Magnetic Jam in the Corona of the Sun

Abstract

The outer solar atmosphere, the corona, contains plasma at temperatures of
more than a million K, more than 100 times hotter that solar surface. How this
gas is heated is a fundamental question tightly interwoven with the structure
of the magnetic field in the upper atmosphere. Conducting numerical experiments
based on magnetohydrodynamics we account for both the evolving
three-dimensional structure of the atmosphere and the complex interaction of
magnetic field and plasma. Together this defines the formation and evolution of
coronal loops, the basic building block prominently seen in X-rays and extreme
ultraviolet (EUV) images. The structures seen as coronal loops in the EUV can
evolve quite differently from the magnetic field. While the magnetic field
continuously expands as new magnetic flux emerges through the solar surface,
the plasma gets heated on successively emerging fieldlines creating an EUV loop
that remains roughly at the same place. For each snapshot the EUV images
outline the magnetic field, but in contrast to the traditional view, the
temporal evolution of the magnetic field and the EUV loops can be different.
Through this we show that the thermal and the magnetic evolution in the outer
atmosphere of a cool star has to be treated together, and cannot be simply
separated as done mostly so far.

Full text

Nature Physics, final version published online 27 April 2015. DOI: 10.1038/nphys3315
Main text of this arXiv version is identical to the originally submitted manuscript.
Magnetic Jam in the Corona of the Sun
F. Chen,1H. Peter,1S. Bingert,2M.C.M. Cheung3
1Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 G¨
ottingen, Germany
2Gesellschaft f¨
ur wissenschaftliche Datenverarbeitung, Am Faßberg 11, 37077 G¨
ottingen, Germany
3Lockheed Martin Solar and Astrophysics Laboratory, Palo Alto, CA 94304, USA
The outer solar atmosphere, the corona, contains
plasma at temperatures of more than a million K,
more than 100 times hotter that solar surface. How
this gas is heated is a fundamental question tightly in-
terwoven with the structure of the magnetic field in
the upper atmosphere. Conducting numerical exper-
iments based on magnetohydrodynamics we account
for both the evolving three-dimensional structure of
the atmosphere and the complex interaction of mag-
netic field and plasma. Together this defines the for-
mation and evolution of coronal loops, the basic build-
ing block prominently seen in X-rays and extreme ul-
traviolet (EUV) images. The structures seen as coro-
nal loops in the EUV can evolve quite differently from
the magnetic field. While the magnetic field continu-
ously expands as new magnetic flux emerges through
the solar surface, the plasma gets heated on succes-
sively emerging fieldlines creating an EUV loop that
remains roughly at the same place. For each snapshot
the EUV images outline the magnetic field, but in con-
trast to the traditional view, the temporal evolution of
the magnetic field and the EUV loops can be differ-
ent. Through this we show that the thermal and the
magnetic evolution in the outer atmosphere of a cool
star has to be treated together, and cannot be simply
separated as done mostly so far.
In the upper atmosphere of the Sun the energy den-
sity of the magnetic field supersedes the density of the
internal or the kinetic energy by far. Thus the magnetic
field can easily provide the energy to heat the plasma in
the corona to its high temperatures, exceeding those on
the surface by more than a factor of 100. The dominance
of the magnetic field also gives rise to the coronal loops
that are seen so nicely in the extreme ultraviolet (EUV) or
X-rays (see Fig. 1): if energy is deposited on a magnetic
fieldline, heat conduction in a fully ionised plasma will
redistribute that energy efficiently along only that field-
line (but not across). Thus the plasma along a fieldline
gets heated and becomes visible in EUV and X-rays. In
this picture the EUV and X-ray emission shows the mag-
netic field in a similar way as iron filings are used in
school to show fieldlines of a permanent magnet. Despite
its pivotal importance, actual measurements of the mag-
netic field in the solar corona are notoriously difficult, and
are not yet possible on a regular basis at all places in the
upper atmosphere1.
Instead of direct measurements, mainly extrapola-
tions of the observed magnetic field at the surface pro-
vide the magnetic information in the corona2. For a
comparison, one can combine stereoscopic observations
to reconstruct the three-dimensional structure of coro-
nal loops, in particular their path in space3. Compar-
ing such stereoscopy to magnetic field extrapolations
provides evidence that the loops seen in EUV indeed
outline fieldlines4. This paradigm underlies both one-
dimensional modeling5, 6 where the thermodynamics of
the coronal plasma is treated in detail along assumed
static fieldlines, and magnetofrictional modeling7where
an instantaneous thermal equilibrium is assumed along
dynamic fieldlines. On the real Sun we will not find
these extreme cases, but a changing magnetic field host-
ing plasma with an evolving thermal structure.
Models accounting for this three-dimensional
structure and evolution of the solar corona already
pointed to a mismatch between magnetic and thermal
structure8, which plays an important role to understand
the cross section of coronal loops9. The thermal evo-
lution, i.e., when plasma gets heated and when a loop
becomes visible in EUV, is not coupled to the fieldlines
as such, but to the heat input along fieldlines. Thus one
can imagine scenarios in which the appearance of coronal
loops decouples from the motions of magnetic field lines.
We show that such scenarios are realistic for situations on
the Sun, and thus our understanding of the structure and
evolution of the solar corona, and ultimately the heating
processes, will have to fully acknowledge the intimate in-
teraction of the thermal evolution of coronal loops and the
changing magnetic structure.
To investigate the relation of the thermal and mag-
netic evolution in the corona above an active regions we
conduct a numerical three-dimensional experiment. For
this we solve the problem of magnetohydrodynamics in
which the induction equation describing the magnetic
field is coupled to the description of a fluid governed
by the conservation of mass, momentum and energy. In
the latter we account for the heat conduction along the
1
arXiv:1505.01174v1 [astro-ph.SR] 5 May 2015

