# Kinetic-scale current sheets in the solar wind at 1 AU: Scale-dependent
properties and critical current density

Ivan Y. Vasko
Space Sciences Laboratory, University of California at Berkeley, California,
CA94720, USA
Space Research Institute of Russian Academy of Sciences, Moscow, 117997,
Russia
<EMAIL_ADDRESS>
Kazbek Alimov
Space Sciences Laboratory, University of California at Berkeley, California,
CA94720, USA
Tai Phan
Space Sciences Laboratory, University of California at Berkeley, California,
CA94720, USA
Stuart D. Bale
Space Sciences Laboratory, University of California at Berkeley, California,
CA94720, USA
Department of Physics, University of California at Berkeley, California,
CA94720, USA
Forrest S. Mozer
Space Sciences Laboratory, University of California at Berkeley, California,
CA94720, USA
Anton V. Artemyev
Institute of Geophysics and Planetary Sciences, University of California, Los
Angeles, California, CA 90095, USA
Space Research Institute of Russian Academy of Sciences, Moscow, 117997,
Russia

###### Abstract

We present analysis of 17,043 proton kinetic-scale current sheets collected
over 124 days of Wind spacecraft measurements in the solar wind at 11
Samples/s magnetic field resolution. The current sheets have thickness
$\lambda$ from a few tens to one thousand kilometers with typical value around
100 km or from about 0.1 to 10$\lambda_{p}$ in terms of local proton inertial
length $\lambda_{p}$. We found that the current density is larger for smaller
scale current sheets, $J_{0}\approx 6\;{\rm nA/m^{2}}\cdot(\lambda/100\;{\rm
km})^{-0.56}$ , but does not statistically exceed critical value $J_{A}$
corresponding to the drift between ions and electrons of local Alvén speed.
The observed trend holds in normalized units, $J_{0}/J_{A}\approx
0.17\cdot(\lambda/\lambda_{p})^{-0.51}$. The current sheets are statistically
force-free with magnetic shear angle correlated with current sheet spatial
scale, $\Delta\theta\approx 19^{\circ}\cdot(\lambda/\lambda_{p})^{0.5}$. The
observed correlations are consistent with local turbulence being the source of
proton kinetic-scale current sheets in the solar wind, while mechanisms
limiting the current density remain to be understood.

_K_ eywords solar wind, $\cdot$ turbulence $\cdot$ current sheets

##  1 Introduction

The understanding of turbulence dissipation and plasma heating in a weakly
collisionless plasma is of fundamental importance for numerous astrophysical
systems [1]. Numerical simulations showed that turbulence dissipation should
be spatially intermittent with substantial plasma heating localized in and
around coherent structures, such as current sheets, which occupy a relatively
small volume [2, 3, 4, 5, 6]. Numerical simulations also showed that magnetic
reconnection in kinetic-scale current sheets produced by turbulence not only
results in plasma heating, but also fundamentally affects development of the
turbulence cascade at sub-proton scales [7, 8, 9, 10]. Modern simulations are
still incapable of entirely reproducing the complex dynamics of realistic
three-dimensional plasma turbulence, and substantial effort has been directed
toward comparing simulation results with observations in the solar wind, a
weakly collisional plasma most accessible for in-situ measurements [11].

Spacecraft measurements showed that solar wind heating should continuously
occur within a few tens of solar radii of the Sun as well as further out in
the heliosphere and dissipation of turbulent magnetic field fluctuations
should be the dominant solar wind heating mechanism [12, 13, 14]. The solar
wind observations revealed turbulence to be dominated by Alfénic fluctuations
highly oblique ($k_{\perp}\gg k_{||}$) to local mean magnetic field,
Kolmogorov-like spectrum $E_{k_{\perp}}\propto k_{\perp}^{-5/3}$ at scales
larger than proton kinetic scales, and a steeper spectrum
$E_{k_{\perp}}\propto k_{\perp}^{-2.8}$ at scales smaller than proton kinetic
scales [15]. Numerical simulations have successfully reproduced these
properties of the solar wind turbulence [16, 17, 8, 18, 10]. The crucial
unresolved problem is the origin of coherent structures and, specifically,
current sheets observed in the solar wind, and the very ability of these
structures to heat solar wind plasma [19]. In this Letter we present a
statistical analysis of proton kinetic-scale current sheets (CS) in the solar
wind at 1 AU, contributing to understanding the origin and dissipation of
coherent structures in plasma turbulence.

The presence of CSs on a wide range of temporal scales was established by
early spacecraft measurements in the solar wind [20, 21, 22, 23]. These
measurements showed that the magnetic field has more or less constant
magnitude, but rotates across a CS through some shear angle. The typical CS
thickness was around ten proton inertial lengths, while the occurrence rate at
1 AU was a few tens per day. The multi-spacecraft analysis showed that the
magnetic field component normal to a CS surface is much smaller (if present at
all) than local magnetic field magnitude [24, 25]. The majority of CS studies
were limited by magnetic field measurements of relatively low resolution (a
few seconds at best) and typically by CSs with shear angles $\gtrsim
30^{\circ}$ [23, 26, 27]. Devoid of these shortcomings, the analysis by [28]
included more than 6,000 CSs collected at 1 AU using magnetic field
measurements at 1/3 s resolution. The typical thickness of the CSs was around
a few proton inertial lengths, and the occurrence rate was about a few hundred
CSs per day. Thus, the higher temporal resolution allowed resolving thinner
CSs, which turned out being much more abundant than larger-scale CSs reported
in the early studies. Based on the log-normal distribution of waiting times of
the CSs, [28] suggested they are produced by local turbulence, in accordance
with earlier theoretical hypothesis [29].

