Coronal loops are the building blocks of the X-ray bright solar corona. They owe their brightness to the dense confined plasma, and this review focuses on loops mostly as structures confining plasma. After a brief historical overview, the review is divided into two separate but not independent parts: the first illustrates the observational framework, the second reviews the theoretical knowledge. Quiescent loops and their confined plasma are considered, and therefore topics such as loop oscillations and flaring loops (except for non-solar ones which provide information on stellar loops) are not specifically addressed here. The observational section discusses loop classification and populations, and then describes the morphology of coronal loops, its relationship with the magnetic field, and the concept of loops as multi-stranded structures. The following part of this section is devoted to the characteristics of the loop plasma and of its thermal structure in particular, according to the classification into hot, warm, and cool loops. Then, temporal analyses of loops and the observations of plasma dynamics and flows are illustrated. In the modeling section some basics of loop physics are provided, supplying some fundamental scaling laws and timescales, a useful tool for consultation. The concept of loop modeling is introduced and models are distinguished between those treating loops as monolithic and static, and those resolving loops into thin and dynamic strands. Then, more specific discussions address modeling the loop fine structure and the plasma flowing along the loops. Special attention is devoted to the question of loop heating, with separate discussion of wave (AC) and impulsive (DC) heating. Finally, a brief discussion about stellar X-ray emitting structures related to coronal loops is included and followed by conclusions and open questions.
The corona is the outer part of the solar atmosphere. Its name derives from the fact that, since it is extremely tenuous with respect to the lower atmosphere, it is visible in the optical band only during the solar eclipses as a faint crown (corona in Latin) around the black moon disk. When inspected through spectroscopy the corona reveals unexpected emission lines, which were first identified as due to a new element (coronium) but which were later ascertained to be due to high excitation states of iron (e.g., Golub and Pasachoff, 1997, 2001). It became then clear that the corona is made of very high temperature gas, hotter than 1 MK. Almost all the gas is fully ionized there and thus interacts effectively with the ambient magnetic field. It is for this reason that the corona appears so inhomogeneous when observed in the X-ray band, in which plasma at million degrees emits most of its radiation. In particular, the plasma is confined inside magnetic flux tubes which are anchored on both sides to the underlying photosphere. When the confined plasma is heated more than the surroundings, its pressure and density increase. Since the tenuous plasma is optically thin, the intensity of its radiation is proportional to the square of the density, and the tube becomes much brighter than the surrounding ones and looks like a bright closed arch: a coronal loop.
When observed in the X-ray band, the bright corona appears to be made entirely by coronal loops that can, therefore, be considered as the building blocks of X-ray bright corona. This review specifically addresses coronal loops as bright structures confining plasma. It first provides an observational framework that is the basis for the second part dealing with modeling and interpretation. There have been several earlier books (Bray et al., 1991; Golub and Pasachoff, 1997; Aschwanden, 2004) and reviews (Vaiana and Rosner, 1978; Peres and Vaiana, 1990; Golub, 1996; Aschwanden et al., 2001; Reale, 2005), in particular on coronal heating (Zirker, 1993; Cargill, 1995; Klimchuk, 2006), that have in general a larger or different scope but include information about coronal loops. Interested readers are urged to survey these other reviews in order to complement and fill in any gaps in topical coverage of the present paper.
First evidence of magnetic confinement came from rocket missions in the 1960s. In particular, in 1965, arcmin angular resolution was achieved with grazing incidence optics (Giacconi et al., 1965). The data analysis led to the first density and temperature diagnostics with wide band filters, to derive high pressure in compact regions with intense bipolar magnetic fields and to propose the magnetic confinement (Reidy et al., 1968). The first coronal loop structures were identified properly after a rocket launch in 1968 which provided for the first time an image of an X-ray flare (Vaiana et al., 1968), with a resolution of a few arcsec.
In the course of collecting the results of all rocket missions of the American Science and Engineering (AS&E) program, Vaiana et al. (1973) proposed a classification of the morphology of the X-ray corona as fundamentally consisting of arch-like structures connecting regions of opposite magnetic polarity in the photosphere. The classification was based on the loop size, on the physical conditions of the confined plasma, and on the underlying photospheric regions. They distinguished active regions, coronal holes, active regions interconnection, filament cavities, bright points, and large-scale structures (Vaiana and Rosner, 1978; Peres and Vaiana, 1990).
The magnetic structuring of the solar corona is evident. However, the magnetic field lines can be traced only indirectly because direct measurements are feasible generally only low in the photosphere through the Zeeman effect on spectral lines. It is anyhow possible to extrapolate the magnetic field in a volume. This was done to derive the magnetic field structure of a relatively stable active region by Poletto et al. (1975) using the Schmidt (1964) method, under the assumption of negligible currents in the corona. This was also useful to derive sufficient magnetic field intensities for hot plasma confinement. Later on, even more reliable magnetic field topologies were derived assuming force-free fields (e.g., Sakurai, 1981), i.e., with currents everywhere parallel to the magnetic field as it is expected in coronal loops.
The rocket missions lacked good time coverage and the information about the evolution of coronal loops was only limited, mostly available from the Orbiting Solar Observatory-IV (OSOIV) mission (Krieger et al., 1972). This satellite had an angular resolution in the order of the arcmin and could not resolve individual loops. In 1973 the X-ray telescope S-054 on-board Skylab monitored the evolution of coronal loops for several months, taking 32 000 X-ray photographs with a maximum resolution of 2 arcsec and an extended dynamic range. It was possible to study the whole evolution of an active region, from the emergence as compact loops filled with dense plasma to its late spreading, a few solar rotations later, as progressively longer and longer loops filled with less and less dense plasma (Golub et al., 1982). It was confirmed that the whole X-ray bright corona consists of magnetic loops, whose lifetime is typically much longer than the characteristic cooling times (Rosner et al., 1978). No exception is made by the coronal holes where the magnetic field opens radially to the interplanetary space and the plasma streams outwards with practically no X-ray emission.
In the same mission, coronal loops were also detected in the UV band at temperatures below 1 MK, by Extreme UltraViolet (EUV) telescopes S-055 (Reeves et al., 1977) and S-082 (Tousey et al., 1977; Bartoe et al., 1977). These loops are invisible in the X-ray band and many of them depart from sunspots, appear coaxial and are progressively thinner for progressively lower temperature ions (Foukal, 1975, 1976), suggesting nested loops hotter toward outer shells, although this suggestion has never been validated. The apparent scale height of the emission is larger than that expected from a static model, but the loops appear to be steady for long times. Foukal (1976) proposed a few explanations including siphon flows and thermal instability of the plasma at the loop apex. New observations of such cool loops were performed several years later with the SOlar and Heliospheric Observatory (SoHO) mission and provided new details and confirmations (Section 3.5).
A different target was addressed by the Solar Maximum Mission (SMM, 1980 – 1989, Bohlin et al., 1980; Acton et al., 1980), which included high-resolution spectrometers in several X-ray lines, i.e., the Bent Crystal Spectrometer (BCS) and the Flat Crystal Spectrometer (FCS), mostly devoted to obtain time-resolved spectroscopy of coronal flares (e.g., MacNeice et al., 1985). Similarly, the Hinotori mission (1981 – 1991, Tanaka, 1983) was dedicated mainly to solar flare observations in the X-ray band. This was also the scope of the later Yohkoh mission, (1991 – 2001, Ogawara et al., 1991) by means of high resolution X-ray spectroscopy, adding the monitoring and imaging of the hot and flaring corona. Hara et al. (1992) found first indications of plasma at 5 – 6 MK in active regions with the Soft X-ray Telescope (SXT, Tsuneta et al., 1991).
Normal incidence optics have been developed in the late 1980s. An early experiment was the Normal Incidence X-ray Telescope (NIXT, Golub and Herant, 1989), which provided a few high resolution coronal images in the EUV band.
More recent space missions dedicated to study the corona have been the Solar and Heliospheric Observatory (SoHO, Domingo et al., 1995), launched in 1995 and still operative, and the Transition Region and Coronal Explorer (TRACE, Handy et al., 1999), launched in 1998 and replaced in 2010 by the Solar Dynamic Observatory instruments. Both SoHO and TRACE were tailored to observe the quiet corona (below 2 MK). SoHO images the whole corona (Extreme ultraviolet Imaging Telescope, EIT, Delaboudinière et al., 1995) and performs wide band spectroscopy (Solar Ultraviolet Measurements of Emitted Radiation, SUMER, Wilhelm et al., 1995) and (Coronal Diagnostic Spectrometer, CDS, Harrison et al., 1995) in the EUV band; TRACE imaged the EUV corona with high spatial (0.5 arcsec) and temporal (30 s) resolution. Both SoHO/EIT and TRACE are based on normal incidence optics and contain three different EUV filters that provide limited thermal diagnostics.
Thanks to their capabilities, both missions allowed to address finer diagnostics, in particular to investigate the fine transverse structuring of coronal loops, both in its geometric and thermal components, and the plasma dynamics and the heating mechanisms at a higher level of detail. SoHO and TRACE have been complementary in many respects and several studies attempted to couple the information from them.
Among other relevant missions, we mention the CORONAS series (Ignatiev et al., 1998; Oraevsky and Sobelman, 2002), with instruments like SPectroheliographIc X-Ray Imaging Telescope (SPIRIT, Zhitnik et al., 2003), Rentgenovsky Spektrometr s Izognutymi Kristalami (ReSIK, Sylwester et al., 1998), and Solar Photometer in X-rays (SPHINX, Sylwester et al., 2008), which have contributed to the investigation of coronal loops.
In late 2006, two other major solar missions started, namely Hinode (Kosugi et al., 2007) and the Solar TErrestrial Relations Observatory (STEREO) (e.g., Kaiser et al., 2008). On-board Hinode, two instruments address particularly the study of coronal loops: the X-Ray Telescope (XRT, Golub et al., 2007) and the Extreme-ultraviolet Imaging Spectrometer (EIS, Culhane et al., 2007). Both these instruments offer considerable improvements on previous missions. The XRT has a spatial resolution of about 1 arcsec, a very low scattering, and the possibility to switch among nine filters and combinations of them. EIS combines well spectral (∼ 2 mA), spatial (2″), and temporal (∼ 10 s) resolution to obtain accurate diagnostics of plasma dynamics and density. One big achievement of the STEREO mission is that, since it consists of two separate spacecrafts getting farther and farther from each other, it allows — through, for instance, its Sun-Earth Connection Coronal and Heliospheric Investigation (SECCHI) package — a first 3D reconstruction of coronal loops (Aschwanden et al., 2009; Kramar et al., 2009).
Hinode and STEREO represent the state-of-art for coronal loop observations, and the coronal loop studies including recent findings of these missions will be overviewed. In 2010, the new big solar mission Solar Dynamic Observatory (SDO) has been launched and new interesting results about the solar corona are expected soon, with instruments like the Atmospheric Imaging Assembly (AIA).
The Observational Framework
Although coronal loops are often well defined and studied in the EUV band, detected by many space mission spectrometers like those on board SoHO and Hinode, and by high resolution imagers such as NIXT and TRACE, the bulk of coronal loops is visible in the X-ray band (Figure 1). Also, the peak of the coronal emission measure of active regions — where the loops are brightest — is above 2 MK, which is best observed in X-rays (e.g., Peres et al., 2000; Reale et al., 2009a).
Coronal loops are characterized by an arch-like shape that recalls typical magnetic field topology. This shape is replicated over a wide range of dimensions. Referring, for the moment, to the soft X-ray band, the main properties of coronal loops are listed in Table 1. The length of coronal loops spans over at least 4 orders of magnitude: bright points (∼ 108 cm), small Active Region (AR) loops (∼ 109 cm), AR loops (∼ 1010 cm), giant arches (∼ 1011 cm) (Figure 2). As already mentioned, the loops owe their high luminosity and variety to their nature of magnetic flux tubes where the plasma is confined and isolated from the surroundings. Magnetized fully-ionized plasma conducts thermal energy mostly along the magnetic field lines. Due to the high thermal insulation, coronal loops can have different temperatures, from ∼ 105 K (cool loops), to a few 106 (X-ray loops), up to a few 107 K (flaring loops). A density of the confined plasma below 107 – 108 cm-3 can be difficult to detect, while more typical values of bright loops are 109 – 1010 cm-3 in quiescent and active regions loops. Flaring loops can be easily a factor 10 denser. The corresponding plasma pressure can typically vary between 10-3 and 10 dyne cm-2 for non-flaring loops, corresponding to confining magnetic fields of the order of 0.1 – 10 G. One characterizing feature of coronal loops is that typically their cross-section is constant along their length above the transition region, at variance from the topology of potential magnetic fields. There is evidence that the cross-section varies across the transition region, as documented in Gabriel (1976).
