11institutetext: School of Mathematics and Statistics, University of St.
Andrews, North Haugh, St. Andrews, Fife, Scotland KY16 9SS, UK
11email<EMAIL_ADDRESS>22institutetext: Department of Mathematical
Sciences, Durham University, Durham, DH1 3LE, UK

# A new technique for observationally derived boundary conditions for space
weather

Paolo Pagano 11 Duncan Hendry Mackay 11 Anthony Robinson Yeates 22

###### Abstract

In recent years, space weather research has focused on developing modelling
techniques to predict the arrival time and properties of coronal mass
ejections (CMEs) at the Earth.

###### keywords:

MHD – Solar Corona – CME

The aim of this paper is to propose a new modelling technique suitable for the
next generation of Space Weather predictive tools that is both efficient and
accurate. The aim of the new approach is to provide interplanetary space
weather forecasting models with accurate time dependent boundary conditions of
erupting magnetic flux ropes in the upper solar corona. To produce boundary
conditions, we couple two different modelling techniques, MHD simulations and
a quasi-static non-potential evolution model. Both are applied on a spatial
domain that covers the entire solar surface, although they extend over a
different radial distance. The non-potential model uses a time series of
observed synoptic magnetograms to drive the non-potential quasi-static
evolution of the coronal magnetic field. This allows us to follow the
formation and loss of equilibrium of magnetic flux ropes. Following this a MHD
simulation captures the dynamic evolution of the erupting flux rope, when it
is ejected into interplanetary space. The present paper focuses on the MHD
simulations that follow the ejection of magnetic flux ropes to 4$R_{\odot}$.
We first propose a technique for specifying the pre-eruptive plasma properties
in the corona. Next, time dependent MHD simulations describe the ejection of
two magnetic flux ropes, that produce time dependent boundary conditions for
the magnetic field and plasma at 4$R_{\odot}$ that in future may be applied to
interplanetary space weather prediction models. In the present paper, we show
that the dual use of quasi-static non-potential magnetic field simulations and
full time dependent MHD simulations can produce realistic inhomogeneous
boundary conditions for space weather forecasting tools. Before a fully
operational model can be produced there are a number of technical and
scientific challenges that still need to be addressed. Nevertheless, we
illustrate that coupling quasi-static and MHD simulations in this way can
significantly reduce the computational time required to produce realistic
space weather boundary conditions.

##  1 Introduction

The solar corona is a highly dynamic environment where magnetic and plasma
structures are continually evolving. It is a key region of the solar
atmosphere that couples the solar photosphere with interplanetary space. This
coupling results in the solar wind and many other perturbations of the
interplanetary space plasma, referred to as space weather. The origin of these
events can be traced back to flux emergence and flows at the solar
photosphere. The corona stores vast amounts of free magnetic energy and the
release of this energy in the form of coronal mass ejections (CMEs) initiates
the largest and most violent perturbations of the near Earth environment,
referred to as space weather or extreme space weather (Hapgood, 2011). This
topic has drawn the attention of national and international institutions due
to the practical consequences that severe space weather events can have on
national infrastructures as well as on space operations and missions
(Schrijver et al., 2015).

CMEs produce major disruptions to the ambient magnetic and plasma properties
of interplanetary space, as their occurrence signifies the violent ejection of
both magnetic flux and plasma from the low solar corona. Often magnetic flux
ropes in the low corona are considered the main progenitors of CMEs, where an
initial period of stability of the flux rope is followed by a fast and sudden
ejection outwards into interplanetary space. This scenario is supported by a
number of observations (Howard and DeForest, 2014; Chintzoglou et al., 2015;
Cheng et al., 2011; Li and Zhang, 2013). Some models explain the flux rope
formation and the subsequent ejection with photospheric flows (Mackay and van
Ballegooijen, 2006a; Xia et al., 2014), others focus on magnetic flux
emergence from underneath the photosphere (Archontis and Hood, 2012), and
finally others rely on the onset of MHD instabilities (Török and Kliem, 2005;
Zuccarello et al., 2015). It should be noted however, that it is not unusual
to observe CMEs bearing no apparent connection with structures in the low
corona (D’Huys et al., 2014; Ouyang et al., 2015). A statistical study carried
out by Hutton and Morgan (2015) showed that not all CMEs can be directly
associated to magnetic flux ropes and Vourlidas et al. (2013) measured that at
least 40% of CMEs are certainly associated with magnetic flux ropes, while
only a minor percentage are certainly not associated with any magnetic flux
ropes. The above discussion illustrates that there is a strong relationship
between CMEs and flux ropes.

While the CME initiation mechanism is still under debate, it is widely
accepted that, no matter the origin of the CME, they travel outwards due to
their own propulsion until about $\sim 4R_{\odot}$ (Gopalswamy et al., 2000),
beyond which they are dragged out by the solar wind. Sachdeva et al. (2015)
claim that the aerodynamic drag only becomes the dominant force after
$15R_{\odot}$, although it can play a role at shorter radial distances. It
therefore seems reasonable to assume that CMEs are not significantly coupled
to the solar wind until after $4R_{\odot}$. Nevertheless, below $4R_{\odot}$
Gopalswamy et al. (2003) and also at larger distances, CMEs can suffer both
deflection or acceleration. This may be due to the background corona, the
interplanetary magnetic field or interaction with other eruptive events.

To mitigate the effects of space weather it is key to predict the arrival time
and properties of CMEs at the Earth’s magnetosphere. The development of these
predicting tools is not trivial and while current attempts have made
significant advances, it is commonly accepted that more precision and more
accuracy are required for effective predictions. The most recent development
of the drag-based model (DBM) (Vršnak et al., 2013; Žic et al., 2015) is used
to predict the arrival time of CMEs assuming that beyond $20R_{\odot}$ the
aerodynamic drag force dominates the dynamics. Moreover, the model ElEvoHI
(Rollett et al., 2016) empowers the DBM with an improved geometrical fitting
of the CME. Tóth et al. (2005) introduced the Space Weather Modelling
Framework (SWMF), a series of highly complex computer simulations that account
for significantly different physical regimes and to this end utilises a number
of communicating codes (Tóth et al., 2012). The flexibility of the SWMF allows
for the coupling of specific models with different codes to develop new
modelling techniques, as been done by Jin et al. (2017). Merkin et al. (2016)
used a two different MHD codes to model the corona and the interplanetary
space and a common spherical shell doma at $20$ $R_{\odot}$ is used to couple
the two codes. Finally, the 3D MHD WSA-ENLIL model describes how the solar
wind and interplanetary plasma act as a background for a kinematically
inserted CME (Odstrčil et al., 1996; Odstrčil and Pizzo, 1999a, b; Odstrcil,
2003; Odstrcil et al., 2004). In particular, the so called CME Cone Model (Xie
et al., 2004; Xue et al., 2005; Michalek, 2006) is widely used in WSA-ENLIL to
model the insertion of a CME into the solar wind. Its parameters can be tuned
by coronagraph images (Millward et al., 2013), or data collected in situ at
Mercury (Baker et al., 2013). Recently it has also been extended to describe
Halo CMEs (Na et al., 2017).

