Electron-scale temperature gradients in kinetic equilibrium: MMS observations and Vlasov–Maxwell solutions

Shuster, J. R.; Bessho, N.; Wang, S.; Ng, J.

doi:10.1063/5.0069559

For an ideal gas, the notion of thermodynamic equilibrium requires that the system's temperature be spatially uniform. We demonstrate that for a collisionless plasma, kinetic equilibrium can be achieved even in the presence of a temperature gradient. Motivated by recent Magnetospheric Multiscale (MMS) magnetopause observations of thin current layers exhibiting electron-scale temperature gradients, here we present an exact solution to the Vlasov–Maxwell system that accounts for the observed electron temperature gradient and reproduces much of the plasma's spatial variation along one dimension. The current layer in the Vlasov–Maxwell model is self-consistently supported by a prominent crescent-shaped electron velocity distribution that naturally arises due to the temperature transition, which agrees with the electron distribution structures measured by MMS during two asymmetric current layers encountered at the magnetopause.

I. INTRODUCTION

In the context of the thermodynamics and statistical mechanics employed to describe the behavior of ideal (i.e., collisional) gases like air, one often considers an initial setup where two gases of different temperatures are placed next to each other and left to interact. Eventually, after enough time has passed, we intuitively expect the two gases to reach the same temperature—once this has happened, we would say the system has reached “thermodynamic equilibrium.” As we demonstrate in this paper, the situation is more nuanced for collisionless plasmas. Here, we discovered an equilibrium (steady-state) kinetic model for a temperature gradient that (1) is an exact solution to the Vlasov equation governing the evolution of collisionless plasmas, (2) self-consistently solves Maxwell's equations of electromagnetism, and (3) agrees with observations of electron-scale plasma measurements gathered by NASA's Magnetospheric Multiscale mission at the interface between the Sun's solar wind and Earth's magnetic field. Thus, for collisionless plasma systems, counter-intuitively this means that two neighboring regions of a collisionless plasma can have different temperatures and maintain that temperature difference indefinitely.

When a system is said to be in thermal equilibrium, for ideal collisional gases this typically means that the temperature is uniform and the velocity distribution function has relaxed to a Maxwellian (Schroeder, 2000; Gurnett and Bhattacharjee, 2005). For a collisionless plasma governed by the Vlasov–Maxwell system of equations, the concept of thermal equilibrium is more nuanced because temperature differences between constituent plasma species can be sustained for long periods of time in the absence of rapid inter-particle collisions. Consequently, collisionless plasma environments are characterized by the formation and persistence of non-Maxwellian velocity distribution functions (e.g., Ng et al., 2011; 2012; Bessho et al., 2014; 2016; 2017; Wang et al., 2016; 2018; Shuster et al., 2014; 2015).

There have been a great deal of studies making analytic progress and presenting detailed solutions to the Vlasov–Maxwell system for various plasma configurations (e.g., Harris, 1962; Alpers, 1969; Channell, 1976; Mynick et al., 1979; Schindler and Birn, 2002; Camporeale and Lapenta, 2005; and references therein). While solutions for thin current sheets with thicknesses smaller than ion spatial scales have been considered previously (e.g., Sitnov et al., 2006), most equilibrium solutions applicable for modeling magnetospheric current sheet observations do not typically consider scales smaller than a characteristic ion gyroradius. Classes of asymmetric equilibria have been developed that feature ion-scale temperature gradients (e.g., Mottez, 2003; Aunai et al., 2013; Allanson et al., 2017). Force-free equilibria have also been considered (e.g., Neukirch et al., 2009), in some cases yielding spatially oscillatory density and temperature profiles (Wilson et al., 2017). Within the broad class of tangential discontinuities (e.g., Roth et al., 1996), many of these previously reported equilibrium current sheet configurations are applicable to symmetric or force-free current layers exhibiting an ion-scale thickness. Here, we consider the case of an asymmetric, non-force-free, electron-scale current layer suitable for modeling recent observations from NASA's Magnetospheric Multiscale (MMS) mission.

MMS has detected very thin current layers exhibiting electron-scale temperature gradients with spatial scales on the order of a few electron gyroradii in the vicinity of Earth's reconnecting magnetopause (e.g., Shuster et al., 2019; 2021). Such filamentary currents have also been reported in association with reconnection X-line and exhaust regions (Phan et al., 2016), and in the turbulent magnetosheath where a form of small-scale electron-only reconnection has been observed independent of large-scale ion coupling (Phan et al., 2018). Motivated by these discoveries, in this paper we present a model for an electron-scale temperature gradient that exists in exact kinetic equilibrium while self-consistently maintaining electron crescent-like velocity distributions. Our solution to the Vlasov–Maxwell system mimics the observed electron temperature gradient and captures most of the plasma's spatial variations. In this Vlasov–Maxwell model, the current density is supported by a pronounced crescent-resembling electron velocity distribution structure that develops as a result of the transition in temperature across the layer, which is qualitatively consistent with the particle measurements from MMS.