Myriads of loops populate the solar corona and constitute statistical ensembles. Attempts to define and classify coronal loops were never easy, and no finally established result exists to-date. Early attempts were based on morphological criteria, i.e., bright points, active region loops, and large scale structures Vaiana et al. (1973), largely observed with instruments in the X-ray band. In addition to such classification, more recently, the observation of loops in different spectral bands and the suspicion that the difference lies not only in the band, but also in intrinsic properties, have stimulated another classification based on the temperature regime, i.e., cool, warm, and hot loops (Table 2). Cool loops are generally detected in UV lines at temperatures between 105 and 106 K. They were first addressed by Foukal (1976) and later explored more with SoHO observations (Brekke et al., 1997). Warm loops are well observed by EUV imagers such as SoHO/EIT and TRACE, and confine plasma at temperature around 1 – 1.5 MK Lenz et al. (1999). Hot loops are those typically observed in the X-ray band and hot UV lines (e.g., Fe xvi), with temperatures around or above 2 MK (Table 1). These are the coronal loops already identified, for instance, in the early rocket missions Vaiana et al. (1973). This distinction is not only due to observation with different instruments and in different bands, but there are hints that it may be more substantial and physical, i.e., there may be two or more classes of loops that may be governed by different regimes of physical processes. For instance, the temperature along warm loops appears to be distributed uniformly and the density to be higher than that predicted by equilibrium conditions. Does this make such loops intrinsically different from hot loops, or is it just the signature that warm loops are a transient conditions of hot loops?
A real progress in the insight into coronal loops is expected from the study of large samples of loops or of loop populations. Systematic studies of coronal loops suffer from the problem of the sample selection and loop identification, because, for instance, loops in active regions overlap along the line sight. Attempts of systematic studies have been performed in the past on Yohkoh and TRACE data (e.g., Porter and Klimchuk, 1995; Aschwanden et al., 2000). A large number of loops were analyzed and it was possible to obtain meaningful statistics. However, it is difficult to generalize the results because of limited samples and/or selection effects, e.g., best observed loops, specific instrument. One basic problem for statistical studies of coronal loops is that it is very difficult to define an objective criterion for loop identification. In fact, loops are rarely isolated; they coexist with other loops which intersect or even overlap along the line of sight. This is especially true in active regions where most of the loops are found. In order to make a real progress along this line, we should obtain loop samples and populations selected on totally objective and unbiased criteria, which is difficult due to the problems outlined above. Some steps are coming in this direction and we will see results in the future.
Morphology and fine structuring
Coronal loops are magnetic structures and might therefore be mapped easily and safely by mapping reliably the coronal magnetic field. Unfortunately, it is well-known that it is very difficult to measure the magnetic field in the corona, and it can only be done in very special conditions, e.g., very strong local field (White et al., 1991). In some cases it is possible to use coronal seismology (first proposed by Uchida, 1970) to determine the average magnetic field strength in an oscillating loop first used by Nakariakov et al. (1999) and Nakariakov and Ofman (2001) on TRACE loops, and recently investigated in a number of studies. The accuracy of this method depends on the correct detection of the temporally and spatially resolved mode of oscillation, and on the details of the loop geometry.
Since we cannot well determine the coronal magnetic field, coronal loop geometry deserves specific analysis. As a good approximation, loops generally have a semicircular shape (Figure 3). The loop aspect, of course, depends on the loop orientation with respect to the line of sight: loops with the footpoints on the limb more easily appear as semicircular, as well as loops very inclined on the surface near the center of the disk. The assumption of semicircular shape can be useful to measure the loop length even in the presence of important deformations due to projection effects: the de-projected distance of the loop footpoints is the diameter of the arc. However, deviations from circularity are rather common and, in general, the detailed analysis of the loop geometry is not a trivial task. The accurate determination of the loop geometry is rather important for the implications on the magnetic field topology and reconstruction. It is less important for the structure and evolution of the confined plasma, which follow the field lines whatever shape they have and change little also with moderate changes of the gravity component along the field lines. First works on the accurate determination of the loop geometry date back to the sixties (Saito and Billings, 1964). More specific ones take advantage of stereoscopic views allowed by huge loops during solar rotation, with the aid of magnetic field reconstruction methods. These studies find deviations from ideal circularity and symmetry, not surprising for such large structures (Berton and Sakurai, 1985). The geometry of a specific loop observed with TRACE was measured in the framework of a complete study including time-dependent hydrodynamic modeling (Reale et al., 2000b,a). In that case, the discrepancy between the length derived from the distance of the footpoints taken as loop diameter and the length measured along the loop itself allowed to assess the loop as elongated. Later, a reconstruction of loop geometry was applied to TRACE observation of medium-sized oscillating loops, to derive the properties of the oscillations. In this case, a semicircular pattern was applied (Aschwanden et al., 2002). The importance of the deviations from circularity on constraining loop oscillations was remarked later (Dymova and Ruderman, 2006).
The STEREO mission is actually contributing much to the analysis of loop morphology and geometry, thanks to its unique capability to observe the Sun simultaneously from different positions. Feng et al. (2007) presented a first stereoscopic reconstruction of the three-dimensional shape of magnetic loops in an active region from two different vantage points based on simultaneously recorded STEREO/SECCHI images. They derived parameters of five relatively long loops and constraints on the local magnetic field, and found reconstructed loops to be non-planar and more curved than field lines extrapolated from SoHO/MDI measurements, probably due to the inadequacy of the linear force-free field model used for the extrapolation. A misalignment of 20 - 40 deg between theoretical model and observed loops has been quantified from STEREO results and discussed (Sandman et al., 2009; DeRosa et al., 2009). Aschwanden et al. (2008b) presented triangulations and 3D reconstructions of 30 coronal loops, using the Extreme UltraViolet Imager (EUVI) telescopes of both STEREO spacecrafts and deriving a series of loop characteristics, such as the loop plane inclination angles, and the coplanarity and circularity. They derived the parameters of seven complete and quite inclined loops and found deviations from circularity within 30%, and less significant from coplanarity. Aschwanden et al. (2009) applied a reconstruction method to an active region observed with STEREO by combining stereoscopic triangulation of 70 loops and addressing mainly density and temperature modeling with a filling factor equivalent to tomographic volume rendering.
Another interesting issue regarding coronal loop geometry is the analysis of the loop cross-section, which also provides information about the structure of the coronal magnetic field. Yohkoh / SXT allowed for systematic and quantitative studies of loop morphology and showed that the cross-section of coronal loops is approximately constant along their length and do not increase significantly. More in detail, a systematic analysis of a sample of ten loops showed that the loops tend to be only slightly (∼ 30%) wider at their midpoints than at their footpoints, while for a bipolar field configuration we would expect expansions by factors. One possible explanation of this effect is the presence of significant twisting of the magnetic field lines, and therefore the development of electric currents and strong deviations from a potential field. The effect might be seen either as a twisting of a single loop or as a “braiding” of a bundle of unresolved thin loops. At the same time it was found that the variation of width along each loop tends to be modest, implying that the cross section has an approximately circular shape (Klimchuk et al., 1992). Implications of these results on the theory of coronal heating are discussed in Klimchuk (2000), but the conclusion is that none of the current models alone is able to explain all observed properties.
Important information about the internal structuring of coronal loops comes from the joint analysis of the photospheric and coronal magnetic field (Figure 4). An analysis of the magnetic field at the footpoints of hot and cool loops showed, among other results, that the magnetic filling factor is lower in hot loops (0.05 – 0.3 out of sunspots) than in warm loops (0.2 – 0.6) (Katsukawa and Tsuneta, 2005). López Fuentes et al. (2006) investigated the magnetic structure of loops observed with TRACE, applying a linear force-free extrapolation model to SoHO/MDI data and comparing the resulting configuration of the magnetic field directly with TRACE images of the same region. They confirmed that, whereas the model predicts a significant expansion of the magnetic field structures, at least a factor two, from the footpoints to the corona, and a significant asymmetry of the structures, because the magnetic field lines starting from the same footpoint can diverge to different other end footpoints, these features are not observed: also TRACE loops are quite symmetric and their cross-section is constant to a good degree of approximation, as it had been already found for Yohkoh loops (Klimchuk et al., 1992). The results in López Fuentes et al. (2006) suggest that the tangling of the magnetic flux strands driven by the photospheric convection might be very strong, and confirm, therefore, that the magnetic field structure is far more complicated than it can be modelled even with linear force-free extrapolation. Schrijver (2007) studied braiding-induced interchange reconnection of the magnetic field and the width of solar coronal loops and showed that loop width observations support the hypothesis that granular braiding is countered statistically by frequent coronal reconnections, which in turn explain the general absence of entangled coronal field structures in high-resolution observations of the quiescent solar corona. DeForest (2007) addressed the apparent uniform cross section of bright threadlike structures in the corona, long apparent scale height, and the inconsistency between loop densities derived by spectral and photometric means. They found that, if coronal loops are interpreted as a mixture of diffuse background and very dense, unresolved stranded structures, this requires a combination of high plasma density within the structures, which greatly increases the emissivity of the structures, and geometric effects that attenuate the apparent brightness of the feature at low altitudes.
New methods to improve the morphological analysis of loops are being developed. For instance, Dudok de Wit and Auchère (2007) made multispectral analysis of solar EUV images and explored the possibility of separating the different solar structures from a linear combination of images. They found source images with more contrast than the original ones.
It has been long claimed (e.g., Gomez et al., 1993) that coronal loops consist of bundles of thin strands, to scales below the current instrumental resolution. The task to investigate this substructuring is not easy because the thickness of the elementary components may be as small as a few km, according to some nanoflare models (e.g., Vekstein, 2009), and the measured one goes down to the resolution limit of the most powerful imaging instruments (e.g., Gomez et al., 1993). First limited evidence of fine structuring was the low filling factor inferred for loops observed with NIXT (Di Matteo et al., 1999) (see Section 3.3.2). The high spatial resolution achieved by the TRACE normal incidence telescope allowed to address the transverse structure of the imaged coronal loops. TRACE images visibly show that coronal loops are substructured (Figure 5).
There were some early attempts to study the structure along the single strands in TRACE observations (Testa et al., 2002). Later, it was shown that in many cases, hot loop structures observed with the Yohkoh/SXT are not exactly co-spatial with warm structures observed with the SoHO/EIT, which is sensitive in the same bands as TRACE, nor they cool down to become visible to the EIT (Nagata et al., 2003; Schmieder et al., 2004).
The detailed morphological comparison of an active region showed that hot loops seen in SXT (T > 3 MK) and warm loops seen in the SoHO/EIT 195 Å band (T ∼ 1 – 2 MK) are located in almost alternating manner (Nagata et al., 2003). The anti-coincidence of the hot and the warm loops is conserved for a duration longer than the estimated cooling timescale. These results suggest that loops are not isothermal in the transverse direction, rather they have a differential emission measure distribution of modest but finite width that peaks at different temperatures for different loops (see Section 3.3).
In apparent alternative, Winebarger and Warren (2005) showed that hot monolithic loops visible with the Yohkoh/SXT are later resolved as stranded cooler structures with TRACE. However, this occurs with a time delay of 1 to 3 hours, much longer than the plasma cooling times, and, therefore, correlation can be hardly established between the plasma detected by Yohkoh and that detected by TRACE. Direct density measurements of loop plasma made from multi-line observations at the solar limb indicate not very dense plasma and relatively high plasma filling factor (0.2 – 0.9, Ugarte-Urra et al., 2005). This is in the direction of a moderate structuring of the loops.
Aschwanden and Nightingale (2005) analyzed specifically and systematically TRACE images to search for the thinnest coherent structures that can be resolved with TRACE. They found that about 10% of the positions can be fitted with an isothermal model and proposed that, since the corresponding structures have a uniform thermal distribution, they should be elementary loop components, with an average width of about 2000 km. Aschwanden et al. (2007) studied statistically a large set of coronal loops and found further evidence of elementary loop strands resolved by TRACE.
Other studies based both on models and on analysis of observations independently suggest that elementary loop components should be even finer, with typical cross-sections of the strands to be of the order of 10 – 100 km (Beveridge et al., 2003; Cargill and Klimchuk, 2004; Vekstein, 2009).
Diagnostics and thermal structuring
The investigation of the thermal structure of coronal loops is very important for their exhaustive physical comprehension and to understand the underlying heating mechanisms. For instance, one of the classifications outlined above is based on the loop thermal regime, and, we remark, it is debated whether the classification indicates a real physical difference.
Diagnostics of temperature are not trivial in the corona. No direct measurements are available. Since the plasma is optically thin, we receive information integrated on all the plasma column along the line of sight. The problem is to separate the distinct contributing thermal components and reconstruct the detailed thermal structure along the line of sight. However, even the determination of global and average values deserves great attention.
Moderate diagnostic power is allowed by imaging instruments, by means of multifilter observations. Filter ratio maps provide information about the spatial distribution of temperature and emission measure (e.g., Vaiana et al., 1973). The emission of an optically thin isothermal plasma as measured in a j-th filter passband is
where T is the temperature and EM is the emission measure, defined as
where n is the particle density and V the plasma volume. The ratio Rij of the emission in two different filters i, j is then independent of the density, and only a function of the temperature:
The inversion of this relationship provides a value of temperature.