For all present and future models the coupling between the outer corona and
the lower corona is key. This includes the coupling of the solar wind streams
with the origin of CMEs. In particular realistic boundary conditions for space
weather need to consider the origin of CMEs in the low corona and how to
insert these CMEs into the solar wind. The cone model performs this key role
in the WSA-ENLIL model and has improved its predictive capabilities (Dewey et
al., 2015), however it injects only a density and velocity perturbation in the
solar wind, neglecting the magnetic flux that is an essential part of the CME
disturbance to space weather. As an alternative to this, Shiota and Kataoka
(2016) introduced the use of a spheromak to model the perturbation from an
ejecting magnetic flux rope. It considers the injection of an idealised
structure described in terms of density and magnetic field. In Merkin et al.
(2016), where two different MHD codes are coupled, the common domain between
the two codes is in fact a time dependent boundary condition for space
weather. Other techniques to input CMEs into space weather models use
coronagraph data directly, such as in the Tappin-Howard (T-H) model (Howard
and Tappin, 2009; Tappin and Howard, 2009). In this model ICMEs are
reconstructed from visible light images. As an alternative, in Bisi et al.
(2013) remote-sensing radio observatios of Interplanetary Scintillation (IPS)
are used to probe the inner heliosphere and to identify density irregularities
in the solar wind. When enhanced by the UCSD time-dependent tomography
technique (Jackson et al., 2010). Harrison et al. (2017) provides an extensive
review on how heliospheric imaging can be used to improve our space weather
forecasts.

Many recent studies have measured our predictive capability. Zhao and Dryer
(2014) estimated the uncertainty in our capability to predict the arrival time
of CME at 12 hours. Also Falkenberg et al. (2011) found a similar result for
WSA-ENLIL. They explain that one of the reasons that prevents better estimates
is our limited capability in reproducing solar transients and how the CMEs are
input in the background solar wind. This seems a general trend, as no major
difference has been found between the various models, where, for instance, the
DBM and ENLIL predicting capabilities differ by less than $10\%$, except for
the case of strong solar activity when ENLIL performs significantly better
(Vršnak et al., 2014). More recently Tucker-Hood et al. (2015) carried out an
extensive survey that proved how much room for improvement there is in the
field. Out of the 60 predictions made, 36 were false alarms and when the
prediction was successful the arrival time was still estimated with an error
of $\sim 16$ hours. Mays et al. (2015) shows came to similar conclusions, but
found that results were slightly more positive for ENLIL. These results show
that significant modelling improvements are needed to reduce the arrival time
error and the property of the geomagnetic perturbations. To improve these
estimations more accurate boundary conditions of the injection of CMEs into
interplanetary space are required.

In the present paper we focus on a specific kind of possible improvement: to
provide realistic and accurate boundary conditions for the next generation of
space weather forecasting models. In particular we need realistic initial and
time dependent boundary conditions that reflect the complexity of the
transition from the solar corona to interplanetary space, in terms of the
injection of both plasma (density and velocity) and magnetic field in the
solar wind. To do this we present a novel approach where simulations of
magnetic flux rope ejections that are derived directly from surface
magnetograms may be used to aid future space weather predictions. While most
magnetic structures in the solar corona are close to equilibrium, it is
essential to identify those that lose equilibrium and then erupt. To this end
Yeates et al. (2007) have opened the way for the use of synoptic magnetograms
to model the quasi-static, non-potential evolution of the global corona. This
modelling technique has proved to be accurate in predicting the formation and
helicity of solar filaments (Yeates et al., 2008), the variation of the Sun’s
open magnetic flux (Yeates et al., 2010b) and to a lesser extent the location
of CMEs (Yeates et al., 2010a). As already described in Mackay and van
Ballegooijen (2006a) and Pagano et al. (2013b), the formation and ejection of
magnetic flux ropes occur over very different time-scales and in different
dynamic regimes. The formation of magnetic flux ropes occurs slowly (days or
weeks), through a series of quasi-equilibrium states (formation times much
longer than the Alfvén time) and in a magnetically-dominated regime ($\beta\ll
1$). In contrast, magnetic flux rope ejections takes place over a few hours,
where no equilibrium exists and where compression and heating create extended
regions of high-$\beta$ plasma. To simulate the global corona during both the
formation and eruption of magnetic flux ropes, we expand upon the work already
carried out in Pagano et al. (2013b, a, 2014). In these studies we have shown
that coupling a quasi-static non-potential model with MHD simulations is a
viable way to describe the full life span of a flux rope: from formation to
ejection. One limitation of the previous studies, was that they only
considered a small wedge-shaped portion of the Sun and simple idealised
magnetic field configurations. We now apply the same approach to realistic
observationally derived magnetic fields that cover the full sphere of the Sun
as simulated in the global non-potential model of Yeates et al. (2010a). The
model of Yeates et al. (2010a) uses bipoles deduced from synoptic magnetograms
to realistically model both the time evolution of photospheric fields and the
quasi-static response of coronal fields to global motions over long periods of
time. One aspect of this is the formation of magnetic flux ropes, which
subsequently lose stability. At this loss of stability we switch the modelling
approach where MPI-ARMVAC is used to follow the MHD evolution. Throughout the
MHD simulation we derive the boundary conditions for space weather forecasting
tools, i.e. the density, velocity and magnetic field distributions that the
flux rope ejections inject in the solar wind and interplanetary space.

In the present paper we present the basic physics of a model that aims to
produce boundary conditions for space weather prediction models. At the
present time we focus on the properties of the model, where in future studies
we will consider its application and viability in the context of an
operational model along with its coupling to interplanetary models. The
significance of this approach relies on its accuracy and efficiency. The
global non-potential model has been extensively tested and has been shown to
be accurate in describing the evolution of the photospheric magnetic field and
subsequent formation of non-potential magnetic fields and flux ropes in the
corona. General MHD simulations (capable of handling multi-$\beta$ domains)
are the most accurate tool to model flux rope ejections and then the evolution
of the flux rope through interplanetary space. While we put forward a two-
stage approach for producing the boundary condition it is of course possible
to use MHD simulations to simulate both the magnetic flux rope formation and
ejections. However, present computational power is insufficient to model the
slow formation of flux ropes over days to weeks. A MHD simulation would
consume unreasonable computational resources and additionally the high number
of time steps required to cover the physical time span would lead to the
accumulation of significant round-off errors. Taking into account these
considerations, we propose a three stage numerical model. For the present
paper we consider the first two steps of formation and eruption. The third and
final stage which is the connection of the ejection stage to an interplanetary
evolution model will be carried out in the future. This will include using the
output from the MHD simulation as a time-dependent boundary condition for
driving a new generation of space weather forecasting tools.

The structure of the paper is as follows: in Sec.2 we present the construction
of the model, in Sec.3 we describe in detail the MHD simulation, in Sec.4 we
explain how this study can be used in a space weather forecasting context and
we finally discuss and give conclusions in Sec.5.

##  2 Model

###  2.1 Overview of Coupled Modelling Technique.

In order to model the ejection of magnetic flux ropes in the global corona, we
employ a dual modelling technique of MHD simulations coupled with a quasi-
static non-potential global model (Yeates et al., 2010a). This approach is an
extension of the technique successfully pioneered in Pagano et al. (2013b) and
then further developed in Pagano et al. (2013a) and Pagano et al. (2014). In
these studies, a magnetic configuration obtained from the global non-potential
model case study of Mackay and van Ballegooijen (2006a) was used as an initial
condition in the MHD simulation. The combined technique allows us to follow
the slow build-up of stress and electric currents over observed solar
timescales (days -months), along with being able to follow the dynamic
eruption timescale (min-hours).

