catastrophic cooling of impulsively heated coronal loops - IOPscience

The Astrophysical Journal, 624:1080–1092, 2005 May 10 # 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.

CATASTROPHIC COOLING OF IMPULSIVELY HEATED CORONAL LOOPS Ce´sar A. Mendoza-Bricenõ Solar Physics and Upper Atmosphere Research Center, Department of Applied Mathematics, University of SheDeld, Hicks Building, Hounsfield Road, SheDeld S3 7RH, England, UK; and Centro de Fı´sica Fundamental, CFF, Facultad de Ciencias, Universidad de los Andes, Apartado Postal 26, La Hechicera, Me´rida 5251, Venezuela; [email protected], [email protected]

Leonardo Di G. Sigalotti Centro de Fı´sica, Instituto Venezolano de Investigaciones Cientı´ficas, IVIC, Apartado Postal 21827, Caracas 1020A, Venezuela; [email protected]

and Robert Erde´lyi Solar Physics and Upper Atmosphere Research Center, Department of Applied Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, England, UK; [email protected] Receivved 2004 September 29; accepted 2005 January 27

ABSTRACT The physical mechanisms that cause the heating of the solar corona are still far from being completely understood. However, recent highly resolved observations with the current solar missions have shown clear evidence of frequent and very localized heating events near the chromosphere that may be responsible for the observable high temperatures of the coronal plasma. In this paper, we perform one-dimensional hydrodynamic simulations of the evolution of a hypothetical loop model undergoing impulsive heating through the release of localized Gaussian energy pulses near the loop’s footpoints. We find that when a discrete number of randomly spaced pulses is released, loops of length L ¼ 5 and 10 Mm heat up and stay at coronal temperatures for the whole duration of the impulsive heating stage, provided that the elapsed time between successive heat injections is P175 and P215 s, respectively. The precise value of the critical elapsed time is sensitive to the loop length. In particular, we find that for increased loop lengths of 20 and 30 Mm, the critical elapsed time rises to about 240 and 263 s, respectively. For elapsed times longer than these critical values, coronal temperatures can no longer be maintained at the loop apex in spite of continued impulsive heating. As a result, the loop apex undergoes runaway cooling well below the initial state, reaching chromospheric temperatures (104 K) and leading to the typical hot-cool temperature profile characteristic of a cool condensation. For a large number of pulses (up to 1000) having a fully random spatiotemporal distribution, the variation of the temperature along the loop is highly sensitive to the spatial distribution of the heating. As long as the heating concentrates more and more at the loop’s footpoints, the temperature variation is seen to make a transition from that of a uniformly heated loop to a flat, isothermal profile along the loop length. Concentration of the heating at the footpoints also results in a more frequent appearance of rapid and significant depressions of the apex temperature during the loop evolution, most of them ranging from 1:5 ; 106 to 104 K and lasting from about 3 to 10 minutes. This behavior bears a tight relation with the strong variability of coronal loops inferred from SOHO observations in active regions of the solar atmosphere. Subject headinggs: hydrodynamics — methods: numerical — Sun: corona — Sun: transition region

1. INTRODUCTION

1994; Vekstein & Jain 2003)—as one likely mechanism for coronal heating. These events release about 109 times the energy of a typical large flare and so they are commonly referred to as nanoflares. Although they occur at small spatial and temporal scales, they can give rise to a coronal filling factor—defined as the ratio of the coronal volume radiating EUV and X-rays to the total volume—of order unity, leading to a relatively uniformly heated corona (Cargill 1993, 1994; Cargill & Klimchuk 2004). In addition to the nanoflare-reconnection model, there are several other mechanisms that have been proposed for explaining the coronal heating. For instance, recent interest has centered on the idea that energy can be transported into the corona by nonlinear Alfveń waves along the magnetic loop arcades (Hollweg et al. 1982; Mariska & Hollweg 1985; Hollweg 1992; Kudoh & Shibata 1999; Moriyasu et al. 2004). In particular, Kudoh & Shibata (1999) performed MHD simulations in which Alfveń waves are excited by photospheric random motions. They found that for motions with amplitudes greater than

In spite of the increasing number of available hydrodynamic and magnetohydrodynamic (MHD) models of coronal loop heating and high-resolution observations of the solar corona with the Soft X-Ray Telescope (SXT ) on board the Yohkoh satellite and the extreme-ultraviolet (EUV) spectrometers and imagers on board the Solar and Heliospheric Observatory (SOHO) and the Transition Region and Coronal Explorer (TRACE ) spacecraft, the heating mechanisms responsible for keeping the coronal plasma at a few millions degrees still remain elusive. Recent space-based observations have unequivocally revealed the presence of arcsecond to subarcsecond wide open (filamentary) and closed (looplike) magnetic structures in the solar corona. Further strong evidence for rapid and localized dynamic events in the corona and the chromosphere has favored the picture of microscale heating—the accumulation of numerous small flares due to reconnection of the magnetic field lines (Parker 1988; Cargill 1080