The limitations of this method are substantial. In particular, one filter ratio value provides one temperature value for each pixel; this is a reliable measurement, within experimental errors, as long as the assumption of isothermal plasma approximately holds for the plasma column in the pixel along the line of sight. If the plasma is considerably multithermal, the temperature value is an average weighted for the instrumental response. Since the response is a highly non-linear function of the emitting plasma temperature, it is not trivial to interpret the related maps correctly. In addition, it is fundamental to know the instrument response with high precision, in order to avoid systematic errors, which propagate dangerously when filter ratios are evaluated. In this sense, broadband filters provide robust thermal diagnostics, because they are weakly dependent on the details of the atomic physics models, e.g., on the presence of unknown or not well-known spectral lines, on the choice of element abundances. Narrowband filters can show non-unique dependencies of filter ratio values on temperature (e.g., Patsourakos and Klimchuk, 2007), due to the presence of several important spectral lines in the bands, but a more general problem can be the bias to detect narrow ranges of temperatures forced by the specific instrument characteristics. This problem can be important especially when the distribution of the emission measure along the line of sight is not simple and highly non-linear (e.g., Reale et al., 2009b). The problem of diagnostics of loop plasma from filter ratios and, more in general, the whole analysis of loop observations, are made even more difficult by the invariable presence of other structures intersecting along the line of sight. A uniform diffuse background emission also affects the temperature diagnostics, by adding systematic offsets which alter the filter ratio values. The task of subtracting this “background emission” from the measured emission is non-trivial and can affect seriously the results of the whole analysis. This problem emerged dramatically when the analysis of the same large loop structure observed with Yohkoh/SXT on the solar limb led to three different results depending mostly on the different ways to treat the background (Priest et al., 2000; Aschwanden, 2001; Reale, 2002b). The amount of background depends on the instrument characteristics, such as the passband and the point response function: it is most of the signal in TRACE UV filterbands, for instance, and its subtraction becomes a very delicate issue (e.g., Del Zanna and Mason, 2003; Reale and Ciaravella, 2006; Aschwanden et al., 2008a; Terzo and Reale, 2010). The problem can be mitigated if one analyzes loops as far as possible isolated from other loops, but this is not easy, for instance, in active regions. If this is not the case, broadband filters may also include contamination from many structures at relatively different temperature and make the analysis of single loops harder. The problem of background subtraction in loop analysis has been addressed by several authors, who apply different subtraction ranging from simple offset, to emission in nearby pixels or subregions, to values interpolated between the loop sides, to whole images at times when the loop is no longer (or not yet) visible (Testa et al., 2002; Del Zanna and Mason, 2003; Schmelz et al., 2003; Aschwanden and Nightingale, 2005; Reale and Ciaravella, 2006; Aschwanden et al., 2008a; Terzo and Reale, 2010).
More accurate diagnostics, although with less time and space resolution, is in principle provided by spectrometers and observations in temperature-sensitive spectral lines, which are being constantly improved to provide better and better spatial information. Early results from UV spectroscopy already recognized the link between transition region and coronal loops, for instance, from Skylab mission (Feldman et al., 1979; Mariska et al., 1980). Together with background subtraction, one major difficulty met by spectroscopic analysis is that, in the UV band, the density of lines is so high that they are often blended and, therefore, it is hard to separate the contribution of the single lines, especially the weak ones. Fine diagnostics, such as Doppler shifts and line broadening, can become very tricky in these conditions and results are subject to continuous revisitation and warnings from the specialized community. The problem of background subtraction is serious also for spectral data, because their lower spatial and temporal resolution determines the presence of more structures, and therefore more thermal components, along the line of sight in the same spatial element.
Care should be paid also when assembling information from many spectral lines into a reconstruction of the global thermal structure along the line of sight. Methods are well-established (e.g., Gabriel and Jordan, 1975) and several approaches are available. The so-called method of the emission measure loci (Jordan et al., 1987) is able to tell whether plasma is isothermal of multithermal along the line of sight (Figure 7), but less able to add details. Detailed emission measure distributions can be obtained from differential emission measure (DEM) reconstruction methods (e.g., Brosius et al., 1996; Kashyap and Drake, 1998), but this is an ill-posed mathematical problem and, therefore, results are not unique and are subject to systematic and unknown errors. Forward modeling and simulations can be ways to escape from these problems, but they require non-trivial computational efforts and programming, and it is not always possible to provide accurate confidence levels. All these approaches are constantly improved and probably the best way to proceed is to combine different approaches and multiband observations and to finally obtain a global consistency.
In addition to the problems intrinsic to diagnostic techniques, we have to consider that loops appear to have different properties in different bands, as mentioned in Section 3.1.1. It is still debated whether such differences originate from an observational bias due to the instruments or from intrinsic physical differences, or both. In view of this uncertainty, in the following we will make a distinction between hot and warm loops, which will generally correspond to loops observed in the X-ray (and hot UV lines, e.g., Yohkoh/SXT, Hinode/XRT, SoHO/CDS Fe xvi line) and in the UV band (e.g., SoHO/EIT, TRACE), respectively. Cool loops are also observed in the UV band. The boundary between hot and warm loops is, of course, not sharp, and it is not even clear whether they are aspects of the same basic structure, or they really are physically different and are heated differently (see also Section 3.3.3). We will devote attention to the comparison between hot and warm loops.
After the pioneering analyses driven by the Skylab X-ray instruments (Section 2), Yohkoh/SXT allowed to conduct large scale studies on the thermal and structure diagnostics of hot loops, and the comparison with other instruments, for instance on-board SoHO, allowed to obtain important cross-checks and additional information. Filter ratio maps of flaring loops were shown early after the mission launch (Tsuneta et al., 1992).
Systematic measurements of temperature, pressure, and length of tens of quiescent and active region coronal loops were conducted on Yohkoh observations (Porter and Klimchuk, 1995) using the filter ratio method. For this sample of loops, selected to be steady and isolated, the lengths were measured with assumptions on the loop geometry and ranged in a decade between 5 × 109 < 2L < 5 × 1010 cm. The temperature measurements were averaged over about half of the loops and also ranged in a decade (2 < T < 30 MK), with a mean of about 6 MK. Therefore, it appears as a sample of particularly hot loops, although the uncertainties in the hot tail of the distribution are very large, probably due to the flat dependence of the temperature on the filter ratio at high temperature. Pressures were derived from the equation of state, after derivation of the density, from the emission measure and from the volume inferred from the length and assumptions on the loop aspect. They ranged in two decades (0.1 < p < 20 dyne cm-2). Overall, it was shown that the temperature and length of this sample of hot loops are uncorrelated, that pressure varies inversely with length (as overall expected for a thermally homogeneous sample from loop scaling laws, see Section 4.1.1), although with a large spread. Such distributions were used as constraints on the loop heating through the derivation of the dependence of the magnetic field intensity on the loop length (Klimchuk and Porter, 1995). They also led to accurate analysis of data uncertainties (Klimchuk and Gary, 1995). Another systematic analysis was made on a sample of about 30 bright steady Yohkoh loops located in active regions (Kano and Tsuneta, 1995). The temperatures were measured after averaging several images and taking the value at the loop top. While this analysis confirmed some of the correlations found in Porter and Klimchuk (1995), it found a correlation between the loop length and the temperature, and showed deviations from RTV scaling laws (Section 4.1.1). It cannot be excluded that correlations between parameters depend on the loop sample, as a single scaling law links three parameters. Yohkoh/SXT loops hotter than 3 MK were found also in another study, the hottest ones with shorter lifetimes (less than few hours), and often exhibiting cusps (Yoshida and Tsuneta, 1996).
A big effort has recently been devoted to the possible detection of hot plasma outside of evident flares. This would be a conclusive evidence of nanoflaring activity in coronal loop (e.g., Klimchuk, 2006, see Section 4.4). Hinode instruments appear to be able to provide new interesting contributions to this topic. The analysis of spectroscopic observations of hot lines in solar active regions from Hinode/EIS allows to construct emission measure distributions in the 1 – 5 MK temperature range, and shows that the distributions are flat or slowly increasing up to approximately 3 MK and then fall off rapidly at higher temperatures (Patsourakos and Klimchuk, 2009). Evidence of emission from hot lines has been found also in other Hinode/EIS observations, and in particular from the analysis of the emission from the Ca xvii at 192.858 Å, formed near a temperature of 6 × 106 K, which has been found in several parts of active regions (Ko et al., 2009). Using Fe lines, Young et al. (2009) has shown very accurate density measurements (≈ 5%) across an active region, with values in the range 8.5 ≤ log (Ne/cm-3) ≤ 11.0. A smaller density range (from 108.5 to 109.5 cm-3 has been found by Watanabe et al. (2009) using the Fe xiii line group, although one pair has been found to reach the high density limit. Density sensitive lines have been used to measure the filling factor of coronal structures. Dere (2008, 2009) has used spectra and images obtained with EIS and comparison with TRACE to determine the volumetric filling factor of bright points. The emission measure and bright point widths have been compared with the electron densities and with TRACE data. The plasma-filling factor has been found to vary from 3 × 10-3 to 0.3 with a median value of 0.04, which may indicate considerable subresolution structure, or the presence of a single completely-filled unresolved loop with subarcsec width.
Thanks to its multifilter observations, also Hinode/XRT is providing useful information about the thermal structure of the bright X-ray corona. Temperature maps derived with combined filter ratios show fine structuring to the limit of the instrument resolution and evidence of multithermal components (Reale et al., 2007). This kind of temperature diagnostics is supported by the evidence of warm structures bright in the TRACE images. Observations including flare filters show evidence of a hot component in active regions outside of flares (Schmelz et al., 2009b) and data in the medium thickness filters appear to constrain better this component of hot plasma as widespread, although minor, and peaking around log T ∼ 6.8 – 6.9, with a tail above 10 MK (Reale et al., 2009b). This may support the hypothesis that active regions are heated impulsively. Evidence of a persistent although small hot plasma component outside of flare is shown also by RHESSI data (McTiernan, 2009), and the comparison between RHESSI and XRT data seem to support this scenario in a consistent way (Reale et al., 2009a). The topic of coronal active region heating is debated. Evidence interpreted in the direction of more gradual heating has been obtained by Warren et al. (2010) (see Section 4.4).
Comparison of hot and warm loops
Before the SoHO/EIT and TRACE observations, warm loops had been imaged in a similar spectral band and with similar optics by the rocket NIXT mission (see Section 2). Studies of NIXT loops including the comparison with hydrostatic loop models (Section 4.1.1) pointed out that bright spots also visible in H.. band were the footpoints of hot high-pressure loops (Peres et al., 1994). This result was confirmed by the comparison of the temperature structure obtained from Yohkoh with NIXT data (Yoshida et al., 1995) (see also Section 3.3.3).
Another comparison of loops imaged with NIXT and Yohkoh/SXT showed that the compact loop structures (length ∼ 109 cm) have a good general morphological correspondence, while larger scale NIXT loops (∼ 1010 cm) have no obvious SXT counterpart (Di Matteo et al., 1999). Comparison with static loop models (see Section 4.1.1) allowed to derive estimates of the loop filling factors, important for the loop fine structure (Section 3.2.2). In the NIXT band, the filling factor of short loops was found to be very low (10-3 – 10-2), but of the order of unity in the SXT band and for the largest structure. Information about the loop filling factor was derived also from the analysis of simultaneous SoHO and Yohkoh observations of a small solar active region, suggesting a volume filling factor decreasing with increasing density and possible differences between emitting material in active regions and the quiet Sun (Griffiths et al., 2000).
Some similarity between loops observed in the TRACE EUV band and Yohkoh X-ray band was found out of the core of active region loops Nitta (2000) and interpreted as evidence of loops with a broad range of temperatures. Core loops were instead observed only in the X-rays and found to be variable, indicating that probably they are not steady.
The thermal distribution across the loop structures, i.e., along the line of sight, can be investigated with the analysis of observations in several spectral lines, as obtained, for instance, with the SoHO/CDS. Information on the validity of the data analysis and of the loop diagnostics can be obtained from the comparison with simultaneous and co-spatial data from imaging instruments.
Density and temperatures in two active regions were accurately determined from SOHO-CDS observations (Mason et al., 1999) and it was confirmed quantitatively that the AR cores are hotter than larger loop structures extending above the limb. From the analysis of a single loop observed on the solar limb with SoHO/CDS, Schmelz et al. (2001) found a bias to obtain flat temperature distributions along the loop from ratios of single lines or narrow band filters (TRACE), while a careful DEM reconstruction at selected points along the loop is inconsistent with isothermal plasma both across and along the loop. A whole line of works started from this study reconsidering and questioning the basic validity of the temperature diagnostics with TRACE and emphasizing the importance of the background subtraction, but also the need to obtain accurate spectral data (Schmelz, 2002; Martens et al., 2002; Aschwanden, 2002; Schmelz et al., 2003). Similar results but different conclusions were reached by Landi and Landini (2004); Landi and Feldman (2004) who analyzed a loop observed with SoHO and, finding it nearly isothermal, considered this evidence as real and invoked a non-constant cross-section to explain it. On the other hand, evidence of nonuniform temperature along loops observed with TRACE was also found (Del Zanna and Mason, 2003; Reale and Ciaravella, 2006), emphasizing that the temperature diagnostics with narrow band instruments is a delicate issue.
