# Turbulent reconnection acceleration

Siyao Xu Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540,
USA<EMAIL_ADDRESS>111NASA Hubble Fellow Alex Lazarian  Department of Astronomy,
University of Wisconsin, 475 North Charter Street, Madison, WI 53706, USA;
<EMAIL_ADDRESS>Centro de Investigación en Astronomía, Universidad
Bernardo O’Higgins, Santiago, General Gana 1760, 8370993,Chile

###### Abstract

The ubiquitous turbulence in astrophysical plasmas is important for both
magnetic reconnection and reconnection acceleration. We study the particle
acceleration during fast 3D turbulent reconnection with reconnection-driven
turbulence. Particles bounce back and forth between the reconnection-driven
inflows due to the mirror reflection and convergence of strong magnetic
fields. Via successive head-on collisions, the kinetic energy of the inflows
is converted into the accelerated particles. Turbulence not only regulates the
inflow speed but also introduces various inflow obliquities with respect to
the local turbulent magnetic fields. As both the energy gain and escape
probability of particles depend on the inflow speed, the spectral index of
particle energy spectrum is not universal. We find it in the range from
$\approx 2.5$ to $4$, with the steepest spectrum expected at a strong guide
field, i.e. a small angle between the total incoming magnetic field and the
guide field. Without scattering diffusion needed for confining particles, the
reconnection acceleration can be very efficient at a large inflow speed and a
weak guide field.

††software: MATLAB (MATLAB, 2021)

##  1 Introduction

Reconnection acceleration is a fundamental acceleration process in solar and
space plasmas, as well as in high-energy astrophysical systems (Comisso &
Sironi, 2018; Marcowith et al., 2020; Lazarian et al., 2020; Shibata, 2020; Li
et al., 2021). It accounts for the particle energization in diverse
environments, such as the solar atmosphere (Turkmani et al., 2005; Chitta &
Lazarian, 2020; Jafari et al., 2021), the Heliosheath (Lazarian & Opher,
2009), pulsar wind nebulae (Lyutikov et al., 2018), gamma-ray bursts (Zhang &
Yan, 2011; Deng et al., 2016; Lazarian et al., 2019), accretion disks and jets
(de Gouveia dal Pino & Lazarian, 2005; Kadowaki et al., 2021).

As demonstrated by both magnetohydrodynamic (MHD) simulations and particle-in-
cell (PIC) simulations, reconnection and reconnection acceleration in 3D space
are intrinsically different from their 2D models (Priest et al., 2003; Kowal
et al., 2011; Parnell et al., 2011; Dahlin et al., 2017; Werner & Uzdensky,
2021; Zhang et al., 2021b, a). As a result of reduction in dimensionality,
reconnection in 2D occurs at single X-type magnetic null points. Reconnection
acceleration in 2D has limited efficiency and energy gain due to the
insufficient global volume-averaged reconnection rate and the trapping of
particles within 2D magnetic islands. In realistic 3D astrophysical plasmas,
magnetic fields are not confined in a closed domain, and the ubiquitous
turbulence regulates the diffusion of magnetic fields. Consequently, the
reconnection occurs continuously throughout a volume determined by the extent
of the diffusion of turbulent magnetic fields (Lazarian & Vishniac 1999; see
Lazarian et al. 2020 for a review). Turbulence makes magnetic field lines
intrinsically stochastic (Eyink et al., 2013), introducing stochasticity in
particle propagation and new diffusion mechanism of particles (Lazarian & Xu,
2021a).

The 3D turbulent reconnection model proposed by Lazarian & Vishniac (1999)
(henceforth LV99) describes a fast reconnection process in the sense that the
reconnection rate does not depend on the microscopic resistivity (Eyink et
al., 2011). It is generally believed that the macroscopic transport properties
of turbulence are independent of microscopic diffusivities (Spiegel, 1971).
This is also found to be true for diffusion of magnetic fields in the presence
of turbulence (Lazarian et al., 2004; Santos-Lima et al., 2010; Lazarian,
2014). Efficient reconnection acceleration requires a sufficiently high
reconnection rate. Although a localized high reconnection rate can be achieved
by adding new microscopic physics at an X-point (e.g., Parker 1993; Mandt et
al. 1994; Drake et al. 2008), it is not clear how to explain a high global
reconnection rate on macroscopic scales required for efficient acceleration of
energetic particles. With turbulence-governed diffusion of magnetic fields,
the LV99 model naturally has a high global reconnection rate when the level of
turbulence is high. The turbulence can be driven by diverse astrophysical
processes, various instabilities, or reconnection itself (Uritsky et al.,
2017; Kowal et al., 2017, 2020). In the case of reconnection-driven
turbulence, the level of turbulence depends on the fluid compressibility
(Kowal et al., 2017), and a reconnection rate on the system size comparable to
$V_{A}$ can be reached (Kowal et al. in prep), where $V_{A}$ is the Alfvén
speed.

On the basis of the LV99 model of turbulent reconnection, de Gouveia dal Pino
& Lazarian (2005) proposed that particles experience successive head-on
collisions with the inflows when they are trapped by converging magnetic
mirrors during turbulent reconnection. Efficient particle acceleration in
simulations of turbulent reconnection was found by Kowal et al. (2011, 2012a);
Zhang et al. (2022). Beresnyak & Li (2016) considered the curvature drift
acceleration during the initial stage of spontaneous turbulent reconnection
with insignificant turbulence and negligible inflows. For the particle
acceleration during turbulent reconnection with a global high reconnection
rate and inflow speed, a quantitative description is still missing. Moreover,
because of the limited ranges of length scales and particle energies, and the
partially developed turbulence in PIC simulations, the effect of turbulence on
the simulated reconnection and reconnection acceleration cannot be fully
investigated, and the energy gain of particles is also limited, e.g., up to
$100$ MeV for electrons even for relativistic reconnection (Sironi &
Spitkovsky, 2014; Guo et al., 2014). These are the motivations of the current
theoretical study on turbulent reconnection acceleration.

The reconnection acceleration of particles with the gyroradii larger than the
thickness of the reconnection region was theoretically proposed by Lazarian et
al. (2012) and numerically demonstrated by Zhang et al. (2021a). Here we focus
on the case with the gyroradii smaller than the thickness of the reconnection
region and the acceleration process inside the reconnection region. In Section
2, we first briefly review the basic physical pictures of turbulent
reconnection and reconnection acceleration. In Section 3, we study in detail
the acceleration of both reflected and unreflected particles, and their
averaged fractional energy gain. In Sections 4 and 5, we analyze the escape
probability and energy spectral index of reconnection accelerated particles,
as well as the acceleration timescale. Discussion and final conclusions are
provided in Sections 6 and 7.

##  2 Turbulent reconnection and reconnection acceleration

###  2.1 Turbulent reconnection of magnetic fields

The reconnection of non-turbulent magnetic fields happens due to the resistive
diffusion of magnetic fields, and thus the reconnection rate is limited by the
resistivity (Parker, 1957; Sweet, 1958), with

$\frac{U_{\text{in}}}{V_{A}}=S^{-\frac{1}{2}}=\frac{\Delta}{L_{x}},$ (1)

where $U_{\text{in}}$ is the inflow speed that characterizes the speed of
reconnection,

$V_{A}=U_{\text{out}}=\frac{B_{\text{in}}\sin\theta}{\sqrt{4\pi\rho}}$ (2)

is the Alfvén speed (equal to the outflow speed) corresponding to the strength
$B_{\text{in}}\sin\theta$ of oppositely directed magnetic fields in the
inflows, $B_{\text{in}}$ is the total magnetic field strength in the inflows,
$\theta$ is the angle between ${\bm{B}_{\text{in}}}$ and the guide field
${\bm{B}_{\text{g}}}$, the guide field strength is
$B_{g}=B_{\text{in}}\cos\theta$, $\rho$ is the plasma density that is constant
in incompressible medium considered in this work,

$S=\frac{L_{x}V_{A}}{\eta}$ (3)

is the Lundquist number, $L_{x}$ and $\Delta$ are the length and thickness of
the reconnection region, and $\eta$ is the resistive diffusivity. Because of
the slow resistive diffusion and thus large disparity between $L_{x}$ and
$\Delta$, the Sweet-Parker (SP) model of reconnection is slow with
$U_{\text{in}}\ll V_{A}$.

For turbulent magnetic fields, their diffusion is regulated by turbulence
rather than resistivity (LV99; Eyink et al. 2011). Therefore, $\Delta$ is
determined by the diffusion distance of turbulent magnetic fields in $y$
direction corresponding to a distance $L_{x}$ in $x$ direction (see Fig. 1).
The resulting reconnection rate is (Lazarian & Vishniac, 1999)

$\frac{U_{\text{in}}}{V_{A}}=\text{min}\bigg{[}\Big{(}\frac{L_{x}}{L_{i}}\Big{)}^{\frac{1}{2}},\Big{(}\frac{L_{i}}{L_{x}}\Big{)}^{\frac{1}{2}}\bigg{]}M_{A}^{2},~{}~{}M_{A}<1,$
(4)

where $L_{i}$ is the correlation length of the externally driven turbulence
that can be larger or smaller than $L_{x}$, and $M_{A}=V_{L}/V_{A}$ is defined
as the Alfvén Mach number with the turbulent velocity $V_{L}$ at $L_{i}$ and
$V_{A}$ corresponding to $B_{\text{in}}\sin\theta$. When $L_{i}\sim L_{x}$ and
$M_{A}\sim 1$, the LV99 model of turbulent reconnection leads to
$U_{\text{in}}\sim V_{A}$. When $L_{i}$ and $L_{x}$ are very different or
$M_{A}\ll 1$ for weakly turbulent magnetic fields, there can be
$U_{\text{in}}\ll V_{A}$. The LV99 model describes a fast reconnection process
in the sense that the reconnection rate is independent of the microscopic
magnetic diffusivity. This is simply because the process of turbulent
(reconnection) diffusion of magnetic fields, frequently referred to as
reconnection diffusion (Lazarian, 2005), is faster than their microscopic
diffusion on scales where turbulence exists. The rigorous mathematical proof
of the above statement is provided in Eyink et al. (2011); Eyink (2015). The
predicted fast reconnection has been confirmed by 3D MHD turbulent
reconnection simulations in both non-relativistic (Kowal et al., 2009, 2012b)
and relativistic (Takamoto et al., 2015) cases.

