4 Loop Physics and Modeling

4.1 Basics

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 β‰ͺ 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, and neglecting gradients across the direction of the field, effects of curvature, non uniform loop shape, magnetic twisting, currents 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 12*).

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Figure 12: The plasma confined in a loop can be described with one-dimensional hydrodynamic modeling, with a single coordinate (s) along the loop. Image: TRACE, 6 November 1999, 2 UT.

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:

dn ∂v ---= − n ---, (4 ) dt ∂s
dv- ∂p- -∂- ∂v- nmH dt = − ∂s + nmHg + ∂s (μ∂s ), (5 )
d πœ– ∂v (∂v )2 ---+ (p + πœ–)--- = H − n2βiP (T) + μ --- + Fc , (6 ) dt ∂s ∂s
with p and πœ– defined by:
3- p = (1 + βi)nkBT , πœ– = 2p + nβiχ , (7 )
and the conductive flux:
∂ ( ∂T ) Fc = --- κT 5βˆ•2--- , (8 ) ∂s ∂s
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., neβˆ•nH, Fc the conductive flux, κ the thermal conductivity (Spitzer, 1962), kB the Boltzmann constant, and χ the hydrogen ionization potential. H (s,t) is a function of both space and time that 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*,a*,a*; MacNeice, 1986*; Gan et al., 1991; Hansteen, 1993*; Betta et al., 1997*; Antiochos et al., 1999*; Ofman and Wang, 2002; Müller et al., 2003; Bradshaw and Mason, 2003; Sigalotti and Mendoza-Briceño, 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. It has been shown that time-dependent loop models must include a relatively thick, cool, and dense chromosphere and the transition region for a correct description of the mass transfer driven by transient heating (e.g., Bradshaw and Cargill, 2013*) and to maintain the necessary numerical stability (Antiochos, 1979; Hood and Priest, 1980; Peres et al., 1982*). The main role of the chromosphere is only that of a mass reservoir and, therefore, in several codes, it is treated as simply as possible, e.g., an isothermal inactive layer that neither emits, nor conducts heat. In other cases, a more accurate description is chosen, e.g., including a detailed chromospheric model (e.g., Vernazza et al., 1981), maintaining a simplified radiative emission and a detailed energy balance with an ad hoc heat input (Peres et al., 1982*; Reale et al., 2000a*). Overall, a typical loop initial condition is a hydrostatic atmosphere with a temperature distribution from ∼ 104 K to > 106 K, basically dictated by a thermal conduction profile (Figure 14*). The lower boundary of the computational domain is typically not involved in the evolution of the loop plasma. Many loop models assume mirror symmetry with respect to the apex and, therefore, describe only half of the loop. The upper boundary conditions are those of symmetry at the loop apex.

The models also require to define an input heating function (see Section 4.4), 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. An insufficient resolution of the transition region can lead to inaccurate description of the loop plasma dynamics, e.g., chromospheric evaporation (Bradshaw and Cargill, 2013, see Section 4.1.2). 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. Some deviations can be possible because of non-local thermal conduction that may lengthen considerably the conduction cooling times and may enhance the chances of observing hot nanoflare-heated plasma (West et al., 2008).

In recent years, time-dependent loop modeling has been revived in the light of the observations with SoHO, TRACE, and SDO 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 13*) 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*).

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Figure 13: Emission in two TRACE filterbands predicted by a model of loop made by several thin strands. Image reproduced with permission from Reale and Peres (2000*), copyright by AAS.

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 independent 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 (Guarrasi et al., 2010*). 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.

An efficient approach to loop modeling is to describe the temporal evolution of average loop quantities (temperature, pressure, and density), i.e., a “0-D” model (Klimchuk et al., 2008; Cargill et al., 2012a,b). This model is useful for the description of loops as collections of myriads of independent strands with a statistical distribution of heating events.

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. Global “ab initio” approaches have been developed (Gudiksen and Nordlund, 2005*; Hansteen et al., 2007; see also Yokoyama and Shibata, 2001 for the case of a flare model) to model – with full MHD – boxes of the solar corona 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. Large-scale MHD modeling has been used to explain the appearance of constant cross-section in EUV observations as due to temperature variations across the loop (Peter and Bingert, 2012). Another global model of the solar corona includes also information from photospheric magnetic field data (Sokolov et al., 2013).

