# The Acceleration of Charged Particles and Formation of Power-law Energy
Spectra in Nonrelativistic Magnetic Reconnection

Xiaocan Li<EMAIL_ADDRESS>Department of Physics and Astronomy,
Dartmouth College, Hanover, NH 03755, USA  Fan Guo  Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  Yi-Hsin Liu  Department of Physics and
Astronomy, Dartmouth College, Hanover, NH 03755, USA

###### Abstract

Magnetic reconnection is a primary driver of particle acceleration processes
in space and astrophysical plasmas. Understanding how particles are
accelerated and the resulting particle energy spectra is among the central
topics in reconnection studies. We review recent advances in addressing this
problem in nonrelativistic reconnection that is relevant to space and solar
plasmas and beyond. We focus on particle acceleration mechanisms, particle
transport due to 3D reconnection physics, and their roles in forming power-law
particle energy spectra. We conclude by pointing out the challenges in
studying particle acceleration and transport in a large-scale reconnection
layer and the relevant issues to be addressed in the future.

††preprint: AIP/123-QED

##  I Introduction

In space, solar, and astrophysical plasmas, magnetic reconnection is one of
the primary mechanisms for converting magnetic energy into plasma kinetic
energy and accelerating high-energy nonthermal particles [1]. One remarkable
example is solar flares, where magnetic reconnection is thought to trigger the
release of a large amount of magnetic energy in the solar corona [2, 3, 4, 5]
and drive the acceleration of a large number of nonthermal electrons and ions.
These particles produce a broad range of nonthermal emissions (hard X-ray,
microwave, and gamma-ray) [6, 7, 8, 9, 10, 11]. Some of the accelerated
particles get released into interplanetary space as impulsive solar energetic
particles [12, 13]. Understanding how particles are accelerated during
reconnection and the resulting particle energy distributions have been central
problems in the study of magnetic reconnection.

One ubiquitous feature revealed by observations is that the accelerated
particles tend to develop power-law energy spectra [14, 15, 16]. Solar flare
emissions often show power-law energy distributions and are presumably
produced by nonthermal particles with power-law distributions [17, 18, 19, 20,
16, 10]. Although other mechanisms may also accelerate particles, reconnection
holds the promise to accelerate a large number of electrons [21], as suggested
by some observations [22, 18]. In addition, reconnection-accelerated particles
may serve as an injection population to be further accelerated by other
mechanisms, such as termination shocks [23, 7, 24], plasma turbulence [25, 26,
14, 27, 28] driven by reconnection outflow and large-scale Alfvén waves in the
flare loops [29]. Thus, it is crucial to understand particle acceleration by
solar flare reconnection. In-situ observations in space plasmas give more
direct evidence of power-law formation in the reconnection regions. In Earth’s
magnetotail, electrons develop power-law energy spectra near the reconnection
diffusion region [30, 31], inside magnetic islands [32, 33, 34], and in the
flux pile-up region [35, 36]. In the solar wind, local enhancements of
energetic particles and the formation of power-law energy spectra have been
associated with contracting and merging small-scale flux ropes. [37, 38, 39,
40, 41, 42]. Motivated by these observations, many studies have been dedicated
to investigating particle acceleration associated with magnetic reconnection.
While the analytical studies have made predictions consistent with
observations [43, 44, 40, 41, 42], the whole process can be very nonlinear and
numerical simulations still have to provide convincing evidence for power-law
formation during reconnection. Test-particle simulations (either in full orbit
or guiding-center motions) usually generate hard power-law energy spectra,
much harder than that observed in space and solar plasmas [45, 46, 47, 48, 49,
50]. Comparing to the kinetic simulations of relativistic reconnection (the
Alfvén velocity approaches the speed of light) that consistently produce hard
power-law energy spectra [51, 52, 53, 54, 55], simulations in the
nonrelativistic regime relevant to space and solar plasmas show persistent
difficulties in obtaining power-law spectra [56, 57, 58, 59, 60].

In this review, we focus on critical issues for understanding energetic
particle acceleration and power-law formation in reconnection. First, what are
the dominant particle acceleration mechanisms during magnetic reconnection?
Traditionally, it is thought that particles are accelerated only by the non-
ideal electric field near the reconnection X-line [61], which cannot explain
the large number of electrons generated during solar flares. Recent studies
have made significant progress showing that Fermi-type mechanisms are
important for particle acceleration during reconnection [21, 52, 58, 59].
While the acceleration mechanisms are similar in different regimes, power-law
energy spectra are only commonly seen in the relativistic simulations,
suggesting that the plasma parameters (e.g., Alfvén velocity, plasma $\beta$,
or guide field–the magnetic field component perpendicular to the reconnection
plane) might play a significant role in the power-law formation. The broad
distributions of these parameters in space and solar plasmas could explain why
the power-law indices range broadly from 2 to 9 [6, 18, 19, 20, 16, 62] (In
the rest of this paper, a power-law energy spectrum is defined as
$f(\varepsilon)=dN(\varepsilon)/d\varepsilon\sim\varepsilon^{-p}$, where
$\varepsilon$ is particle kinetic energy, $dN(\varepsilon)$ is the number of
particles in a spherical energy shell with thickness $d\varepsilon$ around
$\varepsilon$, and $p$ is the spectral index). Second, how do particles
propagate in the reconnection region? To gain more energy, particles have to
be trapped in the acceleration region for a long time or access the
acceleration region multiple times. Both depend on the waves and turbulence in
the reconnection region and will become even more important in the 3D
reconnection layer that can self-generate plasma turbulence [63, 64, 65, 66,
67, 68, 69, 70, 71, 72, 73]. This review summarizes some recent progress on
these two issues and points out several problems to be addressed in the next
step. In Section II, we summarize particle acceleration mechanisms during
reconnection and three closely linked views on the Fermi mechanism. In Section
III, we summarize how 3D physics facilitate particle transport and
acceleration. In Section IV, we present a model for explaining the power-law
formation in 3D reconnection. In Section V, we point out the challenges in
studying particle acceleration in a large-scale reconnection layer and the
relevant issues to be addressed in the future.

