# Laboratory study of the failed torus mechanism in arched, line-tied,
magnetic flux ropes

Andrew Alt<EMAIL_ADDRESS>Department of Astrophysical Sciences, Princeton
University, Peyton Hall, Princeton, NJ 08544, USA  Princeton Plasma Physics
Laboratory, PO Box 451, Princeton, NJ 08543, USA  Hantao Ji  Department of
Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ
08544, USA  Princeton Plasma Physics Laboratory, PO Box 451, Princeton, NJ
08543, USA  Jongsoo Yoo  Princeton Plasma Physics Laboratory, PO Box 451,
Princeton, NJ 08543, USA  Sayak Bose  Princeton Plasma Physics Laboratory, PO
Box 451, Princeton, NJ 08543, USA  Aaron Goodman  Department of Astrophysical
Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA
Princeton Plasma Physics Laboratory, PO Box 451, Princeton, NJ 08543, USA
Masaaki Yamada  Princeton Plasma Physics Laboratory, PO Box 451, Princeton, NJ
08543, USA

###### Abstract

Coronal mass ejections (CMEs) are some of the most energetic and violent
events in our solar system. The prediction and understanding of CMEs is of
particular importance due to the impact that they can have on Earth-based
satellite systems, and in extreme cases, ground-based electronics. CMEs often
occur when long-lived magnetic flux ropes (MFRs) anchored to the solar surface
destabilize and erupt away from the Sun. One potential cause for these
eruptions is an ideal magnetohydrodynamic (MHD) instability such as the kink
or torus instability. Previous experiments on the Magnetic Reconnection
eXperiment (MRX) revealed a class of MFRs that were torus-unstable but kink-
stable, which failed to erupt. These “failed-tori” went through a process
similar to Taylor relaxation where the toroidal current was redistributed
before the eruption ultimately failed. We have investigated this behavior
through additional diagnostics that measure the current distribution at the
foot points and the energy distribution before and after an event. These
measurements indicate that ideal MHD effects are sufficient to explain the
energy distribution changes during failed torus events. This excludes Taylor
relaxation as a possible mechanism of current redistribution during an event.
A new model that only requires non-ideal effects in a thin layer above the
electrodes is presented to explain the observed phenomena. This work broadens
our understanding of the stability of MFRs and the mechanism behind the failed
torus through the improved prediction of the torus instability and through new
diagnostics to measure the energy inventory and current profile at the foot
points.

Submitted to Physics of Plasmas

##  I Introduction

The study of astrophysical phenomena is often accomplished via remote ground-
and space-based observations and numerical simulations [1, 2, 3]. Many
insights can be learned from both of these approaches but they also have their
limitations and difficulties. The remote nature of observations limits the
available diagnostics, and since one does not have control over the events,
individual parameters cannot be adjusted to study the results in isolation.
While simulations can control individual parameters and isolate specific
phenomena, computational considerations require approximations that can limit
the available physics. In this paper, laboratory experiments are used to
create models of astrophysical phenomena so that they can be studied in-situ.
This allows for fine control of experimental parameters and for the use of a
wide array of plasma diagnostics. The experiments presented here focus on the
study of solar eruptions and the resulting coronal mass ejections (CMEs).
These events lead to space weather which can be hazardous to Earth-based
satellites as well as sensitive ground-based equipment [4]. Therefore,
understanding and predicting space weather phenomena has been identified as a
subject of great scientific interest [5]. The experiments presented here allow
for new physics insights that are required for better predictions of space
weather events but are difficult to obtain otherwise.

CMEs often occur when long-lasting magnetic flux ropes (MFRs) protruding from
the Sun suddenly and violently erupt [6, 7]. These arched structures are
bundles of twisted magnetic-field lines anchored to the solar surface via
line-tying to the conductive photosphere [8, 9, 10]. The trigger for an
eruption in an MFR is often cast in terms of two ideal magnetohydrodynamic
(MHD) instabilities, the kink and torus instabilities. These instabilities
have long been studied in asymmetric devices such as tokamaks [11, 12],
however they can also be extended to arched MFRs [13, 14]. Previous
experiments found a class of MFRs that were torus-unstable and kink-stable
which failed to erupt [15]. These “failed torus” ropes would begin to rise,
then the toroidal current would reorganize to become more hollow before the
rope would collapse down, failing to erupt. Understanding the cause of the
failed torus would help in future predictions of confined eruptions on the
Sun. The previous description of the failed torus was that a self-organization
event, such as Taylor relaxation, would occur [16]. This would exchange
toroidal and poloidal fluxes while maintaining the helicity, requiring non-
ideal effects of magnetic reconnection.

The remainder of this paper has the following organizational structure:
Section II describes the relevant forces and MHD instabilities that can occur
in MFRs and then summarizes the previously observed failed torus regime.
Section III describes the experimental setup including the new diagnostics
added for this work. Here, we will also present an overview of the parameter
space that is reachable in our experiment. The remaining sections examine the
Taylor relaxation explanation of the failed torus and find that based on the
energy inventory (Section IV), that ideal MHD must hold during events. This
rules out the possibility of Taylor relaxation which requires reconnection
within the rope. Section V validates the current hollowing throughout a failed
torus rope and compares the foot-point current distribution of different
regimes. Section VI then presents a mechanism through which we can explain the
changes in toroidal flux without violating ideal MHD except for a small region
above the foot points. Section VII examines the change in toroidal field
profile during events. Together, these provide a sufficient mechanism to
explain the failed torus. This section also discusses how these results can be
extended to the conditions on the Sun.

##  II Flux ropes and associated instabilities

Figure 1: A model of an arched, line-tied magnetic flux rope, showing the breakdown of the fields and currents based on the direction and source. The line-tying condition at the conducting photosphere causes the foot points to be anchored with a separation of $2x_{\mathrm{f}}$. The internal fields, $B_{\mathrm{Pi}}$ and $B_{\mathrm{Ti}}$, are generated by the currents in the rope, while the external fields, $B_{\mathrm{s}}$ and $B_{\mathrm{g}}$, are generated by the Sun. Reproduced from Myers et al., Physics of Plasmas 23, 112102 (2016), with the permission of AIP Publishing, and adapted with permission from Chen, Astrophys. J. 338, 453 (1989) and Chen and Krall, J. Geophys. Res. Space 108, 1410 (2003). Copyright 1989 American Astronomical Society and 2003 American Geophysical Union. Force | Symbol | Source Term | Analytical Expression
---|---|---|---
Hoop force (upward) | $F_{\mathrm{h}}$ | $f_{\mathrm{h}}=J_{\mathrm{T}}B_{\mathrm{Pi}}$ | $F_{\mathrm{h}}=\frac{\mu_{0}I_{\mathrm{T}}^{2}}{4\pi R}\left[\ln\left(\frac{8R}{a}\right)-1+\frac{\ell_{\mathrm{i}}}{2}\right]$
Strapping force (downward) | $F_{\mathrm{s}}$ | $f_{\mathrm{s}}=-J_{\mathrm{T}}B_{\mathrm{s}}$ | $F_{\mathrm{s}}=-I_{\mathrm{T}}B_{\mathrm{s0}}$
Tension force (downward) | $F_{\mathrm{t}}$ | $f_{\mathrm{t}}=-J_{\mathrm{P}}B_{\mathrm{T}}$ | $F_{\mathrm{t}}=\frac{-\mu_{0}I_{\mathrm{T}}^{2}}{8\pi R}\left[\frac{\left<B_{\mathrm{T}}^{2}\right>-B_{\mathrm{g0}}^{2}}{B_{\mathrm{P}a}^{2}}\right]\approx-\frac{1}{2}\frac{\mu_{0}I_{\mathrm{T}}^{2}}{4\pi R}$
Table 1: Breakdown of forces on a flux rope. The forces are separated by
contributions to the $J$$\times$$B$ term. The fields and currents used are
shown in Fig. 1 and the analytical expressions are derived in Myers  _et al._
[17]. Reproduced from Myers et al., Physics of Plasmas 23, 112102 (2016), with
the permission of AIP Publishing

In an MFR, the magnetic field, $B$, and electric currents, $J$, of an MFR can
be broken up into components based on their direction and source as shown in
Fig. 1. First, $J$ can be divided into toroidal, T, and poloidal, P,
components. Then, $B$ can be separated into two components: internal and
external. The internal field is generated by currents within the MFR, while
the external field is generated by currents within the Sun. The internal field
is further divided into toroidal and poloidal components. The external field
is divided into the guide field, $B_{\mathrm{g}}$, along the axis of the rope
and the strapping field, $B_{\mathrm{s}}$, perpendicular to this axis.

The dominant force on a low-$\beta$ plasma, such as an MFR, is the
$J$$\times$$B$ force. Here, $\beta\equiv 2\mu_{0}P/B^{2}$ is the ratio of
thermal to magnetic energy in the plasma, $P$ is the plasma pressure, and
$\mu_{0}$ is the permeability of free space. This force can be decomposed
based on the components of the source terms discussed above. These forces are
called the hoop, strapping, and tension forces and are defined in Table 1. The
tension and strapping forces hold a rope down while the hoop force causes it
to expand upward.

###  II.1 The tension force

Of particular importance to this work is the tension force, $F_{\mathrm{t}}$,
caused by the poloidal current and toroidal field,
$f_{\mathrm{t}}=-J_{\mathrm{P}}B_{\mathrm{T}}$. The tension force is caused by
the helical winding of current around a torus. The force in the direction of
the major radius, $R$, is directed inward on the inboard side of the torus and
is directed outward on the outboard side. However, since the number of current
windings must be the same on both sides, both $J_{\mathrm{P}}$ and
$B_{\mathrm{T}}$ are larger on the inboard side. When the $R$-directed force
is averaged over a poloidal cross section, the net force is directed inward to
the center of the torus.

When $F_{\mathrm{t}}$ is simplified using the large-aspect-ratio approximation
in a full torus, it is given by [18]

$F_{\mathrm{t}}=-\frac{\mu_{0}I_{\mathrm{T}}^{2}}{8\pi
R_{0}}\left[\frac{\left\langle B_{\mathrm{T}}^{2}\right\rangle-
B_{\mathrm{g0}}^{2}}{B_{\mathrm{P}a}^{2}}\right],$ (1)

where $\langle\dots\rangle$ denotes cross-section averaging, $R_{0}$ is the
major radius, $I_{\mathrm{T}}$ is the total toroidal current,
$B_{\mathrm{g0}}$ is the edge guide field,
$B_{\mathrm{P}a}\equiv\mu_{0}I_{\mathrm{T}}/(2\pi a$) is the edge poloidal
field, and $a$ is the minor radius. Here we see that a downward tension force
is caused by a paramagnetic toroidal field within the rope.

