CORONAL RAIN AS A MARKER FOR CORONAL HEATING MECHANISMS

Coronal loops are dynamical entities that exhibit heating and cooling processes constantly. Most of them are actually far from being in hydrostatic equilibrium (Aschwanden et al. 2001). The dynamical nature of loops can manifest itself in observations in a variety of forms, among which propagating intensity variations is a common one. Indeed, intensity variations traveling along coronal loops have been frequently observed in many different wavelengths, in hot coronal EUV lines as well as chromospheric cool lines. The agents producing these intensity variations can be either propagating waves or flows along coronal loops, and observers can have a hard time differentiating them despite their very different physical nature. Coronal rain is an example of intensity variations caused by flows, and seems to be a phenomenon of active regions, where loops are dense. Due to the low temperatures, the condensations constituting coronal rain appear as bright emission profiles in chromospheric lines such as Hα or Ca ii H and K when observed at the limb (De Groof et al. 2004; Figure 1) or as dark absorption profiles when observed on disk. If observed in EUV wavelengths, these propagating blobs appear as dark features (Schrijver 2001).

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Although the observational history of coronal rain traces back to the early 1970s (Kawaguchi 1970; Leroy 1972), not much research has been carried out on this spectacular phenomenon, maybe because it was first considered to be linked directly to prominences, as matter detaching and falling, and thus not a separate phenomenon by itself. It was not until recently that coronal rain was shown to correspond to flows of cool chromospheric-like plasma in a hot coronal environment falling down along coronal loops, not linked to prominences or solar flares and neither to waves (De Groof et al. 2004, 2005).

The previous papers report simultaneous observations in EIT 304 Å, Hα images from Big Bear Solar Observatory, and in 171 Å (TRACE) of intensity variations propagating along a coronal loop. Following a detailed analysis, they put forward a series of points through which a differentiation between waves and flows (cool plasma condensations, coronal rain) can be done. Propagating waves frequently reported in SOHO/EIT 195 Å and TRACE 171 Å appear as low min-to-max intensity variations as compared to coronal rain. Most observed waves correspond to sound modes propagating at constant speed corresponding to the local sound speed of the corona, ∼150 km s⁻¹. On the other hand, cool plasma condensations are seen to accelerate while falling along coronal loops to velocities above 100 km s⁻¹. Furthermore, waves have only been seen to propagate upward, within the first 20 Mm of loops, after which they are usually damped, while cool condensations have only been seen to fall down along the loops from coronal heights. The falling speeds of the condensations have been reported to be lower than free fall speeds, due probably to the gas pressure along the loop, which increases considerably below the transition region. In some cases, continuous flows from one footpoint to the other are also observed (Doyle et al. 2006).

Antiochos et al. (1999) propose a common physical mechanism for coronal rain and prominence formation. Ranges for temperatures and densities seem to be shared by both mechanisms, as well as the location of formation, high in the corona. While a prominence forms and is supported by the magnetic field for a timescale of days, coronal rain occurs on a timescale of minutes. It was shown that prominences may not need to rely on the geometry of the magnetic field to form (presence of "dips"), contrary to the general belief. They showed that a coronal loop whose heating is concentrated toward the footpoints is subject to a thermal instability in the corona, which dramatically cools down to chromospheric temperatures in timescales of minutes once the density and the temperatures have reached critical values. This phenomenon was termed "catastrophic cooling" and has so far gained acceptance as possible explanation for coronal rain. Schrijver (2001) estimates that loop bundles in the interior of an active region undergo catastrophic cooling on average once every 2 days, while in a decayed bipolar region that time interval is approximately a week. On the other hand, numerical simulations have pointed out that catastrophic cooling acts in the corona in a cyclic manner whose period is on the timescale of hours. This periodicity depends on geometrical aspects such as the loop length and the heating scale height (Mendoza-Briceño et al. 2002, 2005; Mendoza-Briceño & Erdélyi 2006), and can be obtained even with a time-independent heating (Müller et al. 2003, 2004). In the present paper, we also report short timescales for catastrophic cooling. Furthermore, on-going observational work with the Crisp spectropolarimeter (CRISP; Scharmer et al. 2008) of the Swedish 1 m Solar Telescope (SST; Scharmer et al. 2003) indicates that coronal rain is a fairly more common phenomenon of active regions than previously thought (P. Antolin et al. 2010, in preparation).

These results show that coronal rain can be a rather important phenomenon due to its link with coronal heating. Indeed, it is a characteristic of the heating mechanism itself. At first glance, coronal rain may appear as a random failure of the coronal heating mechanism to heat the loops in active regions. However, as previously stated, this phenomenon seems to act in the corona in a cyclic manner. Furthermore, the simulations point to the necessity of specific spatially dependent heating in order to allow the catastrophic cooling leading to coronal rain. This cooling phenomenon then seems to be deeply linked to the unknown coronal heating mechanism. Since it is a fairly easily observable phenomenon due to the large velocities and density variation (hence clear Doppler-shifted emission or absorption profiles), it may act as a marker of the operating heating mechanism in the loop. It has been shown, for instance, that footpoint-concentrated heating may lead to catastrophic cooling if the heating scale height is sufficiently concentrated toward the footpoints (Müller et al. 2003; Mendoza-Briceño et al. 2005). Uniform heating over the loop on the other hand fails to reproduce the phenomenon since the heating rate per unit mass needs to decrease in time locally in the corona in order to allow the thermal instability to set in. It was further shown that catastrophic cooling does not need time-dependent heating. In other words, it can happen even with a constant heating function. The footpoint-concentrated heating function to which simulations point to matches the observational evidence of coronal loops above active regions for being mainly heated at their footpoints (Aschwanden 2001), which sets most loops out of hydrostatic equilibrium. Further evidence of this fact has been found by Hara et al. (2008) using the Hinode/EIS instrument, which shows that active region loops exhibit upflow motions and enhanced nonthermal velocities. Possible unresolved high-speed upflows were also found, fitting in the footpoint-concentrated heating scenario.

