# Thin current sheets in the magnetotail at lunar distances: statistics of
ARTEMIS observations

###### Abstract

The magnetotail current sheet’s spatial configuration and stability control
the onset of magnetic reconnection - the driving process for magnetospheric
substorms. The near-Earth current sheet has been thoroughly investigated by
numerous missions, whereas the midtail current sheet has not been adequately
explored. This is especially the case for the long-term variation of its
configuration in response to the solar wind. We present a statistical analysis
of 1261 magnetotail current sheet crossings by the Acceleration, Reconnection,
Turbulence and Electrodynamics of Moon’s Interaction with the Sun (ARTEMIS)
mission orbiting the moon ($X\sim-60$ RE), collected during the entirety of
Solar Cycle 24. We demonstrate that the magnetotail current sheet typically
remains extremely thin, with a characteristic thickness comparable to the
thermal ion gyroradius, even at such large distances from Earth’s dipole. We
also find that a substantial fraction ($\sim$ one quarter) of the observed
current sheets have a partially force-free magnetic field configuration, with
a negligible contribution of the thermal pressure and a significant
contribution of the magnetic field shear component to the pressure balance.
Further, we quantify the impact of the changing solar wind driving conditions
on the properties of the midtail around the lunar orbit. During active solar
wind driving conditions, we observe an increase in the occurrence rate of thin
current sheets, whereas quiet solar wind driving conditions seem to favor the
formation of partially force-free current sheets.

JGR: Space Physics

Department of Earth, Planetary, and Space Sciences, University of California,
Los Angeles, USA Space Research Institute of Russian Academy of Sciences,
Moscow, Russia

Sergei<EMAIL_ADDRESS>
We present a statistical analysis of magnetotail current sheets collected by
the ARTEMIS mission for 11 years of observations at $\sim 60$ RE tail

We observe a large population ($\sim 56$%) of ion-kinetic scale current sheets
and a smaller population of partially force-free current sheets ($\sim 24$%)

We show that the occurrence rates of intense current sheets and partially
force-free current sheets correlate with the solar wind parameters

##  1 Introduction

During the energy-storage phase of magnetospheric substorms (the so-called
growth phase), solar wind energy is stored as magnetic energy in the
magnetotail North and South lobes that are separated by the near-equatorial
magnetotail current sheet (CS). In the subsequent substrom expansion and
recovery phases, this energy is transformed into charged particle acceleration
and plasma heating due to magnetic field line reconnection in this CS [Baker .
(1996), Angelopoulos . (2008)]. Observational evidence indicates that during
these energy release phases, a significant portion of plasma from the near-
Earth plasma sheet is expelled tailward in the form of a plasmoid
[<]e.g.,¿Hoshino94,Angelopoulos13,Li13:plasmoid. The substorm growth phase is
characterized by thinning of the magnetotail CS
[<]e.g.,¿Petrukovich07,Runov21:jastp, whereas the expansion phase starts with
a CS instability largely controlled by the CS’s spatial configuration
[<]e.g.,¿Sitnov19. Therefore, the spatial configuration and dynamics of the
magnetotail CS play a key role in the overall magnetospheric dynamics.

The magnetic field in the magnetotail lobes is in nearly-anti-parallel
directions (towards and away from Earth in the North and South lobes,
respectively). Thus, the field configuration enables magnetic reconnection
[Coppi . (1966), Vasyliunas (1975)] – the process which is believed to play an
important role in substorm development [<]see¿[and references
therein]book:Gonzalez&Parker. Considerable effort has been put into
theoretical and observational investigation of magnetotail CS [<]see reviews
by¿Baumjohann07,Zelenyi11PPR,Petrukovich15:ssr,Sitnov19,Runov21:jastp.
Analysis of multi-spacecraft missions, like Cluster [Escoubet . (2001)], Time
History of Events and Macroscale Interactions during Substroms [<]THEMIS,
see¿Angelopoulos08:ssr, and Magnetospheric Multiscale spacecraft [<]MMS,
see¿Burch16 has allowed estimating the typical thickness and spatial profile
of the near-Earth magnetotail CS ($X>-30$ RE). The near-Earth CS is
characterized by current densities of a few $nA/m^{2}$ and consists of a thin
CS embedded into a thicker plasma sheet [Asano . (2004), Runov . (2006),
Artemyev . (2011), Artemyev . (2016), Lu . (2019), Lu . (2020)]. Here, thin CS
refers to a current layer with a thickness comparable to the ion gyroradius,
which is approximately $\approx 400$ km for a 1 keV ion in the presence of a
magnetic field strength of approximately $\approx 10$ nT. This magnetic field
strength is commonly observed in the magnetotail lobes at lunar distances
[<]see¿Xu18:artemis_cs,Runov18. Observations made by ISEE-3 [Bame . (1983),
Slavin . (1983), Slavin . (1985)] and Geotail [Nishida (1994), Nishida Ogino
(1998), Nishida (2000)] in the middle ($-50$ R${}_{E}<X<-12$ RE) and distant
tail ($X<-50$ RE) revealed the presence of similar thin CSs [Pulkkinen .
(1993), Hoshino . (1996), Hoshino . (1997), Vasko . (2015)].

From a theoretical perspective, the presence of ion-gyroscale CSs at various
distances in the magnetotail, ranging from the near-Earth tail
($X\sim-12R_{E}$) to the middle and distant tail regions, poses a significant
challenge. Such thin current sheets are characterized by strong current, and,
thus, necessitate a strong radial gradient in plasma pressure [<]see¿[for
discussion of the stress balances in CSs]Rich72. However, magnetotail models
struggle to incorporate such a strong pressure gradient simultaneously across
a wide radial range. Thus, thin CSs either should form locally, within a
radially confined region of strong pressure gradient [Schindler Birn (2002),
Birn . (2004), Liu . (2014)], or should be balanced by alternative mechanisms
that do not involve strong pressure gradients [<]see discussion
in¿Artemyev21:grl,An22:currentsheet,Arnold&Sitnov23. Further statistical
investigations of the middle and distant magnetotail CSs should shed light on
how thin CSs typically form at these distances. This, in turn, can offer
additional insights supporting one of these two scenarios.

Prior CS statistical studies have been based on measurements of near-Earth
multi-spacecraft missions. Thus far, information about the midtail CS has been
limited to a Geotail statistical survey, covering a short interval of
observations [<]1994–1995, see¿Vasko15:jgr:cs. Due to the lack of spacecraft
missions operating within the range of $X\in[-60,-30]$ RE, we adopt the
approach proposed by Runov18 to compare the properties of the middle tail
region (corresponding to lunar distances; $X\sim-60$ RE) with the well-
established characteristics of the near-Earth current sheet. At such large
distances, the influence of the Earth’s dipole field is negligible, and the
magnetotail CS should be largely controlled by the solar wind properties.
Indeed, in the magnetotail CS at lunar distances, there is a strong
correlation between the ion temperature ($T_{i}$) and the energy of solar wind
protons [Artemyev . (2017), Lukin . (2019)]. This suggests that shocked but
not yet fully thermalized solar wind plasma plays a crucial role in populating
the magnetotail at $X\sim-60$ RE.

In this study we use the measurements made by the two Acceleration,
Reconnection, Turbulence and Electrodynamics of Moon’s Interaction with the
Sun (ARTEMIS) mission probes [Angelopoulos (2011)]. Originally a part of the
THEMIS mission (ThB as P1 and ThC as P2), these identical probes were
subsequently repositioned to orbit the Moon, which passes through the
magnetotail at $X\sim-60$ RE. This provides us with an opportunity to
investigate the properties of the magnetotail at lunar distances. At such
large distances, the CS’s flapping motion is rather typical [Sergeev . (1998),
Sergeev . (2006)], which can be used to estimate the CS’s spatial scale
(thicknesses) and current density. ARTEMIS started operating in 2011 and, by
now, its measurements encompass 12 years of observations [<]see large
statistics of plasma and magnetic field characteristics in¿Liuzzo22. In this
study, we use data collected between 2012 and 2022, which makes 11 years in
total (one full solar cycle). Thus, our CS dataset allows us to reveal how
different solar wind conditions impact the properties of the CSs in the
magnetotail at lunar distances. In this paper, we aim to address the following
two questions: (1) do the characteristics of the near-Earth magnetotail thin
CS, e.g, thickness, current density magnitude and embedding differ from
characteristics of thin CSs in the the middle tail? (2) To what extent are
these characteristics influenced by solar wind conditions? The answer to the
first question should provide important data for new CS models
[<]see¿Sitnov&Arnold22,ArnoldSitnov23,An23:apj. The answer to the second
question should reveal if solar wind is the main driver of thin CS formation
in the middle tail.

