# Successive interacting coronal mass ejections: How to create a perfect
storm?

G. J. Koehn Blackett Laboratory, Imperial College London, London, UK  R. T.
Desai Centre for Fusion, Space and Astrophysics, University of Warwick, UK
Blackett Laboratory, Imperial College London, London, UK  E. E. Davies
University of New Hampshire, New Hampshire, USA  R. J. Forsyth Blackett
Laboratory, Imperial College London, London, UK  J. P. Eastwood Blackett
Laboratory, Imperial College London, London, UK  S. Poedts Katholieke
Universiteit Leuven, Leuven, Belgium  Institute of Physics, University of
Maria Curie-Skłodowska, Lublin, Poland

(Accepted November 10, 2022)

###### Abstract

Coronal mass ejections (CMEs) are the largest type of eruptions on the Sun and
the main driver of severe space weather at the Earth. In this study, we
implement a force-free spheromak CME description within 3-D
magnetohydrodynamic simulations to parametrically evaluate successive
interacting CMEs within a representative heliosphere. We explore CME-CME
interactions for a range of orientations, launch time variations and CME
handedness and quantify their geo-effectiveness via the primary solar wind
variables and empirical measures of the disturbance storm time index and
subsolar magnetopause standoff distance. We show how the interaction of two
moderate CMEs between the Sun and the Earth can translate into extreme
conditions at the Earth and how CME-CME interactions at different radial
distances can maximise different solar wind variables that induce different
geophysical impacts. In particular, we demonstrate how the orientation and
handedness of a given CME can have a significant impact on the conservation
and loss of magnetic flux, and consequently Bz, due to magnetic reconnection
with the interplanetary magnetic field. This study thus implicates
identification of CME chirality in the solar corona as an early diagnostic for
forecasting geomagnetic storms involving multiple CMEs.

Magnetohydrodynamical simulations (1966), Solar coronal mass ejections (310),
Space weather (2037), Heliosphere (711), Magnetic storms (2320), Solar storm
(1526), Solar wind (1534), Space plasmas (1544), Interplanetary shocks (829)

##  1 Introduction

Coronal mass ejections (CMEs) are the largest type of eruption seen from the
solar atmosphere and are the primary cause of severe geomagnetic storms and
disturbances when they arrive at the Earth. While most geomagnetic storms are
created by the impact of a single CME, over a quarter are caused by the
interaction of multiple CMEs (Zhang et al., 2007); and it has been suggested
that the majority of fast complex ejecta in the solar wind are produced by
multiple CMEs (Burlaga et al., 2001). Several studies have examined multiple
successive interplanetary CMEs (ICMEs) and determined that preconditioning of
the solar wind, where an initial CME clears the path for a second CME (Liu et
al., 2014; Temmer & Nitta, 2015; Desai et al., 2020), and CME–CME collisions
(Shiota & Kataoka, 2016; Scolini et al., 2020), can result in a significant
increase in geo-effectivness, i.e. the severity of the impact on Earth.
Quantification of this amplification has the potential to address the question
of what upstream conditions can produce an exceptionally geo-effective event,
i.e. a ‘perfect storm’, at the Earth.

A survey of extreme space weather events reveals that the most severe events
on record have featured multiple successive CMEs. For example, the famous
Carrington event of 1859 involved multiple successive eruptions and four
nights of low-latitude aurora (Tsurutani et al., 2003), and the earlier solar
storm of September 1770 was reported to consist of repeated low-latitude
aurora for nearly 9 days (Hayakawa et al., 2017). Early events in the Space
Age also exhibited this phenomenon, including the August 1972 solar storm
which resulted in the record Sun-1 AU CME transit time of 14.6 hours (Knipp et
al., 2018), and the solar storm of March 1989, which triggered widespread
power grid failures in North America and Europe (Boteler, 2019). Further
notable storms in 2003 (Webb & Allen, 2004) and 2017 (Scolini et al., 2020)
similarly featured multiple CMEs.

For the past decade, in space weather operations, CMEs have mainly been
modeled using the cone model, which has been shown to accurately capture their
shock arrival times (Odstrčil & Pizzo, 1999; Xie et al., 2004; Xue et al.,
2005; Chané et al., 2006). This approach, however, neglects the intrinsic
magnetic field structure and thus fails to predict the magnetic
characteristics at the Earth or capture {the magnetised interactions of a CME
with another. More enhanced CME descriptions include an intrinsic toroidal
flux rope structure, adding complexity in defining initial conditions (Gibson
& Low, 1998; Titov & Démoulin, 1999; Jin et al., 2017; Török et al., 2018).
Recent studies have taken simpler approaches to simulate flux ropes in the
heliosphere utilising a spheromak approximation of a CME (Vandas et al., 1997,
1998) initialised outside of the solar corona where they can be prescribed
initial conditions commensurate with cone-based models (Kataoka et al., 2009;
Shiota & Kataoka, 2016; Verbeke et al., 2019; Singh et al., 2020). In this
study we thus utilise the spheromak CME description to theoretically evaluate
the direct interaction of successive ICMEs and assess how this can enhance CME
geo-effectiveness.

This article is structured as follows: Section 2 provides a description of the
simulation approach employed to model the solar wind and CMEs, as well as
detailing the methods employed to constrain geo-effectiveness. Section 2.5
contains the design and analysis of the parametric study simulations, followed
by a discussion of the results and future prospects in Section 4.

##  2 Magnetohydrodynamic Modelling

###  2.1 Heliospheric Set-up

In this study we use a magnetohydrodynamic (MHD) heliospheric model developed
from the astrophysical PLUTO code. A description of the heliospheric model is
provided by Desai et al. (2020) and the underlying numerics are presented by
Mignone et al. (2007). In this study, we employ the Harten-Lax-van-Leer
discontinuities (HLLC) approximate Riemann solver which is stable for highly
magnetised plasmas (David et al., 2019). In this study we self-consistently
model the solar wind evolving through the inner heliosphere as well as the
insertion and evolution of multiple CMEs.

The ideal MHD equations are solved across a physical domain extending from 0.1
AU ($\approx$21.50 solar radii ($R_{S}$)) out to 1.1 AU ($\approx$ 236.5
$R_{s}$), colatitudes $\theta$ = 30∘ to 150∘ and longitudes $\phi$ = -60∘ to
60∘. This reduction of the global heliosphere to a solid angle volume allows
us to capture all dynamics of the CMEs focused on herein, whilst reducing the
computational requirements. The static computational grid has 60 and 273 cells
along angular and radial axes, respectively, which produces a radial spatial
resolution of 0.788 $R_{S}$ and angular resolution of $2^{\circ}$. This set-up
was validated with an error convergence analysis conducted up to a resolution
3.3x greater than this (not shown). The $\phi$-boundaries, defined by the
coordinate range above, implement periodic boundary conditions to represent a
closed system in azimuth.

###  2.2 Solar Wind & Parker Spiral

To produce a heliosphere we implement a slow solar wind at the inner boundary
of 0.1 AU to represent outflow from the streamer belt located around the Sun’s
equator, and an interplanetary magnetic field (IMF) structure based upon the
Parker (1958) model. The solar wind mostly comprises electrons and protons,
but also some heavier elements, which are reflected in a mean molecular mass
of $\mu=0.6$. The model assumes the radial speed to be only a function of
radius, Faraday’s law and Maxwell’s equation $\nabla\mathbf{\cdot B}=0$ for a
spherically symmetric geometry, leading to the explicit magnetic field
components:

$B_{r}=B_{S}\Big{(}\frac{R_{S}}{r}\Big{)}^{2},~{}~{}~{}~{}~{}~{}~{}~{}B_{\theta}=0,~{}~{}~{}~{}~{}~{}~{}~{}B_{\phi}=-B_{S}\frac{\omega_{S}R_{S}}{v}\frac{R_{S}}{r}\sin\theta,$
(1)

where the magnetic field at the solar surface, $B_{S}$ = 2.5$\cdot$105 nT and
solar rotation rate, $\omega$ = 2.7$\cdot$10-6 rad s-1. At the inner boundary
(0.1 AU) the radial velocity and density are set to 300 km/s and 666.66
mp/cm3, respectively.