Many heating mechanisms have been proposed as candidates for heating the solar corona up to the observed few million degree temperatures. In this context, a large emphasis has been put on the search for Alfvén waves in the solar corona. Theoretically, they can be easily generated in the photosphere by the constant turbulent convective motions, which inputs large amounts of energy into the waves (Muller et al. 1994; Choudhuri et al. 1993). Having magnetic tension as its restoring force, the Alfvén waves can travel less affected by the large transition region gradients with respect to other modes. Also, when traveling along thin magnetic flux tubes, they are cut-off free since they are not coupled to gravity (Musielak et al. 2007).⁴ Alfvén waves generated in the photosphere are thus able to carry sufficient energy into the corona to compensate the losses due to radiation and conduction, and, if given a suitable dissipation mechanism, heat the plasma to the high million degree coronal temperatures (Uchida & Kaburaki 1974; Wentzel 1974; Hollweg et al. 1982; Poedts et al. 1989; Erdélyi & Goossens 1996; Ruderman et al. 1997; Kudoh & Shibata 1999; Antolin & Shibata 2010) and power the solar wind (Suzuki & Inutsuka 2006; Cranmer et al. 2007). The main problem faced by Alfvén wave heating is actually to find a suitable dissipation mechanism. Being an incompressible wave, it must rely on a mechanism which converts the magnetic energy into heat. Several dissipation mechanisms have been proposed, such as parametric decay (Goldstein 1978; Terasawa et al. 1986), mode conversion (Hollweg et al. 1982; Kudoh & Shibata 1999; Moriyasu et al. 2004), phase mixing (Heyvaerts & Priest 1983; Ofman & Aschwanden 2002), or resonant absorption (Ionson 1978; Hollweg 1984; Poedts et al. 1989; Erdélyi & Goossens 1995). The main difficulty lies in dissipating sufficient amounts of energy in the correct time and space scales. For more discussion regarding this issue, the reader can consult, for instance, Aschwanden (2004), Klimchuk (2006), Erdélyi & Ballai (2007), and Taroyan & Erdélyi (2009). Studies considering Alfvén wave heating as coronal heating mechanism have shown that the obtained coronae are uniformly heated (Moriyasu et al. 2004; Antolin et al. 2008; Antolin & Shibata 2010; Suzuki & Inutsuka 2006). In these studies, the heating issues from shocks of longitudinal modes (mainly slow modes) from mode conversion of the Alfvén waves due to the density fluctuation, wave-to-wave interaction, and deformation of the wave shape during propagation. The coronal loops ensuing from Alfvén wave heating are found to satisfy quite well the RTV scaling law (Rosner et al. 1978) which quantifies the heating uniformity in the loops. This result would then point toward an inhibition of coronal rain if Alfvén wave heating is predominant in the loop. It is this idea that is addressed in this work.

Another promising coronal heating candidate mechanism is the nanoflare reconnection heating model. The nanoflare reconnection process was first suggested by Parker (1988), who considered a magnetic flux tube as being composed by a myriad of magnetic field lines braided into each other by continuous footpoint shuffling. Many current sheets in the magnetic flux tube would be randomly created along the tube that would lead to many magnetic reconnection events, releasing energy impulsively and sporadically in small quantities of the order of 10²⁴ erg or less (nanoflares). Parker's original idea was a nanoflare reconnection heating acting uniformly in the corona, but it was later proposed to be concentrated toward the footpoints of loops, where the magnetic canopy lies and magnetic field lines may entangle (Klimchuk 2006; Aschwanden 2001). In the reconnection scenario, waves are also expected to be generated (Roussev et al. 2001) and the energy imparted into Alfvén waves is a matter of debate. The imparted energy may well depend on the location in the atmosphere of the reconnection event. Parker (1991) suggested a model in which 20% of the energy released by reconnection events in the solar corona is transferred as a form of Alfvén wave. Yokoyama (1998) studied the problem simulating reconnection in the corona, and found that less than 10% of the total released energy goes into Alfvén waves. This result is similar to the two-dimensional simulation results of photospheric reconnection by Takeuchi & Shibata (2001), in which it is shown that the energy flux carried by the slow magnetoacoustic waves is 1 order of magnitude higher than the energy flux carried by Alfvén waves. On the other hand, recent simulations by Kigure et al. (2010) show that the fraction of Alfvén wave energy flux in the total released magnetic energy during reconnection in low β plasmas may be significant (more than 50%). Since nanoflares are thought to play an important role in the heating of the corona (Hudson 1991) and since they are generally assumed to be a signature of magnetic reconnection, it is then crucial to determine the energy going into the Alfvén waves during the reconnection process. Moreover, Moriyasu et al. (2004) have shown that the observed spiky intensity profiles due to impulsive releases of energy may actually be a signature of Alfvén waves. It was found that due to nonlinear effects Alfvén waves can convert into slow and fast magnetoacoustic modes which then steepen into shocks and heat the plasma to coronal temperatures balancing losses due to thermal conduction and radiation. The shock heating due to the conversion of Alfvén waves was found to be episodic, impulsive, and uniformly distributed throughout the corona, producing an X-ray intensity profile that matches observations. Hence, Moriyasu et al. (2004) proposed that the observed nanoflares may not be directly related to reconnection but rather to Alfvén waves.

Differentiating Alfvén wave heating from nanoflare reconnection heating during observations is one of the main tasks needed in order to solve the coronal heating problem. Following the work of Moriyasu et al. (2004), Antolin et al. (2008) have compared both heating mechanisms by studying the hydrodynamic response of a loop subject to both kinds of heating. It was found that Alfvén waves lead to a dynamic, uniformly heated corona with steep power law indexes (ensuing from statistics of heating events) while nanoflare reconnection heating leads to lower dynamics (besides the times when catastrophic cooling takes place in the case of footpoint-concentrated heating) and shallow power laws. It was further found that footpoint nanoflare heating (i.e., nanoflare reconnection heating concentrated toward the footpoints of loops) leads to hot upflows (as observed in the Fe xv 284.16 Å line) due to the plasma being heated rapidly toward the footpoints before being ejected into the corona, while Alfvén wave heating leads to hot downflows due to the plasma achieving the maximum temperatures in the corona (rather than at the footpoints) and being carried back to the footpoints by the strong shocks generated there (Antolin et al. 2010). In this work, we propose coronal rain as another observational signature through which both heating mechanisms can be distinguished.

We first start by reporting on limb observations of coronal rain from Hinode/SOT in the Ca ii H line. The velocities and shapes of the falling condensations are analyzed. With a 1.5-dimensional code, we then proceed to model a coronal loop being subject to a heating mechanism that is concentrated toward the footpoints, such as the nanoflare reconnection heating model (as proposed by Parker 1988; Klimchuk 2006). In the case considered, catastrophic cooling happens three times and we select the first event to analyze and compare with our observations. Next, we proceed by generating Alfvén waves at the photospheric level in the previous model and investigate the effect on catastrophic cooling. We conclude by analyzing the important implications of the results on coronal heating.