The paper is organized as follows: In Section 2, we introduce our
observational set consisting of 1261 CS crossings and discuss three
representative examples. In Section 3, we describe the selection criteria that
enable us to focus on CSs with reliably estimated current density and
thickness. Utilizing these criteria, we introduce a subset of 453 CSs from the
complete set of 1261 CS crossings. Using this subset, we examine the CS’s
thickness, current density and force-free character, and discuss relationships
between these characteristics. In Section 4, we analyze the cross-tail
profiles of the main plasma properties surrounding the CSs. In Section 5, we
discuss the relationship of the solar wind parameters on the properties of
these magnetotail CSs at lunar distances. In Section 6, we summarize our
results.

##  2 ARTEMIS Dataset and Methods

ARTEMIS P1 and P2 are equipped with electrostatic analyzers (ESA) that measure
electron ($<30$keV) and ion ($<25$keV) distribution functions which are used
to compute both on-board and on the ground plasma moments (ion density
$n_{i}$, ion and electron temperatures $T_{i}$ and $T_{e}$) at 4 s cadence
[McFadden . (2008)]. Magnetic field data is provided by the Fluxgate
Magnetometer (FGM) at a rate of 128 Samples/s with offset stability $<$ 0.2
nT/12 hr[Auster . (2008)]. While most of our analyses rely on P1 measurements,
P2 measurements are employed to fill gaps in P1 data. These gaps are likely
attributed to operational shadows or resets caused by space weather events
like solar energetic particle (SEP) events, particularly in the years 2016 and
2018. We use an aberrated GSM (AGSM) coordinate system, unless otherwise
stated. AGSM is introduced by rotating the X-axis of the conventional
Geocentric Solar Magnetospheric (GSM) system by approximately 4 degrees. This
adjustment accounts for the Earth’s relative motion and the disparity between
the direction of the solar wind flow and $X_{GSM}$ [Runov . (2005), Vasko .
(2015)]. To achieve this, we align the X-axis with the average direction of
the solar wind flow. Subsequently, the Y-axis is established by orienting the
conventional $Y_{GSM}$ perpendicular to the new X-axis.

We employ the following selection criteria for CS crossings: (1) the
spacecraft crosses the neutral sheet ($B_{x}=0$) and the observed magnitudes
of $B_{0}^{+}=max\;B_{x}>0$ and $B_{0}^{-}=min\;B_{x}<0$ should be larger than
$0.15\;B_{lobe}$, where $B_{lobe}$ is the lobe magnetic field estimated via
the vertical pressure balance
($\mathbb{B}^{2}/2\mu_{0}+n_{i}(T_{e}+T_{i})\equiv B_{lobe}^{2}/2\mu_{0}$)
[Baumjohann . (1990)]; (2) the vertical pressure balance
$B_{lobe}^{2}/2\mu_{0}=const$ should be satisfied with $30\%$ accuracy
throughout the CS crossing. The first criterion will explore us to
sufficiently probe the inner part of the CS, using $B_{x}/B_{lobe}$ as an
effective coordinate to determine cross-tail profiles of plasma properties
surrounding the CS. Employing the aforementioned criteria, we manually
collected 1261 CS crossings in the midtail $X\in[-65,\;-50]$ RE (2012—2022).
Using the CS’s flapping motion, we convert temporal into spatial variations as
explained in the following 2 paragraphs.

To determine the CS’s local coordinate system
($\mathbb{l},\;\mathbb{m},\;\mathbb{n}$), we use the Maximum Variance Analysis
(MVA)[Sonnerup Cahill (1968)]. In the ($\mathbb{l},\;\mathbb{m},\;\mathbb{n}$)
system, $\mathbb{l}$ refers to the direction of maximum variance, $\mathbb{n}$
refers to the direction minimum variance, and $\mathbb{m}$ completes the
right-handed system. The applicability and accuracy of the MVA rely on the
ratios of eigenvalues corresponding to those eigenvector directions,
$\lambda_{1}/\lambda_{2}$ and $\lambda_{2}/\lambda_{3}$: the higher these
ratios are, the more accurate the estimation of CS’s local coordinate system
is.

According to the method developed by Sergeev98, Hoshino96 and Sergeev06, one
can determine the vertical (along the normal) velocity of the CS flapping
motion based on a correlation analysis between the ion bulk velocity component
perpendicular to the CS and the time gradient of the maximum variance magnetic
field component, $\sim B_{x}$. We select a subset of 453 CS crossings where
the ion bulk velocity correlates with the magnetic field time gradient (see
further discussion for details).

The ARTEMIS mission has covered more than a half of Solar Cycle 24 (2008—2019)
and a significant fraction of the ongoing Solar Cycle 25 (2019—today). This
temporal coverage provides us with a unique opportunity to investigate
magnetotail current sheets under various solar wind conditions. As a result,
we can statistically assess the impact of solar wind dynamics on the
magnetotail at lunar distances. Towards that goal we employ the Operating
Missions as Nodes on the Internet (OMNI) [King Papitashvili (2005)] dataset,
at 1min resolution. This includes the Auroral Electrojet (AE) and the
Interplanetary Magnetic Field IMF $\mathbb{B}$ = IMF
($B_{x},\;B_{y},\;B_{z}$). Subsequently we transform all the IMF components
into the AGSM coordinate system. In order to determine this system, we use
OMNI data of solar wind speed $\mathbb{V}_{sw}$. For each CS crossing in our
dataset, the values of $AE$ and $V_{SW}$ are prescribed by averaging OMNI 1min
data over an interval of 3 hours prior to the crossing.

Figures 1—3 show three examples from the collected CS subset. The CS crossing
in Figure 1 occurred on 11th July 2014. Figure 1a shows the magnetic field in
the AGSM coordinate system. A clear reversal of the $B_{x}$ component is
observed, resembling a discontinuity, with a relatively symmetric jump from
$B_{x}\approx-7.9$ nT in the southern lobe to $B_{x}\approx+8.5$ nT in the
northern lobe. The shear component $B_{y}$ remains small in comparison to
$\Delta B_{x}\approx 16$ nT and the normal component $B_{n}$ fluctuates around
zero. We check the pressure balance condition during the entire CS crossing by
calculating
$B_{lobe}=\left(\mathbb{B}^{2}+2\mu_{0}n_{i}(T_{e}+T_{i})\right)^{1/2}$; the
resultant $B_{lobe}$ profile is shown by the gray dashed line. $B_{lobe}$
remains almost constant and only slightly varies between $9$–$11$ nT,
indicating that pressure balance is satisfied with $10\%$ accuracy. Further,
we use the mean value of $B_{lobe}$ as the lobe magnetic field (we use the
same notation $B_{lobe}$ for $\langle B_{lobe}\rangle$). Figure 1a presents
magnetic field components in the local
($\mathbb{l},\;\mathbb{m},\;\mathbb{n}$) coordinate system. Note that for this
CS crossing the local ($\mathbb{l},\;\mathbb{m},\;\mathbb{n}$) coordinate
system is well determined, as the ratios of the three eigenvalues are:
$\lambda_{1}/\lambda_{2}=35$ and $\lambda_{2}/\lambda_{3}=4$, i.e., the three
variance directions are well defined. Figure 1d shows the profiles of ion
density $n_{i}$, electron and ion temperatures $T_{e}$ and $T_{i}$. We observe
bell-shaped profiles for both density and temperatures. However, the
respective maxima are located slightly off the CS’s center, $B_{x}=0$. The ion
density varies less than temperature: $n_{i}$ grows from $0.05$ cm-3 at the
CS’s boundaries to $\approx 0.1$ cm-3 near the CS’s center, whereas $T_{i}$
exhibits a steep increase from $200$ eV at the boundaries to $1.2$ keV near
the center. Additionally, the electron temperature $T_{e}$ increases from $60$
eV to around $320$ eV. Comparison of the density and magnetic field profiles
shows that this CS crossing resembles a Harris-like CS, where
$B_{l}\propto\tanh(z/L)$ and $n_{i}\propto cosh^{-2}(z/L)\propto
1-(B_{l}/B_{lobe})^{2}$. However, this Harris-like CS represents a very
uncommon event in our dataset (less than a few events). For the overwhelming
majority ofthe CSs, the density profiles are observed to be almost flat.
Figure 1 d presents profiles of electron ($\beta_{e}$) and ion ($\beta_{i}$)
beta parameters, which are defined as $\beta_{e}=2\mu_{0}n_{i}T_{e}/B^{2}$ and
$\beta_{i}=2\mu_{0}n_{i}T_{i}/B^{2}$. Both increase towards the CS center,
reaching $\beta_{i}\approx 7-13$ and $\beta_{e}\approx 2-3$, respectively.

