#  Error Analysis of Cooperative NOMA with Practical Constraints: Hardware-
Impairment, Imperfect SIC and CSI

Beddiaf Safia, Khelil Abdellatif, Faical Khennoufa, Ferdi Kara, Hakan Kaya,
Xingwang Li, Khaled Rabie, and Halim Yanikomeroglu B. Safia, K. Abdellatif and
F. Khennoufa are with Department of Electrical Engineering, Echahid Hamma
Lakhdar University, El-Oued, Algeria, email:{beddiaf-safia, abdellatif-khelil,
khennoufa-faical<EMAIL_ADDRESS>Kara is with the Department of Computer
Engineering, Zonguldak Bulent Ecevit University, Zonguldak, Turkey, 67100,
e-mail<EMAIL_ADDRESS>Kaya is with the Electical and Electronics
Engineering, Zonguldak Bulent Ecevit University, Zonguldak, Turkey, 67100,
e-mail<EMAIL_ADDRESS>Li is with School of Physics and Electronic
Information Engineering, Henan Polytechnic University, Jiaozuo, China, e-mail:
<EMAIL_ADDRESS>Rabie is with the Department of Engineering,
Manchester Metropolitan University, Manchester M1 5GD, U.K. e-mail:
<EMAIL_ADDRESS>Yanikomeroglu is with the Department of Systems and
Computer Engineering, Carleton University, Ottawa, K1S 5B6, ON, Canada,
e-mail<EMAIL_ADDRESS>
###### Abstract

Non-orthogonal multiple access (NOMA) has been a strong candidate to support
massive connectivity in future wireless networks. In this regard, its
implementation into cooperative relaying, named cooperative-NOMA (CNOMA), has
received tremendous attention by researchers. However, most of the existing
CNOMA studies have failed to address practical constraints since they assume
ideal conditions. Particularly, error performance of CNOMA schemes with
imperfections has not been investigated, yet. In this letter, we provide an
analytical framework for error performance of CNOMA schemes under practical
assumptions where we take into account imperfect successive interference
canceler (SIC), imperfect channel estimation (ICSI), and hardware impairments
(HWI) at the transceivers. We derive bit error rate (BER) expressions in CNOMA
schemes whether the direct links between source and users exist or not which
is, to the best of the authors’ knowledge, the first study in the open
literature. For comparisons, we also provide BER expression for downlink NOMA
with practical constraints which has also not been given in literature, yet.
The theoretical BER expressions are validated with computer simulations where
the perfect-match is observed. Finally, we discuss the effects of the system
parameters (e.g., power allocation, HWI level) on the performance of CNOMA
schemes to reveal fruitful insights for the society.

###### Index Terms:

BER, cooperative NOMA, HWI, imperfections, imperfect CSI, practical
constraints.

##  I Introduction

Non-orthogonal multiple access (NOMA) is a promising solution for satisfying
the demands for spectrum efficiency and network density in the next
generations of wireless access. Therefore, the integration of the NOMA with
cooperative relaying (CNOMA) networks has attracted recent attention to
improve transmission reliability and extend network coverage and increase
spectral efficiency [1].

In CNOMA schemes, there are two possible configurations depending on whether
direct links are available between source and users or not. In the first, the
relay node(s) improves the key performance indicators (KPIs), e.g., capacity,
outage probability and bit error rate (BER), while in the second, the relay
node(s) extends the coverage. In [2], the authors investigate the capacity and
outage performances for both cases. Nevertheless, almost all literature is
devoted to second scenario where the direct links are blocked by some
obstacles. On the other hand, most of the existing CNOMA schemes take into
account perfect assumptions where practical constraints are disregarded at
whole or in part. However, assuming a perfect successive interference canceler
(SIC) for NOMA or perfect hardware implementation with no imperfection or
ideal channel state information (CSI) in the communication system are not
realistic. In practical scenarios, CNOMA schemes suffer from imperfect SIC,
CSI errors and also from hardware impairments such as oscillator phase noise,
high power amplifier, and in-phase and quadrature-phase imbalance. [3].
Nevertheless, several studies have examined the effects of such imperfections
on the performances of CNOMA schemes. An amplify-and-forward (AF) CNOMA system
with HWI is analyzed in terms of outage probability and intercept probability
over multipath fading channel in [4]. The authors in [5] investigate the
outage probability and ergodic rate performance of the full-duplex (FD) CNOMA
with the presence of HWI over Rician fading channels. The authors in [6] study
the impacts of HWI on the NOMA with imperfect CSI (ICSI), where both
cooperative and non-cooperative NOMA are analyzed in terms of outage
probability, ergodic capacity and energy efficiency. In [7], the outage
probability of the cognitive radio NOMA network is analyzed under HWI, ICSI
and imperfect successive interference cancellation (ipSIC). On the other hand,
BER performances of FD-CNOMA have been investigated in [8] with ipSIC where
perfect CSI and no HWI are considered. Then, the BER and outage probability
performances of a multi-hop decode-and-forward (DF) CNOMA with ICSI has been
investigated in [9]. The pairwise error probability (PEP) is derived for the
AF-CNOMA by considering HWI with no ICSI. [10].

All aforementioned CNOMA studies investigate the performances of CNOMA without
direct links (DL) and CNOMA with DL has not been investigated with
imperfections in terms of any performance metrics. Besides, most of above
papers [1, 2, 4, 5, 6, 7] analyze the effects of one or multiple imperfections
only in terms of information-theoretic perspectives (e.g., achievable rate or
outage probability) whilst the BER performance (i.e., one of the most
important KPIs) is considered in very limited studies [8, 9, 10]. Furthermore,
the effects of HWI on the BER performances of NOMA with/without cooperative
have not been studied well although its effects on capacity and outage
performances have been evaluated in [4, 5, 6, 7]. As discussed above, to the
best of our knowledge, the BER performance of NOMA schemes (with/without
cooperative relaying) has not been investigated with practical constraints
(i.e., ipSIC, ICSI and HWI). Besides, the performance of CNOMA with DL has not
been revealed with any imperfections in the open literature.

Motivated by these discussions, in this letter, we provide an analytical
framework for the BER performances of NOMA schemes under practical constraints
where we consider ipSIC, ICSI and HWI. We derive exact BER expressions for
three NOMA schemes: downlink NOMA, CNOMA without DL and CNOMA with DL. We
validate our analysis through extensive simulations and discuss the effects of
the system parameters (e.g., power allocation or HWI level) on the BER
performances of NOMA schemes

The rest of the letter is organized as follows. The CNOMA schemes with/without
direct links are introduced in Section II. In Section III, we analyze the
exact end-to-end (e2e) BER expressions. The simulation results are presented
in Section IV. Finally, Section V concludes the letter.

##  II System Model

We consider a downlink cooperative NOMA system as presented in Fig. 1. The
model consists of a source (S), DF relay and two users: user 1 (u1) and user 2
(u2). The S, relay (R) and users are equipped with a single antenna and the
relay works in half duplex (HD) mode. We assume three different scenarios of
the considered system which are : $i)\ Downlink\ NOMA:$ S transmits directly
the signal to the users, namely NOMA. $ii)\ Cooperative\ NOMA\ without\
direct\ link:$ S communicates with users with the aid of DF relay without
direct link between the S and users, namely CNOMA. $iii)\ Cooperative\ NOMA\
with\ direct\ link:$ S sends the signal to the user through the DF relay and
the direct links, where the users combine both received signals in two phases,
namely CNOMA-WDL. The communication is completed in a single phase for
downlink NOMA whereas it requires two phases in cooperative schemes. The
complex flat fading channel coefficient between S-users, R-users and S-R are
denoted as $h_{si}\sim\mathcal{CN}(0,\ \sigma_{si}^{2})$,
$h_{ri}\sim\mathcal{CN}(0,\ \sigma_{ri}^{2})$ and $h_{sr}\sim\mathcal{CN}(0,\
\sigma_{sr}^{2})$ where $i=1,\ 2$, respectively. $\sigma_{{h}_{si}}^{2}\
={d}_{h_{si}}^{-a}$, $\sigma_{{h}_{ri}}^{2}\ ={d}_{h_{ri}}^{-a}$ and
$\sigma_{{h}_{sr}}^{2}\ ={d}_{h_{sr}}^{-a}$, where ${d}_{h_{si}}$,
${d}_{h_{ri}}$ and ${d}_{h_{sr}}$ are distances between related nodes and $a$
is the path loss exponent. We assume that the ICSI exists for each node. The
estimated channel coefficients are given as $\tilde{{h}}_{si}={h}_{si}-e$,
$\tilde{{h}}_{ri}={h}_{ri}-e$ and $\tilde{{h}}_{sr}={h}_{sr}-e$, where the
channel estimation error is presented as $e\sim\mathcal{CN}(0,\
\sigma_{\epsilon}^{2})$. Since $\tilde{{h}}_{si}$, $\tilde{{h}}_{ri}$,
$\tilde{{h}}_{sr}$ and ${e}$ are independent, then it can be modeled as
$\sigma_{\tilde{{h}}_{si}}^{2}\ =\sigma_{{h}_{si}}^{2}\
-\sigma_{\epsilon}^{2}$, $\sigma_{\tilde{{h}}_{ri}}^{2}\
=\sigma_{{h}_{ri}}^{2}\ -\sigma_{\epsilon}^{2}$ and
$\sigma_{\tilde{{h}}_{sr}}^{2}\ =\sigma_{{h}_{sr}}^{2}\
-\sigma_{\epsilon}^{2}$ [11].

Figure 1: System model CNOMA schemes with HWI.

As presented in Fig 1, the S transmits a superimposed coding (SC) signal
${s}_{sc}=\sqrt{\alpha_{1}}m_{1}+\sqrt{\alpha_{2}}m_{2}$ to R and users with
different power coefficients according to channel gain
$|{h}_{s1}|^{2}<|{h}_{s2}|^{2}$. The received signals at u1, u2 and R (for
cooperative schemes) are given as

${y}_{si}=(\tilde{{h}}_{si}\
+{e})(\sqrt{P_{s}}{s}_{sc}+\eta_{t,{si}})+\eta_{R,si}+{n}_{si},$ (1)
${y}_{sr}=(\tilde{{h}}_{sr}\
+{e})(\sqrt{P_{s}}{s}_{sc}+\eta_{t,{sr}})+\eta_{R,sr}+{n}_{sr},$ (2)

where $m_{1}$, $m_{2}$ are the message of the u1 and u2, and $\alpha_{1}$,
$\alpha_{2}$ are the PA coefficients of u1 and u2, respectively.
$\alpha_{1}+\alpha_{2}=1$. $P_{s}$ is the transmit power of S. $\eta_{t,si}$,
$\eta_{t,sr}$ and $\eta_{R,si}$, $\eta_{R,sr}$ are distortion noises at the
transmitter and receiver, respectively, which occur due to the HWI at
transceivers such as oscillator phase noise (PN), high power amplifier (HPA)
distortion and in-phase and quadrature-phase imbalance (IQI). We assume
${n}_{si}={n}_{sr}={n}$ and it is the additive white Gaussian noise (AWGN)
which follows $\ {n}\sim\mathcal{CN}(0,\ \frac{N_{0}}{2})$. The distortion
noises are defined as

$\begin{split}&\mathbf{\eta}_{t,{si}}\sim\mathcal{CN}(0,\
{k}_{t,{si}}^{2}{P}_{s}),\ \mathbf{\eta}_{R,{si}}\sim\mathcal{CN}(0,\
{k}_{R,{si}}^{2}{P}_{s}|{h}_{si}|^{2}),\\\
&\mathbf{\eta}_{t,{sr}}\sim\mathcal{CN}(0,\ {k}_{t,{sr}}^{2}{P}_{s}),\
\mathbf{\eta}_{R,{sr}}\sim\mathcal{CN}(0,\
{k}_{R,{sr}}^{2}{P}_{s}|{h}_{sr}|^{2}),\end{split}$ (3)

where ${k}_{t,{si}}^{2}$, ${k}_{t,{sr}}^{2}$ and ${k}_{R,{si}}^{2}$,
${k}_{R,{sr}}^{2}$ represent level of impairment at the transmitter and
receiver, respectively. Then, (1) can be regarded as

${y}_{si}=(\tilde{{h}}_{si}\ +{e})(\sqrt{P_{s}}{s}_{sc}+\eta_{si})+{n},$ (4)
${y}_{sr}=(\tilde{{h}}_{sr}\ +{e})(\sqrt{P_{s}}{s}_{sc}+\eta_{sr})+{n},$ (5)

where $\eta_{si}$ and $\eta_{sr}$ are the independent distortion noise terms
defined as $\mathbf{\eta}_{si}\sim\mathcal{CN}(0,\ {k}_{si}^{2}{P}_{s})$,
$\mathbf{\eta}_{sr}\sim\mathcal{CN}(0,\ {k}_{sr}^{2}{P}_{s})$, $k_{si}^{2}$
and $k_{sr}^{2}$ are the HWI level at the transceiver. It has been
demonstrated in[11] that the impact of the transceiver HWI can be
characterized by the aggregate level of impairments,
$k_{si}^{2}={k}_{t,{si}}^{2}+{k}_{R,{si}}^{2}$ and
$k_{sr}^{2}={k}_{t,{sr}}^{2}+{k}_{R,{sr}}^{2}$.

In the downlink NOMA, the u1 decodes its message directly by maximum-
likelihood detection (MLD), the u2 and relay performs an SIC to detect $m_{1}$
and $m_{2}$.

