# Deciphering the solar coronal heating:
Energizing small-scale loops through surface convection

D. Nóbrega-Siverio Instituto de Astrofísica de Canarias, E-38205 La Laguna,
Tenerife, Spain  Universidad de La Laguna, Dept. Astrofísica, E-38206 La
Laguna, Tenerife, Spain  Rosseland Centre for Solar Physics, University of
Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway  Institute of Theoretical
Astrophysics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway  F.
Moreno-Insertis  Instituto de Astrofísica de Canarias, E-38205 La Laguna,
Tenerife, Spain  Universidad de La Laguna, Dept. Astrofísica, E-38206 La
Laguna, Tenerife, Spain  K. Galsgaard School of Mathematics and Statistics,
University of St Andrews, St Andrews KY16 9SS, Scotland, UK  K. Krikova
Rosseland Centre for Solar Physics, University of Oslo, PO Box 1029 Blindern,
0315 Oslo, Norway  Institute of Theoretical Astrophysics, University of Oslo,
PO Box 1029 Blindern, 0315 Oslo, Norway  L. Rouppe van der Voort Rosseland
Centre for Solar Physics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo,
Norway  Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029
Blindern, 0315 Oslo, Norway  R. Joshi Rosseland Centre for Solar Physics,
University of Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway  Institute of
Theoretical Astrophysics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo,
Norway  M. S. Madjarska Max Planck Institute for Solar System Research,
Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany  Space Research and
Technology Institute, Bulgarian Academy of Sciences, Acad. Georgy Bonchev
Str., Bl. 1, 1113, Sofia, Bulgaria

(Received October 23, 2023; Accepted November 17, 2023)

###### Abstract

The solar atmosphere is filled with clusters of hot small-scale loops commonly
known as Coronal Bright Points (CBPs). These ubiquitous structures stand out
in the Sun by their strong X-ray and/or extreme-ultraviolet (EUV) emission for
hours to days, which makes them a crucial piece when solving the solar coronal
heating puzzle. In addition, they can be the source of coronal jets and small-
scale filament eruptions. Here we present a novel 3D numerical model using the
Bifrost code that explains the sustained CBP heating for several hours. We
find that stochastic photospheric convective motions alone significantly
stress the CBP magnetic field topology, leading to important Joule and viscous
heating concentrated around the CBP’s inner spine at a few megameters above
the solar surface. We also detect continuous upflows with faint EUV signal
resembling observational dark coronal jets and small-scale eruptions when
H$\alpha$ fibrils interact with the reconnection site. We validate our model
by comparing simultaneous CBP observations from SDO and SST with observable
diagnostics calculated from the numerical results for EUV wavelengths as well
as for the H$\alpha$ line using the Multi3D synthesis code. Additionally, we
provide synthetic observables to be compared with Hinode, Solar Orbiter, and
IRIS. Our results constitute a step forward in the understanding of the many
different facets of the solar coronal heating problem.

magnetohydrodynamics (MHD) — methods: numerical — methods: observational —
Sun: atmosphere — Sun: chromosphere — Sun: corona

##  1 Introduction

Understanding why the solar corona temperature is a few hundred times larger
than that at the surface requires studying the heating mechanisms across
different solar regions and phenomena (e.g., Parker, 1972; Galsgaard &
Nordlund, 1996; Gudiksen & Nordlund, 2005; Klimchuk, 2006; Parnell & De
Moortel, 2012). A particularly important case is that of the Coronal Bright
Points or CBPs, structures with projected sizes between 5 and 40 Mm that stand
out with enhanced EUV and X-ray emission against the quiet Sun (Golub et al.,
1974; Madjarska, 2019). CBPs are often observed to appear above photospheric
regions with a strong parasitic polarity surrounded by a network predominantly
of the opposite polarity, a situation that typically leads to a fan-spine
magnetic structure with a nullpoint at several megameters in the corona (Zhang
et al., 2012; Mou et al., 2016; Galsgaard et al., 2017; Madjarska et al.,
2021; Cheng et al., 2023). Their relevance stems from their copious energy
emission, representing the main contributor to high-energy radiation over the
solar disk outside active regions (Mondal et al., 2023); as well as their
nearly uniform presence in the Sun and their long lifetime, from hours to days
(Madjarska, 2019). Considered to be downscaled versions of active regions
(Madjarska, 2019; Gao et al., 2022), CBPs are composed of small-scale loops,
making them a relevant case study for this kind of fundamental magnetic field
topology (Rappazzo et al., 2008; Reale, 2014). Moreover, CBPs can be the
source regions of eruptions of both hot and cool plasma (Hong et al., 2014;
Sterling et al., 2015; Kumar et al., 2019; Madjarska et al., 2022), which may
constitute a significant input of energy and mass for the solar corona
(Lionello et al., 2016; Viall & Borovsky, 2020).