We next check the correlation between the ion bulk velocity and the time
derivative of the magnetic field in order to estimate the CS vertical speed
relative to the spacecraft. In Figure 1e we plot the temporal profile of the
normal component of the ion bulk velocity $V_{n}$ and the time derivative of
the main magnetic field component, $dB_{l}/dt$. These profiles are correlated,
which indicates that the observed crossing was likely caused by the plasma
(and magnetic field) motion relative to the spacecraft, i.e., due to the
flapping. We check the applicability of a linear fit by computing
thecorrelation coefficient, $R=\left|corr(V_{n},dB_{l}/dt)\right|$. We find it
quite high ($R=0.71$) which enables us to use the approximation
$V_{n}=p_{1}dB_{l}/dt+p_{2}$, where $p_{2}[{\rm km/s}]$ is a velocity offset
and $p_{1}[{\rm km/nT}]$ refers to the slope of the fitting. Using $p_{1}[{\rm
km/nT}]$ we can estimate the CS’s spatial scale as
$L_{corr}=0.5\;|p_{1}|(B_{0}^{+}-B_{0}^{-})$. This method works best when when
it is applied to the short sub-interval around the neutral plane, $B_{l}\sim
0$, where the magnetic field spatial profile can be approximated by a linear
function $B_{l}\propto z/L$, and thus $dB_{l}/dt\propto V_{n}/L$. To enhance
the precision of our method, we select a sub-interval centered around the CS
that spans approximately 30-60 seconds. We use data only from this sub-
interval for the linear fit.

We also estimate the CS’s spatial scale by integrating $V_{n}$ over the
crossing: $L_{int}=\int_{B=B_{0}^{-}}^{B=B_{0}^{+}}(V_{n}-V_{n0})dt$. Here we
take into account possible velocity offset $V_{n0}$ \- a steady ion flow which
doesn’t belong to the flapping motion. In order to estimate $V_{n0}$, we
average $V_{n}$ closer to the lobes, defined as $|B_{l}|>0.8B_{0}$ [<]see
similar discussion in¿Sergeev98,Vasko15:jgr:cs. The two approaches of spatial
scale evaluation usually yield results which agree within a factor of $10$. In
order to increase the reliability of our estimations, we only use a subset of
CS crossings for which the discrepancy between these two estimations is within
a factor of $4$. For such CSs, we define the thickness $L$ as the mean value
of $L_{corr}$ and $L_{int}$. $L$ is then used to estimate the current density
peak: $J_{0}=0.5\;\mu_{0}^{-1}(B_{0}^{+}-B_{0}^{-})/L$. For the CS crossing
presented in Figure 1, the linear correlation gives $p_{1}\approx 132$ km/nT
and $L_{corr}=1061$ km. The estimation based on integration gives
$L_{int}=370$ km and thus $L_{corr}/L_{int}\approx 2.37$. This gives $L\approx
753$ km and $J_{0}\approx 8.5$ nA/m2. Current density of a few $nA/m^{2}$
represents a typical value for the near-Earth’s magnetotail current sheet
[<]e.g.,¿Runov06,Petrukovich15:ssr. Thickness of a few thousand kilometers,
however, indicates a rather thin ion-scale CS [<]see discussion
in¿Runov06,Artemyev11:jgr. Such thin CSs are believed to be closely associated
with plasma instabilities, resulting in magnetic reconnection [<]see¿[and
references therein]Petrukovich16:book,Sitnov19. We estimate CS’s thickness in
ion thermal gyroradius $\rho_{i}$, which can be defined either using the
maximum magnetic field $B_{0}=max\;|B_{l}|$ at the CS’s boundaries [<]see for
example¿Artemyev11:jgr,Vasko15:jgr:cs or by using the lobe magnetic field
$B_{lobe}$ [<]see¿Runov06. In this study we use $B_{lobe}$ as a characteristic
value of magnetic field strength and define ion thermal gyroradius as
$\rho_{i}=m_{i}v_{thi}/eB_{lobe}$, where $v_{thi}=(2\langle
T_{i}\rangle/m_{i})^{1/2}$ stands for ion thermal velocity evaluated for the
cross-CS averaged temperature. For the CS crossing from Figure 1, we get
$\rho_{i}\approx 478$ km, which is comparable to the CS’s thickness,
$L/\rho_{i}\approx 1.58$. For $\rho_{i}$ evaluated in $B_{0}$, we would have
$\rho_{i}\approx 478$ km and $L/\rho_{i}\approx 1.60$ [<]see discussion of
these two estimates in¿Artemyev11:jgr,Petrukovich15:ssr.

Figures 2—3 show two CS crossings from our subset. The CS crossing on January
19 2019 (see Fig. 2) represents the most common type of CS in our set. Figure
2a shows magnetic field profiles in the aberrated GSM coordinate system. The
gray dashed line represents $B_{lobe}$ inferred from the pressure balance
condition. We see that the magnetic field magnitude at the CS’s boundaries,
$B_{0}=(\max B_{l}-\min B_{l})/2$, is substantially smaller than the lobe
magnetic field $B_{lobe}$. This effect (the so-called CS embedding) is a
rather typical property of near-Earth’s thin CSs [<]see¿Runov06. For the CS
from Figure 2, $B_{0}/B_{lobe}\approx 0.25$. Thus, the observed magnetic field
strength at the CS’s boundary is only $25\%$ of what we should see for a
Harris-like configuration. Temporal variations of the plasma properties
surrounding the CSs (Fig. 2c) show a relatively flat profile of ion density
and temperature, with $n_{i}$ and $T_{i}$ only slightly fluctuating near the
constant level of around $0.3$ cm-3 and $0.5$ keV, respectively. Therefore,
here we see a specific type of CS’s spatial configuration in which the spatial
distributions of density and temperature are substantially broader than that
of the current density, i.e., current density peak is embedded into the plasma
density profile. Current sheets exhibiting a comparable spatial configuration
have been investigated in the near-Earth magnetotail [Runov . (2006), Artemyev
. (2011), AA. Petrukovich . (2015)]. It has been proposed that the embedding
of these CSs (i.e., $B_{0}/B_{lobe}<1$) can be described as an ion-scale thin
current sheet enveloped within a thicker sheet, which accounts for density and
temperature variations [Sitnov . (2003), Sitnov . (2006), Zelenyi . (2011),
Artemyev Zelenyi (2013)].

In order to estimate CS’s thickness, we compare $V_{n}$ with $dB_{l}/dt$.
Figure 2e demonstrates that there is a strong correlation between these two
quantities, the linear correlation coefficient is rather high ($R\approx
0.9$). Applying the same method as previously described, we end up with the
following estimations of the spatial scale: $L_{corr}=801$ km and
$L_{int}=1030$ km. The two methods of estimation agree within a factor of $2$,
which allows us to evaluate $L=915$ km and $J_{0}=4.5$ nA/m2. A small $L$
suggests that this CS is of ion scale, a typical ion gyroradius for this CS is
$\rho_{i}\approx 493$ km and $L/\rho_{i}\approx 1.86$. This thin CS differs
from the thin CSs observed near the reconnection region for which
$B_{0}/B_{lobe}\sim 1$ [Nakamura . (2006)]. However, it does resemble thin
current sheets detected in the solar wind, where effective $B_{0}/B_{lobe}$
varies within a wide range of values [<]e.g.,¿Vasko21:apjl. A small
$B_{0}/B_{lobe}$ implies that even a weak plasma pressure variation across the
sheet, $\sim\Delta\left(nT_{i}\right)/nT_{i}\approx B_{0}^{2}/B_{lobe}^{2}\ll
1$, is sufficient to balance this CS. Indeed, for this CS,
$\Delta\left(nT_{i}\right)/nT_{i}=0.24$ and $B_{0}^{2}/B_{lobe}^{2}=0.23$.
Such a subtle variation is nearly imperceptible in the density and temperature
profiles presented in Fig. 2.