4.1.1 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, low thermal flux at the base of the transition region, i.e., the lower boundary of the model. Under 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:

1βˆ•3 T0,6 = 1.4 (pL9) (9 )
7βˆ•6 −5βˆ•6 H −3 = 3p L9 , (10 )
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 6 10 K (MK), 9 10 cm and −3 −3 −1 10 erg cm s 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 (1979a). 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*), to different heating functions (Martens, 2010), 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 low-lying, 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 14* shows two examples of solution for different values of heating uniformly distributed along the loop.

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Figure 14: Distributions of temperature, density, and pressure along a hydrostatic loop computed from the model of Serio et al. (1981) for a high pressure loop (AR) and a low pressure one (Empty) with heating uniformly distributed along the loop.

Hydrostatic weighting has an effect on the loop visibility and on the vertical temperature structure of the solar corona (Reale, 1999; Aschwanden and Nitta, 2000). From the comparison of SoHO-CDS observations of active region loops with a static, isobaric loop model (Landini and Landi, 2002), a classical model was not able to reproduce the observations, but ad hoc assumptions are necessary (BrkoviΔ‡ et al., 2002; Landi and Landini, 2004). Loop static models were found to overestimate the footpoint emission by orders of magnitude and non-uniformity in the loop cross section, more specifically a significant decrease of the cross section near the footpoints, was proposed as the most likely solution to the discrepancy (Landi and Feldman, 2004, Section 4.1). On the same line, loop models with steady uniform heating were compared to X-ray loops and EUV moss in an active region core (Winebarger et al., 2008). A filling factor of 8% and loops that expand with height provided the best agreement with the intensity in two X-ray filters, though maintaining still some discrepancies with observations. A simple electrodynamic model was useful to evaluate the connection of electric currents and heating to the loop cross-section in a solar active region (Gontikakis et al., 2008).

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.

4.1.2 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. It is worth mentioning 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; Reale et al., 2012*).

The evolution of single coronal loops or single loop strands subject to impulsive heating was 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 15* tracks this evolution, which maps on the path drawn in the density-temperature diagram of Figure 16* (see also Jakimiec et al., 1992*).

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Figure 15: Scheme of the evolution of temperature (T, thick solid line), X-ray emission, i.e., the light curve (LC, thinner solid line) and density (n, dashed line) in a loop strand ignited by a heat pulse. The strand evolution is divided into four phases (I, II, III, IV, see text for further details). Image reproduced with permission from Reale (2007*), copyright by ESO.
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 timescale 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 timescales, 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:
----L------ --L9-- τsd = ∘ ---------≈ 80∘ ----, (11) 2kBT0 βˆ•m T0,6
where m is the average particle mass. After the evaporation front has reached the loop apex, the loop continues to fill more gently. The timescale 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 timescale (s):
2 2 2 τc = 3nckBT0L--- = 10.5nckBL---≈ 1500n9L-9 , (12) 2βˆ•7 κT07βˆ•2 κT 50βˆ•2 T56βˆ•2
where nc (nc,9) is the particle density (9 −3 10 cm) 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 timescale (s):
3kBTM 3kBTM T3Mβˆ•,26 τr = ---------= ---α---- ≈ 3000 -----, (13) nM P (T ) bT M nM nM,9
where TM (TM,6) is the temperature at the time of the density maximum (106 K), nM (nM,9) the maximum density (109 cm− 3), and P (T ) the plasma emissivity per unit emission measure, expressed as:
α P (T ) = bT ,

with b = 1.5 × 10 −19 and α = − 1βˆ•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 16* 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.

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Figure 16: Scheme of the evolution of pulse-heated loop plasma of Figure 15* in a density-temperature diagram (solid line). The four phases are labeled. The locus of the equilibrium loops is shown (dashed-dotted line, marked with QSS), as well as the evolution path with an extremely long heat pulse (dashed line) and the corresponding decay path (marked with EQ). Image adapted from Reale (2007*), copyright by ESO.

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.

Serio et al. (1991*) derived a global thermodynamic timescale for the pure cooling of heated plasma confined in single coronal loops, which has been later refined to be (s) (Reale, 2007*):

− 4-L--- -L9--- τs = 4.8 × 10 √T0--= 500 ∘T----. (14 ) 0,6

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 16*. 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 16*: 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 [Eq. (11*)], 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 timescale is therefore that reported in Eq. (14*).

