RADIO SIGNATURES OF SOLAR FLARE RECONNECTION

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1 The Astrophysical Journal, 631: , 2005 September 20 # The American Astronomical Society. All rights reserved. Printed in U.S.A. RADIO SIGNATURES OF SOLAR FLARE RECONNECTION M. Bárta and M. Karlický Astronomical Institute, CZ Ondřejov, Czech Republic; barta@asu.cas.cz Receivved 2004 July 13; accepted 2005 June 1 ABSTRACT The time evolution of plasma parameters (density, magnetic field, etc.) in a two-dimensional MHD model of solar flare reconnection is computed numerically. Then assuming plasma radio emission from locations where the double-resonance instability generates upper hybrid waves due to unstable distribution function of suprathermal electrons, the radio spectra and spatial source structures in the reconnection region are modeled. By comparison of the modeled and observed spectra, a remarkable similarity has been found between the computed narrowband emission and the observed lace bursts. Finally, a new radio diagnostic of the reconnection process based on the model is proposed. Subject headinggs: MHD Sun: activity Sun: flares Sun: radio radiation turbulence 1. INTRODUCTION It has been generally accepted that the reconnection of magnetic field lines plays a key role in complex solar active processes, which can be observed as various dynamic phenomena, such as solar flares, plasmoid ejecta and CMEs (see, e.g., Priest & Forbes 2000). On the other hand, it is very difficult to detect the reconnection process itself our information is only due to consequent processes such as accelerated particles, mass motions, plasma heating. Therefore, any observational facts related more directly to the magnetic reconnection would be of great importance. According to reconnection theory, plasma flows, where MHD turbulence is probably generated (Chiueh & Zweibel 1987), represent an intrinsic part of the reconnection process. Moreover, due to acceleration processes, particle beams and other types of nonequilibrium particle distribution functions unstable with respect to growth of various kinds of (electrostatic) plasma waves are expected in this region. Under such conditions, a high level of electrostatic plasma waves can be produced, and due to the turbulent environment (see Benz & Wentzel 1981) they can be effectively converted to the electromagnetic mode. Consequently, strong radio emission from the reconnection region with features typical for the plasma emission from turbulent sources can be expected. This paper aims to model such emission (caused by a particular mechanism) and to search for corresponding radio signatures in the observed data. 2. THE MODEL In the present paper, as a continuation and extension of our previous ideas ( Karlický et al. 2001; Bárta & Karlický 2001, 2003), we propose and numerically test the following hypothesis: in the radio spectra observed in the decimetric range one should be able to find signatures of radiation caused by some kind of plasma emission process acting in the reconnection region. Among various possibilities, the instability generating the upper hybrid (UH) waves due to an excess of the transversal electron temperature is considered in this paper. Namely, such types of distribution function can be easily produced in the reconnection region, e.g., by an expansion of hot flare plasma into a colder environment along magnetic field lines (e.g., Bárta & Karlický 2001) or by betatron acceleration in the collapsing magnetic structures of different spatial sizes near plasmoids (see Karlický & Kosugi 2004). In addition, analysis of the Nobeyama radioheliograph observations made by Melnikov et al. (2002) 612 delivers evidence for a pancake-like electron velocity distribution. Among wave modes excitable under such circumstances we prefer the UH mode, generated by the double-resonance instability, because it leads to very bright coherent radiation; see the model of the zebra patterns (Zheleznyakov & Zlotnik 1975). Moreover, it does not require the cyclotron frequency to be higher than the plasma frequency. The whole process is rather complex and involves in particular the following stages: evolution of the plasma/magnetic field system during the reconnection! plasma (UH) wave generation! transformation of the plasma waves into