[a]Jacob Finkenrath

# Fine grinding localized updates via gauge equivariant flows in the 2D
Schwinger model

###### Abstract

.

###### Abstract

State-of-the-art simulations of discrete gauge theories are based on Markov
chains with local changes in the field space, which however at very fine
lattice spacings are notoriously difficult due to separated topological
sectors of the gauge field. Hybrid Monte Carlo (HMC) algorithms, which are
very efficient at coarser lattice spacings, suffer from increasing
autocorrelation times. This makes simulation of lattice QCD close to the
continuum infeasible even with exa-scale computing.

An approach, which can overcome long autocorrelation times, is based on
trivializing maps, where a new gauge proposal is given by mapping a
configuration from a trivial space to the target one, distributed via the
associated Boltzmann factor. Using gauge equivariant coupling layers, the map
can be approximated via machine learning techniques. However the deviations
scale with the volume in case of local theories and extensive distributions,
rendering a global update unfeasible for realistic box sizes.

In this proceeding, we will discuss the potential of localized updates in case
of the 2D Schwinger Model. Using gauge-equivariant flow maps, a local update
can be fine grained towards finer lattice spacing. Based on this we will
present results on simulating the 2D Schwinger Model with dynamical Nf=2
Wilson fermions at fine lattice spacings using scalable global correction
steps and compare the performance to the HMC.

##  1 Motivation

Monte Carlo simulations in gauge theories show critical slowing down towards
fine lattice spacings, i.e. autocorrelations increases proportional to a
higher inverse power of the lattice spacings [1]. Especially in gauge
theories, this is even more severe due to topological freezing. Due to that it
is currently not feasible to simulate with state-of-the-art algorithms at very
fine lattice spacings below $a<0.04\;\textrm{fm}$ with periodic boundary
conditions, see for a review [2]. However, if possible, simulations at very
fine lattice spacings have a major impact on measurements in lattice QCD,
namely by stabilising continuum extrapolations which would enable a robust way
to make advances at the precision frontier of the standard model of particle
physics.

In this proceedings we will investigate possible algorithmic solutions in case
of the two-dimensional (2D) Schwinger model with two mass degenerated quarks.
For the pure gauge action $S_{g}$ the Wilson plaquette definition and for the
fermion action $S_{f}$ the plain Wilson fermion discretization $D(U)$ is used.
The Boltzmann weight or here also called target distribution is given by

$\rho(U)=Z^{-1}e^{-S_{f}(U)-\beta
S_{g}(U)}=Z^{-1}\textrm{det}D^{2}(U)\cdot\textrm{exp}\\{-\beta\sum_{x=1}^{V}P_{12}(U,x)\\}$
(1)

with $Z$ the partition function and $P_{12}(U,x)$ the plaquette at lattice
point $x$. The Schwinger model shares several properties with the 4D-QCD
model, e.g. the Hybrid Monte Carlo algorithm suffers from topological freezing
towards larger values of $\beta$. It also simplifies implementations and
computations, due to that it is often used as a test bed to develop
algorithms, as we will do here.

Simulations in lattice QCD are based on Markov Chain Monte Carlo (MCMC)
algorithms, which generate an ensemble of configurations distributed via the
Boltzmann weight $\rho(U)$. In general the MCMC procedure consists of two
basic steps. In the first step a new configuration $U^{\prime}$ is proposed
via a proposal probability $T_{0}(U\rightarrow U^{\prime})$ where we denote
$\tilde{\rho}(U^{\prime})$ as the probability distribution of the proposal. In
the second step the weight is corrected via an accept-reject step

$P_{acc}(U\rightarrow
U^{\prime})=\textrm{min}\left[1,\frac{\tilde{\rho}(U)\rho(U^{\prime})}{\rho(U)\tilde{\rho}(U^{\prime})}\right]~{}.$
(2)

Now, the MCMC method works efficiently, if the proposal can decouple the
proposed configuration $U^{\prime}$ effectively from starting configuration
$U$. The decoupling time is measured as the autocorrelation time, where the
longest is usually associated with the topological charge in gauge theories.
Thus, decoupling requires to have a high topological tunnelling rate for the
proposal at a relative high acceptance rate in the accept-reject step.

For the acceptance rate follows [3]

$P_{acc}=\textrm{erfc}\\{\sqrt{\sigma^{2}(\Delta S)/8}\\}$ (3)

with $\Delta
S=\textrm{ln}(\rho(U^{\prime}))-\textrm{ln}(\rho(U))+\textrm{ln}(\tilde{\rho}(U))-\textrm{ln}\tilde{\rho}(U^{\prime}))$
if the distribution
$(\tilde{\rho}(U)\rho(U^{\prime}))/(\rho(U)\tilde{\rho}(U^{\prime}))$ is log-
normal distributed.

To reach high acceptance rate, the variance of $\Delta S$ has to be under
control. This can be done by

  * 1.

Using correlations between the proposal and target distribution, e.g. by
maximizing the covariance between $\rho$ and $\tilde{\rho}$.

  * 2.

Reducing the degrees of freedom within the proposal distribution, e.g. by
localization of updates via domain decomposition techniques.

Both methods can be combined using hierarchical filter steps as outlined in
[4] to reach an even higher acceptance rate.

Now, in order to have a high tunneling rate of the topological charge, the
proposal for a new gauge configuration should allow for topological
transitions. One possible choice of such proposal is given by trivializing
flows [5]. The basic idea is to start from a configuration weighted via a
uniform distribution $r(U_{0})$ and to define a map $U=f^{-1}(U_{0})$, which
transforms the configuration to the target space distributed by $\rho(U)$.
Using an appropriate definition of the map the distribution of the flow
$\tilde{\rho}$ can be calculated via the Jacobian of the transformation, which
is given by

$\tilde{\rho}(U)=r(f(U))\cdot\left|\textrm{det}\frac{\partial f(U)}{\partial
U}\right|.$ (4)

Examples for a tractable maps are given by normalizing discrete flows [6, 7,
8, 9, 10] or continuous flows [11], see also [12]. Here, we will use the
former one [9], where the flow distribution can be trained using convolutional
neural networks within the map transformations also called coupling layers.
The training or optimization of the networks is efficient if the acceptance
rate of the accept-reject step is high or if the corresponding variance
$\sigma^{2}(\Delta S)$ is under control, compare eq. (3) . This is usually
achieved by using the KL-divergence which leads to
$\tilde{\rho}(U)\approx\rho(U)$ [7].

Note the variance $\sigma^{2}(\Delta S)$ is an extensive quantity, i.e. the
lattice action is a function of the volume. In case of a local theory, where
the correlation length decays with $\textrm{cov}(x,y)\propto e^{-d|x-y|}$ and
$d>0$, the variance of an extensive quantity, like the plaquette action, can
be written as $\sigma^{2}=V(a_{0}+a_{1}e^{-d}+a_{2}e^{-d\sqrt{2}}+\ldots)$
with $a_{n}$ the coefficient of the $\sqrt{n}$ distance suppressed via
$e^{-d\sqrt{n}}$ and $V$ the volume. Now, by training the coupling layers the
target distribution is approximated to a certain degree. If we fix this
approximation to $S$, the difference to the target distribution will scale
with the volume. This means that to achieve a reduced scaling behavior the
approximation needs to be improved when increasing the volume. This will
become very hard or computational very challenging if higher order terms or
very small long range correlations are left and are required to be learned or
approximated to further improve the model. Due to that larger volumes with
high acceptance rate are currently out of reach [12, 13, 14] and will require
further advances in the definitions of the maps as well as in the training
procedures.

##  2 Localization

A natural way out of volume scaling in local theories is the localization of
the update. This can be done by decomposing the lattice into domains. Now, we
can freeze the boundary links of each domain, which share loops with
neighbouring domains, and only update links within each domain. This can be
applied to the equivariant flow map, as illustrated in the left panel of Fig.
1. While some modifications compared to the periodic case in the training are
needed, the approach works well for lattice sizes of $L=8$, as demonstrated in
[15]. However, for larger volumes updates become difficult due to the volume
scaling, which makes a direct adaptation to 4D-SU(3) without improvements
difficult.

Figure 1: The left panel illustrates the flow maps which transforms the
plaquettes from a uniform distribution (red) towards the target distribtution
(blue). The upper case shows the approach introduced [15], where the
plaquettes and with it the corresponding links in the boundaries are fixed.
The lower case shows the here introduced fine graining maps, where only the
central plaquette is updated and smoothed out including the neighbouring
plaquettes towards the target distribution. The right panel shows the basic
kernel used to update the active plaquettes (red) involving frozen ones (blue)
but not passive ones (organge) within an application of a coupling layer.

