# Eruptivity Criteria for Two-dimensional Magnetic Flux Ropes in the Solar
Corona

Oliver E. K. Rice and Anthony R. Yeates ∗

###### Abstract

## 1

We apply the magneto-frictional approach to investigate which quantity or
quantities can best predict the loss of equilibrium of a translationally-
invariant magnetic flux rope. The flux rope is produced self-consistently by
flux cancellation combined with gradual footpoint shearing of a coronal arcade
which is open at the outer boundary. This models the magnetic field in
decaying active regions on the Sun. Such a model permits two types of
eruption: episodic small events caused by shearing and relaxation of the
overlying arcade, and major eruptions of the main low-lying coronal flux rope.
Through a parameter study, we find that the major eruptions are best predicted
not by individual quantities but by thresholds in the ratios of squared rope
current to either magnetic energy or relative magnetic helicity. We show how
to appropriately define the latter quantity for translationally-invariant
magnetic fields, along with a related eruptivity index that has recently been
introduced for three-dimensional magnetic fields. In contrast to previous
configurations studied, we find that the eruptivity index has only a weak
predictive skill, and in fact is lower prior to eruption, rather than higher.
This is because the overlying background magnetic field has the same direction
as the arcade itself. Thus we propose that there are a whole class of solar
eruptions that cannot be predicted by a high eruptivity index.

##  2 Keywords:

Sun, Solar corona, Magnetic Fields, Eruptions, Helicity

##  3 Introduction

Flux ropes are twisted bundles of magnetic flux in the solar corona (Liu,
2020). Accurately predicting their behaviour is essential for reliable space
weather predictions, as unstable flux ropes can erupt and lead to large
coronal mass ejections (CMEs; Forbes et al., 2006; Chen, 2011). The causes of
such eruptions are not yet indisputably understood and a variety of mechanisms
have been proposed. These have been explored with approaches ranging from
analytic two-dimensional models to three-dimensional full magnetohydrodynamic
(MHD) simulations.

Analytic two-dimensional models of flux rope behaviour date back to Kuperus
and Raadu (1974) and van Tend and Kuperus (1978), who modelled a horizontal
line current and its interaction with a specified background magnetic field.
They established conditions on the current that allow it to be stable, and
showed that, for an eruption to occur, the background magnetic field strength
must decrease rapidly with increasing height.

Further analytic approaches have introduced the torus instability (e.g., Kliem
and Török, 2006) where the flux rope is modelled as a current ring rather than
a line current. Rather than a condition on the decay of the background field
with altitude, it is instead proposed that for instability to occur the
background field component orthogonal to the torus must decrease sufficiently
quickly. It has since been shown that the conditions for such an instability
are essentially the same as for instability of a line current, and they are
just two special cases of a continuous theory of more general current paths
(Kliem et al., 2014). Although these works provide useful theoretical
background as to the nature of flux rope behaviour, they are difficult to
relate to physical ropes as these tend to be significantly more complex and
much of the physics is by necessity not taken into account. There are indeed
other conflicting explanations for instability, such as that of Ishiguro and
Kusano (2017), who propose that fast magnetic reconnection can occur below a
flux rope even if the overlying magnetic field does not decay with altitude.

Full 3D MHD simulations of flux rope formation and behaviour are also now
possible. Leake et al. (2013, 2014) presented a series of such simulations,
based on magnetic flux emergence into a preexisting field. Some ropes were
found to be eruptive and others stable, depending on the configuration of the
background magnetic field. Pariat et al. (2017) then analysed various
properties of the system for these simulations, looking for properties that
could predict whether or not the flux rope would erupt. Most diagnostics, such
as the magnetic free energy and relative magnetic helicity, were shown not to
have any strong correlation to eruptivity. However, they identified an
“eruptivity index” – the fraction of the relative magnetic helicity that is in
the current-carrying component – which became large only in erupting cases. In
a different set of numerical simulations, where coronal flux ropes are created
by a variety of solar surface footpoint motions, this eruptivity index again
increased prior to eruption and the eruption was found to occur for a fixed
value of the index, irrespective of the pattern of footpoint motions that
injected the energy (Zuccarello et al., 2018). However, while promising, a
complete theoretical understanding of this eruptivity index and its generality
remains lacking.

Our modeling approach is chosen as a compromise between the simple analytic
models of the past and the expensive full MHD simulations now possible. We use
the magneto-frictional model, pioneered by Yang et al. (1986), whereby a
fictional velocity field is determined explicitly from the magnetic field as
opposed to using the fluid equations. For tracking the quasi-static injection
of magnetic energy into the corona through solar surface motions, the model
provides a viable alternative to full MHD simulations, at a fraction of the
computational cost. Magneto-frictional models have previously been coupled
with time-dependent lower boundary conditions based on imposed surface flux
transport (e.g., Yeates et al., 2008) and realistic flux rope formation, as
well as loss of equilibrium, is observed in such simulations (Mackay and van
Ballegooijen, 2006; Yeates and Mackay, 2009; Lowder and Yeates, 2017; Hoeksema
et al., 2020). Flux ropes are also formed on a smaller scale when the model is
driven by high-resolution observations within active regions (e.g., Yardley et
al., 2018). Since the magneto-frictional model neglects the full equation of
motion, it cannot accurately follow the dynamics once a flux rope loses
equilibrium and erupts. Nevertheless, if ropes produced by magneto-friction
are used to initialise full MHD simulations, it is found that the magnetic
flux ropes do indeed lose equilibrium at the same point, and can lead to CME
eruptions (Kliem et al., 2013; Pagano et al., 2013). Thus magneto-friction can
act as an accurate model for the pre-eruption evolution.

In our study, we use the magneto-frictional approach but simplify it to a
2.5-dimensional (translationally invariant) cartesian domain, which
significantly increases computational speed while still exhibiting the
fundamental features of fully 3D models. In particular, both the flux rope and
the (sheared) overlying arcade are formed self-consistently by flux
cancellation and shearing motions on the photosphere. In our model these
shearing motions are purely large-scale, essentially modelling the
differential rotation. As such, we focus on the more gradual evolution in
decaying active regions – a potentially different scenario to the active
region eruptions where the eruptivity index was previously studied, but
nevertheless an important source of solar eruptions.

In a similar manner to the work of Pariat et al. (2017), we attempt to find a
scalar quantity that acts as a proxy for the eruptivity of our 2.5D flux rope.
The large number simulations we are able to run (in the order of $500$) allows
for an extensive parameter study based on the variation of the magneto-
friction coefficient and the rate of photospheric flux cancellation. Using a
probabilistic approach, a large number of eruptive and non-eruptive ropes will
be compared against diagnostic measurements of the system at various points of
their evolution. We note that, in addition to the formation of flux ropes, our
model exhibits “arcade eruptions” whereby we observe periodic reconnection at
the top of a sheared magnetic arcade. It is possible that these represent
streamer blowouts or even “stealth CMEs”, as coined by Webb and Howard (2012),
since they are characterised by the lack of a detectable signature on the
solar surface. The nature and period of these eruptions (around every 25-30
days) matches well between our 2.5D model, global magneto-frictional
simulations and observations (Bhowmik et al., 2021). However, the main focus
of this paper is not on arcade eruptions but “flux rope eruptions”, when we
observe magnetic reconnection below a flux rope and the rope itself moves
upwards out of the domain. Such eruptions are larger than arcade eruptions and
it is likely that in reality a significant number of CMEs occur as a result of
such a mechanism.

