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# Kinetic model of anisotropic force-free current sheets

Ya-Ze Wu, Fan Yang, Xu-Zhi Zhou, Anton V. Artemyev, Xin An, Zhi-Yang

Liu, Shan Wang, Qiu-Gang Zong

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## To cite this version:

Ya-Ze Wu, Fan Yang, Xu-Zhi Zhou, Anton V. Artemyev, Xin An, et al., Kinetic model of anisotropic force-free current sheets. Physics of Plasmas, 2024, 31, 10.1063/5.0213897. insu-04833951

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# Kinetic model of anisotropic force-free current sheets

Ya-Ze Wu✉; Fan Yang✉; Xu-Zhi Zhou✉; Anton V. Artemyev✉; Xin An✉; Zhi-Yang Liu✉; Shan Wang✉; Qiu-Gang Zong✉

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*Phys. Plasmas* 31, 083904 (2024)

https://doi.org/10.1063/5.0213897

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# Kinetic model of anisotropic force-free current sheets

Cite as: Phys. Plasmas **31**, 083904 (2024); doi: 10.1063/5.0213897

Submitted: 14 April 2024 · Accepted: 29 July 2024 ·

Published Online: 23 August 2024

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Ya-Ze Wu,<fnref n="1" /> ID Fan Yang,<fnref n="1" /> ID Xu-Zhi Zhou,<fnref n="1" />,<fnref n="4" /> ID Anton V. Artemyev,<fnref n="2" />,<fnref n="3" /> ID Xin An,<fnref n="2" /> ID Zhi-Yang Liu,<fnref n="4" /> ID Shan Wang,<fnref n="1" /> ID and Qiu-Gang Zong<fnref n="1" />,<fnref n="5" /> ID

## AFFILIATIONS

## ABSTRACT

Force-free current sheets, characterized by field-aligned electric currents and approximately uniform plasma pressures, have been widely observed in the planetary magnetosphere and throughout the heliosphere. Recent observations of force-free current sheets have clearly shown the presence of anisotropic electron distributions with different temperatures perpendicular and parallel to the local magnetic field. In most of the kinetic models for one-dimensional, force-free current sheets, however, the electron distributions are nearly isotropic, which necessitates the construction of new models accounting for the electron temperature anisotropy. In this paper, we develop a model for anisotropic force-free current sheets, by incorporating the magnetic moment as an additional invariant of motion into the nearly isotropic electron distribution function of a previous model. Despite the different electron distributions, the electromagnetic profiles of the new model are often close to those in the nearly isotropic model. The applicability of our model is then validated via a comparison to a typical force-free current sheet in the Jovian magnetodisk, which shows good agreement between the model and the observations.

© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0213897

## I. INTRODUCTION

Current sheets are prevalent plasma structures in space, where dramatic magnetic field variations, including turns or reversals, often occur.<fnref n="1" />⁻<fnref n="3" /> These structures play significant roles across various space regions, such as solar flares,<fnref n="4" />⁻⁶ boundaries of planetary magnetospheres,⁷,⁸ magnetotails of planets,⁹⁻¹¹ and comet environments.¹²,¹³ The instability of current sheets may cause the onset of magnetic reconnection,¹⁰,¹⁴ leading to a conversion of magnetic energy into particle kinetic and thermal energy.¹⁵,¹⁶ Thus, it is crucial to understand the kinetics of instabilities in the current sheets, which in turn requires information on the initial current sheet equilibrium in various space plasma environments.

To model the equilibrium current sheets, one can utilize the magnetohydrodynamics (MHD) approach.¹⁷⁻²¹ However, the kinetic properties of the current sheets can only be described within the

Vlasov-Maxwell (VM) equilibrium framework,²²⁻²⁸ with self-consistent electromagnetic fields and particle distribution functions $f_s(v, r)$ ($s=i, e$ correspond to ions and electrons, respectively). The stationary Vlasov equations can be satisfied by requiring particle distribution functions (DFs) to be expressed as functions of the integrals of motion.²²,²⁹,³⁰ Since the integrals of motion are functions of electromagnetic potentials, the moments of the particle DFs could be substituted into the Maxwell equations to yield self-consistent equations for kinetic equilibria. Many general models of current sheets have been developed based on the VM framework.²⁹⁻³¹ The Harris current sheet model,²² one of the most notable solutions derived within the VM framework, leverages the particle energy, $H$, and canonical momentum in the $y$-direction, $p_y$ (assuming the normal direction of the current sheet is along the $z$ axis, and the magnetic field is along the $x$ axis) as integrals of motion, to choose the distribution functions as follows:

<footnote n="1">¹School of Earth and Space Sciences, Peking University, Beijing 100871, China</footnote>

<footnote n="2">²Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90095, USA</footnote>

<footnote n="3">³Space Research Institute, Russian Academy of Sciences, Moscow 119991, Russia</footnote>

<footnote n="4">⁴Institut de Recherche en Astrophysique et Planetologie (IRAP), CNES-CNRS-Université Toulouse III Paul Sabatier, Toulouse 31062, France</footnote>

<footnote n="5">⁵State Key Laboratory of Lunar and Planetary Sciences, Macau University of Science and Technology, Macau 999078, China</footnote>

<footnote>a) Author to whom correspondence should be addressed: xzzhou@pku.edu.cn</footnote>
$$H_s = \frac{m_s v^2}{2} + q_s \Phi, \qquad (1)$$

$$p_{ys} = m_s v_y + \frac{q_s A_y}{c}, \qquad (2)$$

$$F_{HS,s}(\mathbf{v}, z) = n_0 \left( \frac{m_s}{2\pi T_s} \right)^{\frac{3}{2}} \exp \left( - \frac{H_s - u_{0s} p_{ys} + \frac{1}{2} m_s u_{0s}^2}{T_s} \right), \quad (3)$$

where the subscript HS stands for the Harris sheet model. Here, $m_s$, $q_s$, and $T_s$ ($s = i, e$) represent the mass, charge, and temperatures of ions and electrons, respectively (the Boltzmann constant, $k_B$, has been absorbed into the temperature definitions), $\Phi$ and $A$ denote the electrostatic and magnetic vector potentials, respectively, $n_0$ is the plasma density in the current sheet center, and $u_{0s}$ is the plasma bulk velocity. The substitution of the above DFs into Maxwell's equation, after some mathematical derivations, delivers the following current density, $j$, magnetic field, $B$, and magnetic vector potential, $A$:

$$\dot{j}_{HS} = \frac{cB_0}{4\pi L} \left( 0, \frac{1}{\cosh^2\left(\frac{z}{L}\right)}, 0 \right), \qquad (4)$$

$$\mathbf{B}_{HS} = B_0 \left( \tanh\left(\frac{z}{L}\right), 0, 0 \right), \qquad (5)$$

$$\mathbf{A}_{HS} = B_0 L \left( 0, -\ln\left(\cosh\left(\frac{z}{L}\right)\right), 0 \right), \qquad (6)$$

where $B_0 = [8\pi n_0(T_i + T_e)]^{1/2}$ represents the magnetic field strength at $z \to \pm\infty$, and $L = c/(q_i u_{0i} + q_e u_{0e})[(T_i + T_e)/(2\pi n_0)]^{1/2}$ is the characteristic thickness of the current sheet. In this one-dimensional model, the electrostatic potential $\Phi_{HS}$ remains zero. The model is also characterized by a reversal of the magnetic field at its center and a balance between the magnetic pressures on either side and the plasma thermal pressure at the center.