A significant theoretical effort has been devoted to studying the heating of
CBPs through (a) analytical studies collectively known as Converging Flux
Models (Priest et al., 1994; Dreher et al., 1997; Priest et al., 2018); (b)
one-dimensional models of individual magnetic loops (Reale, 2014); or (c) by
means of multi-dimensional purely magnetohydrodynamical (MHD) models, mainly
analyzing the energy conversion during magnetic reconnection (Galsgaard et
al., 2000; von Rekowski et al., 2006; Santos & Büchner, 2007; Javadi et al.,
2011; Wyper et al., 2018; Syntelis et al., 2019). A drawback of these
approaches is that they rely on ad-hoc driving mechanisms, imposing converging
flows or large-scale surface motions in order to drive magnetic reconnection
at the coronal nullpoint and get a CBP. These mechanisms do not reflect the
actual stochastic convective flows in the solar photosphere, which have been
shown to provide enough energy to heat the corona in previous numerical
simulations (e.g., Gudiksen & Nordlund, 2005; Rempel, 2017; Chen et al.,
2022). However, as we propose in this paper, the energy injection through
surface convection may indeed be the source of the CBP heating. An additional
drawback of previous CBP models is that they did not include an appropriate
radiation treatment, thus ignoring important entropy sources in the lower
atmosphere. This not only precludes the calculation of many basic
observational diagnostics but also restricts the possibility of understanding
the heating and cooling processes in the chromosphere.

In this Letter, we present a comprehensive 3D radiative-MHD model of a CBP
based on a magnetic nullpoint configuration. Our model shows that a CBP can be
powered for at least several hours through the continuous action of the
underlying surface convection. The inclusion of the relevant radiation and
entropy sources in our model allows us to analyze the associated deposition of
energy in different layers of the solar atmosphere and the corresponding
changes in the dynamics, including the study of hot ejections and small-scale
eruptive chromospheric phenomena within the CBP. To validate our theoretical
model, we also present simultaneous observations of a representative CBP from
space (SDO) and from the ground (SST). Through forward modeling, we
successfully reproduce the main observed features from SDO and SST, confirming
the adequacy of our model across different atmospheric layers.

##  2 Results

###  2.1 Overview of the Experiment

The initial configuration consists of a rotationally-symmetric 3D null point
magnetic topology (see Figure 5 in Appendix A, where details concerning the
initial condition and the numerical code are also given). The inner spine of
the structure is rooted in a region with a parasitic positive polarity, while
the fan structure is anchored in the surrounding field of the opposite sign. A
general outlook of the subsequent evolution of the experiment can be gained
through Figure 1 and its associated animation. The initial topology structure
is rapidly disrupted by the stochastic granulation in the photosphere leading
to a magnetic network pattern of the predominant polarity at the surface; the
overall magnetic field topology above is stressed and magnetic reconnection is
triggered at the nullpoint. Simultaneously, the magnetic loops underneath the
fan surface experience intense heating, giving rise to a CBP with
conspicuously enhanced emissivity in EUV, X-rays, and UV, as evidenced in
Figure 1 through the synthetic observables of panels (f), (g), (h), and (i)
(see Appendix C for the calculation details). The diameter of the CBP,
estimated from the EUV response at its base, is approximately 20 Mm, which is
within the typical range reported in observations (5–40 Mm, see Madjarska,
2019, and references therein).

The CBP nullpoint undergoes continuous perturbations, as illustrated in panel
(b) through the time-evolution of its coordinates (xnp, ynp, znp), which can
be shifted by up to a few Mm, similarly to the nullpoint displacements
inferred from observations (Galsgaard et al., 2017). Around $t=111.67$ min,
the time of the snapshots in Figure 1, the current sheet around the nullpoint
experiences a major disturbance, concerning both its location and shape. The
reason for this perturbation is the approach of an elongated cool and dense
structure to the reconnection site visible in panels (c) and (d). This
structure is shown to be an H$\alpha$ chromospheric fibril in Section 2.3.
Throughout the evolution of the CBP, other chromospheric fibrils and spicule-
like structures are found to develop in the lower atmosphere (see animation at
$t=47,\ 107,\ 153,\ 221$ min), reaching the reconnection site and affecting
the CBP dynamics as well as its brightness. Such features could be the result
of the interaction of chromospheric waves and shocks with the CBP magnetic
structure. A detailed analysis will be covered in a follow-up paper.