In addition to the ubiquitous turbulence in astrophysical plasmas,
reconnection itself naturally drives turbulence with the released magnetic
energy converted to turbulent energy. The reconnection-driven turbulence in
turn regulates the reconnection rate (LV99; Lazarian & Vishniac 2000). The
reconnection process with reconnection-driven turbulence is illustrated in
Fig. 1. In the steady state, the antiparallel component of
${\bm{B}_{\text{in}}}$ in the inflows reconnect. The reconnection-driven
turbulence determines $U_{\text{in}}$ and the outflow thickness $\Delta$, and
causes perturbation of the guide field that is initially along the $z$
direction. The resulting turbulent guide field ${\bm{B}_{\text{out}}}$ in the
outflow leaves the system along the $x$ direction with the outflow speed
$U_{\text{out}}=V_{A}$. This process has been numerically demonstrated by
Kowal et al. (2017, 2020) with 3D MHD simulations. The turbulence generated
during reconnection can originate from the Kelvin-Helmholtz instability
induced by velocity shear in reconnecting layers, which dominates over the
tearing instability in driving turbulence (Kowal et al., 2020).

For the magnetic reconnetion in a compressible medium, the conservation of
mass leads to

$\frac{U_{\text{in}}}{V_{A}}=\frac{\rho_{\text{out}}}{\rho_{\text{in}}}\frac{\Delta}{L_{x}},$
(5)

where $\rho_{\text{in}}$ and $\rho_{\text{out}}$ are the densities in the
inflows and outflows, respectively. Given
$\rho_{\text{out}}>\rho_{\text{in}}$, a higher reconnection rate in a
compressible medium is expected compared to that in an incompressible medium.
The density change and density inhomogeneity in a compressible medium can
enhance the growth of instabilities, e.g., the Rayleigh-Taylor instability,
the Richtmyer-Meshkov instability, that do not exist in an incompressible
medium and thus lead to a higher level of turbulence. Kowal et al. (2017)
found that in a compressible medium, the reconnection-driven turbulence and
the resulting $U_{\text{in}}$ have dependence on plasma $\beta_{p}(=\text{gas
pressure}/\text{magnetic pressure})$ given the same free magnetic energy.

Figure 1: Illustration for turbulent magnetic reconnection with reconnection-
driven turbulence. The dashed arrow in (a) indicates the initial guide field.
Via reconnection, the guide field becomes turbulent as seen in the $y-z$ plane
in (b). The region between the dashed lines in (b) is the reconnection region.

In our analysis, we consider

$U_{\text{in}}=fV_{A}=fU_{\text{out}}=f\frac{B_{\text{in}}\sin\theta}{\sqrt{4\pi\rho}}$
(6)

for turbulent magnetic reconnection, where $\theta$ is the angle between the
initial guide field and ${\bm{B}_{\text{in}}}$. The factor
$f=\Delta/L_{x}\lesssim 1$ depends on turbulent properties (see Eq. (4)) and
plasma $\beta_{p}$ in the case of a compressible medium. $f$ in the range
$0.001-0.2$ is indicated by observations of solar flares (Cassak et al.,
2017). While a small $f$ up to $0.02$ is seen in weakly compressible
reconnection simulations with reconnection-driven turbulence (Kowal et al.,
2017), a large $f$ with values above $0.1$ is found in a highly compressible
medium (Kowal et al. in prep).

For simplicity, we consider that the fluctuation component of $B_{\text{out}}$
comes from the reconnection-driven turbulence alone, with

$\delta B_{\text{out}}\sim B_{\text{in}}\sin\theta.$ (7)

We define the local obliquity $\alpha$ of the inflow as the angle between the
inflow direction ($y$ direction) and the local ${\bm{B}_{\text{out}}}$
direction. The original inflow obliquity with respect to the initial guide
field is $\sim 90^{\circ}$ (see Fig. 1(b)). Due to the reconnection-driven
turbulence, the inflow has a range of local obliquities
$(90^{\circ}-\delta\alpha,90^{\circ})$. For the variation of obliquity
$\delta\alpha$, there is

$\tan\delta\alpha=\frac{\delta
B_{\text{out}}}{B_{g}}\sim\frac{B_{\text{in}}\sin\theta}{B_{\text{in}}\cos\theta}=\tan\theta,$
(8)

where $B_{g}$ is the strength of initial guide field. Therefore, we
approximately have $\delta\alpha\sim\theta$.

###  2.2 Reconnection acceleration of particles

Reconnection acceleration of particles happens due to the convergence of flows
driven by the reconnection, irrespective of the compressibility of the medium.
As the particles are confined in the reconnection region by mirror reflection
or gyrating around the converging magnetic fields, unlike the diffusive shock
acceleration (DSA, Axford et al. 1977; Krymsky 1977; Bell 1978; Blandford &
Ostriker 1978; Drury 1983; Jones & Ellison 1991), scattering diffusion is not
a necessary confining mechanism for reconnection acceleration. Efficient
reconnection acceleration requires a sufficiently high reconnection rate and
thus a sufficiently high level of turbulence. The associated electric field
for the acceleration is induced by the inflow motion of plasma, while the
effect of Ohmic/non-ideal electric field in microscopic diffusion regions is
negligible (Kowal et al., 2012a; Guo et al., 2019).

Although there is a divergence of flow associated with the outflows from the
reconnection region, it does not necessarily cause energy loss of particles.
For particles with the gyroradii $r_{g}$ smaller than $\Delta$, they cannot
freely diffuse in the $x$ direction (see Fig. 1) by crossing the guide field
(along $z$) and the transverse magnetic field (along $y$) in the reconnection
region. Drury (2012) considered the case with $r_{g}>\Delta$. One of the main
assumptions in Drury (2012) is that the particles with isotropic distribution
can sample the entire system that encompasses both inflows and outflows. Under
this assumption, compression of plasma becomes a necessary condition for
acceleration to happen. However, as we mentioned, this assumption is not valid
for particles with $r_{g}<\Delta$. For particles with $r_{g}>\Delta$, to
satisfy this assumption, very efficient scattering is required on scales
larger than $\Delta$ to ensure the isotropic distribution and free diffusion
of particles. As reconnection-driven turbulence only exists within the
reconnection region on scales $\lesssim\Delta$, the origin of the scattering
is unclear.

For particles with $r_{g}<\Delta$, they bounce back and forth within the
reconnection region and undergo the first-order Fermi acceleration by
colliding with the inflows many times (de Gouveia dal Pino & Lazarian, 2005).
The particles entrained on the magnetic fields in the outflow are advected
away from the reconnection region in either direction along the $x$-axis (Fig.
1). As the particles cannot freely diffuse and simultaneously sample the
bidirectional outflows, they do not suffer the energy loss due to the flow
divergence. Efficient particle acceleration during turbulent reconnection has
been numerically demonstrated in Kowal et al. (2011, 2012a); Zhang et al.
(2022).

The curvature drift acceleration is the main acceleration mechanism in 2D
reconnection models. The acceleration happens due to the curvature drift of
particles in the direction of the motional electric field, which is induced by
island contraction or merging. Particles are trapped within a contracting
magnetic island or between two merging islands (Drake et al., 2006, 2013; le
Roux et al., 2018). The acceleration stops when contraction or merging comes
to an end. In 3D turbulent reconnection, turbulent magnetic fields do not form
2D islands. The curvature of magnetic fields caused by turbulence results in
stochastic acceleration with both curvature drift acceleration and
deceleration (Beresnyak & Li, 2016). The release of magnetic energy via
relaxation of magnetic tension of curved field lines is found to be
subdominant compared to that via magnetic pressure gradient when the 3D
reconnection evolves to a more turbulent state (Du et al., 2022). Given the
expected stochastic nature of the curvature drift acceleration and subdominant
magnetic energy release via curvature relaxation in turbulent reconnection, we
focus below on the systematic acceleration associated with the magnetic field
gradient in inflows.

Figure 2: Illustration for reconnection acceleration with $r_{g}>\Delta$.
${\bm{E}}=-{\bm{U}_{\text{in}}}\times{\bm{B}_{\text{in}}}/c$ is the electric
field calculated in the rest frame of the plasma in the reconnection region.
The drift of particle is caused by the different magnetic field directions in
the inflows above and below the reconnection region. The drift direction
coincides with the initial guide field direction and ${\bm{E}}$ direction, and
thus the particle is accelerated during every gyration. The original picture
was presented in Lazarian et al. (2012).