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Figure 17: Pressure evolution obtained from a hydrodynamic simulation of a loop strand ignited by heat pulses of different duration (0.5, 1, 3 times the loop decay time, see text) and with a continuous heating. Most of the rise phase can be reasonably described with a linear trend (dashed lines). Image reproduced with permission from Reale (2007*), copyright by ESO.

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 17* (see also Jakimiec et al., 1992), the time to reach flare steady-state equilibrium is:

teq ≈ 2.3τs. (15 )

For t ≥ teq, the density asymptotically approaches the equilibrium value:

2 2 n = --T0----= 1.3 × 106 T0-, (16 ) 0 2a3kBL L
where a = 1.4 × 103 (c.g.s. units), or
T20,6 n9 = 1.3 ----. (17 ) L9

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 17* shows that, after the initial impulsive evaporation on a timescale given by Eq. (11*), 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:

tM nM ≈ n0t--, (18 ) eq
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:

TM = 9 × 10 −4(nM L)1βˆ•2 (19 )
1βˆ•2 TM,6 = 0.9(nM,9L9) .

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

Δt0−M ≈ τcln ψ, (20 )
-T0- ψ = TM

and τc [Eq. (12*)] is computed for an appropriate value of the density nc. A good consistency with numerical simulations is obtained for n = (2βˆ•3 )n c M.

By combining Eqs. (20*) and (18*) we obtain:

Δt0−-M-≈ 1.2lnψ , (21 ) tM
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., 2003b*,a*, 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.

4.2 Fine structuring

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). Small-scale structuring is already involved in the magnetic carpet scenario and flux-tube tectonics model (Priest et al., 2002*, see also Section 4.4). 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. Standard models of single hydrostatic loops with uniform heating were soon found to be unable to explain such indication of uniform temperature distribution (Lenz et al., 1999). A uniform filter ratio could be reproduced by the superposition of several thin hydrostatic strands at different temperatures (Reale and Peres, 2000*). In alternative, also a model of long loops heated at the footpoints leads to mostly isothermal loops (Aschwanden, 2001*). 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 (Mendoza-Briceno and Hood, 1997) 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., 2003b*,a). According to detailed hydrodynamic loop modeling, 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 (Warren et al., 2002*). 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*). An evolution of this approach was to model an entire active region for comparison with a SoHO/EIT observation (Warren and Winebarger, 2006); the modeling includes extrapolating the magnetic field and populating the field lines with solutions to the hydrostatic loop equations assuming steady, uniform heating. The result was the link between the heating rate and the magnetic field and size of the structures, but there were also 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, sensitive to high temperatures (∼ 3 MK) (Tripathi et al., 2009, Section 3.3.2). Short (∼ 1 min) pulses with flare-like intensity (∼ 10 MK) are able to produce loops with high filling factors at ∼ 3 MK and lower filling factors at ∼ 1 MK (Guarrasi et al., 2010). 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. The consequent prediction that loops should show filamented emission for temperature > 3 MK has received confirmations by observations of active region cores in the 94 Å channel with SDO/AIA (Reale et al., 2011) and in the Ca xvii and Fe xviii EUV lines (Testa and Reale, 2012), although the temperature of the emitting plasma is still debated (Teriaca et al., 2012). Low filling factors of warm loops have been predicted also by full MHD modeling (Dahlburg et al., 2012).

The description of loops as bundles of strands applies also to models that include heating by the dissipation of MHD waves (Alfvén/ion-cyclotron waves – particles). One such model addressed the evidence of flat TRACE/EIT filter-ratios along loops that were explained by the multi-filament loop structure (Bourouaine and Marsch, 2010*). Transverse oscillations and flows were observed in multi-stranded loops (Ofman and Wang, 2008*; Wang et al., 2012). Multi-stranded loop models were used in 3D MHD studies of transverse loop oscillations (e.g., Ofman, 2009*) and in MHD normal mode analysis (e.g., Luna et al., 2010).

State-of-the-art approaches to the study of multi-stranded loops are based on the concept that each fibril is independent of the others and that the heating is released randomly presumably with a power-law distribution. Within the limitations of idealized loop models (without magnetic twist, time-dependent thread cross-sections, or oscillation), the coronal loops might then be described as self-organized critical systems with no characteristic timescales (e.g., Bak et al., 1989; Lu and Hamilton, 1991; Charbonneau et al., 2001). This model has had a practical realization specific to reproduce soft X-ray steady-state loops (López Fuentes and Klimchuk, 2010*) and is able to reproduced the loop light curves observed, for instance, with GOES/SXI. Another interesting approach is to use artificial neural networks (Tajfirouze and Safari, 2012*) as mentioned in Section 3.4.