We begin in Section 4 by outlining the mathematical basis of our model,
including the mechanism by which we represent surface shearing (influenced by
the differential rotation of the sun at different latitudes), photospheric
diffusion and the effect of the solar wind. We identify the variable
parameters in the model and discuss which of these we can use to produce an
array of differing flux rope behaviour. We then discuss the system diagnostics
we have chosen to focus on, including a newly-defined measure of the relative
helicity in two dimensions (Section 4.4.8). This measure allows us to compare
our results against 3D equivalents and calculate the eruptivity index. The
results of our parameter study are presented in Section 5, and the findings
discussed in Section 6.

##  4 Methods

###  4.1 Magneto-frictional model

Rather than using full magnetohydrodynamics (MHD), we adopt a simplified
approach that has been developed for global modelling of the solar coronal
magnetic field: the magneto-frictional model (e.g., Mackay and Yeates, 2012).
On a global scale, this technique has been applied with time-dependent lower
boundary conditions from surface flux transport models (e.g., Yeates et al.,
2008; Yeates, 2014). We adopt a 2.5-dimensional (translationally invariant)
version of this approach.

In full MHD models, the velocity field, ${\bf v}$, is determined by the
momentum equation

$\rho\frac{\mathrm{D}{\bf v}}{\mathrm{D}t}={\bf j}\times{\bf B}-\nabla
p-\rho\nabla\Psi,$ (1)

where $\rho$ is the fluid density, $p$ is the fluid pressure, $\Psi$ is the
gravitational potential, and ${\bf j}=\mu_{0}^{-1}\nabla\times{\bf B}$ is the
current density. This is coupled to the induction equation

$\frac{\partial{\bf B}}{\partial t}=\nabla\times({\bf v}\times{\bf
B})-\eta\nabla\times\bf j,$ (2)

along with additional fluid equations to close the system. In the magneto-
frictional method, equation (2) is retained, but the inertial terms, pressure
gradients and gravity are neglected and instead a frictional velocity is
imposed as

$\nu{\bf v}=(\nabla\times{\bf B})\times{\bf B}.$ (3)

Combined with the induction equation, this leads to monotonic relaxation
towards a stationary force-free field with ${\bf j}\times{\bf B}={\bf 0}$. The
friction coefficient $\nu$ is typically given the form $\nu=\nu_{0}|{\bf
B}|^{2}$ (with some minimum value imposed) so that the overall evolution is
independent of the magnitude of ${\bf B}$ and relaxation is not unduly slow
near to magnetic null points (Yang et al., 1986).

In the outer corona, the solar wind outflow prevents the magnetic field from
being force-free, but this effect can be approximated in the magneto-
frictional model by relaxing towards an equilibrium with a specified outflow
${\bf v}_{\rm out}$, thus choosing ${\bf v}$ according to

$\bf v=\frac{(\nabla\times{\bf B})\times{\bf B})}{\nu}+{\bf v}_{\rm out}$ (4)

This ad hoc approach was introduced by Mackay and van Ballegooijen (2006), and
has subsequently been used in global magneto-frictional models of solar and
stellar coronae (e.g. Yeates, 2014; Gibb et al., 2016; Mackay et al., 2018;
Meyer et al., 2020).

###  4.2 Our 2.5-Dimensional Model

In our model, we simplify the problem by using a Cartesian coordinate system,
and removing any dependence of the solution on the $y$ coordinate. The domain
ratio is taken to be $(x:z)=(2:1)$. The resulting 2.5-dimensional system
captures many of the essential features of the evolution in the lower solar
corona, while affording a vast reduction in computational expense. This has
allowed us to run an extensive parameter study, comprising hundreds of
simulations.

The state of the system can be described in terms of a vector potential ${\bf
A}(x,z,t)$, such that

${\bf B}(x,z,t)=\nabla\times{\bf A}(x,z,t).$ (5)

The vector field ${\bf A}$ is then evolved according to

$\frac{\partial{\bf A}}{\partial t}=-{\bf E},$ (6)

where ${\bf E}$ is the electric field, satisfying Ohm’s Law

${\bf E}=\eta{\bf j}-{\bf v}\times{\bf B},$ (7)

in terms of the magnetic field ${\bf B}$, current density ${\bf j}$ and
frictional velocity ${\bf v}$ as given in equation (4). Here $\eta$ is a
constant representing coronal turbulent diffusivity, which is assumed to be
much more significant than ohmic diffusivity in the highly-conducting corona,
though still smaller than the other effects in the model. The outflow velocity
used to represent the solar wind is taken to be

${\bf v}_{\rm out}=v_{1}\left(\frac{z}{z_{1}}\right)^{10}{\bf{e}}_{z},$ (8)

so the effect is minimal near the lower boundary and increases rapidly near
the top boundary at $z=z_{1}$ at which the parameter $v_{1}$ gives the maximum
speed.

The simulations are initialised with an “outflow field” (Rice and Yeates,
2021), a variation on a potential arcade that takes into account the effect of
${\bf v}_{\rm out}$, such that the system is initially in equilibrium. As seen
in Figure 1 these fields are similar to potential arcades, except in the upper
half of the domain where the field lines open out to become more vertical.

The boundary conditions determine the overall behaviour of the system. On the
top boundary ($z=z_{1}$) we have the condition that ${\bf B}\perp{\bf n}=0$,
ensuring that the magnetic field lines are vertical/radial here. This
condition is consistent with observations that the field lines in the real
corona become almost radial above a certain altitude. The notable exception to
this is during eruptions, but these last a relatively short time and occur in
our model even with the radial condition imposed. On the sides of the domain
we set ${\bf B}\cdot{\bf n}=0$, i.e. there is no magnetic flux through the
sides. The lower boundary condition is more complex, and incorporates two
effects: photospheric shearing and photospheric diffusion.

In our model, the photospheric shearing is assumed to originate from the
differential rotation of the solar surface, namely that the Sun rotates more
quickly at the equator than the poles. A magnetic arcade that has footpoints
at different latitudes will thus be sheared, stretching the field lines along
the polarity inversion line (PIL). We represent this shearing by imposing a
velocity in the out-of-plane ($y$) direction at the lower boundary, following
the profile

${\bf v}(x,0)=(0,V_{y0},0),\qquad V_{y0}=\sin(\pi x).$ (9)

This profile is chosen as it ensures symmetry and simplicity while
approximating the qualitative effect of differential rotation. The
photospheric diffusion, $\eta_{0}$ represents the large-scale effect of
supergranular flows, and is also imposed on the lower boundary (Mackay and van
Ballegooijen, 2006). The combined effect of both shearing and diffusion
results in the lower boundary condition

${\bf E}(x,0)=-V_{y0}(x,0)B_{z}(x,0){\bf
e}_{x}-\frac{\eta_{0}}{\mu_{0}}\frac{\partial B_{z}(x,0)}{\partial x}{\bf
e}_{y}.$ (10)

The effect of this is to bring the arcade footpoints closer together,
eventually reconnecting to form a twisted flux rope (van Ballegooijen and
Martens, 1989). Thus the ropes form self-consistently, without the need for
any imposed flux emergence.