##  II Particle Acceleration Mechanisms

###  II.1 Particle Acceleration Sites

The reconnection region is a natural particle acceleration site. Continuous
magnetic flux comes into the reconnection region and deposits its free energy.
At the same time, fresh, low energy particles are brought in through the
reconnection inflow, get accelerated in the reconnection region, and escape as
the outflow expels them out of the reconnection region. It has been
demonstrated that the reconnection proceeds at a speed fast enough [74, 75] to
sustain the efficient magnetic energy conversion and particle acceleration
observed in Earth’s magnetotail and solar flares. During reconnection, the
non-ideal electric field near the reconnection X-point can accelerate a small
fraction of particles (mostly electrons) [76, 77, 78, 79, 80, 81]. More
importantly, reconnection-driven Alfvénic outflow on a much larger scale will
induce motional electric fields that are more broadly distributed than the
non-ideal electric field and can accelerate more particles [54]. Recent
studies have demonstrated that an elongated current sheet will break into a
chain of magnetic islands (plasmoids) due to tearing instability [82, 83, 84,
85, 86] or collapse of the reconnection exhaust [87]. These magnetic islands
tend to contract due to magnetic tension force, and they merge to form larger
islands. Meanwhile, secondary current sheets are expected to form between the
plasmoids, and new plasmoids can further develop in the new current sheets.
This can repeatedly happen that gives rise to a hierarchical, fractal-like
plasmoid structure [82, 88, 89, 90]. This fractal structure enables particles
to gain energy at multiple sites in a reconnection region, such as the
contracting magnetic islands [21, 43, 44] and island merging regions [91, 92,
57, 93]. These acceleration regions have been identified in kinetic
simulations by tracking particle trajectories [21, 91, 52, 60], which include
detailed information (e.g., position, velocity, and electric and magnetic
fields) to characterize the acceleration history. Fig. 1(a) shows the
trajectory of an electron being energized in three contracting magnetic
islands in a 2D particle-in-cell (PIC) simulation [60]. By carefully analyzing
many such trajectories, we could differentiate the roles of these different
acceleration regions. The trajectories have revealed that the parallel
electric field near the reconnection X-line is not enough to accelerate
particles to high energies but may provide an injection mechanism for further
acceleration [94, 54, 95]. High-energy particles are mostly accelerated in
contracting or merging islands, and the island-merging regions seem to be the
most efficient in accelerating electrons to the highest energy [91, 93, 96].
Because of the multiple phases of accelerations, particles can be accelerated
from thermal energy to nonthermal energies by reconnection without requiring
external injection mechanisms like a X-point acceleration [54, 95]. These
findings show that particle trajectories can help illustrate the acceleration
processes. However, since one can only carefully analyze a small number of
them, the statistical information on particle acceleration may be buried in
the billions of other particles.

Figure 1:  Left: an electron trajectory showing the acceleration processes are
associated with particle guiding-centering drift motions. (a) The trajectory
in the simulation $x$–$z$ plane. The background is the out-of-plane electric
field $E_{y}$ at three time slices. The green crosses indicate the positions
of this electron at the three time slices. (b) The electron kinetic energy
$\gamma-1$ (in the unit of $m_{e}c^{2}$) evolving with its $x$-position. (c)
The electron’s $x$-position versus its $y$-position. Reproduced with
permission from Li et al., Astrophys. J, 843.1 (2017): 21. Copyright 2017
Institute of Physics (IOP). Right: Fermi mechanisms in reconnection. Blue
curves indicate the magnetic field lines. Green curves indicate particle
trajectories. $v$ is the particle velocity when entering the acceleration
region. $v_{A}$ is the Alfvénic outflow velocity. (d) A single bounce off the
reconnection outflow. Multiple bounces in (e) a contracting magnetic island,
(f) island merging regions, and (g) a single reconnection exhaust.

###  II.2 Fermi mechanism in reconnection

Although the reconnection region is complex with multiple acceleration regions
and involves kinetic physics, the underlying particle acceleration mechanism
turns out to be quite general. The Fermi mechanism has been proposed and
demonstrated to be the dominant process for high-energy particle acceleration
during reconnection [21, 58, 59]. Particles could gain energy through the
Fermi mechanism when bouncing off the Alfvénic reconnection outflow driven by
contracting magnetic field lines (Fig.1(d)). The net energy gain due to one
bounce is $\Delta\varepsilon=2m(vv_{A}+v_{A}^{2})$, where $v$ is the initial
particle speed, and $v_{A}$ is the Alfvén speed. Assuming the interaction
takes about $\Delta t=2L/v$ ($L$ is the characteristic length scale, e.g., the
typical island diameter), the acceleration rate
$\alpha\equiv\Delta\varepsilon/\varepsilon\Delta t\approx 2v_{A}/L$ if $v\gg
v_{A}$, which is independent of particle energy, as predicted by the Fermi
mechanism. In the non-relativistic regime, one bounce is usually not enough to
accelerate particles to high energies, especially for electrons (which usually
have $v\gg v_{A}$). To gain more energy, particles need to bounce multiple
times off the flows. The multi-scale structures in a reconnection layer enable
this scenario throughout the reconnection layer. Particles trapped in magnetic
islands will continuously gain energy through bouncing off the flow driven by
the contracting magnetic field lines [21] (Fig.1(e)) and merging magnetic
islands [91, 57] (Fig.1(f)). Recent findings have also shown that particles
could interact with a single reconnection outflow multiple times to gain
energies when trapped near the reconnection X-line by a parallel electric
potential or the magnetic bottle [80] (Fig.1(g)). Alternatively, particles may
experience pitch-angle scattering by self-generated turbulence or fluctuations
in reconnection and bounce multiple times within a single reconnection exhaust
[66, 70].

Figure 2:  Time evolution of different fluid energization terms over the
entire simulation box. Left: the energization associated with guiding-center
drift motions for (a) electrons and (b) ions in a simulation with $B_{g}=0$.
Reproduced with permission from Li et al., Astrophys. J, 843.1 (2017): 21.
Copyright 2017 Institute of Physics (IOP). Besides the listed three terms, the
sum also includes the energization associated with polarization drift. $K_{e}$
and $K_{i}$ are particle kinetic energies. Right: the energization associated
with flow compression and shear. Reproduced with permission from Li et al.,
Astrophys. J, 855.2 (2018): 80. Copyright 2018 Institute of Physics (IOP).
$\bm{j}_{e-\text{agy}}\cdot\bm{v}_{E}$ is the energization associated with
agyrotropic pressure tensor. (c) A simulation without a guide field. (d) A
simulation with $B_{g}/B_{0}=0.5$.

When particles are well magnetized, the acceleration processes during
reconnection can be well quantified by three closely linked models [43, 44].
The first one is based on the conservation of the first and second adiabatic
invariants, i.e., magnetic moment $\mu=mv_{\perp}^{2}/2B$ and parallel action
$m\int v_{\parallel}dl$, where $v_{\perp}$ and $v_{\parallel}$ are velocities
perpendicular and parallel to the local magnetic field, respectively. When
magnetic islands contract or merge, the magnetic field lines tend to shorten
[21, 43], increasing $v_{\parallel}$. Depending on whether the island area is
conserved when islands contract, the magnetic field could get stronger or
weaker. If the island area is conserved (circularization or incompressible
case), the magnetic field and $v_{\perp}$ will decrease [21, 57, 97, 98]. If
the island area shrinks (compressible case), the increasing magnetic field
will lead to an increase of $v_{\perp}$. In contrast, the magnetic field and
$v_{\perp}$ tend to decrease when two islands merge [57, 43]. Analytical
models based on the conservation of these two adiabatic invariants have shown
the formation of power-law energy spectra in a “sea” of magnetic islands [99,
57, 43, 100].