In order to establish a self-consistent solution to the solar wind within the
simulation domain, a simulation is set to run for 10 days. After $\approx$7
days a steady state solution is established with values at 1 AU of $v_{r}$ =
477.66 km/s, density, $\rho$ = 4.21 mp/cm3, Br = 5.46 nT, Bθ = 0.0001 nT and
$B_{\phi}$ = -2.57 nT. These values are well within acceptable conditions for
a slow solar wind and the resulting magnetic field structure shown in Figure 1
clearly shows an Archimedean Parker (1958) spiral wrapping clockwise around
the Sun when viewed from above.

###  2.3 Spheromak CME approximation

To model CMEs, we utilise a linear force-free spheromak (LFFS) (Chandrasekhar
& Kendall, 1957; Verbeke et al., 2019) to capture the intrinsic magnetic field
flux rope-like structure. Spheromak CMEs can be thought of as an approximation
of the main frontal lobe of a CME but assume spherical closure which is in
contrast to observations where CMEs are often observed to possess ellipsoidal
or croissant-like shapes with footpoints still connected to the Sun (Janvier
et al., 2013). The model is therefore more appropriate for frontal impacts as
opposed to glancing blows (Maharana et al., 2022).

The implementation and conservation of this complex magnetic field topology
through the inner boundary is non-trivial due to magnetic field divergence
considerations and in this implementation the spheromak is initialised over
multiple grid cells near the inner boundary to ensure numerical stability. The
spheromak defines a spherical region of space and the hydrodynamic quantities
are thus commensurate with those implemented for a cone based model with the
exception of the velocity which features an additional expansion due to the
magnetic pressures. The magnetic fields of the spheromak CME are expressed as

$B_{r^{\prime}}=2B_{0}\Big{(}\frac{j_{1}(\alpha r^{\prime})}{\alpha
r^{\prime}}\Big{)}\cos\theta^{\prime},~{}~{}~{}~{}~{}~{}~{}~{}B_{\theta^{\prime}}=-B_{0}\Big{(}\frac{j_{1}(\alpha
r^{\prime})}{\alpha r^{\prime}}+j_{1}(\alpha
r^{\prime})\Big{)}\sin\theta^{\prime},~{}~{}~{}~{}~{}~{}~{}~{}B_{\phi^{\prime}}=H\cdot
B_{0}j_{1}(\alpha r^{\prime})\sin\theta^{\prime}.$ (2)

Here the coordinate system, r′, $\theta^{\prime}$ and $\phi^{\prime}$, is
defined by the main axis of the spheromak and j1($\alpha r^{\prime}$) is the
vanishing spherical Bessel function of order unity with
$\alpha$r${}_{0}\approx 4.48$ to satisfy the force-free condition with the
magnetic field tending to zero at the boundary of the spheromak. The
handedness, H, defines the direction of toroidal flux and can take values of
$\pm$1.

To demonstrate and verify the implementation within the heliospheric model,
the spheromak was initialised within a local simulation with plasma conditions
representative of 0.1 AU. These conditions were chosen to closely resemble the
conditions close to the inner simulation boundary, at 0.1 AU, of the
subsequent parametric study. The input parameters correspond to those used in
the subsequent parametric study as listed as CME1 in Table 1, and a grid
resolution of 0.75 $R_{S}$ radial, 9∘ azimuthal, and 4∘ polar was used with a
second order Runge-Kutta algorithm for tracing the magnetic field lines.
Figure 2 shows the spheromak shortly after initialisation and then after 60
minutes have elapsed. The structure of the spheromak is well-preserved
although differences can be seen which are attributed to numerical diffusion
and magnetic expansion. The conservation of the global structure demonstrates
the force-free implementation of the spheromak.

###  2.4 Quantifying Geo-effectivness

To diagnose the results of Sun-to-Earth CME simulations we insert a virtual
spacecraft into the simulation domain in Figure 1 at X = 1 AU, at colatitude
$\theta=90^{\circ}$ and longitude $\phi=0^{\circ}$. Multiple latitudes and
longitudes close to the Sun-Earth line were however examined to ensure the
results were representative (not shown). This virtual spacecraft outputs
timeseries of all modeled variables and allows us to identify interplanetary
CME signatures and indicators of geo-effectiveness. The primary controllers of
geo-effectiveness stem from the solar wind velocity and density, which
controls the pressure exerted on the magnetosphere and the southward component
of the IMF, BZ, which drives magnetic reconnection and the Dungey (1961)
cycle.

####  2.4.1 Disturbance Storm Time Index

To provide a more sophisticated estimate of geo-effectiveness based upon these
key solar wind variables, we use the empirical relationship determined by Wu &
Lepping (2005) for the minimum Disturbance Storm Time (Dst) index induced by a
given CME. This provides a geo-effectiveness estimate based on the ring
current content in response to the solar wind BZ and vr. This empirical
relationship is based on measurements of 135 magnetic clouds, defined as
regions of enhanced magnetic field, a smooth rotation of the magnetic field’s
direction and a low plasma beta (Burlaga et al., 1981; Lepping et al., 1990).
This Dst relationship is written as:

$Dst_{min}=-16.48-12.89\cdot(v_{r}\cdot B_{Z})_{max},$ (3)

where $B_{Z}$ = $|B_{GSM_{z}}|$ for $(B_{GSM})_{z}$ $<$ 0 and $B_{Z}$ = 0 for
$(B_{GSM})_{z}$ $\geq$ 0\. $(B_{GSM})_{z}$ is thus equal to $(B_{GSE})_{z}$ as
outputted from the simulations for the case of zero dipole tilt. Note that
this cap of Dst is just required for this relationship: the Dst can be
positive, yet cannot be estimated with this relationship in the positive
regime. It shall be mentioned here that this relationship has been found to
have an error of less than 10% for magnetic cloud regions, yet larger errors
for CME sheath regions. This Dst relationship can be used to categorise storms
as either weak ($\leq$ –30 nT), moderate ($\leq$ –50 nT), strong ($\leq$ –100
nT), severe ($\leq$ –350 nT), and great or extreme ($\leq$ –350 nT), according
to the commonly-used classifications (Loewe & Prölss, 1997). While this model
estimates the minimum Dst, this variable is plotted over time to understand
the location of the minimum Dst. It is also based upon the solar wind electric
field which itself represents a solar wind-magnetosphere coupling function
(Newell et al., 2007). In addition to the extrema, the timeseries thus also
illustrates the time-dependent nature of the solar wind-magnetosphere
coupling.