Although the Harris model and its variants³² can describe many types of tangential discontinuities observed in space,²⁷,³⁰,³¹ the modeled current sheets are always associated with thermal pressure gradients along the normal direction. These gradients are balanced by the reversed gradients of the magnetic pressure in these models. In other words, these models can hardly be applied to force-free current sheets, which have been widely observed in planetary magnetospheres as well as in the heliosphere³³⁻³⁸ with nearly constant magnetic strength throughout the structure. Recently, significant efforts have been made to develop force-free current sheet models,²³,²⁶,³⁹⁻⁴² particularly, a force-free generalization of the Harris model.²³,²⁴,⁴³⁻⁴⁵ In these models, another integral of motion (in addition to those adopted in the Harris model), the canonical momentum in the $x$ direction,²³

$$p_{xs} = m_s v_x + \frac{q_s A_x}{c}, \qquad (7)$$

is also utilized in the construction of particle DFs. For example, in a specific version of the force-free Harris sheet²³,⁴⁵ (hereinafter symbolized by subscript ffHS), the electron distribution is determined through a transform method from the pressure tensor, to be

$$F_{ffHS,e}(\mathbf{v}, z) = n_0 \left(\frac{1}{2} + b\right)^{-1} \left(\frac{m_e}{2\pi T_e}\right)^{\frac{3}{2}} \exp\left(-\frac{H_e}{T_e}\right) \\ \times \left[ \exp\left(\frac{2u_{0e}p_{ye} - m_e u_{0e}^2}{2T_e}\right) + b + \frac{1}{2}\cos\left(\frac{u_{0e}p_{xe}}{T_e}\right) \exp\left(\frac{m_e u_{0e}^2}{2T_e}\right) \right], \qquad (8)$$

which contains three different populations. Besides the Harris-like, shifted-Maxwellian population (the first term on the right-hand side), there are also a background population (the second term, represented by parameter $b$) and a population that carries the current in the $x$ direction (the third term on the right-hand side, in which the new invariant, $p_{xs}$, resides). Note that the third population differs from the Maxwellian-type distributions due to the presence of a cosine function, which indicates that $T_e$, the temperature of the other two electron components, may not represent the accurate temperature of the combined electron population. Moreover, to ensure that the electron distribution function remains positive, the parameter $b$ must be greater than $\exp(m_e u_{0e}^2/2T_e)/2$. In addition, the following constraints are also imposed in Ref. 24, to avoid two-stream instability: $T_e > m_e u_{0e}^2/8$ and $b > \exp(m_e u_{0e}^2/2T_e)(m_e u_{0e}^2/2T_e + 1)/2$. The ion distribution function follows the Maxwell distribution,⁴⁵

$$F_{ffHS,i}(\mathbf{v}, z) = n_0 \left(\frac{m_i}{2\pi T_i}\right)^{\frac{3}{2}} \exp\left(-\frac{H_i}{T_i}\right), \qquad (9)$$

which carries no current. In this model, the $j$, $B$, and $A$ profiles can be expressed as follows:²³

$$\dot{j}_{ffHS} = \frac{cB_0}{4\pi L} \left( \frac{\tanh\left(\frac{z}{L}\right)}{\cosh\left(\frac{z}{L}\right)}, \frac{1}{\cosh^2\left(\frac{z}{L}\right)}, 0 \right), \qquad (10)$$

$$\mathbf{B}_{ffHS} = B_0 \left( \tanh\left(\frac{z}{L}\right), \frac{1}{\cosh\left(\frac{z}{L}\right)}, 0 \right), \qquad (11)$$

$$\mathbf{A}_{ffHS} = B_0 L \left( 2 \arctan \left( \exp\left(\frac{z}{L}\right) \right), -\ln \left( \cosh\left(\frac{z}{L}\right) \right), 0 \right), \qquad (12)$$

in which the $B_0$ and $L$ values are modified to satisfy the following relationship: $4\pi L e n_0 u_{0e} = -cB_0(\frac{1}{2} + b)$ and $B_0 L e u_{0e} = -2cT$. The electrostatic potential also remains zero in this model.

A comparison between these expressions and their Harris sheet counterparts [Eqs. (4)–(6)] indicates identical profiles of $j_y$, $B_x$, and $A_y$ components, while additional components $j_x$, $B_y$, and $A_x$ are included in the force-free model. One can verify from Eq. (11) that $|\mathbf{B}| \equiv B_0$. By integrating the electron and ion DFs, one may also find that the particle density $n_i = n_e = n_0$. Both the magnetic pressure and the thermal pressure remain constant throughout the current sheet, thereby ensuring pressure balance. The comparison of the Harris sheet and the force-free Harris sheet is shown in Fig. 1.