###  2.1 Fine graining flows

Before we discuss the here introduced modification to the localized flow, we
want to point out an observation how flow maps are constructed. The standard
procedure is based on a map at fix number of lattice points or at fixed
lattice extents. In case of a trivializing flow, which is reducing effectively
the lattice spacing during mapping, the physical size of the box is changing.
This implies that during the mapping to the target theory, the projection has
to filter out un-wanted noisy modes and need to build up longer correlations
at the same time to match to the physics at the target space.

A more natural projection is given by keeping the physical box size fixed
during the mapping. This however would imply for a trivializing map to add new
physical degrees of freedoms to new intermediate lattice points in the
discrete lattice during a mapping from a coarse to a finer theory. While with
machine learning techniques this might be in range [16], such operation
requires additional non-smooth steps which are non-trivial to tune.

Here, instead, we will use an effective fine graining approach, by introducing
a map which smoothes out a local defect. This is done by placing a local
defect into a larger lattice or domain and fine grain it into the local
neighbourhood. A similar approach is done using multi-tempering, which is
successfully applied in unfreezing topological charges in 4D-SU(N) pure gauge
models, see [17].

Here, we will use local flow transformations to map between the target
distribution and the defect, which is generated by updating the links within
the defect according to a uniform distribution. This can be also applied
within a multi-tempering algorithm, like proposed in the T-REX algorithm for
4D-SU(3) in [18]. In this proceeding, we will use a uniform update of the
defect to enable topological transitions. This will require to adapt the
training procedure to find an appropriate map.

We will follow [7, 8] by using the gauge equivariant flow. For a local defect,
we introduce a localized map-condition to update the plaquettes. Namely, we
use in 2D a maximal compact map by placing the passive plaquette in the center
and updating all links belonging to it by using the neighbour plaquettes. This
gives an active to passive plaquette ratio of 4:1, see right panel of Fig. 1.
Note, that in principle for a unit N-D cube in higher dimensions N the ratio
decreases, however additional passive links outside the cube can be included
using the procedure outlined in [19]. Now, we can design additional center
symmetric kernels by placing the introduced symmetric kernel around the
defect, as illustrated in the left panel of Fig. 2. Additional we add a
smoothing kernel, which we denote as the 4th kernel, by using nine times the
2nd kernel and shifting with
$\\{(1,0),(0,1),(-1,0),(0,-1),(1,1),(1,-1),(-1,-1),(-1,1),(0,0)\\}$
respectively.

Figure 2: The figure shows the kernels for the maps used for the different
coupling layers. Each link of a passive plaquettes (orange) is associated with
an active plaquettes (red) to achieve a compact layout. The frozen plaquette
(blue) are fixed and used as input to update the active plaquettes.

To ensure that the proposals allows topological tunneling, we modified the
training conditions by only using sets from the batch for the optimization
process of the coupling layers, for which the topology changed. Namely, by
introducing a topological loss function, given by

$L=\textrm{ln}(\tilde{\rho}(U^{\prime})/\rho(U^{\prime}))\cdot|Q(U^{\prime})-Q(U)|$
(5)

with $Q$ the topological charge, which is exactly quantified in the 2D-U(1)
model. Note, if the standard loss-function is used the flow would become most
likely a fancy plaquette update.

Fine graining the defect to the neighbouring links requires to adapt the
proposal and accept-reject procedure of the MCMC chain. Namely, it requires to
first apply a back transformation to the configuration, before the defect can
be updated, see Fig. 3. To stabilize the training of the coupling layers the
back-transformation is fixed and updated after several epochs. In general,
this allows to introduce an additional loss-function given by

$L=\textrm{ln}\\{\rho(U^{\prime})\\}-\textrm{ln}\\{\rho(U)\\}+\textrm{ln}\\{\tilde{\rho}_{\textrm{fixed}}(U)\\}-\textrm{ln}\\{\tilde{\rho}_{\textrm{train}}(U^{\prime})\\}$
(6)

which is used additional to the topological loss-function to increase the
acceptance rate.

Figure 3: The figure illustrate the fine graining update step. In the first
step the reverse mapping is applied to the starting configuration. Then, the
links of the inner plaquette (red) are updated using a uniform distribution.
In the final step the plaquettes are mapped back towards the target
distribution.

The training for a flow map to increase topological tunneling requires a fine
tuning or grinding process. Namely, first the maps are pre-trained on a $L=8$
lattice with periodical boundary conditions using the topological loss. Note,
if we use the 4th kernel plaquettes within the pretraining also the boundary
plaquettes are updated. This map is then used as the fix backwards
transformation on a $L=16$ lattice with fixed boundary condition, i.e. the
flow acts only on the center links close to the placed defect. The training is
then iterated by updating the fixed backward maps and the procedure is further
fine tuned by using smaller optimization steps.

##  3 Results

###  3.1 Fine graining updates

Figure 4: The left panel illustrates the rate of a topological transition of a
fine graining flow update on a fixed $L=16$ lattice at $\beta=11.25$. For
that, the fine graining flow update is repeated 1k times with the same initial
configuration. The right panel shows the acceptance rate of the local block
determinants at $\beta=8.45$ and $m=-0.025$. The block size of the fine
graining updates are given by $V=16\times 16$ (blue). This is compared to the
acceptance rate of the localized flow update [15] using a block size of
$V=8\times 8$ including correlation with the pure gauge action (orange) and
without (yellow).

An example for a successfully trained flow map at $\beta=11.25$ is illustrated
in the left panel of Fig. 4 in case of the topological charge distribution on
a single configuration before and after the fine graining update. For the fine
graining flow map, we used a combination of the kernels
$[1,3,2,1,3,2,1,3,2,4]$, resulting into 18 coupling layers in total.

Compare to the fixed boundary update, introduced in [15], the FG-flow updates
shows also a higher acceptance rate in case of local corrections via the
associated block determinant. This is shown in the right panel of Fig. 4,
where the fine grain (FG)-flow update is done within a $16\times 16$ block and
reached an acceptance rate of $85\%$ while the fixed boundary flow only
reached $65\%$ acceptance rate using a $8\times 8$ domain.

###  3.2 Topological transitions

Now, we can use the FG-flow updates within a MCMC procedure. To achieve an
ergodic algorithm, the FG-flow updates are combined with a Hybrid Monte Carlo
(HMC) update step, i.e. after each FG-flow update a shift and a trajectory via
the Hybrid Monte Carlo algorithm is performed. To include the fermion weight
hierarchical filter steps in combination with the domain decomposed
determinant like discussed in [4, 15] are used, which we will denote as
flowGC. The final step is given by a global correction step (GC) with the
determinants of the corresponding Schur complements [4]. The normalized
topological transition rate $T(Q)\sqrt{\beta}/L$ with
$T(Q)=\langle|Q_{i}-Q_{i+1}|\rangle$ is used in a range of
$\beta\in[3.0;\,11.25]$ as shown in Fig. 5. The comparison between flowGC and
FG-flowGC is done at $L/a=64$ while for the HMC the rate is measured by fixing
$L/\beta\approx 40$ and a mass-scale set to $m_{PS}\sqrt{\beta}=0.4$, where
$m_{PS}$ is the pseudoscalar mass, see [20]. By only randomizing $0.8\%$ of
the links the FG-flow procedure shows a similar transition behaviour like the
flowGC, which both unfreeze the topological charge successfully in contrast to
the pure gauge HMC algorithm.

Figure 5: The topological tunneling rate normalized with $\sqrt{\beta}/L$ is
shown on the left panel obtained for the HMC (yellow squares), for the flowGC
algorithm (blue circles) and the FG-flowGC at fixed lattice size $L/a=64$ (red
diamonds). The right panel shows the volume dependence of global acceptance
rate for the different global correction step algorithms at a gauge coupling
of $\beta=6.0$ and a mass scale of $m_{PS}\sqrt{\beta}=0.4$, see [20]. The
impact of using hierarchical filter are demonstrated, i.e. the acceptance rate
using the full determinant drops fast by increasing the lattice size (blue
(left-pointing) triangles), using a 4-level method with domain decomposition
with $L=8$ blocks increases the rate (yellow (top-pointing) triangles) and
using a 5-level method with recursive domain decomposition increases the rate
(purple (down-pointing) triangles)) even further, see also [15]. The , here
introduced, flow grain update is using a 4-level methods with domain
decomposition of $L=16$ blocks (orange circles).