electromagnetic ones! radiative transfer of the generated radio waves. To test our hypothesis numerically including to some extent all the mentioned subprocesses, the following scheme has been used (see Fig. 1): first, a system of MHD equations (eq. [1]) is solved in two dimensions for the initial Harris-type current sheet. Plasma parameters (mass and energy densities, magnetic field, and velocity) are computed inside the reconnection region in at successively increasing times. The calculated density and magnetic field structures (together with the preassumed electron velocity distribution) are then used as input for the calculation of the double-resonance radio emission. The evolution of the UH modes is actually calculated in the second step, whereas some plausible assumptions have been made in treating their conversion into radio radiation and its subsequent transfer. The final results artificial radio spectra are then compared with observations MHD Model of the Reconnection The plasma evolution in the model is described by a system of compressible resistive one-fluid MHD þ :=(u) ¼ þ (u =:)u ¼ :pþ j < ¼ :<(u < B) þ :=S ¼ 0; where the energy flux S and auxiliary variables (plasma pressure p and current density j ) are given by standard formulae (see, e.g., Kliem et al. 2000). The resistivity is dynamically ð1þ

2 RADIO SIGNATURES OF SOLAR FLARE RECONNECTION 613 and the real part of the dispersion equation (in the limit k k Tk? where the growth rate has a maximum) can be approximated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!ðkþ¼! 2 UH! 2 B e! 2 p e! 2 UH kk 2 k 2 : ð5þ Fig. 1. Scheme of the model. changed to an anomalous value whenever the electron-ion drift velocity exceeds a given threshold. The system is numerically solved using a two-dimensional Lax-Wendroff scheme inside a rectangular box with symmetric boundary conditions on x- andy-axes and with a free outer boundary. The Harris type current sheet perturbed for a finite time by the anomalous resistivity has been used as an initial state. See papers by Karlický (1988) and Kliem et al. (2000) for details Radio Emission Calculation As already mentioned, the radio emission under study is considered to be due to the double-resonance instability of UH waves and their subsequent transformation into the escaping electromagnetic (EM) mode. The EM waves are then transferred through the solar atmosphere and eventually received by the radio telescope Dynamics of the UH Waves The dynamics of the UH mode with a wave vector k is basically described by Here v Tek is the thermal electron velocity along the magnetic field, I s (k? v? /! Be ) is the modified Bessel function, z? ¼ k? 2 k BT? /m e! 2 B e,andtheintegers represents the harmonic number. The density and the magnetic field structures evolving during the reconnection process govern the emission of the UH waves by means of the frequencies! pe and! Be contained implicitly in equation (3). In addition to the linear dynamics (2) two effects are taken into account: To simulate higher order nonlinear effects causing saturation of the instability under study, the growth of a particular unstable mode is stopped at a certain level of the wave energy. Furthermore, the instability is assumed to be suppressed completely if the temporal scale of variations of local background plasma parameters approaches the characteristic growth time of the UH waves. As a consequence of equations (3), (4), and (5), the UH waves are generated only in the close vicinity of the resonant surface given by the equation!! UH (r) s! Be (r) ¼ 0; ð6þ where the spatial dependence of relevant frequencies is indicated Mode Conversion and Radiative Transfer Because it is assumed that the source is located inside an inhomogeneous, turbulent environment, the scattering of the UH waves on the low-frequency ( LF) plasma disturbances represents an efficient conversion mechanism: UH þ LF! EM: Assuming that the inhomogeneities are driven by the plasma flow and that they are practically unaffected by ponderomotive forces of the UH and EM waves, then by omitting the inverse process (EM þ LF! UH) the flux scattered into the EM mode is proportional to the energy density of the UH waves: ð7þ E(k; t þ t) ¼ E(k; t)expf½(k) Štg ð2þ F EM (!) ¼ cw UH (!): ð8þ in each point of the source, where and represent, respectively, the growth rate and the collisional damping. The relevant growth rate in the case under consideration of an equilibrium electron distribution function perturbed by a bi-maxwellian beam with T? > T k and relative density ¼ n beam /n 0 is (Mikhailovskii 1974) pffiffiffi! 