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Astron. Astrophys. 337, 887-896 (1998)

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5. LRC-circuit analog

In the self-consistent model of LRC-circuit both capacitance C and resistance R depend on electric current flowing along the magnetic loop, because the loop structure depends on the current. To find [FORMULA] and [FORMULA] let us consider the plasma velocity perturbations as well as the perturbations of the electric and magnetic fields driven by small oscillations in the current-carrying magnetic loop. We restrict our approach to Alfvénic-type fluctuations when both pressure and density of the plasma remain constant. For slender flux tubes we introduce a local cylindrical reference frame [FORMULA], in which [FORMULA]-axis in each point of the loop is parallel to the loop axis. We introduce also the following designations:

[EQUATION]

where [FORMULA], and the values of [FORMULA], and [FORMULA] satisfy Eqs. (5) describing the structure of a steady-state loop. The linearized equations describing region 1 (photosphere-convection zone where the loop footpoints are located) are expressed as

[EQUATION]

[EQUATION]

where

[EQUATION]

here

[EQUATION]

We assume that the eigenfrequencies of an equivalent LRC-circuit for the coronal loop are small compared with the inverse Alfvén transit time in the scale of the order of the loop thickness. In this case the circuit eigenmodes can be considered as adiabatic for the perturbations of the loop magnetic field. It means that for eigenmodes the magnetic field can be expressed as in the case of a steady-state situation:

[EQUATION]

Substituting Eq. (11) into Eq. (15) and taking Eqs. (14) into account, we obtain the following relations between the components of the magnetic field and currents:

[EQUATION]

Using Eqs. (16) we can write the following expression for the velocity:

[EQUATION]

Excluding the velocity from Eq. (13) and by using Eq. (17) we get:

[EQUATION]

We introduce also the average values for the current as well as the magnetic and electric fields:

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] and [FORMULA] are the constant and deviated z-components of the electric current in the magnetic flux tube. The integration of Eq. (18) over the cross-section of the loop results in the equation for the loop part in the dynamo region (region 1):

[EQUATION]

where

[EQUATION]

and [FORMULA] and [FORMULA] are the conductivity and the plasma density in the region 1.

Consider now the coronal part of a loop (region 2). We will assume that the currents and magnetic fields in region 2 vary in space according to Eqs. (16). This is due to the fact that Eqs. (5), which result in Eq. (15) for adiabatic magnetic field perturbations ([FORMULA]), are true for the coronal part of a loopwhere [FORMULA] = 0 (fully ionized plasma). It should be taken into account that the small diffusion plasma flow runs through the loop surface [FORMULA] in region 2, because both the plasma density and temperature inside and outside the loop are different. An additional argument in favour of the similarity of the space structure of magnetic fields in photospheric and coronal parts of a loop has been given by the observations of an almost constant loop cross-section along the entire loop length (Klimchuk et al. 1992).

Keeping in mind these arguments we obtain the following equation for the coronal part of the electric circuit (region 2):

[EQUATION]

where [FORMULA] and [FORMULA] are the conductivity and the plasma density in the region 2.

Deep in the photosphere where the electric current closure between the loop footpoints take place (region 3), we use the formula

[EQUATION]

Here [FORMULA] and [FORMULA] are the plasma conductivity and the cross-sectional area of the current channel in the region 3.

To describe the global electrodynamics of the current-carrying loop we must integrate [FORMULA] over the entire circuit keeping in mind Eqs. (20), (21), and (22), and the expression

[EQUATION]

where L is the circuit inductance. This results in the global electrodynamics equation:

[EQUATION]

where [FORMULA], and

[EQUATION]

[EQUATION]

[EQUATION]

and [FORMULA], [FORMULA], and [FORMULA] are the lengths of the circuits' parts in the dynamo-region, corona, and the region of the current closure, respectively. Eq. (23) suggests that deviated part of the electric current is small compared the current mean value. It opens a possibility for the interpretation of quasi-periodic low amplitude modulation of mcw-emission in terms of eigen oscillations of an equivalent LRC-circuit. Therewith the oscillation period depends on the current mean value.

Some peculiarities should be noted in Eq. (23). First, the steady-state e.m.f. acting in the photospheric dynamo-region is included in Eq. (23) by a self-consistent way via Eqs. (5). This e.m.f. generates the steady-state component of the electric current [FORMULA], persisting in Eq. (23) as a parameter which determines the main characteristics of the equivalent LRC-circuit. Both the circuit resistance R and capacitance C depend on the total current [FORMULA] because this current determines the self- consistent magnetic field inside the tube and, as a result, the effective conductivity and the Alfvén velocity. Secondly, the resistance R and capacitance C of the circuit with the given electric current [FORMULA] depend on the magnetic field twisting in a loop, [FORMULA]. Thirdly, it is easy to show that the total circuit resistance R is determined mainly by the nonlinear part located in the dynamo-region:

[EQUATION]

This means that the essential steady-state Joule losses for a current-carrying magnetic loop occur mainly in the photosphere.

Slow variations of the current [FORMULA] (for the time [FORMULA]) in the global electric circuit are described by:

[EQUATION]

where

[EQUATION]

[EQUATION]

The right hand side of Eq. (28) denotes e.m.f. driven by the photospheric convection in the loop footpoints:

[EQUATION]

In a self-consistent approach the [FORMULA]-component of the magnetic field [FORMULA] depends on the current [FORMULA], and hence we can write approximately [FORMULA], where [FORMULA] is the average value of the radial velocity inside the tube. A typical time scale of current increase is determined by the minimum value of the two time scales:

[EQUATION]

Variations of the circuit inductance in time can be dealing with, for example, an emerging magnetic loop. Steady-state current value can be found from the equation [FORMULA], and it is easy to show that it coincides with Eq. (10) if the coronal part and the region of current closure in the total circuit resistance are omitted, and the structure coefficient [FORMULA] when [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998
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