James A. Klimchuk

On solving the coronal heating problem

The question of what heats the solar corona remains one of the most important problems in astrophysics. Finding a definitive solution involves a number of challenging steps, beginning with an identification of the energy source and ending with a prediction of observable quantities that can be compared directly with actual observations. Critical intermediate steps include realistic modeling of both the energy release process (the conversion of magnetic stress energy or wave energy into heat) and the response of the plasma to the heating. A variety of difficult issues must be addressed: highly disparate spatial scales, physical connections between the corona and lower atmosphere, complex microphysics, and variability and dynamics. Nearly all of the coronal heating mechanisms that have been proposed produce heating that is impulsive from the perspective of elemental magnetic flux strands. It is this perspective that must be adopted to understand how the plasma responds and radiates. In our opinion, the most promising explanation offered so far is Parkers idea of nanoflares occurring in magnetic fields that become tangled by turbulent convection. Exciting new developments include the identification of the “secondary instability” as the likely mechanism of energy release and the demonstration that impulsive heating in sub-resolution strands can explain certain observed properties of coronal loops that are otherwise very difficult to understand. Whatever the detailed mechanism of energy release, it is clear that some form of magnetic reconnection must be occurring at significant altitudes in the corona (above the magnetic carpet), so that the tangling does not increase indefinitely. This article outlines the key elements of a comprehensive strategy for solving the coronal heating problem and warns of obstacles that must be overcome along the way.

MAGNETIC FIELD AND PLASMA SCALING LAWS: THEIR IMPLICATIONS FOR CORONAL HEATING MODELS

In order to test different models of coronal heating, we have investigated how the magnetic field strength of coronal flux tubes depends on the end-to-end length of the tube. Using photospheric magnetograms from both observed and idealized active regions, we computed potential, linear force-free, and magnetostatic extrapolation models. For each model, we then determined the average coronal field strength, B, in approximately 1000 individual flux tubes with regularly spaced footpoints. Scatter plots of B versus length, L, are characterized by a flat section for small L and a steeply declining section for large L. They are well described by a function of the form log = C1 + C2 log L + C3/2 log(L2 + S2), where C2 ≈ 0, -3 ≤ C3 ≤ -1, and 40 ≤ S ≤ 240 Mm is related to the characteristic size of the active region. There is a tendency for the magnitude of C3 to decrease as the magnetic complexity of the region increases. The average magnetic energy in a flux tube, B2, exhibits a similar behavior, with only C3 being significantly different. For flux tubes of intermediate length, 50 ≤ L ≤ 300 Mm, corresponding to the soft X-ray loops in a study by Klimchuk & Porter (1995), we find a universal scaling law of the form Lδ, where δ = -0.88 ± 0.3. By combining this with the Klimchuk & Porter result that the heating rate scales as L-2, we can test different models of coronal heating. We find that models involving the gradual stressing of the magnetic field, by slow footpoint motions, are in generally better agreement with the observational constraints than are wave heating models. We conclude, however, that the theoretical models must be more fully developed and the observational uncertainties must be reduced before any definitive statements about specific heating mechanisms can be made.

A Nanoflare Explanation for the Heating of Coronal Loops Observed by Yohkoh

The nanoflare model of Cargill (1994a) is used to model active region loops observed by the Yohkoh Soft X-ray Telescope (SXT). Using observed information concerning the dimensions and energy-loss rate of each loop, a range of loop models with different temperatures, emission measures, and filling factors is generated. For hot loops (T > 4 × 106 K), it is shown that filling factors less than 0.1 can fit the data, although the uncertainties can be quite large. For cool loops (T ≈ 2 × 106 K), the model cannot reproduce the observed temperature and emission measure for any value of the filling factor. Earlier work of Porter & Klimchuk suggested that some of these loops cannot be explained by a steady state heating model either. It is proposed that there may exist two distinct classes of loops and that coronal material is injected into the cooler loops by a mechanism that is not directly related to heating (e.g., not chromospheric evaporation).