Figure 3 shows the third CS crossing. It exemplifies another important type of
the CSs that make up a noticeable part of the dataset ($\sim 24\%$). Again, we
observe flat profiles of the main plasma properties surrounding the CSs (see
Figure 3c). Both density $n_{i}\approx 0.15-0.20$ cm-3 and temperature
$T_{i}\approx 1.05-1.1$ keV do not exhibit any variations throughout the CS
crossing. This results in negligible thermal pressure variation within the CS,
$\Delta(n_{i}T_{i})/n_{i}T_{i}<0.1$ for $B_{0}/B_{lobe}\approx 1/2$. Despite
that, the pressure balance $B_{lobe}\approx const$ is well satisfied to within
$2\%$. Thus, we should expect $\Delta\mathbb{B}^{2}\sim 0$ in order to
maintain the full pressure balance $\mathbb{B}^{2}+2\mu_{0}n_{i}T_{i}\approx
const$ (electron contribution can be omitted due to $T_{e}/T_{i}<0.1$).
According to Figure 3, the main magnetic field component $B_{l}$ varies from
$B_{l}\approx-5$ nT to $B_{l}\approx+5$ nT. We also observe that the normal
component remains small through the crossing, $|B_{n}|\lesssim 1$ nT, which is
typical of the majority of the CSs in our dataset ($|B_{n}|/B_{0}<0.3$ for
$95\%$ of CSs and $|B_{n}|/B_{0}<0.1$ for $60\%$ of CSs). The shear component
$B_{m}$ exhibits a bell-shaped profile, growing from a zero-value at the CS
boundaries up to $5$ nT at the CS’s center. Therefore, $B_{m}$ variation
almost precisely compensates for the drop of $|B_{l}|$ and provides
$B_{m}^{2}+B_{l}^{2}\approx const$. For 1D current sheets (with only
$\partial/\partial r_{n}\neq 0$) this implies $j_{l}B_{m}-j_{m}B_{l}={\bf
j}\times{\bf B}=0$. Thus, the CSs with such magnetic field configuration are
referred to as force-free CSs. They have previously been reported in the solar
wind [Artemyev, Angelopoulos Vasko (2019), Neukirch, Vasko . (2020), Lotekar .
(2022)] and planetary magnetospheres
[<]e.g.,¿Artemyev14:pss,DiBraccio15,Rong15:venus. This includes individual
cases in the Earth’s magnetotail at lunar distances [Xu . (2018)] and near-
reconnection region [Nakamura . (2008), Artemyev . (2013)]. Theoretical models
of such force-free CSs can be found in
Artemyev11:pop,Mingalev12,Vasko14:angeo_by,Allanson15,Neukirch20:jpp. In the
simplest model of a force-free CS, the spatial profile of the main magnetic
field component is identical to that of the Harris sheet, $B_{l}\sim
tanh(z/L)$, whereas the shear component profile is $B_{m}\sim cosh^{-1}(z/L)$
[Harrison Neukirch (2009)]. To quantify the number of similar CSs we introduce
$\Delta B_{m}^{2}/B_{0}^{2}=\Delta(B_{m}^{2})/B_{0}^{2}$ parameter, which
characterizes the relative contribution of $B_{m}$ into the CS pressure
balance ($\Delta B_{m}^{2}$ is the difference between $B_{m}^{2}$ value at the
CS’s boundary and at the center). In a 1D CS, $\Delta(B_{m}^{2})/B_{0}^{2}$
parameter equals to integrated cross-sheet ratio of the two tension force
components: $\int{j_{l}B_{m}dr_{n}}/\int{j_{m}B_{l}dr_{n}}$. In the case of a
purely force-free current sheet, where $j_{l}B_{m}-j_{m}B_{l}={\bf
j}\times{\bf B}=0$, this parameter equals one. The same parameter has been
used in [Xu . (2018)]. For the CS crossing from Figure 3, $\Delta
B_{m}^{2}/B_{0}^{2}\approx 1$, a value which corresponds to the purely force-
free CS configuration. Statistically, CSs with a pronounced contribution of
$B_{m}$ (that is $\Delta B_{m}^{2}/B_{0}^{2}\gtrsim 0.1$) make up about $24\%$
of the events in our set. Note $B_{m}$ variation with the maximum at $B_{l}=0$
is generated by local currents $j_{l}$ and differs well from the conventional
flaring $B_{m}$ component, which drops to zero at $B_{l}=0$.

Figure 1: The current sheet observed at 2014-Jun-11 by ARTEMIS P1: magnetic
field in AGSM coordinates (a), magnetic field in local (MVA) coordinates (b),
plasma density, ion and electron temperatures (c), ion and electron $\beta$
(d), ion $V_{n}$ and $dB_{l}/dt$ (e), correlation of $V_{n}$ and $dB_{l}/dt$
(f), current density as a function of time (g). Main current sheet parameters
are: $\lambda_{1}/\lambda_{2}=35.0$, $\lambda_{2}/\lambda_{3}=4.0$, $R=0.71$,
$L=753$ km, $J_{0}=8.5$ nA/m2 Figure 2: The current sheet observed at
2019-Jan-19 by ARTEMIS P1: magnetic field in AGSM coordinates (a), magnetic
field in local (MVA) coordinates (b), plasma density, ion and electron
temperatures (c), ion and electron $\beta$ (d), ion $V_{n}$ and $dB_{l}/dt$
(e), correlation of $V_{n}$ and $dB_{l}/dt$ (f), current density as a function
of time (g). Main current sheet parameters are:
$\lambda_{1}/\lambda_{2}=23.3$, $\lambda_{2}/\lambda_{3}=11.2$, $R=0.85$,
$L=916$ km, $J_{0}=4.52$ nA/m2 Figure 3: The current sheet observed at
2016-May-20 by ARTEMIS P1: magnetic field in AGSM coordinates (a), magnetic
field in local (MVA) coordinates (b), plasma density, ion and electron
temperatures (c), ion and electron $\beta$ (d), ion $V_{n}$ and $dB_{l}/dt$
(e), correlation of $V_{n}$ and $dB_{l}/dt$ (f), current density as a function
of time (g). Main current sheet parameters are:
$\lambda_{1}/\lambda_{2}=13.4$, $\lambda_{2}/\lambda_{3}=11.0$, $R=0.84$,
$L=962$ km, $J_{0}=5.6$ nA/m2

Figure 4 presents the distribution of all 1261 CS crossings in our set. The
spatial distribution of observed CS crossings on the $X_{AGSM}$—$Y_{AGSM}$(in
Earth radii RE units) plane is shown in Figure 4a. The selected set covers
$X\in[-65,\;-50]$ RE and $Y\in[-30,30]$ RE, which uniformly resolves both dusk
and dawn flanks. Figure 4b shows CSs’ distribution in the parametric space of
$AE$ and $V_{SW}$. We observe CSs in a wide range of both $AE$ and $V_{SW}$:
$AE$ can vary by two orders of magnitude from $10$ nT to $1000$ nT, whereas
CSs are observed during both slow solar wind $V_{SW}\approx 300-400$ km/s and
during relatively fast solar wind with $V_{SW}$ reaching up to $800$ km/s.

Figure 4: Distribution of current sheet crossings in $(X,Y)$ AGSM plane (a),
$(AE,V_{sw})$ plane (b), where $AE$ and $V_{SW}$ are averaged over an interval
of 3 hours prior to the CS crossing. Color refers to the year of observation.

##  3 Statistics of current sheet characteristics

Our set consists of 1261 CS crossings that cover 11 years of ARTEMIS
observations (from 2012 to 2022). Within this set we have a subset of 453 CSs
for which we can reliably determine the thickness and current density. For the
subset, both linear correlation between $V_{n}$ and $dB_{l}/dt$ and direct
integration of $V_{n}$ yield similar (within a factor of $4$) estimations of
the spatial scale. Figure 5 shows the distribution of the CSs’ characteristics
from this subset. Typical current density is a few nA/m2 (see Figure 5a).
These values are similar to those previously observed in the near-Earth
magnetotail [Runov . (2006), Artemyev . (2011), Lu . (2019)] as well as in the
distant magnetotail [Vasko . (2015)]. We also notice a substantial population
($24\%$ of the subset) of relatively intense currents ($J_{0}>10$ nA/m2).

Figure 5b presents the distribution of $\beta_{i}=2\mu_{0}n_{0}T_{i}/B^{2}$
averaged over the CS’s center, $|B_{l}|<0.15B_{lobe}$. It is a log-normal
distribution, with the average $\langle\beta_{i}\rangle\approx 100$.
Interestingly, the distribution also contains a substantial number of events
with $\beta_{i}\lesssim 10$. CSs characterized by such $\beta$ values are not
commonly found in the near-Earth magnetotail [Asano . (2003), Asano . (2004)];
however, they are rather ubiquitous in the solar wind environment
[<]e.g.,¿Vasquez07,Artemyev19:jgr:solarwind,Vasko22:apjl. In our dataset, more
than one third (38%) of CSs have $\beta_{i}\lesssim 10$. This population is
substantially larger than what has previously been observed in the near-Earth
magnetotail. This can be attributed to the fact that, compared to the middle
and near magnetotail, ion temperature in the magnetotail at lunar distances is
usually a few times lower.

Distribution of CSs’ thickness $L$ is shown in Figure 5c. On average,
$L\approx 1000$ km, which is comparable to the ion thermal gyroradius for $1$
keV energy and $\sim 5$ nT of $B_{0}$ field. Thinner CSs ($L\lesssim 1000$ km)
are also observed, comprising 34% of the entire subset. The number of thick
CSs, with $L\gtrsim 10^{4}$ km, is negligible ($<1\%$). The bias is due to the
criteria of CSs’ collections: typically flapping velocities don’t exceed
$V_{N}\approx 10-50$ km/s, whereas most of crossings are fast ($<1$ min).

Figures 5d-e show the distribution of normalized CS thickness $L$: we use ion
gyroradius $\rho_{i}$ and ion inertial length $d_{i}$. Both $\rho_{i}$ and
$d_{i}$ are evaluated using plasma parameters at the CS’s center; for
$\rho_{i}$ we use $B_{lobe}$ as the typical value of magnetic field strength.
The majority of CSs are of ion-gyroscale, the average value is $L/\rho_{i}=4$
(see Figure 5d), whereas CSs of sub ion-scale ($L/\rho_{i}<1$) constitute 10%
of the subset. Figure 5f presents the statistical distribution for the lobe
magnetic field $B_{lobe}$. The average value of $B_{lobe}$ is approximately
$10$ nT, which is 2 to 3 times lower than in the near-Earth magnetotail [Runov
. (2006), Artemyev . (2011)]. However, it is similar to the observations made
by Geotail in the distant magnetotail [Vasko . (2015)].