4.3 Flows

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). A complete detailed model of loop siphon flows was developed and used 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).

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Figure 18: Example of solutions of a siphon flow loop model including a shock. Image reproduced with permission from Orlando and Peres (1999*), copyright by Elsevier.

Critical and supersonic flows create the conditions for the presence of stationary shocks in coronal loops (Figure 18*). The shock position depends on the volumetric heating rate of the loop (Orlando et al., 1995a). 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 and Herant, 1989). Including the effect of ionization non-equilibrium, the UV lines are predicted to be blue-shifted by loop models (Spadaro et al., 1990). So non-equilibrium emission from flows cannot explain the observed dominant redshifts (Section 3.5). Non-equilibrium of ionization in UV line emission can be driven by shocked siphon flows (Orlando and Peres, 1999) and by reconnection flows (Imada et al., 2011).

In the 1990s, modeling efforts were devoted to explain specifically the extensive evidence of red-shifted UV lines on the solar disk. A hydrodynamic loop model including the effects of non-equilibrium of ionization showed that the redshifts might be produced by downward propagating acoustic waves, possibly stimulated by nanoflares (Hansteen, 1993). Two-dimensional hydrodynamic simulations showed that the UV redshifts might be produced by downdrafts driven by radiatively-cooling condensations in the solar transition region (Reale et al., 1996, 1997b). Predicted redshifts range from those typical of quiet Sun to active regions and may occur more easily in the higher pressure plasma, typical of active regions.

Explosions below the corona were explored to drive flows (Teriaca et al., 1999*; Sarro et al., 1999) in magnetic loops around the O vi and C iv formation temperature. The observed redshift of mid-low transition region lines as well as the blueshift observed in low coronal lines (T > 6 × 105 K) were compared to numerical simulations of the response of the solar atmosphere to an energy perturbation of 24 4 × 10 erg, including non-equilibrium of ionization (Teriaca et al., 1999). Performing an integration over the entire period of simulations, they found a redshift in C iv, and a blueshift in O vi and Ne viii, of a few km s–1, in reasonable agreement with observations. A similar idea was applied to make predictions about the presence or absence of non-thermal broadening in several spectral lines (e.g., Ne viii, Mg x, Fe xvii) due to nanoflare-driven chromospheric evaporation (Patsourakos and Klimchuk, 2006*). 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 address instead the relative unimportance of flows in coronal loops. In particular, as already mentioned in Section 3.2.2, steady hydrodynamic loop modeling (i.e., assuming equilibrium condition and, therefore, dropping the time-dependent terms in Eqs. (4*), (5*), and (6*)), showed that flows may not be able to explain the evidence of isothermal loops (Patsourakos et al., 2004), as instead proposed by Winebarger et al. (2002c). Flows are able to enhance its density to the levels typically diagnosed from TRACE observations, but they also produce an inversion of the temperature distribution and a structured filter ratio, not observed.

Plasma cooling is a mechanism that may drive significant downflows in a loop (e.g., Bradshaw and Cargill, 2005, 2010). Catastrophic cooling in loops (Müller et al., 2004*, 2005) was proposed 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 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 (O’Neill and Li, 2005*). Analogous results were obtained independently 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 more recently shown to be amplified by the presence of loop flows (Taroyan, 2009).

Large-scale MHD models have also addressed the presence of flows in the low corona. These models show that heat pulses released low in the corona in places of strong magnetic field braiding trigger downflows and slight upflows (Figure 19*, Hansteen et al., 2010*; Zacharias et al., 2011). The corresponding Doppler-shifts are similar to those often observed (see Section 3.5). Most of the mass circulating across the transition region is probably confined in very short loops (∼ 2 × 108 cm) (Guerreiro et al., 2013).

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Figure 19: Maps of intensity (left), Doppler shift (middle) and line width (right) in the C iv line in a 3-D MHD simulation of a box of the upper solar atmosphere. The velocity scale is from –40 km s–1 (blue) to 40 km s–1 (red). Line widths range from narrow black to wide yellow/red with a maximum of 51 km s–1. The average line-shift in the C iv line is 6.6 km s–1 (redshift is positive). Image reproduced with permission from Hansteen et al. (2010), copyright by AAS.