On the other hand, in the cooperative schemes, the R implements an SIC process
to obtain both users’ symbols. Then, R implements a new SC signal and forwards
it by the power of the R to the users in the second phase. Again, R
distributes the values of the power coefficient according to channel gain as
in S. The received signal at u1 and u2 in the second phase is given as

${y}_{ri}=(\tilde{{h}}_{ri}\ +{e})(\sqrt{P_{r}}{s}_{sc}+\eta_{ri})+{n},$ (6)

where $P_{r}$ is the relay transmit power and
$\mathbf{\eta}_{ri}\sim\mathcal{CN}(0,\ {k}_{ri}^{2}{P}_{r})$.

In CNOMA, the users receive only the second phase signal; hence, they decode
their messages according to ${y}_{ri}$ (6) by using MLD or SIC. However, in
CNOMA-WDL, the users receive signals in both phases. Thus, the users first
implement a maximum ratio combining by using ${y}_{si}$ (1) and ${y}_{ri}$
(6), then, they implement a proper detection scheme (e.g., MLD or SIC).

##  III Performance Analysis

In this section, we derive the BER expressions of NOMA, CNOMA and CNOMA-WDL
for both users over Rayleigh fading channels. The binary phase-shift keying
(BSPK) modulation is used to modulate the message of u1 and u2 at S and R. The
e2e BER expressions in the CNOMA for both users are given as [9] by

${P}_{e2e,i}^{(CNOMA)}={P}_{m_{i},sr}\left(1-{P}_{m_{i},ri}\right)+\left(1-{P}_{m_{i},sr}\right){P}_{m_{i},ri},$
(7)

where ${P}_{m_{i},sr}$ is the BER of the u1 and u2 symbols in relay at first
phase, ${P}_{m_{i},ri}$ is BER of u1 and u2 in second phase.

On the other hand, the e2e BER expressions in CNOMA-WDL for both users are
given as [12] by

$\begin{split}&P_{e2e,i}^{(CNOMA-WDL)}=\\\
&\frac{1}{2}\sum\limits_{f}g_{f}[P_{m_{i},prop}\times
P_{m_{i},sr}+(1-P_{m_{i},sr})P_{m_{i},coop}],\ f=z,\ v,\end{split}$ (8)

where $g_{f},\ f=z,v$ is a parameter for u1 and u2, respectively, which is
determined according to modulation order. It will be defined in the following
related subsections. $P_{m_{1},prop}$ is the BER in the presence of error
propagation from R to users achieved by the MRC and $P_{m_{i},coop}$ is the
BER when the symbols of users are detected correctly at the relay and
forwarded to the users and then combined with MRC at the users.

###  III-A BER of NOMA for point-to-point (P2P) communication

In P2P communication, each node (i.e., R, u1 and u2) performs an MLD to detect
$m_{1}$ symbol. Thus, the BER for $m_{1}$ symbols in the presence of the
effects of HWI and ICSI is given as

${P}_{m_{1},j}=\frac{1}{2}\sum\limits_{z=1}^{2}\mathrm{Q}\left(\sqrt{2\delta_{m1,j,z}}\right),\
\text{where}\ j=sr,\ s1,\ r1,$ (9)

where $\delta_{m1,j,z}$ defines the signal-to-interference plus noise ratio
(SINR) for $m_{1}$ symbols in the presence of HWI and ICSI. It is given as
$\delta_{m_{1},sr,z}=\frac{P_{s}\psi_{z}|\tilde{h}_{sr}|^{2}}{N_{0}+2P_{s}k_{sr}^{2}|\tilde{h}_{sr}|^{2}+2(k_{sr}^{2}+\psi_{z})P_{s}\sigma_{\epsilon}^{2}}$,
$\delta_{m_{1},s1,z}=\frac{P_{s}\psi_{z}|\tilde{h}_{s1}|^{2}}{N_{0}+2P_{s}k_{s1}^{2}|\tilde{h}_{s1}|^{2}+2(k_{s1}^{2}+\psi_{z})P_{s}\sigma_{\epsilon}^{2}}$,
$\delta_{m_{1},r1,z}=\frac{P_{r}\psi_{z}|\tilde{h}_{r1}|^{2}}{N_{0}+2P_{r}k_{r1}^{2}|\tilde{h}_{r1}|^{2}+2(k_{r1}^{2}+\psi_{z})P_{r}\sigma_{\epsilon}^{2}}$
where
$\psi_{z}=[(\sqrt{\alpha_{1}}+\sqrt{\alpha_{2}})^{2},(\sqrt{\alpha_{1}}-\sqrt{\alpha_{2}})^{2}]$
is given according to modulation order.

By using the moment generating function (MGF) and alternate form of
$\mathrm{Q}(.)$ function as being in [13, eq. (5.4)-(5.6)], the derivation of
the average BER (ABER) is defined as follows

${P}_{m_{1},j}=\frac{1}{4}\sum\limits_{z=1}^{2}\left(1-\sqrt{\frac{\bar{\delta}_{m1,j,z}}{1+\bar{\delta}_{m1,j,z}}}\right),$
(10)

where $\bar{\delta}_{m_{1},j,z}=E[{\delta}_{m_{1},j,z}]$ is defined and $E[.]$
is an Expectation operator. Thus,
$\bar{\delta}_{m_{1},j,z}=\frac{P_{s}\psi_{z}\sigma_{\tilde{h}_{j}}^{2}}{N_{0}+2P_{s}k_{j}^{2}\sigma_{\tilde{h}_{j}}^{2}+2(k_{j}^{2}+\psi_{z})P_{s}\sigma_{\epsilon}^{2}}$
is given.

As explained above, the (10) gives the BER for a P2P NOMA scheme. Thus, it
refers to BER of $m_{1}$ symbols in downlink NOMA (i.e.,
$P_{m_{1},s1}\triangleq P_{e2e,1}^{(NOMA)}$), the BER of $m_{1}$ symbols in
the first phase of CNOMA and CNOMA-WDL (i.e., $P_{m_{1},sr}$), and the BER of
$m_{1}$ symbols in the second phase of CNOMA (i.e., $P_{m_{1},r1}$).

To detect $m_{2}$ symbols, each node implements a SIC process. The BER of
$m_{2}$ symbols at each node is obtained as the sum of the correct and
erroneous detection of the $m_{1}$ symbols during the SIC process. Therefore,
by also considering the imperfect SIC, the BER of $m_{2}$ at relay and u2 is
given by

${P}_{m_{2},l}=\frac{1}{2}\sum\limits_{v=1}^{6}g_{v}\mathrm{Q}\left(\sqrt{2\delta_{m_{2},l,v}}\right),\
\text{where}\ l=sr,\ s2,\ r2,$ (11)

where $\mathbf{g}_{v}=[1,1,-1,1,1,-1]$,
$\delta_{m_{2},sr,v}=\frac{P_{s}\zeta_{v}|\tilde{h}_{sr}|^{2}}{N_{0}+2P_{s}k_{sr}^{2}|\tilde{h}_{sr}|^{2}+2(k_{sr}^{2}+\xi_{v})P_{s}\sigma_{\epsilon}^{2}}$,
$\delta_{m_{2},s2,v}=\frac{P_{s}\zeta_{v}|\tilde{h}_{s2}|^{2}}{N_{0}+2P_{s}k_{s2}^{2}|\tilde{h}_{s2}|^{2}+2(k_{s2}^{2}+\xi_{v})P_{s}\sigma_{\epsilon}^{2}}$,
$\delta_{m_{2},r2,v}=\frac{P_{r}\zeta_{v}|\tilde{h}_{r2}|^{2}}{N_{0}+2P_{r}k_{r2}^{2}|\tilde{h}_{r2}|^{2}+2(k_{r2}^{2}+\xi_{v})P_{r}\sigma_{\epsilon}^{2}}$,
$\boldsymbol{\zeta}_{v}=[\alpha_{2},\alpha_{2},(\sqrt{\alpha_{1}}+\sqrt{\alpha_{2}})^{2},(2\sqrt{\alpha_{1}}+\sqrt{\alpha_{2}})^{2},(\sqrt{\alpha_{1}}-\sqrt{\alpha_{2}})^{2},(2\sqrt{\alpha_{1}}-\sqrt{\alpha_{2}})^{2}]$
and
$\boldsymbol{\xi}_{v}=[(\sqrt{\alpha_{1}}+\sqrt{\alpha_{2}})^{2},(\sqrt{\alpha_{1}}-\sqrt{\alpha_{2}})^{2},(\sqrt{\alpha_{1}}+\sqrt{\alpha_{2}})^{2},(\sqrt{\alpha_{1}}+\sqrt{\alpha_{2}})^{2},(\sqrt{\alpha_{1}}-\sqrt{\alpha_{2}})^{2},(\sqrt{\alpha_{1}}-\sqrt{\alpha_{2}})^{2}]$.
In definition of $\zeta_{v}$, both correct and erroneous SIC conditions are
considered. In the case of erroneous SIC, the erroneous detected $m_{1}$
symbols are subtracted from the received signal, thus the decision rule for
detection is shifted. Hence, a few terms in $\zeta_{v}$ have the coefficient
of $2$ which occurs due to the ipSIC. Besides, one can motice that the priori
probability for some terms has a negative value in the coefficient $g_{v}$
which occurs due to conditional probability in case of ipSIC.

By using the MGF and alternate form of $\mathrm{Q}(.)$ function, the ABER of
$m_{2}$ is expressed

${P}_{m_{2},l}=\frac{1}{4}\sum\limits_{v=1}^{6}g_{v}\left(1-\sqrt{\frac{\bar{\delta}_{m_{2},l,v}}{1+\bar{\delta}_{m_{2},l,v}}}\right),$
(12)

where $\bar{\delta}_{m_{2},j,v}=E[{\delta}_{m_{2},j,v}]$,
$\bar{\delta}_{m_{2},j,v}=\frac{P_{s}\zeta_{v}\sigma_{\tilde{h}_{j}}^{2}}{N_{0}+2P_{s}k_{j}^{2}\sigma_{\tilde{h}_{j}}^{2}+2(k_{j}^{2}+\xi_{v})P_{s}\sigma_{\epsilon}^{2}}$.

As discussed for u1, the BER in (12) gives the error probability of NOMA for a
P2P communication. Therefore, it refers to BER of $m_{2}$ symbols in downlink
NOMA i.e., $P_{m_{2},s2}\triangleq P_{e2e,2}^{(NOMA)}$), the BER of $m_{2}$
symbols in the first phase of CNOMA and CNOMA-WDL (i.e., $P_{m_{2},sr}$), and
the BER of $m_{2}$ symbols in the second phase of CNOMA (i.e.,
$P_{m_{2},r2}$).

###  III-B BER of cooperative diversity with MRC

1) At u1: Based on the detected signals in the first phase, R implements a new
SC signal again and sends it to u1 and u2 by its power. Thus, u1 and u2
receive two signals from different sources, one from S and one from the R. To
improve the users’ messages, the users perform an MRC to combine the received
signals from the two different phases. The BER of the u1 with diversity
combining of the two branches using MRC is given by [12], [13, pp. 320-321]

$P_{m_{1},coop}=\mathrm{Q}\left(\sqrt{2({\delta}_{m_{1},s1,z}+{\delta}_{m_{1},r1,z})}\right).$
(13)

The ABER for cooperative MRC of two branches at u1 when the mean of SNRs of
branches are different (i.e.,
$\bar{{\delta}}_{m1,s1,z}\neq\bar{{\delta}}_{m1,r1,z}$) is given as in (14)
(see the top of the next page).

$P_{m_{1},coop}=\frac{1}{2}\left(1-\frac{1}{\bar{{\delta}}_{m_{1},s1,z}-\bar{{\delta}}_{m_{1},r1,z}}\left(\bar{{\delta}}_{m_{1},s1,z}\sqrt{\frac{\bar{{\delta}}_{m_{1},s1,z}}{1+\bar{{\delta}}_{m_{1},s1,z}}}-\bar{{\delta}}_{m_{1},r1,z}\sqrt{\frac{\bar{{\delta}}_{m_{1},r1,z}}{1+\bar{{\delta}}_{m_{1},r1,z}}}\right)\right).$
(14)

2) At u2: The u2 detects $m_{1}$ firstly after combining the received signals
with MRC. Thereafter, using the SIC detects its symbols $m_{2}$. The BER for
MRC after the SIC to detect u2 symbols is expressed by [12],[13, 320-321]

$P_{m_{2},coop}=\mathrm{Q}\left(\sqrt{2({\delta}_{m_{2},s2,v}+{\delta}_{m_{2},r2,v})}\right).$
(15)

The ABER for cooperative MRC of two branches at u2 when the mean of the SNRs
of branches are different (i.e.,
$\bar{{\delta}}_{m_{2},s2,v}\neq\bar{{\delta}}_{m_{2},r2,v}$) is expressed by
[12], [14, pp. 846-847] as in (16) (see the top of the next page).