There is an important feature of the force-free Harris sheet. The electron distributions in the model are nearly isotropic as long as the electron thermal gyroradius, $r_e$, calculated based on the velocity $v_\perp = \sqrt{2T_e/m_e}$, is much smaller than the current sheet thickness $L$.
<figure><img src="image_12.png" /><figcaption>FIG. 1. Model comparison between Harris sheet and force-free Harris sheet, with the horizontal axis representing the location (z) along the current sheet normal direction. (a) Normalized current density in the Harris sheet model, characterized by a $j_y$ peak in the center part. (b) Normalized magnetic field in the Harris sheet, with a $B_x$ reversal at the current sheet center. (c) The normalized energy spectrum of the electron fluxes in the Harris sheet, with enhanced fluxes near the thermal energy at the current sheet center. (d)–(f) Profiles of the force-free Harris sheet with the same format as in panels (a)–(c).</figcaption></figure>

<figure><img src="image_13.png" /></figure>

This feature can be demonstrated by examining the components of the temperature tensor,

$$T_{e,ij} = \frac{m_e}{n_0} \int F_{ffHS,e}(\mathbf{v}, z)(v_i - u_i)(v_j - u_j)d^3\mathbf{v}, \quad (13)$$

where $v_i$ and $v_j$ are electron velocity components, and $u_i$ and $u_j$ represent the mean velocities in the corresponding directions. After some straight-forward derivations, the temperature tensor could be expressed as follows:

$$T_{e,xx} = T_e - \left(\frac{r_e}{L}\right)^2 \frac{2T_e}{\left(\frac{1}{2} + b\right)^2} \cdot \left(\frac{1}{2} + b\right) \left(\tanh^2\left(\frac{z}{L}\right) - \frac{1}{2}\right) + \frac{\tanh^2\left(\frac{z}{L}\right)}{\cosh^2\left(\frac{z}{L}\right)}, \quad (14)$$

$$T_{e,yy} = T_e + \left(\frac{r_e}{L}\right)^2 \frac{2T_e}{\left(\frac{1}{2} + b\right)^2} \cdot \left(\frac{1}{2} + b\right) \frac{1}{\cosh^2\left(\frac{z}{L}\right) - \cosh^4\left(\frac{z}{L}\right)}, \quad (15)$$

$$T_{e,zz} = T_e, \quad (16)$$

$$T_{e,xy} = -\left(\frac{r_e}{L}\right)^2 \frac{2T_e}{\left(\frac{1}{2} + b\right)^2} \frac{\tanh\left(\frac{z}{L}\right)}{\cosh^3\left(\frac{z}{L}\right)}, \quad (17)$$

$$T_{e,xz} = T_{e,yz} = 0. \quad (18)$$

One may find although $T_{e,zz} = T_e$, the other two diagonal elements of the temperature tensor, $T_{e,xx}$ and $T_{e,yy}$, both deviate from $T_{e,zz}$ on the order of $(r_e/L)^2$. One of the non-diagonal elements, $T_{e,xy}$, also deviates from 0 on the order of $(r_e/L)^2$. Since the coefficients of $(r_e/L)^2$ in these equations are bounded, the electron distributions in the force-free Harris model are nearly isotropic as long as the thermal electron gyroradius is much smaller than the current sheet thickness (which is usually the case, see discussions in Sec. III).

However, recent observations have identified force-free current sheets in the solar wind and planetary magnetosphere with evident perpendicular-parallel anisotropy in electron distributions,33,35,46 which could potentially lead to the occurrence of firehose instability. Therefore, it is our goal in this paper to incorporate perpendicular-parallel anisotropy into the force-free Harris model, which could thus be treated as a starting point for future studies to explore the stability of anisotropic, force-free current sheets.

This can be done by using integrals of motion involving parallel or perpendicular components of the particle velocity. One of the approximate integrals of motions (adiabatic invariant), the magnetic moment $\mu = m_e v_{\perp}^2 / (2|\mathbf{B}|)$, serves as an ideal candidate for this purpose.47,48 By adding the magnetic moment into the electron DFs, we alter the perpendicular temperature of electrons and construct a new equilibrium. Finally, we validate our model by comparing it with an anisotropic force-free current sheet observed by the Juno satellite in Jupiter's magnetosphere.49

In Sec. II, we will revisit the anisotropic force-free current sheet observed in Jupiter's magnetosphere33 and summarize its main features. In Sec. III, we will provide a detailed description of our anisotropic model and discuss the conditions necessary for its approximation. In Sec. IV, we will compare the profiles of our model to the observations detailed in Sec. II. The summary will be given in Sec. V.

## II. OBSERVATIONS

We revisit the force-free current sheet observed by the Juno spacecraft in the Jovian magnetodisk33 at $r \sim 55R_J$ radial distances
(R<sub>l</sub> represents the radius of Jupiter). The event took place on 2018, day of year 086 between 10:15 and 11:15.

We use the Fluxgate Magnetometer (FGM) and the Jovian Auroral Distributions Experiment (JADE) datasets. The FGM<sup>50,51</sup> instrument provides 1-s resolution data of the magnetic field. JADE<sup>52–54</sup> offers insights into the electron distributions across pitch angles and energy levels. The electron distribution measurements have the temporal resolution of 1 s and the energy range of 0.1 eV –100 keV. Using the 30-s spacecraft spin, we derive the pitch angle distributions with a resolution of 30 s. It is important to note that particles that JADE measure are nearly confined in the plane perpendicular to the spacecraft spin axis.<sup>52</sup> Consequently, the pitch angle coverage during a spin period depends on the angle between the spin axis and the magnetic field direction. Measurements taken at the center of the current sheet are capable of covering the complete pitch angle range from 0° to 180°. However, measurements away from the center on either side cannot cover the pitch angles near 0°/180° and thus are incomplete.

Before we present the data, a Minimum Variance Analysis (MVA)<sup>55</sup> has been carried out on the magnetic field measurements to define a local coordinate system (l, m, n) of the current sheet, where B<sub>n</sub> denotes the magnetic field component with the least variance, B<sub>l</sub> represents the field component with the largest variance, and B<sub>m</sub> is defined through the cross-product of vectors n and l.

Figure 2(a) shows the observed magnetic field profile in the local coordinate system (l, m, n), alongside the magnetic field strength |B|. Remarkably, |B| remains nearly constant throughout the current sheet.

<figure><img src="image_14.png" /><figcaption>FIG. 2. Juno spacecraft observations of an anisotropic, force-free current sheet in the Jovian magnetodisk. (a) Three magnetic field components in the local (l, m, n) coordinate system, together with the magnetic field strength. (b) The electron flux as functions of energy and pitch angle, averaged over the entire current sheet crossing.</figcaption></figure>

The B<sub>l</sub> component changes direction at 10:40, which coincides with the B<sub>m</sub> peak at the current sheet’s center. Moreover, B<sub>m</sub> gradually diminishes as it extends away from the center. These magnetic field profiles are very similar to those in the force-free Harris model<sup>23</sup> shown in Fig. 1(e). Figure 2(b) displays the averaged electron fluxes as a function of energy and pitch angle across the entire current sheet. The data reveal two distinct electron populations. The hotter population (about 1 ~ 20 keV) exhibits significant anisotropy with the perpendicular fluxes much lower than the parallel ones, whereas the colder background population (below 300 eV) appears nearly isotropic.