Rapid upflows can be seen associated with the CBP all along the experiment, as
visualized through the blue-purple volume rendering in panel (a), which show
velocities from 45 to 75 km s-1, and through the vertical velocity $u_{z}$ in
panel (e). These upflows are mainly concentrated around the spines of the CBP;
those along the outer spine are basically reconnection outflows emanating from
the nullpoint, whereas the ones around the inner spine are probably due to
channeling of magnetosonic waves. The flows arising from the CBP nullpoint,
like the one at the time of the figure, are of particular interest for the
corona. These hot (1–2 MK) collimated ejections have low contrast in density
compared to the surrounding plasma, and therefore they are not (or barely)
distinguishable in the synthetic EUV images (see discussion in Section 3).

Figure 1: Experiment overview. (a) 3D magnetic nullpoint topology (red lines)
at $t=111.67$ min superimposed on a horizontal map of $B_{z}$ at $z=0$. Blue-
purple volume rendering corresponds to upflows with velocities from 45-75 km
s-1. The yellow isosurface at $B=1$ G harbors the nullpoint location at the
center of the image. The red-green-blue coordinate system indicates the x-y-z
axis orientation. (b) Time-evolution of the nullpoint coordinates (xnp, ynp,
znp). (c), (d), (e) Temperature, density, and vertical velocity cuts at y$=$16
Mm, respectively. (f), (g), (h), (i) Synthetic response integrated along the
y-axis to mimic a limb observation by SDO/AIA 193 Å, SolO/EUI-HRI 174 Å,
Hinode/XRT Al-poly (reversed color scale), and IRIS/SJI 1400 Å, respectively,
in DN s-1 pix-1 units. The associated animation comprises the whole experiment
evolution from $t=0$ to $t=268.33$ min.
(An animation of this figure is available.)

###  2.2 Energization

To address the fundamental question about the sustained CBP heating and its
location, we have first analyzed the heating per unit mass due to the Joule
and viscous terms. Panels (a) and (b) of Figure 2 contain time-averaged
results over approximately 4.5 hours of the simulation at three different
heights, $z=[8,5,2]$ Mm, thus encompassing the regions in close proximity to
the CBP nullpoint downwards to the low atmosphere. Both the Joule heating
(left column) and viscous heating (right column) primarily occur at lower
heights, at a few megameters above the solar surface, concentrated around the
inner spine, in the region dominated by the strong parasitic positive
polarity. We found that in these regions the electric current vector subtends
a small angle with respect to the concentrated nearly vertical magnetic field,
indicating that the magnetic configuration is close to force-free there.
Heating is also found, albeit to a lesser extent, in the magnetic patches
outside of the fan surface in the surroundings of the CBP at $z=2$ Mm (bottom
panels), as well as in the region near the nullpoint at $z=8$ Mm (top panels).
Thus, our findings indicate that CBP loops are primarily heated in the low
atmosphere, with a secondary contribution at coronal heights near the
reconnection site. Furthermore, there is a remarkable spatial correlation
between the Joule and viscous heating across various heights. This suggests
that regions exhibiting the highest electric current intensity and velocity
gradient typically coincide. As a consequence, our results concerning the
location of the CBP heating could be general, regardless of which mechanism
(Joule or viscous) is dominant, even if in our experiment the Joule heating
term is approximately one order of magnitude greater than the viscous one (see
discussion in Section 3).

Figure 2: Heating in our CBP model. (a), (b) Heating per unit mass averaged
over $\sim 4.5$ hours for the Joule ( $\overline{\\!Q_{J}/\rho\\!}$ ) and
viscous ( $\overline{\\!Q_{V}/\rho\\!}$ ) terms, respectively, at $z=[8,5,2]$
Mm. (c), (d) Heating per unit mass averaged horizontally within the CBP at
$z=[5,2]$ Mm, respectively, for the Joule
($<\\!Q_{J}/\rho\\!>^{\mathrm{CBP}}$, black) and viscous
($<\\!Q_{V}/\rho\\!>^{\mathrm{CBP}}$, orange) terms as a function of time. The
Pearson correlation coefficient $r$ between the curves is shown in red in the
upper left corner of the plot frame. (e) Time average of the horizontal mean
vertical Poynting flux ($\overline{<\\!S_{z}\\!>}$) within the CBP (blue) and
in the coronal hole surroundings (red).