$P_{m_{2},coop}\\{s_{sc}\in
v\\}=\frac{1}{2}\left(1-\frac{1}{\bar{{\delta}}_{m_{2},s2,v}-\bar{{\delta}}_{m_{2},r2,v}}\left(\bar{{\delta}}_{m_{2},s2,v}\sqrt{\frac{\bar{{\delta}}_{m_{2},s2,v}}{1+\bar{{\delta}}_{m_{2},s2,v}}}-\bar{{\delta}}_{m_{2},r2,v}\sqrt{\frac{\bar{{\delta}}_{m_{2},r2,v}}{1+\bar{{\delta}}_{m_{2},r2,v}}}\right)\right).$
(16)

###  III-C BER of cooperative MRC with error propagation

Unless a genie-aided relaying is implemented, the R node forwards the detected
signals to the users. By considering the ipSIC, the relay can also detect
symbols erroneously. Therefore, the erroneous symbols in R are also forwarded
to the users. This phenomenon is called error propagation. These erroneous
symbols are also combined with the signals received from the direct path.
After the implantation of the MRC, we should describe the total received
signal in order to acquire the BER in case of error propagation from u1 and
u2. Without loosing the generality, we assume that the BS sent a symbol (+1)
and a symbol (+1) for u1 and u2, respectively. During the SIC process in R,
the symbol of u1 is detected erroneously as (-1). Thus, after the SIC process,
the symbol of the u2 is detected erroneously also as (-1).

####  III-C1 At u1

Consequently, the sum of the received signals at the u1 by MRC is given as

$\begin{split}&{\varphi}_{m_{1},coop}=\tilde{h}^{*}_{s1}y_{1}+\tilde{h}_{r1}^{*}y_{r1}\\\
&=({\varphi}_{m_{1},s1,z}-{\varphi}_{m_{1},r1,z})+\tilde{n}_{a},\end{split}$
(17)

where ${\varphi}_{m_{1},s1,z}=P_{s}\psi_{z}|\tilde{h}_{s1}|^{2}$ and we define
${\varphi}_{m_{1},r1,z}=P_{r}\psi_{z}|\tilde{g}_{r1}|^{2}$. However,
$\tilde{n}_{a}$ is the effective noise with
$\mathbf{E}[\tilde{n}_{a}]\sim\mathcal{CN}(0,\frac{N_{0}}{2}[\sigma_{\tilde{h}_{s1}}^{2}+\sigma_{\tilde{h}_{r1}}^{2}]+k_{s1}^{2}P_{s}\sigma_{\tilde{h}_{s1}^{2}}{\sigma_{\epsilon}^{2}}+k_{s1}^{2}P_{s}\sigma_{\tilde{h}_{s1}}^{2}+k_{r1}^{2}P_{r}$
$\sigma_{\tilde{h}_{r1}}^{2}{\sigma_{\epsilon}}^{2}+k_{r1}^{2}P_{r}\sigma_{\tilde{h}_{r1}}^{2}+\tau_{m_{1},s1,z}\sigma_{\tilde{h}_{s1}}^{2}\sigma_{\epsilon}^{2}+\tau_{m_{1},r1,z}\sigma_{\tilde{h}_{r1}}^{2}{\sigma_{\epsilon}^{2}})$,
where $\tau_{m_{1},s1,z}=P_{s}\psi_{z}$ and $\tau_{m_{1},r1,z}=P_{r}\psi_{z}$.
Through the MLD decision rule at the u1, $m_{1}=+1$ is declared if
${\varphi}_{m1,coop}\geq 0$. Thus, the BER of the error propagation for the u1
symbols is defined as

$\begin{split}P_{m_{1},prop}&=P({\varphi}_{m_{1},s1,z}-{\varphi}_{m_{1},r1,z}<\tilde{n}_{a})\\\
&=\mathrm{Q}\left(\frac{\varphi_{m_{1},s1,z}-{\varphi}_{m_{1},r1,z}}{\sqrt{\omega_{a}}}\right),\end{split}$
(18)

where
$\omega_{a}=\frac{N_{0}}{2}[\sigma_{\tilde{h}_{s1}}^{2}+\sigma_{\tilde{h}_{r1}}^{2}]+k_{s1}^{2}P_{s}{\sigma_{\tilde{h}_{s1}}^{2}}{\sigma_{\epsilon}^{2}}+k_{s1}^{2}P_{s}\sigma_{\tilde{h}_{s1}}^{2}+k_{r1}^{2}$
$P_{r}\sigma_{\tilde{h}_{r1}}^{2}{\sigma_{\epsilon}}^{2}+k_{r1}^{2}P_{r}\sigma_{\tilde{h}_{r1}}^{2}+\tau_{m_{1},s1,z}\sigma_{\tilde{h}_{s1}}^{2}\sigma_{\epsilon}^{2}+\tau_{m_{1},r1,z}\sigma_{\tilde{h}_{r1}}^{2}{\sigma_{\epsilon}^{2}}$.
From (18), if the relay forwards an incorrect symbol, this has a significant
impact instead of noise on the decision variable MLD. According to [12], we
approximate the BER as $\varphi_{m_{1},s1,z}-\varphi_{m_{1},r1,z}<0$. The BER
under error propagation is given as

$P_{m_{1},prop}=P({\varphi}_{m_{1},s1,z}-{\varphi}_{m_{1},r1,z}<0).$ (19)

By averaging (19) over ${\varphi}_{m_{1},s1,z}$and ${\varphi}_{m_{1},r1,z}$,
the average error probability under error propagation is obtained as

$P_{m_{1},prop}=\frac{{\bar{\varphi}}_{m_{1},r1,z}}{{\bar{\varphi}}_{m_{1},r1,z}+{\bar{\varphi}}_{m_{1},s1,z}},$
(20)

where
${\bar{\varphi}}_{m_{1},s1,z}=E[{\varphi}_{m_{1},s1,z}]=P_{s}\psi_{z}\sigma_{\tilde{h}_{s1}}^{2}$
and
${\bar{\varphi}}_{m_{1},r1,z}=E[{\varphi}_{m_{1},r1,z}]=P_{r}\psi_{z}\sigma_{\tilde{h}_{r1}}^{2}$.

####  III-C2 At u2

Also, after combining the S and R signals at the u2, the total received signal
by u2 is given as

${\varphi}_{m_{2},coop}=\tilde{h}^{*}_{s2}y_{2}+\tilde{h}_{r2}^{*}y_{r2}.$
(21)

After the implementation of the MRC in u2, it detects the u1 symbols first,
then it detects its symbols through the SIC process. Therefore, the sum of the
received signal at u2 can be indicated as

${\varphi}_{m_{2},coop}=({\varphi}_{m_{2},s2,v}-{\varphi}_{m_{2},r2,v})+\tilde{n}_{b},$
(22)

where ${\varphi}_{m_{2},s2,v}=P_{s}\zeta_{v}|\tilde{h}_{s2}|^{2}$ and
${\varphi}_{m_{2},r2,v}=P_{r}\zeta_{v}|\tilde{h}_{r2}|^{2}$. $\tilde{n}_{b}$
is the effective noise with
$\mathbf{E}[\tilde{n}_{b}]\sim\mathcal{CN}(0,\frac{N_{0}}{2}[\sigma_{\tilde{h}_{s2}}^{2}+\sigma_{\tilde{h}_{r2}}^{2}]+k_{s2}^{2}P_{s}{\sigma_{\tilde{h}_{s2}}^{2}}{\sigma_{\epsilon}^{2}}+k_{s2}^{2}P_{s}\sigma_{\tilde{h}_{s2}}^{2}+k_{r2}^{2}P_{r}$,
$\sigma_{\tilde{h}_{r2}}^{2}{\sigma_{\epsilon}}^{2}+k_{r2}^{2}P_{r}\sigma_{\tilde{h}_{r2}}^{2}+\tau_{m_{2},s2,v}\sigma_{\tilde{h}_{s2}}^{2}\sigma_{\epsilon}^{2}+\tau_{m_{2},r2,v}\sigma_{\tilde{h}_{r2}}^{2}{\sigma_{\epsilon}^{2}})$,
where ${\tau}_{m_{2},s2,v}=P_{s}\zeta_{v}$, and
${\tau}_{m_{2},r2,v}=P_{r}\zeta_{v}$. Through the MLD decision rule and the
SIC process at the u2 are declared as $m_{1}=+1$ and $m_{2}=+1$. Thus, the BER
of the error propagation for the u2 symbols is defined as

$\begin{split}P_{m_{2},prop}&=P({\varphi}_{m_{2},s2,v}-{\varphi}_{m_{2},r2,z}<\tilde{n}_{b})\\\
&=\mathrm{Q}\left(\frac{\varphi_{m_{2},s2,v}-{\varphi}_{m_{2},r2,v}}{\sqrt{\omega_{b}}}\right),\end{split}$
(23)

where
$\omega_{b}=\frac{N_{0}}{2}[\sigma_{\tilde{h}_{s2}}^{2}+\sigma_{\tilde{h}_{r2}^{2}}+k_{s2}^{2}P_{s}{\sigma_{\tilde{h}_{s2}}^{2}}{\sigma_{\epsilon}^{2}}+k_{s2}^{2}P_{s}\sigma_{\tilde{h}_{s2}}^{2}+$
$k_{r2}^{2}P_{r}\sigma_{\tilde{h}_{r2}}^{2}{\sigma_{\epsilon}}^{2}+k_{r2}^{2}P_{r}\sigma_{\tilde{h}_{r2}}^{2}+\tau_{m_{2},s2,v}\sigma_{\tilde{h}_{s2}}^{2}\sigma_{\epsilon}^{2}+\tau_{m_{2},r2,v}\sigma_{\tilde{h}_{r2}}^{2}{\sigma_{\epsilon}^{2}}$
From (23), if R forwards an incorrect symbol, this has a significant impact
instead of noise on the decision variable MLD and SIC. Consequently, we
approximate the BER for the SIC to detect $m_{2}$
as${\varphi_{m_{2},s2,v}-{\varphi}_{m_{2},r2,v}}<0$[12], the BER under error
propagation is presented as

$P_{m_{2},prop}=P({\varphi}_{m_{2},s2,v}-{\varphi}_{m_{2},r2,v}<0).$ (24)

By averaging (24) over ${\varphi}_{m_{2},s2,v}$ and ${\varphi}_{m_{2},r2,v}$ ,
the average error probability under error propagation is obtained as

$P_{m_{2},prop}=\frac{{\bar{\varphi}}_{m_{2},r2,v}}{{\bar{\varphi}}_{m_{2},r2,v}+{\bar{\varphi}}_{m_{2},s2,v}}.$
(25)

where
${\bar{\varphi}}_{m_{2},s2,v}=E[{\varphi}_{m_{2},s2,v}]=P_{s}\zeta_{v}\sigma_{\tilde{h}_{s2}}^{2}$
and
${\bar{\varphi}}_{m_{2},s2,v}=E[{\varphi}_{m_{2},r2,v}]=P_{r}\zeta_{v}\sigma_{\tilde{h}_{r2}}^{2}$.

Finally, to find the e2e BER in CNOMA, we obtain $P_{m_{1},sr}$,
$P_{m_{1},r1}$ by using (10) and $P_{m_{2},sr}$, $P_{m_{1},r2}$ by using (12)
and substitute them into (7) for u1 and u2, respectively. On the other hand,
to obtain the e2e BER in CNOMA-WDL, we determine $P_{m_{1},sr}$,
$P_{m_{1},coop}$ and $P_{m_{1},prop}$ by using (10), (14) and (20) and
substitute them into (8) for u1. Similarly, we determine $P_{m_{2},sr}$,
$P_{m_{2},coop}$ and $P_{m_{2},prop}$ by using (12), (16) and (25) and
substitute them into (8) for u2.

##  IV Numerical results

In this section, we validate the analytical BER results with computer
simulations for three schemes (i.e., NOMA, CNOMA, CNOMA-WDL). Unless otherwise
stated, we set the parameters to $d_{s1}=4m$, $d_{s2}=2m$, $d_{sr}=1m$,
$d_{r1}=3m$, $d_{r2}=1m$, $P_{r}=P_{s}$, $a=2$, $\alpha_{1}=0.8$,
$\alpha_{2}=0.2$ and the HWI level is equal for all nodes, i.e.,
$k_{s1}=k_{s2}=k_{r1}=k_{r2}=k$.

In Fig. 2.a, we present the BER performance w.r.t SNR where the ICSI is
considered as $\sigma_{\epsilon}^{2}=0.005$ and the HWI factors are set at
$k=0.175$ representing the worst value of HWI in open literature. First, it is
observed that the numerical results match perfectly with the simulation
results, which proves the correctness of our analysis. Based on the results,
as expected, with the increase of HWI and ICSI levels , all NOMA schemes get
worse performance, which reflect to practical implementations. By comparing
NOMA schemes with the HWI, we obverse that NOMA is superior to CNOMA. This can
be explained as follows. In the presence of HWI, due to the increased HWI in
total (additional HWI at R node), the performance of each phase is drastically
degraded, so that the e2e performance of CNOMA becomes worse than simple
downlink NOMA schemes. On the other hand, CNOMA-WDL outperforms both schemes
whether a HWI is introduced or not since a diversity path is achieved by MRC.
Nevertheless, a full diversity order (i.e., $2$) may not be observed. This is
for two reasons. The first is due to error propagation from relay to users. As
explained in analysis, unless a genie-aided relaying is not considered, the
relay node also forwards erroneous symbols to the users, which causes an error
floor in high SNR regime. The second reason of the error floor is the
imperfections in the system due to HWI or ICSI.

(a)

(b)

(c)

Figure 2: BER performances of NOMA, CNOMA and CNOMA-WDL a) w.r.t SNR when
$\sigma_{\epsilon}^{2}=0.005$ b) w.r.t HWI level when
$\sigma_{\epsilon}^{2}=0.005$ and $SNR=15,\ 40$ dB c) w.r.t PA when
$\sigma_{\epsilon}^{2}=0.005$ and $SNR=20$ dB.

