# First-Principles Theory of the Rate of Magnetic Reconnection in
Magnetospheric and Solar Plasmas

Yi-Hsin Liu  Department of Physics and Astronomy, Dartmouth College, Hanover,
NH 03750, USA  Paul Cassak  Department of Physics and Astronomy, West Virginia
University, Morgantown, WV 26506, USA  Xiaocan Li  Department of Physics and
Astronomy, Dartmouth College, Hanover, NH 03750, USA  Michael Hesse  NASA’s
Ames Research Center, Moffett Field, CA 94035, USA  Shan-Chang Lin  Department
of Physics and Astronomy, Dartmouth College, Hanover, NH 03750, USA  Kevin
Genestreti  Southwest Research Institute, Durham, NH 03824, USA

###### Abstract

Abstract: The rate of magnetic reconnection is of the utmost importance in a
variety of processes because it controls, for example, the rate energy is
released in solar flares, the speed of the Dungey convection cycle in Earth’s
magnetosphere, and the energy release rate in harmful geomagnetic substorms.
It is known from numerical simulations and satellite observations that the
rate is approximately 0.1 in normalized units, but despite years of effort, a
full theoretical prediction has not been obtained. Here, we present a first-
principles theory for the reconnection rate in non-relativistic electron-ion
collisionless plasmas, and show that the same prediction explains why Sweet-
Parker reconnection is considerably slower. The key consideration of this
analysis is the pressure at the reconnection site (i.e., the x-line). We show
that the Hall electromagnetic fields in antiparallel reconnection cause an
energy void, equivalently a pressure depletion, at the x-line, so the
reconnection exhaust opens out, enabling the fast rate of 0.1. If the energy
can reach the x-line to replenish the pressure, the exhaust does not open out.
In addition to heliospheric applications, these results are expected to impact
reconnection studies in planetary magnetospheres, magnetically confined fusion
devices, and astrophysical plasmas.

###### pacs:

52.27.Ny, 52.35.Vd, 98.54.Cm, 98.70.Rz

##  I Introduction

Magnetic reconnection converts magnetic energy into plasma thermal and kinetic
energy in laboratory, space and astrophysical plasmas. Two major and largely
separate endeavors have been pursued to quantitatively predict how the plasma
is energized by reconnection (Wygant  _et al._ , 2005; Drake  _et al._ , 2006;
Birn and Hesse, 2010; Aunai  _et al._ , 2011; Egedal  _et al._ , 2012;
Eastwood  _et al._ , 2013, 2020; Shay  _et al._ , 2014; Bessho  _et al._ ,
2014; Dahlin  _et al._ , 2014; Guo  _et al._ , 2014; Shuster  _et al._ , 2015;
Yamada  _et al._ , 2015; Wang  _et al._ , 2018; Guo  _et al._ , 2020; Li  _et
al._ , 2021) and the rate at which reconnection proceeds (Sweet, 1958; Parker,
1957; Petschek, 1964; Shay and Drake, 1998; Birn  _et al._ , 2001; Rogers  _et
al._ , 2001; Bessho and Bhattacharjee, 2005; Cassak  _et al._ , 2007; Hesse
_et al._ , 2011; Liu  _et al._ , 2014; TenBarge  _et al._ , 2014; Stanier  _et
al._ , 2015a, b; Comisso and Bhattacharjee, 2016; Liu  _et al._ , 2017; Cassak
_et al._ , 2017; Liu  _et al._ , 2018; Nakamura  _et al._ , 2018a; Genestreti
_et al._ , 2018; Nakamura  _et al._ , 2018b; Huang  _et al._ , 2020; Li and
Liu, 2021). Nevertheless, the linkage between these two fundamental aspects of
reconnection is missing. The most critical question in understanding fast
reconnection is what localizes the diffusion region (DR) (Biskamp and Schwarz,
2001) (i.e., what makes it far shorter than the system size), giving rise to
an open geometry of reconnection outflow. Petschek’s model (Petschek, 1964)
provides a valid steady-state solution for such an open outflow geometry, but
it fails to provide a valid localization mechanism in the uniform resistivity
magnetohydrodynamics (MHD) model (Biskamp, 1986; Uzdensky and Kulsrud, 2000);
reconnection in such system always results in a system-size long diffusion
region, known as the Sweet-Parker solution (Sweet, 1958; Parker, 1957). The
idea of a spatially localized anomalous resistivity was later invoked to
explain the localization needed (Ugai and Tsuda, 1977; Sato and Hayashi, 1979;
Kulsrud, 2001), but no clear evidence of such anomalous resistivity during
collisionless reconnection has yet been identified.

Kinetic simulations beyond the MHD model suggest that antiparallel
reconnection with an open outflow geometry occurs when the current sheet thins
down to the ion inertial scale (Jara-Almonte and Ji, 2021; Daughton  _et al._
, 2009; Cassak  _et al._ , 2005; Bhattacharjee, 2004). When this occurs, the
Hall term in the generalized Ohm’s law (Vasyliunas, 1975; Swisdak  _et al._ ,
2008) dominates the electric field in the ion diffusion region (IDR), where
the ions become demagnetized. The correlation between the Hall effect and fast
reconnection was clearly demonstrated in the GEM reconnection challenge study
(Birn  _et al._ , 2001); this study showed that simulation models with the
Hall term in the generalized Ohm’s law (particle-in-cell (PIC), hybrid and
Hall-MHD) realize fast reconnection, while only the uniform resistive-MHD
model, which lacks the Hall term, exhibits a slow rate (Sweet, 1958; Parker,
1957). However, it remains unclear why and how the Hall term localizes the
diffusion region, producing an open geometry. The dispersive property of waves
arising from the Hall term was proposed as an explanation (Mandt  _et al._ ,
1994; Shay  _et al._ , 1999; Rogers  _et al._ , 2001; Drake  _et al._ , 2008),
but the role of dispersive waves derived from linear analysis was called into
question because reconnection can be fast even in systems that lack dispersive
waves (Liu  _et al._ , 2014; TenBarge  _et al._ , 2014; Stanier  _et al._ ,
2015a; Bessho and Bhattacharjee, 2005).

In this work, we illustrate the role of Hall physics in plasma energization
and why this causes the open geometry necessary to achieve fast reconnection
in electron-ion plasmas. The two key points are: (1) the Hall term ${\bf
E}_{{\rm Hall}}={\bf J}\times{\bf B}/n{\rm e}c$, while it dominates the
electric field within the IDR, does not convert energy into plasmas because
${\bf J}\cdot{\bf E}_{{\rm Hall}}={\bf J}\cdot({\bf J}\times{\bf B})/n{\rm
e}c=0$. Thus, the inflowing plasma only gains a small amount of thermal energy
within the IDR. (2) Insufficient pressure buildup at the x-line, where the
magnetic field lines change their connectivity, causes the upstream magnetic
pressure to locally pinch the diffusion region, opening out the exhaust (Liu
_et al._ , 2020). For reconnection of antiparallel magnetic fields, an open
geometry occurs if $P|_{\rm xline}<B_{x0}^{2}/8{\rm\pi}+P_{0}$, where $P|_{\rm
xline}$ is the thermal pressure at the x-line, $P_{0}$ is the asymptotic
thermal pressure and $B_{x0}^{2}/8{\rm\pi}$ is the magnetic pressure based on
the asymptotic magnetic field $B_{x0}$ far upstream from the IDR. These two
results are used to derive a first-principles theory of the reconnection rate
(the phrase “first-principles” refers to a theory that does not rely on
measured empirical inputs from the simulations or observations). In order to
show this pressure depletion during magnetic reconnection in electron-ion
plasmas, we use PIC simulations to investigate the role of Hall
electromagnetic fields in energy conversion and kinetic heating near the
x-line. The cross-scale coupling from the mesoscale upstream MHD region, the
IDR, and down to the electron diffusion region (EDR) is treated to obtain a
prediction of the reconnection rate. Finally, we extend the discussion to
systems without the Hall term, including electron-positron (pair) plasmas and
resistive-MHD reconnection, explaining why the former is fast while the latter
does not have an open outflow and is slow. We show that the same theoretical
approach leads to the Sweet-Parker scaling, and provides the reason of why
Sweet-Parker reconnection has a system-size long diffusion region.

##  II Results

We use 2-D PIC simulations to illustrate the key features of energy conversion
in the diffusion region. Details of the simulation setup are in the “Methods”
section. The units used in the presentation include the ion cyclotron time
$\Omega_{\rm ci}^{-1}$ $\equiv$ $({\rm e}B_{x0}/m_{\rm i}c)^{-1}$, the Alfvén
speed $V_{\rm A0}$ $\equiv$ $B_{x0}/(4{\rm\pi}n_{0}m_{\rm i})^{1/2}$ based on
$B_{x0}$ and the background density $n_{0}$, and the ion and electron inertial
length $d_{s}$ $\equiv$ $c/(4{\rm\pi}n_{0}{\rm e}^{2}/m_{s})^{1/2}$ for
species $s$ $=$ $\rm i$ and $\rm e$, respectively. The ion to electron mass
ratio is $m_{\rm i}/m_{\rm e}=400$ and the background plasma beta is
$\beta=0.01$.