The second model is based on particle guiding-center motions. Particle
trajectories have shown that particles tend to drift in the out-of-plane
direction (e.g., Fig. 1 (c)) when they gain energy (e.g., Fig. 1 (b)),
suggesting that particle acceleration is associated with particle guiding-
center drift motions, which include the parallel guiding-center velocity,
$\bm{E}\times\bm{B}$ drift $\bm{v}_{E}$, grad-$B$ drift, inertial drift,
parallel drift, and polarization drift [101, 102, 103, 44]

$\displaystyle\left<\bm{v}_{g}\right>_{\phi}=v_{\parallel}\bm{b}+\bm{v}_{E}+\frac{M}{q_{s}}\frac{\bm{B}\times\nabla
B}{B^{2}}+\frac{m_{s}v_{\parallel}}{q_{s}B}\bm{b}\times\frac{d\bm{b}}{dt}+\frac{M}{q_{s}}(\nabla\times\bm{b})_{\parallel}+\frac{m_{s}}{q_{s}B}\bm{b}\times\frac{d\bm{v}_{E}}{dt},$
(1)

where $\left<\dots\right>_{\phi}$ indicates gyrophase-average,
$d/dt\equiv\partial/\partial t+(v_{\parallel}\bm{b}+\bm{v}_{E})\cdot\nabla$,
and $M\equiv m_{s}(\bm{v}_{\perp}-\bm{v_{E}})^{2}/2B$ is particle magnetic
moment in the plasma frame where $\bm{v}_{E}=0$. The inertial drift (the
fourth term on the right) includes the curvature drift
$(m_{s}v_{\parallel}^{2}/q_{s}B)\bm{b}\times\bm{\kappa}$, where
$\bm{\kappa}=\bm{b}\cdot\nabla\bm{b}$ is the magnetic curvature. In the
guiding-center approximation, a single particle gains energy at a rate

$\displaystyle\left<\dot{\varepsilon}\right>_{\phi}=q_{s}\bm{E}\cdot\left<\bm{v}_{g}\right>_{\phi}+M\frac{\partial
B}{\partial t},$ (2)

where the second term on the right is due to the conservation of the magnetic
moment (betatron acceleration) and tends to be negative as the magnetic energy
is released ($\partial B/\partial t<0$) in reconnection. Among the
acceleration processes, the term associated with particle curvature drift is
proportional to the particle parallel kinetic energy and is the Fermi
mechanism in reconnection [21]. Since the curvature drift is proportional to
$m_{s}v_{\parallel}^{2}$ ($\approx m_{s}v^{2}/3$ for an isotropic particle
distribution), the resulting acceleration rate
$\left<\dot{\varepsilon}\right>_{\phi}/\varepsilon\sim\bm{v}_{E}\cdot\bm{\kappa}$
is independent of particle energy. To compare the energization with kinetic
simulations, we often integrate $\left<\dot{\varepsilon}\right>_{\phi}$ over
electrons or ions to get the fluid energization terms [104, 58, 59, 60, 81],
obtaining
$\bm{j}_{s}\cdot\bm{E}=\bm{j}_{s\parallel}\cdot\bm{E}_{\parallel}+\bm{j}_{s\perp}\cdot\bm{E}_{\perp}$,
where $\bm{j}_{s}$ is total current density, of which the perpendicular
component is (see Appendix B in Li, Guo, and Li [105] for derivation)

$\displaystyle\bm{j}_{s\perp}=p_{s\parallel}\frac{\bm{B}\times(\bm{B}\cdot\nabla)\bm{B}}{B^{4}}+p_{s\perp}\frac{\bm{B}\times\nabla
B}{B^{3}}-\left[\nabla\times\frac{p_{s\perp}\bm{B}}{B^{2}}\right]_{\perp}+\rho_{s}\bm{v}_{E}-n_{s}m_{s}\frac{d\bm{v}_{s}}{dt}\times\frac{\bm{B}}{B^{2}},$
(3)

where $p_{s\parallel}$ and $p_{s\perp}$ are parallel and perpendicular
pressures to the local magnetic field, respectively, $\rho_{s}$ is the charge
density, $n_{s}$ is particle number density, $m_{s}$ is particle mass,
$\bm{v}_{s}$ is the species flow velocity, and
$d/dt\equiv\partial_{t}+\bm{v}_{s}\cdot\nabla$. The terms on the right show
that the fluid energization is then associated with curvature drift, gradient
drift, magnetization, and flow inertia. $\bm{v}_{E}$ does not contribute to
the energization because $\bm{v}_{E}\cdot\bm{E}=0$. Fig. 2 left shows that the
energization associated with curvature drift is dominant in the weak guide
field limit. As the guide field $B_{g}$ increases,
$\bm{j}_{s\perp}\cdot\bm{E}_{\perp}$ will be suppressed as the curvature of
the magnetic field $|\bm{\kappa}|$ becomes smaller. Instead, the energization
due to the parallel electric field
$\bm{j}_{s\parallel}\cdot\bm{E}_{\parallel}$ will dominate when $B_{g}>B_{0}$
[58, 60], where $B_{0}$ is the strength of the reconnection magnetic field.

The third model attempts to describe the energization in terms of flow
compression and shear [44, 97, 106, 96]. The former is the leading
acceleration mechanism in energetic particle transport theory and diffusive
shock acceleration [107, 108, 109]. While the reconnection layer is assumed
incompressible traditionally [110, 111], recent MHD simulations have shown
that compression effects are important in reconnection in low-$\beta$ and
weak-$B_{g}$ regime [112, 113]. Kinetic models based on adiabatic invariants
for reconnection acceleration have pointed out that compression could also be
important for particle acceleration during reconnection [43, 44, 114]. Drury
[115] presented a reconnection acceleration model showing that the compression
ratio between the outflow and inflow plasmas determines the power-law spectral
index of reconnection-accelerated particles. These findings appear to be in
contradiction with some previous theories that assume the reconnection layer
is incompressible and compression acceleration is not in action [56, 57]. It
turns out that the energization associated with guiding-center drift motions
can be reorganized to

$\displaystyle\bm{j}_{s\perp}\cdot\bm{E}_{\perp}$
$\displaystyle=\nabla\cdot(p_{s\perp}\bm{v}_{E})-p_{s}\nabla\cdot\bm{v}_{E}-(p_{s\parallel}-p_{s\perp})b_{i}b_{j}\sigma_{ij}+n_{s}m_{s}\frac{d\bm{v}_{s}}{dt}\cdot{\bm{v}_{E}},$
(4)

where
$\sigma_{ij}=0.5(\partial_{i}v_{Ej}+\partial_{j}v_{Ei}-(2\nabla\cdot\bm{v}_{E}\delta_{ij})/3)$
is the shear tensor of $\bm{v}_{E}$ flow, and
$p_{s}\equiv(p_{s\parallel}+2p_{s\perp})/3$ is the effective scalar pressure.
The second term on the right is due to flow compression and $\propto$ the
scalar pressure. The third term on the right is due to flow shear and
$\propto$ the pressure anisotropy. When the particle distribution is nearly
isotropic, the flow shear is ineffective in energizing plasma but can provide
a second-order acceleration term for high-energy particles [116], such as
cosmic rays. Using kinetic simulations, Li  _et al._ [106] showed that the
flow compression and shear effects capture the primary plasma energization
(Fig. 2 right) and particle acceleration processes in reconnection.
Compression energization dominates reconnection energization but becomes
comparable with shear energization when there is a moderate guide field
($B_{g}/B_{0}\sim 0.5$). As $B_{g}$ increases, the plasma becomes less
compressible because the guide field has to be also compressed when the plasma
is being compressed. As a result, the compression energization becomes less
efficient as $B_{g}$ increases. On the other hand, the pressure anisotropy
tends to increases with the guide field [117, 106]. As a result, the shear
energization (proportional to the pressure anisotropy) decreases slower with
the guide field than compression energization. Overall, the joint energization
due to flow compression and shear decreases with the guide field, providing an
alternative explanation why the plasma energization is less efficient as
$B_{g}$ increases. These results suggest that one may study particle
acceleration in a large-scale solar flare reconnection site using the
transport theory.

