# Laboratory verification of electron-scale reconnection regions modulated by
a three-dimensional instability

###### Abstract

During magnetic reconnection in collisionless space plasma, the electron fluid
decouples from the magnetic field within narrow current layers, and
theoretical models for this process can be distinguished in terms of their
predicted current layer widths. From theory, the off-diagonal stress in the
electron pressure tensor is related to thermal non-circular orbit motion of
electrons around the magnetic field lines. This stress becomes significant
when the width of the reconnecting current layer approaches the small
characteristic length scale of the electron motion. To aid in situ spacecraft
and numerical investigations of reconnection, the structure of the electron
diffusion region is here investigated using the Terrestrial Reconnection
EXperiment (TREX). In agreement with the closely matched kinetic simulations,
laboratory observations reveal the presence of electron-scale current layer
widths. Although the layers are modulated by a current-driven instability, 3D
simulations demonstrate that it is the off-diagonal stress that is responsible
for breaking the frozen-in condition of the electron fluid.

JGR: Space Physics

University of Wisconsin - Madison Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, USA

Samuel<EMAIL_ADDRESS>
The structure of the electron diffusion region is important to reconnection in
space plasma

The widths of laboratory electron reconnection layers match those of kinetic
simulations

Kinetic simulations show that the electron pressure tensor breaks the electron
frozen flux condition

## Plain Language Summary

“Space weather” describes the conditions of the plasma surrounding Earth,
which can have severe impact on the functionality of spacecraft as well as the
health of human space travelers. Space weather and the dynamics of space
plasmas in general are closely linked to the structure and topology of the
magnetic fields permeating our solar system. By a process called magnetic
reconnection, magnetic field lines can rapidly and suddenly break and
reconfigure their connectivity, allowing for an explosive release of magnetic
energy. This phenomenon is at the origin of explosive events such as solar
flares and is the driver of magnetic storms in the Earth’s magnetosphere
powering the Auroras.

We present new laboratory observations of this near-Earth reconnection process
recreated in the Terrestrial Reconnection EXperiment (TREX). The experiment
provides detailed measurements of the width of the region where the magnetic
field lines break, the electron diffusion region (EDR). Consistent with
supercomputer simulations of reconnection, the width of the EDR is measured to
be set by the fine spatial scale of the electron orbit motions. As such, the
observations provide renewed evidence that the reconnection process is
mediated by forces of thermal stress related to the electron motion within the
reconnection region.

##  1 Introduction

Magnetic reconnection [Dungey (1953)] is the process of changing the topology
of magnetic field lines in the presence of a plasma, often permitting an
explosive release of magnetic energy. Well-known examples include solar flares
[Priest Forbes (2000)] and auroral substorms in the Earth’s magnetosphere
[Vasyliunas (1975)]. Although reconnection often governs the global dynamics
of plasma systems, the reconnection process occurs in localized electron
diffusion regions (EDRs), where the motion of the electron fluid decouples
from the magnetic field, breaking the frozen-in law of magnetohydrodynamics.
The origin of this process in the collisionless regime, where conventional
resistive friction is absent, remains controversial. For example, laminar
kinetic models predict that the EDRs are characterized by intense current
layers with widths as narrow as the kinetic scales associated with the
electron orbit motion [Vasyliunas (1975), Pritchett (2001)]. In other models,
the scattering of electrons by electric field fluctuations associated with
high-frequency instabilities is proposed to widen the current layers and
enhance the anomalous transfer of momentum from the electrons to the ions
[Papadopoulos (1977), Huba . (1977), Hoshino (1991)].

Significant insight into reconnection physics is provided by fully kinetic
numerical models. In 3D configurations it has been argued that turbulence can
cause local suppression of the effective conductivity [Silin . (2005), Che .
(2011), Muñoz . (2017)], but other simulation studies have reported these
effects are relatively small in both low-$\beta$ parameter regimes [Liu .
(2013)] relevant to solar physics and for higher-$\beta$ regimes with
asymmetric layers relevant to the magnetosphere [Roytershteyn . (2012), Hesse
. (2018), Le . (2018)]. Rather, these 2D and 3D kinetic models typically
suggest that fast reconnection can be mediated by electron inertia, and terms
in the electron pressure tensor [Speiser (1965), Vasyliunas (1975), Lyons
Pridmore-Brown (1990), Horiuchi Sato (1994), Cai Lee (1997), Kuznetsova .
(1998)]. These effects require the formation of intense electron current
channels with widths characterized by either the electron inertial length
$d_{e}=c/\omega_{pe}$ or the electron orbit scale [Roytershteyn . (2013)].