###  3.3 Comparison with other global correction approaches

The FG-flowGC has not only a high acceptance rate for the correction of the
local block determinants, it shows also good performance at the global
correction step. This is illustrated in Fig. 6. The different procedures are
given as follows:

  * •

GC-bare with two-levels

    * –

pure gauge update via HMC of the full lattice

    * –

global correct reject step including correlations

  * •

flowGC with $L=8$ and 4 levels:

    * –

flow update within domains by fixing each boundary links (updating only every
4th domain)

    * –

accept reject with local pure action (second level), $L=8$ block determinants
(third level)

    * –

global correction step with determinant of the Schur complement

  * •

flowGC with $L=8$ and 5 levels:

    * –

similar to flowGC, only adding an additional filter level before the global
correction step, by introducing an extended domain with $L=24$ including each
of the updated domains

  * •

FG-flowGC with 4 levels: fine graining updates within a $L=16$ domain

    * –

similar to the flowGC with 4 level only using fine graining updates within
$L=16$ domains.

Note that all filter steps before the global correction steps are local and
can be perform in parallel, making the approach scalable and inline with
current algorithmic trends such as highlighted in [2, 21]. Moreover, the
approach can be easily generalized to the multi-level procedure [22, 23],
which would not only help to fight against long autocorrelation times but also
mitigate exponential increasing noise in the long-distance measurements of
correlators. Additional the procedure can be combined with other updates
schemes, which show success in pure gauge simulations such as multi-tempering
approaches [17, 24].

Figure 6: The sketch illustrates the different filter steps of the global
correction step algorithms. While the 5-level approach, discussed in [15],
using two filter steps including block determinants on $L=8$ and $L=24$, the
4-level flowGC only uses the $L=8$ blocks and the GC-bare only includes the
global determinant. The here used FG-flowGC algorithm is based on a 4-level
approach, using block determinants of $L=16$ as a intermediate filter step.

##  4 Conclusion

Within this proceeding we discussed a new way to design flow maps to decouple
the topological charge by updating only a minimal number of links in the 2D
Schwinger model. The trained flow are given by fine graining updates, which
are updating a local defect using center symmetric kernels. The current status
of the training procedure involves several steps resulting in a fine tuning
procedure or a grinding task. However, using the fine graining updates high
acceptance rates are achieved for the fermion corrections, for the local block
determinants as well as for the global correction steps. Furthermore the
update unfreezes topology, i.e. thus solve the key problem. If the training
procedure is under better control, the next step is its application to
4D-SU(3).

Acknowledgments: J.F. received financial support by the German Research
Foundation (DFG) research unit FOR5269 "Future methods for studying confined
gluons in QCD". The author gratefully acknowledges the Cyprus Institute for
providing computational resources on Cyclone for training the neural networks.

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11institutetext: Department of Physics, Imperial College London, London, UK
11email<EMAIL_ADDRESS>22institutetext: Space Research
Institute, Austrian Academy of Sciences, Graz, Austria 33institutetext:
Institute of Geodesy, Graz University of Technology, Graz, Austria
44institutetext: Space and Atmospheric Electricity Group, Department of
Meteorology, University of Reading, Reading, UK 55institutetext: Institute of
Physics, University of Graz, Graz, Austria 66institutetext: Conrad
Observatory, Zentralanstalt für Meteorologie und Geodynamik, Vienna, Austria
77institutetext: Technical University of Braunschweig, Braunschweig, Germany
88institutetext: STFC-RAL Space, Didcot, UK

# In situ multi-spacecraft and remote imaging observations of the first CME
detected by Solar Orbiter and BepiColombo

E.E. Davies 11 C. Möstl 2233 M.J. Owens 44 A.J. Weiss 223355 T. Amerstorfer 22
J. Hinterreiter 2255 M. Bauer 22 R.L. Bailey 66 M.A. Reiss 2233 R.J. Forsyth
11 T.S. Horbury 11 H. O’Brien 11 V. Evans 11 V. Angelini 11 D. Heyner 77 I.
Richter 77 H-U. Auster 77 W. Magnes 22 W. Baumjohann 22 D. Fischer 22 D.
Barnes 88 J.A. Davies 88 R.A. Harrison 88

(Received: 10 December 2020 — Revised: 1 February 2021 — Accepted: 20 February
2021)

###### Abstract

Context. On 2020 April 19 a coronal mass ejection (CME) was detected in situ
by Solar Orbiter at a heliocentric distance of about 0.8 AU. The CME was later
observed in situ on April 20 by the Wind and BepiColombo spacecraft whilst
BepiColombo was located very close to Earth. This CME presents a good
opportunity for a triple radial alignment study, as the spacecraft were
separated by less than 5∘ in longitude. The source of the CME, which was
launched on April 15, was an almost entirely isolated streamer blowout. The
Solar Terrestrial Relations Observatory (STEREO)-A spacecraft observed the
event remotely from -75.1∘ longitude, which is an exceptionally well suited
viewpoint for heliospheric imaging of an Earth directed CME.

Aims. The configuration of the four spacecraft has provided an exceptionally
clean link between remote imaging and in situ observations of the CME. We have
used the in situ observations of the CME at Solar Orbiter, Wind, and
BepiColombo and the remote observations of the CME at STEREO-A to determine
the global shape of the CME and its evolution as it propagated through the
inner heliosphere.

Methods. We used three magnetic flux rope models that are based on different
assumptions about the flux rope morphology to interpret the large-scale
structure of the interplanetary CME (ICME). The 3DCORE model assumes an
elliptical cross-section with a fixed aspect-ratio calculated by using the
STEREO Heliospheric Imager (HI) observations as a constraint. The other two
models are variants of the kinematically-distorted flux rope (KFR) technique,
where two flux rope cross-sections are considered: one in a uniform solar wind
and another in a solar-minimum-like structured solar wind. Analysis of CME
evolution has been complemented by the use of (1) the ELEvoHI model to compare
predicted CME arrival times and confirm the connection between the imaging and
in situ observations, and (2) the PREDSTORM model, which provides an estimate
of the Dst index at Earth using Solar Orbiter magnetometer data as if it were
a real–time upstream solar wind monitor.

Results. A clear flattening of the CME cross-section has been observed by
STEREO-A, and further confirmed by comparing profiles of the flux rope models
to the in situ data, where the distorted flux rope cross-section qualitatively
agrees most with in situ observations of the magnetic field at Solar Orbiter.
Comparing in situ observations of the magnetic field between spacecraft, we
find that the dependence of the maximum (mean) magnetic field strength
decreases with heliocentric distance as $r^{-1.24\pm 0.50}$ ($r^{-1.12\pm
0.14}$), which is in disagreement with previous studies. Further assessment of
the axial and poloidal magnetic field strength dependencies suggests that the
expansion of the CME is likely neither self-similar nor cylindrically
symmetric.

###### Key Words.:

Sun: coronal mass ejections (CMEs) – evolution – solar-terrestrial relations –
solar wind

##  1 Introduction

Coronal mass ejections (CMEs) are large-scale expulsions of plasma and
magnetic fields that are expelled from the solar atmosphere. The simplest
configuration of a CME observed in visible-light images has a three-part
structure: a bright high-density leading loop enclosing a dark low-density
cavity and a bright core (e.g. Illing & Hundhausen, 1985; Vourlidas et al.,
2013), where the cavity is thought to be a magnetic flux rope structure (Dere
et al., 1999; Wood et al., 1999).

In situ measurements of interplanetary CMEs (ICMEs) often display signatures
of magnetic flux rope structures (Cane & Richardson, 2003), likely the
interplanetary manifestations of the magnetic flux rope associated with the
dark cavity observed remotely (Burlaga et al., 1982). ICMEs are the main
drivers of severe space weather at Earth (e.g. Kilpua et al., 2017) and
therefore their evolution is of great interest in space weather modelling. To
better understand ICME evolution in situ, it is useful to track signatures of
specific ICMEs over large heliocentric distances whilst spacecraft are close
to radial alignment (e.g. Burlaga et al., 1981). Studies of ICMEs observed by
spacecraft separated by less than 5∘ in longitude and separated by more than
0.2 AU in heliocentric distance are relatively rare but provide valuable
insight into the properties, rotation, expansion, and interaction with other
features of the solar wind environment as the ICME propagates (e.g. Good et
al., 2015, 2018; Winslow et al., 2016; Kilpua et al., 2019; Lugaz et al.,
2019; Davies et al., 2020).

The link between CMEs and ICMEs and their global structure and kinematics has
previously been studied by using remote observations made by the Solar
Terrestrial Relation Observatory (STEREO; Kaiser et al., 2008) in combination
with in situ observations (e.g. Davis et al., 2009; Möstl et al., 2009; Wood &
Howard, 2009; Liu et al., 2010; Rouillard et al., 2010). The global structure
of CMEs is a complex issue affected by the interaction of the CME with the
solar wind and other CMEs as it propagates. The kinematic distortion of the
large-scale structure of the CME in a latitudinally structured solar wind
leads to a flattening or ‘pancaking’ of the flux rope cross-section (Odstrcil
& Pizzo, 1999; Riley & Crooker, 2004; Owens, 2006; Owens et al., 2017),
identified in remote observations (e.g. Savani et al., 2010).

