” tablenum SIXtablenum AAPymd AAPmd AAPym hmsUT hmUT AAPymd hmUT

11institutetext: Centre for mathematical Plasma Astrophysics, Department of
Mathematics, KU Leuven, Celestijnenlaan 200B bus 2400, 3001 Leuven, Belgium.
11email<EMAIL_ADDRESS>22institutetext: Institut d’Astrophysique
Spatiale, CNRS, Univ. Paris-Sud, Université Paris-Saclay, Bât. 121, 91405
Orsay cedex, France 33institutetext: LPC2E, CNRS/University of Orléans/CNES,
3A avenue de la Recherche Scientifique, Orléans, France 44institutetext:
Predictive Science, Inc., San Diego, CA 92121, USA 55institutetext: Institute
of Applied Computing & Community Code, Universitat de les Illes Balears, 07122
Palma de Mallorca, Spain

#  The role of asymmetries in coronal rain formation during thermal non-
equilibrium cycles

Gabriel Pelouze  The role of asymmetries in coronal rain formation during
thermal non-equilibrium cycles The role of asymmetries in coronal rain
formation during thermal non-equilibrium cycles The role of asymmetries in
coronal rain formation during thermal non-equilibrium cycles The role of
asymmetries in coronal rain formation during thermal non-equilibrium cycles
Frédéric Auchère  The role of asymmetries in coronal rain formation during
thermal non-equilibrium cycles The role of asymmetries in coronal rain
formation during thermal non-equilibrium cycles Karine Bocchialini  The role
of asymmetries in coronal rain formation during thermal non-equilibrium cycles
The role of asymmetries in coronal rain formation during thermal non-
equilibrium cycles Clara Froment  The role of asymmetries in coronal rain
formation during thermal non-equilibrium cycles The role of asymmetries in
coronal rain formation during thermal non-equilibrium cycles Zoran Mikić  The
role of asymmetries in coronal rain formation during thermal non-equilibrium
cycles The role of asymmetries in coronal rain formation during thermal non-
equilibrium cycles Elie Soubrié  The role of asymmetries in coronal rain
formation during thermal non-equilibrium cycles The role of asymmetries in
coronal rain formation during thermal non-equilibrium cycles The role of
asymmetries in coronal rain formation during thermal non-equilibrium cycles
The role of asymmetries in coronal rain formation during thermal non-
equilibrium cycles Alfred Voyeux  The role of asymmetries in coronal rain
formation during thermal non-equilibrium cycles The role of asymmetries in
coronal rain formation during thermal non-equilibrium cycles

###### Abstract

Context.  Thermal non-equilibrium (TNE) produces several observables that can
be used to constrain the spatial and temporal distribution of solar coronal
heating. Its manifestations include prominence formation, coronal rain, and
long-period intensity pulsations in coronal loops. The recent observation of
abundant periodic coronal rain associated with intensity pulsations by Auchère
et al. allows to unify these two phenomena as the result of TNE condensation
and evaporation cycles. On the other hand, many intensity pulsation events
observed by Froment et al. show little to no coronal rain formation.

Aims.  Our goal is to understand why some TNE cycles produce such abundant
coronal rain, while others produce little to no rain.

Methods.  We reconstruct the geometry of the event reported by Auchère et al.,
using images from STEREO/SECCHI/EUVI and magnetograms from SDO/HMI. We then
perform 1D hydrodynamic simulations of this event, for different heating
parameters and variations of the loop geometry (9000 simulations in total). We
compare the resulting behaviour to simulations of TNE cycles by Froment et al.
that do not produce coronal rain.

Results.  Our simulations show that both prominences and TNE cycles (with and
without coronal rain) can form within the same magnetic structure. We show
that the formation of coronal rain during TNE cycles depends on the asymmetry
of the loop and of the heating. Asymmetric loops are overall less likely to
produce coronal rain, regardless of the heating. In symmetric loops, coronal
rain forms when the heating is also symmetric. In asymmetric loops, rain forms
only when the heating compensates the asymmetry.

###### Key Words.:

Sun: atmosphere – Sun: corona – Sun: oscillations – Sun: UV radiation

##  1 Introduction

Understanding the heating of coronal loops has been a long-standing problem in
solar physics. Numerous physical mechanisms have been proposed (see Cranmer &
Winebarger, 2019, and references therein). The main challenge is to identify
which ones are predominant in the solar atmosphere. This requires strong
observational diagnostics to discriminate between candidate mechanisms. To
that end, it is important to constrain the spatial and temporal properties of
the heating.

Thermal non-equilibrium (TNE) provides several valuable observables to
constrain coronal heating. In the solar corona, TNE occurs in coronal loops
which are subject to a highly stratified heating localised close to the
surface (Antiochos & Klimchuk, 1991; Antiochos et al., 1999, 2000; Karpen et
al., 2001; Klimchuk & Luna, 2019; Klimchuk, 2019; Antolin, 2020). In this
configuration, the heating may not balance the radiative losses at the apex of
the loop. Such loops thus have no thermal equilibrium state. As a result of
the footpoint heating, chromospheric plasma evaporates, and upflows develop in
both legs of the loop. Plasma accumulates at the loop apex, leading to an
increase of the density. The radiative losses increase quadratically with the
density, which at some point, since the heat input is constant, results in an
inevitable cooling. As the temperature drops, the radiative losses further
increase (Pottasch, 1965; McWhirter et al., 1975), and the plasma quickly
starts to condense close to the apex of the loop. Depending on the geometry of
the magnetic field, the condensation may accumulate at coronal heights and
form a prominence (e.g. Mok et al., 1990; Antiochos & Klimchuk, 1991;
Antiochos et al., 1999, 2000; Karpen et al., 2001, 2005; Xia et al., 2011,
2014; Xia & Keppens, 2016). Alternatively, the condensation may be evacuated
from the loop, and fall towards one of its footpoints. In this latter case,
the process starts over, resulting in condensation and evaporation cycles,
also referred to as TNE cycles (Kuin & Martens, 1982; Martens & Kuin, 1983;
Craig & Schulkes, 1985). The characteristics of TNE cycles (e.g. its period,
or the minimum and maximum temperature) depend on the spatial and temporal
properties of the heating, as well as the geometry of the loop. These cycles
thus constitute a valuable tool to constrain the heating of coronal loops.

TNE cycles result in long-period ($2$ to $16\text{\,}\mathrm{h}$) pulsations
of the extreme-ultraviolet (EUV) emission of coronal loops. Such pulsations
were first detected by Auchère et al. (2014) in images from the Extreme-
ultraviolet Imaging Telescope (EIT; Delaboudinière et al., 1995) onboard the
Solar and Heliospheric Observatory (SOHO; Domingo et al., 1995), and by
Froment et al. (2015) in images from the Atmospheric Imaging Assembly (AIA;
Lemen et al., 2012; Boerner et al., 2012) onboard the Solar Dynamics
Observatory (SDO; Pesnell et al., 2012). These long-period intensity
pulsations have been found to be very common in coronal loops (Auchère et al.,
2014; Froment, 2016). This provides a strong indication that a subset of
coronal loops are subject to a stratified heating, localised close to their
footpoints.

Another key consequence of TNE is the formation of coronal rain. It occurs
when plasma at coronal heights condenses to chromospheric temperatures before
being evacuated (Schrijver, 2001; Müller et al., 2003, 2004, 2005; De Groof et
al., 2005; Antolin et al., 2010). This condensed plasma falls along the
coronal loop, forming blob-like structures that are observed in chromospheric
and transition-region lines (Kawaguchi, 1970; Leroy, 1972; Foukal, 1978;
Schrijver, 2001; O’Shea et al., 2007; De Groof et al., 2004, 2005; Antolin et
al., 2010, 2012; Antolin & Rouppe van der Voort, 2012; Vashalomidze et al.,
2015). Coronal rain can also occur in post-flare loops as a result of the
intense transient heating from the flare (Scullion et al., 2016). While the
formation of periodic coronal rain during TNE cycles had long been suspected
(e.g. Antolin et al., 2010), it proved challenging to observe.

The first case of periodic coronal rain associated with EUV intensity
pulsations was reported by Auchère et al. (2018). This event, nicknamed the
“rain bow ”, was observed in long trans-equatorial loops observed off-limb by
SDO/AIA. The authors report the detection of EUV intensity pulsations from
plasma at coronal temperatures, accompanied by periodic coronal rain observed
in the $304\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ channel. The observation
in the $304\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ channel indicates that the
condensations cool down to at least $90\,000\text{\,}\mathrm{K}$ during the
TNE cycle. The rain is also seen in images of the H$\alpha$ coronograph of the
Pic du Midi observatory (Koechlin et al., 2019), bringing the temperature
range all the way down to chromospheric (this was not published by Auchère et
al., 2018). Finally, at the end of the observation sequence, the coronal rain
remains longer at the apex of the loop before falling, looking like the onset
formation of a prominence. This event confirms that coronal rain and EUV
intensity pulsations are indeed two aspects of TNE cycles. It also hints to
the fact that prominences could result from extremely similar heating or
magnetic field configurations. Further observations of coronal rain associated
with EUV intensity pulsations were performed by Froment et al. (2020).

On the other hand, TNE cycles can occur with little to no formation of coronal
rain. This happens when the condensing plasma is reheated before it has had
time to cool down to chromospheric temperatures. Entirely ruling out the
presence of coronal rain in observations of TNE cycles is difficult because
the finer rain drops may not be resolved by the available instruments.
However, Froment et al. (2015) performed a detailed analysis of the thermal
evolution of three EUV intensity pulsation events, and concluded that some of
the observed cycles were unlikely to produce coronal rain. This was later
confirmed by the analysis of off-disk observation of these events by the
Extreme Ultraviolet Imager (EUVI; Wuelser et al., 2004) from the Sun Earth
Connection Coronal and Heliospheric Investigation instrument suite (SECCHI;
Howard et al., 2008) onboard the Solar Terrestrial Relations Observatory
(STEREO; Driesman et al., 2008; Kaiser et al., 2008). Finally, flows develop
during all TNE cycles, even if the plasma remains at coronal temperature
throughout the cycle and no condensation forms (Mikić et al., 2013). The
observation of such flows of plasma at coronal temperature was reported by
Pelouze et al. (2020).