An interesting debate has focussed on the question whether the loops observed with TRACE and CDS have a uniform transverse thermal distribution, i.e., a narrow DEM, or a multi-thermal distribution, i.e., a wide DEM which may group together warm and hot loops. Although tackled from a different perspective, this question also concerns the fine longitudinal structuring of the loops and of their heating and is therefore strictly connected to the subject of Sections 3.2.2 and 4.4. Del Zanna and Mason (2003) found a loop detected with TRACE to be isothermal (with temperatures below 1 MK) along the line of sight from diagnostics of spectral lines obtained with SoHO/CDS. Schmelz et al. (2005) analyzed another loop on the limb observed with SoHO/CDS, with a DEM reconstruction and a careful analysis of background subtraction, and found a multithermal distribution across the loop. From the comparison with the isothermal structure of hot loops derived from CDS data (Di Giorgio et al., 2003; Landi and Landini, 2004), they concluded that there may be two different classes of loops, multi-thermal and isothermal, which they found to be confirmed by a systematic inspection of the CDS atlas (Figure 7).
Multiband observations allow to obtain even more information and constraints. Reale and Ciaravella (2006) analyzed a well-defined loop system detected in a time-resolved observation in several spectral bands, namely three TRACE UV filters, one Yohkoh/SXT filter, two rasters taken with SoHO/CDS in twelve relevant lines (5.4 ≤ log T ≤ 6.4). The data analysis supported a coherent scenario across the different bands and instruments, indicating a globally cooling loop and the presence of thermal structuring. The study overall indicated that the loop analysis can be easily affected by a variety of instrumental biases and uncertainties, for instance due to gross background subtraction. The fact that the loop that could be well analyzed across several bands and lines is a cooling loop may not be by chance (see end of Section 3.3.3).
Specific analyses of SoHO spectrometric data have continued to contribute much to the study of the loop thermal structure up to recently. A Differential Emission Measure (DEM) analysis of coronal loops using a forward-folding technique on SoHO/CDS data has shown different results for two loops, one to be isothermal and the other to have a broad DEM (Schmelz et al., 2007b). Landi and Feldman (2008) have analyzed an extensive active region spectrum observed by the SUMER instrument on board SOHO and found that the plasma is made of three distinct isothermal components, whose physical properties are similar to coronal hole, quiet-Sun, and active region plasmas.
Hinode/EIS observations of active region loops certify that structures which are clearly discernible in cooler lines (∼ 1 MK) become fuzzy at higher temperatures (∼ 2 MK, Tripathi et al., 2009, Figure 8) as already pointed out by Brickhouse and Schmelz (2006).
Comparative studies of active region loops in the transition region and the corona Ugarte-Urra et al. (2009) observed with Hinode seem to point out the presence of two dominant loop populations, i.e., core multitemperature loops that undergo a continuous process of heating and cooling in the full observed temperature range 0.4 – 2.5 MK shown by the X-Ray Telescope, and peripheral loops which evolve mostly in the temperature range 0.4 – 1.3 MK.
The TRACE mission opened new intriguing questions because the data showed new features, e.g., stranded bright structures mostly localized in active regions, name “the moss”, and because the narrow band filters offered some limited thermal diagnostics, but not easy to interpret. Reliable temperatures are in fact found in a very narrow range, and many coronal loops are found to be isothermal in that range.
As mentioned in Section 3.3.2, first loop diagnostics with normal incidence telescopes were obtained from data collected with NIXT Peres et al. (1994). The bright spots with Hα counterparts were identified with the footpoints of high pressure loops, invisible with NIXT because not sensitive to plasma hotter than 1 MK. They have been later addressed as the moss in the TRACE images, which undergo the same effect. Interactions of moss with underlying chromospheric structures were first described by Berger et al. (1999a). Comparison of SOHO/CDS and TRACE observations led to establish that the plasma responsible for the moss emission has a temperature range of about 1 MK and is associated with hot loops at 1 – 2 MK, with a volume filling factor of order 0.1 (Fletcher and De Pontieu, 1999). It was also found that the path along which the emission originates is of order 1000 km long. According to an analytical loop model, a filling factor of about 0.1 is in agreement with the hypothesis of moss emission from the legs of 3 MK loops (Martens et al., 2000).
As for temperature diagnostics with narrow band filters, loops soon appeared to be mostly isothermal with ratios of TRACE filters (Lenz et al., 1999; Aschwanden et al., 2000). Is this a new class of loops? Equivalent SoHO/EIT filter ratios provided analogous results (Aschwanden et al., 1999b). This evidence is intriguing and many investigations have addressed it (see also Section 3.3.2). From the diagnostic perspective, Schmelz et al. (2001) reconstructed DEM distributions along the line of sight from spectral SoHO/CDS data and synthesized EIT count rates from them, which led to almost uniform temperatures along the loop, pointing again to an instrumental bias. Weber et al. (2005) confirmed that, provided they are flat, i.e., top-hat-shaped, even broad DEMs along the line of sight produce constant TRACE filter ratio values. However, we learn from DEM studies made both with spectrometers and from multi-wideband imagers that the DEM of coronal loops is most probably neither isothermal nor broad and flat, instead peaked with components extending both to low and high temperatures (e.g., Peres et al., 2000; Reale et al., 2009b). The critical point becomes the DEM width and its range of variation.
Later, Schmelz et al. (2007a) found that even TRACE triple-filter data cannot, in general, constrain the temperature distribution for plasma in warm loops. On the other hand, Patsourakos and Klimchuk (2007) studied the cross-field thermal structure of a sample of coronal loops from triplefilter TRACE observations, and found that the observations are compatible with multithermal plasma with significant emission measure throughout the range 1 – 3 MK. Schmelz et al. (2009a) used TRACE filter ratios, emission measure loci, and two methods of differential emission measure analysis to examine the temperature structure of three different loops. In agreement with previous studies, they found both isothermal and multithermal cases. This might not be a contradiction, in the view of the presence of at least three possible conditions of warm loops, as discussed at the end of this section. Noglik et al. (2008) compared TRACE to CDS data to measure the temperature along a coronal loop in an active region on the solar limb. Their double filter ratio temperature analysis technique led to temperatures between 1.0 and 1.3 MK. Emission measure loci from CDS lines were consistent with a line-of-sight isothermal structure which increases in temperature from ∼ 1.20 to 1.75 MK along the loop, in contrast with the nearby multithermal background.
Another puzzling issue, certainly linked to the loop isothermal appearance, is that warm loops are often diagnosed to be overdense with respect to the equilibrium values predicted by loop scaling laws (Lenz et al., 1999; Winebarger et al., 2003a, Section 4.1.1). To explain both these pieces of evidence, several authors claimed that the loops cannot be at equilibrium and that they must be filamented and cooling from a hotter state, probably continuously subject to heating episodes (nanoflares, Warren et al., 2002, 2003, Sections 4.2 and 4.4). Other authors proposed that part of the effect might be due to inaccurate background subtraction (Del Zanna and Mason, 2003).
The Hinode mission is stimulating new analyses of warm coronal structures, mostly based on its high quality EIS spectral data. Modeling observations of coronal moss with Hinode/EIS confirmed that the moss intensities predicted by steady, uniformly heated loop models are too intense relative to the observations (Warren et al., 2008b). A nonuniform filling factor is required and must vary inversely with the loop pressure. Observations of active region loops with EIS indicate that isolated coronal loops that are bright in Fe xii generally have very narrow temperature distributions (3 × 105 K), but are not properly isothermal and have a volumetric filling factors of approximately 10% (Warren et al., 2008a).
Schmelz et al. (2008) studied temperatures of loops identified in a TRACE image in three density-sensitive line ratios. While emission measure loci plots indicated that the loop plasma is not isothermal, a more detailed differential emission measure analysis showed that two broad components can reproduce the background-subtracted intensities. They proposed that the two-component DEM distribution represents two ensembles of strands, one for each of the loops seen in the TRACE image.
Density diagnostics through density-sensitive line ratio also led to measure directly density values, for instance in active regions (e.g., Doschek et al., 2007b). Tripathi et al. (2008) found that the hot core of the active region is densest with values as high as 1010.5 cm-3. The electron density estimated in specific regions in the active region moss decreases with increasing temperature. The density within the moss region was highest at log T = 5.8 – 6.1, with a value around 1010 cm-3.
In a cooler regime (4.15 < log T < 5.45) observed in coordination by SOHO spectrometers and imagers, STEREO/EUVI, and Hinode/EIS, active region plasma at the limb has been found to cool down from a coronal hole status with temperatures in the 5.6 < log T < 5.9 range (Landi et al., 2009).
The loop reconstruction analysis described in Aschwanden et al. (2009) was used mainly for density and temperature modeling of the warm loops of an active region observed with STEREO. The rendering reduces the problem of background subtraction. Although based on simple model assumptions, the derived density and temperature distributions are able to reproduce the total observed fluxes within 20%. The modeling extrapolates results quite outside of the range of sensitivity of the STEREO EUV filters, anyhow finding emission measure distributions not very different from those obtained from spectroscopic observations (Brosius et al., 1996) and deviations from hydrostatic values in agreement with other previous studies (Lenz et al., 1999; Winebarger et al., 2003a).
In summary, the current observational framework and loop analysis seems to indicate that for a coherent scenario warm loops are manifestations of at least three different loop conditions: i) in loops consisting of bundles of thin independently-heated strands, few cooling strands of steady hot X-ray loops might be detected as warm overdense loops in the UV band. These warm loops would coexist with hot loops and would show a multithermal emission measure distribution (Patsourakos and Klimchuk, 2007; Warren et al., 2008a; Tripathi et al., 2009); ii) we might have warm loops as an obvious result of a relatively low average heating input in the loop. These loops would be much less visible in the X-rays and thus would not be cospatial with hot loops, and would also be much less multithermal (Di Giorgio et al., 2003; Landi and Landini, 2004; Aschwanden and Nightingale, 2005; Noglik et al., 2008); iii) warm loops might be globally cooling from a status of hot X-ray loop (Reale and Ciaravella, 2006). These loops would also be overdense and cospatial with hot loops but with a time shift of the X-ray and UV light curves, i.e., they would be bright in the X-rays before they are in the UV band. Also these loops would have a relatively narrow thermal distribution along the line of sight.
The solar corona is the site of a variety of transient phenomena. Coronal loops sometimes flare in active regions (see the review by Benz, 2008). However, most coronal loops are well-known to remain in a steady state for most of their life, much longer than the plasma characteristic cooling times (Rosner et al., 1978, see Section 4.1.1). This is taken as an indication that a heating mechanism must be on and steady long enough to bring the loop to an equilibrium condition, and keep it there for a long time. Nevertheless, the emission of coronal loops is found to vary significantly on various timescales, and the temporal analyses of coronal loop data have been used to obtain different kinds of information, and as a help to characterize the dynamics and heating mechanisms. The time variability of loop emission is generally not trivial to interpret. The problem is that the emission is very sensitive to density and less to temperature. Therefore, variations are not direct signatures of heating episodes, not even of local compressions, because the plasma is free to flow along the magnetic field lines. Variations must therefore be explained in the light of the evolution of the whole loop. This typically needs accurate modeling, or, at least, care must be paid to many relevant and concurrent effects.
Another important issue is the band in which we observe. The EUV bands of the normal incidence telescopes are quite narrow. Observations are then more sensitive to variations because cooling or heating plasma is seen to turn on and off rapidly as it crosses the band sensitivity. On the other hand, telescopes in the X-ray band detect hotter plasma which is expected to be more sensitive to heating and, therefore, to vary more promptly, but the bandwidths are large and do not take as much advantage of the temperature sensitivity as the narrow bands. Finally, spectroscopic observations are, in principle, very sensitive to temperature variations because they observe single lines, but their time cadence is typically low and able to follow variations only on large timescales. Time analysis studies can be classified to address two main classes of phenomena: temporal variability of steady structures and single transient events, such as flare-like brightenings.
In spite of limited time coverage, the instrument S-054 on-board Skylab already allowed for early studies of variability of hot X-ray loops. Decay times were studied by Krieger (1978) who found evidence of continued evaporation of coronal plasma in slowly-decaying structures. Sheeley Jr (1980) and Habbal et al. (1985) found timescales of moderate variability of a few hours over a substantial steadiness for observations of active region loops in 2 MK lines such as Fe xv and Si xii. Substantial (but non-flaring) temporal variability was reported by Haisch et al. (1988) in two active region loops observed with SMM in a few relatively hot X-ray lines (∼ 5 MK) on time scales of some minutes. Cooler loops (< 1 MK) first studied in detail by Foukal (1976) (see Section 3.5) were found to be more variable and dynamic (e.g., Kopp et al., 1985).