###  II.2 The torus instability

The torus instability occurs when the net force on an MFR increases as it is
displaced upward away from equilibrium, [12, 19, 14]

$\left.\sum_{i}F_{i}=0\right|_{z=z_{\mathrm{ap}}}\quad\text{and}\quad\left.\sum_{i}\frac{\partial
F_{i}}{\partial z}>0\right|_{z=z_{\mathrm{ap}}},$ (2)

where $F_{i}$ are the constituent forces given in Table 1 and
$z_{\mathrm{ap}}$ is the equilibrium height of the rope’s apex. It is
sometimes described as a loss-of-equilibrium via a “catastrophe” mechanism
[19].

The torus instability was first studied in the context of fusion devices,
where the role of the external strapping field, $B_{\mathrm{s}}$, is played by
the applied vertical field [12]. Since the main inward force is caused by
$B_{\mathrm{s}}$, the torus instability criterion is often cast in terms of
the decay index of this field,111The decay index is defined such that a field,
$B\propto z^{-n}$ has a decay index of $n$.

$n_{\mathrm{s}}=-\frac{z}{B_{\mathrm{s}}}\frac{\partial
B_{\mathrm{s}}}{\partial z}>n_{\mathrm{cr}}^{\mathrm{}},$ (3)

where $z$ is the height above the photosphere and
$n_{\mathrm{cr}}^{\mathrm{}}=1.5$ in an axisymmetric, large aspect ratio, full
torus [14]. When previous MFR experiments were translated to the conditions on
the Sun, a lower value of $n_{\mathrm{cr}}^{\mathrm{}}\approx 0.9$ has been
observed in the partial tori of arched MFRs [21]. If $B_{\mathrm{s}}$ decays
faster than this critical value, the torus instability can occur.

Figure 2: Example data taken during a failed torus event. a: The apex height
during the failed torus event. The points represent heights of the nulls of
the poloidal magnetic field at each time. b, c: 2D plots of the toroidal
current density (top) and toroidal magnetic field (bottom) at the times
indicated in a. The in-plane magnetic field at the probe locations is also
shown via arrows.

###  II.3 The kink instability

The kink instability is another ideal MHD instability that can affect MFRs
[11, 22, 13]. The instability occurs when $I_{\mathrm{T}}$ is too large, or
equivalently, $B_{\mathrm{g}}$ is too small, such that the field lines at the
edge of the rope become too twisted around the axis. When the field lines at
the edge of the rope become too twisted around the axis, the instability can
occur. This twisting can be described in terms of the edge safety factor,
$q_{a}$, or equivalently, the twist number $N$,

$q_{a}\equiv\frac{1}{|N|}=\frac{2\pi
a}{L}\frac{B_{\mathrm{g0}}}{B_{\mathrm{P}a}}<q_{\mathrm{cr}},$ (4)

where $L$ is the length of the MFR. For toroidally symmetric fusion devices,
the critical safety factor is $q_{\mathrm{cr}}=1$ [11, 22]. However in
previous MFR experiments where toroidal symmetry is broken by line-tying, a
lower critical value of $q_{\mathrm{cr}}\approx 0.8$ has been observed [17].
When the kink instability onsets, it causes the axis of a rope to tilt and
begin to rotate. However, the instability saturates nonlinearly at relatively
small amplitudes and therefore cannot be the sole cause of eruptions.

###  II.4 Observations of failed tori in previous experiments

In a previous experimental campaign, a class of ropes that were torus-unstable
but kink-stable were observed failing to erupt [15]. This regime is referred
to as the “failed torus” regime and an example MFR is shown in Fig. 2. Since
the rope is initially unstable to the torus instability, the apex begins to
rise. However, before a full eruption can occur, the toroidal current profile
becomes more hollow, causing the toroidal flux to increase. This flux
increases causes a spike in the tension force (due to Eq. 1), and the rope
collapses back down to a lower height [23, 15].

The reorganization that occurs and increases the toroidal flux simultaneously
decreases the poloidal flux. This can be explained by the conservation of
helicity which describes the linking of these two fluxes. Magnetic helicity
measures the linking of magnetic-field lines and is given by [16, 24]
$H=\int\mbox{\boldmath{$A$}}\mbox{\boldmath{$\cdot$}}\mbox{\boldmath{$B$}}{\rm
d}V$, where $A$ is the vector potential defined such that
$\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$A$}}=\mbox{\boldmath{$B$}}$.
In order for $H$ to be gauge invariant, the integral must be carried out over
a simply connected volume, $V$, whose boundary, $S$, is a magnetic surface
such that
$\left.\mbox{\boldmath{$B$}}\mbox{\boldmath{$\cdot$}}\hat{\mbox{\boldmath{$n$}}}\right|_{S}=0$,
where $\hat{\mbox{\boldmath{$n$}}}$ is the unit normal vector to $S$. Helicity
conservation is a common feature of Taylor relaxation [16, 25] where the
global field-line linking is conserved while the total energy is minimized.
Under this constraint, the minimum energy state is a force-free field where
$\mu_{0}\mbox{\boldmath{$J$}}=\alpha\mbox{\boldmath{$B$}}$, with $\alpha$
being a constant. However, in order to reach this state, field-line breaking
via reconnection must be allowed in the plasma volume.

##  III Experimental setup

Figure 3: The MRX vessel used to create arched, line-tied flux ropes. An arc
discharge is created between two copper electrodes and is separated from the
magnetic-field coils by a glass substrate. The model in Fig. 1 corresponds to
the pink plasma arc in the center of the image. Four coils were inserted into
MRX in order to control the profiles of both the guide and strapping fields.
The orange coils contribute to the guide field along the rope while the blue
coils control the strapping field across it. Control of the vacuum fields
allows for control of the instability parameters for the torus and kink
instabilities. Reproduced from Myers et al., Nature 528, 526 (2015). Copyright
2015 Nature Publishing Group.

Flux ropes have previously been created inside Magnetic Reconnection
Experiment (MRX) in order to study their solar counterparts [26, 15, 17, 21,
18]. The experimental setup is shown in Fig. 3 and is described in further
detail in Myers  _et al._ [17]. The vacuum fields are controlled by four sets
of coils, two for each the guide and strapping fields. By controlling the
magnitude and direction of the current in these coils, the strength and decay
index of these two fields can be controlled independently. The currents evolve
slowly compared to a rope’s lifetime, $\tau_{\mathrm{MFR}}\sim
1\,\mathrm{ms}$. The ropes are created by a discharge between two copper
electrodes with radius $a_{0}=7\,\mathrm{cm}$ with a foot-point half-
separation of $x_{\mathrm{f}}=17\,\mathrm{cm}$. The electrodes form the foot
points of a rope and can be considered to be perfectly conducting on the
timescale of a rope’s lifetime. Therefore the ropes are line-tied to their
foot points and exist within a static background magnetic field. Before the
discharge is initiated, the vessel is filled with neutral hydrogen to a
pressure in the range of $30-40\,\mathrm{mTorr}$.

An important property of solar MFRs is the large separation between both time
and spatial scales. For the storage and release paradigm of CMEs to hold, the
characteristic driving time for energy buildup in the corona,
$\tau_{\mathrm{D}}$, must be much larger than the resistive dissipation time,
$\tau_{\mathrm{R}}\equiv\mu_{0}aL/\eta$. In addition, for MHD to be valid, the
Alfvèn transit time, $\tau_{\mathrm{A}}\equiv L/v_{\mathrm{A}}$, must be much
shorter than both of these timescales, i.e.
$\tau_{\mathrm{R}}\gg\tau_{\mathrm{D}}\gg\tau_{\mathrm{A}}$. In our
experiments, these times are $\tau_{\mathrm{R}}\sim 1\,\mathrm{ms}$,
$\tau_{\mathrm{D}}\sim 150\,\mathrm{\mu s}$, and $\tau_{\mathrm{A}}\sim
3-10\,\mathrm{\mu s}$, and therefore the inequality holds [18].

The primary diagnostic in the flux rope experiments is an array of B-dot
probes with over 300 pickup coils arranged in a 2D array (usually placed in
the $y$-$z$-plane) with two lines of out-of-plane probe triplets along the
center line. The out-of-plane probes are placed parallel to the $z$-axis at
$y=0$ and $x=\pm 3\,\mathrm{cm}$. The out-of-plane probes give better insight
into the instantaneous curvature than was previously possible. The curvature
measurements yield an estimate for the out-of-plane derivatives of $B$ which
are necessary to calculate the in-plane components of $J$.

###  III.1 Rogowski coils for measuring the foot-point current distribution

Figure 4: Diagram of the Rogowski coils used to measure the distribution of
plasma current at the foot points. Left: The circuit that was printed into a
PCB to create the three concentric Rogowski coils. Right: Top and side view of
the positioning of the Rogowski coils below the segmented copper electrodes.
The “panhandle” contains the electrical connections used to measure the
signals which were insulated from the plasma. A small hole was made in the
center of the electrode to allow for gas puffing at the foot points.

Rogowski coils were made via a printed circuit board (PCB) (see Fig. 4) and
were placed below the electrodes located at the foot points of the MFRs [27].
There were three sets of coils printed into each board and arranged to measure
the current passing through the three segments of the electrodes. Each
electrode segment consists of a concentric circle or annulus of copper and
four posts connecting each to a shared base plate which is connected directly
to the capacitor bank driving the flux ropes. The PCBs were then placed
between the copper segments of the electrode and the base plate. Each set of
Rogowski coils consists of two counter-wound coils which is used to eliminate
electrostatic noise. Additionally, flux loop pairs were printed into each PCB
to measure the toroidal magnetic flux at each coil location.

The coils were calibrated by placing them around a wire that was connected to
the driving capacitor bank to create currents similar in magnitude and
timescale to the experimental currents. The calibration current was externally
measured with a commercially available Rogowski coil. Previous measurements of
failed torus ropes were focused on the apex of the rope and could not measure
any potential hollowing of the current at the foot points. The addition of
segmented electrodes and Rogowski coils at the foot points allows for the
measurement of the current distribution at the foot points to determine
whether hollowing is occurring throughout the entire rope or if it is
localized to the apex.