The work is organized as follows. In Section 2, we report observations of coronal rain in the Ca ii H line performed with Hinode/SOT. In Section 3, we introduce the 1.5-dimensional MHD model in which our loop is based on and discuss the heating models of the loop. In Section 4, we present the results of footpoint heating and analyze a typical case of catastrophic cooling. The effect of Alfvén waves on the thermal stability of the loop is also studied. In Section 5, we discuss the results in the context of coronal heating and conclude the work.

In this section, we report on high-resolution observations of coronal rain at the limb performed by the Solar Optical Telescope on board Hinode (Tsuneta et al. 2008) in a 0.3 nm broadband region centered at 396.8 nm, the H-line spectral feature of singly ionized calcium (Ca ii H line). The observation was performed on 2006 November 9 from 19:33 to 20:44 UT with a cadence of 15 s and a resolution of 005448 pixel⁻¹, and focused on NOAA AR 10921 on the west solar limb. The Ca ii H line is a chromospheric line typically showing plasmas with temperatures on the order of 20,000 K. The data set shown in Figure 1 corresponds to a field of view of 80 Mm × 40 Mm, and displays, apart from the active region on disk, many interesting structures over the limb, such as spicules, a prominence, and coronal rain. The same data set was used by Okamoto et al. (2007) to report on observations of transverse magnetohydrodynamic waves propagating in the observed prominence. The latter show complex horizontal threads displaying continuous horizontal motions, and appear to be located on the background of the images. Coronal loops exhibiting coronal rain appear to be located on the foreground of the prominence, and seem not to be linked to it.

Although coronal rain is seen at many different locations surrounding the sunspot, we have focused on a system of loops on the left side of Figure 1 in this study. The cool condensations fall along the loops, and by doing so they trace the magnetic field strands. They seem to form close to the apexes, which, unfortunately, are mostly located outside of the field of view. Also, in most cases, the tracking of the condensations cannot be pursued toward the footpoints, since these are located in brighter intensity regions on the disk. However, we can estimate the loop heights to be between 20 Mm and 30 Mm above the surface, and their lengths to be between 60 Mm and 100 Mm.

The condensations that constitute coronal rain vary greatly in size and shape, being, in general, thin elongated structures whose width (across the field lines) can be as small as the resolution of the telescope (around 40 km per pixel in the present case). Widths vary along the loop, being thicker near the apex, on average around 500 km wide (up to ∼1 Mm wide in some cases), than at the footpoints, where the average lies around 250 km and the structures are more elongated along the strands. Accordingly, the condensations span, in general, several strands of the loops near the apex, detach as they fall, and end up tracing individual strands toward the footpoints. The set of strands constituting the loops that are traced in this way span a region (transversal to the axes of the loops) of about 5 Mm wide at the top of the observed region (and possibly up to ∼8 Mm at the apex).

By fitting the strands with curves, we can trace the condensations along their way down the loops and calculate the corresponding velocities and accelerations. Due to the varying sizes of the condensations along the loop, many curves are needed for the tracking. For our system of loops, a total of 30 strands have been tracked. Two widget-based tools, the CRisp SPectral EXplorer (CRISPEX)⁵ tool and the Timeslice ANAlysis Tool (TANAT), both programmed in the Interactive Data Language (IDL), have been used to that end. Among other things, CRISPEX enables the easy browsing of the image (and if present, also spectral) data, the determination of loop paths and extraction of length–time diagrams. Although CRISPEX has been primarily designed to handle CRISP data, it can just as easily tackle data from other instruments, provided the data are supplied in one of the expected formats. Using TANAT, the dynamics of the observed coronal rain were subsequently obtained from the length–time diagrams by determining the slope of the features in those diagrams, thus yielding both the line-of-sight velocities and accelerations (in case of multiple measurements on the same feature).

Coronal rain is seen to happen constantly in the system of loops. Three main time intervals can however be identified in which coronal rain predominantly occurs. These three events can be clearly seen in Figure 2 and have peaks at roughly 7 minutes, 37 minutes, and 55 minutes from the start of the observation. We have selected and traced all noticeable condensations in each event. In Figure 3 (top), we show length–time diagrams along three strands and of ∼16 minutes time interval covering each one of the three events. In the middle and bottom panels of Figure 3, we show histograms corresponding to the calculations of velocities and accelerations, respectively, that result from the tracking of all the observed condensations in each event. The calculations of the velocities basically involve two kinds of errors. First, an error resulting from the determination of the slope in the length–time diagrams. In this case, the standard deviation in one measurement has been estimated to be ∼5 km s⁻¹ and ∼0.02 km s⁻² for velocities and accelerations, respectively. Since velocities are deduced from intensity patterns that are integrated along the line of sight, the second kind of error involves projection effects. The angle between the line of sight and the normal to the solar surface at the footpoints of the loops is roughly 825. Thus, assuming that the loop plane is perpendicular to the line of sight (which is hard to estimate since only about one fourth of the loop lengths is in the field of view), this gives an underestimation of at least ∼7% in all calculated velocities and accelerations.

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The histograms in the middle panel of Figure 3 show large variances in the measured velocities. This is due to the accelerations the condensations experience during their fall. Most condensations exhibit a constant downward acceleration which resembles movement along a parabolic curve (as seen, for instance, in the top left panel of Figure 3). For these cases, we have calculated the velocity at the topmost location of the strand (close to the apex) and the velocity at the bottom of the strand (close to the footpoint) and deduced the corresponding acceleration. A consequence of this procedure is the presence of two bumps in each velocity histogram, one at 30–40 km s⁻¹ corresponding to the velocities close to the apex, and the other between 60 and 100 km s⁻¹ corresponding to footpoint velocities. Condensations experiencing decelerations along the fall, mainly close to the footpoints, are also observed. This is the case for the main condensation that can be seen in the top middle panel in Figure 3. Furthermore, changes of direction of coronal rain are also observed, as the upward motion shown in the upper part of the top right panel of Figure 3.