NANOFLARE HEATING OF THE CORONA REVISITED

The radiative signatures of the nanoflare model for coronal heating are investigated. If an observed coronal loop is assumed to consist of many small strands that cannot be distinguished spatially by EUV or X-ray observations, we are able to calculate differential emission-measure profiles and filling factors for a range of heating models. In this picture the strands undergo continual heating and cooling, leading to a corona comprising strands with a broad range of temperatures and densities. Thus, observations over a range of temperatures will show a multithermal coronal structure. The cyclical heating-cooling leads inevitably to loops that are underdense and overdense at high and low temperatures, respectively, compared to what would be expected from static equilibrium models, and in addition, we show that differential emission-measure profiles with shallow slopes can be obtained, as reported in recent observations. The differences between filling factors that can be seen by broadband and narrowband instruments are explored. Loops with broadband filling factors near unity can still have small narrowband factors, and the narrowband factor is shown to be a strong function of the local temperature. Nanoflare energy distributions that are constant, flat, or power laws are considered. Power laws lead to wide distributions of temperatures and densities in the corona, and steep power laws lead to larger filling factors.

The possible role of MHD waves in heating the solar corona

The possible role of waves in the heating of the solar corona has been investigated. A general dispersion relation has been derived for waves propagating in a homogeneous plasma subject to dissipation by viscosity and thermal conduction. The dissipation mechanisms have been incorporated self-consistently into the equations, and no assumptions about the strength of the damping have been made. Solutions of the sixth-order dispersion relation provide information on how the damping of both slow and fast mode waves depends upon the plasma density, temperature, field strength, and angle of propagation relative to the background magnetic field. We provide a detailed comparison to the standard approach, which is to solve for the wave quantities in the absence of dissipation and then to use these quantities in expressions for the heating due to viscosity and thermal conduction.

PATTERNS OF NANOFLARE STORM HEATING EXHIBITED BY AN ACTIVE REGION OBSERVED WITH SOLAR DYNAMICS OBSERVATORY/ATMOSPHERIC IMAGING ASSEMBLY

It is largely agreed that many coronal loops---those observed at a temperature of about 1 MK--- are bundles of unresolved strands that are heated by storms of impulsive nanoflares. The nature of coronal heating in hotter loops and in the very important but largely ignored diffuse component of active regions is much less clear. Are these regions also heated impulsively, or is the heating quasi steady? The spectacular new data from the Atmospheric Imaging Assembly (AIA) telescopes on the Solar Dynamics Observatory (SDO) offer an excellent opportunity to address this question. We analyze the light curves of coronal loops and the diffuse corona in 6 different AIA channels and compare them with the predicted light curves from theoretical models. Light curves in the different AIA channels reach their peak intensities with predictable orderings as a function the nanoflare storm properties. We show that while some sets of light curves exhibit clear evidence of cooling after nanoflare storms, other cases are less straightforward to interpret. Complications arise because of line-of-sight integration through many different structures, the broadband nature of the AIA channels, and because physical properties can change substantially depending on the magnitude of the energy release. Nevertheless, the light curves exhibit predictable and understandable patterns consistent with impulsive nanoflare heating.

HINODE X-RAY TELESCOPE DETECTION OF HOT EMISSION FROM QUIESCENT ACTIVE REGIONS: A NANOFLARE SIGNATURE?

The X-Ray Telescope (XRT) on the Japanese/USA/UK Hinode (Solar-B) spacecraft has detected emission from a quiescent active region core that is consistent with nanoflare heating. The fluxes from 10 broadband X-ray filters and filter combinations were used to construct differential emission measure (DEM) curves. In addition to the expected active region peak at log T = 6.3-6.5, we find a high-temperature component with significant emission measure at log T > 7.0. This emission measure is weak compared to the main peak—the DEM is down by almost three orders of magnitude—which accounts of the fact that it has not been observed with earlier instruments. It is also consistent with spectra of quiescent active regions: no Fe XIX lines are observed in a CHIANTI synthetic spectrum generated using the XRT DEM distribution. The DEM result is successfully reproduced with a simple two-component nanoflare model.

