Dynamic interaction of inductively connected current-carrying magnetic loops.

The generalized ponderomotive force of interaction of current - carrying

magnetic loops can be defined as where

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

We consider the modelling system, as shown on Fig.5 The loops are inclined at the angles i to the ver- tical direction.

For the dynamical analy- sis of the system on Fig.5 it is sufficient to consider only a part of the whole po

tential force function U, so called mutual potential

force function:

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

In Fig.6 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.6: 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.5 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 period of

oscillations

Eext - the disturbing external energy input

- amplitude and

velocity of the top of the loop

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.

Abstract

Effects of electromagnetic inductive interaction in groups of slowly growing current-carrying magnetic loops are studied. Each loop is considered as an equivalent electric circuit with variable parameters (resistance, inductive coefficients) which depend on shape, scale, position of the loop with respect to neighbouring loops, as well as on the plasma parameters in the magnetic tube. By means of such a model a process of current generation and temperature change in a growing, initially current-free, loop, as well as dynamical interaction of loops with each other were studied. The data on the 3D structure and dynamics of coronal loops expected from STEREO will provide the necessary information for testing and further development of these models.

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 flowing from below the photosphere into the corona [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 current-carrying mag- netic loops interactsthrough the magnetic

field of one and thecurrent of an other by a

1/c [jxB] force, which couples them dynami-

cally. Recent high re-solution observations (Fig.1) give a nice

view on the coronal loops dynamics: grow

motions, oscillations, meandering, and twis- Fig.1: Coronal loops in EUV (TRACE) ting. The oscillations of the loops are usually modelled as standing, 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.

Iductive currents in coronal magnetic loops

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

(1)

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

- 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;

On the inductive interaction of current-carrying magnetic loops in solar active regions

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

E-mail: Maxim.Khodachenko@oeaw.ac.at

- 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 blow 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

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 in the beginning of the loop rising and its following more slow change (see Figs.3, 4)

Fig.3: Build up of the current in the initially currnt-free magnetic loop with r0 = 5 .107 cm, and R0

loop = 2 . 108 cm, containing the coronal plasma with T = 106 K and growing up with the speed a = 104 cm/s in the background magnetic field B0 = 100 G, orthogonal to the plane of the loop. The dashed line corresponds to the numerical solution of the Eq.(2), and the solid line is the approximate solution (5).

Fig.4: 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 speed of rising of the loop (a = 2.5 .105, 105, 104 cm/s ) are considered.