The Astrophysical Journal, 716:154–166, 2010 June 10
C 2010.
doi:10.1088/0004-637X/716/1/154
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
CORONAL RAIN AS A MARKER FOR CORONAL HEATING MECHANISMS
P. Antolin1,2,3 , K. Shibata2 , and G. Vissers1
1
Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029, Blindern, NO-0315 Oslo, Norway; antolin@astro.uio.no, g.j.m.vissers@astro.uio.no
2 Kwasan Observatory, Kyoto University, Yamashina, Kyoto 607-8471, Japan; shibata@kwasan.kyoto-u.ac.jp
Received 2009 October 7; accepted 2010 April 19; published 2010 May 14
ABSTRACT
Reported observations in Hα, Ca ii H, and K or other chromospheric lines of coronal rain trace back to the days of the
Skylab mission. Corresponding to cool and dense plasma, coronal rain is often observed falling down along coronal
loops in active regions. A physical explanation for this spectacular phenomenon has been put forward thanks to
numerical simulations of loops with footpoint-concentrated heating, a heating scenario in which cool condensations
naturally form in the corona. This effect has been termed “catastrophic cooling” and is the predominant explanation
for coronal rain. In this work, we further investigate the link between this phenomenon and the heating mechanisms
acting in the corona. We start by analyzing observations of coronal rain at the limb in the Ca ii H line performed
by the Hinode satellite, and derive interesting statistical properties concerning the dynamics. We then compare
the observations with 1.5-dimensional MHD simulations of loops being heated by small-scale discrete events
concentrated toward the footpoints (that could come, for instance, from magnetic reconnection events), and by
Alfvén waves generated at the photospheric level. Both our observation and simulation results suggest that coronal
rain is a far more common phenomenon than previously thought. Also, we show that the structure and dynamics of
condensations are far more sensitive to the internal pressure changes in loops than to gravity. Furthermore, it is found
that if a loop is predominantly heated from Alfvén waves, coronal rain is inhibited due to the characteristic uniform
heating they produce. Hence, coronal rain may not only point to the spatial distribution of the heating in coronal loops
but also to the agent of the heating itself. We thus propose coronal rain as a marker for coronal heating mechanisms.
Key words: magnetohydrodynamics (MHD) – Sun: corona – Sun: flares – waves
Online-only material: color figures
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
1. INTRODUCTION
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).
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
3 Also at: Center of Mathematics for Applications, University of Oslo,
P.O. Box 1053, Blindern, NO-0316, Oslo, Norway.
154
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CORONAL RAIN AS A MARKER FOR CORONAL HEATING MECHANISMS
155
Figure 1. High-resolution Hinode/SOT image at the limb in the Ca ii H line. The field of view is 80 Mm × 40 Mm. The observation was performed on 2006 November
9, between 19:33 to 20:44 UT with a cadence of 15 s and a resolution of 0. 05448 pixel−1 . In order to be able to see the fainter corona, the intensity of the photosphere
(disk) has been decreased. Also, the intensity of the limb structures (spicules) is saturated. The sunspot visible on disk corresponds to the main sunspot of NOAA
AR 10921. Above the limb, cloud-like prominence structures and coronal rain can be observed. The intensity of coronal rain in the Ca ii H line is about 1% of the
on-disk photospheric intensity.
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 cutoff 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
4
Verth et al. (2010) have pointed out however that the assertion made by
Musielak et al. (2007) is valid only when the temperature in the flux tube does
not differ from that of the external plasma. When this is not the case, a cut-off
frequency is introduced. This is also the case when temperature gradients are
present in the loop, as shown by Routh et al. (2010).
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ANTOLIN, SHIBATA, & VISSERS
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
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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 1.5dimensional 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.
2. OBSERVATIONS OF CORONAL RAIN WITH
HINODE/SOT
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
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CORONAL RAIN AS A MARKER FOR CORONAL HEATING MECHANISMS
with a cadence of 15 s and a resolution of 0. 05448 pixel ,
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).
5
The actual code and further information can be found at
http://www.astro.uio.no/∼gregal/crispex/crispex_main.html.
157
−1
Figure 2. Histogram showing the time occurrence for coronal rain events in the
system of loops identified in Figure 1 during the 71 minutes of observation time.
For each time unit (15 s), we plot the number of coronal rain events that are
happening in the loops. Coronal rain occurs almost at all times with, however,
three main time intervals of strong occurrence. These three events have peaks
at roughly 7 minutes, 37 minutes, and 55 minutes.
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 82. 5. 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.
