# Role of Parallel Solenoidal Electric Field on Energy Conversion in 2.5D
Decaying Turbulence with a Guide Magnetic Field

Peera Pongkitiwanichakul<EMAIL_ADDRESS>Department of Physics, Faculty of
Science, Kasetsart University, Bangkok 10900, Thailand  David Ruffolo
Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400,
Thailand  Fan Guo  Los Alamos National Laboratory, Los Alamos, NM 87545, USA
New Mexico Consortium, Los Alamos, NM 87544, USA  Senbei Du  Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  Piyawat Suetrong  Department
of Physics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand
Chutima Yannawa  Department of Physics, Faculty of Science, Kasetsart
University, Bangkok 10900, Thailand  Kirit Makwana  Indian Institute of
Technology Hyderabad Kandi-502285, Sangareddy Telangana, India  Kittipat
Malakit  Department of Physics, Faculty of Science and Technology, Thammasat
University, Pathum Thani 12120, Thailand

###### Abstract

We perform 2.5D particle-in-cell simulations of decaying turbulence in the
presence of a guide (out-of-plane) background magnetic field. The fluctuating
magnetic field initially consists of Fourier modes at low wavenumbers (long
wavelengths). With time, the electromagnetic energy is converted to plasma
kinetic energy (bulk flow+thermal energy) at the rate per unit volume of ${\bf
J}\cdot{\bf E}$ for current density ${\bf J}$ and electric field ${\bf E}$.
Such decaying turbulence is well known to evolve toward a state with strongly
intermittent plasma current. Here we decompose the electric field into
components that are irrotational, ${\bf E}_{\rm ir}$, and solenoidal
(divergence-free), ${\bf E}_{\rm so}$. ${\bf E}_{\rm ir}$ is associated with
charge separation, and ${\bf J}\cdot{\bf E}_{\rm ir}$ is a rate of energy
transfer between ions and electrons with little net change in plasma kinetic
energy. Therefore, the net rate of conversion of electromagnetic energy to
plasma kinetic energy is strongly dominated by ${\bf J}\cdot{\bf E}_{\rm so}$,
and for a strong guide magnetic field, this mainly involves the component
${\bf E}_{\rm so,\parallel}$ parallel to the total magnetic field ${\bf B}$.
We examine various indicators of the spatial distribution of the energy
transfer rate J${}_{\parallel}\cdot$Eso,∥, which relates to magnetic
reconnection, the best of which are 1) the ratio of the out-of-plane electric
field to the in-plane magnetic field, 2) the out-of-plane component of the
non-ideal electric field, and 3) the magnitude of the estimate of current
helicity.

##  1  Introduction

The conversion of magnetic field energy into internal energy of plasmas in
weakly collisional or collisionless plasmas is not fully understood and has
important implications for space plasma and astrophysical plasma phenomena
(Parashar et al., 2015). A magnetic field in a non-potential form can serve as
a source of energy for plasma heating and/or particle acceleration in numerous
space, solar and astrophysical processes such as coronal heating (van
Ballegooijen, 1986; Parker, 1988; Pongkitiwanichakul et al., 2015), solar
flares (Li et al., 2017; Chen et al., 2020; Fu et al., 2020), solar wind
acceleration (De Pontieu et al., 2007), and flares in relativistic jets (Zhang
et al., 2015; Guo et al., 2015; Comisso & Sironi, 2018; Zhang et al., 2018,
2020). These events involve multiple phenomena operating on a wide range of
scales. At large scales, the plasma behaves like a fluid and nonlinear
interactions such as turbulence can cause an energy cascade toward small
scales where kinetic physics becomes important (Leamon et al., 1998; Gary &
Borovsky, 2004; Sahraoui et al., 2009; Chandran et al., 2010; Wan et al.,
2012; Perri et al., 2012; TenBarge & Howes, 2013). While the overall energy
conversion rate can be controlled by turbulence at large scales (Wu et al.,
2013; Pongkitiwanichakul et al., 2015), unclear mechanisms at kinetic scales
determine the proportion of energy transferred to ions or electrons. Ions gain
much more energy than electrons in high-energy turbulent astrophysical systems
(Zhdankin et al., 2019). Changing the level of fluctuation at kinetic scales
causes a change in dominant heating mode. For a higher amplitude of
fluctuation, ions gain energy more effectively, while a lower amplitude
supports more electron heating (Gary et al., 2016; Hughes et al., 2017; Shay
et al., 2018).

A number of recent works searched for the responsible mechanisms and the
locations for the energy conversion by using the statistics of properties of
the energy transfer rate per unit volume from electromagnetic energy to plasma
energy, ${\bf J}\cdot{\bf E}$, where ${\bf J}$ is a current density and ${\bf
E}$ is an electric field. Some previous studies explored how different
components of ${\bf J}\cdot{\bf E}$ contribute to the total conversion rate.
For example, one can consider the perpendicular component of ${\bf J}$ as the
sum of multiple currents due to the anisotropic and non-uniform pressure
tensor, such as the grad-B current, the curvature current, and the
polarization current (Dahlin et al., 2014; Li et al., 2017, 2018, 2019a). In
the case of large-scale magnetic reconnection, ${\bf J}\cdot{\bf E}$ from the
curvature current dominates. In the case of decaying turbulence with no
dominant magnetic reconnection site, the currents are also intermittent and
form sheet-like structures (Camporeale et al., 2018). For this case, the main
contribution to ${\bf J}\cdot{\bf E}$ comes from the interactions between
particles and the parallel component of the electric field (Wan et al., 2012;
Makwana et al., 2017). Another line of work has considered the conversion into
internal energy due to the interaction between the pressure tensor and
gradient of the components of bulk flows (Yang et al., 2017a, b; Du et al.,
2018, 2020).

