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Another possibility for MHD-wave-driven QPPs in solar flares is a periodic triggering of the primary energy release, the reconnection of the magnetic field lines, by external MHD waves. These waves may be MHD oscillations in nearby plasma structures, e.g. a neighbouring loop, or may be approaching propagating fast waves generated elsewhere. For example, the periodically varying inflow of the plasma with the frozen-in magnetic field should modulate the reconnection rate in the current sheet or X-point. Moreover, such a mechanism does not require the driving external MHD oscillation to be of large amplitude, due to essentially nonlinear effect of the triggering.

A detailed model of the periodic modulation of magnetic reconnection by transverse MHD waves, such as, in particular, kink or sausage modes, was designed by Nakariakov et al. (2006, see the sketch in Fig. 44). The external oscillation can be either in the trapped or leaky regimes. Hence, the linkage of this oscillation with the flaring site is provided by the evanescent or leaky part of the oscillation, outside the wave guiding plasma non-uniformity. From the point of view of the flaring site, this external trans- verse oscillation looks like periodically incoming and outgoing perpendicular plasma flows. Even although these fast magnetoacoustic waves have relatively small ampli- tudes and periods, prescribed by the parameters of the external loop, approaching the magnetic null point of the flaring site, they rapidly grow up because the decrease in the fast magnetoacoustic speed near the magnetic null point, and also because the effect of focussing by refraction (McLaughlin and Hood 2004). The fast magnetoa- coustic wave evolution in the vicinity of an X-point is accompanied by the dramatic increase in the density of the periodically varying electric current induced by the wave.

When the current density becomes greater than some threshold valuejthres, various

micro-instabilities are excited in the plasma in the vicinity of the X-point. The micro- instabilities lead to the onset of micro-turbulence that can dramatically increase the transport coefficients in the plasma, in particular, producing “anomalous”resistivity. The resistivityηthen can be modelled by the piecewise relation,

η=

ηclass, for|j| ≤jthres,

ηanom,for|j|> jthres,

(71) where ηclass and ηanom ηclass are the classical and anomalous values of the resis-

tivity, respectively. Oscillatory behaviour of the anomalous resistivity causes periodic triggering of the magnetic reconnection that results, in turn, in periodic acceleration of charged particles. The periodic variation of the density, energy and pitch angle distri- bution of non-thermal electrons produces QPPs of hard X-ray, gamma-ray, microwave, and other emissions. The period of these QPPs is prescribed by the external magnetoa- coustic oscillation, and hence is not determined by the geometrical sizes of the flaring region.

A similar mechanism, but associated with slow magnetoacoustic waves, was inde- pendently developed by Chen and Priest (2006). The periodic modulation was proposed to be based on the variation of the plasma density in the wave in the vicinity of the reconnection site. Assuming that the electric current density in the current sheet in the reconnection site remains constant and changes much slower than the QPP period, the density variation in a slow wave leads to variation of the electron drift speed. When this drift speed overcomes some threshold speed, e.g. the ion-acoustic or even electron thermal speeds, ion-acoustic or Langmuir micro-turbulences are excited due to the ion- acoustic or Buneman instabilities. In the current-carrying plasma the micro-turbulence is known to be a cause of the anomalous electric resistivity.

The idea that flaring QPPs could be triggered by external MHD waves is also consistent with the observed progression of the flaring QPPs along the magnetic neutral line that separated the ribbons in large, “two-ribbon” solar flares (Nakariakov and Zimovets 2011). The impulsive energy releases of the hard X-ray and visible light emission on the solar surface are usually observed to propagate along the magnetic neutral line at the speed of a few tens of km s−1. This value of the speed is significantly below the Alfv´en and sound speeds in the corona. It can be interpreted in terms of slow magnetoacoustic waves guided by the plasma arcade that forms the flaring active region in the direction perpendicular to the magnetic field (see the discussion in Sec. 6.3). In a uniform medium slow magnetoacoustic waves are able to propagate either strictly along the magnetic field lines or weakly oblique, in a rather narrow cone along the field. However, in the presence of plasma non-uniformities these oblique slow waves can bounce between two reflecting or refracting boundaries positioned across the field, e.g. the footpoints of the coronal magnetic field lines, and hence move gradually in the direction perpendicular to the field.

In this scenario, a slow magnetoacoustic perturbation excited by an initial energy release somewhere at the top of the arcade propagates obliquely towards the arcade footpoints. There it experiences reflection on the sharp gradient of the sound speed and returns back to the top the arcade. This results into a slow motion of the wave along the arcade axis, across the field. The angle of the most efficient perpendicular propagation is about 25◦–28◦, which corresponds to the perpendicular wave vector defined by the width of the wave guide and the distance between the footpoints along the magnetic field in the arcade. Hence, the wave arrives at the top of the arcade at

the location shifted along the arcade’s axis, because of the obliqueness of the wave vector. Approaching the X-point above the arcade, the slow wave triggers another act of the energy release — another burst of the QPPs. Moreover, the induced energy release reinforces the slow perturbation, compensating its dissipative and scattering losses. The period of the generated QPPs is similar to the second harmonics of the longitudinal mode of a coronal loop that is determined by the speed of the triggering slow magnetoacoustic wave and the travel path, i.e the loop length (Sec. 6.3). For typical parameters of solar flares the period of the induced QPPs is about 10–600 s, which is consistent with observations.

In addition, the discussed scenario can successfully address another frequently ap- pearing interesting feature of flaring QPPs, the presence of double maxima in the elementary bursts (e.g. Zimovets and Struminsky 2009). According to the discussed model the double peaks of the emission may occur due to some asymmetry in the location of the wave source or in the arcade geometry. In this case, the slow pulses, reflected from the opposite footpoints, arrive at the arcade top and trigger the next elementary bursts at slightly different times and locations.