####  2.4.2 Magnetospheric Compression

The complexity of the solar wind-magnetosphere interactions leads to a myriad
space weather impacts, which are complex to quantify with a single
parameterisation. We thus also utilise the Shue et al. (1998) model to predict
the location of the subsolar terrestrial magnetopause. The magnetopause
controls the outer boundary of the magnetosphere and its motion is a
fundamental driver of outer radiation belt dynamics. Moreover, rapid
compression inside of geosynchronous orbits can accelerate new radiation belts
across sub-drift time-scales due to the generation of a compressive front
which propagates through the magnetosphere at speeds compariable to particle
drift speeds. This process led to the highest observed electron energies of
$>$20 MeV in the inner magnetosphere following the interplanetary shock of
March 1991. (Blake et al., 1992; Horne & Pitchford, 2015). This is therefore
used as a further diagnostic of the magnetospheric and radiation belt
response. The Shue et al. (1998) model is based upon the solar wind dynamic
pressure, Dp and BZ, and the subsolar magnetopause is expressed as

$r_{MP_{SS}}=(10.22+1.89\textrm{ tanh}[0.184(B_{Z}+8.14)])(D_{p})^{-1/6.6}.$
(4)

It should be noted that large deviations have been reported inside 8 RE
(Staples et al., 2020), but the underlying dependence on dynamic pressure
provides a consistent estimate across the events studied herein.
Interplanetary shocks at the leading edge of CMEs can also trigger large-scale
oscillations of the magnetopause surface which result in large overshoots
(Freeman et al., 1995; Desai et al., 2021; Cahill & Winckler, 1992) which
might also cause the magnetopause to be temporarily compressed closer to the
Earth than indicated by this steady-state prediction.

###  2.5 Parametric Study

To efficiently progress through a large possible space of CME-CME
interactions, the parametric study was conducted linearly in three stages,
each time choosing the most geo-effective case and moving to the next stage.
The baseline scenario has been chosen to be two CMEs originating from the same
active region and propagating along the centre of the domain outward. The tilt
angle of a single spheromak around the Heliocentric Earth EQuatorial (HEEQ)
$x$-axis was first chosen to be varied within _Stage 1_. The tilt angle
changes the axis of symmetry of the spheromak (0∘ aligned with HEEQ $z$-axis,
90∘ aligned with HEEQ $y$-axis). Once the most geo-effective orientation was
identified, _Stage 2_ proceeded to model two successive CMEs to examine the
effects of CME-CME collisions. The next parameter varied was the waiting time
between the two launches to produce varying CME interaction dynamics. Finally,
the handedness of the CMEs were varied in _Stage 3_ , in order to further
explore magnetic interactions between two CMEs.

###  2.6 Input Parameters of Spheromak

The initial conditions of CME parameters can be constrained from remote-
sensing observations of CMEs and their source region on the Sun. The choice of
parameters here is based on prior estimation made by Scolini et al. (2019) for
the CME-CME event recorded on the 13/14 June 2012. The choice was thus made to
inject two identical CMEs of different velocities one after another. The
initial CME, hereafter referred to as _CME1_ , was given a moderate radial
speed $v_{radial}$ = $v_{r}$ = 500 km/s and the subsequent CME, hereafter
referred to as _CME2_ , would be given a high radial speed of $v_{r}$ = 1500
km/s. When considering magnetic expansion, these speeds correspond to
propagation speeds at the front of the CME of 723 km/s and 1,723 km/s,
respectively. The precise parameters chosen are outlined in Table 1.

##  3 Results

###  3.1 Single CME

CME1 is first individually simulated with a tilt angle $\tau=180^{\circ}$ and
handedness $H$ =+1. The global simulation outputs of density, radial velocity
and southward IMF component, BZ are shown in Figure 3 and the timeseries for
the virtual spacecraft at 1 AU is shown in Figure 4. A fast-forward shock is
visible at the leading edge of the CME and a high density sheath region
follows which contains enhanced magnetic fields components due to a pile-up of
the ambient solar wind. The shock is thus separated from the flux rope CME
where the magnetic field increases and the density falls of to a very low
depression compared to the background solar wind level. The CME exhibits a
plasma beta below unity, $\beta<1$, between 62 h and 74 h, where plasma motion
is dominated by the magnetic field. The magnetic field lines within the CME
cause the temperatures to evolve separately from the ambient solar wind and
the temperatures within the CME are therefore of the right magnitude of less
than 106 K. Importantly, the magnetic field inside the CME exhibits a smooth
rotation indicative of a flux rope and which completes the definition of a
magnetic cloud (Burlaga et al., 1981).

The estimated Dst shows a significant depression in the region of the CME and
reaches a minimum of -63 nT. This event may therefore be classified as a
moderate geomagnetic storm. The subsolar magnetopause distance is predicted to
be compressed to 8 RE following the initial impact, followed by an expansion.

###  3.2 Single CME Stage 1: Tilt Angle

The first stage of the parametric study identified the most geo-effective
single CME tilt angle. The tilt angle describes the main axis of the
spheromak, i.e. +Z in Figure 2. To this end, the magnetic field timeseries of
four simulations are shown in Figure 5 with tilt angles $\tau$ = 0∘, 90∘,
180∘, 270∘ and fixed handedness $H$ = +1. A further two cases of $\tau$ = 135∘
and 225∘ are also simulated to further resolve the primary peak/trough in the
trends. The magnetic field structure of the spheromak is strongly deformed in
the propagation from Sun to Earth, yet the magnetic field direction of the
initial spheromak in the z′-axis can still be recovered at 1 AU.

For the $\boldmath{\tau=0^{\circ}}$ tilt run, the spheromak is inserted in
upright orientation as shown in Figure 2. Thereby, one would expect the main
direction of the magnetic field lines to be in the positive $z$-direction
apart from the very centre of the spheromak. A strong positive $(B_{GSE})_{z}$
component is indeed observable in the virtual spacecraft timeseries between 54
h and 80 h. The $(B_{GSE})_{x}$ component undergoes a rotation from positive
to negative, as expected for the radial component inside the spheromak. The
$(B_{GSE})_{y}$ component however displays some deviation to the spheromaks
structure at 0.1 AU.

The $\boldmath{\tau=180^{\circ}}$ tilt run, as presented in Figures 3 and 4,
displays the converse trends to this which lead to a large and prolonged
southward-directed magnetic field between 50 h and 74 h. A significant twist
in the $\phi$-direction is also visible and appears coherent with the negative
$B_{\phi}$ component of the Parker spiral. Analogous trends are visible for
for the $\boldmath{\tau=90^{\circ}}$ and $\boldmath{\tau=270^{\circ}}$ cases.
The maximum in the selected plasma dynamical variables are shown in Figure 6,
and indicate the tilt angle of $\tau=180^{\circ}$ leads to the largest geo-
effectiveness.

###  3.3 CME-CME Stage 2: Waiting Time

The second stage of the parametric study proceeded to investigate the
interaction of two CMEs at different locations within the simulation domain.
The second CME is initialized with a higher velocity of 1,500 km/s compared to
the first CME’s velocity of 500 km/s and the second CME is thus gaining
quickly on the first CME, see Table 1. The waiting time was varied from 12-36
h. This range encompasses the two boundary cases of 1) a collision at
$\sim$0.5 AU and 2) a pure preconditioning of the solar wind by the first CME
where they subsequently collide beyond the Earth. (e.g. Temmer & Nitta, 2015)
Here a collision of the flux ropes and sheaths of the two CMEs is dubbed as a
full collision, unless otherwise specified. All spheromaks were set to have an
initial tilt angle of $\tau=180^{\circ}$, expecting this would lead to the
highest geo-effectiveness as found in Stage 1, and a handedness $H$ = +1. A
total of 7 cases (plus CME1 and CME2 individually) have been run leading to a
waiting time resolution of 4 h. Individual simulations for CME1 and CME2 allow
us to compare the enhancement induced through the interaction. To elucidate
key features of CME-CME interactions, in the following sections we describe
two distinct cases in detail which correspond to; 1) a full merge of the two
CME including their sheaths (20 h waiting time) and a merge of the magnetic
centres only (28 h waiting time), each occurring at approximately 0.9 AU just
ahead of the Earth.

####  3.3.1 20 hours: Shock-CME Interaction

The waiting time of 20 h resulted in an extended interaction of both the CME
magnetic flux ropes and sheath regions prior to reaching Earth’s position. The
sheath regions of the two CMEs merge at $\sim$0.9 AU. This simulation exhibits
the shock of the latter CME traveling within the ejecta of the first CME as
observed by Lugaz et al. (2015). The virtual spacecraft timeseries is shown in
Figure 7, with the timeseries from the individual simulations for CME1 and
CME2 also shown in dashed and dotted lines, respectively. The $z$-plane
density, radial velocity and southward magnetic field are shown in Figure 8,
30 and 45 h after launch.