Examination of the spatial distribution, species contributions, and
directional components of ${\bf J}\cdot{\bf E}$ is useful for understanding
how a magnetic field loses its energy to particles. Furthermore, we point out
the utility of decomposing ${\bf E}$ into irrotational (${\bf E}_{\rm ir}$)
and solenoidal (divergence-free, ${\bf E}_{\rm so}$) components. The
irrotational component arises from the charge separation between ions and
electrons. Consider the rates of change of magnetic field and electric field
energy according to Maxwell’s equations:

$\displaystyle\frac{\partial}{\partial t}\left(\frac{B^{2}}{2\mu_{0}}\right)$
$\displaystyle=$ $\displaystyle-\frac{1}{\mu_{0}}{\bf B}\cdot\nabla\times{\bf
E}$ (1) $\displaystyle\frac{\partial}{\partial
t}\left(\frac{\epsilon_{0}E^{2}}{2}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\mu_{0}}{\bf E}\cdot\nabla\times{\bf B}-{\bf
J}\cdot{\bf E}.$ (2)

The first term on the right hand side of Equation 2 has the same spatial
average as the negative of the right hand side of Equation 1, which represents
the rate of conversion of magnetic energy to electric energy. From Equation 1,
the irrotational component does not contribute to the time evolution of the
magnetic field. In that sense, it cannot exchange energy with the magnetic
field. Only the solenoidal component ${\bf E}_{\rm so}$ can directly drain
energy from the magnetic field. Figure 1 shows an illustration of how overall
energy is converted between the magnetic field, electric field and particle
kinetic energy (of both bulk flow and thermal energy). The energy density
conversion rate between the magnetic field and the electric field is ${\bf
B}\cdot\nabla\times{\bf E}_{\rm so}/\mu_{0}$, while ${\bf J}\cdot{\bf E}$ is
the energy density conversion rate between the electric field and particles.
The characteristics of ${\bf E}_{\rm so}$ control the energy conversion
between the magnetic field and the electric field. As will be discussed
shortly, our simulation results show that the energy density in the electric
field remains low with a very slow rate of change, compared to the magnetic
energy or kinetic energy.

Furthermore, we find that the spatial average $\langle{\bf J\cdot
E_{ir}}\rangle$ is very low. Therefore, the net energy flow is from the
magnetic field through the solenoidal electric field energy and then, at about
the same rate, through to the particle kinetic energy via ${\bf J\cdot
E_{so}}$. In other words, ${\bf E_{so}}$ controls the flow of magnetic energy
to particle kinetic energy, with a spatial distribution marked by ${\bf J\cdot
E_{so}}$. The analysis of ${\bf J}\cdot{\bf E}$ from previous studies
generally included a major contribution from ${\bf E}_{\rm ir}$, which may
obscure the net energy-conversion mechanism and spatial distribution.

Figure 1: Schematic of the net energy transfer between the magnetic field, the
electric field, and particles via $\langle{\bf B}\cdot\nabla\times{\bf E}_{\rm
so}/\mu_{0}\rangle$ and $\langle{\bf J}\cdot{\bf E}\rangle$. According to our
simulation results, for decaying 2.5D turbulence the two net energy transfer
rates are nearly equal, and the net transfer $\langle{\bf J\cdot E}\rangle$ is
dominated by the solenoidal component $\langle{\bf J\cdot E_{so}}\rangle$.

In this work, we investigate how the magnetic energy is converted into the
kinetic energy of particles. We run particle-in-cell simulations for cases of
2.5-dimensional (2.5D) decaying turbulence with an initially uniform guide
magnetic field. The purpose of studying decaying turbulence is to study the
evolution of turbulence without the strong influence of the turbulent energy
input (forcing) that would have to be added ad hoc to maintain a steady state.
Simulations in 2.5D are useful for effectively utilizing computational
resources, particularly for the case of a strong guide field, where the
turbulent dynamics involve stronger spatial variations along the two
directions perpendicular to that field (Montgomery, 1982), i.e., they have a
quasi-2D nature. We analyze the behavior of ${\bf J}\cdot{\bf E}$ from ions
and electrons separately. We use Helmholtz decomposition (Kida & Orszag, 1992;
Yang et al., 2021) to separate the electric field into the irrotational
component and the solenoidal component. We focus on how the solenoidal
component drains energy out of the magnetic field and find which types of
particles gain energy from this component. We consider multiple magnitudes of
the guide field and observe how the energy conversion is modified. We find
that the simulations have two phases. In the early phase, magnetic islands are
formed. In the later phase, these magnetic islands merge. As plasma dynamics
in the later phase is less sensitive to the initial conditions and more
closely related to astrophysical applications, we focus on the energy transfer
in this phase. In the later phase, the interaction between particles and the
parallel component of the solenoidal electric field is the main mechanism to
drain energy out of the magnetic field in the case of a strong guide field.
The interaction is strongly localized, and we find that three good indicators
of the spatial locations are 1) the ratio of the out-of-plane electric field
to the in-plane magnetic field, 2) the out-of-plane component of the non-ideal
electric field, and 3) the magnitude of the estimate of current helicity.

##  2  Simulation Setup

Simulations presented here were performed using VPIC (Bowers et al., 2008),
which is an explicit particle-in-cell (PIC) code that has been used to study
magnetic reconnection (Daughton et al., 2011; Guo et al., 2020b, a),
turbulence (Wan et al., 2015), and particle energization (Guo et al., 2014,
2015, 2019; Li et al., 2019b). The simulations are 2.5D with $y$ as the
ignorable coordinate. The initial magnetic field is the sum of a uniform guide
magnetic field and an additional in-plane magnetic field in the form of
several Fourier modes with a certain range of wavenumbers:

${\bf B}(x,z)=B_{0}\hat{y}+{\bf\delta b}(x,z),$ (3)

where

${\bf\delta b}(x,z)=\sum_{n=1}^{8}\sum_{m=1}^{8}{\bf b}(n,m)e^{i(2\pi
nx/L+2\pi mz/L+\phi_{\rm mn})},$ (4)

$B_{0}\hat{y}$ is the uniform guide field and $L$ is the dimension of the
cubical box. The wavenumber components along the $x$ and $z$ directions vary
from $2\pi/L$ to $16\pi/L$. The phases $\phi_{\rm mn}$ are randomized. The
Fourier coefficient ${\bf b}(n,m)$ is initially in the $x$-$z$ plane and is
perpendicular to the corresponding wavenumber to maintain $\nabla\cdot{\bf
B}=0$. The magnitude of ${\bf b}(n,m)$ is uniform and is set to give
$\langle{\bf\delta b}^{2}\rangle=b_{0}^{2}$, where $b_{0}$ is the unit of the
magnetic field in the simulations. This form of the magnetic field is
associated with plasma current along the $+y$ and $-y$ directions and contains
a total energy available for energy conversion of equal to $(\langle
B^{2}\rangle-B^{2}_{0})L^{3}/(2\mu_{0})=b^{2}_{0}L^{3}/(2\mu_{0})$. There is
no initial electric field and the plasma pressure is uniform. Therefore at its
early state, the forces in the plasma are extremely unbalanced.