In the acceleration histograms of the bottom panels in Figure 3, the dot-dashed line denotes the value of the solar gravity in the photosphere, namely 0.274 km s⁻². We can clearly see that the average accelerations experienced by the condensations is well below the solar gravity value, as has been reported previously for coronal rain observations (De Groof et al. 2004). Since the condensations fall along coronal loops, they will experience the effective gravity, that is, the component of gravity along the field lines. The average angle that the strands make at the top of the field of view (close to the apex) with the normal to the solar surface is roughly 50°, and becomes rapidly smaller at lower heights (less than 10° toward the footpoints). Hence, the effective gravity in the observed loops takes values roughly between 0.176 km s⁻² and 0.274 km s⁻², values that are considerably higher than the obtained average accelerations. Furthermore, in some cases, we have found condensations accelerating to values even higher than the solar gravity at the surface (0.3–0.4 km s⁻², as shown by the acceleration panels). Although motions can also be apparent, resulting from cooling or heating moving fronts that may act downward or upward along the loop, we consider the observed motions to be real flows produced by other important forces present in the loops, such as gas pressure or wave pressure gradients. In particular, we consider the internal gas pressure changes in the loops to be the main agent explaining the dynamics of the condensations, as suggested by the results of the simulations reported in Section 4. In our simulations, the changes in pressure are a consequence of the spatial distribution of the heating mechanism, which is located toward the footpoints of the loop.

In order to simulate coronal rain, we follow the model of Antiochos et al. (1999), in which it is shown that cool condensations can dynamically form in the corona resulting from footpoint heating of the loops. This mechanism is known as catastrophic cooling and is further explained in this section.

3.1. Model

We consider a magnetic flux tube (loop) of 100 Mm in length, roughly the same length as the loops exhibiting coronal rain in the observations reported here. The geometry of the loop takes into account the cross section area, which considers the predicted expansion of magnetic flux in the photosphere and the chromosphere, displaying an area ratio between the corona and the photosphere of 1000. As discussed in the Introduction, catastrophic cooling, as proposed by Antiochos et al. (1999), needs the heating to be concentrated toward the footpoints of loops. In the case of footpoint heating without the generation of Alfvén waves in the photosphere, the plasma motion is governed by the usual one-dimensional HD equations for the conservation of mass, momentum, and energy. The model in this case is the same as the heating model considered for nanoflare reconnection with heating concentrated toward the footpoints in Antolin et al. (2008), and is further discussed below. When Alfvén waves are considered, the model gains the azimuthal component and is the same model as the model for Alfvén wave heating in Antolin et al. (2008).

We write the one-dimensional HD equations for the conservation of mass, momentum, and energy in the following way: the mass conservation equation:

$$\begin{equation} \frac{\partial \rho }{\partial t}+v\frac{\partial \rho }{\partial s}= -\rho B\frac{\partial }{\partial s}\left(\frac{v}{B}\right); \end{equation} \tag{ 1 }$$

the momentum equation:

$$\begin{equation} \frac{\partial v}{\partial t}+v\frac{\partial v}{\partial s}= -\frac{1}{\rho }\frac{\partial p}{\partial s}-g_{s}; \end{equation} \tag{ 2 }$$

and the energy equation:

$$\begin{eqnarray} \frac{\partial e}{\partial t}+v\frac{\partial e}{\partial s}&=& {-}(\gamma -1)e B\frac{\partial }{\partial s}\left(\frac{v}{B}\right) -\frac{R-\mathcal {S}-\mathcal {H}}{\rho }\nonumber \\ && +\frac{1}{\rho r^{2}}\frac{\partial }{\partial s}\left(r^{2}\kappa \frac{\partial T}{\partial s}\right); \end{eqnarray} \tag{ 3 }$$

where

$$\begin{equation} p=\rho \frac{k_{B}}{m}T, \hspace{0.5cm} e=\frac{1}{\gamma -1}\frac{p}{\rho }. \end{equation} \tag{ 4 }$$

In the above Equations (1)–(3), s measures the distance along the flux tube (central field line) and r is the radius of the tube. ρ, p, v, and e are, respectively, density, pressure, velocity along the loop, and internal energy; B is the magnetic field along the loop and is a function of r alone, B = B₀(r₀/r)², where B₀ is the value of the magnetic field at the photosphere and r₀ = 200 km is the initial radius of the loop; and k_B is the Boltzmann constant and γ is the ratio of specific heats for a monatomic gas, taken to be 5/3. The g_s is the effective gravity along the loop and is given by

$$\begin{equation} g_{s}=g_{\odot }\cos \left(\frac{s}{L}\pi \right), \end{equation} \tag{ 5 }$$

where g_☉ = 2.74 × 10⁴ cm s⁻² is the gravity at the base and L is the total length of the loop.

We assume an inviscid perfectly conducting fully ionized plasma. The effects of thermal conduction and radiation are taken into account, where the Spitzer conductivity corresponding to a fully ionized plasma is considered, and radiative losses are defined as

$$\begin{equation} R(T)=n_{e}n_{p}Q(T)=\frac{n^{2}}{4}Q(T). \end{equation} \tag{ 6 }$$

Here, n = n_e + n_p is the total particle number density (n_e and n_p are, respectively, the electron and proton number densities, and we assume n_e = n_p = ρ/m to satisfy plasma neutrality, with m the proton mass) and Q(T) is the radiative loss function. We follow the model of Sterling et al. (1993) and Hori et al. (1997, see their Table 1) for the treatment of the radiation. We thus assume optically thin radiation for temperatures T>4 × 10⁴ K, for which Q(T) is approximated with analytical functions of the form Q(T) = χT^γ (Landini & Monsignori Fossi 1990), where the parameters χ and γ are empirically determined (Hildner 1974; Rosner et al. 1978; Mariska et al. 1982). For temperatures below 4 × 10⁴ K, we assume that the plasma becomes optically thick. In this case, the radiative losses R can be approximated by R(ρ) = 4.9 × 10⁹ρ erg cm⁻³ K⁻¹, after Sterling et al. (1993), based on the empirical result of Anderson & Athay (1989), that the heating rate per gram over a large part of the chromosphere is roughly constant at 4.9 × 10⁹ erg g⁻¹ s⁻¹. In Equation (3), the heating term $\mathcal {S}$ has a constant non-zero value which is non-negligible only when the atmosphere becomes optically thick. Its purpose is mainly for maintaining the initial temperature distribution of the loop. Here, $\mathcal {H}$ denotes the heating function in the loop, which corresponds to a nanoflare heating model and is presented in the next section.