3 ðkþ (k) ¼ vt 3 ek k 2 k k X 1 ; exp! 2 k 2 k v 2 T ek s¼ 1 where the frequency mismatch is " I s expð z? Þ! T # k! þ s! Be ; ð3þ T?!!(k) k k v k s! Be ð4þ Assuming furthermore a temporally constant energy spectrum of the low-frequency waves, the factor c also does not depend on time. The frequency of the EM waves emitted by the process in equation (7) would be! ¼! UH þ LF! pe þ! 2 B e 2! pe þ LF : ð9þ However, due to the axially symmetric distribution of the UH waves in the k-space (and possibly also due to their reflections on the LF inhomogeneities), one may consider also a coalescence of two UH waves UH þ UH 0! EM ð10þ as a possible process of mode conversion, resulting in a radio emission with frequency higher than 2! pe. The radiative transfer of the emitted radio waves depends on their frequency. If the process in equation (10) applies as the

3 614 BÁRTA & KARLICKÝ Vol. 631 Fig. 2. Snapshot of the reconnection dynamics in the first quadrant of the reconnection at the time t ¼ 920 A. Magnetic field and density structures are expressed by dashed lines and gray scale, respectively. The density fluctuations and secondary plasmoid formation (at X /L A ¼ 40) are clearly visible. conversion mechanism, the waves are practically unaffected by the environment and propagate almost freely. In the case of direct scattering (eq. [7]) the radiated frequency (eq. [9]) is also well above the local plasma frequency at least of 6% of! pe for the third harmonic used (s ¼ 3). Nevertheless, places with a positive deviation of the electron density can still represent obstacles for the propagation of the waves originating in the density depletions. As such obstacles form isolated hills rather than continuous barriers and the collisional damping is quite low in the hot flaring atmosphere, we believe that such effects would cause the radio waves to propagate via scattering or some kind of ducting rather than to be absorbed. Such a nonstraightforward propagation can still influence the received radiation since it causes a temporal dispersion of the emitted signal. However, due to a relatively small source size we expect only minor effects on spectrum formation. 3. RESULTS 3.1. MHD Model The system (1) was solved inside a h0; 64i ; h0; 16i box (length units are in the half-width of the current sheet L A ) with scaling parameters (see Kliem et al. 2000) for the magnetic field and electron density B 0 ¼ 0:025 T and n 0 ¼ 3 ; m 3,respectively, and plasma beta parameter ¼ 0:15. An example of MHD reconnection modeling is shown in Figure 2. It shows a snapshot of the reconnection dynamics taken at time t ¼ 920 A (Alfvén time; A ¼ L A /V A ); only the plasma parameters important for the subsequent radio emission calculations (electron density and magnetic field) are displayed. Fast mass flows between primary and secondary plasmoids generate variations of plasma parameters. The resulting electron density fluctuations were analyzed by the Fourier method and results are shown in Figure 8 (left). Variations exhibit a typical turbulent power-law spectrum with spectral index 2.Thesameresulthasalso been found for the magnetic field (vector potential) Radio Emission The B-field and density structures obtained are then passed into the radio emission code. The mode conversion process (eq. [7]) implying the relation in equation (8) for the radio flux and emission on the s ¼ 3 harmonic was considered here. However, besides the quantities above, the growth rate (eq. [3]) depends on the parameters and spatial structure of the perturbed distribution function, which cannot be derived from the MHD simulation. Therefore two in some sense extreme cases were studied: (1) the distribution function confined along the thin magnetic flux-tube carried away by the reconnection flow this approach was applied in our previous work (Bárta & Karlický Fig. 3. Radio brightness integrated over frequencies (gray scale) inside the reconnecting magnetic structure expressed by the field lines (dashed lines; see Fig. 2) at the time t ¼ 920 A. The emission for s ¼ 3 is located within a thin area around the resonant surface defined by condition (eq. [6]). The solid contours mark out isosurfaces of constant radiated frequency; labels are in GHz. Rectangular boxes represent examples of the source areas contributing to the background continuum (B) and to the narrowband (N) component of the radio spectrum. 