Chen, Peter, Bingert & Cheung: Magnetic Jam in the Corona of the Sun, Nature Physics (2015). DOI: 10.1038/nphys3315
3D MHD model synthesized 304 Åobservation
(a) (b)
SOO / AIA 304 Å / He II
Figure 1: The upper atmosphere of the Sun seen in light emitted by about 100 000 K hot plasma. Panel (a) shows an
observation from space with the Solar Dynamics Observatory (SDO) taken in the 304 ˚
A band dominated by emission from singly
ionised He. The limb of the Sun is indicated by the dashed line. Coronal loops are mostly seen edge-on rising some 40000 km
above the limb. Panel (b) shows a numerical simulation as described in this paper. It shows the synthesised emission in the same
304 ˚
A channel integrated horizontally through the computational box of a numerical experiment. Similar to the real Sun, loops arch
above the surface (dashed line).
magnetic field, optically thin radiative losses and heating
through Ohmic dissipation. Our model follows the phi-
losophy of previous studies in which the magnetic field is
driven at the surface of the Sun, which is the lower bound-
ary of the model10–12. In contrast to these earlier models,
we drive our system at the lower boundary by a separate
model of an emerging sunspot pair13. This way coronal
loops form in the emerging active region in response to
the enhanced Poynting flux into the corona at locations
where magnetic field is pushed around, similar to flux
braiding14 or flux-tube tectonics15. This new study on
the evolution of thermal and magnetic properties is based
on the same simulation as used before to investigate the
formation of active region loops16.
In order to study the relation of the magnetic field
to the coronal loops seen in emission we have to follow
the temporal evolution of both in the simulation. The
procedure to follow (a bundle of) magnetic fieldlines as
well as an EUV loop is described in the Supplemen-
tary Material (SM) S2. From the output of the MHD
simulation, we use the temperature and density at each
grid point to evaluate the coronal emission. Integrating
along a line-of-sight one then obtains synthetic obser-
vations that can be treated as real ones17, which shows
the same typical structures as real observations12. The
example comparison shown in Fig. 1 underlines that 3D
models now capture the essential observational signatures
found for emerging loops, which is a significant step for-
ward in understanding the structure, dynamics and heat-
ing of the corona, one of the enigmatic problems in as-
trophysics. Here we synthesise the coronal emission as
it would be seen by the Atmospheric Imaging Assembly
(AIA)18 onboard NASA’s Solar Dynamics Observatory.
For our analysis we synthesise the 193 ˚
A filter which is
dominated by emission from Fe XII that forms at around
1.5 MK, following well established procedures9.
In Fig. 2 we show the synthesised 193 ˚
A observa-
tion when integrating horizontally through the computa-
tional domain. The snapshot shown here clearly shows
a coronal loop hosting million K hot plasma. Following
the temporal evolution in the movie that is available in the
online edition (further snapshots in Fig. S4 in SM S3) it
is evident that the EUV loop forms, becomes bright and
then starts fading over the course of a good fraction of an
hour. Most importantly, the EUV loop, i.e., the pattern
visible in the 193 ˚
A channel, remains at more or less the
same place. In particular the EUV loop is not expanding
upwards.
This is in contrast to the evolution of the structure
of the magnetic field. Also in Fig. 2, we overplot one sin-
gle magnetic fieldline at different times. It can be seen
that this fieldline moves upwards while the active region
is emerging (see the movie or Fig. S4 in SM S3). In Fig. 2
(and the movie) we also show the coronal emission in a
vertical slab in the middle of the loop (and perpendicu-
lar to the loop) to emphasise how differently the pattern
of the EUV emission evolves compared to the magnetic
structure. There is no mass flow across fieldlines. We
emphasise that at each snapshot the EUV loop is roughly
following a fieldline, but at each time it is a different field-
line that is aligned with the EUV loop.
To understand this behaviour one has to investi-
gate the heat input and the resulting temperature, den-
sity and emission structure along individual fieldlines, the
details of which are described in SM S3. We find that
each individual fieldline shows an increased Ohmic heat-
ing for about one hundred seconds. This heats up the
plasma and through evaporation of gas from the lower at-
2