Figure 1: Hall electromagnetic fields and the generalized Ohm’s law. a) The
Hall magnetic field $B_{y}$ and b) the Hall electric field $E_{z}$ (normalized
by $B_{x0}$) overlaid with Poynting vector ${\bf S}$ streamlines (yellow) at
time $48/\Omega_{\rm ci}$. c) The out-of-plane component of terms in the
generalized Ohm’s law (normalized by $B_{x0}V_{\rm A0}/c$) across the x-line
in the inflow direction. Variables ${\bf E}$, ${\bf B}$, ${\bf V}_{\rm i}$,
${\bf V}_{\rm e}$, ${\bf J}$, $\mathbb{P}_{\rm e}$, $n$, ${\rm e}$, $m_{\rm
e}$ and $c$ are electric field, magnetic field, ion velocity, electron
velocity, current density, electron pressure tensor, density, proton charge,
electron mass and the speed of light, respectively. The vertical red
transparent band marks the electron diffusion region (EDR).

###  II.1 The role of Hall electromagnetic fields

Figure 1a shows the out-of-plane magnetic field $B_{y}$ at time
$48/\Omega_{\rm ci}$, which is the Hall quadrupole field within the IDR of
magnetic reconnection in collisionless electron-ion plasmas Sonnerup (1979).
Importantly, this Hall quadrupole magnetic field $B_{y}$ along with the
inward-pointing Hall electric field $E_{z}$, shown in Fig. 1b, constitute a
Poynting vector $S_{x}=-cE_{z}B_{y}/4{\rm\pi}$ in the $x$-direction. This
component diverts the inflowing electromagnetic energy toward the outflow.
This is shown by the streamlines of ${\bf S}=c{\bf E}\times{\bf B}/4{\rm\pi}$
in yellow, which bend in the $x$ direction significantly before reaching
$z=0$. These Hall electromagnetic fields arise from the Hall term in the
generalized Ohm’s law Vasyliunas (1975); Swisdak  _et al._ (2008); Hesse  _et
al._ (2011), ${\bf E}+{\bf V}_{\rm i}\times{\bf B}/c={\bf J}\times{\bf
B}/n{\rm e}c-\nabla\cdot\mathbb{P}_{\rm e}/n{\rm e}+(m_{\rm e}/{\rm
e}^{2})d\left({\bf J}/n\right)/dt$ where $d/dt\equiv\partial_{t}-({\bf
J}/n{\rm e})\cdot\nabla$. The left-hand side (LHS) is the ideal electric field
that becomes finite when the ion frozen-in condition is violated. Terms on the
right-hand side (RHS) contribute to this violation in kinetic plasmas,
including the Hall term, the electron pressure divergence term, and the
electron inertia term. Fig. 1c shows the terms in the out-of-plane ($y$)
component of Ohm’s law in a vertical cut through the x-line; the Hall term
$({\bf J}\times{\bf B})_{y}/n{\rm e}c$ (in purple) is the dominant term
supporting the reconnection electric field $E_{y}$ (in red) between the ion
inertial scale $d_{\rm i}$ and the electron inertial scale $d_{\rm e}$. The
Hall term arises because of the decoupling of the relatively immobile ions
from the motion of electrons that remain frozen-in to the magnetic fields
(Sonnerup, 1979). Electrons, the primary current carrier within the IDR (i.e.,
${\bf J}\simeq-{\rm e}n{\bf V}_{\rm e}$), then drag (both reconnected and not-
yet reconnected) magnetic field lines out of the reconnection plane, producing
the out-of-plane quadrupolar Hall magnetic field (Drake  _et al._ , 2008;
Mandt  _et al._ , 1994; Ren and Yamada et al., 2005; Burch  _et al._ , 2016).

Figure 2: Transport patterns of electromagnetic energy in Hall reconnection
and Sweet-Parker reconnection. a) Hall reconnection, where $\nabla\cdot{\bf
S}\simeq-{\bf J}\cdot{\bf E}_{{\rm Hall}}=0$ within the ion diffusion region
(IDR). It requires a Poynting vector ${\bf S}$ streamline pattern that leads
to an energy void around the x-line, limiting the difference in the
$zz$-component of the pressure tensor between the x-line and the far upstream
asymptotic region $\Delta P_{zz}^{\rm xline}$ (green arrow), and thus
localizes the diffusion region. $B_{x0}^{2}/8{\rm\pi}$ is the asymptotic
magnetic pressure far upstream from the IDR. b) Sweet-Parker reconnection,
where $\nabla\cdot{\bf S}<0$ within the diffusion region (DR). The ${\bf S}$
streamlines thus can end uniformly on the outflow symmetry line, producing an
elongated diffusion region.

Since the Hall term dominates the electric field ${\bf E}\simeq{\bf E}_{{\rm
Hall}}$ $=$ ${\bf J}\times{\bf B}/n{\rm e}c$ inside the IDR, then
$\nabla\cdot{\bf S}$ $=$ $-{\bf J}\cdot{\bf E}$ $\simeq 0$ per Poynting’s
theorem in the steady state. Along the inflow symmetry line ($x=0$) toward the
x-line magnetic energy $B^{2}/8{\rm\pi}\rightarrow 0$ since $|B_{x}|$
decreases. Also, $B_{z}=0$ and $B_{y}=0$ (in antiparallel reconnection) due to
symmetry. Consequentially, $\nabla\cdot{\bf S}$ $\simeq$ $0$ requires the
${\bf S}$ streamlines to be diverted to the outflow direction as illustrated
in Fig. 2a (also shown in Fig. 1a-b, consistent with the presence of $S_{x}$
$=$ $-cE_{z}B_{y}/4{\rm\pi}$). Since Poynting flux transports electromagnetic
energy, this ${\bf S}$ streamline pattern implies an energy void centered
around the x-line. This pattern introduces the localization to the diffusion
region, even in an initially planar current sheet. In contrast, in resistive-
MHD, $\nabla\cdot{\bf S}$ $=$ $-{\bf J}\cdot{\bf E}$ $\simeq$ $-\eta
J_{y}^{2}$ $<$ $0$ where $\eta$ is the resistivity. Thus, as illustrated in
Fig. 2b, ${\bf S}$ streamlines do not need to bend (i.e., $S_{x}\simeq 0$),
instead ending and distributing energy uniformly on the outflow symmetry line
($z=0$). This is why the diffusion region in Sweet-Parker reconnection is not
localized.

To quantify the degree of localization, we need to estimate the thermal
pressure at the x-line. The key is that ${\bf J}\cdot{\bf E}$ $\simeq$ $0$
inside the Hall dominated IDR limits the energy conversion to particles and
thus also limits the difference in the $zz$-component of the pressure tensor
between the x-line and the far upstream asymptotic region $\Delta P_{zz}^{\rm
xline}\equiv P_{zz}|_{\rm xline}-P_{0}$ (illustrated in Fig. 2a). Given that
magnetic pressure $B^{2}/8{\rm\pi}=0$ at the antiparallel reconnection x-line,
if $\Delta P_{zz}^{\rm xline}<B_{x0}^{2}/8{\rm\pi}$, the reconnecting field
bends toward the x-line as it approaches the x-line due to the force-balance
condition $\nabla(P+B^{2}/8{\rm\pi})=({\bf B}\cdot\nabla){\bf B}/4{\rm\pi}$
(Liu  _et al._ , 2020). This bending makes the outflow exhausts open out. This
fact will be used to develop a first-principles theory of the reconnection
rate.

###  II.2 Non-vanishing kinetic heating within the IDR

Even though total plasma heating is limited within the IDR because ${\bf
J}\cdot{\bf E}_{{\rm Hall}}=0$, there is a nonzero energy conversion arising
from the convection electric field $-{\bf V}_{\rm i}\times{\bf B}/c$ (the grey
curve in Fig. 1c) that is critical for modeling the thermal pressure at the
x-line. To reveal the kinetic heating process within the IDR along the inflow
direction, we show phase space diagrams in Fig. 3. Figures 3a and 3b show the
initial (reduced) distributions $f(v_{z},z)$ of electrons and ions,
respectively. The density profiles are shown by $n_{s}/n_{0}-1$ in pink where
$s={\rm e}$ and ${\rm i}$. The initial denser and hotter populations are
visible within the current sheet ($|z|\lesssim 1d_{\rm i}$); they balance the
magnetic pressure across the initial Harris sheet. At time $48/\Omega_{\rm
ci}$, the phase space diagrams of electrons and ions through the x-line show
rich structures in Fig. 3c and 3d, respectively. The pink profiles therein
show that the density around the x-line ($z=0$) essentially matches the
upstream value. On the other hand, while quasi-neutrality remains valid, ions
in Fig. 3d form a phase space hole centered around the x-line.

The ion distribution in Fig. 3d arises because the Hall electric field $E_{z}$
(Fig. 1b) ballistically accelerates ions from both sides of the current sheet
toward the x-line. They penetrate across the mid-plane, forming counter-
streaming ion beams and thus this phase space hole (Wygant  _et al._ , 2005;
Chen  _et al._ , 2008; Aunai  _et al._ , 2011). Note that the acceleration is
“ballistic” for inflowing ions because ions are already demagnetized within
the IDR and they see the Hall $E_{z}$ as a DC field. Importantly, this kinetic
effect increases the ion $zz$ pressure component above the asymptotic value,
as quantified by $\Delta P_{{\rm i}zz}\equiv P_{{\rm i}zz}(z)-P_{0}$.
Kinetically, the Hall $E_{z}$ arises from charge-separation due to the
relative inflow motion between lighter, faster electrons and heavier, slower
ions. This $E_{z}$ then speeds up ions and slows down electrons, which self-
regulates its magnitude. Thus, the associated $\Delta P_{{\rm i}zz}$ buildup,
even though effective, does not dominate the incoming energy budget during
reconnection.

