THERMAL NON-EQUILIBRIUM REVISITED: A HEATING MODEL FOR CORONAL LOOPS

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1 C The American Astronomical Society. All rights reserved. Printed in the U.S.A. doi: / x/773/2/134 THERMAL NON-EQUILIBRIUM REVISITED: A HEATING MODEL FOR CORONAL LOOPS Roberto Lionello 1, Amy R. Winebarger 2, Yung Mok 3, Jon A. Linker 1, and Zoran Mikić 1 1 Predictive Science, Inc., 9990 Mesa Rim Rd., Ste. 170, San Diego, CA , USA; lionel@predsci.com, linkerj@predsci.com, mikicz@predsci.com 2 NASA Marshall Space Flight Center, ZP 13, Huntsville, AL 35812, USA; amy.r.winebarger@nasa.gov 3 Department of Physics and Astronomy, University of California, 4129 Reines Hall, Irvine, CA 92697, USA; ymok@uci.edu Received 2013 May 31; accepted 2013 June 25; published 2013 August 2 ABSTRACT The location and frequency of events that heat the million-degree corona are still a matter of debate. One potential heating scenario is that the energy release is effectively steady and highly localized at the footpoints of coronal structures. Such an energy deposition drives thermal non-equilibrium solutions in the hydrodynamic equations in longer loops. This heating scenario was considered and discarded by Klimchuk et al. on the basis of their onedimensional simulations as incapable of reproducing observational characteristics of loops. In this paper, we use three-dimensional simulations to generate synthetic emission images, from which we select and analyze six loops. The main differences between our model and that of Klimchuk et al. concern (1) dimensionality, (2) resolution, (3) geometrical properties of the loops, (4) heating function, and (5) radiative function. We find evidence, in this small set of simulated loops, that the evolution of the light curves, the variation of temperature along the loops, the density profile, and the absence of small-scale structures are compatible with the characteristics of observed loops. We conclude that quasi-steady footpoint heating that drives thermal non-equilibrium solutions cannot yet be ruled out as a viable heating scenario for EUV loops. Key words: Sun: corona Sun: UV radiation Online-only material: color figures 1. INTRODUCTION Early X-ray observations of the Sun indicated that hot coronal plasma was collimated to closed magnetic field lines creating coronal loops. The heating mechanism that creates the million-degree plasma is still unknown. Observational properties of coronal loops from X-ray or EUV images and spectra can be compared to solutions of hydrodynamic simulations to determine information about the coronal heating mechanism. For instance, Rosner et al. (1978, hereafter RTV) found general agreement between the X-ray observations of coronal loops from Skylab with solutions to the hydrodynamic equations for steady, uniform heating. However, EUV observations of coronal loops from Solar and Heliospheric Observatory (SOHO)/EIT and Transition Region and Coronal Explorer (TRACE) indicate that a more complex heating scenario is required (Lenz et al. 1999). Klimchuk et al. (2010) summarized five observational characteristics of EUV loops that must be re-created with hydrodynamic models for a successful heating scenario. (1) The observed density of loops can be much higher than the value predicted by steady, uniform heating (Aschwanden et al. 1999, 2001; Winebarger et al. 2003a). Winebarger et al. (2003a) showed that the ratio of the observed apex density to the apex density predicted by steady, uniform heating depended on loop length and ranged from 1 to (2) The density of warm loops is found to decrease with height much more slowly than expected for a gravitationally stratified plasma at the measured temperature (Aschwanden et al. 2001). (3) The ratio of two EUV channel intensities is flat along the loop, which could imply that the temperature is uniform along the loop; such uniform temperature is not predicted for steady, uniform heating (Lenz et al. 1999; Aschwanden et al. 1999, 2001; Reale & Peres 2000). (4) The loops do not appear to have small-scale intensity structures, but rather the brightness transitions smoothly (Kano & Tsuneta 1996; Klimchuk 2000; López Fuentes et al. 2008). (5) The lifetimes of these loops in a single EUV channel are typically s (Winebarger et al. 2003b; Winebarger & Warren 2005; Ugarte-Urra et al. 2009). To this list of features, we add the following: (6) EUV loops are cooling; they first appear in channels sensitive to higher temperatures and later appear in channels sensitive to lower temperatures (Winebarger et al. 2003b; Mulu-Moore et al. 2011; Viall & Klimchuk 2012). (7) For some EUV loops, the measured delay from a hotter channel to a cooler channel can be consistent with the measured lifetime of the loop. These observations have been interpreted as a single cooling loop. Other EUV loops are found to have measured lifetimes much longer than expected. These loops have been modeled as many sub-resolution strands, each impulsively heated and evolving independently (e.g., Warren et