####  2.1.1 Quasi-static Model

In this paper, we use a magnetic configuration obtained from a simulation run
of the global non-potential model, where the model simulated the entire time
span of Cycle 23 (Yeates and Mackay, 2012). A magnetic configuration is chosen
near the end of the run, where the structure and connectivity of the corona
was produced by the combined effects of differential rotation, meridional
flow, surface magnetic diffusion and magnetic flux emergence. The properties
of the flux emergence events were deduced from NSO/KP and NSO/SOLIS
magnetograms and were included to maintain the accuracy of the photospheric
field compared to that found on the Sun. This application of the Global Model
has proved to be very successful in reproducing the chirality of solar
filaments (Yeates and Mackay, 2012) throughout the solar cycle.

Figure 1: Representation of the magnetic configuration imported from the
Global Non-potential Model. The two panels show maps of the radial component
of the magnetic field on the surface from two opposite points of view. Black
lines are some representative magnetic field lines to highlight existing
structures.

Additionally, we also took care by selecting an initial condition for the MHD
simulation where there were two relatively isolated magnetic flux ropes that
had formed and both had accumulated enough stress that they were going to
erupt. In the global model we have developed an automated technique for the
identification of erupting flux ropes. The technique is described in Yeates
and Mackay (2009a). The global magnetic configuration under investigation can
be seen from two viewpoints, $180^{\circ}$ in longitude apart, in Fig.1. The
magnetic field lines exhibit a number of features characteristic of those on
the Sun, both at the photosphere and in the magnetic field connectivity in the
low corona. At several locations opposite polarity patches lie close to one-
another, and these are the preferred locations of flux cancellation where flux
ropes are found. However, at only two of these locations (FR1 in the northern
hemisphere and FR2 in the southern hemisphere) is the magnetic field sheared
enough along the polarity inversion line to show a twisted magnetic flux rope.
At FR1 a flux rope sits above a PIL in a bipole that extends over a region of
$0.4R_{\odot}\times 0.3R_{\odot}=0.12R_{\odot}^{2}$. The PIL is mainly North-
South directed. At the location FR2 there is a smaller flux rope and bipole,
where now the bipole extends only for $0.23R_{\odot}\times
0.24R_{\odot}=0.06R_{\odot}^{2}$. The polarities of the bipole are separated
by a PIL which lies directed in a South-West to North-East direction. For both
of the active regions the flux rope lie above the PIL therefore have a
slightly different orientation. The rest of the coronal magnetic field is
mostly close to potential and the other bipoles that are present in the domain
show no significant shear above their polarity inversion lines.

####  2.1.2 Coupling to MHD simulation

To use the magnetic configuration illustrated in Fig.1 as the initial
condition for the MHD simulation, we import into the MHD simulation all three
components of the magnetic field from the global non-potential model. A full
description of how this is carried out, where both the stability (or
instability) and magnetic connectivity of the configuration is preserved, is
given in Pagano et al. (2013b) and Pagano et al. (2013a) where a number of
test cases are described.

For the present paper we have slightly modified the way in which the 3D
interpolation is performed. In spherical coordinates, let $A(r,\theta,\phi)$
be the value of a function that we want to have interpolated to the position
$(r,\theta,\phi)$, where $r$ is the radial distance from the centre of the
Sun, $\theta$ is the polar angle and $\phi$ is the azimuthal angle. We know
that it lies in the cell defined by the indexes $[i:i+1,j:j+1,k:k+1]$ where
the original function $a$ is defined, we compute $A(r,\theta,\phi)$ as

$A(r,\theta,\phi)=\sum_{i,j,k}^{i+1,j+1,k+1}a[i,j,k]V[i,j,k]/V$ (1)

where $V[i,j,k]$ is the volume defined by the point $(r,\theta,\phi)$ and the
cell corner opposite to the position $[i,j,k]$ and V is the sum of all the
volumes. This approach guarantees the continuity of the interpolated solution
and its smoothness, independent of the spatial resolution of the grid where
$A(r,\theta,\phi)$ is defined.

###  2.2 MHD Simulation

####  2.2.1 Plasma distribution

As the global non-potential model only provides a magnetic configuration, to
produce a complete set of MHD variables in the initial condition we need to
determine distributions of plasma density, velocity and temperature. In
specifying these we aim at a realistic and general representation of the solar
corona, where the distribution of plasma takes into account the heterogeneity
of the structures of the solar corona. This heterogeneity can be simplified by
representing the solar corona as a combination of dense active regions and
less dense quiet Sun regions. In addition, there are dominantly horizontal
magnetic fields (such as prominences or filaments) which are generally two
orders of magnitude cooler and denser than typical coronal values.

We define the following proxy function to link the plasma temperature and
density to the magnetic field:

$\omega=\sqrt{\omega_{r}^{2}+\omega_{\theta}^{2}+\omega_{\phi}^{2}}$ (2)

where

$\omega_{r}=\frac{\left|\vec{B}\times\vec{\nabla
B_{r}}\right|^{2}}{\left|\vec{\nabla
B_{r}}\right|^{2}},~{}~{}\omega_{\theta}=\frac{\left|\vec{B}\times\vec{\nabla
B_{\theta}}\right|^{2}}{\left|\vec{\nabla
B_{\theta}}\right|^{2}},~{}~{}\omega_{\phi}=\frac{\left|\vec{B}\times\vec{\nabla
B_{\phi}}\right|^{2}}{\left|\vec{\nabla B_{\phi}}\right|^{2}}$ (3)

The function $\omega$ is positive definite and peaks where the magnetic field
exhibits a complex twisted field, e.g. near the axis of a magnetic flux rope.
As the value of $\omega$ is also proportional to the magnetic field intensity,
it is higher near the solar surface (where the magnetic field is more intense)
and lower at further radial distances from the solar surface.

In order to effectively use $\omega$ to model the solar atmosphere we define
the functions:

$\Omega=\frac{\Omega_{B}+\Omega_{\theta}}{2}+\frac{\left|\Omega_{B}-\Omega_{\theta}\right|}{2},$
(4)
$\Omega_{\theta}(\theta)=\frac{\arctan\left(\frac{\theta-(\pi-\theta^{\star})}{\Delta\theta}\right)}{\pi}-\frac{\arctan\left(\frac{\theta-\theta^{\star}}{\Delta\theta}\right)}{\pi}+1,$
(5)
$\Omega_{B}(\omega)=\frac{\arctan{\left(\frac{\omega-\omega^{\star}}{\Delta\omega}\right)}}{\pi}+0.5,$
(6)

where $\Omega_{\theta}$ and $\Omega_{B}$ are functions bound between $0$ and
$1$ while $\Omega$ is defined to pick the higher between $\Omega_{\theta}$ and
$\Omega_{B}$. $\Omega_{\theta}$ is a function that only depends on the
coordinate $\theta$ that defines two regions near the poles where we quench
any dynamics close to the boundaries of the simulation. Using $\Omega$ the
temperature is defined by:

$T=\Omega(T_{fluxrope}-T_{corona})+T_{corona}.$ (7)

Next, the thermal pressure is independently specified by the solution for
hydro-static equilibrium with a uniform temperature set equal to $T_{corona}$,

$p=\frac{\rho_{LB}}{\mu m_{p}}k_{B}2T_{corona}\exp\left({-\frac{M_{\odot}G\mu
m_{p}}{2T_{corona}k_{B}R_{\odot}}}\right)\exp\left({\frac{M_{\odot}G\mu
m_{p}}{2T_{corona}k_{B}r}}\right),$ (8)

where $\rho_{LB}$ is the density at $r=R_{\odot}$ when $|B|=0$, $\mu=1.31$ is
the average particle mass in the solar corona, $m_{p}$ is the proton mass and
$k_{B}$ is Boltzmann constant. Finally, the density is simply given by the
equation of state applied to Eq.7 and Eq.8:

$\rho=\frac{p}{T(\vec{B})}\frac{\mu m_{p}}{k_{B}}.$ (9)

This produces an inhomogeneous solar corona of cool dense flux ropes and
hotter emptier corona arcades.