The plasma consists of ion-electron pairs with the ion-to-electron mass ratio
$m_{i}/m_{e}$ of $25$ or $100$. The initial particle distributions are
Maxwellian with a uniform density $n_{0}$ and a uniform temperature for ions
and electrons. The initial electron thermal speed is $0.2c$. The initial bulk
flow velocities of ions and electrons give a net-zero mass flow but provide
the current density ${\bf J}=\nabla\times{\bf B}/\mu_{0}$. The unit of time is
the nominal inverse ion cyclotron frequency for the in-plane magnetic field,
$\Omega^{-1}_{\rm i}=m_{\rm i}/(q_{\rm i}b_{0})$ and the unit of length is the
initial ion inertial length $d_{i}=c/\omega_{\rm pi}$, where $\omega_{\rm pi}$
is the initial ion plasma frequency. The initial density $n_{0}$ is set so
that the initial electron plasma frequency is $\omega_{\rm pe}=\Omega_{\rm
e}=q_{\rm e}b_{0}/m_{\rm e}$.

For $m_{i}/m_{e}=25$, we ran simulations with $B_{0}/b_{0}$ =$0.1$, $0.5$, $1$
and $2$. For $m_{i}/m_{e}=100$, the simulation is with $B_{0}/b_{0}=2$.
Therefore, the initial values of plasma beta range from $0.016$ to $0.08$. The
domain size is $L\times L\times L$ = $102.4d_{\rm i}\times 102.4d_{\rm
i}\times 102.4d_{\rm i}$. The resolution of the simulations is $N_{x}\times
N_{y}\times N_{z}=2048\times 1\times 2048$ for $m_{i}/m_{e}=25$ and
$4096\times 1\times 4096$ for $m_{i}/m_{e}=100$. The number of macroparticles
for each species per cell is 400. Periodic boundary conditions are used.
Initially, the magnetic fluctuation in all simulations contains the total
energy of $5.4\times 10^{5}\,b^{2}_{0}d^{3}_{i}/\mu_{0}$. Simulations were run
until the magnetic free energy dropped to $\sim 12\%$ of the total energy from
the initial magnetic fluctuation. All our reported results are for
$m_{i}/m_{e}$ = 25 except where noted.

##  3 Results

Figure 2: The $y$-component of the current density at $\Omega_{i}t=0$, $4$,
$150$, and $300$ from the simulations with $m_{i}/m_{e}=25$ and
$B_{0}/b_{0}=0.1$ (top) and $B_{0}/b_{0}=2$ (bottom).

###  3.1 Plasma Dynamics

Figure 2 shows the $y$-component of the current density, $j_{y}$, at
$\Omega_{i}t=0$, $4$, $150$ and $300$ from the simulations with
$B_{0}/b_{0}=0.1$ and $2$. The initial plasma configuration is unstable since
the forces are extremely imbalanced. At this early time, the plasma flow is
highly compressible as current structures contract to form magnetic islands.
Figures 3(a) shows the compressible fraction of the ion flow energy $\langle
U^{2}_{\rm i,ir}\rangle/\langle U^{2}_{\rm i}\rangle$, which is the ratio of
the kinetic energy of the compressible (irrotational) ion flow $U_{\rm i,ir}$
to the total kinetic energy of the ion flow $U_{\rm i}$. This ratio quickly
rises at the beginning and reaches a maximum value greater than 0.8 after a
few ion cyclotron times in all simulations. Later, the ratio declines and
remains fairly stationary after $\Omega_{\rm i}\tau=5$ (dashed line). The
current contraction also builds free energy in $B_{y}$ as shown in Figure
3(b). From the simulations, the magnitude of $B_{y}-B_{0}$ is high at the
centers of the magnetic islands. After a few ion cyclotron times, the current
within each flux rope contracts, reducing energy in the magnetic field.
Multiple major magnetic islands become obvious after the contraction of plasma
currents inside each island as shown in Figure 2 at $\Omega_{\rm i}t=4$,
$150$, and $300$.

The conditions at $t\lesssim\tau$, the island-forming phase, involve
imbalanced forces that reflect the initial condition and are unlikely to
appear in nature.

Figure 3: (a) Compressible fraction of the ion flow energy $\langle U^{2}_{\rm
i,ir}/U^{2}_{\rm i}\rangle$ as a function of time. (b) Spatial average of the
square of the magnetic fluctuation in the $y$ direction ,i.e., the free energy
of the guide magnetic field. Dashed line indicates time $\tau$ between the
island-forming and decay phases.

At $t\gtrsim\tau$ , the current-merging or decaying phase, the forces are only
weakly imbalanced, and the root-mean-square of current density reaches its
maximum value at $\Omega_{\rm i}t=25$. In this phase, multiple magnetic
islands merge as the currents sharing the same direction attract one another.
Before the merging of two islands is complete, a current sheet emerges between
them and becomes a site for magnetic reconnection accompanied with a number of
small islands. This phase is more applicable to natural situations in space
plasmas. For stronger $B_{0}$, there are more of these tiny islands; see the
plots of $j_{y}$ at $\Omega_{i}t=150$ and $300$ in Figure 2. These emerging
plasmoids from the reconnection have a short lifetime as they are destroyed by
smaller-scale magnetic reconnection after colliding with the surrounding
plasma. As the current merging continues, there are fewer sites for magnetic
reconnection and associated plasma activity

###  3.2 Energy Conversion Profiles

####  3.2.1 Overview

Figure 4 shows the magnetic fluctuation energy, electric field energy, total
ion energy, total electron energy, and total transferable energy as time
series from the simulations with $B_{0}/b_{0}=0.1$ and $2$. The electric
energy is tiny and nearly constant, so it does not serve as a significant
source or sink of energy, and refering to Figure 1, ${\bf J}\cdot{\bf E}$ can
accurately represent both electric and magnetic energy conversion rates per
unit volume. When $B_{0}$ changes, the overall amount of energy gained by ions
and electrons also changes. Electrons eventually gain more energy for higher
$B_{0}$ (Gary et al., 2016; Hughes et al., 2017). In our 2.5D simulations, all
quantities depend on $x$ and $z$ while $y$ is an ignorable coordinate. By
applying Faraday’s law to our scenarios, only the $y$ component of the
electric field can modify the in-plane magnetic field while the in-plane
electric field can only modify $B_{y}$. Since the initial magnetic fluctuation
is in the $x-z$ plane, the overall conversion of energy from the magnetic
field is well represented by the integral of $J_{y}E_{y}$ over space and time.
This integration includes magnetic energy lost during the island-forming and
the decaying phases.

At $t\lesssim\tau$, energy in $B_{x}$ and $B_{z}$ drops via $E_{y}$ but the
energy in $B_{y}$ increases. During the island-forming phase, $\langle
J_{y}E_{y}\rangle$ is higher than $\langle{\bf J}\cdot{\bf E}\rangle$ because
part of energy from $\langle J_{y}E_{y}\rangle$ is used for building strong
free energy $\langle(B_{y}-B_{0})^{2}\rangle$ at centers of the magnetic
islands via the in-plane solenoidal electric field ${\bf E}_{{\rm so},xz}$.
Therefore, $\langle J_{y}E_{y}\rangle$ must be less than $\langle{\bf
J}\cdot{\bf E}\rangle$ during the decaying phase.