Understanding the evolution of CMEs through the inner heliosphere is one of
the objectives of the Solar Orbiter mission (Müller et al., 2013, 2020), which
launched on 2020 February 10. Solar Orbiter provides an opportunity to study
CMEs both remotely and in situ, at closer heliocentric distances to the Sun
(0.28 AU perihelion) and higher latitudes out of the ecliptic (up to 33∘). In
this study, we present the magnetic field (MAG; Horbury et al., 2020)
observations of the first CME measured in situ at Solar Orbiter on 2020 April
19 whilst Solar Orbiter was in close radial alignment with both the Wind and
BepiColombo (Benkhoff et al., 2010) spacecraft. During this period of time,
BepiColombo was completing a flyby of Earth whilst on its cruise phase to
Mercury, where the mission plans to investigate the interior, surface,
exosphere, and magnetosphere to better understand the origin and evolution of
the planet. The data are not public so far but available by request from the
PI D. Heyner.

Solar Orbiter Energetic Particle Detector (EPD; Rodríguez-Pacheco et al.,
2020) observations of the Forbush decrease associated with the same CME are
presented by von Forstner et al. (in revision at A&A, 2021). STEREO-A was well
positioned to observe the CME remotely using Heliospheric Imager (HI; Eyles et
al., 2009) data, providing an exceptional link to the in situ measurements to
study the dynamical evolution and propagation of the CME. Section 2 presents
the spacecraft observations and the flux rope models used to investigate the
properties and large-scale structure of the CME as it propagates, the results
of which are presented in Section 3 where both the radial and non-radial
evolution of the CME has been assessed. Additionally, a modelling of the Dst
index is demonstrated by using Solar Orbiter magnetometer data as if they were
used as a real–time upstream solar wind monitor.

##  2 Spacecraft data and methods

###  2.1 In situ observations

Table 1: ICME properties at Solar Orbiter, Wind, and BepiColombo, for 2020 April 19-21. ICME times $t$ are given, where the subscripts stand for (1) shock arrival time, (2) flux rope start time, (3) flux rope end time, (4) unperturbed magnetic flux rope (UMFR) start, (5) UMFR rope end. Also shown are the duration $\Delta$ for the full ICME, the sheath, the magnetic obstacle (MO) and the UMFR, and the mean total magnetic field in those 4 intervals. The spacecraft heliocentric distance $R$ is given at 3 different times, corresponding to the subscripts 1-3 above, as well as the HEEQ longitude $lon$ and HEEQ latitude $lat$. | unit | Solar Orbiter | Wind | BepiColombo
---|---|---|---|---
$t_{1}$ | UT | Apr 19 05:06 | Apr 20 01:34 | Apr 20 03:09
$t_{2}$ | UT | Apr 19 08:59 | Apr 20 07:56 | Apr 20 08:05
$t_{3}$ | UT | Apr 20 09:15 | Apr 21 11:18 | Apr 21 10:08
$t_{4}$ | UT | Apr 19 13:56 | Apr 20 11:28 | Apr 20 14:03
$t_{5}$ | UT | Apr 20 04:36 | Apr 21 02:10 | Apr 21 03:18
$\Delta_{ICME}$ | h | 28.15 | 33.73 | 30.98
$\Delta_{sheath}$ | h | 3.88 | 6.37 | 4.93
$\Delta_{MO}$ | h | 24.27 | 27.37 | 26.05
$\Delta_{UMFR}$ | h | 14.67 | 14.70 | 13.25
$B_{ICME}$ | nT | 17.2 | 12.3 | 12.7
$B_{sheath}$ | nT | 9.9 | 6.8 | 7.2
$B_{MO}$ | nT | 18.4 | 13.6 | 13.7
$B_{UMFR}$ | nT | 19.2 | 14.9 | 14.6
$R_{1}$ | AU | 0.809 | 0.996 | 1.011
$R_{2}$ | AU | 0.808 | 0.996 | 1.011
$R_{3}$ | AU | 0.802 | 0.996 | 1.011
$lon_{1}$ | deg | -4.02 | 0.19 | -1.25
$lon_{2}$ | deg | -3.98 | 0.18 | -1.27
$lon_{3}$ | deg | -3.70 | 0.18 | -1.41
$lat_{1}$ | deg | -3.94 | -5.15 | -5.53
$lat_{2}$ | deg | -3.92 | -5.13 | -5.52
$lat_{3}$ | deg | -3.80 | -5.03 | -5.47
Figure 1: Overview of spacecraft positions and in situ solar wind data. (a)
WSA/HUX ambient solar wind speed in the ecliptic plane and spacecraft
positions on 2020 April 20 00:00 UT in Heliocentric Earth Equatorial (HEEQ)
coordinates. (b) Solar Orbiter magnetic field components in Radial-Tangential-
Normal (RTN) coordinates, (c) BepiColombo magnetic field data in Heliocentric
Earth Ecliptic coordinates (HEE), (d) Wind magnetic field data in HEE
coordinates, and (e) the Wind proton speed and density (solid lines) and the
WSA/HUX solar wind speed at Earth and Solar Orbiter (dashed lines). Vertical
lines indicate from left to right the shock arrival time and the beginning and
end times of the flux rope, determined visually.

Figure 1 gives an overview of spacecraft locations and in situ magnetic field
and plasma data, taken by the Solar Orbiter magnetometer (MAG; Horbury et al.,
2020), the magnetometer (MAG) on the Mercury Planetary Orbiter (MPO) of the
BepiColombo spacecraft (MPO-MAG; Glassmeier et al., 2010; Heyner et al.,
2020), and the Magnetic Field Investigation (MFI; Lepping et al., 1995) and
Solar Wind Experiment (SWE; Ogilvie et al., 1995) of Wind.

Table 1 summarises the arrival times of the shock, the start of the magnetic
obstacle, which in this case is a clean flux rope with a rotating magnetic
field, the end of the magnetic obstacle, the spacecraft position at those
times and basic ICME parameters for the sheath, magnetic obstacle and the full
ICME interval, consisting of the sheath and the magnetic obstacle (Rouillard,
2011) combined. The Solar Wind Analyzer (SWA; Owen, C. J. et al., 2020)
instrument onboard Solar Orbiter was not yet fully commissioned during the
time period the CME occurred, and therefore solar wind plasma data were only
available at Wind, where it arrived with a low mean ICME speed of 346 km s-1.
The selected boundaries will be discussed in more detail in the results
section.

At the ICME shock arrival time on 2020 April 19 05:06 UT, Solar Orbiter was
positioned at a heliocentric distance of 0.809 AU, and at $-4.02$∘ longitude
and $-3.94$∘ latitude in Heliocentric Earth Equatorial (HEEQ) coordinates.
Wind was near the Sun–Earth L1 point at 0.996 AU, 0.19∘ longitude, and -5.15∘
latitude at the shock arrival time on April 20 01:34 UT, about 20.5 hours
after the shock arrived at Solar Orbiter. The shock then reached BepiColombo
on April 20 03:09 UT, about 1.5 hours after it impacted Wind, when BepiColombo
was located at 1.011 AU, $-1.25$∘ longitude and $-5.53$∘ latitude, still near
Earth a few days after a flyby event. Additionally, STEREO-A was imaging the
event from -75.1∘ longitude, which is an exceptionally well suited viewpoint
for heliospheric imaging of an Earth directed CME.

Figure 1a shows the ambient solar wind conditions in the ecliptic plane on
2020 April 20 00:00 UT calculated with the Wang-Sheeley-Arge/Heliospheric
Upwind eXtrapolation (WSA/HUX) model (Reiss et al., 2019, 2020). The WSA/HUX
model is based on magnetic maps of the photospheric field from the Global
Oscillation Network Group (GONG) provided by the National Solar Observatory
(NSO). Specifically, we use the magnetic maps as an inner boundary condition
to the Potential Field Source Surface (PFSS; Altschuler & Newkirk, 1969) and
the Schatten current sheet model (SCS; Schatten, 1971) to compute the large-
scale coronal magnetic field. Using the global magnetic field topology, we
specify the solar wind conditions near the Sun based on the established Wang-
Sheeley-Arge relation (WSA; Arge et al., 2003). To evolve the solar wind
conditions from near the Sun to the Earth, we use the Heliospheric Upwind
eXtrapolation model (HUX; Riley & Lionello, 2011; Owens & Riley, 2017; Reiss
et al., 2020) which simplifies the fluid momentum equation as much as
possible. Figure 1a shows the computed large-scale solar wind conditions in
interplanetary space. The time interval under scrutiny reveals typical solar
wind conditions during solar minimum with extended polar coronal holes and
streamers confined within a narrow latitude band near the equator. Our model
results show that a higher speed stream in the ambient solar wind is expected
to arrive right after the ICME impacts Earth, and indeed it is seen in the
solar wind speed time series at Wind (Figure 1e) that this higher speed stream
interacts with the rear of the ICME when it passes 1 AU. It is also seen that
a rise in the HUX model speed matches to a good degree the observed rise in
the in situ speed at Wind.