####  2.2.2 MHD simulation

Using the approach described in Sec.2.2.1, we construct the initial condition
for the MHD simulation, where Table 1 shows the value used in our model for
all parameters.

Table 1: Parameters Parameter | value | Units
---|---|---
$\rho_{LB}$ | $2.77\times 10^{-15}$ | $g/cm^{3}$
$\theta^{\star}$ | $0.13$ | radiants
$\Delta\theta$ | $0.005$ | radiants
$\omega^{\star}$ | $30$ | $G$
$\Delta\omega$ | $3$ | $G$
$T_{fluxrope}$ | $10^{4}$ | K
$T_{corona}$ | $2\times 10^{6}$ | K

With this initial condition we use the MPI-AMRVAC software (Porth et al.,
2014), to solve the MHD equations where external gravity is included as a
source term,

$\frac{\partial\rho}{\partial t}+\vec{\nabla}\cdot(\rho\vec{v})=0,$ (10)
$\frac{\partial\rho\vec{v}}{\partial
t}+\vec{\nabla}\cdot(\rho\vec{v}\vec{v})+\nabla
p-\frac{(\vec{\nabla}\times\vec{B})\times\vec{B}}{4\pi}=\rho\vec{g},$ (11)
$\frac{\partial\vec{B}}{\partial
t}-\vec{\nabla}\times(\vec{v}\times\vec{B})=0,$ (12) $\frac{\partial
e}{\partial t}+\vec{\nabla}\cdot[(e+p)\vec{v}]=\rho\vec{g}\cdot\vec{v},$ (13)

where $t$ is time, $\rho$ is density, $\vec{v}$ velocity, $p$ thermal pressure
and $\vec{B}$ the magnetic field. The total energy density $e$ is given by

$e=\frac{p}{\gamma-1}+\frac{1}{2}\rho\vec{v}^{2}+\frac{\vec{B}^{2}}{8\pi},$
(14)

where $\gamma=5/3$ denotes the ratio of specific heats. The expression for
solar gravitational acceleration is given by

$\vec{g}=-\frac{GM_{\odot}}{r^{2}}\vec{\hat{r}},$ (15)

where $G$ is the gravitational constant, $M_{\odot}$ denotes the mass of the
Sun, and $\vec{\hat{r}}$ is the unit vector.

The computational domain is composed of $256\times 256\times 512$ cells,
distributed on a uniform spherical grid. With this resolution the simulation
domain extends over $3$ $R_{\odot}$ in the radial direction starting from
$r=R_{\odot}$. The co-latitude, $\theta$, spans from $\theta=0.75^{\circ}$ to
$\theta=179.25^{\circ}$ and the longitude, $\phi$, spans $360^{\circ}$. The
boundary conditions are treated with a system of ghost cells and match those
used in Yeates et al. (2010a). Open boundary conditions are imposed at the
outer boundary, reflective boundary conditions are set at the $\theta$
boundaries and the $\phi$ boundaries are periodic. The reflective $\theta$
boundary condition does not allow any plasma or magnetic flux to pass through.
The stability at this boundary is reinforced through the use of the function
$\Omega_{\theta}$, which results in plasma near the $\theta$ boundary having a
density significantly higher than that prescribed by gravitational
stratification. This enhancement quenches any upward motion at this boundary.
At the lower boundary which represents the photosphere, we impose a fixed
boundary condition taken from the first two $\theta$-$\phi$ planes of cells
derived from the global non-potential model.

The interpolation technique used to import the magnetic field configuration
from the spatial grid of the global non-potential model, to the alternative
grid that we use in the MHD simulation only applies to $r<2.5R_{\odot}$. In
the region beyond $r=2.5R_{\odot}$ we assume a purely radial field where
magnetic flux is conserved:

$B_{r}(r>2.5R_{\odot},\theta,\phi)=B_{r}(2.5R_{\odot},\theta,\phi)\frac{\left(2.5R_{\odot}\right)^{2}}{r^{2}}.$
(16)

Figure 2: Mollweide projections (central meridian at longitude
$\phi=180^{\circ}$) on the lower boundary of the MHD simulation of (a) Lorentz
Force, (b) $\omega$, (c) $Log_{10}(\rho[g/cm^{3}])$ and (d) $Log_{10}(T[K])$.

The key properties of the initial condition in the MHD simulation can be seen
in Fig.2 which shows in Mollweide projection distributions of (a) the radial
component of the Lorentz force at the lower boundary, (b) the function
$\omega$ derived from the magnetic configuration and finally the resulting
distributions of (c) plasma density and (d) temperature. The Mollweide
projection has the advantage of showing in a single field of view the whole
solar disk and therefore clearly illustrates the connectivity between
different regions. The Lorentz force exhibits several areas of strong outward
radial force shown as the white regions. In particular two structures show a
significantly larger positive Lorentz force compared to surrounding areas. One
lies in the northern hemisphere to the left of central meridian and the other
is in the southern hemisphere to the right of central meridian. These are at
locations where large flux ropes have formed and can no longer be fully
contained by the overlying fields. Close to where the flux ropes lie, with
their positive radial component of Lorentz force, smaller regions where the
Lorentz force is negative can be seen.

The map of $\omega$ (Fig.2b) follows a similar distribution, where several
small structures are visible along the active latitudes. Three zones exist
where $\omega$ is significantly larger with peak values nearly $\sim 100$
times greater than ambient background values. Flux ropes show higher values of
$\omega$, as close to flux ropes all the three terms $\omega_{r}$,
$\omega_{\theta}$, and $\omega_{\phi}$ are important at different locations.
Namely $\omega_{r}$ peaks at the center of the magnetic flux rope axis, while
$\omega_{\theta}$ and $\omega_{\phi}$ peak near the footpoints. Finally, the
maps of density (Fig.2c) and temperature (Fig.2d) show that the locations with
a more complex magnetic configuration are denser and cooler. With this the
magnetic flux ropes in the simulation appear as regions with density and
temperature one or two orders of magnitude denser and colder than the
surrounding ambient values. Thus the initial conditions produce an
inhomogeneous corona that exhibits many features found on the Sun.

The initial plasma $\beta$ in the simulation ranges between $\beta\sim
10^{-3}$ at the flux ropes to $\beta\sim 1$ in confined regions where the
magnetic field is weak. While there is a wide range, normally the value of
$\beta$ lies between $0.1$ and $0.01$ throughout the vast majority of the
volume. Due to the low $\beta$ values at the flux ropes, the strongest
unbalanced force in the initial condition is the radially directed Lorentz
force at these locations. In addition to the unbalanced Lorentz force the
radial profile of density and pressure also does not prescribe a balance
between the thermal pressure gradient and gravity. While this pressure force
imbalance exists, the consequences have been discussed in detail in Pagano et
al. (2013a), where it is shown that any evolution due to these imbalances
occurs over timescales much longer than that of the eruption dynamics
triggered by the Lorentz force. As such we can neglect these effects.

##  3 Simulation

The MHD simulation produces two individual eruptions that originate from the
two twisted magnetic flux ropes shown in Fig.1 and Fig.2. In each case, the
radial Lorentz force excess due to a local non-equilibrium, triggers the
dynamic evolution of the eruption that occurs in the simulation. Fig.3 shows
images of the column density where the central meridian longitude is chosen to
be $\phi=90^{\circ}$ (left-hand side column) and $\phi=270^{\circ}$ (right-
hand side column) at (a) $t=0$ $min$, (b) $t=33.6$ $min$ and (c) $t=60.3$
$min$ where magnetic field lines (green) have been drawn from the photospheric
boundary.