On the whole, typical values of $10\;nT$ at $X\in[-60,-50]$ RE are in a good
agreement with predictions of empirical models for the $B_{lobe}$ radial
profile [Nakai . (1991), Shukhtina . (2004)]. Figure 5g presents the
distribution of the embedding ratio $B_{0}/B_{lobe}$, i.e., the ratio of
magnetic field strength at the CS’s boundary $B_{0}$ to the total (lobe)
magnetic field $B_{lobe}$. For a Harris-like CS this quantity always equals to
one and, therefore, any significant deviation of $B_{0}/B_{lobe}$ from unity
would likely indicate an embedded CS configuration. According to Figure 5g,
the majority of CSs at lunar distances are weakly embedded. Typically, $B_{0}$
can reach $60\%-90\%$ of the total magnetic field. Note that the observed
$B_{0}/B_{lobe}$ ratio is generally lower in the near-Earth magnetotail, where
$B_{0}/B_{lobe}\sim 0.3-0.4$ [<]see¿Artemyev08:angeo,Artemyev10:jgr, i.e., CSs
at lunar distances are less embedded and closer to the Harris CS
configuration.

Figure 5h shows the distribution of ion to electron temperature ratio
$T_{i}/T_{e}$, computed at the CS’s center. In the near Earth’s magnetotail
CS,for $X\in[-30,-10]$ RE, $T_{i}/T_{e}$ typically varies within $[1,5]$ [Wang
. (2009), Artemyev . (2016), Runov . (2015)]. For the CSs at lunar distances,
the $T_{i}/T_{e}$ ratio is generally higher than $5$. This indicates that
$T_{i}/T_{e}$ decreases earthward and, as a result, electrons should be heated
more effectively than ions during earthward transport [<]see discussion
in¿Runov18.

Figure 5i presents the analyses of the force-free measure $\Delta
B_{m}^{2}/B_{0}^{2}$, where $\Delta B_{m}$ stands for the variance of the
shear magnetic field component $B_{m}$ from the CS’s center to the boundary.
For most of the CSs, the $B_{m}$ variation remains negligible ($\Delta
B_{m}^{2}/B_{0}^{2}<0.05$ for 65% of CSs). Nevertheless, 24% of CSs in our
dataset are characterized by a substantial contribution of $B_{m}$ into the
pressure balance($\Delta B_{m}^{2}/B_{0}^{2}>0.1$). These CSs can be
classified as partially or even fully force-free
[<]see¿Neukirch20,Neukirch20:jpp.

Figure 5: Distributions of current sheet characteristics: current density
magnitude $J_{0}$ (a), ion $\beta_{i}$ (b), current sheet thickness $L$ (d)
current sheet thickness normalized to ion thermal gyroradius $L/\rho_{i}$, (d)
current sheet thickness normalized on ion inertial length $L/d_{i}$ (e), lobe
magnetic field magnitude $B_{lobe}$ (f), embedding parameter $B_{0}/B_{lobe}$
(g), ion to electron temperature ratio $T_{i}/T_{e}$, force-free measure
$\Delta B_{m}^{2}/B_{0}^{2}$ (i)

Using the subset of 453 CSs with reliably determinable current density $J_{0}$
and thickness $L$, we can statistically explore the potential relationship
between $J_{0}$ and $L$ in order to reveal the role of $B_{0}$ in the CS
configuration. There are two main scenarios: (1) $B_{0}$ is approximately the
same for all CSs, and thus $J_{0}L\approx const$, (2) $B_{0}$ is proportional
to $J_{0}$ magnitude, while $L$ remains almost the same for all CSs [<]see
discussion in¿Petrukovich11. Figure 6a presents the distribution of CSs in
$J_{0}-L$ space with color bar indicating the number of CSs within each bin.
The fitting $J_{0}\propto 1/L$ describes the observed distribution (the best
power-law fitting yields the exponent between -0.9 and -0.8). This indicates
that $B_{0}$ remains constant (actually around a fraction of $B_{lobe}$) for
all CSs, which implies that the configuration of thin CSs is primarily
governed by the boundary conditions, set by $B_{lobe}$. In this context, the
local plasma currents are spatially distributed in such a way that
$J_{0}L\approx const$. Thus, we do not observe any intense, thick CSs nor do
we observe any low-intensity, thin CSs. Note that the near-Earth magnetotail,
where $B_{0}/B_{lobe}\ll 1$, does contain a population of low-intensity, thin
CSs [<] see¿Petrukovich15:ssr.

Figure 6b shows the number of CSs in $J_{0}-E_{y}$ space, where
$E_{y}=|\langle V_{x}B_{z}\rangle|$ is the convection electric field. $V_{x}$
denotes the ion bulk-velocity along $x$-direction, $B_{z}$ is the $z$
component of the magnetic field (in AGSM system), $\langle..\rangle$ denotes
averaging over the CS crossing. The distribution of CSs in $J_{0}-E_{y}$ space
indicates that intense CSs (large $J_{0}$) are usually associated with larger
values of convection electric field $E_{y}$. This correlation can be
understood within the framework of an embedded CS, supported by a relatively
small population of demagnetized Speiser ions [Speiser (1965)] moving along
open trajectories and contributing significantly to the cross-tail current
density [<]see¿Burkhart92TCS,Sitnov00,Zelenyi00. These ions are effectively
accelerated by the convection field $E_{y}$ [Lyons Speiser (1982), Ashour-
Abdalla . (1993), Ashour-Abdalla . (1994), XZ. Zhou . (2011), XZ. Zhou .
(2012)]; this acceleration increases the Speiser ion current density. Thin CSs
supported by such transient ion populations have been detected in the near-
Earth magnetotail [X. Zhou . (2009), Artemyev . (2010)]. They may, however, be
more common in the distant tail, where the equatorial magnetic field is
comparatively weaker, allowing for a larger ion population to be demagnetized
[<]see discussion in¿Zelenyi13:UFN.

To provide an alternative measure of the current intensity, we introduce the
ion-electron drift velocity $J_{0}/en_{0}$ ($n_{0}$ – plasma density). Figure
6 c provides the number of CS crossings in the
$J_{0}/en_{0}-sign(V_{x})\;\log_{10}(|V_{x}|)$ space. Using
$sign(V_{x})\;\log_{10}(|V_{x}|)$ allows us to employ a logarithmic scale
while still distinguishing between tailward flows ($V_{x}<0$) and earthward
flows ($V_{x}>0$). Magenta lines indicate $J_{0}/en_{0}=\pm\;V_{x}$. These
results clearly indicate that the most intense CSs occur within fast flows
[<]see also¿Vasko15:jgr:cs. This could be the result of larger $E_{y}\propto
V_{x}$ and larger hot ion population, transported by fast plasma flow and
supporting the thin CSs. Interestingly, it appears that CSs at the lunar orbit
are more often observed within earthward flows (for $\sim 60\%$ of events we
have $V_{x}>0$), whereas statistical studies of plasma flows reveal an equal
number of tailward and earthward flows [Kiehas . (2018)].

Typical plasma conditions in the magnetotail at lunar distances (low-$\beta$)
seem to favor the formation of partially or even entirely force-free CS
configurations. Figure 6d demonstrates that there is an inverse correlation
between force-free measure $\Delta B_{m}^{2}/B^{2}_{0}$ and ion beta
$\beta_{i}$. The best power-law fitting yields $\Delta
B_{m}^{2}/B^{2}_{0}\propto 1/\beta_{i}^{0.55}$(note that we also show $\Delta
B_{m}^{2}/B_{0}^{2}\propto 1/\beta_{i}$ for comparison). This relationship can
be further verified for space plasma systems with a wide range of $\beta$,
e.g., Earth’s magnetosheath.

In addition to exploring the distributions of CSs’ characteristics, we have
also investigated potential connections with solar wind parameters and Auroral
Electrojet Index ($AE$). For each CS crossing, we determine the value of the
solar wind speed $\mathbb{V}_{sw}$, interplanetary magnetic field IMF
$\mathbb{B}$, $AE$ averaged over three hours prior to the crossing. Figure 6e
shows that there is a degree of correlation between the lobe magnetic field
$B_{lobe}$ and $AE$. According to the previous studies, $AE$ correlates with
the solar wind speed $V_{sw}$ [Hajra . (2014)]. Therefore, the observed
correlation between $AE$ and $B_{lobe}$ can likely be attributed to the
relationship between $B_{lobe}$ and the solar wind dynamic pressure
$P_{D}\propto V_{sw}^{2}$, as discussed in Nakai91.

We were unable to find any significant correlation between $B_{lobe}$ and
$V_{sw}$ (not shown). On the other hand, a similar relationship between the
equatorial plasma pressure (approximately $B_{lobe}^{2}/2\mu_{0}$) and the
$AE$ index has already been reported for the near-Earth magnetotail [<]see
Fig. 5 in¿Baumjohann90.

