From general considerations it is understandable that in such a non-equilibrium MHDS there must arrive a comparatively simple intermittent motion that returns, within a period of the oscillations, z~o, to its original state at rest.

At the initial moment of time, each magnetic island is subjected to a force along the X-axis toward the centre. Supposing that p = p(A), we will still further specify the situation. The electromagnetic force (attraction of parallel currents) will then only act on the island at the initial moment. From Equation (1) we have:

porcL 2 d V - 1 ~ ( V A ) 2 d l = - - - 1 ~ 2 dt 8re 8~ B s dl , (5)

where V is the mean velocity of the magnetic island as a whole along the X-axis, and B s is the magnetic field along the separatrix that bounds a single island. If at the initial

THE FLARE PROCESS IN A SYSTEM 155

moment of time, the scale of magnetic field increase in the vicinity of the X-point from a zero value to the quantity Bo is in the direction of both the X- and Y-axes, of order of magnitude equal to the radius of the unreconnected part of the magnetic loop, L, we then estimate from (5) the island's acceleration as:

100 7~L 2 dV B 2 - - ~ - - - rcL. (6) dt 8re

Under the effect of a force acting on the island, the MHDS region in the vicinity of the X-point starts to deform. With this deformation, along the Y-axis there starts the formation of a current sheet that is infinitely thin in thickness (because a = ~) , whose width increases with time with increasing scale of the region being deformed. The scale of the deformation region increases in proportion to the length traversed by the centre of mass of the magnetic island along the X-axis under the action of the force. A qualitative picture of the deformation is shown in Figure 3. If at the initial moment of

(a) (b)

I I

S ~0-S Do

Fig. 3. Picture of deformations near the X-point. (a) Configuration before the deformation, D O - from the center of mass of the island to the X-point. (b) Configuration after the island has displaced a distance S.

time the scale of field enhancement in the vicinity of the X-point is Dx along the X-axis and Dy along the Y-axis, then as the island's centre of mass displaces a distance s, we have a deformation region along the X-axis s:, ~ 2s and along the Y-axis sy ~ sxDy/D x . The width of the newly formed current sheet is of the order of sy while the magnetic field at the current sheet boundary B ( s y ) ~ BosSDy. The ambient pressure on the separatrix inside an infinitely thin current sheet increases by an amount B2(sy)/8~ compared to the pressurep(A,) on the separatrix outside the current sheet. This excess pressure leads to two effects. Firstly, the differential pressure leads to the formation of an ejection along the Y-axis with the speed CA(B ) of a plasma mass which originally was located outside the separatrix (a dashed volume in Figure 3). The mass of the ejection at the time when the island has displaced a distance on the order of magnitude is poSSy. Secondly, the excess ambiental pressure within the current sheet gives rise to

156 v.M. BARDAKOV

a force directed inwards from the centre along the X-axis, that is applied to the magnetic island:

dV polrL 2 - - , ~ - ~zL + sy. (7)

dt 87z 8=

The first stage of the process will cease when the magnetic island ceases to be accelerated toward the centre. At that moment, the right-hand side of Equation (7) will become equal to zero. For the case when D x ~ Dy ~ L, the centre of mass of the magnetic island will then be displaced a distance sm ,~ L. The duration of the first stage, on the order of magnitude, is estimated from (7):

h ~ (Smlam) m , (8)

where a,,, is the maximum acceleration of the island, being C2(Bo)/L in this case. Then z~ ~ L/CA(Bo) ~ Za, where ZA = Lo/CA(Bo) is the Alfv6n time of the MHDS. The redistribution of magnetic and hydrodynamical energy at the end of the first stage is as follows: due to a decrease in magnetic energy, part of the MHDS with a reconnected magnetic flux (reduction in field lines), there appears a hydrodynamical energy of magnetic islands Wm ~ poL2C2A(Bo) ~ L2B2o/8~ as well as the plasma ejection hydrody- namical energy of the same order of magnitude along the Y-axis Wej ,~ L2B2o/8~.