Numerically, equation (6) is solved on a finite-difference staggered grid
(Yee, 1966) similarly to (Cheung and DeRosa, 2012), with typical resolutions
of $256\times 128$ cells. The code is initialised with Python but runs using
Fortran 90, making use of MPI parallelisation.

###  4.3 Parameters and Units

There are a number of free parameters in our model that can be easily varied.
The aim is to produce a set of simulations that model realistic behaviour
well, but provide enough variation to be able to make predictions of future
behaviour. The parameters in the model are:

  * •

$\eta$ : The coronal diffusion.

  * •

$\nu_{0}$ : The magneto-frictional friction coefficient.

  * •

$v_{1}$ : The imposed solar wind outflow speed.

  * •

$\eta_{0}$: The photospheric diffusion.

  * •

$V_{y0}$: The photospheric shearing velocity.

However, in our analysis we fix some of these parameters. Firstly, we take the
coronal diffusion to be very small, $\eta\approx 1\times 10^{-6}$, so that its
effect on short-term flux rope behaviour is negligible. (The corresponding
diffusion time across the height of the domain would be $10^{6}$ time units.)
Secondly, we effectively set the time unit by fixing the maximum shearing
velocity $V_{y0}$ to unity. Finally, we fix the outflow speed $v_{1}=50$. This
is reasonable since the outflow velocity has little effect on flux rope
behaviour; the solar wind mainly serves to induce currents in the upper
corona, at the top of the domain and far from the flux rope formation region.
The only notable effect of an increased outflow velocity is a slight increase
in the frequency of arcade eruptions. We choose $v_{1}=50$ to reflect the fact
that the solar wind outflow is faster than the shearing velocity from
differential rotation.

This leaves two remaining parameters: $\nu_{0}$ and $\eta_{0}$. These can be
varied considerably (between certain bounds) and still exhibit realistic
behaviour. By running simulations with different combinations of these
parameters we observe different flux rope behaviours, with a good mix of
eruptive and non-eruptive simulations. The results from this parameter study
are described in Section 5.

Note that we adopt dimensionless units throughout this paper, with a maximum
photospheric shearing velocity of unity according to equation (9). For
comparison to the real corona, one would choose the length unit – equivalently
the height of the domain, $z_{1}$ – and specify an observed shearing velocity
caused by the differential rotation of the coronal arcade. These would then
fix the time that corresponds to one dimensionless time unit in our paper. For
example, if we take the angular velocity of differential rotation at latitude
$\lambda$ on the Sun to be
$\Omega(\lambda)=0.18-2.396\sin^{2}\lambda-1.787\sin^{4}\lambda$ degrees per
day (Snodgrass, 1983) and choose the latitudinal limits of our domain to be
$10^{\circ}$ to $40^{\circ}$, this results in a maximum shearing velocity
$|v_{\phi}|\approx 0.086\,\mathrm{km}\,\mathrm{s}^{-1}$. Taking
$z_{1}=1.8\times 10^{5}\,\mathrm{km}$ (half of the the latitudinal extent)
would then imply that a code time unit is of the order $\approx
25\,\mathrm{days}$.

###  4.4 System Diagnostics

In this section we briefly describe the diagnostic measurements of the system
used to identify events and ultimately try to make predictions of future
behaviour such as eruptions. There are innumerable measurements that could
theoretically be taken of the state of the magnetic field, but for practical
purposes we have selected nine, which are as follows:

####  4.4.1 Open Flux

The open flux is the sum of the unsigned magnetic flux through the top
boundary of the domain. (Due to the symmetry of the system, the sum of the
signed flux is zero.) The open flux increases due to the effect of the solar
wind, as the field lines become more vertical near the top of the domain and
fewer of them loop back down in the arcade.

####  4.4.2 Maximum Current

For practical computation we set $\mu_{0}=1$ and define the current density as
the curl of the magnetic field: ${\bf j}=\nabla\times{\bf B}$. As a
diagnostic, we take the maximum value that this attains in the domain. A
potential field has zero current and an outflow field only has current
concentrated near the top boundary, due to the effect of the solar wind.

####  4.4.3 Rope Current

We also measure the integral of the current within the rope, in the direction
of the rope axis (the $y$ direction). In our two-dimensional model the rope is
easily identifiable as it consists of the infinitely-long field lines that
never reach either the photospheric or outer boundary. As the rope is in
general in the lower half of the domain, this current is unaffected by the
behaviour at the top boundary.

####  4.4.4 Magnetic Energy

The magnetic energy is defined in our unitless system as $E_{\rm
M}=\int_{0}^{1}\int_{-1}^{1}\frac{1}{2}|{\bf B}|^{2}\,\mathrm{d}x\mathrm{d}z$.
A potential arcade corresponds to a minimum energy solution for given normal-
field boundary conditions, and an outflow field has only a slightly increased
energy (less than 1%). There is a large increase in magnetic energy during
flux rope formation as the system evolves far from potentiality.

####  4.4.5 Free Magnetic Energy

In addition to the overall magnetic energy, we can calculate the “free
magnetic energy”, defined as f $E_{\rm F}=E_{\rm
M}-\int_{0}^{1}\int_{-1}^{1}\frac{1}{2}|{\bf
B}_{P}|^{2}\,\mathrm{d}x\mathrm{d}z$, where ${\bf B}_{P}$ is a potential
magnetic field as calculated in Section 4.4.8 below. This quantity has the
advantage that the contribution to the overall energy from the background
magnetic field is lessened, such that the contribution from the non-potential
flux rope is more significant.

####  4.4.6 Poloidal Rope Flux

This is a measure of the poloidal (in-plane) magnetic flux contained within
the flux rope (the region with infinitely-long field lines), defined as the
flux intersecting a chord between the centre of the rope and the edge of the
rope (usually the lower edge of the domain).

####  4.4.7 Axial Rope Flux

This is defined as the integral of the magnetic flux in the rope (the region
with infinitely-long field lines) in the $y$ direction, parallel to the axis
of the rope itself. This appears to be roughly correlated to the rope current.

####  4.4.8 Relative Helicity

The classical helicity within a volume $V$ would be defined as
$h(V)=\int_{V}{\bf A}\cdot{\bf B}\,dV,$ where ${\bf A}$ is the vector
potential of ${\bf B}$. This quantity is dependent in general on the gauge of
${\bf A}$, and so we use the alternative relative helicity instead (Berger and
Field, 1984). In a 3D domain, this would be calculated by finding a potential
field ${\bf B}_{P}$ matching the original magnetic field on the boundary, and
a corresponding vector potential, ${\bf A}_{P}$. The relative helicity would
then be

$H_{R}=\int_{V}({\bf A+A}_{P})\cdot({\bf B-B}_{P})\,dV.$ (11)

Care is required to define the relative helicity for our 2.5D field. A two-
dimensional helicity measure for $h(V)$ has been proposed before (Hu et al.,
1997), but we are not aware of a previously published two-dimensional analogue
for the relative helicity.