The different consequences of TNE in coronal loops have also been investigated
in a number of simulation studies. The first simulation studies of TNE focused
on the evolution of plasma inside a fixed coronal loop, as a response to an
imposed heating. A common approach is to reduce a 3D magnetohydrodynamic (MHD)
problem to a 1D hydrodynamic problem, where the plasma evolves along a fixed
magnetic field line. Such simulations produced both TNE cycles with formation
of coronal rain (Kuin & Martens, 1982; Martens & Kuin, 1983; Karpen et al.,
2001, 2005; Müller et al., 2003, 2004, 2005; Antolin et al., 2010; Xia et al.,
2011), and prominences (Antiochos & Klimchuk, 1991; Antiochos et al., 1999,
2000; Karpen et al., 2001, 2005, 2006; Karpen & Antiochos, 2008; Xia et al.,
2011). However, simulations by Mikić et al. (2013) showed that the
characteristics of TNE cycles depend more strongly on the geometry of the loop
than assumed in previous works. In particular, they demonstrated that TNE
cycles without formation of coronal rain (incomplete condensations) could
develop in loops with a non-uniform cross-sectional area. Froment et al.
(2017) later performed simulations that accurately reproduced the TNE cycles
with little coronal rain observed by Froment et al. (2015). Froment et al.
(2018) then investigated how the heating and the geometry of the loop interact
to produce TNE cycles. To that end, they simulated the response of three loops
of different geometries to a range of heating functions. They concluded that
TNE cycles only occur for a fraction of the explored heating parameters, and
require a specific match between the loop geometry and the heating function.

However, it is still unclear why some TNE cycles produce little to no coronal
rain (e.g. Froment et al., 2015), while some produce abundant coronal rain
(e.g. the rain bow, Auchère et al., 2018). Furthermore, the rain bow forms
very abundant rain showers compared to the other periodic coronal rain event
(Froment et al., 2020), and the observations even suggests the onset formation
of a prominence. Froment et al. (2018) have shown that cycles with and without
coronal rain can occur within the same loop, given different heating
functions. Our goal is to better understand which geometry and heating
parameters lead to the formation of coronal rain during TNE cycles, and to the
extreme behaviour of the rain bow. To that end, we perform numerical
simulations of the rain bow event (Auchère et al., 2018). We derive eight
variants of the geometry, and study how the plasma evolves in each loop as a
response to a large number of heating functions. We then compare the results
to the parametric study realised by Froment et al. (2018) for different loop
geometries, in particular that of a loop experiencing TNE cycles with little
coronal rain. This means comparing an event with abundant periodic coronal
rain forming in trans-equatorial loops (the rain bow), to TNE cycles with
little to no rain occurring in long loops located at the edge of an active
region (Froment et al., 2018, loop B).

We first determine the geometry of the rain bow, which is required to perform
the simulations (Sect. 2). We then present the numerical setup (Sect. 3.1),
and the exploration of the geometries and heating parameter space (Sect. 3.2).
This parametric study shows that all evolution scenarios of TNE (pulsation at
coronal temperatures, periodic coronal rain, and prominence formation) can
occur in the rain bow loop, depending on the heating function (Sect. 3.3).
Additionally, we estimate the spatial distribution of the heating in the
observed loops (Sect. 3.4). We then compare our simulations to the parametric
study performed by Froment et al. (2018) for different loop geometries, in
order to understand how the asymmetry of the loop and the heating result in
the formation of coronal rain during TNE cycles (Sect. 4). Finally, we
summarise our conclusions in Sect. 5.

##  2 Geometry of the “rain bow ”

Figure 1:  Geometries of the “rain bow ” from the stereoscopic reconstruction
(S) and from the magnetic field extrapolation (E13 and E98), viewed from the
position of SDO (a), STEREO B (b), and near the solar South pole (c). The
heliographic grid is spaced by $10\text{\,}\mathrm{\SIUnitSymbolDegree}$. The
cross indicating the North and East directions on the solar sphere is located
at the latitude of $-20\text{\,}\mathrm{\SIUnitSymbolDegree}$ and the
Carrington longitude of $220\text{\,}\mathrm{\SIUnitSymbolDegree}$.  Figure 2:
Altitude (a), projected gravity (b), and relative loop cross-sectional area
(c) as a function of the position along the loop $s$ divided by the loop
length $L$, for the geometries shown in Fig. 1, and the six loop area profiles
generated for the $S$ geometry. The maximum values of the altitude and
relative loop area are marked on the corresponding y-axes. The position $s=0$
corresponds to the North footpoint of each loop.

To perform numerical simulations of the rain bow event (Auchère et al., 2018),
it is necessary to know the 3D geometry of the loop. For loops observed on-
disk, the geometry can be reconstructed using a magnetic field extrapolation
(e.g. Mikić et al., 2013; Froment et al., 2017). However, this is not possible
for loops observed off-limb such as the rain bow, as there exist no cotemporal
magnetogram. The closest magnetograms suitable for extrapolation are
necessarily out of date, having been taken when the loop was observed on-disk
from Earth. In the case of the rain bow which is observed at the East limb,
that is either at the previous solar rotation (about 20 days before), or once
the structure reached the disk centre (about six days later).

However, the rain bow was also observed on-disk by SECCHI/EUVI onboard STEREO
B. We combine these observations with images from SDO/AIA to get a
stereoscopic view of the loop and reconstruct its geometry (Sect. 2.1 and
Appendix A).

Then, we use synoptic magnetograms from the Helioseismic and Magnetic Imager
(HMI; Scherrer et al., 2012) onboard SDO to get an estimation of the magnetic
field expansion in the loop (Sect. 2.2 and Appendix B).

###  2.1 Stereoscopic reconstruction

We use stereoscopic observations of SDO/AIA (near Earth), and STEREO
B/SECCHI/EUVI (separation of $None$ with Earth) to reconstruct the
tridimensional geometry of the rain bow loop. The reconstruction is summarised
in Appendix A. The resulting 3D loop shape is shown in Fig. 1 (line S). It has
a length of $635\text{.}36\text{\,}\mathrm{M}\mathrm{m}$, and is oriented in
the North-South direction. While it appears to be in a plane when seen from
SDO (Fig. 1 a), the loop is clearly non-planar and leans towards the East of
the solar sphere. As a result, it is much longer than it appears to be in AIA
images (Auchère et al., 2018 report a length of
$438\text{\,}\mathrm{M}\mathrm{m}$ in the plane of the sky). Fig. 2 (a) and
(b) show the altitude $z$ and the projected gravity $g_{\parallel}$ as a
function of the position along the loop $s$ divided by the loop length $L$
($s=0$ corresponds to the North footpoint of the loop). These plots reveal the
presence of a dip near the apex, which is an unusual feature for a coronal
loop. The dip has a depth of $2\text{\,}\mathrm{M}\mathrm{m}$, and a width of
$53\text{\,}\mathrm{M}\mathrm{m}$. The loop legs are otherwise rather
symmetric, both having comparable altitude profiles. In addition, both
footpoints are almost vertical, with a projected gravity equal to the solar
surface gravity.

The stereoscopic reconstruction has a few limitations. The first one is the
that loop footpoints are not well constrained. The second limitation is that
the geometry was only reconstructed at a given time, and therefore does not
reflect the possible time evolution of the magnetic field. Finally, it only
provides us with the shape of the loop, but not its cross-sectional area.

###  2.2 Magnetic field extrapolation

The hydrodynamic simulations also need as an input the relative cross-
sectional loop area. The relative loop area $A$ is inversely proportional to
the magnetic field strength $B$: $A(s)=B(0)/B(s)$. We normalise the cross-
sectional area to the area of the North footpoint (i.e. $A(0)=1$). To estimate
the loop expansion of the rain bow, we perform a potential magnetic field
extrapolation from a low-resolution synoptic line-of-sight magnetogram
constructed using SDO/HMI data. The construction of the synoptic magnetogram
and the magnetic extrapolation are detailed in Appendix B.

We select two magnetic field lines extrapolated from this magnetogram. The
shape of these lines is shown in Fig. 1 (lines E13 and E98). The altitude,
projected gravity, and relative loop area are shown in Fig. 2. Both loops are
non-planar and inclined towards the East (albeit less than the
stereoscopically-reconstructed shape S), and have no apex dip. Line E98 has a
length of $628\text{.}86\text{\,}\mathrm{M}\mathrm{m}$, and is overall similar
to the stereoscopic shape S with no apex dip. The dip present in shape S could
result from the deformation of the magnetic field by the abundant coronal
rain, because the stereoscopic reconstruction was performed late in the
observing sequence, after a lot of rain had already formed. Such deformation
would not be visible in the result of the potential extrapolation. Line E13 is
shorter, with a length of $465\text{.}54\text{\,}\mathrm{M}\mathrm{m}$.

As discussed in the appendix, the potential extrapolation yields a very coarse
estimation of the magnetic field. To mitigate this, we build six cross-
sectional area profiles from four extrapolated magnetic field lines, and
combine them with the shape from the stereoscopic reconstruction S. Thus, we
can later verify whether the simulation results strongly depend on the loop
area expansion, and assess the importance of the uncertainties associated with
the extrapolation. We label the resulting geometries with their maximum
relative area expansion: S10, S25, S47, S60, S73, and S98. The cross-sectional
area profiles of these geometries are shown in Fig. 2 (c).

We perform numerical simulations for the six geometries using the stereoscopic
loop shape, as well as for the two extrapolated geometries.

##  3 Numerical simulations of TNE cycles

###  3.1 Numerical setup

We use the 1D hydrodynamic setup described by Mikić et al. (2013). This setup
allows us to compute the evolution of the plasma inside a fixed magnetic field
tube of non-uniform cross-sectional area. To that end, the equations for the
conservation of mass, momentum, and energy are solved. The energy equation
includes radiative losses, thermal conduction, and a constant, non-uniform
imposed heating term $H(s)$. The radiative losses are computed using the
Chianti atomic database (version 7.1; Dere et al., 1997; Landi et al., 2013),
coronal abundances (Feldman et al., 1992; Feldman, 1992; Grevesse & Sauval,
1998; Landi et al., 2002), and the Chianti collisional ionization equilibrium
model (Dere et al., 2009). The thermal conduction is computed using the
Spitzer conductivity $\kappa_{\parallel}(T)$ (Spitzer, 1962) above the cut-off
temperature $T_{c}=$250\,000\text{\,}\mathrm{K}$$, and a constant conductivity
$\kappa_{\parallel}(T_{c})$ at lower temperatures (Lionello et al., 2009;
Mikić et al., 2013). This artificially broadens the transition region,
allowing for larger numerical grid cells and thus a significant reduction of
the computation time. Mikić et al. (2013) showed that the use of the modified
conductivity, as well as the choice of the radiative loss function and of the
abundances, do not significantly change the simulation results. In this setup,
it is assumed that the plasma is composed of fully ionised hydrogen (thus the
electron and ion number densities are equal), and that the ion and electron
temperatures are equal.