We note that the reconnection electric field $E_{y}$ is $\sim 6$ times smaller
than the peak $E_{z}$ in our simulation, and it accelerates both species in
the out-of-plane direction ($\pm y$ for ions and electrons, respectively).
Some energy may be imparted into $\Delta P_{zz}$ through particle meandering
motions, but the ballistic acceleration by $E_{y}$ is more efficient in
imparting its energy into bulk kinetic energy of the current carriers in the
$y$-direction (Hesse  _et al._ , 2018, 2011) rather than to thermal energy.
Thus, it is not expected to greatly alter $\Delta P_{zz}$ and is ignored here.

Figure 3: Phase space diagrams, thermal pressures, and force-balance. The
phase space diagrams (inflow velocity $v_{z}$ versus $z$-location) of the
initial a) electrons and b) ions along the inflow symmetry line across the
x-line at $(x,z)=(0,0)$. The structured but more tenuous c) electrons and d)
ions in the same phase space at time $48/\Omega_{\rm ci}$. Particles are
collected within $x/d_{\rm e}$ $\in$ $[-5,5]$ with velocities normalized to
their background thermal speed $v_{{\rm th}s}$ for species $s={\rm e}$ and
${\rm i}$. Profiles of the magnetic field $B_{x}$ (white) and the density
relative to the background value $n_{s}/n_{0}-1$ (pink) are overlaid for
reference and their values are given by the right axis. The colormaps in
panels a and b are capped to better show the hot current sheet components,
thus the cold dense background plasmas appear as dark red at $v_{sz}\simeq 0$.
e) The diagonal components of ion pressure relative to the far upstream value
in solid lines ($\Delta P_{{\rm i}xx},\Delta P_{{\rm i}yy},\Delta P_{{\rm
i}zz}$) at time $48/\Omega_{\rm ci}$ along the same inflow symmetry line, and
that of electrons in dotted lines (near the bottom). For comparison, the
initial scalar pressures at the x-line ($\Delta P_{\rm i0}$, $\Delta P_{\rm
e0}$) are marked by the grey dashed horizontal line at value 0.5. The magnetic
pressure at the ion-inertial ($d_{\rm i}$) scale, $B_{x{\rm
i}}^{2}/8{\rm\pi}$, is indicted by the red dot. f) The $z$-component of the
integrated forces in Eq. (1). The vertical red transparent band in panel e
marks the electron diffusion region (EDR), while the vertical blue transparent
band in panel f marks the ion diffusion region (IDR). Predictions from Eqs.
(4), (6) and (12) are marked by the magenta horizontal lines in panels e and
f.

In Fig. 3e, we show diagonal elements of the pressure tensor of electrons and
ions (normalized to $B_{x0}^{2}/8{\rm\pi}$) through the x-line at time
$48/\Omega_{\rm ci}$. For comparison, the initial pressure of each species,
which completely balances the upstream magnetic pressure
$B_{x0}^{2}/8{\rm\pi}$, is marked by the grey-dashed horizontal line (note
that $\mathbb{P}_{\rm e}=\mathbb{P}_{\rm i}$ initially for this simulation).
In the steady state, the ion $\Delta P_{{\rm i}zz}$ (solid green) is slightly
reduced while the electron $\Delta P_{{\rm e}zz}$ (dotted green) is almost
completely depleted. Other significant pressure depletion occurs in ion
$\Delta P_{{\rm i}xx}$ (solid blue), electron $\Delta P_{{\rm e}xx}$ (dotted
blue) and electron $\Delta P_{{\rm e}yy}$ (dotted orange). Ion $\Delta P_{{\rm
i}yy}$ (solid orange) is roughly half of $\Delta P_{{\rm i}zz}$.

Our interest is in $P_{zz}$, since it affects force-balance in the inflow
direction. Along the inflow, the ram pressure ($\sum_{s}^{{\rm i},{\rm
e}}n_{s}m_{s}V_{sz}^{2}$) is small, so the force-balance condition in the $z$
direction is $({\bf J}\times{\bf B})_{z}/c\simeq(\nabla\cdot\mathbb{P})_{z}$,
where the total pressure $\mathbb{P}\equiv\sum_{s}^{{\rm i},{\rm
e}}\mathbb{P}_{s}$. Using ${\bf J}\times{\bf B}/c={\bf B}\cdot\nabla{\bf
B}/4{\rm\pi}-\nabla B^{2}/8{\rm\pi}$ and integrating along the inflow ($z$)
direction, the force-balance condition reads

$\frac{B^{2}}{8{\rm\pi}}+\Delta P_{zz}-\int^{z}\frac{{\bf B}\cdot\nabla
B_{z}}{4{\rm\pi}}dz^{\prime}\simeq{\rm constant}.$ (1)

Initially, the magnetic pressure is totally balanced by the thermal pressure
with no contribution from curvature. In the steady-state shown in Fig. 3f, the
thermal pressure (green) at the x-line drops significantly, and the upstream
magnetic tension (blue) at the mesoscale develops to counter balance the
magnetic pressure (red). This tension force is realized through the bending of
upstream field lines, which produces the Petschek-type open exhaust with a
localized diffusion region (Liu  _et al._ , 2020, 2017).

###  II.3 Estimating the pressure within the IDR

Within the IDR and outside the EDR, the electric field is ${\bf E}$ $=$ $-{\bf
V}_{\rm e}\times{\bf B}/c$ $=$ $-{\bf V}_{\rm i}\times{\bf B}/c+{\bf
J}\times{\bf B}/n{\rm e}c$. Thus the local energy conversion rate is

${\bf J}\cdot{\bf E}=-{\bf J}\cdot{\bf V}_{\rm i}\times{\bf B}/c={\bf V}_{\rm
i}\cdot{\bf J}\times{\bf B}/c={\rm e}n{\bf V}_{\rm i}\cdot{\bf E}_{{\rm
Hall}}.$ (2)

All of the work is done on the ions since the rate of work done on electrons
is $-{\rm e}n{\bf V}_{\rm e}\cdot{\bf E}={\rm e}n{\bf V}_{\rm e}\cdot({\bf
V}_{\rm e}\times{\bf B}/c)=0$. Note that even though ${\bf J}\cdot{\bf
E}_{{\rm Hall}}=0$, the ions do gain energy by ${\bf E}_{{\rm Hall}}$,
consistent with the ballistic acceleration displayed in Fig. 3d.

Near the inflow symmetry line ($x=0$), $J_{x}$, $B_{y}$, $B_{z}$ and $V_{ix}$
are negligible because of the symmetry in antiparallel reconnection. Thus
${\bf J}\cdot{\bf E}\simeq-V_{{\rm i}z}J_{y}B_{x}/c+V_{{\rm i}y}J_{z}B_{x}/c$.
The first term is work done by the Hall $E_{z}\simeq-J_{y}B_{x}/n{\rm e}c$,
while the second term is work done by the reconnection electric field
$E_{y}\simeq J_{z}B_{x}/n{\rm e}c$. Following the discussion of Fig. 3, the
effective $\Delta P_{{\rm i}zz}$ buildup primarily arises from ballistic
acceleration by the Hall $E_{z}$ that results in counter-streaming ions near
the x-line. We calculate the contribution of $-V_{{\rm i}z}J_{y}B_{x}/c$ to
the total rate of energy conversion $\int{\bf J}\cdot{\bf E}dV$ for the volume
enclosed by the Gaussian surface (1-2-3-4) in Fig. 4a. The right surface (3-4)
is chosen to coincide with an ion streamline near the inflow symmetry line, so
there is no ion energy flux through that surface. The $x$-location of this
surface relative to $x=0$ is quantified by $\ell(z)$. Then

$\begin{split}\int_{1234}-\frac{V_{{\rm
i}z}J_{y}B_{x}}{c}dV&=\int_{1234}-V_{{\rm
i}z}\frac{c}{4{\rm\pi}}\left(\frac{\partial B_{x}}{\partial
z}\right)\frac{B_{x}}{c}dxdydz\\\ &\simeq\ell_{y}\int_{d_{\rm e}}^{d_{\rm
i}}\ell(z)V_{{\rm i}z}(z)\frac{\partial}{\partial
z}\left(\frac{B_{x}^{2}}{8{\rm\pi}}\right)dz\\\ &\simeq\left(\frac{B_{x{\rm
i}}^{2}-B_{x{\rm e}}^{2}}{8{\rm\pi}}\right)V_{{\rm i}z}(d_{\rm i})\ell(d_{\rm
i})\ell_{y}.\\\ \end{split}$ (3)

Here we assumed $|\partial_{x}B_{z}|$ $\ll$ $|\partial_{z}B_{x}|$ in writing
$J_{y}$ $\simeq$ $(c/4{\rm\pi})\partial_{z}B_{x}$. We take the
$\ell(z)\rightarrow 0$ limit, so the variables within the integrand are
approaximately independent of $x$. We defined $B_{x{\rm i}}\equiv
B_{x}|_{d_{\rm i}}$ and $B_{x{\rm e}}\equiv B_{x}|_{d_{\rm e}}$, and used
$\ell_{y}$ as the dimension of the integrated surface in the out-of-plane
(translationally invariant) direction. We also assumed that the density is
nearly incompressible along the inflow so $V_{{\rm i}z}(z)\ell(z)\simeq
V_{{\rm i}z}(d_{\rm i})\ell(d_{\rm i})$.