The high time coverage and resolution of Yohkoh triggered studies of brightenings on short time scales. The Yohkoh/SXT resolution and dynamic range allowed to study the interaction of differently bright hot loops and to show, for instance, that X-ray bright points often involve loops considerably larger than the bright points themselves, and that they vary on timescales from minutes to hours (Strong et al., 1992). The analysis of a large set (142) of macroscopic transient X-ray brightenings indicated that they derive from the interaction of multiple loops at their footpoints (Shimizu et al., 1994). Some other more specific loop variations were also observed, e.g., the shrinkage of large-scale non-flare loops (Wang et al., 1997). This was interpreted not as an apparent motion but as a real contraction of coronal loops that brighten due to heating at footpoints followed by gradual cooling. Fine-scale motions and brightness variations of the emission were found on timescales of 1 minute or less, with dark inclusions corresponding to jets of chromospheric plasma seen in the wings of Hα. Such small scale variations are associated with the fine structure and dynamics of the conductively heated upper transition region between the solar chromosphere and corona (Berger et al., 1999b).
Loop variability was specifically studied in several UV spectral lines observed with SoHO/CDS for about 3 hours by Di Giorgio et al. (2003). In the hottest lines, within the limited time resolution of about 10 minutes, a few brightenings of a hot loop (∼ 2 MK) were detected but they are minor perturbations over a steadily high emission level. The observation of the whole life of a cool loop (log T ∼ 5.3) on a time lapse of a few hours confirmed the highly transient nature of cool loops, probably linked to the presence of substantial flows (Section 3.5).
Variability analyses have been conducted also on warm loops present in TRACE data. The brightening of a single coronal loop was analyzed in detail by Reale et al. (2000b) in an observation of more than 2 hours with a cadence of about 30 s. The loop evolves coherently in the rise phase and brightens from the footpoints to the top, allowing for detailed hydrodynamic modeling (Reale et al., 2000a) (see also Section 4.4). Active region transient events, i.e., short-lived brightenings in small-scale loops, were observed with TRACE with a high cadence of 35 s over half an hour (Seaton et al., 2001). Several brightenings detected over a neutral line in a region of emerging flux were interpreted as reconnection events associated with flux emergence, possible EUV counterparts to active region transient brightenings. The fast evolution probably implies high speed flows and high coronal densities. Shimojo et al. (2002) noticed apparent shrinking and expansion of brightening warm loops and proposed heating and cooling of different concentric strands, leading to coronal rain visible in the Hα line. Plasma condensations in hot and warm loops were detected also in the analysis of line intensity and velocity in temporal series data from SOHO/CDS (O’Shea et al., 2007). Antiochos et al. (2003) found no significant variability of the moss regions observed with TRACE. This has been taken as part of the evidence toward steady coronal heating in active region cores (Warren et al., 2010, Section 4.4).
The analysis of temporal series from various missions has been used, more recently, to investigate the possible presence of continuous impulsive heating by nanoflares. The temporal evolution of hot coronal loops was studied in data taken with GOES Solar X-ray Imager (SXI), an instrument with moderate spatial resolution and spectral band similar to Yohkoh/SXT (López Fuentes et al., 2007). The durations and characteristic timescales of the emission rise, steady, and decay phases were found to be much longer than the cooling time and indicate that the loop-averaged heating rate increases slowly, reaches a maintenance level, and then decreases slowly (Figure 9), not in contradiction with the early results of Skylab (Section 2). This slow evolution is taken as an indication of a single heating mechanism operating for the entire lifetime of the loop. If so, the timescale of the loop-averaged heating rate might be roughly proportional to the timescale of the observed intensity variations.
Joint TRACE and SOHO/CDS observations allowed to study temperature as a function of time in active region loops (Cirtain et al., 2007). In many locations along the loops, the emission measure loci were found consistent with an isothermal structure, but the results also indicated significant changes in the loop temperature (between 1 and 2 MK) over the 6 h observing period. This was interpreted as one more indication of multistranded loops, substructured below the resolution of the imager and of the spectrometer. Further support to fine structuring comes from the analysis of the auto-correlation functions in SXT and TRACE loop observations (Sakamoto et al., 2008). The duration of the intensity fluctuations for the hot SXT loops was found to be relatively short because of the significant photon noise, but that for the warm TRACE loops agrees well with the characteristic cooling timescale. This may support loops to be continuously heated by impulsive nanoflares. The energy of nanoflares is estimated to be 1025 erg for SXT loops and 1023 erg for TRACE loops. The occurrence rate of nanoflares is about 0.4 and 30 nanoflares s-1 in a typical hot SXT loop and a typical warm TRACE loop, respectively.
A recent study on time series has been performed on data taken with the Hinode mission. Hinode’s Solar Optical Telescope (SOT) magnetograms and high-cadence EIS spectral data were taken to study the relationship between chromospheric, transition region, and coronal emission and the evolution of the magnetic field (Brooks et al., 2008). The data have allowed to distinguish hot, relatively steadily emitting warm coronal loops from isolated transient brightenings and to find that they are both associated with highly dynamic magnetic flux regions. Brightenings have been typically found in regions of flux collision and cancellation, while warm loops are generally rooted in magnetic field regions that are locally unipolar with unmixed flux. The authors suggest that the type of heating (transient vs. steady) is related to the structure of the magnetic field, and that the heating in transient events may be fundamentally different from that in warm coronal loops.
Diagnosing the presence of significant flows in coronal loops is not an easy task. Apparently moving brightness variations may not be a conclusive evidence of plasma motion, since the same effect may be produced by the propagation of thermal fronts or waves. Conclusive evidence of plasma motion comes from measurements of Doppler shifts in relevant spectral lines. However, the detection of significant Doppler shifts requires several conditions to be fulfilled altogether, e.g., significant component along the line of sight, amount of moving plasma larger than amount of static plasma, plasma motion comparable to typical line broadening effects.
In general, we can distinguish two main classes of mass bulk motions inside coronal loops: siphon flows, due to a pressure difference between the footpoints, and loop filling or draining, due to transient heating and subsequent cooling, respectively. Some other evidence of bulk motions, such as systematic redshifts in UV lines, has been difficult to interpret.
Siphon flows have been mainly invoked to explain motions in cool loops. The existence of cold loops has been known for a long time (Foukal, 1976) (see Section 2) and SoHO has collected highquality data showing the presence of dynamic cool loops (Brekke et al., 1997). A well-identified detection was found in SoHO/SUMER data, i.e., a small loop showing a supersonic siphon-like flow (Teriaca et al., 2004) and in SoHO/CDS data (Di Giorgio et al., 2003).
Redshifts in transition region UV lines have been extensively observed on the solar disk (e.g., Doschek et al., 1976; Gebbie et al., 1981; Dere, 1982; Feldman et al., 1982; Klimchuk, 1987; Rottman et al., 1990; Brekke, 1993; Peter, 1999). Some mechanisms have been proposed to explain these redshifts: downward propagating acoustic waves (Hansteen, 1993), downdrafts driven by radiatively-cooling condensations in the solar transition region (Reale et al., 1996, 1997b), nanoflares (Teriaca et al., 1999b); a conclusive word is still to be given with the better and better definition of the observational framework.
Blue-shifts in the transition region are also studied but not necessarily associated with coronal loops (e.g., Dere et al., 1986). More localized and transient episodes of high velocity outflows, named explosive events, have been observed in the transition lines such as Civ, formed at 100 000 K (e.g., Dere et al., 1989; Chae et al., 1998b; Winebarger et al., 1999, 2002a,b). However, Teriaca et al. (2002) found indications that such EUV explosive events are not directly relevant in heating the corona, are characteristic of structures not obviously connected with the upper corona, and have a chromospheric origin.
Chae et al. (1998a) found Doppler shifts increasing and then decreasing with increasing temperature and explained them by the dominance of emission from plasma flowing downward from the upper hot region to the lower cool region along flux tubes with varying cross section (a factor about 30). Teriaca et al. (1999a) confirmed these results with the exception of blueshifts at higher temperature.
EUV spectra of coronal loops above an active region show clear evidence of strong dynamical activity. In the Ov 629 Å line, formed at 240 000 K, line-of-sight velocities greater than 50 km s-1 have been measured with the shift extending over a large fraction of a loop (Brekke et al., 1997). Active region loop structures appear to be extremely time variable and dynamic at transition region temperatures, with large Doppler-shifts (Brekke, 1999). Di Giorgio et al. (2003) showed the direct observation and identification of the birth, evolution, and cooling of one of such transient cool loops, and measured a blue-shifted up-flow all along the loop, probably a one-direction siphon flow.
Winebarger et al. (2002c) analyzed co-aligned TRACE and the SoHO/SUMER observations of warm active region loops. Although these loops appear static in the TRACE images, SUMER detects line-of-sight flows along the loops of up to 40 km s-1. Apparent motions were also detected in TRACE images (Winebarger et al., 2001).
In SoHO/EIT high-cadence 304 Å images, De Groof et al. (2004) analyzed systematic intensity variations along an off-limb half loop structure propagating from the top towards the footpoint and reported several arguments supporting that these intensity variations are due to flowing/falling plasma blobs and not to slow magneto-acoustic waves (Section 4.4). This evidence has been the object of modeling studies (see Section 4.3).
The high spectral resolution of Hinode/EIS is allowing for very detailed studies of persistent loop plasma motions, which are very important to assess whether the loops are to be treated more as static or dynamic structures as a whole. More specifically, nonthermal velocities were detected in solar active regions (Doschek et al., 2007a). The largest widths seem to be located more in relatively faint zones, some of which also show Doppler outflows. Doppler flows in active region loops observed by Hinode EIS were explicitly addressed by Del Zanna (2008), who found a multifaceted scenario (Figure 10). Persistent redshifts, stronger in cooler lines (about 5 – 10 km s-1 in Fe xii and 20 – 30 km s-1 in Fe viii), were observed in most loop structures. Persistent blueshifts, stronger in the hotter lines (typically 5 – 20 km s-1 in Fe xii and 10 – 30 km s-1 in Fe xv), were present in areas of weak emission, in a sharp boundary between the low-lying “hot” 3 MK loops and the higher “warm” 1 MK loops. Strong localized outflows (∼ 50 km s-1) in a widespread downflow region were clearly visible in Doppler-shifts maps obtained with EIS (Doschek et al., 2008). The outflows might be tracers of long loops and/or open magnetic fields.
Ofman and Wang (2008) used the high resolution Hinode SOT observations and detected cool plasma flowing in multi-threaded coronal loops with speeds in the range 74 – 123 km s-1. In addition to flows, the loops exhibited transverse oscillations.
Further analysis of coronal plasma motions near footpoints of active region loops showed again a strong correlation between Doppler velocity and nonthermal velocity (Hara et al., 2008). Significant deviations from a single Gaussian profile were found in the blue wing of the line profiles for the upflows. These may suggest that there are unresolved high-speed upflows. Tripathi et al. (2009) found that an active region was comprised of redshifted emissions (downflows) in the core and blueshifted emissions (upflows) at the boundary. All these results have to be matched with the recent finding of extensive blueshifts correlated with spicules upflows and with coronal emission intensity (De Pontieu et al., 2009) (Section 4.4).
Loop Physics and Modeling
The basics of loop plasma physics are well established since the 1970s (e.g., Priest, 1978). In typical coronal conditions, i.e., ratio of thermal and magnetic pressure β ≪ 1, temperature of a few MK, density of 108 – 1010 cm-3, the plasma confined in coronal loops can be assumed as a compressible fluid moving and transporting energy only along the magnetic field lines, i.e., along the loop itself (e.g., Rosner et al., 1978; Vesecky et al., 1979). In this configuration, the magnetic field has only the role of confining the plasma. It is also customary to assume constant loop cross-section (see Section 3.2.1). In these conditions, neglecting gradients across the direction of the field, effects of curvature, and transverse waves, the plasma evolution can be described by means of the one-dimensional hydrodynamic equations for a compressible fluid, using only the coordinate along the loop (Figure 11).
The time-dependent equations of mass, momentum, and energy conservation typically include the effects of the gravity component along the loop, the radiative losses from an optically thin plasma, the plasma thermal conduction, an external heating input, the plasma compressional viscosity:
with p and ∈ defined by
and the conductive flux
where n is the hydrogen number density, s the spatial coordinate along the loop, v the plasma velocity, mH the mass of hydrogen atom, μ the effective plasma viscosity, P(T) the radiative losses function per unit emission measure (e.g., Raymond et al., 1976), βi the fractional ionization, i.e., ne/nH, Fc the conductive flux, κ the thermal conductivity (Spitzer Jr, 1962), κB the Boltzmann constant, and χ the hydrogen ionization potential. H(s, t) is a function of both space and time which describes the heat input in the loop.