CAN THERMAL NONEQUILIBRIUM EXPLAIN CORONAL LOOPS

Any successful model of coronal loops must explain a number of observed properties. For warm (~1 MK) loops, these include (1) excess density, (2) flat temperature profile, (3) super-hydrostatic scale height, (4) unstructured intensity profile, and (5) 1000-5000 s lifetime. We examine whether thermal nonequilibrium can reproduce the observations by performing hydrodynamic simulations based on steady coronal heating that decreases exponentially with height. We consider both monolithic and multi-stranded loops. The simulations successfully reproduce certain aspects of the observations, including the excess density, but each of them fails in at least one critical way. Monolithic models have far too much intensity structure, while multi-strand models are either too structured or too long-lived. Our results appear to rule out the widespread existence of heating that is both highly concentrated low in the corona and steady or quasi-steady (slowly varying or impulsive with a rapid cadence). Active regions would have a very different appearance if the dominant heating mechanism had these properties. Thermal nonequilibrium may nonetheless play an important role in prominences and catastrophic cooling events (e.g., coronal rain) that occupy a small fraction of the coronal volume. However, apparent inconsistencies between the models and observations of cooling events have yet to be understood.

EVIDENCE OF WIDESPREAD HOT PLASMA IN A NONFLARING CORONAL ACTIVE REGION FROM HINODE/X-RAY TELESCOPE

Nanoflares, short and intense heat pulses within spatially unresolved magnetic strands, are now considered a leading candidate to solve the coronal heating problem. However, the frequent occurrence of nanoflares requires that flare-hot plasma be present in the corona at all times. Its detection has proved elusive until now, in part because the intensities are predicted to be very faint. Here we report on the analysis of an active region observed with five filters by Hinode/XRT in November 2006. We have used the filter ratio method to derive maps of temperature and emission measure both in soft and hard ratios. These maps are approximate in that the plasma is assumed to be isothermal along each line-of-sight. Nonetheless, the hardest available ratio reveals the clear presence of plasma around 10 MK. To obtain more detailed information about the plasma properties, we have performed Monte Carlo simulations assuming a variety of non-isothermal emission measure distributions along the lines-of-sight. We find that the observed filter ratios imply bi-modal distributions consisting of a strong cool (log T ~ 6.3-6.5) component and a weaker (few percent) and hotter (6.6 < log T < 7.2) component. The data are consistent with bi-modal distributions along all lines of sight, i.e., throughout the active region. We also find that the isothermal temperature inferred from a filter ratio depends sensitively on the precise temperature of the cool component. A slight shift of this component can cause the hot component to be obscured in a hard ratio measurement. Consequently, temperature maps made in hard and soft ratios tend to be anti-correlated. We conclude that this observation supports the presence of widespread nanoflaring activity in the active region.

Evidence for Widespread Cooling in an Active Region Observed with the SDO Atmospheric Imaging Assembly

A well known behavior of EUV light curves of discrete coronal loops is that the peak intensities of cooler channels or spectral lines are reached at progressively later times than hotter channels. This time lag is understood to be the result of hot coronal loop plasma cooling through these lower respective temperatures. However, loops typically comprise only a minority of the total emission in active regions. Is this cooling pattern a common property of active region coronal plasma, or does it only occur in unique circumstances, locations, and times? The new SDO/AIA data provide a wonderful opportunity to answer this question systematically for an entire active region. We measure the time lag between pairs of SDO/AIA EUV channels using 24 hours of images of AR 11082 observed on 19 June 2010. We find that there is a time-lag signal consistent with cooling plasma, just as is usually found for loops, throughout the active region including the diffuse emission between loops for the entire 24 hour duration. The pattern persists consistently for all channel pairs and choice of window length within the 24 hour time period, giving us confidence that the plasma is cooling from temperatures of greater than 3 MK, and sometimes exceeding 7 MK, down to temperatures lower than ~ 0.8 MK. This suggests that the bulk of the emitting coronal plasma in this active region is not steady; rather, it is dynamic and constantly evolving. These measurements provide crucial constraints on any model which seeks to describe coronal heating.