The loop is initialised in hydrostatic equilibrium, with a parabolic
temperature profile reaching $2\text{\,}\mathrm{M}\mathrm{K}$ at the loop
midpoint, and a $3\text{.}5\text{\,}\mathrm{M}\mathrm{m}$-thick chromosphere
at a constant temperature of $20\,000\text{\,}\mathrm{K}$. Throughout the
simulation, the constant chromospheric temperature $T_{\mathrm{ch}}$ and
number density $N_{\mathrm{ch}}$ are imposed at the domain boundaries ($s=0$
and $s=L$). We set $T_{\mathrm{ch}}=$20\,000\text{\,}\mathrm{K}$$ and
$N_{\mathrm{ch}}=$3\text{\cdot}{10}^{19}\text{\,}\mathrm{m}^{-3}$$. This
boundary condition allows us to maintain a layer of chromospheric plasma at
both footpoints, which serves as a mass reservoir for evaporation, and absorb
falling condensations. Previous studies using this numerical setup imposed
$N_{\mathrm{ch}}=$6\text{\cdot}{10}^{18}\text{\,}\mathrm{m}^{-3}$$ (Mikić et
al., 2013; Downs et al., 2016; Froment et al., 2017, 2018). However, some
simulations of the rain bow produce very large condensations that are not
absorbed by the resulting chromospheric layer. Such condensations cause
numerical issues that interrupt the simulation when they reach the boundaries
of the computation domain. To reduce the number of interrupted simulations, we
increased the chromospheric density to
$3\text{\cdot}{10}^{19}\text{\,}\mathrm{m}^{-3}$, thus increasing the
thickness of the chromospheric layer. While this value may be larger than
typical chromospheric densities, it does not affect the solution in the
coronal part of the loop (see discussion in Mok et al., 2005). We verified
that the behaviour in the coronal part of the simulation did not change when
using $N_{\mathrm{ch}}=$6\text{\cdot}{10}^{18}\text{\,}\mathrm{m}^{-3}$$ and
$3\text{\cdot}{10}^{19}\text{\,}\mathrm{m}^{-3}$.

The loop is subject to a constant stratified heating, localised near both
footpoints. The heating consists of a uniform background term, and two
exponentially decreasing terms, one for each leg of the loop. It is given by:

$H(s)=H_{0}+H_{1}\left(\mathrm{e}^{-g(s)/\lambda_{1}}+\mathrm{e}^{-g(L-s)/\lambda_{2}}\right)$
(1)

where $H_{0}$ is the constant background volumetric heating rate, $H_{1}$ the
stratified volumetric heating rate, and $\lambda_{1}$ and $\lambda_{2}$ are
the scale heights of the stratified heating at both footpoints. The term
$g(s)=\max(s-\Delta,0)$ allows for a constant heating in the chromosphere, up
to $\Delta=$5\text{\,}\mathrm{M}\mathrm{m}$$ away from both footpoints. To
obtain a stratified heating, we choose $H_{1}\gg H_{0}$, and
$\\{\lambda_{1},\lambda_{2}\\}<L$. A greater stratification is obtained with
smaller heating scale height values, or higher values of $H_{1}/H_{0}$. We
note that more energy is deposited in the loop as either $\lambda_{1}$ or
$\lambda_{2}$ increase. A symmetric heating is obtained when
$\lambda_{1}=\lambda_{2}$.

We use $10\,000$ grid points along the loop, with variable cell sizes of
$4\text{\,}\mathrm{k}\mathrm{m}$ in the chromosphere,
$40\text{\,}\mathrm{k}\mathrm{m}$ in the transition region, and
$400\text{\,}\mathrm{k}\mathrm{m}$ around the loop apex. We let the system
evolve for $72\text{\,}\mathrm{h}$.

###  3.2 Parameter-space scan: loop geometry and heating

Table 1: Explored parameter space: simulated geometries with their loop length $L$, maximum relative area expansion $A_{\text{max}}$, and constant background heating $H_{0}$, as well as the explored values of the stratified heating intensity $H_{1}$ and the heating scale heights $\lambda_{1}$ and $\lambda_{2}$.  Geometry |  $L$ [Mm] |  $A_{\text{max}}$ |  $H_{0}$ [$\mathrm{nW}\text{\leavevmode\nobreak\ }{\mathrm{m}}^{-3}$] |  $H_{1}/H_{0}$ |  $\\{\lambda_{1},\lambda_{2}\\}/L$
---|---|---|---|---|---
S10 |  635.36 |  10.0 |  20.3 |  Five values each: 1280, 2560, 5120, 10240, 20480. |  $15$ values each, from 1% to 20%.
S25 |  24.6 |  19.0
S47 |  46.7 |  15.5
S60 |  59.5 |  14.5
S73 |  72.5 |  14.0
S98 |  98.4 |  13.0
E98 |  628.86 |  98.4 |  14.0
E13 |  465.54 |  13.3 |  53.0

We investigate how the evolution of the plasma in the loop depends on the
parameters of the heating function (Eq. 1), the cross-sectional area of the
loop, and the presence of a dip at the apex of the loop. To that end, we
perform simulations for the eight geometries described in Sect. 2 and shown in
Figs. 1 and 2. Geometries S10, S25, S47, S60, S73, and S98 share the
stereoscopically reconstructed loop shape, with a length of
$635\text{.}36\text{\,}\mathrm{M}\mathrm{m}$ and a dip at the apex. Their
maximum area expansions vary from $10$ to $98$. Geometries E13 and E98 are
obtained from magnetic extrapolations and do not have apex dips. Geometry E98
is the closest to the stereoscopic geometries, with a length of
$628\text{.}86\text{\,}\mathrm{M}\mathrm{m}$. E13 is shorter, with a length of
$465\text{.}54\text{\,}\mathrm{M}\mathrm{m}$.

For each of these geometries, we choose $H_{0}$ such that the apex temperature
is approximately $1\text{\,}\mathrm{M}\mathrm{K}$ when the loop is subject to
a uniform heating (i.e. $H_{1}=0$). We then choose five values for $H_{1}$,
expressed as multiples of $H_{0}$: $(1280,2560,5120,10240,20480)\times H_{0}$.
We also select 15 values for each heating scale height $\lambda_{1}$ and
$\lambda_{2}$, ranging from $1\text{\,}\mathrm{\char 37\relax}$ to
$20\text{\,}\mathrm{\char 37\relax}$ of the loop length $L$. While the most
extreme heating functions may no longer correspond to realistic coronal
conditions (typically functions with high values of all three parameters),
this choice of parameters allows us to not neglect some realistic functions
where only one or two parameters reach extreme values. In particular, we will
show in Sect. 3.4 that a high value of $H_{1}/H_{0}$ is required in order to
reproduce the observed behaviour of the rain bow.

We perform simulations for all possible combinations of these heating
parameters. This corresponds to $1125$ simulations for each loop geometry, for
a total of $9000$ simulations. The properties of the loop geometries and of
the explored heating parameters are summarised in Table 1.

###  3.3 Results: all TNE consequences in the same simulation set

Figure 3:  Temperature evolution for the 225 simulations of the rain bow using
the geometry S47 (stereoscopically reconstructed shape, with a maximum area
expansion $A_{\text{max}}=47$), a constant heating rate
$H_{0}=$15\text{.}5\text{\,}\mathrm{\SIUnitSymbolMicro
W}\text{\leavevmode\nobreak\ }{\mathrm{m}}^{-3}$$, and a stratified heating
rate $H_{1}=2560H_{0}=$39\text{.}7\text{\,}\mathrm{\SIUnitSymbolMicro
W}\text{\leavevmode\nobreak\ }{\mathrm{m}}^{-3}$$. (a) Overview of the
simulations performed with $\lambda_{1}$ and $\lambda_{2}$ each varying from
$1\text{\,}\mathrm{\char 37\relax}$ to $20\text{\,}\mathrm{\char 37\relax}$ of
the loop length $L$. Each sub-plot shows the evolution of temperature as a
function of the position along the loop $s$ (x-axis, ranging from $0$ to
$L=$635\text{.}36\text{\,}\mathrm{M}\mathrm{m}$$) and of time (y-axis, ranging
from $0$ to $72\text{\,}\mathrm{h}$). Event types are marked by coloured
squares: TNE cycles with no coronal rain ($\blacksquare$), cycles with rain
($\blacksquare$), and prominence-like structures ($\blacksquare$). The white
dashed line corresponds to $\lambda_{1}=\lambda_{2}$. (b) Sample simulation
showing TNE cycles without coronal rain. (c) Sample simulation showing TNE
cycles with coronal rain. (d) Sample simulation showing the formation of a
prominence-like structure.

Figure 4:  Characteristic quantities for the $9000$ simulations of the rain
bow. (a) Maximum temperature, (b) maximum electron number density, (c) average
velocity, computed around the loop apex ($z>$0\text{.}9$z_{\mathrm{max}}$)
during the second half of each simulation ($t>$36\text{\,}\mathrm{h}$$). (d)
Event classification: no TNE behaviour ($\square$), formation of a prominence-
like structure ($\blacksquare$), TNE cycles with ($\blacksquare$) and without
($\blacksquare$) coronal rain, and simulations that failed because of
numerical issues ($\blacksquare$). Each sub-figure contains 40 squares,
corresponding to the eight line geometries (columns), and the five values of
$H_{1}/H_{0}$ (rows). Each square contains 225 simulations for the different
values of $\lambda_{1}$ (y-axis) and $\lambda_{2}$ (x-axis), ranging from
$1\text{\,}\mathrm{\char 37\relax}$ of the loop length (top-left) to
$20\text{\,}\mathrm{\char 37\relax}$ of the loop length (bottom-right). The
solid lines on (d) correspond to $\lambda_{1}=\lambda_{2}$. The dashed lines
correspond to $\lambda_{1}=0.73\lambda_{2}$. The square marked with a star
($\star$) corresponds to the 225 simulations shown in Fig. 3 (a).