##  III Particle spatial transport

###  III.1 Artificial Confinement in 2D Simulations

Most kinetic simulations of reconnection were carried out in a 2D geometry due
to computational constraints. In 2D reconnection, high-energy particles are
mostly confined in magnetic islands due to the restricted particle motion
across the magnetic field lines [118, 119, 120]. Since particles are usually
well magnetized and coupled with the field lines except in the diffusion
regions, onion-like particle distributions form in the islands [60]. Particles
entering the reconnection region earlier stay in an island’s central region,
while those entering later will be in the outer region (Fig.3 (a)). Particles
are adiabatically compressed and develop thermal-like energy distributions in
each layer (Fig. 3 (b)). Those in the central region experience compression
longer and thus have a higher temperature than that in the outer layers.
Although some particles can still be accelerated to hundreds of times initial
thermal energy, the particle distribution in the reconnection region is
actually the superposition of these thermal-like distributions [60].
Furthermore, due to the confinement, high-energy particles cannot access the
regions where the acceleration is most active (Fig.3 (c) & (d)). The newly
reconnected field lines in those regions are strongly bent and drive strong
outflow ($\sim\bm{v}_{E}$) well aligned with the magnetic curvature
$\bm{\kappa}$. Lower-energy particles accessing these regions will then have a
higher acceleration rate associated with curvature drift
$\bm{v}_{E}\cdot\bm{\kappa}$ than high-energy particles confined in the
central region of magnetic islands. One may expect that merging islands will
break the confinement and drive stronger particle acceleration and pitch-angle
scattering through forming multiple X-lines. However, to access particles in a
large island’s central region, this island has to merge with one containing
more magnetic flux, which will become less likely as fewer islands grow to
large sizes.

Figure 3:  Artificial particle trapping in 2D simulations. (a) The contour of
the out-of-plane component of the vector potential $A_{y}$ in a run with
$\beta_{e}=0.02$. The region between the black lines is the reconnection
layer. (b) Electron energy spectra in these layers. The black dashed line
shows the spectrum inside the reconnection layer bounded by the black lines in
panel (a). The dashed line is the fitted Maxwellian for the spectrum in the
innermost region. (a) & (b): Reproduced with permission from Li et al.,
Astrophys. J, 843.1 (2017): 21. Copyright 2017 Institute of Physics (IOP). (c)
The spatial distribution of high-energy electrons in the 2D simulation. (d)
The acceleration rate associated with curvature drift
$\bm{v}_{\bm{E}}\cdot\bm{\kappa}$ in a 2D simulation. Note that we multiply
the rate by 100 just to make the color bar range from -1 to 1. (e) A slice of
high-energy electron distribution in a 3D simulation. The arrow points out
some electrons in the large flux rope that do not actively participate in the
acceleration. (f) The corresponding acceleration rate associated with
curvature drift in the 3D simulation. The black contour indicates the boundary
of the major acceleration regions. (c)–(f): Reproduced with permission from Li
et al., Astrophys. J, 884.2 (2019): 118. Copyright 2019 Institute of Physics
(IOP).

###  III.2 Fast Spatial Transport in 3D Reconnection

Including 3D physics will help resolve these issues because magnetic flux
ropes (the counterpart of magnetic islands in 3D) could be broken apart or
even destroyed by plasma instabilities [63, 64, 92, 71, 121], such as oblique
tearing and kink instabilities. As most of these instabilities require the
third dimension, they are artificially suppressed in 2D simulations. The
nonlinear growth and interaction of these instabilities will lead to a
fragmented current layer filled with secondary flux ropes and current sheets
(Fig. 4(a)). Magnetic field lines passing these regions will exponentially
separate from each other and thus become chaotic [122, 67, 68, 69, 70, 123,
124]. For example, the neighboring field lines in Fig. 4(a) quickly diverge
from each other and access a broad region within just one crossing of the
reconnection layer. The chaotic field lines enable different regions (e.g.,
flux ropes and reconnection exhausts) to be magnetically connected, allowing
particles to access broader regions. The mixing of particles with different
energies helps to develop truly nonthermal distributions locally (see next
section for a discussion). Additionally, a 3D reconnection layer produces
fluctuations that are likely to undergo both forward and inverse cascade [63,
123], resulting in broadband turbulent fluctuations [122, 71, 73, 123]. Fig.
4(b) shows that the magnetic power spectrum resembles a power-law distribution
with a spectral index of about -2.7, consistent with that for reconnection-
mediated turbulence in the collisionless regime [125]. Other 3D simulations
(either MHD or kinetic) have observed similar magnetic power spectra [122, 53,
71] steeper than the classical Kolmogorov spectrum with a spectral index -5/3.
If the energetic particles resonate with waves with wavelengths near the
kinetic scales, such a steep spectrum will result in particle diffusion
coefficients different from the quasi-linear theory assuming a Kolmogorov-like
magnetic power spectrum [126, 127].

High-energy particles in a 3D reconnection layer will resonate with the
turbulent fluctuations and experience pitch-angle scattering [126], preventing
them from being confined in a flux rope for a long time. The snapshots of a
test-particle simulation shown in Fig. 5 (a) & (b) show how particles are
quickly transported along the chaotic magnetic field lines in a turbulent
reconnection layer. Instead of being trapped in the large flux rope, the
electrons initially close to each other quickly fill the whole simulation
domain after about two Alfvén crossing times.

The fast particle transport in 3D enables high-energy particles to access the
regions where the magnetic energy is actively converted into plasma kinetic
energy. Fig. 3(e) shows the distribution of high-energy electrons in the 3D
simulation shown in Fig. 4, and Fig. 3(f) shows the corresponding acceleration
rate associated with curvature drift acceleration. In contrast to the 2D
results (Fig. 3(c) & (d)), high-energy electrons can access regions and have a
large acceleration rate associated with curvature drift
$\sim\bm{v}_{E}\cdot{\kappa}$. Thus, it is reasonable to expect that high-
energy particle acceleration will become more efficient in 3D than in 2D. Fig.
5(c) & (d) show one electron trajectory traced in the 3D kinetic simulation.
The electron gets three Fermi bounces (1–3) early in the simulation. It is
then trapped in the large flux rope, and its acceleration gets slower and
slower. In a 2D simulation, the electron would be confined in the large
magnetic island until the island merges with other islands. However, in the 3D
simulations, the electron can easily escape due to the chaotic magnetic field
lines and self-generated turbulence. The trajectory shows it gets out of the
flux ropes and experiences two more Fermi bounces in other regions (indicated
by 4 & 5 in Fig. 5(c) & (d)). These results show that the fast spatial
transport due to chaotic field lines and self-generated turbulence helps
particle acceleration in 3D simulations.