Large-scale chromospheric upflows (type-II spicules, see Section 3.5) are explored as viable mechanisms of mass and energy supply to coronal loops with loop modeling (Judge et al., 2012). However, theoretical estimates suggest that a corona dominated by this scenario would lead to large discrepancies with observations, therefore confining the possible action of this mechanism in a limited number of structures (Klimchuk, 2012).

4.4 Heating

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, the magnetic field plays an active role in heating the coronal loops (Golub et al., 1980). The observation of a magnetic carpet (Schrijver et al., 1998) suggests that current sheets at the boundary of the carpet cells can dissipate and heat the corona, acting in analogy to geophysical plate tectonics (Priest et al., 2002*). A common scenario is that the field lines are wound continuously by the photospheric convective motions and the generated non-potential component is dissipated into heating. Several 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).

It is known that twisted loops can become kink unstable above a critical twist; however, according to Parker (1988*), as soon as the magnetic field lines are tangled at an angle of ∼ 15∘, enough magnetic energy can be released to power a loop by rapid reconnection across the spontaneous tangential discontinuities. The dissipation destroys the cross-component of the magnetic field as rapidly as it is produced by the motion of the footpoints, reaching a steady state. The twisting of coronal loops has been studied in the framework of kink instability (Hood and Priest, 1979b; Velli et al., 1990) with resistive effects (Baty, 2000), of loop cross-section (Klimchuk, 2000), of flux emergence (Hood et al., 2009a), of cellular automaton loop modeling (López Fuentes and Klimchuk, 2010*), and in connection with transverse oscillations (Ofman, 2009*). Loop twisting or braiding and kink instability can lead to magnetic reconnection with the formation and fragmentation of thin current sheets and their dissipation through resistivity (Hood et al., 2009b; Wilmot-Smith et al., 2010; Bareford et al., 2010*; Pontin et al., 2011; Wilmot-Smith et al., 2011; Bareford et al., 2011, 2013).

The comparison of TRACE and Hinode time sequences of active region loops with magnetic field reconstruction models have allowed to measure the changes in the magnetic topology and energy with the time. A high variability corresponds to a high number of magnetic separation lines (Priest et al., 2002*) where the energy can be released in short timescales (Lee et al., 2010). A model of nonlinear force-free field traced that magnetic energy is built up in the core of active regions by small-scale photospheric motions (Mackay et al., 2011). 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, although there are some indications of anomalously high dissipation coefficients (Martens et al., 1985; Nakariakov et al., 1999*; Fuentes-Fernández et al., 2012). 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 (see Section 4.4.2).

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 on a TRACE observation of a brightening coronal loop (Reale et al., 2000a,b, 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 difficult for 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) split 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*).

4.4.1 DC heating

Heating by nanoflares has a long history 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*; Mendoza-Briceño et al., 2002; 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*; Taroyan et al., 2006; Vekstein, 2009). The coronal tectonics model (Priest et al., 2002) is an updated version of Parker’s nanoflare theory, for which the motions of photospheric footpoints continually build up current sheets along the separatrix boundaries of the flux coming from each microscopic source (Priest, 2011).

Models of loops made of thousands of nanoflare-heated strands were developed and applied to derive detailed predictions (Cargill, 1994*). In particular, whereas the loop total emission measure distribution should steepen above the canonical T 1.5 (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). The nanoflare model was early applied to the heating of coronal loops observed by Yohkoh (Cargill and Klimchuk, 1997). 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). An application of both static and impulsive models to solar active regions 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 (Patsourakos and Klimchuk, 2008). As a further improvement, the simulation of an entire active region with an impulsive heating model reproduced the total observed soft X-ray emission in all of the Yohkoh/SXT filters (Warren and Winebarger, 2007). However, once again, at EUV wavelengths the agreement between the simulation and the observation was only partial.

Nanoflares have been studied also in the framework of stellar coronae. Intermittent heating by relatively intense nanoflares deposited at the loop footpoints makes the loop stable on long timescales (Testa et al., 2005*; Mendoza-Briceño et al., 2005) (loops infrequently 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 (Cargill, 1994; Testa et al., 2005). Therefore, this is an alternative way to obtain a steep temperature dependence of the loop emission measure distribution in the low temperature range.