For particles with $r_{g}>\Delta$, they gyrate around the reconnection region.
As illustrated in Fig. 2, because of the opposite directions of reconnecting
magnetic fields, particles drift along the $z$ direction, parallel to the
motional electric field
${\bm{E}}=-{\bm{U}_{\text{in}}}\times{\bm{B}_{\text{in}}}/c$, where $c$ is the
light speed. This mechanism was first proposed in Lazarian et al. (2012) and
recently confirmed by Zhang et al. (2021a) with 3D PIC simulations of
relativistic reconnection.

In this work, we will focus on the particle acceleration with $r_{g}<\Delta$
during turbulent reconnection.

##  3 Particle energy gain

###  3.1 Reflected particles

####  3.1.1 Energy gain per reflection

As mentioned above, with respect to the turbulent guide field
${\bm{B}_{\text{out}}}$, the inflow velocity along the local magnetic field
direction becomes $U_{\text{in}}/\cos\alpha$, where $U_{\text{in}}$ is the
inflow speed along the $y$ direction. Similar to the shock drift acceleration
(SDA) at an oblique shock (Sonnerup, 1969; Wu, 1984; Armstrong et al., 1985;
Ball & Melrose, 2001; Xu & Lazarian, 2022), because of the change of magnetic
field strength from $B_{\text{out}}$ in the reconnection region to
$B_{\text{in}}$ in the inflow, a particle incident on the inflow experiences a
gradient-B drift along the induced electric field
$-{\bm{U}}\times{\bm{B}_{\text{in}}}/c$ in the inflow frame, where ${\bm{U}}$
is the flow velocity seen in the inflow frame. We term this acceleration
process “reconnection drift acceleration (RDA)” (see Fig. 3). During the RDA,
the kinetic energy of the inflow is converted to the particle kinetic energy.

Figure 3: Illustration for the RDA with $r_{g}<\Delta$. Left and right panels
show one encounter of a particle with the inflow seen from different viewing
angles. $U(=U_{\text{in}})$ is the plasma speed in the reconnection region
seen in the reference frame of the inflow.
${\bm{E}}=-{\bm{U}}\times{\bm{B}_{\text{in}}}/c$ is the motional electric
field. The magnetic field gradient $\nabla B$ is caused by the transition from
the strong magnetic field in the inflow to the weak magnetic field in the
reconnection region. The $\nabla B$ drift of a particle with the drift speed
$v_{D}$ is along the ${\bm{E}}$ direction. Its trajectory is shown in the
right panel (red). A particle in the reconnection region experiences multiple
encounters with the inflows and thus is repeatedly accelerated.

For shock acceleration, the SDA, i.e., the gradient-B drift acceleration, is
equivalent to the acceleration via reflection by approaching mirrors, i.e.,
the Fermi mechanism (Lee & Fisk, 1982; Krauss-Varban & Wu, 1989). As the shock
front, i.e., the magnetic field gradient/magnetic mirror is not at rest, it is
convenient to adopt a special reference frame for which the intersection point
of the shock front and the magnetic field line is at rest, i.e., the de-
Hoffmann-Teller (HT) frame (de Hoffmann & Teller, 1950). The fractional energy
gain can then be obtained by transformation back to the rest frame of the
upstream medium (Ostrowski, 1988; Xu & Lazarian, 2022).

Similarly, for reconnection acceleration, the RDA, i.e., the gradient-B drift
acceleration, is equivalent to the acceleration via reflection by approaching
magnetic mirrors. Despite the similarity in the acceleration mechanism, the
magnetic mirror does not form due to shock compression, but the annihilation
of oppositively directed magnetic fields. It creates the magnetic pressure
gradient that drives inflows and approaching magnetic mirrors. To derive the
energy gain during the interaction with the inflow, we also adopt the HT frame
as long as the inflow is subluminal. In the HT frame, the intersection point
between the inflow and the local guide field is at rest and the fluid motion
is parallel to the magnetic field. Therefore, the electric field vanishes, and
the particle energy remains unchanged in the HT frame. Given the similarity
between RDA and SDA, here we follow a similar analysis to that in Ostrowski
(1988) for SDA. The Lorentz factor $\gamma$ of a particle in the rest frame of
the reconnection region is related to $\gamma^{\prime}$ in the HT frame by

$\gamma=\Gamma\gamma^{\prime}(1-\beta\mu^{\prime}),$ (9)

where $\Gamma=1/\sqrt{1-\beta^{2}}$, and

$\beta=\frac{U_{\text{in}}}{c\cos\alpha}.$ (10)

The cosine of the particle pitch angle $\mu$ in the rest frame of the
reconnection region is related to $\mu^{\prime}$ in the HT frame by

$\mu=\frac{\mu^{\prime}-\beta}{1-\beta\mu^{\prime}}.$ (11)

A particle incident from the reconnection region on the inflow can be
reflected by the mirror force due to the increase of magnetic field strength
from the reconnection region to the inflow. Under the consideration of
conserved magnetic moment in the HT frame, the relation

$\frac{{v^{\prime}}_{\perp}^{2}}{B_{\text{out}}}=\frac{{v^{\prime}}^{2}}{B_{\text{in}}}$
(12)

constrains the smallest pitch angle in the HT frame for reflection to happen,
where ${v^{\prime}}_{\perp}$ is the particle velocity perpendicular to the
magnetic field in the HT frame, and $v^{\prime}$ is the total particle
velocity in the HT frame. We find that particles with $\mu^{\prime}$ in the
range

$0<\mu^{\prime}<\sqrt{1-\frac{B_{\text{out}}}{B_{\text{in}}}}$ (13)

can be reflected. It corresponds to (Eq. (11))

$-\beta<\mu<\frac{\sqrt{1-\frac{B_{\text{out}}}{B_{\text{in}}}}-\beta}{1-\beta\sqrt{1-\frac{B_{\text{out}}}{B_{\text{in}}}}}$
(14)

for the range of $\mu$. Compared to the mirroring effect due to shock
compression of magnetic fields, at a large $\theta$ with a weak guide field,
the mirroring effect caused by magnetic reconnection is much stronger, and
thus particles with a large range of $\mu$ can be reflected. For the
unreflected particles in the loss cone with

$\frac{\sqrt{1-\frac{B_{\text{out}}}{B_{\text{in}}}}-\beta}{1-\beta\sqrt{1-\frac{B_{\text{out}}}{B_{\text{in}}}}}<\mu<1,$
(15)

unlike the case of an oblique shock, instead of being transmitted, they are
attached to the incoming magnetic fields via gyromotion and further advected
back to the reconnection region together with the converging magnetic fields
carried by the inflow. Both reflected and unreflected particles undergo the
RDA. Here we will first analyze the acceleration of reflected particles and
move to the acceleration of unreflected particles in Section 3.2.

Given the symmetry of the inflow, when

$\frac{\sqrt{1-\frac{B_{\text{out}}}{B_{\text{in}}}}-\beta}{1-\beta\sqrt{1-\frac{B_{\text{out}}}{B_{\text{in}}}}}>0,$
(16)

that is

$\beta<\sqrt{1-\frac{B_{\text{out}}}{B_{\text{in}}}},$ (17)

we consider the range of $\mu$

$0<\mu<\frac{\sqrt{1-\frac{B_{\text{out}}}{B_{\text{in}}}}-\beta}{1-\beta\sqrt{1-\frac{B_{\text{out}}}{B_{\text{in}}}}}$
(18)

for reflection to happen. For a reflected particle, the ratio of the energies
after ($E_{f}$) and before ($E_{0}$) the reflection is (Eq. (9))

$\Big{(}\frac{E_{f}}{E_{0}}\Big{)}_{\text{r}}=\frac{1+\beta\mu^{\prime}}{1-\beta\mu^{\prime}},$
(19)

where $\mu^{\prime}$ and $-\mu^{\prime}$ are the pitch angle cosines before
and after the reflection in the HT frame. By using the relation in Eq. (11),
Eq. (19) leads to

$\Big{(}\frac{E_{f}}{E_{0}}\Big{)}_{\text{r}}=\frac{1+2\beta\mu+\beta^{2}}{1-\beta^{2}}.$
(20)

In the limit of a small $\beta$, the above expression approximately becomes

$\Big{(}\frac{E_{f}}{E_{0}}\Big{)}_{\text{r}}\approx 1+2\beta\mu.$ (21)

At a large $\beta$, the quadratic term in Eq. (20) is not negligible, and the
energy gain after one reflection can be very significant. The acceleration of
reflected particles can happen over a range of pitch angles and inflow
obliquities. To obtain the particle energy spectrum, we will need both pitch-
angle and obliquity averaged fractional energy gain of reflected particles
(see Section 3.3).

####  3.1.2 Pitch angle distribution

As a particle reverses its direction in the HT frame after reflection, by
using Eq. (11), we can obtain the pitch angle cosine after reflection

$\mu_{f}=\frac{-\mu^{\prime}-\beta}{1+\beta\mu^{\prime}}=-\frac{\mu+2\beta+\beta^{2}\mu}{1+2\beta\mu+\beta^{2}}.$
(22)

We find that the ratio of the particle momentum perpendicular to the magnetic
field after and before the reflection as (Eqs. (20) and (22))

$\frac{p_{\perp,f}}{p_{\perp,0}}=\Big{(}\frac{E_{f}}{E_{0}}\Big{)}_{\text{r}}\frac{\sqrt{1-\mu_{f}^{2}}}{\sqrt{1-\mu^{2}}}=1.$
(23)

It means that the acceleration of a reflected particle only causes increase of
its momentum parallel to the magnetic field and thus decrease of its pitch
angle.