During the second stage of the process, the island will be inhibited by an opposing force arising due to the continued increase in the width of the current sheet. The hydrodynamical energy of the island will decrease while the magnetic energy will continue to increase. The ejection's hydrodynamical energy will be transmitted to the ambient medium, on the one hand, in the form of the energy itself of the hydrodynamical motion (involvement of a vaster region in the motion) and, on the other, the magnetic energy of part of the MHDS lying outside the separatrix will again increase during the second stage (the field lines will stretch out under the action of the ejection). Of course the overall hydrodynamical energy of this part of the MHDS will be decreasing in this case. The end of the second stage corresponds to there being no hydrodynamical motion in the MHDS and to the reverting of the magnetic energy to the initial level but with a different distribution of magnetic field at which there exists a thin current sheet along the Y-axis. The length of this stage is also of the order of h. The next two stages will proceed in reverse order and the system will revert to its original state.

Thus, we have described qualitatively the MHDS evolution in the approximation of the total absence of hydrodynamical and magnetic energy dissipation. In this case, the MHDS will find itself in an oscillatory regime with a period of the oscillations % ~ CA.

AS a next step, we will take account of the hydrodynamieal energy dissipation and see how the oscillatory regime just considered is altered and how the MHDS evolves to an asymptotic, non-physical equilibrium state treated in the preceding section.

If the medium possesses compressibility, then the ejection's hydrodynamical energy will be transferred to the surrounding medium with an irreversible transformation of part of the energy into heat due to the dissipation processes occurring in the shock wave produced due to the action of the ejection. Compressibility is substantial for a finite ratio

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of the ejection's characteristic velocity to Alfv6n velocity CA(Bt), determined from the toroidal field. The higher the ratio VeJCA(Bt), the more effectively the ejection's hydrodynamical energy dissipation must proceed. Along the path of the MHDS evolution to the asymptotic state, as will become apparent in the discussion to follow, the characteristic velocity of the ejection Vej will be well above that, V, of the magnetic island. Precisely for this reason we restrict ourselves to consideration of only the dissipation of the ejection's hydrodynamical energy. We confine our consideration to the inclusion of the loss coefficient e. We will assume that for one period of oscillation of the MHDS, a quantity eWej of the ejection's hydrodynamical energy, Wej , is transformed into the thermal form.

With due regard for the attenuation of the hydrodynamical motion of the ejection, the MHDS, after the first period of oscillation, will not revert to its original state, described at the beginning of this section. Since we are, for the moment, interested in general characteristics of the MHDS evolution, we will proceed in the following way. We suppose that after the first and subsequent oscillations, the MHDS returns to its quiescent state that is similar to the original one in that the thin current sheet is lacking along the Y-axis in this state. The differences of each of such states from the original one are, however, thus: since the MHDS starts evolving toward as asymptotic, non- physical equilibrium. Then the distance between the X- and Y-points l', decreases with each oscillation and the free magnetic energy becomes still smaller compared with maximum energy and one may write WFr(l', L )~ I?V(L)(I'/L) n, where n is still an unknown degree. From the conservation of magnetic flux between the X- and O-points, it follows

BI' ~ BoL, (9)

implying that the magnetic field, at least on the X-axis, is increased. The localization scale along the Y-axis of the increased magnetic field coincides with the scale of field increase, l, along the Y-axis in the vicinity of the X-point from zero to the value of B. Let the scale of magnetic field increase along the X-axis in the vicinity of the X-point from zero to the magnitude of B be equal to l".

Note that in each such a quiescent state, the basic physical content of MHDS non-equilibrium, the attraction of the islands, must be maintained. Then, applying Equation (5) to this state, we obtain a restriction on the scale l:

B2l ~ B2oL. (10)

The remaining unknown characteristics of the state are the scale 1" and degree n, which will be expressed in terms of l' later in this text. We will thereby characterize each new quiescent state through a single parameter, l'. Let us make one more assumption. We will assume that during the oscillations with %(l') ~ z A the major part of the magnetic islands will be solid body (then the island will respond to the applied force at once) while the non-sold body region is a region of deformation of scale l" in the vicinity of the X-point. We will examine the relaxation regime wihtout such an assumption in a separate paper. Note that in this case the MHDS will involve oscillations with a period ~a that modulate and slightly modify the oscillatory regime with %(l') ~ ~a"

158 V. M. BARDAKOV

The M H D S evolution may now be represented thus. Starting from the quiescent island described above, the system executes a subsequent oscillation with a period %(l'). During this period, part of the ejection's hydrodynamical energy eWej ( l ' ) is lost and the system returns to its quiescent state in which l' is still smaller. Such an oscillatory regime continues until an asymptotic state with l' --, 0 and, as we will see now, %(l ' ) ---, 0 when l' ~ 0.