Figure 3: Maps of the column density seen from either side of the Sun in the
MHD simulation along with some superimposed magnetic field lines traced from
the lower boundary (green) at $t=0$ $min$ (a), $t=33.6$ $min$ (b), and
$t=60.3$ $min$ (c). Central meridian for left-hand side panels is at
$\phi=90^{\circ}$ and at $\phi=270^{\circ}$ for right-hand sided panels and
the FOVs are chosen to be same as in Fig.1

The two flux ropes are located such that one is in the northern hemisphere,
while the other is in the southern hemisphere, and they lie approximately
$180^{\circ}$ in longitude ($\phi$) apart. When the ejections occur they
displace dense plasma to larger radii, which results in an increased column
density around the locations of each flux rope. This increased density can be
seen as the two circular structures expanding at either limb, one towards
North and $\phi=0^{\circ}$, the other towards South and $\phi=180^{\circ}$.
During the ejections the global magnetic field undergoes a rapid evolution.
Even though there is a rapid evolution, it is possible to follow the evolution
of the twisted magnetic field lines of the flux ropes. After $t=60.3$ $min$
the two circular expanding structures of density have expanded sufficiently
that they cross the outer boundary of the domain.

To better understand the evolution of the two flux rope ejections we plot
radial cuts of density, temperature, radial velocity and $\omega$ from the
photospheric boundary to the outer boundary (Fig.4 and Fig.5). These plots are
taken along a radial line intersecting with the initial position of the
centers of the two magnetic flux ropes.

Figure 4: (a) Cuts of $Log_{10}(\rho[g/cm^{3}])$ along the radial direction
from solar surface to the outer boundary at the locations of FR1 and FR2 at
$t=10.4$ $min$ (blue line) and $t=27.8$ $min$ (red line). (b) Cuts of
$Log_{10}(T[K])$ along the radial direction from solar surface to the outer
boundary at the locations of FR1 and FR2 at $t=10.4$ $min$ (blue line) and
$t=27.8$ $min$ (red line).

Fig.4(a) shows the density at $t=0$ $min$ (black line), $t=10.4$ $min$ (blue
line) and $t=27.8min$ (red line) for both Flux Rope 1 and 2. In each plot the
black line shows the initial density profile with the dense flux rope at low
heights along with the stratified atmosphere. At later times the presence of
the ejection is identified by an excess in density compared to that of the
initial stratified atmosphere. At $t=10.4$ $min$ the density excess of the
ejected plasma has reached, $\sim 1.8$ $R_{\odot}$ for FR1 and $\sim 2$
$R_{\odot}$ for FR2 showing that they are traveling at different speeds.
Similar estimations can be made with the fronts of the two ejections and at
different times. In all cases, we consistently find that FR2 travels about
$25\%$ faster than FR1.

At the lead front of the density excess there is a very sharp temperature
increase to $T\sim 10^{7}$ $K$ (Fig.4b). Beneath the front the temperature
profile also shows significant structure. At $t=10.4$ $min$ the temperature
profile in the atmosphere exhibits a dip at the location of the axis for both
FR1 ($1.2R_{\odot}$) and FR2 ($1.4R_{\odot}$). It however should be noted that
the temperature at the center of the flux rope is comparable to that of the
background external corona at $2MK$. While this value is high, it is still
cooler than that of the surrounding ejection.

Figure 5: (a) Cuts of radial velocity along the radial direction from solar
surface to the outer boundary at the locations of FR1 and FR2 at $t=10.4$
$min$ (blue line) and $t=27.8$ $min$ (red line). (b) Cuts of
$Log_{10}(\omega)$ along the radial direction from solar surface to the outer
boundary at the locations of FR1 and FR2 at $t=10.4$ $min$ (blue line) and
$t=27.8$ $min$ (red line).

At $t=10.4$ $min$ the radial velocity profiles in Fig.5(a) for both FR1 and
FR2 exhibit a very similar behavior, where there is an initial sharp rise at
the photosphere, followed by a slight dip before becoming roughly level
between 1.5 - 2$R_{\odot}$. In both cases at the lead edge there is a sharp
fall to zero. Combining the information in Fig.4 and Fig.5 it can be seen that
the ejection of both flux ropes can be characterized by a front (density
excess, sharp increase in temperature and radial velocity) and a core (density
excess, temperature dip, slower than the front). While this is characteristic
of the early stages of the ejection similar structures are visible in the
plots at $t=27.8$ min. However, there are some differences from the earlier
times which include: i) the volume of density excess increases and thus the
density at the flux rope decreases, ii) the temperature at the front
increases, while the temperature dips increase in number and become colder and
iii) the radial velocity at the front slightly increases and with this the
speed of the flux rope ejection. In addition to the features above, we also
find that for both FR1 and FR2 $\omega$ shows an increase in value with
respect to that of the initial condition. This occurs both in the region
beyond the front but more prominently near the location of the flux rope
center.

The occurrence of the two ejections has a significant effect on the energy
budget of the solar corona. Fig.6 shows the total energy as a function of time
in the MHD simulation (black line). The total energy is approximately
conserved and after $t=68.44$ $min$ the total energy has increased by less
than $9\%$. The increase can be accounted for as there is a net flux of energy
into the simulation after taking into account the energy flux across the lower
and outer boundaries. At the start, the magnetic energy accounts for about
$90\%$ of the total energy and the thermal energy for the remaining $10\%$. By
the end of the simulation the kinetic energy is $6\%$ of the total energy
where its increase is due to the conversion of magnetic energy which falls to
$80\%$ of the total energy, while the thermal energy slightly increases.

Figure 6: Energy in the MHD simulation integrated over the whole spatial
domain as a function of time. Black line shows total energy, blue magnetic
energy, red kinetic energy, and green thermal energy.

##  4 Toward a space weather application

One of the goals of the present work is to provide accurate boundary
conditions of the outer solar corona that can be used in future space weather
forecasting tools, such as those for the solar wind and the evolution of
Interplanetary Coronal Mass Ejections (ICMEs). To achieve a realistic model of
space weather conditions at 1 AU, it is key that accurate initial conditions
for the injection of plasma and magnetic flux into the solar wind are
specified close to the Sun. These boundary conditions in turn require accurate
time-dependent modelling of low coronal magnetic fields during the build up to
and occurrence of an eruption.

Our technique of coupling the global non-potential model with the MHD
numerical solver MPI-AMRVAC is suitable for space weather forecasting purposes
because it accurately models observed magnetic field configurations on the Sun
and is very computationally efficient. It allows us to model the fast dynamic
ejection of magnetic flux ropes, maintaining the accuracy and generality of a
full MHD model, while allowing us to model the slow quasi-static formation of
flux ropes and the global corona in a numerically cheaper way. Initial
estimations suggest that the global non-potential model is between $10^{4}$
and $10^{5}$ times faster than the MPI-AMRVAC in advancing the simulation in
physical time. This therefore allows for the near real time simulation of
observed magnetic fields on the Sun. We now discuss how the dual approach
described above may provide useful boundary conditions for space weather
models.

###  4.1 Injection of the CME into interplanetary space

In this section, we present some exploratory results on how our combined model
injects a CME into interplanetary space. As we have set the outer boundary
condition at $4$ $R_{\odot}$ we focus on the conditions of the plasma and
magnetic field as a consequence of the eruption before it starts to interact
with the solar wind.

