AN INVESTIGATION OF TIME LAG MAPS USING THREE-DIMENSIONAL SIMULATIONS OF HIGHLY STRATIFIED HEATING

2016. The American Astronomical Society. All rights reserved. doi:10.3847/0004-637x/831/2/172 AN INVESTIGATION OF TIME LAG MAPS USING THREE-DIMENSIONAL SIMULATIONS OF HIGHLY STRATIFIED HEATING Amy R. Winebarger 1, Roberto Lionello 2, Cooper Downs 2, Zoran MikiĆ 2, Jon Linker 2, and Yung Mok 3 1 NASA Marshall Space Flight Center, ZP 13, Huntsville, AL 35812, USA; amy.r.winebarger@nasa.gov 2 Predictive Science, Inc., 9990 Mesa Rim Rd., Ste. 170, San Diego, CA 92121-2910, USA; lionel@predsci.com,cdowns@predsci.com,mikicz@predsci.com, linkerj@predsci.com 3 Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA; ymok@uci.edu Received 2015 June 23; revised 2016 July 29; accepted 2016 August 12; published 2016 November 4 ABSTRACT The location and frequency of coronal energy release provide a significant constraint on the coronal heating mechanism. The evolution of the intensity observed in coronal structures found from time lag analysis of Atmospheric Imaging Assembly (AIA) data has been used to argue that heating must occur sporadically. Recently, we have demonstrated that quasi-steady, highly stratified (footpoint) heating can produce results qualitatively consistent with the evolution of observed coronal structures. The goals of this paper are to demonstrate that time lag analysis of 3D simulations of footpoint heating are qualitatively consistent with time lag analysis of observations and to use the 3D simulations to further understand whether time lag analysis is a useful tool in defining the evolution of coronal structures. We find the time lag maps generated from simulated data are consistent with the observed time lag maps. We next investigate several example points. In some cases, the calculated time lag reflects the evolution of a unique loop along the line of sight, though there may be additional evolving structures along the line of sight. We confirm that using the multi-peak AIA channels can produce time lags that are difficult to interpret. We suggest using a different high temperature channel, such as an X-ray channel. Finally, we find that multiple evolving structures along the line of sight can produce time lags that do not represent the physical properties of any structure along the line of sight, although the cross-correlation coefficient of the lightcurves is high. Considering the projected geometry of the loops may reduce some of the line-of-sight confusion. Key words: Sun: corona Sun: UV radiation 1. INTRODUCTION Solar coronal loops have been observed to cool (i.e., appear in images sensitive to higher temperature plasma before appearing in images sensitive to lower temperature plasma ( 2003; Ugarte-Urra et al. 2006; Mulu-Moore et al. 2011)). Viall & Klimchuk (2012) developed a technique to measure the time lag between different EUV channels from data observed by the Solar Dynamics Observatory s SDO Atmospheric Imaging Assembly (AIA). In this method, they analyze each AIA pixel of a 12- (or 2-) data cube individually. They calculate the cross-correlation coefficient as a function of time lag between two AIA channel lightcurves and produce a map of the time lags at the peak of the cross-correlation function. The map indicates that most pixels in the active region show evidence of cooling, even when no discernible loop is present. In many of the channel pairs, they find a wide range of time lags, with the longest time lags appearing in the longest loops. In Viall & Klimchuk (2013), they demonstrated that an arcade of loops suffering so-called nanoflare storms (Klimchuk 2006) can be used to understand these lightcurves. In this heating scenario, the loops are formed of sub-resolution strands; the strands are heated impulsively at different times. Summing the emergent intensity for all strands along the line of sight produces time lags of hundreds of seconds between pairs of EUV channels. Bradshaw & Viall (2016) recently modeled an entire active region with many loops along the line of sight. They heated the loops at a variety of frequencies and demonstrated that time lag maps can discriminate between different heating frequencies. Lionello et al. (2016), however, investigated whether simple impulsive heating could explain the longest time lags observed by Viall & Klimchuk (2012). They completed a parameter space study of loops with lengths in the range of lengths in the Viall & Klimchuk (2012) active region. They calculated how the magnitude of the heating event, the length of the loop, the abundance of the plasma, and whether the loop expanded or not impacted the time lag. They found that some of these measured time lags could not be explained by impulsive heating. One possible explanation for the inability of the impulsive heating to reproduce the long time lags is that the time lags retrieved by the cross-correlation technique are not representative of actual loop evolution. Individual loops may be cooling through the channels at a faster rate, but for some reason, the cross-correlation process detects a longer period. Another is that a different heating scenario, such as footpoint heating, is at work in these loops. Quasi-steady and highly stratified (i.e., concentrated close to the loop footpoints) heating has been investigated to determine if it could explain the properties of EUV loops (Klimchuk et al. 2010; Lionello et al. 2013; Mikić et al. 2013). Under some conditions, this heating scenario does not have a steady 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; such a 1