To observationally address this issue, a primary goal of NASA’s Magnetospheric
Multiscale (MMS) Mission is to characterize the structures of EDRs for
reconnection sites in the Earth’s magnetosphere [Burch, Moore . (2016)].
However, the in situ observations have not as yet provided conclusive insight
to the role of anomalous resistivity. For example, the initial magnetotail
observations are consistent with laminar kinetic reconnection [Torbert .
(2018), Egedal . (2019), Genestreti . (2018), TKM. Nakamura . (2018)].
Meanwhile, for a magnetopause reconnection layer crossing [Burch, Torbert .
(2016)] evidence for anomalous resistivity was identified near an EDR [Torbert
. (2016)], but a separate analysis concluded the electron dynamics were in
agreement with a 2D kinetic model (without anomalous resistivity) [Egedal .
(2018)].

Dedicated laboratory experiments can provide complementary methods to study
EDRs. Contrary to spacecraft measurements, laboratory experiments allow the
controlled and reproducible study of reconnection layers with well understood
upstream conditions and magnetic geometry. Results from the Magnetic
Reconnection Experiment (MRX) at Princeton find that the current layer widths
are much wider (by approximately a factor of four) compared to the predictions
by kinetic models [Ji . (2008)]. This disagreement [Dorfman . (2008)] remains
unresolved as it persists even when accounting for collisions [Roytershteyn .
(2010)], and 3D instabilities [Roytershteyn . (2013)].

In this paper, we report on experimental and numerical investigations of the
EDR width using the Terrestrial Reconnection EXperiment (TREX) at the
University of Wisconsin-Madison [Forest . (2015), Olson . (2016), Olson . (in
press, 2021)]. While TREX and MRX both have similar normalized system sizes,
$\simeq 10c/\omega_{pi}$ where $\omega_{pi}$ is the ion plasma frequency, TREX
is physically larger and operates at lower plasma density and a more
collisionless regime (Lundquist number, $S\simeq 10^{4}$), where anisotropic
and non-gyrotropic electron pressure tensor effects can begin to develop
around the EDR [Le . (2015)]. Furthermore, the TREX experiment applies a
unique jogging method, where the reconnection layer is swept across magnetic
sensors, yielding high spatial resolution measurements of the magnetic
structures. To elucidate the experimental findings, fully kinetic 2D and 3D
simulations were performed with plasma profiles and reconnection drives that
are comparable to the experiment.

##  2 The Terrestrial Reconnection EXperiment (TREX)

The applied TREX configuration is presented by the engineering schematic in
Fig. 1(a). The vacuum vessel, provided by the Wisconsin Plasma Physics
Laboratory (WiPPL) [Forest . (2015)], is a $3$ meter diameter sphere that uses
an array of permanent magnets embedded in the chamber wall to limit the plasma
loss area to a very narrow fraction of the total surface area while keeping
the bulk of the plasma unmagnetized. The setup includes a set of internal
drive coils and an exterior Helmholtz coil that provides a near-uniform axial
magnetic field with a magnitude up to $100$\mathrm{mT}$$ [Olson . (2016),
Olson . (in press, 2021)]. The current through the three internal drive coils
(purple) ramps up to create a magnetic field that opposes and reconnects with
the background Helmholtz field, resulting in an anti-parallel magnetic
configuration (e.g. no significant guide field). The plasma source is a set of
plasma guns located at the machine’s pole (shown in yellow). TREX can operate
in hydrogen, deuterium and helium plasmas; results presented in this paper
will focus on hydrogen and deuterium. As will be described in the following
sections, the interpretation of the experimental results are aided by 2D
simulations at the full hydrogen/electron mass ratio ($m_{i}/m_{e}=1836$). For
numerical tractability, 3D simulations were implemented with a mass ratio of
$m_{i}/m_{e}=400$.

Figure 1: (a) Engineering sketch of TREX. The internal drive coils (purple)
drive a magnetic field that opposes the external Helmholtz coil’s field. The
plasma source is a polar array of plasma guns (yellow). (b) A cross-section of
the top half of the TREX vessel showing a theoretical example of the typical
experimental geometry. The magnetic field lines are shown in cyan. The
reconnection region (light orange) is driven down from the drive coils to the
central axis, as indicated by the arrows. The layer is measured during this
transit by the three probes shown: the 3-axis linear $\mathbf{\dot{B}}$ probe
array (blue), the speed probe (long red), and the multi-tip Langmuir
temperature/density probe, known as the $T_{e}$ probe (short red). The hook
probe, another array of 3-axis $\mathbf{\dot{B}}$ probes can be scanned
through the shaded green area, allowing for the compilation of data from
multiple experimental shots. These probes operate on sampling frequencies on
the order of $10$\mathrm{MHz}$$, while the elapsed time between the layer’s
generation and its arrival at the central axis is on the order of
$20$\mathrm{\SIUnitSymbolMicro s}$$

.