Fig.7 shows Mollweide projections of (a) density and (b) the radial velocity
at the outer boundary of the simulation at a time near when both flux ropes
cross this boundary ($t=60.3$ $min$).

Figure 7: Mollweide projections (central meridian at longitude
$\phi=180^{\circ}$) at the outer boundary of the MHD simulation at $t=60.3$
$min$ of (a) $\rho[g/cm^{3}]$ and (b) radial velocity of the plasma $[cm/s]$.

The two flux rope ejections appear as density excesses covering an area of
some tens of squared degrees where FR1 involves a wider area than FR2. One
predominately lies in the northern hemisphere and the other in the southern
hemisphere, but both straddle the equator. The density of the flux ropes is
around $100$ times larger than that of the background corona, which has a
density of around $\sim 10^{-19}$ $g/cm^{3}$. Due to the two orders of
magnitude difference it appears that at the outer boundary the flux rope has a
near homogeneous density. However, on closer inspection of the data internal
variations in the density are also visible, where higher density structures
lie in the centre of the ejection for FR1, and at the southern boundary for
the ejection of FR2. A similar pattern can also be seen in the radial velocity
maps, where both ejections exhibit an outward velocity in the order of
hundreds of $km/s$. Outside of the region corresponding to the erupting flux
ropes the corona has a significantly smaller outward velocity. It should be
noted that in this simulation the fastest speeds are not found at the flux
rope locations, but rather along the inversion line at the outer boundary,
where it is possible that the eruptions trigger some magnetic reconnection
processes that accelerate the plasma. At this location where the radial
component of the magnetic field changes sign, the radial outward velocity
reaches $\sim 1000$ $km/s$. This outflow is related to a streamer-like
phenomenon, where plasma accelerates in conjunction with reconnection at a
current sheet. This is partly due to the magnetic field from the global
simulation relaxing at this location in the high corona when the MHD
simulation starts. This is an artefact of the coupling of the two codes that
we will tackle in the future. Whilst this occurs, the low density associated
with this feature means that it does not contribute significantly to the
momentum flux and is essentially negligible. The internal structure of the
radial velocity patterns at both flux ropes differs. For FR1 there are higher
radial velocities at the borders of the region involved in the ejection and
the region of high positive radial velocity extends to near the polarity
inversion line . In contrast the radial velocity pattern of FR2 shows more
modest internal structuring and remains isolated from the polarity inversion
line.

Figure 8: Cuts of (a) $Log_{10}(\rho[g/cm^{3}])$, (b) radial velocity [cm/s],
and (c) temperature [K] along $\phi$ at $t=0$ $min$ (black line), $t=33.6$
$min$ (blue line) and $t=60.3$ $min$ (red line) at three different co-
latitudes: the center of FR1 ($\theta=71.4^{\circ}$), the equator
($\theta=90^{\circ}$), and the center of FR2 ($\theta=110^{\circ}$.)

Fig.8 shows cuts along the $\phi$ direction at the outer boundary at three
different co-latitudes for (a) density, (b) radial velocity and (c)
temperature. In each of the plots different co-latitudes are considered
corresponding to the center of FR1 ($\theta=71.5^{\circ}$, top), the equator
($\theta=90^{\circ}$, middle) and finally the center of FR2
($\theta=110^{\circ}$, bottom). In each plot values are shown at three
different times: $t=0$ (black line), $t=33.6$ $min$ (blue line) and $t=60.3$
$min$ (red line). The plots allow us to show more quantitatively the
inhomogeneous nature at the outer boundary due to the two ejections. The
latter two times correspond approximately to the times when the fronts of the
ejections reach the outer boundary and when the centres of each flux rope
cross it. It is only approximate as the two fronts and the two centres cross
the boundary at slightly different times. The plots of (a) density and (b)
radial velocity both show that the perturbation of the quiet corona increases
in amplitude as the flux ropes reach and cross the outer boundary. When the
front of each ejection first reaches the outer boundary ($t=33.6$ $min$) the
density increases by less than an order of magnitude and is localised in
space. However once the flux rope crosses the surface ($t=60.3$ $min$) the
density increases by two orders of magnitude. In contrast, the radial velocity
exhibits a number of different features. When the fronts from both ejections
reach the outer boundary a peak of $\sim 1000$ $km/s$ occurs, after which the
radial velocity settles to a more plateau structure of around $500$ $km/s$.
Variations in both the density and velocity occur over a wide angular degree.
Even though both of the initial ejections occur far from the equator, their
spatial extension at the outer boundary is wide enough to cross into the
opposite hemisphere. From the cuts taken at the latitudes of either FR1 or FR2
there are clear indications of the occurrence of a CME in the opposite
hemisphere. In contrast the equatorial cut shows a similar contribution from
both FR1 ($\phi\sim 50^{\circ}$ in Fig.8) and FR2 ($\phi\sim 250^{\circ}$) in
Fig.8), as well as from the polarity inversion line plasma flows ($\phi\sim
140^{\circ}$). Finally the temperature of the plasma crossing the outer
boundary qualitatively changes, as first the front crosses the equator and
then the flux rope core. As the fronts from both ejections cross the boundary,
a temperature increase is found. However as the flux rope cores pass through
the boundary this changes to a temperature decrease. This behavior seems
reasonable from a qualitative point of view, as we expect to find higher
temperatures at the front due to the compression and heating of the plasma and
lower temperatures at the flux ropes core due to the cooler denser material
that is present within it. While we see these temperature features we should
also acknowledge that the present model is not designed to model accurately
the temperature evolution as it lacks crucial terms such as thermal conduction
and radiative losses in Eq.13. Even though these terms are not included, they
are not essential for the modelling of the thermal structure of the front and
core of the flux rope ejections.

Figure 9: Mollweide projections (central meridian at longitude
$\phi=180^{\circ}$) at the lower boundary of the MHD simulation at $t=0$ $min$
(left column) and at the outer boundary at $t=0$ $min$ (central column) and
$t=60.3$ $min$ (right column) for the three magnetic field components $B_{r}$,
$B_{\theta}$, $B_{\phi}$.

Fig.9 shows Mollweide projections of the three components of the magnetic
field for (a)-(c) the initial condition of the MHD simulation at the level of
the photosphere and (d)-(f) at the outer boundary, and (g)-(i) at the outer
boundary at the time when both flux ropes cross the outer surface ($t=60.3$
$min$). The first row shows $B_{r}$, the middle row $B_{\theta}$ and the
bottom row $B_{\phi}$. Upon comparing the individual field components at the
different heights and at the different times, it is not surprising that the
complexity and structure of the magnetic field at the solar surface (Fig.9a-c)
is not carried out to the outer corona (Fig.9d-f). It is however essential to
notice that the configuration of the magnetic field at the outer boundary is
the result of the interaction between the outward travelling structures and
the background solar corona. This interaction simplifies the configuration of
the magnetic field and alters the surviving structures in a way that is not
predictable a-priori. In particular while many magnetic features are visible
in Fig.9(a) only two distinct ones appear at the outer boundary during the
eruption (Fig.9(g)). At the outer boundary the flux rope PILs are of different
shape and size, even though the two flux ropes are initially of the same size.
Finally, both ejections show a large negative $B_{\theta}$ patch at the outer
boundary, but opposite and different $B_{\phi}$ patches showing that they
undergo a different rotation.