This hypothesis was further supported by observations of similar distributions
of magnetic field rotations and waiting times of coherent structures in the
solar wind and MHD turbulence simulations [30, 31, 32]. One of the
alternatives is that magnetic field rotations through angles $\gtrsim
30^{\circ}$ are boundaries between flux tubes originating at the Sun, while
smaller rotations are produced by local turbulence [33, 34]. However, this
interpretation can hardly explain the universal log-normal distribution of
magnetic field rotations at various temporal scales [32, 35]. The ability of
coherent structures in heating of solar wind plasma was questioned by [36],
but there is currently a growing evidence that plasma heating does occur
around coherent structures in plasma turbulence [37, 38, 39, 40].

In this Letter we present the most extensive dataset of proton kinetic-scale
CSs collected at 1 AU using magnetic field measurements at 1/11 s resolution.
Due to the higher resolution, the CSs with thickness from a few to ten times
smaller than in the previous studies could be resolved. We reveal distinct
scale-dependencies of the current density and shear angle, as well as the
critical current density that is not exceeded statistically. The results
indicate that proton kinetic-scale CSs in the solar wind are indeed produced
by turbulence cascade, and advance understanding of turbulence dissipation.

##  2 Data and methodology

We use measurements of the Wind spacecraft that is located at the L1
Lagrangian point, about $200$ Earth radii from the Earth [41]. We use
continuous magnetic field measurements at $1/11$ s resolution provided by the
Magnetic Field Instrument [42], and proton densities and flow velocities at 3s
cadence provided by Wind 3DP instrument [43]. Note that ion densities coincide
within a few tens of percent with electron densities provided at about 9s
cadence by Solar Wind Experiment instrument [44]. We consider a period of 124
days, from October 1, 2010 to February 2, 2011, except the first 16 hours on
December 7, 2010, when magnetic field measurements at $1/11$ s resolution were
not available.

CSs were selected using the Partial Variance Increment (PVI) method [45]. We
computed PVI index, ${\rm PVI}=\left(\sum_{\alpha}\Delta
B_{\alpha}^{2}(t,\tau)/\sigma_{\alpha}^{2}\right)^{1/2}$, where $\Delta
B_{\alpha}(t,\tau)=B_{\alpha}(t+\tau)-B_{\alpha}(t)$ are magnetic field
increments of three magnetic field components ($\alpha=X,Y,Z$) and
$\sigma_{\alpha}$ are standard deviations of $\Delta{B}_{\alpha}(t,\tau)$
computed over 2h intervals, that is over a few outer correlation scales of the
solar wind turbulence [46]. Coherent structures at various temporal scales
correspond to non-Gaussian fluctuations with, for example, PVI$>$5\. We used
only PVI index computed at $\tau=1/11$ s to focus on the thinnest resolvable
coherent structures. This methodology is essentially equivalent to that of
[47], who used current density estimates $J\propto|\Delta{\bf
B}(t,\tau)|/\tau$ with $\tau=1/11$ s to identify the most intense currents in
the solar wind. The analysis of synthetic magnetic field signals with spectra
typical of the solar wind showed that $J\propto|\Delta{\bf B}(t,\tau)|/\tau$
with $\tau=1/11$ s provides reasonable current density estimates [47]. Note
that the instrument noise does not affect the current density estimates and
selection of coherent structures in our study, because the magnetometer noise
level at frequencies of 0–10 Hz is less than 0.006 nT [42], which is at least
six times smaller than the standard deviation,
$\sigma=\left(\sum_{\alpha}\sigma_{\alpha}^{2}\right)^{1/2}$, characterizing
magnetic field increments $\Delta{\bf B}(t,\tau)$ at $\tau=1/11$ s in our
interval.

There are different types of coherent structures among the non-Gaussian
fluctuations [48]. To select CSs, each of about $2\cdot 10^{5}$ continuous
clusters of points with PVI$>$5 was considered over intervals of $\pm 1$s,
$\pm 2$s, $\pm 3$s and $\pm 4$s around its center and unit vector
$\textbf{\emph{x}}^{\prime}$ along the direction of the magnetic field
component with the largest variation was computed for each interval using the
Maximum Variance Analysis method [49]. We visually inspected all profiles of
${\bf B}\cdot\textbf{\emph{x}}^{\prime}$ and selected clusters of points with
${\bf B}\cdot\textbf{\emph{x}}^{\prime}$ reversing the sign within at least
one of the intervals. We then manually adjusted the boundaries, that is the
regions to the left and to the right of ${\bf
B}\cdot\textbf{\emph{x}}^{\prime}$ reversal, so that each boundary has a
duration of at least half a second, and excluded events with substantial
relative variations of the magnetic field at the boundaries. We also excluded
about 10% of events with bifurcated magnetic field profiles [50], to focus on
CSs with relatively smooth magnetic field rotation.