In the planar cut of TREX shown in Fig. 1(b), the cyan lines are
theoretically-derived magnetic flux contours meant to illustrate the typical
magnetic geometry of an experimental run. As the current through the drive
coils ramps up, the reconnection region is pushed from underneath the drive
coils radially inward (orange arrows in Fig. 1(b)). During this transit, the
reconnection layer “jogs” past the electrostatic and magnetic probes. Given
the near constant speed of the reconnection layer, this facilitates high
spatial resolution measurements of the entire layer geometry over the course
of a single experimental shot; this type of measurement is referred to as the
jogging method. These probes and their locations are represented by the blue
and red rectangles in Fig. 1(b). In addition to these jogging method probes, a
different array of 3-axis $\mathbf{\dot{B}}$ probes can be moved between
shots, allowing for the creation of multi-shot datasets. The coverage area of
this probe is given by the light green rectangle in Fig. 1(b). By compiling
data from multiple shots taken at different location, this probe provides
information about the reconnection geometry without relying on the jogging
method.

Figure 2: Example of experimental data. Plots (a) and (b) show the data from
the hook probe recorded in a scan including $34$ different probe positions
covering the green region in Fig. 1(b). The black lines are contours of the
flux function, $\Psi$, which map to the magnetic field lines. (a) shows the
reconnecting magnetic fields, and (b) shows the out-of-plane current layer.
(c) shows data from the $T_{e}$ probe (short red in Fig. 1(b)); the shaded
regions represent the $95\%$ confidence interval for the values of density and
temperature based on the fit of the probe’s IV curve. The lower $R$ side of
the layer is closer to the plasma sources and thus has a higher density than
the other side of the layer. There is a jump in the plasma temperature when
the layer passes the probe. The data in (c) is compiled using the jogging
method to convert the time signal into a measurement of the $R$-coordinate.

An example of data collected from a typical set of experimental shots is
provided in Fig. 2, where Fig. 2(a-b) shows data from $34$ shots combined into
one picture; for each shot, the hook probe is at a different position within
the green region in Fig. 1(b). The black lines are contours of the flux
function $\Psi$ to illustrate the in-plane magnetic field lines. Fig. 2(c)
shows the temperature and density data measured by the $T_{e}$ probe. Note
that in Fig. 2(c), the time signals from the $T_{e}$ probe are converted into
position data using the jogging method described above. Typical plasma
parameters include $T_{i}\ll T_{e}\simeq 5-20$\mathrm{eV}$$, $n_{e}\simeq
2\cdot 10^{18}$\mathrm{m}$^{-3}$, $B_{rec}\simeq 4$\mathrm{mT}$$, yielding
$\beta_{e}\simeq 0.4$ and $S\simeq 10^{4}$.

##  3 2D and 3D Kinetic Simulations of TREX

Figure 3: Comparison of TREX experimental data from a scan of the hook probe
(green in Fig. 1(b)) with cylindrical VPIC runs in both 2 and 3 dimensions.
The 2D run is at full mass ratio, and the 3D run is at a mass ratio of 400.
The 3D run plots show data taken from a single value of $\phi$. Each row of
plots for a different magnetic feature shows contours of the magnetic flux
function $\psi$ in black. Each row has been scaled relative to the same
magnitude, shown in the leftmost plot of each row. (a-c) compare the in-plane
($B_{z}$); (d-f) show out-of-plane ($B_{\phi}$) magnetic fields. (g-i) show
the out-of-plane current density ($J_{\phi}$); (j-l) show the in-plane
($J_{z}$) current structures. Plots (m) and (n) show experimental data from
the linear probe (blue in Fig. 1(b)) using the jogging method. The jogging
method provides a high spatial resolution of $\sim$ $0.4$ $\mathrm{cm}$ in the
$R$ direction. Plot (m) shows a mostly laminar layer with some bifurcation,
whereas plot (n) shows a plasmoid being ejected from the reconnection region.
Plot (o) is a standard Ohm’s Law analysis of the 2D VPIC simulation, along the
path indicated by the magenta line in plot (k). The dashed cyan line is the
location of the $B_{z}$ $=$ $0$ point along the line; the dashed black line is
the location of the maximum out-of-plane current density.

