A nanoflare distribution generated by repeated relaxations triggered by kink instability

Abstract

Context. It is thought likely that vast numbers of nanoflares are responsible for the corona having a temperature of millions of degrees. Current observational technologies lack the resolving power to confirm the nanoflare hypothesis. An alternative approach is to construct a magnetohydrodynamic coronal loop model that has the ability to predict nanoflare energy distributions. Aims. This paper presents the initial results generated by a coronal loop model that flares whenever it becomes unstable to an ideal MHD kink mode. A feature of the model is that it predicts heating events with a range of sizes, depending on where the instability threshold for linear kink modes is encountered. The aims are to calculate the distribution of event energies and to investigate whether kink instability can be predicted from a single parameter. Methods. The loop is represented as a straight line-tied cylinder. The twisting caused by random photospheric motions is captured by two parameters, representing the ratio of current density to field strength for specific regions of the loop. Instability onset is mapped as a closed boundary in the 2D parameter space. Dissipation of the loop's magnetic energy begins during the nonlinear stage of the instability, which develops as a consequence of current sheet reconnection. After flaring, the loop evolves to the state of lowest energy where, in accordance with relaxation theory, the ratio of current to field is constant throughout the loop and helicity is conserved. Results. There exists substantial variation in the radial magnetic twist profiles for the loop states along the instability threshold. These results suggest that instability cannot be predicted by any simple twist-derived property reaching a critical value. The model is applied such that the loop undergoes repeated episodes of instability followed by energy-releasing relaxation. Hence, an energy distribution of the nanoflares produced is collated. This paper also presents the calculated relaxation states and energy releases for all instability threshold points. Conclusions. The final energy distribution features two nanoflare populations that follow different power laws. The power law index for the higher energy population is more than sufficient for coronal heating.

Coronal heating by the partial relaxation of twisted loops

“…The fields must be continuous at the inner radial boundaries, R 1 , R 2 and R b . (The positions are R 1 = 0.5, R 2 = 0.9 and R b = 1, so that most of the loop is similar to the one described by Bareford et al (2010), but with a thin current neutralisation layer between R 2 and R b .) Therefore, the coefficients B j and C j ( j = 2, 3, 4) are determined by the requirement of continuity of the magnetic field at all interfaces (Bareford et al 2011).…”

“…The linear kink instability threshold for this currentneutralised loop was determined by the CILTS code (Van der Linden 1991; Browning & Van der Linden 2003) -it uses a bicubic Hermite finite element method to calculate the growth rates and eigenfunctions for specific line-tied α-configurations. In contrast to the stability space for a loop of net current (Bareford et al 2010), the instability threshold is open, see Fig. 2.…”

“…A finer sampling and/or higher resolution is required to bring this ratio down to below 0.1. (2003), Browning et al (2008) and Bareford et al (2010), a loop with net current was assumed to relax such that it expanded to fill the entire potential envelope: the α-profile was invariant between r = 0 and r = 3 (R B ). The relaxed alpha was identified by assuming that ψ (axial flux) and K/ψ 2 (the normalised helicity) were conserved over the loop and envelope, in accordance with Taylor 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.…”