Chen, Peter, Bingert & Cheung: Magnetic Jam in the Corona of the Sun, Nature Physics (2015). DOI: 10.1038/nphys3315
50 60 70 80 90
x [Mm]
0
5
10
15
20
25
z [Mm]
(a)
fieldline through the center of the AIA 193Å emission
center fieldline at t = 600 s
t = 130 s
t = 0 s
500 1000 1500
AIA 193Å [DN/pixel/s]
20 25 30
y [Mm]
0
5
10
15
20
25 center of the emission
center of the fieldlines
t = 130 s
(b)
Figure 2: Snapshot of the coronal loop in the numerical simulation. This shows synthesised emission as seen in a wave-
length band at 193 ˚
A dominated by Fe XII forming near 1.5 MK. Panel (a) displays the loop from the side with the emission integrated
through the computational box at time 130 s. The emission pattern remains more or less at the same place (cf. Fig. S4 in SM S3).
In contrast, the fieldlines expand, here indicated by the same fieldline shown at three different times (0s, 130 s, and 600 s). For
comparison the blue line shows the fieldline through the center of the emission structure at 130 s (see SM S2 for a more precise
definition of the red and blue lines). To get a better impression of the 3D structure, panel (b) shows the middle part of the loop
integrated along the loop (from x=70 Mm to 77 Mm as indicated by the dotted lines in panel a). Here the image shows again
the 193 ˚
A channel emission, the blue diamond the center of the EUV loop and the red triangles the position of the fieldline in the
x=74 Mm plane at the same three times as in panel (a). These plots cover only part of the computational domain (≈150 ×75 ×
50 Mm3in the x,y,and zdirections). A movie showing the temporal evolution is available in the online edition.
It is also available at http://www2.mps.mpg.de/data/outgoing/peter/papers/2015-magnetic-jam/movie-fig-2.mp4
mosphere the density of the loop increases. Because the
EUV passbands are sensitive to a limited range of temper-
atures only, the plasma along each expanding fieldline is
brightening up only for some 50 s to 100 s. The fieldlines
get heated in succession, i.e., one after the other while
moving upwards. This creates a more or less stationary
pattern of increased emission, while the structure of the
magnetic field is constantly moving upwards. This might
be compared to a traffic jam triggered by at a construction
site on a highway. Here all cars (defining the structure)
are moving forward, but the construction site (heating up
not the cars but the tempers) and thus the pile-up of cars
(defining the pattern) remain at the same location.
The reason for the transient enhancement of the
heating along individual fieldlines is found at their roots
in the photosphere. The coalescent flow that forms the
sunspot drives magnetic patches towards the strong mag-
netic field of the sunspot13, 19. This is shown by the ar-
rows in Fig. 3 that display the horizontal flow field in
the photosphere. Consequently, at the outer edge of the
spot, there will be a region of enhanced (vertical) Poynt-
ing flux, i.e., the upward directed flux of magnetic en-
ergy. This is similar to the flux-tube-tectonics model15
where (horizontal) shuffling of magnetic patches leads
to an upward directed flux of magnetic energy, which
is then available to heat the coronal plasma. Because
each fieldline is pushed into the spot and thus transverses
the region of the enhanced Poynting flux, the heat in-
put into the corona along individual fieldlines is transient
(see detailed discussion in SM S3 and movie attached to
Fig. S2). In short, one would expect the coronal EUV
loops to show up wherever there are strong (horizontal)
gradients of the magnetic field at the footpoints, which
are the locations of strong currents and hence strong
Poynting flux.
This mechanism is illustrated by the cartoon in
Fig. 4. During the emergence of magnetic flux forming a
sunspot pair the field is pushed upwards and to the sides.
In sunspots, where the magnetic field is very strong, con-
vection is suppressed, and thus the flows driving the coa-
lescence of the magnetic field come to a halt. Whenever
a fieldline is crossing the region of the enhanced Poynt-
ing flux, energy is deposited along that fieldline and the
plasma on it is heated. Consequently this fieldline be-
comes visible in EUV for a short time. With successive
fieldlines passing the ”hot spot” of Poynting flux they
all brighten roughly at the same place, creating the illu-
3

Chen, Peter, Bingert & Cheung: Magnetic Jam in the Corona of the Sun, Nature Physics (2015). DOI: 10.1038/nphys3315
t = 130 s
85 90 95 100
x [Mm]
25
30
35
40
y [Mm]
(a)
−2
−1
0
1
2
vertical magnetic field [kG]
in the photoshpere
85 90 95 100
x [Mm]
25
30
35
40
(b)
105
106
107
108
vertical Poynting flux [W m−2]
Figure 3: Evolution at the solar surface while the coronal loop forms. Panel (a) shows the vertical magnetic magnetic field
and panel (b) the vertical component of the Poynting flux, both in the photosphere. These snapshots are taken at the time t=130 s.
The concentration of magnetic field in panel (a), seen in red, shows the location of one of the two sunspots that form the active
region in this simulation. For the time t=−600 s we indicate a number of positions by the diamonds that are located at the footpoints
of fieldlines that transverse the bright coronal loop later. The asterisks and triangles show the position of these locations at later
times t=+130 s, and +1200 s, when they are carried with the coalescent flow forming the sunspot. The field-of-view covers only a
small fraction of the whole computational domain (≈150 ×75 Mm2in the horizontal directions).
An animation showing the temporal evolution over 30 minutes from t=−600 s to +1200 s is available in the online edition.
It is also available at http://www2.mps.mpg.de/data/outgoing/peter/papers/2015-magnetic-jam/movie-fig-3.mp4
sion of a static emission pattern forming a loop, while the
magnetic field is moving.
In our 3D numerical experiment we find that the
temporal evolution of the structure of the magnetic field
in the corona of the Sun can be radically different from
that of the patterns seen in the coronal emission. This im-
plies that modeling the temporal evolution of EUV loops
as 1D structures following a fieldline is a problematic
concept in regions where the magnetic field is evolving,
i.e. whenever the Sun gets dynamic — and interesting.
Thus the credibility of many of the time-dependent 1D
loop models that have been used as the workhorse in the-
oretical coronal studies over the last two decades need to
be reconsidered. Still, at any given snapshot the coro-
nal EUV loops in our model outline magnetic field lines.
Therefore EUV observations should provide useful when
implemented into procedures to recover a snapshot of the
coronal magnetic field through extrapolation methods20.
In summary, we have to treat the magnetic and the ther-
mal evolution of the corona as one single problem —
this requires to have a more holistic view of the mag-
netic and thermal properties of the corona when address-
ing the question of the structure, dynamics and heating of
the corona.
Figure 4: Cartoon showing the interplay between mag-
netic field expansion and the EUV loop. A coalescent flow
forming the sunspot drags the magnetic field in the photosphere
near the solar surface into the sunspot. In response a hot spot
of enhanced upward directed Poynting flux, S, forms (red ar-
row). The expanding fieldlines (blue) move upwards and to the
side. When they transverse the hot spot of Poynting flux, the
plasma on that fieldline gets heated and brightens up. As the
fieldline expands further, it leaves the hot spot and gets darker
again. In consequence a bright coronal EUV loop forms (or-
ange) and remains rather stable while the successively heated
fieldlines move through.
4

Chen, Peter, Bingert & Cheung: Magnetic Jam in the Corona of the Sun, Nature Physics (2015). DOI: 10.1038/nphys3315
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Correspondence
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Acknowledgements
We thank R. Cameron for discussions and comments on the
manuscript. This work was supported by the International
Max-Planck Research School (IMPRS) for Solar System Sci-
ence at the University of Gttingen. It was was partially funded
by the Max-Planck/Princeton Center for Plasma Physics. The
computations were done at GWDG in G¨
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perMUC in at LRZ in Munich. We acknowledge PRACE for
awarding us the access to SuperMUC based in Germany at the
Leibniz Supercomputing Centre (LRZ).
Individual contributions to the paper
The numerical experiment was designed by H.P. and S.B, the
numerical simulation has been conducted by F.C. and S.B., the
analysis of the data has been done by F.C., H.P., and S.B., the
boundary conditions have been provided and implemented by
M.C. and F.C., and H.P. and F.C. wrote the text.
6

