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

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5. Discussion and concluding remarks

The Metsähovi data at 22 and 37 GHz the flares have very different time histories. (i) Pulsations with large amplitudes (Fig. 2a,b). (ii) Single pulse with fast increase and relative slow decay Fig. 3). (iii) Pulsations superimposed upon bulk burst (Fig. 4a,c), and pulsations followed by explosive phase (Fig. 4b). These examples show that the temporal dynamics of flare energy release should contain pulsating and explosive regimes. The two-colliding- loop model (Tajima et al. 1987) explains to some extent the time profile of the types (i) and (iii) bursts similar to the event in Fig. 2b, and 4a,b, but do not explain the time profile of type (ii). Moreover the model of Tajima et al. (1987) holds in the special case of two identical current loops. It is not clear, however, what happens in the more realistic case of non-symmetric loops. From the other side, recent Yohkoh and Nobeyama observations provide strong evidence that flares often occur in a simple loop configuration.

Attempts to explain the pulsation structure of the flare emission were made by several authors (see review of Aschwanden 1987). In order to search the periodic modulation of the loop magnetic field, a MHD-approach was commonly used. Then the effect of field oscillations on the time behaviour of flare emission in different wave bands was took into account. This approach is non-self-consistent because the reaction of magnetic field variation on the rate of energy release is ignored. Thus, in the frame of that approach it is impossible to get an explosive energy release or the combination of pulsating and explosive regimes.

In this paper, we took into account the response of the energy release rate in a loop to magnetic field variations driven by a tongue of surrounding plasma. As flare trigger we considered the flute instability producing the intrusion of cold partially ionized plasma of the prominence (near the loop top) or chromosphere plasma (near the loop foot points) into the current channel of a loop. In this case the energy release is accompanied by plasma heating due to Joule dissipation of electric current driven by ion-atom collisions under non-steady-state regime of plasma injection into a magnetic loop, and by electron acceleration in DC-electric field. The current dissipation required for a flare is provided by the increase of the loop plasma resistance and is [FORMULA] orders higher than in the pre-flare stage.

In present paper, we took into consideration the response of the Ampère force arising from the deformation of initially force-free magnetic flux tube by the plasma intrusion into the current channel. This self-consistent approach helps us to understand why flare energy release driven by flute instability occurs in initially force-free flux tubes with [FORMULA]. A plasma tongue entering into a force-free current channel [FORMULA] deforms the tube magnetic field and causes a Ampère force (Eq. (7)). This force prevents the penetration of a tongue, and its magnitude grows from the tube axis towards the periphery up to the surface at [FORMULA]. The plasma is heated faster near the channel surface than in the tube axis (see Eqs. (3)and (7)). This heating becomes more effective due to the repeated (oscillating) penetration of a plasma tongue into the current channel. The gradient of gas pressure directed towards the flux tube surface compensates the Ampère force completely and causes an explosive energy release. Wheatland & Melrose (1995) omitted the effect of inhomogeneous heating of the current channel in their analysis of energy release in a force-free current-carrying loop and obtained a resistance four orders less than required for a flare. Our self-consistent model yields a powerful energy release in initially force-free magnetic loop.

The Joule heating rate per unit volume is given by Eq. (14). Let us estimate the total heating rate in the case when a flare develops near the foot-points of a loop with radius [FORMULA] cm in the column of [FORMULA] cm. The energy release volume is about [FORMULA] cm3. From Eq. (12) for a typical plasma tongue velocity [FORMULA] cm s-1 and burst duration [FORMULA] s we obtain a maximum value of [FORMULA]. Supposing that at the loop foot-points [FORMULA] cm-3 and [FORMULA] K we find the Joule heating rate: [FORMULA] erg s-1. The heating rate depends strongly on the loop magnetic field and on the relative density of neutrals, F. In order to explain the hard X-ray emission, the accelerator has to supply [FORMULA] el s-1 (Miller et al. 1977). From Eq. (23) we obtain [FORMULA], and using Eq. (22) we find [FORMULA]. Here we suppose [FORMULA] i.e. [FORMULA].

Now we can compare the heating rate Q and acceleration rate [FORMULA] in our single-loop flare model. For medium energy electrons (20 keV) Eq. (23) gives [FORMULA] erg s-1, and [FORMULA]. Depending on the magnetic field value in the loop, Joule heating can be either more or less effective then electron acceleration. For [FORMULA] G the heating prevails over acceleration. In comparatively week magnetic fields particle acceleration is more effective. Observations of solar flares provide evidence that the ratio [FORMULA] varies from event to event (Wu et al. 1986). Note that the strong dependence of the Joule heating on the magnetic field was to be expected from Eq. (3): [FORMULA]. The Joule heating rate obtained by Holman, Kundu & Kane (1989) gives [FORMULA].

We have shown that the gas pressure-gradient is very important in the flare model and is essential for the temporal behavior of the energy release. The following modes of energy release are possible in the frame of our self-consistent approach.

  1. If the initial gradient of the gas pressure in a loop is negative, [FORMULA], e.g. gas pressure drops from the loop axis, a pulsating energy release occurs with typical period [FORMULA] which is of the order of the Alfvén transit time. This case fits well for the explanation of the time profiles consisting of the pulses with deep modulation (type `i' in our classification) or single pulse (type `ii'). Single pulse occurs when pulsation period is of order of damping time of pulsations [FORMULA].

  2. When the initial gradient of the gas pressure is positive, [FORMULA], e.g. the gas pressure grows towards the periphery of the tube, flute instability leads to co-existing pulsating and explosive regimes. The lifetime of the pulsation phase [FORMULA] can be shorter or longer than the time scale of the explosive phase [FORMULA]. In the first case ([FORMULA]) pulsations appear before the explosive phase (Fig. 4a). In the second case pulsations are either superimposed upon the explosive burst (Fig. 4c) or followed by the explosive phase (Fig. 4b).

We have not considered the energy losses (electron heat conductivity, viscosity, radiation losses etc.) from the energy release volume. It should be noted also that the additional saturation and switch-off of the energy release may be connected with an increasing ionization rate in the flare process, which gives a decreasing ratio of Joule heating driven by ion-atom collisions. The electron heat conductivity and radiation losses have been accounted for in the numerical analysis. We found that both factors do not introduce any noticeable changes of the conclusions concerning the pulsating and explosive energy release.

During the pulsating and explosive regimes of Joule heating of the flare plasma, electron acceleration by DC-electric field operates simultaneously and with the same time history. This is in favor of the paradigm that heating and acceleration are two aspects of the flare process (Güdel & Benz 1993; Linsky 1996). This circumstance explains also the in-phase time profiles of microwave and hard X-ray emission of solar flares (Lee & Wang 1998). Finally, we have to note that our interpretation of mm-wave emission of solar flares under various dynamical regimes should be considered as tentative because we can not determine the emission mechanism (bremsstrahlung or gyrosynchrotron) nor the optical depth of a source in each case. Depending on the radio emission mechanism and on the optical thickness of the source the observed flux of mm-wave emission reflects both the plasma heating and particle acceleration processes in the flare region in different ways.

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Online publication: June 5, 2000