Klimchuk (2006) splits the heating problem into six steps: the identification of the source of energy, its conversion into heat, the plasma response to the heating, the spectrum of the emitted radiation, the final signature in observables. Outside of analytical approaches, the source and conversion of energy are typically studied in detail by means of multi-dimensional full MHD models (e.g., Gudiksen and Nordlund, 2005), which, however, are still not able to provide exhaustive predictions on the plasma response and complete diagnostics on observables. On the other hand, the plasma response is the main target of loop hydrodynamic models, which, instead, are not able to treat the heating problem in a self-consistent way (Section 4.1).
In the investigation of the source of energy, Golub et al. (1980) already pointed out that the magnetic field plays an active role in heating the coronal loops. They assumed that the field lines are wound continuously by the photospheric convective motions and the generated non-potential component is dissipated into heating. Several following studies were devoted to the connection and scaling of the magnetic energy to the coronal energy content (Golub et al., 1982) and to the rate of energy release through reconnection (Galeev et al., 1981). The photospheric motions are, therefore, the ultimate energy source and stress the field or generate waves depending on whether the timescale of the motion is long or short compared to the end-to-end Alfvén travel time. Following Klimchuk (2006), dissipation of magnetic stresses can be referred to as Direct Current (DC) heating, and dissipation of waves as Alternating Current (AC) heating.
The question of the conversion of the magnetic energy into heat is also challenging, because dissipation is predicted to occur on very small scales or large gradients in the corona by classical theory, unless anomalous dissipation coefficients are invoked. As reviewed by Klimchuk (2006), large gradients may be produced in various ways, involving either magnetic field patterns and their evolution, magnetic instabilities such as the kink instability, or velocity pattern, such as turbulence. For waves, resonance absorption and phase mixing may be additional viable mechanisms.
The problem of plasma response to heating has been kept historically well separated from the primary heating origin, although some attempts have been made to couple them. For instance, in Reale et al. (2005) the time-dependent distribution of energy dissipation along the loop obtained from a hybrid shell model was used as heating input of a time-dependent hydrodynamic loop model (see below). A similar concept was applied to search for signatures of turbulent heating in UV spectral lines (Parenti et al., 2006).
As already mentioned, studies using steady-state or time-dependent purely hydrodynamic loop modeling have addressed primarily the plasma response to heating, and also its radiative emission and the detailed comparison with observations. A forward-modeling including all these steps was performed by Reale et al. (2000b,a). They first analyzed a TRACE observation of a brightening coronal loop (see also Section 3.4). The analysis was used to set up the parameters for the forward modeling, and to run loop hydrodynamic simulations with various assumptions on the heating location and time dependence. The comparison of the TRACE emission predicted by the simulations with the measured one constrained the heat pulse to be short, much less than the observed loop rise phase, and intense, appropriate for a 3 MK loop, and its location to be probably midway between the apex and one of the footpoints.
The investigation of the heating mechanisms through the plasma response is made difficult by a variety of reasons. For instance, the problem of background subtraction can be crucial in the comparison with observations, as shown by the three analyses of the same large loop structure observed with Yohkoh/SXT on the solar limb, mentioned in Section 3.3. More specifically, Priest et al. (2000) tried to deduce the form of the heating from Yohkoh observed temperature profiles and found that a uniform heating best describes the data, if the temperature is obtained from the ratio of the total filter intensities, with no background subtraction. Aschwanden (2001) splitted the measured emission into two components and found a better agreement with heating deposited at the loop footpoints. Reale (2002b) revisited the analysis of the same loop system, considering conventional hydrostatic single-loop models and accounting accurately for an unstructured background contribution. With forward-modeling, i.e., synthesizing from the model observable quantities to be compared directly with the data, background-subtracted data are fitted with acceptable statistical significance by a model of relatively hot loop ( 3.7 MK) heated at the apex, but it was pointed out the importance of background subtraction and the necessity of more specialized observations to address this question. More diagnostic techniques to compare models with observations were proposed afterwards (e.g., Landi and Landini, 2005).
Independently of the adopted numerical or theoretical tool, many studies have been addressing the mechanisms of coronal loop heating clearly distinguishing between the two main classes, i.e., DC heating through moderate and frequent explosive events, named nanoflares (e.g., Parker, 1988) and AC heating via Alfvén waves (e.g., Litwin and Rosner, 1998).