Finally, figure 6f shows the distribution of the CSs in the space of IMF
$B_{y}$ and CS $B_{y}$: IMF $B_{y}$ stands for $Y$ component of the
interplanetary magnetic field, whereas $B_{y}$ is computed at the CS’s center.
The magenta line indicates IMF $B_{y}=B_{y}$. Note that the ultimate origin of
$B_{y}$ component in the Earth’s magnetotail is still unclear. A number of
studies have speculated that, at least to some extent, the origin of $B_{y}$
can be traced down to the penetration of the IMF $B_{y}$ to the Earth’s
magnetosphere [Nakamura . (2008), AA. Petrukovich Lukin (2018)]. Other
studies, however, have suggested that this mechanism for penetration may not
be effective in the magnetotail at lunar distances [Xu . (2018)]. Indeed,
Figure 6f shows that there is no significant correlation between IMF $B_{y}$
and $B_{y}$.

Figure 6: Distributions of observed CSs in different parametric spaces:
current density magnitude $J_{0}$ versus current sheet thickness $L$, magenta
line indicates power law-fit $J[nA/m^{2}]\approx(7.4/L[10^{3}km])^{0.8}$(a),
current density magnitude $J_{0}$ versus magnetotail convectional electric
field $E_{y}$(b), ion-electron drift velocity $J_{0}/en_{0}$ versus
tailward/earthward ion bulk velocity $V_{x}$ , magenta lines show
$J_{0}/en_{0}=\pm V_{x}$ (c), force-free measure $\Delta B_{m}^{2}/B_{0}^{2}$
versus ion beta $\beta_{i}$, lines show two power-law fits: $\Delta
B_{m}^{2}/B_{0}^{2}\approx(0.11/\beta_{i})^{0.55}$(magenta), $\Delta
B_{m}^{2}/B_{0}^{2}\approx 0.5/\beta_{i}$(black)(d), lobe magnetic field
$B_{lobe}$ versus Auroral Electrojet index $AE$(e), $B_{y}$ component of the
magnetic field within the CS versus $IMFB_{y}$, magenta line indicates
$B_{y}=IMF\;B_{y}$.

##  4 Profiles of current sheet characteristics

An essential aspect of the CS spatial configuration is the relationship
between spatial (along normal to the neutral plan) distributions of plasma
characteristics and current density [<]see results and discussion for the
near-Earth magnetotail CSs in¿Runov06,Artemyev11:jgr,Lu19:jgr:cs. To determine
if a CS is embedded (i.e., CS configuration deviates from a single-component
Harris CS), we compare the current density and plasma density profiles. In the
case of an embedded thin current sheet (CS), there are typically two distinct
characteristic scales that define its structure. The larger scale is
associated with the variations in density and temperature profiles across the
CS. Meanwhile, the smaller scale is associated with a intense peak in the
current density around the CS’s center.

We use $B_{l}$ component of the magnetic field (normalized to the lobe
magnetic field $B_{lobe}$) as a proxy for the distance to the CS neutral
plane, and used $B_{l}/B_{lobe}$ to shape spatial profiles of plasma
characteristics across CSs. Figures 7(a,b,c) show profiles of current density
$J$, ion density $n_{i}$, and ion temperature $T_{i}$. For each CS we
normalize these profiles to their maximum values and centre the profile at the
position of the maximum current density, $B_{l}-B_{l}^{*}$ with $\max
J=J(B_{l}^{*})$. Comparing Figures 7a-7c reveals that the majority of CSs
exhibit a thin current density peak (with a typical thickness of
$0.5\;B_{lobe}$), embedded into thicker density and temperature profiles
(covering an entire range of $0<|B_{l}|<B_{lobe}$). Similar to the results for
the near-Earth CSs, the ion temperature varies more substantially across the
CS compared to the plasma density [<]see discussion
in¿Artemyev17:jgr:THEMIS&Geotail.

Figure 7d presents profiles of the shear magnetic field component
$B_{m}(B_{l})$, with colors indicating averaging over CSs in different
$\beta_{i}$ ranges. As $\beta_{i}$ decreases, the relative contribution of
$B_{m}$ component increases and the $B_{m}(B_{l})$ profile more closely
resembles a bell-shaped curve. For CSs with $\beta_{i}<10$, the relative
contribution of $B_{m}$ can reach up to 40% of $B_{0}$. A magnetic field
configuration with a shear component that has a bell-shaped profile, is
typically attributed to a force-free equilibrium. There are series of
theoretical models describing such an equilibrium for CSs with $B_{n}=0$
[<]e.g.,¿[and references therein]Neukirch20,Neukirch20:jpp and with a finite
$B_{n}\neq 0$ [<]e.g.,¿Artemyev11:pop,Vasko14:angeo_by,An23:apj.

Figure 7: Averaged profiles of current sheet characteristics versus
$B_{l}/B_{0}$ for different $\beta_{i}$ ranges: current density $J$ normalized
on $max(J)$ (a), plasma density $n_{i}$ normalized on it’s peak value (b), ion
temperature $T_{i}$ normalized on it’s peak value (c), shear magnetic field
component $B_{m}$ normalized to the magnetic field at the CS’s boundary
$B_{0}$ (d).

Figure 8 reproduces the profiles previously shown in Figure 7, but in this
case, the spatial coordinate along the normal to the current sheet, denoted as
$Z(t)$, is used: $Z(t)=\int_{t_{0}}^{t}V_{N}dt$ with $t_{0}$ indicating the
time when $B_{l}=0$. We use $Z-Z^{*}$ in order to account for the offset of
the current density peak location $Z^{*}$ relative to the $B_{l}=0$. We also
normalize $Z-Z^{*}$ to the CS thickness $L$. Figure 8a presents profiles of
$J(Z)$, normalized to the current density peak value for each CS. A typical
scale of the central current density peak is $\sim L$, however, a thin CS
(thin current layer) is embedded into a larger scale (thicker) sheet. The
intensity of this sheet is usually lower ($J\sim 0.3\;J_{0}$) and its typical
thickness is $\sim 3-4\;L$. These double-scale current density profiles have
also been observed in the near-Earth magnetotail [AA. Petrukovich . (2011),
Zelenyi, Artemyev, Malova . (2010)].

Figures 8 b and 8 c show spatial profiles of the $B_{l}$ component normalized
either to its maximum value $B_{0}$ (Fig. 8(b)) or to the lobe magnetic field
$B_{lobe}$ (Fig. 8(c)). For this figure, we orient each $B_{l}$ profile in
such a way that $B_{l}<0$ for $Z-Z^{*}<0$ and $B_{l}>0$ for $Z-Z^{*}>0$.
Comparison between Figs. 8b and 8c shows that $B_{0}$ is quite close to
$B_{lobe}$, and the normalized CS configuration does not vary with plasma
$\beta$.

Profiles of ion density $n_{i}$ and temperate $T_{i}$ are shown in Figures 8 d
and 8 e, respectively. Typical spatial scales of density and temperature
variations are significantly larger than $L$ and thus cannot be directly
associated with the observed thin current density peak. Such broad density and
temperature profiles are often detected in the near-Earth magnetotail, and
represent a typical attribute of the embedded thin CS configuration [Runov .
(2006), Artemyev . (2011), Lu . (2019)].

Figure 8 f shows profiles of the normalized shear magnetic field component
$B_{m}/B_{0}$. For smaller $\beta_{i}$ values, there is a significant
contribution from $B_{m}$ component into the total pressure balance.
Interestingly, the $B_{m}/B_{0}$ peak scales with the current density peak,
but does not show a large-scale variation. Thus, $B_{m}$ may contribute to the
pressure balance in the thin CS, whereas plasma pressure is responsible for
the balance of thick embedded CS. Note that a similar spatial configuration is
observed for force-free CSs in the solar wind, where $B_{m}$ profile does not
have a small-scale peak, but rather envelopes a thin current layer
[<]see¿Artemyev19:jgr:solarwind.

Figure 8: Averaged profiles of current sheet characteristics in $Z/L$ space
for different $\beta_{i}$ ranges: current density $J$ normalized to the peak
value (a), main magnetic field component $B_{l}$ normalized to the magnetic
field magnitude at the CS’s boundary $B_{0}$ (b), main magnetic field
component $B_{l}$ normalized to the lobe magnetic field $B_{lobe}$ (c), plasma
density $n_{i}$ normalized in peak value (d), ion temperature $T_{i}$
normalized in peak value (e), normalized shear magnetic field $B_{m}/B_{0}$
(f).