Figure 1. Response functions of the AIA channels. The three channels shown in the left panel are the cooler channels and are characterized by strong narrow peaks at low temperatures, although there are some transition region lines in the pass bands, increasing the response slightly at 0.3 MK. The three hot channels are shown in the right panel, along with the Hinode XRT response calculated using the thin beryllium filter. Although the AIA channels do have high temperature (>2 MK) response, they are broader and have secondary peaks at lower ( 1 MK) temperatures. phenomenon has been observed and is called coronal rain (Schrijver 2001; Antolin et al. 2010, 2012, 2015). If the heating remains unchanged over many hours, multiple cycles on the same loop would occur. Long cycle oscillations have recently been discovered in EUV images data (Auchère et al. 2014, 2016; Froment et al. 2015). On the basis of 1D plasma simulations, Klimchuk et al. (2010) argued that highly stratified heating could not explain EUV loop observations, stating that it fails to reproduce the observed loop properties. However, Mok et al. (2008, 2016) presented 3D plasma simulations showing that quasi-steady, footpoint heating can lead to the formation of realistic looking loops. Mikić et al. (2013) showed that highly stratified heating on realistic field line geometries can generate temperature and density evolution similar to those observed in coronal loops. Lionello et al. (2013) showed that the properties of such simulated 3D loops well match the observed properties of the EUV loops. (2014) showed the diagnostics determined from the simulated data well match true loop evolution. In this paper, we continue using the simulated images based on a 3D hydrodynamic simulation to investigate the time lag diagnostic. First, we show that time lag maps from the simulated images are qualitatively consistent with the time lag results of Viall & Klimchuk (2012). We then choose several example points in the time lag maps and use plasma temperature and density along the line of sight to investigate whether the time lag captures properties of a unique structure along the line of sight. We find there are cases where the time lag represents evolution of a single structure along the line of sight, even when multiple evolving structures are present. We confirm the difficulty in using the high temperature channels of the AIA instrument in calculating the time lag due to the multiple peaks in the response functions, as first discussed by Viall & Klimchuk (2012). We also find locations where the measured time lag does not represent the evolution of a single structure. Instead, the lightcurves of multiple structures along the line of sight are correlated. In this case, the time lag does not represent a true measure of a cooling time of an individual structure. We suggest that additional constraints such as loop geometry could be added to improve the results of time lag analysis and eliminate these false positives. 2. TIME LAG ANALYSIS OF SIMULATED DATA In this section we perform time lag analysis on a series of simulated images from a 3D plasma model of an active region. The model has been fully described in Mok et al. (2005, 2008, 2016). The model is derived from the photospheric field measurements of Active Region (AR) 7986 made on 1996 August 30. The heating function, based on simulations by Rappazzo et al. (2007, 2008), is Hch ( x, y, z, t) = h B7 4 0 ( x, y, z) n1 8( x, y, z, t) L3 4erg cm 3 s 1, where h0 = 0.273 is a constant, B ( x, y, z) is the magnetic field strength, n ( x, y, z, t) is the density, and L is the length of the field line that passes through spatial position ( x, y, z). The units of all parameters are cgs. In this equation, only the density is timedependent and H ch is only weakly dependent on density; hence the resultant heating function is quasi-steady and highly stratified. The result of the 3D model is the calculated temperature and density as a function of time and space. From these quantities, we compute synthetic images of the active region in the SDO/ AIA channels. The response functions of the AIA 171, 193, and 211 Å channels, thought of as the cool channels, are shown in the left panel of Figure 1 and are strongly peaked at 0.9, 1.6, and 1.8 MK, respectively. The response functions of the AIA 335, 94, and 131 Å channels, thought of as the hot channels, are shown in the right panel of Figure 1 and have multiple peaks at both high and low temperatures, which complicate the analysis of the intensity in those channels (see O Dwyer et al. 2010 for a complete discussion). We integrate the intensity over the z dimension (analogous to looking at an active region at disk center); Figure 2 shows the simulated active region in each of the AIA channels at a single time. For each channel, we calculate sequences of images at the cadence of the simulation, 288.6 s (4.8 minutes), for a total time of 20.8 hr; hence there were 260 images for each passband. We only consider the last 12 hr of the simulation, to match one of the time windows chosen by Viall & Klimchuk (2012). Although the cadence of the simulated images is significantly slower than that of actual SDO/AIA observations, the slow evolution of these loops ( 2014) indicates that the low cadence will not affect the result. We follow closely the analysis described in detail in Viall & Klimchuk (2012) and described briefly here. We evaluate each 2

Figure 2. Simulated observations in six EUV channels from AIA are shown above. The intensities are scaled logarithmically. pixel of the simulated data individually. For each pixel, we resample the 12 hr lightcurves of that pixel in the 6 AIA channels onto a grid with a 30 s cadence. We then find the crosscorrelation as a function of time lag using the IDL routine C_CORRELATE.pro for each pair of AIA channels. In Viall & Klimchuk (2012), they considered a window of ±2 hr for the time lag, but for the simulated data, we found that some crosscorrelation functions peaked at larger time lags so we allow time lags as large as ±4 hr. We record both the time lag and peak of the cross-correlation function. We calculate the time lag so that if the hotter channel peaks before the cooler channel, the time lag will be positive. The maps of the time lags for the pairs of AIA EUV channels are shown in Figures 3 and 4 with the same color scale used by Viall & Klimchuk (2012). In these images, if the peak of the cross-correlation function is less than 0.5, we set the color of the pixel to white. This tends to occur in the center of the active region. We will show examples of two data points with peaks of their crosscorrelation function less than 0.5 in Section 3. Additionally, if the intensity is always less than 1% of the maximum intensity in either channel, we set the color to white. This occurs near the edges of the simulation, where the heating rate falls off rapidly so that it is no longer representative of active region heating. The time lag maps shown in Figures 3 and 4 are qualitatively consistent with those calculated by Viall & Klimchuk (2012) from AIA data. The magnitudes and distribution of the time lags are similar to those of observations. For instance, in the AIA 211 171 Å time lag map, the loops in the center of the active region are color-coded red, while loops outside the core are color-coded yellow. This was explained by Viall & Klimchuk (2012) as the difference in cooling time between the shorter core loops and the longer arcade loops. In the time lag maps including the hotter AIA channels (94, 131, and 335 Å), there is a mix of positive (red yellow) and negative (blue green) time lags. This was explained in Viall & Klimchuk (2012) as being due to the multi-peaked temperature response of these channels. There are also some differences between the time lag maps for simulations and those for observations. For instance, in Viall & Klimchuk (2012), the AIA 94 335 Å time lag was primarily positive in the shortest core loops and negative in the longest loops, while in our simulation, the pattern is reversed. 3. EXPLORING EXAMPLE POINTS For the quasi-steady footpoint heating, such as that applied in this simulation, it is expected that some loops will cycle through high and low temperatures (e.g., Mikić et al. 2013), while others will remain effectively steady. The cycles occur on timescales of hours, so over this 12 hr period, we can capture multiple cycles of the same loop. Additionally, because the line of sight of each pixel in the integrated image samples loops with different lengths and heating parameters, it is very likely that multiple loops will be cycling along the line of sight and those cycles will not necessarily be correlated. In this section we investigate the time lag for a series of example points selected from the AIA 335 193 Å and AIA 211 171 Å time lag maps. We choose two points (A and D) that appear normal when compared to the other points in the time lag map or in Viall & Klimchuk (2012), two points (B and E) that appear abnormal because of the sign or magnitude of the time lag, and two points (C and F) that have a peak cross-correlation coefficient less than 0.5. The selected points are shown in the appropriate panels of Figures 2 4. 3.1. 335 193 Å Points The AIA 335 Å channel has a broad temperature response with three distinct peaks at log T = 5.25, 5.9, and 6.4 (or 0.18, 3