CORONAL LOOP HEATING 1 km s1, the plasma is heated to temperatures of 106 K through the transport of an energy flux by torsional Alfveń waves. More recently, Moriyasu et al. (2004) extended Kudoh & Shibata’s (1999) MHD simulations by including the nonlinear propagation of Alfveń waves along with the effects of heat conduction and radiative cooling in an emerging flux loop. They found that dissipation of Alfveń waves via mode-coupling with slow- and fast-mode waves and shock dissipation may reproduce the properties of observed micro/nanoflare events, suggesting that nanoflare heating may not be the result of magnetic reconnection events but rather of MHD shock events generated by nonlinear Alfveń waves. Clues to the coronal heating processes have also been inferred from the detection of nonthermal motions in the transition region and corona at both disk center (e.g., Chae et al. 1998a) and above the limb (e.g., Tu et al. 1998; Erde´lyi et al. 1998; Harrison & Hood 2002). Although the origin of nonthermal motions, as inferred from the broadening of the spectral emission lines, is unknown, it is often assumed that they are due to Alfveńic turbulence. Models of coronal loops heated by turbulence-driven Alfveń waves have recently been proposed by Dmitruk & Go´mez (1997, 1999), Chae et al. (2002), and Li & Habbal (2003). In particular, the latter authors have shown that if the nonthermal motions detected in the transition region are due to Alfveń waves and if these waves are dissipated by a turbulent cascade process, then they are capable of reproducing the coronal electron densities, temperatures, and flow speeds in the range inferred from observations (see Aschwanden et al. 2000; Chae et al. 2000; Winebarger et al. 2002). Further recent hydrodynamic simulations also include studying the structure and dynamical behavior of TRACE coronal loops undergoing transient heating on timescales comparable to the coronal radiative cooling time (3000 s) for these loops (Spadaro et al. 2003). It was found that transient heating may be the cause of persistent downflows with speeds from 5 to 20 km s1 at transition region temperatures, producing a net redshift at around 0.1 MK, in agreement with observations (Brekke et al. 1997; Chae et al. 1998b; Peter 1999). In addition, strong transient heating may induce catastrophic cooling to temperatures well below the equilibrium value, as reported by Schrijver (2001) from TRACE observations of quiescent solar coronal loops over active regions. The cooling loops frequently show Ly and C iv emission lines arising initially near the loop tops by the time strong downflows develop. The cool material often forms clumps that move at speeds of up to 100 km s1. Since it is very likely that actual loops do consist of a bundle of thin strands (Warren et al. 2003), the cooling is seen to progress with delays of the order of 103 s between neighboring strands, although it is unlikely that every single one of them is heated impulsively and sequentially. In this way, hot and cool loops are transiently outlined at essentially the same location. In contrast to previous models, recent calculations by Mu¨ller et al. (2004) have shown that catastrophic cooling, accompanied by strong downflows and transient brightenings of transition region lines, can also result from a loss of thermal equilibrium at the loop apex as a consequence of heating at the footpoints, even if the heating is constant in time. Hydrodynamic simulations of loop plasma subject to transient heating by Hansteen (1993), Sarro et al. (1999), Teriaca et al. (1999), and Roussev et al. (2001a, 2001b), and more recently by Bradshaw & Mason (2003), have shown that nonequilibrium ionization and energy loss through a time-dependent radiation function may explain the thermal broadening observed in some emission lines during explosive events, as well as reconcile theoretical models with observa-

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tions concerning the longevity of some loops detected in the TRACE filters. The actual timescale of the heating is unknown and may depend on the physical mechanism that is at work. Shorter timescales corresponding to highly impulsive heating are also supported by high-resolution SOHO and TRACE observations, which have revealed the occurrence of very small scale activity at transition region temperatures such as explosive events (e.g., Innes et al. 1997; Pe´rez & Doyle 2000; Patsourakos & Vial 2001), blinkers (e.g., Harrison 1997), and micro/nanoflares (Porter et al. 1987; Pe´rez et al. 1999; Erde´lyi et al. 2001). Aschwanden et al. (2001) measured the thermal structure of many loops using the TRACE satellite and showed that their observed temperature profiles are essentially isothermal. A comparison with calculated profiles using a static loop model led to the conclusion that heat deposition toward the loop footpoints could be responsible for the coronal plasma heating. Recently, Winebarger & Warren (2004) investigated the evolution of a set of cooling loops, finding that solutions with the same total energy deposition but different heating parameters share the same equilibrium point, defined as the time in the evolution of the cooling loop when its structure resembles that of a uniformly heated loop. They found that only before this time, when the loop is underdense and heats up, can its structure provide information on the heating parameters, while after this time, when the loop cools and becomes overdense, its evolution is essentially independent of the details of the heating. Since most TRACE loops are overdense, the above results imply that the heating parameters (i.e., the loop’s magnitude, location, and duration) cannot be determined from these observations. Numerical models aimed at studying the nature of explosive events and their contribution to the coronal heating mechanisms have been proposed by Sarro et al. (1999), whereas Walsh & Galsgaard (2001) studied the response of the coronal plasma to dynamical heat input generated by the flux-braiding model. More recently, Mendoza-Bricenõ et al. (2002, 2003) investigated the hydrodynamic behavior of closed magnetic loops undergoing impulsive heating near the footpoints. It was found that when a discrete number (P10) of pulses are injected either periodically or randomly in space with constant elapsed times of either 60 or 120 s, the average plasma temperature stays over a million kelvins for the duration of the impulsive heating, with approximate isothermal profiles along the upper, hot loop segments. These temperature profiles are consistent with a heating function that decays exponentially from the loop’s footpoints toward the apex, in good agreement with observations of TRACE loops (Aschwanden et al. 2001). In this paper, we extend our previous work to investigate the response of the coronal loop plasma to spatiotemporal microscale heating near the footpoints. First, we explore the effects of increasing the constant elapsed time between successive energy inputs on runaway cooling for a discrete number of randomly spaced pulses and varied total length of the loop. Second, we consider models with a large number (up to 1000) of pulses in which the energy releases are now applied randomly in space and time as well as in their occurrence at one or both footpoints. This gives rise to loop evolutions in which the heat injections are fully random and asymmetric. Finally, for these latter models we also consider the effects of varying the length of the bottom loop segments along which the localized pulses are randomly distributed. This allows for hypothetical loop models undergoing impulsive heating at the footpoints and across the transition region. In x 2 we write down the basic equations that are solved and describe the loop models. In x 3 we briefly