###  III.2 Langmuir probes in the flux rope experiments

Triple Langmuir probes have been used extensively in previous MRX operation to
measure the plasma density, temperature, and potential [28]. However, in
previous experience with the flux rope setup, it has been seen that the probe
tips quickly became dirty and it was not feasible to remove and clean them
often enough to maintain data quality. To combat this, a new type of probe tip
was created. These new probe tips consist of a loop of 0.005" diameter
tungsten wire that is held in place by a double-bore alumina tube. Each end of
the loop is connected to a twisted pair that runs out of the vacuum vessel to
the probe circuit. During measurement, each twisted pair is shorted together,
allowing each loop to be biased as a uniform probe tip. Periodically, a
current of about $2\,\mathrm{A}$ was passed through each loop until it began
to faintly glow from black-body radiation. This heating removed surface
impurities via outgassing, improving performance. The area of each probe tip
was measured by cross-calibrating with a reference Langmuir probe in the
regular MRX configuration with a quiet, toroidally symmetric plasma [28, 18].

###  III.3 Experimental parameter space

Figure 5: The experimental parameter space of nearly 2,500 shots. Each point
represents a shot and is placed based on its value of the two ideal
instability parameters, $q_{a}$ and $n$. The color represents the instability
amplitude based on the magnitude of repeated deviations of $z_{\mathrm{ap}}$,
normalized by $x_{\mathrm{f}}$. Shots that experienced multiple eruptions are
represented with triangles while non-eruptive shots are circles. The magenta
crosses represent specific parameters that will be considered in Section IV.
Four different regions of stability can be seen, though their boundaries are
not perfectly defined by the parameters.

In Fig. 5, each shot taken during three separate experimental campaigns is
plotted based on its value of the stability parameters for the torus and kink
instabilities, $n$ and $q_{a}$ as described in Sections II.3 and II.2. The
results of nearly 2,500 discharges are represented in the plot. The color of
each point represents the instability amplitude, $\langle\delta
z\rangle/x_{\mathrm{f}}$, which is based on the time-averaged size of an
envelope around the apex height changes normalized by the foot-point half-
separation, $x_{\mathrm{f}}$. (The instability amplitude is described in
further detail in Myers  _et al._ [17].) Shots that erupted multiple times are
shown by triangles while non-eruptive shots are circles.

####  III.3.1 The quadrants of the experimental parameter space

The four quadrants in Fig. 5 can be described by qualitatively different
stability properties, as labeled. The boundaries of these quadrants are
denoted by the gray bars centered on $q_{a}=0.8$ and $n=0.8$. Ropes in the
“Stable” quadrant are stable to both the kink and torus instabilities and do
not erupt. Ropes in the “Failed Kink” quadrant have a lower $q_{a}$ and are
unstable to the kink instability. However, since the kink instability
saturates at a relatively low amplitude, these ropes kink and rotate without
erupting. In the “Eruptive” quadrant, ropes have high $n$ and low $q_{a}$ and
therefore are unstable to both the torus and kink instabilities. These ropes
are able to erupt. The last regime is the “Failed Torus” quadrant, which
contains ropes that are torus-unstable but kink-stable. Despite being torus-
unstable, these ropes often fail to erupt (see Section II.4 for more detail).

The apex height (Defined as nulls of the poloidal magnetic field.) of an
example eruptive rope is presented in Fig. 6. Here it can be seen that
$z_{\mathrm{ap}}$ is multivalued at certain times. This occurs due to
experimental constraints caused by the large external inductance in the
circuit driving the plasma current. This external inductance causes the total
toroidal current to remain constant on the timescale of an eruption.
Therefore, instead of the erupting rope dissipating as it rises, a new rope is
formed at a lower height. This rope exists in the same vacuum fields, and is
therefore also unstable. The new rope then erupts, creating a repeating
pattern of eruptions. At times when two ropes coexist, $z_{\mathrm{ap}}$ is
presented as multivalued and usually consists of three values, two O-points
corresponding to the two ropes’ apexes and an X-point between them.

##  IV Evolution of the energy inventory during events

In this section we are primarily focused on the energy inventory before and
after a failed torus event. Since our experimental plasma is low-$\beta$, the
majority of the stored energy is in the magnetic fields, however the thermal
and kinetic energies are significant enough to be considered. The thermal
properties were not able to be measured on previous flux rope experiments
conducted on MRX due to difficulties of implementing Langmuir probes in the
flux rope environment. The triple Langmuir probe design described in Section
III.2 allowed for a direct measurement of the electron thermal properties.

###  IV.1 Event identification and timing

Figure 6: The data overlaying procedure for an example eruptive rope. The apex
height, $z_{\mathrm{ap}}$, at each time is shown by the blue points. Whenever
the rope crosses a set trigger height, $z_{\mathrm{trig}}=20\,\mathrm{cm}$
(horizontal dashed line), the crossing time, $t_{\mathrm{trig}}$ (vertical
dashed lines), is noted. The data in a fixed window around each
$t_{\mathrm{trig}}$ is then averaged together to build the behavior of an
“average event.”

In this section, we will be focused on what happens during an event, which
will be defined as either an eruption or a failed torus event. In order to
investigate the behavior of an “average event,” we look first at the data
gathered from many shots with the same experimental conditions. However, due
to the nature of the instabilities, even under the same experimental
conditions, events can occur at very different times in the shot. While the
events may occur at very different times, they are often very similar if
correctly shifted in time. The procedure for averaging event data is shown
with the values of $z_{\mathrm{ap}}$ presented in Fig. 6. First a fixed
height, $z_{\mathrm{trig}}=20\,\mathrm{cm}$, is picked.222The specific value
of $z_{\mathrm{trig}}$ is not important to the final results as long as it
falls within the range of most events. Then each time, $t_{\mathrm{trig}}$,
where the rope passes through this height is found, i.e.
$z_{\mathrm{ap}}(t=t_{\mathrm{trig}})=z_{\mathrm{trig}}$. After each
$t_{\mathrm{trig}}$ is found, the value of $z_{\mathrm{ap}}(t+\Delta t)$ is
recorded for time shifts, $\Delta t\in[-20\,\mathrm{ms},20\,\mathrm{ms}]$.
This data is then averaged for each $\Delta t$ across events and shots to
create an average event. The averaging procedure can be repeated for any other
experimentally measured quantity such as the results of the Langmuir probes.

The averaged apex height for a selected failed torus condition and an eruptive
condition are shown in the bottom row of Fig. 7. The times where there is no
apex for the eruptive condition correspond to times where many of the events
have multiple apexes and a good average cannot be created. The repetitive
nature of the eruptive events can be seen in the events both preceding and
following the event occurring at $\Delta t=0$. These events survive the
averaging due to the coherent nature of the eruptive events across shots. In
contrast to this, only one event can be seen in the failed torus condition.
This is because the events are not as coherent and the delay between
subsequent events varies significantly between shots, even with the same
experimental conditions. However, if we limit our analysis to around
$-5\,\mathrm{\mu s}<\Delta t<5\,\mathrm{\mu s}$, the events remain more
coherent and averaging can be used to build an average event.

###  IV.2 Measurements of an average event

We first want to investigate how the total energy stored within a flux rope
changes during an event. The stored energy density is given by333Due largely
to experimental constraints, we have chosen to neglect the ion pressure in
this equation. The consequences of this are discussed in Section IV.3.

$w=\frac{P_{e}}{\gamma-1}+\frac{B^{2}}{2\mu_{0}},$ (5)

where $P_{e}$ is the electron pressure and $\gamma$ is the adiabatic index and
we will use $\gamma=5/3$. Therefore the total energy per unit length in a
wedge area around the apex is given by

$W=\int h_{T}\left[\frac{P_{e}}{\gamma-1}+\frac{B^{2}}{2\mu_{0}}\right]{\rm
d}A,$ (6)

where the integral is carried out in the poloidal plane over the flux surface
that contains 50% of the current and $h_{T}$ is the scale factor defining the
width of the wedge volume around the apex (highlighted in Fig. 1) as described
in Alt [18]. The flow energy of a moving flux rope can be defined as the
kinetic energy of the ions moving with the flow velocity,

$K=\int h_{T}\left[\frac{1}{2}m_{i}nv_{\mathrm{ap}}^{2}\right]{\rm d}A,$ (7)

where $m_{i}$ is the ion mass, $n$ is the particle density, and
$v_{\mathrm{ap}}$ is the apex velocity.

The average value of $W$ and the two terms separately are presented in the top
row of Fig. 7 along with the flow energy for both a failed torus and eruptive
condition. The pressure and density in the rope are measured via the triple
Langmuir probe as described in Section III.2 and the magnetic energy is
measured with the B-dot probe array. In the failed torus case, we see a large
change in the total energy stored in the rope during the collapse, while there
is a more moderate change in the eruptive case. At times where the eruptive
ropes split and there is no well defined apex height, the flow energy cannot
be defined.

Figure 7: Energy inventory during an average failed torus (left) and eruptive
(right) event. Top: The internal pressure energy is shown with a blue dashed
line, the magnetic energy is a blue dotted line, and the sum of these is a
solid blue line. The flow energy of the moving apex is shown in red. Middle:
Compressive power done by the rope, defined by Eq. 24. The two terms of this
equation are plotted separately. The compression due to the thermal pressure
is shown in blue in the top row and the compression of the magnetic field is
shown in red. Bottom: The average apex height.

The scale factor (and therefore wedge width) used in Fig. 7 is normalized by
setting $h_{T}(z_{\mathrm{ap}})=1$ so that the integral is scaled by a unit
length. In addition to this scaling, the changing width with height is
compensated by setting the nominal wedge width to be a fixed fraction of the
total rope length using the shifted circle model described in Alt  _et al._
[21]. This fixed fraction compensates for changes in energy per unit length
that would be caused by the changing of the total rope length.

In addition to the stored energy within a rope, we also want to investigate
the work done on a rope as events occur. When a failed torus rope collapses,
it moves into a region of larger external magnetic field, which causes the
ropes to decrease in cross-sectional area. The change in volume of the wedge
around the apex causes work to be done on the rope via Eq. 24 as shown in
Appendix A. One of the assumptions made in Appendix A is that the integration
volume follows the plasma velocity. Since we are interested in a volume
centered on the apex, this assumption is equivalent to neglecting axial flow.
We are interested in Alfvénic timescales, therefore the flow is negligible as
long as $v_{\mathrm{flow}}\ll v_{\mathrm{A}}$, which is expected in our
experiments based on the frequency of the kink instability [31, 32].

The compressive power done by a rope is then given by

$P_{\mathrm{comp}}=\left(P_{e}+\frac{B^{2}}{2\mu_{0}}\right)\frac{{\rm
d}V}{{\rm d}t},$ (8)

where ${\rm d}V/{\rm d}t$ is the rate of change of the wedge volume around the
apex defined by $h_{T}$, described in Alt [18]. The measured compressive power
is shown in the middle row of Fig. 7. As a failed torus rope rises, it does a
small amount of work in order to expand in cross-sectional area. However, when
the rope quickly collapses during a failed torus event, a large amount of work
is done on the rope. The same values of $B$ is used in the calculation of the
magnetic terms of both $W$ and $P_{\mathrm{comp}}$.

