# Presentation: 752800

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

**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 (*x**i** *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 - U**123* , of *U *appears just as a constant under the derivative in the expression for *F**i*
- In Fig.6 the mutual potential force function *U**123* of the system of three loops with *d=5 **.**10**8** cm*, *r**0  i**=5 **.**10**7** cm*, *T**i** = 10**6** K*, *i=1,2,3*, *I**1**=I**3**= **-**0.5 **.**10**10* *A*, and *I**2** = 10**10** A* is shown as a function of the angles *2* = * *, and *1* = - *3* =  for different relations between the size parameters of the central, *b**2**=h**2**** s**c*=*5**.**10**9** cm*, and lateral, *b**1**=h**1**=b**3**=h**3**=s**l*, loops.
- Fig.6:*  **Mutual potential force function **U**123 **(**** for different relations betwe-en the size parameters of the central and lateral loops:  a) s**c**/s**l** = 50; b) s**c**/s**l** = 10; c) s**c**/s**l** = 5; d) s**c**/s**l** = 5/3. *
- Vertical position of the central loop is offten unstable (*U**123 **(***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 *(h**i**(t), b**i**(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 *U**123 **(***(Fig.7)
**means  the  possibility  of  oscillations of
-                                                            the central loop near the vertical position
- 									  - the period of
- 									    oscillations
- 					   *E**ext** *- the disturbing external energy input
- 								         -  amplitude  and
- 					  velocity of the  top of the loop
-    Table I. *Parameters of the central loop oscillations for U**123 **(****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 *rot****B*** 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 *10**12 *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 *V**A*. 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 interacts								through the magnetic 							field of one and the								current of an other by a 							*1/c [****j***x***B****] *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 (*R**loop** >> r**0*) loop
-                       - resistance, where *(T*) is conductivity of plasma;
*U**0**  - *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. Rucker*
- Space Research Institute, Austrian Academy of Sciences, Schmiedlstr.6, A-8042 Graz, Austria
- E-mail:  Maxim.Khodachenko@oeaw.ac.at

-                           - inductive electromotive force;
*R**loop** *and *S**loop**  *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  *M**ij**, *mutual inductances.
- Characteristic time of the current change *t**c**=L /R** **c**2* in the coronal elect-ric circuit is very large (~ *10**4* 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 *R**loop**(t) = R**0**loop** + a t * the Eq.(3) can be solved analytically
-                                                                                                               (4)
- For *b**1**t = t/t**c** << 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 r**0** = 5 **.**10**7** cm, and R**0**loop** = 2 **.** 10**8** cm, containing the coronal plasma with T = 10**6** K and growing up with the speed a = 10**4** cm/s in the background magnetic field B**0** = 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 (r**0** = 5 **.**10**7** cm, R**0**loop** = 2 **.** 10**8** cm, T = 10**6** K) **growing up** in the background magnetic field B**0** = 100 G. Different values of the speed of rising of the loop (a = 2.5 **.**10**5**, 10**5**, 10**4** cm/s ) are considered.*