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Astron. Astrophys. 357, 1105-1114 (2000)

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3. Single-loop flare model

Single-loop flare (or compact flare) models are described in detail in several reviews (e.g. Priest 1985; Sakai & de Jager 1996). These models can be divided into two classes: (1) magnetic reconnection models and (2) electric current dissipation models. To the first class belongs the model of Heyvaerts et al. (1977), in which the loop-like emerging flux reconnects with the overlaying magnetic field. Spicer's (1977) model suggests that twisting magnetic field lines of a coronal arch leads to the tearing instability which is responsible for the conversion of the magnetic field energy into plasma heating and particle acceleration. However the resistive tearing-mode instability is too slow to explain the impulsive phase of a flare. Sakai & de Jager (1996) proposed that the flare may either by due to fluxthread interaction, or else be provoked by forces from outside such as emerging flux. Yohkoh observations induced single-loop cusp-type model described by Tsuneta et al.(1992) and Masuda (1994). One serious problem with the cusp-type model is that the magnetic field is too low ([FORMULA] G) in reconnection region to explain the observed flare power. A magnetodynamic flare mechanism based on the interaction of two sweeping pinches at the top of a symmetric loop leading to the annihilation of the magnetic energy has been proposed by Uchida & Shibata (1988).

The Alfvén-Carlqvist flare model is based on the dissipation of electric current in a coronal loop, which forms an equivalent electric circuit. The current is generated by photospheric convection, and the sudden increase of the loop resistance (current interruption) is due to formation of electric double layers. In the phenomenological model of Colgate (1978) the flare occurs due to dissipation of the azimuthal magnetic field [FORMULA] that is induced by the electric current along the loop. Zaitsev & Stepanov (1992) suggested that the partially ionized plasma and non-steady-state conditions play a decisive role in the flare energy release. Partially ionized plasma can penetrate into the loop current channel from the prominence located near the loop top (Pustil'nik 1973; Zaitsev & Stepanov 1992) or otherwise from the chromosphere (Zaitsev & Khodachenko 1997). In both cases the flare trigger is the flute instability.

3.1. Current-carrying steady-state loop

In studying the reasons for various kinds of the temporal dynamics of flare emission during impulsive phase we consider a coronal magnetic loop with footpoints embedded into the photosphere and formed by the converging flow of photospheric plasma. This structure can be formed when the loop footpoints are located in the nodes of supergranulation cells (Bray et al. 1984). The equivalent electric circuit consists of three domains (Fig. 5). The magnetic field and the associated electric currents are generated in region 1 located in the photosphere and lower chromosphere. In this region the electrons are more closely bound to the magnetic field lines than the positive ions. Consequently, the neutrals hydrogen atoms in the convective flow are better able to carry the ions. A radial electric field [FORMULA] develops due to charge separation. Together with the initial magnetic field [FORMULA] it generates a Hall current [FORMULA] and strengthens [FORMULA] (Sen & White 1972). In this region the electro-motive force [FORMULA] driven by the photospheric convection (Zaitsev et al. 1998) is

[EQUATION]

with H the thickness of region 1, [FORMULA] the flux tube radius, [FORMULA] the azimuthal magnetic field, [FORMULA] km s-1 the horizontal velocity of convective flow (Bray et al. 1984). This [FORMULA] supports the electric current in the flux tube, which flows from one footpoint to the other and closes in the photosphere (region 2) at the level where the conductivity becomes isotropic, corresponding to the level [FORMULA]. Region 3 is the coronal part of the loop. Here plasma beta [FORMULA], and the loop magnetic field is force-free, i.e. the electric currents flow along magnetic field lines.

[FIGURE] Fig. 5. Cartoon model of current-carrying magnetic loop. Converging convective plasma flows generate the electric current in region 1. This current flows trough coronal part of a loop from one foot-point to another and closes just under photosphere (slightly below the level of [FORMULA]) where conductivity becomes isotropic (region 2). In region 1 located between chromosphere and photosphere the flute instability arises. The resistance of current channel in region 1 grows by many orders due to flute instability and it leads to the flare. The same situation can be realized also near the loop top due to the loop-prominence interaction (region 3)

The amplification of the loop magnetic field due to the generation of Hall currents [FORMULA] continues until the field enhancement caused by the converging convective flow is compensated by the magnetic field diffusion due to finite plasma conductivity in region 1. A steady-state magnetic loop is formed. The loop magnetic field is determined by the total energy input of the convective flow during the time of loop formation (of the order of [FORMULA], [FORMULA] is the scale of supergranulation cell). The electric current value in a flare magnetic loop is of the order of [FORMULA] A (Zaitsev et al. 1998). For the energy release rate in a solar flare [FORMULA] W with this currents one needs a resistance [FORMULA] Ohm. Classical Spitzer resistivity gives [FORMULA] Ohm in the solar corona and photosphere (Kan et al. 1983; Zaitsev & Stepanov 1992). Hence, for the solar flare the resistance should increase by [FORMULA] orders of magnitude.