### III. ANISOTROPIC KINETIC MODEL

#### A. Anisotropic model

For the purpose of introducing an anisotropy to the particle distributions, we choose to add the magnetic moment,<sup>47,48</sup> $\mu = m_e v_\perp^2 / (2|B|)$, into the electron distribution function. We first consider the condition that the magnetic moment, $\mu$, can be considered an approximate integral of motion. In general, it is required that the radius of particle gyration is much shorter than the length scale of the magnetic field variations ($r_e \ll L$). This condition is questionable in the original Harris model since the magnetic field is very weak near the current sheet center where the gyroradius becomes extremely large. In the force-free current sheets, however, the constant magnetic field strength indicates that the thermal electron gyroradius is constant across the current sheet and much smaller than the typical current sheet thickness (which, according to observations,<sup>35,56,57</sup> is usually at the scale of ion inertial length).

We keep the ion distribution function unchanged from that in the force-free Harris model [Eq. (9)] and incorporate the magnetic moment $\mu$ only into the electron distribution function [Eq. (8)]. Since there is an isotropic cold population in the observations (Fig. 2), in our model, we also include a cold isotropic background with Maxwell distribution.<sup>32</sup> The modified electron distribution function can thus be expressed as follows:

$$F_e(\mathbf{v}, z) = n_0 \left(\frac{1}{2} + b_{\text{hot}} + b_{\text{cold}}\right)^{-1} [f_{e1}(\mathbf{v}, z) + f_{e2}(\mathbf{v}, z)], \quad (19)$$

$$f_{e1}(\mathbf{v}, z) = b_{\text{cold}} \left(\frac{m_e}{2\pi T_{e1}}\right)^{\frac{3}{2}} \exp\left(-\frac{H_e}{T_{e1}}\right), \quad (20)$$

$$f_{e2}(\mathbf{v}, z) = \left(\frac{m_e}{2\pi T_{e2}}\right)^{\frac{3}{2}} \left(\frac{B_0 + b_1 T_{e2}}{B_0}\right) \exp\left(-\frac{H_e}{T_{e2}} - b_1 \mu\right) \\ \cdot \left[b_{\text{hot}} + \exp\left(\frac{2u_{0e}p_{ye} - m_e u_{0e}^2}{2T_{e2}}\right) + \frac{1}{2}\cos\left(\frac{u_{0e}p_{xe}}{T_{e2}}\right) \exp\left(\frac{m_e u_{0e}^2}{2T_{e2}}\right)\right], \quad (21)$$

where $b_{hot}$ and $b_{cold}$ represent the hot and cold background electron populations, with their temperatures represented by $T_{e1}$ and $T_{e2}$, respectively. As discussed before, $b_{hot}$ should be larger than $1/2 \cdot \exp[m_e u_{0e}^2 / (2T_{e2})]$ to ensure positive values of the electron velocity distribution. Note that this role cannot be fulfilled by the cold electron population, since the associated phase space densities drop off rapidly in the high energy range. To characterize the anisotropy within the electron distribution, we introduce the parameter $b_1$ to quantify the
degree of anisotropy. In this case, the perpendicular and parallel temperatures of the hot background population can be defined by $T_{e2\perp} = B_0 T_{e2}/(B_0 + b_1 T_{e2})$ and $T_{e2\|} = T_{e2}$, respectively. Moreover, the anisotropy of the current-carrying populations can be modified accordingly.

We next determine, from the particle distributions given above, the self-consistent electromagnetic field profiles of the current sheet. This could be done by substituting the moments of the particle distributions to Ampère's law and the quasi-neutrality equation and solving the resultant differential equations. A major difficulty, however, stems from the $\mu$-dependence of the electron distribution function. Since $\mu$ depends on the magnetic field, it is required to have the magnetic field profile (the output of the model, not yet determined at this stage) before the moments of the electron distributions can be calculated.

This difficulty could be tackled via an iteration procedure, starting from an approximate profile [the force-free Harris model, Eqs. (11) and (12), and $\Phi \equiv 0$] and updating the profiles at each step until the solution converges. After testing a few sets of parameters, we realize that the converged solutions are usually associated with very weak electric fields, and the magnetic field profile is also close to that of the force-free Harris model. These results encourage us to examine the conditions under which our anisotropic model yields approximately the same electromagnetic field profiles as the force-free Harris model.

## B. Conditions for approximation

Before the detailed analysis, we first introduce a parameter,

$$k = \left(\frac{r_e}{L}\right)^2 \left(1 - \frac{T_{e2\perp}}{T_{e2\|}}\right), \qquad (22)$$

to simplify the mathematical expressions that we are going to show below. If we substitute the force-free Harris solution into the electron distribution function (19), the integrated electron density $n_e$ equals

$$n_e = n_0 \left(\frac{1}{2} + b_{\text{cold}} + b_{\text{hot}}\right)^{-1} \left[ b_{\text{hot}} + b_{\text{cold}} + \left(\tanh^2\left(\frac{z}{L}\right) - \frac{1}{2}\right) \exp\left(k \cosh^{-2}\left(\frac{z}{L}\right)\right) + \cosh^{-2}\left(\frac{z}{L}\right) \exp\left(-k \tanh^2\left(\frac{z}{L}\right)\right) \right], \qquad (23)$$

which can be structured as the product of $n_0$ and some complicated functions of $z$ and $k$. As discussed above, the conservation of $\mu$ in our model requires a small ratio between $r_e$ and $L$, which is valid in most force-free current sheets. The small ratio also indicates that the parameter $k$, defined in Eq. (22), is much smaller than 1 as long as $T_{e2\perp}$ and $T_{e2\|}$ are not vastly different.

We next utilize the approximation $|k| \ll 1$ to simplify the above expressions. Since both the functions $\tanh^2(z/L)$ and $1/\cosh^2(z/L)$ are bounded between 0 and 1, their exponentials $\exp(k\tanh^2(z/L))$ and $\exp(k/\cosh^2(z/L))$ are bounded between $\exp(-k)$ and $\exp(k)$. With $|k| \ll 1$, we rewrite the coefficient of $n_e$ as a Taylor series of $k$,

$$\frac{n_e}{n_0} = 1 + \frac{b_{\text{hot}} + b_{\text{cold}} - \frac{1}{2} \cosh^{-2}\left(\frac{z}{L}\right)}{\frac{1}{2} + b_{\text{cold}} + b_{\text{hot}}} k^1 + o(k^2), \qquad (24)$$

in which the coefficient of $k^1$ is also bounded, between $\frac{b_{\text{hot}}+b_{\text{cold}}-1/2}{b_{\text{hot}}+b_{\text{cold}}+1/2}$ and 1. Therefore, the deviation of $n_e$ from $n_i \equiv n_0$ is of the order of $k$. The minor deviation indicates that quasi-neutrality can be achieved relatively easily, without the need to introduce a large electrostatic potential in the electron distribution function (19) that adjusts the electron density.