For each simulation, we obtain the temperature $T$, electron number density
$N_{e}$, pressure $p$, and velocity along the loop $v_{\parallel}$ as a
function of time and position along the loop. In Fig. 3 (a), we show the
evolution of the simulations performed with the geometry S47, a heating rate
of $H_{1}=2560H_{0}=$39\text{.}7\text{\,}\mathrm{\SIUnitSymbolMicro
W}\text{\leavevmode\nobreak\ }{\mathrm{m}}^{-3}$$, and all explored values of
the heating scale heights $\lambda_{1}$ and $\lambda_{2}$. This corresponds to
225 of the total 9000 simulations. Different TNE behaviours are obtained
depending on the values of $\lambda_{1}$ and $\lambda_{2}$:

  * •

TNE cycles without formation of coronal rain, where the plasma in the coronal
part of the loop remains at coronal temperatures throughout the simulation
(Fig. 3 b);

  * •

TNE cycles with formation of coronal rain (Fig. 3 c);

  * •

the formation of a prominence-like structure, where condensed111The
temperature and electron-number density reach chromospheric values, typically
$T\sim$0\text{.}1\text{\,}\mathrm{M}\mathrm{K}$$ and
$N_{e}\sim${10}^{16}\text{\,}\mathrm{m}^{-3}$$. plasma remains trapped in the
dip at the apex of the loop (Fig. 3 d);

  * •

finally, a few cases with no TNE where the loop reaches a steady state, that
is either a static equilibrium or continuous siphon flows (e.g. at
$\lambda_{1}=$6\text{.}35\text{\,}\mathrm{M}\mathrm{m}$,\lambda_{2}=$127\text{\,}\mathrm{M}\mathrm{m}$$).

The cycles without coronal rain correspond to the cycles with incomplete
condensations (IC) described by Mikić et al. (2013) and Froment et al. (2018).
Cycles with coronal rain correspond to cycles with complete condensations
(CC).

The beginning of simulations forming a prominence-like structure is very
similar to that of simulations showing TNE cycles with coronal rain. Both
begin with an initial relaxation, followed by a plasma condensation. However,
in the case of the prominence-like structures, the condensation remains
trapped in the dip at the apex of the loop. In the case of TNE cycles with
coronal rain, the condensation is free to fall along one of the loop legs.

These simulations show the three behaviours that can result from TNE in
coronal loops: evaporation and condensation cycles with coronal rain, cycles
without coronal rain, and formation of a prominence-like structure. As the
examples in Fig. 3 show, different events can occur within the same loop
geometry, and with similar heating parameters. Simulations by Mikić et al.
(2013), Mok et al. (2016), and Froment et al. (2018) have already shown that
cycles with and without coronal rain can occur in the same loop. However, the
current parametric study is, to our best knowledge, the first to show that the
three consequences of TNE can occur within the same loop geometry.

####  3.3.1 Analysis of simulation results

To reduce the large amount of simulations, we summarise each simulation to a
few characteristic quantities shown in Fig. 4. The maximum temperature and
electron number density reached around the apex of the loop
($z>$0\text{.}9$z_{\mathrm{max}}$) during the second half of the simulation
($t>$36\text{\,}\mathrm{h}$$) are shown in Fig. 4 (a) and (b), respectively.
The velocity averaged at the loop apex during the second half of the
simulation is shown in Fig. 4 (c). Finally, the type of behaviour obtained in
each simulation is presented in Fig. 4 (d).

Determining the behaviour in each simulation first requires to identify
simulations with TNE cycles, and then those which produce coronal rain. It
also requires to identify the simulations that form prominence-like
structures. Froment et al. (2018) detected simulations with pulsations in the
Fourier space. However, many of the cycles produced in our simulations have
very long periods, ranging from $20$ to $30\text{\,}\mathrm{h}$. This gives
between $2\text{.}5$ and $3\text{.}5$ periods in a simulation of
$72\text{\,}\mathrm{h}$, which is not easily detected in the Fourier space. We
thus identify pulsations manually, by looking at the evolution of the
temperature in the loops. We then apply the same criterion as Froment et al.
(2018) to detect coronal rain: a simulation is considered to produce coronal
rain if the temperature in the coronal part of the loop (defined as
$z>$20\text{\,}\mathrm{M}\mathrm{m}$$) locally drops below
$0\text{.}5\text{\,}\mathrm{M}\mathrm{K}$ after the first
$10\text{\,}\mathrm{h}$ of the simulation. Similarly, prominence-like
structures are identified manually based on the temperature evolution.

####  3.3.2 Stereoscopically reconstructed loop

We first present the result of the simulations performed with the geometries
S10 to S98. These geometries all use the stereoscopically reconstructed loop
shape, combined with different cross-sectional area profiles.

The event classification for all simulations is shown in Fig. 4 (d). TNE
occurs in a large fraction of the explored parameter space, in particular for
smaller values of $H_{1}/H_{0}$. This includes both TNE cycles (with and
without coronal rain) and the formation of prominence-like structures.
Prominence-like structures (in blue in Fig. 4 d) form for relatively symmetric
heating functions, with scale heights close to the line
$\lambda_{1}=0.73\lambda_{2}$. In contrast, TNE cycles with coronal rain (in
red) occur for more asymmetric heating functions, and cycles without coronal
rain (in yellow) for even more asymmetric functions. In a small fraction of
the parameter space, the loop can reach a steady state, and thus shows no TNE
behaviour (in white). Finally, some simulations fail prematurely because of
numerical issues (in grey), and therefore we cannot determine their behaviour.
Theses numerical issues occur when massive condensations fall through the
chromospheric layer, and reach a boundary of the computation domain (see Sect.
3.1). Such condensations form for relatively strong and symmetric heating
functions. The strong heating causes important evaporation, while its symmetry
allows for the condensations to gain significant mass before falling.

Fig. 4 (a) shows the maximum temperature reached around the loop apex during
the second half of each simulation. Simulations with TNE cycles reach higher
temperatures than the neighbouring simulations without cycles, which is
consistent with the results of Froment et al. (2018). In addition, loops with
a larger area expansion reach slightly lower maximum temperatures (the maximum
temperatures reached with the S98 geometry are approximately
$10\text{\,}\mathrm{\char 37\relax}$ lower than those reached with the S10
geometry). This effect has been explained for static loops by Vesecky et al.
(1979): a larger area expansion increases the upwards thermal conduction,
which is, in turn, compensated by larger radiative losses in the corona.
Because the radiative losses are a decreasing function of the temperature in
the corona, this results in lower coronal temperatures. However, the
temperature variation remains small compared to the cross-sectional area
variation. The maximum temperature also increases when the heating rate
$H_{1}/H_{0}$ increases.

Fig. 4 (b) shows the maximum electron number density reached around the loop
apex during the second half of the simulations. The maximum electron density
is higher in simulations close to the diagonal $\lambda_{1}=\lambda_{2}$.
However, there is no clear-cut relationship between the simulations that
produce prominence-like structures, and adjacent simulations that produce TNE
cycles with coronal rain. The maximum electron density increases with the
heating rate $H_{1}/H_{0}$ (as expected from loop scaling laws), and slightly
decreases with the loop area expansion. Similarly to the temperature, the
density variation is small compared to the cross-sectional area variation
(approximately $25\text{\,}\mathrm{\char 37\relax}$ difference between S10 and
S98). In some simulations, the plasma $\beta$ (i.e. the ratio of gas and
magnetic pressures) is greater than one, which breaks the assumption that the
magnetic field does not evolve. This occurs for simulations with large heating
rate ($H_{1}/H_{0}$) and scale heights ($\lambda_{1}$ and $\lambda_{2}$). In
the most extreme case, the plasma $\beta$ locally reaches $500$. However, such
deviations only occur locally, when complete condensations are formed. As a
result, their effect is limited to the local dynamics and morphology of the
coronal rain or prominence. Thus, we do not expect them to have an effect on
whether condensations are formed in the first place. In addition, the actual
plasma $\beta$ is likely to be lower than the aforementioned value, because
the intensity of the extrapolated magnetic field is probably under-estimated
(see discussion in Appendix B). Finally, while beyond the assumptions of the
simulation, a high plasma $\beta$ is consistent with the observations. It
would allow the massive condensations to deform the magnetic field and form
the observed apex dip. The gradual formation of the apex dip would also
explain why coronal rain is only seen to linger at the end of the sequence
observed by Auchère et al. (2018).

Fig. 4 (c) shows the average velocity around the loop apex during the second
half of the simulations. For low heating rates $H_{1}/H_{0}$, simulations with
prominence-like structures and neighbouring simulations forming TNE cycles
display similar apex velocities. In this case, the average velocities are of
the order $+5\text{\,}\mathrm{km}\text{\leavevmode\nobreak\
}{\mathrm{s}}^{-1}$ when $\lambda_{1}>0.6\lambda_{2}$ (i.e. simulations for
which $\lambda_{1}>\lambda_{2}$, plus simulations with prominence-like
structures), and of the order of
$-5\text{\,}\mathrm{km}\text{\leavevmode\nobreak\ }{\mathrm{s}}^{-1}$
otherwise. For higher heating rates, simulations with prominence-like
structures have higher average apex velocities than neighbouring simulations
with TNE cycles, reaching values of
$\sim$8\text{\,}\mathrm{km}\text{\leavevmode\nobreak\ }{\mathrm{s}}^{-1}$$. In
simulations that form a prominence-like structure, upflows develop in both
legs of the loop and feed material into the prominence, while the velocity in
the apex dip is zero. The North edge of the dip (where velocities are
positive) has a higher altitude (see Fig. 2), and thus contributes more than
the South edge to the velocity averages shown in Fig. 4 (c). This explains why
the averaged velocities around the apex are positive in simulations that form
prominence-like structures.

####  3.3.3 Extrapolated geometries

We now present the results of the simulations performed with the geometries
E13 and E98, obtained through magnetic field extrapolation. The goal of these
simulations is to explore how the presence of the dip at the apex of the
stereoscopically-reconstructed plasma affects the evolution of the loops.
Geometry E98 is similar to geometry S98, but with no apex dip: it has a length
of $629\text{\,}\mathrm{M}\mathrm{m}$ ($635\text{\,}\mathrm{M}\mathrm{m}$ for
S98), a relative area expansion of $98$ (identical to that of S98), and a
similar altitude profile in the loop legs. Geometry E13 has an area expansion
of $13$ (thus comparable to that of S10), but a much shorter length
($465\text{.}54\text{\,}\mathrm{M}\mathrm{m}$).