Nature Physics, final version published online 27 April 2015. DOI: 10.1038/nphys3315
Supplementary Material for
Magnetic Jam in the Corona of the Sun
F. Chen,1H. Peter,1S. Bingert,2M.C.M. Cheung3
1Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 G¨
ottingen, Germany
2Gesellschaft f¨
ur wissenschaftliche Datenverarbeitung, Am Faßberg 11, 37077 G¨
ottingen, Germany
3Lockheed Martin Solar and Astrophysics Laboratory, Palo Alto, CA 94304, USA
S1 Three-dimensional coronal models compared to observational properties.
S2 Following (a bundle of) magnetic field lines and an EUV loop in time.
S3 Thermal evolution and coronal emission along individual fieldlines.
Movies for the evolution of the snapshots displayed in Figs.2 and 3 of the main text and in Fig. S2 of the supplementary material
are available in the online edition or at http://www2.mps.mpg.de/data/outgoing/peter/papers/2015-magnetic-jam/.
S1 Three-dimensional coronal models compared to observational properties
Heat input in the model
Our three-dimensional (3D) model of the corona above
an emerging active region is driven by the convective
motions in the photosphere which are prescribed at the
lower boundary. To set the velocity field as well as the
density, temperature, and magnetic field at the bottom
boundary of our computational box, we use the time-
dependent output of a model of a magnetic fluxtube
emerging through the surface13,19 . The upper bound-
ary of that flux-emergence model is located about 1 Mm
above the solar surface (where the optical depth is unity)
and thus does not include the evolution of the corona
above. We use the conditions of the flux-emergence
model at the solar surface to prescribe the lower boundary
condition of our coronal model.
The horizontal motions at the bottom boundary to-
gether with the magnetic field produce a Poynting flux
that on average is upward directed. This is similar to the
concept of fieldline braiding14 and fluxtube tectonics15
(see also SM S3 for the Poynting flux at the coronal
base). The disturbances of the magnetic field propagate
upwards, the currents induced by this are dissipated, and
consequently magnetic energy is converted to internal en-
ergy through Ohmic heating.
When solving the basic equations of magnetohy-
drodynamics (MHD)21, we assume a constant value of
the magnetic resistivity ηin the induction equation and
the energy equation. For ηwe choose a value so that
the magnetic Reynolds number Rm=UL/η is of or-
der unity when we choose the grid spacing as the length
scale L(and the sound speed as the typical velocity U).
This ensures that the magnetic energy is mostly dissipated
at the smallest scales resolved by the numerical simula-
tion and that at the same time the numerical dissipation is
small compared to the Ohmic dissipation. In the present
simulation we used η= 1010 m2/s. Previous studies have
shown that for such a setup the energy input into the sys-
tem will be independent of the choice of η, as long as η
is adapted to the grid spacing of the simulation22–24. To
perform the numerical simulation we employed the Pen-
cil code11, 25 (http://pencil-code.googlecode.com/).
One particular feature of the energy input of this
model is that there is a continuous distribution of heat
deposition into the upper solar atmosphere, all the way
from the chromosphere to the corona. Typically the en-
ergy input per volume through Ohmic heating shows a
monotonic decrease with height. In the coronal part this
decrease is roughly exponential with a scale height rang-
ing from some 5 Mm to 10 Mm, depending on the struc-
tures on the surface. This applies to the average drop10 as
well as for the variation along individual fieldlines26. The
heating per particle peaks in the transition region from the
chromosphere to the corona11, 27 because the scale height
of the heating is in-between the pressure scale heights of
the chromosphere and the corona.
Comparison to observational properties
The type of model used here has been employed for
a decade and successfully reproduced and explained a
number of puzzling observations in the solar corona. Us-
SM 1