Substitution of the force-free current sheet profile also yields the current density,

$$j_x = \frac{cB_0}{4\pi L} \frac{\tanh\left(\frac{z}{L}\right)}{\cosh\left(\frac{z}{L}\right)} (B_0 + b_1 T_e)^{-1} \left[ B_0 \exp\left(k \cosh^{-2}\left(\frac{z}{L}\right)\right) + b_1 T_e \left[ \tanh^2\left(\frac{z}{L}\right) \exp\left(k \cosh^{-2}\left(\frac{z}{L}\right)\right) + \cosh^{-2}\left(\frac{z}{L}\right) \exp\left(-k \tanh^2\left(\frac{z}{L}\right)\right) \right] \right], \qquad (25)$$

$$j_y = \frac{cB_0}{4\pi L} \frac{1}{\cosh^2\left(\frac{z}{L}\right)} (B_0 + b_1 T_e)^{-1} \left[ B_0 \exp\left(-k \tanh^2\left(\frac{z}{L}\right)\right) + b_1 T_e \left[ \tanh^2\left(\frac{z}{L}\right) \exp\left(k \cosh^{-2}\left(\frac{z}{L}\right)\right) + \cosh^{-2}\left(\frac{z}{L}\right) \exp\left(-k \tanh^2\left(\frac{z}{L}\right)\right) \right] \right], \qquad (26)$$

$$j_z = 0, \qquad (27)$$

in which the $B_0$ and $L$ values are modified, once again, to satisfy the following relationship: $4\pi L n_0 u_{0e} = -cB_0(1/2 + b_{\text{hot}} + b_{\text{cold}})$ and $B_0 L e u_{0e} = -2cT_{e2}$. These expressions can also be structured as the product of the original force-free Harris sheet expressions [Eq. (10)] and some complicated functions of $z$ and $k$. A similar Taylor series analysis can also be done to indicate their minor deviations from the original model. In other words, if the two conditions mentioned above ($r_e \ll L$, $T_{e\perp} \sim T_{e\|}$) are satisfied, the revised model of the current sheet with anisotropic electron distributions [Eq. (19)] would have approximately the same electromagnetic profiles [Eqs. (11) and (12), and $\Phi \equiv 0$] as the force-free Harris sheet.

We also comment on the physics underlying the fact that the introduction of $\mu$ in the electron distributions, or equivalently the change of perpendicular temperature, barely modifies the associated electromagnetic profile of the force-free current sheet. The absence of electron density gradient or temperature gradient implies the absence of the electron diamagnetic motion, or equivalently, the electron gyro-motion does not contribute to the current density. Furthermore, the magnetic field strength remains constant, eliminating the presence of a gradient B drift. Hence, variations in the perpendicular temperature can hardly affect the current density.

## IV. DATA-MODEL COMPARISON

We next utilize our model to reconstruct the force-free current sheet observed in Sec. II. The adopted parameters are mostly derived from observations, which include $T_{e1} = 20$ eV, $T_{e2} = 6384$ eV, $|B| = 6nT$, $T_{\|}/T_{\perp} = 2.5$, $b_{\text{hot}} = 0.8$, $b_{\text{cold}} = 4$. The selection of these parameters also ensures that $b_{\text{hot}}$ exceeds $1/2 \cdot \exp(m_e u_{0e}^2/(2T_{e2})) = 0.50$.
so that $F_{e2}$ remains positive. We next estimate the current sheet thickness, $L = 4000$ km, based on the crossing time of 4.5 min and the measured corotation velocity$^{33}$ (with the radial distance of $r = 55R_J$ and the angle of magnetodisk inclination of about $\theta \approx 9.5^\circ$). We also verify the conditions for magnetic moment conservation and for the simplification of $j$ and $n$. Since $r_e/L \sim 0.01$ and the parameter $k \sim 0.0001$, the approximations discussed in Sec. IIIB are well satisfied.

In Figs. 3(a) and 3(b), we present the magnetic field profile and the electron fluxes from our model, facilitating a direct comparison with the observations depicted in Fig. 2. The most notable characteristics include

1. the uniform magnetic field strength and the profile of magnetic components in $(x, y, z)$: note that the $(x, y, z)$ axes can be replaced by $(l, m, n)$ in local coordinate system;

<figure><img src="image_15.png" /><figcaption>**FIG. 3.** Profiles of current sheet models, with input parameters found from the case shown in Fig. 2. (a) Magnetic field profile in both the original force-free Harris and the anisotropic force-free models. (b) Pitch angle and energy dependence of the electron fluxes in our anisotropic model and, for comparison purposes, (c) in the original force-free current sheet model. All panels share the same format as Fig. 2.</figcaption></figure>

2. the anisotropy observed in the hot electron population, with higher fluxes in the parallel direction; and

3. the presence of an isotropic cold electron background.

The features of electron flux as discussed in Sec. II are well reproduced by our model. Moreover, the observed profiles of the $B_x$ and $B_y$ field are in excellent agreement with our model, although the nonzero $B_n$ field ($\sim 1nT$) in the observations cannot be captured. To include the $B_n$ field in the model, a new dimension is required, e.g., by having a $x$-dependent $A_y$. This will bring significant complexity to our model, and we make no attempt to do so in this paper. This may be the subject of a future study.

The data-model comparison also involves the original force-free Harris model, with the associated electron fluxes given in Fig. 3(c) hardly displaying any anisotropic features. This is consistent with our expectations, which further justifies our motivation in the development of the anisotropic, force-free model.

## V. SUMMARY AND DISCUSSION

In this article, we aim to model the observed perpendicular-parallel anisotropy of force-free current sheets, by modifying the existing model$^{23,24}$ of force-free Harris sheet. We introduce the magnetic moment, $\mu = m_e v_{\perp}^2 / (2|\mathbf{B}|)$, as an additional integral of motion to the electron distribution function of the model. The reason why $\mu$ can be treated as an invariant of motion is that the electron gyroradius, constant across the entire force-free current sheet, is much smaller than the characteristic thickness of the current sheet in most cases. Based on the condition of $\mu$ conservation, together with the fact that the electron perpendicular and parallel temperatures are usually comparable in magnitude, we show that the modified electron distributions, given by Eq. (19), yield magnetic field and density profiles nearly identical to those of the force-free Harris sheet.