al. 2003). We summarize this list in Table 1, where we refer to the steady, uniform heating case as the RTV case. These seven observational characteristics set constraints to three basic observations, the intensities along the loops, the densities along the loops, and the evolution of the loops. We note that the above seven characteristics have been derived from many different studies, which have selected loops based on very different criteria. To date, there has not been a comprehensive study of coronal loops where the above characteristics have been measured for a single set of loops. One possible explanation of these loop characteristics is that each loop consists of bundles of unresolved strands that are impulsively heated during a finite interval of time and then evolve independently from its neighbors (see Klimchuk 2009 and references therein). Such a heating scenario is called a nanoflare storm. Another possible explanation is that the heating of the loops is effectively steady and localized at the loop footpoints. Such a heating scenario may not have a steady 1

2 Table 1 EUV Loop Characteristics Observable Characteristic # Density larger than RTV (1) Scale height longer than RTV (2) Intensity Constant intensity ratio along loop (3) Intensity varies smoothly along loop (4) Evolution Lifetimes in EUV channels of (5) Appears in hotter channels before cooler channels (6) Observed lifetimes sometimes longer than expected (7) solution; instead, the heating drives thermal non-equilibrium solutions to the hydrodynamic equations (Kuin & Martens 1982; Martens & Kuin 1983; Antiochos & Klimchuk 1991; Antiochos et al. 1999; Karpen et al. 2001, 2003; Müller et al. 2003, 2004; Karpen et al. 2006). Depending on the heating parameters and geometry of the loop, dense, cold condensations can be periodically formed in the corona and slide down the field lines into the photosphere. Mok et al. (2008) presented threedimensional (3D) plasma simulations showing that thermal nonequilibrium can lead to the formation of realistic looking loops. However, Klimchuk et al. (2010) rejected the possibility that non-equilibrium solutions may explain coronal observations on the basis of one-dimensional (1D) plasma simulations, stating that it fails to reproduce the observed properties we have mentioned above. In this article we revisit the viability of thermal nonequilibrium solutions to explain the observational properties of EUV loops. Unlike previous works with 1D models such as that of Klimchuk et al. (2010), our study is fully 3D, takes into account of the geometry of the coronal magnetic field, and uses a more realistic heating mechanism. We first use the model of Mok et al. (2005) to solve for the density and temperature distribution for the coronal plasma of Active Region 7986, which was observed in 1996 August. We prescribe a heating function inspired by the work of Rappazzo et al. (2007, 2008), who determined the rate of dissipation of magnetic energy in the Parker field-line tangling problem (Parker 1972, 1988). From the thermodynamic properties of the plasma, we then construct synthetic EUV and X-ray images in several instrument channels and spectral lines. These images are then analyzed as observations: we select six loops and determine the characteristics of their intensity, density, and temporal evolution. As in real loop studies, we find it difficult to measure all the necessary properties of each loop mainly due to line-of-sight confusion. We find examples of all the observational characteristics listed in Table 1 in these modeled loops. This analysis indicates that effectively steady footpoint heating cannot be ruled out as a mechanism through which warm coronal loops are heated. Additional analysis is necessary to differentiate the non-thermal equilibrium solutions from the nanoflare storm scenario. This paper is the second in a series of three papers. In Mikić et al. (2013, hereafter Paper I), we explore the differences in the 3D solution, first presented by Mok et al. (2005) and analyzed in this paper, and the 1D solutions, presented in Klimchuk et al. (2010). We determine that the differences can be attributed to both loop geometry and symmetry. Klimchuk et al. (2010) considered semi-circular loops without magnetic field expansion. In Paper I, we show that when the field expands and the loop is not symmetric, the non-thermal equilibrium solutions are very different from those of Klimchuk et al. (2010). In A. R. Winebarger et al. (2013, in preparation, hereafter Paper III), we examine whether typical diagnostic techniques used to analyze coronal loops, such as those used in this paper, return physically accurate information. This paper is organized as follows. In Section 2 we describe our 3D plasma model, in Section 3 we show how we calculate synthetic emission images, in Section 4 we analyze the results of our simulation, and in Section 5 we draw our conclusions. 2. 