The characteristic values and event classification of the simulations using
these geometries are shown in Fig. 4. Simulations of E98 mainly differ from
those of S98 in the fact that they do not form prominence-like structures.
This is the direct result of the fact that E98 has no apex dip. As a result,
all condensations fall along the loop legs, resulting in TNE cycles. Apart
from this, simulations of E98 and S98 have overall similar characteristics,
including the period of the TNE cycles. This suggests that the presence of an
apex dip leads to the formation of prominence-like structures when the heating
is relatively symmetric, but does not otherwise change the characteristics of
TNE cycles. On the other hand, simulations of E13 have very different
characteristics. This is primarily explained by the fact that geometry E13 is
significantly shorter than the other geometries.

###  3.4 Reproducing the observations of the rain bow

Figure 5:  Simulation that best reproduce the observations of the rain bow,
obtained with geometry S60, $H_{1}/H_{0}=20480$, and
$\lambda_{1}=\lambda_{2}=$15\text{\,}\mathrm{M}\mathrm{m}$$. (a) Temperature,
(b) electron number density, (c) velocity, as a function of time and position
along the loop. (d) Temperature (orange), electron number density (blue), and
velocity (purple) averaged close to the apex
($$370\text{\,}\mathrm{M}\mathrm{m}$<s<$380\text{\,}\mathrm{M}\mathrm{m}$$,
solid lines) and in the leg
($$500\text{\,}\mathrm{M}\mathrm{m}$<s<$510\text{\,}\mathrm{M}\mathrm{m}$$,
dashed lines). The apex and leg regions are also shown in sub-figures (a),
(b), and (c). The figure shows times from $35$ to $51\text{\,}\mathrm{h}$ out
of the $72\text{\,}\mathrm{h}$ of the simulation. The temporal evolution of
the temperature, electron number density, and velocity along the loop can be
seen in the online movie.

Figure 6:  Synthetic intensities for the simulation that best reproduces the
observations of the rain bow. (a) Intensity in the
$171\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ (red),
$193\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ (green), and
$335\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ (blue) bands of SDO/AIA as a
function of time and position along the loop. The black contour shows regions
where the temperature is lower than $0\text{.}5\text{\,}\mathrm{M}\mathrm{K}$.
(b) Intensities in the six coronal channels of AIA, integrated close to the
apex,
($$370\text{\,}\mathrm{M}\mathrm{m}$<s<$380\text{\,}\mathrm{M}\mathrm{m}$$).
Each light curve is normalised by subtracting its average and dividing by its
standard deviation. (c) Temperature-response function of the SDO/AIA channels
shown on panel (b).

Additionally, we use the simulations to determine the heating parameters and
the cross-sectional area expansion of the rain bow. To that end, we search for
a simulation that uses the extrapolated geometry (S10–S98), has TNE cycles
with abundant formation of coronal rain, and has the closest period to that of
the observed pulsations ($6\text{.}7\text{\,}\mathrm{h}$). The simulation
which best matches these criteria has a relative area expansion of $60$
(geometry S60), and a symmetric, highly stratified heating
($\lambda_{1}=\lambda_{2}=$15\text{\,}\mathrm{M}\mathrm{m}$=$0\text{.}024$L$)
with a high heating rate
($H_{1}=$20\,480$H_{0}=$307\text{\,}\mathrm{\SIUnitSymbolMicro
W}\text{\leavevmode\nobreak\ }{\mathrm{m}}^{-3}$$). It displays TNE cycles
with a period of $8\text{\,}\mathrm{h}$ and abundant formation of coronal
rain. The simulations using the extrapolated geometries S10 to S60 and the
aforementioned heating parameters ($\lambda_{1}$, $\lambda_{2}$ and $H_{1}$)
all have similar behaviour and period. However, simulations that use other
heating parameters display periods of at least $10\text{\,}\mathrm{h}$. Hence,
the heating parameters are well constrained, while the loop area expansion is
not.

In Fig. 5, we show the evolution of the temperature (a), electron number
density (b), and velocity (c) as a function of position along the loop and
time, during two pulsation periods. The temperature and density profiles show
that a large condensation periodically forms at the top of the South leg
(around $s=$375\text{\,}\mathrm{M}\mathrm{m}$$, which is about
$25\text{\,}\mathrm{M}\mathrm{m}$ away from the South edge of the apex dip).
This condensed plasma is then evacuated along the same footpoint, thus
resulting in coronal rain. While the condensation falls, a zone of lower
density and higher velocity forms at the opposite footpoint. When the
condensation reaches the chromosphere, a smaller condensation bounces back up,
but does not reach the loop apex. A perturbation travelling at the sound speed
is also created, which reaches the opposite footpoint.

In Fig. 5 (d), we show the evolution of the temperature, electron number
density, and velocity at the location where the condensation forms (near the
apex, for
$$370\text{\,}\mathrm{M}\mathrm{m}$<s<$380\text{\,}\mathrm{M}\mathrm{m}$$),
and in the leg along which it falls
($$500\text{\,}\mathrm{M}\mathrm{m}$<s<$510\text{\,}\mathrm{M}\mathrm{m}$$).
Near the apex, the cycle starts with a temperature of
$0\text{.}8\text{\,}\mathrm{M}\mathrm{K}$ and a density of
${10}^{14}\text{\,}\mathrm{m}^{-3}$. The plasma starts to condense at the
apex, with the temperature decreasing to
$0\text{.}1\text{\,}\mathrm{M}\mathrm{K}$ while the density increases to
$7\text{\cdot}{10}^{14}\text{\,}\mathrm{m}^{-3}$. The condensed plasma then
starts falling along the loop (the apex velocity reaches
$25\text{\,}\mathrm{km}\text{\leavevmode\nobreak\ }{\mathrm{s}}^{-1}$). As it
falls along the loop, the plasma continues to condense and its velocity
increases. In the loop leg, the temperature decreases to
$0\text{.}05\text{\,}\mathrm{M}\mathrm{K}$, the density increases to
$1\text{\cdot}{10}^{16}\text{\,}\mathrm{m}^{-3}$, and the velocity increases
to $100\text{\,}\mathrm{km}\text{\leavevmode\nobreak\ }{\mathrm{s}}^{-1}$.

The condensation starts falling with a constant acceleration of about
$5\text{\,}\mathrm{m}\text{\leavevmode\nobreak\ }{\mathrm{s}}^{-2}$ along the
loop. The acceleration then increases in the loop leg (for
$s>$420\text{\,}\mathrm{M}\mathrm{m}$$, where the loop is steeper), and
reaches a maximum value of $40\text{\,}\mathrm{m}\text{\leavevmode\nobreak\
}{\mathrm{s}}^{-2}$. Auchère et al. (2018) measured coronal rain acceleration
in the rain bow loop, and reported values ranging from
$5\text{\,}\mathrm{m}\text{\leavevmode\nobreak\ }{\mathrm{s}}^{-2}$ to
$10\text{\,}\mathrm{m}\text{\leavevmode\nobreak\ }{\mathrm{s}}^{-2}$ on the
plane of the sky222The angle between the loop and the plane of the sky (POS)
about $None$ in this section of the loop. Thus, the accelerations projected
onto the POS is about the same as along the loop.. While the simulation
features a larger acceleration in the lower part of the loop, it is overall
consistent with the observations. In particular, these values are about five
times lower than the gravitational acceleration projected along the loop
($26\text{\,}\mathrm{m}\text{\leavevmode\nobreak\ }{\mathrm{s}}^{-2}$ where
the condensation starts falling, and
$235\text{\,}\mathrm{m}\text{\leavevmode\nobreak\ }{\mathrm{s}}^{-2}$ it
reaches maximum acceleration). The observed and simulated acceleration
profiles are typical of coronal rain, which is known to have velocities lower
than the free-fall speed (see e.g. Schrijver, 2001; De Groof et al., 2005;
Antolin et al., 2010; Vashalomidze et al., 2015). This happens because coronal
rain is slowed down by forces that oppose gravity, such as pressure gradient
(Oliver et al., 2014), or drag (in multidimensional simulations, Martínez-
Gómez et al., 2020).

The period of the TNE cycles in this simulation is $20\text{\,}\mathrm{\char
37\relax}$ longer than the one observed in the rain bow. However, this is the
simulation using stereoscopically-reconstructed geometries with the smallest
period. Johnston et al. (2019) have shown that the period of TNE pulsations
can increase if the background heating ($H_{0}$ in our case) is too high. They
report a period increase for background heating values above
${10}^{3}\text{\,}\mathrm{nW}\text{\leavevmode\nobreak\ }{\mathrm{m}}^{-3}$.
Although we did not include different values of $H_{0}$ in our parametric
study, the background heating values that we used
($<$55\text{\,}\mathrm{nW}\text{\leavevmode\nobreak\ }{\mathrm{m}}^{-3}$$)
should thus be low enough to not modify the TNE periods.