Similar compressive power calculations can be done for average eruptive ropes.
However, the decrease in apex size of an eruptive rope before an event is due
to the creation of a new, lower rope (as discussed in Section III.3.1) and not
due to an actual decrease in size. Therefore, the change in apex size cannot
be considered a true compression and the values plotted in Fig. 7 for these
times are not accurate.

###  IV.3 Energy inventory during a failed torus event

Work or energy change | Source term | Energy ($10^{16}\,\mathrm{eV/cm}$)
---|---|---
Pressure compressive | $\int P_{e}\frac{{\rm d}V}{{\rm d}t}{\rm d}t$ | $-8.1$
Magnetic compressive | $\int\frac{B^{2}}{2\mu_{0}}\frac{{\rm d}V}{{\rm d}t}{\rm d}t$ | $-57$
Thermal energy | $\int\frac{P_{e}}{\gamma-1}{\rm d}V$ | $12$
Magnetic energy | $\int\frac{B^{2}}{2\mu_{0}}{\rm d}V$ | $36$
Flow energy | $\int\frac{1}{2}\rho v^{2}{\rm d}V$ | $13$
Total work |  | $-65$
Total energy change |  | $62$
Table 2: The energy inventory during an averaged failed torus event. The work
done on the rope by the changing volume as well as the energy within a wedge
around the apex are considered. The work is integrated over the same time
window the difference in energies are taken, $3.2\,\mathrm{\mu s}\leq\Delta
t\leq 8.0\,\mathrm{\mu s}$. The work done and energy change balance within 6%,
implying that ideal MHD is sufficient to describe the energy change during a
failed torus event.

The energy inventory during an average failed torus event can be found via the
plots in Fig. 7. By comparing the peak energy in a rope during an event to the
value before, we can determine the change in energy during an event. This is
shown in Table 2. In addition to the change in energy within a rope, we can
look at the compressive work done on a rope. When a failed torus rope
collapses, it moves into a region of larger external magnetic field, which
causes the ropes to decrease in cross-sectional area. The change in volume of
the wedge around the apex (See Alt [18] for a description of the wedge
volume.) causes work to be done on the rope via Eq. 24. The total work done on
a rope during this compression can be found by integrating the curve shown in
Fig. 7 over the time period of an event.

The total work done on a rope can then be compared to the total change in
energy to validate the ideal MHD model as a description of the energy
inventory. When these two values are compared, it is seen that they agree to
within 6%. This is evidence that failed tori can be explained within the
framework of ideal MHD without the need for non-ideal effects such as
reconnection. Previous work has seen that around 7-8% of the magnetic energy
is dissipated during Taylor relaxation [33]. However, this value compares the
energy dissipated to the total magnetic energy, while our 6% agreement is
relative to the measured energy change. This means that as long as the error
in our thermal energy measurements is $\lesssim 50\%$, our agreement still
falls well below the expected energy dissipation.

In Fig. 7, it can be seen that the total energy along with its components,
magnetic and internal, drop by a factor of almost 2. This is a much larger
change than would be expected in a device with fixed walls such as in Ji  _et
al._ [33]. This energy change is understood via the large change in the
geometry of the system when a rope undergoes a failed torus event. During an
event, the volume of a rope changes dramatically, causing a large change in
energy. However, it is notable that even a large energy change of $\sim 50\%$
can be explained through purely ideal MHD without resorting to non-ideal
effects such a reconnection.

In this section, we have neglected the changes in the ion pressure, $P_{i}$.
Due to the difficulty of measuring the ion temperature, we do not have any
direct measurements of $P_{i}$ in our experiments. The characteristic
timescale for energy transfer between electrons and ions is $\tau_{E,ei}\sim
3\,\mathrm{\mu s}$ while the dynamic time of an event is
$\tau_{\mathrm{event}}\sim 2\,\mathrm{\mu s}$. Since these two values are
comparable, the electrons should be able to transfer some energy to the ions
during an event, but do not transfer energy fast enough to bring them into
equilibrium at all times. Therefore, the actual situation is somewhere between
$P_{i}={\rm Const.}$ and $P_{i}=P_{e}$. The latter assumption would double the
values of both the pressure compressive work and internal energy presented in
Table 2. This decreases the total work by $8\times 10^{16}\,\mathrm{eV/cm}$
while increasing the total energy change by $12\times
10^{16}\,\mathrm{eV/cm}$, bringing the two values closer to agreement.
Therefore, the assumption of $P_{i}={\rm Const.}$ does not weaken the main
conclusion of this chapter that ideal MHD is sufficient to explain the failed
torus events.

##  V Foot-point current measurements

In this section, we will investigate the changing current distribution at the
foot points of the ropes measured via the Rogowski coils described in Section
III.1. The total current varies slowly over the lifetime of a rope
($\tau_{\mathrm{D}}\sim 150\,\mathrm{\mu s}$) which is also measured by the
outermost Rogowski coil. Therefore, fast changes in the current measured by
the other coils represent current redistribution rather than actual change in
the current. From these measurements, we can observe how the current profile
changes at the foot points of ropes during events. The change in current
distribution at the foot points during failed torus events can be a further
indication of hollowing current throughout the rope, strengthening the
conclusions previously made based on measurements at the apex.

As a note on notation: we will define the current measured by one of the
Rogowski coils with a symbol such as $I_{\mathrm{A1}}$. The subscript will
start with either an “A” or a “C” to refer to coils on the anode or cathode
side, respectively. The number can be any of $\\{0,1,2\\}$ and refers to
specific coils printed into each PCB. For example, $I_{\mathrm{A0}}$ refers to
the current measured by the outermost coil on the anode side (which should be
the same as the total plasma current), and $I_{\mathrm{C2}}$ refers to the
innermost coil on the cathode side. We will also use $I_{\mathrm{Xx}}$ to
refer to an arbitrary coil.

###  V.1 Oscillating foot-point current envelope definition

Figure 8: The currents measured by the middle Rogowski coils,
$I_{\mathrm{A1}}$ and $I_{\mathrm{C1}}$, are plotted in color compared to the
total plasma current in black for both an example failed torus (top) and
eruptive rope (bottom). For comparison with negative currents,
$-I_{\mathrm{p}}$ is also shown. The envelopes around the oscillating current
is shown with dashed lines. The time-averaged width of this envelope is used
to determine how much the current distribution changes at each foot point.

In order to investigate the change in current at the foot points of the rope,
we define an envelope around the oscillating currents. This is similar to the
procedure used to define the instability amplitude of the apex height,
$\langle\delta z\rangle/x_{\mathrm{f}}$. An example of these envelopes based
on $I_{\mathrm{X1}}$ for two example ropes can be seen in Fig. 8. Here the
current in the middle Rogowski coils at both the anode and cathode are plotted
for an example failed torus and eruptive rope. The total current passing
through the electrodes is also plotted in black for reference. A normalized
oscillation amplitude, $\langle\delta
I_{\mathrm{Xx}}\rangle/I_{\mathrm{p,max}}$, is defined to measure the change
in current normalized to the peak plasma current. Here $\langle
I_{\mathrm{Xx}}\rangle$ is envelope width time-averaged over a period where
the plasma current is within 80% of its maximum value and $I_{\mathrm{p,max}}$
is the peak plasma current.

The values of $\langle\delta I_{\mathrm{A1}}\rangle/I_{\mathrm{p,max}}$ and
$\langle\delta I_{\mathrm{C1}}\rangle/I_{\mathrm{p,max}}$ for each shot are
shown in color in Fig. 9. As in Fig. 5, each shot is plotted based on its
value of the stability parameters for the kink and torus instabilities,
$q_{a}$ and $n$ as described in Sections II.3 and II.2, respectively. The
results of almost 2,000 shots are shown. The electrons in the anode sheath are
less mobile than those near the cathode, which causes more resistivity and
thus weaker line-tying at the anode [31]. The stronger line-tying effects at
the cathode cause the cathode envelopes to be smaller than the anode
envelopes. To allow for easier visual inspection, the range represented by the
color bar in Fig. 9 is slightly smaller than in Fig. 9.

Figure 9: The normalized foot-point current envelope of the middle Rogowski
coils, $\langle\delta I_{\mathrm{X1}}\rangle/I_{\mathrm{p,max}}$, (in color)
for both the anode (top) and the cathode (bottom). As in Fig. 5, each point is
placed based on its value of the two ideal instability parameters, $q_{a}$ and
$n$ and shots that erupted are represented by triangles while non-eruptive
shots are circles. The envelope sizes for the anode are significantly larger
than those of the cathode due to the stronger line-tying there. The shots with
the largest changes are in the failed torus regime rather than the most
violently eruptive shots.

The largest values of $\langle\delta
I_{\mathrm{A1}}\rangle/I_{\mathrm{p,max}}$ are in the failed torus regime.
Even though the eruptive ropes rise and erupt much more dramatically, the
redistribution of current at the anode is not as significant. This is an
indication that the current hollowing discussed in Section II.4 occurs
throughout the rope in failed torus events, but is not significant in purely
eruptive events. The reason why there is not significant current hollowing in
eruptive events is discussed in Section VII.1. Similarly, we see that
$\langle\delta I_{\mathrm{C1}}\rangle/I_{\mathrm{p,max}}$ is also larger in
the failed torus regime than in the eruptive regime. However, the largest
values of $\langle\delta I_{\mathrm{C1}}\rangle/I_{\mathrm{p,max}}$ occur in
kinked ropes. It is currently unclear why these ropes in particular had larger
changes at the cathode than the anode. But since the focus of this work is not
on the pure kink instability, we will focus on the top two quadrants. We now
have evidence for current hollowing occurring during failed torus events at
the apex and at both electrodes simultaneously.

Figure 10: A comparison of the normalized foot-point current envelopes at the
anode vs. the cathode for: (a): each shot, (b): failed torus ropes, and (c):
eruptive ropes. The points in (b) and (c) are a subset of the points in (a).
The line $y=x$ has been added to aid comparison of the two quantities. The
instability amplitude is also shown in color using the same scale as in Fig.
5. Here we see that for the majority of ropes, $\langle\delta
I_{\mathrm{A1}}\rangle/I_{\mathrm{p,max}}>\langle\delta
I_{\mathrm{C1}}\rangle/I_{\mathrm{p,max}}$. The failed torus ropes have a
generally larger $\langle\delta I_{\mathrm{A1}}\rangle/I_{\mathrm{p,max}}$
while the spread on $\langle\delta I_{\mathrm{C1}}\rangle/I_{\mathrm{p,max}}$
is similar.