Supplementary Material for Chen, Peter, Bingert & Cheung, Nature Physics (2015). DOI: 10.1038/nphys3315
ing spectral line profiles synthesised from 3D models of
the corona above an active region, the comparison of the
average properties of model and observations showed a
good match. This applies to the redshifts in the tran-
sition region and of the average differential emission
measure17, 28, as well as to the temporal fluctuations of
Doppler shift and intensity29. A later 3D model gave
an explanation also of the coronal blueshifts27. Inves-
tigations of individual loops appearing in the emission
synthesised from the model show an intensity variation
along the loop comparable to observations, and a life-
time comparable to observed solar loops9. These studies
also provided a new explanation for the puzzling obser-
vational finding that many loops seem not to expand with
height9, 16. Furthermore, the 3D models show envelopes
of groups of loops similar to observations30. Many of the
loops appearing in the models have densities higher than
expected from hydrostatic equilibrium, just as found in
observations31
The above models have not been fine-tuned to re-
produce specific features on the Sun; sample loops from
the models have been picked to compare them to actual
observation, or average properties between the 3D model
and observations have been compared. In one particular
numerical experiment a setup was chosen in which the
magnetic field and the horizontal velocities at the bottom
boundary of the model were taken from an actual obser-
vation of an active region in the photosphere. The com-
parison of the resulting coronal structures shows that the
loops forming in the model appear at the same locations
as the coronal loops have been observed on the real Sun,
which was confirmed by a 3D analysis of stereoscopic
solar observations12.
S2 Following (a bundle of) magnetic field lines and an EUV loop in time
Conventionally, it is agreed that coronal loops seen
in the EUV and X-ray images approximately indicate the
magnetic field lines. This is because the magnetic en-
ergy is dominating in the corona, the emitting plasma is
confined by magnetic field, and the heat conduction is
very sufficient along magnetic field lines. Stereoscopic
observations4and 3D simulations11, 32, 33 of active regions
with a gradual evolution of the (photospheric) magnetic
field support this concept. However, if the magnetic field
would be dynamic, for example, during the formation of
an active region, this concept has to be questioned, in par-
ticular if the timescales of the magnetic evolution and the
thermal evolution of the plasma would be different.
To investigate the spatial relation between coronal
loops and magnetic field lines, we need to follow the field
lines in time. For this we assume that plasma elements
are frozen-in to the magnetic field lines, which is true if
the magnetic Reynolds number is large, and which is the
case in most of the corona. Then we follow the motion
of a selected plasma element and calculate the magnetic
field line through the plasma element at each instant time.
In our numerical model, we have to employ a certain
magnetic diffusivity, which unfortunately allows plasma
to move across the magnetic field lines. However, the
average diffusion speed across 10Mm is of the order of
1 km/s. This is much smaller than the typical perpendicu-
lar velocity (of well above 10 km/s) due to the expansion
of the magnetic field, as will be detailed below.
The snapshot cadence of the numerical simula-
tion is 30 s, which sufficiently captures the evolution of
MHD variables in our simulation. When we follow the
field lines as outlined above, the selected plasma element
might move across several grid points in one time step.
However, because the evolution of the MHD variables is
smooth, we can safely use a cubic spline method to inter-
polate the snapshots to a sufficiently high cadence of 1 s
and use this to follow the gas packages on the fieldlines.
We tested this interpolation for part of the time series by
writing snapshots with 1 s cadence and found the same
results.
Axis of a magnetic tube
To follow the magnetic field, we select twelve points in
the vertical middle plane of the simulation box as seeds.
This plane is in the middle between the two sunspots in
the photosphere and perpendicular to the connecting line
between the spots. Thus it is also roughly perpendicu-
lar to the EUV loops that form and connect the opposite
polarities of the emerging active region. Of these seeds,
eleven form a circle of roughly 2 Mm in diameter and the
twelfth is in the middle of that circle. The initial selec-
tion of these twelve points is chosen so that at some time
SM 2

Supplementary Material for Chen, Peter, Bingert & Cheung, Nature Physics (2015). DOI: 10.1038/nphys3315
(t=130 s) these points roughly encircle the EUV loop that
forms. Basically the 11 points define a magnetic tube and
the twelfth point is on the axis of that tube.
We follow these fieldlines in time (backwards and
forwards). First we trace each fieldline from each of the
initial points. Then we follow the fieldline in time by as-
suming that the fieldlines are frozen-in with the gas. In
practice, we follow the gas parcel near the apex of the
fieldline using the plasma velocity at that location, and
then use the new location of that gas parcel at the next
time step as the new point for tracing the fieldline at that
next time step. This is done forward and backward in
time until the time period of interest is covered. The elec-
tric conductivity in the model is sufficiently high so that
the (numerical) diffusion speed of the plasma through the
magnetic field is small compared to the actual speed of
the rising magnetic field lines. For each of the fieldlines
we calculate the positions of their respective intersections
with the middle plain, ~ri, where iis the index from 1 to
12. The position of the center of the magnetic tube in the
middle plan, ~cmag, we define as
~cmag =1
12
12
X
i=1
~ri.
Tracing the field line for each time step from this point
provides us with the fieldline of the axis of the mag-
netic tube. We chose this procedure because the magnetic
tube will change its shape while expanding. In general, a
tube with a circular cross section will get deformed into
a more elongated (or even more strangely shaped) cross
section16. By using the axis of the magnetic tube we get
a better representation of the evolution of the magnetic
tube independent of the shape of the cross section of the
tube. The center fieldlines plotted in red color in Fig. 2 in
the main text and its attached movie, as well as in Fig. S4
are these axis of the magnetic tube.
The vertical speed associated with the upward ex-
pansion of the apex of the fieldline is about 30 km/s, as
can be seen by inspection of the movie attached to Fig. 2
in the main text (or in Fig. S4).
Axis of an EUV loop
To follow (the axis of) the EUV loop we use the emission
synthesised in the 193 ˚
A band as it would be observed
with AIA34. This shows plasma at temperatures of about
1.5 MK. We calculate the center-of-gravity of the emis-
sion in the vertical midplane and calculate the magnetic
fieldline through this point.
If the emission at each gridpoint in the midplane is
εi, and the position of that gridpoint is ~ri, then the center-
of-gravity of the emission is
~cemiss =Piεi~ri
Piεi
.
For convenience (with no impact on the result) we carry
out the summations only over those gridpoints with an
emissivity above a certain threshold (20DN/pixel/s/Mm).
For each timestep we now calculate this center po-
sition of the EUV loop in the midplane and follow the
magnetic fieldline through it. This fieldline we define as
the axis of the EUV loop (plotted in blue color in Fig. 2
in the main text and its attached movie, as well as in
Fig. S4).
S3 Thermal evolution and coronal emission along individual fieldlines
In order to understand the difference in temporal
evolution of the EUV structures and the magnetic field
lines we first investigate the actual heat input on individ-
ual fieldlines. For this we use the fieldlines mentioned
in SM S2. We then study how this heat input establishes
the density and temperature structure along the fieldlines.
Eventually, this sets the EUV emission along the field-
lines. Finally by relating the evolution of the fieldlines
to the Poynting flux at the coronal base we can under-
stand what causes the magnetic field to apparently move
through the EUV loop.
For the study of the temporal evolution we choose
an arbitrary zero time, t=0. At this time the loop as seen
in EUV is just about to form. We use this zero time
throughout the manuscript, so negative times refer to the
temporal evolution before the EUV loop formed. All the
times given in figures and movies are with respect to this
zero time.
Heat input for individual fieldlines
The density and temperature structure along each field-
line is set by the heat input. To describe the temporal
SM 3