Figure 4: Time series of the total magnetic fluctuation energy, total electric
energy, total ion energy, total electron energy, total transferable energy and
(a) $B_{0}/b_{0}=0.1$ and (b) $B_{0}/b_{0}=2$. Figure 5: Fraction of electric
energy in the (a) $y-$component and (b) irrotational component as a function
of time. Dashed line indicates time $\tau$ between island-forming and decaying
phases.

Since the island-forming phase and decaying phase have distinct mechanisms of
energy conversion, these can be analyzed separately. As the decaying phase is
more applicable to astrophysical situations, we focus more on the energy
conversion during this later phase of the simulation. We examine the mechanism
that takes energy out of $B_{y}$ while $E_{y}$ continues to take energy out of
the in-plane components of the magnetic field.

Figures 5(a) shows the ratio of the energy in $E_{y}$ to the total electric
field energy at times up to $30\Omega^{-1}_{i}$. As $B_{0}$ is increased,
$E_{y}$ provides a lower energy fraction at $t\gtrsim\tau$. For
$B_{0}/b_{0}=2$, the energy in $E_{y}$ is around $5\%$ of the total electric
field energy at $t\gtrsim\tau$. Meanwhile, the energy in ${\bf E}_{\rm ir}$
can be as high as $80\%$ of the total electric field energy for
$B_{0}/b_{0}=2$ at $t\gtrsim\tau$, as shown in Figure 5(b). Since ${\bf
E}_{\rm ir}$ provides a significant fraction of the electric energy, which
even dominates at large $B_{0}$, a large portion of ${\bf J}\cdot{\bf E}$
represents energy exchange between ions and electrons without directly
involving the conversion of magnetic energy. The analysis of how ${\bf
J}\cdot{\bf E}$ take energy out of the magnetic field must consider the large
contribution from ${\bf E}_{\rm ir}$ into account.

####  3.2.2 Island-forming Phase

First consider $\langle{\bf J}\cdot{\bf E}\rangle$ at $t<\tau$. We separately
consider the energy transfer to ions and electrons as well as the
contributions from ${\bf E}_{\rm ir}$, ${\bf E}_{{\rm so},xz}$ and $E_{y}$ to
$\langle{\bf J}\cdot{\bf E}\rangle$ (Figure 6). (Note that ${\bf E}_{y}$ is
purely solenoidal.) At $t\lesssim\tau$, simulations with $B_{0}/b_{0}=0.1$ and
$2$ both show that electrons are mainly responsible for taking energy out of
the in-plane magnetic field via ${\bf E}_{y}$. Electrons then transfer most of
this energy to ions via ${\bf E}_{\rm ir}$ (recall that ${\bf E}_{\rm ir}$ is
associated with charge separation) and ions gain more energy than electrons in
this phase. The profiles of the energy transfer rates to electrons and ions
via ${\bf E}_{\rm ir}$ are very similar but with opposite signs. This implies
that the particle interactions with ${\bf E}_{\rm ir}$ mostly do not store or
extract electric energy in ${\bf E}_{\rm ir}$ but only cause the energy
exchange between particles of opposite charges.

Figure 6: Panels (a) and (b) show the rates of energy transfer to ions and
electrons, respectively, for various electric field components with
$B_{0}/b_{0}=0.1$ for $t<\tau$. Panels (c) and (d) show the same rates to ions
and electrons, respectively, with $B_{0}/b_{0}=2$.

For simulations with $m_{i}/m_{e}=25$, and $B_{0}/b_{0}=0.1$, $0.5$, $1$ and
$2$, the total energy $W_{25,1}$ gained by particles during the $1^{st}$
(island-forming) phase from the beginning to $t=\tau$ was $2.5\times 10^{5}$,
$1.9\times 10^{5}$, $1.3\times 10^{5}$ and $1.0\times 10^{5}$, respectively in
units of $b^{2}_{0}d^{3}_{\rm i}/\mu_{0}$. Less energy is transferred to
particles during this phase when $B_{0}$ is higher. Table 1 shows the fraction
of the total electromagnetic energy transfer to particles that go to ions and
electrons via ${\bf E}_{\rm ir}$, ${\bf E}_{{\rm so},xz}$, $E_{y}$ and the
total eletric field ${\bf E}$ from the beginning to $t=\tau$, for each
simulation.

Table 1:  The fractions of the total electromagnetic energy transfer to
particle energy that go to ions and electrons during the island-forming phase
$(t<\tau)$ via interactions with ${\bf E}_{\rm so,xz}$, $E_{y}$, ${\bf E}_{\rm
ir}$, and ${\bf E}$.

$B_{0}/b_{0}$ | Species | ${\bf J}_{\rm s}\cdot{\bf E}_{{\rm so},xz}$ | ${\bf J}_{\rm s}\cdot{\bf E}_{y}$ | ${\bf J}_{\rm s}\cdot{\bf E}_{\rm ir}$ | ${\bf J}_{\rm s}\cdot{\bf E}$
---|---|---|---|---|---
0.1 | Ion | -0.01 | 0.28 | 0.52 | 0.79
Electron | -0.05 | 0.78 | -0.52 | 0.21
0.5 | Ion | -0.08 | 0.35 | 0.55 | 0.82
Electron | -0.17 | 0.92 | -0.57 | 0.18
1 | Ion | -0.18 | 0.30 | 0.76 | 0.88
Electron | -0.26 | 1.17 | -0.79 | 0.12
2 | Ion | -0.24 | 0.19 | 0.88 | 0.83
Electron | -0.13 | 1.28 | -0.98 | 0.17

In all simulations, electrons take energy from the in-plane components of the
magnetic field and transfer a large amount of energy to ions via ${\bf E}_{\rm
ir}$. Both ions and electrons lose some energy to form strong $|B_{y}-B_{0}|$
at the centers of magnetic islands. As $B_{0}/b_{0}$ is increased up to 1,
more energy is transferred to $B_{y}$ and the energy transfer via ${\bf
E}_{\rm ir}$ is also stronger. At $t=\tau$, ions have gained more energy than
electrons even though electrons are the main species interacting with $E_{y}$
and reducing the magnitude of the in-plane magnetic field.