The combined simulations result in a first shock arriving at the time the CME1
shock normally would have. This is however significantly greater in magnitude
and indicates that CME2 travelled faster within the reduced density wake of
CME1, effectively slip-streaming before impacting and merging with CME1. The
sheath densities, velocities and temperatures show a clear double-step
increase and that CME2 maintains an individual front despite the impact and
merge. This is also visible in Figure 8. What is most notable is the large
enhancement in the sheath density between CME2 and CME1-CME2 of +98% from 17.8
to 34.8 mp/cm3. The magnetic field is significantly enhanced within this
region, BZ also displaying a double peak structure, and reveals that both flux
ropes now arrive several hours earlier than when individually simulated and
are now located close to what would have been the high density former ‘sheath’
region for CME1. This effect is visible within Figure 8 where at 30 h the BZ
reveals two distinct adjacent CMEs, but at 45 h, the minima in BZ are located
close to the front of the combined event. What is also noticeable is that as
CME2 has travelled significantly faster, it is stretched and diffused to
encompass a greater area and, also due to the greatly enhanced combined sheath
in-front, shows as a reduced magnetic field intensity behind the second shock
compared to when simulated individually.

The Dst timeseries for the single CME in Figure 4 demonstrate the general
concept that within CMEs, both the sheath and flux rope induce Dst minima. In
Figure 7, we can now see that the CME-CME impact has resulted in two such
minima combining across the same region. The minimal Dst thus reaches -267 nT,
an 137 % increase in geo-effectiveness with respect to CME2 alone. This event
would therefore be classified as a severe storm. This Dst trend is reflected
in the underlying variables BZ which sees an 91 % increase from -11.2 nT to
-21.4 nT and vr being 340 km/s greater. The enhanced density and dynamic
pressure also results in the subsolar magnetopause being compressed to 5.32
$R_{E}$, a decrease of 25% from 7.11 to 5.32 $R_{E}$. The region following the
minima in BZ and Dst shows an extraordinary low density down to 0.15 mp/cm3.
Here the magnetic field is stretched radially, as expected for preconditioned
solar wind.

####  3.3.2 28 hours: CME-CME Interaction

The waiting time of 28 h results in a later merge of the magnetic flux ropes
at $\sim$0.9 AU. The corresponding virtual spacecraft timeseries is shown in
Figure 9 with the individual CMEs once again shown in dashed and dotted lines.

Due to the later merge, the arrival of the forward shock and sheath associated
with CME1 remains unchanged. Following this, the flux rope for CME1 is
coincident with the shock and sheath from CME2 which itself has arrived
earlier due to passing through the preconditioned solar wind behind CME1. This
results in a double peak in the density and a highly compressed magnetic field
signature. The most striking feature of this virtual spacecraft timeseries is
thus the strong and sharp increase in BZ due to strong decrease in the
$(B_{GSE})_{z}$. The minimal $(B_{GSE_{z}})$ for the CME1-CME2 timeseries is
-31 nT representing a 174 % change with respect to the CME2 timeseries. The
minimal Dst for the CME2-CME1 timeseries is predicted to be -401 nT, again
representing a 245% change and an even more severe storm than in the 20 h
waiting time scenario examine previously. The subsolar magnetopause position
is compressed by 14% from 7.11 $R_{E}$ for CME2 to nearly 6.09 $R_{E}$ for the
combined CME1-CME2. The subsolar magnetopause is not as compressed as in the
20 h merge due to a low peak in dynamic pressure. This highlights the need for
differing measures of geo-effectiveness which represent difference facets of
geomagnetic storms at the Earth.

####  3.3.3 Overarching Trends

To draw conclusions about the relationship between geo-effectiveness and
waiting time, the absolute maxima within the CME1-CME2 simulations for
selected physical variables at 1 AU are summarised in Figure 10.
To identify the type of event we distinguish between the interaction of the
CME sheath regions and the slower traveling CME flux rope centres. Here the
clearly identifiable merge of sheath regions is taken as the defining
characteristic of an interaction. The range of waiting times of 12 h, 16 h, 20
h and 24 h correspond to full merges of the CME sheath regions at $\approx$0.5
AU, 0.7 AU, 0.9 AU and just after Earth, respectively. All other runs do not
exhibit a merger within the simulation domain, extending up to 1.1 AU. Thus
one may identify the first three simulations, 12-20 h, as collision events and
the last two, 32-36 h, as mostly preconditioning cases. The events in the
middle of these exhibit features of both regimes.

The maximal and delta velocity show a plateau of a maximal velocity of
approximately 1200 km/s and a relative increase from the faster CME2 of 400
km/s, due to the low densities behind CME1 and thus lower drag forces acting
on CME2. Short waiting times do not appear to allow sufficient time for
propagation of CME1 to impact on the speed of propagation of CME2. The 36 h
waiting time case already shows a recovery of the solar wind speed toward its
nominal state.

The maximal density shows an increase up to the 20 h waiting time case due to
the sheaths merging and becoming the most concentrated just ahead of the
Earth, as is discussed in Section 3.3.2. This translates into the most
compressed magnetopause position. After 20 hours the maximal density rapidly
falls away due to the CME sheath regions no longer fully merging. The minimum
southward magnetic field component shows a smooth depression centred at 28 h.
This is due to the 28 h case, as discussed in Section 3.3.2, featuring a merge
of the magnetic flux ropes close to 1 AU which produces a highly concentrated
BZ. This effect must also occur during earlier mergers but the lack of such a
strong BZ, and indeed two distinct CME signatures in Figure 7 can be
attributed to the conversion of magnetic energy into kinetic and thermal
components (Scolini et al., 2020) or the relaxation of this concentrated flux
due to the elastic nature of CME-CME interactions previously identified by
Shen et al. (2013). For longer waiting times of 26 hours and beyond, a
recovery to the solar wind normal starts to appear, see also in Desai et al.
(2020, Figure 5) for the extension to this trend where the initial CME has no
impact on the subsequent event for a sufficiently long waiting time between
their eruptions.

The maximum BZ translates into a maximum Dst close to the 28 h case, with a
Dst of -401 nT, which would be a great or extreme storm. It must be noted here
that the Dst relationship used is only linear in velocity and is not a good
predictor for high velocity events as investigated here. The reflected trends
in the underlying variables, however, appear to justify this characterisation
of an extreme event.

###  3.4 CME-CME Stage 3: Handedness

The third and final stage of this parametric study concerned itself with the
handedness, $H$, the results of which are displayed in Table 2. The handedness
defines the sign of the $B_{\phi}^{\prime}$ component. A positive handedness
$H=+1$ results in the magnetic field lines turning counterclockwise looking
down on the spheromak and a negative handedness correspondingly a clockwise
rotation. The intention of this stage was to test whether a particular
handedness combination would result in altered structural properties of the
CMEs at 1 AU and therefore potentially a more geo-effective interaction. It
must be emphasized that the initial $(B_{GSE})_{z}$ component of the spheromak
is kept constant. A total of four simulation runs were run for the 20 hour
case to capture all $H_{1}=\pm 1$ and $H_{2}=\pm 1$ combinations, referring to
CME1 and CME2 respectively, and each CME is also simulated individually. This
20 h case captured a balance of an extended interaction and significantly
enhanced BZ and geo-effectiveness. In the following we shall refer to each run
as H:[$H_{1}$, $H_{2}$] = H:[$\pm$1, $\pm$1].