Figure 4:  Turbulence and chaotic magnetic field lines in the 3D simulation.
(a) 3D reconnection layer showing an isosurface of the current density with
$|\bm{J}|/J_{0}=0.3$, and magnetic field lines starting from uniformly
distributed points along a line of length $2d_{i}$. The field lines are color-
coded with their seed identification numbers (IDs). The local electron energy
spectra in the four whites cubes of $(2.3d_{i})^{3}$ will be shown in Fig. 6
(b). (b) Magnetic power spectra at five time slices indicated by the crosses
in the embedded plot. The black dashed line indicates a power-law $\propto
k_{\perp}^{-2.7}$. The inset also shows the time evolution of the magnetic
energy $\varepsilon_{B}$ for both 2D and 3D simulations. $\varepsilon_{B0}$ is
the initial magnetic energy. Reproduced with permission from Li et al.,
Astrophys. J, 884.2 (2019): 118. Copyright 2019 Institute of Physics (IOP).
Figure 5:  Fast particle transport leads to stronger particle acceleration in
a 3D reconnection layer. Panels (a) & (b) are results from a test-particle
simulation with the fields from the 3D PIC simulation as background. The
particles are uniformly distributed in a ball with a radius of $d_{e}$ in the
center of the flux rope (panel (a)). The magnetic field lines in green all
pass through the ball. The red balls in (b) indicate their final positions.
Panels (c) & (d) show on electron tracer trajectory in the 3D PIC simulation.
(c) The trajectory’s projection on the $x$–$z$ plane. It is color-coded by its
kinetic energy. The numbers 1–5 indicate five phases of acceleration. The
arrow points out when the electron escapes from being trapped in the large
flux rope. (d) $x$-position versus the electron kinetic energy. Reproduced
with permission from Li et al., Astrophys. J, 884.2 (2019): 118. Copyright
2019 Institute of Physics (IOP).

##  IV The formation of power-law spectra

###  IV.1 Formation of Power-law Spectra in 3D

While plasma dynamics and reconnection rate have been the focus of most 3D
simulations of magnetic reconnection [64, 72, 71], a few recent large-scale 3D
kinetic simulations have been dedicated to studying particle acceleration in
3D reconnection [53, 124, 66, 67, 70]. In contrast to the relativistic
simulations showing similar particle energy spectra in 2D and 3D reconnection
[51, 52, 53, 128], nonrelativistic simulations have shown stronger particle
acceleration in 3D than in 2D because of the fast transport [66, 67, 70]. Fig.
6(a) shows the time evolution of the electron energy spectrum in a 3D kinetic
simulation [70]. The spectrum has a power-law tail $\propto\varepsilon^{-4}$,
and it persists throughout the nonlinear reconnection phase until saturation.
This result provides the first convincing evidence of forming the power-law
energy spectrum in 3D nonrelativistic reconnection. Comparing with 2D
reconnection, 3D reconnection accelerates electrons to higher energies. The
high-energy tail keeps growing in the 3D simulation. In contrast, the maximum
energy in the 2D simulation does not change much, and the flux keeps piling up
at lower energies, causing the spectrum to become softer as the simulation
evolves. This explains why the spectrum becomes very steep to the end of the
2D simulation shown in Fig. 3(a) & (b). The fast particle transport not only
helps particle acceleration but also enhances particle mixing. Fig. 6(b) shows
the local electron spectra at 4 different locations have similar power-law
tails instead of different thermal-like distributions in 2D (Fig. 3(b)).

Figure 6:  (a) Time evolution of the global electron energy spectrum in the 3D
simulation with the inset comparing with the corresponding 2D simulation at
three time slices. The black solid line indicates a power-law distribution
with a spectral index $p=4$. We normalize $\varepsilon$ by the initial thermal
energy $\varepsilon_{\text{th}}\approx 0.015m_{e}c^{2}$. (b) Electron energy
spectra for electrons in the four local boxes shown in Fig.4 (a) at
$t\Omega_{ci}=150$. Reproduced with permission from Li et al., Astrophys. J,
884.2 (2019): 118. Copyright 2019 Institute of Physics (IOP).

###  IV.2 Power-law Formation Model

Before looking into the simulation data, let us briefly review the formation
of power-law spectra due to the Fermi mechanisms (see Guo  _et al._ [129] for
a more detailed review). In the standard Fokker-Planck approach, the particle
distribution evolves according to

$\partial_{t}f+\partial_{\varepsilon}(\alpha\varepsilon
f)=\partial_{\varepsilon}^{2}(D_{\varepsilon\varepsilon}f)-\frac{f}{\tau_{\text{esc}}}+\frac{f_{\text{inj}}}{\tau_{\text{inj}}},$
(5)

where $D_{\varepsilon\varepsilon}$ is the energy diffusion coefficient,
$\tau_{\text{esc}}$ is the escape time scale, $f_{\text{inj}}$ is the injected
particle distribution, and $\tau_{\text{inj}}$ is the particle injection time
scale. The acceleration rate
$\alpha(\varepsilon)\equiv(d\varepsilon/dt)\varepsilon^{-1}=(\partial_{t}\varepsilon+\partial_{\varepsilon}D_{\varepsilon\varepsilon})\varepsilon^{-1}$
could describe first-order Fermi processes, direct acceleration in the
reconnection diffusion region, and a first-order drift in energy
$\partial_{\varepsilon}D_{\varepsilon\varepsilon}$ associated with second-
order Fermi mechanisms. Since the first-order Fermi mechanism is thought to
dominate particle acceleration processes in magnetic reconnection [21], we can
ignore the energy diffusion $D_{\varepsilon\varepsilon}$ and the associated
energy drift term ($\sim\partial_{\varepsilon}D_{\varepsilon\varepsilon}$) in
$\alpha$. For particles with much higher energies than the injected particles,
their energy distribution resembles a power-law with a spectral index [130]

$p=1+\frac{1}{\alpha\tau_{\text{esc}}}+\frac{\partial\ln\alpha}{\partial\ln\varepsilon},$
(6)

which simplifies to $1+(\alpha\tau_{\text{esc}})^{-1}$ when both rates are
independent of particle energy [52]. The spectral index is for particles in
the acceleration region. For the energetic particles escaped from the
acceleration region, the flux $f/\tau_{\text{esc}}$ has a power-law index