The final dataset includes 17,043 CSs, which is the most extensive dataset of
proton kinetic-scale CSs at 1 AU. Since the CSs were selected using only PVI
index with $\tau=1/11$ s, the distributions of CS parameters presented in the
next section are biased toward the thinnest resolvable CSs, but this bias does
not affect the key conclusions of this study (Section 4). Each CS will be
considered in local coordinate system xyz most suitable for describing local
CS structure [25, 51, 52]: unit vector z is along the CS normal determined by
the cross-product of magnetic fields at the CS boundaries; unit vector x is
along
$\textbf{\emph{x}}^{\prime}-\textbf{\emph{z}}\cdot(\textbf{\emph{x}}^{\prime}\cdot\textbf{\emph{z}})$;
unit vector y completes the right-handed coordinate system,
$\textbf{\emph{y}}=\textbf{\emph{z}}\times\textbf{\emph{x}}$. Note that
vectors x and y determine the local CS surface, while vector z is directed
across the CS surface.

Figure 1 presents a CS observed on February 1, 2011. Panel (a) presents the
magnetic field magnitude and three components in the Geocentric Solar Ecliptic
coordinate system. The magnetic field rotates across the CS through shear
angle $\Delta\theta\approx 60^{\circ}$ (the angle between magnetic fields at
the CS boundaries), while the magnetic field magnitude remains more or less
constant. We characterize the magnetic field magnitude by $\langle B\rangle$
that is the mean value of magnetic field magnitudes at the CS boundaries, and
$\Delta B_{max}$ that is the difference of maximum and minimum values of the
magnetic field magnitude within the CS. For the considered CS we have $\langle
B\rangle\approx 10$ nT and $\Delta B_{max}\approx 1.4$ nT. Panel (b) presents
three magnetic field components in local CS coordinate system xyz. The $B_{x}$
component varies from about $5$ to $-5$ nT across the CS, $B_{y}$ has similar
values at the CS boundaries and a bit larger value around the $B_{x}$
reversal. The normal component $B_{z}$ is around zero at the CS boundaries in
accordance with definition of the CS normal and remains small within the CS.
We characterize the CS asymmetry by $\langle B_{x}\rangle$ that is the mean of
$B_{x}$ values at the CS boundaries, and the CS amplitude by the absolute
value of their difference denoted as $\Delta B_{x}$. The considered CS is
rather symmetric with $\langle B_{x}\rangle\approx-0.04$ nT and $\Delta
B_{x}\approx 10.5$ nT. The CS central region highlighted in panel (b)
corresponds to $|B_{x}-\langle B_{x}\rangle|<0.2\;\Delta B_{x}$

Panel (c) presents current densities $J_{x}$ and $J_{y}$ estimated as follows

$\displaystyle
J_{y}=-{\color[rgb]{0,0,0}\frac{1}{\mu_{0}V_{n}}}\frac{dB_{x}}{dt},\;\;\;J_{x}={\color[rgb]{0,0,0}\frac{1}{\mu_{0}V_{n}}}\frac{dB_{y}}{dt}$
(1)

where $\mu_{0}$ is the vacuum permeability and $V_{n}$ is the normal component
of proton flow velocity at the moment closest to the CS. In this estimate we
took into account that solar wind CSs are locally planar structures [25, 27],
so that only the normal component of the proton flow velocity matters, and the
Taylor frozen-in hypothesis is valid at spatial scales larger than electron
kinetic scales [53]. The spatial coordinate along the normal, $z=-V_{n}t$ with
$t=0$ corresponding to $B_{x}=\langle B_{x}\rangle$, is shown in Figure 1. We
estimate the CS thickness as follows

$\displaystyle\lambda={\color[rgb]{0,0,0}\frac{\Delta B_{x}}{2\mu_{0}\langle
J_{y}\rangle}},$ (2)

where $\langle J_{y}\rangle$ is the absolute value of the current density
$J_{y}$ averaged over the CS central region, $|B_{x}-\langle
B_{x}\rangle|<0.2\;\Delta B_{x}$, highlighted in Figure 1. For the considered
CS we have $\lambda\approx 80$ km or about 1.5$\lambda_{p}$, where
$\lambda_{p}$ is proton inertial length. We refer to $\lambda$ as CS
thickness, but underline that strictly-speaking $\lambda$ is a half-thickness,
because according to Eq. (2) the magnetic field and current density profiles
can be approximated as $B_{x}\approx\langle B_{x}\rangle+0.5\Delta
B_{x}\tanh(z/\lambda)$ and $J_{y}\approx\langle J_{y}\rangle{\rm
sech}^{2}(z/\lambda)$.

Panel (d) presents current densities parallel and perpendicular to local
magnetic field computed as $J_{||}=(J_{x}B_{x}+J_{y}B_{y})/B$ and
$J_{\perp}=(J_{y}B_{x}-J_{x}B_{y})/B$. Since the perpendicular current density
is statistically much smaller than the parallel current density (Section 4),
we quantify the CS intensity by $J_{0}$ that is the absolute value of parallel
current density $J_{||}$ averaged over the CS central region and $J_{peak}$
that is the absolute peak value of parallel current density $J_{||}$ within
the CS. For the considered CS we have $J_{0}\approx 52$ nA/m2 and
$J_{peak}\approx 68$ nA/m2. We compare the CS intensities to local Alfvén
current density $J_{A}$ and defined as the current density corresponding to
the drift between protons and electrons of local Alfén speed $V_{A}$

$\displaystyle J_{A}=eN_{p}V_{A},\;\;\;V_{A}=\frac{\langle
B\rangle}{\left({\color[rgb]{0,0,0}\mu_{0}}N_{p}m_{p}\right)^{1/2}}$ (3)

where $e$ and $m_{p}$ are proton charge and mass, $N_{p}$ is the closest
measurement of proton density. For the considered CS, proton density was
around 21 cm-3 and we obtain $J_{0}\approx 0.3\;J_{A}$ and $J_{peak}\approx
0.4\;J_{A}$. In the next section we present results of the statistical
analysis performed using the described methodology.