Heating by nanoflares has a long story as a possible candidate to explain the heating of the solar corona and, in particular, of the coronal loops (e.g., Peres et al., 1993; Cargill, 1993; Kopp and Poletto, 1993; Shimizu, 1995; Judge et al., 1998; Mitra-Kraev and Benz, 2001; Katsukawa and Tsuneta, 2001; Warren et al., 2002, 2003; Spadaro et al., 2003; Cargill and Klimchuk, 1997, 2004; Müller et al., 2004; Testa et al., 2005; Reale et al., 2005; Vekstein, 2009).
More specifically, Cargill (1994) provided detailed predictions from a model of loops made of thousands of nanoflare-heated strands. In particular, whereas the loop total emission measure distribution should steepen above the canonical T1.5 (Jordan, 1980; Orlando et al., 2000; Peres et al., 2001) dependence for temperature above 1 MK. Moreover, it was stressed the importance of the dependence of effects such as the plasma dynamics (filling and draining) on the loop filling factor driven by the elemental heat pulse size (Section 4.1.2). In Cargill and Klimchuk (1997) the nanoflare model was applied to the heating of coronal loops observed by Yohkoh. A good match was found only for hot (4 MK) loops, with filling factors less than 0.1, so that it was hypothesized the existence of two distinct classes of hot loops.
Although there is evidence of intermittent heating episodes, it has been questioned whether and to what extent nanoflares are able to provide enough energy to heat the corona (e.g., Aschwanden, 1999). On the other hand, loop models with nanoflares and, in particular, those considering a prescribed random time distribution of the pulses deposited at the footpoints of multi-stranded loops have been able to explain several features of loop observations, for instance, of warm loops from TRACE (Warren et al., 2002, 2003), (see Section 3.2.2).
Hydrodynamic loop modeling showed also that different distributions of the heat pulses along the loop have limited effects on the observable quantities (Patsourakos and Klimchuk, 2005), because most of the differences occur at the beginning of the heat deposition, when the emission measure is low, while later the loop loses memory of the heat distribution (see also Winebarger and Warren, 2004). Patsourakos and Klimchuk (2008) applied both static and impulsive models to solar active regions and showed that the latter ones are able to simultaneously reproduce EUV and SXR loops in active regions, and to predict radial intensity variations consistent with the localized core and extended emissions. Cargill and Klimchuk (2004) showed with a semi-analytical loop model that the cycle of loop heating/cooling naturally leads to hot-underdense/warm-overdense loop (Section 4.1.2), as observed (Winebarger et al., 2003b, Section 3.3.3), and that the width of the DEM of a nanoflare-heated loop can depend on the number of strands which compose the loop: a relatively flat DEM or a peaked (isothermal) DEM are obtained with strands of diameter about 15 km or about 150 km, respectively. This is of relevance for the diagnostics both of the loop fine structure (Section 3.2.2) and of the DEM reconstruction (Section 3.3). As a further improvement, Warren and Winebarger (2007) added an impulsive heating model to the simulation of an entire active region and found that it is possible to reproduce the total observed soft X-ray emission in all of the Yohkoh/SXT filters. However, once again, at EUV wavelengths the agreement between the simulation and the observation is only partial.
Nanoflares have been studied also in the framework of stellar coronae. Testa et al. (2005) showed that intermittent heating by relatively intense nanoflares deposited at the loop footpoints make the loop stable on long time scales (loops continuously heated at the footpoints are unstable) and, on the other hand, produces a well-defined peak in the average DEM of the loop, similar to that derived from the DEM reconstruction of active stars, and also to those shown in Cargill (1994). Therefore, this is an alternative way to obtain a steep temperature dependence of the loop emission measure distribution in the low temperature range.
An alternative approach to study nanoflare heating is to analyze intensity fluctuations (Shimizu and Tsuneta, 1997; Vekstein and Katsukawa, 2000; Katsukawa and Tsuneta, 2001; Vekstein and Jain, 2003) and to derive their occurrence distribution (Sakamoto et al., 2008, 2009). From the width of the distributions and autocorrelation functions, it has been suggested that nanoflare signatures are more easily found in observations of warm TRACE loops than of hot Yohkoh/SXT loops. It is to be investigated whether the results change after relaxing the assumption of temperature-independent distribution widths. Also other variability analysis of TRACE observations was found able to put constraints on loop heating. In particular, according to Antiochos et al. (2003), in TRACE observations, the lack of observable warm loops and of significant variations in the moss regions implies that the heating in the hot moss loops should not be truly flare-like, but instead quasi-steady and that the heating magnitude is only weakly varying. Further evidence in this direction has been found more recently by Warren et al. (2010).