Another important question for CBPs, and for the corona in general, concerns
whether the heating occurs on short or long timescales (Reale, 2014). To
pursue this question, panels (c) and (d) of Figure 2 illustrate, as a function
of time, the horizontal average within the CBP fan surface of the Joule
(black) and viscous (orange) heating per unit mass at $z=[2,5]$ Mm,
respectively. Both heating mechanisms seem to be continuously at work, showing
fluctuations with different time scales. For instance, the pronounced peak
observed mainly in the viscous heating profile at $z=5$ Mm at $t\in[110,120]$
min corresponds to the H$\alpha$ fibril eruption explained in the previous
section. In general, there is a clear time correlation between the Joule and
viscous dissipation mechanisms. In fact, the Pearson correlation coefficient
$r$ between them is greater than $0.8$ for the different heights we have
analyzed.

As a final point, we consider the energy injection through the Poynting flux.
Panel (e) of Figure 2 contains the mean vertical Poynting flux computed within
the CBP fan surface and averaged in time, $\overline{<\\!S_{z}\\!>}$, as a
function of height (blue curve). This plot strongly suggests that the heating
of the plasma within the CBP domain is due to the deposition of the upward
Poynting flux, which, itself, results from the stochastic photospheric motions
stressing the CBP magnetic field. This implies that no major organized
photospheric flows (leading, e.g., to spine dragging or convergence of
magnetic polarities) are necessary to maintain the heating of the CBPs.
Importantly, the average $\overline{<\\!S_{z}\\!>}$ within the CBP is larger
at all the heights than the corresponding quantity in the surroundings (red
curve), and the same applies to the corresponding gradient in height (in
absolute value). The greater vertical Poynting flux obtained within the CBP is
due to the strong magnetic concentration in the parasitic polarity. The
Poynting flux values and the height gradient are in agreement with those
indicated by other radiative-MHD numerical experiments as necessary to sustain
heating in the quiet Sun corona (Rempel, 2017; Chen et al., 2022). In
addition, these values seem to fulfill the energy requirements for a hot
corona inferred from observations (e.g., $0.8\times 10^{6}$ erg cm-2 s-1 for
coronal holes, Withbroe & Noyes, 1977). Moreover, computing the surface
integral of the vertical Poynting flux within the fan surface domain over two
horizontal planes in the corona (e.g., $z=2$ Mm and $z=5$ Mm), we obtain a
magnetic energy deposition in the CBP of $\approx 7\times 10^{24}$ erg s-1,
which is more than enough to explain the rough estimate of the CBP energy
losses obtained through observations ($10^{23}$ to $10^{24}$ erg s-1, Habbal &
Withbroe, 1981). The loss of electromagnetic energy through the fan surface is
most probably small by comparison, at any rate when averaging in time.

###  2.3 Confronting Observations and Modeling

To reinforce the previous results, we present here SDO and SST observations
(see Appendix B) of a CBP that occurred on 2022 July 01 located at
heliocentric coordinates $(x,y)=(177\arcsec,147\arcsec)$. The million-K
coronal response of the CBP, shown in the SDO/AIA images (panels (a) and (b)
of Figure 3), has a compact and bright rowel-like shape that covers a
horizontal domain of $\sim 40\arcsec\times 40\arcsec$. The photospheric line-
of-sight magnetic field image from SST/CRISP in panel (d) shows a parasitic
negative polarity at the center of the CBP, surrounded by a dominant opposite
polarity magnetic region, a configuration that can naturally lead to a fan-
spine configuration with a nullpoint at coronal heights (Zhang et al., 2012;
Mou et al., 2016; Galsgaard et al., 2017; Madjarska et al., 2021; Cheng et
al., 2023). The chromospheric H$\alpha$ core image from SST/CRISP in panel (c)
indicates that the CBP contains a bunch of dark and long H$\alpha$ fibrils
that connect opposite polarities, analogous to fibrils reported in lower
resolution observations (Madjarska et al., 2021); and there are strong
brightenings mainly located at the base of the fibrils, possibly indicative of
chromospheric heating. The spectral profiles at different locations of the
fibrils and brightenings are shown in panel (e).