##  3 Statistical Results

Figure 2 presents statistical distributions of various CS parameters. Panel
(a) shows that the magnetic field magnitude across the CSs is almost constant,
$\Delta B_{max}$ is less than $0.15\langle B\rangle$ for more than 95% of the
CSs, that is in accordance with the previous studies of larger-scale CSs [20,
28, 27]. The CSs are typically asymmetric, that is the values of $B_{x}$ at
the CS boundaries are different, only about 50% of the CSs satisfy $\langle
B_{x}\rangle\lesssim 0.1\;\Delta B_{x}$. Panel (b) shows that the current
densities $J_{0}$ are within 15 nA/m2 for more than 95% of the CSs. The most
probable value of the current densities $J_{peak}$ is around 5 nA/m2 that is
in accordance with current density estimates in the solar wind by [47]. Panel
(c) shows that the thickness of the CSs is in the range between about 20 and
1000 km with the most probable value around 100 km. Note that resolving the
CSs with thickness of a few tens of kilometers was possible due to relatively
small normal component $V_{n}$ of the proton flow velocity for those CSs.

Figure 3 shows that the CSs are proton kinetic-scale structures with some
fraction on sub-proton scales. Panel (a) demonstrates that the CS thickness
$\lambda$ is statistically larger in solar wind plasma with larger proton
inertial length $\lambda_{p}$. Similar trend is observed between the CS
thickness and thermal proton gyroradius $\rho_{p}=\lambda_{p}\beta_{p}^{1/2}$,
because proton beta $\beta_{p}$ is in the range between 0.4 and 2 for more
than $80\%$ of the CSs in our dataset as well as in the solar wind in general
[54]. Panel (b) presents a statistical distribution of $\lambda/\lambda_{p}$
and shows that the thickness of the CSs is in the range between about 0.1 and
10$\lambda_{p}$ with the most probable value around $\lambda_{p}$. Similar
distribution of $\lambda/\rho_{p}$ shows that the most probable value of the
CS thickness is around $\rho_{p}$. Thus, the typical thickness of the CSs is
from a few to ten times smaller than of those in the previous statistical
studies [20, 28, 27]. The resolution of these kinetic-scale CSs became
possible due to higher temporal resolution of magnetic field measurements
aboard Wind spacecraft.

Figure 4 presents analysis of the CS current density and its dependence on the
CS thickness. Panel (a) shows the scatter plot of $J_{0}$ versus $\lambda$. We
sorted the CSs into bins corresponding to different spatial scales and
computed the median current density value within each bin. Panel (a) shows the
median current density profile along with error bars corresponding to 15th and
85th percentiles of the current density within each bin. The number of the CSs
within each bin is shown at the bottom of panel (a). The median profile shows
that the CSs with smaller thickness tend to be more intense. The least squares
fitting of all the scattered data by a power-law function reveals the
following best fit

$\displaystyle J_{0}=6\;{\rm nA\;m^{-2}}\cdot\left(\frac{\lambda}{100\;{\rm
km}}\right)^{-0.56},$ (4)

which is shown in panel (a). The fitting of the 15th, 50th and 85th percentile
profiles by power law functions reveals the slopes between $-0.44$ and $-0.6$,
indicating thereby that the uncertainty of the best fit slope in Eq. (4) is
less than 20%. Panel (b) presents the comparison between the peak current
densities $J_{peak}$ and local Alfvén current densities $J_{A}$. First, the
CSs tend to be more intense in solar wind plasma with larger Alfvén current
density. Second, the peak current densities are statistically below local
Alfvén current density, $J_{peak}<J_{A}$ for more than 99% of the CSs and
$J_{peak}<J_{A}/2$ for 97% of the CSs.

Panel (c) shows there is a positive correlation between local Alfvén speed
$V_{A}$ and $J_{peak}/eN_{p}$ that is the peak value of the drift velocity
between ions and electrons. Thus, the positive correlation between $J_{peak}$
and $J_{A}$ in panel (b) is not a trivial effect of plasma density variation.
The physics behind the correlation of the two seemingly unrelated quantities
in panel (c) will be discussed in the next section. The trends and
correlations similar to those in panels (b) and (c) are also observed for the
averaged current densities $J_{0}$. It is noteworthy there is no any
correlation between $J_{peak}/eN_{p}$ or $J_{0}/eN_{p}$ and local ion-acoustic
speed $(T_{e}/m_{p})^{1/2}$, where $T_{e}$ is local electron temperature. The
ion-acoustic speed in our dataset is in the range from 40 to 60 km/s for more
than 95% of the CSs, while the drift velocities $J_{peak}/eN_{p}$ vary over
almost two orders of magnitude.