By comparing Fig.9(g)-(i) with Fig.9(d)-(f), it is clear that the magnetic
field carried by the flux ropes significantly perturbs the outer boundary. The
radial component of the magnetic field exhibits at $t=0$ $s$ a clearly defined
distribution, where there are two regions with opposite polarity that are
separated by a polarity inversion line. Within these locations when the flux
rope ejection occurs and reaches the outer boundary it alters the magnetic
field distribution. The distribution becomes more in-homogeneous and
structured. Within each region that was originally of a single polarity,
opposite polarity patches now appear indicating the flux rope ejections. In
addition to the changes in the radial field, as the initial magnetic field is
prescribed to be only radial at the outer boundary, the $\theta$ and $\phi$
components at the outer boundary show a major change once the erupting flux
rope reaches it. Both erupting flux ropes carry a negative $\theta$ component
creating two areas (green) clearly visible in a more uniform and less intense
background. Finally, a clear signature of the flux ropes can also be seen in
the $\phi$ component of the magnetic field, where once again two very visible
regions are present. An interesting feature is that around FR1 a distinct ring
structure of positive $B_{\phi}$ is present, within which there is contained a
weak $B_{\phi}$. In contrast around FR2, there is a more uniform negative
$B_{\phi}$. The origin of this variation is still under investigation, however
it illustrates the importance of modelling realistic photospheric magnetic
field configurations both before and during the eruption as variations at the
outer boundary must ultimately be due to variations in the low coronal field.
Thus within the present simulation due to the low coronal properties of the
field and the regions in which the flux ropes form, we find significant
differences in the field distribution at the outer boundary. If the low
coronal field is not accurately modelled then such a variation in ICME’s will
not be taken into account.

Figure 10: Cuts of $B_{\theta}$ along $\phi$ at three different latitudes: the
center of FR1 ($\theta=71.4^{\circ}$), the equator ($\theta=90^{\circ}$), and
the center of FR2 ($\theta=110^{\circ}$). a) At $t=0$ and lower boundary. b)
At $t=0$ $min$ (black line), $t=33.6$ $min$ (blue line) and $t=60.3$ $min$
(red line) and outer boundary

In Fig.10 we focus on the time evolution of $B_{\theta}$ at the latitudes of
FR1 (top), the equator (middle) and FR2 (bottom). Fig.10(a) shows the quantity
at the level of the photosphere at time $t=0.0$ just before the eruption
starts. Fig.10(b) shows the results for the outer boundary at the times of
$t=0.0$ (black line), $t=33.6$ (blue line) and $t=60.3$ (red line). Again
comparisons between Fig.10(a) and Fig.10(b) show no evident correlation
between the $B_{\theta}$ values at the lower boundary before the onset of
ejections to that of the resulting configuration at the outer boundary as the
eruption passes through it.

In Fig.10(b) the $\theta$ component of the magnetic field is particularly
important as it gives the out-of-ecliptic plane component at the equator. For
space weather consequences the intensity and orientation of this component
when the ICME encounters the Earth magnetosphere determines the geo-
effectiveness of the disturbance. At the present time it is not clear if the
out-of-ecliptic component of the magnetic field at 4 $R_{\odot}$ is preserved
during the transit of the CME to Earth. However in order to produce accurate
predictions at Earth of its intensity and orientation accurate initial
conditions are required when it is ejected into interplanetary space. In
Fig.10(b) we find that the ejection front generates a modest enhancement in
the $B_{\theta}$ component at all three latitudes. However the overall effect
of each front is quite wide. At the equator when the two fronts reach the
outer boundary we find a preference for negative $B_{\theta}$ at the
longitudes corresponding to the flux ropes. A more significant $B_{\theta}$
enhancement occurs when the flux ropes themselves reach the outer boundary.
This is most intense at the center of the flux rope locations, but still
visible at other longitudes. At the equator, we find a strong negative
$B_{\theta}$ value at the longitudes of the two flux ropes, with a positive
peak between them.

Figure 11: Mollweide projection of $\omega$ at the outer boundary of the MHD
simulation at $t=60.3$ $min$ (central meridian at longitude
$\phi=180^{\circ}$) .

Finally in Fig.11 we show a Mollweide projection of $\omega$ at the outer
boundary at $t=60.3$ $min$ when the flux rope crosses the surface. This is to
determine whether or not the function $\omega$ is appropriate to identify the
flux ropes at higher radial distances. Analysis indicates that the locations
involved with the flux rope ejections show up as visible features in the
$\omega$ distribution. These generally occur at high $\omega$ values even if
it is not possible to define a simple threshold that isolates the flux rope.
It is also clear from Fig.11 that the polarity inversion line is also a very
clear feature due to the relatively lower intensity and less twisted
configuration.

##  5 Discussion, Outlook, and Conclusions

The work presented here is the first step of a longer-term effort to improve
our predictive capability for the occurrence of events on the Sun relevant to
space weather. As part of this process we will also improve our understanding
of the evolution of non-potential magnetic structures in the solar corona and
how they lead to CMEs.

###  5.1 Present Results

In this paper we have carried out global simulations of the Sun where the
entire extent of the corona is considered from the photosphere at $R_{\odot}$
out to $4R_{\odot}$. Two models have been coupled to follow the full life
cycle of flux ropes from formation to eruption. Initially, a quasi-static non-
potential model is used to consider the slow evolution of the coronal magnetic
field and formation of flux ropes up until the point of eruption. To follow
the dynamics of the eruption an MHD simulation is then carried out. As the
quasi-static non-potential simulation does not provide the MHD simulation with
a density profile, we put forward a new technique to prescribe the density
using the twisted nature of the magnetic field. This allows us to produce an
inhomogeneous corona, along with flux ropes that have a density 1-2 orders of
magnitude greater than that of the surrounding corona. The simulation
presented in this paper shows the ejection of two separate magnetic flux ropes
that have formed in the global solar corona at different locations. These flux
ropes have been formed self-consistently in magnetic field configurations
determined directly from observations. In the simulations the onset of
eruption occurs once the tension of the overlying arcades is insufficient to
hold down the underlying flux rope. This occurs depending on the size of the
flux rope and the strength/topology of the overlying arcades and represents
the occurrence of a non-equilibrium. When a non-equilibrium occurs and the
flux rope starts to rise, reconnection then occurs under it which results in a
radially outward Lorentz force which ejects the flux rope. This can be seen as
a two step process of initial slow rise, followed by a faster ejection (see
Mackay and van Ballegooijen, 2006a, b, for a description).

In the present model the flux ropes are ejected out of the computational
domain as the Lorentz forces generated in the corona due to the surface
magnetic reconnection and shear are too large to be contained by the overlying
arcades (Yeates and Mackay, 2009b). During the ejections, both flux ropes show
velocity values along with magnetic and density structures compatible with
standard CME parameters at $4$ $R_{\odot}$.

When the ejections reach the outer boundary at $4$ $R_{\odot}$ we discuss how
the CME passes over this boundary and is injected into the solar wind. We find
that the flux rope ejection into the solar wind occurs over an area of tenths
of square degree, where the ejected plasma is denser compared to that of the
quiet corona. In addition the ejected plasma and magnetic field show a highly
inhomogeneous structure. We highlight here the importance, for this study, of
the approximation we have made to construct a realistic atmosphere around a
magnetic configuration that contains flux ropes (given by Eq. 2,3 and 7). This
allows us to account for the diversity of density in the coronal environment
taking into account the importance of the magnetic field in shaping the
structures in the solar corona, including dense horizontal structures.