In order to prove that our model satisfactorily reproduces the observations,
Figure 4 and its associated animation contain the forward-modeling results
from the simulations. Panel (a) shows that we indeed obtain a rowel-like
structure composed of hot loops with enhanced EUV response similar to the
SDO/AIA 193 Å observations, although our CBP is smaller than the observed one.
Given its current interest, in panel (b) we also add the EUV response as would
be observed by Solar Orbiter’s EUI instrument, with its higher spatial
resolution. The comparison to observations is also successful for the
chromospheric counterpart, as seen in panel (c). Our CBP model is realistic
enough to reproduce a pattern of elongated dark fibrils, as observed in the
H$\alpha$ images; and the existence of strong brightenings at the base of
these fibrils, surely a manifestation of the location of the heating in the
lower atmosphere, as discussed in the previous section. The agreement is
further illustrated by comparing panels (e) of Figures 3 and 4. The
observational and synthetic spectral profiles show a similar behaviour
regarding the H$\alpha$ line core intensity at different locations of the
fibrils and brightenings, namely, the fibrils are characterized by a less
bright H$\alpha$ core than the average with a small Doppler shift, while the
brightenings show an enhanced H$\alpha$ core. The smaller width in the
synthetic profiles may be due to the modest numerical resolution not capturing
the highly dynamic complexity of the chromosphere (Hansteen et al., 2023). At
the photosphere, the vertical magnetic field of the model (panel (d) of Figure
4) shows that the elongated H$\alpha$ fibrils connect the central parasitic
polarity with the strong, opposite-polarity magnetic patches surrounding the
CBP, as shown in our observations (panel (d) of Figure 3) as well as in other
CBP observations (Madjarska et al., 2021).

Figure 3: Observations of a representative CBP on 2022 July 01 at 08:08:52 UT.
(a), (b) Hot coronal response in EUV detected by SDO/AIA 193 Å and SDO/AIA 171
Å, respectively. (c) Cool chromospheric structure of the CBP in the H$\alpha$
line core observed with SST/CRISP. (d) Photospheric line-of-sight (LOS)
magnetic field (BLOS) from SST/CRISP. (e) H$\alpha$ profiles for different
regions of the dark fibrils (cool colors) and core brightenings (warm colors)
marked in panel (c). The average profile over the whole field-of-view is shown
as a black dashed line. The intensity of the profiles has been normalized to
the intensity value at $50$ km s-1.  Figure 4: Forward modeling from our 3D
CBP numerical experiment as observed on-disk. (a), (b) Synthetic EUV images
for SDO/AIA 193 Å and SolO/EUI-HRI 174 Å, respectively. (c) Synthetic
H$\alpha$ response in the core of the line as observed by SST/CRISP. (d)
Vertical magnetic field $B_{z}$ at the solar surface ($\tau_{500\thinspace{\rm
nm}}=1$). (e) H$\alpha$ profiles for different regions of the dark fibrils
(cool colors) and core brightenings (warm colors) marked in panel (c). The
average profile over the whole box is shown as a black dashed line. The
intensity of the profiles has been normalized to the intensity value at $50$
km s-1. The associated animation shows the synthetic results for $t=111.67$,
$202.50$, and $264.17$ min.
(An animation of this figure is available.)

##  3 Discussion

In this Letter, through a 3D radiative-MHD model using the Bifrost code, we
have shown that CBPs can be energized by the action of the continuous surface
convection motions, thus obviating the need for major, organized photospheric
flows such as converging flows (Priest et al., 1994; Dreher et al., 1997;
Priest et al., 2018; Syntelis et al., 2019) or large-scale surface flows
(Wyper et al., 2018). Our investigation reveals that the CBP loops are heated
by both Joule and viscous heating predominantly in the lower atmosphere, with
a secondary contribution at coronal heights close to the reconnection site.
This is a major difference with 2D models (e.g., Syntelis et al., 2019;
Nóbrega-Siverio & Moreno-Insertis, 2022), which can only capture the heating
at the reconnection site since mechanisms, such as magnetic fieldline
braiding, are intrinsically three-dimensional. The heating rate from both the
resistive and viscous heating terms is intermittent, which seems to be
consistent with findings from simulations focused on individual coronal loops
(Breu et al., 2022). Furthermore, both the spatial and time correlation found
between the Joule and viscous heating within the CBP suggests that the heating
results obtained in this Letter may be widely independent of the value of the
magnetic Prandtl number, in line with the conclusions reached in recent
coronal numerical simulations (Rempel, 2017; Chen et al., 2022). This implies
that, even though current numerical models are far from reproducing the
Prandtl numbers of the solar atmosphere, our results may well be applicable to
understand the heating of small-scale loops (CBPs) and their properties.