TREX was simulated using VPIC, a kinetic particle-in-cell code [K. Bowers .
(2009), Daughton . (2018), Bowers (2020)]. The TREX boundary conditions were
implemented in the new Cylindrical VPIC code with conducting walls at
$R=1.5$\mathrm{m}$$ and $Z=\pm 1.5$\mathrm{m}$$, as well as an additional
conducting wall at an adjustable minimum (nonzero) $R$ near the central axis.
Within the simulation domain, current sources with the same dimensions as the
TREX drive coils were added at the drive coil locations. The current density
at these locations is increased as a function of time to mimic the ramping
current injection utilized in the experiment. Using density data from TREX,
initial density and magnetic field profiles were set at the simulation start
time to balance the magnetic and kinetic pressures for a given applied
Helmholtz field. Electron-electron, electron-ion, and ion-ion collisions were
implemented in some 2D runs with a Monte-Carlo collision operator for binary
Coulomb collisions [Takizuka Abe (1977)]. The collision frequencies were
calculated from TREX data. Not all simulations implemented collisions; results
from testing a range of 2D simulation collision parameters at relevant
experimental levels showed very little difference between runs with and
without collisions. The 3D run discussed in this paper did not include
collisions.

The number of grid-points in the 2D simulation described here was $1512$ by
$3600$ in the $R$ and $Z$ directions, respectively; in 3D, these increased to
$1024$ and $2048$ respectively with another $256$ grid divisions in the $\phi$
direction. The system size in 2D was about $193$ by $399$ electron skin depths
in the $R$ and $Z$ directions, respectively. In 3D, these values were $87$ and
$186$ electron skin depths, respectively. The average number of super-
particles per cell was $500$ in 2D and $100$ in 3D. In both simulations, the
ratio of the electron cyclotron frequency to the electron plasma frequency was
$1$.

Both 2D $(RZ)$ and 3D $(R\phi Z)$ simulations of TREX can be compared to
experimental results; one such comparison is shown in Fig. 3. Experimental
data in subplots (a,d,g,j) come from combining multiple shots worth of
experimental data from the hook probe (green in Fig. 1(b)).

The 2D simulation (b,e,h,k) was obtained at full mass ratio while the 3D
simulation (c,f,i,l) applies $m_{i}/m_{e}=400$. For numerical tractability,
the 3D simulation is limited to a $60^{\circ}$ wedge with periodic boundaries
in $\phi$. Both the experimental and simulation profiles in Fig. 3(a-l) are
displayed with the domains normalized by the local ion skin depths. Here the
local ion skin depth is obtained from the value of $n_{e}$ in the high-density
inflow (e.g., the density value shown in Fig. 2(c) at
$R^{\prime}\sim-0.1$\mathrm{m}$$). The scaling of simulation variables
relative to experimental ones was implemented using the technique described in
egedal:2019, where temperature and magnetic field profile matching occurred
near the X-line during the reconnection process. Further similarities between
TREX and 3D Cylindrical VPIC will be discussed later in this paper.

##  4 Inferring the Reconnection rate in the TREX geometry

In 3D geometries the rate of reconnection is not always trivial to define
[Hesse Birn (1993)]. However, given the nominal 2D experimental setup we can
define the reconnection rate as the rate at which flux upstream of the
reconnection region reconnects and moves downstream. Fig. 4(a) shows the
different “categories” of magnetic flux in the TREX cross-section. The red
region contains field lines from the Helmholtz coil that go through the drive
loop area and are upstream of the reconnection region. The blue lines are also
upstream of the reconnection, but these represent the new magnetic flux
injected into the system by the drive coils. Reconnection results in the
downstream field lines, shown in green. The magenta field lines are those from
the external Helmholtz coil that are initially above the internal drive coils
and thus do not take part in the reconnection process. We can then define the
remaining unreconnected flux, $\Psi_{B}$, as the (red) magnetic flux between
$R=0$ and the reconnection layer
$\Psi_{B}=\int_{0}^{2\pi}\mathrm{d}\phi\int_{0}^{R_{x}(\phi)}RB_{z}dR$, where
the integral is taken at a constant $Z$ that matches the location of the
$X$-line and from $R=0$ to $R=R_{x}$. Here $R_{x}(\phi)$ is the radius of the
center of the current layer which is moving radially inwards and which in the
simulation is observed to be a function of $\phi$. This path of this integral
is represented by the cyan line in Fig. 4(a).

Again, because $\Psi_{B}$ is the remaining un-reconnected magnetic flux, it is
clear that $-d\Psi_{B}/dt$ is the rate at which magnetic flux is being
reconnected, i.e. the reconnection rate. There exists of course ambiguity in
how to define the center of the reconnection layer $R_{x}(\phi)$, but as long
as $R_{x}(\phi)$ correctly characterizes the inward motion of the reconnection
layer it turns out that $-d\Psi_{B}/dt$ is largely unaffected by the
differences between any reasonable choice of $R_{x}(\phi)$. By Faraday’s law
it is also clear that

$-\frac{d\Psi_{B}}{dt}=\oint_{R_{x}}(\mathbf{E}+\mathbf{v}_{R_{x}}\times\mathbf{B})\cdot
d\mathbf{l}\simeq\oint_{R_{x}}\mathbf{E}\cdot d\mathbf{l}\quad.$ (1)

Note that in 3D for a particular choice of $R_{x}(\phi)$ the value of
$B_{z}(R_{x}(\phi))$ could be finite and oscillate along $R_{x}(\phi)$, but it
is reasonable to impose that for a valid choice of $R_{x}(\phi)$, the average
value of $B_{z}(R_{x}(\phi))$ must be small. Meanwhile $\mathbf{v}_{R_{x}}$
will be near constant and directed radially inward such that the average value
of $(\mathbf{v}_{R_{x}}\times\mathbf{B})\cdot d\mathbf{l}$ also becomes small
and can be neglected, as expressed in Eq. 1.