Figure 3. Maps of the time lag associated with the peak in the cross-correlation function for pairs of AIA channels. The lightcurves were over 12 hr of simulated data. Although we allow for longer time lags than Viall & Klimchuk (2012), we use the same color scaling as Viall & Klimchuk (2012) for ease of comparison. Additionally the color of the image has been set to white if either the peak of the cross-correlation function is less than 0.5 or the peak intensity in the lightcurve of either channel is less than 1% of the maximum intensity in that channel. 0.79, and 2.5 MK). The AIA 193 Å channel has a relatively narrow temperature response that peaks at log T = 6.2 (or 1.6 MK). Because the AIA 335 Å channel has temperature response at temperatures both higher and lower than the 193 Å channel, it provides an interesting case study to understand the information in the time lag maps. The first three example points, labeled A, B, and C, are shown in the AIA 335 and 193 Å images in Figure 2 and the AIA 335 193 Å time lag map in Figure 3. Point A is closest to the center of the active region; the time lag associated with it is 3360 s, which is in a similar range to the surrounding data points and also in the typical range of values found by Viall & Klimchuk (2012). The time lag is positive, indicating the 335 Å channel intensity peaks before the 193 Å intensity. Point B is selected because it has a negative time lag of 3990 s. Although there are several locations of negative time lags in this pair of channels, both in the simulated results and in Viall & Klimchuk (2012), they are more difficult to understand given that AIA 335 Å has a higher temperature response than AIA 193 Å. Finally, we select Point C because the peak in the crosscorrelation function is 0.24. The time lag in this pixel is also negative ( 1890 s). The lightcurves associated with these three points in the AIA 335 (solid) and 193 Å (dashed) channels are shown in the top panels of Figure 5. The lightcurves for Point A appear to be relatively simple to understand. It appears that the lightcurve is dominated by a single loop that completes two full cycles during the 12 hr of simulation time considered here. The brightenings in the AIA 335 Å lightcurve precede brightenings in the AIA 193 Å lightcurve; hence we would conclude this 4

Figure 4. Same as Figure 3 for the remaining image pairs. The color scale of the A211-A193 and A171-A131 time lag maps has been reduced to scale between 2000 and 2000 s. Figure 5. Lghtcurves for Points A C are shown in top panels for AIA 335 (solid) and 193 Å (dashed) channels. Asterisks indicate local maxima of the AIA 335 Å intensity, and diamonds indicate the local maxima shifted by the time lag. The cross-correlations as a function of time lag for the two lightcurves are shown in the lower panels. Point A has a positive time lag of 3360 s with a cross-correlation of 0.67, Point B has a negative time lag of 3990 s with a cross-correlation of 0.67, and Point C has a time lag of 1890 s and peak cross-correlation of 0.24. Note that the cross-correlations functions have multiple peaks; these can be explained by the cycles that occur as a result of the footpoint heating. 5

Figure 6. Temperature and density (left) along the line of sight of Point A as a function of height above the solar surface and time. The emergent intensities in the AIA 335 and 193 Å (right) channels are also shown. The times of the local maxima of the AIA 335 Å channel are shown as dashed horizontal lines. The dotted vertical line shows the approximate height of the structures responsible for the peaks observed in the AIA 335 Å lightcurve. Figure 7. Same as Figure 6 for Point B. loop is cooling. Two local maxima in the AIA 335 Å lightcurve are marked with asterisks. The cross-correlation as a function of time lag for Point A is shown in the lower left panel of Figure 5. The correlation function peaks at 3360 s with a value of 0.67. To illustrate the relative time, we shifted the location of the two local maxima in the AIA 335 Å lightcurve by this time lag and plotted those points as diamonds. These shifted maxima roughly align with the local maxima in the AIA 193 Å lightcurve. From the lightcurve, the time lag in this case appears to be representative of the typical delay time of a single structure along the line of sight. In Figure 6, we show the plasma properties and emergent intensity in the AIA 335 and 193 Å channels along the line of sight of Point A as a function of height above the lower boundary of the simulation and time. Although the lightcurve appeared to show one single loop cycling along the line of sight, there are actually many structures along the line of sight contributing to the intensity. There are at least two unique cycle times along the line of sight, a compact loop cycling at higher frequency at a height of 15 Mm, and a more extended structure between 30 and 60 Mm cycling at lower frequency. Note that at each position in z (height), the line of sight intersects a unique loop so this extended structure is actually several adjacent field lines that are cycling at roughly the same frequency. The horizontal dashed lines across all the panels show the times of the local maxima in the AIA 335 Å lightcurve (i.e., the time associated with the