###  V.2 Comparison of the anode and cathode foot-point current envelopes

In Fig. 10 we have again plotted each of the shots shown in Fig. 9, but now
each shot is plotted based on its value of $\langle\delta
I_{\mathrm{C1}}\rangle/I_{\mathrm{p,max}}$ on the $x$-axis and $\langle\delta
I_{\mathrm{A1}}\rangle/I_{\mathrm{p,max}}$ on the $y$-axis, along with the
instability amplitude, $\langle\delta z\rangle/x_{\mathrm{f}}$, in color. This
allows us to directly compare the changing current at the anode and cathode
for each shot. From this plot, it is immediately obvious that $\langle\delta
I_{\mathrm{A1}}\rangle/I_{\mathrm{p,max}}>\langle\delta
I_{\mathrm{C1}}\rangle/I_{\mathrm{p,max}}$ for most shots, which is easily
explained by the stronger line tying at the cathode. We also see that the most
eruptive shots do not have the largest value of $\langle\delta
I_{\mathrm{A1}}\rangle/I_{\mathrm{p,max}}$. Despite having the largest change
in apex height, they do not have the most current redistribution at the foot
points. The reason why eruptive ropes have less current redistribution than
failed torus ropes is discussed in Section VII.1.

In order to investigate this, it is helpful to separate the shots in Fig. 10
based on stability quadrant. In Figs. 10 and 10, we recreate the same plot but
with shots only in the failed torus regime (Fig. 10) or in the eruptive regime
(Fig. 10). Here, we see that the failed torus ropes have larger values of
$\langle\delta I_{\mathrm{A1}}\rangle/I_{\mathrm{p,max}}$ than the eruptive
ropes. This indicates that there is more current redistribution occurring
during failed torus events, which is indicative of the current hollowing that
is occurring in these events. We also see that the ropes with $\langle\delta
I_{\mathrm{A1}}\rangle/I_{\mathrm{p,max}}<\langle\delta
I_{\mathrm{C1}}\rangle/I_{\mathrm{p,max}}$ are much more likely to be eruptive
than failed torus. This could be due to the fact that new ropes are created
during eruptive events and the profile change is new rope formation rather
than current hollowing.

###  V.3 Foot-point current redistribution during an event

Figure 11: Top: The current measured by the middle Rogowski coils,
$I_{\mathrm{X1}}$, on both the cathode (solid lines) and the anode (dashed
lines) for both eruptive events (blue) and failed torus events (red). Bottom:
The average apex height for each condition for comparison with the measured
current. A larger current (in absolute value) represents a more peaked current
distribution at the electrode.

We will now investigate the specific behavior of the measured foot-point
current on the Rogowski coils during an average event. An average failed torus
or eruptive event is created the same way as it was done in Section IV.1. The
results of this for the middle coils are shown in Fig. 11. The results of the
smallest, center coils are qualitatively similar as the middle coils and can
be seen in Alt [18]. This means that the redistribution of the current is not
localized to a particular radius but occurs throughout the foot point. Since
the middle Rogowski coil measures the current in both the center electrode
segment and the middle ring, changes in $I_{\mathrm{X2}}$ also cause changes
in $I_{\mathrm{X1}}$. However, since the changes seen in $I_{\mathrm{X1}}$ are
about twice the size of the changes seen in $I_{\mathrm{X2}}$, we can still
conclude that the current redistribution is occurring throughout the entire
minor radius.

The change in currents during an average eruptive event in Fig. 11 look
remarkably similar between the electrodes. However, since the nominal current
to each electrode has a different sign, we should actually be focused on
absolute value changes instead. In absolute value, $I_{\mathrm{A1}}$ and
$I_{\mathrm{C1}}$ actually have opposite behavior. Leading up to an event,
$|I_{\mathrm{A1}}|$ increases, (Implying that the current is becoming more
peaked at the anode.) while $|I_{\mathrm{C1}}|$ decreases. (Implying that the
current is becoming less peaked at the cathode.) The trend then reverses while
a new rope is formed and the cycle begins again. This means that there is not
a consistent current redistribution during these events and the field lines
that are gaining current intersect the cathode and anode at different minor
radii.

The foot-point current change during an eruptive event can be contrasted with
that of a failed torus event. During the rise, both $|I_{\mathrm{A1}}|$ and
$|I_{\mathrm{C1}}|$ decrease. This means that there is less current going to
the center two electrode segments444Since the total plasma current is
constant, the decrease of the center currents also implies an increase to the
current of the outer ring., implying that the current is becoming less peaked
at both electrodes. Since the failed torus events are less coherent than the
eruptive events (see Section IV.1), the return to the initial state is not as
dramatic, but the current can still be seen returning to a more peaked state.
This is the first evidence that the hollowing current previously seen in
failed torus events occurs throughout the rope and is not localized to the
apex where previous measurements were focused [15]. With this new evidence, we
will develop a theory of how the current hollowing and subsequent toroidal
flux enhancement can occur without violating the ideal MHD assumptions except
in a small region around the electrodes.

##  VI Toroidal flux change during failed torus events

Figure 12: Top: The average toroidal magnetic field measured by the outer flux
loop on both the cathode (solid lines) and the anode (dashed lines) for both
eruptive events (blue) and failed torus events (red). Bottom: The average apex
height for each condition for comparison with the measured flux. The majority
of the toroidal flux in a rope is created on timescales that are too long to
be measured by the flux loops. Therefore, the measurements here only show
dynamic changes rather than absolute values.

In addition to the measurements of the foot-point current distribution made
possible by the PCBs, we can measure the toroidal flux through flux loops also
printed into the boards. An example of such a measurement is shown in Fig. 12
where the measured flux is normalized by the flux loop area to yield an
average magnetic field, $\langle\delta B\rangle$. The majority of the toroidal
flux in a rope is created by the external coils on timescales that are too
long to be measured by the flux loops. Therefore, the measurements here only
show dynamic changes rather than absolute values. In these plots we see that
in both conditions, $\langle\delta B\rangle$ changes more on the anode than
the cathode, consistent with the stronger line-tying there. The measured
changes also lie within a relatively narrow range of $\langle\delta
B\rangle\sim 2-5\,\mathrm{G}$. Since the flux loops lie below the electrodes,
which can be considered to be perfectly conducting on eruption time scales,
the measured fluxes do not directly correspond to the toroidal flux in the
ropes. In the rest of this section, we will devise a model to describe how
these measurements would correspond to the changes in the ropes near the foot
points.

###  VI.1 Toroidal flux change in a rising MFR

Figure 13: Top: The measured total, internal, and vacuum toroidal fluxes at
the apex of a rope during an example failed torus event. The internal and
total flux both increase during the rope’s rise. Bottom: The apex height of
the rope for comparison. During the failed torus event, the apex height is
briefly multi-valued.

Figure 14: Cartoon of the changing toroidal fluxes in a rising rope. The
fluxes are considered at three locations, the apex, the top of a non-ideal
sheath layer, and at the electrode surface. Terms that increase are marked in
green while terms that decrease are marked in red. (a): The initial state of
the rope with guide field, $B_{\mathrm{g0}}$. (b): The rope rises into a
region with smaller guide field, $B_{\mathrm{g1}}$, and so $J_{\mathrm{P}}$ is
enhanced to increase $\Phi_{\mathrm{a,v}}$ so that toroidal flux is conserved.
(c): Since the current follows field lines, $J_{\mathrm{P}}$ is enhanced
throughout the rope, increasing $\Phi_{\mathrm{s,i}}$. (d): Eddy currents are
generated in the electrode to conserve $\Phi_{\mathrm{f}}$, this bends field
lines into the rope and increases $\Phi_{\mathrm{s}}$ and therefore
$\Phi_{\mathrm{a}}$.

The toroidal flux in a rope has been seen to increase during failed torus
events as shown in Fig. 13. The cartoon in Fig. 14 illustrates a series of
events wherein this can happen. In Fig. 14, we have the initial state of our
rope. The background guide field is $B_{\mathrm{g0}}$ and we will discuss the
toroidal flux in three locations: at the apex, $\Phi_{\mathrm{a}}$, at the
sheath edge $\Phi_{\mathrm{s}}$, and at the electrode surface,
$\Phi_{\mathrm{f}}$. The surface for $\Phi_{\mathrm{s}}$ is the edge of a non-
ideal layer at the base of the rope where non-ideal MHD effects cannot be
ignored, such as a Debye sheath. Other potential non-ideal layers are
discussed in Section VI.4. Since the rope is line-tied at the electrode,
$\Phi_{\mathrm{f}}$ is a constant. The three fluxes can also be broken up
based on contributing fields. We will separately consider the internal flux
created by currents in the rope, the vacuum flux created by the external
magnetic field coils, and the eddy flux created by eddy currents in the copper
electrode that exist to maintain the line-tied condition. These sources will
be denoted with the subscripts “i”, “v”, and “e”, respectively. The fluxes can
then be written as

$\displaystyle\Phi_{\mathrm{a}}$
$\displaystyle=\Phi_{\mathrm{a,i}}+\Phi_{\mathrm{a,v}},$ (9a)
$\displaystyle\Phi_{\mathrm{s}}$
$\displaystyle=\Phi_{\mathrm{s,i}}+\Phi_{\mathrm{s,v}}+\Phi_{\mathrm{s,e}},$
(9b) $\displaystyle\Phi_{\mathrm{f}}$
$\displaystyle=\Phi_{\mathrm{f,i}}+\Phi_{\mathrm{f,v}}+\Phi_{\mathrm{f,e}},$
(9c)

where the eddy contribution to $\Phi_{\mathrm{a}}$ has been ignored since the
eddy fields should be localized around the electrode. We can consider the
initial values of $\Phi_{\mathrm{s,e}}$ and $\Phi_{\mathrm{f,e}}$ to be zero
by including any initial eddy current fields in the vacuum fluxes.

