RRethinking the solar flare paradigm
D. B. MELROSE ∗ SIfA, School of Physics, The University of Sydney, NSW 2006, Australia (Dated: March 29, 2018)
Abstract
It is widely accepted that solar flares involve release of magnetic energy stored in the solarcorona above an active region, but existing models do not include the explicitly time-dependentelectrodynamics needed to describe such energy release. A flare paradigm is discussed that includesthe electromotive force (EMF) as the driver of the flare, and the flare-associated current thatlinks different regions where magnetic reconnection, electron acceleration, the acceleration of massmotions and current closure occur. The EMF becomes localized across regions where energyconversion occurs, and is involved in energy propagation between these regions.
Key words : solar flares, time-dependent electrodynamics ∗ Email address: [email protected] a r X i v : . [ a s t r o - ph . S R ] M a r Introduction
It is widely accepted that solar flares are due to explosive release (over tens of minutes) ofmagnetic energy that has been built up and stored above an active region in the corona over alonger time (days to weeks). The released energy appears in various forms of kinetic energy:mass motions, energetic particles and heat. Such release of magnetic energy is referred to asa magnetic explosion [1, 2]. However, most models for solar flares do not include an essentialfeature of the physics of a magnetic explosion: the driver of a flare involves explicitly time-dependent electrodynamics. The history of models for solar flares, over the decades sinceGiovanelli’s 1948 model [3], is well summarized by an archive of the cartoons used to describeeach model ( http://solarmuri.ssl.berkeley.edu/~hhudson/cartoons/ ). As discussedin Section 2, most of these models may be included in one of three classes: CSHKP models,circuit models or quadrupolar models, none of which is explicitly time-dependent. Themissing ingredients are the electromotive force (EMF) and the flare-associated current. Thetime-changing coronal magnetic field implies a large-scale inductive electric field, whoseline-integral along any closed path is interpreted as an EMF, Φ, which tends to drive aflare-associated current along a closed path that needs to be identified in a specific flaremodel. The EMF is either ignored or replaced by a proxy, such as a photospheric dynamo,obscuring the essential role it plays as the driver of the flare-associated current. The EMFis the electrodynamic driver of the flare in the sense that it drives the current that linkswidely-separated regions, with the power released is due to the rate work is done by theEMF against the flare-associated current in localized regions along this current path. Notethat although both an inductive electric field and a parallel electric field (associated withfinite conductivity) appear in MHD models for flares [4], these are local fields, whereas theEMF and the current that it drives are on a global scale, and the parallel electric field ariseswhen the EMF is localized along the path of the driven current. MHD, which is neededto model local regions along the flare-driven current path, needs to be complemented bythe global electrodynamics associated with the EMF and the flare-associated current thatit drives. It is this global perspective that is obscured in many models for solar flares.In discussing the energetics of solar flares, it is important to distinguish between theenergy going into mass motions and that going into energetic electrons in the impulsivephase. A subset of flares is “eruptive” in the sense that the flare is associated with a2oronal mass ejection (CME) that carries away a large fraction of the energy released.All flares have an “impulsive” phase in which a large fraction of the energy released goesinto 10–20 keV electrons: the electrons that precipitate into the denser regions of the solaratmosphere produce hard X-ray bursts (HXBs) [5, 6] and H α brightenings, and the electronsthat escape produce type III solar radio bursts. Explicit time-dependence is included in somemagnetohydrodymanic (MHD) models for the acceleration of a CME [7–9], but this is notthe same as including the EMF as the driver of the flare. In the discussion of the energetics inthis paper, emphasis is given to the acceleration of energetic electrons, as a defining featureof the impulsive phase of any flare. The general problem of the acceleration of solar energeticparticles has been studied extensively [10, 11], with most acceleration mechanisms applyingto a small fraction of already suprathermal particles. In early literature, the acceleration ofthe electrons that produce HXBs was called “bulk energization” to indicate it involves all theelectrons in a localized region having their mean energy increased by a large factor [12–14].It has since been confirmed that the number density of the precipitating electrons in HXBsis comparable with the ambient electron number density [15]. More recent observations [16]suggest that the precipitating electrons are confined to very narrow current channels wherethey are accelerated by a parallel electric field [17, 18]. A flare model needs to include thisbulk energization as the main form of dissipation for the released magnetic energy.The neglect of the EMF and the flare-associated current obscures some important ques-tions that need to be answered in a realistic time-dependent flare model: What is theflare-associated current and along what path does it flow? Where does the flare-associatedcurrent close across field lines? Where does the EMF become localized along the currentpath? Where does magnetic reconnection occur and what role does it play in the global elec-trodynamics? Where is the dissipation/acceleration region located? How is the magneticenergy transported along the current path to the acceleration/dissipation region?Existing flare models are classified and discussed briefly in Section 2. The requirementsof a time-dependent flare model are discussed in Section 3, and some remarks on the accel-eration/dissipation are given in Section 4. The conclusions are summarized in Section 5. Most flare models can be classified as “standard” (or CSHKP) models, circuit models or3uadrupolar models. How the explicit time-dependence is avoided in each of these classes isdiscussed in this Section.