####  3.2.3 Decaying Phase

At $t>\tau$, the current contraction is less active. Magnetic energy drops due
to the merging of magnetic islands. We carry on the analysis of energy
conversion by considering the contributions from ions and electrons, again
separating effects of ${\bf E}_{\rm ir}$ from ${\bf E}_{\rm so}$. In this
phase, we found that separating the solenoidal field into ${\bf E}_{y}$ and
${\bf E}_{{\rm so},xz}$ is not the most effective way to understand the energy
conversion. In this phase, energy conversion often takes place where both
$B_{y}$ and the in-plane magnetic field lose energy together. We found that
separating the solenoidal electric field into the components parallel ${\bf
E}_{\rm so,\parallel}$ and perpendicular ${\bf E}_{\rm so,\perp}$ to the local
magnetic field is more suitable for a strong guide field. 111We have
considered the relevance of components parallel and perpendicular to B vs.
out-of-plane ($y$) and in-plane ($x$-$z$) components for the decaying phase.
We found that resolving components parallel and perpendicular to B provides a
clearer physical distinction for particle acceleration (direct parallel
acceleration vs. perpendicular drift motion) and also a clearer distinction
among components of the pressure tensor.

Figures 7(a) and 7(c) show the energy conversion rates to electrons and ions
via ${\bf E}_{\rm so}$ for $B_{0}/b_{0}=0.1$ and $2$, respectively, at
$t>\tau$. For $B_{0}/b_{0}=0.1$, $\langle{\bf J}_{i,\perp}\cdot{\bf E}_{\rm
so}\rangle$ is positive and dominates other rates. For $B_{0}/b_{0}=2$,
$\langle{\bf J}_{e,\perp}\cdot{\bf E}_{\rm so}\rangle$ and $\langle{\bf
J}_{i,\perp}\cdot{\bf E}_{\rm so}\rangle$ fluctuate with large magnitudes,
share similar shapes and seems to have the opposite signs. On the other hand,
the rates $\langle{\bf J}_{e,\parallel}\cdot{\bf E}_{\rm so}\rangle$ and
$\langle{\bf J}_{i,\parallel}\cdot{\bf E}_{\rm so}\rangle$ remain mostly
positive with smaller magnitudes. The rates from ${\bf E}_{\rm ir}$ are shown
in Figure 7(b) for $B_{0}/b_{0}=0.1$ and Figure 7(d) for $B_{0}/b_{0}=2$.
These rates fluctuate with stronger magnitude when $B_{0}$ is higher. The
rates for ions and electrons are nearly equal and opposite almost all the
time, representing energy transfer among particle species with little net
conversion of electromagnetic energy to particle motion.

Figure 7: The rates of energy conversion to ions and electrons by interacting
with (a) components of ${\bf E}_{\rm so}$ and (b) ${\bf E}_{\rm ir}$ from the
simulations with $B_{0}/b_{0}=0.1$ while panels (c) and (d) shows the energy
conversion rates to ions and electrons from interaction with components of
${\bf E}_{\rm so}$ and ${\bf E}_{\rm ir}$, respectively, with $B_{0}/b_{0}=2$.

Let $W_{25,2}$ be the total energy gained by particles in the $2^{nd}$
(decaying) phase after $t=\tau$ for each simulation with $m_{i}/m_{e}=25$. The
values are $2.3\times 10^{5}$, $2.8\times 10^{5}$, $3.4\times 10^{5}$ and
$3.7\times 10^{5}$ in units of $b^{2}_{0}d^{3}_{\rm i}/\mu_{0}$ for
$B_{0}/b_{0}=0.1$, $0.5$, $1$ and $2$, respectively. Table 2 shows the
fractions of the total energy gain that go to ions and electrons via ${\bf
E}_{\rm so,\parallel}$, ${\bf E}_{\rm so,\perp}$, ${\bf E}_{\rm ir}$ and the
total electric field ${\bf E}$. The values in the table are fractions of
$W_{25,2}$. For $B_{0}/b_{0}=0.1$, the most effective motion to transfer
energy via ${\bf E}_{\rm so}$ is the perpendicular motion of ions. This motion
takes $0.51W_{25,2}$ of the converted energy while the parallel and the
perpendicular motions of electrons take $\sim 0.20W_{25,2}$ each. The
interaction between ions and electrons via ${\bf E}_{\rm ir}$ is weak. The
total amounts of energy gained by ions and electrons for $t>\tau$ are
$0.65W_{25,2}$ and $0.35W_{25,2}$, respectively. For small $B_{0}$ at
$t>\tau$, the energy transferred via ${\bf E}_{y}$ still dominates and $92\%$
of electromagnetic energy conversion is via ${\bf E}_{y}$. Therefore,
separating the rates from ${\bf E}_{\rm so,\parallel}$ and ${\bf E}_{\rm
so,\perp}$ is not particularly useful. As $B_{0}$ increases, the parallel
motion of ions becomes the most effective motion to take energy from ${\bf
E}_{\rm so}$. The particle interaction through ${\bf E}_{\rm ir}$ is also
stronger as ${\bf E}_{\rm ir}$ is stronger for higher $B_{0}$. For
$B_{0}/b_{0}=2$, the parallel motion of ions takes $0.91W_{25,2}$ from ${\bf
E}_{\rm so}$ while ${\bf E}_{\rm ir}$ takes $0.67W_{25,2}$ from ions and gives
$0.68W_{25,2}$ to electrons. For $t>\tau$, the energy gained by electrons is
$0.73W_{25,2}$ while ions gain only $0.27W_{25,2}$. As $B_{0}$ increases, the
energy conversion by the parallel electric field becomes stronger. For
$B_{0}/b_{0}=2$, ${\bf E}_{\rm so,\parallel}$ is responsible for almost
$100\%$ of $W_{25,2}$ while $E_{y}$ takes $88\%$ of $W_{25,2}$ out of the
magnetic field. The dominance of ${\bf E}_{\rm so,\parallel}$ implies that
energy in $B_{y}$ and the in-plane magnetic field are likely drained at the
same locations via ${\bf E}_{\rm so,\parallel}$.

For comparison, we calculate the energy conversion via ${\bf E}_{\rm
so,\parallel}$ and ${\bf E}_{\rm so,\perp}$ in Table 3 for $t<\tau$. For
$B_{0}/b_{0}=0.1$, the magnetic field lies mostly in the $x$-$z$ plane
initially and the perpendicular direction is nearly along the $y$ direction.
Therefore, in the first phase, the sum of ${\bf J}_{\perp}\cdot{\bf E}_{\rm
so,\perp}$ from both species, with $E_{y}$ as the main component in ${\bf
E}_{\rm so,\perp}$, is responsible for $95\%$ of the energy conversion. For
the case of $B_{0}/b_{0}=2$, the initial magnetic field tends to be more
aligned with the $y$ direction. As $E_{y}$ is the main component of ${\bf
E}_{\rm so,\parallel}$, the sum of ${\bf J}_{\parallel}\cdot{\bf E}_{\rm
so,\parallel}$ from both species is responsible for $89\%$ of the energy
conversion. Thus these results are consistent with our conclusion from the
previous section that $E_{y}$ plays the dominant role in energy conversion for
$t<\tau$ (see Table 1).