In 3D configurations variations are permitted and likely present in
$\mathbf{E}$ along $R_{x}(\phi)$. However, the local electric field may always
be expressed on the form $\mathbf{E}=-\nabla\Phi-\partial\mathbf{A}/\partial
t$, and because for any $\Phi$ we have $\oint_{R_{x}}\nabla\Phi\cdot
d\mathbf{l}=0$, it becomes clear that the reconnection electric field defined
as

$E_{\rm
rec}\equiv-\frac{1}{2\pi\left<R_{x}\right>}\frac{d\Psi_{B}}{dt}=\frac{1}{2\pi\left<R_{x}\right>}\oint_{R_{x}}-\frac{\partial\mathbf{A}}{\partial
t}\cdot d\mathbf{l}\quad,$ (2)

is a measure of the average toroidal inductive electric field, not directly
dependent on any electrostatic electric fields $-\nabla\Phi$ which may be
present in the reconnection region.

The physics that allows the electron fluid, with bulk velocity
$\mathbf{v_{e}}$, to decouple from the motion of the magnetic field can be
analyzed using the momentum equation of the electron fluid (the generalized
Ohm’s law), which takes the form:

$\displaystyle\mathbf{E}=-\mathbf{v_{e}}\times\mathbf{B}+\eta\mathbf{J_{e}}-\frac{1}{ne}\bm{\nabla}\cdot\mathbf{P_{e}}-\frac{m_{e}}{e}\frac{\mathrm{d}\mathbf{v_{e}}}{\mathrm{d}t}$
(3)

Here $\mathbf{P_{e}}$ is the electron pressure tensor, with elements
$p_{ij}=m\int\left(u_{i}-v_{e,i}\right)\left(u_{j}-v_{e,j}\right)f\mathrm{d}^{3}u$,
$\mathbf{v_{e}}$ is the bulk electron fluid velocity, and
$\mathrm{d}/\mathrm{d}t$ is the total convective derivative,
$\mathrm{d}/\mathrm{d}t=\partial/\partial t+\mathbf{v_{e}}\cdot\bm{\nabla}$.

In particular for the pressure tensor, we may split its contributions into its
scalar and off-diagonal parts $\mathbf{P_{e}}=p_{e}\mathbf{I}+\bm{\pi}$, where
trace$(\bm{\pi})=0$. As discussed above, the reconnection rate $E_{\rm rec}$
is proportional to the $R_{x}(\phi)$-average of $\mathbf{E}$, and it becomes
clear that while $-\nabla p_{e}$ contributions can be important to balance
local electrostatic components of $\mathbf{E}$, the total contribution is $0$
because $\oint\nabla p_{e}\cdot d\mathbf{l}=0$. Thus, the
$\bm{\nabla}\cdot\mathbf{P_{e}}$ term only contributes to reconnection through
the off-diagonal stress in $\bm{\pi}$.

An analysis of the generalized Ohm’s Law for a 2D VPIC simulation of TREX is
shown in Fig. 3(o), where the terms of Ohm’s Law are evaluated along the path
defined by the magenta line in Fig. 3(k). Given the boundary conditions of
this simulation, this analysis does not include any form of spatial averaging,
resulting in some fluctuation in the net Ohm’s law term (the black line in
Fig. 3(o)); nonetheless, the pressure tensor divergence term (green) is
clearly the dominant contributor to the reconnection electric field (red),
consistent with prior 2D simulations of low-collisionality asymmetric
reconnection, including egedal:2018.