Let us describe the parameters of the M H D S oscillatory regime in the stage when the distance between the X- and O-points is l' ~ L. The character of the M H D S evolution during one period of oscillation, we believe, remains nearly the same as in the dissipationless case. The ejection velocity will be of the order of Alfvrnic in the field magnitude B, i.e., CA(B). The ejection energy W e j ( l ' ) ~ po l l"CZ(B) . The maximum attraction force that acts on the island in the quiescent state, is obtained from the equation

pozrL2am = F m ~ - - - W z r ( l ' , L ) (11) ~l'

assuming that the free magnetic energy plays the role of potential energy. The time %(l ' ) is determined from (8), where s m ~ l" . The system (8)-(11) is closed by two relations: (1) %(l ' ) ~ zej ~ l /CA(B). This is the time required for an ejection to form or the time taken by the plasma to outflow from the region between the two halves of the separatrix (see Figure3) and is comparable with the M H D S period of oscillation; (2) W e j ( l ' ) ~ Whe~, porcL2V 2. This is the hydrodynamical energy of the island as a whole, and compares with the plasma ejection energy along the Y-axis. Using these two relations, together with (8)-(11), we readily obtain: n = 1, l " / L ~ ( l ' / L ) 6. The para- meters of the M H D S oscillatory regime will thus be

% ( l ' ) ~ ZA(I 'IL) 3 , V CA(Bo) ( r l L ) 3 , (12)

Wej Wh, Wo(r/L) 6 , v Vej CA(B).

It is apparent that % ~ 0, Wej ~ 0 if l' ~ 0. The oscillation-averaged decrease in freemagnetic energy due to attenuation of the

hydrodynamical energy of the ejection may be written as

The characteristic time of free magnetic energy decrease, as is evident from (13), is

( L ) 2 8 -1 (14) ~ H ~ Z A 7

Thus, in the presence of hydrodynamical energy dissipation alone, the system, while remaining in an oscillatory regime, evolves to an asymptotic, non-physical equilibrium state and the oscillation period is thereby decreased while the characteristic time of free magnetic energy decrease grows.

THE FLARE PROCESS IN A SYSTEM 159

Let us consider now the reconnection process due to finite conductivity. During each

oscillation, a current sheet is formed along the Y-axis, having a width of order l. Since the lifetime of this sheet % ~ l/CA(B), then on the applicability limit the sheet may be considered stationary (Parker-Sweet's flow; Sweet, 1958; Parker, 1957) throughout the entire period of oscillation %. On this basis we find the fraction of the reconnected magnetic flux for the period of one oscillation to be

~Aoj ~ BCA(B)%JR)/o 2 ~ Bo~L, R,o = CA(B)Z/eo,

where bL is the increment of the reconnected flux for the time % in fractions of radius of the original unperturbed loop. The increase in reconnected flux provides the access for a new fraction of the free magnetic energy. From the definition bA~o we will obtain bL ~ Lo(l ' /L)llZ/R~ 12 and from (4) we have the increment:

blY(L) ~ Wo(I' /L)I/2/R~/2 . (15)

We can write the increment of the free magnetic energy in such a form if the bWo that is transformed into thermal form during the reconnection in the current sheet is small compared with ~I~(L). The value of bW~ is readily determined: bW~ ~ ( l ' /L)~l~, i.e., when l ' /L ,~ 1 the desired inequality is satisfied.