asterisks in the upper left panel of Figure 5). These times coincide with the cycle of the more extended structure. The vertical dotted lines show the approximate height of this structure. For this structure, the AIA 335 Å channel is peaking at the time and location where the temperature is 2.5 MK. The 193 Å intensity follows the 335 Å intensity when the temperature drops. In this case, the time lag method does detect the evolution of a loop cooling from the hot 335 Å temperature (2.5 MK) to 193 Å temperature range (1.6 MK). There are additional, evolving structures along the line of sight that are not captured by the time lag analysis. The lightcurve for Point B is shown in the top middle panel of Figure 5; the corresponding cross-correlation function is shown in the lower middle panel. In this example, the AIA 193 Å lightcurve appears to precede the AIA 335 Å lightcurve. 6

Figure 8. Same as Figure 6 for Point C. The measured time lag is negative, confirming this result. Figure 7 shows the plasma properties and AIA 335 and 193 Å intensity along the line of sight of Point B. Unlike Point A, there appears to be a single dominant structure along this line of sight in the AIA 335 Å channel between 15 and 40 Mm. The horizontal dashed lines show the times of the peaks in the AIA 335 Å lightcurve; the vertical dotted lines show the average height of the structure associated with this cycle. In this example, the AIA 335 Å channel peaks when the temperature is <1 MK, not at 2.5 MK. Hence, as expected from the lightcurve analysis, the AIA 335 Å does peak after the 193 Å channel due to the low temperature response in the AIA 335 Å channel. The lightcurve for Point C is shown in the top right panel of Figure 5; the corresponding cross-correlation function is shown in the lower right panel. It is not immediately clear how to interpret this lightcurve; there is no clear relationship between the peaks in the AIA 335 Å channel and 193 Å channel. The measured time lag is negative, but the peak cross-correlation coefficient is low. The plasma properties and emergent intensity along this line of sight are shown in Figure 8. In the AIA 335 Å intensity, there appears to be one dominant structure at heights between 20 and 40 Mm that dominate the cycle time; the average height of this structure is illustrated with a dotted vertical line. Examining the cycle that occurs near 8 hr, the intensity in AIA 335 Å is double peaked, meaning it peaks at approximately 7 hr, fades briefly, then peaks again shortly after 8 hr. The AIA 193 Å intensity peaks at roughly 8 hr. The temperature at this height shows that the temperature goes from 2.5 MK at 7 hr and is <1 MK shortly after 8 hr. Therefore, the double peak in the AIA 335 Å intensity is due to the double peak of the temperature response function. Reexamining the integrated lightcurve (top right panel of Figure 5), we see that in both cycles shown, the AIA 335 Å intensity both precedes and follows the AIA 193 Å intensity. Despite the fact that there is a single dominant structure along the line of sight, time lag analysis is not adequate to capture the evolution of this structure because of the multiple peaks in the AIA 335 Å response function. 3.2. 211 171 Å Points In this section, we examine three points selected from the AIA 211 171 Å time lag map. Unlike the AIA 335 Å channel, discussed above, the responses of the 211 and 171 Å channels are fairly narrow and strongly peaked at a log T of 6.25 and 5.95 (or 1.8 and 0.9 MK), respectively. Despite this fact, there remain many points in the time lag map that have low correlation coefficients. Additionally, some of the time lags measured for this channel pair are larger than found by Viall & Klimchuk (2012). For this channel pair, we chose three lines of sight to investigate. Point D is near the core of the active region and has a peak cross-correlation of 0.70 at a time lag of 1770 s. This value appears representative of both this region in the image and also similar to values that Viall & Klimchuk (2012) found. Point E is in the arcade, far away from the active region core. It has a peak cross-correlation of 0.69 at a time lag of 8670 s. These values are larger than those measured by Viall & Klimchuk (2012). Finally, Point F is in the arcade. It has a peak cross-correlation of 0.47 at a time lag of 10,560 s. We investigate these points in detail below. The integrated lightcurves measured at these three positions are shown in Figure 9. Point D (upper left panel) appears to be a relatively simple lightcurve. The AIA 211 Å channel precedes the AIA 171 Å channel. The local maxima in the AIA 211 Å lightcurve are shown with asterisks. The cross-correlation function (lower left panel) is strongly peaked at 1770 s. The shift of the local maxima of the AIA 211 Å lightcurve by this time lag (shown as diamonds) roughly lines up with the local maxima in the AIA 171 Å lightcurve. The temperature, density, and emergent intensity along this line of sight are shown in Figure 10. There is one extended structure that is cycling along this line