To further evaluate the effect of HWI, in Fig. 2.b, we present BER
performances versus the HWI level for $SNR=15,\ 40$ dB when
$\sigma_{\epsilon}^{2}=0.005$. As expected, with increasing HWI, all schemes
have a higher BER. In Fig. 2.b, one can easily see that the CNOMA-WDL is
superior to both schemes for all HWI values. For $SNR=15$ dB, the CNOMA
outperforms the NOMA regardless of HWI level. However, for $SNR=15$ dB, NOMA
becomes better than CNOMA after $k\approx 0.1$ since, as explained in the
previous figure, with increasing HWI, an error floor occurs for both phases
(i.e., $S-R$ and $R-Ui,\ i=1,2$). Hence, the performance of CNOMA decreases
dramatically, so NOMA provides better performance. In this regard, we can say
that when the HWI is high, using NOMA rather than CNOMA is more beneficial.

Fig. 2.c shows the effect of power allocation (PA) on the BER performances
with different levels of HWI. It is observed that the PA affects the BER
performance of one user at the expense of the other. Nevertheless, increasing
$\alpha_{2}$ too much does always not mean an increase in performance of u2
due to SIC process. Thus, PA should be carefully chosen not to cause an
unfairness for users.

##  V Conclusion

In this letter, we investigate the BER performance of three NOMA schemes
(i.e., downlink NOMA, cooperative NOMA with and without direct links) under
practical assumptions. We consider all imperfections (i.e., ipSIC, ICSI, HWI)
and derive the exact e2e BER expressions for all schemes. All theoretical
deductions are validated with computer simulations. We reveal that NOMA can
outperform CNOMA when the HWI is relatively high. Besides, we discuss the
effect of PA on the BER performances and present that the PA has an important
role; thus, it should be chosen wisely no to cause an error propagation
whether from relay to users in CNOMA-WDL or an error in SIC processes in all
schemes. We believe that this letter can provide fruitful insights for the
practical implementation of NOMA schemes and the analysis in this work help
researchers to make further analysis for other NOMA schemes under practical
assumptions.

## References

  * [1] M. Zeng, W. Hao, O. A. Dobre, and Z. Ding, “Cooperative NOMA: State of the art, key techniques, and open challenges,” _IEEE Netw._ , vol. 34, no. 5, pp. 205–211, 2020.
  * [2] H. Liu, Z. Ding, K. J. Kim, K. S. Kwak, and H. V. Poor, “Decode-and-forward relaying for cooperative NOMA systems with direct links,” _IEEE Trans. Wirel. Commun._ , vol. 17, no. 12, pp. 8077–8093, 2018.
  * [3] L. Smaini, _RF Analog Impairments Modeling for Communication systems Simulation: Application to OFDM-based Transceivers_. John Wiley $\&$ Sons, 2012.
  * [4] M. Li, B. Selim, S. Muhaidat, P. C. Sofotasios, M. Dianati, P. D. Yoo, J. Liang, and A. Wang, “Effects of residual hardware impairments on secure NOMA-based cooperative systems,” _IEEE Access_ , vol. 8, pp. 2524–2536, 2019.
  * [5] C. Deng, M. Liu, X. Li, and Y. Liu, “Hardware impairments aware full-duplex NOMA networks over Rician fading channels,” _IEEE Syst. J._ , vol. 15, no. 2, pp. 2515–2518, 2020.
  * [6] X. Li, J. Li, Y. Liu, Z. Ding, and A. Nallanathan, “Residual transceiver hardware impairments on cooperative NOMA networks,” _IEEE Trans. Wirel. Commun._ , vol. 19, no. 1, pp. 680–695, 2019.
  * [7] S. Arzykulov, G. Nauryzbayev, A. Celik, and A. M. Eltawil, “Hardware and interference limited cooperative CR-NOMA networks under imperfect SIC and CSI,” _IEEE Open J. Commun. Soc._ , vol. 2, pp. 1473–1485, 2021.
  * [8] A. A. Hamza, I. Dayoub, I. Alouani, and A. Amrouche, “On the error rate performance of full-duplex cooperative NOMA in wireless networks,” _IEEE Trans. Commun._ , vol. 70, no. 3, pp. 1742–1758, 2022.
  * [9] F. Khennoufa, K. Abdellatif, and F. Kara, “Bit error rate and outage probability analysis for multi-hop decode-and-forward relay-aided NOMA with imperfect SIC and imperfect CSI,” _AEU-Internat. J. Electron. Commun._ , vol. 147, p. 154124, 2022.
  * [10] L. Mohjazi, L. Bariah, S. Muhaidat, P. C. Sofotasios, O. Onireti, and M. A. Imran, “Error probability analysis of non-orthogonal multiple access for relaying networks with residual hardware impairments,” in _IEEE 30th Annual Internat. Symp. Pers., Indoor Mobile Radio Commun. (PIMRC)_ , 2019.
  * [11] A. Afana, N. Abu-Ali, and S. Ikki, “On the joint impact of hardware and channel imperfections on cognitive spatial modulation MIMO systems: Cramer–Rao bound approach,” _IEEE Syst. J._ , vol. 13, no. 2, pp. 1250–1261, 2018.
  * [12] F. Kara and H. Kaya, “On the error performance of cooperative-NOMA with statistical CSIT,” _IEEE Commun. Lett._ , vol. 23, no. 1, pp. 128–131, 2018\.
  * [13] M. K. Simon and M.-S. Alouini, _Digital Communications over Fading Channels_. John-Wiley $\&$ Sons, 2004\.
  * [14] J. G. Proakis and M. Salehi, _Digital Communications_. McGraw-Hill, 2008.


# Energy Dissipation in Turbulent Reconnection

R. Bandyopadhyay<EMAIL_ADDRESS>Department of Astrophysical Sciences,
Princeton University, Princeton, NJ 08544, USA  A. Chasapis  Laboratory for
Atmospheric and Space Physics, University of Colorado Boulder, Boulder, CO
80309, USA  W. H. Matthaeus  Department of Physics and Astronomy, University
of Delaware, Newark, DE 19716, USA  Bartol Research Institute, University of
Delaware, Newark, DE 19716, USA  T. N. Parashar  School of Chemical and
Physical Sciences, Victoria University of Wellington, Wellington 6012, New
Zealand  C. C. Haggerty  University of Hawaii, Institute for Astronomy,
Honolulu 96822, USA  M. A. Shay  Department of Physics and Astronomy,
University of Delaware, Newark, DE 19716, USA  Bartol Research Institute,
University of Delaware, Newark, DE 19716, USA  D. J. Gershman  NASA Goddard
Space Flight Center, Greenbelt, Maryland 20771, USA  B. L. Giles  NASA Goddard
Space Flight Center, Greenbelt, Maryland 20771, USA  J. L. Burch  Southwest
Research Institute, San Antonio, Texas 78238, USA

###### Abstract

We study the nature of pressure-strain interaction at reconnection sites,
detected by NASA’s Magnetospheric Multiscale (MMS) Mission. We employ data
from a series of published case studies, including a large-scale reconnection
event at the magnetopause, three small-scale reconnection events at the
magnetosheath current sheets, and one example of the recently discovered
electron-only reconnection. In all instances, we find that the pressure-strain
shows signature of conversion into (or from) internal energy at the
reconnection site. The electron heating rate is larger than the ion heating
rate and the compressive heating is dominant over the incompressive heating
rate in all cases considered. The magnitude of thermal energy conversion rate
is close to the electromagnetic energy conversion rate in the reconnection
region. Although in most cases the pressure-strain interaction indicates that
the particle internal energy is increasing, in one case the internal energy is
decreasing. These observations indicate that the pressure-strain interaction
can be used as an independent measure of energy conversion and dynamics in
reconnection regions, in particular independent of measures based on the
electromagnetic work. Finally, we explore a selected reconnection site in a
turbulent Particle-in-Cell (PIC) simulation which further supports the
observational results.

††preprint: AIP/123-QED

##  I Introduction

Magnetic reconnection is a fundamental process of energy conversion in
plasmas. During reconnection, magnetic energy is converted into particle
energy. Energy dissipation during reconnection has been a topic of increasing
interest because of its potentially significant effects on the plasma. This is
of central interest in several problems of space physics, such as
understanding the heating of the solar corona and solar wind, and acceleration
of energetic particles Montgomery and et al. (1980); Bruno and Carbone (2005);
Matthaeus and Velli (2011); Verscharen, Klein, and Maruca (2019). Despite its
importance and compelling nature, the nature of energy conversion in magnetic
reconnection in weakly-collisional plasmas remains incompletely understood.
Several questions remain such as what fraction of energy is converted to bulk
velocity of the particles and what fraction of energy is dissipated to the
internal energy of each charged particle species? Several previous works,
using data from experiments and simulations, have addressed this point (see
Yamada  _et al._ (2015); Yoo  _et al._ (2017); Fadanelli  _et al._ (2021) and
references therein.)

There have been various efforts to identify the specific mechanisms involved
in collisionless dissipation. Several heating mechanisms such as
Landau/transit-time resonances Leamon  _et al._ (1999); Howes  _et al._
(2008), stochastic heating Chandran  _et al._ (2010), cyclotron resonances
Hollweg and Markovskii (2002), kinetic instabilities Gary (1993), and magnetic
pumping Lichko  _et al._ (2017) have been proposed as contributing candidates.
Another approach, followed in this paper, focuses more broadly on the general
discussion of the pathways leading to dissipation: transfer across scales,
transfer between particles and electromagnetic fields, and transfer between
flow energies and internal energies for each species.

This approach Yang  _et al._ (2017a, b) eliminates the simplifying assumptions
that accompany the identification of specific dominant mechanisms. All
available channels, apart from collisions, are formulated based on the multi-
species Vlasov equation. Therefore, an identification of a dominant pathway to
dissipation includes all kinetic processes that lead to dissipation, and
consequently, is more general and broadly applicable.

##  II Energy conversion channels

For a collisionless plasma consisting of charged species labeled by $\alpha$,
the total energy density at each point $\mathbf{x}$ at a given time $t$ can be
separated into the electromagnetic energy density,

$\displaystyle\mathcal{E}^{m}({\bf x},t)={\frac{1}{8\pi}}\left({\bf
B}^{2}({\bf x},t)+{\bf E}^{2}({\bf x},t)\right)$ (1)

and the sum over species of the individual particle kinetic energy density

$\displaystyle\mathcal{E}_{\alpha}={\frac{1}{2}}m_{\alpha}\int{|{\bf{v}}|^{2}f_{\alpha}\left({\bf
x},{\bf v},t\right)d{\bf v}}.$ (2)

Here ${\bf B}$ and ${\bf E}$ are magnetic and electric fields, respectively,
$m_{\alpha}$ is the mass of particles of species $\alpha$, and $f_{\alpha}$ is
the velocity distribution function of particles of type $\alpha$, varying in
position and time. The collective motion is quantified by the fluid velocity
${\bf u_{\alpha}}$ defined by $n_{\alpha}{\bf u}_{\alpha}=\int{\bf
v}f_{\alpha}d{\bf{v}}$, where $n_{\alpha}=\int f_{\alpha}d{\bf v}$ is the
number density of species $\alpha$. Hereafter we refer to these energy
densities, for brevity, simply as energies.

Separating the kinetic energy into average and random parts facilitates the
understanding the energy conversion processes. The fluid flow kinetic energy
of species $\alpha$ is

$\displaystyle\mathcal{E}^{f}_{\alpha}={\frac{1}{2}}n_{\alpha}m_{\alpha}|\bf{u}_{\alpha}|^{2}$
(3)

and the corresponding thermal (or internal, or random) energy is

$\displaystyle\mathcal{E}^{th}_{\alpha}={\frac{1}{2}}m_{\alpha}\int{\left(\bf{v}-\bf{u}_{\alpha}\right)^{2}f_{\alpha}\left({\bf
x},{\bf v},t\right)d\bf{v}},$ (4)

we have
$\mathcal{E}_{\alpha}=\mathcal{E}^{f}_{\alpha}+\mathcal{E}^{th}_{\alpha}$.

The time evolution of the energies follows directly from manipulations of the
Vlasov equation and Maxwell’s equations. The fluid flow energy evolves in time
according to

$\displaystyle\partial_{t}\mathcal{E}^{f}_{\alpha}+\nabla\cdot\left(\mathcal{E}^{f}_{\alpha}\bf{u}_{\alpha}\right)+\nabla\cdot\left(\bf{P}_{\alpha}\cdot{\bf
u}_{\alpha}\right)$
$\displaystyle=\left(\bf{P}_{\alpha}\cdot\nabla\right)\cdot{\bf
u}_{\alpha}+n_{\alpha}q_{\alpha}{\bf u}_{\alpha}\cdot\bf{E}.$ (5)

Similarly one obtains the time evolution equation for the internal energy
111Some readers may prefer to use terminology such as “thermal energy” or
“random kinetic energy” to refer to the quantity $\mathcal{E}^{th}_{\alpha}$.
of species $\alpha$,

$\displaystyle\partial_{t}\mathcal{E}^{th}_{\alpha}+\nabla\cdot\left(\mathcal{E}^{th}_{\alpha}\bf{u}_{\alpha}\right)+\nabla\cdot\bf{h}_{\alpha}$
$\displaystyle=$
$\displaystyle-\left(\bf{P}_{\alpha}\cdot\nabla\right)\cdot\bf{u}_{\alpha}.$
(6)

where $\bf{h}$ is the heat flux vector.