The rate of increase of the free magnetic energy during the reconnection in a current sheet may be written thus:

( dW,,~ ,,~ 6W/'r,o ~ Wfr/ZA (I' /L) 7'2 R~/2 ~ Wfr/v~ , (16)

where z r g ZA(I'/L) 7/2 R~/2 is the characteristic time of the increase in free magnetic energy. Note that in the absence of the hydrodynamical energy dissipation (l' ~ L), the time

~o "~ "CA R1/2 (17)

is the characteristic time of merging of two loops. With an essential dissipation of the ejection's hydrodynamical energy (~- 1 ,~ R~/2), an intensive decrease in free magnetic energy proceeds first in the MHDS. In this, the magnetic islands are deformed so that l ' /L decreases with time. The process of decreasing I ' /L and free magnetic energy with a characteristic time z~ ceases when z~ becomes equal to ~. This will happen when

' Z , , (l / ), ~ 1/~ 2/11 ~ = eR~/2 (18)

with r ~ , ~ ~ , ~ Zo/~ 7/zl. The subsequent evolution of the MHDS proceeds under conditions of dynamical balance, when the rate of increase in free magnetic energy during the reconnection is comparable with the rate of its decrease during the dissipation of the ejection's hydrodynamical energy. In this case, there is a decrease in total energy of the MHDS (magnetic plus hydrodynamical) with a rate

- - ~ - ~ , ( 1 9 )

dt Fir ~ (I" / L ) . "freer

where Zm~r ~ r0/r 5/11 >> Z~, ~ Zr,"

160 V. M. BARDAKOV

Plt P.

I "~Pz

Fig. 4. Rate of energy release in a relaxation process.

t

The parameter (l'/L), itself and the free magnetic energy Wfr(l', L) ~ I,V(L) (I'/L), will be changing in the dynamical balance regime with this characteristic time. The time of merging of the loops is also of the order of "truer:

~'mer ~ "C0/r 5/11 ~ "CO" (20)

Thus, an attenuation of hydrodynamical motions in the MHDS leads to a con- sequence of fundamental importance. The reconnection process of the magnetic flux that eliminates the topological non-equilibrium of the MHDS, is substantially accelerat- ed (20). The fraction of the magnetic energy that is transformed into thermal forms during the reconnection process itself within the current sheet is small compared with the fraction of the magnetic energy released during the merger of the loops to produce a thermal form in a step-wise fashion, through excitation and dissipation of the hydrodynamical motions:

~ ~ ~ ~2/11 ~ 1. W \ d / l < r / L ) ~_ ~1,/L), L ,

This implies that a point-type (l ~ L) current sheet intermittently produced in the MHDS, serves merely as a means for a rapid reconstruction of the topologically non-equilibrium magnetic configuration, making a minor contribution to the rate of energy release during the merger of the magnetic loops.

4. Relaxation Process

We now consider a general temporal evolution of a non-equilibrium MHDS, starting from reconnection of a minor fraction of the poloidal flux under the action of an external force.

Suppose that after the two loops make contact, the external force has slightly pressed the loops to each other, thereby performing on the system a work AWex t. During the

THE FLARE PROCESS IN A SYSTEM 161

pressing process, part of the magnetic flux, ALb/Lo, has reconnected. Suppose further that during the reconnection, an energy equal to A Wext has been released in a thermal form. As a result, let at the initial moment of time there be an MHDS at rest, as depicted in Figure l(b). The magnetic energy of such a system is 2Wo. Assume that A L J L o ~ 1 and that the external forces at the subsequent moment of time do not act on the MHDS. The maximum free magnetic energy at this initial moment of time, according to (4), is VV(ALb) ~ WoALb/L o. The scale of field enhancement in the vicinity of the X-point along the Y-axis is of order Dy ,,~ AL b. Since this state of the MHDS is non-equilibrium and the physical nature of non-equilibrium is the attraction of magnetic islands to each other (see Section 3), then in a time ALb/CA(Bo) along the Y-axis there will form a current sheet of width of order AL b.