We start by considering the 3D formula (11) on a finite volume $V_{y_{1}}$,
where $-1<x<1$, $-y_{1}<y<y_{1}$ and $0<z<1$. Although ${\bf B}(x,z)$ in our
2.5D field is independent of $y$, the corresponding potential reference field
${\bf B}_{P}$ will, in general, vary in the $y$ direction. This arises from
the fact that it is potential, coupled with the need to match $B_{Py}(x,\pm
y_{1},z)=B_{y}(x,z)$ on $y=\pm y_{1}$. We define the relative helicity per
unit length to be

$H_{R}^{(2.5D)}=\lim_{y_{1}\to\infty}\frac{1}{2y_{1}}\int_{V_{y_{1}}}({\bf
A}+{\bf A}_{P})\cdot({\bf B}-{\bf B}_{P})\,\mathrm{d}V,$ (12)

where ${\bf B}_{P}$ and its vector potential ${\bf A}_{P}$ are calculated on
$V_{y_{1}}$. However, a physically meaningful helicity measure for our 2.5D
field cannot possibly require integration in $y$. We will show that
$H_{R}^{(2.5D)}$ not only converges as $y_{1}\to\infty$ but can indeed be
calculated by a two-dimensional integral in $x$ and $z$.

To do this, we decompose ${\bf B}_{P}$ into three components,

${\bf B}_{P}=\nabla\phi_{0}(y)+\nabla\phi_{1}(x,z)+\nabla\phi_{2}(x,y,z),$
(13)

where the first component is a uniform field accounting for the net flux in
the $y$ direction,

$\phi_{0}=\Phi_{0}y,\qquad\Phi_{0}=\frac{1}{2}\int_{0}^{1}\int_{-1}^{1}B_{y}(x,z)\,\mathrm{d}x\mathrm{d}z,\\\
$ (14)

and the other two components are both potential fields satisfying
$\Delta\phi_{1}=\Delta\phi_{2}=0$, with corresponding boundary conditions

$\displaystyle\frac{\partial\phi_{1}}{\partial x}(\pm
1,z)=0,\qquad\frac{\partial\phi_{1}}{\partial
z}(x,0)=B_{z}(x,0),\qquad\frac{\partial\phi_{1}}{\partial z}(x,1)=B_{z}(x,1),$
(15) $\displaystyle\frac{\partial\phi_{2}}{\partial x}(\pm
1,y,z)=\frac{\partial\phi_{2}}{\partial
z}(x,y,0)=\frac{\partial\phi_{2}}{\partial
z}(x,y,1)=0,\qquad\frac{\partial\phi_{2}}{\partial y}(x,\pm
y_{1},z)=B_{y}(x,z)-\Phi_{0}.$ (16)

Notice that $\nabla\phi_{0}$ and $\nabla\phi_{1}$ are independent of both $y$
and $y_{1}$. The $y$ dependence is concentrated only in $\nabla\phi_{2}$.

The potential $\phi_{2}$ has the important property that, as $y_{1}$
increases, it becomes more and more concentrated near to the end boundaries
$y=\pm y_{1}$, irrespective of $B_{y}(x,z)$. To see this, note that in the
Cartesian domain $V_{y_{1}}$, the solution for $\phi_{2}$ may be written as a
Fourier series

$\phi_{2}(x,y,z)=\sum_{m,n\neq
0}c_{m,n}\cos\left(\frac{m\pi(x+1)}{2}\right)\cos\left(n\pi z\right)\sinh(k\pi
y),$ (17)

where $m$, $n$ are integers, $k^{2}=(m/2)^{2}+n^{2}$, and the sum includes all
terms except $m=n=0$ (which has been separated as the $\phi_{0}$ component).
The coefficients are then determined by the boundary condition on
$\partial\phi_{2}/\partial y$, which gives

$c_{m,n}=\frac{2}{k\pi\cosh(k\pi
y_{1})}\int_{0}^{1}\int_{-1}^{1}B_{y}(x,z)\cos\left(\frac{m\pi(x+1)}{2}\right)\cos\left(n\pi
z\right)\,\mathrm{d}x\mathrm{d}z.$ (18)

Now consider the value of $\phi_{2}(x,ay_{1},z)$ for some fixed fraction
$|a|<1$. Then

$\phi_{2}(x,ay_{1},z)=\sum_{m,n}\frac{2\sinh(k\pi ay_{1})}{k\pi\cosh(k\pi
y_{1})}F(x,z),$ (19)

where $F(x,z)$ contains the $x$ and $z$ dependence from (17) and (18). Noting
that

$\lim_{y_{1}\to\infty}\frac{\sinh(k\pi ay_{1})}{\cosh(k\pi
y_{1})}=\lim_{y_{1}\to\infty}\frac{\cosh(k\pi ay_{1})}{\cosh(k\pi y_{1})}=0,$
(20)

we see that the non-zero part of $\nabla\phi_{2}$ becomes an increasingly
smaller fraction of the domain length as $y_{1}\to\infty$. This is illustrated
in Figure 2. It follows that the contribution from $\nabla\phi_{2}$ to
$H_{R}^{(2.5D)}$ in equation (12) vanishes in the limit, so that

$H_{R}^{(2.5D)}=\int_{0}^{1}\int_{-1}^{1}({\bf A}+{\bf
A}_{P}^{(2.5D)})\cdot({\bf B}-{\bf B}_{P}^{(2.5D)})\,\mathrm{d}x\mathrm{d}z,$
(21)

where ${\bf B}_{P}^{(2.5D)}(x,z)=\nabla\phi_{1}(x,z)+\Phi_{0}{\bf e}_{y}$. The
gauge of ${\bf A}_{P}^{(2.5D)}$ does not affect the integral (as usual for
$H_{R}$), so one is free to choose an ${\bf A}_{P}^{(2.5D)}$ that is
independent of $y$ and hence evaluate $H_{R}^{(2.5D)}$ with a purely 2D
integral. This is our approach in this paper.

####  4.4.9 Eruptivity Index

The 3D expression that has been proposed (Pariat et al., 2017) as an
eruptivity index is $\lvert H_{J}/H_{R}\rvert$, where

$H_{J}=\int_{V}({\bf A-A}_{P})\cdot({\bf B-B}_{P})\,dV$ (22)

is the helicity of the current-carrying part of the field (Berger, 1999) and
$H_{R}$ is the relative helicity as above. We define the 2.5D version of
$H_{J}$ analogously to $H_{R}^{(2.5D)}$, thus we consider the ratio $\lvert
H_{J}^{(2.5D)}/H_{R}^{(2.5D)}\rvert$ with

$H_{J}^{(2.5D)}=\int_{0}^{1}\int_{-1}^{1}({\bf A}-{\bf
A}_{P}^{(2.5D)})\cdot({\bf B}-{\bf B}_{P}^{(2.5D)})\,\mathrm{d}x\mathrm{d}z.$
(23)

####  4.4.10 Ratios

We also consider the ratios between the above quantities, generally defined
such that the ratios are independent of the magnetic field strength.

##  5 Results

###  5.1 Qualitative Behaviour

We first illustrate the two types of eruption that can occur in the system.

####  5.1.1 Zero Photospheric Diffusion

When we have no photospheric diffusion ($\eta_{0}=0$), the effect of the
shearing causes the arcade footpoints to move only in the $y$ direction. The
magnetic energy increases as the system is no longer in a relaxed state, but
it is not possible for flux ropes to form in the low corona as the footpoints
are not brought closer together and there is no flux cancellation. In this
case, the only free parameter is the friction coefficient $\nu_{0}$.

