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:et al., 1976), the fractional ionization, i.e., , the conductive flux, the thermal conductivity (Spitzer Jr, 1962), the Boltzmann constant, and the hydrogen ionization potential. 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.
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 laws6 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 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 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.
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 density-temperature diagram of Figure 15 (see also Jakimiec et al., 1992).
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.
with and . 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 density vs 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.
Serio et al. (1991) derived a global thermodynamic time scale for the pure cooling of heated plasma confined in single coronal loops, which has been later (Reale, 2007) refined to be ():
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 () and, as shown for instance in Figure 16 (see also Jakimiec et al., 1992), the time to reach flare steady-state equilibrium is
For , the density asymptotically approaches the equilibrium value
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
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 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 (Equation (11)) is computed for an appropriate value of the density . A good consistency with numerical simulations is obtained for .
By combining Equations (19) and (17) we obtain
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.
Living Rev. Solar Phys. 7, (2010), 5
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