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Pulsating and explosive energy release in solar ares

V.V. Zaitsev1, S. Urpo2, and A.V. Stepanov3
2

Applied Physics Institute, Ulyanova 46, N.Novgorod 603600, Russia Metsahovi Radio Observatory, Metsahovintie 114, FIN-02540 Kylmala, Finland 3 Pulkovo observatory, St.Petersburg 196140, Russia
1

Radio and hard X-ray observations of solar ares reveal various time behaviour of the energy release. Fig.1 presents the time pro le of 37 GHz emission of the event of 1991 May 11 observed in Metsahovi Urpo et al., 1992 which starts from six 10 s pulses with enhanced amplitude at pre- ash phase and then an explosive non-exponential energy release occurs. The ux reaches its maximum value of about 530 sfu after four seconds. This event is similar to the are of 2 November 1991 Fig.2 observed at microwaves and in hard X-ray emission Lee & Wang, 1998. Fine time structure, in particular pulse structure, interpreted usually in terms of two colliding current-currying loops Sakai & de Jager, 1996. However recent X-ray and microwave data suggest that the ares can occur in simple loop. Flare origin in a single loop was considered by many authors see, e.g. Alfven & Carlqvist, 1967; Zaitsev & Stepanov,1992; Sakai & de Jager, 1996. To explain pulses the MHD-oscillations of a magnetic loop have been used usually and only the in uence of the variation of the loop magnetic eld on the radiation modulation was considered non-self-consistent approach. Moreover, such a models don't explain an explosion energy release. We interpret here both pulsating and explosive energy releases in terms of single current loop and the advanced circuit model Zaitsev & Stepanov, 1992. In accordance to this model the are energy release occurs due to the penetration into the loop currentchannel of partially ionised plasma from the prominence or from the chromosphere. The loop resistance grows by many orders and e ective current dissipation leads to the are. In our self-consistent model the feedback of the magnetic eld variations is taking on the energy release rate into account.

Loop Model
Let us consider a coronal magnetic loop with footpoints imbedded into the photosphere in the nodes of supergranulation cells and formed hence by the converging ows. The equivalent electric circuit for such a loop can be represented as three domains. The loop magnetic eld and associated electric current are generated in the dynamo-region in the photosphere. In this region !e ea, !i ia where !e and !i are gyrofrequencies of electrons and ions, ea and ia are the frequencies of electron-atom and ion-atom collisions. Consequently, the electrons are magnetised, and ions are entrained by the neutral component of the plasma. The radial electric eld is excited due to charge imbalance, which together with the initial magnetic eld Bz generates the Hall current and which, in turn, strengthens Bz . In this region the e.m.f. driven by the photospheric convection exists which supports the electric current owing in the loop from one footpoint to the other and closes in the photosphere at the level 5000 =1 where conductivity is isotropic Zaitsev et al.,1998. In the coronal part plasma beta 1 and the loop magnetic


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eld is force-free. The strengthening of the loop magnetic eld continues until the eld enhancement caused by the converging convective ow is compensated by the magnetic eld di usion due to the nite plasma conductivity in the dynamo-region. The electric currents in a are loops usually of the order of 1011 , 1012 A and the magnetic eld at the loop axis can be as high as 2000 G Zaitsev et al., 1998.

Temporal dynamics of energy release
Penetration of partially ionised plasma driven by ute instability into the loop current channel gives an e ective electric current dissipation caused by ion-atom collisions. Joule dissipation is described based on the generalised Ohm's law and the rate is Zaitsev & Stepanov, 1992
2 ergs q =E + 1 V B j = nmFc2 j B2 cm3 s 1 c i ia where F is the relative density of neutrals. In the steady-state situation the currentcarrying coronal magnetic loop is force-free j B = 0, for example

B'0 = rr 1+B0 =r2 ; Bz0 = 1+B0 =r r2 0 r2 0

2 0

2

and no energy release exists. The ute instability disturbs the force-free situation. As a result the Ampere force appears
2 1 j B = B0 r e4y e2y , 1 e 2 r2 c 2r0 1 + r0 e2y 3

0

3 4

where

1 Zt V t0 + F 2 @V0 dt0 y = ,r 0 nmiia @t0 0 0

is the relative magnitude of the penetrating `tang' of partially ionized plasma, n is the plasma density. Ampere force in the case of converging plasma owV0t 0 pushes the penetrating plasma out the current channel. At the same time as it follows from Eq. 1 the Ampere force produces strong Joule heating inside the current loop. The deviation of gas pressure in a loop is described by the formula
4 1 dp = F 2B0 r2e8y e2y , 12 4 , 1 dt nmi ia 42 r0 1 + r022 e2y 6 r