By means of X-ray, EUV, and UV forward modeling, we have shown that the CBP’s
multi-wavelength response and corresponding brightness exhibit fluctuations on
the scale of minutes (see animation of Figure 1), resembling intensity
variations reported in observations (Ugarte-Urra et al., 2004; Doschek et al.,
2010; Kumar et al., 2011; Ning & Guo, 2014; Zhang et al., 2014; Chandrashekhar
& Sarkar, 2015; Kayshap & Dwivedi, 2017; Gao et al., 2022). Our results also
indicate that chromospheric phenomena, such as H$\alpha$ fibrils and spicule-
like features, can perturb the CBP, leading to significant changes in the EUV
intensity. These findings support the recent results about the impact of
spicules in the CBP brightness variations (Nóbrega-Siverio & Moreno-Insertis,
2022; Bose et al., 2023). Additionally, our model demonstrates that CBPs can
produce collimated hot ejections that do not show appreciable emission in EUV
due to the slight density contrast with the surroundings. This may offer
theoretical support to the observed features rooted in CBPs named dark jets
that have been detected in EUV spectroscopic data using Hinode/EIS, but with
either weak or absent signatures in SDO/AIA 193 Å images (Young, 2015). This
may be also the key to explaining the lack of clear observable jetting
activity in EUV images in the early stages of CBPs (Kumar et al., 2019).
Further observational evidence using off-limb examples and spectroscopic
observations is needed. In this vein, our results could provide valuable
insights for the near future missions Multi-slit Solar Explorer (MUSE, De
Pontieu et al., 2022; Cheung et al., 2022) and Solar-C/EUVST (Shimizu et al.,
2020).

In this Letter, we have also provided simultaneous observations of a
representative CBP from SDO and SST. Thus, we not only address the scarcity of
observational results about the chromosphere below CBPs (Madjarska et al.,
2021; Bose et al., 2023), but also we impose a case-study to test our model.
The similarities between the synthetic observables from the model and the
observations are striking, even reproducing chromospheric features such as the
brightenings in the core of the H$\alpha$ line, the dark and elongated
H$\alpha$ fibrils, as well as their magnetic photospheric connections. By
increasing the numerical resolution, we expect to achieve even greater
agreement with the observations in aspects such as the width of the H$\alpha$
line and the intricate fine structure within CBPs. All the results described
above are a good indication that we are including the most important physical
ingredients to explain the evolution and heating of CBPs as well as
reproducing their main identifying signatures.

This research has been supported by the European Research Council through the
Synergy Grant number 810218 (“The Whole Sun”, ERC-2018-SyG); by the Spanish
Ministry of Science, Innovation and Universities through project
PGC2018-095832-B-I00; and by the Research Council of Norway (RCN) through its
Centres of Excellence scheme, project number 262622. The authors acknowledge
the computer resources at the MareNostrum supercomputing installation and the
technical support provided by the Barcelona Supercomputing Center (BSC, RES-
AECT-2021-1-0023, RES-AECT-2022-2-0002). The use of UCAR’s VAPOR software (Li
et al., 2019; Pearse et al., 2023) is gratefully acknowledged. This work also
benefited from discussions at the International Space Science Institute (ISSI)
in Bern, through ISSI International Team project #535 Unraveling surges: a
joint perspective from numerical models, observations, and machine learning.
L.R.v.d.V. is supported by RCN project number 325491. M.M. acknowledges
financial support by DFG grant WI 3211/8-1. The Swedish 1-m Solar Telescope is
operated on the island of La Palma by the Institute for Solar Physics of
Stockholm University in the Spanish Observatorio del Roque de Los Muchachos of
the Instituto de Astrofísica de Canarias. The Institute for Solar Physics is
supported by a grant for research infrastructures of national importance from
the Swedish Research Council (registration number 2017-00625). SDO
observations are courtesy of NASA/SDO and the AIA, EVE, and HMI science teams.
The authors thank Dr. Frédéric Auchère for his help to compute the synthetic
observables for SolO/EUI-HRI.

##  Appendix A NUMERICAL EXPERIMENT

###  A.1 Code

The experiment was carried out using the radiation-magnetohydrodynamics
Bifrost code (Gudiksen et al., 2011). This code includes, in a self-consistent
manner, radiative transfer from the photosphere to the corona; approximate
recipes for the main radiative losses in the chromosphere due to neutral
hydrogen, singly-ionized calcium, and singly-ionized magnesium; field aligned
thermal conduction, which tends to an isotropic conductivity prescription for
regions with weak magnetic field such as nullpoints; optically thin losses;
and an equation of state considering the $16$ most relevant atomic elements
for the solar atmosphere either because of their high abundance or because of
their importance as electron donors at low temperatures. Explicit resistivity
and viscosity are included in Bifrost through hyper-diffusion terms (Gudiksen
et al., 2011), following the idea originally described by Nordlund & Galsgaard
(1995). A comprehensive description of the resistive and viscous hyper-
diffusion operator and the associated free parameters can be found in the
papers by Færder et al. (2023a, b), including a comparative analysis of the
hyper-diffusive resistivity versus other resistivity models.