of sight around an average height of 40 Mm. The temperature cycles from 3 to 1 MK along this structure. The lightcurves of Point E appear more complicated. The peaks in the AIA 211 and 171 Å channels are broad, and there is no clear indication that the AIA 211 Å directly precedes the AIA 171 Å intensity, although their peaks are offset in time. The time lag calculated from this lightcurve is longer than that found by Viall & Klimchuk (2012). The shift of the local maxima of AIA 211 Å by this large time lag (shown as diamonds) only roughly lines up with the local maxima in the AIA 171 Å lightcurve. The temperature, density, and emergent intensity along this line of sight is shown in Figure 11. The structure responsible for the peaks in the AIA 211 Å intensity is at the height of 40 Mm; this is shown with vertical dotted lines. 7

Figure 9. Lightcurves for Points D F are shown in the top panels for the AIA 211 (solid) and 171 Å (dashed) channels. Asterisks indicate local maxima of the AIA 211 Å intensity and diamonds indicate the local maxima shifted by the time lag. The cross-correlations as a function of time lag for the two lightcurves are shown in the lower panels. Point D has a positive time lag of 1770 s with a cross-correlation of 0.70, Point E has a time lag of 8670 s with a cross-correlation of 0.69, and Point F has a time lag of 10,560 s and peak cross-correlation of 0.47. Figure 10. Same as Figure 6 for Point D. At this height, the temperature varies slowly between 1.2 and 1.5 MK. The height at which 171 Å intensity is emitted is 27 Mm. At this height, the temperature varies slowly from 0.8 to 1.2 MK. The temperature does not get high enough to emit in 211 Å. Hence, although we calculated a time lag with an acceptable correlation coefficient, the lightcurves we were correlating originate in two different structures along the line of sight. The time lag we calculated was, in this case, not related to the cooling time or to anything physical in a single structure. Finally, the lightcurves associated with Point F are shown in the top right panel of Figure 9, and the corresponding correlation function is shown in the lower right panel of Figure 9. The lightcurves show multiple peaks in both 211 and 171 Å. It appears that the AIA 211 Å intensity precedes the AIA 171 Å intensity. The correlation function has several peaks, the largest of which is at 10,560 s. The properties along the line of sight are shown in Figure 12. In this case, there are many structures with different properties along the line of sight emitting in the AIA 211 Å temperature range. The small pulsations seen in the integrated lightcurve are largely due to the structure at 30 Mm. At this height, the temperature is varying between 1.4 and 1.7 MK, hence it does not get down to AIA 171 Å temperatures. There are pulsations in both the 211 and 171 Å intensities at a height of 20 Mm, but they are weaker in 211 Å. We conclude this poor correlation is due to line-ofsight confusion of multiple structures. 8

Figure 11. Same as Figure 6 for Point E. Figure 12. Same as Figure 6 for Point F. 4. DISCUSSION In this paper, we have applied the time lag analysis, introduced in Viall & Klimchuk (2012), to simulated active region observations. The simulation applies a quasi-steady, highly stratified heating; this type of heating can drive thermal non-equilibrium solutions to the hydrodynamic equations in some loops. We find that the time lag maps from the simulated data are qualitatively consistent with the observed time lag maps. The magnitude and distribution of the time lags are similar to the measured time lags. Hence, although Viall & Klimchuk (2012, 2013) have argued that the observed time lag maps strongly indicate impulsive, sporadic heating and subsequent cooling, we suggest that the time lag maps alone cannot rule out quasi-steady, highly stratified heating. In addition, because we are applying this analysis to simulated data with knowledge of the true conditions of the plasma along the line of sight, we can determine how well this diagnostic captures the evolution of those structures. To do this, we select six example lines of sight for a detailed analysis. Two of the lines of sight (Points A and D) were thought of as relatively normal and simple. In both cases, the time lag captured the evolution of plasma in one structure along the line of sight, although in Point A, there were two distinct structures along the line of sight with different properties. In these two cases the time lag can then be related to the physical properties of a single loop. We also confirmed the conclusion of Viall & Klimchuk (2012) that negative time lags when using the AIA hot channels can be explained by the presence of multiple peaks in these channels response functions. In Point B, we found that the negative time lag between the AIA 335 and AIA 193 Å channels was explained because the AIA 335 Å emission was originating in cooler plasma. Because of the multi-peak nature of the hot AIA channels, the lightcurves lead to confusion when the emission in the channel originates from both the high and low temperature plasma. At point C, the low crosscorrelation coefficient was attributed to this reason, and in the lightcurve the AIA 335 Å intensity appeared to both precede and follow the AIA 193 Å intensity. This makes a strong argument for including a true high-temperature channel in the time lag analysis, such as images from the Hinode X-ray Telescope (XRT). The XRT response function for the thin beryllium filter is shown as a red line in the right panel of Figure 1. We also found locations (Points E and F) where the time lag measured between two channels was purely coincidental, 9