$p^{\prime}=1+\frac{1}{\alpha\tau_{\text{esc}}}+\frac{\partial\ln\alpha}{\partial\ln\varepsilon}-\frac{\partial\ln\alpha_{\text{esc}}}{\partial\ln\varepsilon},$
(7)

where $\alpha_{\text{esc}}=\tau_{\text{esc}}^{-1}$ is the escape rate. Both
$\alpha$ and $\alpha_{\text{esc}}$ (and thus $\tau_{\text{esc}}$) can be
energy-dependent in general, but their energy-dependence should be the same
for $p$ and $p^{\prime}$ to be energy-independent. In reconnection, it is
often assumed that $\tau_{\text{esc}}\sim L/v_{A}$ due to the advection loss
by the reconnection outflow [57, 52, 131], where $L$ is the characteristic
length scale (e.g., the typical island diameter). Since $\tau_{\text{esc}}$
does not depend on particle energy, $\alpha$ needs to be independent of
particle energy as well. Therefore, to explain the formation of power-law
spectra in nonrelativistic reconnection, one needs to demonstrate that the
acceleration rate is indeed constant in the kinetic simulations. The model for
explaining power-law formation in relativistic reconnection has shown that
$\alpha\tau_{\text{inj}}$ should be $>1$ to obtain an extended power-law
spectrum [52, 53, 129], where $\tau_{\text{inj}}$ is particle injection time
scale. This result indicates that simulations sustaining large $\alpha$ and
long $\tau_{\text{inj}}$ will help power-law formation. Since the acceleration
rate associated with the Fermi mechanism is $\sim\bm{v}_{E}\cdot\bm{\kappa}$,
a larger Alfvén speed or magnetic curvature (or compression, see Section II.2)
will lead to stronger particle acceleration and the potential formation of
power-law energy spectra in kinetic simulations. When the thermal speed is
fixed, a larger Alfvén speed gives a lower plasma $\beta$. As the magnetic
curvature decreases with the guide field, a larger magnetic curvature means a
lower guide field. Thus, boosting $\alpha$ by performing simulations in
low-$\beta$ and weak guide-field regimes will help obtain power-law spectra in
a finite simulation time [70].

###  IV.3 Acceleration and Escaped Rates in Reconnection

Since kinetic simulations include self-consistent electric and magnetic fields
at all particle positions, we can evaluate the energy-dependent particle
acceleration rate
$\alpha(\varepsilon)\equiv\left<\dot{\varepsilon}/\varepsilon\right>$, where
$\left<\dots\right>$ is the ensemble average for particles in different energy
bands, where $\dot{\varepsilon}=q\bm{v}\cdot\bm{E}$, and $\bm{v}$ is the
particle velocity. We can also decompose the velocity into different guiding-
center motions and calculate the acceleration rates associated with different
acceleration mechanisms. Fig. 7(a) shows the total and decomposed acceleration
rates for electrons in a 3D simulation. The primary acceleration mechanism is
a Fermi-type mechanism associated with particle curvature drift, the same as
2D results [58, 52, 59]. Both parallel electric field and grad-$B$ drift
contribute much less than the curvature drift and even decelerate high-energy
electrons. More importantly, since particles with different energies can
access to the same acceleration regions due to the fast spatial transport and
mixing, the total acceleration rate is nearly a constant in the 3D simulation.
In contrast, due to the artificial confinement of high-energy electrons in 2D
simulations, the electron acceleration rate sharply decreases with particle
energy (Fig. 7(b)) and even becomes negative for high-energy electrons later
in the simulations. The energy dependence explains quantitatively why the
electron energy spectrum keeps getting steeper in the 2D simulations (Fig. 6
(a)).

Figure 7:  Energy-dependent electron acceleration rate $\alpha$. (a) The
contributions to $\alpha$ by the acceleration associated with curvature drift,
gradient drift, and the parallel electric field in a 3D simulation. (b)
$\alpha$ for the 2D (orange) and 3D (blue) simulations at two time slices.
Reproduced with permission from Li et al., Astrophys. J, 884.2 (2019): 118.
Copyright 2019 Institute of Physics (IOP).

Particle escape processes are often not considered because most kinetic
simulations have periodic or closed boundary conditions. A common assumption
is that $\tau_{\text{esc}}\to\infty$ in these simulations because particles
cannot leave the simulation box. However, if $\tau_{\text{esc}}\to\infty$, the
power-law spectral index $p=1+(\alpha\tau_{\text{esc}})^{-1}$ will approach 1,
which cannot explain the simulation results (Fig. 6). In reconnection, as the
primary acceleration regions are only a fraction of the reconnection layer
(Fig. 3(d) & (f)), effective escape is still possible when particles are out
of these regions and do not actively participate in the acceleration processes
(Fig. 3(e)). Particles in large inactive magnetic structures (islands or flux
ropes) are slowly accelerated and behave similarly to the escaped particles.
Although some particles can be further accelerated when they are scattered out
of these structures by turbulence [66, 70], not all particles can be scattered
and escape from the flux rope at the same time. Those escaped particles could
be further accelerated when these structures merge, but the time scale between
merging can be as long as the dynamical time scale. Thus, these large inactive
magnetic structures serve as escape regions. By separating the acceleration
and escape regions (Fig. 3(f)), we can calculate the acceleration rate of the
particles in the acceleration regions and their escape rate from the
acceleration regions. The power-law spectrum is a dynamical balance between
particle acceleration and escape, as predicted by Eq. 6. Fig. 8 shows that
$\tau_{\text{esc}}^{-1}\approx 3\alpha$ early in the simulation, and the
resulted power-law index $p=1+(\alpha\tau_{\text{esc}})^{-1}\approx 4$ is
similar to the simulation results. However, the rates deviate from each other
after the reconnection outflows collide at the boundary at $t\Omega_{ci}\sim
150$. The acceleration rate sharply decreases after that, revealing why the
power-law is short in the 3D simulation. Kinetic simulations with larger
domains will likely sustain a more extended power-law spectrum, and
simulations that can bypass the kinetic scales will also help the formation of
extended power-law spectra [113, 131].

Figure 8:  An estimate of the power-law index by evaluating the acceleration
rate $\alpha$ and the escape rate $1/\tau_{\text{esc}}$ for high-energy
electrons ($\varepsilon>40\varepsilon_{\text{th}}$) in the major acceleration
region, where $|\bm{v}_{\bm{E}}\cdot\bm{\kappa}|$ is larger than a threshold,
as indicated in Figure 3 (f). (a) Time evolution of $3\alpha$ and
$1/\tau_{\text{esc}}$. The embedded plot compares the distributions of the
regions with negative and positive $\bm{v}_{\bm{E}}\cdot\bm{\kappa}$ at
$t\Omega_{ci}=100$. The vertical dashed line indicates the chosen threshold
0.001 for $|\bm{v}_{\bm{E}}\cdot\bm{\kappa}|$. Regions with
$|\bm{v}_{\bm{E}}\cdot\bm{\kappa}|<0.001$ do not contribute to the high-energy
particle energization because positive $\bm{v}_{\bm{E}}\cdot\bm{\kappa}$
balance with negative values. Reproduced with permission from Li et al.,
Astrophys. J, 884.2 (2019): 118. Copyright 2019 Institute of Physics (IOP).

##  V Outlooks

Recent kinetic simulations have made outstanding progress in understanding
particle acceleration and power-law formation during magnetic reconnection.
However, it is still incredibly challenging to address particle acceleration
at global scales that are usually much larger than the kinetic scales. For
example, Fig. 9 shows that the flare scale is $\sim 5$ orders of magnitude
larger than kinetic simulations. The entire simulation domain of the largest
PIC simulation is even smaller than the finest grid-scale of state-of-the-art
MHD simulations, not to mention the observation scales. The large scale
separation in typical space and solar plasmas (e.g., solar flares) prohibits
using a single simulation to study particle acceleration and transport in a
large-scale reconnection layer. To explain the observations, we need to
incorporate the large-scale acceleration mechanisms learned from kinetic
simulations (e.g., compression) into macroscopic energetic particle transport
models to study particle acceleration and transport at observable scales [113,
131]. Here, we discuss several issues to address for tackling the global-scale
acceleration problem in the future.