An analogous approach is to analyze the intensity distributions. The distribution of impulsive events vs their number in the solar and stellar corona is typically described with a power law. The slope of the power law is a critical parameter to establish weather such events are able to heat the solar corona (Hudson, 1991). In particular, a power law index of 2 is the critical value above or below which flare-like events may be able or unable, respectively, to power the whole corona (e.g., Aschwanden, 1999). Unfortunately, due to the faintness of the events, the distribution of weak events is particularly difficult to derive and might even be separate from that of proper flares and microflares. Parenti and Young (2008) used a hydrodynamic model to simulate the UV emission of a loop system heated by nanoflares on small, spatially unresolved scales. The simulations confirm previous results that several spectral lines have an intensity distribution that follows a power-law, in a similar way to the heating function (Hudson, 1991). However, only the high temperature lines best preserve the heating function’s power law index (especially Fe xix).
Loop oscillations, modes, and wave propagation deserve a review by themselves, and are outside of the scope of the present one. Here we account for some aspects which are relevant for the loop heating. A recent review of coronal waves and oscillations can be found in Nakariakov and Verwichte (2005). New observations from SDO AIA provide ample evidence of wave activity in the solar corona (Title, 2010). These observations are currently the subject of intensive analysis and will be reported on in the future.
As reviewed by Klimchuk (2006), MHD waves of many types are generated in the photosphere, e.g., acoustic, Alfvén, fast and slow magnetosonic waves. Propagating upwards, the waves may transfer energy to the coronal part of the loops. The question is what fraction of the wave flux is able to pass through the very steep density and temperature gradients in the transition region. Acoustic and slow-mode waves form shocks and are strongly damped, fast-mode waves are strongly refracted and reflected (Narain and Ulmschneider, 1996).
Ionson (1978, 1982, 1983) devised an LRC equivalent circuit to show the potential importance of AC processes to heat the corona. Hollweg (1984) used a dissipation length formalism to propose resonance absorption of Alfvén waves as a potential coronal heating mechanism. A loop may be considered as a high-quality resonance cavity for hydromagnetic waves. Turbulent photospheric motions can excite small-scale waves. Most Alfvén waves are strongly reflected in the chromosphere and transition region, where the Alfvén speed increases dramatically with height. Significant transmission is possible only within narrow frequency bands centered on discrete values where loop resonance conditions are satisfied (Hollweg, 1981, 1984; Ionson, 1982). The waves resonate as a global mode and dissipate efficiently when their frequency is near the local Alfvén waves frequency . By solving the linearized MHD equations Davila (1987) showed that this mechanism can potentially heat the corona, as further supported by numerical solution of MHD equations for low beta plasma (Steinolfson and Davila, 1993), and although Parker (1991) argued that Alfvén waves are difficult to be generated by solar convection.
Evidence for photospheric Alfvén waves was obtained from magnetic and velocity fluctuations in regions of strong magnetic field (Ulrich, 1996) and from granular motions in the quiet Sun (Muller et al., 1994) with fluxes of the order of 107 erg cm–2 s–1, which might contribute to heating if transmitted efficiently to the corona.
Hollweg (1985) estimated that enough flux may pass through the base of long (> 1010 cm) active region loops to provide their heating, but shorter loops are a problem, since they have higher resonance frequencies and the photospheric power spectrum is believed to decrease exponentially with frequency in this range. Litwin and Rosner (1998) suggested that short loops may transmit waves with low frequencies, as long as the field is sufficiently twisted. Hollweg and Yang (1988) proposed that Alfvén resonance can pump energy out of the surface wave into thin layers surrounding the resonant field lines and that the energy can be distributed by an eddy viscosity throughout large portions of coronal active region loops.