3D PLASMA MODEL The plasma model used in the present analysis is that of Mok et al. (2005, 2008), which solves the 3D hydrodynamic equations along magnetic field lines and includes thermal conduction parallel to the magnetic field, radiative cooling, and a functional form for coronal heating. We here present only its main characteristics and refer the reader interested in more details to the aforementioned works. We decided not to use the 3D MHD thermodynamic model of (2009) because simulations with this model are very time consuming and often impractical. Fortunately, the plasma β of active regions is very low (i.e., the magnetic pressure dominates over the thermal pressure), which means that the magnetic structure is unaffected by the plasma. Therefore, as explained in Mok et al. (2005, 2008), we first obtain a 3D zero-β field structure of Active Region 7986; then we solve the hydrodynamic equations along fixed magnetic field lines. We here describe this two-step procedure Magnetic Field Model To model the magnetic field of an active region, we solve a simplified form of the MHD equations in Cartesian geometry ( 2002; Mok et al. 2005): B = 0, (1) B = 4π c E = 1 c J, (2) B t, (3) E + v B = ηj, (4) c ( ) v ρ t + v v = 1 J B + (νρ v). (5) c E and B are, respectively, the electric and magnetic fields, v is the plasma velocity, c is the speed of light, η is the resistivity, ν is the viscosity, and ρ is the plasma density assumed to be constant. When Equations ((2) (5)) are integrated to steady state, the solution represents a force-free, non-constant α solution for a given set of boundary conditions. For the present calculation, we used a non-uniform grid to discretize a cubic computational domain of 1 R side. The Lundquist number, S, is defined as the ratio of the resistive diffusion time τ R = 4πL 2 η 1 c 2 to the Alfvén time τ A = L/V A (at z = 70 Mm above the lower boundary, with a typical number density of n cm 3 and magnetic field B 40 G, V A 3000 km s 1 ). If we consider the typical length scale in our model to be L 0.25 R,wehaveS , which is much lower than the value in the solar corona. This is necessary to dissipate structures that cannot be resolved since they are 2

3 ( ) v ρ t + v ˆb v = ˆb ( p + ρg + (νρ v ˆb)), (8) 500 B z [G] Mm Figure 1. Magnetic field used to calculate the plasma parameters. On the surface, the value of the normal component of the magnetic field, B z,isshown.somefield lines have been traced in random colors to illustrate the magnetic configuration. smaller than the cell size. It does not, however, influence the large-scale structures in which we are interested. For the same reason, we choose ν such that the ratio of the viscous dissipative time versus the Alfvén time is τ ν /τ A = 200. As in Mok et al. (2005, 2008), we started from a minimally smoothed magnetogram from SOHO/MDI, which was used as boundary conditions for the magnetic field, and of which we calculated the potential magnetic field extrapolation. Since there were no vector magnetograms available, we could not use the method of Mikić & McClymont (1994). Therefore, we began twisting the magnetic field by specifying a photospheric surface flow that preserves the magnetic flux ( 2002), until the twist parameter α = J z /B z was between 0.8 and Mm 1, which is comparable with the values determined by Mandrini et al. (2000) and Démoulin et al. (2002). This range of values for α can be explained considering that the magnetic field in the simulation was not perfectly settled down. On the other hand, this is only an approximate model and is not meant to faithfully reproduce the real active region. The final magnetic configuration is illustrated in Figure 1, which has the distribution of B z on the lower boundary and some magnetic field lines traced in random colors Hydrodynamic Model Having obtained the magnetic field configuration, we advanced the following 3D hydrodynamic equations along fixed magnetic field lines: ( 1 T γ 1 t ρ t + (v ˆbρ) = 0, (6) ) + v ˆb T = T vˆb + m 2kρ ( κ ˆbˆb T n e n p Q(T )+H ch ), (7) where ˆb is the normalized vector along the direction of B; T, p, and ρ are the plasma temperature, pressure, and density; m is the hydrogen mass; k is Boltzmann s constant; κ = κ 0 T 5/2, where κ 0 is the Spitzer coefficient for thermal conduction; γ = 5/3 is the ratio of specific heats; Q is the radiation loss function from CHIANTI (Dere et al. 1997, 2009). The special treatment of the thermal conduction and radiation loss function as explained in (2009) was used to lower the gradients in the transition region without significantly affecting the coronal solution. n e and n p are the electron and proton number density and are equal in our calculation (this is the so-called coronal approximation for full ionization), g is the gravitational acceleration, m p is the proton mass, and ρ = m p n p. We used an approximately steady and highly stratified heating function H ch = h 0 B 7/4 n 1/8 e /L 3/4 erg cm 3 s 1, where L is the length of loops and h 0 = is a constant. According to the simulations of Rappazzo et al. (2007, 2008), this function represents coronal heating through the