Looking at the magnetic field data in Figure 1b-d, we can see that when each
of the three spacecraft are in the magnetic obstacle (between the second and
third vertical lines), the normal component $B_{n}$ or $B_{z}$ of the magnetic
field is bipolar and turns from negative to positive, and the $B_{t}$ or
$B_{y}$ component is mostly unipolar and negative, which in the HEE (used for
the BepiColombo and Wind data) and Radial-Tangential-Normal (RTN) coordinate
systems (used for Solar Orbiter) points towards solar east (note that
differences in the axis directions are small between HEE and RTN). The
unipolar component $-B_{t}$ or $-B_{y}$ (east) is also the direction of the
flux rope axial field. This field rotation corresponds to a left-handed south-
east-north (SEN) type of flux rope (Bothmer & Schwenn, 1998; Mulligan et al.,
1998), which is of left-handed chirality and has a low inclination of the flux
rope axial field to the solar equatorial plane.

The event is essentially a triple radial ICME lineup event observed by Solar
Orbiter, Wind, and BepiColombo. The spacecraft separation between Solar
Orbiter and Wind of 0.19 AU makes it possible to study the radial evolution of
the ICME flux rope, as the difference in longitude from Wind of 4.2∘ is well
below the estimated full longitudinal extent of ICME flux ropes of 60∘ (e.g.
Bothmer & Schwenn, 1998; Good & Forsyth, 2016). BepiColombo, with a 1.3∘
difference in longitude and 0.015 AU in radial distance from Wind, is too
close to Wind to see significant differences in the large-scale shape or
structure of the flux rope (Lugaz et al., 2018). However, this does help in
setting flux rope boundaries for Solar Orbiter and Wind to identify its
coherent part, and is very well suited for multi-point studies of small scale
features of the shock and sheath (Kilpua et al., 2021, in prep.).

###  2.2 Heliospheric imaging

Figure 2: Evolution of the CME in STEREO-A HI1. Upper three panels: Running
difference images of the CME, with green arrows indicating the CME leading
edge. The ecliptic plane corresponds to the horizontal centre of the image,
and the elongation angles marked on the x-axis pertain to the ecliptic. The
orange crosses mark the extent of the cavity or void, the black lines indicate
the angular width of the flux rope. Lower four panels: The propagation of the
CME leading edge in the ecliptic plane modelled with an ELEvoHI ensemble
simulation, shown for different times, up to the impact of the CME at Solar
Orbiter (green square) and Earth. The red tangent corresponds to the
elongation measurements from HI.

Figure 2 shows the evolution of the CME in STEREO-A HI1 images. The first HI1
observation of this CME is at 2020 April 15 20:49 UT, when the leading edge is
already at around 10∘ elongation. Due to a data gap, no earlier HI1
observations from this event are available. The leading edge of the CME in HI1
images is relatively faint compared to the core following the cavity, which
makes accurate tracking of the leading edge of the density structure quite
challenging. At 2020 April 16 18:09 UT, the leading edge enters the HI2 field
of view at 18.8∘ elongation. Within HI2, the Milky Way is the most prominent
feature hindering the tracking of the faint CME front. However, the flattening
of the CME cross-section during evolution is perfectly captured by HI1 and
enables measurement of the evolution of the aspect ratio of the cross-section
for about 22 hours.

The density pile up at the back of the flux rope from the faster solar wind
stream helps in increasing the density of the shell wrapping around the flux
rope, and thus highlighting its shape in HI images. The CME is clearly viewed
as edge-on by HI (e.g. Thernisien et al., 2006; Wood & Howard, 2009), which
means that the observer looks along the flux rope axis direction. We note that
this already gives a seldom made, direct and consistent connection between the
CME shape observed in HI observations and in situ observations (cf. Möstl et
al., 2009), where the SEN flux rope type exhibits a low inclination to the
solar equatorial plane, consistent with the edge-on view.

We ran ensembles of the ELEvoHI model (Rollett et al., 2016; Amerstorfer et
al., 2018) based on the tracked elongation values of the CME front in the
ecliptic plane in HI1 and HI2 images. ELEvoHI models the evolution of the CME
leading edge in the ecliptic plane, which is in this case the CME shock, as an
ellipse that reacts to the ambient solar wind via drag forces (Vršnak et al.,
2013). ELEvoHI was set up using the Ecliptic cut Angles from the graduated
cylindrical shell (GCS) model for ELEvoHI tool (EAGEL; Hinterreiter et al.,
2021), an ambient solar wind speed range of $425\pm 200$ km s-1 (for more
information on that approach see Amerstorfer et al., 2020) and a total number
of ensemble members of 220. The number of ensemble members comes from running
all possible combinations of the range of input parameters (propagation
direction, half-width, and inverse ellipse ratio) and the step size used. This
results in a median arrival time error between the model results and the
observations of $1.8\pm 1.9$ hours (Solar Orbiter) and $-3.3\pm 1.9$ hours
(Earth), demonstrating the unambiguous connection between the imaging and in
situ observations. The consistency in modelled arrival time at both spacecraft
also demonstrates that no large local distortions of the shock front from the
ellipse shape in the ecliptic plane are needed to adequately model the CME
propagation between Solar Orbiter and Earth.

A comparison to the results of the graduated cylindrical shell (GCS;
Thernisien et al., 2006) model by von Forstner et al. (in revision at A&A,
2021) also gives a good overall consistency of the HI results with the HEEQ
longitude of $18\pm 6$∘ given by GCS fitting, which used both STEREO COR2 and
LASCO C2 coronagraphs. Limiting the ELEvoHI ensemble to the members with an
exact agreement of modelled and observed arrival time at Earth, that is $\pm
0.0$ hours, results in an HEEQ longitude of $5\pm 2$∘ and an angular width
(within the ecliptic) of $70\pm 10$∘ given by the ELEvoHI model. These results
are broadly consistent with the shock normal at Wind, calculated by Kilpua et
al. (2021) using the mixed mode method which resulted in a HEEQ longitude of
-1.8∘ . The shock normal points radially away from the Sun and is therefore
consistent with a central impact of the shock at Earth.

###  2.3 Flux rope modelling

In this study, we use three variations of magnetic flux rope models to
interpret the large-scale structure of the ICME from the in situ magnetic
field profile. As the three models make different assumptions about the flux
rope morphology, comparing their correspondence with the observations will
enable a greater understanding of the physical processes affecting the ICME
evolution.

The first two models are variants of the kinematically-distorted flux rope
(KFR) technique (Owens et al., 2006). Here, the flux rope is assumed to have a
circular cross-section close to the Sun, thus beginning life as a force-free,
constant-$\alpha$ solution (Lundquist, 1950). Initial CME shapes can vary due
to factors such as the underlying magnetic field geometry (Krall & Cyr, 2006),
however, the assumed circular cross-section is similar to that observed in
STEREO COR1-A images of the CME (not shown). The cross-section is then
kinematically distorted by radial propagation at the ambient solar wind speed
and cross-sectional expansion of the flux rope from internal pressure. This
leads to a ‘pancaking’ of the flux rope cross-section (Riley & Crooker, 2004)
that is sometimes inferred from HI observations (Savani et al., 2010). Two
forms of this KFR model are considered: a KFR in a uniform solar wind (Owens
et al., 2006) and a KFR in a solar-minimum-like solar wind, wherein slow solar
wind occurs at low latitude and fast wind at high latitude (Owens, 2006).

For the KFR models, global parameters, such as the flux rope helicity and axis
orientation, are determined by minimising the mean squared error (MSE) between
the observed and model magnetic field components for a range of model
solutions. The start and end boundaries can be free parameters in this
minimisation process, but that significantly increases parameter space and the
possibility of finding only a local minimum. Instead, here we identify (by
eye) discontinuities in the observed magnetic field profile and assess which
boundaries produce the maximum eigenvalue ratios in the variance directions
(see Owens, 2008, for more info). This likely selects only the portion of the
flux rope which is unperturbed by strong interaction with ambient solar wind.