Utilizing parameters derived from observations, we plot the magnetic field profiles in the revised model. Moreover, the electron fluxes as functions of energy and pitch angle are self-consistently plotted. Our results replicate the phenomena observed in the current sheets. This study contributes to our understanding of the structures and kinetic properties of anisotropic force-free current sheets and may help better elucidate mechanisms behind current sheet instabilities, especially the firehose instability, which can be induced by the different perpendicular and parallel temperatures and the two-stream instability mentioned in Sec. I. Discussions about the stabilities and the marginal behaviors of our model are beyond the scope of this paper and can be the subject of a future study.

## ACKNOWLEDGMENTS

This study was supported by NSFC Grant No. 42174184. The authors would like to thank Sergei Kamaletdinov from UCLA for participating in the discussions of our work.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.
## Author Contributions

Ya-Ze Wu and Fan Yang contributed equally to this work.

**Ya-Ze Wu:** Conceptualization (equal); Data curation (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Software (lead); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). **Fan Yang:** Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (lead); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Xu-Zhi Zhou:** Conceptualization (lead); Formal analysis (equal); Investigation (lead); Methodology (equal); Supervision (lead); Visualization (equal); Writing – original draft (equal); Writing – review & editing (lead). **Anton V. Artemyev:** Conceptualization (lead); Formal analysis (equal); Investigation (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). **Xin An:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). **Zhi-Yang Liu:** Formal analysis (equal); Validation (equal). **Shan Wang:** Formal analysis (equal); Validation (equal). **Qiu-Gang Zong:** Formal analysis (equal); Supervision (equal).

## DATA AVAILABILITY

The data that support the findings of this study are openly available in Planetary Data System at https://doi.org/10.17189/2775-4623, Ref. 58.

## REFERENCES

¹E. J. Smith, "Identification of interplanetary tangential and rotational discontinuities," *J. Geophys. Res.* **78**, 2054–2063, https://doi.org/10.1029/2017i013p02054 (1973).

²M. Neugebauer, "The structure of rotational discontinuities," *Geophys. Res. Lett.* **16**, 1261–1264, https://doi.org/10.1029/GL016i011p01261 (1989).

³L. F. Burlaga and N. F. Ness, "Tangential discontinuities in the solar wind," *Sol. Phys.* **9**, 467–477 (1969).

⁴P. J. Cargill, L. Vlahos, G. Baumann, J. F. Drake, and Å. Nordlund, "Current fragmentation and particle acceleration in solar flares," *Space Sci. Rev.* **173**, 223–245 (2012).

⁵L. S. Ledentsov and B. V. Somov, "On discontinuous plasma flows in the vicinity of reconnecting current sheets in solar flares," *Astron. Lett.* **37**, 131–140 (2011).

⁶B. C. Low and R. Wolfson, "Spontaneous formation of electric current sheets and the origin of solar flares," *Astrophys. J.* **324**, 574 (1988).

⁷J. Burch, H. Hietala, J. Eastwood, C. Norgren, B. Lavraud, P. Tenfjord, S. Fadanelli, K. Trattner, S. Eriksson, V. Constantinescu et al., "Characteristics of the flank magnetopause: MMS results," *J. Geophys. Res.: Space Phys.* **125**, e27623, https://doi.org/10.1029/2019JA027623 (2020).

⁸A. Lukin, E. Panov, A. Artemyev, A. Petrukovich, S. Haaland, R. Nakamura, V. Angelopoulos, A. Runov, E. Yushkov, L. Avanov et al., "Comparison of the flank magnetopause at near-Earth and Lunar distances: MMS and ARTEMIS observations," *J. Geophys. Res.: Space Phys.* **125**, e2020JA028406, https://doi.org/10.1029/2020JA028406 (2020).

⁹A. Petrukovich, A. Artemyev, I. Vasko, R. Nakamura, and L. Zelenyi, "Current sheets in the earth magnetotail: Plasma and magnetic field structure with cluster project observations," *Space Sci. Rev.* **188**, 311–337 (2015).

¹⁰M. Sitnov, J. Birn, B. Ferdousi, E. Gordeev, Y. Khotyaintsev, V. Merkin, T. Motoba, A. Otto, E. Panov, P. Pritchett et al., "Explosive magnetotail activity," *Space Sci. Rev.* **215**, 1–95 (2019).

¹¹L. Zelenyi, H. Malova, A. Artemyev, V. Y. Popov, and A. Petrukovich, "Thin current sheets in collisionless plasma: Equilibrium structure, plasma instabilities, and particle acceleration," *Plasma Phys. Rep.* **37**, 118–160 (2011).

¹²M. Volwerk, G. H. Jones, T. Broiles, J. Burch, C. Carr, A. J. Coates, E. Cupido, M. Delva, N. J. T. Edberg, A. Eriksson, C. Goetz, R. Goldstein, P. Henri, H. Madanian, H. Nilsson, I. Richter, K. Schwingenschuh, G. Stenberg Wieser, and K.-H. Glassmeier, "Current sheets in comet 67P/Churyumov-Gerasimenko's coma," *J. Geophys. Res.: Space Phys.* **122**, 3308–3321, https://doi.org/10.1002/2017JA023861 (2017).

¹³M. Volwerk, "Currents in cometary comae," in *Electric Currents in Geospace and Beyond*, Geophysical Monograph Series (AGU, 2018), pp. 513–533.

¹⁴W. Gonzalez and E. Parker, "Magnetic reconnection," *Astrophys. Space Sci. Libr.* **427**, 542 (2016).

¹⁵J. Birn, A. Artemyev, D. Baker, M. Echim, M. Hoshino, and L. Zelenyi, "Particle acceleration in the magnetotail and aurora," *Space Sci. Rev.* **173**, 49–102 (2012).

¹⁶M. Oka, J. Birn, M. Battaglia, C. Chaston, S. Hatch, G. Livadiotis, S. Imada, Y. Miyoshi, M. Kuhar, F. Effenberger et al., "Electron power-law spectra in solar and space plasmas," *Space Sci. Rev.* **214**, 82 (2018).

¹⁷I. M. Podgorny, A. I. Podgorny, S. Minami, and M. Morimoto, "An MHD model for a heliospheric current sheet," *Astron. Rep.* **48**, 433–438 (2004).

¹⁸J. Loizu, S. R. Hudson, A. Bhattacharjee, S. Lazerson, and P. Helander, "Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets," *Phys. Plasmas* **22**, 090704 (2015).