Finally, using the Maxwell’s equations, the equation governing
$\mathcal{E}^{m}$ can be written as

$\displaystyle\partial_{t}\mathcal{E}^{m}+{\frac{c}{4\pi}}\nabla\cdot\left(\bf{E}\times\bf{B}\right)$
$\displaystyle=$ $\displaystyle-\bf{j}\cdot\bf{E}$ (7)

where $\bf{j}=\sum_{\alpha}\bf{j}_{\alpha}$ is the total electric current
density, and ${\bf j}_{\alpha}=n_{\alpha}q_{\alpha}\bf{u}_{\alpha}$ is the
electric current density of species $\alpha$.

Several features of energy transfer can be seen from Eqs. (5), (6) and (7).
First, all the divergence terms on the left-hand sides are transport terms
that do not change the total amount of energy of the respective types, but
simply move energy from one location to another. These transport terms
integrate to zero for suitable boundary conditions and may be important in
reconciling the energy balance at any point in space and time Eastwood  _et
al._ (2020). However, here we are not concerned with transport effects as the
emphasis is on quantifying conversion between different types of energy222We
do not suggest that transport effects such as heat flux and convective heat
transport are small, as in many circumstances these are significant or even
dominant contributions to the balance of equation 6; however these terms do
not exchange energy between different forms, and it is the exchange between
different pathways or channels that is our main interest here.. Therefore, we
will only discuss the terms on the right-hand sides of these equations that
are responsible for conversion of energy from one form to another.

Examining these terms, it is evident that the $\bf{j}_{\alpha}\cdot{\bf E}$
terms convert electromagnetic energy into or out of flow energy for each
species $\alpha$. All changes of internal energy of each species are
accomplished solely by the pressure-stress interaction, $\left({\bf
P}_{\alpha}\cdot\nabla\right)\cdot{\bf
u}_{\alpha}=P^{(\alpha)}_{ij}\nabla_{i}\,u^{(\alpha)}_{j}$.

Quantities such as $\bf{j}\cdot{\bf E}$ and $\left({\bf
P}\cdot\nabla\right)\cdot{\bf u}$ are not single signed, as energy may be
transferred into, or out of, the electromagnetic fields, and likewise, into or
out of the collective fluid motion of each species $\alpha$. While the
distributions of these quantities are not sign-definite, the expectation is
that when there is net dissipation and heating, the appropriate sign
indicating net transfer into random motions will be favored. This has been
seen in magnetosheath observations Bandyopadhyay  _et al._ (2020a) and in
plasma simulations in decaying turbulence Yang  _et al._ (2018). Therefore,
with some care, these quantities may be used to trace the flow of energy
through different channels leading to dissipation.

Further decomposition of the pressure-strain is convenient. A standard
procedure for decomposing the pressure tensor $P^{(\alpha)}_{ij}$ and the
stress tensor $S^{(\alpha)}_{ij}=\nabla_{i}u^{(\alpha)}_{j}$ is to separate
out the trace. One then defines
$P^{(\alpha)}_{ij}=p_{\alpha}\delta_{ij}+\Pi^{(\alpha)}_{ij}$ where
$p_{\alpha}=\frac{1}{3}P^{(\alpha)}_{jj}$ and ,
$\Pi_{ij}=P_{ij}-p\delta_{ij}$. Similarly the stress tensor is conveniently
decomposed as
$S^{(\alpha)}_{ij}=\frac{1}{3}\theta_{\alpha}\delta_{ij}+D^{(\alpha)}_{ij}+\Omega^{(\alpha)}_{ij}$
where $\theta=\nabla\cdot{\mathbf{u}}$,
$D^{(\alpha)}_{ij}=\frac{1}{2}\left(\nabla_{i}u^{(\alpha)}_{j}+\nabla_{j}u^{(\alpha)}_{i}\right)$,
and
$\Omega^{(\alpha)}_{ij}=\frac{1}{2}\left(\nabla_{i}u^{(\alpha)}_{j}-\nabla_{j}u^{(\alpha)}_{i}\right)$.
Then we see that the pressure stress interaction separates as

$\displaystyle\left(\bf{P}_{\alpha}\cdot\nabla\right)\cdot{\bf
u}_{\alpha}=p^{(\alpha)}\theta^{(\alpha)}+\Pi^{(\alpha)}_{ij}D^{(\alpha)}_{ij}.$
(8)

Below, we study quantities that participate in termination of the inertial-
range cascade, while opening channels for production of velocity space
structures (higher-order moments) Servidio  _et al._ (2017); Schekochihin  _et
al._ (2016) that lead eventually to collisional effects. In this sense the
${\bf j}\cdot{\bf E}$ and $\left({\bf P}\cdot\nabla\right)\cdot{\bf u}$
transfer channels provide the gateways between fluid scale effects and kinetic
dissipation.

Based on this discussion, there are at least two key motivations for studying
the pressure-strain interaction: Changes in the random energy,
$\mathcal{E}^{th}_{\alpha}$, take place only through the pressure-strain term
333We emphasize again that the transport terms such as heat flux can change
the local value of $\mathcal{E}^{th}_{\alpha}$ too, but here we are concerned
only with conversion of energy. (Eq. 6); and the pressure-strain interaction
advances the ohmic energy dissipation $\mathbf{j}\cdot\mathbf{E}$, in that
this approach tracks the electron versus proton heating, as well as
compressive $(-p\theta)$ versus incompressive $(-\Pi_{ij}D_{ij})$ heating.

However, we note that an evaluation of $\left({\bf
P}\cdot\nabla\right)\cdot{\bf u}$ requires accurate measurement of the full
pressure tensor as well as computation of spatial derivatives down to kinetic
scales (Dunlop  _et al._ , 1988; Paschmann and Daly, 1998). This requires the
availability of multi-point measurement with very high resolution, which was
not possible before the Magnetospheric Multiscale (MMS) Mission (Burch  _et
al._ , 2016). The MMS data set provides the first opportunity to study these
quantities using observational data.

The dissipation through the pressure-strain interaction is agnostic of
dissipation mechanism. Although several kinetic processes can participate in
the energy dissipation process, here we only investigate one of them: magnetic
reconnection, which plays a central role in the dynamics of Earth’s
magnetosphere Sonnerup (1979, 2013).

##  III MMS Observations

MMS data provide the dual advantages of high-time cadence and simultaneous
multi-spacecraft measurements at very small inter-spacecraft separations. This
combination enables the study of the nature of kinetic plasma at small scales
with an unprecedented level of accuracy and resolution. In this study, we use
magnetic field $(\mathbf{B})$ data from the Fluxgate Magnetometers (FGM)
Russell  _et al._ (2016), electric field $(\mathbf{E})$ data from the Electric
Field Double Probes (EDP) Ergun  _et al._ (2016); Lindqvist  _et al._ (2016),
and ion and electron densities $(n_{\mathrm{i}},n_{\mathrm{e}})$, bulk
velocities $(\mathbf{u}_{\mathrm{i}},\mathbf{u}_{\mathrm{e}})$, and pressure
tensor $(\mathbf{P}_{\mathrm{i}},\mathbf{P}_{\mathrm{e}})$ from the Fast
Plasma Investigation (FPI) Pollock  _et al._ (2016), using the computed
moments of the ion and electron distribution functions. We mainly use MMS
burst resolution data which provide magnetic-field measurements (FGM) at $128$
samples/second, ion density and velocity (FPI-DIS) at $150$ ms, electron
density and velocity (FPI-DES) at $30$ ms, and electric field data at $8192$
samples/second. For the electron-only reconnection case (see Section VI) we
use the FPI quarter-moments data product, which generates $7.5$ ms electron
and $37.5$ ms ion data products by separating the individual energy sweeps
Rager  _et al._ (2018). We employ a multi-spacecraft technique, similar to the
curlometer technique Dunlop  _et al._ (1988); Paschmann and Daly (1998),
enabling evaluation of the velocity strain tensor. We average the pressure
tensor over the four MMS spacecraft. To calculate the Zenitani measure
Zenitani  _et al._ (2011) $\mathbf{j}\cdot\mathbf{E}^{\prime}$, where
$\mathbf{E}^{\prime}=\mathbf{E}+\mathbf{V}_{\mathrm{e}}\times\mathbf{B}$, we
use particle density and velocity,
$\mathbf{j}=qn_{\mathrm{e}}(\mathbf{V}_{\mathrm{i}}-\mathbf{V}_{\mathrm{e}})$
($q$ is the electronic charge) and average over the 4 spacecraft.

##  IV Large-Scale Reconnection

Magnetic reconnection at the large scale, such as those in the Earth’s
magnetopause or magnetotail, can act as a “driver", injecting energy into the
system and drive turbulence Pucci  _et al._ (2017). Magnetopause reconnection
is understood to be an important process for coupling solar wind mass and
momentum to the Earth’s magnetosphere. Since its launch in 2015, MMS has
observed numerous reconnection events along the sunward boundary of Earth’s
magnetosphere where the interplanetary magnetic field reconnects with the
terrestrial magnetic field Burch  _et al._ (2016); Burch and Phan (2016); Chen
_et al._ (2016). Here, we first investigate the nature of the pressure-strain
interaction at an MMS-identified magnetopause reconnection site to gain
insights into the nature of dissipation during the large-scale reconnection
process at the magnetopause.

We begin with an example of the MMS measurements of the reconnection diffusion
region around an X-line within the magnetopause current sheet, reported in
Burch  _et al._ (2020). On 15 April 2018, the MMS tetrahedron encountered a
reconnection site on the dayside magnetopause and observed the conversion of
magnetic energy to particle energy. The details are already given in Burch
_et al._ (2020), so here we only focus on the dissipation measures at the
reconnection site.

The top panel of Fig. 1 plots the magnetic field components in geocentric
solar ecliptic (GSE) coordinates Franz and Harper (2002), measured by MMS1.

Figure 1: Magnetopause reconnection from Burch  _et al._ (2020).

The second panel compares the energy conversion rate from electromagnetic
energy, as quantified by the Zenitani measure (in narrow black), and the
energy conversion rate to random energy, measured by the total pressure-strain
(in broad blue). Both conversion rates show an enhanced signal at the
reconnection region. Further, the two quantities are close, but the value of
$\left({\bf P}\cdot\nabla\right)\cdot{\bf u}$ is somewhat higher than ${\bf
J}\cdot{\bf E}^{\prime}$. This observation suggests that the interval energy
may come from sources other than the electromagnetic energy, possibly the
fluid kinetic energy.

The third panel plots the total pressure-strain rate from electron and ions.
This panel shows that the electrons are responsible for the majority of the
heating.

The bottom two panels show the pressure-strain interaction decomposed into the
compressive and incompressive channels (Eq. 8), for electrons and ions
separately. The two panels show that for both charged species, the compressive
channel is responsible for most of the energy conversion.

##  V Small-Scale Reconnection

Strong current sheets form between the margins of magnetic islands in a
turbulent system, and reconnection occurs in these current sheets.
Reconnecting X-lines near thin current sheets are an important element of
turbulent cascade Carbone, Veltri, and Mangeney (1990); Servidio  _et al._
(2009); Cothran  _et al._ (2003); Retinò  _et al._ (2007); Comisso  _et al._
(2018). In addition to the standard reconnection in large-scale current sheets
at the magnetopause, MMS has identified many reconnection events in the
magnetosheath current sheets, downstream of the bow shock Vörös  _et al._
(2017); Wilder  _et al._ (2017, 2018). Similar to the previous section, here
we study the nature of pressure-strain interaction in the MMS-identified
reconnection sites in the turbulent magnetosheath to find the incompressive
versus compressive conversion of energy and also ion versus electron heating
in turbulent reconnection. We focus on three samples with varying out-of-plane
guide magnetic-field strengths, reported previously by Wilder  _et al._ (2018)

###  V.1 Weak Guide Field

We start with a sample of magnetosheath reconnection in presence of a very
small guide field $(B_{\mathrm{g}}\approx 0.1)$. Figure 2 shows the magnetic
field components and energy conversion measures in the same format as the
previous section.

Figure 2: Magnetosheath reconnection with small guide field, reported by
Wilder  _et al._ (2017).

From the second panel, both $\mathbf{j}\mathbf{\cdot}\mathbf{E}^{\prime}$ and
$-\mathbf{P}\mathbf{\cdot}\mathbf{\nabla}\mathbf{u}$ are of positive sign and
similar magnitude. We interpret this as particles being energized by
converting energy from the electromagnetic energy and then almost all of that
energy is being dissipated into thermal degree of freedom. The heating rate is
more than the rate of electromagnetic energy dissipation, which suggests that
the internal energy originates from flows as well. We note here that there is
a peak of $-\mathbf{P}\mathbf{\cdot}\mathbf{\nabla}\mathbf{u}$ with a value
twice as large $(\sim 40\mathrm{nW}/\mathrm{m}^{3})$, very close to the actual
magnetic reversal point. However, the large but narrow peak is immediately
followed by an almost equally large and narrow peak in the negative direction.
Thus, in considering the dissipation rate in the region surrounding the peak
in $\mathbf{j}$ that captures the full current structure, the primary
contribution comes from the $(\sim 20\mathrm{nW}/\mathrm{m}^{3})$ peak.
Another point worth remarking is that the peaks of the two quantities are
slightly shifted in time (space). Although this could be due a physical effect
as the pressure strain interaction is shown to be active near and not in the
same location as the current sheets Parashar and Matthaeus (2016); Yang  _et
al._ (2017a); Bandyopadhyay  _et al._ (2020b). However, the limitations of the
curlometer technique and other approximations in data analysis could also be
responsible for it.

From the third panel, similar to the previous section, electrons contribute to
most of the heating. In fact, ions show a signature of cooling rather than
heating. Finally, the fourth panel indicates that majority of the electron
heating occurs through the compressive channel. Similarly, the last panel
shows that although ions are cooling, most of this effect occurs through the
compressive channel.