˜ O, SIGALOTTI, & ERDE´LYI MENDOZA-BRICEN

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summarize the results of previous simulations for a discrete number of pulses and describe the results of the new model calculations, while the conclusions are given in x 4. 2. HYDRODYNAMIC MODELS A coronal loop is modeled here as a semicircular magnetic flux tube of constant cross section anchored in the photosphere and oriented perpendicular to the solar surface. Motion of the plasma along the tube can be approximately described by solving numerically the standard set of transport equations for mass, momentum, and energy in one dimension, including the effects of heat conduction, radiative cooling, and heating. In Lagrangian form these equations are d ln @v þ ¼ 0; dt @s

dv @p ¼ þ g(s); dt @s

ð1Þ ð2Þ

dT ( 1) @v @ @T 2 ¼ p þ Q(T ) H(s; t) ; dt Rg @s @s @s ð3Þ where t is time, s is the curvilinear coordinate along the field line, is the (mass) density, v is the fluid velocity, T is the plasma temperature, p is the gas pressure, g(s) is the component of gravity along the field line, Q(T ) is the optically thin radiationloss function (Cook et al. 1989), H(s; t) is the coronal heating function, is the mean molecular weight, Rg is the gas constant, (=5/3) is the ratio of specific heats, and ¼1:0 ; 106 T 5= 2 ergs cm1 s1 K1 is the coefficient of thermal conductivity parallel to the magnetic field (Braginskii 1965). The gravitational acceleration term on the right-hand side of equation (2) is assumed to depend only on position along the semicircular loop and is given by s ; ð4Þ g(s) ¼ g cos L where g 2:74 ; 104 cm s2 is the solar surface gravity and L is the total length of the loop. Note that in the present models no reflection symmetry is enforced at the apex and so the evolution of the whole semicircle is calculated. Equations (1)–(3) are solved in coupled form with the equation of state p ¼ Rg T /. In all simulations the energy pulses are assumed to have a Gaussian shape along with an exponentially decaying amplitude in time, while their location is chosen randomly within a segment of length L extending from the loop’s footpoints (at s ¼ 0 and L). With these prescriptions, the spatial and temporal dependence of the heat deposition is in general given by H(s; t) ¼ h0 þ H0

n X

exp ½(t i )

i¼1

(s sl; i ) 2 (s sr; i )2 exp ; kl; i exp þ k ; r; i 2 2 ð5Þ where h0 ¼ 3:6 ; 104 ergs cm3 s1 is the uniform background heating rate, H0 ¼ 30 ergs cm3 s1 determines the maximum amplitude of the impulsive heating, 3:6 ; 106 cm is the spatial width of the heating, and ¼ ln (0:1)/t so that

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90% of the total energy is deposited during a finite time taken to be t ¼ 150 s. For the calculations of this paper, we use heat pulses of total energy Etot 1025 ergs. The elapsed time i between successive pulses is chosen randomly within the interval 20 s i 190 s. In addition, the pulses are centered at distances sl; i ¼ L ; RANl; i and sr; i ¼ L(1 L ; RANr; i ) from the left and right footpoints, respectively, where RANl and RANr define different sequences of random numbers between 0 and 1. This allows for asymmetrical heat injections near the footpoints in the sense that the spatial distribution of the pulses on one loop segment is never the same as that over the opposite loop segment. Finally, the parameters kl, i and kr, i are randomly chosen to be either 0 or 1 so that the injections can arbitrarily happen at one or both footpoints. Note that for each model calculation four distinct sequences of n ¼ 5000 random numbers each were employed to generate 5000 realizations of the parameters i , sl, i , sr, i , and (kl, i , kr, i) in equation (5). In all models with constant i (xx 3.1 and 3.2 below), we take kl; i ¼ kr; i ¼ 1 so that a pulse injection occurs simultaneously near both footpoints. In these cases, the first pair of pulses is always released at the beginning (t ¼ 0) of the calculation. To solve the hydrodynamic loop equations, we use the onedimensional finite-difference code employed in previous models of impulsively heated loops (Mendoza-Bricenõ et al. 2002, 2003) and adopt many of the same parameters and assumptions that were made in those earlier simulations with this code. We refer the reader to Sigalotti & Mendoza-Bricenõ (2003) for a detailed account of the numerical methods and tests. All of our model calculations begin with an initial loop configuration in hydrostatic equilibrium. The initial cool atmosphere (0.55 MK) is such that the base pressure is always at 0.1 dynes cm2 and is consistent with the value chosen for the background volumetric heating rate h0. As in previous models, the total length of our basic loop model is L ¼ 1:0 ; 109 cm (10 Mm), excluding the chromosphere. Additional calculations are presented for loops with total length L ¼ 5, 20, and 30 Mm. According to KjeldsethMoe & Brekke (1998), typical active region loops occur with altitudes above the limb of 25 to 100 Mm, corresponding to loop arcades of approximate lengths between 78 and 314 Mm. The lengths used in the present calculations are all below the above observable range and so they apply only to small transition region loops, which can just barely be spatially resolved with the currently available instruments. With the choice of L ¼ 5, 10, 20, and 30 Mm, the Gaussian width of the pulses is always a very small fraction (7:2 ; 103 for L ¼ 5 Mm and 1:2 ; 103 for L ¼ 30 Mm) of the total loop length, implying a very localized heating. Appropriate boundary conditions are applied at both footpoints by fixing the density and temperature there to their initial equilibrium values. Given that p / T , this results in a constant pressure at s ¼ 0 and L. In this way, the presence of a deep chromosphere is mimicked by evolving the velocity at the loop ends, thereby allowing for a flow of mass across the footpoints. All models are initialized the same way and the background heating is always maintained, in which case we expect the loop model to return to the equilibrium density and temperature associated with this heating rate. Models with a discrete number (=10) of randomly spaced pulses are all identical except for the constant elapsed time i between consecutive heat injections, which is varied from 60 to 240 s for the L ¼ 5 and 10 Mm loops and to higher values (up to 300 s) for the L ¼ 20 and 30 Mm cases. Finally, the model calculations with a large number (up to 1000 or more) of impulsive injections distributed randomly in space, time, and in their occurrence close to the footpoints all start with L ¼ 10 Mm, the only variations in the

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CORONAL LOOP HEATING

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Fig. 1.—Time variation of the apex temperature for loop models of total length (a) L ¼ 10 Mm and (b) L ¼ 20 Mm, undergoing impulsive heating through the release of 10 randomly spaced pulses near both footpoints. The elapsed time i between successive pulses is of 60 s (solid line), 120 s (dotted line), 180 s (dashed line), 220 s (dot-dashed line), and 240 s (triple-dot–dashed line) in (a) and 120 s (solid line), 180 s (dotted line), 240 s (dashed line), 245 s (dot-dashed line), and 260 s (triple-dot– dashed line) in (b). Catastrophic cooling to temperatures P104 K is evident for i ¼ 220 and 240 s in (a) and i ¼ 245 and 260 s in (b).