turbulent dissipation of magnetic energy in the Parker field-line tangling problem (Parker 1972, 1988). This formula can be derived considering that the dissipated turbulent power density is D = δb2 8π 1 τ NL, (9) where δb is the amplitude of the magnetic perturbations perpendicular to the guide field of a loop. For a loop of length L, Rappazzo et al. (2008) estimate the non-linear dissipation time τ NL as τ NL λ ( ) α8 4πρ λ 4πρ B, (10) δb δb 4πρL where λ is the length scale of the turbulence and α 8 depends on the magnitude of the Poynting flux. The B 7/4 dependence in H ch is obtained for α 8 = 2, corresponding to a case of mediumstrength turbulence. In the chromosphere the heating is cut off: this has only a small effect on the extension of the temperature plateau, without affecting the coronal part of the solution ( 2009). The normal velocity was set to zero at all boundaries, except at the base where we solved the gas characteristic equation (Mok et al. 2005). At the base we also imposed a fixed number density (n e = cm 3 ) and fixed temperature (T = 20,000 K). As in Mok et al. (2005), we prescribed as the initial condition a stratified atmosphere, which was designed to approximate the plasma properties of the lower corona and ensure a short transient phase. We then advanced Equations (6) (8) in time for about 22 hr, the first 1.2 hr of which were occupied by the transient phase and not used for analysis. 3. GENERATING IMAGES From the temperature and density calculated in 3D, we computed synthetic emission images of the active region in several instrument passbands and spectral lines. As explained in Mok et al. (2005) and (2009), we can obtain emission images in optically thin emission bands by calculating the following integral along the line of sight, s: I λ = n 2 e (s)ɛ λ(t (s),n e (s)) ds, (11) 3

4 Figure 2. Top (from left to right): synthetic EIT 171 Å image showing Loops 1 and 2; synthetic EIS Fe xiii 202 Å image showing Loop 3 and 4; synthetic EIS Fe xiii 202 Å limb image of Loops 5 and 6. Bottom (from left to right): the same with loop boundaries underlined. Both images are scaled logarithmically. Only the central portion of the computational domain is shown in these images. Table 2 Generated Images Instrument Channel/Spectral Line Log T max EIT 171 Å 5.9 EIT 195 Å 6.1 EIT 284 Å 6.3 EIS Fe xiii and Å 6.2 XRT Al-mesh 6.5 where I λ designates the intensity registered by an instrument in a given passband or spectral line, ɛ λ (T,n e ) is the response function associated with the instrument passband or the emissivity function associated with a spectral line. The dependence on the density of the response function, which is calculated using the CHIANTI database (Dere et al. 1997; Landi et al. 2013), is generally negligible (see Figure 11 of Mok et al. 2005, which also gives more details on how to calculate ɛ λ ). For this paper, we focus on the intensities in EUV channels and densities measured through density-sensitive line ratios. Hence, we only consider the passbands and spectral lines given in Table 2. We calculated sequences of images in each of these channels or spectral lines for each instant recorded in our simulation. Our imaging cadence was s and the total time was 20.8 hr; hence there were 260 images for each channel or spectral line. We generate the images for two different lines of sight. We consider the perspective of looking down on the active region from above, which would represent disk center observations. To generate these images, we integrated over the z-dimension. We also considered the perspective of looking at the side of the active region, which would represent limb observations. For this perspective we integrated over the y-dimension. Example images of the active regions are shown in Figure LOOP ANALYSIS 4.1. Loop Selection and Background Subtraction The loop characteristics summarized in Table 1 have been derived from many different types of observations. Loops were selected according to different criteria in different channels or spectral lines. For instance, density measurements tend to occur on loops observed at the limb to better measure the loop height, while the evolution of the loops is typically measured using disk observations where there is less line-of-sight confusion. To date, there has been no complete observational description of a statistically significant set of coronal loops. The analysis of these simulated loops provided many of the same challenges as observed loops, such as line-of-sight confusion. Hence, for each characteristic, we use only loop data that best match the criteria of that characteristic. To select the loops we first generate movies of the simulated active region in each passband and spectral line. Visual inspection of the EUV movies reveals distinct loops can be observed 4