The third model we apply is the 3DCORE flux rope modelling technique (Weiss et
al., 2021). It uses a 3D torus-like structure, with an embedded Gold-Hoyle
like magnetic field that is based on an analytical solution derived by Vandas
& Romashets (2017). The torus-like structure is anchored on the surface of the
Sun and expands self-similarly during its propagation throughout the
heliosphere. ‘Pancaking’ or flattening of the cross-section is approximated by
using an elliptical cross-section with an aspect-ratio that is either set as
constant or a free fitting parameter. Here, we use a value of the aspect ratio
that is fixed using the HI observations as a constraint. The flux rope
expansion is modelled using a simple power law where the radius of the cross-
section expands as $r^{1.14}$ and the magnetic field decreases as $r^{-1.64}$
(Leitner et al., 2007). The model parameters are inferred using an approximate
Bayesian computation (ABC) algorithm that generates an ensemble of solutions
with the root-mean-square error (RMSE) serving as a summary statistic. The
fitting is performed on a variable interval within two fixed boundaries which
are chosen by manual inspection. In order to compare 3DCORE with the two other
models these boundaries were chosen to be very similar. For a more detailed
description of the model and the fitting procedure we refer the reader to
Weiss et al. (2021).

For KFR, the model time series is generated by time evolving the flux rope
structure past a fixed point in the heliosphere, approximating a spacecraft
encounter with a magnetic cloud. For 3DCORE, the simulation includes a moving
observer with the CME propagating within an inertial reference frame. As both
KFR and 3DCORE models include flux rope expansion, this leads to an asymmetric
magnetic field time series.

##  3 Results

###  3.1 Radial evolution

The small longitudinal (within 5∘) and latitudinal separation between Solar
Orbiter, Wind, and BepiColombo during the passage of the CME provides an
opportunity to study the radial evolution of the CME using in situ
measurements of the magnetic field. However, caution must still applied when
interpreting results as significant differences in CME magnetic flux rope
properties have previously been observed over small longitudinal separations
(Kilpua et al., 2011; Winslow et al., 2016; Davies et al., 2020) and therefore
differences in observations and properties may also be due to spatial
variation of the CME. Similarly to the KFR and 3DCORE models, this analysis
focuses on the section of the flux rope which is unperturbed (the UMFR) by the
interaction with the following higher speed stream and the physical processes
occurring towards the leading edge (Kilpua et al., 2021, in prep.).

Figure 3: Magnetic field measurements at Wind (black), overlaid with the
scaled magnetic field measurements of Solar Orbiter (red) and BepiColombo
(blue). The vertical solid line indicates the time the shock front was
measured at Wind, and the dashed lines constrain the flux rope. The vertical
dotted lines within constrain the UMFR, where the magnetic field measurements
are well correlated between the three spacecraft.

Figure 3 compares the magnetic field profiles at Solar Orbiter, Wind, and
BepiColombo. The UMFR has been selected by considering matching features
towards the leading and trailing edges in the magnetic field data from the
three spacecraft. We compare the durations between different features at each
spacecraft to scale the times of the magnetic field time series and maximise
the Pearson correlation coefficients. For scaled Solar Orbiter and Wind data
the correlation coefficient was found to be 0.87, and for scaled BepiColombo
and Wind data the correlation coefficient was found to be 0.80. The start and
end times of the identified UMFR are given in Table 1. Comparing the UMFR
duration at Solar Orbiter and BepiColombo to that of Wind gives a scaling
factor of 0.998 and 0.901, respectively. The scaled times for each spacecraft
are aligned with the start of the UMFR, delineated by the first dotted line in
Figure 3. The scaling factor between Solar Orbiter and Wind is close to unity,
which would indicate that the UMFR has undergone little to no expansion as it
propagated beyond 0.8 AU to L1.

The magnetic field strengths of the flux rope at Solar Orbiter and BepiColombo
have been scaled using a power law relationship found by fitting the observed
maximum magnetic field strength against heliocentric distance at each
spacecraft. The magnetic field of the UMFR achieved maximum values of
$B_{max}$ = 21.2, 16.3, and 16.1 nT and mean values of $B_{mean}$ = 18.9,
14.9, and 14.6 nT at Solar Orbiter, Wind, and BepiColombo, respectively.
Considering the location of each spacecraft, we find the relationships of
maximum and mean magnetic field strength with increasing heliocentric
distance, $r$, to be $B_{max}\propto$ $r^{-1.24\pm 0.50}$ and
$B_{mean}\propto$ $r^{-1.12\pm 0.14}$, respectively. Previous studies that
have used similar fitting to derive relationships of magnetic field strength
with heliocentric distance include Leitner et al. (2007); Gulisano et al.
(2010); Winslow et al. (2015), and Good et al. (2019). Both Leitner et al.
(2007) and Gulisano et al. (2010) used Helios 1 and 2 observations; Gulisano
et al. (2010) considered the mean magnetic field to obtain $\langle
B\rangle\propto r^{-1.85\pm 0.07}$ and Leitner et al. (2007) fitted magnetic
field observations to obtain an axial field relationship of $B_{0}\propto
r^{-1.64\pm 0.40}$ for CMEs observed in the inner heliosphere. Good et al.
(2019) considered 18 CMEs observed by radially aligned spacecraft including
MESSENGER, Venus Express, STEREO A/B, and Wind to find the axial field
strength relationship, $B_{0}\propto r^{-1.76\pm 0.04}$, and similarly,
Winslow et al. (2015) considered CMEs observed by MESSENGER and ACE to obtain
a relationship of $\langle B\rangle\propto r^{-1.95\pm 0.19}$. Neither the
$B_{max}$ nor $B_{mean}$ relationship calculated in this study agree with
previous relationships obtained suggesting that this event is an outlier to
overall trends, perhaps due to a more complex interaction with the solar wind
and distortion of the flux rope and the possible spatial variation of the
magnetic flux rope over the small longitudinal/latitudinal separation between
spacecraft.

For a self-similarly expanding cylindrical force-free magnetic flux rope, one
would expect the axial field to decrease as $r^{-2}$ and the poloidal field to
decrease with distance as $r^{-1}$ (Farrugia et al., 1993). Calculating the
RMSE between the scaled Solar Orbiter and Wind time series over the UMFR
interval for the ideal axial and poloidal field relationships results in
errors of 1.98 and 1.97, respectively. By minimising the RMSE between the time
series, the axial field scales as $r^{-1.52}$ and the poloidal field scales as
$r^{-0.75}$, with RMSE values of 1.75 and 1.92, respectively. This result
implies that the expansion of the CME is not self similar nor cylindrically
symmetric, and that the twist distribution changes as the CME propagates.

Using the solar wind speed time series at Wind, we can assess the local
expansion of the CME at 1 AU and compare this to the global expansion of the
CME. The dimensionless expansion parameter (for more detail, see Démoulin &
Dasso, 2009; Gulisano et al., 2010) given by Equation 1 provides a measure of
local expansion, typically around 0.8 for non-perturbed magnetic clouds
(Gulisano et al., 2010). The speed at the mid-point of the UMFR, or the cruise
velocity, $V_{c}$ is measured to be 343 km s-1 and the rate of expansion,
$\Delta V/\Delta t$, or in this case, $\Delta V_{UMFR}/\Delta_{UMFR}=7.5\times
10^{-4}$ km s-2. The heliocentric distance, $d$, of Wind at the mid-point of
the flux rope is 0.996 AU.

$\zeta=\dfrac{\Delta V}{\Delta t}\dfrac{d}{V_{c}^{2}}.$ (1)

Gulisano et al. (2010) found that the dimensionless expansion parameter is
independent of radial distance and therefore could be used to relate to the
global expansion of CMEs with the magnetic field strength. The magnetic field
strength would be expected to decrease as $r^{-2\zeta}$ and radial size
expected to increase as $r^{\zeta}$. For the UMFR at Wind, we find that
$\zeta$ = 0.95, and therefore would expect the magnetic field strength to
decrease with heliocentric distance as $r^{-1.91}$ and the size of the UMFR to
scale as $r^{0.95}$.

The expansion velocity at Wind, calculated as the difference between the
velocities at the front and the back of the UMFR is $\Delta V_{UMFR}$ = 39.8
km s-1, and therefore assuming the expansion speed remained constant between
Solar Orbiter and Wind, the radial diameter of the flux rope would be expected
to increase by 0.02 AU or as $r^{0.87}$. The observed relationship for the
radial expansion agrees well with the expected relationship using the
dimensionless expansion parameter, however, the magnetic field strength
relationships are not in agreement ($r^{-1.91}$ using the dimensionless
expansion parameter, in comparison to $r^{-1.12}$/$r^{-1.24}$ found using the
observed mean/maximum magnetic field). This discrepancy may be due to
compression of the flux rope as a whole by the higher speed stream following
the CME.

###  3.2 Non-radial evolution

Figure 4: Solar Orbiter magnetic field data during the ICME event, overlaid
with the three flux rope model time series. The observed 1-minute cadence
magnetic field is shown in black, with flux rope model profiles shown in
colour. Pink: The 3DCORE fit, which assumes an elliptical flux rope cross-
section. Orange: The kinematically-distorted flux rope (KFR) model, assuming a
uniform solar wind. Blue: the KFR model assuming a solar-minimum-like
latitudinal solar wind speed profile.