simulations being the seeds generating the four distinct sequences of random numbers employed in equation (5) and the length L/L (=0.1, 0.2, 0.3, or 0.5) of the bottom loop segments along which the pulses are injected. Models with varying sequences of the random numbers may well represent loops that are being impulsively heated at different levels of the solar activity. 3. RESULTS AND DISCUSSION 3.1. Previous Results for Impulsively Heated Loops In an earlier paper (Mendoza-Bricenõ et al. 2003), we investigated the evolution of a 10 Mm loop when 10 randomly spaced pulses were released near the footpoints over loop segments of length 0.1L and with constant elapsed times between successive injections of either 60 or 120 s. In these models, as well as in the models of x 3.2 below, we mean by a pulse two independent and simultaneous heat injections, one for each footpoint. The initial stage of the evolution is always dominated by the impulsive heating that causes the loop’s temperature to rise steeply toward typical coronal values, with the loop plasma being maintained at such temperatures for the entire duration of the impulsive heating (i.e., 540 s for the i ¼ 60 s model and 1100 s for the i ¼ 120 s case). It was found that the instantaneous temperature profiles of the evolving loops were characterized by the appearance of localized thermal bumps along their hot coronal segments. Such bumps bear a strong resemblance with the intermittent behavior detected by Patsourakos & Vial (2002) from their analysis of light curves obtained in the O iv and Ne viii transition region and low corona emission lines, as recorded by SOHO SUMER in a quiet-Sun region. These two lines form at temperatures of 105.2 and 105.8 K, respectively, and hence they provide a good temperature coverage of the transition region and lower corona. The observed bursts exhibit a rather random temporal variation and are presumably due to intermittent energy release followed by its dissipation. In this way, Patsourakos & Vial (2002) concluded that the intermittence of the examined signals is related to well-known types of transition events in the corona such as explosive events, blinkers, and micro/nanoflares. The variation of the apex temperature with time for these two model calculations is shown in Figure 1a by the solid line for

the i ¼ 60 s evolution and the dotted line for the i ¼ 120 s case. We can see that for the entire duration of the heating, the apex temperature oscillates with time around typical coronal values. Once the impulsive heating is turned off, the loop cools down and returns to its initial stationary equilibrium state because of the applied uniform background heating. The results also indicate that when the elapsed time i is increased, the loop achieves comparatively lower temperatures during the impulsive heating stage as shown by the upper two curves in Figure 2, which depict the integrated temperature variation along the loop length for the two evolutions described here. This behavior clearly suggests a dependence of the overall loop temperature on the time interval separating consecutive pulses. In other words, there should exist a critical elapsed time beyond which the loop can no longer be maintained at coronal temperatures (see x 3.2 below). Note that the profiles in Figure 2 were obtained by integrating the loop temperature over the duration of the impulsive

Fig. 2.—Time-integrated temperature profiles for the evolutions shown in Fig. 1a, when the elapsed time between successive heat injections is of 60 s (solid line), 120 s (dotted line), 180 s (dashed line), and 240 s (dot-dashed line). The temperature is given in units of 106 K. In all cases, the time integration was performed over the entire duration of the impulsive heating stage.

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Fig. 3.—Evolution of the impulsively heated loop models of Figs. 1a and 2 in the ( apex , Tapex )-plane. The dashed line in each panel defines the locus of a uniformly heated, static loop of total length 10 Mm as given by eq. (6). The tracks to the right of this line give the loop evolution during its heating phase, while those to the left correspond to the loop evolution during its cooling phase. The starting point of the evolution, corresponding to the initial state, is indicated by the arrow in each panel. When the elapsed time exceeds the critical value i 215 s, the loop apex undergoes catastrophic cooling (bottom right).

heating stage (i.e., 540 s for i ¼ 60 s, 1080 s for i ¼ 120 s, 1620 s for i ¼ 180 s, and 2160 s for i ¼ 240 s). In a recent paper, Winebarger & Warren (2004) calculated the cycle that an impulsively heated loop goes through when only one pulse (of duration 500 s) is released in terms of a phase diagram, where the apex density is plotted as a function of the apex temperature (see also Brown 1996, where a similar analysis is carried out for loops evolving under a temporally varying heating rate). In particular, the former authors found that once the impulsive heating is turned off, the loop begins to cool mainly as a result of the establishment of a conductive flux through the loop’s footpoints. During this conductive cooling phase, the evolution will eventually pass through an equilibrium point in which both the apex density and temperature are consistent with the structure of a uniformly heated loop (see their Fig. 1). Physically, this time marks the precise instant in the evolution when the conductive and radiative cooling times become the same. Past this point, the loop energetics is mainly dominated by radiative cooling processes. As a consequence the loop becomes overdense and continues to cool down until it eventually returns to its initial atmosphere, leading to completion of a closed path or cycle in the (apex , Tapex )-plane. In addition, they showed that loops heated with the same total energy share the same equilibrium point and that changes in the

heating parameters will affect their path in the phase-plane evolution only during the initial heating and conductive cooling stages. In contrast with Winebarger & Warren (2004), our simulations apply to multiple heating events and so we cannot expect the loop evolution to be described by a single closed contour in the (apex , Tapex )-plane. Also, note that for the present simulations the pulses have a duration of 150 s each, much shorter than that of the single pulse employed by Winebarger & Warren (2004). The top left and top right panels of Figure 3 show the phase-plane tracks for the 10 Mm loop evolutions with i ¼ 60 and 120 s, respectively. Starting from the initial equilibrium point (indicated by the arrow), the evolution always proceeds to the right of this point. The dashed line defines the locus for a static loop of the same length that is being heated uniformly and steadily. A relationship for this line can be derived by combining the scaling law given in Serio et al. (1981), relating the temperature to the loop length, with relation (4) of Winebarger & Warren (2004) to give m T2 kp 2 1 10 H apex exp 0:06L þ ; apex 7:29 ; 10 s H kp kB L L ð6Þ