###  V.2 Moderate Guide Field

Next, we consider another magnetosheath reconnection with medium guide field
strength $(B_{\mathrm{g}}\approx 0.5)$. The MMS observations are shown in Fig.
3.

Figure 3: Magnetosheath reconnection with moderate guide field from Wilder
_et al._ (2018).

In this case, both electrons and ions show signatures of heating. The total
heating rate is almost exactly equal to the rate of electromagnetic energy
dissipation rate (second panel), indicating only a small fraction of the
magnetic energy, after conversion, persists as fluid kinetic energy. Again, we
find that the electron heating is much stronger than the ion. Further, the
compressive channel of heating is much stronger than the incompressive one for
both charged species.

###  V.3 High Guide Field

As a final case of magnetosheath reconnection, we show an example with high
guide field $(B_{\mathrm{g}}\approx 1.3)$ in Fig. 4. Here, we see that the
electrons are cooling instead of heating, while the ions show a signature of
slight heating. The electron cooling is the more prominent effect, and so the
rate of change of total particle internal energy indicates cooling.

Figure 4: Magnetosheath reconnection with high guide field Wilder  _et al._
(2018). The particles are losing internal energy although electromagnetic
energy is being dissipated.

Although this electron cooling is only seen in this high-guide field case,
with this limited number of case studies, it is not possible to conclude
whether it is the high guide field strength or some other parameters such as
the upstream Alfvén flow speed that is responsible for the electron cooling.
However, this example shows that the pressure-strain interaction can be used
as an independent diagnostic at reconnection sites to see whether particles
are heating or cooling. In this case, the electron pressure is enhanced near
the reconnection site (see Fig 6 of Wilder  _et al._ (2018)), and so the
mechanical pressure may possibly cause a local expansion of the plasma. This
expansion, in turn, results in local cooling of the plasma, which is
manifested through the $p\theta$ term. We note that on either side of the
central electron cooling region, there are nearby regions of electron heating,
separated by several ion inertial scales and coinciding with the flanks of the
central current enhancement (not shown; see Wilder  _et al._ (2018)). It
appears in this case that the central electron diffusion region is cooling
while the broader ion-scale diffusion region may be increasing its internal
energy, resulting in overall heating. On the other hand, the electromagnetic
energy is dissipating in the central region and gaining in the surrounding
region. This suggests that in the electron diffusion region, both
electromagnetic energy and internal energy are contributing to flow speed and
outside the flow speed is approximately equally partitioned to electromagnetic
and internal energy.

##  VI Electron-only Reconnection

In 2018, MMS observed reconnection events where only the electrons participate
in the reconnection process, but the ions do not Phan  _et al._ (2018).
Although electron-only reconnection was already well known from numerical
simulations Shay  _et al._ (1998), the 2018 events Phan  _et al._ (2018) were
the first time that such events were observed in natural plasmas by
spacecraft. These novel “electron-only" reconnection are different from the
more traditional “ion-coupled" reconnection in many different aspects Stawarz
_et al._ (2019). Therefore, as a final case study, we present the energy
conversion diagnostics in an electron-only reconnection event detected by MMS.
An analysis of pressure-strain dissipation in electron-only reconnection
events has not been presented previously.

Figure 5: Electron-only reconnection in the magnetosheath, found by Phan  _et
al._ (2018).

In Fig. 5, we show analysis of the ion and electron pressure-strain
dissipation at an electron-only reconnection site, observed by MMS. Both
$\mathbf{j}\cdot\mathbf{E}$ and
$-\mathbf{P}\mathbf{\cdot}\mathbf{\nabla}\mathbf{u}$ are positive. Similar to
the previous examples of magnetopause and magnetosheath reconnections, the
magnitude of electromagnetic energy dissipation rate and the rate of change of
internal energy are close to each other, but not exactly equal. The internal
energy increase rate is somewhat higher than the electromagnetic energy
dissipation rate. This observation supports the idea that increase of internal
energy happens via flow energy and not directly from the electromagnetic
energy. During this event, likely the flow is losing more energy than is being
pumped in by the electromagnetic interactions.

The present analysis shows that this occurs through the intermediary of
electron shear flows that couple through the pressure tensor directly into
heat. In the previous samples we found that electrons contribute majority of
the rate of change in internal energy. However, here we see that the ions do
not show any enhancement at the reconnection site, suggesting their non-
participation. The electron contribution alone accounts for nearly the total
dissipation at the reconnection site. Nevertheless, we see that most of the
energy conversion is being facilitated through the compressive channel, as we
found in the other reconnection cases.

##  VII PIC Simulation

To further examine the nature of dissipation in turbulent reconnection sites,
we perform fully kinetic turbulent PIC simulations of a fully ionized
proton–electron plasma using the P3D Zeiler  _et al._ (2002) code. We perform
the simulations in a 2.5D setup, which means that all field vectors have three
components, but they vary only in a two-dimensional spatial grid $(X,Y)$ and
there is no variation in the $Z$ direction.

All the results are in dimensionless units in which number density is
normalized to a reference number density $n_{r}$, mass to ion mass $m_{i}$,
charge to ion charge $q_{i}$, and magnetic field to a reference value $B_{r}$.
Length is normalized to the ion inertial length $d_{i}$, time to the ion
cyclotron time $\omega_{ci}^{-1}$, velocity to a reference Alfvén speed
$v_{A}=B_{r}/\sqrt{4\pi m_{i}n_{r}}$, and temperature to
$T_{r}=m_{i}v_{A}^{2}$. The simulation was performed in a square periodic
spatial domain represented by $4096^{2}$ grid points and initialized with 3200
(macro-) particles of each species per cell. The length of each side is
$L=149.5648\,d_{i}$, so the grid size is $\Delta
X=0.037\,d_{i}\approx\lambda_{D}$, where $\lambda_{D}$ is the Debye length.
The time step is $\Delta t=0.005\,\omega_{ci}^{-1}$. We use an ion to electron
mass ratio of $m_{i}/m_{e}=25$ and a ratio of speed of light to Alfvén speed
of $c/v_{A}=15$ in this run.

The simulation represents a decaying turbulence system, starting with uniform
density $(n_{0}=1.0)$ and temperature of ions and electrons $(T_{0}=0.3)$. The
mean magnetic field is $B_{0}=1.0$ directed out of the plane of the simulation
box. The ion and electron beta (ratio of thermal to magnetic pressure) values
are $\beta_{i}=0.6$, $\beta_{e}=0.6$, respectively.

We perform a case study using a snapshot after the time of maximum root mean
square current density ($220\,\omega_{ci}^{-1}$). The system is in highly non-
linear state at this point, well past the initial transient phase; yet, the
turbulent fluctuation amplitudes have not decayed to very low values (see
Parashar, Matthaeus, and Shay (2018) for context and more details). To achieve
sufficient statistical smoothness, we reran the PIC code for
$0.6\,\omega_{ci}^{-1}$ starting from time $220\,\omega_{ci}^{-1}$, and save
the full outputs at every timestep (i.e., $0.005\,\omega_{ci}^{-1}$). We
average all the quantities over the $120$ time steps. Further, we perform a
spatial smoothing by averaging over 3 neighboring points in each direction.
The method described in Haggerty  _et al._ (2017) (also see Servidio  _et al._
, 2009, 2010) is used to find reconnection X-points. There are some subtleties
associated with the number of X-points in numerical simulations, in that one
needs also to understand whether the identified X-points are physical Comisso
_et al._ (2016), or if they are numerical artifacts Matthaeus and Lamkin
(1985); Wan  _et al._ (2010). Haggerty  _et al._ (2017) performed a careful
analysis and found that a Fourier filter applied at the Debye scale is an
optimal choice for eliminating the possible effects of numerical noise in
identification of X-points in PIC data, and here we follow the same
methodology. After performing these procedures, we find 273 critical saddle
points. The top panel in Fig. 6 shows the out-of plane current in the whole
simulation box and the X-points identified within. Most X-points are clustered
around strong current sheets near magnetic island boundaries.

Figure 6: First panel: Color plot of out-of plane current ($j_{\mathrm{z}}$)
together with X points (black crosses). Second: Enlarged view of the out-of
plane current; Third: electromagnetic energy conversion rate; Fourth: the
total pressure-strain around a chosen X point. The direction of the one-
dimensional trajectory through the X point, shown in more detail in Fig. 7, is
indicated by the dashed lines.

We focus on a particular X-point with strong current, at position
$(45.37\,d_{i},148.92\,d_{i})$, denoted by a black rectangle in the top panel
of Fig. 6. The particular X-point is chosen because it has a high value of
current, and the X-point is distinctly positioned in the current sheet between
two magnetic islands with no other X-points nearby 444We performed the same
analysis on 2 other strong current X points and found essentially the same
behavior (not shown).. The bottom three panels show an enlarged view of the
region surrounding the X-point. The second panel clearly shows current
enhancement near the X-point. The third panel shows enhancement of
electromagnetic energy dissipation rate, and the bottom panel shows
enhancement of internal energy conversion rate near the X-point. To further
investigate the energy conversion channels into charged species, we record the
measurements along a one dimensional sample through the X point. The direction
of this sample is shown with dashed lines in the right panels of Fig. 6. The
magnetic field and energy conversion measures along this trajectory are shown
in Fig. 7.

Figure 7:  Magnetic field components and energy dissipation measures along the
1 dimensional cut through the X point in a 2.5D PIC simulation, shown in the
right panels of Fig. 6.

From Fig. 7, we see very similar characteristics as the MMS observations
presented in the previous sections. First, from the second panel we see that
the charged particles are heating and electromagnetic energy is being
depleted. This behavior is consistent with the scenario of dissipating all
particle energy, gained from electromagnetic fields, into heat. The third
panel shows that electron heating rate is much greater than the ion heating
rate, as seen in most MMS cases. Notably, this X-point has a high guide field.
Unlike the MMS high-guide field, the electrons are not cooling here. This
observation supports our previous suggestions that the cooling of the electron
is probably controlled by some other factors and not only the guide field
strength. The plots in the fourth panel indicates both compressive and
incompressive channels of heating are effective for the electrons. However,
the ions are heating mostly through the compressive channel, as shown in the
last panel. This result contrasts with all the previously shown MMS results,
which feature a large contribution from the compressive electron heating (or
cooling) component, while ions are not significantly heated via the
compressive term. The artificial electron mass used in the simulation could be
the source of this discrepancy; however, a larger statistical study would be
required to reach a more definite conclusion.

##  VIII Conclusions and Discussion

Previous observational studies, performed using MMS data in the Earth’s
magnetosheath, have focused on individual current sheets (Chasapis  _et al._ ,
2018) or global turbulence statistics (Bandyopadhyay  _et al._ , 2020a).
However, the behavior of pressure-strain interaction in reconnection sites, a
crucial link in production of internal energy, has not been studied in detail,
to the best of our knowledge. We note here that other approaches, such as
tracking the net energy transfer between turbulent fields and plasma particles
in velocity space with the field-particle correlation technique Klein and
Howes (2016); Howes, Klein, and Li (2017) has the advantage of detecting a
specific dissipation mechanism. The pressure-strain-based measures, on the
other hand, are agnostic of dissipation mechanisms. However, the pressure
strain based measures can differentiate between the transfer of energy between
fields and fluid flow, and between fluid-flow and internal energy. In this
paper, we demonstrate the use of $\bf{J}\cdot{\bf E}$, $-p\theta$, and
$-\Pi_{ij}D_{ij}$ as independent observational diagnostics in reconnection
using several case studies.

Selected cases of different types of reconnection are presented. We find that
the pressure-strain interaction can be used as an independent diagnostic of
particle energization at reconnection sites. Although electromagnetic energy
dissipation rate is positive in all the reconnection sites, pressure-strain
shows particles can be either losing or gaining internal energy at the
reconnection regions. However, in all cases the magnitudes of both measure are
not very different. The rate of change of internal energy is larger for the
electrons compared to the ions. This stronger electron energization may be due
to the lower mass of the electrons. It is possible that the electrons, being
lighter in mass, respond to the reconnection process quickly and get energized
faster than the ions. Interestingly, simulations of kinetic plasma turbulence
Parashar and Matthaeus (2016); Matthaeus  _et al._ (2016); Yang  _et al._
(2018) as well as observations in the magnetosheath Bandyopadhyay  _et al._
(2020a) and solar wind Cranmer  _et al._ (2009) frequently report a higher
global heating rate of ions compared to that of electrons. An explanation for
this disparity remains elusive at present and is deferred to future work.

We also find that the compressive channel of energy conversion is much
stronger than the incompressive channel overall, and most of the time
individually for each charged species. This observation is consistent with a
recent result, obtained in a current sheet within a turbulent simulation Pezzi
_et al._ (2021). This behavior was not observed in ‘stand-alone’ spontaneous
reconnection simulations, and the compressive channel can evidently become
stronger in a turbulent environment. Therefore, we here adopt the explanation
offered by Pezzi  _et al._ (2021) \- the reconnection events occurring in thin
current sheets, are often at the boundaries of magnetic islands. The magnetic
islands compressing against each other due to external driving by turbulent
motions might be the reason the compressive channel is stronger. Another
factor could be due to the dependence of the plasma compressibility with the
plasma beta. Larger plasma beta indicates a more compressible plasma. However,
the MMS observations are all in the magnetosheath plasma, which have high
beta. Therefore, since the larger compressive heating/cooling is observed, at
least for ions, in both low beta (simulation) and high beta (MMS) systems, the
role of plasma beta is not clear here. There are several control parameters,
such as the plasma beta, guide-field strength, artificial mass of electrons,
upstream parameters that may influence the ion to electron heating rate as
well as the relative magnitude of compressive versus incompressive heating
rates. With these limited number of case studies that we present here, it is
not possible to scrutinize these possibilities.