We choose $\nu_{0}$ of the order unity, in our code units. In general, the
equivalent friction coefficents used in 3D global magneto-frictional
simulations (e.g., Yeates and Hornig, 2016) are slightly smaller than this if
directly compared, but this is by no means a precise measurement. There is
thus a compromise between these more realistic values and the significant
improvements in computational speed gained from increasing $\nu_{0}$. Altering
the domain shape also has a significant effect on the ideal $\nu_{0}$, but for
our $(x,z)=(2:1)$ ratio setting $\nu_{0}\approx 0.5$ produces realistic
behaviour. For $\nu_{0}$ significantly larger than this, the frictional
relaxation is unrealistically slow relative to the footpoint shearing.

Consequently, our parameter study is confined to $0.5<\nu_{0}<2$. For these
$\nu_{0}$ values, the magnetic arcade becomes more sheared in the $y$
direction, as expected. After a certain period during which the magnetic
energy and open flux increase gradually, we observe fast magnetic reconnection
at the top of the domain accompanied by a sharp decrease in these diagnostics.
This is what we refer to as an “arcade eruption” (Linker and Mikic, 1995). A
sequence of snapshots of one of these eruptions is shown in Figure 3. The
first eruption is shown in the centre pane at time $t=2.92$.

In general this process then repeats, leading to periodic eruptions occuring
every time unit or so, as shown clearly by the oscillation of the diagnostic
measures in Figure 4. We observe that the arcade becomes more sheared up to
time $t=2.76$, at which point there is an eruption, denoted by a blue circle.
The process then repeats, building up to another eruption at $t=4$ and so on,
with the times of eruptions (as determined by the mid-point of the drop in
open flux) shown by blue circles.

During an arcade eruption there is a rapid decrease in open flux as the
reconnection at the top of the domain results in the closing down of field
lines at the top of the arcade. The corresponding decrease in magnetic energy
occurs as these newly closed field lines are in a more potential state
immediately after the eruption, although the free energy has a non-zero
“floor” because the background field is non-potential due to the outflow
velocity. The peaks in current during an eruption occur at the top boundary at
the current sheet that temporarily forms above the PIL. The relative helicity
does not follow the same pattern as the other diagnostics and instead
increases during an eruption. This increase results from the fact that the
newly-closed potential field lines at the top of the arcade now have a mutual
“linkage” with the still sheared field lines beneath.

####  5.1.2 Nonzero Photospheric Diffusion

When the photospheric diffusion $\eta_{0}$ is nonzero, we observe the
formation of flux ropes. These are characterised by twisted bundles of
magnetic field lines that do not meet the boundary of the domain. As a result,
the flux ropes are effectively infinitely long as there is no variation in the
$y$ direction. At the same time, we observe similar periodic buildups and
releases in magnetic energy to the case with no photospheric diffusion. These
overlying arcade eruptions do not affect the amount of poloidal magnetic flux
within the flux rope, but they do cause the rope to oscillate vertically.

However, in addition to these eruptions, we also observe more significant
events whereby the flux rope itself erupts, which in a 2D setting causes it to
move rapidly upward out of the domain. Unlike with arcade eruptions, we
observe magnetic reconnection below the flux rope, between it and the polarity
inversion line on the lower boundary. There is a significant decrease in free
magnetic energy, open flux and relative helicity, and in most cases there is
no longer any flux contained within the rope after an eruption (although this
is not always the case).

Snapshots showing a flux rope eruption are shown in Figure 5, and diagnostics
corresponding to the same simulation in Figure 6. Here the red circles denote
the flux rope eruptions, shown at the time when the poloidal rope flux drops.
In most cases, after a flux rope eruption a new rope will form and the process
will repeat, although depending on the specific parameters the system may just
relax into a steady state. Compared to Figure 4, Figure 6 includes new plots
of the rope flux and current as well as the current-carrying helicity
$H_{J}^{(2.5D)}$ and eruptivity index $\lvert
H_{J}^{(2.5D)}/H_{R}^{(2.5D)}\rvert$. The figure shows both the poloidal/in-
plane flux and the axial/out-of-plane flux, contained within the region of
infinitely-long magnetic field lines. In our model, prior to eruption, there
is no null point beneath the rope. So the poloidal rope flux is simply the
difference in the vector potential, $\max(A_{y}(x,z))-A_{y}(0,0)$.

The effect of a flux rope eruption on the diagnostic measurements is more
significant than an arcade eruption. We still observe rapid decreases in open
flux and magnetic energy, but unlike with arcade eruptions there is now a
significant decrease in the relative helicity during the eruption, rather than
an increase. Arcade eruptions do not affect the poloidal flux in the rope at
all as can be seen in Figure 6, but naturally as the rope no longer exists
after a flux rope eruption these fluxes – as well as the axial rope current –
immediately fall to zero, before increasing again if flux cancellation is
ongoing. The axial rope current and axial rope flux are affected by arcade
eruptions, but not significantly. The eruptivity index will be discussed
below.

###  5.2 Dependence on Photospheric Diffusion

In the parameter study we primarily focus on the effect of changing the
photospheric diffusion parameter $\eta_{0}$. For very small $\eta_{0}$ the
system exhibits periodic arcade eruptions as described above, and the flux
ropes form too slowly to erupt within the timeframe of the simulation, which
we set at $50$ time units (corresponding to several years on the real Sun). It
is likely, however, that these ropes will eventually erupt.

When the photospheric diffusion is in the range $2\times
10^{-3}<\eta_{0}<1\times 10^{-1}$ we observe both arcade and flux rope
eruptions before $t=50$. For runs where $\eta_{0}$ is not too large, there are
several arcade eruptions between each flux rope eruption that cause the rope
to oscillate. At some point, seemingly arbitrarily, one of these arcade
eruptions coincides with reconnection below the rope and causes the flux rope
to move rapidly upwards and out of the top of the domain. Usually after one of
these eruptions the system returns to a potential-like state and the process
repeats, although the out-of plane magnetic strength usually diminishes after
each eruption. This repeats until the system relaxes to a steady state, with
no rope present.

Varying the value of the photospheric diffusion ($\eta_{0}$) produces a wide
range of flux rope behaviours. Figure 7 shows 100 simulations with
$\nu_{0}=0.5$ and varying $\eta_{0}$. For low $\eta_{0}$ (the left side of the
figure), we observe regular arcade eruptions (indicated by the blue circles)
similarly to the cases with no flux ropes. Small ropes do form in this region
(indicated by the red lines), and it is likely that they would erupt
completely given a sufficiently long integration time.

In the region $\eta_{0}>2\times 10^{-3}$ we observe flux rope eruptions before
$t=50$ (the end of the simulations). In general, these eruptions occur sooner
as $\eta_{0}$ increases, although there is not a simple, continuous dependence
of the eruption time on $\eta_{0}$ for all $\eta_{0}$. They are usually
preceded by a number of arcade eruptions, varying in magnitude. In most cases,
after a flux rope eruption a second flux rope forms, and we observe further
eruptions of both types.