¹⁹P. L. Israelevich, T. I. Gombosi, A. I. Ershkovich, K. C. Hansen, C. P. T. Groth, D. L. DeZeeuw, and K. G. Powell, "MHD simulation of the three-dimensional structure of the heliospheric current sheet," *Astron. Astrophys.* **376**, 288–291 (2001).

²⁰C. Matsuoka, K. Nishihara, and T. Sano, "Nonlinear dynamics of non-uniform current-vortex sheets in magnetohydrodynamic flows," *J. Nonlinear Sci.* **27**, 531–572 (2017).

²¹V. Merkin, M. Sitnov, and J. Lyon, "Evolution of generalized two-dimensional magnetotail equilibria in ideal and resistive MHD," *J. Geophys. Res.: Space Phys.* **120**, 1993–2014, https://doi.org/10.1002/2014J A020651 (2015).

²²E. G. Harris, "On a plasma sheath separating regions of oppositely directed magnetic field," *Il Nuovo Cimento* (1955–1965) **23**, 115–121 (1962).

²³M. G. Harrison and T. Neukirch, "A one-dimensional Vlasov-Maxwell equilibrium for the force-free Harris sheet," *Phys. Rev. Lett.* **102**, 135003 (2009).

²⁴T. Neukirch, F. Wilson, and M. G. Harrison, "A detailed investigation of the properties of a Vlasov-Maxwell equilibrium for the force-free Harris sheet," *Phys. Plasmas* **16**, 122102 (2009).

²⁵B. Abraham-Shrauner, "Force-free Jacobian equilibria for Vlasov-Maxwell plasmas," *Phys. Plasmas* **20**, 102117 (2013).

²⁶T. Neukirch, F. Wilson, and O. Allanson, "A family of Vlasov-Maxwell equilibrium distribution functions describing a transition from the Harris sheet to the force-free Harris sheet," *J. Plasma Phys.* **86**, 825860302 (2020).

²⁷P. H. Yoon and A. T. Lui, "A class of exact two-dimensional kinetic current sheet equilibria," *J. Geophys. Res.: Space Phys.* **110**, A01202, https://doi.org/10.1029/2003JA010308 (2005).

²⁸V. Kocharovsky, V. V. Kocharovsky, and V. J. Martyanov, "Self-consistent current sheets and filaments in relativistic collisionless plasma with arbitrary energy distribution of particles," *Phys. Rev. Lett.* **104**, 215002 (2010).

²⁹M. Sitnov and V. Merkin, "Generalized magnetotail equilibria: Effects of the dipole field, thin current sheets, and magnetic flux accumulation," *J. Geophys. Res.: Space Phys.* **121**, 7664–7683, https://doi.org/10.1002/2016JA023001 (2016).

³⁰V. V. Kocharovsky, V. V. Kocharovsky, V. Y. Martyanov, and S. V. Tarasov, "Analytical theory of self-consistent current structures in a collisionless plasma," *Phys.-Usp.* **59**, 1165 (2016).

³¹M. Roth, J. de Keyser, and M. M. Kuznetsova, "Vlasov theory of the equilibrium structure of tangential discontinuities in space plasmas," *Space Sci. Rev.* **76**, 251–317 (1996).

³²P. H. Yoon and A. T. Y. Lui, "Model of ion- or electron-dominated current sheet," *J. Geophys. Res.: Space Phys.* **109**, A11213, https://doi.org/10.1029/2004JA010555 (2004).

³³A. V. Artemyev, Q. Ma, R. W. Ebert, X. Zhang, and F. Allegrini, "Force-free current sheets in the Jovian magnetodisk: The key role of electron field-aligned anisotropy," *J. Geophys. Res.: Space Phys.* **128**, e2022JA031280, https://doi.org/10.1029/2022JA031280 (2023).
<footnote n="34">³⁴I. Y. Vasko, K. Alimov, T. Phan, S. D. Bale, F. S. Mozer, and A. V. Artemyev, “Kinetic-scale current sheets in the solar wind at 1 au: Scale-dependent properties and critical current density,” *Astrophys. J. Lett.* **926**, L19 (2022).</footnote>

<footnote n="35">³⁵A. V. Artemyev, V. Angelopoulos, and I. Y. Vasko, “Kinetic properties of solar wind discontinuities at 1 AU observed by ARTEMIS,” *J. Geophys. Res.: Space Phys.* **124**, 3858–3870, https://doi.org/10.1029/2019JA026597 (2019).</footnote>

<footnote n="36">³⁶Z. Rong, S. Barabash, G. Stenberg, Y. Futaana, T. Zhang, W. Wan, Y. Wei, X. Wang, L. Chai, and J. Zhong, “The flapping motion of the Venusian magnetotail: Venus express observations,” *J. Geophys. Res.: Space Phys.* **120**, 5593–5602, https://doi.org/10.1002/2015JA021317 (2015).</footnote>

<footnote n="37">³⁷S. Kamaletdinov, A. Artemyev, A. Runov, and V. Angelopoulos, “Thin current sheets in the magnetotail at Lunar distances: Statistics of ARTEMIS observations,” *J. Geophys. Res.: Space Phys.* **129**, e2023JA032130, https://doi.org/10.1029/2023JA032130 (2024).</footnote>

<footnote n="38">³⁸A. B. Lotekar, I. Y. Vasko, T. Phan, S. D. Bale, T. A. Bowen, J. Halekas, A. V. Artemyev, Y. V. Khotyaintsev, and F. S. Mozer, “Kinetic-scale current sheets in near-sun solar wind: Properties, scale-dependent features and reconnection onset,” *Astrophys. J.* **929**, 58 (2022).</footnote>

<footnote n="39">³⁹O. Allanson, “One-dimensional nonlinear force-free current sheets,” in *Theory of One-Dimensional Vlasov-Maxwell Equilibria: With Applications to Collisionless Current Sheets and Flux Tubes*, edited by O. Allanson (Springer International Publishing, Cham, 2018), pp. 69–112.</footnote>

<footnote n="40">⁴⁰X. An, A. Artemyev, V. Angelopoulos, A. Runov, and S. Kamaletdinov, “Kinetic equilibrium of two-dimensional force-free current sheets,” *Astrophys. J.* **952**, 36 (2023).</footnote>

<footnote n="41">⁴¹I. Vasko, A. Artemyev, A. Petrukovich, and H. Malova, “Thin current sheets with strong bell-shape guide field: Cluster observations and models with beams,” *Ann. Geophys.* **32**, 1349–1360 (2014).</footnote>