Since catastrophic cooling events happen in the timescale of minutes and high-speed flows and shocks ensue, the plasma constituting coronal rain is in a highly dynamical state, which complicates the full numerical treatment of coronal rain substantially. For instance, if we are interested in reproducing the observed spectral features of coronal rain (considered, for instance, in Müller et al. 2003, 2004), non-equilibrium ionization effects become important. Indeed, the timescale needed to reach ionization or excitation equilibrium may be longer than the dynamic timescale of the plasma, case in which the population rate equations vary in time. This is the case of hydrogen in the chromosphere and transition region as shown by Carlsson & Stein (2002). Non-equilibrium ionization effects are also important for the study of blueshifts and redshifts in chromospheric and transition region lines as shown by Hansteen (1993). In this paper, we assume that the radiation fields in all directions and all frequencies and the level populations do not affect the ionization level of the plasma. Also, the plasma in the condensation which may become optically thicker may not considerably affect the energy equation, since a condensation falls down a loop on a timescale of minutes. We justify our approach since we focus on the mechanism through which coronal rain is achieved (i.e., catastrophic cooling) rather than on the radiating properties of coronal rain.

For generating Alfvén waves in the loop, we follow the same model as in Antolin et al. (2008). Random torque motions are produced in the photosphere, which generate Alfvén waves with a white noise spectrum in frequency. We adopt this model instead of a monochromatic wave generator since we consider the buffeting of magnetic field lines by convective motions to have a turbulent nature, thus leading to random motions. For further details about this model, please refer to Antolin et al. (2008).

3.2. Nanoflare Heating Function

As shown by Antiochos et al. (1999), in order for catastrophic cooling to happen, we have to apply a heating mechanism that is concentrated toward the footpoints of the loop. There may be many proposed heating mechanisms that can act preferentially toward the footpoints of loops. Here, we will assume that the loop is subject to "footpoint nanoflare" heating from the nanoflare reconnection model described in Antolin et al. (2008). In this picture, we assume that the energy imparted onto Alfvén waves from reconnection events is low and can be neglected relative to the imparted energy on the slow modes. Hence, in this picture, the corona would be heated mainly by the accumulation of numerous nanoflares coming from reconnection events and by slow magnetoacoustic shocks. Hydrodynamic modeling of nanoflare heating has already been done in the past (Hansteen 1993; Walsh et al. 1997; Cargill & Klimchuk 2004; Patsourakos & Klimchuk 2005; Taroyan et al. 2006; Mendoza-Briceño et al. 2005). The nanoflare model considered here is similar to the model of Taroyan et al. (2006) with respect to the heating function $\mathcal {H}$ in Equation (3). In the present case, we assume that heating events simulating reconnection events (leading to nanoflares) occur toward the footpoints of the loop. These are modeled as artificial perturbations in the internal energy of the gas (thus generating only slow modes), which are randomly input in space and time intervals specified below. The heating rate due to the nanoflares is represented as

$$\begin{equation} \mathcal {H}=\sum _{i=1}^{n}\mathcal {H}_{i}(t,s), \end{equation} \tag{ 7 }$$

where $\mathcal {H}_{i}(t,s)$, i = 1,...,n are the discrete episodic heating events, and n is the total number of events, and

$$\begin{equation} \mathcal {H}_{i}(t,s)= \left\lbrace \begin{array}{@{}ll} E_{0}\sin \big(\frac{\pi (t-t_{i})}{\tau _{i}}\big) \exp \big(-\frac{|s-s_{i}|}{s_{h}}\big),\!\! & \hbox{$t_{i}<t<t_{i}+\tau _{i}$} \\ 0,\!\! & \hbox{otherwise} \end{array}\!\!\! ; \right. \end{equation} \tag{ 8 }$$

where E₀ is the maximum volumetric heating and s_h is the heating length scale. The offset time t_i, the maximum duration τ_i, and the location s_i of each event are uniform deviates, that is, random numbers with a uniform probability distribution which lie in the following ranges:

$$\begin{eqnarray} && t_{i}\in [0, t_{\rm total}],\hspace{0.3cm}\tau _{i}\in [0,\tau _{\max }],\nonumber \\ && s_{i}\in [s_{\min }, s_{\max }]\bigcup [L - s_{\max }, L - s_{\min }], \end{eqnarray} \tag{ 9 }$$

where t_total is the total simulation time and s_min (s_max) define the lower (upper) boundaries of the range in the loop where heating events occur. The random numbers have been obtained with a random number generator that has passed the most important statistical tests (the "ran1" routine of Numerical Recipes, Press et al. 1992).

In order to set the values to the parameters of the heating function, Equations (7)–(9), an estimate of the nanoflare duration time is needed. One of the hardest parameters to estimate in magnetic reconnection theory is the thickness of the current sheet, i.e., the length across the reconnection region. If this parameter is of the order of ∼1000 km, the timescale of a (small) reconnection event leading to a nanoflare should oscillate between 1 and 10 s, since the order of the Alfvén speed in the chromosphere and in the corona is, respectively, ∼100 and ∼1000 km s⁻¹. This value, however, is not established. Different values have been tried for the parameters of the heating function defined in Equations (7)–(9). Since the purpose of this work is not to study the ranges in which catastrophic cooling happens, we will limit ourselves in the present model to present a typical case in which it happens. For this case, we have the maximum duration time of a heating event τ_max = 40 s, the heating length scale s_h = 1000 km, the maximum volumetric heating E₀ = 0.5 erg cm⁻³ s⁻¹, an average occurrence of one heating event each 50 s, the upper and lower boundaries of the ranges in which heating occurs {s_min = 1, s_max = 10} Mm. These parameters set a mean energy per event of 1.9 × 10²⁶ erg and a mean energy flux of 2.5 × 10⁷ erg cm⁻² s⁻¹.

3.3. Initial Conditions and Numerical Code

We first perform the simulation of the loop being heated only by the events from the "footpoint nanoflare" model, that is, without Alfvén waves. We then allow the generation of Alfvén waves at the footpoints of the loop and analyze the effect on the catastrophic cooling events.