We observe that the time of the first flux rope eruption, if any, is
negatively correlated to the diffusion $\eta_{0}$. This is visible in the
pattern of red squares in Figure 7, which roughly follow a curve from the top-
left of the diagram, flattening out as $\eta_{0}$ increases. We are unsure
whether this curve tends to an asymptote at $\eta_{0}=0$ or at a finite,
minimum value of $\eta_{0}$ below which flux rope eruptions do not occur. The
time period between successive flux rope eruptions is also roughly negatively
correlated to $\eta_{0}$.

For $\eta_{0}\gtrapprox 0.1$ the flux ropes form and erupt very quickly, and
there is no time for arcade eruptions (which have a frequency roughly
independent of $\eta_{0}$). The flux ropes are small but have a high poloidal
flux (resulting in the thicker vertical lines on Figure 7).

###  5.3 Diagnostic Measurements as Predictors of Eruptivity

Here we aim to quantify which, if any, of the considered diagnostic
measurements in Section 4.4 can be used as a proxy for the eruptivity of a
flux rope. In this section we only consider flux rope eruptions and not arcade
eruptions, which are present at almost all times. We calculate the usefulness
of each of the diagnostic measures using a probabilistic approach.

The data used here are from 500 simulation runs up to time $t=50$, covering
the parameter space $5\times 10^{-4}<\eta_{0}<0.5$ and $0.5<\nu_{0}<2.0$. For
reference, a realistic value for $\nu_{0}$ is likely of order unity. For
$\nu_{0}>2$, the relaxation is unrealistically slow relative to the driving,
and eruptions become very frequent and less well-defined, with flux ropes not
having time to form properly. Each simulation logs $5000$ data points, for a
total of $2.5\times 10^{6}$ measurements for each diagnostic. We separate each
point according to whether it precedes a flux rope eruption, within a certain
specified time cutoff.

With multiple flux rope eruptions observed within each simulation, we are able
to observe a large distribution of eruption magnitudes and rope
sizes/strengths. None of the diagnostics alone are enough to reliably predict
imminent eruptions. The magnetic energy, helicity, and open flux all decrease
significantly during flux rope eruptions, but in the preceding time interval
there is no significant indication of an imminent eruption, and in fact most
of the variation in these quantities is due to arcade eruptions. In contrast,
the rope fluxes and currents build up from zero with each new rope, until they
reach a threshold at which there is an eruption, making them more promising
predictors. However, this threshold does not have a constant value, as we can
see by observing in Figure 7 that there is large variation in the size of flux
rope eruptions. Figure 6 also shows that the peak levels of the rope fluxes
and currents tend to reduce for each successive eruption, even in a single
simulation.

However, by calculating the ratios of (squared) rope fluxes and current to the
other diagnostics, we observe that some of these ratios do indeed appear to
have an eruptivity threshold with a roughly constant value. To motivate this
observation, Figure 8 shows scatter plots of the points immediately preceding
eruptions, with the appropriate diagnostics plotted against one other. The
diagnostics on each axis are weighted to be proportional to the square of the
magnetic field strength.

In particular, we observe good linear correlations between the square of the
axial rope current immediately prior to eruption, and both the magnetic energy
and relative helicity of the system. This suggests that the ratios
${I_{y}^{2}}/{H_{R}}$, ${I_{y}^{2}}/{H_{J}}$, ${I_{y}^{2}}/{E_{M}}$ and
${I_{y}^{2}}/{E_{F}}$, where $I_{y}$ is the axial rope current and
$E_{M},E_{F}$ are the total and free magnetic energies respectively, should be
good indicators of an imminent flux rope eruption. These particular ratios
seem to work because the denominators are relatively steady between each
successive flux rope eruption, but have a different level each time (this is
visible in Figure 6). Since $I_{y}^{2}$ increases prior to a flux rope
eruption, the ratios also increase but reach a common, normalized threshold
between different eruptions. Note that $H_{R}$, $H_{J}$, $E_{M}$ and $E_{F}$
all include important contributions from the sheared background field
surrounding the rope. Quantities derived only from the rope itself are not
found to be successful denominators.

The distributions of eruptive and non-eruptive points for these ratios and a
selection of the diagnostics are shown in Figure 9. At any point in the
simulations when a flux rope is present, we check whether the rope will erupt
within a time cutoff $t=0.3$ (roughly equivalent to $8$ days). If so, the
diagnostics at that time are added to the histograms coloured red. The points
corresponding to ropes that will not erupt are coloured blue. The areas under
the histograms are corrected for an equal weighting of eruptive and non-
eruptive ropes, but in principle this could be altered if the overall
prior/uninformed probability of eruption were known. Thus the diagnostics with
least red/blue overlap are better predictors of eruptivity than those with a
large overlap.

This can be quantified by assigning a “probability” of eruption to each value
of each diagnostic. This is essentially the height of the red histogram curve
on Figure 9 divided by the combined height of both curves, and is output as a
number $0<P_{e}<1$ for each data point. The probability $P_{e}$ can then be
compared against whether the rope will actually erupt or not to produce a
skill score, defined as

$E=\frac{\sum_{\begin{subarray}{c}\rm Eruptive\\\ \rm
Points\end{subarray}}P_{e}+\sum_{\begin{subarray}{c}\rm Non-\\\ \rm
Eruptive\\\ \rm Points\end{subarray}}(1-P_{e})}{{\rm
Total\,Number\,of\,Points}}.$ (24)

If a diagnostic is a perfect indicator of eruptivity then it would have a
skill score $E=1$. If the diagnostic is a no better predictor than random
chance then it would have skill score $E=0.5$. The values of $E$ for each
diagnostic are given in the headings on Figure 9. We observe that the free
energy and open flux are not suitable predictors, as they are little better
than chance. The relative helicity and eruptivity index are slightly better,
correctly predicting around $60\%$ of eruptions within the time frame. We
note, however, that these diagnostics are inversely correlated to the
probability of eruption. This is likely because eruptions are more frequent
for lower values of the helicity, open flux and energy, as smaller ropes tend
to erupt more frequently in our model.

In contrast, the ratios of the squared axial current to the helicities and
energies are excellent predictors in this time frame, all with skill scores
above $E=0.86$ and significantly better than the constituent diagnostics on
their own. The correlations are also all positive, indicating that these
ratios increase until they reach a given threshold – represented by the peaks
in the red curves on Figure 9. The best predictor for $t=0.3$ before an
eruption (the time at which the predictors are best) is ${I_{y}^{2}}/{H_{R}}$,
with a skill score of $E=0.93$, or a predictive accuracy of $93.0\%$.

If we disregard the effect of arcade eruptions, the free energy and helicity
are roughly constant during the periods between successive flux rope
eruptions. However, the ratio between these quantities has also been
identified as a suitable predictor, as seen in Figure 9. Immediately before an
eruption the relative helicity decreases significantly, before the
corresponding decrease in free energy, and thus there is a brief peak in the
ratio between the two quantities.

The chosen time cutoff naturally has an effect on the skill scores of each of
these quantities. This is illustrated in Figure 10, where the skill scores for
each of the quantities are plotted against the time cutoff, up to $t=1.0$
before the eruption. Of the raw diagnostics, the relative helicity is
consistently the best predictor, with skill scores higher than $E=0.65$ at all
times. At almost all times the eruptivity index does not perform as well, and
indeed becomes little better than chance for time cutoffs approaching $t=1.0$.
The free energy and open flux fare little better, with skill scores around
$E=0.6$.