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Fig. 4.—Evolution of the temperature when 10 pairs of heat pulses are injected randomly over a segment of length 0.1L from each footpoint in a loop model of total length L ¼ 10 Mm. The time interval between successive injections is of 180 s, with the first pair of pulses being released at the beginning of the evolution (t ¼ 0). A projected contour plot is shown for the temperature variations, where only contour lines for 0.5, 1.0, and 1.5 MK are shown. After the energy injections are turned off, the loop returns on an oscillatory way to its initial equilibrium state.

where mH is the mass of the hydrogen atom, kB is Boltzmann’s constant, kp ¼ 2mH kB T g(s) is the pressure scale height, and sH is the heating scale height, assumed to be infinite for a uniformly heated loop. In all panels of Figure 3, the region to the left of the dashed line is for a cooling loop and that to the right is for a heated loop. When the first pair of simultaneous pulses is turned on (at t ¼ 0), the loop is heated in much the same way as described by Winebarger & Warren (2004). That is, the temperature rises rapidly to typical coronal values, accompanied by an increased pressure gradient that drives an upflow of material into the corona, a process known as chromospheric evaporation. As a result, the density begins to increase. As soon as the first pulse is turned off, the loop cools because of heat conduction until the advent of the second pair of pulses, which then reheats the loop. During the conductive cooling stage, the path turns away from the equilibrium point and descends while remaining in the heated region to the right of the dashed line. This causes the first cycle to overshoot the initial equilibrium state and bifurcate into a new cycle. This process will then be repeated until the last pair of pulses is released. This repetitive cycling explains the oscillatory behavior of the apex temperature (around 106 K) with time evident in Figure 1a. Note that the shape of the paths after successive bifurcations changes. This occurs because of the spatial random distribution of the heat injections that causes the cycles to bifurcate at different points in the phase plane. Only when the last pair of pulses is turned off does the path go through the equilibrium point and enter the cooling region where radiative cooling dominates over conductive cooling. During this phase a downflow sets in through the footpoints and the loop becomes underdense. The downflowing plasma increases the radiative losses and cools the loop. Because of the background heating, the loop returns to its initial atmosphere and so the evolution path closes. Note that when the elapsed time is increased, the last two (for i ¼ 120 s) and three (for i ¼ 180 s) cycles go through the equilibrium point as radiative cooling becomes sig-

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Fig. 5.—Same as Fig. 4, but with an elapsed time of 220 s between successive heat injections. Inspection of the temperature contour plot shows the development of a cool segment around the apex loop region. Only contour lines for 0.5, 1.0, and 1.5 MK are shown.

nificantly more important close to the end of the impulsive heating stage for longer elapsed times. Since all loop models are heated with pulses of the same constant maximum amplitude (H0) and share the same spatial profiles of the heating, it follows from Figure 3 that the shape of the evolution paths is sensitive to the elapsed time in both the underdense and overdense phases of the loop evolution. Therefore, in the presence of multiple heating events, observations of the variations in the local thermodynamic properties can provide adequate information to discriminate between different heating scenarios, as opposed to the conclusion of Winebarger & Warren (2004) for a single heating event. 3.2. Catastrophic Cooling of Impulsively Heated Loops In order to investigate the existence of a critical value of i , we have performed further simulations of the 10 Mm loop model along a sequence of increasing constant elapsed times up to i ¼ 240 s. In particular, Figure 4 shows the evolution of the loop temperature when the time interval between the heat pulses is of 180 s. In this case, the impulsive heating can maintain the overall loop temperature at around 1.5 MK for about 1700 s. As expected, the loop temperature achieved is lower compared to that of the previous two models having shorter elapsed times, as we can see from the resulting integrated temperature profile corresponding to the dashed line in Figure 2. After 1700 s, the loop cools down and returns to the initial equilibrium atmosphere, which is maintained until the termination of the calculation at 4000 s. The contour plot in Figure 4 depicts the temperature variation during the impulsive heating phase. For the sake of clarity, we show contour lines only for temperatures of 0.5, 1.0, and 1.5 MK. It is interesting to note the oscillatory behavior of the 0.5 MK contour line after about 2000 s, when the perturbations are relaxed and the loop returns to its equilibrium state. Similar qualitative trends are found for longer elapsed times up to the critical value of i 215 s. For values higher than this, the loop is maintained at coronal temperatures only for the duration of the first two pulses. Soon after, the top temperature suddenly drops below the initial state toward typical chromospheric values. Figure 5 displays the details of the evolution when i ¼ 220 s. We can see from the contour plot that a

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Fig. 6.—Same as Figs. 4 and 5, but with an elapsed time of 240 s between successive heat injections. The contours clearly indicate that about 20% of the entire loop has suffered catastrophic cooling around the apex region. Only contour lines for 0.5, 1.0, and 1.5 MK are shown.

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temperature depression around the apex occurs after about 550 s in the evolution. The runaway cooling of the top regions is better visualized in Figure 1a (dot-dashed line), where the time variation of the apex temperature is depicted for models with increasing elapsed times. For the i ¼ 220 s case, cooling at chromospheric temperatures is kept for approximately 1000 s, during which seven pulses have been released. However, the last three pulses are able to reheat the plasma and maintain it at coronal temperatures for 350 s (see Fig. 1a) before returning to the stationary initial equilibrium, which is then maintained for the remainder of the evolution. When the elapsed time is further increased to i ¼ 240 s, the evolution produced is qualitatively similar (see Fig. 6). In this case, however, the cooling of the top loop segments lasts for a longer period (1900 s) until the last pair of pulses is released (see Fig. 1a). The spatial extent of the cool region around the loop apex is clearly evidenced by the temperature contour plot. We estimate that about 20% of the loop undergoes runaway cooling. Figures 7 and 8 show details of the temperature and velocity profiles at selected times in the evolution of Figure 6. After the injection of the first pair of pulses at t ¼ 0, the plasma heats up and in tens of seconds it reaches temperatures of 1:5 ; 106 K (see the snapshot at t ¼ 36:48 s in Fig. 7). This rapid rise of the overall loop temperature is accompanied by an increased pressure gradient along the loop legs that drives a

Fig. 7.—Temperature profiles at selected times for the evolution shown in Fig. 6.