3.2. Joule heating

What is the origin of the flare energy release in single current-carrying coronal loops? One of the possible triggers of the flare is the flute instability (Pustil'nik 1973; Zaitsev & Stepanov 1992; Zaitsev & Khodachenko 1997). Penetration of partially ionized plasma from the prominence or the surrounding chromosphere into the current channel of a loop gives a very effective electric current dissipation caused by ion-atom collisions. In this case the rate of Joule heating can be described by the generalized Ohm's law (see e.g. Cowling 1957)

[EQUATION]

Here [FORMULA] and [FORMULA] are the neutral gas and electron pressures, respectively, [FORMULA] is the Coulomb conductivity, [FORMULA], [FORMULA] and [FORMULA] are electron-ion, electron-atom and ion-atom collision frequencies, respectively, [FORMULA] is the velocity of bulk plasma [FORMULA], which is determined from the motion equation

[EQUATION]

with [FORMULA] and [FORMULA] being the total gas pressure and the total plasma density, respectively, [FORMULA] is the relative density of neutrals. In the case [FORMULA] we obtain from Eqs. (1) and (2) the formula for the rate of Joule dissipation:

[EQUATION]

Here and below we use the definition [FORMULA]. In steady state the current-carrying coronal magnetic loop is force-free. Here the gas pressure inside the loop is much less than the magnetic field pressure, [FORMULA]. Thus, the electric current [FORMULA] is approximately parallel to the magnetic field [FORMULA] [FORMULA], and the energy release is very small. When a "tongue" of surrounding chromosphere plasma penetrates into the current channel of the loop due to the flute instability, the loop magnetic field is deformed in accord with the equation

[EQUATION]

As a result the loop resistance can be as high as [FORMULA] Ohm (Zaitsev et al. 1998).

The threshold of the ballooning mode of the flute instability (Mikhailovskii 1975) can be represented as [FORMULA], where H is approximately equal to the scale of the transition region between the photosphere and the chromosphere (L is the loop length). For [FORMULA] km, [FORMULA] km and [FORMULA] km which gives [FORMULA]. The plasma beta in the chromosphere can be estimated as [FORMULA], hence the flute instability threshold can be easily reached due to, for instance, local heating or plasma compression. We suppose the following approximation for the radial component of the velocity field of the tongue (Henoux & Somov 1991): [FORMULA]. In such a case, both [FORMULA] and [FORMULA] inside the flux tube, can be represented as (Zaitsev & Khodachenko 1997):

[EQUATION]

If the initial magnetic flux tube is force-free [FORMULA], for example (Priest 1982)

[EQUATION]

the force-free field is disturbed by the flute instability according to Eq. (5). Ampère's force becomes

[EQUATION]

This force for [FORMULA], pushes the penetrating plasma out of the current channel. At the same time it follows from Eq. (3) that the Ampère force produces strong Joule heating inside the current loop. The variation of the gas pressure in the loop is described by

[EQUATION]

where [FORMULA] is the ratio of specific heats. Hence we obtain the self-consistent set of equations for the plasma velocity and gas pressure. It provides information about the time evolution of the energy release in the impulsive phase of the flare.

Consider now the case, when the velocity of penetrating plasma is small compared to the Alfvén velocity [FORMULA]. We investigate the region of the flux tube near its axis [FORMULA]. The distribution of the gas pressure can be approximated by

[EQUATION]

We can use also [FORMULA] instead of [FORMULA], because the velocity of the penetrating plasma is small. As a result, we obtain the following set of equations

[EQUATION]

We will use the approximation [FORMULA] which means that the oscillation amplitude of a tongue is much less than the loop radius. With this assumption Eqs. (10)-(12) read

[EQUATION]

Combining Eqs. (13)-(15) yields

[EQUATION]

Here we introduced the following definitions:

[EQUATION]

The ratio [FORMULA] is the parameter of the effectiveness of Joule heating of the plasma. It varies in a wide interval depending on the magnetic field value, the tube radius as well as number density, temperature, and ionization ratio of the plasma which penetrates into the loop due to the flute instability. [FORMULA] describes the efficiency of dissipation of MHD-oscillations of the current channel by ion-atom collisions. From Eq. (14) it follows that temporal dynamics of plasma Joule heating depends on the function [FORMULA] which, in turn, determines from Eq. (16).