Figure 5 reveals remarkable scale-dependencies of normalized intensity,
normalized amplitude and shear angle of the CSs. To demonstrate the scale
dependence of a specific quantity, we sorted the CSs into bins corresponding
to different spatial scales, and computed the median as well as 15th and 85th
percentile values of the quantity within each bin. The number of the CSs
within each bin is shown at the bottom panels. Panel (a) presents the scatter
plot of $J_{0}/J_{A}$ versus $\lambda/\lambda_{p}$. Similarly to the trend
given by Eq.(4) in physical units, the median profile shows that the CSs with
smaller normalized thickness have larger normalized current densities. The
least squares fitting of all the scattered data by a power-law function
reveals the following best fit

$\displaystyle
J_{0}/J_{A}=0.17\;\cdot\left(\lambda/\lambda_{p}\right)^{-0.51},$ (5)

which is shown in panel (a) and rather well describes the median profile. The
fitting of the 15th, 50th and 85th percentile profiles by power law functions
reveals the slopes in the range from $-0.45$ to $-0.55$, so that the
uncertainty of the best fit slope in Eq. (5) is within about 10%. Panel (b)
presents the scatter plot of $\Delta B_{x}/\langle B\rangle$ versus
$\lambda/\lambda_{p}$. The median profile reveals a clear scale-dependence,
the CSs with larger normalized thickness have larger normalized amplitudes.
The least squares fitting of all the scattered data by a power-law function
reveals the following best fit

$\displaystyle\Delta B_{x}/\langle
B\rangle=0.33\;\cdot\left(\lambda/\lambda_{p}\right)^{0.49},$ (6)

which again well describes the median profile. Note that this scaling relation
could be foreseen based on Eq. (5), because $J_{0}\approx\Delta
B_{x}/2\mu_{0}\lambda$ and $J_{A}=eN_{p}V_{A}=\langle
B\rangle/\mu_{0}\lambda_{p}$. The fitting of the 15th, 50th and 85th
percentile profiles reveals the slopes in the range from $0.4$ to $0.5$, so
that the uncertainty of the best fit slope in Eq. (6) is within 20%. Since in
a CS with more or less constant magnetic field magnitude, $\Delta
B_{x}/\langle B\rangle$ is unambiguously related to the shear angle, we expect
to observe a positive correlation and similar trend between $\Delta\theta$ and
$\lambda/\lambda_{p}$. Panel (c) confirms the scale-dependence of the shear
angle with the following best fit of the scattered data

$\displaystyle\Delta\theta=0.33\;\cdot\left(\lambda/\lambda_{p}\right)^{0.5}\approx
19^{\circ}\;\cdot\left(\lambda/\lambda_{p}\right)^{0.5},$ (7)

which well describes the median profile. The uncertainty of the best fit slope
in Eq. (7) is again within 20%. There is a scale-dependent upper threshold on
the shear angles, $\Delta\theta\lesssim 2\;\lambda/\lambda_{p}$ for more than
99% of the CSs and $\Delta\theta\lesssim\lambda/\lambda_{p}$ for about 97% of
the CSs.

##  4 Theoretical interpretation and discussion

We presented the analysis of proton kinetic-scale CSs in the solar wind based
on the most extensive dataset collected at 1 AU. Certainly, this dataset does
not include all CSs present in the considered interval. We selected the CSs
using the PVI index with the minimum time increment $\tau=1/11$ s and,
therefore, our dataset and statistical distributions in Figures 2 and 3 are
biased toward the thinnest resolvable CSs. The use of PVI indexes at larger
increments would be essentially equivalent to the use of magnetic field
measurements at a lower resolution. Therefore, the previous studies based on a
lower resolution magnetic field may demonstrate the results of using larger
increments in our procedure. The studies based on magnetic field measurements
at a few second resolution resolved CSs with typical thickness around 1000 km
[20, 22, 23, 27], while the use of magnetic field measurements at 1/3 s
resolution allowed resolving CSs with typical thickness around a few hundred
kilometers and these CSs turned out to be one order of magnitude more abundant
than CSs resolved at a few second resolution [28]. We used magnetic field
measurements at 1/11 s resolution and resolved CSs with typical thickness
around 100 km and averaged occurrence rate ($17,043/124\approx 140$ CSs/day)
comparable with that reported by [28]. Thus, the higher resolution of magnetic
field measurements allowed us to collect truly kinetic-scale CSs, which are
thinner than those reported in the previous studies. Although our dataset does
not include CSs of all possible spatial scales, which collection would require
repeating our selection procedure with different increments $\tau$, this
dataset allows us to address the structure and origin of kinetic-scale CSs in
the solar wind.

###  4.1 Local structure of solar wind current sheets

Since solar wind CSs are locally planar structures [25, 27], we can describe
the local magnetic field of the CSs as follows

$\displaystyle{\bf
B}=B(z)\sin\theta(z)\;\textbf{\emph{x}}+B(z)\cos\theta(z)\;\textbf{\emph{y}}+B_{z}\;\textbf{\emph{z}},$
(8)

where $\theta(z)$ and $B(z)$ determine respectively magnetic field rotation
and magnetic variation within CS, the normal component $B_{z}$ is much smaller
than $B(z)$. The shear angle $\Delta\theta$ estimated in our experimental
analysis corresponds to the difference of $\theta$ values at the CS
boundaries. Note that Eq. (8) provides the most general description of a CS
with non-zero $B_{y}$. The specific models, such as Harris CS with a constant
$B_{y}$ or a force-free Harris CS, often used in reconnection simulations and
theoretical studies [55] are special cases of Eq. (8) corresponding to
particular profiles of $B(z)$ and $\theta(z)$.