In Fig. 14, the rope has risen to a height with reduced guide field,
$B_{\mathrm{g1}}$. This causes $\Phi_{\mathrm{a,v}}$ to decrease (indicated by
the red text) and therefore $\Phi_{\mathrm{a,i}}$ must increase (indicated by
the green text) to conserve $\Phi_{\mathrm{a}}$. This is accomplished by
increasing the poloidal current, $J_{\mathrm{P}}$, at the apex. Since the
plasma is force-free, the current must flow along field lines. Therefore, an
increase in $J_{\mathrm{P}}$ at the apex leads to an increase along the entire
rope, leading to Fig. 14. The increase in $J_{\mathrm{P}}$ leads to an
increase in both $\Phi_{\mathrm{s,i}}$ and $\Phi_{\mathrm{f,i}}$. Since
$\Phi_{\mathrm{f}}$ must be exactly conserved, this means that
$\Phi_{\mathrm{f,e}}$ and $\Phi_{\mathrm{s,e}}$ must become negative. The
fields due to the eddy currents are highly localized to the electrode,
therefore $|\Phi_{\mathrm{s,e}}|<|\Phi_{\mathrm{f,e}}|$. This means that the
total flux, $\Phi_{\mathrm{s}}$, will increase as in Fig. 14. The flux
increase is caused by field line bending and finite $J$$\times$$B$ is allowed
in the non-ideal layer.555The amount of field line bending allowed is examined
via a simplified model in Section VI.3. The increase in $\Phi_{\mathrm{s}}$
also leads to an increase in $\Phi_{\mathrm{a}}$ since ideal MHD can be
applied to the rest of the rope.

The increase in toroidal flux throughout the rope does not violate the frozen-
in condition of ideal MHD because the new flux comes from incorporating new
flux surfaces into the rope rather than changing the flux in a fluid element.
The new field lines do not terminate on either electrode. However, they are
able to carry plasma current because this current is able to cross field lines
in the non-ideal layer and therefore still reach the electrode.

The increase in $\Phi_{\mathrm{a,i}}$ requires an increase in $J_{\mathrm{P}}$
away from the center of the rope. This causes the current profile to be less
peaked in the center, the extreme case of which is a hollowing of the current
where it is largest at the edges of the rope as seen in Fig. 2. While this
current profile is only required of $J_{\mathrm{P}}$, the profile must also be
mirrored in $J_{\mathrm{T}}$ because the rope should be force free, i.e. the
current must flow along field lines.

###  VI.2 Fields in a straight MFR with decreasing guide field

One assumption made in Section VI.1 is that $\Phi_{\mathrm{a,v}}$ decreases
when the rope enters a region with lower guide field. However, it is not
immediately obvious that this must be the case. One could imagine a rope
entering a low $B_{\mathrm{g}}$ region and expanding proportionally to
maintain $\Phi_{\mathrm{a,v}}$ rather than enhance the toroidal field. In this
section, we will argue that this cannot be the case if flux and energy are to
be conserved. We will do this using a simplified 1D straight flux rope model.

Consider a straight flux rope with radius, $a_{0}$, constant guide field,
$B_{\mathrm{g0}}$, and constant internal toroidal field, $B_{\mathrm{Ti0}}$,
so that its total toroidal field is
$B_{\mathrm{T0}}=B_{\mathrm{Ti0}}+B_{\mathrm{g0}}$. Then the toroidal flux and
magnetic energy are given by

$\displaystyle\Phi_{\mathrm{T0}}$ $\displaystyle=\pi
a_{0}^{2}\left(B_{\mathrm{Ti0}}+B_{\mathrm{g0}}\right),$ (10a) $\displaystyle
W_{0}$ $\displaystyle=\frac{\pi
a_{0}^{2}}{2\mu_{0}}\left(B_{\mathrm{Ti0}}+B_{\mathrm{g0}}\right)^{2}.$ (10b)

Now suppose we reduce the guide field to $B_{\mathrm{g1}}$, where
$B_{\mathrm{g1}}<B_{\mathrm{g0}}$. We will then determine the values for the
other resulting rope parameters, $a_{1}$ and $B_{\mathrm{Ti1}}$. Using
toroidal flux conservation, $\Phi_{\mathrm{0}}=\Phi_{\mathrm{1}}$, we have,

$\pi a_{0}^{2}(B_{\mathrm{Ti0}}+B_{\mathrm{g0}})=\pi
a_{1}^{2}(B_{\mathrm{Ti1}}+B_{\mathrm{g1}}).$ (11)

Due to energy conservation, the change in magnetic energy must be equal to the
work done by the rope on the external (guide) fields.666We have assumed zero
$\beta$ so the internal energy cannot change and have neglected changes in the
poloidal magnetic field energy. A poloidal energy term can be added to the LHS
of Eq. 13, however this term is generally negative for an arched MFR and
increases the resulting $B_{\mathrm{Ti1}}$. The compressive work done by the
external fields is caused by the magnetic pressure,

$W_{\mathrm{comp,g}}=\int P_{\mathrm{comp,g}}{\rm d}t=\int
B_{\mathrm{g}}^{2}\frac{{\rm d}V}{{\rm d}t}{\rm d}t,$ (12)

where $V\propto a^{2}$ is the volume of the rope. This yields777In general,
$W_{\mathrm{comp,g}}$ will depend on the form of $B_{\mathrm{g}}(V)$. If one
assumes an ansatz of $B_{\mathrm{g}}=\alpha V^{-n}$ for some $\alpha$ and
$n>1$, then the RHS of Eq. 13b is multiplied by $1/(2n-1)$. This changes the
specific values but not the general conclusions of this section. The condition
of $n>1$ is equivalent to the rope expanding slower than a vacuum rope with
$B_{\mathrm{Ti}}=0$.

$\displaystyle W_{1}-W_{0}$ $\displaystyle=W_{\mathrm{comp,g}},$ (13a)
$\displaystyle
a_{1}^{2}(B_{\mathrm{Ti1}}+B_{\mathrm{g1}})^{2}-a_{0}^{2}(B_{\mathrm{Ti0}}+B_{\mathrm{g0}})^{2}$
$\displaystyle=a_{1}^{2}B_{\mathrm{g1}}^{2}-a_{0}^{2}B_{\mathrm{g0}}^{2}.$
(13b)

These two equations can be solved for $a_{1}$ and $B_{\mathrm{Ti1}}$,

$\displaystyle\left(\frac{a_{1}}{a_{0}}\right)^{2}$
$\displaystyle=\frac{-B_{\mathrm{Ti0}}^{2}-2B_{\mathrm{Ti0}}\pm\sqrt{D}}{2B_{\mathrm{g1}}^{2}},$
(14a) $\displaystyle B_{\mathrm{Ti1}}$
$\displaystyle=\frac{B_{\mathrm{Ti0}}^{2}-2B_{\mathrm{g1}}B_{\mathrm{Ti0}}-2B_{\mathrm{g1}}+2B_{\mathrm{Ti0}}\pm\sqrt{D}}{2(B_{\mathrm{Ti0}}+1)},$
(14b)

where the magnetic fields have been normalized to $B_{\mathrm{g0}}$ and $D$ is
defined as

$D=B_{\mathrm{Ti0}}^{2}(B_{\mathrm{Ti0}}+2)^{2}+4B_{\mathrm{g1}}^{2}(B_{\mathrm{Ti0}}+1)^{2}.$
(15)

We are only interested in the positive root of these equations in order to
maintain $a_{1}^{2}>0$. It can be seen that there are only two free parameters
in these equations, $B_{\mathrm{g1}}$ and $B_{\mathrm{Ti0}}$.

Figure 15: Contour plots of the normalized change in the minor radius (left)
and internal toroidal field (right) in a straight flux rope with decreasing
guide field. When quantities are normalized to $a_{0}$ and $B_{\mathrm{g0}}$,
there are only two free parameters left, $B_{\mathrm{g1}}$ and
$B_{\mathrm{Ti0}}$ which are represented on the $x$ and $y$-axes,
respectively. Since both quantities are non-negative, both the minor radius
and internal toroidal field increase when the guide field is decreased.

Contour plots of $a_{1}/a_{0}-1$ and
$(B_{\mathrm{Ti1}}-B_{\mathrm{Ti0}})/B_{\mathrm{g0}}$ are presented in Fig.
15. It can be seen in these plots that both of these quantities are
nonnegative for all values of $B_{\mathrm{g1}}$ and $B_{\mathrm{Ti0}}$.
Therefore, the rope expands and the internal toroidal field is enhanced by the
decrease in external guide field. While this simplified model without toroidal
effects cannot be effectively used to quantitatively describe the changing
vacuum flux during the rise of an arched rope, it does indicate that we should
expect $\Phi_{\mathrm{a,v}}$ to decrease as a rope rises into areas of smaller
$B_{\mathrm{g}}$. Since this simplified model does not include any effects of
the field profile, (i.e. it assumes constant $B_{\mathrm{T}}$) it cannot
accurately predict the enhancement of the tension force. A comparison to
experimental profiles during an event with constant flux similar to this model
are presented in Section VII.

###  VI.3 Toroidal flux enhancement by field-line bending around the
electrodes

Figure 16: The field lines of a current loop located $\Delta z$ above the
electrodes (shown in brown) along with an image current to maintain the
frozen-in condition at $z=0$. Left: The field lines of just the current loops.
Right: The current loops embedded in a background field in the $z$-direction.
The electrode thickness and the sheath thickness have both been exaggerated
for clarity.

In order to better study the field-line bending occurring in Fig. 14, a simple
model of enhanced poloidal current is devised. We will model the current as a
infinitesimal current loop with current $I_{\mathrm{loop}}$ of radius $a$
located at a height $z=\Delta z$ above the electrode. This height can be taken
as the edge of the non-ideal layer where the plasma current is able to cross
field lines. In order to maintain the frozen-in condition at the electrode
surface, $\Delta B(z=0)=0$, a mirror current loop of current
$-I_{\mathrm{loop}}$ is located at $z=-\Delta z$. The field lines of this
configuration are shown in Fig. 16. Fig. 16 is not to scale. In particular,
the ratio $\Delta z/a$ has been increased to ease visibility of the field
lines in the non-ideal layer.

These field lines are only meant to represent the field due to the enhanced
poloidal current, and so Fig. 16 shows the field lines of these current loops
embedded in a straight field representing the guide field. Here one can see
that field lines that would not be in the rope without the loop currents have
been bent into the rope. In reality the the enhanced $J_{\mathrm{P}}$ should
exist along the entire length of the rope rather than being localized to the
base of the rope. While this simplified model can reasonably describe the
field-line bending in the non-ideal layer, it cannot be used above the current
loop at $z=\Delta z$. So, while the additional lines are later bent back out
of the rope, in reality we expect that the enhanced toroidal flux will be
frozen into the rope

Figure 17: The average field at the flux loops calculated from the current
loop model and using the actual experimental geometry. The average field,
$\langle\Delta B\rangle$, is normalized to the loop current,
$I_{\mathrm{loop}}$. The sheath thickness, $\Delta z$, sets the location of
the current loop above the electrode. For $\Delta z\ll a_{0}$ this relation is
approximately linear, where $a_{0}=7\,\mathrm{cm}$ is the electrode radius.