In the current sheet due to finite conductivity, the poloidal flux is reconnected: ~ BoCA(Bo)/R~/~, RAL ~ CA(Bo)AL/~Io. This increases the maximum free magnetic

energy, as well as the width of the current sheet:

d W ( A L ) , ~ W ~ 1 7 6 1 (21)

dt qJA \ A L l R 1/2

The characteristic time of the increase of I~(AL), depending on AL, ensues from (21) to be:

( A L ~ ~/2 R~/2 . (22) z r ~ z A \ L o /

The period of the possible oscillations of the MHDS in which a current sheet is intermittently produced in the vicinity of the X-point (see Section 3), is equal to % ( A L ) ~ zAAL/L o. These oscillations will be manifest only after %(AL) becomes smaller than % i.e., when AL/L o > (AL/Lo) ~ ,~ R o 1. In this way, when ALb/L o < R o 1

in a time tf ~ ZAALb/Lo a current sheet is formed which, while increasing in its width, persists until the moment of time to, ,~ ZARo 1. The fraction of the reconnected flux by that time will be of order R o-

After that, the system enters the oscillatory regime which will continue during a certain period of time without deformation of the magnetic islands ( l ' /L ~ 1). The period of the oscillations grows with the increasing fraction of the reconnected flux and the oscilla- tion-averaged characteristic time of the increase in maximum free energy during this phase of MHDS evolution is determined by (22). The parameters of the oscillatory regime are determined in the same way as was done in Section 3:

AL zo~(AL) ~ z A - - ,

Lo 2LoJ

( L)4 VC~j ~ Wh, ~ WO \ Lo ! �9 (23)

162 V. M. BARDAKOV

The characteristic time of the decrease in free magnetic energy due to dissipation of the ejections (hydrodynamical energy) will be obtained using (13): z H ~ Zae-I(Lo/AL)2. The regime of reconnection of the magnetic flux without defor- mation of the magnetic islands will continue until the time zr that grows with increasing AL/L o is equal to the time z/~ that decreases with increasing AL/L o. This will happen when (AL/Lo)pr ~ 4- 2/7. The time elapsed from the onset of the process to this moment of time will be:

tpr ~ z~[(AL/Lo)~ (AL/Lo)pr ] ~ ~ A R 1 / 2 / ~ 3/7 . (24)

It may be readily shown that when AL/Lo > 4- 2/5 or, equivalently, when t > Zo/43/5, the rate of energy release in the system starts to be primarily determine d by dissipation of the ejection's hydrodynamical energy. Therefore, the oscillation-averaged rate of energy release will be calculated in the following way:

P(t)~ (dWfr~ ~ gWej (25)

Before time tp~, the rate of energy release, according to (25), grows with increasing

AL/Lo: P(t) ..~ ePo(AL/Lo) 3, where Po = WO/'CA �9 At time tpr, the rate of energy release is

Ppr ~ P o e / ~ 6/7 " (26)

The MHD S evolution at t > tp~ proceeds with deformation of the islands (l ' /L < 1). Indeed, if l ' /L g 1 is formally fixed, then for AL/L o > (AL/Lo)pr we will have z/~ < g while for such an inequality between the characteristic times, deformation of the islands will necessarily arise (see Section 3). Since at time t = tp, the characteristic time is zn g % then during the course of the subsequent evolution, the dynamical balance regime will immediately be realized when the rate of free magnetic energy increase is comparable with the rate of its decrease during dissipation of the ejection's hydrody- namical energy. A treatment of this regime is similar to the one presented in Section 3.

Proceeding in the same way as in Section 3, we will obtain:

( L o ) 2 ( t o ~ 2 1 (AL~3/2(~) 7,2

In the dynamical balance regime z~ ~ g we have from (27) the relation for the definition of ( l ' /L ) , , depending on the fraction of the reconnected flux AL/Lo:

Loo .,~, \ALl I 42/11 (28)

The oscillation period of the MHDS in this regime is % ~ ZA(l'/Lo) 3 (AL/Lo). Being averaged over the oscillations, the rate of increase of the reconnected magnetic flux is governed by the equation:

dz 45/11Z 12/11 AL - ( 2 9 )

dt Zo L o

THE FLARE PROCESS IN A SYSTEM 163

The rate of energy release, according to (25), will thus be:

e ( t ) : eoZ 12/1 / 6/11. (30)