Figure 9:  The large scale separation in solar flares. $d_{i}$ is the ion
inertial length. The scale for the largest PIC is based on 2D simulations
($\sim 10^{3}d_{i}$) [58, 59] and reduced plasma parameters (e.g., the mass
ratio and light speed). Constrained by the computational cost, 3D simulations
are even smaller than those 2D simulations [70]. The MHD grid scale is
estimated based on some state-of-the-art MHD simulations of solar flares [132,
133, 134]. The observation resolution depends on the emission band.

###  V.1 Plasma Parameters and Simulation Setup

Although kinetic simulations have shown evidence of the power-law formation in
the nonrelativistic reconnection [70], we have just started to explore the
broad plasma parameter space (e.g., plasma $\beta$ and guide field). While 3D
simulations in the low-$\beta$ and weak guide-field regimes have shown the
power-law formation [70], the results are only suitable for explaining power-
law formation in solar flares and Earth’s magnetotail. They are not proper for
studying reconnection acceleration in the solar wind, where the plasma $\beta$
is considerably higher. Earlier studies have shown no evidence of substantial
nonthermal particle acceleration in the reconnection exhausts observed in the
solar wind [135]. However, recent studies have revived this topic and shown
that reconnection-associated processes (e.g., contracting and merging magnetic
flux ropes) can accelerate particles to develop nonthermal power-law energy
spectra in the solar wind [39, 41, 42]. While analytical theories have shown
that it is possible to develop power-law energy spectra in a “sea” of
contracting and merging flux ropes even in high-$\beta$ plasmas [57, 43, 44],
numerical simulations are yet to demonstrate power-law energy spectra can
indeed form in such plasmas. To test reconnection acceleration in the
high-$\beta$ plasmas, performing simulations in these regimes and making
$\tau_{\text{inj}}$ as large as possible is essential. Since
$\tau_{\text{inj}}$ is limited by the reconnection duration and constrained by
the boundary conditions (e.g., periodic boundaries), simulations with larger
domains and open-boundary conditions will help. Considering the requirements
of 3D simulations to achieve fast particle transport [66, 67, 70], studying
power-law formation in high-$\beta$ regimes poses a challenge. For
reconnection with a stronger guide field, it appears that the spectrum will
become softer because the acceleration rate decreases with the guide field
[58, 60, 131]. Whether this explanation is complete remains unclear due to the
lack of studies on the escape mechanisms during reconnection and how the
escape changes with different guide field.

Constrained by the computational cost, most kinetic simulations used reduced
proton-to-electron mass ratio $m_{i}/m_{e}$ (e.g., 25) and ratio between the
light speed and Alfvén speed. Such setup essentially reduces the scale-
separation between electron and proton scales. While the reconnection rate is
insensitive to the mass ratio [105], particle acceleration mechanisms and
power-law formation could be modified. For example, using a set of 2D
simulations, Li, Guo, and Li [105] showed that a reduced mass ratio leads to a
stronger electron acceleration by artificially enhancing the acceleration rate
associated with curvature drift. If the mass-ratio dependence holds in 3D, the
power-law formation processes and the power-law spectral index could deviate
from previous results [70]. Therefore, we should be cautious when using the
results from kinetic simulations to interpret the observations.

###  V.2 Turbulence Properties and Particle Transport

Despite the encouraging results from 3D kinetic simulations suggesting that
fast particle transport is essential for the power-law formation in
nonrelativistic reconnection [70], no simulations have been dedicated to
studying particle transport in the turbulent reconnection layer. To understand
particle transport, it is important to quantify the turbulence properties
relevant to particle acceleration first, including turbulence amplitude, power
spectrum, anisotropy, compressibility, and correlation length [136]. The
turbulence amplitude $\delta B^{2}/B_{0}^{2}$ is the ratio of the fluctuating
magnetic energy and the energy of the mean magnetic field. It represents how
strong the magnetic field fluctuates and determines how strongly particles are
scattered and whether second-order particle acceleration is significant [28].
The turbulence amplitude might decrease with the guide field because the guide
field is not dissipated in magnetic reconnection and because the magnetic
energy conversion becomes less efficient as the plasma becomes less
compressible when the guide field increases [137, 106, 105]. The other
turbulence properties are subjects of intense debate [71, 72, 138, 73, 139,
125, 140, 141, 142, 143]. In the MHD regime, there is no conclusion on whether
these results are consistent with classical Goldreich–Sridhar theory for
anisotropic turbulence in a homogeneous plasma permeated by a uniform magnetic
field [144, 145, 146]. Compared with the MHD studies, studying turbulence
properties in 3D kinetic simulations is even more challenging because the
computation cannot cover a similar dynamical range as in MHD simulations due
to the multi-scale nature. Although the turbulence spectrum is routinely shown
in recent kinetic simulations [122, 147, 70], a good understanding of the
anisotropy level, compressibility, and turbulence correlation length still
needs more studies.

If a much larger PIC simulation is accessible, the particle diffusion
coefficients can then be evaluated using these turbulence properties according
to particle transport theories [126, 148, 127, 149, 150]. As a complement to
this analysis, the diffusion coefficients can also be evaluated following the
Taylor-Green-Kubo (TGK) forms using test particles in the 3D kinetic
simulations. The comparison will tell us whether current transport theories
can explain particle transport in reconnection-driven turbulence or new kinds
of transport models are required [43, 100, 44, 97]. The results will guide the
efforts in modeling particle acceleration and transport in large-scale
reconnection layers, for example, by solving Parker’s transport equation
[113].

###  V.3 Ion Acceleration

Ion acceleration can be as efficient as electron acceleration during
reconnection. For example, solar flare observations have shown well-correlated
electron-generated hard X-ray flux and ion-generated gamma-ray flux [151],
suggesting a similar acceleration mechanism for both electrons and ions. Also,
heavy ions (especially Fe/O) often show substantial enhancement from their
coronal abundance during solar flares [152]. Compared to electron
acceleration, ion acceleration associated with magnetic reconnection is much
less studied, especially in kinetic simulations. Although ions can reach a few
thousands of times of the initial thermal energy [60, 105], the simulation
domains are usually too small ($\sim 100$ ion kinetic scales) for ions to have
enough dynamical scale for developing power-law energy distributions. To study
power-law formation for ions, hybrid kinetic simulations (kinetic ions and
fluid electrons) [153, 154] that do not resolve the electron kinetic scales
are a better choice. The simulation box can be 5–10 times larger than the
fully kinetic simulations, providing enough dynamical scales for ion
acceleration.