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
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ANTOLIN, SHIBATA, & VISSERS
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Figure 3. Top: length–time diagrams for a system of loops exhibiting coronal rain on Figure 1. Three time intervals in which coronal rain predominantly occurs are
selected (see Figure 2). Each one of these events is composed of several falling condensations tracing magnetic field strands of the loops. We have chosen and traced
three of such strands using the CRISPEX analysis tool. The measurement of velocities and accelerations are performed with the TANAT tool using the timeslices
issued by CRISPEX. Middle and bottom: histograms of calculated velocities and accelerations with TANAT for the three events, respectively. Dotted and dashed
lines denote extremum and mean values, respectively. The dot-dashed line in the bottom panel denotes the value of the solar gravity at the photosphere, gsun = 0.274
km s−2 . The standard deviation in one measurement is estimated to be ∼5 km s−1 and ∼0.02 km s−2 for velocities and accelerations, respectively. The error from
projection effects is estimated to be ∼7%. See the text for details.
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.
3. SIMULATION SETUP
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
No. 1, 2010
CORONAL RAIN AS A MARKER FOR CORONAL HEATING MECHANISMS
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:
∂ρ
∂ρ
∂ v
;
+v
= −ρB
∂t
∂s
∂s B
(1)
the momentum equation:
∂v
∂v
1 ∂p
+v
=−
− gs ;
∂t
∂s
ρ ∂s
(2)
and the energy equation:
∂e
∂ v R−S −H
∂e
+v
= −(γ − 1)eB
−
∂t
∂s
∂s B
ρ
1 ∂
∂T
r 2κ
;
+ 2
ρr ∂s
∂s
where
p=ρ
kB
T,
m
e=
1 p
.
γ −1ρ
(3)
(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 kB is the Boltzmann constant
and γ is the ratio of specific heats for a monatomic gas, taken
to be 5/3. The gs is the effective gravity along the loop and is
given by
s gs = g cos
π ,
(5)
L
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
R(T ) = ne np Q(T ) =
n2
Q(T ).
4
(6)
Here, n = ne + np is the total particle number density (ne and np
are, respectively, the electron and proton number densities, and
we assume ne = np = ρ/m to satisfy plasma neutrality, with
m the proton mass) and Q(T ) is the radiative loss function. We
159
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 ) = χ T 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 × 109 ρ 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 S has a constant non-zero value which is nonnegligible only when the atmosphere becomes optically thick.
Its purpose is mainly for maintaining the initial temperature
distribution of the loop. Here, 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
160
ANTOLIN, SHIBATA, & VISSERS
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 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
H=
n
Hi (t, s),
(7)
i=1
where Hi (t, s), i = 1,. . .,n are the discrete episodic heating
events, and n is the total number of events, and
i|
i)
exp − |s−s
, ti < t < ti + τi
E0 sin π(t−t
τi
sh
;
Hi (t, s) =
0,
otherwise
(8)
where E0 is the maximum volumetric heating and sh is the
heating length scale. The offset time ti , the maximum duration
τi , and the location si of each event are uniform deviates, that is,
random numbers with a uniform probability distribution which
lie in the following ranges:
ti ∈ [0, ttotal ],
τi ∈ [0, τmax ],
si ∈ [smin , smax ] [L − smax , L − smin ],
(9)
where ttotal is the total simulation time and smin (smax ) 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 τmax = 40 s, the heating length
scale sh = 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 {smin = 1, smax = 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 .
Vol. 716
3.3. Initial Conditions and Numerical Code
For the model including Alfvén waves, a sub-photospheric
region is considered by adding a 2 Mm section at each footpoint
of the loop, in which the radius of the loop is kept constant (hence
keeping a constant magnetic flux). We take the origin s = z = 0
as the top end of this region. The loop is assumed to follow
hydrostatic pressure balance in the sub-photospheric region and
in the photosphere up to a height of 4H0 = 800 km, where
H0 is the pressure scale height at z = 0. The inclusion of the
sub-photospheric region avoids unrealistic density oscillations
due to the reflection of waves at the boundaries, thus avoiding
any influence from the boundary conditions on the coronal
dynamics. For the rest of the loop, density decreases as ρ ∝ h−4 ,
where h is the height from the base of the loop. This is based on
the work by Shibata et al. (1989a, 1989b), in which the results
of two-dimensional MHD simulations of emerging flux by
Parker instability exhibit such pressure distribution. The initial
temperature all along the loop is set to T = 104 K. The density at
the photosphere (z = 0) is set to ρ0 = 2.53 × 10−7 g cm−3 , and,
correspondingly, the photospheric pressure is p0 = 2.09 × 105
dyn cm−2 . The plasma β parameter is chosen to be unity at
z = 0, setting the photospheric magnetic to B0 = 2.29 × 103 G.
The value of the magnetic field at the top of the loop is then
Btop = 2.29 G.
