Heating the corona by nanoflares: simulations of energy release triggered by a kink instability

Abstract

Context. The heating of solar coronal plasma to millions of degrees is likely to be due to the superposition of many small energyreleasing events, known as nanoflares. Nanoflares dissipate magnetic energy through magnetic reconnection. Aims. A model has been recently proposed in which nanoflare-like heating naturally arises, with a sequence of dissipation events of various magnitudes. It is proposed that heating is triggered by the onset of ideal instability, with energy release occurring in the nonlinear phase due to fast magnetic reconnection. The aim is to use numerical simulations to investigate this heating process. Methods. Three-dimensional magnetohydrodynamic numerical simulations of energy release are presented for a cylindrical coronal loop model. Initial equilibrium magnetic-field profiles are chosen to be linearly unstable, with a two-layer parameterisation of the current profile. The results are compared with calculations of linear instability, with line-tying, which are extended to account for a potential field layer surrounding the loop. The energy release is also compared with predictions that the field relaxes to a state of minimum magnetic energy with conserved magnetic helicity (a constant α force-free field). Results. The loop initially develops a helical kink, whose structure and growth rate are generally in accordance with linear stability theory, and subsequently a current sheet forms. During this phase, there is a burst of kinetic energy while the magnetic energy decays. A new relaxed equilibrium is established that corresponds quite closely to a constant α field. The fraction of stored magnetic energy released depends strongly on the initial current profile, which agrees with the predictions of relaxation theory. Conclusions. Energy dissipation events in a coronal loop are triggered by the onset of ideal kink instability. Magnetic energy is dissipated, leading to large or small heating events according to the initial current profile.

A nanoflare distribution generated by repeated relaxations triggered by kink instability

“…The helical deformation of the kink instability generates current sheets in the nonlinear regime, in which fast magnetic reconnection rapidly dissipates magnetic energy. Recently, it has been demonstrated using 3D MHD simulations that this process causes the field to relax towards a state which is closely-approximated as a constant-α state, and the energy release is in good agreement with relaxation theory (Browning et al 2008;Hood et al 2009). These papers have investigated field profiles, based on the model of Browning & Van der Linden (2003), in the unstable region of parameter space and have shown that fast reconnection develops in a current sheet, with dissipation of magnetic energy and relaxation to a new, lower-energy state.…”

“…This is unrealistic in the context of the solar corona; the loop would be more stable than it might be otherwise. Browning et al (2008) plotted the relationship between the growth rate of the instability and the distance to the outer surface of the potential envelope (i.e., a more distant conducting wall). They found that for six unstable loop states the growth rate was invariant once the outer surface of the envelope (R 3 ) exceeded 3 2 R 2 .…”

“…Essentially, helical current sheets become the site of Ohmic dissipation, resulting in a heating event. Further simulations have revealed the appropriate correlation between magnetic energy and Ohmic heating (Browning & Van der Linden 2003;Browning et al 2008;Hood et al 2009). The energy released by an instability depends on the complex dynamics of the magnetic reconnection events that occur inside the loop.…”

“…Helicity is still affected by global resistive diffusion, however, it can still be treated as a conserved quantity, if the change in helicity is substantially smaller than the change in magnetic energy. The results of one aforementioned MHD simulation (Browning et al 2008) have shown that δK/K ∼ 10 −4 and δW/W ∼ 10 −2 (it should be noted that these figures are limited by the coarseness of the numerical grid used in the simulation). During relaxation, helicity is redistributed more evenly within the loop; as a consequence α becomes invariant with radius.…”

Shock heating in numerical simulations of kink-unstable coronal loops

“…When the two speeds are equal, the ions and electrons are about to separate, and this will lead to the higher (anomalous) resistivities associated with collisionless reconnection. For an ionized hydrogen plasma, Z = 1 and m i = m p , and, since the plasma is still (although only just) a single fluid, T e = T. Equation (3.11) therefore simplifies to 12) which can expressed in code normalized units, 13) through the use of the appropriate normalizing constants,…”