In contrast, the ratios of the axial current to the chosen diagnostics perform
consistently well, although their skill score decreases as the time cutoff
increases. The quantities $E_{F}/H_{R}$ and ${I_{y}^{2}}/{H_{R}}$ perform best
at small time cutoffs, with maximum skill scores of $E=0.938$ and $E=0.937$
respectively. The ratios of the axial current to magnetic energy also perform
well, and indeed for time cutoffs greater than $t=0.5$ these ratios perform
equally well or better than the other ratios. The maximum skill scores for
most of the ratios occurs at a time cutoff of around $t=0.3$, which
corresponds to several days in reality. As such these predictors have
potential for predicting solar eruptions a useful time before they occur.

##  6 Discussion

In an extensive parameter study, we have analysed the behaviour of two-
dimensional magnetic flux ropes to examine which, if any, properties of the
system can be used to predict whether or not the rope will erupt, an event
which in reality is likely to cause a CME. For our simulations we used the
magneto-frictional model rather than full MHD, reducing computational
complexity further and allowing us to run thousands of simulations over a wide
parameter space. We observe repeated arcade eruptions as in equivalent 3D
simulations, as well as the formation of flux ropes. The main focus of our
results is the prediction of the eruption of the flux ropes, rather than
eruptions within the overlying arcade.

We have observed that the behaviour depends greatly on the values of the
magneto-friction coefficient $\nu_{0}$ and the photospheric diffusion
$\eta_{0}$. In particular, flux ropes form and erupt more quickly for higher
$\eta_{0}$, whereas the frequency of arcade eruptions within the overlying
field is roughly independent of $\eta_{0}$. By comparing the probability of a
rope erupting within a certain time to a number of diagnostic measurements, we
have assigned a “skill-score” to each of them and the ratios between them. Of
particular interest are the relative helicity (which we have newly defined for
2.5D systems) and its current-carrying component, $H_{J}$. The ratio of
$H_{J}$ to the total relative helicity constitutes the eruptivity index of
Pariat et al. (2017).

We found that none of the diagnostics considered were by themselves good
predictors of eruptivity, with skill scores not significantly greater than
random chance. Of note, the eruptivity index has similar predictive skill to
the other diagnostics, and is in fact negatively correlated to the likelihood
of a flux rope eruption, unlike in the previous MHD simulations (Pariat et
al., 2017; Zuccarello et al., 2018). However, when we consider the ratios
between the diagnostics, weighted so as to be independent of the overall
magnetic field strength, we find that certain of these can be good predictors.

Of particular note are the ratios with the axial current as the numerator, and
either a helicity or energy measure as a denominator. The axial current
indicates the “strength” of the rope, and the helicity/energy is in effect a
measure of the strength of the background field. These denominator are roughly
constant in between eruptions, whereas the axial current steadily increases as
the rope becomes larger. Upon the ratio reaching a certain threshold, the rope
will erupt. Notably, this threshold appears to be independent of both
$\eta_{0}$ and $\nu_{0}$. The ratio of free energy to relative helicity is
also a very good predictor for eruptions in the immediate future, as a rapid
decrease in relative helicity is often followed by an eruption.

###  6.1 Effect of Background Magnetic Field on the Eruptivity Index

In this section we propose a straightforward explanation for why Pariat et al.
(2017) observed a high eruptivity index prior to the flux rope eruption,
whereas we do not. We propose that this difference is due to the direction of
the overlying magnetic field. We will consider a simple analytical model,
which shows that the eruptivity index will naturally be higher when the
background/overlying horizontal field direction is opposite to that of the
arcade.

Two magnetic field configurations are presented in Figure 11. The left pane
shows a configuration similar to the fields we generate naturally by shearing
a potential field, where the background magnetic field is orientated in the
same direction as the arcade. The right pane has the background field in the
opposite direction, leading to a magnetic null point above the arcade. When
flux ropes emerge into the second type of field, they are more likely to
erupt. This was shown clearly by Pariat et al. (2017), who compared
simulations with both orientations of the overlying field. We endeavour to
show here that the right-hand field configuration fundamentally results in a
higher eruptivity index, for a given sheared field component. By contrast, our
simulations correspond to the left-hand field configuration, so even though
they do erupt, this is not accompanied by a high eruptivity index.

The model magnetic field plotted in Figure 11 comes from the analytical
expression ${\bf B}=\widetilde{\bf B}+B_{0}{\bf e}_{x}$, where

$\displaystyle\widetilde{B}_{x}$ $\displaystyle=4ze^{-\xi}$ (25)
$\displaystyle\widetilde{B}_{y}$ $\displaystyle=2(1-\xi)e^{-2\xi}$ (26)
$\displaystyle\widetilde{B}_{z}$ $\displaystyle=-4xe^{-\xi},$ (27)

with $\xi=4(x^{2}+z^{2})$. The sheared, out-of-plane component, $B_{y}$, is
fixed, and the only parameter is the strength of the background magnetic
field, given by the (constant) parameter $B_{0}$. The left pane in Figure 11
has $B_{0}=0.25$, the right $B_{0}=-0.25$. We proceed to observe the
dependence on $B_{0}$ of $H_{R}^{(2.5D)}$ and $H_{J}^{(2.5D)}$, as defined in
Section 4.4.8, and the resultant effect on the eruptivity index
$\lvert{H_{J}^{(2.5D)}}/{H_{R}^{(2.5D)}}\rvert$.

Following Section 4.4.8, in order to calculate the relative helicity we choose
a vector potential ${\bf A}=\widetilde{\bf A}(x,z)-B_{0}z{\bf e}_{y}$, where
$\nabla\times\widetilde{\bf A}=\widetilde{\bf B}$ and $\widetilde{\bf A}$ is
independent of $B_{0}$. Since the $B_{0}{\bf e}_{x}$ component of ${\bf B}$ is
a potential field, we can similarly choose ${\bf
A}_{P}^{(2.5D)}=\widetilde{\bf A}_{P}(x,z)-B_{0}z{\bf e}_{y}$. It follows
that, for these fields, the “current-carrying” helicity
$H_{J}^{(2.5D)}=\iint({\bf A-A}_{P}^{(2.5D)})\cdot({\bf
B-B}_{P}^{(2.5D)})\,\mathrm{d}x\mathrm{d}z$ has no dependence on the
background field $B_{0}$, because the $B_{0}$ terms from ${\bf A}$ and ${\bf
A}_{P}^{(2.5D)}$ will cancel. For the relative helicity, however,
$H_{R}^{(2.5D)}=\int_{V}({\bf A+A}_{P}^{(2.5D)})\cdot({\bf
B-B}_{P}^{(2.5D)})\,\mathrm{d}V$, so the two terms add together and it does
depend on $B_{0}$. Thus we can write

$H_{R}^{(2.5D)}=H_{0}^{(2.5D)}-B_{0}\iint(2z{\bf e}_{y})\cdot({\bf
B-B}_{P}^{(2.5D)})\,\mathrm{d}x\mathrm{d}z,$ (28)

where $H_{0}^{(2.5D)}$ is the relative helicity with $B_{0}=0$. Since
$H_{0}^{(2.5D)}$ and $H_{J}^{(2.5D)}$ have no dependence on $B_{0}$, the
eruptivity index can simply be expressed as