meaning the dominant structure emitting in the AIA 211 Å channel was different from the dominant structure emitting in the AIA 171 Å channel. For Point F, the peak of the crosscorrelation function was low enough to ignore this point, but for Point E, the high cross-correlation value indicated a true match. This false positive is difficult to detect in this time lag analysis, although it may be easier if the spatial location of the loops is considered. In standard loop analysis, the observer selects a loop that looks identical in multiple channels along its length. If these two structures along the line of sight had different projected geometries, it would reduce some of the line of sight confusion that occurs in time lag analysis. In both these cases, the time lag was long (greater than 5000 s). These long time lags are difficult to explain through impulsive heating alone (Lionello et al. 2016). In Viall & Klimchuk (2012), they argue that the positive time lags when comparing hot cold channel pairs indicate cooling due to impulsive heating. In this paper, we demonstrate that a similar time lag pattern can be achieved with quasisteady, highly stratified heating. Such heating drives the plasma in a loop to first be at high temperatures and low density. The temperature then slowly decreases as the density (and corresponding radiated energy) increases until the heating can no longer maintain the density and temperature in the loop. At this point, the plasma in the loop then dramatically cools and drains. The ordering in the lightcurves from high to low temperatures is due to the offset, in time, of the density and temperature cycles. The observed lightcurves will be dominated by the time when the temperature is decreasing and the density is high (for additional information, see Lionello et al. 2013; Mikić et al. 2013; 2014). The simulation used in this paper utilizes highly stratified heating. This type of heating is expected to be prevalent in the corona. Although highly stratified heating does not necessarily lead to thermal non-equilibrium solutions (it can also lead to steady solutions), along each line of sight shown in this image, we expect to cross at least one structure (more likely many structures) undergoing thermal non-equilibrium cycles, creating periodicities in the lightcurves. These types of periodicities have been observed in EUV images (Auchère et al. 2014, 2016; Froment et al. 2015). If there are periodicities, we should calculate high correlation coefficients between an integer number of cycles. The only reason we do not is because we limit the window of allowed time lags. In Viall & Klimchuk (2012), they limited the allowed time lag to ±2 hr, whereas in this paper we limit it to ±4 hr. The clear periods shown in Figures 5 and 9 are on the order of 6 hr. It is important to stress that the properties of AR 7986 that forms the basis of the 3D simulation are quite different from the properties of AR 11082 studied in the Viall & Klimchuk (2012) paper. The area where the magnetic field is larger than 50 G and less than 500 G is 2000 Mm 2 in AR 11082, while in AR 7986 it is 26,000 Mm 2. The maximum loop length in AR 11082 is roughly 400 Mm (determined from potential field extrapolations; see Lionello et al. 2016), while the loop lengths in the simulation reach 1100 Mm. Unfortunately, AR 7986 was observed before the launch of AIA, so similar time lag maps cannot be generated. For these reasons, the comparisons between the simulation and observations can only be qualitative. In the future, we plan to run a similar simulation of an active region observed with both SDO/AIA and Hinode/ XRT to make quantitative comparisons. The authors would like to thank the referee for helpful comments and Jim Klimchuk, Nicki Viall, and Frederic Auchère for many useful discussions. This work was supported by NASA s LWS, Theory, and Heliophysics Supporting Research Programs, NSF s Strategic Capabilities Program and the Center for Integrated Space Weather Modeling, and AFOSR. This paper is an outgrowth of the participation of A. W., Z.M., and R.L. in the 2011 Loops Workshop. REFERENCES Antiochos, S. K., MacNeice, P. J., Spicer, D. S., & Klimchuk, J. A. 1999, ApJ, 512, 985 Antiochos, S. K., & Klimchuk, J. A. 1991, ApJ, 378, 372 Antolin, P., Shibata, K., & Vissers, G. 2010, ApJ, 716, 154 Antolin, P., Vissers, G., Pereira, T. M. 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