Supplementary Material for Chen, Peter, Bingert & Cheung, Nature Physics (2015). DOI: 10.1038/nphys3315
in the middle vertical plane
1
2
3
4
5
6
Ohmic heating [10−4 W m−3]
(a)
at the corona base (z = 2.9 Mm)
0 200 400 600 800
time [s]
0.2
0.4
0.6
0.8
1.0
1.2
vertical Poynting flux [107 W m−2]
(b)
Figure S1: Heating along individual field lines. The coloured
lines show the temporal variation of the heating for the twelve
fieldlines as defined in SM S2. Panel (a) shows the volumetric
energy deposition due to Ohmic dissipation at the cross section
of the respective field line and the vertical midplane between
the footpoints. This is close to the apex of the respective field-
line. The dotted line displays an envelope for the heat input.
Panel (b) shows the magnetic energy flux into the loop, viz. the
vertical component of the Poynting flux as defined in Eq. (1),
at the base of the corona. For comparison the dotted envelope
from panel (a) is plotted in panel (b), too, just scaled to roughly
match the peak of the Poynting flux.
evolution of the heat input we investigate two aspects, the
volumetric heat input due to Ohmic dissipation near the
loop apex, and the flux of magnetic energy into the loop
at the coronal base.
For the volumetric heating we investigate the
Ohmic heating of the crossing point of the respective
fieldline with the vertical midplane between the two
sunspots. Because the setup is quite symmetric it is close
(but not identical) to the heat input at the apex of the
fieldline. Following the fieldline in time as outlined in
SM S2 we find the time variation of the heat input near
the apex for that particular fieldline. In Fig. S1a we show
this for the twelve fieldlines discussed in SM S2 that co-
incide with the bright EUV loop at t≈130 s. It is clear
that each fieldline is heated for some 50 s, with all the
fieldlines defining the magnetic tube of roughly 2 Mm di-
ameter peaking over times from t≈130 s to 160 s.
To investigate the flux of magnetic energy into
the coronal part of the fieldline we study the vertical
Poynting flux at be base of the corona for each field-
line. For simplicity we use the height of z=2.9 Mm,
which is the average height where the temperature rises
above 105K. In general the Poynting flux is defined as
S=η j×B−(v×B)×B/µ0,with the current j, the
magnetic field B, velocity v, magnetic resistivity η, and
the magnetic permeability µ0. At the base of the corona,
where the magnetic energy density already dominates
the thermal energy density, the first term involving the
currents is negligible. The (v×B)×Bterm contains
the contribution from emerging horizontal fields and (al-
most) vertical fields being shifted around, e.g., following
the concept of braiding14 or the tectonics15. Because we
consider only the energy input into the fieldlines reaching
coronal heights, we consider only the latter part. Finally,
we are left with the contribution to the vertical Poynt-
ing flux involving the velocity v⊥perpendicular to the
magnetic field. So in terms of the components along the
horizontal x- and y-directions and the vertical z-direction
the vertical Poynting flux is
Sz=−1
µ0v⊥xBx+v⊥yByBz.(1)
We evaluate this quantity for each of the fieldlines at the
base of the corona (at each of its legs). The temporal evo-
lution of this quantity (for the “right” leg at x≈95 Mm; cf.
Fig. S2) is shown in Fig. S1b for the set of twelve field-
lines. Just as the Ohmic heating at the apex of the field-
lines, this energy input shows a clear (double) peak in
time. Comparing the two panels of Fig. S1 shows that the
energy input at the base of the corona precedes the heat-
ing rate at the apex by about 30 s (the dotted lines in both
panels). For a typical Alfv´
en speed of some 500 km/s this
is the time delay expected for the magnetic disturbances
traveling up the half loop length of some 15 Mm from the
coronal base.
SM 4

Supplementary Material for Chen, Peter, Bingert & Cheung, Nature Physics (2015). DOI: 10.1038/nphys3315
t = 130 s at the corona base (z = 2.9 Mm)
85 90 95 100
x [Mm]
25
30
35
40
y [Mm]
104
105
106
107
vertical Poynting flux [W m2]
Figure S2: Hot spot of Poynting flux at the base of the
corona. The image shows the vertical Poynting flux at the
base of the corona as defined in Eq. (1). The red region in
the middle of the image shows the location of the enhanced
Poynting flux. This is the hot spot of energy flux into the
corona. The white lines show the projection of the magnetic
fieldlines of the magnetic tube defined in SM S2 at time
t=130 s. The diamonds and the triangles indicate the position
of the footpoints of these fieldlines at times t=0 s and 600 s.
The pattern of the Poynting flux remains rather stable over
10 minutes. The field-view and the polygon roughly encircling
the hot spot are the same as in Fig. 3 in the main text.
The temporal evolution is shown in a movie available online.
The movie is also available at:
http://www2.mps.mpg.de/data/outgoing/peter/papers/2015-
magnetic-jam/movie-fig-s2.mp4
Hot spot of energy input at the coronal base
The discussion above shows clearly the increase and sub-
sequent decrease of the heat input on individual expand-
ing fieldlines. To investigate the cause for this transient
heating on a fieldline, we follow the footpoints of the
fieldlines at the base of the corona and relate it to the ver-
tical Poynting flux at the base of the corona as defined in
Eq. (1).
In Fig. S2 we display the vertical Poynting flux at
the time t=130 s along with the projection of the twelve
selected field lines as defined before in SM S2. The pat-
tern of the vertical Poynting flux is relatively stable over
the course of more than 10 minutes (cf. the movie at-
tached to Fig. S2). In particular, the increased Ponyting
flux in the middle of the panel remains roughly at the
same position forming some sort of hot spot. This hot
spot at the base of the corona is roughly co-spatial with
the increased Poynting flux at the solar surface (the black
polygon in Fig. S2 is at the same position as the polygon
in Fig. 3 in the main text showing the Poynting flux at the
surface).
While the fieldlines evolve and rise into the atmo-
sphere they move (roughly) horizontally at low heights.
Thus at the base of the corona they transverse the hot spot
of the Poynting flux. This is evident by inspection of the
movie attached to Fig. S2. Of course, while the fieldline
transverses the hot spot, the Poynting flux at the coronal
base changes slowly. This is the reason why a dip is seen
between two peaks of the Poynting flux in Fig. S1b. How-
ever, it is not the temporal change of the Poynting flux at
the base of the corona that is responsible for the increase
of the heating. Instead, the main effect for this is the foot-
point of the fieldline transversing a hot spot of Poynting
flux at the base of the corona.
Temperature and density along individual fieldlines
While a fieldline is rising upwards through the lower
atmosphere (with up to 10 km/s to 30 km/s vertically),
the density decreases continuously (before time t≈130 s;
Fig. S3a). This is because the rising fieldline is lifting up
the cool material and the upwards directed pressure gra-
dient is no longer able to counteract gravity. Thus the
plasma drains downwards along the fieldline. In part, the
loss of mass for individual fieldlines is also due to nu-
merical imperfections of the simulation: (hyper) diffu-
sion that is needed to smooth out numerical instabilities
allows the plasma also to diffuse across fieldlines. In this
early phase, when the fieldlines rise through the chromo-
sphere, this can account for up to about half of the mass
loss of individual fieldlines. However, this might not be
too unrealistic considering that on the real Sun in the cool
chromosphere there will be significant cross-field diffu-
sion of mass, because the plasma is only partially ionised
there. Once the temperature on the fieldline starts to rise
(around t≈130 s) the cross-field mass diffusion no longer
plays a role.
After the heating on the fieldline sets in, the tem-
perature will rise (for the set of fieldlines considered here,
SM 5