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Fig. 8.—Velocity profiles at selected times for the evolution shown in Fig. 6.

transient upflow of chromospheric material into the corona, with a consequent increase of the apex density. After about 37 s, the upper loop temperature drops to 5:0 ; 105 K because of conductive cooling (see the snapshots at t ¼ 78:30 and 240.40 s in Fig. 7). Since heat conduction is very efficient at high temperatures, the temperature profile in the upper, hot parts of the loop is kept approximately flat. During the conductive cooling phase, the pressure gradients around the loop apex decrease and eventually become much smaller than the corresponding hydrostatic equilibrium values. As a result the gravitational forces are no longer balanced by the pressure gradients and so the apex material is accelerated on its way down along the loop legs, resulting in flow speeds that continually increase toward the transition region with peak values up to v 200 km s1 (see the snapshots at t ¼ 36:48 and 78.30 s in Fig. 8). Note that a downflow in the left loop leg has v < 0, while a downflow in the right loop leg is characterized by v > 0. The converse is true when an upflow sets in. When the downflow encounters the transition region, it is strongly decelerated by the pressure gradients of the underlying plasma, as shown by the rapid drop of the velocity close to loop’s footpoints. A zone of compression is also evident at s ¼ 2 and 8 Mm at t ¼ 36:48 s, which soon dissipates. When the second pair of pulses is switched on, the loop reheats and an upflow sets in again because of the rapidly increasing pressure gradients. The snapshot at t ¼ 240:40 s in Figure 8

shows the onset of the upflow when the peak velocity is v P 100 km s1 close to the footpoints. Note that at this time, the central plasma is still downflowing with maximum velocities of v 80 km s1 close to the zone of compression around s ¼ 2 and 8 Mm. After having reached coronal temperatures, the loop cools down because of heat conduction (t ¼ 313:89 and 430.49 s in Fig. 7) and a downflow is established again (t ¼ 313:89 s in Fig. 8). Therefore the loop density decreases as the plasma drains through both footpoints. Note that by 430.49 s the central part of the loop has cooled down to temperatures of 5:0 ; 105 K. In this way, a process involving chromospheric evaporation, plasma condensation, and draining repeats at relatively high temperatures and low densities (P1:0 ; 1014 g cm3) during the first heating phase, which lasts approximately 480 s. After the release of the third pair of pulses, only the plasma close to the footpoints is heated (see the snapshot at t ¼ 486:61 s in Fig. 7). A downflow behind and an upflow ahead of the heated region is induced by the strong temperature gradients on both sides of it, with the consequent formation of two dense and cool blobs that move up with a mean velocity of v 25 km s1. The snapshot at t ¼ 486:61 s in Figure 8 clearly shows that the flow speed immediately behind the ascending blobs (at s 0:57 and 9.43 Mm) reaches peak values of 80 km s1, while ahead of them (at s 1:14 and 8.86 Mm) it is very close to zero. Because

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of these rather strong velocity gradients the blobs contract as they move up. By about 650 s they meet the loop apex, causing it to undergo runaway cooling to temperatures of T P 104 K (see Fig. 1a). The further evolution is characterized by the central blob contracting and expanding repeatedly as the dynamics along the loop legs alternate between upflows and downflows due to the impulsive heating, which lasts until 2160 s. In particular, the cool condensation contracts because of the large temperature gradients at its edges. Meanwhile, the blob survives because radiative cooling dominates over loop heating. The pressure built during the contraction is rapidly lost because of the radiative cooling, and the blob expands (see the snapshots at t ¼ 830:65 and 1582.62 s in Fig. 7). During this process the density of the cool summit oscillates between 3:2 ; 1013 and 4:0 ; 1014 g cm3, whereas its temperature oscillates between 3:2 ; 103 and 1:0 ; 104 K. As long as the heating is switched off, the pressure gradients in the upper parts of the loop quickly decrease because of the continued radiative losses and the cool plasma slides down on both sides of the loop, draining through the footpoints with flow velocities in the wake of the falling plasma of up to v 20 km s1. While being evacuated, the overall loop cools to chromospheric temperatures (see the snapshot at t ¼ 2430:55 s in Fig. 7). After about 2400 s, it recovers from catastrophic cooling (see the snapshots at t ¼ 2880:20 and 3000 s in Fig. 7) and enters a quiet, warm phase, during which the flow velocity never exceeds v 15 km s1 (see the snapshots at t ¼ 2430:55 and 2880.20 s in Fig. 8). Immediately after the two dense blobs have drained through the footpoints, the loop starts reheating because its heat capacity is very low. A weak reflected shock forms on both sides, followed by a phase of chromospheric evaporation that refills the depleted loop with plasma. At about 3000 s, it returns to its initial equilibrium state, which is then maintained for the remainder of the evolution because of the applied background heating. The evolutionary tracks in the phase plane for the i ¼ 180 and 240 s models are shown in the bottom left and bottom right panels, respectively, of Figure 3. Since i ¼ 180 s is below the critical value for catastrophic cooling, the paths for this model are qualitatively similar to those displayed in the top panels for i ¼ 60 and 120 s. Conversely, the evolution of the i ¼ 240 s loop model, which undergoes catastrophic cooling, follows a qualitatively different track in the phase plane. In this case, as the loop is heated by the first two pairs of pulses, the path cycles once to the right of the dashed line without crossing it and bifurcates far away from the initial state. Because of the longer elapsed time for this model, conductive cooling has enough time to make the path cross the dashed line and enter into the radiative cooling region for the next two cycles by the time the second pair of pulses has already been released. Note that the location of the equilibrium point for these two cycles is different because of the randomness in the spatial distribution of the pulses. At this point, radiative cooling proceeds faster than impulsive heating and the loop apex cools down below the initial state. As a consequence, the path leaves the cycling zone and rises in the phase plane, following an almost straight line toward chromospheric densities and temperatures. During this cool phase, the oscillations of the apex density and temperature with time (see Fig. 1a) are represented in the phase plane by the tracks turning repeatedly before descending to the initial equilibrium state as a result of the background heating. The critical elapsed time beyond which runaway cooling occurs is seen to depend on the total loop length. In particular, we find that loop models with the same heating parameters and increasing length experience catastrophic cooling at pro-