The spatial resolution in the numerical scheme is set to 5 km
up to a height of ∼16,000 km above the photosphere. Then,
the grid size slowly increases until it reaches a size of 20 km
in the corona. The size is then kept constant up to the apex
of the loop. We assume rigid wall boundary conditions at the
photosphere. The numerical scheme adopted is the MOC-CT
scheme for solving the magnetic induction equation (Evans &
Hawley 1988; Stone & Norman 1992) and the CIP scheme
(cubic interpolated propagation; Yabe & Aoki 1991) for solving
the others. Please refer to Kudoh et al. (1998) for details about
the application of these scheme. The total time of the simulation
is 568 minutes.
4. SIMULATION RESULTS
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
No. 1, 2010
CORONAL RAIN AS A MARKER FOR CORONAL HEATING MECHANISMS
Figure 4. Phase diagram of mean temperature and mean density of the corona
in the case of footpoint nanoflare heating. Arrows show the time direction; and
solid, dashed, and dot-dashed curves denote the limit cycles (blue, green, and
red in the online version). The circle, triangle, square, and lozenge denote the
end of these cycles, respectively. The dotted curve corresponds to the start of
the simulation.
(A color version of this figure is available in the online journal.)
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.
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
161
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.
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.
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
2
is vφ,ph
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
6
It should be noted that the temperatures in the condensation drop as low as
104 K. This does not show up in the phase diagram of Figure 4, since the
averages are taken over the entire corona.
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ANTOLIN, SHIBATA, & VISSERS
Vol. 716
Figure 5. Density (left) and pressure (right) maps along the loop. Catastrophic cooling characterized by a temperature and pressure drop occurs (locally in this case)
forming a cool and dense condensation close to the apex of the loop, which falls down with increasing speed due to gravity. The motion of the condensation is also
subject to the large gas pressure changes inside the loop, to the siphon flows that ensue, and to the strong acoustic shocks from the heating events at the footpoints.
The traces of the propagating shocks are clearly observed. Note the strong deceleration of the blob as it enters the chromosphere, and the following oscillation of the
transition region.
Figure 7. Phase diagram of the mean temperature and density in the corona
for the case of a loop heated by Alfvén waves. The photospheric rms velocity
amplitude of the waves is 1.5 km s−1 . The arrow indicates the time direction.
The times corresponding to the circle and the triangle (end of simulation) are
indicated. Limit cycles are absent in this case. The corona reaches a uniform
energy state, which acts as an attractor in the diagram.
Figure 6. Upper panel: velocity of the condensation in Figure 5 along the
loop with respect to time. Lower panel: length along the loop of the same
condensation with respect to time.
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).
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.
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
No. 1, 2010
CORONAL RAIN AS A MARKER FOR CORONAL HEATING MECHANISMS
Figure 8. Photospheric azimuthal (rms) velocity amplitude with respect to
time. The loop is subject to both nanoflare reconnection heating and Alfvén
wave heating. The amplitude of the Alfvén waves increases with time becoming
a non-negligible energy source in the second half of the simulation. Symbols
indicate the same times as the corresponding same symbols in Figure 9.
Figure 9. Phase diagram of the mean temperature and density in the corona for
the case of a loop with both nanoflare reconnection heating and Alfvén wave
heating. Arrows indicate the time direction; and solid, dashed, and dot-dashed
curves (blue, green, and red curves in the online version) denote limit cycles
(the circle, triangle, and square denote the start of these cycles, respectively).
The dotted curve corresponds to the initial stage of the simulation. In this case,
the amplitude of the Alfvén waves increases in time as indicated in Figure 8.
When the energy from the latter becomes non-negligible the corona reaches
thermal equilibrium and correspondingly the cycles converge to a uniform state.
(A color version of this figure is available in the online journal.)
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,
ρvφ 2 Bφ2
Bs Bφ vφ
F =−
+
+
vs
4π
2
4π
ρvs2
γ
p+
+ ρgz vs .
+
(10)
γ −1
2
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
163
Figure 10. Phase diagram of mean temperature and mean density of the corona
for the case of footpoint nanoflare heating with increased heating rate. Arrows
show the time direction; and solid, dashed, dot-dashed, and long dashed curves
denote the limit cycles (blue, green, red, and pink in the online version). The
circle, triangle, square, and lozenge denote the end of the first three cycles,
respectively. The asterisk denotes the end of the simulation. The dotted curve
corresponds to the start of the simulation.
(A color version of this figure is available in the online journal.)
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.
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
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ANTOLIN, SHIBATA, & VISSERS
Vol. 716
Figure 11. Density (left) and pressure (right) maps along the loop for the case of footpoint nanoflare heating with increased heating rate. Catastrophic cooling occurs
forming two cool and dense condensations one after another. The subsequent motion of the condensation is determined mainly from the internal pressure changes in
the loop rather than gravity.
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.
5. DISCUSSION AND CONCLUSIONS
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 highresolution 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
No. 1, 2010
CORONAL RAIN AS A MARKER FOR CORONAL HEATING MECHANISMS
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 threedimensional 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
165
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.
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