5 Figure 3. Example of background subtraction using spatial pixels surround loop. The left column shows the background subtraction from frame 195, when the loop is bright in the synthetic EIS Fe xiii 202 Å image. The right column shows the background subtraction from frame 190 when the loop is not bright in this spectral line. The top row shows the original emission; middle, the calculated background emission; bottom, processed emission with removed background. Color scales are different for the three images. fading into and out of the active region images. We focus mainly on loops late in the active region evolution to limit the influence of the initial conditions of the simulation. Because we were specifically trying to isolate EUV loops, we select loops only from the EUV images. We choose six loops to analyze in this paper: two loops from disk images of the synthesized EIT 171 Å channel (frame 195), two loops from disk images of the synthesized EIS 202 Å spectral line (frame 195), and two loops from limb images of the synthesized EIS 202 Å spectral line (frame 210). These loops are shown in Figure 2. The loops were initially selected by hand. A computer program was then used to determine the center of the loop at every position along the loop. The lines drawn in Figure 2 outline the loop; the outline that is drawn is 31 pixels wide for Loops 1 and 3 and 21 pixels wide for Loops 2 and 4 (1 pixel is approximately 1 ). We extracted intensities from the images much the same way as we process actual data (see Winebarger et al. 2003b). From the loop trace, we extract a region along the loop including several pixels to either side of the loop. We straighten the loop to run along the image. The top left panel of Figure 3 shows Loop 3 extracted from frame 195. The dotted lines from Figure 2 are also shown in Figure 3. There are two methods of background subtraction we consider in this paper. The first is to determine the background by using the pixels surrounding the loop; we refer to this as the spatially defined background. For each column of pixels in the straightened loop image, we fit a polynomial to the 5 pixels on either edge of the image. We then replace the interior pixels using the polynomial fit. An example of this type of background is shown in Figure 3. The spatially defined background for Loop 3 is shown in the middle left panel of Figure 3. To determine the loop intensities, we subtract the background from the original image, shown in the bottom left panel of Figure 3. This type of background subtraction works well when the background is smoothly varying and no other structures brighten near the loop during the loop s evolution. In previous loop analyses of observations, only loops that were isolated during their entire evolution were analyzed. From the simulation data, however, it is not possible to find loops that are isolated throughout their entire evolution. Instead, loops brighten nearby both before and after the target loop brightens. This is likely due to the smoothly varying nature of the assumed heating function. As an example, the same region of Loop 3 was extracted from frame 190; this is shown in the top right panel of Figure 3. Loop 3 is not bright at this time, but another structure that lies parallel to Loop 3 is bright. This loops falls along the edge of the box and is used to calculate the background intensities in the box. This results in an overestimate of the background, demonstrated by the middle right panel of Figure 3. The narrow loop that was on the edge of the box is now fatter and extending into the central portion of the box. When we subtract the background image from the original, the pixels at the location of target loop become negative (bottom right panel of Figure 3), even though they should probably be zero (or a very small number). Hence we use a second, simpler background subtraction method when calculating the intensities of the loops as a function of time which we will refer to as the temporally defined background. We first find the average intensity along the central 3 pixels of the loop as a function of time. We define the time range of the loop, and then find the minimum intensity of the loop over that time range and use this single value as background. This is demonstrated in Figure 4. The average intensity in the center 3 pixels of Loop 3 in the Fe xiii 202 5

6 Table 3 Measured Delays between the EIT 195 Å and EIT 171 Å Channels, Calculated Cooling Times, Predicted Cooling Times, and Measured Lifetimes for Loops 1 4 Shown in Figure 2 Loop Measured Measured Cooling Calculated Ratio Number Delay Lifetime Time Lifetime τ195 meas /τ calc 195 (s) (s) (s) (s) Figure 4. Example of background subtraction using the evolution of the loop. The solid line is the synthetic EIS Fe xiii 202 Å intensity averaged over the center three pixels of the loop. The background is assumed to be the minimum in this intensity, shown as a dotted line. spectral line is shown as the solid line and the background (minimum average intensity) is shown as a dotted line. We apply the temporally defined background subtraction method only when determining the light curves of the loop (Section 4.2). When analyzing the intensity and density along the loops in single frames, we