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gressively higher critical elapsed times. For loops of length L ¼ 5 Mm, the critical elapsed time is i 175 s. This value rises to about 215, 240, and 263 s for loops of length 10, 20, and 30 Mm, respectively. Figure 1b shows the time evolution of the apex temperature in the 20 Mm loop model for varied constant elapsed times between consecutive pulses. Loops with the same subcritical value of i and varied length evolve in a qualitatively similar fashion, as we can see by comparing the time variations of the apex temperature depicted in Figures 1a and 1b for i ¼ 120 and 180 s. Since the timescale for heat conduction is cond / L2 , conductive cooling is much more effective in the shorter than in the longer loops. This explains why during the heating stage the apex temperature in the 20 Mm loop oscillates about higher coronal values compared to the 10 Mm model. Similar trends are also seen when comparing the other model evolutions for L ¼ 5 and 30 Mm. Larger values of the critical elapsed time are therefore required in the longer loops to allow conductive cooling to have enough time to bring the loop apex past the equilibrium point, where radiative losses dominate and eventually induce the runaway cooling of the loop summit. The present model calculations show that for given heating parameters, increasing the elapsed time beyond a certain critical value may induce the impulsively heated loop to evolve dynamically toward a hot-cool temperature profile similar to that calculated by Steele & Priest (1990) for prominence formation. This is shown by the dot-dashed curve in Figure 2, which depicts the integrated temperature variation along the loop length for the 10 Mm model with i ¼ 240 s. This curve reproduces the typical hot-cool profile characteristic of a cool condensation. 3.3. Multiple Spatiotemporal Impulsive Heating We now consider the evolution of the 10 Mm loop model when it is heated by a large number of pulses distributed randomly in space and time near the footpoints. In contrast with the previous models, the pulses may not be simultaneous in the sense that they could appear randomly at one or both footpoints. The randomness in time is obtained by choosing arbitrarily the elapsed time between consecutive injections from the interval 20 s i 190 s. These values are all below the critical one for which we would expect the loop to undergo catastrophic cooling to chromospheric temperatures. The next calculations in this series will explore the effects of extending the above interval to include higher elapsed times and changing the semicircular geometry into a semielliptical one. Three model calculations were first tried with the variant that the four sets of random numbers used to determine the parameters i , sl,i , sr,i , and (kl,i , kr,i) in equation (5) were all different. In all three cases a segment of length 0.1L from the footpoints was employed to randomly distribute the pulses. Figure 9 shows the resulting temperature evolution for one of these loop models up to 15,000 s, when more than about 1000 pulses have been released. We can see that the hottest segments of the loop reach temperatures higher than 1.5 MK, which are maintained until the termination of the calculation. In addition, numerous localized thermal bumps are visible, indicating the location of the energy pulses. When the randomness of the impulsive heating is varied, the evolution undergoes only quantitatively small changes. For instance, an obvious change involves differences in the spatiotemporal distribution of the thermal bumps that certainly modify the instantaneous shape of the temperature profiles when compared at identical evolutionary times. The variation of the apex density and temperature with time for the three model calculations is shown in Figure 10. One important feature in these plots is the sporadic appearance of temperature drops accompanied

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Fig. 9.—Evolution of the loop temperature when a large number (up to 1000 or more) of heat pulses with a random distribution in space and time and in their occurrence at one or both footpoints are injected over a segment of length 0.1L from the feet in a loop model of total length L ¼ 10 Mm.

by corresponding density rises. That is, during its impulsive heating the loop apex suddenly cools to temperatures k104 K and reheats to coronal temperatures in a very short timescale. These variations in the local thermodynamic properties may be related to the observed rapid time variability of coronal loops detected in active regions of the solar atmosphere, which involves temperature variations in the interval from 104 to

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Fig. 11.—Time-integrated temperature profiles for the model evolution of Fig. 9, when the length of the bottom loop segments over which the pulses are randomly distributed is 0.1L (solid line), 0.3L (dotted line), and 0.5L (dashed line). The temperature is given in units of 106 K. In each case, the time integration was performed over the whole evolution (15,000 s).

2:7 ; 106 K (Kjeldseth-Moe & Brekke 1998). The predicted timescales for these rapid variations are of 3–10 minutes, which is toward the lower end of the range inferred observationally (10–30 minutes). This implies that a loop at a given temperature that is missing in one location at a particular time may be present at another time. Also note that the number of the temperature depressions is related to the spatiotemporal dependence of the impulsive heating.

Fig. 10.—Variation of the apex density (left) and apex temperature (right) with time for three loop models of total length L ¼ 10 Mm each, undergoing impulsive heating through the release of a large number (1000 or more) of pulses. The spatiotemporal random distribution of the pulses differ for each model. In all cases, the pulses were injected over segments of length 0.1L from the loop’s footpoints. The top panels correspond to the model evolution shown in Fig. 9.

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Fig. 12.—Same as Fig. 9, but with the heat injections now happening over a segment of length 0.3L from the loop footpoints.

We next consider the effects of varying the length L of the bottom loop segments along which the pulses are randomly distributed for the same evolution model of Figure 9. In particular, Figure 11 depicts the integrated temperature profiles when L/L ¼ 0:1 (solid line), 0.3 (dotted line), and 0.5 (dashed line). We can see that the form of the temperature variation is highly sensitive to the spatial distribution of the heating. As a consequence of distributing the pulses on a broader region, hotter loops are produced, as confirmed by the L/L ¼ 0:3 and 0.5