Figure 9 presents dawn-dusk distributions of CS characteristics. We use the
same AGSM coordinate system defined above. We employ five $10$ RE wide bins,
centered at $Y=-20,-10,0,10,20$ RE. We introduce normalized current density
$J_{0}/J_{th}$, where $J_{th}=en_{0}v_{th}$ ($e$ is the elementary charge,
$n_{0}$ is ion density in the CS center, $v_{th}$ is ion thermal velocity) and
use $J_{0}/J_{th}$ to divide our subset of 453 CSs into 3 subgroups depending
on the relative current intensity: (1) low-intensity CSs with
$J_{0}/J_{th}<0.25$, (2) intermediate-intense CSs with $0.25<J_{0}/J_{th}<0.8$
and (3) intense CSs with $J_{0}/J_{th}>0.8$. Based on these parameters, low
and intermediate intense CSs accounted for $\approx 45\%$ of our subset, each
($\approx 90\%$, cumulatively). Intense CSs represent the residual $10\%$ of
the subset. Figure 9 a shows the dawn-dusk distribution of the average current
density $J_{0}$, with colors indicating different sub-groups. Low and
intermediate intense CSs do not exhibit any noticeable dawn-dusk asymmetry.
For intense CSs, the current density $J_{0}$ trends towards larger $J_{0}$ at
the dusk flank, increasing from $10$ nA/m2 at the dawn side to $>20$ nA/m2 at
the dusk side. Such dawn-dusk asymmetry is a common feature observed in the
near-Earth magnetotail [Artemyev . (2011), Lu . (2016)], where it is
attributed to duskward drift of hot ions [<]see discussion in¿Walsh14.
However, this phenomenon cannot be easily explained at lunar distances, as a
significant sukward drift would require a strong $\partial B_{z}/\partial x$
gradient, absent at such distances. Importantly, a similar duskward growth has
also been reported in middle-tail [Vasko . (2015)]. Note that in CSs with
$\partial/\partial x\approx 0$, the classical diamagnetic plasma drift,
$\sim{\bf B}\times{\nabla(nT_{i})}/B^{2}\propto
B_{x}(\partial(nT_{i})/\partial z)/(B_{x}^{2}+B_{z}^{2})$, becomes zero around
$B_{x}=0$ ($B_{l}=0$). Therefore, this diamagnetic drift cannot account for
duskward drift of hot ions in the presence of a small $B_{z}\neq 0$ [<]see
discussion in¿Zelenyi10GRL,Artemyev&Zelenyi13.

Figure 9 b shows the dawn-dusk profile of $\beta_{i}$. For intermediate
intense CSs this profile seems to be relatively symmetric, whereas intense
CSs’ $\beta_{i}$ grows ($\times 10$) towards the dusk flank. This could be a
result of ion temperature increase at dawn flank.

Figure 9 c shows the dawn-dusk distribution of the normalized thickness
$L/\rho_{i}$. The thinnest (and most intense) CSs have typical $L/\rho_{i}\sim
1$ and do not show any dawn-dusk asymmetry of thickness. On the average,
observed CSs have $L/\rho_{i}\approx 3-4$, with a slightly larger population
of thinner CSs around the midnight sector ($Y\sim 0$), where we expect most of
tailward plasma flows with hot ions from near-Earth reconnection region (see
Kiehas18 and the dawn-dusk temperature profiles below).

Figures 9 d presents the dawn-dusk distribution of the lobe magnetic field
$B_{lobe}$. For low and intermediate intense CSs, there is only a weak growth
of $B_{lobe}$ from $10$ nT at the dusk flank to $11-12$ nT at dawn flank. For
the intense CSs, however, there is a local minimum $B_{lobe}\approx 9$ nT in
the midnight sector.

Figures 9 e shows the distribution of $B_{0}/B_{lobe}$, which characterizes
CSs’ embedding. In all ranges of $J_{0}/J_{th}$, $B_{0}/B_{lobe}$
distributions are relatively symmetric and show only a subtle increase in
$B_{0}/B_{lobe}$ around the midnight sector. Interestingly, it appears that
larger values of $B_{0}/B_{lobe}\approx 0.8$ are more typical of intense CSs.

Figure 9 f shows the average dawn-dusk distributions of the force-free measure
$\Delta B_{m}^{2}/B^{2}_{0}$. Observed profiles for intense and intermediate
intense CSs exhibit a modest $\Delta B_{m}^{2}/B^{2}_{0}$ increase towards
flanks, reaching levels of $\Delta B_{m}^{2}/B^{2}_{0}\approx 0.1-0.2$.
However, no such trend is observed in $\Delta B{m}^{2}/B^{2}_{0}$ for low-
intensity CSs. This suggests that force-free CSs don’t have any strong
preferences in terms of dawn-dusk occurrence.

Figure 9 g shows the distribution of ion density in the observed CSs. Density
profiles are generally symmetric and feature a minimum at the midnight, with a
gradual increase towards flanks. This is similar to what has been observed in
the Geotail statistics of CSs [Vasko . (2015)]; the flankward density increase
is likely caused by magnetosheath plasma penetration across the equatorial
magnetopause. Compared with intense and intermediate intense CSs
($n_{0}\approx 0.2\;cm^{-3}$)., low-intensity CSs seem to be denser
($n_{0}\approx 0.3\;-0.4\;cm^{-3}$).

Figure 9 h presents dawn-dusk profiles of ion temperature $T_{i}$.
intermediate-intense CSs are characterized by a steady growth of $T_{i}$, from
$0.8$keV at dawn flank to $1.3$ keV at dusk flank. For low-intensity CSs, a
similar growing trend is followed by a sudden drop closer to the dusk flank.
This drop is also observed for intense CSs, and seems to concur with a density
increase (see Fig. 9 g). The temperature decrease, and the corresponding
density growth closer to the flank, should be attributed to the entry of cold
magnetosheath plasma. Figure 9 i shows dawn-dusk distributions of ion to
electron temperature ratio $T_{i}/T_{e}$. For all types of CSs, we see a
similar pattern: $T_{i}/T_{e}$ increases from around $4-6\;$ at $Y\approx-15$
RE to $7$ at $Y=20$ RE. So far, we do not have a good explanation for the
observed dawn-dusk asymmetry of ion temperature and $T_{i}/T_{e}$ ratio in the
magnetotail at lunar distances, where $\partial B_{z}/\partial x\approx 0$
(see previous discussion about the effects of the diamagnetic drift).

Figure 9 j presents the occurrence rate of intense CSs, with
$J_{0}/J_{th}>0.8$. The highest relative number of such CSs is detected near
the midnight sector, where up to $20\%$ of observed CSs have current densities
comparable to $J_{th}$. This number gradually decreases to $5-10\%$ toward the
flanks.

Figure 9 k shows the dawn-dusk distribution of ion-electron drift velocity
$J_{0}/en_{0}$. Profiles of $J_{0}/en_{0}$ for low and intermediate intensity
CSs do not exhibit any noticeable dawn-dusk asymmetry and $J_{0}/en_{0}$ is
relatively constant across the tail ($40$ km/s for low-intensity CSs and $150$
km/s for moderately intense CSs). For intense CSs, the drift velocity profile
is slightly asymmetric and exhibits a local maximum at the dawn flank $Y=-20$
RE, reaching $700$ km/s. Similar asymmetry is observed for the dawn-dusk
profile of the tailward/earthward velocity $|V_{x}|$ collected in intense CSs’
crossing (see Fig.9 i). Indeed, Figure 9 l also shows that intense CSs are
usually observed within faster flows. In the case of low-intense CSs, the
maximum of $|V_{x}|$ is $\approx 100$ km/s, whereas for intense currents this
maximum, and $|V_{x}|$ profile itself, shows almost twofold increase to
$|V_{x}|\approx 200$ km/s. This result suggests that CS’s intensity is somehow
correlated with the ion flow speed [<]see discussion in¿Maha94,Zelenyi13:UFN.

Figure 9: Dawn-dusk distribution of current sheet characteristics in different
current density ranges, low-intensity $J_{0}/J_{th}<0.25$(green),
intermediate-intensity $0.25<J_{0}/J_{th}<0.8$(black), intense
$J_{0}/J_{th}>0.8$(blue), and the average profile for the entire data set
(dashed-red): magnitude of the current density $J_{0}$ (a), ion-beta
$\beta_{i}$ (b), current sheet thickness normalized to ion thermal gyroradius
$L/\rho_{i}$ (c), magnetic field strength at the lobes $B_{lobe}$ (d),
embedding parameter $B_{0}/B_{lobe}$ (e), force-free measure $\Delta
B_{m}^{2}/B_{0}^{2}$ (f), ion density $n_{i}$ (g), ion temperature $T_{i}$
(h), ion to electron temperature ratio $T_{i}/T_{e}$ (i), occurrence rate of
intense current sheets (j), ion-electron drift velocity $J_{0}/en_{0}$ (k),
tailward/earthward ion bulk velocity (l).