use the spatially defined background (Sections 4.3 and 4.4). In Paper III, we will test how well both methods approximate the true background of the loop Temporal Evolution of Loops Typically, the temporal evolution of a loop is determined for loops on the disk to minimize the plasma along the line of sight. Additionally, isolated sections of the loops are chosen to minimize the interference of surrounding structures on the light curve. For these reasons, we choose to calculate the evolution of Loops 1 4 only in the regions between the dotted lines shown in Figure 2. Since B and L are held constant in the simulation, in H ch there is only one possible time dependence, which is caused by n 1/8 e and, given the small exponent, is very weak. Thus the total heating is quasi-steady and, during the cycles shown in Figure 2, varies by less than 0.7%. We then average the intensity over selected region and along the three central pixels of the loop to obtain an intensity in that channel or spectral line as a function of time. We complete background subtraction by assuming the minimum intensity is the background intensity. The resulting light curves are shown in Figure 5. We have normalized all the intensities. The frame number is shown on the bottom x-axis and the time in minutes is shown on the top x-axis of each plot. The temporal evolution in the emission channels can be compared with Figure 6, which shows the integral of the heating injected into whole computational domain, H ch dv, as a function of the frame number. An initial qualitative examination of the light curves shows that all loops appear in the hotter channels or spectral line before appearing in the cooler channels or spectral lines, i.e., all the loops appear to be cooling. In Loops 1 and 3, the loops appear to be bright in each channel at distinct times, while in Loops 2 and 4, there is significant overlap between the different channels. If we assume the plasma in the loop is cooling and, at any given time, can be characterized by a single temperature and density, then the difference between the appearance times of a loop in 171 and 195 Å channels, as well as the time a loop remains bright in a single channel, are both controlled by the cooling time of the plasma. We can determine the cooling time from the observations and see if the evolution of the light curve is consistent with this interpretation. We adopt many of the same assumptions as in previous investigations, which we summarize below (Winebarger et al. 2003b; Mulu-Moore et al. 2011). In the previous analyses, however, only the TRACE instrument was used, hence we calculate here the relationship between the different EIT channels and the cooling time. To determine the appearance time of the loops in each channel as well as the lifetime of the loop in a single channel, we fit rise and decline phases of the EIT 171 and 195 Å light curves with polynomial functions. We define the time the loop appears in each channel as the time in which the intensity in the channel is larger than half the maximum intensity. The differences between the appearance times in the EIT 195 and 171 Å channels, which we term the measured delays, are given in the first column of Table 3. We define the lifetime of the loop in a single channel as the time between the half-maximum values. The measured lifetimes of the loops in the EIT 195 Å channel are given in the second column of Table 3. These simulated loops have lifetimes in the range s. Next we use the measured delay to calculate cooling times, assuming that the loops are cooling exponentially (Winebarger et al. 2003b). For the EIT channels, we find the relationship between the delay, Δt , and the cooling time, γ,is γ T 2.47Δt (12) The cooling times calculated from the delays are given in Table 3. Finally, we calculate the expected lifetime, τ, in the EIT 195 Å channel for the calculated cooling time. For the EIT 195 Å channel, we find the relationship between the cooling time and lifetime to be τ calc γ T. (13) The calculated lifetimes are given in the fourth column and the ratios of the measured to calculated lifetimes are given in the final column of Table 3. For Loops 1 and 3, the calculated and measured lifetimes are approximately the same; hence, these loops could be consistent with a single cooling loop model. Loops 2 and 4, however, have measured lifetimes that are significantly longer than the calculated lifetimes. Both of these characteristics are observed in real data. In previous studies, these were interpreted as sub-resolution cooling strands or multiple loops along the line of sight. 6

