Fig.4: The system consisting of one main loop in the center and two minor lateral loops.

For the dynamical analysis of the system on Fig.4 it is sufficient to consider only a part of the whole potential force function U, so-called mutual potential force function:

(8)

The rest part, U - U123 , of U appears just as a constant under the derivative (6) in the expression for Fi

In Fig.5 the mutual potential force function U123 of the system of three loops with d=5 .108 cm, r0 i=5 .107 cm, Ti = 106 K, i=1,2,3, I1=I3= -0.5 .1010 A, and I2 = 1010 A is shown as a function of the angles 2 = , and 1 = - 3 = for different relations between the size parameters of the central, b2=h2 sc=5.109 cm, and lateral, b1=h1=b3=h3=sl, loops.

Fig.5: Mutual potential force function U123 ( for different relations betwe-en the size parameters of the central and lateral loops: a) sc/sl = 50; b) sc/sl = 10; c) sc/sl = 5; d) sc/sl = 5/3.

Vertical position of the central loop is offten unstable (U123

(has a maxi-mum). Thus, the external disturbances (shocks from neighboring flares) can cause a quick reconfiguration of the system.

Oscillations of magnetic loops.

Let´s suppose in the system in Fig.4 the fixed angles 2 = 0, 1 = - 3 = and consider a linear temporal grow of the size (hi(t), bi(t), i=1,3) of the initially current-free lateral loops.

The dynamics of currents in the loops is defined by a set of equations

(i=1...3).

Existence of the dip in U123 ((Fig. 7)means the possibility of

oscillations of the central loop near the vertical position. The quantity

is a period of oscillations,

where

Eext - the disturbing external energy input.

- amplitude

Fig.6: Evolution of mutual potential and velocity of the top of the

loop.force function U123 ( of the system with growing lateral loops

Table I. Parameters of the central loop oscillations for U123 (t=16000s

References[1] Gary, G.A., Demoulin, P., 1995, ApJ, 445, 982.[2] Leka, K.D., Canfield, R.C., McClymont, A.N., van Driel-Gesztelyi, L., 1996, ApJ, 462, 547.[3] Canfield, R.C., de La Beaujardiere, J.-F., Fan, Y., Leka, K.D., McClymont, A.N., Metcalf, T.R., Mickey, D.L., Wuelser, J.-P., Lites, B.W., 1993, ApJ, 411, 362. [4] Melrose, D.B., 1995, ApJ, 451, 391.[5] Spicer, D.S., 1982, Space Sci. Rev., 31, 351.[6] Aschwanden, M.J., Fletcher, L., Schrijver, C.J., Alexander, D., 1999, ApJ, 520, 880.[7] Khodachenko, M.L., Haerendel, G., Rucker, H.O., 2003, A&A, 401, 721.

Abstract

Inductive interaction of longitudinal currents in groups of slowly growing magnetic loops is considered. Each loop is modelled by an equivalent electric circuit with variable resistance and inductive coefficients. These parameters of the electric circuit are defined by geometry and position of the loop with respect to neighbouring loops, as well as on the plasma temperature and density in the magnetic tube. Ponderomotoric interaction of the electric circuits of the current-carrying magnetic loops is studied. Possible dynamics and conditions for stability of the interacting magnetic loops are considered. Special attention is paid to the possibility of oscillations or fast change of an inclination of the loops, which could result in their coalescence and reconnection. The main characteristics of the oscillatory dynamics of a loop were calculated and compared with the observed ones.

Introduction

Observations of a vector magnetic field on the Sun provide a sufficient information to determine a vertical component of rotB and hence to identify the vertical component of a current [1], [2]. The observed currents in active regions can reach values up to several times 1012 A. The distribution of the current, which is deduced from vector magnetograms, can be presented as a set of current-carrying loops centered on a neutral line [3]. The data indicate that the current flows from one footpoint of the magnetic loop to the other with no evidence for a return current, which should naturally appear in the case if the current along the loop is generated by a subphotospheric twisting motion [4]. Up to now there appears to be no theory, explaining how such unneutralized currents could be set up in a magnetic loop. Below we present a mechanism which could cause and influence the currents flowing in the coronal magnetic loops. It is based on the effects of the inductive electromagnetic interaction of relatively moving (rising and growing) neighboring magnetic loops. We pay our attention to the fact that in any realistic geometry of a current-carrying loop in which the current is confined to a current channel, it generates a magnetic field outside the channel. This implies that magnetic loops should interact with each other through their magnetic fields and currents. The simplest way to take into account this interaction consists in application of the equivalent electric circuit model of a loop which includes a time-dependent inductance, mutual inductance, and resistance.