The difference between the runs is most apparent in tabular form of the
maximal values, as we show in Table 2. Even though all runs are initialized
with the same (BGSE)z structure, a large variation in the signatures at Earth
is found. A clear trend from positive to negative handedness is identified and
the handedness of the first CME has a higher influence than the handedness of
the second CME. The maximal southward magnetic field varies between -12.7 to
-21.8 nT. This is a 71% difference over which the dynamic variables show no
significant trend. The resulting Dst consequently also exhibits this same
trend, increasing from -163 to -275 nT, a 68% change. Note again that the
minimum in Dst are found in sheath/shock regions due to the merge of the high
density sheath and subsequent flux ropes. These differences in geo-
effectiveness would already change the classification of the geomagnetic storm
from strong to severe. It is thus a significant finding that the handedness of
spheromak in CME-CME interactions can have such a dramatic effect on the geo-
effectiveness.

To investigate the cause of this dramatic impact, Figure 11 shows the
$y$-plane profiles of $B_{\phi}$ and the 3D magnetic field lines projected
onto that plane. It only includes the most and least geo-effective cases,
H:[+1,+1] and H:[-1,-1] respectively at 40 h. This figure indicates the
potential cause for this significant difference in geo-effectiveness. The
H:[+1,+1] case exhibits a greater winding of magnetic field lines in the CME-
CME region in comparison to the H:[-1,-1] where in this case the CME structure
appears to merge with the joint sheath region. We therefore propose that this
behaviour may be explained as follows. For a spheromak of $\tau=180^{\circ}$,
a handedness of $H=+1$ means that the internal magnetic field lines in the
toroidal direction flow in the same sense as the Parker spiral on the radially
outward propagating side. This is also referred to as an inverse CME as
defined by Low & Zhang (2002). This results in a greater conservation of field
lines within the spheromak. This preferentially directed toroidal flux lends
stability to the magnetic ejecta and results in a prolonged compression of
field lines. A negative handedness, $H=-1$, on the other hand leads to field
lines of spheromak and Parker spiral flowing in the opposing directions on the
outward radial side. This results in greater erosion of the spheromak due to
magnetic reconnection (McComas et al., 1994; Schmidt & Cargill, 2003; Gosling
et al., 2005, 2007) and explains the de-facto erosion of magnetic flux for the
H:[-1,-1] case. The above explanation is supported by the observation that the
first CME’s handedness has a larger effect than that of the second one, see
the progressive trend in Table 2. This is because only the first CME strongly
interacts with the Parker spiral as it is compressed by the subsequent ejecta,
whereas the second CME travels within the preconditioned solar wind where the
magnetic field is more stretched in the radial direction. This is consistent
with the single CME simulations where only the faster CME has an altered BZ at
1 AU and interestingly also an altered velocity. The results of the single CME
simulations are therefore also consistent with high-resolution 2.5D MHD
simulations (Hosteaux et al., 2019, 2021) which found that the CME polarity
had an effect on the CME flux and that increased velocities resulted in
greater erosion.

##  4 Discussion & Conclusions

This study has conducted a parametric evaluation of successive interacting
coronal mass ejections in a representative heliospheric environment. We
employed a 3D heliospheric MHD model (Desai et al., 2020) and implemented a
force-free spheromak CME description (Verbeke et al., 2019). The force-free
implementation was first verified within a static simulation representative of
0.1 AU and then the CME was propagated to 1 AU where a virtual spacecraft
demonstrated the modelled CME fulfilled the criteria for a magnetic cloud
(Burlaga et al., 1981).

We then conducted a parametric evaluation of successive interacting coronal
mass ejections using CME parameters defined by Scolini et al. (2019) for the
two eruptions identified on 13-14 July 2012. Two CMEs were thus simulated, a
slow and then fast one, and the collision and the resulting characteristics at
1 AU evaluated using timeseries from a virtual spacecraft. As over 90 % of
CMEs have only been observed by a single spacecraft, this provided a clear
interpretation of the simulation results in context with the literature,
whilst the global simulations provided context to explain these timeseries.
The results at 1 AU were thus used to assess the geo-effectiveness through
quantification of the relevant solar wind variables and an empirical measure
of the Disturbance Storm Time index and subsolar magnetopause standoff
distance.

The CME tilt angle was first varied and the BZ component specified during
initialisation was found to be well conserved within this idealised
environment with a tilt angle of 180∘, thus providing the most geo-effective
event at 1 AU. The launch time between the successive CMEs was then varied to
produce full CME-CME interactions starting at 0.5 AU out to beyond the Earth,
the latter of which represents a case of pure solar wind preconditioning. The
various merges produced a diverse range of geophysical impacts with different
solar wind variables peaking for merges at different launch times. For
example, a waiting time of 20 h produced a collision at 0.5 AU and the
prolonged interaction resulted in the maximum dynamic pressures due to the
merge of the CME sheath regions. This resulted in the minimum estimated
subsolar magnetopause position of just 5.12 RE. A waiting time of 28 h
produced a less complete merge close to the Earth. This however resulted in
the two spheromaks colliding near to the Earth which produced the most
concentrated and greatest BZ signature which translated into the most geo-
effective event according to the Dst estimates. The Dst here was predicted to
be less than -400 nT and, due to similarly amplified underlying solar wind
variables, led us to classify this as an extreme event. These results thus
demonstrate how two CMEs, with an optimal waiting time between them, can
convert CMEs which would typically result in only moderate storms, into severe
and extreme storms. The final stage then took the 20 h case, due to its
prolonged interaction, and varied the handedness, also referred to as
chirality, of the CMEs to determine whether this induced any effect at 1 AU.
While the handedness only had a negligible effect for a single CME, the CME-
CME interaction simulations showed a clear trend of increasing BZ, and thus
Dst, for handedness changes from H=+1 to H=-1. This was attributed to a
positive handedness resulting in the spheromak field lines pointing in the
opposite direction to the Parker spiral which resulted in erosion of the CME’s
magnetic structure due to magnetic reconnection.

Magnetic reconnection in the solar wind was first reported to exist upstream
of CMEs (McComas et al., 1994; Gosling et al., 1995; Schmidt & Cargill, 2003),
and indeed potentially at the interface between two magnetic clouds and
further thin current sheets associated with large changes in the magnetic
field direction (Gosling et al., 2005). Since then, reconnection associated
with CMEs has been identified by several studies (e.g. Ruffenach et al., 2012;
Lavraud et al., 2014). The findings of this study build upon these results and
highlight that within CME-CME interactions, this process can potentially have
an enhanced effect on the global CME structures and thus CME geo-effectiveness
at the Earth. The explanation that we propose here highlights that early
identification of the chirality of a given CME in the solar atmosphere (e.g.
DeForest et al., 2017; Palmerio et al., 2017) presents a potentially
significant diagnostic for identifying geo-effective events. It is important
to note, however, that magnetic reconnection in ideal MHD codes is dependent
on numerical resistivity. While this has been shown to be appropriate for
simulations of the global dynamics of the Earth’s magnetosphere (Toffoletto &
Siscoe, 2009) which exists within a comparable range of plasma betas and
Lundquist numbers to the CME simulations conducted herein, such an
approximation of kinetic physics might well influence the results. Further
studies of the microphysics of the interaction between magnetised solar wind
structures and the ambient IMF are therefore required to further understand
the large-scale trends discovered herein.

This effect of large-scale magnetic field erosion appeared dominant for the
leading CME which predominantly interacted with the Parker spiral. The final
stage of this study thus found that the handedness of a CME within a CME-CME
event can alter the geo-effectiveness from a moderate to severe event.