##  5 Ohm’s Law Results from 3D Simulations of TREX

Figure 4:  a) Block diagram of TREX’s theoretical magnetic geometry,
demonstrating the different regions of flux relative to the location of the
reconnection layer. Blue and red are upstream of the reconnection region,
starting from the drive coils and the Helmholtz coil, respectively. The green
lines are downstream of the reconnection region, and the magenta lines are
those that originate from the Helmholtz coil but are above the drive coils and
thus do not impact the reconnection process. The cyan line represents the path
of the integral used to define the flux function $\Psi$. b) Cuts of the
current density, $|{\bf J}_{e}|$, in a 3D kinetic simulation of TREX
implemented as a $60^{\circ}$ degree wedge with periodic boundaries in $\phi.$
The drive coils are shown as gold surfaces. (c) Simulation profiles of of
${\bf E}_{\phi}$ mapped onto the $R\phi$-plane at a single value of $Z$. The
black line represents the $B_{z}$ $=$ $0$ line. Fluctuations in ${\bf
E}_{\phi}$ have the characteristics of the electrostatic LHDI; notably, they
are stronger above the layer where the particle density is lower. The average
electron cyclotron radius, $\overline{\rho}_{e}$, is calculated along the path
marked by the small white Xs; this is one of the primary scale lengths use to
describe the two modes of the LHDI. (d) Profile of $|{\bf J}_{e}|$ mapped onto
the $R\phi$-plane at a single value of $Z$; the black line is still the
$B_{z}$ $=$ $0$ line. Also shown in white are two different paths of
integration, a straight line (labeled SL) that simply cuts through the entire
$\phi$ domain and the optimized integration curve $C$. The results of
integrating the different terms of Ohm’s Law along path $C$ are shown in (e);
the Lorentz term (red) is almost completely matched by the pressure divergence
term (green), and the net Ohm’s Law term (black) is consistently negligible.
Also displayed are the results of integrating solely the $\mathbf{v_{e}}$
$\times$ $\mathbf{B}$ term over the straight path $SL$ (shown in magenta).

The 3D simulation domain can also be seen in Fig. 4(b). Taking a cut through
the layer reveals the development of a toroidal instability which is inferred
to be the lower hybrid drift instability (LHDI). Because the LHDI is driven by
diamagnetic currents, it can be more vigorous within asymmetric reconnection
layers which feature strong pressure gradients in the central portion
[Roytershteyn . (2012), Le . (2017), Le . (2018)], and several of the
characteristics of the LHDI [Daughton (2003), Davidson . (1977)] matched the
numerical layer fluctuations. The fastest growing LHDI modes are short
wavelength ($\rho_{e}k_{\perp}\sim 2.9$), primarily electrostatic, and are
localized on the edge of the layer; these can be seen in Fig. 4(c). However,
the LHDI features a rich spectrum [Daughton (2003)], with longer wavelength
$\sqrt{\rho_{e}\rho_{i}}k_{\perp}\sim 1.4$ electromagnetic modes that
penetrate into the center, giving rise to a global rippling of the layer in
the toroidal direction; this matches the characteristics of the kinking in the
$B_{z}=0$ and the current layer, as shown in Fig. 4(b)-(d).

Given the definition of the reconnection rate in Eq. 1, we may now use the
generalized Ohm’s Law in Eq. 3 to examine which term is responsible for making
$\oint_{C}{\bf E}\cdot d{\bf l}$ finite and thereby breaking the frozen-in
condition for the electrons. While the value of $\oint_{C}{\bf E}\cdot d{\bf
l}$ is largely insensitive to the choice for $R_{X}(\phi)$, we will show that
the magnitude of the terms on the RHS of Eq. 3 greatly depend on this contour
of integration; this has important implications for the amplitude of the
aforementioned anomalous terms due to correlated spatial fluctuations.

As mentioned, choosing a path over which to integrate is nontrivial; prior
publications analysing Ohm’s Law in similar parameter regimes have returned
different results based on different choices of how to spatially average the
relevant variables [Price . (2017), Le . (2018)]. A simple average over a
single dimension (usually one analogous to what is here defined as our $\phi$
dimension) may pick up data from outside the diffusion region, leading to
dominant terms that do not appear when an average is made over a path that has
been adapted to fit the shape of the layer and any constituent instabilities
[Che . (2011), Price . (2017), Le . (2018)].

Keeping this in mind, we chose to integrate over a layer-specific path while
showing the potential consequence of selecting a simple single-variable path.
The simple path is shown in Fig. 4(d) as a white dashed straight line labeled
$SL$. The black path represents the contour along which $B_{z}=0$; this path
was used as a starting point for an iterative process that determined the
layer-specific contour $C$ (solid white curve) by minimizing the contributions
of the $-en\mathbf{v_{e}}\times\mathbf{B}$ term to the path integral.

Integrating Eq. 3 along the path C produces the terms in Fig. 4(e). Note that
collisionality was set to $0$ in this simulation, so the resistivity term of
Eq. 3 is also $0$. The left side of Eq. 3 (the red line) is almost completely
matched at every location by the pressure-tensor-divergence term (green line).
The net Ohm’s Law term (black) line remains near zero at every location.