FIG. 1: A cartoon describing a version of the CSHKP model in which the upflow from the reconnec-tion region becomes a CME and the downflow from the reconnection region leads to accelerationof energetic electrons at a shock front where downflow encounters the closed-field region. Theexplicit time-dependence is replaced by the flows at implicit boundaries of the model. (From [19],reproduced by permission of the AAS.)
Early versions of the “standard” model for a flare, also known as the CSHKP (initialsof the authors of [20–23]) model, are two-dimensional (2D) with a bipolar magnetic fieldforming closed-loops below a vertical current sheet which separates regions of oppositely-directed nearly-vertical magnetic field (Figure 1). The magnetic energy release is describedin terms of a stationary inflow of (frozen-in) magnetic field and plasma to the current sheet,where (2D) magnetic reconnection occurs, and a stationary outflow of reconnected magneticfield and plasma both up and down from the reconnection site.There is no explicit time-dependence in a CSHKP model, and hence there is no EMF.The time-dependence is implicit in the boundary conditions, involving inflows and outflows.The only electric field is a potential field, and the line-integral of this field along any closedpath inside the boundary is zero. In 2D versions of the model the only allowed current isalong the axis perpendicular to the 2D structures. Although the 2D assumption is relaxed4
IG. 2: A circuit model for a solar flare: left, assumed current path; right, circuit diagram showinginductance, L , capacitance, C , in the corona, and a voltage source V G and resistance in the (shaded)photosphere. (From [25].) in more recent 3D versions [16], the flows remain stationary in such models, which thereforehave no EMF. In a circuit model [24, 25] the current is assumed to flow along a specified path, usuallyincluding a direct and return path along a coronal loop, with current closure across field linesat the two footpoints in the photosphere (Figure 2). Stored magnetic energy is identifiedas LI , where L is the inductance of the loop. The current is assumed to be driven by avoltage source located in one of the foopoints. Thus, in this case the EMF is replaced by aphotospheric dynamo. Also shown in Figure 2 is a capacitance, C : the kinetic energy in amass motion is simulated by capacitive energy Q / C = C Φ /
2, with the charge Q buildingup due to the flare-associated current during the flare.In the current-interruption version of a circuit model, the onset of a flare and the asso-ciated dissipation is attributed to the turning on of a coronal resistance, leading to “inter-ruption” of the coronal current [24]. Both direct and return current paths in the corona areneeded to allow the postulated resistive region to short out the current between these twopaths. Such a circuit/current-interruption model is not consistent with the electrodynamicsof a magnetic explosion. IG. 3: A quadrupolar model for a flare showing two (lightly shaded) pre-flare magnetic loops, two(darkly shaded) post-flare loops, with the locations of the sources of microwaves and hard X-raysindicated. In the text, the North (N-pol) and South (S-pol) poles of the larger and smaller pre-flareloops are labeled as footpoints 1 ± and 2 ± , respectively, and the Reconnection Region is labelled C . (From [27], reproduced by permission of the AAS.) In a quadrupolar model [26–30] two bipolar magnetic loops come together and reconnectforming two new loops connecting the four footpoints (Figure 3). The reconnection betweenthe magnetic fields in the two loops involves transfer of currents as well as magnetic fluxbetween the loops. The release of magnetic energy is attributed to the change in the currentpaths, described by the change in the self- and mutual-inductances [28], rather than achange in the currents. The energy release is estimated by comparing before and afterstates. Because the time-dependent changes during the flare are not considered, there is noEMF in these models.One way of including the time-dependence in such a model is by allowing the inductanceto be time-dependent. The simplest example involves a single current-carrying flux tubethat implodes [31], such that its length and hence its inductance decreases with time. Insuch a model the magnetic flux is LI , implying an EMF − ( dL/dt ) I as L decreases duringthe implosion. In a more general