This parametric study of CME-CME interactions demonstrates how two moderate
CME events, which individually induce an estimated Dst of just -63 nT can
combine to increase their characteristics and geo-effectiveness. The variety
of possible merges examined induced a range of Dst estimated up extreme cases
of less than -400 nT. The complexities of the interactions between the Sun and
the Earth, highlight the difficulties in providing accurate long-term space
weather forecasts based on the solar corona and highlight the need for self-
consistent physics-based modelling approaches to capture the magnetised
interactions within our heliosphere.

## Acknowledgements

GJK carried out this masters research project at Imperial College London. RTD
acknowledges an STFC Ernest Rutherford Fellowship ST/W004801/1, and NERC
grants NE/P017347/1 and NE/V003062/1. EED is supported by NASA grant
80NSSC19K0914. JPE was supported by NERC grant NE/V003070/1. SP acknowledges
support from the projects C14/19/089 (C1 project Internal Funds KU Leuven),
G.0D07.19N (FWO-Vlaanderen), SIDC Data Exploitation (ESA Prodex-12), and
Belspo project B2/191/P1/SWiM. This study used the Imperial College High
Performance Computing Service (doi: 10.14469/hpc/2232).

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Figure 1: Density profile and magnetic field lines (red) in the steady-state heliosphere from (a) the equatorial plane and (b) showing both the meridional and equatorial planes.  Figure 2: Magnetic field lines of (a) the initialised spheromak and (b) after 60 min of evolution, in static heliospheric conditions representative of conditions 0.1 AU. The colour is proportional to the normalised magnetic field magnitude. Table 1: Parameters chosen as the base CME for LFFS model in this paramedic study. Parameters in light grey cells are fixed throughout the study. Parameters in bold have been adapted directly from observational estimations by Scolini et al. (2019) for realistic modelling. All plain cells are varied throughout _Stages 1-3_. $\dagger$ indicates a derived quantity not directly inputted, i.e. implied by other parameters.  | Variable | CME1 | CME2
---|---|---|---
| Insertion time of CME [in h] | 0 | 12-36
LightGrey | Initial radius ${r_{0}}$ [in $R_{S}$] | 10.5 | 10.5
LightGrey Initial position | Radial $r_{CME}$ |  0.1 AU + $r_{0}$ |  0.1 AU + $r_{0}$
LightGrey | Polar $\theta_{CME}$ |  90∘ |  90∘
LightGrey | Azimuthal $\phi_{CME}$ |  0∘ |  0∘
LightGrey Velocity | Total $v_{3D}^{\dagger}$ [in km/s] | 723 | 1723
LightGrey | Radial $v_{r}$ [in km/s] | 500 | 1500
LightGrey | Magnetic expansion ${v_{exp}^{\dagger}}$ [in km/s] | 223 | 223
LightGrey Magnetic field | Field strength ${B_{0}}$ [in nT] | 1400 | 1400
| Tilt angle $\tau_{CME}$ |  0-270∘ |  0-270∘
| Handedness $H$ | +1 (or -1) | +1 (or -1)
Figure 3:  CME1 Spheromak simulation with initial tilt angle $\tau$ = 180∘ and handedness $H$ = +1, 30 h after launch. (a) Shows the normalised density, (b) the radial velocity and (c) (BGSE)z in the equatorial plane. Further CME1 parameters are specified in Table 1.  Figure 4: Virtual spacecraft timeseries data at 1 AU of CME1 shown in Figure 3. CME1 has a tilt angle $\tau$ = 180∘ and handedness $H$ = +1. Further parameters for CME1 are specified in Table 1. The blue shaded region indicates the region of the shock and sheath and the yellow shaded region the ejecta/flux rope. Figure 5: Magnetic field timeseries at 1 au for CME1 for different initial tilt angles.  Figure 6: Maximal values for the range of tilt angles CME1 is simulated for. (a) shows vr, (b) shows the density, $\rho_{r}$, (c) the dynamic pressure, Pram, (d) $(B_{GSE})_{z}$, (e) the predicted Dst and (f) the predicted subsolar magnetopause standoff distance, Rmp.  Figure 7: Virtual spacecraft data at for the CME1-CME2 interaction with waiting time of 20 h between their respective launches. This corresponded to a merge of the CME sheaths at approximately 0.9 AU. The timeseries in solid blue, dotted yellow and dashed green refer to the signatures of CME1-CME2, CME1, and CME2, respectively. Figure 8:  CME1-CME2 interaction with waiting time of 20 h between their launches. (a) shows the normalised density, (b) the radial velocity and (c) the (BGSE)z in the equatorial plane. Figure 9: Virtual spacecraft data at for the CME1-CME2 interaction with waiting time of 28 h between their respective launches. This corresponded to a merge of the CME flux ropes at approximately 0.9 AU. The timeseries in solid blue, dotted yellow and dashed green refer to the signatures of CME1-CME2, CME1, and CME2, respectively. Figure 10:  Maximal values for the range of waiting times CME1 and CME2 are simulated for. (a) shows vr, (b) shows the density, $\rho_{r}$, (c) the dynamic pressure, Pram, (d) $(B_{GSE})_{z}$, (e) the predicted Dst and (f) the predicted subsolar magnetopause standoff distance, Rmp. Table 2: Summary of extrema of handedness CME-CME runs, accompanied by single CME runs of CME1 and CME2 for comparison. Handedness | Max/Min values
---|---
$H_{1}$ | $H_{2}$ |  $(\rho)_{max}$ [mp/cm3] |  $(v_{r})_{max}$ [km/s] |  $((B_{GSE})_{z})_{min}$ [nT] |  $(Dst)_{min}$ [nT] |  $(R_{mp})_{min}$ [Rmp]
1 | 1 | 33.7 | 1144 | -21.8 | -275 | 5.23
1 | -1 | 33.0 | 1166 | -20.1 | -243 | 5.38
-1 | 1 | 34.4 | 1132 | -17.6 | -212 | 5.59
-1 | -1 | 32.4 | 1180 | -12.7 | -163 | 5.80
1 | - | 12.0 | 609.9 | -6.93 | -62.5 | 8.34
-1 | - | 12.8 | 611.8 | -6.97 | -62.0 | 8.24
- | 1 | 17.3 | 866.9 | -10.7 | -111.0 | 7.06
- | -1 | 17.7 | 820.2 | -12.0 | -119.0 | 7.15
Figure 11: Meridional plane profiles and selected magnetic field lines for the
collision of CME1 and CME2, 40 h after CME1 initialisation. Each handedness
variation is shown (a-b) for a waiting time of 20 h and tilt angles of $\tau$
= 180∘.


Further author information: (Send correspondence to Gilles Orban de Xivry)
G. Orban de Xivry: E-mail<EMAIL_ADDRESS>
# ALF: an asymmetric Lyot wavefront sensor for the ELT/METIS vortex
coronagraph

G. Orban de Xivry  Space sciences, Technologies, and Astrophysics Research
(STAR) Institute, Université de Liège, allée du Six Août 19c, 4000 Liège,
Belgium  O. Absil  Space sciences, Technologies, and Astrophysics Research
(STAR) Institute, Université de Liège, allée du Six Août 19c, 4000 Liège,
Belgium  C. Delacroix  Space sciences, Technologies, and Astrophysics Research
(STAR) Institute, Université de Liège, allée du Six Août 19c, 4000 Liège,
Belgium  P. Pathak  Space sciences, Technologies, and Astrophysics Research
(STAR) Institute, Université de Liège, allée du Six Août 19c, 4000 Liège,
Belgium  Indian Institutes of Technology, Kanpur, India  M. Quesnel  Space
sciences, Technologies, and Astrophysics Research (STAR) Institute, Université
de Liège, allée du Six Août 19c, 4000 Liège, Belgium  T. Bertram  Max-Planck-
Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany

###### Abstract

Non-common path quasi-static and differential aberrations are one of the big
hurdles of direct imaging for current and future high-contrast imaging
instruments. They increase speckle and photon noise thus reducing the
achievable contrast and lead to a significant hit in HCI performance. The Mid-
infrared ELT Imager and Spectrograph (METIS) will provide high-contrast
imaging, including vortex coronagraphy in L, M and N bands, with the ultimate
goal of directly imaging temperate rocky planets around the nearest stars.
Ground-based mid-infrared observations are however also impacted by water
vapor inhomogeneities in the atmosphere, which generate additional chromatic
turbulence not corrected by the near-infrared adaptive optics. This additional
source of wavefront error (WFE) significantly impacts HCI performance, and
even dominates the WFE budget in N band. Instantaneous focal plane wavefront
sensing is thus required to mitigate its impact. In this context, we propose
to implement a novel wavefront sensing approach for the vortex coronagraph
using an asymmetric Lyot stop and machine learning. The asymmetric pupil stop
allows for the problem to become solvable, lifting the ambiguity on the sign
of even Zernike modes. Choosing the Lyot plane instead of the entrance pupil
for this mask is also not arbitrary: it preserves the rejection efficiency of
the coronagraph and minimizes the impact of the asymmetry on the throughput.
Last but not least, machine learning allows us to solve this inversion problem
which is non-linear and lacks an analytical solution. In this contribution, we
present our concept, our simulation framework, our results and a first
laboratory demonstration of the technique.

###### keywords:

high-contrast imaging, coronagraph, mid-infrared imaging, observational, focal
plane wavefront sensing

##  1 INTRODUCTION

The Mid-infrared ELT imager and spectrograph (METIS) is a first generation
instrument of the Extremely Large Telescope currently under construction at
Cerro Armazones. It will provide L, M, N band imaging and high dispersion
spectroscopy, with high-contrast capabilities. METIS recently passed its final
design review. It is now well in its construction phase, and is expected to
see its first light towards the end of the decade[1].

Among a wide array of science topics, METIS is expected to bring major
breakthroughs in exoplanet detection and characterisation, and in our
understanding of protoplanetary disks. To this end, METIS is specifically
designed with high-contrast imaging capabilities in mind, including a high-
performance single-conjugated adaptive optics module (see e.g. Bertram et
al.[2]) and several advanced coronagraphic concepts (see e.g. Absil et
al.[3]): vortex coronagraphs to reach the smallest inner working angle and
high throughput, and grating-vector apodising phase plates implemented for
more robustness to aberrations.

The HCI performance is influenced by a large range of instrumental and
environmental effects, which has been the subject of detailed analyses[4, 5].
Overall, non-common path aberrations (NCPA) are expected to play a major role
in the overall performance budget, and our abilitiy to measure and correct
them will be a key driver for the final HCI performance. In the framework of
METIS, we can distinguish two main time-varying NCPA contributors: i) quasi-
static aberrations essentially due to slight variation of the beam footprint
on the optics, ii) dynamical aberration caused by water vapor in the
atmosphere. Indeed water vapor becomes increasingly dispersive at infrared
wavelengths leading to differential aberrations between the METIS SCAO working
at K band and the science channel working at L, M or N band. Specifically, we
expect about 300nm rms wavefront error due to water vapour seeing at
11.5$\mu$m. We refer to Absil et al.[6] for thorough discussion of water
vapour seeing effect and how it is modeled for METIS. Because we expect water
vapour to be displaced by wind, similarly to the dry air seeing, this
contribution is highly dynamic and requires an instantaneous focal plane
wavefront sensing approach to effectively mitigate it.

In this contribution, we propose to implement a novel wavefront sensing
approach for the vortex coronagraph of METIS using an asymmetric Lyot stop and
machine learning. In Sec. 2, we shortly introduce our concept. In Sec. 3, we
present the simulation framework we have implemented for this work. In Sec. 4,
we present several simulation results for classical imaging, vortex
coronagraph, and our first laboratory results. We conclude in Sec. 5 with our
perspectives.

##  2 ASYMMETRIC LYOT MASK WAVEFRONT SENSOR

Martinache et al.[7] first proposed the asymmetric pupil wavefront sensor
(APF-WFS) method for classical imaging. It relies on the Fourier properties of
the images acquired after an asymmetric mask has been placed in the entrance
pupil. In particular, he showed that, in the low-aberration regime, the phase
$\Phi$ in the Fourier plane (or OTF plane) of an image can be related linearly
to the entrance pupil phase $\phi$. Specifically, for a point source, we have

$\Phi=\mathbf{A}\phi,$ (1)

where $\mathbf{A}$ corresponds to the phase transfer matrix. This matrix
establishes the mapping between the discretized representations of the two
spaces and is thus generally rectangular. By pseudo-inversion (whose existence
is ensured by the presence of the asymmetry in the pupil), the pupil phase can
be derived.

This linear formalism is only valid for normal imaging and cannot be applied
to focal plane coronagraphs which, by suppressing the on-axis light, break the
shift-invariance properties implicitly required by this Fourier approach[8]
Nevertheless an asymmetric (pupil or Lyot) mask, by breaking the centro-
symmetry, should make the problem solvable, lifting the ambiguity on the sign
of even Zernike modes. What is missing is thus a solution to this non-linear
problem, i.e. retrieving the entrance pupil phase aberration from a focal
plane post-coronagraphic image. Since we lack an analytical solution, we
resort to deep learning to determine one, and more specifically use
convolutional neural networks (CNNs) in a supervised learning approach.

For the vortex coronagraphic mode, we choose the Lyot plane instead of the
entrance pupil to insert the asymmetric mask. It allows us to preserves the
coronagraphic rejection efficiency and to minimise the impact of the asymmetry
on the performance. Hence, our proposed concept is named the Asymmetric Lyot
mask waveFront sensor, or ALF.

##  3 SIMULATION FRAMEWORK

In the context of the METIS final design review, we have developed a
simulation framework that includes system-level knowledge such as the
instrument mode, the SCAO residuals, the water vapour seeing, and the quasi-
static NCPAs. The framework is coded in Python and uses the HCIPy[9] package
to define the optical model of the instrument and to perform optical
propagations.

The framework is based essentially on two types of classes: the sensor on one
hand, and the instrument on the other hand. The sensors we have implemented so
far include the kernel (or APF-WFS), a generic deep learning sensor
(particularly for ALF), and phase sorting interferometry (PSI, see, for
example, Codona et al. [10]). In this contribution, we focus on the first two
sensors, i.e. using the kernel approach or deep learning.

Our framework is illustrated in Fig. 1. The adaptive optics residuals and
water vapour seeing phase screens are generated off-line using the METIS-AO
simulator based on COMPASS[11]. The quasi-static NCPA, due to chromatic beam
wander, were obtained via a dedicated modeling strategy of the atmospheric
refraction using ZEMAX[12]

Figure 1: Sketch of our simulation framework. Our simulations take a series of
phase screen buffers as input. An instrument simulator allows to simulate
specific observing mode with ELT/METIS, such as classical imaging or classical
vortex coronagraphy. The produced science images are then fed to a sensor
which will derive a phase map used as a correction in closed-loop.

##  4 RESULTS & PERFORMANCE

We focus our simulations on the N band imaging case, where the impact of water
vapour seeing is most severe. First, we investigate the normal imaging case,
exploring the impact of the asymmetric mask configuration on performance and
establishing a global error budget. Next, we turn our attention to the vortex
coronagraph, exploring different mask configurations, and performing a
representative closed-loop simulation. Finally, we present our initial attempt
to experimentally test ALF on our VODCA bench.

###  4.1 Normal imaging

For normal imaging, we use the kernel formalism, see Sec. 2, and rely on the
XARA package[7]111see also https://github.com/fmartinache/xara/ which we
integrate to our simulation framework.