An alternative approach to study nanoflare heating is to analyze intensity fluctuations (Shimizu and Tsuneta, 1997; Vekstein and Katsukawa, 2000; Katsukawa and Tsuneta, 2001; Vekstein and Jain, 2003) and to derive their occurrence distribution (Sakamoto et al., 2008, 2009). From the width of the distributions and autocorrelation functions, it has been suggested that nanoflare signatures are more easily found in observations of warm TRACE loops than of hot Yohkoh/SXT loops. It is to be investigated whether the results change after relaxing the assumption of temperature-independent distribution widths. Also other variability analysis of TRACE observations was found able to put constraints on loop heating. In particular, 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 (Antiochos et al., 2003; Warren et al., 2010b*).

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; Bareford et al., 2010; Tajfirouze and Safari, 2012). 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. A hydrodynamic model was used to simulate the UV emission of a loop system heated by nanoflares on small, spatially unresolved scales (Parenti and Young, 2008). The simulations confirmed 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).

The shape of the emission measure distribution is, in principle, a powerful tool to constrain the heating mechanisms. The width in temperature provides information about the temporal distribution of a discontinuous heating mechanism: for a broad (multi-thermal) distribution the simultaneous presence of many temperature components along the line of sight may be produced by many strands randomly heated for a short time and then spending most of the time in the cooling, thus “crossing” many different temperatures. A peaked distribution, i.e., plasma closer to an isothermal condition, indicates a plasma sustained longer at a certain temperature, with a heating much more uniform in time than for multi-thermal loops. A semi-analytical loop model of a cycling heating/cooling (Cargill and Klimchuk, 2004) naturally led to hot-underdense/warm-overdense loop (Section 4.1.2), as observed (Winebarger et al., 2003b, Section 3.3.3), and showed 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). In general, a broad emission measure distribution would be a signature of a low-frequency heating, whereas a peaked distribution would be a signature of high-frequency heating (Warren et al., 2010b; Susino et al., 2010). The timescale is basically dictated by the cooling times. High frequency heating seems to explain several debated evidence in warm loops of active regions, i.e., loop lifetime, high density, and the narrow differential emission measure, but not the higher temperature loops detected in the X-rays (Warren et al., 2010a). It is remarked that overdense plasma would be emphasized also by deviations from equilibrium of ionization due to impulsive heating (Bradshaw and Klimchuk, 2011), and that the predicted cool side of the emission measure distribution might steepen using updated radiative losses (Reale et al., 2012). However, the constraints on heating from emission measure distribution are largely debated; broad and peaked emission measure distributions of hot 3 MK loops might be compatible with steady heating models (Winebarger et al., 2011). This debate has been specifically addressed and all evidence has been collected and analysed through loop modeling. In particular, the consistency of the DEM slopes on the cool side with low frequency nanoflare heating has been tested. It has been found that, although heating by single pulses might explain the majority of DEMs derived in the literature (Bradshaw et al., 2012) and that trains of nanoflares might explain practically all of them (Reep et al., 2013), the uncertainties in the data analysis and DEM reconstruction are too large reach conclusive answers. Radiative losses are important to the existence of small and cool loops (height ≤ 8 Mm, 5 T ≤ 10 K) that determine the cool side of the emission measure distribution (Sasso et al., 2012).

Support to dynamic heating comes from modeling loops with steady heating located at the footpoints. It is known that such heating is not able to keep loop atmosphere in steady equilibrium because they are thermally unstable (Antiochos and Klimchuk, 1991; Antiochos et al., 1999; Müller et al., 2004; Karpen and Antiochos, 2008; Mok et al., 2008). Catastrophic cooling occurs along the loops some time after the heating is switched on and might explain deviations from hydrostatic equilibrium, and some features of the light curves measured in the EUV band (Peter et al., 2012). However, the timescales required by this scenario seem too long compared to the measured loop lifetimes (Klimchuk et al., 2010).

4.4.2 AC heating

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 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), as reported on in Section 3.5.2.

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 βˆ•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.

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 some evidence 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 non-thermal broadening of soft X-ray lines. The observed non-thermal 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 non-thermal line broadening could be produced by unresolved loop flows that are unrelated to waves (e.g., Patsourakos and Klimchuk, 2006). Ofman et al. (1998) included inhomogeneous density structure 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). A multi-strand loop model where the heating is due to the dissipation of MHD waves was applied to explain filter-ratios along loops (Bourouaine and Marsch, 2010, see Sections 3.3.3, 4.2). 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., 2007b). 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 structure has been recognized to be important in possible wave dissipation and loop twisting (Ofman, 2009). Long-period (> 10 s) chromospheric kink waves might propagate into the corona by transformation into Alfvén waves and be dissipated there (Soler et al., 2012).