The ratio of the perpendicular and parallel components of particle momentum is
then

$\Big{(}\frac{p_{\perp}}{p_{\|}}\Big{)}_{f}=\frac{\sqrt{1-\mu_{f}^{2}}}{\mu_{f}}.$
(24)

In Fig. 4, we compare the above ratio after reflection to its value before
reflection $(p_{\perp}/p_{\|})_{0}=\sqrt{1-\mu^{2}}/\mu$ over the range of
$\mu$ given in Eq. (18), where we adopt
$B_{\text{out}}/B_{\text{in}}\sim\cos\theta$ and $\theta=45^{\circ}$.
Naturally, we see that toward a larger $\beta$, the acceleration of reflected
particles leads to a more anisotropic pitch angle distribution. This finding
can explain the numerical result in Kowal et al. (2012a), where they found
that particles are preferentially accelerated in the direction parallel to the
magnetic field during turbulent reconnection.

With the increase of anisotropy in pitch angle distribution, more particles
would fall in the loss cone. The acceleration of unreflected particles acts to
suppress the anisotropy and allows the particles to leave the loss cone (see
Section 3.2). In addition, turbulent scattering also tends to isotropize the
particle distribution. These effects are important for suppressing the
anisotropy in pitch angle distribution and sustaining the mirror reflection.
In our analysis through the paper, we will consider isotropic pitch angle
distribution for simplicity.

Figure 4: The ratio of perpendicular to parallel momentum of a particle after
reflection $-(p_{\perp}/p_{\|})_{f}$ (with a negative sign added) vs. that
before reflection $(p_{\perp}/p_{\|})_{0}$ at different $\beta$.
$\theta=45^{\circ}$ is used. The dashed line indicates the equalization
between the two.

####  3.1.3 Pitch-angle averaged energy gain

The fractional energy gain $(E_{f}/E_{0})_{\text{r}}-1$ of reflected particles
has $\mu$ dependence (Eq. (20)). At fixed $fB_{\text{in}}/\sqrt{4\pi\rho}$,
for isotropic pitch angle distribution, we find the pitch-angle averaged
fractional energy gain during reflection at given $\theta$ and inflow
obliquity $\alpha$ as

$d_{\alpha}=\frac{\int
V_{\text{rel}}\Big{[}\Big{(}\frac{E_{f}}{E_{0}}\Big{)}_{\text{r}}-1\Big{]}d\mu}{\int
V_{\text{rel}}d\mu},$ (25)

where $V_{\text{rel}}$ is the rate for particles to approach the inflow,

$V_{\text{rel}}=\frac{\mu\cos\alpha+\frac{U_{\text{in}}}{c}}{1+\frac{U_{\text{in}}\mu\cos\alpha}{c}},$
(26)

and the integral range is given by Eq. (18). A similar expression of
$d_{\alpha}$ can be found in Ostrowski (1988) for studying the SDA at an
oblique shock. By considering
$V_{\text{rel}}\approx\mu\cos\alpha+U_{\text{in}}/c$, we find approximately

$d_{\alpha}\approx\frac{2\beta[(\beta+\mu_{c})^{3}-\beta^{3}]}{3(\beta\mu_{c}+\frac{\mu_{c}^{2}}{2})(1-\beta^{2})},$
(27)

where

$\mu_{c}=\frac{\sqrt{1-\frac{B_{\text{out}}}{B_{\text{in}}}}-\beta}{1-\beta\sqrt{1-\frac{B_{\text{out}}}{B_{\text{in}}}}}.$
(28)

At a given $\theta$, $d_{\alpha}$ is only a function of $\beta$. As shown in
Fig. 5, where we assume $B_{\text{out}}/B_{\text{in}}\sim\cos\theta$,
$d_{\alpha}$ can significantly exceed unity when $\theta$ and $\beta$ are
large. At a large $\theta$ with a weak guide field, due to the strong
mirroring effect, the reflection can happen at a large $\beta$ (see Eq. (17)),
leading to a significant energy gain. In the low-$\beta$ limit, by using
$(E_{f}/E_{0})_{\text{r}}\approx 1+2\beta\mu$ and
$V_{\text{rel}}\approx\mu\cos\alpha$, we find

$d_{\alpha}\approx\frac{4\beta}{3}\mu_{c}.$ (29)

It provides a good approximation when $\beta$ is small (see Fig. 5).

Figure 5:  (a) Pitch-angle averaged fractional energy gain $d_{\alpha}$ as a
function of $\beta$ for reflected particles. The thick and thin solid lines
are calculated using Eq. (25) for different $\theta$. The dashed lines are
their approximation in the low-$\beta$ limit given by Eq. (29). (b) Pitch-
angle weighted fractional energy gain $d_{\text{r},\alpha}$ as a function of
$\beta$ for reflected particles. The thick and thin solid lines are calculated
using Eq. (32) for different $\theta$.

At given $\theta$ and $\alpha$, the probability of reflection is (Ostrowski,
1988)

$P_{\text{r},\alpha}=\frac{S_{\text{r}}}{S_{\text{r}}+S_{\text{nr}}},$ (30)

where

$S_{\text{r/nr}}=\int V_{\text{rel}}d\mu$ (31)

with the integral range given by Eq. (18) for $S_{\text{r}}$ and Eq. (15) for
$S_{\text{nr}}$. Therefore, we have the pitch-angle weighted fractional energy
gain for reflection at given $\theta$ and $\alpha$ as

$\displaystyle d_{\text{r},\alpha}$
$\displaystyle=P_{\text{r},\alpha}d_{\alpha}$ (32)
$\displaystyle=\frac{\int_{0}^{\mu_{c}}V_{\text{rel}}\Big{[}\Big{(}\frac{E_{f}}{E_{0}}\Big{)}_{\text{r}}-1\Big{]}d\mu}{\int_{0}^{1}V_{\text{rel}}d\mu}$
$\displaystyle\approx\frac{2\beta[(\beta+\mu_{c})^{3}-\beta^{3}]}{3(\beta+\frac{1}{2})(1-\beta^{2})}.$

As shown in Fig. 5, at the maximum $\beta$ for reflection, $\mu_{c}=0$, and
thus $d_{\text{r},\alpha}$ becomes $0$. We find that $P_{\text{r},\alpha}$ is
approximately equal to

$P_{\text{r},\alpha}\approx\frac{\mu_{c}(\beta+\frac{\mu_{c}}{2})}{\beta+\frac{1}{2}},$
(33)

which can be reduced to $P_{\text{r},\alpha}\approx\mu_{c}^{2}$ at a small
$\beta$. At a given $\theta$, as the range of $\mu$ for reflection decreases
with increasing $\beta$, $P_{\text{r},\alpha}$ becomes small at a large
$\beta$, resulting in the drop of $d_{\text{r},\alpha}$ at the maximum $\beta$
for reflection. At a given $\beta$, the range of $\mu$ for reflection
increases with increasing $\theta$. When $\theta$ is large,
$d_{\text{r},\alpha}$ is close to $d_{\alpha}$ except when $\beta$ approaches
unity.

###  3.2 Unreflected particles

The unreflected particles in the loss cone cannot penetrate deep into the
inflow because of the strong incoming magnetic fields. After entering the
inflow, they are attached to the incoming magnetic fields threading the inflow
and advected back into the reconnection region. For a particle entering and
exiting from the inflow, the ratio of the energies after ($E_{f}$) and before
($E_{0}$) the interaction is

$\Big{(}\frac{E_{f}}{E_{0}}\Big{)}_{\text{ur}}=\frac{1+2\beta_{\text{in}}\xi+\beta_{\text{in}}^{2}}{1-\beta_{\text{in}}^{2}},$
(34)

where $\beta_{\text{in}}=U_{\text{in}}/c$, and $\xi$ is the cosine of the
angle $\Phi$ between the particle velocity and the inflow direction ($y$
direction). The derivation of Eq. (34) is similar to that of Eq. (20) for a
reflected particle, but the Lorentz transformation is between the rest frames
of the reconnection region and the inflow. In the above calculation we assume
that the particle enters the inflow with $\xi^{\prime}$ and exits with
$-\xi^{\prime}$, where $\xi^{\prime}$ in the rest frame of the inflow is
related to $\xi$ by

$\xi=\frac{\xi^{\prime}-\beta_{\text{in}}}{1-\beta_{\text{in}}\xi^{\prime}}.$
(35)

Although the form of Eq. (34) is similar to Eq. (20) of a reflected particle,
the energy gain of an unreflected particle does not depend on the local
obliquity of the inflow. As $\beta_{\text{in}}<\beta$, the energy gain of an
unreflected particle after a single encounter with the inflow can be much
smaller than that of a reflected particle. Moreover, the return of an
unreflected particle to the reconnection region is not due to the mirror
reflection, but the advection of the inflow.