The maximum rate of energy release attainable in the MHDS when z ~ 1 will be

PM ~ ePo/~ 6/11. (31)

The characteristic time of the increase of the rate of energy release up to a maximum value PM is determined from Equation (29) to be Zim ~ ZO/~ 5/11

Thus, after a time of order tpr + Zim, the MHDS reaches a state with a fraction &the reconnected flux comparable with the unreconnected flux. In this, a maximum rate of energy release will be achieved. It is possible, in a similar manner, to examine the decreasing phase of the rate of energy release which corresponds to a decrease in the fraction of the unreconnected flux L/L o down to zero, i.e., until a complete merging of two magnetic loops. The characteristic time of the decrease in the rate of energy release become also of order Zo/~ 5/11. The overall qualitative picture for the rate of energy release in the MHDS, when two magnetic loops are merging, is depicted in Figure 4.

The ratio of time tpr to the characteristic time zero during which the maximum rate of energy release is attained and maintained and almost all free magnetic energy of the MHDS is released, is 0 -- tpr/%, ~ ~5/77. The difference in the rate of energy release which is attained during a relaxation process in a time "tim is J = PM/Pp~ ~ ~ 2 4 / 7 7 .

During the phase of maximum rate of energy release, a maximum increase in poloidal magnetic field is reached: BM/B o ~ (Lo/l ') ,M ~ ~2/11. The enhancement of the poloidal magnetic field is localized in a segment of a size of SB ~ ( I I ' ) ,~ ~ L 2 / ~ 6/11 of each of

the magnetic islands. Thus, we have carried out an analysis of the entire relaxation process during merging

of two magnetic loops. We obtained the basic characteristics of the process: characteris- tic times, changes in the rate of energy release and in the structure of the MHDS magnetic field.

5. Conclusions

Let us outline the most important features of a relaxation process which may be present in any topologically non-equilibrium MHDS having a very large Reynolds number.

(1) Dissipation of the hydrodynamical energy of the motions of a medium arising due to topological non-equilibrium of the MHD S is of decisive importance. In this case, the energy release in the system corresponds to a step-wise transformation of magnetic energy.

(2) The energy release has a well defined flare-like evolution in time. The essence of the flare evolution implies that the relaxation process breaks into two phases: a pre-flare phase with a slow increase in the rate of energy release, and a phase of rapid increase of energy release with maximum rate which may be called the impulsive phase. The duration of the pre-flare phase is of the same order of magnitude as the duration of the

164 V.M. BARDAKOV

impulsive phase during which almost all free magnetic energy of a topologically non- equilibrium MHDS is released.

Investigation of the process ofhydrodynamical energy transformation into the thermal form provides an insight into the particular content of the thermal form of energy in the MHDS in a relaxation process and, therefore, into the possible manifestations (obser- vationaUy) of the flare process. This would make it possible to represent a model of a solar flare based on the step-like transformation of the magnetic field energy within a topologically non-equilibrium MHDS. These issues will all be treated in a subsequent paper.

When analyzing the relaxation process of a particular topologically non-equilibrium MHDS in Sections 3 and 4, simplifying assumptions were made concerning the solid body conditions of magnetic islands and the constancy of magnetic viscosity qo. Note that abandoning these assumptions would not affect the existence of fundamental features of the relaxation process mentioned above; this will be the subject of future papers.

In conclusion, we would like to emphasize that the relaxation process in a topological- ly non-equilibrium MHDS appears to be a natural mechanism for magnetic field energy release in solar flares. Also, the trigger mechanism for flare energy release is quite simple, it is the appearance of the topologically non-equilibrium MHDS itself while the main temporal characteristic of the relaxation process (the evolution of the energy release rate, see Figure 4) correlates reasonably well with various observational manifestations of a flare process in the Sun.

Acknowledgements

The author expresses his sincere gratitute to Drs I. G. Shukhman and S. M. Churilov for invaluable discussions. Special thanks are due to Mr V. G. Mikhalkovsky for his assistance with preparing the English version of the manuscript and for typing and retyping the text.

References

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