These equations can be solved numerically and several specific codes have been used extensively to investigate the physics of coronal loops and of X-ray flares (e.g., Nagai, 1980; Peres et al., 1982; Doschek et al., 1982; Nagai and Emslie, 1984; Fisher et al., 1985a,b,c; MacNeice, 1986; Gan et al., 1991; Hansteen, 1993; Betta et al., 1997; Antiochos et al., 1999; Müller et al., 2003; Ofman and Wang, 2002; Bradshaw and Mason, 2003; Bradshaw and Cargill, 2006).
The concept of numerical loop modeling is to use simulations, first of all, to get insight into the physics of coronal loops, i.e., the reaction of confined plasma to external drivers, to describe plasma evolution, and to derive predictions to compare with observations. One major target of modeling is, of course, to discriminate between concurrent hypotheses, for instance, regarding the heating mechanisms and to constrain the related parameters.
The models require to be provided with initial loop conditions and boundary conditions. In view of our ignorance of the specific heating mechanisms (see Section 4.4), the models require to define an input heating function, specifying its time-dependence, for instance it can be steady, slowly or impulsively changing, and its position in space. The output typically consists of distributions of temperature, density, and velocity along the loop evolving with time. From simulation results, some modelers derive observables, i.e., the plasma emission, which can be compared directly to data collected with the telescopes. The model results are, in this case, to be folded with the instrumental response. This forward-modeling allows to obtain constraints on model parameters and, therefore, quantitative information about the questions to be solved, e.g., the heating rate and location (e.g., Reale et al., 2000a).
Loop codes are typically based on finite difference numerical methods. Although they are one-dimensional, and therefore typically less demanding than other multi-dimensional codes that study systems with more complex geometry, and although they do not include the explicit description of the magnetic field, as full MHD codes, loop codes require some special care. One of the main difficulties consists in the appropriate resolution of the steep transition region (1 – 100 km thick) between the chromosphere and the corona, which can easily drift up and down depending on the dynamics of the event to be simulated. The temperature gradient there is very large due to the local balance between the steep temperature dependence of the thermal conduction and the peak of the radiative losses function (Serio et al., 1981). The density is steep as well so to maintain the pressure balance. The transition region can become very narrow during flares.
Also a fine temporal resolution is extremely important, because the highly efficient thermal conduction in a hot magnetized plasma can lead to a very small time step and make execution times not so small even nowadays. Another important issue is the necessary presence of a relatively thick, cool, and dense region under the transition region, i.e., a chromosphere, otherwise the atmosphere gets unstable. The main role of the chromosphere in this context is only that of a mass reservoir and, therefore, in several codes, it is chosen to treat it as simply as possible, e.g., an isothermal inactive layer which neither emits, nor conducts heat. In other cases, a more accurate description is chosen, e.g., to include a detailed model (e.g., Vernazza et al., 1981) and to maintain a simplified radiative emission and a detailed energy balance with an ad hoc heat input (Peres et al., 1982; Reale et al., 2000a).
In recent years, time-dependent loop modeling has revived in the light of the observations with SoHO and TRACE for the investigation of the loop dynamics and heating. The upgrade driven by the higher quality of the data has consisted in the introduction of more detailed mechanisms for the heating input, for the momentum deposition, or others, e.g., the time-dependent ionization and the saturated thermal conduction (Bradshaw and Cargill, 2006; Reale and Orlando, 2008). Some codes have been upgraded to include adaptive mesh refinement for better resolution in regions of high gradients, such as in the transition region, or during impulsive events (e.g., Betta et al., 1997). Another form of improvement has been the description of loops as collections of thin strands. Each strand is a self-standing, isolated and independent atmosphere, to be treated exactly as a single loop. This approach has been adopted both to describe loops as static (Reale and Peres, 2000) (Figure 12) and as impulsively heated by nanoflares (Warren et al., 2002). On the same line, collections of loop models have been applied to describe entire active regions (Warren and Winebarger, 2006).
One limitation of current 1D loop models is that they are unable to treat conveniently the tapering expected going down from the corona to the chromosphere (or expansion upwards) through the transition region. This effect can be neglected in many circumstances, but it is becoming increasingly important with the finer and finer level of diagnostics allowed by upcoming observational data. For instance, the presence of tapering changes considerably the predicted distribution of emission measure in the low temperature region (Section 4.1.1).
Possible deviations from pure 1D evolution might be driven by intense oscillations or kinks, as described in Ofman (2009). The effect of the three-dimensional loop structure should then be taken into account to describe the interaction with excited MHD waves (McLaughlin and Ofman, 2008; Pascoe et al., 2009; Selwa and Ofman, 2009).
However, the real power of 1D loop models, that makes them still on the edge, is that they fully exploit the property of the confined plasma to evolve as a fluid and practically independently of the magnetic field, and that they can include the coronal part, the transition region, and the photospheric footpoint in a single model with thermal conduction. In this framework, we may even simulate a multi-thread structure only by collecting many single loop models together, still with no need to include the description and interaction with the magnetic field. We should however be aware that the magnetic confinement of the loop material is not as strong and the thermal conduction is not as anisotropic below the coronal part of the loop as it is in the corona.
Klimchuk et al. (2008) illustrate a new efficient model of dynamic coronal loops called “Enthalpy-Based Thermal Evolution of Loops” (EBTEL), which accurately describes the evolution of the average temperature, pressure, and density along a coronal strand with a “0-D”, very fast approach. This model is particularly useful for the description of loops as collections of myriads of strands. In more detailed modeling, it has been recently shown that non-local thermal conduction may lengthen considerably the conduction cooling times and may enhance the chances of observing hot nanoflare-heated plasma (West et al., 2008).
Alternative approaches to single or multiple loop modeling have been developed more recently, thanks also to the increasing availability of high performance computing systems and resources. A global “ab initio” approach was presented by Gudiksen and Nordlund (2005) and by Hansteen et al. (2007) (see also Yokoyama and Shibata, 2001, for the case of a flare model). They model a small part of the solar corona in a computational box using a three-dimensional MHD code that span the entire solar atmosphere from the upper convection zone to the lower corona. These models include non-grey, non-LTE (Local Thermodynamic Equilibrium) radiative transport in the photosphere and chromosphere, optically thin radiative losses, as well as magnetic field-aligned heat conduction in the transition region and corona. Although such models still cannot resolve well fine structures, such as current sheets and the transition region, they certainly represent the first important step toward fully self-consistent modeling of the magnetized corona.
Monolithic (static) loops: scaling laws
The Skylab mission remarked, and later missions confirmed (Figure 9), that many X-ray emitting coronal loops persist mostly unchanged for a time considerably longer than their cooling times by radiation and/or thermal conduction (Rosner et al., 1978, and references therein). This means that, for most of their lives, they can be well described as systems at equilibrium and has been the starting point for several early theoretical studies (Landini and Monsignori Fossi, 1975; Gabriel, 1976; Jordan, 1976; Vesecky et al., 1979; Jordan, 1980). Rosner et al. (1978) devised a model of coronal loops in hydrostatic equilibrium with several realistic simplifying assumptions: symmetry with respect to the apex, constant cross section (see Section 3.2.1), length much shorter than the pressure scale height, heat deposited uniformly along the loop, and low thermal flux at the base of the transition region, i.e., the lower boundary of the model. In these conditions, the pressure is uniform all along the loop, which is then described only by the energy balance between the heat input and the two main losses mentioned above. From the integration of the equation of energy conservation, one obtains the well-known scaling laws
where T0,6, L9, and H-3 are the loop maximum temperature T0, length L and heating rate per unit volume H, measured in units of 106 K (MK), 109 cm, and 10-3 erg cm-3 s-1, respectively. These scaling laws were found in agreement with Skylab data within a factor 2.
Analogous models were developed in the same framework (Landini and Monsignori Fossi, 1975) and equivalent scaling laws were found independently by Craig et al. (1978), and more general ones by Hood and Priest (1979). They have been derived with a more general formalism by Bray et al. (1991). Although scaling laws could explain several observed properties, some features such as the emission measure in UV lines and the cool loops above sunspots could not be reproduced and, although the laws have been questioned a number of times (e.g., Kano and Tsuneta, 1995) in front of the acquisition of new data, such as those by Yohkoh and TRACE, they anyhow provide a basic physical reference frame to interpret any loop feature. For instance, they provide reference equilibrium values even for studies of transient coronal events, they have allowed to constrain that many loop structures observed with TRACE are overdense (e.g., Lenz et al., 1999; Winebarger et al., 2003a, Section 4.1.2) and, as such, these loops must be cooling from hotter status (Winebarger and Warren, 2005) (see Section 3.3.3, and so on). They also are useful for density estimates when closed with the equation of state, and for coronal energy budget when integrated on relevant volumes and times.
Scaling laws have been extended to loops higher than the pressure scale height (Serio et al., 1981) and limited by the finding that very long loops become unstable (Wragg and Priest, 1981). According to Antiochos and Noci (1986), the cool loops belong to a different family and are lowlying, and may eventually explain an evidence of excess of emission measure at low temperature.
The numerical solution of the complete set of hydrostatic equations allowed to obtain detailed profiles of the physical quantities along the loop, including the steep transition region. Figure 13 shows two examples of solution for different values of heating uniformly distributed along the loop.
Reale (1999) and Aschwanden and Nitta (2000) investigated in detail the effect of hydrostatic weighting on the loop visibility and on the vertical temperature structure of the solar corona. From the comparison of SOHO-CDS observations of active region loops with a static, isobaric loop model (Landini and Landi, 2002), Brković et al. (2002) showed that a classical model is not able to reproduce the observations, but ad hoc assumptions might be needed. Further improvements of this approach including flows did not improve the agreement between the model and the observations (Landi and Landini, 2004). Using static models, Landi and Feldman (2004) found that the loop models overestimate the footpoint emission by orders of magnitude and proposed that non-uniformity in the loop cross section, more specifically a significant decrease of the cross section near the footpoints, is the most likely solution to the discrepancy (Section 4.1). On the same line, Winebarger et al. (2008) modelled X-ray loops and EUV moss in an active region core with steady uniform heating and found that a filling factor of 8% and loops that expand with height provide the best agreement with the intensity in two X-ray filters, though maintaining still some discrepancies with observations. Gontikakis et al. (2008) studied the distribution of coronal heating in a solar active region using a simple electrodynamic model and attributed the observed small coronal-loop width expansion to both the preferential heating of coronal loops of small cross-section variation and the cross-section confinement due to the random electric currents flowing along the loops.
The strength of scaling laws is certainly their simplicity and their easy and general application, even in the wider realm of stellar coronae. However, increasing evidence of dynamically heated, fine structured loops is indicating the need for improvements.
Structured (dynamic) loops
In the scenario of loops consisting of bundles of thin strands, each strand behaves as an independent atmosphere and can be described as an isolated loop itself. If the strands are numerous and heated independently, a loop can be globally maintained steady with a sequence of short heat pulses, each igniting a single or a few strands (nanoflares). In this case, although the loop remains steady on average for a long time, each strand has a continuously dynamic evolution. The evolution of a loop structure under the effect of an impulsive heating is well-known and studied from observations and from modeling (e.g., Nagai, 1980; Peres et al., 1982; Cheng et al., 1983; Nagai and Emslie, 1984; Fisher et al., 1985a,b,c; MacNeice, 1986; Betta et al., 2001), since it resembles the evolution of single coronal flaring loops. We have to mention here that there have been attempts to model even flaring loops as consisting of several flaring strands (Hori et al., 1997, 1998; Reeves and Warren, 2002; Warren, 2006).
The evolution of single coronal loops or single loop strands subject to impulsive heating has been recently summarized in the context of the diagnostics of stellar flares (Reale, 2007). A heat pulse injected in an inactive tenuous strand makes chromospheric plasma expand in the coronal section of the strand, and become hot and dense, X-ray bright, coronal plasma. After the end of the heat pulse, the plasma begins to cool slowly. In general, the plasma cooling is governed by the thermal conduction to the cool chromosphere and by radiation from optically thin conditions. In the following we outline the evolution of the confined heated plasma into four phases, according to Reale (2007). Figure 14 tracks this evolution which maps on the path drawn in the densitytemperature diagram of Figure 15 (see also Jakimiec et al., 1992).
Phase I: From the start of the heat pulse to the temperature peak (heating). If the heat pulse is triggered in the coronal part of the loop, the heat is efficiently conducted down to the much cooler and denser chromosphere. The temperature rapidly increases in the whole loop, with a time scale given by the conduction time in a low density plasma (see below). This evolution changes only slightly if the heat pulse is deposited near the loop footpoints: the conduction front then propagates mainly upwards and on timescales not very different from the evaporation time scales, also because the heat conduction saturates (e.g., Klimchuk, 2006; Reale and Orlando, 2008). In this case the distinction from Phase II is not clearly marked.