model involving multiple loops, the time-dependence isincluded in mutual inductances [32]. I are needed, and theseare discussed in Sections 3.1 and 3.2, respectively. An early estimate of the EMF due to the time-varying magnetic field of a star [33] gaveΦ of order 10 V, and this was supported by later estimates for solar flares [24, 34]. Forexample, an estimate of Φ from the rate of change of the magnetic flux during a flare is asfollows. Equating the energy released, say 10 J in a moderate-sized flare, to the changein the magnetic flux times the current, which is estimated below to be of order 10 A,one infers that a change in flux of 10 Wb is needed. Such a change over 10 s implies anEMF of 10 V. There is an independent indirect argument that supports such a large Φ inan eruptive flare. The capacitor model for the kinetic energy of a CME involves equatingthe kinetic energy in the CME to the energy Q / C = C Φ / J. With C of order 10 F the required charge is Q = 10 C, and a current I = 10 A can build up this charge over the duration 10 s of a flare. The capacitive energymay also be written as Q Φ /
2, again suggesting Φ of order 10 V. Other variants of suchestimates also lead to Φ of order 10 –10 V.There is no direct signature for this enormous EMF, posing the question as to how itcan seemingly be hidden in the corona during a flare. There is direct evidence from HXBs[5, 6] that a large fraction of the energy released goes into bulk energization of ≈ ε = e ∆Φ with ∆Φ of order 10 V. However,to explain a power of 10 W with an EMF ∆Φ = 10 V would require I = 10 A, which isimpossible in the solar corona [35]. Assuming I = 10 A requires that ∆Φ = Φ /M appearsacross M regions in series along the current path. A suggested geometry [35] is M pairs ofup- and down-current paths with precipitating electrons being accelerated by ∆Φ on eachof the up-current paths. A global model for a flare needs to incorporate this requirement. The integrated form of Faraday’s equation implies that the line-integral of the electric7eld along any path that encloses a changing magnetic flux is nonzero. However, most closedpaths are irrelevant because a current cannot flow along them. The only relevant closed pathis the one along which the flare-associated current actually flows, and this path needs tobe identified. The EMF is the line-integral of the electric field along the actual path of theflare-associated current.Energetically important coronal currents flow through the photosphere and cannot changesignificantly on the time scale of a flare [28]. The flare-associated current can be attributedto redirection or reconfiguration of currents that are flowing in the corona prior to the flare.It has long been known from vector magnetogram data [36] that the currents flowing throughthe photosphere in a flaring region are of order 10 –10 A. This suggests that appropriatefiducial values for a moderate-size flare are Φ = 10 V and I = 10 A. With these valuesone can account for the power I Φ = 10 W and the energy 10 J released over 10 s.This suggests that the flare-associated current is set up by Alfv´en waves transferringa pre-flare current across field lines. Steady-state currents in the corona are nearly force-free implying that current lines are parallel to magnetic field lines. An Alfv´en wave hasa cross-field current, allowing transfer of a parallel current from one field line to another.A sequence of Alfv´en waves, propagating between two end points can set up a closed-loopcurrent over a time scale of many Alfv´en propagation times between the end points [37].However, this is possible only if the (quasi-steady-state) current can close across field lines atthe two ends. The current flows in the opposite direction to the initial current along one fieldline, thereby partially canceling the initial current, and in the direction of the current alonganother field line which becomes the final current path. The flare-associated current mayinvolve several such closed-loop currents in the corona. It also includes additional currentloops set up between a reconnection region in the corona and a current-closure region inthe chromosphere. With the flare-associated current due to reconfiguration of pre-existingcurrents, it follows that the (maximum) flare-associated current should be comparable inmagnitude with the current flowing in the coronal flux tube prior to the flare, assumed hereto be I = 10 A.Currents can flow freely along field lines in the corona, but can flow across field linesonly under special conditions. In a specific model for the flare-associated current the specialconditions that allow cross-field closure at the two ends need to be identified. There arethree possibilities for such cross-field closure.8