Waves may be generated directly in the corona and evidence for their presence was found (e.g., Nakariakov et al., 1999; Aschwanden et al., 1999a; Berghmans and Clette, 1999; De Moortel et al., 2002). It is unclear whether coronal waves carry a sufficient energy flux to heat the plasma (Tomczyk et al., 2007). Ofman et al. (1995) studied the dependence on the wavenumber for comparison with observations of loop oscillations and found partial agreement with velocity amplitudes measured from nonthermal broadening of soft X-ray lines. The observed nonthermal broadening of transition region and coronal spectral lines implies fluxes that may be sufficient to heat both the quiet Sun and active regions, but it is unclear whether the waves are efficiently dissipated (Porter et al., 1994). Furthermore, the nonthermal line broadening could be produced by unresolved loop flows that are unrelated to waves (Patsourakos and Klimchuk, 2006). Ofman et al. (1998) included inhomogeneous density structures and found that a broadband wave spectrum becomes necessary for efficient resonance and that it fragments the loop into many density layers that resemble the multistrand concept. The heat deposition by the resonance of Alfvén waves in a loop was investigated by O’Neill and Li (2005) By assuming a functional form first proposed by Hollweg (1986), hydrodynamic loop modeling showed that, depending on the model parameters, heating by Alfvén waves leads to different classes of loop solutions, such as the isothermal cool loops indicated by TRACE, or the hot loops observed with Yohkoh/SXT. Specific diagnostics are still to be defined for the comparison with observations.
Efficient wave dissipation may be allowed by enhanced dissipation coefficients inferred from fast damping of flaring loop oscillations in the corona (Nakariakov et al., 1999), but the same effect may also favor efficient magnetic reconnection in nanoflares. Alfvén waves required for resonant absorption are relatively high frequency waves. Evidence for lower frequency Alfvén waves has been found in the chromosphere with the Hinode SOT (De Pontieu et al., 2007). Such waves may supply energy in the corona even outside of resonance with different mechanisms to be explored with modeling. Among dissipation mechanisms phase mixing with enhanced resistivity was suggested by Ofman and Aschwanden (2002) and supported by the analysis of Ofman and Wang (2008). Also multistrand structures has been recognized to be important in possible wave dissipation and loop twisting, as recently modelled by Ofman (2009).
Intensity disturbances propagating along active region loops at speeds above 100 km s–1 were detected with TRACE and interpreted as slow magnetoacoustic waves (Nakariakov et al., 2000). These waves probably originate from the underlying oscillations, i.e., the 3-minute chromospheric / transition-region oscillations in sunspots and the 5-minute solar global oscillations (p-modes). Slow magnetosonic waves might be good candidates as coronal heating sources according to quite a detailed model by Beliën et al. (1999), including the effect of chromosphere and transition region and of the radiative losses in the corona. Such waves might be generated directly from upward propagating Alfvén waves. Contrary conclusions, in favor of fast magnetosonic waves, have been also obtained, but with much simpler modeling (Pekünlü et al., 2001). Slow magnetosonic waves with periods of about 5 minutes have been more recently detected in the transition region and coronal emission lines by Hinode/EIS at the footpoint of a coronal loop rooted at plage, but found to carry not enough energy to heat the corona (Wang et al., 2009).
Investigation of AC heating has been made also through comparison with DC heating. Antolin et al. (2008) compared observational signatures of coronal heating by Alfvén waves and nanoflares using two coronal loop models and found that Hinode XRT intensity histograms display power-law distributions whose indices differ considerably, to be checked against observations. Lundquist et al. (2008a,b) applied a method for predicting active region coronal emissions using magnetic field measurements and a chosen heating relationship to 10 active regions. With their forward-modeling, they found a volumetric coronal heating rate proportional to magnetic field and inversely proportional to field-line loop length, which seems to point to, although not conclusively, the steady-state scaling of two heating mechanisms: van Ballegooijen’s current layers theory (van Ballegooijen, 1986), taken in the AC limit, and Parker’s critical angle mechanism (Parker, 1988), in the case where the angle of misalignment is a twist angle.
Some studies have investigated loop heating by including also the magnetic field in the analysis and modeling, and have tried to discriminate between different mechanisms using global approaches and scalings inferred from modeling. Based on a previous study of the plasma parameters and the magnetic flux density (Mandrini et al., 2000), Démoulin et al. (2003) derived the dependence of the mean coronal heating rate on the magnetic flux density from the analysis of an active region. By using the scaling laws of coronal loops, they found that models based on the dissipation of stressed, current-carrying magnetic fields are in better agreement with the observations than models that attribute coronal heating to the dissipation of MHD waves injected at the base of the corona. Schrijver et al. (2004) considered a similar approach applied to the whole corona, by populating magnetic field lines taken from observed magnetograms with quasi-static loop atmospheres and obtaining the best match to X-ray and EUV observation with a heating that scales as it would be expected from DC reconnection at tangential discontinuities. These approaches will certainly be very useful when they will provide more detailed predictions and constraints.