By performing the average over $\Phi$, we find the mean fractional energy gain
of unreflected particles as

$d_{\text{ur}}=\frac{\int_{0}^{1}V_{\text{rel,ur}}\Big{[}\Big{(}\frac{E_{f}}{E_{0}}\Big{)}_{\text{ur}}-1\Big{]}d\xi}{\int_{0}^{1}V_{\text{rel,ur}}d\xi},$
(36)

where

$V_{\text{rel,ur}}=\frac{\xi+\beta_{\text{in}}}{1+\beta_{\text{in}}\xi}.$ (37)

Here we adopt the range $0<\xi<1$ for unreflected particles in the loss cone
with different pitch angles with respect to the turbulent magnetic fields in
the reconnection region. By taking
$V_{\text{rel,ur}}\approx\xi+\beta_{\text{in}}$, we approximately have

$d_{\text{ur}}\approx\frac{2\beta_{\text{in}}(3\beta_{\text{in}}^{2}+3\beta_{\text{in}}+1)}{3(\beta_{\text{in}}+\frac{1}{2})(1-\beta_{\text{in}}^{2})}.$
(38)

When $\beta_{\text{in}}$ is small, it can be further simplified to

$d_{\text{ur}}\approx\frac{4\beta_{\text{in}}}{3}.$ (39)

We note that the result in Eq. (39) is similar to the average fractional
energy gain $(4/3)V/c$ after one round trip crossing and recrossing the shock
in parallel shock acceleration, where $V=(3/4)U_{\text{sh}}$, and
$U_{\text{sh}}$ is the shock speed (Longair, 1997). Despite the similarity, in
the case of shock acceleration, the energy gain comes from the kinetic energy
of the converging flows due to shock compression. A particle crossing the
shock is scattered and isotropized within the moving downstream flow, and then
returns to the upstream medium. For the reconnection acceleration of an
unreflected particle, the energy gain comes from the kinetic energy of
reconnection-driven inflows. The particle is coupled to the moving flow not by
frequent scattering, but by attaching to the inflow magnetic field.

Following the similar analysis as that in Section 3.1.2, we see that after an
encounter of an unreflected particle with the inflow, $|\xi|$ increases (see
Eq. (22)). With respect to the turbulent magnetic field with a small $y$
component, this causes the increase of particle perpendicular momentum and
pitch angle, leading to the suppression of the anisotropy caused by
acceleration of reflected particles.

###  3.3 Averaged energy gain of reflected and unreflected particles

The reconnection-driven turbulence induces a range of local obliquities of the
inflow with respect to the turbulent magnetic field ${\bf B}_{\text{out}}$.
The range of pitch angles for reflection to happen depends on the local inflow
obliquity (see Section 3.1.1). Under the consideration of the acceleration of
both reflected and unreflected particles, we define the averaged fractional
energy gain for reconnection acceleration as

$d=P_{\text{r}}d_{r}+(1-P_{\text{r}})d_{\text{ur}},$ (40)

where

$d_{\text{r}}=\frac{\int_{\cos(\alpha_{\text{max}})}^{\cos(\alpha_{\text{min}})}\int_{0}^{\mu_{c}}V_{\text{rel}}\Big{[}\Big{(}\frac{E_{f}}{E_{0}}\Big{)}_{r}-1\Big{]}d\mu
d\cos\alpha}{\int_{\cos(\alpha_{\text{max}})}^{\cos(\alpha_{\text{min}})}\int_{0}^{\mu_{c}}V_{\text{rel}}d\mu
d\cos\alpha}$ (41)

is the pitch-angle and obliquity averaged fractional energy gain of reflected
particles,

$P_{\text{r}}=\frac{\int_{\cos(\alpha_{\text{max}})}^{\cos(\alpha_{\text{min}})}\int_{0}^{\mu_{c}}V_{\text{rel}}d\mu
d\cos\alpha}{\int_{0}^{\cos(\alpha_{\text{min}})}\int_{0}^{1}V_{\text{rel}}d\mu
d\cos\alpha}$ (42)

is the probability of reflection, and $d_{\text{ur}}$ as the averaged
fractional energy gain of unreflected particles is given by Eq. (36). At a
given $\theta$, $d_{r}$ depends on the range of $\beta$. The smallest
obliquity depends on the level of turbulence and is
$\alpha_{\text{min}}=90^{\circ}-\delta\alpha\sim 90^{\circ}-\theta$ (see
Section 2.1). As both the level of reconnection-driven turbulence and
$U_{\text{in}}$ depend on $\theta$, the corresponding $\beta$ is (Eq. (6))

$\beta_{\text{min}}\approx\frac{U_{\text{in}}}{c\cos(\pi/2-\theta)}=\frac{fB_{\text{in}}}{c\sqrt{4\pi\rho}},$
(43)

which is independent of $\theta$. The largest obliquity $\alpha_{\text{max}}$
allowed for reflection is limited by the condition in Eq. (17), where we
assume $B_{\text{out}}/B_{\text{in}}\sim\cos\theta$. So the corresponding
$\beta_{\text{max}}$ only depends on $\theta$. Toward a larger $\theta$, the
mirroring effect becomes stronger, and thus the reflection can happen at
larger $\beta$ values, leading to a larger $d_{r}$. $d_{\text{ur}}$ only
depends on $\beta_{\text{in}}$ (see Eq. (38)) and thus also increases with
$\theta$. The first term in Eq. (40) corresponding to the acceleration of
reflected particles only exists when $\beta_{\text{max}}>\beta_{\text{min}}$,
i.e., $\cos\theta<1-\beta_{\text{min}}^{2}$.

When $\beta_{\text{max}}\gg\beta_{\text{min}}$, we approximately have

$\displaystyle\int_{\cos(\alpha_{\text{max}})}^{\cos(\alpha_{\text{min}})}\int_{0}^{\mu_{c}}V_{\text{rel}}d\mu
d\cos\alpha$ (44) $\displaystyle\approx$
$\displaystyle\frac{U_{\text{in}}^{2}}{c^{2}}\int_{\beta_{\text{min}}}^{\beta_{\text{max}}}\frac{1}{\beta^{3}}\Big{(}\frac{\mu_{c}^{2}}{2}+\beta\mu_{c}\Big{)}d\beta$
$\displaystyle\approx$
$\displaystyle\frac{\beta_{\text{max}}^{2}U_{\text{in}}^{2}}{2c^{2}}\int_{\beta_{\text{min}}}^{\beta_{\text{max}}}\frac{d\beta}{\beta^{3}}$
$\displaystyle\approx$
$\displaystyle\frac{\beta_{\text{max}}^{2}U_{\text{in}}^{2}}{4c^{2}\beta_{\text{min}}^{2}},$

and

$\displaystyle\int_{0}^{\cos(\alpha_{\text{min}})}\int_{0}^{1}V_{\text{rel}}d\mu
d\cos\alpha$ (45) $\displaystyle\approx$
$\displaystyle\frac{U_{\text{in}}^{2}}{c^{2}}\int_{\beta_{\text{min}}}^{\infty}\frac{1}{\beta^{3}}\Big{(}\frac{1}{2}+\beta\Big{)}d\beta$
$\displaystyle\approx$
$\displaystyle\frac{U_{\text{in}}^{2}}{4c^{2}\beta_{\text{min}}^{2}},$

where $\mu_{c}\approx\beta_{\text{max}}-\beta$ is used, and the integral is
dominated by small $\beta$. Therefore, we have

$P_{r}\approx\beta_{\text{max}}^{2}=1-\frac{B_{\text{out}}}{B_{\text{in}}},$
(46)

which only depends on $\theta$ and increases with $\theta$. To derive the
approximate expression of $d_{r}$, we also find

$\displaystyle\int_{\cos(\alpha_{\text{max}})}^{\cos(\alpha_{\text{min}})}\int_{0}^{\mu_{c}}V_{\text{rel}}\Big{[}\Big{(}\frac{E_{f}}{E_{0}}\Big{)}_{r}-1\Big{]}d\mu
d\cos\alpha$ (47) $\displaystyle\approx$
$\displaystyle\frac{2U_{\text{in}}^{2}}{3c^{2}}\int_{\beta_{\text{min}}}^{\beta_{\text{max}}}\frac{(\mu_{c}+\beta)^{3}-\beta^{3}}{\beta^{2}(1-\beta^{2})}d\beta$
$\displaystyle\approx$
$\displaystyle\frac{2U_{\text{in}}^{2}\beta_{\text{max}}^{3}}{3c^{2}\beta_{\text{min}}},$

where we consider that the integral is dominated by small $\beta$. Combining
Eqs. (44) and (47) yields

$d_{r}\approx\frac{8\beta_{\text{min}}\beta_{\text{max}}}{3}.$ (48)

Together with the approximate expression of $d_{\text{ur}}$ at a small
$\beta_{\text{in}}$ in Eq. (39), we obtain

$d\approx\frac{8\beta_{\text{min}}\beta_{\text{max}}^{3}}{3}+\frac{4(1-\beta_{\text{max}}^{2})\beta_{\text{in}}}{3}.$
(49)

It can be further simplified to

$d\approx d_{\text{ur}}\approx\frac{4\beta_{\text{in}}}{3}$ (50)

at a small $\theta$, and

$d\approx d_{\text{r}}\approx\frac{8\beta_{\text{min}}}{3}$ (51)

at a large $\theta$.

In Figs. 6 and 6, we present $d_{\text{r}}$, $d_{\text{ur}}$, and $d$
calculated at $fB_{\text{in}}/(c\sqrt{4\pi\rho})=0.07$. At a small $\theta$,
$d_{\text{r}}$ and $d_{\text{ur}}$ are comparable, but $P_{r}$ is small
because of the small ranges of obliquities and pitch angles allowed for
reflection. Therefore, $d$ is dominated by $d_{\text{ur}}$. With the increase
of $\theta$, $P_{r}$ increases and reaches a large value at a large $\theta$.
Hence $d$ is dominated by $d_{\text{r}}$, which is larger than $d_{\text{ur}}$
due to the larger energy gain at a larger $\beta$ for reflection acceleration.
We also present the approximate results in the limit of
$\beta_{\text{min}}\ll\beta_{\text{max}}$ derived above. At a small $\theta$
with a small $\beta_{\text{max}}$, the condition
$\beta_{\text{min}}\ll\beta_{\text{max}}$ is not well satisfied. At a large
$\theta$, the consideration that the integral in Eq. (47) is dominated by
small $\beta$ is not well justified. Therefore, we see some deviations in
$d_{r}$, $P_{r}$, and $d$ despite the generally good agreement.