Table 2: Event type counts in all simulations of the rain bow. For each geometry, we give the number of simulations showing each event type (Count column), the fraction compared to the total number of simulations using the geometry (Frac. tot. column), and the fraction compared to the number of simulations showing TNE (Frac. TNE column).  |  S10 to S98 | E98 | E13
---|---|---|---
| Count | Frac. tot. | Frac. TNE | Count | Frac. tot. | Frac. TNE | Count | Frac. tot. | Frac. TNE
TNE behaviours: $\blacksquare$ |  |  |  |  |  |  |  |  |
prominence $\blacksquare$ | $1063$ | $15\text{.}7\text{\,}\mathrm{\char 37\relax}$ | $20\text{.}0\text{\,}\mathrm{\char 37\relax}$ | – | – | – | – | – | –
cycle with rain $\blacksquare$ | $3658$ | $54\text{.}2\text{\,}\mathrm{\char 37\relax}$ | $69\text{.}0\text{\,}\mathrm{\char 37\relax}$ | $665$ | $59\text{.}1\text{\,}\mathrm{\char 37\relax}$ | $87\text{.}3\text{\,}\mathrm{\char 37\relax}$ | $344$ | $30\text{.}6\text{\,}\mathrm{\char 37\relax}$ | $88\text{.}2\text{\,}\mathrm{\char 37\relax}$
cycle without rain $\blacksquare$ | $582$ | $8\text{.}6\text{\,}\mathrm{\char 37\relax}$ | $11\text{.}0\text{\,}\mathrm{\char 37\relax}$ | $97$ | $8\text{.}6\text{\,}\mathrm{\char 37\relax}$ | $12\text{.}7\text{\,}\mathrm{\char 37\relax}$ | $46$ | $4\text{.}1\text{\,}\mathrm{\char 37\relax}$ | $11\text{.}8\text{\,}\mathrm{\char 37\relax}$
(all) $\blacksquare$ |  ($5303$) |  ($78\text{.}5\text{\,}\mathrm{\char 37\relax}$) |  |  ($762$) |  ($67\text{.}7\text{\,}\mathrm{\char 37\relax}$) |  |  ($390$) |  ($34\text{.}7\text{\,}\mathrm{\char 37\relax}$) |
No TNE $\square$ | $1227$ | $18\text{.}2\text{\,}\mathrm{\char 37\relax}$ |  | $300$ | $26\text{.}7\text{\,}\mathrm{\char 37\relax}$ |  | $735$ | $65\text{.}3\text{\,}\mathrm{\char 37\relax}$ |
Failed simulations $\blacksquare$ | $220$ | $3\text{.}3\text{\,}\mathrm{\char 37\relax}$ |  | $63$ | $5\text{.}6\text{\,}\mathrm{\char 37\relax}$ |  | – | – |

To better compare to the observations of Auchère et al. (2018), we compute
synthetic intensities for this simulation. The intensities are computed using
the SDO/AIA temperature response functions from Chianti 8 (Dere et al., 1997;
Del Zanna et al., 2015), and the method described in Froment et al. (2017,
Section 3.2.1). In Fig. 6 (a), we show the evolution of the intensity in the
intensity in the $171$, $193$, and
$335\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ bands of AIA, as a function of
the position along the loop and time. In Fig. 6 (b), we plot the evolution of
the intensity integrated close to the apex (between $370$ and
$380\text{\,}\mathrm{M}\mathrm{m}$) for the six coronal channels of AIA333We
note that the choice of the integration region does not significantly
influence the order in which different bands peak. . The temperature-response
function of these bands is shown in Fig. 6 (c). The intensity first peaks in
the $171\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ band (maximum temperature
response at $0\text{.}9\text{\,}\mathrm{M}\mathrm{K}$), which is consistent
with the fact that the loop reaches a maximum temperature of
$0\text{.}8\text{\,}\mathrm{M}\mathrm{K}$ during the cycle. The intensity then
peaks in the other bands in the following order:
$94\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,
$131\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,
$193\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,
$211\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, and
$335\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. These peaks, except
$94\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, correspond to local maximums of
each band’s temperature response function below
$0\text{.}8\text{\,}\mathrm{M}\mathrm{K}$, peaking in order of decreasing
temperature. This behaviour is consistent with the numerous reports that
coronal loops (undergoing TNE or not) are generally observed in their cooling
phase (Warren et al., 2002; Winebarger et al., 2003; Winebarger & Warren,
2005; Ugarte-Urra et al., 2006, 2009; Mulu-Moore et al., 2011; Viall &
Klimchuk, 2011, 2012; Froment et al., 2015). However, the bands peak in a
different order in the observations of the rain bow, with
$193\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,
$94\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,
$211\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, and
$335\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ peaking before
$171\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ (Auchère et al., 2018). While
this order is also consistent with loops observed in their cooling phase, it
indicates that the maximum temperature observed in the rain bow ranges between
$1$ and $1\text{.}5\text{\,}\mathrm{M}\mathrm{K}$, thus higher than in our
simulation. This temperature difference would explain why the simulated AIA
bands peak in a different order than in the observations.

The simulation that best matches the observed behaviour of the rain bow
reproduces its main observed characteristics: the development of TNE cycles
with abundant formation of coronal rain. While it does not precisely reproduce
some quantitative details of the observation (the pulsation period and the
maximum temperature), the qualitative behaviour is similar. This allows us to
conclude that a highly stratified, symmetric heating is required in order to
produce the periodic coronal rain observed in the rain bow. We can also
conclude that these loops must have a large cross-sectional area expansion.

##  4 The role of asymmetries in the formation of coronal rain

###  4.1 Simulation statistics

To get an overview of the different kind of events obtained in the simulations
of the rain bow, we count the number of simulations showing each kind of
behaviour: TNE cycles with and without rain, formation of a prominence-like
structure, relaxation to a steady-state (i.e. no TNE), and failed simulation.
The counts are given in Table 2. In this table, we also include the fraction
of the explored parameter space corresponding to each type of event. Because
these values depend on the choice of the parameter space, they do not
constitute a precise measurement of the expected event type. Rather, they
offer a synthetic view of the trends visible in Fig. 4.

We first consider the $6750$ simulations performed with the six
stereoscopically-reconstructed geometries (S10 to S98). These simulations
undergo TNE in a large fraction of the explored parameter space
($78\text{.}5\text{\,}\mathrm{\char 37\relax}$). They are split in three
groups: cycles with coronal rain ($69\text{.}0\text{\,}\mathrm{\char
37\relax}$ of simulations with TNE, or $86\text{.}3\text{\,}\mathrm{\char
37\relax}$ of simulations with TNE cycles), formation of a prominence-like
structure ($20\text{.}0\text{\,}\mathrm{\char 37\relax}$), and cycles without
rain ($11\text{.}0\text{\,}\mathrm{\char 37\relax}$). In
$18\text{.}2\text{\,}\mathrm{\char 37\relax}$ of the parameter space,
simulations relax to a steady state, thus showing no TNE behaviour. Finally,
$3\text{.}3\text{\,}\mathrm{\char 37\relax}$ of the simulations fail because
of the numerical issues discussed in Sect. 3.1.

The $1125$ simulations performed with the extrapolated geometry E98 have
relatively similar statistics. A slightly smaller fraction of the parameter
space experiences TNE ($67\text{.}7\text{\,}\mathrm{\char 37\relax}$), and no
simulation forms a prominence-like structure. However,
$87\text{.}3\text{\,}\mathrm{\char 37\relax}$ of TNE cycles produce coronal
rain, which is very close to the value obtained for geometries S10 to S98
($86\text{.}3\text{\,}\mathrm{\char 37\relax}$). Besides, more simulations,
albeit still a small fraction, fail because of numerical issues
($5\text{.}6\text{\,}\mathrm{\char 37\relax}$).

Finally, the $1125$ simulations using the extrapolated geometry E13 have
rather different statistics. A much smaller fraction of the parameter space
experiences TNE ($34\text{.}7\text{\,}\mathrm{\char 37\relax}$). Nonetheless,
a similar fraction of cycles produce coronal rain
($88\text{.}2\text{\,}\mathrm{\char 37\relax}$). None of the simulations using
this geometry fail because of numerical issues. These differences are
explained by the fact that the length and altitude profile of geometry E13
strongly differ from those of the other geometries (see Sect. 2).

Overall, a large fraction of the parameter space that we explored experiences
TNE. Periodic coronal rain is easily formed in the rain bow, with on average
$86\text{\,}\mathrm{\char 37\relax}$ of TNE cycles forming coronal rain, for
the parameters we explored.

###  4.2 Interplay between the asymmetry of the loop and of the heating

We have shown that TNE cycles with coronal rain occur preferentially in the
simulations of the rain bow. Indeed, they are the most common behaviour, found
in $51\text{\,}\mathrm{\char 37\relax}$ of all simulations, and
$86\text{\,}\mathrm{\char 37\relax}$ of simulations with TNE cycles. The
simulated behaviour is consistent with the observation of abundant periodic
coronal rain showers (the “monsoon”) in the rain bow loop (Auchère et al.,
2018).

While the rain bow loop has characteristics similar to those of the loop
studied by Froment et al. (2015), this latter had little or no rain. In order
to understand this different behaviour, we compare the simulations of the rain
bow to the simulations performed for three different geometries by Froment et
al. (2018). The authors used the same simulation setup and heating function as
in the current paper. They consider a semi-circular loop geometry with a
length of $367\text{\,}\mathrm{M}\mathrm{m}$ (loop A), a very asymmetric loop
obtained from magnetic field extrapolation (loop B, matching the observations
for which no rain was detected), and a relatively symmetric loop also obtained
from extrapolation (loop C). Loop B corresponds to a loop bundle in which TNE
cycles with little to no coronal rain had been previously identified (Froment
et al., 2015). Its eastern leg (at $s=0$) has a high inclination, resulting in
a very small projected gravity in the first $10\text{\,}\mathrm{\char
37\relax}$ of the loop length (Froment et al., 2018, Figure 3). Loop C is a
loop in which no TNE behaviour was observed. The legs of this loop are
relatively symmetric. The classification of events Froment et al. (2018)
obtained for these loops is reproduced in Fig. 7 for easier comparison with
our simulations of the rain bow (Fig. 4 d).

Figure 7:  Event classification for the simulations performed by Froment et
al. (2018), shown with a similar layout as Fig. 4 (d). Loop A is a
semicircular with a length of $367\text{\,}\mathrm{M}\mathrm{m}$, heated with
10 values of $\lambda_{1}$ and $\lambda_{2}$, both ranging from $2$ to
$11\text{\,}\mathrm{\char 37\relax}$ of the loop length. Loop B is a highly
asymmetric loop with a length of $367\text{\,}\mathrm{M}\mathrm{m}$, heated
with 12 values of $\lambda_{1}$ and $\lambda_{2}$, ranging respectively from
$7$ to $18\text{\,}\mathrm{\char 37\relax}$, and from $2$ to
$13\text{\,}\mathrm{\char 37\relax}$ of the loop length. Loop C is a
moderately asymmetric loop with a length of
$139\text{\,}\mathrm{M}\mathrm{m}$, heated with 12 values of $\lambda_{1}$ and
$\lambda_{2}$, both ranging from $4$ to $15\text{\,}\mathrm{\char 37\relax}$
of the loop length. The diagonal lines in each square correspond to
$\lambda_{1}=\lambda_{2}$.

The simulations performed by Froment et al. (2018) for the two symmetric
geometries preferentially produce coronal rain
($84\text{.}6\text{\,}\mathrm{\char 37\relax}$ TNE of cycles in loop A and
$99\text{.}4\text{\,}\mathrm{\char 37\relax}$ of cycles in loop C produce
rain). However, less rain forms in the very asymmetric loop B (only
$50\text{.}7\text{\,}\mathrm{\char 37\relax}$ of the cycles produce rain).