Although the pressure-strain rate is weaker for ions compared to electrons in
the reconnection cases studied, the electron-only reconnection case shows no
signals in the ions at all. So, electron-only reconnection may be completely
due to electron energization without any participation from the ions, unlike
the traditional reconnection, where the ion response is small, but non-
vanishing.

This paper presents case studies selected from prior analyses that did not
employ the pressure strain diagnostics. The MMS mission has now observed many
more large-scale magnetopause reconnection, as well as small-scale
magnetosheath reconnection X-lines. A detailed survey with a larger number of
samples will reveal the statistical nature of ion and electron energization
and their relationship with electromagnetic field dissipation.

###### Acknowledgements.

We wish to acknowledge NASA Heliospheric GI grant 80NSSC21K0739 at Princeton
and support from the MMS project, NASA NNX14AC39 under Theory and Modeling
team grant 80NSSC19K0565 at Delaware.

## Data Availability

All MMS data are available at https://lasp.colorado.edu/mms/sdc/.

## References

  * Montgomery and et al. (1980) D. Montgomery and et al.,  _Report of the NASA Plasma Turbulence Explorer Study Group_ , Vol. 715-78 (NASA, Jet Propulsion Laboratory, Pasadena, CA, 1980).
  * Bruno and Carbone (2005) R. Bruno and V. Carbone, “The solar wind as a turbulence laboratory,” Living Reviews in Solar Physics 2, 4 (2005).
  * Matthaeus and Velli (2011) W. H. Matthaeus and M. Velli, “Who Needs Turbulence?. A Review of Turbulence Effects in the Heliosphere and on the Fundamental Process of Reconnection,” Space Science Reviews 160, 145–168 (2011).
  * Verscharen, Klein, and Maruca (2019) D. Verscharen, K. G. Klein, and B. A. Maruca, “The multi-scale nature of the solar wind,” Living Reviews in Solar Physics 16, 5 (2019).
  * Yamada  _et al._ (2015) M. Yamada, J. Yoo, J. Jara-Almonte, W. Daughton, H. Ji, R. M. Kulsrud, and C. E. Myers, “Study of energy conversion and partitioning in the magnetic reconnection layer of a laboratory plasma,” Physics of Plasmas 22, 056501 (2015).
  * Yoo  _et al._ (2017) J. Yoo, B. Na, J. Jara-Almonte, M. Yamada, H. Ji, V. Roytershteyn, M. R. Argall, W. Fox, and L.-J. Chen, “Electron heating and energy inventory during asymmetric reconnection in a laboratory plasma,” Journal of Geophysical Research: Space Physics 122, 9264–9281 (2017).
  * Fadanelli  _et al._ (2021) S. Fadanelli, B. Lavraud, F. Califano, G. Cozzani, F. Finelli, and M. Sisti, “Energy conversions associated with magnetic reconnection,” Journal of Geophysical Research: Space Physics 126, e2020JA028333 (2021).
  * Leamon  _et al._ (1999) R. J. Leamon, C. W. Smith, N. F. Ness, and H. K. Wong, “Dissipation range dynamics: Kinetic alfvén waves and the importance of $\beta_{e}$,” Journal of Geophysical Research: Space Physics 104, 22331–22344 (1999).
  * Howes  _et al._ (2008) G. G. Howes, S. C. Cowley, W. Dorland, G. W. Hammett, E. Quataert, and A. A. Schekochihin, “A model of turbulence in magnetized plasmas: Implications for the dissipation range in the solar wind,” J. Geophys. R. 113, A05103 (2008), arXiv:0707.3147 .
  * Chandran  _et al._ (2010) B. D. G. Chandran, B. Li, B. N. Rogers, E. Quataert, and K. Germaschewski, “Perpendicular ion heating by low-frequency alfvén-wave turbulence in the solar wind,” The Astrophysical Journal 720, 503–515 (2010).
  * Hollweg and Markovskii (2002) J. V. Hollweg and S. A. Markovskii, “Cyclotron resonances of ions with obliquely propagating waves in coronal holes and the fast solar wind,” Journal of Geophysical Research: Space Physics 107, SSH 1–1–SSH 1–7 (2002).
  * Gary (1993) S. P. Gary,  _Theory of Space Plasma Microinstabilities_ (Cambridge University Press, Cambridge, UK, 1993).
  * Lichko  _et al._ (2017) E. Lichko, J. Egedal, W. Daughton, and J. Kasper, “Magnetic Pumping as a Source of Particle Heating and Power-law Distributions in the Solar Wind,” The Astrophysical Journal 850, L28 (2017).
  * Yang  _et al._ (2017a) Y. Yang, W. H. Matthaeus, T. N. Parashar, C. C. Haggerty, V. Roytershteyn, W. Daughton, M. Wan, Y. Shi, and S. Chen, “Energy transfer, pressure tensor, and heating of kinetic plasma,” Phys. Plasmas 24, 072306 (2017a), arXiv:1705.02054 [physics.plasm-ph] .
  * Yang  _et al._ (2017b) Y. Yang, W. H. Matthaeus, T. N. Parashar, P. Wu, M. Wan, Y. Shi, S. Chen, V. Roytershteyn, and W. Daughton, “Energy transfer channels and turbulence cascade in Vlasov-Maxwell turbulence,” Phys. Rev. E 95, 061201 (2017b).
  * Note (1) Some readers may prefer to use terminology such as “thermal energy” or “random kinetic energy” to refer to the quantity $\mathcal{E}^{th}_{\alpha}$.
  * Eastwood  _et al._ (2020) J. P. Eastwood, M. V. Goldman, T. D. Phan, J. E. Stawarz, P. A. Cassak, J. F. Drake, D. Newman, B. Lavraud, M. A. Shay, R. E. Ergun, J. L. Burch, D. J. Gershman, B. L. Giles, P. A. Lindqvist, R. B. Torbert, R. J. Strangeway, and C. T. Russell, “Energy flux densities near the electron dissipation region in asymmetric magnetopause reconnection,” Phys. Rev. Lett. 125, 265102 (2020).
  * Note (2) We do not suggest that transport effects such as heat flux and convective heat transport are small, as in many circumstances these are significant or even dominant contributions to the balance of equation 6; however these terms do not exchange energy between different forms, and it is the exchange between different pathways or channels that is our main interest here.
  * Bandyopadhyay  _et al._ (2020a) R. Bandyopadhyay, W. H. Matthaeus, T. N. Parashar, Y. Yang, A. Chasapis, B. L. Giles, D. J. Gershman, C. J. Pollock, C. T. Russell, R. J. Strangeway, R. B. Torbert, T. E. Moore, and J. L. Burch, “Statistics of Kinetic Dissipation in the Earth’s Magnetosheath: MMS Observations,” Physical Review Letters 124, 255101 (2020a).
  * Yang  _et al._ (2018) Y. Yang, M. Wan, W. H. Matthaeus, L. Sorriso-Valvo, T. N. Parashar, Q. Lu, Y. Shi, and S. Chen, “Scale dependence of energy transfer in turbulent plasma,” Monthly Notices of the Royal Astronomical Society 482, 4933–4940 (2018).
  * Servidio  _et al._ (2017) S. Servidio, A. Chasapis, W. H. Matthaeus, D. Perrone, F. Valentini, T. N. Parashar, P. Veltri, D. Gershman, C. T. Russell, B. Giles, S. A. Fuselier, T. D. Phan, and J. Burch, “Magnetospheric multiscale observation of plasma velocity-space cascade: Hermite representation and theory,” Phys. Rev. Lett. 119, 205101 (2017).
  * Schekochihin  _et al._ (2016) A. A. Schekochihin, J. T. Parker, E. G. Highcock, P. J. Dellar, W. Dorland, and G. W. Hammett, “Phase mixing versus nonlinear advection in drift-kinetic plasma turbulence,” Journal of Plasma Physics 82, 905820212 (2016).
  * Note (3) We emphasize again that the transport terms such as heat flux can change the local value of $\mathcal{E}^{th}_{\alpha}$ too, but here we are concerned only with conversion of energy.
  * Dunlop  _et al._ (1988) M. Dunlop, D. Southwood, K.-H. Glassmeier, and F. Neubauer, “Analysis of multipoint magnetometer data,” Advances in Space Research 8, 273–277 (1988).
  * Paschmann and Daly (1998) G. Paschmann and P. W. Daly, “Analysis methods for multi-spacecraft data. issi scientific reports series sr-001, esa/issi, vol. 1. isbn 1608-280x, 1998,” ISSI Scientific Reports Series 1 (1998).
  * Burch  _et al._ (2016) J. L. Burch, T. E. Moore, R. B. Torbert, and B. L. Giles, “Magnetospheric multiscale overview and science objectives,” Space Science Reviews 199, 5–21 (2016).
  * Sonnerup (1979) B. U. Ö. Sonnerup, “Magnetic field reconnection,” in  _Solar System Plasma Physics_, edited by L. J. Lanzerotti, C. F. Kennel, and E. N. Parker (North-Holland Publishing Co., 1979) pp. 45–108.
  * Sonnerup (2013) B. U. O. Sonnerup, “Magnetic field reconnection at the magnetopause: An overview,” in  _Magnetic Reconnection in Space and Laboratory Plasmas_ (American Geophysical Union (AGU), 2013) pp. 92–103, https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/GM030p0092 .
  * Russell  _et al._ (2016) C. T. Russell, B. J. Anderson, W. Baumjohann, K. R. Bromund, D. Dearborn, D. Fischer, G. Le, H. K. Leinweber, D. Leneman, W. Magnes, J. D. Means, M. B. Moldwin, R. Nakamura, D. Pierce, F. Plaschke, K. M. Rowe, J. A. Slavin, R. J. Strangeway, R. Torbert, C. Hagen, I. Jernej, A. Valavanoglou, and I. Richter, “The magnetospheric multiscale magnetometers,” Space Science Reviews 199, 189–256 (2016).
  * Ergun  _et al._ (2016) R. E. Ergun, S. Tucker, J. Westfall, K. A. Goodrich, D. M. Malaspina, D. Summers, J. Wallace, M. Karlsson, J. Mack, N. Brennan, B. Pyke, P. Withnell, R. Torbert, J. Macri, D. Rau, I. Dors, J. Needell, P. A. Lindqvist, G. Olsson, and C. M. Cully, “The Axial Double Probe and Fields Signal Processing for the MMS Mission,” Space Science Review 199, 167–188 (2016).
  * Lindqvist  _et al._ (2016) P. A. Lindqvist, G. Olsson, R. B. Torbert, B. King, M. Granoff, D. Rau, G. Needell, S. Turco, I. Dors, P. Beckman, J. Macri, C. Frost, J. Salwen, A. Eriksson, L. Åhlén, Y. V. Khotyaintsev, J. Porter, K. Lappalainen, R. E. Ergun, W. Wermeer, and S. Tucker, “The Spin-Plane Double Probe Electric Field Instrument for MMS,” Space Science Review 199, 137–165 (2016).
  * Pollock  _et al._ (2016) C. Pollock, T. Moore, A. Jacques, J. Burch, U. Gliese, Y. Saito, T. Omoto, L. Avanov, A. Barrie, V. Coffey, J. Dorelli, D. Gershman, B. Giles, T. Rosnack, C. Salo, S. Yokota, M. Adrian, C. Aoustin, C. Auletti, S. Aung, V. Bigio, N. Cao, M. Chandler, D. Chornay, K. Christian, G. Clark, G. Collinson, T. Corris, A. De Los Santos, R. Devlin, T. Diaz, T. Dickerson, C. Dickson, A. Diekmann, F. Diggs, C. Duncan, A. Figueroa-Vinas, C. Firman, M. Freeman, N. Galassi, K. Garcia, G. Goodhart, D. Guererro, J. Hageman, J. Hanley, E. Hemminger, M. Holland, M. Hutchins, T. James, W. Jones, S. Kreisler, J. Kujawski, V. Lavu, J. Lobell, E. LeCompte, A. Lukemire, E. MacDonald, A. Mariano, T. Mukai, K. Narayanan, Q. Nguyan, M. Onizuka, W. Paterson, S. Persyn, B. Piepgrass, F. Cheney, A. Rager, T. Raghuram, A. Ramil, L. Reichenthal, H. Rodriguez, J. Rouzaud, A. Rucker, Y. Saito, M. Samara, J.-A. Sauvaud, D. Schuster, M. Shappirio, K. Shelton, D. Sher, D. Smith, K. Smith, S. Smith, D. Steinfeld, R. Szymkiewicz, K. Tanimoto, J. Taylor, C. Tucker, K. Tull, A. Uhl, J. Vloet, P. Walpole, S. Weidner, D. White, G. Winkert, P.-S. Yeh, and M. Zeuch, “Fast plasma investigation for magnetospheric multiscale,” Space Science Reviews 199, 331–406 (2016).