In Figs. 6 and 6, we present $d_{\text{r}}$, $d_{\text{ur}}$, and $d$
calculated at $fB_{\text{in}}/(c\sqrt{4\pi\rho})=0.7$. Given a larger
$\beta_{\text{min}}$, the condition for reflection cannot be satisfied over a
large range of $\theta$ with $\beta_{\text{min}}>\beta_{\text{max}}$, where
the energy gain is solely contributed by acceleration of unreflected particles
with $d=d_{\text{ur}}$. At a large $\theta$, large $\beta$ values with a large
$\beta_{\text{min}}$ give rise to a large $d_{\text{r}}$. $d_{\text{ur}}$ also
becomes significantly large with a large $\beta_{\text{in}}$. As the range of
pitch angles for reflection is limited at a large $\beta$, $P_{r}$ is small
even at a large $\theta$, and thus $d$ is dominated by $d_{\text{ur}}$.
$d_{\text{ur}}$ is well approximated by Eq. (38) (see Fig. 6).

We see that $d_{\text{r}}$ is in general larger than $d_{\text{ur}}$ when
reflection can happen. However, sufficiently strong mirroring and moderate
$\beta$ are required for a large $P_{r}$ and for $d$ to be dominated by
$d_{\text{r}}$. As a result, $d$ is usually dominated by $d_{\text{ur}}$ at a
small $\theta$. When $\beta_{\text{min}}$ is large, $d$ is basically
determined by $d_{\text{ur}}$ over all $\theta$. By comparing our result with
that in de Gouveia dal Pino & Lazarian (2005), where they derived
$(4/3)U_{\text{in}}/c$ as the pitch-angle averaged fractional energy gain per
crossing of the reconnection region, we find that their expression only
applies to unreflected particles in the case with $U_{\text{in}}\ll c$.

Figure 6: (a) Averaged fractional energy gain of reflected particles
$d_{\text{r}}$ (Eq. (41)) and unreflected particles $d_{\text{ur}}$ (Eq.
(36)). $P_{r}$ (Eq. (42)) is the probability of reflection. Filled circles,
open circles, and crosses correspond to the approximate expressions of
$d_{\text{r}}$ (Eq. (48)), $d_{\text{ur}}$ (Eq. (39)), and $P_{\text{r}}$ (Eq.
(46)) in the limit of $\beta_{\text{min}}\ll\beta_{\text{max}}$. (b) Total
averaged fractional energy gain $d$ (Eq. (40)) during reconnection
acceleration. The filled circles correspond to the approximate expression of
$d$ (Eq. (49)). (a) and (b) correspond to
$fB_{\text{in}}/(c\sqrt{4\pi\rho})=0.07$. (c) and (d) are the same as (a) and
(b), but for $fB_{\text{in}}/(c\sqrt{4\pi\rho})=0.7$. The open circles in (c)
correspond to $d_{\text{ur}}$ given by Eq. (38).

##  4 Probability of escape and particle energy spectrum

###  4.1 Probability of escape

Under the assumption of isotropic particle distribution in the local fluid
frames, in the reference frame of the reconnection region (hereafter R frame),
we find the number of particles that enter the reconnection region per unit
time as

$\displaystyle~{}~{}~{}~{}~{}~{}L_{x}L_{z}\int c\cos\phi fd^{3}{\bm{p}}$ (52)
$\displaystyle=2\pi L_{x}L_{z}\int c\cos\phi fp^{2}dpdcos\phi$
$\displaystyle=2\pi
cL_{x}L_{z}\int_{0}^{\infty}g{p^{\prime}}^{2}dp^{\prime}\int_{0}^{1}\frac{\cos\phi
d\cos\phi}{\Gamma_{\text{in}}^{3}(1-\beta_{\text{in}}\cos\phi)^{3}}$
$\displaystyle=\pi
cL_{x}L_{z}\int_{0}^{\infty}g{p^{\prime}}^{2}dp^{\prime}\frac{1}{\Gamma_{\text{in}}^{3}(1-\beta_{\text{in}})^{2}}$
$\displaystyle=\frac{n_{\text{in}}c}{4}L_{x}L_{z}\frac{1}{\Gamma_{\text{in}}^{3}(1-\beta_{\text{in}})^{2}}$
$\displaystyle=\frac{n_{R}c}{4}L_{x}L_{z}\frac{1}{\Gamma_{\text{in}}^{4}(1-\beta_{\text{in}})^{2}},$

where $\beta_{\text{in}}=U_{\text{in}}/c$,
$\Gamma_{\text{in}}=1/\sqrt{1-\beta_{\text{in}}^{2}}$, $n_{\text{in}}$ is the
number density of particles in the inflow frame,
$n_{R}=\Gamma_{\text{in}}n_{\text{in}}$ is the number density of particles in
the R frame, $g$ is the distribution function in the inflow frame, which is
equal to the distribution function $f$ in the R frame, $p^{\prime}$ and $p$
are particle momentum in the fluid frame and R frame, $\phi$ is the angle
between ${\bf p}$ and the inflow direction in the R frame, and $L_{z}$ is the
length along the $z$ direction. Here we apply the Lorentz transformation
between the inflow frame and the R frame with
$p^{\prime}=\Gamma_{\text{in}}p(1-\beta_{\text{in}}\cos\phi)$ following the
approach in Naito & Takahara (1995), and the particle distribution becomes
anisotropic in the R frame. When $\beta_{\text{in}}\ll 1$, the above
expression is approximately equal to

$\frac{n_{\text{in}}c}{4}L_{x}L_{z}.$ (53)

Particles can escape from the reconnection region by gyrating around the
turbulent guide field, which is advected away by the outflow from the
reconnection region. Of the particles that enter the reconnection region, the
number of escaping particles per unit time in the R frame is

$\displaystyle~{}~{}~{}~{}~{}~{}\Delta L_{z}\int c\cos\phi fd^{3}{\bm{p}}$
(54) $\displaystyle=2\pi c\Delta
L_{z}\int_{0}^{\infty}g{p^{\prime}}^{2}dp^{\prime}\int_{-1}^{1}\frac{\cos\phi
d\cos\phi}{\Gamma_{\text{out}}^{3}(1-\beta_{\text{out}}\cos\phi)^{3}}$
$\displaystyle=4\pi c\Delta
L_{z}\int_{0}^{\infty}g{p^{\prime}}^{2}dp^{\prime}\Gamma_{\text{out}}\beta_{\text{out}}$
$\displaystyle=n_{\text{out}}c\Delta
L_{z}\Gamma_{\text{out}}\beta_{\text{out}}$ $\displaystyle=n_{R}c\Delta
L_{z}\beta_{\text{out}},$

where $\beta_{\text{out}}=U_{\text{out}}/c$,
$\Gamma_{\text{out}}=1/\sqrt{1-\beta_{\text{out}}^{2}}$, and $n_{\text{out}}$
is the number density of particles in the outflow frame. Similar to the above
calculation, here $g$ and $p^{\prime}$ are the distribution function and
particle momentum in the outflow frame, and $\phi$ is the angle between ${\bf
p}$ and the outflow direction in the R frame. When $\beta_{\text{out}}\ll 1$,
the above expression approximately becomes

$n_{\text{out}}U_{\text{out}}\Delta L_{z}.$ (55)

The escape probability is the ratio of Eq. (54) to Eq. (52),

$P_{\text{esp}}=\frac{\Delta}{L_{x}}\frac{4\beta_{\text{out}}}{(1+\beta_{\text{in}})^{2}}.$
(56)

As $\Delta/L_{x}=f$ and $\beta_{\text{in}}=f\beta_{\text{out}}$, we have

$P_{\text{esp}}=\frac{4\beta_{\text{in}}}{(1+\beta_{\text{in}})^{2}},$ (57)

which is approximately

$P_{\text{esp}}\approx 4\beta_{\text{in}}$ (58)

when $\beta_{\text{in}}\ll 1$. The probability for a particle to remain in the
reconnection region is

$P=1-P_{\text{esp}}=1-\frac{4\beta_{\text{in}}}{(1+\beta_{\text{in}})^{2}},$
(59)

which is small when $\beta_{\text{in}}$ is large. Fig. 7 shows
$P_{\text{esp}}$ as a function of $\theta$. We see that with the increase of
$fB_{\text{in}}/\sqrt{4\pi\rho}$ and $\theta$, $P_{\text{esp}}$ can approach
unity at a sufficiently large $\beta_{\text{in}}$.