A cross-field current implies a J × B force, which must be balanced by another forceor associated with the acceleration of mass. Only the latter seems plausible for theLorentz force due to the flare-associated current in the low-beta plasma in the solarcorona. • A cross-field current can flow at a footpoint of a coronal loop, due to the cross-fieldelectrical conductivity (Pedersen conductivity) of the (partially ionized) chromosphericplasma. Due to the large inertia of the chromosphere, such current closure may leadto only a small acceleration of plasma. Although the finite conductivity implies somedissipation, this is unimportant in the overall energy budget for a flare. • The third possibility may be called reconnection-associated current closure . Whencurrent-carrying flux tubes intersect and reconnect, both magnetic flux and (field-aligned) current are partly transferred to two new flux tubes. The field-aligned currentpaths are changing as a function of time during such reconnection, complicating theclassification into field-aligned and cross-field currents.In a single-loop model there are two regions of cross-field current closure. One region ofcross-field closure is in the corona where the Lorentz force produces mass motion driven bythe flare. An argument invoked for both a laboratory plasma [38] and for the terrestrialmagnetosphere during a substorm [39] suggests that the other cross-field closure is at afootpoint; in a flare this corresponds to closure due to the cross-field (Pedersen) conductivityin the chromosphere.For example, in a quadrupolar model in Figure 3 reconnection-associated current closurein the corona is assumed to occur at C . The net effect of the flare-associated currentmust be to change the initial current configuration to the final current configuration. Thisquadrupolar model requires (partial) cancelation of the currents from 1+ to 1 − and from2+ to 2 − and creation of currents from 1+ to C to 2 − and from 2+ to C to 1 − , where C ,1 ± and 2 ± are defined in the caption to Figure 3. In this case two flare-associated currentpaths are needed, one from 1 − to C to 2+ to 1 − and the other from 2 − to C to 1+ to2 − . The newly formed currents are transferred to their post-flare locations, as illustratedin Figure 3, as the (reconnected) magnetic field lines along which the currents flow move totheir post-flare locations. 9he flare-associated current must pass through acceleration/dissipation regions, wherethe magnetic energy released in the corona is transferred to energetic electrons. The sourceof the energetic electrons and the locations of the acceleration/dissipation regions are con-strained by the “number problem”. A long-standing difficulty in interpreting hard X-ray bursts (HXBs) in solar flares is thatthe number of electrons precipitating into the denser regions of the solar atmosphere, whereHXBs are generated, greatly exceeds the number of electrons in the flaring flux tube prior tothe flare [5, 6, 40]. Electrons must be resupplied to the corona at the same rate as energeticelectrons precipitate from the corona. This requirement can be satisfied by invoking areturn current [41], involving electrons flowing up from the chromosphere into the corona.Early models for the return current [42–46] invoked co-located direct and return currents.An alternative is that the electrons drawn up from the chromosphere at one footpoint ofa flaring flux tube are accelerated along their path before precipitating at the conjugatefootpoint [47].An up-current of I = 10 A at the photosphere corresponds to electrons flowing downat
I/e ≈ s − . The thick-target model for HXBs [5] requires energetic electrons precipi-tating at a rate ˙ N of order 10 s − or greater. The two rates, I/e and ˙ N differ by a factorof the same order as the mismatch M = Φ / ∆Φ between the energy e Φ and the typicalenergy ε = e ∆Φ of order 10 eV of the energetic electrons that generate HXBs and type IIIbursts. Both factors are of order 10 . This approximate equality implies that the power I Φ, estimated from the electrodynamics, and the power in precipitating electrons, ˙