Supplementary Material for Chen, Peter, Bingert & Cheung, Nature Physics (2015). DOI: 10.1038/nphys3315
2
4
6
8
10
ne [109 cm−3]
particle density
(a)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
T [MK]
temperature
(b)
0 200 400 600 800
time [s]
50
100
150
200
[DN/pixel/s/Mm]
(c)
AIA 193Å emission
Figure S3: Temporal evolution at the apex of fieldlines. Pan-
els (a) and (b) show the temporal variation of the temperature
and density near the apex of each of the evolving fieldlines.
These are the same fieldlines as in Fig. S1 with the same color
coding. Panel (c) displays the synthesised emission, also at the
intersection of the respective fieldline with the vertical mid-
plane. The normalised temperature response curve (for the cen-
tral fieldline) of the AIA 193 ˚
A channel dominated by Fe XII
is overplotted as a dotted line in panel (b). The emission of
the whole EUV loop (integrated over the vertical midplane) is
shown as a dashed line in panel (c).
this happens at t≈130 s; see Fig. S3b). The draining of
the mass from the fieldline and the increase of the heating
rate together leads to a very strong increase of the heat-
ing per particle, which is responsible for the very sharp
increase in temperature. Within some 50 s the peak tem-
perature along the expanding fieldline is rising from ba-
sically chromospheric temperature to well above 1 MK,
eventually reaching some 3 MK.
In response to the heating of the plasma along the
fieldline, heat conduction back to the surface together
with the enhanced energy input in the low parts of the at-
mosphere leads to heating and evaporation of cool plasma
into the upper atmosphere. The resulting upflows cause
a gradual increase of the density (from time t≈130 s to
250 s; Fig. S3a).
Once the heating on that fieldline ceases (around
t≈200 s; Fig. S1), the temperature remains high, because
the coronal cooling time is of the order of the better part
of an hour. However, the density starts dropping soon
after the heating stopped (from time t≈250 s onwards;
Fig. S3a). This can be illustrated with the help of long-
known equilibrium considerations, even though the vari-
ability in the modelled system is more complex. The tem-
perature Tand the pressure p(and thus the density ρ) are
basically set by the heat input H; under equilibrium con-
ditions they follow power laws35,T∝H2/7,ρ∝H4/7,
i.e., the density is more sensitive to changes of the heat
input than the temperature. Therefore the density adjusts
faster to the drop of the heat input after t≈200 s. (The
density drop for each individual fieldline occurs after the
temperature passed through the temperature of maximum
response of the EUV passband and is thus not of major
relevance for the phenomenon described here; see below).
This filling and draining along fieldlines has been
described earlier for this 3D model16, and the average rate
of change of the mass in the top 20 Mm of the fieldline is
consistent with the (mostly vertical) mass flow across the
chromosphere-corona boundary. In Fig. 3 of our previ-
ous study16 the mass exchange is summarised: increased
heating causes an evaporative upflow in the bottom part,
and later when the fieldline expanded further, the velocity
pattern in the upper part reverses and the loop starts drain-
ing. (Note that the velocity in that Fig 3 is along the loop,
e.g., red on the left side and blue on the right side does
not imply a siphon flow, but evaporation into the corona).
In addition to this, the lower transition region is pushed
down due to the increase in pressure following the heat-
ing in the upper layers, similar to what has been found in
quiet Sun network model27. Together with the expansion
of the fieldlines, this would produce a pattern of net red-
shifts in the transition region and blueshifts in the hotter
regions as found in observations36.
SM 6