###  A.2 Initial Condition

The initial condition was built using a weakly magnetized (0.2 G at the
corona) preexisting 3D numerical experiment that had reached a statistically
stationary equilibrium. In the vertical direction, the numerical box covers a
region from the uppermost layers of the solar interior to the corona, that is,
$-2.9$ Mm $\leq z\leq 32.3$ Mm, with $z=0$ corresponding to the solar surface
(more precisely the horizontal surface where
$<\hskip-2.0pt\tau_{500\thinspace{\rm nm}}\hskip-2.0pt>$=1). The horizontal
extent is $32\times 32.0$ Mm2, with both $x$ and $y$ starting at the origin of
coordinates $(0,0)$. The domain is resolved with $512\times 512\times 512$
cells, employing a uniform grid in the horizontal directions with spacing
$\Delta x=\Delta y=62.5$ km; and a non-uniform mesh for $z$, using $\Delta
z=50$ km from the interior up to 12 Mm in the corona, and then progressively
decreasing the resolution down to $\Delta z=150$ km at the top of the corona.
The boundary conditions are periodic in the horizontal direction. In the
vertical direction, at the top, characteristic boundary conditions are
applied; at the bottom, constant entropy is set for the incoming plasma to
keep solar-like convection, while the rest of the variables are extrapolated.
Panel (a) of Figure 5 shows the horizontal averages of the temperature $T$ and
mass density $\rho$ of the background stratification.

On top of the relaxed snapshot mentioned above, we added a potential magnetic
nullpoint distribution as illustrated in panel (b) of Figure 5. The potential
configuration was calculated from a prescribed axisymmetric distribution at
the bottom boundary containing a circular positive polarity (the parasitic
polarity) embedded in a negative background. This imposed configuration is
such that (a) there is a nullpoint at $(x_{np},y_{np},z_{np})=(16,16,8)$ Mm;
(b) at large heights, the magnetic field becomes asymptotically vertical and
uniform with $B_{z}\to-10$ G, thus mimicking a coronal hole structure; and (c)
at $z=0$, the total positive flux is $\Phi^{+}=4.9\times 10^{19}$ Mx, the
maximum positive vertical field strength is $B_{z}=141.8$ G, the parasitic
polarity covers a circular patch of radius $6.3$ Mm, and the intersection of
the fan surface with the horizontal plane has a radius of $12.4$ Mm.

Figure 5: Initial condition. (a) Initial horizontal average stratification for
the temperature $T$ and density $\rho$. (b) Imposed nullpoint magnetic field
configuration (red lines) embedded within a coronal-hole like magnetic field
(blue). The yellow isosurface at $B=1$ G delimits the nullpoint location.
Other features of this configuration are also highlighted following the 3D
nullpoint terminology (Priest & Titov, 1996). The solar granulation pattern of
the initial model is shown through the vertical velocity $u_{z}$ at $z=0$. The
red-green-blue coordinate system indicates the orientation of the x-y-z axis.

##  Appendix B OBSERVATIONS

###  B.1 The Swedish 1-m Solar Telescope (SST) Data.

We have used observations from the SST (Scharmer et al., 2003) obtained on
2022 July 01 from 08:04:57 to 08:13:03 UT centered at a CBP located at
heliocentric coordinates $(x,y)$ = (176″,135″). To cover the whole CBP and
surroundings within the field-of-view (FOV) of the SST instrumentation, we
have created a mosaic constructed from $3\times 3$ telescope pointings. The
SST data set contains the following spectral scans from the CRisp Imaging
Spectropolarimeter (CRISP, Scharmer et al., 2008): the Fe I 6173 Å line in
spectropolarimetric mode sampled in 14 positions between $-320$ mÅ and $+680$
mÅ from the line core, and H$\alpha$ 6563 Å in spectroscopic mode sampled in
31 positions ranging from $-1500$ mÅ to $+1500$ mÅ from the line core. Each of
the CRISP scans of the mosaic has a FOV of $\sim 58\arcsec\times 58\arcsec$
and a spatial sampling of 0$\farcs$057 per pixel. The CRISP data were
processed using the SSTRED data reduction pipeline (de la Cruz Rodríguez et
al., 2015; Löfdahl et al., 2021), which includes Multi-Object Multi-Frame
Blind Deconvolution (MOMFBD, van Noort et al., 2005) image restoration. Line-
of-sight (LOS) magnetograms were obtained by Milne-Eddington inversions of the
Fe I 6173 Å Stokes profiles (de la Cruz Rodríguez, 2019).