Using the experimental values for $a$ and the flux loop location in this
model, we can predict the expected measured flux change (Shown as an average
field change, $\langle\Delta B\rangle$.) as a function of $I_{\mathrm{loop}}$
and $\Delta z$, as shown in Fig. 17. For small separations where $\Delta z\ll
a$, $\langle\Delta B\rangle$ is approximately linear and is described by

$\langle\Delta B\rangle\leavevmode\nobreak\
(\mathrm{G})\approx-1.06\leavevmode\nobreak\
I_{\mathrm{loop}}(\mathrm{kA})\leavevmode\nobreak\ \Delta
z\leavevmode\nobreak\ (\mathrm{cm}),$ (16)

where $\langle\Delta B\rangle$ is measured in G, $I_{\mathrm{loop}}$ is
measured in kA, and $\Delta z$ is measured in cm. As a rough estimate, we
expect that $I_{\mathrm{loop}}\lesssim 10\,\mathrm{kA}$ since it should not be
significantly larger than the toroidal current, $I_{\mathrm{P}}\approx
12\,\mathrm{kA}$. As seen in Section VI, the measured change in flux by the
flux loops is $\langle\Delta B\rangle\sim 2-5\,\mathrm{G}$. Based on these
numbers, we expect that the non-ideal layer has a height of $\Delta z\sim
0.2\,\mathrm{cm}$. While this simplified model can only be trusted for an
order-of-magnitude estimate of $\Delta z$, we can still compare the expected
non-ideal based on the plasma parameters to this value.

###  VI.4 Possible non-ideal layers between an MFR and electrode

We expect the non-ideal layer between the electrodes and the base of the flux
rope to have a width of around $\Delta z\sim 0.2\,\mathrm{cm}$. Three
potential layers will be considered which are summarized in Table 3. The
simplest potential layer is a Debye sheath with a width based on the Debye
length, $\lambda_{\mathrm{D}}=\sqrt{{\epsilon_{0}T}/{(ne^{2})}}$, where
$\epsilon_{0}$ is the permittivity of free space. However, based on the
experimental parameters, $\lambda_{\mathrm{D}}\approx 2\times
10^{-4}\,\mathrm{cm}$ which is far too small to explain the expected non-ideal
layer.

Layer type | Experimental value | Solar value
---|---|---
Debye sheath | $2\times 10^{-4}\,\mathrm{cm}$ | $1\times 10^{-4}\,\mathrm{cm}$
$\lambda_{\mathrm{D}}=\sqrt{\epsilon_{0}T/(ne^{2})}$ |  |
Hartmann layer | 0.35 cm | $0.086\,\mathrm{cm}$
$\delta_{\mathrm{Ha}}={\sqrt{m_{i}n\nu\eta}}/{B}$ |  |
Resistive layer | 0.052 cm | $2.1\,\mathrm{cm}$
$\delta_{\mathrm{S}}={\eta\sqrt{\mu_{0}m_{i}n}}/{B}$ |  |
Table 3: A comparison of the potential non-ideal layers considered for the
region between the electrode and the ideal rope plasma. Values for both the
experimental conditions and the photosphere are presented. Sources of
anomalous resistivity and viscosity have not been considered.

The Debye length describes the scale under which the quasi-neutrality
approximation holds. However, we can consider the finite viscosity, $\nu$, and
resistivity, $\eta$, in our plasma to find a characteristic scale length under
which the ideal MHD approximations are violated. One example of this is the
Hartmann layer where magnetic forces are balanced by resistive and viscus
forces as described by the Hartmann number
$\mathrm{Ha}={B\delta}/{\sqrt{m_{i}n\nu\eta}}$, where $\delta$ is a
characteristic scale length [38, 39]. In Hartmann flow of a magnetized plasma
over a stationary surface, a characteristic length scale over which
$J$$\times$$B$ forces are balanced by viscus and resistive forces is given by
$\delta_{\mathrm{Ha}}={\sqrt{m_{i}n\nu\eta}}/{B}$. Based on our experimental
plasma parameters, this length scale is $\delta_{\mathrm{Ha}}\approx
0.35\,\mathrm{cm}$, which is similar to the expected $\Delta z$.

In addition to $\delta_{\mathrm{Ha}}$, we could consider a length scale where
magnetic forces are balanced entirely by resistive forces. This scale can be
found by considering the Lundquist number,
$\mathrm{S}=\mu_{0}v_{\mathrm{A}}\delta/\eta$. Similar to how
$\delta_{\mathrm{Ha}}$ is the length scale under which $\mathrm{Ha}=1$, we can
find $\delta_{\mathrm{S}}$ as a length scale where $\mathrm{S}=1$,
$\delta_{\mathrm{S}}={\eta\sqrt{\mu_{0}m_{i}n}}/{B}$. Based on our
experimental plasma parameters, this length scale is
$\delta_{\mathrm{S}}\approx 0.052\,\mathrm{cm}$. This is slightly too small to
explain the expected $\Delta z$, so we expect the Hartmann layer to be the
significant cause of the non-ideal layer.

The solar length scales for these non-ideal layers is also presented in Table
3. Compared to the typical length scales of solar MFRs, we see that these
layers are all too small to be able to explain any significant flux change in
a rope. However, we have only considered classical Spitzer resistivity and
viscosity in these calculations. The addition of anomalous sources of
resistivity and viscosity may increase the solar values significantly.

##  VII Experimentally measured changes in the toroidal field profile

The simple 1D straight flux rope model presented in Section VI.2 can help
explain the increase in $\Phi_{\mathrm{a,i}}$. However, it fails to accurately
predict the value of the enhanced tension force, $F_{\mathrm{t}}$, because it
assumes the fields are constant in the rope. The $F_{\mathrm{t}}$ change we
are interested in depends on the fields via Eq. 1. Since the relevant quantity
is the average squared toroidal field, $\langle B_{\mathrm{T}}^{2}\rangle$,
the exact profile is necessary to predict the changes in $F_{\mathrm{t}}$.

Figure 18: Various rope parameters related to flux and field profile during a
sample failed torus event. Here the rope has been defined by a constant total
toroidal flux, $\Phi_{\mathrm{a}}$ rather than by the toroidal current. The
tension force increases by a factor of more than two, consistent with
experimental measurements of $F_{\mathrm{t}}$. (a): The flux due to the vacuum
and internal fields. (b): The average value of the square of the total
toroidal field compared with the vacuum and internal values. (c): The tension
force derived from $\langle B_{\mathrm{T}}^{2}\rangle$ normalized to the value
at a reference time before the event. (d): The apex height for reference.
During the failed torus event, the apex height is briefly multi-valued.

To that end, we have analyzed the changes in $\langle
B_{\mathrm{T}}^{2}\rangle$ and the predicted $F_{\mathrm{t}}$ during an
example failed torus event in Fig. 18. Rather than defining the boundary of
the rope by a fixed fraction of the total current, we have held the total
toroidal flux, $\Phi_{\mathrm{a}}$, constant in time. This means that the new
field lines that are added to the rope are not being considered in this
analysis, but rather a consistent flux surface is being considered. As the
rope rises, $\Phi_{\mathrm{a,v}}$ decreases and $\Phi_{\mathrm{a,i}}$
increases as expected. We also see that $\langle B_{\mathrm{T}}^{2}\rangle$
increases sharply during the collapse. This causes $F_{\mathrm{t}}$ (defined
here via Eq. 1) to increase dramatically, which in turn causes the rope to
collapse and the eruption to fail. By comparing $\langle
B_{\mathrm{Ti}}^{2}\rangle$ and $\langle B_{\mathrm{Tv}}^{2}\rangle$
separately in Fig. 18, we see that this increase is largely driven by the
change of the internal profile rather than the decrease in $B_{\mathrm{g}}$.
The value of $F_{\mathrm{t}}$ increases by a factor of more than two from the
initial state to the peak during the collapse. This is consistent with
experimental measurements where $F_{\mathrm{t}}$ is calculated by integrating
the $J$$\times$$B$ forces rather than using Eq. 1 [15].

###  VII.1 Effects of the kink instability on failed torus events

By analyzing the change in the tension force, $F_{\mathrm{t}}$, in a rising
MFR while holding the total toroidal flux at the apex, $\Phi_{\mathrm{a}}$,
constant, we have shown that failed torus effects can be explained entirely by
ideal MHD without the need for adding flux in a non-ideal layer. We are left
with the question as to why these effects are absent in the eruptive regime
(the lower left quadrant of Fig. 5) when the safety factor, $q_{a}$, is small.
When a rope is kink-unstable, its axis is able to tilt in the $x$-$y$-plane,
mixing the roles of the strapping and guide fields. In particular, if a rope’s
axis is tilted an angle of $\theta$, its effective strapping and guide fields
would be

$\displaystyle B_{\mathrm{s,eff}}$ $\displaystyle=B_{y}\cos\theta-
B_{x}\sin\theta,$ (17a) $\displaystyle B_{\mathrm{g,eff}}$
$\displaystyle=B_{y}\sin\theta+B_{x}\cos\theta,$ (17b)

where $B_{y}$ and $B_{x}$ are, respectively, the strapping and guide fields
when there is no tilt. The change in a rope’s guide field becomes conflated
with the change in strapping field, and so the rope is able to erupt by
slipping through the field lines in the arcade above it.

If instead a rope is kink-stable and cannot twist significantly, it must rise
up through the arcade without tilting and experience a changing guide field
based on the decay index of the guide field, $n_{\mathrm{g}}$. In order to
begin the rise however, it must be initially unstable to the torus instability
with $n_{\mathrm{s}}>n_{\mathrm{cr}}^{\mathrm{}}$ (limiting us to the upper
right quadrant of Fig. 5). Then when the guide field decays, the profile can
change, enhancing $F_{\mathrm{t}}$. However, this assumes that
$B_{\mathrm{g}}$ decays fast enough for this effect to happen. For example, if
$n_{\mathrm{g}}=0$, then $B_{\mathrm{g}}$ would be a constant and we would
expect no $F_{\mathrm{t}}$ enhancement from these effects. This suggests that
there may be a critical value of $n_{\mathrm{g}}$ where this effect could
happen, i.e. it occurs when $n_{\mathrm{g}}>n_{\mathrm{g,cr}}$. However, due
to experimental constraints, we were not able to probe to sufficiently small
values of $n_{\mathrm{g}}$ to see these effects. Furthermore, on the Sun,
$B_{\mathrm{g}}$ is generated by external currents below the photosphere.
Therefore small values of $n_{\mathrm{g}}$ may not be realizable on the Sun
either.