2003), and (2) a widespread unstable distribution function analyzed here. In this approach the anisotropic beam perturbing an equilibrium distribution function is assumed to fill homogeneously whole the reconnection box. The key parameters of the unstable bi-maxwellian beam are ¼ 10 6, v Tek ¼ 5 ; 10 6 ms 1, v k ¼ 0ms 1 and v Te? ¼ 3 ; 10 7 ms 1. The saturation level of the unstable UH waves has been fixed at 10 6 times the thermal noise. To simulate the radio spectra comparable directly with our observations the emission was sorted to 256 frequency channels, each of 4 MHz bandwidth. The results thus obtained are shown in Figures 3, 4, and 5. The left part of Figure 5 shows the dynamic spectrum of the modeled radio emission from the reconnection. A narrowband emission, rapidly and chaotically changing in peak frequency during the studied time interval, is clearly visible, sometimes accompanied by weak background continua. It is pronounced even more distinctly in the right part of the figure, where the instantaneous spectrum taken at time t ¼ 920 A is displayed. In order to analyze the spatial structure of the radio emission from the reconnection further (see below) the instantaneous spectrum has been divided into narrowband and background components. A threshold at 70% of the peak flux was arbitrarily chosen radiation at frequencies exceeding this level was considered as the narrowband spectral component. In our case it is in the MHz range with a maximum at 1420 MHz. An analysis of the spatial structure of the radio sources within the reconnecting magnetic structure at the same time instant (t ¼ 920 A ) is presented in Figures 2, 3, and 4. Figure 3 shows the radio brightness distribution integrated over all frequencies. Fig. 4. Locations of sources of the narrowband component of the radio emission (gray scale) at the time t ¼ 920 A. Only the sources emitting in the MHz range, where the threshold (70% of the peak flux; see Fig. 5) was exceeded, were retained in the radio brightness distribution from Fig. 3. The thick dashed line represents the resonant surface defined by eq. (6) and s ¼ 3. The solid lines connect the locations, whose possible contributions to the emission radiate at 1420 MHz the frequency of the maximum radiation (see Fig. 5).

4 No. 1, 2005 RADIO SIGNATURES OF SOLAR FLARE RECONNECTION 615 Fig. 5. Left: Dynamic spectrum of the modeled radio emission from the reconnection. The radio flux is expressed in a gray scale (black: low value; white: high value). Homogeneous (in coordinate space) electron velocity distribution was used as a drive for the instability of the UH waves at third (s ¼ 3) harmonic. The narrowband emission with rapid frequency variations dominates. Right: Modeled instantaneous spectrum of the radio emission from the reconnection at the time t ¼ 920 A (a time slice of the spectrum shown right). Again, the narrowband emission represents the main feature, accompanied by weak background continuum. The dashed line at 70% of peak flux shows the threshold level chosen for an identification of the narrowband spectral component. Only the emission in the frequency range MHz exceeds the given threshold and is considered to be narrowband (corresponding to those in Fig. 4). It is confined in a thin region around the surface fulfilling the resonant condition (eq. [6]). The magnetic field lines (dashed curves) indicate the source locations inside the reconnection region; see Figure 2. It is plausible to assume that almost monochromatic emission given by the most unstable UH mode dominates in each point in the source. Therefore, a radiated frequency was unambiguously ascribed to each point inside the reconnection box as the UH mode frequency with the highest growth rate (eq. [3]), determined according to the real part of the dispersion relation (eq. [5]). However, a really significant growth rate (and therefore also the emission) is located only in the vicinity of the resonant surface (eq. [6]). The