The Flare-Energy Distributions Generated by Kink-Unstable Ensembles of Zero-Net-Current Coronal Loops

Abstract

It has been proposed that the million-degree temperature of the corona is due to the combined effect of barely detectable energy releases, called nanoflares, that occur throughout the solar atmosphere. Unfortunately, the nanoflare density and brightness implied by this hypothesis means that conclusive verification is beyond present observational abilities. Nevertheless, we investigate the plausibility of the nanoflare hypothesis by constructing a magnetohydrodynamic (MHD) model that can derive the energy of a nanoflare from the nature of an ideal kink instability. The set of energy-releasing instabilities is captured by an instability threshold for linear kink modes. Each point on the threshold is associated with a unique energy release; thus we can predict a distribution of nanoflare energies. When the linear instability threshold is crossed, the instability enters a nonlinear phase as it is driven by current sheet reconnection. As the ensuing flare erupts and declines, the field transitions to a lower energy state, which is modelled by relaxation theory; i.e., helicity is conserved and the ratio of current to field becomes invariant within the loop. We apply the model so that all the loops within an ensemble achieve instability followed by energyreleasing relaxation. The result is a nanoflare energy distribution. Furthermore, we produce different distributions by varying the loop aspect ratio, the nature of the path to instability taken by each loop and also the level of radial expansion that may accompany loop relaxation. The heating rate obtained is just sufficient for coronal heating. In addition, we also show that kink instability cannot be associated with a critical magnetic twist value for every point along the instability threshold.

Coronal heating by the partial relaxation of twisted loops

“…Loops of uniform twist (loops A-C) are a more likely result of correlated convective driving (Bareford et al 2011), where the threshold is approached via a series of steps with δα 1 ≈ δα 2 -this is the reason why the part of the threshold curve where α 2 > 0 is more finely sampled compared to α 2 < 0. Note, the section of threshold on the left of Fig.…”

“…This contrasts with previous results for loops carrying net current (Browning et al 2008), in which the disturbances in the nonlinear phase of the kink generally extended to the boundary of the simulation. Bareford et al (2011) maintained zero net current by fixing a neutralising loop surface at a specified radius (r = R l ). The neutralising surface is imposed by fixing the field coefficients of the potential envelope, so that they do not change during relaxation.…”

“…However, it is simply not feasible to run such a computationallyintensive process for more than a few examples. Hence, Bareford et al (2010Bareford et al ( , 2011, used relaxation theory to identify the relaxed states for all of the threshold (i.e., marginally unstable) configurations.…”

“…The original intention of relaxation theory was to explain laboratory plasma phenomena; but latterly, it has been frequently applied to the solar corona (Heyvaerts & Priest 1984;Browning et al 1986;Vekstein et al 1993;Zhang & Low 2003;Priest et al 2005). The relative ease with which relaxation theory can be applied meant that Bareford et al (2010Bareford et al ( , 2011 were able to generate heating-event distributions from ensembles of idealised coronal loops, representing, albeit crudely, the population of coronal loops that exist within an active region. Each energy release is determined by where a loop crosses the instability threshold; this location is the outcome of a defined stochastic process.…”

“…The purpose of this paper is to elucidate further (Browning et al 2008;Hood et al 2009) the relaxation process and to understand how it can be applied to coronal loops that lack a conducting wall. We simulate a set of zero-net-current coronal loops that sample the linear instability threshold calculated by Bareford et al (2011). This is a larger set than the one investigated by Hood et al (2009), and futhermore, the loops represented here feature a current-neutralisation layer that maintains zero net current even if the currents inside the loop are predominantly single signed.…”