The equivalent electric circuit model is of course an idealization of the real coronal magnetic loops. It usually involves a very simplified geometry assumptions and is obtained by integrating an appropriate form of Ohm's law for a plasma over a circuit [4],[5]. A simple circuit model ignores the fact that changes of the magnetic field propagate in plasma at the Alfven speed VA. Therefore the circuit equations correctly describe temporal evolution of the currents in a solar coronal magnetic current-carrying structure only on a timescale longer than the Alfven propagation time. This should always be taken into account when one applies the eqivalent electric circuit approach for the interpretation of real processes in solar plasmas.

Besides, each pair of cur-

rent - carrying magnetic loops interacts through the magnetic field of one and the current of an other by a 1/c [jxB] force

which couples them dy- namically. Recent high

resolution observations (Fig.1) give a nice

view on the coronal loops dy- namics : grow motions,

oscillations, meandering, and twisting. The oscil-

lations of the loops are Fig.1: Coronal loops in EUV (TRACE) usually modelled as stan- ding or propagating MHD wave modes [6]. At the same time the oscillatory dynamics of coronal loops can as well be interpreted in terms of the ponderomotoric interaction of their currents [7].

Inductive currents in coronal magnetic loops

The equation for the electric current I in the coronal circuit of a sepa-rate (but not isolated from surroundings) magnetic loop can be written in the following form

(1)

- inductance of the thin loop (Rloop >> r0)

- resistance, where (T) is conductivity of plasma;

U0 - drop of potential between the loop’s foot-points;

- external magnetic flux through the circuit of the loop;

- inductive electromotive force;

Rloop and Sloop are, respectively, the main radius of the loop and the area covered by the loop.

For the multiple loop systems ind appears as EMF of mutual inductan-

ce , where i,j are the loop numbers, and Mij,

mutual inductances.

Characteristic time of the current change tc=L /R c2 in the coronal elect-

ric circuit is very large (~ 104 years), so the dynamics of the current is defined by the loops motion (emergence, submergence, etc.) resulting in the evolution of the inductive coefficients with the time scales

and .

To show how the inductive electromotive force ind, caused by the tem-poral change of an external magnetic flux ext through the circuit of a loop, can result in the appearance of a significant longitudinal current in the loop, we consider a loop, rising in a constant homogeneous background magnetic field. It is not very important which particular process is responsible for the temporal change of ext. In principle, it could be, that the magnetic loop doesn't move and only a new magnetic flux emerges below it. Equally, one can consider a situation when the magnetic loop grows up and the space below it is filled with a magnetic field newly emerging from below the photosphere. In our particular case the Eq.(1) can be written as

(2)

where

Taking account of slow variation of

the term the Eq.(2) can

be reduced to

(3)

where

Fig.2: Loop rising in constant magnetic field.

For the initial condition I(t=0) = 0 and Rloop(t) = R0loop + a t the Eq.(3)

can be solved analytically

(4)

For b1t = t/tc << 1 from (4) follows the approximate formula:

(5)

The model explains typical relatively fast build up of the current inthe beginning of the loop rising and its following more slow change:

Fig.3: Dynamics of the current generated in the initially current-free loop (r0 = 5 .107 cm, R0

loop = 2 . 108 cm, T = 106 K) growing up in the background magnetic field B0 = 100 G. Different values of the loop rising speed are con- sidered (a = 2.5 .105, 105, 104 cm/s ).

Dynamic interaction of inductively connected current-carrying magnetic loops.

We consider the modelling system, as shown on Fig.4. The loops are inclined at the angles i to the vertical direction.

The generalized ponderomotive force of interaction of current-carrying

magnetic loops can be defined as

(6)

where

(7)

is a potential force function of a system of currents (xi is a generalized coordinate).

Inductive Interaction of Coronal Currents as a Possible Source for Magnetic Loops Oscillations

in Solar Active Regions

M. L. Khodachenko, and H. O. Rucker Space Research Institute, Austrian Academy of Sciences, Schmiedlstr.6, A-8042 Graz, Austria