N ε , areof the same order of magnitude. A model is needed to provide an interpretation as to why M is of order 10 . An explanation of the mismatch M of order 10 involves several different ingredients. Oneingredient is the assumption that the direct and return currents are set up in pairs throughpropagation of Alfv´enic fronts between the regions of cross-field current flow in the coronaand in the (partially ionized) chromosphere [48]. The Alfv´en waves transport energy (and10otential) downward from the corona along field lines. Dissipation is essential in providing asink for this energy. If there is no dissipation there is no release of magnetic energy, no EMFand no energy transport. At the start of the flare, the dissipation is required to build upallowing magnetic energy release to build up, and Φ to increase. A second ingredient is thatthere is a maximum Φ max that can be supported by Alfv´en waves [49]. Once Φ reaches Φ max ,any further increase in the rate of energy release, energy transport and energy dissipationrequires a further pair of direct and return currents to develop in series with the first. Thisresults in many pairs of direct and return currents that develop in series, with ∆Φ ≈ Φ max across field lines between direct and return currents in each pair. Such multiple pairs ofcurrent paths has been described as a “picket-fence” model [48]. Magnetic reconnection is an essential ingredient in any model for a magnetic explosion:it is required to allow the magnetic configuration to change. During a flare, current-carryingmagnetic field structures in the corona must change from an initial configuration with ahigher stored magnetic energy to a final configuration with a lower stored magnetic energy.Although there have been calculations of reconnection involving twisted flux ropes [50],how such models relate to the magnetic and current configuration on a global scale, forexample as illustrated in Figure 3, is unclear. Magnetic reconnection in the solar corona canoccur only in localized regions, usually assumed to be associated with magnetic nulls andother special structures [51]. Compared with a solar flare on a global scale, the spatial andtemporal scales of a reconnection event are microscopic, and a statistically large number oflocalized reconnection regions is needed to have a macroscopic effect. One model involvingmany magnetic nulls is referred to as turbulent reconnection [52].As in a CSHKP model, magnetic energy is assumed to flow into the reconnection regionin the form of frozen-in magnetic flux. In a time-dependent model, the rate of change ofmagnetic energy density, − ∂ ( B / µ ) /∂t , is locally balanced by the divergence of the energyflux ( B /µ ) u , where u is the fluid velocity. This inflow builds up over a large volume aroundthe reconnection region. The magnetic energy is partly converted into other forms in thereconnection region, and the energy inflow is balanced by an outflow that includes kineticenergy, energetic particles and heat. However, while this general description applies to most11odels that involve reconnection, the role played by the reconnection region is different indifferent models.In a CSHKP model, the reconnection region is also identified as the region where the con-version of energy into mass motion in a CME and into energetic particles occurs. In earlyCSHKP models, a CME was identified as an upward fast outflow from the reconnection re-gion. The acceleration of particles was assumed to occur either during the reconnection itself,or at a shock wave where the downward fast outflow encounters the underlying closed-fieldregion (Figure 1). Although these features of a CSHKP model are qualitatively appealing,such a model is no longer favored for the generation of a CME, and it encounters seeminglyinsurmountable quantitative difficulties (the “number problem”) when compared with dataon HXBs.A different interpretation of the energetics of the reconnection region is adopted here.It is assumed that the energy conversion in the reconnection region is the first stage in amulti-stage energy conversion process. The second stage includes an energy outflow fromthe reconnection region, as a Poynting flux in Alfv´enic form. There is observational evidencefor such a model for geomagnetic substorms [53, 54], and a similar model has been suggestedfor solar flares [31, 55, 56]. Although no detailed model for the conversion