The energy transfer processes in the phases $t<\tau$ and $t>\tau$ are very
different. Electrons are the main species to draw energy from the magnetic
field in the early phase but lose most of the energy to ions via the charge
separation and ${\bf E}_{\rm ir}$. In the later phase, ions are more
responsible for draining energy from the magnetic field. For
$B_{0}/b_{0}=0.5$, $1$ and $2$, these ions lose energy to electrons via the
charge separation and ${\bf E}_{\rm ir}$. The interaction between ions and
electrons via ${\bf E}_{\rm ir}$ is stronger as $B_{0}$ increases. Our
analysis is insensitive to choices of the boundary time from $\Omega_{\rm
i}\tau=5$ to $10$. By using $\Omega_{\rm i}\tau=10$, the results change
numerically but remain qualitatively similar. The results for
$m_{i}/m_{e}=100$ are also qualitatively similar to those for
$m_{i}/m_{e}=25$.

Table 2:  The fraction of the total electromagnetic energy transfer to
particle energy that go to ions and electrons via interactions with with ${\bf
E}_{\rm so,\parallel}$, ${\bf E}_{\rm so,\perp}$, ${\bf E}_{\rm ir}$, and
${\bf E}$ during the decaying phase ($t>\tau$).

$B_{0}/b_{0}$ | Species | ${\bf J}_{\rm s,\parallel}\cdot{\bf E}_{\rm so,\parallel}$ | ${\bf J}_{\rm s,\perp}\cdot{\bf E}_{\rm so,\perp}$ | ${\bf J}_{\rm s}\cdot{\bf E}_{\rm ir}$ | ${\bf J}_{\rm s}\cdot{\bf E}$
---|---|---|---|---|---
0.1 | Ions | 0.04 | 0.51 | 0.10 | 0.65
Electrons | 0.20 | 0.23 | -0.08 | 0.35
0.5 | Ions | 0.63 | 0.02 | -0.07 | 0.58
Electrons | 0.16 | 0.18 | 0.08 | 0.42
1 | Ions | 1.21 | -0.01 | -0.68 | 0.52
Electrons | -0.27 | 0.05 | 0.70 | 0.48
2 | Ions | 0.91 | 0.03 | -0.67 | 0.27
Electrons | 0.10 | -0.05 | 0.68 | 0.73

Table 3:  The fraction of the total electromagnetic energy transfer to
particle energy that go to ions and electrons via interactions with ${\bf
E}_{\rm so,\parallel}$ and ${\bf E}_{\rm so,\perp}$ for the island-forming
phase ($t<\tau$).

$B_{0}/b_{0}$ | Species | ${\bf J}_{\rm s,\parallel}\cdot{\bf E}_{\rm so,\parallel}$ | ${\bf J}_{\rm s,\perp}\cdot{\bf E}_{\rm so,\perp}$
---|---|---|---
0.1 | Ions | 0.06 | 0.21
Electrons | $\approx 0$ | 0.73
0.5 | Ions | 0.01 | 0.26
Electrons | 0.23 | 0.52
1 | Ions | -0.02 | 0.14
Electrons | 0.48 | 0.43
2 | Ions | 0.03 | -0.08
Electrons | 0.86 | 0.29

###  3.3 Indicators of Energy Conversion Locations

From the previous section, the overall rate of energy transfer to particles in
the decaying phase is approximately equal to the integration of
$J_{\parallel}E_{\rm so,\parallel}$ over $t>\tau$. In all the simulations,
$\nabla\times{\bf B}\approx\mu_{0}{\bf J}$ is valid at most locations so we
can write

$J_{\parallel}E_{\rm so,\parallel}\approx\frac{1}{\mu_{0}}{\bf E}_{\rm
so,\parallel}\cdot\nabla\times{\bf B}=\frac{1}{\mu_{0}}{\bf
B}\cdot\nabla\times{\bf E}_{\rm so,\parallel},$ (5)

noting that ${\bf E}_{\rm so,\parallel}$ is parallel to ${\bf B}$, so there is
no Poynting vector associated with ${\bf E}_{\rm so,\parallel}$. The last term
is the rate of change of the local magnetic energy density due to ${\bf
E}_{\rm so,\parallel}$ from Faraday’s law. Therefore, the energy transferred
via the parallel motion is local and the magnetic energy quite instantly
becomes the kinetic energy of particles where $\nabla\times{\bf
B}\approx\mu_{0}{\bf J}$.

Figure 8(a) shows the color plot of $J_{\parallel}E_{\rm so,\parallel}$ over
the entire domain at $t=22\Omega^{-1}_{i}$ from the simulations with
$B_{0}/b_{0}=2$. Contours of vector potential $A\hat{y}$ are shown as black
$(A<0)$ and grey $(A>0)$. In 2D simulations, these curves are also the
projection of the magnetic field lines on the $x$-$z$ plane. The magnetic
islands are surrounded by circular closed curves while the current sheets are
located at boundaries between magnetic islands that share the same rotation
direction. The rate $J_{\parallel}E_{\rm so,\parallel}$ fluctuates strongly
and is highly intermittent in space. The energy is transferred strongly both
ways, either from fields to particles ($J_{\rm parallel}E_{\rm
so,\parallel}>0$, red color) or from particles to fields ($J_{\rm
parallel}E_{\rm so,\parallel}<0$, blue color). From the previous sections we
have learned that the spatial average of $J_{\parallel}E_{so,\parallel}$ is
positive, indicating a net conversion of energy from fields to particle
kinetic energy. We can see two types of structures with strong (positive or
negative) values of $J_{\parallel}E_{so,\parallel}$: 1) Near the centers of
magnetic islands, there are bipolar structures, with positive and negative
$J_{\parallel}E_{so,\parallel}$. We have confirmed that the positive and
negative values nearly cancel, and these regions make a tiny contribution to
the net energy conversion rate $\langle J_{\parallel}E_{so,\parallel}\rangle$.
2) Near the edges of magnetic islands, there are elongated regions with strong
$J_{\parallel}E_{so,\parallel}$, including regions of magnetic reconnection.
The positive values of $J_{\parallel}E_{so,\parallel}$ dominate and these
regions account for the overall positive energy conversion.Therefore, it would
be useful to find quantities that indicate these regions near the edges of
magnetic islands, especially if they provide some insight into why the energy
conversion mostly occurs in these specific locations.