Recent modeling has been able to explain the ignition of warm loops from primary energy release mechanisms, although it remains unclear how the same mechanisms could produce hot loops. A large scale approach (see also Section 4.1) is by “ab initio” modeling, i.e., with full MHD modeling of an entire coronal region (Gudiksen and Nordlund, 2005). Observed solar granular velocity pattern, a potential extrapolation of a SOHO/MDI magnetogram, and a standard stratified atmosphere were used as initial conditions. The simulation showed that, at steady state, the magnetic field is able to dissipate (3 – 4) × 106 erg cm–2 s–1 in a highly intermittent corona, at an average temperature of 106 K, adequate to reproduce typical warm loop populations observed in TRACE images. Warm loops were also obtained with time-dependent loop modeling including the intermittent magnetic dissipation in MHD turbulence due to loop footpoint motions (Reale et al., 2005). The dissipation rate along a loop predicted with a hybrid-shell model (Nigro et al., 2004) was used as heating input (see Equation (5) in a proper time-dependent loop model, the Palermo-Harvard code (Peres et al., 1982). It was shown that the most intense nanoflares excited in an ambient magnetic field of about 10 G can produce warm loops with temperatures of 1 – 1.5 MK in the corona of a 30 000 km long loop.
More recently, Rappazzo et al. (2007) used reduced MHD (rMHD) to identify MHD anisotropic turbulence as the physical mechanism responsible for the transport of energy from the large scales, where energy is injected by photospheric motions to the small scales, where it is dissipated. Strong turbulence was found for weak axial magnetic fields and long loops. The predicted heating rate is appropriate for warm loops, in agreement with Reale et al. (2005). Buchlin and Velli (2007) also used shell models of rMHD turbulence to analyze the case of a coronal loop heated by photospheric turbulence and found that the Alfvén waves interact non-linearly and form turbulent spectra. They derived an intermittent heating function, on average able to sustain the corona and proportional to the aspect ratio of the loop to the 1.5 power. Buchlin et al. (2007) added in the modeling a profile of density and/or magnetic field along the loop, showing that differences are found in the heat deposition, in particular in the low part of the loop.
There are new efforts to include magnetic effects in the loop modeling. Haynes et al. (2008) studied observational properties of a kink unstable coronal loop, using a fluid code and finding potentially observable density effects. Browning et al. (2008) studied coronal heating by nanoflares triggered by a kink instability using three-dimensional magnetohydrodynamic numerical simulations of energy release for a cylindrical coronal loop model. Magnetic energy is dissipated, leading to large or small heating events according to the initial current profile.
Interesting perspectives are developing from models in which self-organized criticality triggers loop coronal heating. For Uzdensky (2007) and Cassak et al. (2008) coronal heating is self-regulating and keeps the coronal plasma roughly marginally collisionless. In the long run, the coronal heating process may be represented by repeating cycles that consist of fast reconnection events (i.e., nanoflares), followed by rapid evaporation episodes, followed by relatively long periods ( 1 h) during which magnetic stresses build up and the plasma simultaneously cools down and precipitates. Morales and Charbonneau (2008) proposed an avalanche model for solar flares, based on an idealized representation of a coronal loop as a bundle of magnetic flux strands wrapping around one another. The system is driven by random deformation of the strands, and a form of reconnection is assumed to take place when the angle subtended by two strands crossing at the same lattice site exceeds some preset threshold. For a generic coronal loop of length 1010 cm and diameter 108 cm the mechanism leads to flare energies ranging between 1023 and 1029 erg, for an instability threshold angle of 11 degrees between contiguous magnetic flux strands.
Given the difficulty to find a conclusive answer about the heating of coronal loops, even in the presence of considerable observational and theoretical efforts, there has been recently the attempt to propose different or radically alternative scenarios. For instance, it has been suggested that warm and hot loops may be heated by different mechanisms, impulsive the former, much more steady the latter (Warren et al., 2010). New evidence from Hinode satellite indicates the presence of significant upflows in the form of widespread spicules which have correspondence in coronal observations (De Pontieu et al., 2009), This evidence suggests that the interaction between the chromosphere and the corona in the heating processes might be important and may receive support also by some modeling approaches (Gudiksen and Nordlund, 2005). These scenarios are intriguing but need additional investigation and support both from the observational and theoretical point of view.
Living Rev. Solar Phys. 7, (2010), 5
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