  * Rager  _et al._ (2018) A. C. Rager, J. C. Dorelli, D. J. Gershman, V. Uritsky, L. A. Avanov, R. B. Torbert, J. L. Burch, R. E. Ergun, J. Egedal, C. Schiff, J. R. Shuster, B. L. Giles, W. R. Paterson, C. J. Pollock, R. J. Strangeway, C. T. Russell, B. Lavraud, V. N. Coffey, and Y. Saito, “Electron crescent distributions as a manifestation of diamagnetic drift in an electron-scale current sheet: Magnetospheric multiscale observations using new 7.5 ms fast plasma investigation moments,” Geophysical Research Letters 45, 578–584 (2018).
  * Zenitani  _et al._ (2011) S. Zenitani, M. Hesse, A. Klimas, and M. Kuznetsova, “New measure of the dissipation region in collisionless magnetic reconnection,” Phys. Rev. Lett. 106, 195003 (2011).
  * Pucci  _et al._ (2017) F. Pucci, S. Servidio, L. Sorriso-Valvo, V. Olshevsky, W. H. Matthaeus, F. Malara, M. V. Goldman, D. L. Newman, and G. Lapenta, “Properties of turbulence in the reconnection exhaust: Numerical simulations compared with observations,” The Astrophysical Journal 841, 60 (2017).
  * Burch  _et al._ (2016) J. L. Burch, R. B. Torbert, T. D. Phan, L. J. Chen, T. E. Moore, R. E. Ergun, J. P. Eastwood, D. J. Gershman, P. A. Cassak, M. R. Argall, S. Wang, M. Hesse, C. J. Pollock, B. L. Giles, R. Nakamura, B. H. Mauk, S. A. Fuselier, C. T. Russell, R. J. Strangeway, J. F. Drake, M. A. Shay, Y. V. Khotyaintsev, P. A. Lindqvist, G. Marklund, F. D. Wilder, D. T. Young, K. Torkar, J. Goldstein, J. C. Dorelli, L. A. Avanov, M. Oka, D. N. Baker, A. N. Jaynes, K. A. Goodrich, I. J. Cohen, D. L. Turner, J. F. Fennell, J. B. Blake, J. Clemmons, M. Goldman, D. Newman, S. M. Petrinec, K. J. Trattner, B. Lavraud, P. H. Reiff, W. Baumjohann, W. Magnes, M. Steller, W. Lewis, Y. Saito, V. Coffey, and M. Chandler, “Electron-scale measurements of magnetic reconnection in space,” Science 352, aaf2939 (2016).
  * Burch and Phan (2016) J. L. Burch and T. D. Phan, “Magnetic reconnection at the dayside magnetopause: Advances with mms,” Geophysical Research Letters 43, 8327–8338 (2016).
  * Chen  _et al._ (2016) L.-J. Chen, M. Hesse, S. Wang, D. Gershman, R. Ergun, C. Pollock, R. Torbert, N. Bessho, W. Daughton, J. Dorelli, B. Giles, R. Strangeway, C. Russell, Y. Khotyaintsev, J. Burch, T. Moore, B. Lavraud, T. Phan, and L. Avanov, “Electron energization and mixing observed by mms in the vicinity of an electron diffusion region during magnetopause reconnection,” Geophysical Research Letters 43, 6036–6043 (2016).
  * Burch  _et al._ (2020) J. L. Burch, J. M. Webster, M. Hesse, K. J. Genestreti, R. E. Denton, T. D. Phan, H. Hasegawa, P. A. Cassak, R. B. Torbert, B. L. Giles, D. J. Gershman, R. E. Ergun, C. T. Russell, R. J. Strangeway, O. Le Contel, K. R. Pritchard, A. T. Marshall, K.-J. Hwang, K. Dokgo, S. A. Fuselier, L.-J. Chen, S. Wang, M. Swisdak, J. F. Drake, M. R. Argall, K. J. Trattner, M. Yamada, and G. Paschmann, “Electron inflow velocities and reconnection rates at earth’s magnetopause and magnetosheath,” Geophysical Research Letters 47, e2020GL089082 (2020), e2020GL089082 2020GL089082, https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2020GL089082 .
  * Franz and Harper (2002) M. Franz and D. Harper, “Heliospheric coordinate systems,” Planetary and Space Science 50, 217–233 (2002).
  * Carbone, Veltri, and Mangeney (1990) V. Carbone, P. Veltri, and A. Mangeney, “Coherent structure formation and magnetic field line reconnection in magnetohydrodynamic turbulence,” Physics of Fluids A: Fluid Dynamics 2, 1487–1496 (1990).
  * Servidio  _et al._ (2009) S. Servidio, W. H. Matthaeus, M. A. Shay, P. A. Cassak, and P. Dmitruk, “Magnetic reconnection in two-dimensional magnetohydrodynamic turbulence,” Phys. Rev. Lett. 102, 115003 (2009).
  * Cothran  _et al._ (2003) C. D. Cothran, M. Landreman, M. R. Brown, and W. H. Matthaeus, “Three-dimensional structure of magnetic reconnection in a laboratory plasma,” Geophysical Research Letters 30 (2003), https://doi.org/10.1029/2002GL016497.
  * Retinò  _et al._ (2007) A. Retinò, D. Sundkvist, A. Vaivads, F. Mozer, M. André, and C. J. Owen, “In situ evidence of magnetic reconnection in turbulent plasma,” Nature Physics 3, 236–238 (2007).
  * Comisso  _et al._ (2018) L. Comisso, Y.-M. Huang, M. Lingam, E. Hirvijoki, and A. Bhattacharjee, “Magnetohydrodynamic Turbulence in the Plasmoid-mediated Regime,” The Astrophysical Journal 854, 103 (2018).
  * Vörös  _et al._ (2017) Z. Vörös, E. Yordanova, A. Varsani, K. J. Genestreti, Y. V. Khotyaintsev, W. Li, D. B. Graham, C. Norgren, R. Nakamura, Y. Narita, F. Plaschke, W. Magnes, W. Baumjohann, D. Fischer, A. Vaivads, E. Eriksson, P.-A. Lindqvist, G. Marklund, R. E. Ergun, M. Leitner, M. P. Leubner, R. J. Strangeway, O. Le Contel, C. Pollock, B. J. Giles, R. B. Torbert, J. L. Burch, L. A. Avanov, J. C. Dorelli, D. J. Gershman, W. R. Paterson, B. Lavraud, and Y. Saito, “Mms observation of magnetic reconnection in the turbulent magnetosheath,” Journal of Geophysical Research: Space Physics 122, 11,442–11,467 (2017).
  * Wilder  _et al._ (2017) F. D. Wilder, R. E. Ergun, S. Eriksson, T. D. Phan, J. L. Burch, N. Ahmadi, K. A. Goodrich, D. L. Newman, K. J. Trattner, R. B. Torbert, B. L. Giles, R. J. Strangeway, W. Magnes, P.-A. Lindqvist, and Y.-V. Khotyaintsev, “Multipoint measurements of the electron jet of symmetric magnetic reconnection with a moderate guide field,” Phys. Rev. Lett. 118, 265101 (2017).
  * Wilder  _et al._ (2018) F. D. Wilder, R. E. Ergun, J. L. Burch, N. Ahmadi, S. Eriksson, T. D. Phan, K. A. Goodrich, J. Shuster, A. C. Rager, R. B. Torbert, B. L. Giles, R. J. Strangeway, F. Plaschke, W. Magnes, P. A. Lindqvist, and Y. V. Khotyaintsev, “The role of the parallel electric field in electron-scale dissipation at reconnecting currents in the magnetosheath,” Journal of Geophysical Research: Space Physics 123, 6533–6547 (2018).
  * Parashar and Matthaeus (2016) T. N. Parashar and W. H. Matthaeus, “Propinquity of current and vortex structures: Effects on collisionless plasma heating,” The Astrophysical Journal 832, 57 (2016).
  * Bandyopadhyay  _et al._ (2020b) R. Bandyopadhyay, W. H. Matthaeus, T. N. Parashar, Y. Yang, A. Chasapis, B. L. Giles, D. J. Gershman, C. J. Pollock, C. T. Russell, R. J. Strangeway, R. B. Torbert, T. E. Moore, and J. L. Burch, “Statistics of Kinetic Dissipation in the Earth’s Magnetosheath: MMS Observations,” Physical Review Letters 124, 255101 (2020b).
  * Phan  _et al._ (2018) T. D. Phan, J. P. Eastwood, M. A. Shay, J. F. Drake, B. U. Ö. Sonnerup, M. Fujimoto, P. A. Cassak, M. Øieroset, J. L. Burch, R. B. Torbert, A. C. Rager, J. C. Dorelli, D. J. Gershman, C. Pollock, P. S. Pyakurel, C. C. Haggerty, Y. Khotyaintsev, B. Lavraud, Y. Saito, M. Oka, R. E. Ergun, A. Retino, O. Le Contel, M. R. Argall, B. L. Giles, T. E. Moore, F. D. Wilder, R. J. Strangeway, C. T. Russell, P. A. Lindqvist, and W. Magnes, “Electron magnetic reconnection without ion coupling in Earth’s turbulent magnetosheath,” Nature 557, 202–206 (2018).
  * Shay  _et al._ (1998) M. A. Shay, J. F. Drake, R. E. Denton, and D. Biskamp, “Structure of the dissipation region during collisionless magnetic reconnection,” J. Geophys. Res. 103, 9165 (1998).
  * Stawarz  _et al._ (2019) J. E. Stawarz, J. P. Eastwood, T. D. Phan, I. L. Gingell, M. A. Shay, J. L. Burch, R. E. Ergun, B. L. Giles, D. J. Gershman, O. L. Contel, P.-A. Lindqvist, C. T. Russell, R. J. Strangeway, R. B. Torbert, M. R. Argall, D. Fischer, W. Magnes, and L. Franci, “Properties of the turbulence associated with electron-only magnetic reconnection in earth’s magnetosheath,” The Astrophysical Journal Letters 877, L37 (2019).
  * Zeiler  _et al._ (2002) A. Zeiler, D. Biskamp, J. F. Drake, B. N. Rogers, M. A. Shay, and M. Scholer, “Three-dimensional particle simulations of collisionless magnetic reconnection,” Journal of Geophysical Research (Space Physics) 107, 1230 (2002).
  * Parashar, Matthaeus, and Shay (2018) T. N. Parashar, W. H. Matthaeus, and M. A. Shay, “Dependence of kinetic plasma turbulence on plasma $\beta$,” The Astrophysical Journal 864, L21 (2018).
  * Haggerty  _et al._ (2017) C. C. Haggerty, T. N. Parashar, W. H. Matthaeus, M. A. Shay, Y. Yang, M. Wan, P. Wu, and S. Servidio, “Exploring the statistics of magnetic reconnection x-points in kinetic particle-in-cell turbulence,” Physics of Plasmas 24, 102308 (2017).
  * Servidio  _et al._ (2010) S. Servidio, W. H. Matthaeus, M. A. Shay, P. Dmitruk, P. A. Cassak, and M. Wan, “Statistics of magnetic reconnection in two-dimensional magnetohydrodynamic turbulence,” Physics of Plasmas 17, 032315 (2010).
  * Comisso  _et al._ (2016) L. Comisso, M. Lingam, Y.-M. Huang, and A. Bhattacharjee, “General theory of the plasmoid instability,” Physics of Plasmas 23, 100702 (2016).
  * Matthaeus and Lamkin (1985) W. H. Matthaeus and S. L. Lamkin, “Rapid magnetic reconnection caused by finite amplitude fluctuations,” The Physics of Fluids 28, 303–307 (1985).
  * Wan  _et al._ (2010) M. Wan, S. Oughton, S. Servidio, and W. H. Matthaeus, “On the accuracy of simulations of turbulence,” Phys. Plasmas 17, 082308 (2010).
  * Note (4) We performed the same analysis on 2 other strong current X points and found essentially the same behavior (not shown).
  * Chasapis  _et al._ (2018) A. Chasapis, Y. Yang, W. H. Matthaeus, T. N. Parashar, C. C. Haggerty, J. L. Burch, T. E. Moore, C. J. Pollock, J. Dorelli, D. J. Gershman, R. B. Torbert, and C. T. Russell, “Energy conversion and collisionless plasma dissipation channels in the turbulent magnetosheath observed by the magnetospheric multiscale mission,” The Astrophysical Journal 862, 32 (2018).
  * Klein and Howes (2016) K. G. Klein and G. G. Howes, “Measuring Collisionless Damping in Heliospheric Plasmas using Field-Particle Correlations,” The Astrophysical Journal Letters 826, L30 (2016).
  * Howes, Klein, and Li (2017) G. G. Howes, K. G. Klein, and T. C. Li, “Diagnosing collisionless energy transfer using field–particle correlations: Vlasov–poisson plasmas,” Journal of Plasma Physics 83, 705830102 (2017).
  * Matthaeus  _et al._ (2016) W. H. Matthaeus, T. N. Parashar, M. Wan, and P. Wu, “Turbulence and proton-electron heating in kinetic plasma,” The Astrophysical Journal Letters 827, L7 (2016).
  * Cranmer  _et al._ (2009) S. R. Cranmer, W. H. Matthaeus, B. A. Breech, and J. C. Kasper, “Empirical constraints on proton and electron heating in the fast solar wind,” Astrophys. J. 702, 1604–1614 (2009).
  * Pezzi  _et al._ (2021) O. Pezzi, H. Liang, J. L. Juno, P. A. Cassak, C. L. Vásconez, L. Sorriso-Valvo, D. Perrone, S. Servidio, V. Roytershteyn, J. M. TenBarge, and W. H. Matthaeus, “Dissipation measures in weakly collisional plasmas,” Monthly Notices of the Royal Astronomical Society 505, 4857–4873 (2021), https://academic.oup.com/mnras/article-pdf/505/4/4857/38835622/stab1516.pdf .