In this section, we examine various quantities as possible indicators of
energy conversion locations, and specifically of regions with strong and
positive $J_{\parallel}E_{\rm so,\parallel}$ on average. We test these
indicators using the following steps. For any continuous indicator $X$, we
identify the median $X_{m}$ of $X$ values at all grid points over the spatial
domain as a time series (for $t>\tau$). We integrate the total energy
$\epsilon_{X,h}$ transferred to particle kinetic energy by
$J_{\parallel}E_{\rm so,\parallel}$ from the region with $X>X_{m}$ over the
time domain of interest and compare $\epsilon_{X,h}$ with the total energy
$\epsilon_{X}$ from $J_{\parallel}E_{\rm so,\parallel}$ over the spatial and
time domains. We then calculate a measure of predictive power,
$S_{X}=2|\epsilon_{X,h}/\epsilon_{X}-0.5|$. Because a monotonic transformation
of $X$ does not change its percentiles, with this construction, $S_{x}$ is
invariant upon any monotonic transformation of the indicator $X$, and $0\leq
S_{X}\leq 1$. If $S_{X}$ is close to $1$, then $X$ is a good indicator of
energy conversion locations.

We have tested many indicators using the data from simulations with
$B_{0}/b_{0}=2$. Table 4 shows a list of indicators and their predictive power
regarding energy transferred via $J_{\parallel}E_{\rm so,\parallel}$ for all
times $t>\tau$. We include $|E_{\rm so,\parallel}|$ and $|J_{\parallel}|$,
which multiply to make $|J_{\parallel}||E_{\rm so,\parallel}|$, only for
comparison. We found three good indicators in addition to $|E_{\rm
so,\parallel}|$ and $|J_{\parallel}|$ themselves.

Table 4:  Indicators $X$ and a measure of their predictive power, $S_{X}$,
based on the fraction of the energy conversion $J_{\parallel}E_{\rm
so,\parallel}$ that occurs in the region with $X$ greater than its median
$X_{m}$. $|J_{\parallel}|$ & $|E_{\rm so,\parallel}|$ themselves, indicated in
italics, are included for comparison purposes.

Indicator | Formula | $S_{X}$
---|---|---
Magnitude of parallel current | $|{\bf J}_{\parallel}|$ | 0.9
Magnitude of ratio of out-of-plane electric field to in-plane magnetic field | $|E_{y}|/|B_{xz}|$ | 0.82
Magnitude of parallel component of solenoidal electric field | $|{\bf E}_{\rm so,\parallel}|$ | 0.8
Magnitude of $y$-component of non-ideal electric field | $\delta E_{y}=|{\bf E}_{y}+({\bf U}\times{\bf B})_{y,\rm so}|$ | 0.8
Estimate of current helicity | $H=|{\bf B}\cdot\nabla\times{\bf B}/\mu_{0}|$ | 0.76
Magnitude of in-plane component of magnetic fluctuation | $|B^{2}_{x}+B^{2}_{z}|$ | 0.46
Magnitude of vorticity | $|\nabla\times{\bf U}|$ | 0.46
Magnitude of solenoidal component of non-ideal electric field | $|{\bf E}_{\rm so}+({\bf U}\times{\bf B})_{\rm so}|$ | 0.36
Magnitude of non-ideal electric field | $|{\bf E}+{\bf U}\times{\bf B}|$ | 0.34
Divergence of bulk velocity | $\nabla\cdot{\bf U}$ | 0.26
Magnitude of cross helicity | $|{\bf U}\cdot({\bf B}-B_{0}\hat{y})|$ | 0.1
Magnitude of $y$-component of magnetic fluctuation | $|B_{y}-B_{0}|$ | 0.06
Magnitude of curvature of magnetic field | $|\hat{B}\cdot\nabla\hat{B}|$ | 0.06
Magnitude of magnetic helicity | $|{\bf A}\cdot{\bf B}|$ | 0.06
Magnitude of magnetic field | $|B|$ | 0

The best such indicator is $|E_{y}|/|B_{xz}|$, which is based on an indicator
suggested by Lapenta (2021). That work actually examined the formal
transformation velocity ${\bf v}_{L}=c^{2}{\bf B}\times{\bf E}/E^{2}$ such
that a Lorentz transformation would nullify the two components of the magnetic
field in the plane perpendicular to the electric field, as an indicator of
reconnection regions in 3D space. The magnitude of ${\bf v}_{L}$ is almost
always superluminal, with $v_{L}<c$ only in the vicinity of reconnection
sites. For our configuration, with a guide magnetic field component $B_{y}$,
reconnection sites should have ${\bf B}_{xz}\approx 0$ and we can simply use
$v_{L}=c^{2}|B_{xz}|/|E_{y}|$. As a further simplification, we divide $v_{L}$
by $c^{2}$ and invert to obtain $|E_{y}|/|B_{xz}|$, which has units of
velocity and is usually close to zero, but is large near null points of ${\bf
B}_{xz}$ that have non-zero $E_{y}$, i.e., near reconnection regions.

We found that $S_{|E_{y}|/|B_{xz}|}$ is 0.82,i.e., $91\%$ of the energy
transferred via $J_{\parallel}E_{\rm so,\parallel}$ occurs in regions with
$|E_{y}|/|B_{xz}|$ higher than its median value, which are typically near
reconnection sites. This indicator has higher predictive power for
$J_{\parallel}E_{\rm so,\parallel}$ than $|E_{\rm so,\parallel}|$ itself.

The second best indicator is the magnitude of the $y-$component of the non-
ideal electric field, $\delta E_{y}=\hat{y}\cdot({\bf E}+{\bf U}\times{\bf
B})$, where ${\bf U}$ is the plasma bulk velocity. At locations with a high
magnitude of ${\bf E}+{\bf U}\times{\bf B}$, the frozen-in approximation is no
longer valid. However, we find that the magnitude of ${\bf E}+{\bf
U}\times{\bf B}$ is not as good indicator as $\delta E_{y}$. We found that
$S_{\delta E_{y}}$ is 0.8, implying that $90\%$ of the energy transferred via
$J_{\parallel}E_{\rm so,\parallel}$ occurs in regions with $\delta E_{y}$
above its median value. This indicator has a predictive power for
$J_{\parallel}E_{\rm so,\parallel}$ equal to $|E_{\pm so,\parallel}|$ itself.
This implies that strong energy conversion often takes place where the frozen-
in approximation fails which also typically occurs near reconnection sites.

