Context. Relaxation theory offers a straightforward method for estimating the energy that is released when continual convective driving causes a magnetic field to become unstable. Thus, an upper limit to the heating caused by ensembles of nanoflaring coronal loops can be calculated and checked against the level of heating required to maintain observed coronal temperatures (T 10 6 K). Aims. We present new results obtained from nonlinear magnetohydrodynamic (MHD) simulations of idealised coronal loops. All of the initial loop configurations discussed are known to be linearly kink unstable. The purpose of this work is to determine whether or not the simulation results agree with Taylor relaxation, which will require a modified version of relaxation theory applicable to unbounded field configurations. In addition, we show for two cases how the relaxation process unfolds. Methods. A three-dimensional (3D) MHD Lagrangian-remap code is used to simulate the evolution of a line-tied cylindrical coronal loop model. This model comprises three concentric layers surrounded by a potential envelope; hence, being twisted locally, each loop configuration is distinguished by a piecewise-constant current profile, featuring three parameters. Initially, all configurations carry zero-net-current fields and are in ideally unstable equilibrium. The simulation results are compared with the predictions of helicityconserving relaxation theory. Results. For all simulations, the change in helicity is no more than 2% of the initial value; also, the numerical helicities match the analytically-determined values. Magnetic energy dissipation predominantly occurs via shock heating associated with magnetic reconnection in distributed current sheets. The energy release and final field profiles produced by the numerical simulations are in agreement with the predictions given by a new model of partial relaxation theory: the relaxed field is close to a linear force free state; however, the extent of the relaxation region is limited, while the loop undergoes some radial expansion. Conclusions. The results presented here support the use of partial relaxation theory, specifically, when calculating the heating-event distributions produced by ensembles of kink-unstable loops. The energy release increases with relaxation radius; but, once the loop has expanded by more than 50%, further expansion yields little more energy. We conclude that the relaxation methodology may be used for coronal heating studies.

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.

An analysis of the importance of shock heating within coronal magnetic fields has hitherto been a neglected area of study. We present new results obtained from nonlinear magnetohydrodynamic simulations of straight coronal loops. This work shows how the energy released from the magnetic field, following an ideal instability, can be converted into thermal energy, thereby heating the solar corona. Fast dissipation of magnetic energy is necessary for coronal heating and this requirement is compatible with the time scales associated with ideal instabilities. Therefore, we choose an initial loop configuration that is susceptible to the fast-growing kink, an instability that is likely to be created by convectively driven vortices, occurring where the loop field intersects the photosphere (i.e. the loop footpoints). The largescale deformation of the field caused by the kinking creates the conditions for the formation of strong current sheets and magnetic reconnection, which have previously been considered as sites of heating, under the assumption of an enhanced resistivity. However, our simulations indicate that slow mode shocks are the primary heating mechanism, since, as well as creating current sheets, magnetic reconnection also generates plasma flows that are faster than the slow magnetoacoustic wave speed.

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.

We investigate magnetic reconnection in twisted magnetic fluxtubes, representing coronal loops. The main goal is to establish the influence of the field geometry and various thermodynamic effects on the stability of twisted fluxtubes and on the size and distribution of heated regions. In particular, we aim to investigate to what extent the earlier idealised models, based on the initially cylindrically symmetric fluxtubes, are different from more realistic models, including the large-scale curvature, atmospheric stratification, thermal conduction and other effects. In addition, we compare the roles of Ohmic heating and shock heating in energy conversion during magnetic reconnection in twisted loops. The models with straight fluxtubes show similar distribution of heated plasma during the reconnection: it initially forms a helical shape, which subsequently becomes very fragmented. The heating in these models is rather uniformly distributed along fluxtubes. At the same time, the hot plasma regions in curved loops are asymmetric and concentrated close to the loop tops. Large-scale curvature has a destabilising influence: less twist is needed for instability. Footpoint convergence normally delays the instability slightly, although in some cases, converging fluxtubes can be less stable. Finally, introducing a stratified atmosphere gives rise to decaying wave propagation, which has a destabilising effect.