The simple calculations show that current densities parallel and perpendicular
to local magnetic field,
$J_{||}=(J_{x}B_{x}+J_{y}B_{y})/{\color[rgb]{0,0,0}B}$ and
$J_{\perp}=(J_{y}B_{x}-J_{x}B_{y})/{\color[rgb]{0,0,0}B}$, determine
respectively magnetic field rotation and magnitude variation within CS

$\displaystyle
J_{||}=\frac{B}{{\color[rgb]{0,0,0}\mu_{0}}}\frac{d\theta}{dz},\;\;\;\;\;\;J_{\perp}=\frac{1}{{\color[rgb]{0,0,0}\mu_{0}}}\frac{dB}{dz}$
(9)

The amplitudes of parallel and perpendicular current densities can be then
estimated as $J_{||}\approx(c/4\pi)\langle B\rangle\Delta\theta/2\lambda$ and
$J_{\perp}\approx(c/4\pi)\Delta B_{max}/2\lambda$. First, these estimates show
that the current density limitation $J_{||}\lesssim J_{A}$ (Figures 4b) is
equivalent to the scale-dependent upper threshold on shear angles,
$\Delta\theta\lesssim 2\lambda/\lambda_{p}$ (Figure 5c). Second, they show
that the relatively small variations of the magnetic field magnitude within
the CSs (Figure 2a) imply that the current density is dominated by the
parallel component. The ratio $J_{\perp}/J_{||}\approx\Delta B_{max}/\langle
B\rangle\Delta\theta$ is less than 0.3 for more than 95% of the CSs (not
shown). The small variations of the magnetic field magnitude also imply that
the plasma pressure variation across the CSs is much smaller than the magnetic
field pressure. The pressure balance $p+B^{2}/8\pi={\rm const}$ shows that the
maximum variation of the plasma pressure across CS is $\Delta p_{\rm
max}\approx\langle B\rangle\Delta B_{max}/4\pi$, while its ratio to the
typical magnetic field pressure is $\Delta p/p_{B}\approx 2\Delta
B_{max}/\langle B\rangle$. According to Figure 2a, this ratio does not exceed
0.3 for more than 95% of the CSs. The dominance of the parallel current
density and relatively small variations of the plasma pressure show that the
kinetic-scale CSs in the solar wind are statistically more or less force-free.

###  4.2 Current sheets and turbulence

The turbulence in the solar wind is dominated by magnetic field fluctuations
highly oblique to mean magnetic field, with a power law spectrum
$E_{k_{\perp}}\propto k_{\perp}^{-\nu}$, where $\nu\approx 5/3$ in the
inertial range, $k_{\perp}\lambda_{p}\lesssim 1$, and $\nu\approx 2.8$ at
$k_{\perp}\lambda_{p}\gtrsim 1$ [15]. The root-mean-square amplitude $\delta
b_{\lambda}$ of turbulent fluctuations at spatial scale $\lambda$ can be then
estimated as $\delta b_{\lambda}\propto(E_{k_{\perp}}\Delta
k_{\perp})^{1/2}\propto\lambda^{(\nu-1)/2}$, where we took into account that
$\Delta k_{\perp}\propto k_{\perp}\propto\lambda^{-1}$. The corresponding
current density is $j_{\lambda}\propto\delta
b_{\lambda}/\lambda\propto\lambda^{(\nu-3)/2}$. In the inertial range,
$k_{\perp}\lambda_{p}\lesssim 1$, we expect

$\displaystyle\delta
b_{\lambda}\propto\lambda^{1/3},\;\;\;j_{\lambda}\propto\lambda^{-2/3}$ (10)

while at $k_{\perp}\lambda_{p}\gtrsim 1$ we expect

$\displaystyle\delta
b_{\lambda}\propto\lambda^{{\color[rgb]{0,0,0}0.9}},\;\;\;j_{\lambda}\propto\lambda^{-{\color[rgb]{0,0,0}0.1}}$
(11)

Thus, the root-mean-square amplitudes of turbulent magnetic field and current
density fluctuations should be scale-dependent [56, 16]. Since magnetic shear
angle $\delta\theta_{\lambda}$ is proportional to $\delta b_{\lambda}$, it
should be similarly scale-dependent.

We have found that amplitudes, current densities and shear angles of proton
kinetic-scale CSs are scale-dependent in a fashion similar to that of
turbulent fluctuations. Moreover, the CS amplitude scales with the CS
thickness as $\Delta B_{x}/\langle B\rangle\propto(\lambda/\lambda)^{0.49}$,
so that the power law index is between $1/3$ and $0.9$ expected for turbulent
fluctuations at scales above and below proton kinetic scales. Similarly, the
CS intensity scales with the CS thickness as
$J_{0}/J_{A}\propto(\lambda/\lambda_{p})^{-0.51}$ with the power law index
between $-2/3$ and $-0.1$. These scale-dependencies are strong indications
that proton kinetic-scale CSs are produced locally by plasma turbulence. This
conclusion is not affected by 20% uncertainty of the slopes characterizing the
scale-dependent properties of the CSs.