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calculation models. We also note that the upper, hot loop segments become progressively less flat when the impulsive heating is more broadly distributed along the loop. In particular, the form of the temperature variation for the L/L ¼ 0:5 loop model is similar to that corresponding to a uniformly heated loop, as described by Priest et al. (1998) from X-ray observations of the diffuse corona. Moreover, the results also imply that a quasiisothermal temperature distribution along the loop length is a clear signature of the heating being more strongly concentrated at the footpoints, as deduced by Aschwanden et al. (2001) from observations of TRACE loops. In addition, the temperature evolution of the L/L ¼ 0:3 model is shown in Figure 12. A comparison with Figure 9 confirms the much higher overall loop temperatures achieved by this latter model. Finally, we note that when the heating is less concentrated at the footpoints, the occurrence of the rapid temperature depressions strongly diminishes, as we can readily see by comparing the evolutions depicted in Figure 13. This result is consistent with the observational lack of detected strong variability in the X-ray and EUV emission lines at higher coronal temperatures (Kjeldseth-Moe & Brekke 1998; Schrijver 2001). 4. CONCLUSIONS In this paper we have presented one-dimensional hydrodynamic simulations of the evolution of an initially intermediatetemperature (0.55 MK), equilibrium coronal loop model that has been heated impulsively near the footpoints. Five independent sequences of model calculations were considered. The first four sequences define a collection of loop models with a discrete number (=10) of randomly spaced pulses and varying constant elapsed time between successive pulses, which differ

Fig. 13.—Variation of the apex density (left) and apex temperature (right) for the same model of Fig. 9, but with varying lengths of the bottom loop segments along which the pulses are randomly spaced. Three model calculations are shown for 0.1L (top), 0.2L (middle), and 0.3L (bottom). The bottom panels correspond to the model evolution shown in Fig. 12.

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only in the total loop length (L ¼ 5, 10, 20, or 30 Mm). The fifth sequence uses a 10 Mm loop model with a larger number (up to 1000) of heat pulses distributed randomly in space and time and in their occurrence at one or both footpoints, with the only variations being either the form of the spatiotemporal pulse distribution or the length of the bottom loop segments along which the pulses are injected. Therefore, the models may apply to coronal loops undergoing multiple localized heating at their footpoints and across the transition region. We start by summarizing the relevant conclusions for the first four sequences of models. It was found that successive microscale energy inputs are quite capable of heating up the loop plasma to typical coronal temperatures (106 K), which are, in general, maintained for the whole duration of the impulsive heating. The time integration of the evolving loop temperature results in profiles that are approximately isothermal along the hot loop segments with exponential decays toward both footpoints, in good agreement with observations of TRACE loops (Aschwanden et al. 2001). In addition, the presence of randomly distributed thermal bumps along the hot segments of the evolving loops may well explain the intermittent behavior detected by Patsourakos & Vial (2002) from SOHO SUMER observations of O iv and Ne viii transition regions and low corona emission lines formed at temperatures of 105.2 and 105.8 K, respectively. As long as the elapsed time between successive pulses is increased in the 5 and 10 Mm models to 175 and 215 s, respectively, the loop heats up toward progressively lower temperatures. For elapsed times longer than these critical values, coronal temperatures can no longer be maintained as radiative cooling proceeds faster than impulsive heating. As a consequence, the loop apex undergoes catastrophic cooling well below the initial state to typical chromospheric temperatures (104 K). The precise value of the critical elapsed time is seen to increase with increasing total loop length. In particular, catastrophic cooling in the 20 and 30 Mm loop models occurs at about 240 and 263 s, respectively. Further calculations in this series will focus on exploring how the critical elapsed time scales with loop length for much longer loops. One implication of the present results is that for fixed heating parameters, microscale heating events separated by time intervals longer than a certain critical value may induce impulsively heated loops to evolve dynamically into the hot-cool temperature profiles that are typical of a cool condensation (Steele & Priest 1990), thus providing a viable mechanism for prominence formation. In contrast with recent findings of Winebarger & Warren (2004) for a single heating event, we found that in the presence of multiple heating events, not only the initial heating and conductive cooling phases but also the radiative cooling phase of the loop’s evolution are sensitive to the details of the impulsive heating. In particular, the loop evolutionary tracks in the (apex,

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Tapex )-plane are strongly dependent on the spatiotemporal distribution of the heat pulses. In this way, TRACE observations of a loop during its radiative cooling phase, when it becomes overdense, can still give specific information on the heating parameters, provided that the loop was heated impulsively by multiple events. Making a direct analogy with the observations, the above statements are equivalent to saying that all the simulations carried out here would produce apex intensities as a function of time that will look different in the hotter (195 or 286 8) and cooler (171 8) TRACE filters. Finally, loop simulations with a large number (1000 or more) of pulses having a fully random spatiotemporal distribution confirm previous findings that the plasma would stay at coronal temperatures during the impulsive heating stage. Variations in the randomness of the heat releases produce qualitatively similar evolutions, differing mainly in the spatiotemporal distribution of the localized thermal bumps that appear randomly along the hottest loop segments. The model calculations also predict the occurrence of sporadic and very rapid temperature depressions near the loop apex, which are always accompanied by equally rapid rises of the apex density. These depressions may involve strong temperature variations, most of them from 1:5 ; 106 to 104 K, which may last from about 3 to 10 minutes, and their number may be sensitive to the details of the spatiotemporal distribution of the microscale heating. This behavior may be related to the observed rapid time variability of coronal loops inferred from SOHO-CDS observations in active regions of the solar atmosphere (Kjeldseth-Moe & Brekke 1998; Schrijver 2001). Moreover, when the pulses are less concentrated near the loop’s footpoints, the evolution produces hotter loops and progressively less flat temperature profiles in the upper parts of the loop, along with an appreciably reduced number of the temperature depressions. This latter feature is consistent with the observational lack of strong variability at very high coronal temperatures (Kjeldseth-Moe & Brekke 1998; Schrijver 2001).

We thank the referee, D. Mu¨ller, for his comments and suggestions that have improved the style and quality of the manuscript. C. A. M.-B. thanks the CDCHT of the Universidad de los Andes for financial support. C. A. M.-B. and L. G. S. are indebted to the Venezuelan Fondo Nacional de Ciencia, Tecnologıá, e Innovacioń (Fonacit) for partial support, while R. E. acknowledges M. Ke´ray for patient encouragement and the NSF, Hungary (OTKA, reference number T043741). We also acknowledge the Royal Society (UK) and the Particle Physics and Astronomy Research Council (PPARC), UK. This paper is financially supported by the Instituto Venezolano de Investigaciones Cientı´ficas (IVIC).

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