Supplementary Material for Chen, Peter, Bingert & Cheung, Nature Physics (2015). DOI: 10.1038/nphys3315
This behaviour of the temperature and density
along the fieldline is consistent with one-dimensional
models of coronal loops with variable prescribed heating
rates37. However, here we self-consistently describe the
heat input along each fieldline in the 3D model as deter-
mined by the fieldlines being moved across the hot spot
of Poynting flux at the base of the corona.
EUV emission along individual fieldlines
What we see of the corona is neither the temperature
nor the density, but the photons that are emitted by
the plasma. Thus we synthesise the emission from the
model as it would be seen by an EUV instrument. Here
we concentrate on the 193 ˚
A channel of AIA/SDO that
images emission from mainly Fe XII forming at about
1.5 MK. For this we use the same procedures as outlined
before9, 18, implicitly assuming ionisation equilibrium.
The 193 ˚
A channel has a temperature response
function G(T)that peaks sharply at about 1.5 MK. Con-
sequently, when following an individual fieldline that is
heated in time, the contribution to the 193 ˚
A channel will
be significant just during the time the fieldline is at the
matching temperature (cf. Fig. S3b). The actual emission
is then given by n2G(T), where nis the (electron) den-
sity. To characterize the emission from any given field-
line, we show the emission at the intersection of the field-
line with the midplane used before in Fig. S3c. This re-
flects the emission near the apex of the fieldline.
The increase of the coronal emission on an indi-
vidual fieldline peaks sharply (Fig. S3c) when the tem-
perature is close to the peak of the contribution function
G(T). The timescale for the brightening of an individual
fieldline is thus determined by the rise time of the temper-
ature, and is of the order of 50 s. Figure S3 shows that the
peak of the coronal emission for each individual fieldline
is during the phase of rising density, well before the den-
sity drops because the heating for the respective fieldline
ceased. Thus the temporal variability of the coronal emis-
sion for each individual fieldline is mainly governed by
the evolution of the temperature: Each fieldline brightens
shortly after it was heated and it temperature rose quickly.
So ultimately each of the expanding fieldlines is
brightening according to the time when the footpoint of
the fieldline transverses the hot spot of the Poynting flux
at the base of the corona causing the enhanced heat input.
Consequently the fieldlines lighten up in succession ac-
cording to their expansion. This is evident from the set of
fieldlines shown color coded in Figs. S1 and S3c for the
heat input and emission.
The above discussion concentrates on the results
relating to EUV instruments, e.g., the Atmospheric Imag-
ing Assembly (AIA)34. Currently EUV imaging provides
the highest spatial resolution in the corona, significantly
higher than X-ray observations, e.g. with the recent XRT
instrument38. However, the response in temperature for
X-ray instruments is quite different than EUV instru-
ments. The EUV bands typically show plasma over a
temperature range of 0.3 in log10 T[K] (FWH M of the
response function)18, i.e. a factor of 2. In contrast, the
X-ray instruments typically image plasma at higher tem-
perature over wider range of temperature (peak of re-
sponse function near 8 MK, covering a factor of 4 in
temperature)38. This different response might change the
situation quite a bit, in particular because the tempera-
tures in the model loop discussed here reach peak tem-
peratures of about 3 MK. Work including the synthesis of
X-ray emission to discuss this in more detail is underway.
Magnetic fieldlines moving
through stationary EUV loop
In Fig. S4 we show the temporal evolution of the result-
ing EUV loop when seen from the side together with the
position of the center fieldline of the magnetic tube se-
lected in SM S2. The magnetic tube is constantly moving
upwards, from an apex height of ≈12.5 Mm at t=50 s to
≈17 Mm at 200 s corresponding to a speed of 30 km/s.
The expansion of the fieldlines slows down at greater
heights because they are now running into the fieldlines
that emerged before. Until the end of the time series
shown in Fig. S4 at t=600 s the magnetic tube expanded
only another 2.5 Mm, corresponding to an average speed
of about 5 km/s.
While each individual fieldline brightens up for
only some 50 s to 100 s the successive brightening of
the expanding fieldlines causes a comparably stationary
bright loop visible in coronal EUV emission (Fig. S4).
In particular the apex height of the center of the EUV
loop (blue line in Fig. S4) varies only slightly between
z≈14.5 Mm and ≈15.5 Mm, while over the same time the
magnetic tube rose by more that 7 Mm.
The EUV loop formed by the expanding fieldlines
shows some variability of its brightness at the apex, but
it remains bright for well over 10 min (dashed line in
Fig. S3c). However, after about t≈550 s the loop starts
fading away because the hot spot of the Poynting flux at
the base of the corona gets weaker.
The formation of the EUV loop and the apparent
motion of the fieldlines though the loop can be summa-
rized as follows. The fieldlines expand during the flux
SM 7

Supplementary Material for Chen, Peter, Bingert & Cheung, Nature Physics (2015). DOI: 10.1038/nphys3315
0
5
10
15
20
25
fieldline through the emission centercenter fieldline
t = 50 s
t = 100 s
t = 150 s
0
5
10
15
20
25
t = 200 s
t = 250 s
t = 300 s
0
5
10
15
20
25
t = 350 s
t = 400 s
t = 450 s
50 60 70 80 90
x [Mm]
0
5
10
15
20
25
z [Mm]
t = 500 s
50 60 70 80 90
t = 550 s
50 60 70 80 90
t = 600 s
500 1000 1500
AIA 193 channel [DN/pixel/s]
Figure S4: Magnetic fieldline moving through EUV loop. This image sequence shows the evolution of the synthesized EUV
emission in the 193 ˚
A band of AIA dominated by plasma at 1.5 MK radiating in Fe XII. The red line shows the fieldline in the
expanding magnetic field, the blue line the fieldline through the center of the EUV emission pattern at each snapshot. This is
similar to panel (a) in Fig. 2 in the main text.
The movie showing the full temporal evolution is available online with Fig. 2 of the main text.
It is also available at: http://www2.mps.mpg.de/data/outgoing/peter/papers/2015-magnetic-jam/movie-fig-2.mp4
emergence process and start rising. In this process they
loose mass because the cool plasma drains along the field-
lines. During the expansion the fieldlines also move hor-
izontally into the forming sunspot. In this process they
transverse a hot spot of enhanced Poyntig flux that is
formed because of the coalescent flow forming the spots.
Thus each individual fieldline gets a burst of heating and
when it reaches the response temperature of the EUV
channel it brightens up for less than a minute. The ex-
panding fieldlines transverse the hot spot in close succes-
sion and thus light one after the other. This then forms
an EUV loop being stable for a longer time, basically
as long as the hot spot of Poynting flux is sustained by
the (horizontal) flows at the base of the corona. Because
these fieldlines get bright at a similar geometric height,
this creates the illusion that the loop stays at the same po-
sition. In reality, the stable-looking EUV loop is formed
by a dynamically evolving magnetic field.
SM 8