###  4.2 Spectral index of accelerated particles

The shape of particle energy spectrum $N(E)\propto E^{-\zeta}$, where $E$ is
the particle energy, is determined by the competition between acceleration and
escape of particles. It has the power-law index as (Longair, 1997)

$\zeta=1-\frac{\ln P}{\ln\epsilon},$ (60)

where $\epsilon=1+d$, $d$ is given by Eq. (40), and $P$ is given by Eq. (59).
Fig. 7 shows $\zeta$ as a function of $\theta$ at different values of
$fB_{\text{in}}/\sqrt{4\pi\rho}$. We see that the reconnection-accelerated
particles have a non-universal energy spectral index. At a small $\theta$ and
a small $U_{\text{in}}$, the energy gain mainly comes from the acceleration of
unreflected particles, and $P_{\text{esp}}$ is small. There is approximately
(Eqs. (50) and (58)),

$\zeta\approx
1-\frac{\ln(1-4\beta_{\text{in}})}{\ln(1+\frac{4\beta_{\text{in}}}{3})}\approx
4,$ (61)

as shown in Fig. 7. This corresponds to the steepest energy spectrum of
reconnection-accelerated particles. With the increase of $\theta$, $\zeta$
decreases because of the larger energy gain toward larger $\theta$. $\zeta$ at
a large $\theta$ depends on the value of $U_{\text{in}}$. When $U_{\text{in}}$
is large, although the energy gain is significant, the large $P_{\text{esp}}$
leads to a steep spectrum. When $U_{\text{in}}$ is small (relative to $c$),
the energy gain is mainly contributed by the acceleration of reflected
particles, and $P_{\text{esp}}$ is small. We approximately have (Eqs. (51) and
(58))

$\zeta\approx
1-\frac{\ln(1-4\beta_{\text{in}})}{\ln(1+\frac{8\beta_{\text{min}}}{3})}\approx
2.5.$ (62)

This estimate is a bit larger than $\zeta$ at a large $\theta$ calculated for
$fB_{\text{in}}/(c\sqrt{4\pi\rho})=0.07$ in Fig. 7 due to the underestimate of
$d$ (see Fig. 6).

Based on the above analysis, we see that $\zeta$ of reconnection-accelerated
particles depends on $U_{\text{in}}$. It approximately falls in the range
$2.5\lesssim\zeta<4$.

Figure 7:  (a) $P_{\text{esp}}$ calculated using Eq. (57). Solid and dashed
lines correspond to $fB_{\text{in}}/(c\sqrt{4\pi\rho})=0.07$ and
$fB_{\text{in}}/(c\sqrt{4\pi\rho})=0.7$, respectively. (b) Spectral index
$\zeta$ (Eq. (60)) as a function of $\theta$. The same symbols as for (a) are
used. Filled circles correspond to the approximate result with $d$ estimated
by Eq. (49). Open circles correspond to the approximate result using $d\approx
d_{\text{ur}}$, where $d_{\text{ur}}$ is given by Eq. (38).

##  5 Acceleration time

As scattering diffusion of particles is not a necessary process for confining
particles, particles can ballistically move along magnetic field lines in the
reconnection region. As particles cannot penetrate deep into the inflows, we
consider the crossing time of a particle as its residence time in the
reconnection region. For isotropic pitch angle distribution, the crossing time
of a particle is estimated as

$\tau=\frac{\Delta}{c\langle\mu\rangle\langle\cos\alpha\rangle},$ (63)

where

$\langle\cos\alpha\rangle=\frac{\int_{0}^{\cos(\alpha_{\text{min}})}\cos\alpha
d\cos\alpha}{\int_{0}^{\cos(\alpha_{\text{min}})}d\cos\alpha}\approx\frac{\sin\theta}{2},$
(64)

and

$\langle\mu\rangle=\frac{\int_{0}^{1}\mu d\mu}{\int_{0}^{1}d\mu}=\frac{1}{2}.$
(65)

The turbulence-perturbed magnetic fields in the reconnection region have their
orientations deviate from the original guide field direction. With the
increase of $\theta$, a higher level of turbulence introduces a larger range
of magnetic field orientations. As a result, $\tau$ decreases with increasing
$\theta$.

The acceleration time is then

$t_{\text{acc}}=\frac{\tau}{d},$ (66)

where $d$ is given by Eq. (40). At a small $\theta$, $d$ is dominated by
$d_{\text{ur}}$. By using the approximate expression of $d_{\text{ur}}$ (Eq.
(39)) at a small $\beta_{\text{in}}$, we find

$t_{\text{acc}}\approx\frac{3\Delta}{\sin\theta
U_{\text{in}}},~{}(\text{small}~{}\theta).$ (67)

With the increase of $\theta$, $\tau$ decreases and $d$ increases. So the
resulting $t_{\text{acc}}$ decreases. At a large $\theta$, when
$U_{\text{in}}$ is small (relative to $c$), $d$ is dominated by
$d_{\text{r}}$. By using the approximate expression of $d_{\text{r}}$ (Eq.
(51)), we find

$t_{\text{acc}}\approx\frac{3\Delta}{2c\beta_{\text{min}}}\approx\frac{3\Delta}{2U_{\text{in}}},~{}(\text{large}~{}\theta,~{}\text{small}~{}\beta_{\text{in}}).$
(68)

As shown in Fig. 8, at a large $\theta$ and a large $U_{\text{in}}$,
$t_{\text{acc}}$ is close to the light crossing time of $\Delta$, which has an
approximate expression

$t_{\text{acc}}\approx\frac{4\Delta}{cd_{\text{ur}}},~{}(\text{large}~{}\theta,~{}\text{large}~{}\beta_{\text{in}}),$
(69)

where $d_{\text{ur}}$ is given by Eq. (38).

In the above estimate of $\tau$, we consider that particles can ballistically
cross the reconnection region with inefficient scattering by turbulence, as
suggested by the measured particle trajectories in turbulent reconnection
simulations (Zhang et al., 2022). We note that the turbulent superdiffusion
perpendicular to the mean guide field (e.g., Xu & Yan 2013; Hu et al. 2022a)
can shorten $\tau$ at a small $\theta$, while the nonresonant interaction with
turbulent magnetic compressions can cause slow diffusion (Lazarian & Xu,
2021b) and a larger $\tau$ at a large $\theta$.

Figure 8: $t_{\text{acc}}$ (normalized by $\Delta/c$) vs. $\theta$ calculated
using Eq. (66). Solid and dashed lines correspond to
$fB_{\text{in}}/(c\sqrt{4\pi\rho})=0.07$ and
$fB_{\text{in}}/(c\sqrt{4\pi\rho})=0.7$, respectively. Filled circles
correspond to the approximate result with $d$ estimated by Eq. (49). Open
circles correspond to the approximate result using $d\approx d_{\text{ur}}$,
where $d_{\text{ur}}$ is given by Eq. (38).

##  6 Discussion

###  6.1 Comparison between reconnection acceleration and oblique shock
acceleration

At an oblique shock, particles are energized by a combination of SDA and DSA
(Jokipii, 1982; Decker, 1988; Ostrowski, 1988; Kirk & Heavens, 1989; Xu &
Lazarian, 2022). They both have the same nature of Fermi acceleration via
bouncing between approaching magnetic mirrors, including the mirror created by
shock compression or the “mirrors” created by amplified magnetic
irregularities upstream and downstream of the shock. The increase of shock
obliquity causes a stronger shock compression of magnetic fields and thus a
stronger mirroring effect, as well as a larger fractional energy gain.
Isotropic particle distribution and particle diffusion are needed for the
particles to repeatedly return to the shock in both upstream and downstream
regions. The isotropic particle distribution is traditionally related to
efficient pitch-angle scattering without specifying the scattering process. It
can also be realized when particles follow isotropically distributed turbulent
magnetic fields in super-Alfvénic turbulence (Xu & Lazarian, 2022), which are
amplified by nonlinear turbulent dynamo (Xu & Lazarian, 2016, 2017; Hu et al.,
2022b).

For reconnection acceleration, particles undergo head-on collisions with the
inflows driven by magnetic reconnection. Unlike the shock acceleration, the
compression of gas and density change does not play an important role for the
reconnection acceleration. The mirror reflection of particles is caused by the
annihilation of the antiparallel components of magnetic fields. The resulting
change of magnetic field strength is not limited by the gas compression ratio
as in the case of shock acceleration. At a large $\theta$ with a weak guide
field, a much stronger mirroring effect than that of a highly oblique shock is
expected. Therefore, when the condition for reflection is satisfied, a single
encounter with the inflow can result in a much larger fractional energy gain
compared to the interaction with an oblique shock. Unlike the shock
acceleration, particles with loss-cone pitch angles are not transmitted
through the inflow. The unreflected particles are attached to the incoming
magnetic fields and sent back to the reconnection region as the incoming
magnetic fields reconnect. They gain energy during this process.

Compared with nonrelativistic shock acceleration, it becomes much more
challenging for particles to return to the shock from downstream for
relativistic shock acceleration. For relativistic reconnection acceleration,
the return of particles to the reconnection region occurs naturally, allowing
efficient acceleration.

Both magnetic mirroring and convergence of magnetic fields contribute to the
confinement of particles in the reconnection region. Therefore, scattering
diffusion is not a necessary confining mechanism for reconnection
acceleration. For the shock acceleration, the anisotropic diffusion of
particles is important for determining the acceleration efficiencies at
different shock obliquities (Jokipii, 1987; Xu & Lazarian, 2022). For the
reconnection acceleration, instead of having a slow diffusive motion,
particles can ballistically move along magnetic field lines in the
reconnection region when the scattering of particles by turbulence is
inefficient, leading to a much shorter time between successive accelerations
than that for diffusive shock acceleration. Both the significant energy gain
and short residence time at a large $\theta$ can result in much shorter
acceleration timescale and more efficient acceleration than shock
acceleration.