isolines of the constant radiated frequency are shown as the solid contours labeled by corresponding values in GHz. Figure 4 shows the locations of the narrowband emission sources in the reconnecting magnetic structure (field lines; the thin dashed curves). The brightness distribution of the narrowband component (that in the MHz range) is displayed in gray scale. The resonant surface given by equation (6) and s ¼ 3 is shown by the thick dashed line. In addition, a solid line connecting the points contributing ( possibly) to the radio flux at 1420 MHz (the peak frequency) is drawn. Comparing Figures 3 and 4, one can easily identify the source of the narrowband component it originates at places where the resonant surface is locally tangent to the surface of the constant radiated frequency. This leads to accumulation of contributions in one frequency channel and hence to high radio flux at that frequency. Examples of such a situation are shown inside boxes N in Figure 3. On the other hand, box B in the same figure represents a quite different configuration the resonant surface is almost perpendicular to the radiated frequency isolines and the radio flux from such a place (despite the apparently higher brightness) is Fig. 6. Outline of contributions to the radio flux on a given frequency channel centered around the angular frequency! r.theline! r (r) ¼ constant represents a surface in the real space (reconnection box) at which the possible contribution to the radio flux is emitted just on! r. The receiver with the halfbandwidth BW will receive contributions from whole the belt between lines! r (r) BW ¼ constant. The line!(r) ¼ 0 represents the resonant surface exactly fulfilling condition (6). But only the points in the close vicinity of this surface with the frequency mismatch (eq. [4]) less than RW (the resonance width) here represented by the belt between lines!(r) ¼RW contribute significantly to the radio flux. As a result, the significant contribution to the flux at given frequency channel! r comes from the cross-section of both belts. Compare with Fig. 3: the gray thin area shows the position of the resonant belt, isocontours of the constant radiated frequency correspond to the lines! r (r) ¼ constant for four given frequency channels (1.4, 1.6, 1.8, and 2.0 GHz). thus distributed over a broad range of frequencies. The net result is a broadband, weak emission forming background continua in the modeled spectra. In other words, the source in the given frequency channel (1.42 GHz in this case) is much more elongated (along the resonant surface) in the case shown in box N than in the case displayed in box B. Besides the length, the source volume is given also by its width across the resonant surface. It obviously depends on the gradient of the frequency mismatch (eq. [4]) at the resonant surface if it is low, even relatively distant points can still be in resonance. This is again the case inside box N, wherethegray source area is apparently thicker. These ideas, in a generalized form, are schematically sketched in Figure 6. Let BW be the half-bandwidth of one given frequency channel of a radio receiver centered around the angular frequency! r. Then the outlined belt around the isosurface! r (r) ¼ constant represents the source volume that possibly contributes to the radiation in this channel. On the other hand, let RW be the characteristic frequency half-width of the double resonance. Then, significant contributions to the emission come only from the belt around the surface, where the resonant condition (eq. [6]) is fulfilled. The radio flux F at the given channel can thus be estimated as some factor times the cross section of both belts: F / ðbwþðrwþ j:! <:! r j : This relation ensures that the principal contribution to the radio flux comes from (rare) regions where the resonant surface and the surface of constant radiated frequency are locally parallel. As the tangent points between the resonant surface and the surfaces of constant radiated frequency are rare, the resulting spectrum is dominated by narrowband emission centered around some frequency or several discrete frequencies/several spectral branches Relation to Observations The model presented predicts that reconnection is associated with narrowband radio emission with rapid frequency variations.