<footnote n="42">⁴²D. Y. Kolotkov, I. Y. Vasko, and V. M. Nakariakov, “Kinetic model of force-free current sheets with non-uniform temperature,” *Phys. Plasmas* **22**, 112902 (2015).</footnote>

<footnote n="43">⁴³Y. D. Yoon, D. E. Wendel, and G. S. Yun, “Equilibrium selection via current sheet relaxation and guide field amplification,” *Nat. Commun.* **14**, 139 (2023).</footnote>

<footnote n="44">⁴⁴Y. D. Yoon, G. S. Yun, D. E. Wendel, and J. L. Burch, “Collisionless relaxation of a disequilibrated current sheet and implications for bifurcated structures,” *Nat. Commun.* **12**, 3774 (2021).</footnote>

<footnote n="45">⁴⁵T. Neukirch, I. Y. Vasko, A. V. Artemyev, and O. Allanson, “Kinetic models of tangential discontinuities in the solar wind,” *Astrophys. J.* **891**, 86 (2020).</footnote>

<footnote n="46">⁴⁶Y. Shen, A. Artemyev, V. Angelopoulos, T. Z. Liu, and I. Vasko, “Comparing plasma anisotropy associated with solar wind discontinuities and Alfvénic fluctuations,” *Astrophys. J.* **961**, 41 (2024).</footnote>

<footnote n="47">⁴⁷J.-H. Li, F. Yang, X.-Z. Zhou, Q.-G. Zong, A. V. Artemyev, R. Rankin, Q. Shi, S. Yao, H. Liu, J. He, Z. Pu, C. Xiao, J. Liu, C. Pollock, G. Le, and J. L. Burch, “Self-consistent kinetic model of nested electron- and ion-scale magnetic cavities in space plasmas,” *Nat. Commun.* **11**, 5616 (2020).</footnote>

<footnote n="48">⁴⁸F. Yang, X.-Z. Zhou, J.-H. Li, Q.-G. Zong, S.-T. Yao, Q.-Q. Shi, and A. V. Artemyev, “Kinetic-scale flux ropes: Observations and applications of kinetic equilibrium models,” *Astrophys. J.* **926**, 208 (2022).</footnote>

<footnote n="49">⁴⁹F. Bagenal, A. Adriani, F. Allegrini, S. Bolton, B. Bonfond, E. Bunce, J. Connerney, S. Cowley, R. Ebert, G. Gladstone et al., “Magnetospheric science objectives of the Juno mission,” *Space Sci. Rev.* **213**, 219–287 (2017).</footnote>

<footnote n="50">⁵⁰J. E. P. Connerney, M. Benn, J. B. Bjarno, T. Denver, J. Espley, J. L. Jorgensen, P. S. Jorgensen, P. Lawton, A. Malinnikova, J. M. Merayo, S. Murphy, J. Odom, R. Oliversen, R. Schnurr, D. Sheppard, and E. J. Smith, “The Juno magnetic field investigation,” *Space Sci. Rev.* **213**, 39–138 (2017).</footnote>

<footnote n="51">⁵¹J. E. P. Connerney, A. Adriani, F. Allegrini, F. Bagenal, S. J. Bolton, B. Bonfond, S. W. H. Cowley, J.-C. Gerard, G. R. Gladstone, D. Grodent, G. Hospodarsky, J. L. Jorgensen, W. S. Kurth, S. M. Levin, B. Mauk, D. J. McComas, A. Mura, C. Paranicas, E. J. Smith, R. M. Thorne, P. Valek, and J. Waite, “Jupiter’s magnetosphere and aurorae observed by the Juno spacecraft during its first polar orbits,” *Science* **356**, 826–832 (2017).</footnote>

<footnote n="52">⁵²D. J. McComas, N. Alexander, F. Allegrini, F. Bagenal, C. Beebe, G. Clark, F. Crary, M. I. Desai, A. D. Los Santos, D. Demkee, J. Dickinson, D. Everett, T. Finley, A. Gribanova, R. Hill, J. Johnson, C. Kofoed, C. Loeffler, P. Louarn, M. Maple, W. Mills, C. Pollock, M. Reno, B. Rodriguez, J. Rouzaud, D. Santos-Costa, P. Valek, S. Weidner, P. Wilson, R. J. Wilson, and D. White, “The Jovian auroral distributions experiment (JADE) on the Juno mission to Jupiter,” *Space Sci. Rev.* **213**, 547–643 (2017).</footnote>

<footnote n="53">⁵³D. J. McComas, J. R. Szalay, F. Allegrini, F. Bagenal, J. Connerney, R. W. Ebert, W. S. Kurth, P. Louarn, B. Mauk, M. Reno, M. F. Thomsen, P. Valek, S. Weidner, R. J. Wilson, and S. Bolton, “Plasma environment at the dawn flank of Jupiter’s magnetosphere: Juno arrives at Jupiter,” *Geophys. Res. Lett.* **44**, 4432–4438, https://doi.org/10.1002/2017GL072831 (2017).</footnote>

<footnote n="54">⁵⁴T. K. Kim, R. W. Ebert, P. W. Valek, F. Allegrini, D. J. McComas, F. Bagenal, J. E. P. Connerney, G. Livadiotis, M. F. Thomsen, R. J. Wilson, and S. J. Bolton, “Survey of ion properties in Jupiter’s plasma sheet: Juno JADE-I observations,” *J. Geophys. Res.: Space Phys.* **125**, e2019JA027696, https://doi.org/10.1029/2019JA027696 (2020).</footnote>

<footnote n="55">⁵⁵B. Sonnerup and L. Cahill, Jr., “Explorer 12 observations of the magnetopause current layer,” *J. Geophys. Res.* **73**, 1757–1770, https://doi.org/10.1029/JA073i005p01757 (1968).</footnote>

<footnote n="56">⁵⁶M. Hesse, D. Winske, and J. Birn, “On the ion-scale structure of thin current sheets in the magnetotail,” *Phys. Scr.* **1998**, 63.</footnote>

<footnote n="57">⁵⁷Y. H. Liu, L. M. Kistler, C. G. Mouikis, V. Roytershteyn, and H. Karimabadi, “The scale of the magnetotail reconnecting current sheet in the presence of $O^+$,” *Geophys. Res. Lett.* **41**, 4819–4827, https://doi.org/10.1002/2014GL060440 (2014).</footnote>

<footnote n="58">⁵⁸F. Allegrini, R. J. Wilson, R. W. Ebert, and J. Hanley (2022). “Juno JADE 584 derived science data,” NASA Planetary System. https://doi.org/10.17189/2775-4623</footnote>