4.1. Footpoint Nanoflare Heating

Due to the large energy flux from the heating events, the corona is formed rapidly in the considered footpoint nanoflare model (in about 20 minutes). The mean temperature in the corona over the entire simulation time is 〈T〉 ∼ 1.4 × 10⁶ K, with a maximum temperature of 〈T〉 ∼ 3.9 × 10⁶ K. The mean density in the corona is high, 〈n〉 ∼ 1.3 × 10⁹ cm⁻³, characteristic of a dense active region loop. As the heating events have a uniform probability distribution in time, we have a uniform heating input in time into the loop. Since the heating events occur close to the footpoints, chromospheric matter is constantly being pushed upward into the loop by the heating events themselves and also from thermal conduction (chromospheric evaporation), increasing the density in the corona. Figure 4 is a phase diagram of the mean temperature and the corresponding mean density in the corona in time. Arrows indicate the time direction and curve styles (and colors in the online version) indicate different cycles the loop experiences. In the present case, three cycles can be distinguished, each one lasting roughly 170 minutes. The dotted curve in Figure 4 corresponds to the initial phase of the simulation, then the three cycles start, denoted by solid, dashed, and dot-dashed curves (blue, green, and red curves in the online version), respectively, in time. A cycle is composed of four distinctive phases. First, a phase in which the temperature of the corona increases rapidly and the density is roughly constant. Then follows a phase of constant temperature and slow density increase. In the third phase, the temperature in the corona slowly decreases and the density is roughly constant. The last phase is marked by a dramatic decrease of temperature which can happen either locally in the corona or globally (entire collapse of corona), accompanied by a dramatic increase of density (at one or more locations in the corona). These cycles have been termed "limit cycles" by Müller et al. (2003) and can be understood as follows.

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Initially, the density in the corona is low, since gravity depletes the loop before having a considerable injection of mass from the heating mechanism. The corona is thus easily heated to high temperatures from the footpoint heating events (phase 1). These events continuously inject material into the corona, thus slowly increasing its density. Since the energy flux to the corona is kept constant, the high temperatures can only be kept for a specific density range (phase 2). Then the mean temperature starts decreasing due to the steadily decreasing heating rate per unit mass (phase 3). The lower temperature also increases the (optically thin) radiative losses, accelerating the cooling of the corona. The loop then reaches a critical state in which its density is too high and the temperature too low. A thermal instability follows due to the large radiation increase at low temperatures (becoming optically thick) in just the same way the transition region forms (phase 4). The temperature and the pressure then rapidly decrease to chromospheric values on a timescale of minutes, thus leading to a catastrophic cooling event. The low pressures induce high-speed siphon flows from the footpoints into the corona that in some cases can have velocities higher than 200 km s⁻¹. The low pressure regions will also be compressed thus forming condensations that rapidly increase in density. The thermal instability can happen locally in the corona, thus forming a dense and cool blob which subsequently falls down due to gravity, or can have a more global character, case in which the entire corona collapses and several dense blobs are formed. The loop is then evacuated and gets rapidly reheated due to the low density and the constant heating input. The cycle thus restarts.

In the cycle denoted by the solid line (blue line in the online version) of Figure 4, the catastrophic cooling occurs locally in the corona, while in the subsequent cycles it occurs globally. At the end of the first cycle, we notice an excursion to the right which does not happen in the subsequent cycles. The reason for this is the fast density increase of the condensation during its fall, which happens mainly when the catastrophic cooling is local. When the condensation leaves the corona (which is the region where the averages are calculated), the loop gets rapidly evacuated, corresponding to the end of the cycle.⁶ The local low pressure inducing compression, the siphon flows and the subsequent density increase of the first condensation can be clearly seen in Figure 5, where the evolutions in time of the density (left panel) and pressure (right panel) along the loop are shown. The (acoustic) shocks created by the heating events can be followed from the transition region. Some of them also form small condensations while propagating, before colliding with the bigger original condensation. It can be noticed that the blob experiences a change of direction at the loop top, going first toward the left footpoint and then toward the right footpoint. A shock collides with the blob and is reflected just at the time in which the blob changes direction. Another factor of the change of direction of the blob is the lower gas pressure region on the right side of the blob, as compared to the left side. Hence, these condensations are subject to not only gravity but also the local changes of gas pressure in the loops. This mechanism may explain the observed deceleration motion and upward motion of coronal rain in Figure 3. When the blob falls down to the chromosphere, it experiences a strong deceleration by the higher density region, the transition region is left oscillating up and down a couple of times.

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In the upper panel of Figure 6, we track the condensation as it falls down to the chromosphere and plot its velocity along the loop in time. The initial motion toward the left footpoint has a maximum speed of almost ∼30 km s⁻¹ before changing direction and accelerating toward the right footpoint under the action of gravity and pressure gradient. The maximum speed is almost 120 km s⁻¹ close to the footpoint, before being decelerated by the high gas pressure of the chromosphere. The lower panel shows the length along the loop of the condensation as it falls down the loop. We can see a general tendency to elongate, passing from a length of 2 Mm at the apex to a length of 7 Mm close to the footpoints. We will discuss these results in Section 5.

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4.2. Footpoint Nanoflare Heating and Alfvén Waves

In order to analyze correctly the effect of Alfvén waves on the catastrophic cooling events, we first consider the case in which we only have Alfvén waves and no nanoflare reconnection heating. The Alfvén waves are generated in the photosphere and have a white noise spectrum. Figure 7 shows the phase diagram of mean temperature and mean density of the corona for a case in which the photospheric rms azimuthal velocity field is $\langle v_{\phi, \rm ph}^{2}\rangle ^{1/2}=1.5$ km s⁻¹. We can see that as the simulation evolves the mean temperature and density in the corona converge rapidly to roughly constant values, seen as an attractor in the phase diagram. Indeed, after one fifth of the total simulation time (142 minutes), the temperatures and densities of the corona stay roughly constant. As shown in Antolin et al. (2008) and Antolin & Shibata (2010), the coronae that ensue from Alfvén wave heating are uniform and steady, and satisfy the RTV scaling law (Rosner et al. 1978).

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Next, we consider a hybrid model in which both heating mechanisms are present. Heating events simulating reconnection events happen close to the footpoints, and Alfvén waves with a white noise spectrum are generated in the photosphere. For the former heating mechanism, the same footpoint nanoflare heating model as in the previous section is used. For the Alfvén wave heating model, we allow the amplitude of the waves to increase in time, as shown in Figure 8. At the beginning of the simulation, the energy flux from the waves is negligible with respect to the energy flux from the nanoflare reconnection events. Indeed, Alfvén waves have an amplitude smaller than 0.3 km s⁻¹, which from the study in Antolin et al. (2008; see Figure 4 in that paper) we know is not enough to produce a hot corona. In the second half of the simulation the waves have an amplitude larger than 1 km s⁻¹ (reaching ∼2 km s⁻¹ by the end of the simulation), which is enough to produce a hot corona. In Figure 9, we plot the corresponding phase diagram for this case. We can see that as the amplitude of the waves increases the limit cycles get smaller and disappear. Mean temperatures and densities finally converge to roughly constant values. Hence, the loop becomes uniformly heated as the heating from the Alfvén waves is no longer negligible.