Figure 4: Diagrams of diffusion regions for theoretical modeling. a) The
Gaussian surface (1-2-3-4) in the $\ell(z)\rightarrow 0$ limit that is used to
calculate the ion pressure buildup $P_{{\rm i}zz}|^{d_{\rm e}}_{d_{\rm i}}$
between the electron inertial scale ($d_{\rm e}$) and ion inertial scale
($d_{\rm i}$). The x-line is located at the lower-left corner (point 5).
$\ell(z)$ is the distance between the ion streamline and the inflow symmetry
line (line 1-5) as a function of $z$. The downward purple, green and orange
arrows represent the incoming Poynting flux $S_{z}$, enthalpy flux $H_{z}$,
and ion velocity $V_{{\rm i}z}$, respectively. b) The two-scale diffusion
region structure used to derive $B_{x{\rm e}}/B_{x{\rm i}}$. The blue box
represents the ion diffusion region (IDR) and the red box represents the
electron diffusion region (EDR). $V_{\rm in,i}$ ($V_{\rm in,e}$) is the ion
(electron) inflow velocity at the ion (electron) inertial scale where the
local magnetic field is $B_{x{\rm i}}$ ($B_{x{\rm e}}$). $V_{\rm Ai}$ and
$V_{\rm Ae}$ are the ion Alfvén speed and electron Alfvén speed based on the
local quantities, respectively. c) The region used to derive the slope of the
separatrix, $S_{{\rm lope}}$. The blue (red) box represents the IDR (EDR). The
blue solid line depicts an upstream magnetic field $\bf B$ line adjacent to
the separatrix shown by diagonal dashed lines.

From the discussion of Fig. 3d, we expect that most of this energy is
converted to $P_{{\rm i}zz}$, which contributes to the ion enthalpy flux ${\bf
H}_{\rm i}$ $=$ $(1/2)\mbox{Tr}(\mathbb{P}_{\rm i}){\bf V}_{\rm i}$ $+$
$\mathbb{P}_{\rm i}\cdot{\bf V}_{\rm i}$ Birn and Hesse (2010) that enters the
EDR in the vicinity of the x-line. The associated net enthalpy flux difference
$\oint{\bf H}\cdot d{\bf A}$ through the Gaussian surface 1-2-3-4 is
$(3/2)P_{{\rm i}zz}|_{d_{\rm e}}V_{{\rm i}z}(d_{\rm e})\ell(d_{\rm
e})\ell_{y}$ $-$ $(3/2)P_{{\rm i}zz}|_{d_{\rm i}}V_{{\rm i}z}(d_{\rm
i})\ell(d_{\rm i})\ell_{y}$ $\simeq$ $(3/2)P_{{\rm i}zz}|_{d_{\rm i}}^{d_{\rm
e}}V_{{\rm i}z}(d_{\rm i})\ell(d_{\rm i})\ell_{y}$, where $P_{{\rm
i}zz}|_{d_{\rm i}}^{d_{\rm e}}$ $\equiv$ $P_{{\rm i}zz}|_{d_{\rm e}}$ $-$
$P_{{\rm i}zz}|_{d_{\rm i}}$. Equating this quantity with the RHS of Eq. (3)
gives

$P_{{\rm i}zz}|_{d_{\rm i}}^{d_{\rm e}}\simeq\frac{2}{3}\left(\frac{B_{x{\rm
i}}^{2}-B_{x{\rm e}}^{2}}{8{\rm\pi}}\right).$ (4)

Since no work is done on electrons outside the EDR, $P_{{\rm e}zz}|_{d_{\rm
i}}^{d_{\rm e}}$ $\simeq$ $0$ and thus the total thermal pressure difference
$P_{zz}|_{d_{\rm i}}^{d_{\rm e}}$ $\simeq$ $P_{{\rm i}zz}|_{d_{\rm i}}^{d_{\rm
e}}$. This thermal pressure increase is smaller than the magnetic pressure
drop $(B_{x{\rm i}}^{2}-B_{x{\rm e}}^{2})/8{\rm\pi}$ between the $d_{\rm i}$\-
and $d_{\rm e}$-scale, so there is insufficient pressure to balance forces in
the $z$-direction without the bending of field lines, and Hall reconnection
opens into a Petschek-type geometry. This predicted value of Eq. (4),
calculated using the measured $B_{x{\rm i}}$ and $B_{x{\rm e}}$, is plotted as
a horizontal magenta line in Fig. 3e and compares well with the measured
$P_{{\rm i}zz}|_{d_{\rm i}}^{d_{\rm e}}$ (green).

###  II.4 Available magnetic energy at the EDR scale

In order to estimate the relative magnetic pressure (energy) at the EDR, we
write

$\frac{cE_{y{\rm i}}}{B_{x{\rm i}}V_{\rm Ai}}=\frac{V_{\rm in,i}}{V_{\rm
Ai}}\simeq\frac{d_{\rm i}}{L_{\rm i}}\sim\frac{d_{\rm e}}{L_{\rm
e}}\simeq\frac{V_{\rm in,e}}{V_{\rm Ae}}=\frac{cE_{y{\rm e}}}{B_{x{\rm
e}}V_{\rm Ae}}.$ (5)

The quantities are defined and illustrated in Fig. 4b. The first and last
equalities come from the frozen-in conditions $E_{ys}=V_{{\rm in},s}B_{xs}/c$
at the inflow edges of the IDR and EDR for $s={\rm i}$ and ${\rm e}$,
respectively. We use incompressibility for the second and fourth equalities.
For the third equality, we use a geometrical argument that the magnetic field
line threading the x-line and the corners of the EDR and the IDR is
approximately straight, resulting in a similar aspect ratio for the EDR and
the IDR. At the ion-scale, the outflow speed is the ion Alfvén speed $V_{\rm
Ai}\equiv B_{x{\rm i}}/(4{\rm\pi}nm_{\rm i})^{1/2}$. In contrast, the electron
outflow speed is the electron Alfvén speed based on the local conditions,
$V_{\rm Ae}\equiv B_{x{\rm e}}/(4{\rm\pi}nm_{\rm e})^{1/2}$, since ions
decouple from the motion of magnetic field lines in the electron-scale inside
the IDR (Shay and Drake, 1998).

By equating the first and last terms and noting that $E_{y}$ is uniform in 2D
steady-state per Faraday’s law (seen in Fig. 1c), we find

$\frac{B_{x{\rm e}}^{2}}{B_{x{\rm i}}^{2}}\simeq\left(\frac{m_{\rm e}}{m_{\rm
i}}\right)^{1/2}.$ (6)

Note that the equality between the first and the last terms is consistent with
the high-cadence observation of Magnetospheric Multiscale Mission (MMS) (Burch
_et al._ , 2020). For $m_{\rm i}/m_{\rm e}=400$ as in the simulation,
$B_{x{\rm e}}^{2}/B_{x{\rm i}}^{2}\simeq 0.05$. The predicted $B_{x{\rm
e}}^{2}/8{\rm\pi}$ based on the measured $B_{x{\rm i}}^{2}/8{\rm\pi}$ in Fig.
3e compares well with the small $B_{x}^{2}/8{\rm\pi}$ ($\simeq
B^{2}/8{\rm\pi}$ in black) value at the $d_{\rm e}$-scale. For the real proton
to electron mass ratio $m_{\rm i}/m_{\rm e}=1836$, $B_{x{\rm e}}^{2}/B_{x{\rm
i}}^{2}\simeq 0.023$. The smallness of $B_{x{\rm e}}^{2}/B_{x{\rm i}}^{2}$
makes the contribution of the pressure depletion within the EDR negligible.
However, this imbalanced pressure becomes critical in pair plasmas where the
EDR is the same as the IDR, as discussed later.

###  II.5 Cross-scale coupling and the rate prediction

To predict the reconnection rate, we use the force-balance condition $\nabla
B^{2}/8{\rm\pi}+\nabla\cdot\mathbb{P}={\bf B}\cdot\nabla{\bf B}/4{\rm\pi}$ and
geometry to couple the solutions at the IDR, EDR and the upstream MHD region.
First, we discretize this equation at point 7 of Fig. 4c. In the
$z$-direction,

$\frac{B_{x{\rm i}}^{2}-B_{x{\rm e}}^{2}}{8{\rm\pi}(d_{\rm i}-d_{\rm
e})}-\frac{P_{zz}|^{d_{\rm e}}_{d_{\rm i}}}{d_{\rm i}-d_{\rm
e}}\simeq\left(\frac{B_{x{\rm i}}+B_{x{\rm
e}}}{2}\right)\frac{2B_{z8}}{4{\rm\pi}L_{\rm i}(d_{\rm i}-d_{\rm e})/d_{\rm
i}}.$ (7)

From geometry, the slope of the separatrix $S_{{\rm lope}}$ $\simeq$ $d_{\rm
i}/L_{\rm i}$ $\simeq$ $B_{z8}/[(B_{x{\rm i}}+B_{x{\rm e}})/2]$. Solving for
$S_{{\rm lope}}$ gives

$S_{{\rm lope}}^{2}\simeq\frac{B_{x{\rm i}}^{2}-B_{x{\rm e}}^{2}}{(B_{x{\rm
i}}+B_{x{\rm e}})^{2}}-\frac{8{\rm\pi}P_{zz}|^{d_{\rm e}}_{d_{\rm
i}}}{(B_{x{\rm i}}+B_{x{\rm e}})^{2}}.$ (8)

Physically, this expression relates the opening angle of the separatrix to the
thermal pressure difference. Plugging in $P_{zz}|_{d_{\rm i}}^{d_{\rm e}}$
using Eq. (4), we get

$S_{{\rm lope}}^{2}\simeq\frac{1}{3}\left[\frac{1-(B_{x{\rm e}}/B_{x{\rm
i}})}{1+(B_{x{\rm e}}/B_{x{\rm i}})}\right].$ (9)

A similar analysis on the force-balance across the EDR gives

$S_{{\rm lope}}^{2}\simeq 1-\frac{8{\rm\pi}P_{zz}|^{0}_{d_{\rm e}}}{B_{x{\rm
e}}^{2}},$ (10)

where the slope of the separatrix is assumed similar inside and outside the
EDR because the magnetic tension straightens out the field lines.