The critical question is whether the scale-dependencies revealed in Figure 5
would be affected if we included all kinetic-scale CSs present in the
considered interval, rather than used the dataset biased toward the thinnest
resolvable CSs. The strong indication that the revealed scale-dependencies
would not be affected is that they are consistent with the scale-dependent
properties of magnetic field rotations in the solar wind [32, 35]. These
studies showed that statistical distributions of angles $\alpha(\tau)$ between
magnetic fields ${\bf B}(t)$ and ${\bf B}(t+\tau)$ in the solar wind behave
similarly at various temporal scales. More precisely, $\alpha(\tau)$
normalized to its mean value $\langle\alpha\rangle$ is described by a
universal log-normal distribution independent of $\tau$. In turn, the mean
value scales with $\tau$ as $\langle\alpha\rangle\propto\tau^{0.46}$, so that
magnetic field rotation angles are larger at larger spatial scales (see also
[57]). This is qualitatively and even quantitatively consistent with the
scale-dependence of magnetic shear angles across the CSs,
$\Delta\theta\propto(\lambda/\lambda_{p})^{0.5}$, revealed in our analysis.
Thus, we believe that inclusion of kinetic-scale CSs, which could be selected
using PVI indexes at larger increments, would not affect our conclusion that
kinetic-scale CSs are produced by turbulence cascade.

###  4.3 Current density limitation mechanisms

We have found that the peak value of the current density within CS is
correlated with local Alfvén current density $J_{A}$, while the drift velocity
between ions and electrons is correlated with local Alfvén speed $V_{A}$. In
principle, the positive correlation between these quantities could be
expected, because $J_{A}$ and $V_{A}$ are natural units of the current density
and velocities in MHD and Hall-MHD turbulence simulations [10]. The intriguing
property though is that the peak value $J_{peak}$ of the current density does
not statistically exceed $J_{A}$ and, actually, even $J_{A}/2$. There are
several scenarios capable of explaining the observed current density
limitation. The first scenario is that once the parallel current density
exceeds local Alfvén current some instability may lead to current density
limitation or a CS destruction. One of the known instabilities is the so-
called Alfén instability that was considered by [58] and [59] for a force-free
CS at low plasma betas. The relevance of this instability to solar wind CSs at
realistic plasma betas requires a separate analysis. The ion-acoustic
instability is highly unlikely to be relevant to current density limitation,
because we have found no correlation between the ion-electron drift velocity
$J_{peak}/eN_{p}$ and local ion-acoustic speed (see also [Boldyrev15:apj]).
The second scenario is that once the ion-electron drift velocity becomes
comparable with local Alfvén speed, more electrons can be in the Landau
resonance with ambient turbulence [60], which potentially results in electron
scattering and current density limitation.

###  4.4 Our dataset and assumptions of the single-spacecraft analysis

Wind spacecraft allowed us to select the most extensive dataset of proton
kinetic-scale CSs in the solar wind not disturbed by the Earth’s bow shock. In
our single-spacecraft analysis CS normals were determined by the cross-product
of magnetic fields at the CS boundaries, which works relatively well according
to the multi-spacecraft study by [25]. Multi-spacecraft studies would
certainly allow more accurate estimates of CS normals, but the available
multi-spacecraft missions often probe the solar wind disturbed by the Earth’s
bow shock and require careful data selection.

We repeated the analysis presented in this Letter for the current densities
and thicknesses estimated via Eqs. (1) and (2), but using proton flow velocity
magnitude in the yz plane rather than along CS normals z. The so-obtained
quantities represent lower estimates of the current densities and upper
estimates of the CS thicknesses, which are independent of the exact knowledge
of CS normals. We found similar scale-dependencies with only slightly
different fitting parameters, $J_{0}\approx 6\;{\rm
nA\;m^{-2}}\cdot(\lambda/100\;{\rm km})^{-0.49}$, $J_{0}/J_{A}\approx
0.14\cdot(\lambda/\lambda_{p})^{-0.4}$, $\Delta B_{x}/\langle B\rangle\approx
0.28\cdot(\lambda/\lambda_{p})^{0.6}$, and $\Delta\theta\approx
15^{\circ}\cdot(\lambda/\lambda_{p})^{0.62}$. Thus, we feel confident that the
results of our single-spacecraft analysis reflect realistic properties of
proton kinetic-scale CSs in the solar wind.

##  5 Conclusions

The results of this Letter can be summarized as follows

  1. 1.

The proton kinetic-scale CSs in the solar wind are statistically force-free
and typically asymmetric.

  2. 2.

The current density within the CSs is scale-dependent with CSs of smaller
thickness being more intense. The magnetic field amplitude and magnetic shear
angle are scale-dependent as well.

  3. 3.

The drift velocity between electrons and ions in the CSs tends to be larger in
the solar wind with larger Alfvén speed. The current density does not
statistically exceed local Alfvén current density.

Based on these observations we argue that proton kinetic-scale CSs in the
solar wind are produced locally by turbulence, and some mechanism, either CS
instability or scattering of electrons by ambient turbulence, should keep the
current density below local Alfvén current density.