This comparison between the simulations of the rain bow and the simulations of
Froment et al. (2018) suggests that the geometry of the loop has, on its own,
a significant effect on the formation of coronal rain during TNE cycles. In a
relatively symmetric loop, coronal rain forms for a wider range of heating
functions. On the other hand, cycles without rain occur more easily in very
asymmetric loops.444We verified that these conclusions are unchanged if we
only consider the simulations by Froment et al. (2018) and of the rain bow
with overlapping heating parameters. However, the heating function also plays
a key role on the formation of rain. In the corona, a given loop is only
subject to one of the many explored heating functions. The development of TNE
thus results from a specific match between the loop geometry and the heating
function, as noted by Froment et al. (2018). Our simulations of the rain bow
support this conclusion, and further suggest that symmetric loops more
frequently produce cycles with coronal rain, while asymmetric loops lead to
more cycles without rain. In the case of the rain bow, cycles with no rain
only occur in a small volume of the parameter space. This makes such cycles
very unlikely to occur in a real coronal loop, as it would require the heating
and geometry to be stable over several days. On the other hand, cycles with
rain could be produced even with varying heating or geometry, provided that
the variations are slow enough.

Overall, coronal rain is formed if the condensing plasma cools down to
chromospheric temperatures before being evacuated from the loop. In this case,
the condensed plasma then falls along the loop as coronal rain. On the
contrary, no rain is formed if the condensations are destabilised earlier, and
fall while still at coronal temperatures. In relatively symmetric loops (e.g.
the rain bow), condensations can remain longer at the loop apex for a wide
variety of heating functions. Only very asymmetric heating functions lead to
cycles with no rain. As a result, such loops preferentially form cycles with
rain. In asymmetric loops (e.g. loop B of Froment et al., 2018), condensations
can only remain at the apex if an oppositely asymmetric heating creates
upflows that compensate the loop’s asymmetry. In such loops, rain is overall
less likely to occur, as it can only form under specific heating conditions.

Our simulations of the rain bow and the simulations of Froment et al. (2018)
are also consistent with the qualitative predictions of the analytical
formulas of Klimchuk & Luna (2019) which provide conditions for TNE (with or
without rain) to occur in coronal loops. We compared the simulations to the
overall predictions of these formulas, but did not perform a quantitative
comparison. Their first condition (Klimchuk & Luna, 2019, equation 13) is that
the heating must be concentrated at low altitudes. This is the case in all
simulations. The second condition (Klimchuk & Luna, 2019, equation 34) is that
the product of the asymmetries in the heating and the cross-sectional area
must not be too important. In the case of the rain bow, the loop shape and
area expansion are relatively symmetric. As a result, a wider variety of
heating functions, including highly asymmetric ones, may lead to TNE. On the
other hand, loop B of Froment et al. (2018) has a very asymmetric shape and
area expansion. This imposes a larger constraint on the heating function: in
order to have TNE, the heating must also be asymmetric in order to compensate
the geometric asymmetry of the loop.

###  4.3 Other parameters that may influence the formation of coronal rain

In the current paper, we explore different loop geometries and heating
parameters, and their influence on the development of TNE in coronal loops.
However, other factors may influence the evolution of plasma in a loop. In
particular, we did not consider any variation of the heating or the geometry
over time. Variations of the heating over short time scales (i.e. less than
the loop cooling time) have been considered in a few studies of TNE. Antolin
et al. (2010) and Susino et al. (2010) have shown that complete plasma
condensations can occur in impulsively heated loops. Winebarger et al. (2018)
and Johnston et al. (2019) have studied the influence of impulsive heating on
some characteristics of TNE cycles. Most of these studies consider heating
functions with a constant waiting time between the pulses. It would also be
worth considering stochastic heating functions, as these better describe
nanoflare heating mechanisms in the solar corona. Stochastic heating functions
have been considered in several simulations of coronal loops (e.g., in 0D by
Cargill, 2014; Cargill et al., 2015; Barnes et al., 2016, and in 1D by Cargill
et al., 2015; Bradshaw & Viall, 2016), but only in a few simulations of TNE
cycles (Antolin et al., 2010; Antolin, 2020). Including such impulsive
stochastic heating functions in large scale parametric studies is an essential
next step to understand the relationship between heating and TNE cycles.

As noted in previous studies of TNE cycles (Antolin et al., 2010; Auchère et
al., 2018; Froment et al., 2018), the heating parameters and loop geometry are
also likely to vary over longer time scales (i.e. longer than the loop cooling
time), thus resulting in a variation of the TNE conditions over time. This
could result in non-periodic TNE cycles, cycles only lasting for a few
periods, or intermittent production of coronal rain from one cycle to another.
Understanding the range of behaviours that can stem from such long-term
variations would require the exploration of a much larger parameter space than
considered in the current study. In the case of the rain bow, one could take
into account the evolution of the loop geometry over time. Doing so would
require the reconstruction of the loop geometry at several time intervals,
which would be an even greater challenge than the reconstruction presented in
Sect. 2.

The conclusions presented in this paper are based on simulations of realistic
loop geometries, which were derived from observations. The symmetric and
asymmetric loops result in clearly distinct TNE evolutions. However, it is
unclear whether the formation of coronal rain is constrained by asymmetries of
specific geometric features (such as the cross-sectional area, or the slope of
the loop legs), or results from any geometric asymmetry. We note that the
asymmetric loop B of Froment et al. (2018) has a unique geometry, with an
asymmetric cross-sectional area and one very inclined leg This leg is almost
horizontal over its first $30\text{\,}\mathrm{M}\mathrm{m}$, and partly lays
in the chromosphere. We also note that the loops considered by Froment et al.
(2018) and in the current paper have very different area expansion profiles
(Fig. 2 c). However, this is unlikely to have a significant influence compared
to the loop asymmetry, because loops A and C also preferentially produce
coronal rain, despite having different expansion profiles than the rain bow
loops. In order to understand whether the formation of coronal rain is
influenced by asymmetries in specific parts of the loop, it would be
interesting to realise simulations for a wider variety of loop geometries. The
use of synthetic geometries would allow to better control the different
geometric features.

##  5 Summary

In this paper, we present 1D hydrodynamic simulations of the “rain bow ”, a
periodic coronal rain event observed by Auchère et al. (2018). This event
produces abundant coronal rain showers, which strongly contrasts with previous
observations of TNE cycles that did not show evidence of coronal rain (Froment
et al., 2015).

To understand which parameters influence the formation of coronal rain during
TNE cycles, we performed 9000 simulations of the rain bow, exploring different
heating parameters (volumetric heating rate and asymmetry), and variants of
the geometry (shape and area expansion of the loop). To perform these
simulations, we reconstructed the three-dimension geometry of the loop.

These simulations display the three behaviours that can result from TNE in
coronal loops: evaporation and condensation cycles with coronal rain, cycles
without coronal rain, and formation of a prominence-like structure. We thus
show that prominences and TNE cycles can develop in similar conditions (same
geometry, and similar heating parameters). This constitutes a new result for
the understanding of TNE in the solar corona.

By searching for the simulation that best reproduces the observed
characteristics of the rain bow, we estimate the heating conditions of this
event. The rain bow event results from an intense and symmetric heating
localised near the footpoints of the loop (heating scale-height of
$15\text{\,}\mathrm{M}\mathrm{m}$ in both legs). Furthermore, a small
variation of the heating function results in the formation of a prominence-
like structure. This would also be consistent with the onset prominence
formation reported by Auchère et al. (2018).

Finally, we use the large parametric study to explore the relationship between
the geometry of the loop, the asymmetry of the heating, and the formation of
coronal rain during TNE cycles. We compare the simulations of the rain bow
(symmetric loop) with another set of simulations performed by Froment et al.
(2018) (two symmetric loops, and one asymmetric loop). We conclude that
coronal rain is overall more likely to occur in relatively symmetric loops. In
such loops, rain forms for a wide range of heating functions. In asymmetric
loops, however, rain can only form if the heating precisely compensates the
geometric asymmetry, so that condensations can remain in the corona long
enough to reach chromospheric temperatures. When the heating does not
compensate the loop asymmetry, condensations are evacuated from the loop
before they can cool down to chromospheric temperatures, resulting in cycles
with no coronal rain.

Froment et al. (2018) already noted that TNE requires a specific match between
the loop geometry and the heating conditions. The simulations of the rain bow
suggest that a symmetric loop allows for a looser match between the geometry
and the heating. Both behaviours are consistent with the qualitative
predictions of the formulas of Klimchuk & Luna (2019).

###### Acknowledgements.

We thank Peter Cargill for his in depth referee review, which significantly
improved the manuscript. G.P. has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program (grant agreement No. 724326). We acknowledge support from the
International Space Science Institute (ISSI), Bern, Switzerland to the
International Team 401 “Observed Multi-Scale Variability of Coronal Loops as a
Probe of Coronal Heating”. AIA and HMI data are courtesy of NASA/SDO and the
AIA and HMI science teams. EUVI data are courtesy of the STEREO SECCHI team.
This work used data provided by the MEDOC data and operations centre
(CNES/CNRS/Univ. Paris-Sud), medoc.ias.u-psud.fr. CHIANTI is a collaborative
project involving George Mason University, the University of Michigan (USA),
University of Cambridge (UK) and NASA Goddard Space Flight Center (USA).
Software: Astropy (Astropy Collaboration et al., 2013, 2018), ChiantiPy (Dere,
2013), JHelioviewer (Müller et al., 2017), SolarSoft (Freeland & Handy, 2012).

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##  Appendix A Stereoscopic reconstruction

We reconstructed the shape of the rain bow loop (Auchère et al., 2018) using
stereoscopic observations of SDO/AIA and STEREO/SECCHI/EUVI. In this appendix,
we summarise the method and the results pertinent to the present work. More
details will be given in a forthcoming companion paper.

The rain bow event was observed at the limb by AIA during
$2\text{.}5\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$ starting from
2012-07-23, 00:00:00. During this period, the STEREO B probe had a separation
of $None$ with Earth, and observed the event on-disk. The event was not
visible from STEREO A. The EUVI data consist of images in the
$171\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ band with a
$120\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ cadence, and in the
$304\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ band with a cadence of
$2\text{.}5\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ for the first
$200\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ and
$10\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ afterwards. The temperature
response function of the $171\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ band
peaks at $0\text{.}9\text{\,}\mathrm{M}\mathrm{K}$, which allows the
observation of plasma at coronal temperature. The
$304\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ band peaks at
$0\text{.}09\text{\,}\mathrm{M}\mathrm{K}$, allowing the observation of
coronal rain. During the same time period, AIA images are available in these
two bands with a cadence of $12\text{\,}\mathrm{s}$.