of the inflowingmagnetic energy into an outflowing Alfv´en flux is available, some features of such a modelare evident. The flare-associated current flowing along field lines in the corona must beredirected at the reconnection region, so that the current flows along field lines to and fromthe chromosphere, where it flows across field lines due to the Pedersen conductivity. Thereis a cross-field electric field, E ⊥ , between the direct and return current paths, and this maybe attributed to the EMF becoming localized across the reconnection region. (Specifically, E ⊥ = ∆Φ /L ⊥ , where L ⊥ is the distance between the up- and down-current paths.) Themagnetic field due to the up- and down-currents combined with E ⊥ gives the Poynting fluxin Alfv´enic form. The cross-field potential is also transported towards the chromosphere.As in the acceleration of auroral electrons in a geomagnetic substorm, it is assumed thatan acceleration/dissipation region develops on the up-current path, with electrons drawn upfrom the chromosphere in the downward or return current path.Such a model involves magnetic energy release in at least three stages. The first stage isthe conversion of magnetic energy into a Poynting flux that gives the energy inflow into thereconnection region. The second stage is the energy outflow from this region into a downward12lfv´enic Poynting flux towards the chromosphere. The third stage is the conversion of thisAlfv´enic energy flux into energetic electrons in the acceleration/dissipation region. A furtherstage is needed to explain the energy transfer to a CME in an eruptive flare. A long-standing problem in the physics of solar flares is the “bulk energization” of theenergetic (10–20 keV) electrons that produce HXBs and type III solar radio bursts in theimpulsive phase [12–14]. The location of the acceleration region, and the acceleration mech-anism are different in different models.A CSHKP model favors acceleration along the neutral plane separating oppositely di-rected magnetic field lines, cf. Figure 1. One suggestion is that the electron accelerationis associated with contracting magnetic islands during reconnection [57]. Such accelera-tion involves a parallel electric field, E (cid:107) , in the collapsing magnetic island. However, thenumber problem provides a strong argument against this being the bulk-energization mech-anism: it seems impossible for such a model to account for the large number of electronsaccelerated. However, this is an argument against the specific model, rather than againstacceleration by E (cid:107) in general. The quantitative argument favors acceleration in or near thechromosphere, where there is a copious supply of electrons. Other arguments in favor ofacceleration relatively low in the solar atmosphere include a suggested analogy with the ac-celeration of auroral electrons in a geomagnetic substorm [55, 56], an Alfv´en-wave model forenergy transport [31] and observational evidence for the number density of the precipitatingelectrons being of order the ambient number density [15].The fact that the details of the acceleration mechanism remain uncertain is not surprising:acceleration by E (cid:107) is poorly understood in any space or astrophysical context. In particular,the acceleration of auroral electrons is attributed to the potential ∆Φ changing from acrossfield lines to along field lines, producing E (cid:107) [53]. However, despite the acceleration regionbeing probed by spacecraft, the detailed plasma physics remains inadequately understood.Various specific models for E (cid:107) (cid:54) = 0 have been suggested including double layers, anomalousconductivity and phase-space holes [58]. Although it is desirable that the microphysics beunderstood, there is an argument that suggests that a detailed understanding may not beessential. Each of the structures suggested for E (cid:107) (cid:54) = 0 is on a very small spatial and temporal13cale, and a statistically large number of such structures is needed to have a macroscopiceffect. Such a statistical model, perhaps based on network theory or some related model,may be insensitive to the microphysics and allow the acceleration/dissipation to be describedin terms of a macroscopic quantity, such as an equivalent resistance. As suggested earlier [2], it is important to introduce widely different scales in order tomodel magnetic energy release in a solar flare. It is on the