Also shown Fig. 4(e) is the $-en\mathbf{v_{e}}\times\mathbf{B}$ term for the
$SL$ result; this is given by the magenta line. The kinking in the layer
results in $SL$ including locations that are outside the dissipation region,
resulting in large contributions of $-en\mathbf{v_{e}}\times\mathbf{B}$ which
is the primary term for balancing $E_{\phi}$ outside the layer. By focusing on
an average defined by $C$, we can minimize the
$-en\mathbf{v_{e}}\times\mathbf{B}$ contributions and thus avoid drawing
conclusions about the relevant terms of Ohm’s Law in a manner that includes
contributions from the non-reconnecting plasma regions. In previous analyses,
contributions from the correlated fluctuations of the
$-en\mathbf{v_{e}}\times\mathbf{B}$ term are combined with those from the
pressure divergence to form the anomalous viscosity [Le . (2018), Che .
(2011)]. These anomalous terms, when present, have been proposed as a
mechanism that broaden the dissipation region and thus increase the
reconnection rate [Parker (1957)]; however, more recent results from
reconnection dominated by kinetic effects show that this can be achieved
without these anomalous terms becoming significant [Le . (2018)]. While there
has been some disagreement about whether or not these anomalous terms are
dominant in 3D kinetic reconnection [Che . (2011), Price . (2017), Le .
(2018)], our conclusion is that the answer will depend on the path used to
define the spatial average. We find that the path that stays inside the
diffusion region where $-en\mathbf{v_{e}}\times\mathbf{B}$ is small is the
more physical choice. Furthermore, the dominant term breaking the electron
frozen-in condition in the diffusion region is observed to be
$\bm{\nabla}\cdot\mathbf{P_{e}}$ and is in agreement with recent spacecraft
observations [Egedal . (2019), R. Nakamura . (2019)].

##  6 Layer Width Results from Experiment and Simulation

Figure 5: Measuring layer width. (a) method for inferring the current layer
half-width, $\delta_{J}$, defined as the half-width of the layer at $40\%$ of
the maximum. (b) shows an $R\phi$ contour plot of the current layer in TREX
recorded by a curved $\mathbf{\dot{B}}$ array sampling multiple toroidal
angles. Three example slices at fixed $\phi$ are represented by the white
dashed lines documenting variations in the layer widths. Figure 6: Results of
measuring experimental layer width over several different parameter sets. The
colors highlight the difference between “2 gun” (lower density) and a “6 gun”
(higher density) plasmas. The orange region shows the range of current layer
widths measured in the 3D simulation, and the solid black line shows the
result from a 2D axisymmetric simulation (without the instability). The spread
in the experimental and 3D Cylindrical VPIC layer widths is attributed to the
toroidal instability described in Fig. 5.

In the above analysis of the kinetic simulation it was shown that the off-
diagonal stress of $\bm{\nabla}\cdot\mathbf{P_{e}}$ is mainly responsible for
breaking the electron frozen-in condition. From theory, this off-diagonal
stress scales roughly as $1/\delta_{J}^{2}$, where $\delta_{J}$ is the half-
width of the reconnection current layer [Hesse . (1999)]. Prior simulation
results have established that when the current layer half-width $\delta_{J}$
is on the order of a few electron scale lengths (here taken to be the electron
skin depth, $d_{e}=c/\omega_{pe}$, which at electron $\beta$ near unity is
nearly equivalent to both the electron meandering width and the electron
gyroradius), the pressure tensor divergence term will be sufficiently large to
become the dominant term in Ohm’s Law [Roytershteyn . (2013)]. This
characteristic defines the regime of collisionless reconnection; as such, the
measurement of layer widths on the order of the several $d_{e}$ can be used to
establish the regime in which the reconnection is operating [Vasyliunas
(1975), Pritchett (2001)].

To address whether $\bm{\nabla}\cdot\mathbf{P_{e}}$ breaks the frozen-in
condition in the experiment as it does in the simulation, we compare the
widths of the current layers observed in TREX and the simulation. TREX is not
yet able to measure the value of $\bm{\nabla}\cdot\mathbf{P_{e}}$ directly,
but as described above, the existence of thin current layers implies that the
$\bm{\nabla}\cdot\mathbf{P_{e}}$ contribution to Ohm’s Law dominates the
breaking of the frozen-in condition. The method of characterizing the layer
half-width, $\delta_{J}$, is illustrated in Fig. 5(a), where the current
density is recorded by the linear $\mathbf{\dot{B}}$ probe array (blue in Fig.
1(b)) at a set $\phi$ location for a range of $Z$ values. In addition, a
toroidally curved $\mathbf{\dot{B}}$ probe array is used to characterize
toroidal variations in the TREX current layers. An example dataset is shown in
the top half of Fig. 5(b), documenting the intensity of the toroidal current
density, with a range of different radial layer widths indicated below. This
toroidal variation is similar to instability in the simulation, though the
experimental observation is limited by the spatial resolution of the probe.