Phase II: From the temperature peak to the end of the heat pulse (evaporation). The temperature settles to the maximum value (T0). The chromospheric plasma is strongly heated, expands upwards, and fills the loop with much denser plasma. This occurs both if the heating is conducted from the highest parts of the corona and if it released directly near the loop footpoints. The evaporation is explosive at first, with a timescale given by the isothermal sound crossing time (s), since the temperature is approximately uniform in the highly conductive corona:
where m is the average particle mass. After the evaporation front has reached the loop apex, the loop continues to fill more gently. The time scale during this more gradual evaporation is dictated by the time taken by the cooling rate to balance the heat input rate.
Phase III: From the end of the heat pulse to the density peak (conductive cooling). When the heat pulse stops, the plasma immediately starts to cool due to the efficient thermal conduction (e.g., Cargill and Klimchuk, 2004), with a time scale (s):
where nc (nc,10) is the particle density (1010 cm-3) at the end of the heat pulse, the thermal conductivity is κ = 9×10-7 (c.g.s. units). Since the plasma is dense, we expect no saturation effects in this phase.
The heat stop time can be generally traced as the time at which the temperature begins to decrease significantly and monotonically. While the conduction cooling dominates, the plasma evaporation is still going on and the density increasing. The efficiency of radiation cooling increases as well, while the efficiency of conduction cooling decreases with the temperature.
Phase IV: From the density peak afterwards (radiative cooling). As soon as the radiation cooling time becomes equal to the conduction cooling time (Cargill and Klimchuk, 2004), the density reaches its maximum, and the loop depletion starts, slowly at first and then progressively faster. The pressure begins to decrease inside the loop, and is no longer able to sustain the plasma. The radiation becomes the dominant cooling mechanism, with the following time scale (s):
where TM (TM,6) is the temperature at the time of the density maximum (107 K), nM (nM,9) the maximum density (109 cm-3), and P(T) the plasma emissivity per unit emission measure, expressed as:
with b = 1.5 × 10-19 and α = -1/2. The density and the temperature both decrease monotonically.
The presence of significant residual heating could make the decay slower. In single loops, this can be diagnosed from the analysis of the slope of the decay path in the density-temperature diagram (Sylwester et al., 1993; Reale et al., 1997a). The free decay has a slope between 1.5 and 2 in a log density vs log temperature diagram; heated decay path is flatter down to a slope ∼ 0.5. In non-flaring loops, the effect of residual heating can be mimicked by the effect of a strong gravity component, as in long loops perpendicular to the solar surface. The dependence of the decay slope on the pressure scale height has been first studied in Reale et al. (1993) and, more recently, in terms of enthalpy flux by Bradshaw and Cargill (2010).
As clear from Figure 15 the path in this phase is totally below, or at most approaches, the QSS curve. This means that for a given temperature value the plasma density is higher than that expected for an equilibrium loop at that temperature, i.e., the plasma is “overdense”. Evidence of such overdensity (Section 3.3.3) has been taken as an important indication of steadily pulse-heated loops.
This is the evolution of a loop strand ignited by a transient heat pulse. Important properties of the heated plasma can be obtained from the analysis of the evolution after the heating stops, i.e., when the plasma cools down.
This decay time was obtained assuming that the decay starts from equilibrium conditions, i.e., departing from the locus of the equilibrium loops with a given length (hereafter QSS line, Jakimiec et al., 1992) in Figure 15. It is, therefore, valid as long as there is no considerable contribution from the plasma draining to the energy balance. The link between the assumption of equilibrium and the plasma evolution is shown in Figure 15: if the heat pulse lasts long enough, Phase II extends to the right, and the heated loop asymptotically reaches equilibrium conditions, i.e., the horizontal line approaches the QSS line. If the decay starts from equilibrium conditions, Phase III is no longer present, and Phase II links directly to Phase IV. Therefore, there is no delay between the beginning of the temperature decay and the beginning of the density decay: the temperature and the density start to decrease simultaneously. Also, the decay will be dominated by radiative cooling, except at the very beginning (Serio et al., 1991).
The presence of Phase III implies a delay between the temperature peak and the density peak. This delay is often observed both in solar flares (e.g., Sylwester et al., 1993) and in stellar flares (e.g., van den Oord et al., 1988; van den Oord and Mewe, 1989; Favata et al., 2000; Maggio et al., 2000; Stelzer et al., 2002). The presence of this delay, whenever observed, is a signature of a relatively short heat pulse or, in other words, of a decay starting from non-equilibrium conditions.
According to Reale (2007), the time taken by the loop to reach equilibrium conditions under the action of a constant heating is much longer than the sound crossing time (Equation (10)), which rules the very initial plasma evaporation. As already mentioned, in the late rise phase the dynamics become much less important and the interplay between cooling and heating processes becomes dominant. The relevant time scale is, therefore, that reported in Equation (13).
Hydrodynamic simulations confirm that the time required to reach full equilibrium scales as the loop cooling time (τs) and, as shown for instance in Figure 16 (see also Jakimiec et al., 1992), the time to reach flare steady-state equilibrium is
For t ≥ teq, the density asymptotically approaches the equilibrium value
where a = 1.4 × 103 (c.g.s. units), or
If the heat pulse stops before the loop reaches equilibrium conditions, the loop plasma maximum density is lower than the value at equilibrium, i.e., the plasma is underdense (Cargill and Klimchuk, 2004, Section 4.4). Figure 16 shows that, after the initial impulsive evaporation on a time scale given by Equation (10), the later progressive pressure growth can be approximated with a linear trend. Since the temperature is almost constant in this phase, we can approximate that the density increases linearly for most of the time. We can then estimate the value of the maximum density at the loop apex as
where tM is the time at which the density maximum occurs.
Phase III ranges between the time at which the heat pulse ends and the time of the density maximum. The latter is also the time at which the decay path crosses the locus of the equilibrium loops (QSS curve). According to Reale (2007), the temperature TM at which the maximum density occurs is
We can also derive the duration of Phase III, i.e., the time from the end of the heat pulse to the density maximum, as
and τc (Equation (11)) is computed for an appropriate value of the density nc. A good consistency with numerical simulations is obtained for nc = (2/3)nM.
By combining Equations (19) and (17) we obtain
which ranges between 0.2 and 0.8 for typical values of ψ (1.2 – 2).
These scalings are related to the evolution of a single strand under the effect of a local heat pulse. The strands are below the current instrument spatial resolution and, therefore, we have to consider that, if this scenario is valid, we see the envelope of a collection of small scale events. The characteristics of the single heat pulses become, therefore, even more difficult to diagnose, and the question of their frequency, distribution, and size remains open. Also from the point of view of the modeling, a detailed description of a multistrand loop implies a much more complex and demanding effort. A possible approach is to literally build a collection of 1-D loop models, each with an independent evolution (Guarrasi et al., 2010). One common approach so far has been to simulate anyhow the evolution of a single strand and to assume that, in the presence of a multitude of such strands, in the steady state we would see at least one strand at any step of the strand evolution. In other words, a collection of nanoflare-heated strands can be described as a whole with the time-average of the evolution of a single strand (Warren et al., 2002, 2003; Winebarger et al., 2003a,b, see also Section 4.2). Another issue to be explored is whether it is possible, and to what extent, to describe a collection of independently-evolving strands as a single effective evolving loop. For instance, how does the evolution of a single loop where the heating is decreasing slowly compare to the evolution of a collection of independently heated strands, with a decreasing rate of ignition? To what extent do we expect coherence and how is it connected to the degree of global coherence of the loop heating? Is there any kind of transverse coherence or ordered ignition of the strands? It is probably reasonable to describe a multi-stranded loop as a single “effective” loop if we can assume that the plasma loses memory of its previous history. This certainly occurs in late phases of the evolution when the cooling has been going on for a long time.
The description and role of fine structuring of coronal loops is certainly a challenge for coronal physics, also on the side of modeling, essentially because we have few constraints from observations (Section 3.2.2). One of the first times that the internal structuring of coronal loops have been invoked in a modeling context was for the problem of the interpretation of the uniform filter ratio distribution detected with TRACE along warm loops. Lenz et al. (1999) claimed that standard hydrostatic loop models with uniform heating could not explain such indication of uniform temperature distribution. Reale and Peres (2000) showed that a uniform filter ratio could be produced by the superposition of several thin strands described by standard static models of loops at different temperatures. In alternative, Aschwanden (2001) showed that long loops heated at the footpoints result to be mostly isothermal. The problem with this model is that footpoint-heated loops (with heating scale height less than 1/3 of the loop half-length) had been shown to be thermally unstable and, therefore, they cannot be long-lived, as instead observed. A further alternative is to explain observations with steady non-static loops, i.e., with significant flows inside (Winebarger et al., 2001, 2002c, see below). Also this hypothesis does not seem to answer the question (Patsourakos et al., 2004).
A first step to modeling fine-structured loops is to use multistrand static models. Such models show some substantial inconsistencies with observations, e.g., in general they predict too large loop cross sections (Reale and Peres, 2000). Such strands are conceptually different from the thin strands predicted in the nanoflare scenario (Parker, 1988), which imply a highly dynamic evolution due to pulsed-heating. The nanoflare scenario is approached in multi-thread loop models, convolving the independent hydrodynamic evolution of the plasma confined in each pulse-heated strand (see Section 4.3). These are able to match some more features of the evolution of warm loops observed with TRACE (Warren et al., 2002, 2003; Winebarger et al., 2003a,b). By means of detailed hydrodynamic loop modeling, Warren et al. (2002) found that an ensemble of independently heated strands can be significantly brighter than a static uniformly heated loop and would have a flat filter ratio temperature when observed with TRACE. As an extension, time-dependent hydrodynamic modeling of an evolving active region loop observed with TRACE showed that a loop made as a set of small-scale, impulsively heated strands can generally reproduce the spatial and temporal properties of the observed loops, such as a delay between the appearance of the loop in different filters (Warren et al., 2003). As an evolution of this approach, Warren and Winebarger (2006) modelled an entire active region for comparison with a SoHO/EIT observation. They made potential field extrapolations to compute magnetic field lines and populate these field lines with solutions to the hydrostatic loop equations assuming steady, uniform heating. As a result, they constrained the link between the heating rate and the magnetic field and size of the structures. However, they also found significant discrepancies with the observed EIT emission.
More recently modeling a loop system as a collection of thin unresolved strand with pulsed heating has been used to explain why active regions look fuzzier in harder energy bands, i.e., X-rays, and/or hotter spectral lines, e.g., Fe xvi (Tripathi et al., 2009, Section 3.3.2). The basic reason is that in the dynamic evolution of each strand, the plasma spends a relatively longer time and with a high emission measure at temperature about 3 MK (Guarrasi et al., 2010).
Although multistrand models appear much more complex than single loop models and need further refinements to match all the observational constraints, as mentioned in Section 4.1.2, they certainly represent an important issue for the future of coronal loop comprehension.
A generalization of static models of loops (Section 4.1.1) is represented by models of loops with stationary flows, driven by a pressure imbalance between the footpoints (siphon flows). The properties of siphon flows have been studied by several authors (Cargill and Priest, 1980; Priest, 1981; Noci, 1981; Borrini and Noci, 1982; Antiochos, 1984; Thomas, 1988; Montesinos and Thomas, 1989; Noci et al., 1989; Thomas and Montesinos, 1990; Spadaro et al., 1990; Thomas and Montesinos, 1991; Peres et al., 1992; Montesinos and Thomas, 1993). Orlando et al. (1995a) developed a complete detailed model of loop siphon flows and used it to explore the space of the solutions and to derive an extension of RTV scaling laws to loops containing subsonic flows.
Orlando et al. (1995b) explored the conditions for the presence of stationary shocks in critical and supersonic siphon flows in coronal loops (Figure 17), finding that the shock position depends on the volumetric heating rate of the loop, and devising related scaling laws. The presence of massive flows may alter the line emission with respect to static plasma, because of the delay of the moving plasma to settle to ionization equilibrium (Golub et al., 1989). Spadaro et al. (1990) modelled that, even including the effect of ionization non-equilibrium, the UV lines are predicted to be blue-shifted by loop models. So non-equilibrium emission from flows cannot explain the observed dominant redshifts (Section 3.5). The effects of non-equilibrium of ionization in UV line emission from shocked siphon flows are further discussed in Orlando and Peres (1999).
Modeling efforts were devoted in the 1990s to explain specifically the extensive evidence of redshifted UV lines on the solar disk. Hansteen (1993) used a hydrodynamic loop model including the effects of non-equilibrium of ionization to show that the redshifts might be produced by downward propagating acoustic waves, possibly stimulated by nanoflares. By means of two-dimensional hydrodynamic simulations, Reale et al. (1996, 1997b) proposed that the UV redshifts might be due to downdrafts driven by radiatively-cooling condensations in the solar transition region. In the exploration of the parameter space, they found redshifted components at speeds of several km s-1 for ambient pressure values ranging from those typical of quiet Sun to active regions and predicted that redshifts may occur more easily in the higher pressure plasma, typical of active regions.