We now relate $B_{x{\rm i}}$ back to the upstream asymptotic field $B_{x0}$
using

$\frac{B_{x{\rm i}}}{B_{x0}}\simeq\frac{1-S_{{\rm lope}}^{2}}{1+S_{{\rm
lope}}^{2}},$ (11)

which was previously derived using the force-balance condition upstream of the
IDR at the mesoscale Liu  _et al._ (2017). We again assume the slope of the
separatrix is similar inside and outside the IDR.

Finally, using Eqs. (4), (10) and (11), we can rewrite the total thermal
pressure buildup $\Delta P_{zz}^{\rm xline}\simeq P_{zz}|_{d_{\rm i}}^{d_{\rm
e}}+P_{zz}|_{d_{\rm e}}^{0}$ in terms of the upstream asymptotic magnetic
pressure $B_{x0}^{2}/8{\rm\pi}$ as

$\Delta P_{zz}^{\rm xline}\simeq\left[\frac{2}{3}+\left(\frac{1}{3}-S_{{\rm
lope}}^{2}\right)\frac{B_{x{\rm e}}^{2}}{B_{x{\rm
i}}^{2}}\right]\left(\frac{1-S_{{\rm lope}}^{2}}{1+S_{{\rm
lope}}^{2}}\right)^{2}\frac{B_{x0}^{2}}{8{\rm\pi}}.$ (12)

Using Eqs. (12), (9) and (6), for $m_{\rm i}/m_{\rm e}=400$ we get $\Delta
P_{zz}^{\rm xline}$ $\simeq$ $0.283(B_{x0}^{2}/8{\rm\pi})$. This prediction is
within $\simeq 20\%$ of the simulated $\Delta P_{zz}^{\rm
xline}/(B_{x0}^{2}/8{\rm\pi})$ as indicated by the magenta horizontal line in
Fig. 3f. The predicted $\Delta P_{zz}^{\rm xline}$ is considerably less than
$B_{x0}^{2}/8{\rm\pi}$, which from Eq. (1) is consistent with there being an
open outflow geometry.

Once the separatrix slope is determined, we can also obtain a first-principles
prediction of the reconnection rate using the $R-S_{{\rm lope}}$ relation in
Liu et al. Liu  _et al._ (2017),

$R=S_{{\rm lope}}\left(\frac{1-S_{{\rm lope}}^{2}}{1+S_{{\rm
lope}}^{2}}\right)^{2}\sqrt{1-S_{{\rm lope}}^{2}},$ (13)

which is $0.157$ for $m_{\rm i}/m_{\rm e}=1836$ and $0.172$ for $m_{\rm
i}/m_{\rm e}=400$, consistent with the measured rate on the order of
$\mathcal{O}(0.1)$ as shown in Fig. 1c and literature (Birn  _et al._ , 2001;
Shay  _et al._ , 1999; Hesse  _et al._ , 1999).

###  II.6 Pair Plasmas and Resistive-MHD

In systems without the Hall effect, such as pair plasmas and resistive-MHD,
energy conversion inside the diffusion region is totally different. There is
no two-scale structure to the diffusion region. Thus, for the following
discussion, we denote the microscopic thickness of the diffusion region as
$d_{\rm m}$ (corresponding to $d_{\rm i}$ in Fig. 4a) and define $\ell_{\rm
in}\equiv\ell(d_{\rm m})$, $V_{\rm in}\equiv|V_{{\rm i}z}(d_{\rm m})|$,
$B_{x{\rm m}}\equiv B_{x}(d_{\rm m})$ and the inflowing Poynting vector
$S_{\rm in}\equiv|S_{z}(d_{\rm m})|$. We consider the energy conversion rate
integrated over the gray rectangular box (1-5-6-4) in Fig. 4a in the
$\ell_{\rm in}\rightarrow 0$ limit to illustrate the critical difference
compared to Hall reconnection. Near the inflow symmetry line ($x=0$) in
antiparallel reconnection, $J_{x}=0$, thus ${\bf J}\cdot{\bf E}$ $=$
$J_{y}E_{y}+J_{z}E_{z}$. Without the Hall effect, $E_{z}$ vanishes while
$E_{y}$ is uniform in 2D steady-state, thus

$\begin{split}\int_{1564}{\bf J}\cdot{\bf E}dV&\simeq\ell_{y}\ell_{\rm
in}E_{y}\int_{0}^{d_{\rm m}}\frac{c}{4{\rm\pi}}\left(\frac{\partial
B_{x}}{\partial z}\right)dz\\\ &=c\frac{E_{y}B_{x{\rm m}}}{4{\rm\pi}}\ell_{\rm
in}\ell_{y}=S_{\rm in}\ell_{\rm in}\ell_{y}.\end{split}$ (14)

Here we assumed $|\partial_{x}B_{z}|\ll|\partial_{z}B_{x}|$ (i.e., weak
localization) again, so $J_{y}\simeq(c/4{\rm\pi})\partial_{z}B_{x}$.
Importantly, the last equality of Eq. (14) indicates that the Poynting flux
entering the top surface (1-4) is all converted to plasma energy within this
narrow rectangular box. In other words, no Poynting flux is diverted to the
outflow direction, as illustrated in Fig. 2b. The outflowing enthalpy flux
$H_{x}$ and the bulk flow kinetic energy flux ${\bf K}\equiv\sum_{s}^{{\rm
i},{\rm e}}(1/2)nm_{s}V_{s}^{2}{\bf V}_{s}$ (Birn and Hesse, 2010; Eastwood
_et al._ , 2020) in the $x$-direction $K_{x}$ compete for the inflowing
energy. For example, in the low background-$\beta$ limit where both $H_{z}$
and $K_{z}$ at the inflow surface (1-4) are negligible, the energy conversion
is $S_{\rm in}\ell_{\rm in}$ $\simeq$ $\int_{6}^{4}H_{x}dz$ $+$
$\int_{6}^{4}K_{x}dz$.

In electron-positron ($m_{\rm i}=m_{\rm e}$) pair plasmas Bessho and
Bhattacharjee (2005); Liu  _et al._ (2017, 2015), near the inflow symmetry
line $K_{x}$ primarily comes from the bulk flow kinetic energy of the current
carriers $\sum_{s}^{{\rm i},{\rm e}}(1/2)nm_{s}V_{sy}^{2}V_{sx}$. In the
magnetically-dominated relativistic regime Guo  _et al._ (2020), to sustain
the extreme current density, $K_{x}$ can be large and it limits the energy
available for the enthalpy $H_{x}$. This leads to a depleted pressure at the
x-line and fast reconnection Liu  _et al._ (2020). A similar competition could
occur in non-relativistic low-$\beta$ pair plasmas. Without the inward-
pointing Hall $E_{z}$, the counter-streaming ions that efficiently build up
$\Delta P_{zz}$ in Hall reconnection are absent. The only other potential
source of heating is through the reconnection electric field $E_{y}$, which
efficiently increases the current carrier drift speed, but not $\Delta P_{zz}$
because the acceleration is primarily in the $y$-direction. Therefore, $\Delta
P_{zz}$ should be less than $B_{x0}^{2}/8{\rm\pi}$ and fast reconnection with
an open outflow should occur.

For (isotropic) resistive-MHD, $K_{x}$ near the inflow symmetry line vanishes
because the current in MHD is not associated with any kinetic energy. All the
inflowing energy is then converted into enthalpy. Using $E_{y}=V_{\rm
in}B_{x{\rm m}}/c$ in Eq. (14) then $S_{\rm in}\ell_{\rm in}$ $=$ $(B_{x{\rm
m}}^{2}/4{\rm\pi})V_{\rm in}\ell_{\rm in}$ $\simeq$ $\int_{6}^{4}H_{x}dz$ $=$
$(5/2)\int_{6}^{4}PV_{x}dz$ $<$ $(5/2)P|_{6}\int_{6}^{4}V_{x}dz$, where the
inequality arises from a reasonable thermal pressure profile that peaks at
point 6 on the 4-6 line. In the incompressible limit, $\int_{6}^{4}V_{x}dz$
$=$ $V_{\rm in}\ell_{\rm in}$, so we get $P|_{6}$ $>$ $(4/5)B_{x{\rm
m}}^{2}/8{\rm\pi}$, indicating that a balanced-pressure $P|_{4}^{6}$ $=$
$B_{x{\rm m}}^{2}/8{\rm\pi}$ becomes possible. Numerical simulations confirms
that the thermal pressure at the x-line nearly balances the upstream magnetic
pressure in Sweet-Parker reconnection. This implies $S_{{\rm lope}}\rightarrow
0$ from Eq. (8) and explains why Sweet-Parker reconnection is slow; note that
the Sweet-Parker theory Sweet (1958); Parker (1957) itself does not address
why the diffusion region length extends to the system size (i.e., $S_{{\rm
lope}}\rightarrow 0$), but follows naturally in the present model.