The escape of particles in the far downstream region of a shock is caused by
the advection of particles with the flow due to isotropic particle
distribution. The escape of particles from the reconnection region is caused
by the advection of the guide field with the outflow, irrespective of the
particle distribution. For a highly oblique shock, $P_{\text{esp}}$ becomes
large due to the large fluid speed along the magnetic field. For reconnection
acceleration, at a large $\theta$, $P_{\text{esp}}$ becomes large when the
inflow speed (relative to $c$) is large. Both shock acceleration and
reconnection acceleration have non-universal indices of particle energy
spectra. Acceleration at a quasi-parallel shock with a small shock speed leads
to a spectral index equal to $2$ (Bell, 1978). A steep particle energy
spectrum with the spectral index much larger than $2$ is expected at a large
shock obliquity and a large shock speed (Xu & Lazarian, 2022; Xu, 2022). For
reconnection acceleration, we find the particle spectral index dependent on
$U_{\text{in}}$ and $\theta$ and falling in the range $2.5\lesssim\zeta<4$.

For both shock acceleration and reconnection acceleration, turbulence plays a
critical role. For the former case, the amplification of magnetic fields via
the nonlinear turbulent dynamo is important for confining particles near the
shock and determining the largest achievable particle energy. Turbulence
introduces variations of shock obliquities, enables the combined acceleration
mechanism of SDA and DSA, and affects the diffusion of particles in directions
parallel and perpendicular to magnetic fields and acceleration efficiency (Xu
& Lazarian, 2022). For the latter case, turbulence enables fast magnetic
reconnection. The efficient magnetic reconnection is the prerequisite for
efficient reconnection acceleration. Turbulence introduces variations of
inflow obliquities, enables the RDA of reflected particles, and affects the
inflow speed and acceleration efficiency.

Drury (2012) applied the “box” model for shock acceleration (Drury et al.,
1999) to reconnection acceleration. For the reconnection model, Drury (2012)
considered a thin reconnection layer rather than a turbulence-broadened
reconnection region. Accordingly, the acceleration with $r_{g}\gg\Delta$
happens outside the reconnection layer for the “box” model. Under the
assumption that the accelerated particles can sample both the converging
inflows and diverging outflows, the acceleration occurs only by virtue of
fluid compression. With a constant density in an incompressible medium, the
“box” model leads to zero energy gain. As discussed in Section 2.2, the
assumption of efficient scattering responsible for isotropizing the particle
distribution in the entire system that encompasses both inflows and outflows
is invalid, as turbulence only exists within the reconnection region. In
comparison, the turbulent reconnection acceleration with $r_{g}<\Delta$
considered in this work happens within the reconnection region. Particles gain
energy from the converging inflows, irrespective of the fluid compressibility.

###  6.2 Comparison with other earlier studies on reconnection acceleration

It is generally accepted that the acceleration by the motional electric field
associated with reconnection-driven bulk flows dominates over the acceleration
by the non-ideal electric field at reconnection X-points (Guo et al., 2019).
As discussed in Section 2.2, in 2D reconnection models, the curvature drift
acceleration due to the motional electric field induced by contraction and
merging of magnetic islands is identified as the primary acceleration
mechanism. In 3D reconnection, magnetic islands do not exist. The curvature
drift acceleration associated with the motional electric field induced by
outflows is only important when the reconnection is in the linear stage with
insignificant turbulence and slow inflows (Beresnyak & Li, 2016). As the
outflow speed only approaches $V_{A}$ at the reconnection exhausts, only a
small fraction of particles in the outflows far away from the central
reconnection region can gain energy from the kinetic energy of the outflows.
When the reconnection enters the nonlinear stage with significant turbulence
and thus inflows, i.e., high reconnection rate, the magnetic energy release
related to magnetic pressure gradients dominates over that related to magnetic
curvature relaxation (Du et al., 2022). The Fermi acceleration associated with
the motional electric field induced by inflows becomes important, as indicated
by test particle simulations (Zhang et al., 2022).

We do not consider the 2D reconnection with tearing instability and plasmoids,
which are suppressed by other fluid instabilities, e.g., Kelvin-Helmholtz
instability, and turbulence in 3D reconnection (Beresnyak, 2017; Kowal et al.,
2020). The difference between 2D and 3D reconnection entails the difference
between 2D and 3D reconnection acceleration. Unlike 2D reconnection where the
reconnection sites, i.e., X-points, occupy a vanishing fraction of a large-
scale system (Dahlin et al., 2014), 3D turbulent reconnection has the volume-
filling reconnection region, which drives significant inflows. Particles gain
energy from the kinetic energy of the inflows (de Gouveia dal Pino & Lazarian,
2005), and thus the acceleration is strongly dependent on the turbulent
reconnection rate. The LV99 model we adopt is applicable to describing the
fast reconnection events in solar flares (Chitta & Lazarian, 2020) and other
high-energy astrophysical phenomena (Lazarian et al., 2019). The corresponding
reconnection rate is $\gtrsim 0.1$. Under the consideration of fast
reconnection, we expect that the acceleration arising from particles
interacting with the inflows becomes the dominant acceleration mechanism. Its
dominance over the curvature drift acceleration in 3D fast turbulent
reconnection will be further numerically tested in our future work.

In addition to the reconnection-driven turbulence, when there is an external
source of turbulence, the reconnection rate can be further increased (LV99;
Kowal et al. 2009). The externally driven turbulence can increase the range of
inflow obliquities for more particles to undergo reflection at a small
$\theta$.

Although the medium compressibility is not required for reconnection
acceleration (see Section 2.2), the increase of the turbulence driven by
reconnection in a highly compressible medium (Kowal et al. in prep) is
important for increasing the reconnection rate and thus the acceleration
efficiency.

##  7 Conclusions

Efficient reconnection acceleration of energetic particles requires a global
high reconnection rate on macroscopic scales. The macroscopic diffusion of
turbulent magnetic fields is independent of microscopic magnetic
diffusivities, and thus the 3D turbulent reconnection is a fast reconnection
process. The macroscopic reconnection rate is high when the level of
turbulence is high (LV99).

Without invoking the compression of gas, particles that bounce between
converging magnetic fields are accelerated by head-on collisions with the fast
inflows, with the kinetic energy of the inflows converted to the particle
energy. Furthermore, the reconnection-driven turbulence introduces a variety
of local inflow obliquities. With the large inflow speed along the local
magnetic field in the reconnection region, a reflected particle can
occasionally gain significant energy after a single encounter with the inflow.
The effect of fluid compressibility on acceleration is via its effect on the
reconnection rate.

Our study applies to the reconnection events with a high reconnection rate,
where the acceleration arising from the interaction of particles with the
inflows is important. We note that in the linear stage of reconnection with a
low reconnection rate, the acceleration via the interaction with the outflows,
i.e., the curvature drift acceleration, may become dominant (Beresnyak & Li,
2016).

At a small $\theta$ with a strong guide field, the mirroring effect caused by
magnetic reconnection is weak, and the acceleration is dominated by that of
unreflected particles. The unreflected particles are confined in the
reconnection region due to the convergence of strong incoming magnetic fields.
At a large $\theta$ with a weak guide field, the mirroring effect becomes
strong, allowing particles to be reflected at large inflow obliquities. The
acceleration is dominated by that of reflected particles unless the inflow
speed is a significant fraction of light speed. In the latter case, the
acceleration at a large $\theta$ is also dominated by that of unreflected
particles. The confinement of reflected particles in the reconnection region
is due to the mirror reflection. Both reflected and unreflected particles
undergo the reconnection drift acceleration, which is caused by the gradient
drift of particles along the motional electric field induced by the inflows.

Particles attached to turbulent magnetic fields can escape from the
reconnection region by advection with the outflow. The escape probability
$P_{\text{esp}}$ turns out to depend on the inflow speed because of the mass
conservation. Given the dependence of the averaged fractional energy gain $d$
and $P_{\text{esp}}$ on the inflow speed, the energy spectrum of accelerated
particles does not have a universal spectral index. We find that it
approximately falls in the range $2.5\lesssim\zeta<4$. The steepest spectrum
with $\zeta=4$ is expected at a small $\theta$.

Scattering diffusion is not a necessary mechanism for confining particles in
the reconnection region. Given a short crossing time of a particle through the
reconnection region by ballistically following the turbulent magnetic field,
the reconnection acceleration at a large $\theta$ and a large inflow speed can
be very efficient, with the acceleration timescale comparable to the light
crossing time of the thickness $\Delta$ of the reconnection region. Moreover,
the maximum particle energy is not limited by $\Delta$, as particles with
$r_{g}>\Delta$ can also be accelerated (Lazarian et al., 2012; Zhang et al.,
2021a). Given the macroscopic size of $\Delta$ of turbulent reconnection and
strong magnetic fields in the inflows, particles can be accelerated to very
high energies. For relativistic reconnection acceleration, particles are
naturally trapped in the reconnection region by strong incoming magnetic
fields and undergo successive acceleration with very significant energy gain.
With all above factors considered, we see that 3D turbulent reconnection
acceleration can be a very promising mechanism for producing ultra-high-energy
cosmic rays in strongly magnetized astrophysical plasmas.

S.X. acknowledges the support for this work provided by NASA through the NASA
Hubble Fellowship grant # HST-HF2-51473.001-A awarded by the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, Incorporated, under NASA contract NAS5-26555. A.L.
acknowledges the support of NASA ATP AAH7546.

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