Note that there is also some minor variation in the layer structure in the
plane perpendicular to the toroidal direction; this can be seen in the
existence of several smaller peaks in the current densities plotted in the
bottom of Fig. 5(b). In this plane, the layer occasionally experiences some
small amount of bifurcation or other minor irregularities. Examples of such
behaviour are shown in Fig. 3(m) and (n). In this analysis, only the primary
peaks in the electron current density cuts are taken into consideration.

Repeating the width measurement process for a range of experimental settings
we obtain mean values for $\delta_{J}$ and $d_{e}$. The results are plotted in
Fig. 6, where each data point represents the average result for $\delta_{J}$
and $d_{e}$ for a given set of drive potential, Helmholtz field, gun number,
and ion species. There are approximately ten different experimental shots
averaged for each data point, which include estimates for the experimental
uncertainties. Also plotted is the line corresponding to the previously
reported experimental results in MRX [Ji . (2008)]. The orange region
represents the mean of the numerical layer widths from the 3D simulation in
Fig. 4(b), $\pm$ a standard deviation. Similar to the experimental measurement
process shown in Fig. 5(b), the numerical width results were obtained from
radial cuts at different $\phi$ values relative to the phase of the
instability. Finally, the width recorded in a corresponding 2D simulation in
the $RZ$-plane is given by the solid black line.

The width of the current layer is physically limited by the electron
meandering scale, which is slightly larger than the electron inertial scale
and has only a weak dependence on the precise $\beta_{e}$ [Dorfman . (2008)];
in these experiments this parameter is nearly constant ($\beta_{e}\sim
10^{-1}$). There is a clear divide between the blue (2 guns, lower density)
and red (6 guns, higher density) datapoints, as expected from $d_{e}\propto
n_{e}^{-1/2}$. Additionally, even though the lower density points have larger
skin depths, they also have larger layer widths, keeping them on the same
scaling as the higher density datapoints. There is not a particular
relationship between ion species and the experimental width scaling. Most
notably, there is a general spread in measured layer widths, both relative to
different parameter sets and within a given parameter set itself (demonstrated
by the vertical uncertainties). This is consistent with the presence of the
toroidal instability measured in TREX and demonstrated in the 3D simulations.
Crucially, both the absolute values and spread of the measured current layer
widths is in good agreement between the simulation and the experiment.

##  7 Conclusions

To summarize, reconnection in TREX is characterized by thin electron current
layers, consistent with kinetic simulation results. The widths include a
notable spread, $\delta_{J}\sim(1.5-3)d_{e}$, which can be attributed to the
development of a toroidal instability in the current layer. Compared to
previous experiments in MRX, the TREX temperature ratio $T_{i}/T_{e}\ll 1$ may
be more favorable to this instability [Roytershteyn . (2012)]. Nevertheless,
the TREX current layers are in good agreement with those observed in a 3D
kinetic simulation, and are much thinner than those observed previously in the
MRX experiment.

The new Cylindrical VPIC code has allowed the TREX reconnection experiments to
be modeled in a way that preserves its nominal cylindrical symmetry. Both 2D
and 3D simulations reproduce the magnetic geometry measured in TREX, and 3D
Cylindrical VPIC also shows the development of a toroidal instability that
produces the same spread in the layer width scaling. The narrow current layers
observed in TREX and their match to 3D kinetic simulation results validates
the numerical result that off-diagonal stress in the electron pressure tensor
is responsible for breaking the frozen-in condition for low collisionality
configurations relevant to reconnection in the Earth’s magnetosphere.

###### Acknowledgements.

We gratefully acknowledge DOE funds DE-SC0019153, DE-SC0013032, and DE-
SC0010463 and NASA fund 80NSSC18K1231 for support of the TREX experiment. In
addition, the experimental work is supported through the WiPPL User Facility
under DOE fund DE-SC0018266. Simulation work was supported by the DOE Basic
Plasma Science program and by a fellowship from the Center for Space and Earth
Science (CSES) at LANL. CSES is funded by LANL’s Laboratory Directed Research
and Development (LDRD) program under project number 20180475DR. This work used
resources provided by the Los Alamos National Laboratory Institutional
Computing Program, which is supported by the U.S. Department of Energy
National Nuclear Security Administration under Contract No. 89233218CNA000001.
The 3D kinetic simulation was performed at the National Energy Research
Scientific Computing Center (NERSC), a U.S. Department of Energy Office of
Science User Facility operated under Contract No. DE-AC02-05CH11231. Data
availability statement: Data from the 2D simulation is available at
greess_samuel_2021_4554697. Data from the 3D VPIC simulation is available at
greess_samuel_2021_4556518. Full figure data (both simulation and
experimental) is available at greess_samuel_2021_4837721.

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