In fact, we can recover the Sweet-Parker scaling using the framework laid out
here. Using $E_{y}$ $=$ $\eta J_{y}|_{\rm xline}$ $\simeq$
$\eta(c/4{\rm\pi})(B_{x{\rm m}}/d_{\rm m})$ in Eq. (14), we get

$\begin{split}\int_{1564}{\bf J}\cdot{\bf
E}dV\simeq\eta\left(\frac{c}{4{\rm\pi}}\right)^{2}\frac{B_{x{\rm
m}}^{2}}{d_{\rm m}}\ell_{\rm in}\ell_{y}.\end{split}$ (15)

Since this energy is all converted into the enthalpy
$(5/2)\ell_{y}\int_{6}^{4}PV_{x}dz$ $\simeq$ $(5/2)\langle
P\rangle\ell_{y}\int_{6}^{4}V_{x}dz$ $\simeq$ $(5/2)\langle P\rangle V_{\rm
in}\ell_{\rm in}\ell_{y}$, we can solve for the average pressure $\langle
P\rangle$. The pressure buildup can be estimated as

$\begin{split}P|_{d_{\rm m}}^{0}\simeq\langle
P\rangle\simeq\frac{4}{5}\left(\frac{L_{\rm i}}{d_{\rm
m}}\right)^{2}\left(\frac{\eta c^{2}}{4{\rm\pi}V_{\rm Ai}L_{\rm
i}}\right)\frac{B_{x{\rm m}}^{2}}{8{\rm\pi}},\end{split}$ (16)

where we used plasma continuity $V_{\rm in}$ $\simeq$ $V_{\rm Ai}d_{\rm
m}/L_{\rm i}$ over the entire diffusion region. Using this $P|_{d_{\rm
m}}^{0}$ in Eq. (10) though noting $P_{zz}$ $=$ $P$ since the pressure is
isotropic, $d_{\rm e}=d_{\rm i}$ $=$ $d_{\rm m}$, $B_{x{\rm e}}=B_{x{\rm
i}}=B_{x{\rm m}}$ and using $L_{\rm i}/d_{\rm m}$ $\simeq$ $S_{{\rm lope}}$ as
before, we get

$\begin{split}S_{{\rm lope}}^{2}\simeq 1-\frac{4}{5}\left(\frac{1}{S_{{\rm
lope}}}\right)^{2}\left(\frac{\eta c^{2}}{4{\rm\pi}V_{\rm Ai}L_{\rm
i}}\right).\end{split}$ (17)

Looking for a solution in the $S_{{\rm lope}}\rightarrow 0$ limit, as
justified above, we get the reconnection rate from Eq. (13) to be

$\begin{split}R_{\rm SP}\simeq S_{{\rm
lope}}\simeq\sqrt{\frac{4}{5}\left(\frac{\eta c^{2}}{4{\rm\pi}V_{\rm Ai}L_{\rm
i}}\right)}.\end{split}$ (18)

Up to a numerical factor near unity, this result is the Sweet-Parker scaling
(Sweet, 1958; Parker, 1957) of the reconnection rate in uniform resistivity
MHD.

To see the compatibility of the model developed here with the result of a
spatially localized resistivity in MHD (Ugai and Tsuda, 1977; Sato and
Hayashi, 1979; Biskamp, 1986; Uzdensky and Kulsrud, 2000; Lin  _et al._ ,
2021), one needs to retain $-\partial_{x}B_{z}$ in the estimate of $J_{y}$.
This leads to $\int_{1564}{\bf J}\cdot{\bf E}dV<S_{\rm in}\ell_{\rm
in}\ell_{y}$, instead of them being equal as in Eq. (14). Physically, a finite
$\partial_{x}B_{z}$ indicates a localized diffusion region and gives rise to
an $x$ component of the Poynting vector $cE_{y}B_{z}/4{\rm\pi}$. This $S_{x}$
also diverts the inflowing energy to the outflow direction, analogous to
$-cE_{z}B_{y}/4{\rm\pi}$ arising from the Hall effect (Fig. 1). Hence, a
localized diffusion region with a depleted pressure at the x-line takes place.
However, in this case, the diffusion region localization is introduced by hand
or some unidentified mechanisms Kulsrud (2001).

##  III Conclusions

We have shown that, counter-intuitively, a lower energy conversion rate ${\bf
J}\cdot{\bf E}$ along the inflow toward the x-line makes reconnection faster
because the lower pressure requires upstream magnetic field lines to bend to
enforce force balance, therefore opening the outflow exhaust. We predict that
the high thermal pressure required to get an elongated planar current sheet is
not energetically sustainable at the x-line of electron-proton plasmas because
${\bf J}\cdot{\bf E}_{{\rm Hall}}$$=0$. A significant portion of incoming
electromagnetic energy is not transported to the x-line, but is diverted to
the outflow by Hall fields. The theory presented here directly links the Hall
effect to diffusion region localization, and the linkage is the pressure
depletion at the x-line. The predicted pressure drop compares well with
simulations of collisionless magnetic reconnection. Through cross-scale
coupling between the EDR, IDR and the upstream MHD region, the fast
reconnection rate of order $\mathcal{O}(0.1)$ is derived (Eqs. (13),(9),(6))
from first-principles for the first time to the best of our knowledge. A
closer agreement can be made after considering the reconnection outflow speed
reduction by thermal pressure effects (Li and Liu, 2021; Haggerty  _et al._ ,
2018; Liu  _et al._ , 2012); a predicted $R\simeq 0.075$ can be read off from
the $R-S_{{\rm lope}}$ relation (similar to Eq. (13)) in Fig. 5(c) of Li and
Liu Li and Liu (2021) using the same $S_{{\rm lope}}$, which is in excellent
quantitative agreement with the simulated $E_{y}$ here in Fig. 1c. In
addition, the competition between different forms of energy flux explains why
Sweet-Parker reconnection does not have an open exhaust and is slow, while
reconnection in electron-positron (pair) plasma is fast. The same theoretical
framework recovers the Sweet-Parker scaling law (Eq. (18)).

This work is dedicated to explaining the primary localization mechanism for a
stable single x-line in collisionless plasmas relevant to magnetospheric,
solar and laboratory applications. If a stable single x-line can be realized
(i.e., the steadiness is justified by the nearly uniform $E_{y}$ and
negligible $\partial_{t}(J_{y}/n)$ shown in Fig. 1c), the open outflow
geometry can suppress the generation of secondary tearing modes (Shi  _et al._
, 2019; Shepherd and Cassak, 2010; Daughton  _et al._ , 2009) and time-
dependent dynamics become less important. However, if the localization of a
single x-line is weak, the reconnecting layer is very thin, or pressure-
balance across the opened outflow exhausts can not be established (Liu  _et
al._ , 2020), then secondary tearing modes will be triggered. A cycle of fast
generation and ejection of secondary tearing islands provides additional
localization mechanism to increase the average reconnection rate (Liu  _et
al._ , 2020; Comisso  _et al._ , 2016; Pucci and Velli, 2014; Huang and
Bhattacharjee, 2010; Loureiro  _et al._ , 2007; Daughton and Karimabadi,
2007).

Finally, the Hall effect arises whenever the current sheet thins down to the
ion kinetic scale, thus even in a thin 3D current sheet, the argument based on
the two key points in the “Introduction” still holds. Large-scale 3D PIC
simulations also suggest that broader turbulent current sheets can collapse to
thin reconnecting layers in the kinetic scale (Guo  _et al._ , 2021). Kinetic
reconnecting layers also persist and dominate a current sheet that is filled
with self-generated turbulence (Daughton  _et al._ , 2011, 2014; Liu  _et al._
, 2014; Le  _et al._ , 2018; Stanier  _et al._ , 2019; Price  _et al._ ,
2016). Importantly, the reconnection process is often quasi-2D in nature. This
has been ascertained from the abundance of data from the MMS mission, which
has unprecedented spatial and temporal resolutions. In particular, 2D
simulations have done an excellent job reproducing detailed reconnection
dynamics (Torbert  _et al._ , 2018; Nakamura  _et al._ , 2018b, a; Genestreti
_et al._ , 2018; Burch  _et al._ , 2020). Nevertheless, it remains an open
question to explore whether fast reconnection can proceed in Nature without
eventually forming a dominating kinetic current sheet in three-dimensional
plasmas.