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The obtained thermal equilibrium state and corresponding disappearance of the limit cycles may be due to the overall heating rate increase from the addition of Alfvén waves, rather than to the characteristic uniform spatial distribution of Alfvén wave heating. To discard this possibility, we have conducted one more simulation with footpoint nanoflare heating having a heating rate larger than the resulting heating rate in our hybrid model. To do this, we first estimate in our hybrid model the mean heating rate per unit volume 〈∇ · F〉, where F denotes the total energy flux,

$$\begin{eqnarray} F&=&{-}\frac{B_{s}B_{\phi }v_{\phi }}{4\pi }+\left(\frac{\rho v_{\phi ^{2}}}{2}+\frac{B_{\phi }^{2}}{4\pi }\right)v_{s}\nonumber \\ &&+\left(\frac{\gamma }{\gamma -1}p+\frac{\rho v_{s}^{2}}{2}+\rho gz\right)v_{s}. \end{eqnarray} \tag{ 10 }$$

Here, the first two terms on the right-hand side account for the energy flux from the Alfvén waves. The last term includes thermal, kinetic, and potential energy fluxes. In the region where the heating events from nanoflare heating occur, we find an exponential decrease of 〈∇ · F〉 with a heating scale height of ∼3.4 Mm, and a mean value of ∼0.72 erg cm⁻³ s⁻¹. The total energy flux that thus goes into heating is roughly 2.45 × 10⁸ erg cm⁻² s⁻¹. With this value at hand, we choose our parameters for our second footpoint nanoflare heating simulation. We have chosen a total number of 3600 heating events, each with a maximum volumetric heating of E₀ = 1 erg cm⁻³ s⁻¹. The other parameters remain unchanged. The energy flux input is thus set to ∼2.7 × 10⁸ erg cm⁻² s⁻¹ (cf. Equation (17) in Antolin et al. 2008).

Results for the case of footpoint nanoflare heating with increased heating rate are shown in Figures 10 and 11. In Figure 10, we show the phase diagram of mean temperature and mean density of the corona. We also distinguish the presence of three full limit cycles (almost four) in this case, setting a period of ∼140 minutes, thus confirming the fact that the disappearance of catastrophic cooling, when Alfvén waves are included, is due to the characteristic spatial heating and not the overall increase of the heating rate. We notice that in this case we only have local catastrophic cooling events, seen by the excursions to the right of the cycles in Figure 10, as explained in the previous section. This is most likely due to thermal conduction, which, having very large values (mostly toward the footpoints) does not allow a global collapse of the temperature structure of the corona. The period between catastrophic cooling events seems to slightly decrease with increasing heating rate. This can be understood by considering the heating rate per unit mass which stays roughly constant in the corona despite the larger energy input. The later is due to the large density increase in the corona from chromospheric evaporation and from the gas pressure of the heating events toward the footpoints.

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It is interesting to analyze the dynamics of the first catastrophic cooling event. In Figure 11, we show the density (left) and pressure (right) maps for the first limit cycle. We notice the formation of two condensations, one after the other, exhibiting very different dynamic behavior. While the first condensation falls down at a rather constant speed of ∼30 km s⁻¹ instead of accelerating downward as in the case of Figure 5, the second condensation bounces up and down in the corona three times before falling. This behavior can be understood by analyzing the pressure map (right panel) of Figure 11. Indeed, we can clearly see a high pressure region forming all the way down to the chromosphere beneath the condensations. In the oscillatory case, we can see a compression and rarefaction pattern in phase with the up and down movement. The high density of the condensation creates strong downward propagating shocks which contribute to create this high pressure region. Also, the matter ejected upward into the corona by thermal conduction and from the constant occurrence of heating events at the footpoint stays in the coronal part below the condensation. The high density and low temperature of the condensation act therefore as a wall for waves, flows, and thermal conduction. The present case reinforces our hypothesis that internal pressure changes in the loop, rather than gravity, determine the dynamics of the condensation.

Previous observations in Hα, and in EUV bands such as the EIT 304 Å, and the TRACE 1216 Å and 1600 Å bands seem to show that coronal rain is a phenomenon exclusively of active regions (Kawaguchi 1970; Leroy 1972; Schrijver 2001; De Groof et al. 2004, 2005), where loops are dense and heating appears to be concentrated toward the footpoints (Aschwanden et al. 2001; Aschwanden 2001; Hara et al. 2008). Although predicted by simulations, a periodicity of occurrence of coronal rain has not yet been observed. However, an estimate of its occurrence rate has been estimated to be on the order of once every 2 days at most (Schrijver 2001).

In this work, we have reported on unprecedented high-resolution observations of coronal rain over an active region with the Hinode/SOT telescope in the Ca ii H line, and compared those with results of simulations of loops that undergo catastrophic cooling. The observed system of loops exhibits a wide range of velocities for the falling condensations, both low speed (∼30–40 km s⁻¹) and high-speed (∼80–120 km s⁻¹) coronal rain are detected. The accelerations are found to be on average substantially lower than the solar gravity component along the loops, implying the presence of other forces in the loops, presumably gas pressure gradients as suggested by the simulations. The sizes of the condensations vary along the loop, generally spanning several strands in thickness toward the apex, and being thin and elongated toward the footpoints. In particular, the falling condensations can be as thin as the telescope resolution (∼40 km) making them very faint and hard to detect.

During the observation time, coronal rain is detected in the system of loops almost at all times. Although the observation time is fairly short (70 minutes), Figure 2 may suggest a periodicity of roughly 25 minutes. Longer observation runs are however needed in order to clearly address this point. It should be mentioned that loops on the other side of the observed sunspot also exhibit coronal rain, making it a rather ubiquitous phenomenon in the present case. The very fine structure of coronal rain makes high-resolution observation in chromospheric lines necessary, which was not available until recently. This may explain the low frequency in occurrence reported in the work of Schrijver (2001). On-going observational work with the Crisp spectropolarimeter of the Swedish Solar Telescope also indicates a common occurrence of coronal rain. If this is indeed the case, it may have other strong implications for coronal heating. We will address this point more thoroughly in a future paper (P. Antolin et al. 2010, in preparation).

