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

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1. Introduction

The last twenty-five years of solar observations in optical, radio, and X-ray bands have revealed the geometry and main characteristics of solar flares. The flare energy release occurs in a coronal magnetic loop or loop system with typical length of a few times 109 cm, the cross-sectional area [FORMULA] 1017 cm2, the plasma temperature [FORMULA]-107 K, and the number density n [FORMULA] 1010-1011 cm-3 (Bray et al. 1991). Severny (1965) has evaluated the vertical currents at the photospheric level using magnetograph measurements, indicating currents [FORMULA] 1011-1012 A in an active region. With this current the resistance R [FORMULA] 10-4-10-3 [FORMULA] is required to explain the flare energy release rate W = [FORMULA] = 1019-1021 W. Nevertheless, there are no adequate methods now for the diagnostics of electric currents in the coronal magnetic loops.

However, there are a lot of indications on long-term quasiperiodical modulations in flare emission during the flares. Zarro et al. (1987) have concluded that there were quasiperiodic (12-20 s) modulations in the soft X-ray emission in the three flares observed by SMM. Such [FORMULA] 20% modulations were interpreted as local variations of the plasma temperature ([FORMULA] 5[FORMULA]106 K), caused by a repeated energy injection to a coronal loop, and due to the periodical processes of the magnetic field reconnection. The solar event of 7 August 1972, observed by OSO-7 in 2-8 Å and 8-16 Å bands, revealed over 700 cycles of extremely regular 1.6 s oscillations which remained remarkably stable in phase and period, throughout the entire flare (Thomas et al., 1987). Such an effect was explained as the periodic kink mode instability within the magnetic arch. Wülser & Kämpfer (1987) have observed long enduring oscillations of H[FORMULA] flare emission, with time scales of 7-10 s. Kaufmann et al. (1977) have reported on the 10%-modulation at 7 GHz in the flare burst on 28 March 1976, with the period 4.7 s. The modulation was persistent throughout the entire event, e.g., more than 80 min. The VLA and Trieste observations of type I and type IV radio bursts indicated 12 s pulsations during 20 min (Zlobec et al. 1992). Short-duration quasi-periodic ([FORMULA] 10-20 s) light fluctuations beginning after the onset of an intense flare on the Hyades star II Tau and lasting for about 13 min, were reported by Rodono (1974). Andrews (1990) has found quasi-periodic variations in the U-band on AD Leo with periods of the order of 10-40 s, following optical flares for up to 15 min.

Hence in both solar and stellar flares the long-term quasi-periodic variations of the optical, radio and X-ray emission exist. Periodic modulation of solar and stellar flare emission is explained usually in terms of Alfvénic and fast magnetosonic oscillations of coronal magnetic loops (Zaitsev & Stepanov 1982, 1989, Stepanov et al. 1992, Mullan et al. 1992) or as the result of periodical motions of a filament (Zaitsev & Stepanov 1988, Houdebine et al. 1991, Doyle et al. 1990). However, the quality (Q)-factor of MHD-oscillations as well as filament oscillations is not large (Q [FORMULA] 10) under coronal conditions. It is not clear either how one can get high-Q oscillations driven by kink mode instability or magnetic field line reconnection process. Thus there are difficulties in explaining the high quality (Q [FORMULA] 100) oscillations described above. On the other side, the circuit model for a flare offered by Alfvén & Carlqvist (1967) gives us the possibility to use an equivalent LRC-circuit analog in order to resolve the high-Q problem.

Phenomenological electric circuit approach has been applied to the different problems of solar and stellar physics including flares (e.g., Alfvén & Carlqvist 1967, Spicer 1976, Kan et al. 1983, Melrose & McClymont 1987, Melrose 1991, 1995, Zaitsev & Stepanov 1992), filaments (e.g., Kuperus & Raadu 1974, Van Tend & Kuperus 1978, Martens 1987), loop transients (Anzer 1978), heating of the flux tubes (Ionson 1982), as well as the electrodynamics of hot stars (Conti & Underhill 1988), and disk-accreting magnetic neutron star (Miller et al. 1994). Ionson (1982) has considered the electrodynamic coupling between [FORMULA]1 coronal loop plasma and the underlying region with [FORMULA] 1, where [FORMULA] = 8[FORMULA] [FORMULA]. In this model the main potential magnetic field [FORMULA] of a loop is generated by a primary dynamo. The resonant coupling of noise e.m.f. driven by the photospheric convection with loop plasma, using LRC-circuit analog, was investigated. Therefore the energy is transferred from convective photospheric noise into the loop plasma, by the Joule dissipation of noise electric currents excited in a loop circuit.

This article addresses the problem of electrodynamics of current-carrying magnetic loop in terms of LRC-circuit. In this case the magnetic field [FORMULA] of a loop is essentially nonpotential, and is determined by the electric currents flowing throughout the loop but not a primary dynamo that is external to the loop's local mechanical driver as Ionson (1982) had suggested. The reason for the appearance of a current-currying loop can be connected with the photospheric convection (Henoux & Somov 1987). The loop footpoints are located usually at the border of a few supergranules where convective motions produce thin magnetic flux tubes with cross-sectional scales of 100-1000 km and the magnetic field B [FORMULA] 1000 G. An independent evidence in favour of strong electric currents in a coronal loop is the optical and X-ray data suggesting that the cross-sections of coronal loops vary only 10-20% along the entire length of a loop (Klimchuk et al. 1992).

We show that the main peculiarities of LRC-analog of current-carrying magnetic loops are as follows:

  • (i) Effective resistance and capacitance of an equivalent LRC-circuit depend on both the mean current value and the screw rate of the magnetic field. There were no evidence on such dependence in previous articles dealing with the calculations of resistance and capacitance of a current-carrying magnetic loop (Spicer 1976).

  • (ii) The main factor which determines the loop resistance is the loop footpoit area, where effective Joule current dissipation due to ion-atom collisions in partially-ionized plasma, is realized.

As an application of our model we consider here the examples of the diagnostics of the electric currents in coronal loops, using the high-Q oscillations in the Metsähovi mm-wave flare data.

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© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998