##  5 Relation between current sheet characteristics and solar wind parameters

Figure 10 compares yearly-distributions of averaged CSs’ characteristics and
solar wind parameters. Figure 10a shows annual occurrence rate of intense CSs,
which we define as the relative number of CSs with $J_{0}/J_{th}>0.8$ within a
particular year. Figure 10b presents annual occurrence rate of high values of
solar wind electric field $E_{sw,y}>0.05$ mV/m, where we define
$E_{sw,y}=\langle V_{sw,x}\times IMF\;B_{z}\rangle$ with the averaging
$\langle....\rangle$ over one full magnetotail crossing by the ARTEMIS mission
(a few days long time). Figure 10 c shows the occurrence rate of (partially)
force-free CSs, i.e., the relative number of CSs with $\Delta
B_{m}^{2}/B_{0}^{2}>0.2$ within a year. Finally, Figure 10 d presents the
occurrence rate of slow solar wind flows $V_{sw}<400$ km/s. Comparison between
Figures 10 a and 10 b highlights a similarity in occurrence rates of intense
CSs and strong magnetosphere driving. Both occurrence rates maximize in
2012—2013 and undergo a abrupt drop in 2014, which is then followed by a slow
increase until 2017. According to Figure 10 d, this abrupt drop in $E_{sw}$
can be attributed to the decrease in the average solar wind speed. The latter
also manifests itself in a higher occurrence rate of slow solar wind flows in
2014 (see Fig. 10 d). Interestingly, the drop in $E_{sw,y}$ coincides with the
start of the early declining phase of the 24th solar cycle (SC-24) and the
abrupt reduction of the geomagnetic activity [Rawat . (2018)]. The declining
phase of a solar cycle is usually characterized by a growing number of high-
speed streams ($V_{sw}\sim 700-800$ km/s) in the solar wind [Bame . (1976),
Gosling . (1976), Grandin . (2019)], which are thought to have a substantial
impact on the geomagnetic activity [Tsurutani . (2006)].

Recently, Grandin19HSS analyzed the yearly distribution of high-speed streams
during solar cycles 22-24 and found that the early declining phase of SC-24
was characterized by an abrupt drop in the number of high-speed streams. This
may help us to explain the apparent decrease of the average $V_{sw}$ in 2014
and the subsequent reduction of magnetosphere driving, $E_{sw,y}\propto
V_{sw}$. As the occurrence rate of $E_{sw,y}>0.05$ mV/m starts to increase in
2015 and 2016, this increase is accompanied by a similar rise in the relative
number of intense CSs in the magnetotail at lunar distances. Both occurrence
rates reach maxima during the late declining phase in 2017-2018 years. This
observation is not surprising and is, in fact, consistent with the idea that
the number of high-speed streams reaches its maximum just before the SC-24
minimum [Grandin . (2019)]. A pronounced decrease in the occurrence rate of
low solar wind flows in 2016 (see Fig. 10d) supports this explanation.
Overall, obtained distribution of $E_{sw,y}$ looks similar to the results from
Rawat2018SC23_24, in which two local maxima of SC-24 $E_{sw,y}$ were found
around 2013 and 2016.

A substantial number of observed CSs in our dataset can be characterized as
partially force-free magnetic field configurations. According to Figure 10c,
the yearly-distribution of CSs with $\Delta B_{m}^{2}/B_{0}^{2}>0.2$ varies
from $15\%$ to $35\%$. Comparing Figures 10c and 10d, we notice a weak but
rather persistent correlation between the occurrence rate of force-free CSs
and the occurrence rate of slow solar wind flows: both rates reach maximum
during the yearly declining phase of SC-24 in 2014-2015 and during the
transition from SC-24 to SC-25. The source of this correlation remains
unclear. One potential explanation could be that the ion temperature $T_{i}$
in the magnetotail at lunar distances correlates with the energy of solar wind
protons $m_{p}V_{sw}^{2}/2$, [Artemyev . (2017), Lukin . (2019)]. By invoking
this correlation $T_{i}\propto V_{sw}^{2}$, one can speculate that lower
values of the average $V_{sw}$ should result in statistically lower
$\beta_{i}\propto T_{i}$ in the distant magnetotail. This, in turn, should
favor formation of force-free configurations of the magnetic field at lunar
distances (see Fig. 6d).

Figure 10: Yearly-distribution of CS characteristics and solar wind
parameters: occurrence rate of intense current sheets $J_{0}/J_{th}>0.8$ (a),
occurrence rate of intense solar wind electric field $E_{sw}=\langle
V_{sw,x}*IMF\;B_{z}\rangle>0.05$ mV/m (b), occurrence rate of CSs with
significant contribution of the shear magnetic field component $\Delta
B_{m}^{2}/B_{0}^{2}>0.2$ (c), occurrence rate of slow solar wind $V_{sw}<400$
km/s (d).

##  6 Conclusions

In this study we have presented a statistical analysis of 1261 current sheet
crossings in the Earth’s magnetotail at distances $\sim 60\;R_{E}$. We use the
data provided by the Acceleration, Reconnection, Turbulence and
Electrodynamics of Moon’s Interaction with the Sun mission and combine them
with the technique proposed by Hoshino96,Sergeev98, thereby making it possible
to reliably estimate current density and thickness for the subset of 453
current sheets. Our main findings pertaining to the properties of these CSs
can be summarized as follows:

  1. 1.

Current densities are typically a few $\sim$ nA/m2, which is similar to the
results previously reported in the near-Earth magnetotail. However, we do
observe a fraction $\sim 23\%$ of intense current sheets with current density
$J_{0}>10$ nA/m2

  2. 2.

Most of the observed current sheets are relatively thin, with a typical
thickness $L\sim 10^{3}$ km that is usually about the ion thermal gyroradius
$\rho_{i}$. Statistically, the average thickness is approximately a few ion
gyroradii $L/\rho_{i}\sim 1-5$.

  3. 3.

The comparison between the ion-electron drift velocity $J_{0}/en_{0}$ and the
tailward/earthward flow velocity $V_{x}$ indicates that more intense current
sheets are observed within faster tailward/earthward flows. The distribution
of $J_{0}/en_{0}$ versus $V_{x}$ exhibits a small earthward asymmetry, which
might indicate that some fraction of observed currents is associated with
distant tail reconnection.

  4. 4.

Spatial profiles of the main plasma properties surrounding the current sheets
show that typical current sheets are weakly embedded with
$B_{0}/B_{lobe}\approx 0.8$, which is higher than $B_{0}/B_{lobe}\approx
0.3-0.5$ reported for the near-Earth magnetotail.

  5. 5.

As many as a quarter of the observed current sheets are characterized by a
significant variation in the shear magnetic field component $B_{m}$, with the
$B_{m}$ peak at the current sheet center, i.e., this population of current
sheets are partially force-free.

Thanks to the duration of the ARTEMIS mission, we were able to address the
question of how different solar wind conditions may influence the properties
of the magnetotail current sheet at the lunar distances. Here we report the
following findings

  1. 1.

The occurrence rate of intense current sheets ($J_{0}/J_{th}>0.8$) is
correlated with the occurrence rate of strong magnetosphere driving,
$E_{sw,y}=\langle V_{sw,x}*IMF\;B_{z}\rangle>0.05$ mV/m. By recalling the
established inverse relation between the current density and the thickness, we
argue that fast solar wind should drive the formation of thin current sheets.

  2. 2.

The occurrence rate of current sheets with significant variation of $B_{m}$
component (partially force-free current sheets) is correlated with the
occurrence rate of slow solar wind flows.

Previously published results have shown an abundance of ion-gyroscale (thin)
CSs in the near-Earth magnetotail. An important implication of this study is
the similar substantial population of thin current sheets observed in the
middle tail, extending as far as $X\sim-65$ RE. All of this poses a
significant challenge for the current CS models: how to explain the formation
of thin CS within the entirety of $x\in[-10,-65]$ RE domain? The most
developed class of theoretical CS models with the dawn-dusk current
$\propto\partial(nT_{i})/\partial x$ [<]e.g.,¿[and references
therein]SB02,Birn04 cannot explain the formation of such a CS with
$L/L_{x}\approx 1000$ km $/50$ R${}_{E}\ll B_{z}/B_{0}$ [<]see discussion
in¿Sitnov&Merkin16,Sitnov19:jgr,Artemyev19:jgr:ions,Artemyev21:grl. The
prospective approach for modeling such a long, thin CS includes incorporating
anisotropic and non-gyrotropic terms into the CS pressure balance [<]see
discussion in¿An22:currentsheet,Sitnov&Arnold22,ArnoldSitnov23. Therefore, our
results confirm the vital need for non-ideal ion physics in modeling of the
magnetotail CS.

###### Acknowledgements.

We acknowledge support by NASA awards 80NSSC20K1788, 80NSSC22K0752, and
80NSSC18K1122. We acknowledge NASA contract NAS5-02099 for use of data from
the THEMIS mission. We thank K. H. Glassmeier, U. Auster, and W. Baumjohann
for the use of FGM data provided under the lead of the Technical University of
Braunschweig and with financial support through the German Ministry for
Economy and Technology and the German Aerospace Center (DLR) under contract 50
OC 0302.

## Open Research

ARTEMIS data is available at http://themis.ssl.berkeley.edu. Data access and
processing was done using SPEDAS V4.1 Angelopoulos19 available at
https://spedas.org/.

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