3.3. Particle acceleration

The time-dependence of [FORMULA] also determines the temporal dynamics of charged particle acceleration by the large-scale electric fields in a simple magnetic loop. The electric field of the current in a plasma with classical and/or anomalous conductivity (see e.g. Holman 1985; Tsuneta 1985) is small. In order to supply sufficiently high current densities very thin current sheets are required. Here we consider electron acceleration by the electric fields driven by charge separation which arises in a current-carrying flux tube. Such charge separation electric fields are larger than the electric fields from plasma conductivity.

It is well known that in the case of [FORMULA] only the electric field directed along the magnetic field takes part in electron acceleration. From the generalized Ohm's law (Eq. 1) one can find the electric field component parallel to the magnetic field:

[EQUATION]

The terms participating in the parallel component of the electric field are a projection of the electric current on the magnetic field with the Coulomb conductivity, and the projection of the electron gas pressure gradient onto the magnetic field direction. The largest electric fields are generated at the foot-points of the current-carrying magnetic loops where the converging flows exists. Charge separation arises from the strongly magnetized electrons and weak ion magnetization. The electric field stipulated for the conductivity current [FORMULA] on the right part of Eq. (18) is much less as a rule.

In a vertical axial-symmetric magnetic flux tube with convective plasma flow converging towards the tube axis, the gradient of electron gas pressure has radial direction and its value is (Zaitsev & Khodachenko 1997)

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA] (Golant et al. 1977). Hence

[EQUATION]

Here [FORMULA] is the radial component of the magnetic field in the flux tube. If no radial component of the tube magnetic field exists [FORMULA] the large-scale electric field is perpendicular to the magnetic field [FORMULA] except for a small component [FORMULA] which is unable to provide the required electron acceleration. Flute instability yields the penetration of plasma tongues into the current channel. As a result a radial component of the magnetic field appears and the electron acceleration in the quasi-static electric field of the magnetic flux tube becomes possible. From the induction equation (4) it follows that the acceleration process starts when the velocity of the penetrating plasma, [FORMULA], depends on the altitude z, i.e., the plasma tongue should be restricted in altitude. Then from Eq. (4)

[EQUATION]

and for [FORMULA]

[EQUATION]

The number of runaway electrons per second accelerated in DC-electric field is described by well-known formula (Knoepfel & Spong 1979)

[EQUATION]

where V is the volume of the acceleration region, [FORMULA] is the Dreicer field, i.e., the field at which the critical velocity for electron acceleration out of thermal distribution equals the thermal velocity [FORMULA]. If for example the flute instability develops near the foot-point of the magnetic loop with the plasma parameters [FORMULA] cm-3, [FORMULA] K, [FORMULA] cm in the column of [FORMULA] cm, the observed values of [FORMULA] el s-1 (Miller et al. 1997) reach at [FORMULA]. This gives [FORMULA] V/cm, and the energy of accelerated electrons at the scale of [FORMULA] cm is about 120 keV. Hybrid thermal/nonthermal models require [FORMULA] el s-1 (Holman & Benka 1992), and for the same plasma parameters we obtain [FORMULA].

Therefore the intrusion of partially ionized plasma driven by flute instability into the current channel of a coronal magnetic loop simultaneously gives both the plasma heating and electron acceleration in the electric fields. For acceleration near the loop foot points these electric fields have sub-Dreicer values. In this case significant numbers of electrons are accelerated and even with sub-Dreicer electric fields no problem exists with the electron fluxes required for the hard X-ray emission observed. The productivity of the electron accelerator is thus in accord with observations. It is most important that the temporal dynamics of the plasma heating and electron acceleration depends on the time behaviour of the parameter [FORMULA]. The physical sense of this parameter is the relative depth of the penetration of partially ionized plasma into the current channel of a loop. Eq. (16) determines various regimes of the energy release in a current-carrying magnetic loop.

Note, that Dennis et al. (1994) studied the evidence for both electron acceleration and plasma heating from DC-electric fields [FORMULA] in solar flares. They did, however, not address the origin of the electric field.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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