Figure 5: Three-dimensional reconstruction of the event using a single
representative parameter combination from the 3DCORE ensemble solution. Top
panel a): top-down view of the 3DCORE geometry within the solar system up to 1
AU with two embedded magnetic field lines (dashed). Bottom panel b): side view
of the 3DCORE geometry which shows the chosen cross-section aspect-ratio of
$2$.

We now discuss the non-radial evolution of the CME flux rope, first by fits of
three different models to the Solar Orbiter magnetic field data, and secondly
by considering the STEREO HI observations to demonstrate the amount of
flattening of the cross-section for this CME flux rope.

Figure 4 shows the three model solutions for the flux rope cross-section in
comparison to the magnetic field time series at Solar Orbiter, in RTN
coordinates. In all cases, the global flux rope parameters are reasonably
similar, with an axial field lying very close to the $-T$ direction, a left-
handed rotation of the flux rope magnetic field about the axis, and a
spacecraft encounter close to the axis.

The cross-sectional shapes of the uniform-wind KFR and 3DCORE flux ropes,
shown at the top of Figure 4 in orange and pink respectively, are generally
quite similar, particularly close to the flux rope axis. For this reason,
these two models show the same overall variations in the magnetic field
components. Firstly, the $B_{N}$ component shows a strong bipolar signature,
passing through zero at the closest approach to the flux rope axis (around Apr
19 18:00 UT). Qualitatively, this is in agreement with the observations.
Secondly, the $B_{T}$ variations are unipolar with the maximum magnitude at
the closest approach to the axis. Again, this is in qualitative agreement with
the observed variation. Thirdly, the $B_{R}$ component is significantly
smaller in magnitude than $B_{T}$ and $B_{N}$, and shows a weak unipolar
variation. This is in qualitative disagreement with the observations, which
show a dipolar variation in opposition to that of $B_{N}$. As the uniform-wind
KFR assumes cylindrical symmetry and the 3DCORE model allows for a curved flux
rope axis, the relative agreement between the two models suggests there is
relatively little contribution from axial curvature for this encounter. This
is in agreement with an approximately perpendicular spacecraft intersection
with the flux rope axis, and the closest approach of the spacecraft being near
to the axis (Owens et al., 2012).

The KFR in the solar-minimum-like solar wind shows a very different flux rope
cross-section, with a concave-outward profile resulting from faster solar wind
at high latitude than the equator. Perhaps non-intuitively, there is little
change to the $B_{T}$ and $B_{N}$ components for this model compared with the
uniform-wind KFR. The most notable difference is to the $B_{R}$ component,
where a bipolar variation is produced, in qualitative agreement with the
observed $B_{R}$. Thus, there is evidence for flux rope distortion by the
ambient solar wind.

Figure 6: Measurements of the aspect ratio $\delta$ and angular width $a_{w}$
of the CME void in STEREO-A HI1 images, plotted against heliocentric distance.
(a) Observed $\delta$ (black dots), mean and standard deviation (red solid and
dashed lines), and a blue line at $\delta=1$ indicates the value for a
circular cross–section. (b) Angular width $a_{w}$ derived from STEREO-A HI1
images (black dots), with mean and standard deviation as solid and dashed red
lines.

Figure 5 shows a 3D visualisation of a single flux rope fitting solution
generated by applying the 3DCORE model to the Solar Orbiter MAG data with a
Bayesian minimisation algorithm (Weiss et al., 2021). The representative
solution is drawn from the full ensemble by a naive attempt at estimating the
mode of the multi-dimensional posterior distribution. The agreement with the
Solar Orbiter magnetic field profile is demonstrated by the pink fit in Figure
4 which shows the generated mean magnetic field profile (solid line) and the
95% confidence interval that is computed from the ensemble (colored area).

The main results of this analysis are a direction for the flux rope axis of
$13\pm 5$∘ longitude (HEEQ), $-5\pm 5$∘ latitude (HEEQ), and a $349\pm 13$∘
inclination of the axis, with the inclination measured from 0∘ (east) towards
90∘ (south). This means the flux rope axis at Solar Orbiter is elevated
roughly 11∘ to the solar equatorial plane, with the eastern part of the flux
rope as seen from Earth slightly above this plane, and the western part below
it. However, the main result is that the flux rope axis lies almost in the
solar equatorial plane. The model flux rope radial diameter at the torus apex
is $D_{1AU}=$0.114 AU at 1 AU, and the axial field strength at 1 AU is
$B_{1AU}=14.3\pm 0.9$ nT.

The full number of magnetic field line turns is $\tau=-3.7\pm 0.6$ over the
full torus ($\tau<0$ means negative chirality), for which the magnetic field
structure is assumed as a uniform-twist field in the model (for details, see
Section 2.3). However, the assumption that the flux rope structure extends
over the full torus is unlikely to be the case (e.g. Owens, 2016). We
propagated the model that resulted from a best fit to the Solar Orbiter data
to the Earth L1 point, where, in comparison to Wind data, an inconsistency in
the timing is seen, but otherwise the modelled profiles of the magnetic field
components match the Wind observations well (see von Forstner et al., 2021,
revised at A&A). The results of the 3DCORE fitting at Solar Orbiter, Wind, and
BepiColombo are also presented by Weiss et al. (in prep. 2021).

We now proceed to show how we determined the fixed value for the aspect ratio
$\delta_{3DCORE}=2.0$ that was used for the fitting procedure, which would
otherwise be a free parameter, to show how the HI1 observations can constrain
flux rope model parameters in this event.

Figure 6a shows measurements of the flux rope aspect ratio by manual
selection, in each STEREO-A HI1 image, of the horizontal and vertical extent
of the void (see Fig. 2), which represents the flux rope (Davis et al., 2009;
Möstl et al., 2009) as it has low density and thus low intensity in HI images.
We are able to do this because the CME is seen edge-on, thus the HI1 camera
looks along the flux rope axis and we see the cross-section of the flux rope
directly in HI1 data. The elongation of each observed value in HI1 was
converted to a heliocentric distance by use of the ELEvoHI model. For ELEvoHI,
values for the longitude to the observer of 82∘, an angular width of 70∘, and
an inverse aspect ratio of the ellipse front shape of $f=0.7$ were taken (see
Amerstorfer et al., 2018, 2020). After converting to distance, we see in Fig.
6a that the HI1 observations covered 0.11 to 0.25 AU. As there is no trend
observed for $\delta$ in Fig. 6a, we simply derive a mean value of
$<\delta_{obs}>=1.68\pm 0.29$ valid for that distance range.

Given that these direct measurements of the aspect ratio $\delta$ are
constrained to distances ¡ 0.25 AU, we shall make another estimate for this
parameter that is likely more appropriate further out at 1 AU, as we include
the observed radial size for the UMFR observed in situ in the calculation.

Figure 6b presents measurements of the angular width $a_{w}$ (see Fig. 2,
upper right panel) of the CME void in STEREO-A HI1 images. Again, there does
not seem to be a consistent trend with increasing radial distance and we
simply take a mean value of $<a_{w}>=13.8\pm 1.2$∘. We note that a constant
angular width means that the CME expands self-similarly in the non-radial
direction. Assuming $a_{w}$ stays constant out to 1 AU, we calculate how the
extent in angular width would map to a distance along the circumference of a
circle with a radius of 1 AU: $d_{latitude}=2\pi/(360/a_{w})=0.261$ AU.
Dividing $d_{latitude}$ by the UMFR radial size of $d_{UMFR}=0.1209$ AU
observed at Wind, we find a $\delta_{obs;a_{w}}=1.99\pm 0.17$, with the error
arising from the standard deviation in the angular width measurement. This
value is quite similar to the previous value determined directly from the HI1
measurements, but it may be more appropriate for 1 AU given that it includes
in situ information on the UMFR radial size. Thus we arrive at a value for the
aspect ratio $\delta=2.0$, which we used in the 3DCORE ensemble runs (as
previously described).

###  3.3 Dst prediction

An ICME observed near the Sun–Earth line at $<1$ AU provides the opportunity
to study the efficacy of using a spacecraft at such a location as an upstream
solar wind monitor to predict the upcoming geomagnetic effects (Lindsay et
al., 1999; Kubicka et al., 2016). We use the PREDSTORM model (Bailey et al.,
2020) to conduct an analysis of the $Dst$ that would have been forecast at
Earth using Solar Orbiter data. This would have provided a lead time of about
20 hours, being the time for the shock to reach Earth after it impacted Solar
Orbiter. The method for calculating the $Dst$ index for geomagnetic activity
from the solar wind time series follows the same procedure used in Bailey et
al. (2020), partly based on the method from Temerin & Li (2006).