7 Figure 5. Normalized intensity curves as a function of the frame number for the four loops of Figure 2 in five bands: EIT 171 Å, EIT 195 Å, EIS Fe xiii 202 Å, EIT 284 Å, and XRT Al-mesh. The intensity was averaged over the central 50% of the loops between the dashed lines in Figure 2. Standard background subtraction was applied. The loop lifetimes were measured as the FWHM of the emission in the EIT 195 Å channel. These simulated loops have lifetimes of s, compatible with observed EUV loops erg/s H ch dv Frame Number Figure 6. Power injected into the whole computational domain by the H ch heating term of Equation (7) as a function ofthe frame number. During the first 100 frames, while the system is still abandoning the initial condition to settle into the complex cycles driven by thermal non-equilibrium, the heating power decreases by approximately 1.6%. Afterward, it remains constant within 0.7% Intensity Along the Loops As summarized in Table 1, the intensity along warm EUV loops does not generally present small-scale structure. Additionally, the ratio of the intensity in two EUV channels is typically flat. In this section, we investigate these characteristics in the simulated loops. Figure 7 shows the spatially determined backgroundsubtracted, extracted loops along the entire length shown in Figure 2. The loop intensities are taken from the same channel or spectral line and frame number as shown in Figure 2, e.g., Loops 1 and 2 intensities are from the 195th frame of the EIT 171 Å channel disk image, Loops 3 and 4 intensities are from the 195th frame of the Fe xiii 202 Å spectral line disk image, and Loops 5 and 6 are from the 210th frame of the Fe xiii 202 Å spectral line limb image. The loops show a mix of both smooth and sharp variations in their intensities as a function of distance along the loop. For example, Loops 1 4 appear to show only smooth transitions along the loop, while Loops 5 and 6 show sharp transitions. These transitions tend to be close to the loop footpoints and may be due to the difficulty in subtracting background in these regions. For instance, in Loop 6 a second structure crosses the loop at the second footpoint, making the 7

8 Figure 7. Intensity in EIS Fe xiii along loops at some moment during their evolution. Both smooth and sharp intensity variations are visible, especially at the loop footpoints. These may be due to the difficulty in subtracting the background in these regions. intensity in the loop fall to zero. Figure 7 can be compared with Figure 4 of Aschwanden et al. (2000), which shows the observed flux for TRACE loops in a logarithmic scale. It was difficult to find good candidate loops to measure the ratio of the two EIT channels along the loop. First, the loop must have to be bright in both channels simultaneously. Examination of the light curves of the loops shows that only Loops 2 and 4 are bright in both EIT 171 and 195 Å channels co-temporally. However, both Loops 2 and 4 occur in the center of the active region over bright structures we assume represent moss (see Figure 2). Only the inter-moss portion of these loops was used to measure the light curves, in large part due to the difficulty of extracting the background from the moss region. In the end, we determine the channel ratio only for Loop 2, which has a significant fraction of its length in the inter-moss region. The calculated ratio of the EIT 195 Å to the EIT 171 Å intensity as a function of the distance along the loop is shown in Figure 8. The ratio is approximately flat in the inter-moss region Density Along the Loop Another important observational evidence is that EUV loops exhibit a much higher density than is expected, given their temperature and length, and the density scale height is larger than expected. To calculate the density, we use only Loops 5 and 6, which are observed in the limb images. We find that we can measure the spatially determined background-subtracted intensity along the entire length of Loop 5, while we can only use the upper portion of Loop 6 (between the dotted lines shown Figure 8. Temperature profile, i.e., ratio between the EIT 195 Å channel and the EIT 171 Å channel, as a function of the distance along the loop for Loop 2. The ratio is fairly uniform as it is the case in observations. in Figure 2) due to difficulty in extracting the intensity near the footpoints (see Figure 7). We extract the loop intensities in both the Fe xiii 202 and 203 Å spectral lines and use the ratio to calculate the density. These densities are shown as red lines in Figure 9 as a function of the projected distance along the loop. We assume the loop length is the same as the projected loop length. Using the loop length, we then calculate a steady-state solution with uniform heating and an apex temperature of 1.6 MK (the peak emitting temperature of Fe xiii). The densities associated with these solutions are plotted as black lines in 8