In the more general context of coronal heating, after several previous works, follow-up modeling and analytical effort has been devoted to the dissipation of Alfven waves through phase mixing (e.g., Heyvaerts and Priest, 1983; Nakariakov et al., 1997; Botha et al., 2000; Ofman and Aschwanden, 2002) and ponderomotive force (Verwichte et al., 1999) in a nonideal inhomogeneous medium, finding effects on very long timescales (> 1 month, McLaughlin et al., 2011).

Intensity disturbances propagating along active region loops at speeds above 100 km s–1 were detected with TRACE and interpreted as slow magnetosonic 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 a detailed model, including the effect of chromosphere and transition region and of the radiative losses in the corona (Beliën et al., 1999). 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). Slow magnetosonic waves might be coupled to upflows and produced by impulsive events at the base of active region loops (Ofman et al., 2012).

Investigation of AC heating has been made also through comparison with DC heating. Antolin et al. (2008) compared observational signatures of coronal heating by Alfvén waves and nanoflares using two coronal loop models and found that Hinode XRT intensity histograms display power-law distributions whose indices differ considerably, to be checked against observations. Lundquist et al. (2008a,b) applied a method for predicting active region coronal emissions using magnetic field measurements and a chosen heating relationship to 10 active regions. With their forward-modeling, they found a volumetric coronal heating rate proportional to magnetic field and inversely proportional to field-line loop length, which seems to point to, although not conclusively, the steady-state scaling of two heating mechanisms: van Ballegooijen’s current layers theory (van Ballegooijen, 1986), taken in the AC limit, and Parker’s critical angle mechanism (Parker, 1988), in the case where the angle of misalignment is a twist angle. As interesting points of contacts with the models of impulsive heating, it has been proposed that loops can be heated impulsively by Alfvén waves dissipated on reasonable timescales through turbulent cascade that develops when the waves are transmitted from the photosphere to the corona (van Ballegooijen et al., 2011; Asgari-Targhi et al., 2013), using reduced MHD equations.

4.5 Large-scale modeling

Coronal loops have been studied also with models that include the magnetic field. We can distinguish several levels of treatment of the magnetic effects. One basic level is to use global scalings to discriminate between different heating mechanisms. 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. A similar approach was applied to the whole corona, by populating magnetic field lines taken from observed magnetograms with quasi-static loop atmospheres (Schrijver et al., 2004). The best match to X-ray and EUV observation was obtained with a heating that scales as expected from DC reconnection at tangential discontinuities.

Large-scale modeling has been able to explain the ignition of warm loops from primary energy release mechanisms. 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; Gudiksen et al., 2011). Observed solar granular velocity pattern, a potential extrapolation of a SoHO/MDI magnetogram, and a standard stratified atmosphere are used as initial conditions. The first simulations 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 Eq. (6*)] 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, reduced MHD (rMHD) was used 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 (Rappazzo et al., 2007, 2008). 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). Shell models of rMHD turbulence were used to analyze the case of a coronal loop heated by photospheric turbulence and found that the Alfvén waves interact nonlinearly and form turbulent spectra (Buchlin and Velli, 2007). An intermittent heating function is active, on average able to sustain the corona and proportional to the aspect ratio of the loop to the ∼ 1.5 power. Adding a profile of density and/or magnetic field along the loop somewhat change the heat deposition, in particular in the low part of the loop (Buchlin et al., 2007). These models also predict the formation of current sheets that can be dissipated on these small scales and impulsively through turbulent cascades (Rappazzo et al., 2010; Rappazzo and Velli, 2011). Transient current sheets are also found from large-scale full MHD modeling (Bingert and Peter, 2011). In the same framework loops have been described as partially resonant cavities for low-frequency fluctuations transmitted from the chromosphere (Verdini et al., 2012).

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 (e.g., López Fuentes and Klimchuk, 2010). 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 hr) during which magnetic stresses build up and the plasma simultaneously cools down and precipitates. An avalanche model was proposed for solar flares (Morales and Charbonneau, 2008), 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.

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