The third best indicator is an estimate $H=|{\bf B}\cdot\nabla\times{\bf
B}|/\mu_{0}$ of the current helicity $|{\bf J}\cdot{\bf B}|$. We can write

$J_{\parallel}E_{\rm so,\parallel}=\frac{E_{\rm so,\parallel}|{\bf J}\cdot{\bf
B}|}{B},$ (6)

so the current helicity is directly linked with $J_{\parallel}E_{\rm
so,\parallel}$. We use $H=|{\bf B}\cdot\nabla\times{\bf B}|/\mu_{0}$ as an
estimate of the current helicity that only involves magnetic field
measurements, as may be more readily available from spacecraft data. We found
that $S_{H}$ is 0.76, implying that $88\%$ of the energy transferred via
$J_{\parallel}E_{\rm so,\parallel}$ occurs in the region with $H$ above its
median value.

Figure 8: (a) 2D color plot of $J_{\parallel}E_{\rm so,\parallel}$ at
$t=22\Omega^{-1}_{i}$. (b)-(d) Spatial regions (black) where the indicator is
(b) $|E_{y}|/|B_{xz}|$ less than its 10th percentile, or (c) $\delta E_{y}$ or
(d) $H$ greater than its 90th percentile.

To demonstrate how $J_{\parallel}E_{\rm so,\parallel}$ is sensitive to the
indicators, we numerically calculate the average value of $\langle
J_{\parallel}E_{\rm so,\parallel}\rangle$ for each percentile of $X$. The
percentile is calculated over the whole domain for $t>\tau$. Figure 9 shows
$\langle J_{\parallel}E_{\rm so,\parallel}\rangle$ normalized by its average
value over the whole domain at $t>\tau$ as a function of the percentile of
$\delta E_{y}$, $|E_{y}|/|B_{xz}|$ and $H$. The value of $\langle
J_{\parallel}E_{\rm so,\parallel}\rangle$ becomes larger when the percentile
of $|E_{y}|/|B_{xz}|$, $\delta E_{y}$ or $H$ is higher. The regions with one
of $|E_{y}|/|B_{xz}|$ higher than $89$ percentile, $\delta E_{y}$ higher than
$90$ percentile, or $H$ higher than $93$ percentile are responsible for $50\%$
of the energy conversion. Figures 8(b)-(d) show the regions selected by the
values of $|E_{y}|/|B_{xz}|$ (b), $\delta E_{y}$ (c) and $H$ (d) higher than
their $90$ percentile by black coloring. The conversion sites selected by
these indicators have either strongly positive or negative
$J_{\parallel}E_{\rm so,\parallel}$. The regions selected by these three
indicators are quite different. The regions selected by $|E_{y}|/|B_{xz}|$ are
relatively contiguous. They are concentrated near the reconnection sites and
away from the centers of magnetic islands. This is useful because, as noted
above, the magnetic islands provide very little net energy conversion to the
particles. $H$ selects both regions near the reconnection sites and centers of
magnetic islands, though the latter make very little contribution to the net
energy conversion. The regions selected by $\delta E_{y}$ are scattered quite
broadly but statistically concentrated around the reconnection sites.

Figure 9: The average of $J_{\parallel}E_{\rm so\parallel}$ normalized by its
average value as a function of percentile of (a) $|E_{y}|/|B_{xz}|$, (b)
$\delta E_{y}$, and (c) $H$ with $B_{0}/b_{0}=2$ for $t>\tau$.

##  4 Discussion and Summary

We have run 2.5D simulations of decaying turbulence. There are two phases of
the simulations that give two distinct mechanisms of magnetic energy
conversion. These phases are significantly modified by the magnitude of the
guide field. As $B_{0}$ is higher, plasma becomes less compressible as the
magnetic field partially confines the particles. Another effect of higher
$B_{0}$ is the larger amplitude of the irrotational electric field. This field
component is not directly responsible for the magnetic energy conversion and
we therefore focus on the solenoidal electric field when searching for
indicators of energy conversion locations.

For $t<\tau$ or the island-forming phase, plasma current structures contract
to form magnetic islands. In this phase, electron motions take energy out of
the magnetic field but also lose a large amount of energy to ions via the
irrotational electric field. While the in-plane magnetic field becomes weaker,
the out-of-plane component becomes stronger at centers of magnetic islands by
taking energy from the in-plane particle motions. The current contraction
obviously depends on the magnitude of the guide magnetic field. As $B_{0}$
gets stronger, the fractional energy conversion in this phase is weaker, and
the ratio of energy taken out of the in-plane magnetic field by electrons to
ions becomes higher. The irrotational electric field becomes stronger and is
responsible for more energy transfer from electrons to ions. Both ions and
electrons lose more energy to build strong $B_{y}$ at centers of magnetic
islands.

At $t>\tau$ or the decaying phase, the magnetic islands merge. During island
merging, the magnetic reconnection inevitably occurs. As $B_{0}$ is higher,
the particle parallel motion becomes dominant at taking energy out of the in-
plane magnetic field (Wan et al., 2012; Makwana et al., 2017), and while ions
gain more energy from the magnetic field, they lose a large amount of energy
to electrons via the irrotational electric field. Electrons eventually gains
more energy in this phase. Overall electrons become more effective at
extracting the electromagnetic energy when $B_{0}$ is higher (Shay et al.,
2018). This result is similar to the beta effect on the particle heating
(Parashar et al., 2018).

The energy conversion is mainly mediated by the parallel component of the
solenoidal electric field via $J_{\parallel}E_{\rm so,\parallel}$, which is
highly localized. Three particularly useful indicators of the regions of
strong energy conversion are found. They are 1) the ratio of the out-of-plane
electric field to the in-plane magnetic field (related to the suggestion by
Lapenta, 2021), 2) the out-of-plane component of the non-ideal electric field,
and 3) the magnitude of an estimate of current helicity. All of them select
regions near the magnetic reconnection sites as locations of net energy
conversion. Among them, $|E_{y}|/|B_{xz}|$ provides regions that are most
contiguous and tied with multiple reconnection sites. We propose that any
mechanisms involving these indicators should receive particular attention in
order to gain more insight on how magnetic energy is converted to bulk and
thermal motions in turbulence with a strong guide field.

P.P. and D.R. would like to thank Thailand Science Research and Innovation and
Ministry of Higher Education, Science, Research and Innovation (Thailand) for
support through grants RTA 6280002 and RGNS 63-045. This work was partly
supported by the International Atomic Energy Agency (IAEA) under Contract No.
22785.

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