To explain heavy-ion enhancement in flares, while previous theories mostly
rely on wave-particle resonance in reconnection-driven turbulence [155, 27],
Drake  _et al._ [156] proposed a novel mechanism by magnetic reconnection.
When there is a finite guide field, the ions can be energized in a pickup
process [157] as they enter reconnection exhaust regions, and the heating of
heavy ions (with low Q/M) is more efficient than protons. Although fully
kinetic simulations have demonstrated the mechanisms to be effective [158,
159, 160], earlier works have been focused on ion heating instead of the
acceleration of suprathermal ions and the formation of power-law spectra.

###  V.4 Large-scale Acceleration Models

Recently, significant efforts have been made in understanding particle
acceleration in magnetic reconnection through macroscopic kinetic transport
theories that incorporate the acceleration and transport processes learned
from the fully kinetic simulations [115, 57, 136, 44, 113, 161]. Inspired by
these results, Li  _et al._ [113] have studied compression particle
acceleration in magnetic reconnection by solving the Parker
(diffusion–advection) transport equation using velocity and magnetic fields
from 2D MHD simulations of a low-$\beta$ reconnection layer. The compressible
reconnection layer can give significant particle acceleration, leading to
power-law particle energy distributions. These results show that one can use
transport theories to study particle acceleration in a large-scale flare
region. To improve the model, we need to incorporate more acceleration terms
(e.g., flow shear and second-order acceleration) into the transport model and
study whether they are essential for particle acceleration at large scales. In
addition, the diffusion coefficients evaluated based on the turbulence
properties will also help improve modeling particle transport in large-scale
reconnection layers.

One potential limitation of the transport models (see le Roux, Zank, and
Khabarova [97] for an exception) is that energetic particles do not feedback
to the background plasma. Solar flare observations indicate that magnetic
reconnection can accelerate over 10% of the entire electron population (more
than $10^{36}$ electrons) in the flare region into nonthermal distributions
[19, 20, 18, 162]. These energetic electrons thus contain a significant amount
of energy that can affect the plasma dynamics. Consequently, the energetic
particles’ feedback to the plasma might be crucial for evolving the
reconnection layer and associated particle acceleration. Recent results by the
kglobal model [161, 163, 131], which evolves the guiding-center motions of
electrons and includes the feedback to the MHD fluids, have shown the
formation of power-law energy spectra in macroscale reconnection systems
[131]. The power-law spectral indices are softer than the results in similar
models but without feedback [46, 47] and are similar to solar flare
observations. Interestingly, these power-law spectra and their dependence on
the guide field are similar to those obtained by solving Parker’s transport
equation [113]. It suggests that the transport model, which does not feedback
to the background plasmas but includes transport effects, is still very
valuable for studying electron acceleration. To reveal the potential
connection between these two approaches will require additional studies.

##  VI Summary

More and more evidence suggests that magnetic reconnection could be
responsible for the nonthermal particle acceleration and power-law energy
particle energy spectra observed in space and solar plasmas. Motivated by the
observations, many studies (either analytical or numerical) have been focused
on understanding how particles are accelerated in magnetic reconnection. We
summarize the recent progress in particle acceleration mechanisms and spatial
transport in the 3D reconnection and their roles in forming the power-law
energy spectrum.

We reviewed how the multi-scale structure developed in a reconnection layer
enables reconnection to be a natural site for accelerating particles from
thermal to nonthermal energies. In 2D, the contracting magnetic islands and
regions between merging magnetic islands are the most important for
accelerating high-energy particles. The dominant particle acceleration
mechanism has been proposed and demonstrated to be the Fermi mechanism using
kinetic simulations. We summarized the three closely linked models for
studying the Fermi mechanism. One is based on the two adiabatic invariants of
magnetic moment and the parallel action and has become the foundation of many
energetic particle models. The second one is based on individual particle’s
guiding-center drift motions and has revealed the energization associated with
curvature drift as the dominant mechanism for high-energy particle
acceleration. The third one is based on flow compression and shear and is
equivalent to the other two models. It reveals the flow compression is
critical for high-energy particle acceleration during reconnection and links
reconnection acceleration with the classical energetic particle transport
theories.

Although 2D studies have advanced our understanding of particle acceleration
associated with reconnection, they are limited in studying nonthermal particle
acceleration and power-law formation due to the artificial trapping of
energetic particles. We reviewed recent progress on 3D physics that are
relevant to particle acceleration and transport in reconnection. Both self-
generated turbulence and chaotic magnetic field lines in a 3D reconnection
layer can prevent particles from being trapped in local regions and enhance
the spatial transport of energetic electrons. Consequently, energetic
particles can access broader acceleration regions, gain more energy, and
efficiently mix with other particles. The resulted particle energy
distributions resemble power-laws globally and locally. We summarized a model
for explaining the power-law formation in 3D based on the Fermi mechanism. The
fast spatial transport enables particles with different energies to access the
same acceleration regions, leading to the energy-independent acceleration
rate, as often assumed for the Fermi mechanism. The power-law energy spectrum
is due to the dynamical balance between particle acceleration and an effective
escape caused by particles that do not actively participate in the
acceleration.

We pointed out the challenges in addressing particle acceleration and
transport in large-scale reconnection layers (e.g., solar flares) and several
relevant issues to be addressed in the future to tackle the global-scale
acceleration problem. First, how do the simulation results depend on the
plasma parameters and simulation setup? We anticipate that both plasma $\beta$
and guide field will change the power-law formation processes. We should be
cautious about using the simulation results based on reduced physics constants
(e.g., proton-to-electron mass ratio and light speed) to interpret the
observations. Second, what are the turbulence properties relevant to particle
acceleration and transport in 3D reconnection? We anticipate that a good
understanding of the turbulence amplitude, spectrum, anisotropy level,
compressibility, and turbulence correlation length is critical in addressing
this problem. Third, how are ions are accelerated, and what are the resulted
ion energy distributions? We point out that the fully kinetic simulations
might be too small to study ion acceleration, and hybrid kinetic simulations
will be useful. One critical issue of studying ion acceleration is how the
acceleration depends on the charge-to-mass ratio. Last but not least, what
models do we have for studying particle acceleration in a large-scale
reconnection layer? We point out recent progress in developing energetic
particle transport models and including feedback from particles to plasma
fluids for reconnection. Future studies should include more acceleration
physics and transport effects in these models and perform cross-checking
between different models.

###### Acknowledgements.

We thank the anonymous referees for very helpful and constructive reviews. We
gratefully acknowledge discussions with Hui Li, Patrick Kilian, Qile Zhang,
Joel Dahlin, Jim Drake, Adam Stanier, and William Daughton. Contribution from
X.L. and Y.L. is based upon work funded by the National Science Foundation
Grant No. PHY-1902867 through the NSF/DOE Partnership in Basic Plasma Science
and Engineering and NASA MMS 80NSSC18K0289. F. G. acknowledges the support
from Los Alamos National Laboratory through the LDRD/ER program and its Center
for Space and Earth Science (CSES), and NASA programs through grant
NNH17AE68I, 80HQTR20T0073, 80NSSC20K0627, and 80HQTR21T0005. Simulations were
performed at the National Energy Research Scientific Computing Center (NERSC),
at the Texas Advanced Computing Center (TACC) at The University of Texas at
Austin, and with LANL institutional computing.

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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