##  IV Methods

We carry out 2D PIC simulations of magnetic reconnection in a low background
$\beta$ $\equiv$ $8{\rm\pi}P_{0}/B_{x0}^{2}$ plasma. The simulations are
performed using the VPIC code (Bowers  _et al._ , 2008), which solves
Maxwell’s equations and the relativistic Vlasov equation. The simulation
employs the Harris current sheet equilibrium (Harris, 1962) that has the
initial magnetic profile ${\bf B}$ $=$ $B_{x0}\tanh(z/\lambda)\hat{x}$ and the
density profile $n$ $=$ $n_{0}\mbox{sech}^{2}(z/\lambda)$ $+$ $n_{0}$; here
$\lambda$ $=$ $1d_{\rm i}$ is the initial half-thickness of the current sheet,
where $d_{\rm i}$ $\equiv$ $c/(4{\rm\pi}n_{0}{\rm e}^{2}/m_{\rm i})^{1/2}$ is
the ion-inertial scale based on the background plasma density $n_{0}$, which
is also the peak density of the current sheet population in our setup; this
choice avoids current sheet expansion due to density depletion as has been
seen (Bessho and Bhattacharjee, 2010; Wu  _et al._ , 2011). The simulation
size is $L_{x}\times L_{z}=76.8d_{\rm i}\times 38.4d_{\rm i}$ that spans the
domain $[-L_{x}/2,L_{x}/2]\times[-L_{z}/2,L_{z}/2]$ with $n_{x}\times
n_{z}=12288\times 6144$ cells. There are $\simeq 15$ billion macro-particles.
The $x$-direction boundaries are periodic, while the $z$-direction boundaries
are conducting for fields and reflecting for particles. We use the ion-to-
electron mass ratio $m_{\rm i}/m_{\rm e}=400$, the temperature ratio $T_{\rm
i}/T_{\rm e}=1$, the plasma $\beta=0.01$ based on the background temperature
$T_{0}$ and background density $n_{0}$, and $\omega_{\rm pe}/\Omega_{\rm
ce}=4$, where the electron plasma frequency $\omega_{\rm
pe}=(4{\rm\pi}n_{0}{\rm e}^{2}/m_{\rm e})^{1/2}$ and the electron cyclotron
frequency $\Omega_{\rm ce}={\rm e}B_{x0}/m_{\rm e}c$. These result in the
background electron thermal speed $v_{\rm the}\equiv(T_{0}/m_{\rm
e})^{1/2}=0.0125c$, ion thermal speed $v_{\rm thi}\equiv(T_{0}/m_{\rm
i})^{1/2}=0.000625c$ and Alfvén speed $V_{\rm A0}\equiv
B_{x0}/(4{\rm\pi}n_{0}m_{\rm i})^{1/2}=0.0125c$, well within the non-
relativistic regime for solar and magnetospheric applications. A localized
initial magnetic field perturbation of amplitude $\delta B_{z}=0.03B_{x0}$ is
added to induce single x-line reconnection at the center of the simulation
domain. To reduce noise in the generalized Ohm’s law analysis, the presented
data is time-averaged over an interval $0.085/\Omega_{\rm ci}$, where
$\Omega_{\rm ci}={\rm e}B_{x0}/m_{\rm i}c$ is the ion cyclotron frequency.

##  V Data availability

Access to the simulation data and scripts used to plot the figures are
available at Zenodo (https://doi.org/10.5281/zenodo.6258255). All other data
are available from the corresponding author on reasonable request.

##  VI Code availability

The simulations are conducted using an open source Vector Particle-In-Cell
(VPIC) project, which is available at https://github.com/lanl/vpic. The
simulation data are analyzed using IDL, Matlab and Python. The scripts are
available at the data storage site.