5

In the case when the velocity of penetrating plasma is small compared to the Alfven 2 velocity jV0j VA for the region of a magnetic ux tube near its axis r2 r0 we obtain the following equation for y:

@ 3 y + " @ 2 y + @y , 2"" y @y = "y 1 @ 3 1@ 2 @ @

2

6


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Here tc =2 , 1F 2 n + na 1 ; ta = 2r0 ; = tt ; " = ttc ; "1 = 2 " 1 n ia VA , A A The ratio of " = tc=tA is the parameter of the e ectiveness of Joule heating and vary in wide interval depending on magnetic eld value, tube radius as well as number density, temperature, and ionization rate of the plasma which penetrates into a loop. Parameter "1 describes the e ectiveness of dissipation of MHD-oscillations of the current channel due to ion-atom collisions. From Eq. 5 it is follows that temporal dynamics of plasma Joule heating determines entirely by the function yt which, in turn, determines from the Eq. 6. Depending on parameter " = tc=tA various regimes of the energy release are possible. In the case of y 1we can omit the last term "y2 as well as the term of 2""1y@ y=@ in Eq. 6. Supposing that y = 0 and the second derivative @ 2 y=@ 2 =0 for =0 and "1 1 we can take the integral from Eq. 6 and obtain the oscillating solution y = A sin , where A = ,V00=2VA. For the second approximation we obtain more exact formula:
i Pulsating energy release.
2 y = A sin + "A , 4 sin + 1 sin 2 2 3 6

7

We see from Eq. 7 that together with the fundamental mode the second harmonic in the oscillations appears. Just in this way we can explain double sub-peak structure observed in the ares on June 7, 1980 Naka jima et al., 1983. Therewith the period of the fundamental mode is equal to T = r0=VA.
ii Explosive energy release. From Eq. 7 we see that from the moment 2="A the value of y and consequently gas pressure start grow in time. Physically it means that in the ux tube the gradient of gas pressure is equal to the Ampere force. In this case we can omit the third order derivative in Eq. 6 and rewrite it in the form: @ y=@ = "y2. This equation has the explosive solution

1 y = , " 0

8

where 0 =1="A. Hence, characteristic time of explosive energy release is more than two times less then the duration of the pulsating phase as we can see in Fig.1. Time-dependence of yt determines also the temporal dynamics of charged particle acceleration by DC-electric elds in a current loop. It is well known that only the electric eld directed along the magnetic eld takes a part in the electron acceleration. From generalized Ohm's law one can nd Ek = 5peB=enB . In vertical axial-symmetric magnetic ux tube with the convective plasma ow converging to the tube axis the we have 1,F V0B 2 Br Ek 2 , F e2 nc1 + B 2 B 9


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2 2 Here Br is the radial component of the magnetic eld, B 2 = B' + Bz ; = F 2=2 , F c2nmi ia . For solar are condition the estimations have shown that the acceleration _ rate is N 1035 el s at ED =Ek 26 ED is Dreicer eld which gives Ek 2 10,3 V cm for 200 keV electrons. From Eq. 9 it follows that such eld value generates easily in the current loop with Br =Bk 10,3.

Figure 1: The time pro le of 37 GHz emission on 1991 May 11, 1321 UT with enhanced pulses at pre- ash phase followed by the explosive phase.

Figure 2: Flare of 2 November 1991, 1612 UT observed in hard X-rays 50 kev and in 5 GHz radio emission Lee & Wang, 1998


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Therefore the penetration of partially ionised plasma driven by ute instability into the current channel of a coronal magnetic loop gives simultaneously both plasma heating by Joule dissipation and electron acceleration in sub-Dreicer electric elds. As a result we can see radiation signatures of temporal dynamics of energy release described by Eqs. 7 and 8 in microwaves and in hard X-ray emission.

References
Alfven H. & Carlqvist P. 1967, Sol.Phys., 1, 220 Lee C.-Y. &Wang H. 1998 BBSO Preprint 1032 Marsh K.A. & Hurford G.J. 1980, ApJ Lett., 240, L111 Naka jima H., Kosugi T., Kai K., & Enome S. 1983, Nature, 305, 292 Sakai J.-I. &de Jager C. 1996, Space Sci. Rev., 77, 1 Urpo S., Pohjolainen S., & Terasranta H. 1992, Solar Radio Flares 1989-1991, HUT Report 11, Ser. A Zaitsev V.V. & Stepanov A.V. 1992, Sol. Phys., 139, 343 Zaitsev V.V., Stepanov A.V., Urpo S., & Pohjolainen S. 1998, A&A, 337, 887