We compare two asymmetric mask configurations: one with a thicker spider, and
one with two thicker half-spiders. For those two configurations, we explore
different thicknesses. The sensing performance is evaluated for different N
band magnitudes on static aberrations. The results are shown in Fig. 2. The
figure illustrates the impact of the mask configuration and the thickness of
the asymmetry. The best results are obtained for a mask with two half spiders
of thicknesses $\geq$ 10% of the pupil diameter, leading to a reduction of
transmission of $\geq$4.8%.

Figure 2: Residual rms wavefront error as a function of N-band magnitude. The
WFE is measured on the controlled 20 Zernike modes. The different colors
corresponds to different asymmetric spider thicknesses of 2.5, 5, 10, and 15%
of the pupil diameter. (Left) one asymmetric spider. (Right) two half spiders.

We also analyse the impact of polychromaticity and the response in closed-
loop, allowing us to derive a global error budget, see Tab. 1. The fitting and
temporal errors are analyzed analytically, as described in the following two
paragraphs.

Assuming phase Kolmogorov spectrum with pre-corrected tip-tilt, the modal
fitting error can be approximated by[13] $\sigma_{\rm fitting}^{2}\sim
0.134\left(0.2944N_{modes}^{-\sqrt{3}/2}(D/r_{0}^{wv})^{5/3}\right)$. Assuming
an equivalent characteristics scale $r_{0}^{wv}=95m$ for water vapor
(equivalent to 300nm rms of variance, excluding tip-tilt), the residual rms
fitting error for 20 corrected Zernike modes amounts to about 120nm rms and
about 60nm rms for 100 corrected modes.

The temporal error can be estimated with
$\sigma_{temp}^{2}=\left(f_{G}^{wv}/f_{BW}\right)^{2}$. The Greenwood
frequency is given by $f_{G}^{wv}=0.427V_{wind}/r_{0}^{wv}\sim 0.04$ Hz
assuming an equivalent wind speed of $V_{wind}\sim 8.8$ m/s. The bandwidth
frequency $f_{BW}$ is the sensor framerate divided by $\sim 12$, following
standard practices. As we are only interested in the variance of the first 22
modes excluding tip-tilt, we scale the temporal variance by a factor[13]
$(\Delta_{3}-\Delta_{22})\sim 0.11$. Altogether, we obtain a temporal error of
about 50nm rms for a framerate of 10 Hz and $>$300 nm rms at 1 Hz. Note that
the temporal error estimation at 1 Hz is likely overestimated since the
analytical formula (the definition of the Greenwood frequency in particular)
assumes a constant power law and ignores the low-frequency breaks due to the
removal of average phase[14].

Table 1: Focal plane wavefront error budget at N band for normal imaging. The first column provides our baseline, correcting 20 Zernike modes and running at 10 Hz. The second column provides variations to the baseline, by increasing the number of modes (impacting the sensor noise and the fitting error) and decreasing the frame rate (impacting the temporal error). Error term | Baseline [20 modes, 10 Hz] | [100modes, 10 Hz]
---|---|---
Sensor noise | 20 | 50
Chromatic error (20% bandwidth) | 48 | 48
Fitting error | 120 | 60
Temporal error | 50 |  $\sim$300
Total [nm rms] | 140 | 330

###  4.2 Classical vortex coronagraph

Since we lack an analytical solution, we use a deep learning approach to
retrieve the phase from focal plane images. We implement a supervised learning
approach with a workflow as follows:

  1. 1.

we generate a large labelled dataset consisting of science images and the
corresponding Zernike coefficients,

  2. 2.

we train a neural network using a typical 90%-10% split for training and
validation,

  3. 3.

we use our trained model for evaluation on a different dataset.

For the neural network architecture, we use a ResNet[15] with 18 layers. Our
typical training datasets are based on 10,000 random phase screens.

Once again, we compare different mask configurations. On top of the two masks
used for classical imaging, we also test a mask with two half spiders starting
from the central obscuration rather than the external diameter: with a vortex
coronagraph, the central obscuration impacts the light scattered in the
downstream Lyot plane which appears as a monotonically decreasing flux moving
away from the central obscuration. Hence we expect more flux in this region
and by placing an asymmetry there, we might obtain a stronger signature in the
focal plane. The three masks are compared in Fig. 3 by decomposing the
residual wavefront error on the Zernike modal basis. We observe typically a
larger error on even modes (which would not be sensed without asymmetry), and
the mask configuration significantly impacts the sensing. The mask with two
half spiders starting from the central obscuration perform best.

Figure 3: Modal rms wavefront error for different Lyot stop configurations.

Finally, using our best neural network model, we test our approach in closed-
loop, i.e. injecting the residual AO phase screens and the variable water
vapor seeing. This closed-loop simulation is a 1-min simulation with science
DIT of 100msec, residual AO phase screens sampled every 10 msec (10
realisations per science image), and water vapour seeing phase screens sampled
every 100msec (1 realisation per science image). We use the expected METIS N
band photometry, and produce realistic classical vortex images with a
resolution of 9 px/$\lambda/D$, a field-of-view of $5\lambda/D$, and a pupil
plane resolution of 256 pixels. The results are shown in Fig. 4. The input
water vapour seeing is about 330nm rms, with 270 nm rms on the first 20 modes.
With ALF, we obtain a residual rms wavefront error of about 140nm rms on all
modes, and below 50nm rms on the first 20 modes. The 140nm rms is on par with
our error budget Tab. 1 and suggests that we reach similar performance to
normal imaging.

Figure 4: Closed-loop simulation in the presence of atmospheric residuals,
with 0.1s DIT time. The rms WFE on the first 20 modes reduced from $\sim$260nm
to $<$50nm.

###  4.3 First laboratory tests

To validate our concept experimentally, we use our mid-infrared coronagraphic
VODCA bench[16]. It features a vortex coronagraph and an ALPAO deformable
mirror with 97 actuators in the pupil plane. For this first test, we use
circular apertures and insert an asymmetric Lyot stop with a single thick bar.
We acquire large datasets (from 1,000 to 10,000 different realisations), and
train our ResNet-18 on those experimental data. Finally, we test the trained
model by injecting manually aberrations with our DM. The model retrieves
correctly the injected aberrations, and works also well when applied
iteratively. At this stage, we have not yet precisely quantified the
performance of our approach. Our current focus is to demonstrate that the
sensor works for the correct reasons, i.e., to explicitly show that sensing is
enabled by the asymmetry and not due to bench imperfections, such as imperfect
DM calibration or non-uniform pupil illumination.

##  5 CONCLUSION & PERSPECTIVES

In this contribution, we have presented a new concept for performing focal
plane wavefront sensing with the vortex coronagraph, i.e., ALF, the asymmetric
Lyot plane wavefront sensor. We have demonstrated a proof-of-concept by
implementing a full simulation framework representative of the METIS HCI
modes. Specifically, we have explored the impact of the asymmetric mask
configuration on performance, established an error budget for our sensor, and
evaluated performance in closed-loop.

Future work will focus on

  * •

Further optimizing the asymmetric Lyot mask and the number of corrected modes.

  * •

Analyzing other HCI modes beyond the classical vortex coronagraph, such as the
ring-apodizer vortex coronagraph.

  * •

Re-evaluating the global HCI performance in terms of post-processed contrast,
using the achieved residual error after focal plane wavefront sensing and
correction.

  * •

Investigating more advanced predictive control to boost the rejection
bandwidth of our sensor.

  * •

Improving our laboratory tests with better bench calibration and using ELT-
like pupils to obtain more representative results.

  * •

Exploring possibilities to perform an on-sky demonstration on a 8-m class
telescope.

###### Acknowledgements.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 819155), and from the Wallonia-Brussels Federation (grant
for Concerted Research Actions).

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