The coronal rain is easily observed at the limb in the AIA
$304\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ images. We use a combination of
intensity thresholding and manual clean-up to identify the regions of the
images which contain coronal rain. The coronal rain is harder to see on-disk
in the EUVI $304\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ images. To better
detect it, we divide the intensity of each image by the average of the 20
previous and 20 following images. The resulting intensity maps are then used
to manually select the regions where coronal rain is observed.

For each pair of $304\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ images from SDO
and STEREO B, we thus obtain two binary masks, containing the regions in which
coronal rain is observed. In order to reconstruct the longest possible
fraction of the loop, we select a pair of masks in which the rain occupies a
large fraction of the loop seen in the
$171\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ images. The selected images were
captured on 2012-07-25, 03:06:47, near the end of the observing sequence. We
then use the two masks to reconstruct the volume occupied by the coronal rain.
To that end, we trace the lines of sight of each rain pixel in the masks, and
compute the intersection between the lines of sight from SDO and from STEREO
B. Such a stereoscopic reconstruction technique assumes that both instruments
observe the same opaque object. However, that is not entirely the case here:
only the dense core of the rain is visible on-disk in absorption from STEREO
B, while a larger volume of the rain is seen off-limb in emission from SDO.
Additionally, ambiguities occur when rain is detected in adjacent loop
threads. When both instruments observe rain in two adjacent threads, four
distinct threads are present in the volume reconstruction. But only two of
these threads are real. We resolve such ambiguities manually, by ensuring the
continuity of threads along the loop, and in time. The loop footpoints are
inferred manually, as they are partially hidden behind the limb in the AIA
image, and are difficult to see in the EUVI images in which the rain forms
very thin strands. To that end, we extend the volume occupied by the coronal
rain down to the solar surface, while trying to preserve the overall shape of
the loop. This affects the lowest $5\text{\,}\mathrm{M}\mathrm{m}$ of the
northern leg, and the lowest $20\text{\,}\mathrm{M}\mathrm{m}$ of the southern
leg. Finally, we fit a spline to the remaining voxels. This yields an
approximation of the shape of the rain bow loop. The resulting 3D shape is
shown in Fig. 1. The altitude and projected gravity and are shown in Fig. 2
(a) and (b).

The stereoscopic reconstruction reveals a dip near the apex of the loop. While
this feature is unusual for coronal loops, it is consistent with some
prominence models (e.g. Priest et al., 1989). Such dip is consistent with the
observations of Auchère et al. (2018), who report that the condensations
linger at the apex before falling, and suggest a possible link with prominence
formation. This dip appears to be a robust result from the reconstruction, and
we are rather confident that it is not an artefact. The dip is located at the
centre of the loop ($s=$319\text{\,}\mathrm{M}\mathrm{m}$$), and at the
altitude $z=$118\text{\,}\mathrm{M}\mathrm{m}$$. The southern edge of the dip
is located very close to the dip, both in position along the loop
($32\text{\,}\mathrm{M}\mathrm{m}$ away, at
$s=$351\text{\,}\mathrm{M}\mathrm{m}$$), and in altitude
($2\text{\,}\mathrm{M}\mathrm{m}$ higher, at
$z=$120\text{\,}\mathrm{M}\mathrm{m}$$). The northern edge is further from the
dip along the loop ($87\text{\,}\mathrm{M}\mathrm{m}$ away, at
$s=$232\text{\,}\mathrm{M}\mathrm{m}$$) and in altitude
($13\text{\,}\mathrm{M}\mathrm{m}$ higher, at
$z=$131\text{\,}\mathrm{M}\mathrm{m}$$). On this northern side, the altitude
reaches $120\text{\,}\mathrm{M}\mathrm{m}$ (the height of the South edge)at
$s=$298\text{\,}\mathrm{M}\mathrm{m}$$. Overall, the dip has a depth of
$2\text{\,}\mathrm{M}\mathrm{m}$ and a width of
$53\text{\,}\mathrm{M}\mathrm{m}$.

##  Appendix B Magnetic field extrapolation

Figure 8:  Magnetogram used for the magnetic field extrapolation. The vertical dashed line at the longitude of $None$ corresponds to the limit between the synoptic magnetograms from rotations CR2125 (left) and CR2126 (right). The quadrupolar structure mentioned in the text is marked by grey circles. The black star indicates the solar disk centre as seen from Earth at the time for which geometry S was reconstructed (2012-07-25, 03:06:47). The loop geometries, the heliographic grid, and direction cross shown in Fig. 1 are reproduced for easier comparison.  Table 3:  Properties of the extrapolated field lines used to generate area expansion profiles for the stereoscopic loop shape S. Length $L$, magnetic field at the North footpoint $B(0)$, at the South footpoint $B(L)$, and minimum value close to the apex $B_{\mathrm{min}}$ .  Line |  $L$ [Mm] |  $B(0)$ [G] |  $B_{\mathrm{min}}$ [G] |  $B(L)$ [G]
---|---|---|---|---
E08 | 430.63 | 5.06 | 0.60 | 44.78
E13 | 465.54 | 8.59 | 0.65 | 47.02
E25 | 492.24 | 23.45 | 0.95 | 48.37
E98 | 628.86 | 41.92 | 0.43 | 62.68

To determine the area expansion of the rain bow loop (Auchère et al., 2018),
we perform a potential magnetic field extrapolation from a synoptic line of
sight magnetogram of SDO/HMI.

The synoptic magnetogram has a resolution of
$0\text{.}5\text{\,}\mathrm{\SIUnitSymbolDegree}\text{\leavevmode\nobreak\
}{\mathrm{\text{pixel}}}^{-1}$ in Carrington coordinates. The magnetic field
at each longitude was measured when the corresponding meridian was as the
centre of the disk as seen from SDO (i.e. approximately from Earth). For the
Carrington longitudes between $None$ and $None$, we use magnetograms recorded
at the rotation before the observation of the rain bow (CR2125), that is about
20 days before. For longitudes between $None$ and $None$, we use magnetograms
from the following rotation (CR2126), recorded a few days after the
observation of the rain bow. The boundary between the magnetograms from both
rotations is thus located between the footpoints of the stereoscopically-
reconstructed shape S (at Carrington longitudes $None$ and $None$,
respectively). The resulting magnetogram is shown in Fig. 8. This composite
magnetogram contains the polarities required to obtain extrapolated lines with
a shape resembling that of the rain bow: a magnetic quadrupole at the East of
the loops (marked by grey circles in Fig. 8), which explains their
inclination. This quadrupole is not present when only using data from rotation
CR2125 (because the South-West negative polarity had not emerged yet), nor
when only using data from rotation CR2126 (because the North-West positive
polarity has decayed too much).

We then perform a potential extrapolation, where it is assumed that the
magnetic field satisfies the condition $\@vec{\nabla}\times\@vec{B}=\@vec{0}$
(see, e.g. Régnier, 2013; Wiegelmann et al., 2014). We select four magnetic
field lines extrapolated from this magnetogram: E08, E13, E25, and E98. The
properties of these extrapolated lines are summarised in Table 3. Lines E13
and E98 are shown on Figs. 1, 2, and 8. Similarly to the stereoscopically
reconstructed loop shape (Appendix A), the extrapolated lines are non-planar
and inclined towards the East of the solar sphere.

The footpoint magnetic field of these loops ranges from $5$ to
$60\text{\,}\mathrm{G}$, which is somewhat smaller than typical active region
values (see e.g. Aschwanden et al., 1999 who report a range of
$20$–$230\text{\,}\mathrm{G}$ for a selection of 30 active region loops). This
can be explained by the low resolution of the magnetogram, which reduces and
spreads out the magnetic field intensity. Line E98 has an area expansion of
$98$, which is higher than typical active region values (Dudík et al., 2014
report a maximum value of $80$ in a given active region). Overall, potential
magnetic field extrapolation from a low resolution magnetogram only gives a
very coarse estimation of the magnetic field, especially in active regions.
However, the main goal of these extrapolations is to estimate the relative
loop cross-sectional area. (A more precise estimation of the loop shape is
already obtained from the stereoscopic reconstruction.) Nonetheless, because
of the uncertainty of this extrapolation method, we explore different cross-
sectional area profiles.

We thus generate six relative loop area profiles from the extrapolated lines
(E08, E13, E25, and E98). We combine these six area profiles with the shape
from the stereoscopic reconstruction S. The resulting geometries share the
same shape, but have different area profiles, with maximum values ranging from
$10$ to $98\text{.}4$. These new geometries are labelled with their maximum
relative area expansion: S10, S25, S47, S60, S73, and S98. Because the lines
have different lengths, we scale each area expansion profile to the loop
length. The new area expansions profiles are generated from the extrapolated
area expansion profiles using the following relations:

$\begin{split}&A_{\text{{S10}}}(s/L_{\text{{S10}}})=\left[A_{\text{{E08}}}(s/L_{\text{{E08}}})+A_{\text{{E13}}}(s/L_{\text{{E13}}})\right]/2\\\
&A_{\text{{S25}}}(s/L_{\text{{S25}}})=A_{\text{{E25}}}(s/L_{\text{{E25}}})\\\
&A_{\text{{S47}}}(s/L_{\text{{S47}}})=\left[2A_{\text{{E25}}}(s/L_{\text{{E25}}})+A_{\text{{E98}}}(s/L_{\text{{E98}}})\right]/3\\\
&A_{\text{{S60}}}(s/L_{\text{{S60}}})=\left[A_{\text{{E25}}}(s/L_{\text{{E25}}})+A_{\text{{E98}}}(s/L_{\text{{E98}}})\right]/2\\\
&A_{\text{{S73}}}(s/L_{\text{{S73}}})=\left[A_{\text{{E25}}}(s/L_{\text{{E25}}})+2A_{\text{{E98}}}(s/L_{\text{{E98}}})\right]/3\\\
&A_{\text{{S98}}}(s/L_{\text{{S98}}})=A_{\text{{E98}}}(s/L_{\text{{E98}}})\end{split}$
(2)

These cross-sectional area profiles are shown in figure Fig. 2 (c).