###  B.2 The Solar Dynamics Observatory (SDO) Data.

The coronal response of the targeted CBP is obtained from the Atmospheric
Imaging Assembly (AIA, Lemen et al., 2012) onboard SDO (Pesnell et al., 2012)
around 08:08:52 UT: the time at which the center of the SST mosaic was taken.
The photospheric magnetic field is obtained from the Helioseismic and Magnetic
Imager (HMI, Scherrer et al., 2012). Regarding the alignment, cross-alignment
is carried out in all the AIA channels to the HMI continuum; then, the SDO
images are manually aligned with the SST ones.

##  Appendix C FORWARD MODELING

###  C.1 EUV and UV Forward Modeling.

For the forward modeling in the EUV and UV, the expressions of optically thin
radiative transfer are used, assuming, additionally, coronal abundances
(Feldman, 1992) and ionization equilibrium. The emissivity corresponding to
any given transition between electron configuration levels $i$ and $j$ of a
given ionic species of an atomic element (e.g., Fe IX, or Si IV) is given by

$\epsilon=n_{H}\ n_{e}\ G(T,n_{e},\nu_{ij})$ (C1)

where $n_{e}$ and $n_{H}$ are the electron and hydrogen number densities,
respectively; $\nu_{ij}$ is the radiation frequency of the emitted photon; and
$G(T,n_{e},\nu_{ij})$ is the corresponding gain function (or contribution
function), which already contains the abundance of the atomic element. The
emitted intensity is calculated by integrating Equation C1 along the LOS in
the numerical box.

For the SDO/AIA coronal synthetic images, we have used CHIANTI (Del Zanna et
al., 2021) from the Solar Soft package, calculating the gain functions for the
atomic transitions in the relevant spectral range of the different AIA filters
and weighing with the corresponding effective Area Function. The result is
then transformed into SDO/AIA count numbers. To account for the possible
obscuration effects from cool and dense features in the EUV images, we have
included absorption by neutral hydrogen, neutral helium, and singly-ionized
helium in the integration of the emissivity along the LOS adding an absorption
factor $e^{-\tau}$ following the recipes and procedures in the literature (De
Pontieu et al., 2009; Anzer & Heinzel, 2005).

For the synthetic 174 Å images of the Extreme Ultraviolet Imager of the High
Resolution Imager (EUI-HRI) (Rochus et al., 2020); on Solar Orbiter (SoLO)
(Müller et al., 2020), we have employed a contribution function privately
provided by Dr. Frédéric Auchère, member of the Solar Orbiter team.

For the synthetic IRIS (De Pontieu et al., 2014) transition region
counterpart, the contribution function of the Si IV 1393.755 Å line is
obtained through a simple procedure (Chianti Solar Soft routine
ch_synthetic.pro with the flag /goft), multiplying the output by the silicon
abundance relative to hydrogen. This procedure and the conversion to IRIS
count numbers are detailed in a recent paper (Nóbrega-Siverio et al., 2017).

After obtaining the intensity images, we have degraded the results to match
the spatial resolution of the instruments, namely, 1$\farcs$5 for SDO/AIA
(Lemen et al., 2012); 0$\farcs$33 for IRIS (De Pontieu et al., 2014); and
$200$ km for EUI-HRI at perihelion (Rochus et al., 2020).

###  C.2 X-ray Forward Modeling.

The X-ray synthetic images have been computed to mimic observations taken by
the X-ray telescope (XRT) (Golub et al., 2007) onboard Hinode (Kosugi et al.,
2007). In particular, we have combined the make_xrt_wave_resp.pro and
make_xrt_temp_resp.pro routines from Solar Soft to obtain the spectral and
temperature response for the Al-poly channel of the telescope. The result is
correspondingly multiplied by the squared electron number density and then
degraded to the 2$\farcs$0 instrument resolution (Golub et al., 2007).

###  C.3 Chromospheric Forward Modeling.

To obtain H$\alpha$ synthetic images, we have used the 3D non-LTE radiative
transfer code Multi3D (Leenaarts & Carlsson, 2009). Multi3D evaluates the
radiation field in full 3D taking the horizontal structure of 3D model
atmospheres into account. To reduce the 3D radiative transfer computation
cost, we have halved the horizontal resolution of the model atmosphere. We
further optimized the depth scaling of the atmospheric quantities to improve
the convergence time and to reduce the effect of an insufficiently sampled
optical depth scale onto the synthesized H$\alpha$ line profiles. H$\alpha$
was synthesized from a five-level plus continuum hydrogen model atom assuming
Doppler line profiles for the Ly$\alpha$ and Ly$\beta$ lines which affect the
level populations of the H$\alpha$ transition (Leenaarts et al., 2012). We
have reduced the spatial resolution of the synthetic images to the SST/CRISP
instrumental values, namely, 0.”13 at 6301 Å (Scharmer et al., 2008).

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