$\lvert{H_{J}^{(2.5D)}}/{H_{R}^{(2.5D)}}\rvert=\left\lvert\frac{H_{J}^{(2.5D)}}{H_{0}^{(2.5D)}-B_{0}\iint(2z{\bf
e}_{y})\cdot({\bf B-B}_{P}^{(2.5D)})\,\mathrm{d}x\mathrm{d}z}\right\rvert.$
(29)

For a sheared field with $B_{y}$ non-zero (and non-uniform), we can clearly
see that there will be a particular background field strength $B_{0}$ where
the eruptivity index will become infinite as the denominator vanishes. For the
magnetic field specified in Equations (25) to (27) the constants take the
values

$\displaystyle H_{J}^{(2.5D)}$ $\displaystyle\approx-0.0109$ (30)
$\displaystyle H_{0}^{(2.5D)}$ $\displaystyle\approx 0.0307$ (31)
$\displaystyle\int_{V}(2z{\bf e}_{y})\cdot({\bf B-B}_{P}^{(2.5D)})\,dV$
$\displaystyle\approx-0.2026,$ (32)

which results in a peak in the eruptivity index at $B_{0}\approx-0.15$, when
in particular the overlying magnetic field is oppositely directed to the
magnetic field in the upper part of the arcade (as in the right pane of Figure
11). By contrast, if the overlying magnetic field has the same direction as
that in the arcade, so that $B_{0}>0$ (as in the left pane of Figure 11), then
the denominator of (29) will not become very small so the eruptivity index
will not become large.

In all of our simulations – where the flux rope is formed by shearing of a
pre-existing potential arcade – the background field has the same direction as
that of the arcade, whether or not the flux rope erupts. Generalising from the
analytical model with $B_{0}>0$, this explains why our eruptions are not
preceded by a high eruptivity index.

The simulations of (Pariat et al., 2017), which were driven by flux emergence,
included cases with both directions of background field. The eruptivity index
behaved as predicted by the simple model in this section, but in that case
only the cases with oppositely-directed field (and high eruptivity index)
erupted. Our work shows that there is a whole class of eruptions that will not
have high eruptivity index owing to the fact that they occur despite having
the same direction of overlying field.

###  6.2 Implications for space weather forecasting

The simplified nature of our 2.5D system means that any quantitative
predictions will not be valid in 3D or indeed for any differing domain size or
shape. However, the qualitative patterns of behaviour that we observe (such as
those of the ratios of axial rope current to overall helicity/energy) should
be equally valid in all systems, including global coronal models, where flux
ropes are formed by footpoint shearing from differential rotation. Moreover,
since these flux ropes are formed by gradual shearing over days to weeks, and
are located in the magnetically-dominated low corona, we do not expect that
the general conclusions would change significantly if we were to move from the
magneto-frictional model to full MHD. For example, it has been shown that the
linear stability criteria in magneto-frictional and MHD systems are the same
(Craig and Sneyd, 1986). Nevertheless, to make quantitative predictions about
specific 3D magnetic configurations on the Sun will require further work to
understand how the behaviour of the diagnostics depends on the local coronal
magnetic stucture.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

OEKR did all of the calculations and worked out the proof in Section 4.4.8,
under the guidance of ARY who conceived the project. Both authors contributed
to writing the paper. We thank the two referees for valuable comments that
have greatly improved the paper.

## Funding

OEKR was supported by a UKRI/STFC PhD studentship, and ARY by UKRI/STFC
research grant ST/S000321/1.

## Acknowledgments

## Supplemental Data

## Data Availability Statement

The datasets generated in this study are available from the corresponding
author on reasonable request.

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## Figure captions

Figure 1: Examples of initial conditions for the 2.5D magneto-frictional code
in a cartesian domain. The left pane is a potential field and the right pane
is an equivalent outflow field, with the same lower boundary condition. The
black lines represent magnetic field lines. Figure 2: Plot of the function
$f(a)=\frac{\cosh(k\pi ay_{1})}{\cosh(k\pi y_{1})}$ for $k=1$ and various
values of $y_{1}$. Figure 3: Snapshots showing the shearing of the potential
arcade resulting in an arcade eruption and subsequent reformation preceding
another eruption at $t=4$. The black lines are magnetic field lines projected
onto the $(x,z)$ plane, and the heatmap represents the magnetic field strength
into the page (in the $y$ direction). In this simulation, $\eta_{0}=0$ and
$\nu_{0}=0.5$. Figure 4: A selection of diagnostics for a simulation
exhibiting repeated arcade eruptions, which are represented by blue circles.
In this simulation, $\eta_{0}=0$ and $\nu_{0}=0.5$. The time of an eruption is
taken to be the midpoint in time between the maximum and minimum open flux
values either side of the eruption. Figure 5: Snapshots showing the formation
of a flux rope and its subsequent eruption at time $t=8$. This is followed by
the formation of a second flux rope, which experiences an arcade eruption at
$t=15.8$, before the process repeats. The black lines are magnetic field lines
projected onto the $(x,z)$ plane, and the heatmap represents the magnetic
field strength into the page (in the $y$ direction). In this simulation,
$\eta_{0}=7\times 10^{-3}$ and $\nu_{0}=0.6$. Figure 6: A selection of
diagnostics for a simulation exhibiting flux rope eruptions (red circles)
interspersed with multiple arcade eruptions (blue circles). In this
simulation, $\eta_{0}=7\times 10^{-3}$ and $\nu_{0}=0.6$. The time of a flux
rope eruption is taken to be the time of the maximum poloidal rope flux before
this rapidly decreases. Figure 7: Overview of 100 flux rope simulations, for
$\nu_{0}=0.5$ and varying $\eta_{0}$. Each simulation (for a given $\eta_{0}$)
is represented by a vertical red line, and the thickness of this line is
proportional to the poloidal flux in the rope at that time. Arcade eruptions
are represented by blue circles and flux rope eruptions are represented by red
squares. The size of these points is proportional to the decrease in open flux
and poloidal rope flux during the eruption, respectively. Figure 8: Pairwise
scatter plots of the diagnostic values for all snapshots with later flux rope
eruptions, in order to establish whether the ratios of the diagnostics are
good predictors. The sizes of the points are weighted based on proximity to
the eruption, such that the larger points are close to eruptions and vice
versa. Figure 9: Histograms of five of the diagnostic parameters, and five of
the ratios between them. The points that precede an eruption within $t=0.3$
time units are shown in red, and those that do not are shown in blue. The
diagnostics with less red/blue overlap are better predictors of eruptions and
vice versa. The eruptivity skill score $E$ is given for each diagnostic. The
curves are normalised to have an area of unity. Figure 10: Variation of the
skill scores for each of the measured quantities, depending on the time cutoff
within which an eruption must occur. The raw diagnostics and eruptivity index
are plotted as solid lines, and the ratios are plotted as dashed lines. Figure
11: Comparison between two magnetic arcades with overlying magnetic fields in
opposite directions. The black lines represent the magnetic field projected
into the $(x,z)$ plane and the heatmap represents the magnetic field strength
out of this plane.