Acknowledgments: The work of I.V. was supported by the Russian Science
Foundation, grant No. 21-12-00416. The work of T.P. was supported by NASA
Living With a Star grant #80NSSC20K1781. The work of A.A. was supported by
NASA Living With a Star grant #80NSSC20K1788. The work of K.A. was supported
by National Science Foundation grant No. 2026680. I.V. thanks Ajay Lotekar and
Rachel Wang for discussions. The data used in the analysis are publicly
available at https://cdaweb.gsfc.nasa.gov/pub/data/wind/.

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Figure 1: An example of a current sheet observed aboard Wind spacecraft on
February 1, 2011 around 05:22 UT: (a) magnetic field magnitude and three
magnetic field components measured at 1/11 s resolution (the Geocentric Solar
Ecliptic coordinates); (b) three magnetic field components in the local
coordinate system xyz defined in Section 2; (c) current densities $J_{x}$ and
$J_{y}$ computed using Eqs.(1); (d) current densities parallel and
perpendicular to local magnetic field computed as
$J_{||}=(J_{x}B_{x}+J_{y}B_{y})/B$ and $J_{\perp}=(J_{y}B_{x}-J_{x}B_{y})/B$.
The central region of the CS, $|\;B_{x}-\langle B_{x}\rangle\;|<0.2\;\Delta
B_{x}$, is indicated in panels (b)–(d), where $\langle B_{x}\rangle$ is the
mean of $B_{x}$ values at the CS boundaries, and $\Delta B_{x}$ is the
absolute value of their difference. The bottom horizontal axis presents the
spatial coordinate across the CS, $z=-V_{n}t$, where $t=0$ corresponds to
$B_{x}=\langle B_{x}\rangle$ and $V_{n}$ is the normal component of proton
flow velocity measured at the moment closest to the CS. Figure 2: Statistical
distributions of various parameters of 17,043 solar wind CSs: (a) $\Delta
B_{max}/\langle B\rangle$, the maximum relative variation of the magnetic
field magnitude within CS, and parameter $\langle B_{x}\rangle/\Delta B_{x}$
characterizing the CS asymmetry, where $\langle B_{x}\rangle$ is the mean of
$B_{x}$ values at the CS boundaries, and $\Delta B_{x}$ is the absolute value
of their difference; (b) absolute value $J_{0}$ of parallel current density
$J_{||}$ averaged over the CS central region (see, e.g., Figure 1) and peak
value $J_{peak}$ of parallel current density $J_{||}$ within CS; (c) CS
thickness $\lambda$. The bottom panels present cumulative distribution
functions (CDF) of the statistical distributions in the upper panels. Figure
3: Panel (a) presents a scatter plot of the CS thickness $\lambda$ versus
local proton inertial length $\lambda_{p}$. The green curve shows the trend
that is obtained by sorting the CSs into bins corresponding to various spatial
scales and computing the averaged thickness of the CSs within each bin. The
number of CSs within each bin is shown in the bottom panel. The black dashed
line represents $\lambda=\lambda_{p}$ for reference. Panel (b) presents the
statistical distribution of the CS thickness in units of proton inertial
length. The bottom panel presents the corresponding cumulative distribution
function (CDF). Figure 4: The scatter plots of (a) averaged current density
$J_{0}$ versus CS thickness $\lambda$; (b) peak current density $J_{peak}$
versus local Alfén current density $J_{A}=eN_{p}V_{A}$, where $e$ is the
proton charge, $N_{p}$ is proton density, $V_{A}$ is local Alfvén speed; (c)
the peak value of the drift velocity between ions and electrons
$J_{peak}/eN_{p}$ versus local Alfvén speed $V_{A}$. For each of the scatter
plots, we sorted the CSs into bins corresponding to different values of the
quantity on the $x-$axis and computed median values of the quantities on the
$y-$axis for the CSs within each bin. The CSs were sorted in such a way that
each bin contains a sufficiently large (more than 100) number of CSs and the
number of CSs within each bin is presented in the bottom panels (only the last
bin in panel (a) contains less than 100 CSs). The corresponding median
profiles are presented by green curves in panels (a)–(c), while error bars
correspond to 15th and 85th percentiles of the quantities within each bin. The
red line in panel (a) presents the best fit of the scattered data by a power
law function and the best fit parameters are indicated in the panel. Figure 5:
The scatter plots (a) the current density normalized to Alfvén current density
$J_{0}/J_{A}$ versus normalized CS thickness $\lambda/\lambda_{p}$; (b) the CS
amplitude normalized to the mean magnetic field magnitude $\Delta
B_{x}/\langle B\rangle$ versus $\lambda/\lambda_{p}$; (c) magnetic shear angle
$\Delta\theta$ versus $\lambda/\lambda_{p}$. The CSs were sorted into bins
corresponding to different values of normalized CS thickness and median values
of the quantities in panels (a)–(c) were computed within each bin. The CSs
were sorted in such a way that each bin contains a sufficiently large (more
than 100) number of CSs within each bin and the number of CSs within each bin
is presented in panel (d) (only the last bin contains less than 100 CSs).
Panels (a)–(c) present the median profiles (green curves) along with error
bars corresponding to 15th and 85th percentiles within each bin. The panels
also indicate the best power law fits of the scattered data (red curves) along
with the best fit parameters. The dashed line in panel (c) corresponds to
$\Delta\theta=2\lambda/\lambda_{p}$, which corresponds to $J_{||}=J_{A}$.