Kein Folientitel

1
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_at_oeaw.ac.at
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 qi 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 d5 .108 cm, r0 i5 .107 cm, Ti 106 K,
i1,2,3, I1I3 -0.5 .1010 A, and I2 1010 A is
shown as a function of the angles q2 q , and q1
- q3 a for different relations between the
size parameters of the central, b2h2 sc5.109
cm, and lateral, b1h1b3h3sl, loops.
Fig.6 Mutual potential force
function U123 (q,a) 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 (q,a)
has a maxi-mum). Thus, the external disturbances
(shocks from neighboring flares) can cause a
quick reconfiguration of the system.
Oscillations of magnetic loops. Lets suppose
in the system in Fig.5 the fixed angles q2 0,
q1 - q3 p/4, and consider a linear temporal
grow of the size (hi(t), bi(t), i1,3) of the
initially current-free lateral loops.
The dynamics of currents in the loops is
defined by a set of equations


(i1...3). Existence of the dip
in U123 (q) (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 (q,
t16000s) 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
interacts through the magnetic
field of one and the current 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 gtgt r0) loop
- resistance, where s(T) is
conductivity of plasma U0 - drop of potential
between the loops 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 Eind appears as EMF of mutual inductan-ce
, where
i,j are the loop numbers, and Mij, mutual
inductances. Characteristic time of the current
change tcL /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 Eind, caused by
the tem-poral change of an external magnetic flux
Yext 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
Yext. 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(t0)
0 and Rloop(t) R0loop a t the Eq.(3) can be
solved analytically

(4) For b1t
t/tc ltlt 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 R0loop 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, R0loop 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.