In our simulations, the loops are preferentially heated toward the footpoints producing a heating scale height that allows catastrophic cooling to happen. By calculating the mean heating rate per unit volume 〈∇ · F〉, where F denotes the total energy flux, we find that it decreases roughly exponentially toward the footpoints. In the footpoint nanoflare heating simulation of Section 4.1, we find a heating length scale of ∼2.5 Mm (this number is termed damping length in Müller et al. 2003). The loops are subject to cycles ("limit cycles," as termed by Müller et al. 2003) in which they rapidly cool down and then reheat, they get dense and then deplete. The constant heating input at the footpoints of the loops produces coronae out of hydrostatic equilibrium. The coronal density increases in time causing a gradual decrease of the temperature and increase of the radiative losses. When the temperatures are sufficiently low, radiative losses are dramatically increased since the condensation becomes optically thick and radiates considerably more. Catastrophic cooling sets in, either locally in the corona or globally, case in which the entire corona is cooled down to chromospheric temperatures. This is accompanied by a pressure drop, which, if local, causes a local compression of the plasma (condensation). The low pressures also induce high-speed siphon flows (leading to shocks) which increase considerably the density of the condensation. Dense condensations of cool plasma form at coronal heights, which subsequently fall down by gravity. The loop is then depleted and the cycle starts again. In our simulations, the obtained cycles have periods below ∼180 minutes, reinforcing our belief that coronal rain is a common phenomenon. The period of the cycles increases with loop length and heating scale height, as shown by Müller et al. (2003, 2004). Our results show that this parameter has a slight but nonetheless important dependence on the heating rate, decreasing for larger values of the later due to the higher coronal densities that ensue (which do not allow an increase in the heating per unit mass).

The characteristics of our condensations match well the main characteristics of the coronal rain from the reported observations. The velocities are in the same range as would be expected from coronal rain forming at roughly the same locations of loops with similar length. We obtain an elongation of the condensation in the simulation as it falls along the loop and gets denser. This may be explained by considering the effective gravity along the loop, which increases as the condensation falls down the loop. We consider, however, the internal pressure changes that ensue from catastrophic cooling to play a more important role, not only in the elongation but also in the overall dynamics of the condensations. Indeed, as our simulations show, the pressure changes cause decelerations and even upward motions. The strong siphon flows from catastrophic cooling and the high-speed shocks from the heating events at the footpoints propagate along the loop, increasing the density and altering the shape of the condensation, thus contributing to its elongation. This elongation is observed as well in the present Hinode observations and has previously also been reported (Schrijver 2001). The internal pressure changes may thus be a possible explanation for the upward motion of coronal rain reported in the present observations.

The temperatures and densities of the resulting condensations have chromospheric values, which suggest that they would emit radiation in lines such as Hα or Ca ii H or K, as in our observations with Hinode/SOT. It should be noted, however, that our empirical approach for the treatment of optically thick plasma may influence the temperatures and densities achieved during the catastrophic cooling. For instance, our optically thick regime is modeled by setting the radiative losses proportional to the density (see the discussion following Equation (6)), an approach supported by observations and models of the mean chromosphere, but which does not take into account the real highly dynamic and complex nature of the later. In our model, the radiation losses readjust instantly to the local thermodynamic state of the plasma, and we may thus expect shorter timescales in the onset of the catastrophic cooling, resulting in faster upflows due to the local low pressure region in the corona that ensues the temperature drop. This may lead to a higher condensation of plasma in the corona (thus also contributing to the elongation) than in a self-consistent approach for radiation, and may explain the fast increase in density of the condensation along its fall in our simulations, which is not found in the present observations. Our approach may be a critical issue when the focus is on radiation, for instance, when we seek to reproduce spectral features of coronal rain, since non-ionization equilibrium effects become important, as discussed in Section 3.1. The study of the radiative aspects of coronal rain by considering a three-dimensional radiative MHD code will be the subject of a future paper. However, in this paper, we are more interested in the dynamical aspect of coronal rain, in the catastrophic cooling mechanism leading to the phenomenon, and mainly in the influence of Alfvén waves on it.

If coronal rain is indeed the consequence of the catastrophic cooling mechanism, we then must have a heating mechanism acting preferentially toward the footpoints in loops where coronal rain is observed, i.e., active region loops. We have seen that Alfvén waves generated at the footpoints of loops produce uniform coronae and are thus unable to reproduce phenomena such as coronal rain. Furthermore, when Alfvén waves are present in a loop and have enough energy flux to heat and maintain a hot corona, the catastrophic cooling events are inhibited, thus avoiding coronal rain to form. The loop then converges to a uniform and steady state. Coronal loops in active regions seem to show a recurrent occurrence of coronal rain. They are dynamical entities showing heating and cooling processes at all times. Our results indicate then that Alfvén wave heating cannot be the principal heating mechanism for coronal loops in active regions.

Hara et al. (2008), using the Hinode/EIS instrument, have found upflow motions and enhanced nonthermal velocities in the hot lines of Fe xiv 274 and Fe xv 284 in active region loops. Possible unresolved high-speed upflows were also found. In Antolin et al. (2008, 2010), we found that footpoint or uniform heating coming from nanoflare reconnection exhibits hot upflows, thus fitting to the observational scenario of active regions, while Alfvén wave heating was found to exhibit hot downflows, which may fit to the observational scenario of quiet Sun regions (Chae et al. 1998; Brosius et al. 2007). Furthermore, in Antolin & Shibata (2010), we have found that Alfvén wave heating is effective only in thick loops (with area expansions between photosphere and corona higher than 500) and long loops (with lengths above 80 Mm), a scenario which may not fit to active regions, where loops exhibit low area expansions due to the high magnetic field filling factors in those regions (Peter 2001; Marsch et al. 2004; Tian et al. 2009a, 2009b). Hence, Alfvén waves may play an important role in the heating of quiet Sun regions rather than in active regions.

This work was supported by the Research Council of Norway. P.A. thanks T. J. Okamoto, K. Ichimoto, S. Tsuneta, and the people at NAOJ where the observational part of this study originated. Grateful acknowledgement also goes to H. Isobe and T. Yokoyama, for many fruitful discussions regarding the simulations, and to the people at the Department of Astronomy and Kwasan Observatory of Kyoto University, where a large part of this study was carried out. Likewise, this study benefited of many fruitful discussions at the Institute of Theoretical Astrophysics of the University of Oslo. In particular, we thank M. Carlsson, V. Hansteen, L. Rouppe van der Voort, L. Heggland, and B. Gudiksen. Special thanks to R. Erdélyi for valuable comments regarding Alfvén wave heating. P.A. acknowledges S. F. Chen for patient encouragement. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). The numerical calculations were carried out on Altix3700 BX2 at YITP in Kyoto University.