largest (“global”) scale thatour thinking about flares needs to be revised to include time-dependent electrodynamicsexplicitly: specifically, the time-changing magnetic field, the EMF and the flare-associatedcurrent, with the release of magnetic energy due to the work done by the EMF against theflare-associated current. There are several questions that need to be answered but which areobscured by the neglect of the EMF and the flare-associated current in most flare models,as discussed in Section 2. One question is how the very large EMF, Φ of order 10 V,can apparently be hidden in the corona: the suggested answer given here is essentially asoriginally proposed by Holman in 1985 [35]. Answers to other questions related to the flare-associated current suggest that it is due to redirection of pre-flare coronal currents, andthat it involves more than one current loops in the corona, and multiple ( M ) pairs of up-and down-current paths between a reconnection region and the current closure region inthe chromosphere. Answers to further questions relating to energy transport and energydissipation (through bulk energization) are suggested. Although the detailed answers tosuch questions are important, the main point made here is that a new way of thinking aboutflares is needed in order to recognize that such questions are relevant and to address them.This new way of thinking must be based on time-dependent electrodynamics. Acknowledgments
I thank Alpha Mastrano and Mike Wheatland for helpful comments on the manuscript.I also acknowledge helpful discussions with members of the team on “Magnetic Waves in14olar Flares” at the International Space Science Institute, Bern, Switzerland. [1] Moore R L et al
Astrophys. J. , 883[2] Melrose D B 2012
Astrophys. J. , 58[3] Giovanelli R G 1948,
Mon. Not. Roy. Astron. Soc. , 163[4] Shibata K and Magara T 2011
Living Rev. Solar Phys. , 6[5] Brown J C et al Solar Phys. , 489[6] Brown J C 1976 Phil. Trans. Roy. Soc. , 1304[7] Gibson S E and Low B C 1998
Astrophys. J. , 460[8] Wu S T et al
Space Sci. Rev. , 191[9] Chen P F 2011 Living Rev. Solar Phys. , 1[10] Fletcher L et al Space Sci. Rev. , 19[11] Zharkova V V et al
Space Sci. Rev. , 357[12] Ramaty R et al
Solar Flares , Colorado Associated Press, p 117[13] Benz A O 1987
Solar Phys. , 1[14] Melrose D B 1990
Aust. J. Phys. , 703[15] Krucker S et al Astrophys. J. , 96[16] Janvier M et al
Astrophys. J. , 60[17] Haerendel G 2017a
Astrophys. J. , 113[18] Haerendel G 2017a
Astrophys. J. , 143[19] Forbes T G and Malherbe J M 1986
Astrophys. J. , L67[20] Carmichael H 1964
NASA Special Publ. , 451[21] Sturrock P A 1966 Nature , 695[22] Hirayama T 1974
Solar Phys. , 323[23] Kopp R A and Pneuman G W 1976 Solar Phys. , 85[24] Alfv´en H and Carlqvist P 1967 Solar Phys. , 220[25] Spicer D S 1982 Space Sci. Rev. , 351[26] Uchida, Y 1980 in Sturrock P A (ed) Solar Flares p 67[27] Nishio M et al
Astrophys. J. , 976[28] Melrose D B 1997
Astrophys. J. , 523
29] Zaitsev V V et al
Astron. Astrophys. , 887[30] Uchida Y et al
Publ. Astron. Soc. Japan , 553[31] Fletcher L and Hudson H S 2008 Astrophys. J. , 1645[32] Khodachenko M L et al
Space Sci. Rev. , 83[33] Swann W F G 1933
Phys. Rev. , 217[34] Colgate S A 1978, Astrophys. J. , 1068[35] Holman G D 1985
Astrophys. J. , 584[36] Moreton G E and Severny A B 1968
Solar Phys. , 282[37] Melrose D B 1992 Astrophys. J. , 403[38] Simon A 1955
Phys. Rev. , 317[39] McPherron R L 1973 J. Geophys. Res. , 3131[40] MacKinnon A L and Brown J C 1989 Solar Phys. , 303[41] Hoyng P, Brown J C and Van Beek H F 1976
Solar Phys. , 197[42] Knight J W and Sturrock P A 1977 Astrophys. J. , 306[43] Brown J C and Melrose D B 1977
Solar Phys. , 117[44] Brown J C and Bingham R 1984 Astron. Astrophys. , 993[45] Spicer D S and Sudan R N 1984
Astrophys. J. , 448[46] van den Oord G H J 1990
Astron. Astrophys. , 496[47] Emslie A G and H´enoux J C 1995
Astrophys. J. , 371[48] Melrose D B and Wheatland M S 2013
Solar Phys. , 223[49] Melrose D B and Wheatland M S 2014
Solar Phys. , 881[50] Linton M G 2001
Astrophys. J. , 905[51] Longcope D W 2005
Living Rev. Solar Phys. , 7[52] Lazarian A and Vishniac E T 1999 Astrophys. J. , 700[53] Ergun R E et al
Phys. Plasmas , 3685[54] Chaston C C et al Geophys. Res. Lett. , 1535[55] Haerendel G 1994 Astrophys. J. Suppl. , 765[56] Haerendel G 2012 Astrophys. J. , 166[57] Drake J F et al
Nature , 553[58] Main D S et al
Phys. Rev. Lett. , 185001, 185001