Additionally, the movement of the apex may not be small when considering the
onset of the failed torus, so rather than using the local derivative (or
equivalently, $n_{\mathrm{g}}$), a finite difference should be considered
instead. We empirically observe that the failed torus occurs when the change
in guide field is around half of its initial value, i.e. $|\Delta
B_{\mathrm{g}}|/B_{\mathrm{g0}}\sim 1/2$. For a given configuration with
decaying $B_{\mathrm{g}}$ this will occur at some finite displacement, $\Delta
z_{\mathrm{ap}}$, away from its initial apex height. The failed torus will be
able to disrupt the eruption if $\Delta z_{\mathrm{ap}}$ is small enough such
that some other non-ideal effect is not able to disrupt this mechanism. For
example, once a rope rises far enough, tether cutting reconnection can occur
below the rope, disrupting the initial topology. This reasoning can lead to
estimates for $n_{\mathrm{g,cr}}$. However, due to the complexity of the
fields and the onset of reconnection, and analytical solution is intractable
and it warrants future numerical study.

###  VII.2 Application to solar conditions

The toroidal flux increase seen in our experiments in Fig. 13 may not be
possible in solar flux ropes. In our experiments, a non-ideal layer above the
electrodes allows for field-line bending to increase the total flux in a rope.
However, using classical values for resistivity and viscosity in the
photosphere, yields layers that are far too small to contribute a significant
amount of flux (see Table 3). Despite the lack of flux increase, the results
of Section VII and Fig. 18 show that redistribution of $B_{\mathrm{Ti}}$ can
account for a significant increase of $F_{\mathrm{t}}$ without any increase of
$\Phi_{\mathrm{a}}$. Since $F_{\mathrm{t}}$ depends on $\langle
B_{\mathrm{T}}^{2}\rangle$ rather than the total flux, $F_{\mathrm{t}}$ can be
increased by making $B_{\mathrm{T}}$ less uniform, i.e. a $B_{\mathrm{T}}$
profile that is either more peaked or more hollow will increase $\langle
B_{\mathrm{T}}^{2}\rangle$.

We have discussed the differing roles of $n_{\mathrm{s}}$ and $n_{\mathrm{g}}$
in the onset of the torus instability and subsequent failed torus events and
how these roles get mixed when kinking occurs at the apex. $n_{\mathrm{s}}$ is
important for driving the torus instability while $n_{\mathrm{g}}$ can cause
the necessary changes to create a failed torus and end the eruption. However,
this may prove challenging for future predictions of solar events because
observations are often unable to determine the direction of measured magnetic
fields and therefore cannot independently measure $n_{\mathrm{s}}$ and
$n_{\mathrm{g}}$ [40].

##  VIII Discussion and conclusions

The prediction of space weather requires a detailed understanding of the ideal
MHD behavior of arched, line-tied, magnetic flux ropes. Models that are widely
used to describe the behavior of toroidally symmetric systems such as tokamaks
break down when extended to the line-tied system of an MFR. Laboratory
experiments have been carried out to investigate the behavior of ideal MHD
instabilities, particularly the torus instability, in MFRs.

In addition to the standard torus instability occurring at small values of
$q_{a}$, a class of “failed torus” MFRs that were torus-unstable but kink-
stable where observed failing to erupt [15]. These ropes often begin to rise
due to the torus instability, but then the current profile becomes hollow, the
toroidal field (and therefore the tension force) is increased and the eruption
ultimately fails. We have investigated the energy inventory before and after
these events and found that ideal MHD can explain the energy change to within
6%, well within experimental error. In reversed field pinches, Taylor
relaxation is often considered as the mechanism through which the current
profile can change, resulting in a flat twist density profile,
$\alpha=\mbox{\boldmath{$J$}}\mbox{\boldmath{$\cdot$}}\mbox{\boldmath{$B$}}/(\mu_{0}B^{2})$
[16]. However, Taylor relaxation only conserves the total helicity and not the
helicity of each flux surface. This means that internal reconnection (and
therefore non-ideal MHD effects) is required for the plasma to reach its
lowest energy state. This means that Taylor relaxation cannot be the cause of
the current profile change in failed torus ropes.

By measuring the current distribution at the foot points during failed torus
events, we have confirmed that the current redistribution is a global
phenomenon and not localized to the apex where previous measurements were
focused. We have also seen that, in our experiments, the total toroidal flux,
$\Phi_{\mathrm{a}}$, at the apex of a failed torus rope increases as it rises
during an event. We have developed a model for how this can occur without
violating the ideal MHD assumptions in the bulk of the rope. As a rope rises,
the external guide field, $B_{\mathrm{g}}$, at the apex decreases. This causes
the poloidal current, $J_{\mathrm{P}}$, to increase away from the center. The
resulting currents interacting with the frozen-in condition at the conducting
electrodes, causes flux to be pulled into the rope thorough a thin non-ideal
layer above the electrodes. The current is able to cross field lines in this
layer and therefore new field lines are added to the rope, increasing
$\Phi_{\mathrm{a}}$. This flux increase then causes the tension force,
$F_{\mathrm{t}}$, to increase, ultimately causing the eruption to fail. The
expected flux changes during these events have been confirmed by measurements
of flux loops placed below the electrodes.

Additionally, changes in the toroidal field profile have been measured during
failed torus events in our experiments. The changes in $\langle
B_{\mathrm{T}}^{2}\rangle$ when holding $\Phi_{\mathrm{a}}$ constant have been
shown to be significant enough to cause $F_{\mathrm{t}}$ to increase by a
factor of more than two. Due to the thin non-ideal layers on the Sun, this
redistribution mechanism is expected to be far more significant for solar flux
ropes than any $\Phi_{\mathrm{a}}$ increase. This mechanism for failed
eruptions could be used to predict failed CME events on the Sun.

###### Acknowledgements.

The authors would like to thank P. Sloboda and A. Jones for their help with
the experimental setup. This research is supported by Department of Energy
contract numbers DE-SC0019049 and DE-AC02-09CH11466 and NASA grant number
80HQTR17T0005.

## Data availability

Data underlying the results presented in this paper will be made available via
Princeton University’s data repository upon publication.

##  Appendix A Conservation of energy in a flux rope under ideal MHD

We would like to analyze the energy inventory of an MFR before and after an
eruptive or failed torus event. We will consider the energy equation using the
assumptions of ideal MHD and then will compare to our experimental results in
Section IV.3.888The ideal MHD equations presented in this section neglect
thermal conduction by assuming an adiabatic equation of state. However, the
adiabatic assumption is confirmed to be valid in our experiments during the
timescales discussed in Section IV.3. The agreement (or disagreement) will
help validate (or invalidate) our assumptions. A substantial difference in the
energy between the experiment and theory would indicate the need for non-ideal
processes such as reconnection to explain the events.

The conservative form of the ideal MHD energy equation is given by [42]

$\frac{\partial w}{\partial
t}+\mbox{\boldmath{$\nabla$}}\mbox{\boldmath{$\cdot$}}\mbox{\boldmath{$s$}}=0,$
(18)

where $w$ and $s$ are given by

$\displaystyle w$ $\displaystyle=\frac{1}{2}\rho
v^{2}+\frac{P}{\gamma-1}+\frac{B^{2}}{2\mu_{0}},$ (19a) $s$
$\displaystyle=\left(\frac{1}{2}\rho
v^{2}+\frac{P}{\gamma-1}\right)\mbox{\boldmath{$v$}}+P\mbox{\boldmath{$v$}}+\frac{1}{\mu_{0}}\mbox{\boldmath{$E$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}},$
(19b)

where $\rho$ is the plasma mass density, $P$ is the plasma pressure, $E$ is
the electric field, $\gamma$ is the adiabatic index, and $v$ is the plasma
velocity. The terms in $w$ represent the density of the kinetic energy, the
internal energy, and the magnetic energy, respectively. Whereas the terms of
$s$ represent the flux of energy density (both kinetic and internal), the
compressive work done on the plasma, and the Poynting vector, respectively. We
will be assuming ideal MHD holds and so will use the ideal form of Ohm’s Law,

$\mbox{\boldmath{$E$}}+\mbox{\boldmath{$v$}}\mbox{\boldmath{$\times$}}\mbox{\boldmath{$B$}}=0,$
(20)

i.e. we will assume that there is no resistivity.

The total energy in the system is given by

$W=\int_{V}w{\rm d}^{3}r,$ (21)

where the integral is carried out over the entire plasma volume, $V$. We then
take a time-derivative of this equation,

$\frac{{\rm d}W}{{\rm d}t}=\int_{V}\frac{\partial w}{\partial t}{\rm
d}^{3}r+\int_{S}w\mbox{\boldmath{$v$}}\mbox{\boldmath{$\cdot$}}\mbox{\boldmath{${\rm
d}S$}},$ (22)

where $S$ is the surface of the plasma volume. The surface, $S$, moves with
the plasma velocity, $v$, and so the rate of change of the volume is given by

$\frac{{\rm d}V}{{\rm
d}t}=\int_{S}\mbox{\boldmath{$v$}}\mbox{\boldmath{$\cdot$}}\mbox{\boldmath{${\rm
d}S$}}.$ (23)

We can then substitute Eqs. 18 and 19 into Eq. 22 and use Eq. 20 to get (See
Alt [18] for a more detailed derivation.)

$\frac{{\rm d}W}{{\rm
d}t}=-\int_{S}\left(\mbox{\boldmath{$s$}}-w\mbox{\boldmath{$v$}}\right)\mbox{\boldmath{$\cdot$}}\mbox{\boldmath{${\rm
d}S$}}=-\int_{S}\left(P+\frac{B^{2}}{2\mu_{0}}\right)\mbox{\boldmath{$v$}}\mbox{\boldmath{$\cdot$}}\mbox{\boldmath{${\rm
d}S$}}.$ (24)

This equation implies that the change in total energy in the system is
equivalent to the work done by compression of both the thermal and magnetic
pressures.

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  * Lingwood and Alboussiere [1999] R. Lingwood and T. Alboussiere, Physics of Fluids 11, 2058 (1999).
  * Liu [2008] Y. Liu, The Astrophysical Journal Letters 679, L151 (2008).
  * Note [8] The ideal MHD equations presented in this section neglect thermal conduction by assuming an adiabatic equation of state. However, the adiabatic assumption is confirmed to be valid in our experiments during the timescales discussed in Section IV.3.
  * Freidberg [2014] J. P. Freidberg,  _Ideal MHD_ (Cambridge University Press, 2014).