5 616 BÁRTA & KARLICKÝ Vol. 631 Fig. 7. Top: Dynamic spectrum of the lace burst observed by the Ondřejov radio spectrograph on 1999 August 17. The spectrum is formed by several discrete narrowband emission lines (several tangent points or different harmonics with different s in our model) rapidly changing their peak frequency with time. The narrowband emission is accompanied by weak background continua. Bottom: Instantaneous radio spectrum at 15:08:15.5 UT (a time slice of the spectrum shown on top). The narrowband emission is clearly visible. Compare this figure with Fig. 5. Very similar features are observed during the lace bursts a distinct spectral type identified by Karlický et al. (2001) which is correlated with solar flares as well. An example of a lace burst observation is shown in Figure 7. Besides obvious similarities in instantaneous spectra and their dynamics, another common feature was found in both modeled and observed (lace burst) emissions: in Karlický et al. (2001) temporal variations of the emission peak frequency were studied during the lace burst observed in 1998 August 10. The time series of the electron density inside the lace burst source was inferred from the observations and subsequently analyzed by the Fourier method. The results are shown in the middle panel in Figure 8. Assuming an advection in the flow as the major cause of temporal variations in source, one can estimate also spatial power spectra using the relation l v A t: ð11þ This rescaling is expressed at the top abscissa in the middle panel (Fig. 8). The density variations inferred from the lace burst observation exhibit power-law spectra with a spectral index of roughly 2. But the same result has been found for the density fluctuations produced in the present MHD model (left). This agreement is a convincing indication of the connection between the modeled radio emission and the observed lace burst at a deeper level of the source parameter dynamics. An identification of lace bursts with the radio emission modeled in the present paper (Fig. 5) would have applicable consequences for radio diagnostics of solar flare reconnection. If true, it represents the first possibility of a direct detection and analysis of one of the key parts of the reconnection process turbulence in the reconnection flows. 4. APPLICATIONS FOR DIAGNOSTICS The proposed diagnostic method is based on careful analysis of power spectra such as in the middle panel of Figure 8 that can be directly inferred from lace burst observations. Besides the direct study of turbulence properties, the analogy with the power spectra of the hydrodynamic turbulence was employed here to mine further information about the reconnection flows: the right panel of Figure 8 (see, e.g., Sreenivasan 2000) shows schematically the spatial power spectrum of the hydrodynamic turbulence. It exhibits significant breakpoints at the energy input and dissipation scales. According to hydrodynamic turbulence theory, the energy input scale corresponds to the transversal dimension of the flow, the dissipation scale is, in the case of fully developed turbulent cascade, connected with the energy dissipation rate and in the steady state thus also with the power released into the kinetic energy of the reconnection flows. Identifying similar breakpoints in the power spectra inferred from lace-burst observations would provide unique tools for the estimation of these important parameters. If we look at the Fourier spectrum in the middle panel, it seems that there is some indication of a breakdown in the power law around wave number 4 ; 10 7 m 1, although this is not very convincing. Nevertheless, applying the ideas presented to this case leads to an upper limit of the transversal dimension of the reconnection flow 15,000 km. On the other hand, no high-frequency cutoff corresponding to the dissipation scale has been found in observation-based power spectra, whose domain ends due to the limited temporal resolution of the spectrograph. In fact, one should expect it at much higher wave numbers well above this limit. Despite this Fig. 8. Left: Fourier power spectrum of density variations inside the modeled reconnection box at t ¼ 920 A. The spectrum exhibits the power-law distribution typical for a turbulent state with the index 2.0. Middle: Fourier power spectra of density variations inside the lace burst source inferred from observations of the 1998 August 10 event (see Karlický et al. 2001). Temporal variations of central frequency of the emission line were recalculated into the electron density variations. Temporal (lower abscissa) dependence was rescaled into the spatial one (top abscissa) according to eq. (11) with the Alfvén velocity V A ¼ 3:2 ; 10 6 ms 1 (inferred from the scaling parameters B 0 and n 0 ). The solid straight line corresponds to the power-law distribution with the spectral index 2. Right: Scheme of the (spatial) power spectrum of the hydrodynamic turbulence is presented here for comparison. The inertial (power-law) range is bounded between two breakpoints corresponding to the energy input and dissipation scales.