9 Loop 5 Loop 6 Figure 9. Comparison between the density calculated from the emission images (pink) and a uniform heating solution with the same apex temperature and loop length (black) for Loops 5 and 6. Densities are larger and flatter than for steady, uniform solutions of the same length and temperature, shown in black. Figure 9. For both these examples, the steady-state solution has a lower density than found from the thermal non-equilibrium model. The ratio of the apex densities is 4.9 for Loop 5 and 3.8 for Loop 6. The density measured along the loop is also flatter than the density predicted by the steady-state solution in both cases (i.e., the scale height of the simulated density is larger than that of the steady, uniform case). 5. CONCLUSIONS In this paper, we have analyzed loops from a 3D active region simulation to determine if they have seven fundamental characteristics of observed warm loops. Regrettably, these seven observed features have never been studied in a single set of loops. There is clearly a need for a consistent study of these observed loop properties. The heating function in the simulation was highly stratified and quasi-steady, which caused the loops to undergo thermal non-equilibrium, (i.e., when no steady state solution can be found), which, according to Klimchuk et al. (2010), should not be a viable mechanism to explain loops observations. Our method in pursuing this investigation has been quite novel: since the seven characteristics are all deduced from observations, we have striven to follow the same procedures that observers have used in determining them. First, we have construed synthetic emission images from our 3D simulation. Then, we have proceeded to analyze them by applying standard observational procedures as if the images had been obtained from an instrument. After thus determining the properties of the loops, we have compared them with observations. Given the wide variety of observed properties, we find that loops from our simulation largely present characteristics in agreement with observations. One limitation of this analysis is our inability to complete time-dependent background subtraction due to loops being closely packed in the simulation. We address the accuracy of both spatially and temporally determined backgrounds in Paper III. The stark divergence between our conclusions and those of Klimchuk et al. (2010) calls for an explanation. There are several differences to be considered between the model of Klimchuk et al. (2010) and ours. 1. 1D calculation versus 3D calculation. 2. High resolution versus medium resolution. 3. Limited to perfectly symmetric, non-expanding, semicircular loops versus considering the multifarious geometrical properties of loops. 4. Simple exponential heating function versus heating function depending on the magnitude of the magnetic field. 5. Radiation loss function as in Klimchuk et al. (2008) versus radiation loss function as in Athay (1986). Fortunately, the results presented in Paper I helpustodis- criminate which differences between the models are substantial in explaining the discrepancies in the final results and which are merely accidental. Paper I starts by solving the hydrodynamic loop equations in a constant cross-section, semi-circular configuration, with exponentially decaying heating function having symmetric deposition scales. Given these assumptions, the first solution of Paper I is consistent with its corresponding one in Klimchuk et al. (2010). Then the authors gradually remove the geometrical constraints and study how the characteristics of the solutions vary and become gradually more similar with those presented in this work. In particular, allowing for a variable expansion factor along a loop is a condition that favors the development of thermal non-equilibrium. During the thermal cycle long-lived condensations, which are neither frequently observed in nature nor in our 3D simulations, are regularly formed. However, when the uniform scale-length constraint in the heating function is relaxed, siphon flows appear that flush away the condensations in an early stage of formation. In our 3D model (Equations ((6) (8))), as in nature, the expansion factor or cross-sectional area is simply A(s) = 1/B(s) and varies accordingly along the loop. This highly favors the onset of thermal non-equilibrium. The variation in the expansion factor is stronger in a short interval at the base of the corona and considerably less in the main portion of the loop (Paper I). This, in conjunction with the fact that the linear cross-section of the loop is proportional to B(s) 1/2 and thus varies even more slowly, contributes to the observed apparent constant width of coronal loops in Figure 2. Second, our heating function has a strong dependence on the magnetic field ( B 7/4 ), which creates asymmetries in the length scales of the heating and, consequently, generates siphon flows that flush away condensations before they have time to grow. This explains why full-grown condensations do not constitute a prominent feature of observations of coronal loops. Since neither variable expansion factors nor asymmetric heating deposition scales were examined by Klimchuk et al. (2010), it is not surprising that their thermal non-equilibrium solutions had characteristics far different from those of observations. On the contrary, all our model loops were found to have properties not dissimilar from observed loops. We therefore conclude that thermal non-equilibrium heating cannot be ruled out as the 9

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