Figure 7 shows the $Dst$ predicted using Solar Orbiter data mapped to L1 as
well as the Wind L1 data for comparison. The hindcast using Solar Orbiter data
is done under the assumption that there is no L1 data to work with, and apart
from Solar Orbiter MAG data, only the CME speed derived from HI images is
used. The Solar Orbiter magnetic field data is first mapped to L1 with a
constant speed and converted from RTN to GSM coordinates for the $Dst$
calculation. Due to the lack of plasma measurements at Solar Orbiter during
this time period (including solar wind speed), the value taken was the
propagation speed of the CME from HI images with the simpler SSEF30 modelling
(339 km s-1) (Davies et al., 2012). Ideally the data would be mapped using the
varying speed across the ICME. A constant proton density of 5 cm-3 was used
throughout the ICME interval to compute the $Dst$ index.

The output from the 3DCORE modelling of the flux rope was also included to
demonstrate a forecast from only partial measurements of the flux rope. As can
be seen in Fig. 7, the $Dst$ forecast from Solar Orbiter compares well with
the observed Kyoto $Dst$, although the initial climb and the eventual depth of
the trough are both underestimated. The observed $Dst$ minimum of $-60$ nT is,
nevertheless, still closely approximated by the modelled minimum of $-47$ nT.
The timing of the $Dst$ minimum, despite a lead time of 20 hours and only an
estimate of the solar wind expansion speed to use, is off by 0.5–1 hour. The
differences in shape likely result from the use of constant speed and density
values (i.e. no defined shock front or sheath region), and could be
approximated better if the solar wind structures associated with a CME were
accounted for, even if only artificially.

Figure 7:  Prediction of $Dst$ at Earth using Solar Orbiter data mapped to L1
under the assumption of a constant CME speed (339 km s-1). The solid black
line shows the observed Kyoto $Dst$, the solid blue line the $Dst$ predicted
by the model using Wind data, while the dashed green line shows the $Dst$ that
would have been predicted using a forecast from Solar Orbiter. The dotted red
line shows the results for the flux rope magnetic field taken from 3DCORE,
which were used to predict the Dst resulting from the second half of the flux
rope observed by Solar Orbiter.

##  4 Conclusions

The CME launched on the 2020 April 15 was the first CME to be detected in situ
by Solar Orbiter on April 19. The CME was also observed in situ by both the
Wind and BepiColombo spacecraft on April 20, whilst BepiColombo was
longitudinally separated from the Earth by just 1∘. The radial separation of
0.19 AU between Solar Orbiter and Wind/BepiColombo and a longitudinal
separation of less than 5∘ has provided an excellent opportunity to study the
radial evolution of the CME.

STEREO-A was located at an exceptionally well suited viewpoint
(75.1∘longitude) for heliospheric imaging of an Earth directed CME. The source
of the CME was an almost entirely isolated streamer blowout which was pushed
from behind by a higher speed stream. This caused a density pile up at back of
the flux rope, confirmed by in situ measurements of the density at Wind. This
is evidenced in STEREO-A HI1 images by the increased intensity of the density
shell wrapping around the flux rope, highlighting the shape of the CME. A
clear flattening of the CME as it propagated has been observed in the HI1
images. The CME was viewed edge-on by HI1, thus the camera was aligned along
the flux rope axis direction. This gives a direct and consistent connection
between the CME shape observed by HI1 with magnetic field instruments in situ,
where the SEN flux rope type indicates a low inclination to the solar
equatorial plane, consistent with the edge-on view from STEREO-A. The
remarkably clear HI1 images of the CME have allowed direct measurements of the
flux rope cavity to be made. These measurements have been used to calculate an
aspect ratio of 2, further confirming the flattening of the flux rope cross-
section observed as the CME propagated from the Sun to 1 AU.

We investigate the non-radial evolution of the CME flux rope by fitting three
flux rope models to the Solar Orbiter magnetic field data. The magnetic flux
rope models each make different assumptions about the flux rope morphology to
interpret the large-scale structure of the ICME. The 3DCORE model assumes an
elliptical cross-section with a fixed aspect-ratio calculated by using the HI1
observations as a constraint. The cross-sectional shape is relatively similar
to that of the KFR model in the uniform solar wind and therefore, these two
models show similar general variations in their output magnetic field
components. Despite the cylindrical symmetry of the uniform-wind KFR model in
comparison to the 3DCORE model, the latter of which allows for a curved flux
rope axis, the agreement between models suggests that there is relatively
little contribution from axial curvature, in agreement with an approximately
perpendicular spacecraft intersection with the flux rope axis, and the
spacecraft crossing the flux rope close to its axis. The KFR model profile in
a latitudinally structured solar-minimum-like solar wind shows the most
notable differences in results, where a bipolar variation in the radial
magnetic field component is produced. This is in qualitative agreement with
the observed radial magnetic field at Solar Orbiter and thus, provides strong
evidence for flux rope distortion by the ambient solar wind.

The radial evolution of the CME has been assessed by scaling and overlaying
the magnetic field data from Solar Orbiter, Wind, and BepiColombo. The
boundaries of the CME are ambiguous, both towards the leading edge of the flux
rope due to ongoing processes and features in the sheath, and towards the
trailing edge due to the higher speed stream following the CME. Analysis
therefore focuses on the section of the flux rope which is unperturbed. The
so-called unperturbed magnetic flux rope, UMFR, has been identified
independently in both the KFR model technique where (1) boundaries in the
observed magnetic field were identified and assessed to maximise eigenvalue
ratios in the variance directions, and (2) in the magnetic field mapping
technique, where different durations between magnetic features towards the
leading and trailing edge were compared to maximise the Pearson correlation
coefficient between spacecraft time series. We find that the magnetic field
measurements are well correlated between spacecraft for the UMFR interval.
Comparing in situ observations of the magnetic field from the different
spacecraft, we find the dependence of the maximum (mean) magnetic field
strength with heliocentric distance to decrease as $r^{-1.24}$ ($r^{-1.12}$),
in disagreement with previous studies where one would expect the magnetic
field to decrease at a faster rate than observed in this study. Further
assessment of the axial and poloidal magnetic field strength dependencies
suggest that the expansion of the CME is likely not self-similar nor
cylindrically symmetric, in agreement with the evidence from the HI1 imagery
that the flux rope cross-section was distorted as it propagated due to a more
complex interaction with the solar wind.

The CME propagated close to the Sun-Earth line providing a good opportunity to
demonstrate the efficacy of using Solar Orbiter, whilst in such an ideal
location, as an upstream monitor to predict the Dst index observed at Earth.
The in situ observations of the magnetic field at Solar Orbiter have been
mapped using the PREDSTORM model, producing a relatively accurate estimate of
the Dst index, with a Dst minimum of $-47$ nT comparable to observations at
Earth of -60 nT, despite the model assumption that there was no data available
at L1.

The configuration of Solar Orbiter, STEREO-A and spacecraft near the Earth
during the propagation of the CME between 2020 April 15 and April 20 has
provided highly useful observations to determine the large-scale shape of the
CME using both in situ and remote observations in combination with modelling.
This study contributes towards our understanding of the radial and non-radial
evolution of CMEs through the inner heliosphere, whilst questioning the
traditional picture of CME morphology and flux rope model profiles. Further
studies are necessary to determine the commonality of CME flux rope distortion
and the effects of CME interaction with varying solar wind structure. As we
enter Solar Cycle 25 and solar activity increases, the Solar Orbiter mission,
as it approaches the Sun, will contribute to an exciting opportunity for
future multi-spacecraft studies utilising both in situ and remote observations
of CMEs at various radial distances.

###### Acknowledgements.

We have benefited from the availability of Solar Orbiter, STEREO, Wind, and
BepiColombo data, and thus would like to thank the instrument teams, and the
SPDF CDAWeb and Solar Orbiter (http://soar.esac.esa.int/soar/) data archives
for their distribution of data. The Solar Orbiter magnetometer was funded by
the UK Space Agency (grant ST/T001062/1). This research was supported by
funding from the Science and Technology Facilities Council (STFC) studentship
ST/N504336/1 (E.D.) and STFC grants ST/S000361/1 (T.H.) and ST/R000921/1
(M.O.). C.M., A.J.W, R.L.B, M.A.R., T. A., J.H., M.B. thank the Austrian
Science Fund (FWF): P31521-N27, P31659-N27, P31265-N27. D.H., I.R., H.-U. A.
were supported by the German Ministerium für Wirtschaft und Energie and the
German Zentrum für Luft- und Raumfahrt under contract 50 QW 1501. D.B., J.D.,
R.H. recognise the support of the UK Space Agency for funding STEREO/HI
operations in the UK.

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