6 No. 1, 2005 RADIO SIGNATURES OF SOLAR FLARE RECONNECTION 617 drawback, one can still put some constraints on the dissipation scale: in Bárta & Karlický (2003) a radio emission model similar to that presented here was performed inside a one-dimensional source representing a thin flux-tube carried away by the reconnection flow. To simulate the presence of small scales in the MHD turbulence the authors basically extended artificially the spectrum of the plasma density and the B-field variations (Fig. 8, left) to higher values, beyond the limit imposed by the finite mesh size used in the MHD simulations. It was found that the consequent fast changes (at scales comparable with the growth time of the UH waves UH ¼ 1/ UH ) of background parameters can efficiently stop the instability of the UH waves. The resulting emission was identified with narrowband dm spikes. Assuming again advection of inhomogeneities at the Alfvén speed, the transition between spikes and lace bursts should occur when scales shorter than l crit ¼ v A / UH are present in the power spectrum of density/magnetic field variations. In this view, the lace bursts represent the radio emission from a source with a limited spectrum of plasma parameter variations most likely due to a not yet fully developed turbulent cascade. This is probably the reason for their rareness, compared to much more frequent dm spikes. 5. CONCLUSIONS In the present combined model we calculated the radio emission caused by the double-resonant instability of a pancake-type distribution function with respect to UH wave growth inside a two-dimensional reconnecting structure representing a flare core. As a result, the dynamics of the radio spectrum as well as the spatial structure of sources were found. The modeled narrowband emission with rapid frequency variations was identified with observed lace bursts due to apparent similarities in their spectra and, above all, in their dynamics. Moreover, a deeper connection between the modeled and observed emissions was found when analyzing source properties. Of course, one cannot exclude other sources of lace burst emission in fact, every similar situation in the solar corona (an unstable distribution function in background plasmas with chaotically varying parameters) would probably lead to a similar kind of radio emission. Nevertheless, the authors believe that these conditions are most favorably fulfilled just in the flare reconnection. Also, the decimetric frequency range, where the lace bursts occur, corresponds to the expected densities of the primary flare energy release sites. The present paper brings for the first time a direct connection between a key part of solar flare theory (magnetic reconnection) and observed radio bursts. Moreover, based on the model, a unique diagnostic of reconnection flows, and variations inside them, is proposed. At present such a diagnostic is limited by the rarity of lace bursts observations. Nevertheless, there is hope that with an increase of the sensitivity of the radio spectrographs their number will increase. This work was supported by grants A and S of the Academy of Sciences of the Czech Republic. The authors thank Lyndsay Fletcher (University of Glasgow) for discussion and also the anonymous referee for helpful comments and suggestions. Bárta, M., & Karlický, M. 2001, A&A, 379, , in Solar Variability as an Input to the Earth Environment, ed. A. Wilson ( ESA-SP 535; Noordwijk: ESA), 471 Benz, A. O., & Wentzel, D. G. 1981, A&A, 94, 100 Chiueh, T., & Zweibel, E. 1987, ApJ, 317, 900 Karlický, M. 1988, Bull. Astron. Czechosl., 39, 13 Karlický, M., Bárta, M., Jiřička, K., Mészárosová, H., Sawant, H. S., Fernandes, F. C. R., & Cecatto, J. R. 2001, A&A, 375, 638 Karlický, M., & Kosugi, T. 2004, A&A, 419, 1159 REFERENCES Kliem, B., Karlický, M., & Benz, A. O. 2000, A&A, 360, 715 Melnikov, V. F., Shibasaki, K., & Reznikova, V. E. 2002, Proc. 10th European Solar Physics Meeting ( ESA SP-506; Noordwijk: ESA), 257 Mikhailovskii, A. B. 1974, Theory of Plasma Instabilities ( New York: Consultants Bureau) Priest, E., & Forbes, T. 2000, Magnetic Reconnection (Cambridge: Cambridge Univ. Press) Sreenivasan, K. R. 2000, in New Trends in Turbulence ( Paris: Springer), 56 Zheleznyakov, V. V., & Zlotnik, E. Ya. 1975, Sol. Phys., 43, 431