An essential result from our modelling work is that the propagation of the
ejection and the consequent perturbation at $4$ $R_{\odot}$ is highly
sensitive to the local pre-eruption conditions as the two analysed ejections
carry different kinematic and plasma properties. Both also show no simple
connection between the flux rope configuration at the lower boundary and its
following evolution. Due to this, each flux rope ejection should be considered
a unique phenomenon and consequently its associated space weather perturbation
will also be unique.

In the present paper we have shown the potential capability of this technique
through a specific example. An essential follow up to this work will be to
compare models of magnetic flux rope ejections with observations. The quasi-
static non-potential global model has been thoroughly tested for the
identification of actual magnetic flux rope formation and times of eruptions.
In the future we will also compare the magnetic flux rope evolution as
described by the MHD simulation with observations of specific events.

###  5.2 Producing an Operational Model

The present model shows many interesting features that with further
development may be a useful tool for application to space weather prediction.
The technique presented could provide realistic time dependent inhomogeneous
boundary conditions that couple the outer corona to the heliosphere. This
coupling still requires development and testing within a heliospheric model.
In particular, we have shown that from a single magnetic configuration we may
produce multiple eruptions, where each of these eruptions have an
inhomogeneous structure to the magnetic field and density as they pass through
the interface between the corona and interplanetary space. Another interesting
outcome is that the magnetic field and density properties of the two eruptions
have significant differences in size and distribution. Therefore to some
degree each eruption is unique, based on the initial properties of the flux
rope and the surrounding non-potential field produced in the quasi-static
evolution model.

For practical applications any space weather forecasting tool needs to have
two key qualities: accuracy and efficiency. Accuracy is required in order to
avoid false warnings and to ensure key parameters such as the arrival time of
the CME at Earth and the orientation of this magnetic field and density are
reliably determined. In terms of this our model sets high standards of
accuracy as firstly it evolves the global corona from magnetic configurations
determined from the time evolution of observed magnetograms. This produces
non-potential magnetic configurations very close to observations and the
global non-potential coronal model has proved highly reliable in predicting
the locations, helicity and timing of flux rope formations, especially after
the model has taken into account the emergence of new magnetic flux. Secondly,
it solves the full MHD equations to follow the dynamical eruption itself. For
time critical predictions efficiency is essential to run useful simulations
that can output results before or close to when an event has taken place. In
our tests, the global non-potential model is more than $10^{4}$ times
computationally faster than a full MHD simulation. This is mostly due to the
lower number of equations to solve and to the higher time steps involved when
Alfvén waves do not need to be resolved. If a single MHD model was used, then
the vast majority of the physical time of the simulation would be during the
formation of the flux rope (days to weeks) compared to hours for the eruption.
Thus use of the global non-potential model to produce the initial eruptive
configuration is twofold, first it solves fewer equations and secondly it is
much more efficient. This means that only a small part of the physical time
(once the ejection sets in) needs to be considered and modeled with a full MHD
simulation. The combined model is in principle capable of predicting flux rope
formation and subsequent onset of ejection days before it occurs, where the
MHD simulation would only need to simulate a few hours of physical time which
is well within current computational capacity. While this is a positive aspect
of this approach, before such a technique can become viable as a semi-
autonomous, factually accurate, operational space weather forecasting model, a
number of practical and scientific obstacles need to be overcome. These
include

  1. 1.

Flux emergence accuracy.
The current global non-potential model is highly dependent on a semi-automated
determination of flux emergence from observations. This is key as the
emergence of new flux can significantly alter the magnetic configuration of
the solar corona. Therefore a more accurate treatment must be developed.
Techniques are under development and testing that allow for the direct
assimilation of normal component magnetograms across all longitudes and
latitudes (see Weinzierl et al. (2016b) and Weinzierl et al. (2016a)). One
issue with the direct assimilation of magnetogram data into the simulation, is
that only a portion of the global photospheric field may be observed at any
given time. However, if magnetogram data were also available from an L5 based
magnetograph this would significantly enhance the accuracy of the models. In
the recent paper by Mackay et al. (2016) it has been shown that such L5
observations could increase the accuracy of global quantities relevant to
eruptions on the Sun by anywhere from 26-40$\%$. Pevtsov et al. (2016) reports
similar improvements for the estimations of solar wind and plasma outflows as
well.

  2. 2.

Identification of erupting flux ropes.
Although the global non-potential model is able to describe the formation of
magnetic flux ropes, it cannot account for the whole physics of an eruption.
For the latter MHD simulations are employed. Therefore during the global non-
potential simulations automated quantitative analysis of the properties of the
coronal magnetic field must be carried out to first identify the existence of
flux ropes and then the onset of ejections. A number of candidates for this
task include instability analysis, measurement of thresholds, or recognizing
critical indices in the parameters of the magnetic field configuration.

  3. 3.

Coupling AMRVAC back to the Global Non-potential Model.
One of the main strengths of the global non-potential modelling technique is
the continuous nature of the simulations over days to years, in which the
transport of magnetic flux and magnetic helicity from low to high latitudes is
followed. Once magnetic field configuration are imported from the global non-
potential model into ARMVAC the continunity of the global non-potential
simulations ends. Thus in order to allow a continuous modelling of the solar
coronal evolution after the eruption, it is vital to couple the related MHD
solution back to the global non-potential model. This will then be used to
start a new segment of the quasi-static evolution. Currently this is not
possible due to technical limitations where the primary variable in the Global
Non-potential Model is the vector potential $\vec{A}$, while in MPI-AMRVAC it
is the magnetic field $\vec{B}$. To resolve this issue a new MPI-AMRVAC module
that solves the MHD equations using $\vec{A}$ is under development.

  4. 4.

Matching the simulations with the lower boundary of space weather models.
Most space weather forecast tools focus on the physical domain extending from
$0.1AU$ to $>1$ $AU$. The lower boundary of this range corresponds to about
$21$ $R_{\odot}$ and is significantly higher than the outer boundary of the
simulation shown here. To bridge the gap between the models a number of
options will be investigated in the future. These include using a simple solar
wind model above $4$ $R_{\odot}$ (for a review: Echim et al., 2011) or to
scale the plasma and velocity fluxes from $4$ $R_{\odot}$ up to $21$
$R_{\odot}$. In the latter approach we would consider it as a perturbation to
a background solar wind that is included in the applied space weather forecast
tool. In future studies we will consider the validity and technical aspects of
both of these approaches.

In conclusion, the present work shows that in the future the coupling of the
Global Model with MHD simulations may become a useful and essential tool to
bridge magnetogram observations with space weather modellling. This is due to
the complexity of the magnetic configurations generated in the solar corona
and the interaction between propagating flux ropes and the coronal
environment. The technique we put forward is a viable one for future space
weather forecasting tools and has the potential not only to improve our
forecasting capabilities, but also to lead to a better understanding of the
physics of CME origin.

###### Acknowledgements.

This research has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 647214). This work used the DiRAC Data Centric system at
Durham University, operated by the Institute for Computational Cosmology on
behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk. This equipment was
funded by a BIS National E-infrastructure capital grant ST/K00042X/1, STFC
capital grant ST/K00087X/1, DiRAC Operations grant ST/K003267/1 and Durham
University. DiRAC is part of the National E-Infrastructure. We acknowledge the
use of the open source (gitorious.org/amrvac) MPI-AMRVAC software, relying on
coding efforts from C. Xia, O. Porth, R. Keppens. D.H.M. would like to thank
STFC and the Leverhulme Trust for their financial support. ARY was supported
by STFC consortium grant ST/N000781/1 to the universities of Dundee and
Durham. The editor thanks two anonymous referees for their assistance in
evaluating this paper.

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