Teriaca et al. (1999b) explored the idea that the occurrence of nanoflares in a magnetic loop around the Ovi formation temperature could explain the observed redshift of mid-low transition region lines as well as the blueshift observed in low coronal lines (T > 6 × 105 K). Observations were compared to numerical simulations of the response of the solar atmosphere to an energy perturbation of 4 × 1024 erg, including non-equilibrium of ionization. Performing an integration over the entire period of simulations, they found a redshift in Civ, and a blueshift in Ovi and Ne viii, of a few km s-1, in reasonable agreement with observations. A similar idea was applied by Patsourakos and Klimchuk (2006) to make predictions about the presence or absence of nonthermal broadening in several spectral lines (e.g., Ne viii, Mgx, Fe xvii) due to nanoflare-driven chromospheric evaporation. Clearly, the occurrence of such effects in the lines depends considerably on the choice of the heat pulse parameters. Therefore, more constraints are needed to make the whole model more consistent. In other words, modeling should address specific observations to provide more conclusive results.
Theoretical reasons indicate that flows should be invariably present in coronal loop systems, although they may not be necessarily important in the global loop momentum and energy budget. For instance, it has been shown that the presence of, at least, moderate flows is necessary to explain why we actually see the loops (Lenz, 2004). The loop emission and detection is in fact due to the emission from heavy ions, like Fe. In hydrostatic equilibrium conditions, gravitational settlement should keep the emitting elements low on the solar surface, and we should not be able to see but the loop footpoints. Instead, detailed modeling shows that flows of few km s-1 are enough to drag ions high in the corona by Coulomb coupling and to enhance coronal ion abundances by orders of magnitudes. Incidentally, the same modeling shows that, for the same mechanisms, no chemical fractionation of coronal plasma with respect to photospheric composition as a function of the element First Ionization Potential (FIP) should be present in coronal loops.
Other studies point instead to the relative unimportance of flows in coronal loops. In particular, as already mentioned in Section 3.2.2, by means of steady hydrodynamic loop modeling (i.e., assuming equilibrium condition and, therefore, dropping the time-dependent terms in Equations (3), (4), and (5), Patsourakos et al. (2004) showed that flows may not be able to explain the evidence of isothermal loops, as instead proposed by Winebarger et al. (2002c). They found that a heating deposited asymmetrically in a loop is able to drive significant flows in the loop and to enhance its density to the levels typically diagnosed from TRACE observations, but it also produces an inversion of the temperature distribution and, consequently, a highly structured distribution of the relevant filter ratio along the loop, which is not observed.
Plasma cooling is a mechanism that may drive significant downflows in a loop (e.g., Bradshaw and Cargill, 2005, 2010). As an extension of studies on modeling catastrophic cooling in loops (Müller et al., 2004), Müller et al. (2005) tried to explain the evidence of propagating intensity variations observed in the He ii 304 Å line with SoHO/EIT (De Groof et al., 2004, Section 3.5). Two possible driving mechanisms had been proposed: slow magnetoacoustic waves or blobs of cool downfalling plasma. A model of cool downfalling blob triggered in a thermally-unstable loop heated at the footpoints gave a qualitative agreement with measured speeds and predicted a significant braking in the high-pressure transition region, to be checked in future high cadence observations in cool lines.
Plasma waves have been more recently proposed to have an important role in driving flows within loops. Acoustic waves excited by heat pulses at the chromospheric loop footpoints and damped by thermal conduction in corona are possible candidates (Taroyan et al., 2005). Even more attention received the propagation of Alfvén waves in coronal loops. Hydrodynamic loop modeling O’Neill and Li (2005) showed that Alfvén waves deposit significant momentum in the plasma, and that steady state conditions with significant flows and relatively high density can be reached. Analogous results were independently obtained with a different approach: considering a wind-like model to describe a long isothermal loop, Grappin et al. (2003, 2005) showed that the waves can drive pressure variations along the loop which trigger siphon flows. Alfvén disturbances have been recently shown to be amplified by the presence of loop flows (Taroyan, 2009).
As listed above, models predict the development of flows inside coronal loops in a wide variety of situations, namely evaporation, draining, siphon flows, waves. The challenge will be to distinguish clearly among them and to assess the appropriate weight and importance to them both in the spatial and temporal distribution.
The problem of what heats coronal loops is essentially the problem of coronal heating, and is a central issue in the whole solar physics. Although the magnetic origin of coronal heating has been well-established since the very first X-ray observations of the corona, the detailed mechanism of conversion of magnetic energy into thermal energy is still under intense debate, because a series of physical effects conspire to make the mechanism intrinsically elusive.
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 temperatureindependent 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 ωA ≈ 2πvA/L. 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 powerlaw 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 forwardmodeling, 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 steadystate 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.
Modeling including the magnetic field
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.
Stellar Coronal Loops
Non-solar X-ray missions since Einstein and European X-ray Observatory SATellite EXOSAT have established that most other stars have a confined corona, often much more active that the solar one. The level of activity is ruled by several factors but, first of all, the age of the star is important: young fast-rotating stars are more active (e.g., Telleschi et al., 2005). The topic of stellar coronal loops deserves a review by itself (e.g., Rosner et al., 1985) and here only a few relevant issues are discussed. A complete and more recent review of stellar coronae, with an extensive part regarding loops, is by Güdel (2004). In the framework of the solar-stellar connection, it is very important the comparison of what we know about the spatially resolved but single solar corona and what about the unresolved but numerous stellar coronae, which offer a variety of different environments. The lack of spatial resolution inhibits to obtain direct information about the size and appearance of the loops, and the general aspect of the corona. We, therefore, have to rely on indirect evidence. One possible approach to get information is to benefit from transient X-ray events, such as flares, which provide estimations of the loop scale length from their dependence on the decay and rise timescales (Reale, 2002a, 2003, and Section 4.1.2 for reviews). Detailed hydrodynamic modeling can provide even more constraints, for instance, on the heat deposition (e.g., Reale et al., 1988, 2004). The study of stellar X-ray flares allowed, for instance, to constrain that most stellar flares involve plasma confined in closed structures (Reale et al., 2002), and to infer the presence both of loops with size similar to those observed on the Sun (e.g., Reale et al., 1988) and of giant loops (Favata et al., 2005; Getman et al., 2008), with length exceeding the stellar radius.
Another approach is to use the entire solar X-ray corona as a template and “Rosetta stone” to interpret stellar coronae. A detailed implementation of this approach was devised and applied extensively using Yohkoh data over its entire life, which covers a whole solar cycle (Orlando et al., 2000; Peres et al., 2000). It was shown that the solar corona indeed provides a pattern of components, i.e., quiet structures, active regions, active region cores, flares, which can be identified in stellar coronal data and which can explain stellar activity giving different weights to the components (Orlando et al., 2001; Peres et al., 2004). The method was also applied to describe stellar coronae in terms of loop populations and to extract general information and constraints on coronal heating (Peres et al., 2004). It was applied to flares (Reale et al., 2001) and to describe the evolution of active regions (Orlando et al., 2004). More recently it was shown that a continuous unresolved flaring activity may explain the most active coronae, but also that the coronal heating appears to follow different scaling for quiet regions and for active and flaring regions across the cycle (Argiroffi et al., 2008).
Cargill and Klimchuk (2006) realized that the strong hot peaks in the emission measure-temperature distributions in the coronae of some binary stars (Sanz-Forcada et al., 2003) are similar to those expected for an impulsively-heated solar corona. A coronal model comprised of many impulsively heated strands shows that the evidence may be compatible with coronae made of many very small loops (length under 103 km) heated by microflares.
The recent deeper investigation of solar coronal heating mechanisms through the evidence of hot plasma and of variability makes even more important future tighter links to the study of stellar coronae, which show very strong evidence of very hot steady components (e.g., Schmitt et al., 1990; Scelsi et al., 2005).
Conclusions and Perspectives
Coronal loops have been the subject of in-depth studies for about 50 years. Since they owe their identity to the brightness of the confined plasma, most of the studies have addressed the physics of the confined plasma, i.e., its structure, dynamics, and evolution. Most of the basic laws that rule the confined plasma, such as scaling laws, were developed early after the discovery of loops and are now well-established. Although the observational scenario is ever-enriching with the progress of the solar coronal missions, a variety of questions remain still open, at all levels, starting from loop identification itself. The lack of operative definitions and automatic identification tools, and the difficulty to isolate loops from other surrounding and intersecting structures have prevented systematic studies on large and unbiased loop populations. Detailed morphological and theoretical analyses converge to a fine loop substructuring, which is critical to understand the basic mechanisms but is below the resolution limit of present-day instruments. Also the large scale structure of loops leaves room for further developments and, in particular, the link with the confining magnetic field, which is difficult to measure in the corona. One specific question to be addressed is the role of the loop tapering in the transition region.
Looking inside the loops, open questions involve the detailed thermal structure of the confined plasma. This issue is important to assess the basic loop heating mechanism: a broad multi-component thermal distribution would indicate a structured heating, a simpler distribution a monolithic mechanism. This question is still open for many reasons; spectroscopic methods provide very detailed thermal information, but mostly concentrated in the regime of warm loops. Moreover, the methods of analysis do not appear to provide unique answers yet. Also filter ratio diagnostics from broad-band X-ray and UV imaging telescopes have not been able to provide conclusive results so far. As a consequence, we are unable, right now, to assess the problem of the apparently different nature of warm (TRACE, SoHO/EIT) and hot (Yohkoh/SXT, Hinode/XRT) loops, and whether it is simply a matter of different heating rate or the heating mechanisms are radically different.
The role of the dynamics of plasma confined inside loops is also still under investigation. The measurement of plasma motions is made difficult by the possible ambiguity with the apparent motion of thermal fronts. It will be important in the future to evaluate the relative weight of the different flows, i.e., downflows, evaporation and draining, siphon flows, and their influence on the overall loop budget.
Also the investigation of temporal variations deserves attention. In particular, in narrow band instruments thermal variations might be confused with intensity variations, and make the interpretation difficult. On the other hand, the analysis of emission variations is very important, because it can potentially shed light on heating mechanisms based on short impulsive events (nanoflares) or on wave-like phenomena (Alfvén waves).
On the theoretical side, 1-D loop models are well-established and have provided a wealth of sound and important results. Today, their evolution consists essentially in their transformation into “strand” models, and so a collection of them describes a proper loop. Moreover, loops are now seen much more dynamic than they were in the past, either as the site of flows or of wave perturbations, or of heat pulses. So they need more and more time-dependent modeling to address, for instance, the importance of flows and the relative weight between evaporation-draining flows and siphon flows. The modeling is, therefore, becoming more and more demanding, although computing resources are now much more powerful than in the past and although some approaches are able to squeeze the spatial dimension, yet maintaining the temporal description. At the same time, new data seem to require other model refinements, such as the description of loop expansion in the transition region. The improvement of numerical and computing resources is also allowing more complex and complete modeling of whole loop regions including the 3-D magnetic field structure “ab initio”. This approach is very promising and will surely provide important results in the next future, to complement those obtained with basic single loop models. We all look forward self-consistent descriptions including the generation of heat from magnetic field rearrangements and perturbations.
Special attention still deserves the problem of what heats coronal loops, which means basically what heats the whole corona. This problem has revealed to be particularly difficult, essentially because of intrinsic physical reasons and, in particular, i) the highly effective thermal conduction, which inhibits the identification of the heating site, and ii) the expected small scales of the heating processes, which require so far prohibitive spatial/temporal resolution. Nevertheless, coronal loops remain an obvious excellent laboratory to investigate coronal heating mechanisms, because the dense plasma confined therein make them bright and easy to observe. The most recent challenge offered by coronal loops is probably the increasing evidence that the thin strands they are made of are ignited by small scale, rapid, but intense pulses. Most current efforts are devoted to study this aspect both from the theoretical/modeling and from the observational point of view. However, alternative explanations are actively explored and strongly proposed lately.
The study of coronal loops is very alive and is the subject of Coronal Loop Workshops, taking place every two years, which are site of debate, inspiration of new investigations, and school for young investigators. We look forward further great improvements in our knowledge of coronal loops from the new mission Solar Dynamic Observatory.
The author thanks P. Testa, S. Orlando, G. Peres, J. Klimchuk, and the anonymous referees for suggestions. The author acknowledges support from the Italian Ministero dell’Università e della Ricerca. The Transition Region and Coronal Explorer, TRACE, is a mission of the Stanford- Lockheed Institute for Space Research, and part of the NASA Small Explorer program. The solar X-ray images of Figure 2 is from the Yohkoh mission of ISAS, Japan. The X-ray telescope was prepared by the Lockheed-Martin Solar and Astrophysics Laboratory, the National Astronomical Observatory of Japan, and the University of Tokyo with the support of NASA and ISAS.
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