##  VII References

## References

  * Wygant  _et al._ (2005) J. R. Wygant, C. A. Cattell, R. Lysak, Y. Song, J. Dombeck, J. McFadden, F. S. Mozer, C. W. Carlson, G. Parks, E. A. Lucek, A. Balogh, M. Andre, H. Reme, M. Hesse, and C. Mouikis, J. Geophys. Res. 110, A09206 (2005).
  * Drake  _et al._ (2006) J. F. Drake, M. Swisdak, H. Che, and M. A. Shay, Nature 442, 553 (2006).
  * Birn and Hesse (2010) J. Birn and M. Hesse, Phys. Plasmas 17, 012109 (2010).
  * Aunai  _et al._ (2011) N. Aunai, G. Belmont, and R. Smets, J. Geophys. Res. 116, A09232 (2011).
  * Egedal  _et al._ (2012) J. Egedal, W. Daughton, and A. Le, Nature Physics 8, 321 (2012).
  * Eastwood  _et al._ (2013) J. P. Eastwood, T. D. Phan, J. F. Drake, M. A. Shay, A. L. Borg, B. Lavraud, and M. G. G. T. Taylor, Phys. Rev. Lett. 110, 225001 (2013).
  * Eastwood  _et al._ (2020) J. P. Eastwood, M. V. Goldman, T. D. Phan, J. E. Stawarz, P. A. Cassak, J. F. Drake, D. Newman, B. Lavraud, M. A. Shay, R. E. Ergun, J. L. Burch, D. J. Gershman, B. L. Giles, P. A. Lindqvist, R. B. Torbert, R. J. Strangeway, and C. T. Russell, Phys. Rev. Lett. 125, 265102 (2020).
  * Shay  _et al._ (2014) M. A. Shay, C. C. Haggerty, and T. D. Phan et al., Phys. Plasmas 21, 122902 (2014).
  * Bessho  _et al._ (2014) N. Bessho, L. J. Chen, J. R. Shuster, and S. Wang, J. Geophys. Lett. 41, 8688 (2014).
  * Dahlin  _et al._ (2014) J. T. Dahlin, J. F. Drake, and M. Swisdak, Phys. Plasmas 21, 092304 (2014).
  * Guo  _et al._ (2014) F. Guo, H. Li, W. Daughton, and Yi-Hsin Liu, Phys. Rev. Lett. 113, 155005 (2014).
  * Shuster  _et al._ (2015) J. R. Shuster, L. J. Chen, M. Hesse, M. R. Argall, W. Daughton, R. B. Torbert, and N. Bessho, Geophys. Res. Lett. 42, 2586 (2015).
  * Yamada  _et al._ (2015) M. Yamada, J. Yoo, J.-A. Jonathan, W. Daughton, H. Ji, R. M. Kulsrud, and C. E. Myers, Phys. Plasmas 22, 056501 (2015).
  * Wang  _et al._ (2018) S. Wang, L.-J. Chen, N. Bessho, M. Hesse, J. andM. Yamada, Y.-H. Liu, D. J. Gershman, B. L. Giles, and T. E. Moore, J. Geophys. Res. 123, 8185 (2018).
  * Guo  _et al._ (2020) F. Guo, Y.-H. Liu, X. Li, H. Li, W. Daughton, and P. Kilian, Phys. Plasmas 27, 080501 (2020).
  * Li  _et al._ (2021) X. Li, F. Guo, and Y.-H. Liu, Phys. Plasmas 28, 052905 (2021).
  * Sweet (1958) P. A. Sweet, in  _IAU Symp. in Electromagnetic Phenomena in Cosmical Physics, ed. B. Lehnet (New York: Cambridge Univ. Press)_ (1958) p. 123.
  * Parker (1957) E. N. Parker, J. Geophys. Res. 62, 509 (1957).
  * Petschek (1964) H. E. Petschek, in  _Proc. AAS-NASA Symp. Phys. Solar Flares_ , NASA-SP, Vol. 50 (1964) pp. 425–439.
  * Shay and Drake (1998) M. A. Shay and J. F. Drake, Geophys. Res. Lett. 25, 3759 (1998).
  * Birn  _et al._ (2001) J. Birn, J. F. Drake, M. A. Shay, B. N. Rogers, R. E. Denton, M. Hesse, M. Kuznetsova, Z. W. Ma, A. Bhattacharjee, A. Otto, and P. L. Pritchett, J. Geophys. Res. 106, 3715 (2001).
  * Rogers  _et al._ (2001) B. N. Rogers, R. E. Denton, J. F. Drake, and M. A. Shay, Phys. Rev. Lett. 87, 195004 (2001).
  * Bessho and Bhattacharjee (2005) N. Bessho and A. Bhattacharjee, Phys. Rev. Lett. 95, 245001 (2005), 10.1103/PhysRevLett.95.245001.
  * Cassak  _et al._ (2007) P. A. Cassak, J. F. Drake, and M. A. Shay, Phys. Plasmas 14, 054502 (2007).
  * Hesse  _et al._ (2011) M. Hesse, T. Neukirch, K. Schindler, M. Kuznetsova, and S. Zenitani, Space Sci. Rev. 160, 3 (2011).
  * Liu  _et al._ (2014) Y.-H. Liu, W. Daughton, H. Karimabadi, H. Li, and S. P. Gary, Phys. Plasmas 21, 022113 (2014).
  * TenBarge  _et al._ (2014) J. M. TenBarge, W. Daughton, H. Karimabadi, G. G. Howes, and W. Dorland, Phys. Plasmas 21, 020708 (2014).
  * Stanier  _et al._ (2015a) A. Stanier, A. N. Simakov, L. Chacoń, and W. Daughton, Phys. Plasmas 22, 010701 (2015a).
  * Stanier  _et al._ (2015b) A. Stanier, W. Daughton, L. Chacón, H. Karimabadi, J. Ng, Y.-M. Huang, A. Hakim, and A. Bhattacharjee, Phys. Rev. Lett. 115, 175004 (2015b).
  * Comisso and Bhattacharjee (2016) L. Comisso and A. Bhattacharjee, J. Plasma Phys. 82, 595820601 (2016).
  * Liu  _et al._ (2017) Y.-H. Liu, M. Hesse, F. Guo, W. Daughton, H. Li, P. A. Cassak, and M. A. Shay, Phys. Rev. Lett. 118, 085101 (2017).
  * Cassak  _et al._ (2017) P. A. Cassak, Yi-Hsin Liu, and M. A. Shay, J. Plasma Phys. 83, 715830501 (2017).
  * Liu  _et al._ (2018) Y.-H. Liu, M. Hesse, P. A. Cassak, M. A. Shay, S. Wang, and L.-J. Chen, Geophys. Res. Lett. 45, 3311 (2018).
  * Nakamura  _et al._ (2018a) T. K. M. Nakamura, K. J. Genestreti, Y. H. Liu, R. Nakamura, W. L. Teh, H. Hasegawa, W. Daughton, M. Hesse, R. B. Torbert, J. L. Burch, and B. L. Giles, J. Geophys. Res. 123, 9150 (2018a).
  * Genestreti  _et al._ (2018) K. J. Genestreti, T. K. M. Nakamura, R. Nakamura, R. E. Denton, R. B. Torbert, L. J. Burch, F. Plaschke, S. A. Fuselier, R. E. Ergun, B. L. Giles, and C. T. Russell, J. Geophys. Res. 123, 9130 (2018).
  * Nakamura  _et al._ (2018b) R. Nakamura, K. J. Genestreti, T. K. M. Nakamura, W. Baumjohann, A. Varsani, T. Nagai, N. Bessho, J. L. Burch, and R. E. Denton et al., J. Geophys. Res. 124, 1173 (2018b).
  * Huang  _et al._ (2020) K. Huang, Y.-H. Liu, Q. Lu, and M. Hesse, J. Geophys. Lett. 47, e2020GL088147 (2020).
  * Li and Liu (2021) X. Li and Y.-H. Liu, Astrophys. J. 912, 152 (2021).
  * Biskamp and Schwarz (2001) D. Biskamp and E. Schwarz, Physics of Plasmas 8, 4729 (2001).
  * Biskamp (1986) D. Biskamp, Phys. Fluids 29, 1520 (1986).
  * Uzdensky and Kulsrud (2000) D. Uzdensky and R. M. Kulsrud, Phys. Plasmas 7, 4018 (2000).
  * Ugai and Tsuda (1977) M. Ugai and T. Tsuda, J. Plasma Phys. 17, 337 (1977).
  * Sato and Hayashi (1979) T. Sato and T. Hayashi, Phys. Fluids 22, 1189 (1979).
  * Kulsrud (2001) R. M. Kulsrud, Earth, Planets and Space 53, 417 (2001).
  * Jara-Almonte and Ji (2021) J. Jara-Almonte and H. Ji, Phys. Rev. Lett. 127, 055102 (2021).
  * Daughton  _et al._ (2009) W. Daughton, V. Roytershteyn, B. J. Albright, H. Karimabadi, L. Yin, and K. J. Bowers, Phys. Rev. Lett. 103, 065004 (2009).
  * Cassak  _et al._ (2005) P. A. Cassak, M. A. Shay, and J. F. Drake, Phys. Rev. Lett. 95, 235002 (2005).
  * Bhattacharjee (2004) A. Bhattacharjee, Annual Rev. Astron. Astrophys. 42, 365 (2004).
  * Vasyliunas (1975) V. M. Vasyliunas, Rev. Geophys. Space Phys. 13, 303 (1975).
  * Swisdak  _et al._ (2008) M. Swisdak, Yi-Hsin Liu, and J. F. Drake, Astrophys. J. 680, 999 (2008).
  * Mandt  _et al._ (1994) M. E. Mandt, R. E. Denton, and J. F. Drake, Geophys. Res. Lett. 21, 73 (1994).
  * Shay  _et al._ (1999) M. A. Shay, J. F. Drake, B. N. Rogers, and R. E. Denton, Geophys. Res. Lett. 26, 2163 (1999).
  * Drake  _et al._ (2008) J. F. Drake, M. A. Shay, and M. Swisdak, Phys. Plasmas 15, 042306 (2008).
  * Liu  _et al._ (2020) Y.-H. Liu, S.-C. Lin, M. Hesse, F. Guo, X. Li, H. Zhang, and S. Peery, Astrophys. J. Lett. 892, L13 (2020).
  * Sonnerup (1979) B. U. O. Sonnerup, in  _Solar System Plasma Physics, ed. L. Lanzerotti (Amsterdam: North Halland Pub.)_ , Vol. 3 (1979) p. 46.
  * Ren and Yamada et al. (2005) Y. Ren and M. Yamada et al., Phys. Rev. Lett. 95, 055003 (2005).
  * Burch  _et al._ (2016) J. L. Burch, R. B. Torbert, T. Phan, L. J. Chen, and T. E. Moore et al., Science 352, 6290 (2016).
  * Chen  _et al._ (2008) L.-J. Chen, N. Bessho, B. Lefebvre, H. Vaith, and A. Fazakerley et al., J. Geophys. Res. 113, A12213 (2008).
  * Hesse  _et al._ (2018) M. Hesse, Y. H. Liu, L. J. Chen, N. Bessho, S. Wang, L. J. Burch, T. Moretto, C. Norgen, K. J. Genestreti, T. Phan, and P. Tenfjord, Phys. Plasmas 25, 032901 (2018).
  * Burch  _et al._ (2020) J. L. Burch, J. M. Webster, M. Hesse, K. J. Genestreti, and R. E. Denton et al.,, Geophy. Res. Lett. 47, e2020GL089082 (2020).
  * Hesse  _et al._ (1999) M. Hesse, K. Schindler, J. Birn, and M. Kuznetsova, Phys. Plasmas 6, 1781 (1999).
  * Liu  _et al._ (2015) Y.-H. Liu, F. Guo, W. Daughton, H. Li, and M. Hesse, Phys. Rev. Lett. 114, 095002 (2015).
  * Lin  _et al._ (2021) S.-C. Lin, Y.-H. Liu, and X. Li, Phys. Plasmas 28, 072109 (2021).
  * Haggerty  _et al._ (2018) C. Haggerty, M. Shay, A. Chasapis, T. Phan, J. Drake, K. Malakit, P. Cassak, and R. Kieokaew, Phys. Plasmas 25 (2018), 10.1063/1.5050530.
  * Liu  _et al._ (2012) Y.-H. Liu, J. F. Drake, and M. Swisdak, Phys. Plasmas 19, 022110 (2012), 10.1063/1.3685755.
  * Shi  _et al._ (2019) C. Shi, A. Tenerani, M. Velli, and S. Lu, Astrophys. J. 883, 172 (2019).
  * Shepherd and Cassak (2010) L. S. Shepherd and P. A. Cassak, Phys. Rev. Lett. 105, 015004 (2010).
  * Comisso  _et al._ (2016) L. Comisso, M. Lingam, Y.-M. Huang, and A. Bhattacharjee, Phys. Plasmas 23, 100702 (2016).
  * Pucci and Velli (2014) F. Pucci and M. Velli, Astrophys. J. Lett. 780, L19 (2014).
  * Huang and Bhattacharjee (2010) Y. M. Huang and A. Bhattacharjee, Phys. Plasmas 17, 062104 (2010).
  * Loureiro  _et al._ (2007) N. F. Loureiro, A. A. Schekochihin, and S. C. Cowley, Phys. Plasmas 14, 100703 (2007).
  * Daughton and Karimabadi (2007) W. Daughton and H. Karimabadi, Phys. Plasmas 14, 072303 (2007), 10.1063/1.2749494.
  * Guo  _et al._ (2021) F. Guo, X. Li, W. Daughton, H. Li, P. Kilian, Y.-H. Liu, Q. Zhang, and H. Zhang, Astrophys. J. 919, 111 (2021).
  * Daughton  _et al._ (2011) W. Daughton, V. Roytershteyn, H. Karimabadi, L. Yin, B. J. Albright, B. Bergen, and K. J. Bowers, Nature Physics 7, 539 (2011).
  * Daughton  _et al._ (2014) W. Daughton, T. K. M. Nakamura, H. Karimabadi, V. Roytershteyn, and B. Loring, Phys. Plasmas 21, 052307 (2014).
  * Le  _et al._ (2018) A. Le, W. Daughton, O. Ohia, L. J. Chen, Yi-Hsin Liu, W. Nystrom, and R. Bird, arXiv:1802.10205 (2018).
  * Stanier  _et al._ (2019) A. Stanier, W. Daughton, A. Le, X. Li, and R. Bird, Phys. Plasmas 26, 072121 (2019).
  * Price  _et al._ (2016) L. Price, M. Swisdak, J. F. Drake, P. A. Cassak, J. T. Dahlin, and R. E. Ergun, Geophys. Res. Lett. 43, 6020 (2016).
  * Torbert  _et al._ (2018) R. B. Torbert, J. L. Burch, T. D. Phan, M. Hesse, M. R. Argall, J. Shuster, R. E. Ergun, L. Alm, R. Nakamura, and K. J. Genestreti et. al., Science 362, 1391 (2018).
  * Bowers  _et al._ (2008) K. J. Bowers, B. J. Albright, L. Yin, B. Bergen, and T. J. T. Kwan, Phys. Plasmas 15, 055703 (2008).
  * Harris (1962) E. G. Harris, Nuovo Cimento 23, 115 (1962).
  * Bessho and Bhattacharjee (2010) N. Bessho and A. Bhattacharjee, Plasma and Fusion Research 5, S2017 (2010).
  * Wu  _et al._ (2011) P. Wu, M. A. Shay, T. D. Phan, M. Oieroset, and M. Oka, Phys. Plasmas 18, 111204 (2011).

##  VIII Acknowledgments

Y.L. is grateful for supports from NSF grant PHY- 1902867 through the NSF/DOE
Partnership in Basic Plasma Science and Engineering, NASA’s MMS mission
80NSSC18K0289, 80NSSC21K2048, and NSF Career Award 2142430. P.A.C. gratefully
acknowledges support from NSF Grants AGS-1602769 and PHY-1804428, NASA Grants
NNX16AG76G and 80NSSC19M0146 and DOE Grant DE-SC0020294. X.L.’s contribution
is in part supported by NASA under grants 80NSSC18K0289 and 80NSSC21K1313.
Simulations were performed with NERSC Advanced Supercomputing and LANL
institutional computing.

##  IX Author contributions

Y. L. performed and analyzed the simulations, derived the theory and wrote the
paper; P. C. helped develop the theory and improved the writing; X. L. helped
analyze the simulation data and develop the theory; M. H. helped develop the
theory and interpreted the simulation data. S. L. provided inputs based on
resistive MHD results. K. G. provided observational inputs and some insights.
All authors discussed and commented on the paper.

##  X Competing interests

The authors declare no competing interests.