Jack Reid
A long-standing question in solar physics, the coronal heating problem asks why the solar atmosphere becomes millions of degrees hotter further away from the Sun's surface.
Eugene Parker conjectures that the corona may be heated by the collective action of huge numbers of 'nanoflares', namely undetectably small events, each of which entails the dissipation of magnetic energy.
In a minimally stable state, a local instability can cause dissipation to propagate through a series of neighbouring, stable regions.
Avalanches are these chains of recurring events.
Coronal loops, composed of several thin, luminous strands, visible in emission, act as the clustered components of the system through which an avalanche can propagate.
MHD simulations show that the slow, continuous vortical driving of a uniform atmosphere in a straight magnetic field, can create an instability in a magnetic flux tube.
Reconnection of the magnetic field causes a succession of multiple heating events to occur, above a steady background level, as this flux tube destabilizes others within a coronal loop.
Impulsive heating events emerge in a spectrum of sizes and magnitudes.
Viscous and Ohmic dissipation, acting on plasma flows and on strong electric currents, combine to heat the coronal loop.
Magnetic field lines are traced and local heating evaluated along them.
Thermodynamic modelling of individual strands, which follow the magnetic field, verifies the capacity of heating from MHD avalanches self-consistently to maintain hot coronal temperatures.
Intense nuclear fusion within the solar interior generates incredible energy. From its multi-million-degree core, the Sun’s temperature diminishes, moving further out, towards the surface, the photosphere, where it is a few thousand degrees. Counter-intuitively, temperature then increases, climbing higher into the solar atmosphere, further out from the reactive core and into the sparser heights of the atmosphere, the corona, where it can exceed one million degrees. Explaining this requires a way to transport energy into, and release it in, the corona.
Self-organized criticality
Self-organized criticality (SOC) studies complex systems that organize themselves into stable states, which are critical to some external perturbations. One small disturbance may leave the system stable, or could trigger instability in one place, which then rapidly spreads. Wider destabilization is called an ‘avalanche’, as when one small snowball starts a sliding of snow that accumulates and gathers momentum. Best known among examples of this phenomenon is a sand-pile: adding a single grain may leave the system unchanged, or cause a massive slide.
Magnetohydrodynamics
Magnetohydrodynamics studies electrically charged material in the presence of magnetic fields, combining Maxwell’s electromagnetic laws, and those governing fluids. Extreme solar temperatures break atoms apart and make the material there into plasma, which MHD can study.
MHD model
Arced coronal loops, with curved magnetic fields and gravity, are computationally difficult to model. Instead, the model here sites three adjacent constituent strands within a coronal loop, aiming to exploring how these interact with each other. Each end is fixed in boundary planes, representing the dense photosphere, where plasma bubbles up to the surface, twisting these footpoints. Numerical simulations solve the governing MHD equations, and study these slow surface motions. Dynamic plasma behaviour is easily resolved, but prohibitively short time-steps are needed to handle thermal conduction, so the thermodynamic response to heating is excluded from the study.
Instability begins
When one strand become sufficiently twisted and the magnetic field strained, an instability occurs, such as an MHD kink mode, in which a helical current sheet forms and the cylinder ‘kinks’. Then, the unstable strand breaks, through a process of magnetic reconnection. Breaking and rejoining field lines, this produces heating and rapid jets of plasma.
Avalanche: instability spreads
Neighbouring, stressed but stable regions of magnetic field are disrupted in the aftermath. Instability spreads, in an ‘avalanche’, whereby magnetic flux from different strands becomes complexly inter-tangled.
Heating
Shifting from large to shorter scales, a network of local current sheets replaces the previous monolithic formation. These facilitate continual reconnection, dissipating magnetic energy in a recurring series of smaller heating events. Ongoing vortical motions replenish this energy, and the magnetic field powers events of a range of sizes.
Field-aligned heating
In a strong magnetic field, heat is conducted most rapidly along field lines. How heating is distributed along them is thus key in determining whether strands, which trace the magnetic field, can be maintained at the high temperatures documented. Field lines are tracked and the magnitude and composition of heating evaluated along them.
Coronal equilibrium
One-dimensional numerical simulations can more easily handle thermodynamics. From the MHD avalanche, the heating produced is injected as an energy input to models of single coronal strands. In response, individual strands achieve steady temperatures and densities, in line with those observed on the Sun.
Conclusions
Here, MHD avalanches have been shown to be viable. Most notably, smooth, constant motions can readily generate a complex magnetic field geometry, which facilitates constant and variable heating, the energy of which suffices to heat the corona.