In Sec. II, we present a detailed description of the model and its relevant parameters with an initial comparison to MMS data. Direct comparisons of the plasma moments and velocity-space structure of electron distributions measured by MMS and predicted by our Vlasov–Maxwell solution are discussed in Sec. III. Our summary and conclusions are addressed in Sec. IV.

II. VLASOV–MAXWELL EQUILIBRIUM FOR AN ELECTRON-SCALE TEMPERATURE GRADIENT

Motivated by recent MMS magnetopause observations (Shuster et al., 2019; 2021), here we seek self-consistent solutions to the Vlasov–Maxwell system suitable for modeling an asymmetric current layer with the following properties: (1) electron-scale electron temperature gradient $T_{e 2} > T_{e 1}$ ⁠, (2) crescent-shaped electron velocity distributions f_e, (3) asymmetric plasma parameters across the layer including $B_{2} < B_{1}$ and $n_{2} > n_{1}$ ⁠, (4) strong electron bulk velocity U_ey supporting the current density J_y, and (5) hot (⁠ $T_{i} / T_{e} \approx 10$ ⁠), non-drifting (U_i = 0) Maxwellian ion distributions f_i, where “1” denotes the low density (high magnetic field) magnetosphere (MSP) side of the layer, and “2” denotes the high density (low magnetic field) magnetosheath (MSH) side. We note that for these localized electron-scale current layers observed to be embedded within the larger-scale magnetopause transition itself, the local electron temperature gradient with $T_{e 2} > T_{e 1}$ differs from the overall trend of the larger-scale magnetopause transition, where the temperature on the MSP side is higher than on the MSH side with T_MSP > T_MSH due primarily to the contribution of electrons of magnetospheric origin with energies greater than a few keV (e.g., Wang et al., 2017). Nevertheless, throughout this paper we use the abbreviations “MSP” and “MSH” to indicate “toward the MSP proper” and “toward the MSH proper,” respectively. We restrict our consideration to steady-state systems (⁠ $\partial / \partial t \to 0$ ⁠) with spatial variations occurring along only one dimension, which we choose to be the x-direction. Following the approach of Camporeale and Lapenta (2005), we allow the magnetic field $B$ and corresponding magnetic flux function $A$ to have only one component so that $B = B_{z} \hat{z} = (\partial A_{y} / \partial x) \hat{z}$ ⁠. In the present case, we require A_y to be monotonically increasing so that $B_{z} > 0$ ⁠. For species s, we make use of two constants of particle motion within this system: (1) the total energy and (2) the momentum in the y-direction

c_{s 1} = \frac{1}{2} m_{s} v^{2} + q_{s} ϕ (x),

(1)

c_{s 2} = m_{s} v_{y} + q_{s} A_{y} (x),

(2)

where $v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}$ ⁠, and $ϕ (x)$ specifies the spatially varying electric potential. For ions (s = i), $q_{i} = + e$ ⁠, while for electrons (s = e), $q_{e} = - e$ ⁠. Any function of these constants of motion $f_{s} (c_{s 1}, c_{s 2})$ will satisfy the steady-state Vlasov equation

v \cdot \nabla f_{s} + \frac{q_{s}}{m_{s}} (E + v \times B) \cdot \nabla_{v} f_{s} = 0.

(3)

Because the ions are non-drifting, after choosing a particular form for f_e, a self-consistent solution to the Vlasov–Maxwell system is obtained by imposing quasineutrality (n_i = n_e) and solving Ampère's law

\nabla \times B = μ_{0} J \to \frac{d B_{z}}{d x} = - μ_{0} J_{y} = μ_{0} e \int_{- \infty}^{+ \infty} v_{y} f_{e} d^{3} v .

(4)

For the ion distribution function f_i, we choose a Maxwellian population specified by the initial temperature ratio $T_{i} / T_{e 1}$ ⁠, where the ion temperature T_i is constant throughout the layer. For the electron distribution function f_e, we choose a form that resembles a Maxwellian with a spatially varying temperature controlled by the function $T (c_{e 2})$

f_{i} = f_{i 0} exp [\frac{- c_{i 1}}{T_{i}}] = n_{1} {(\frac{m_{i}}{2 π T_{i}})}^{3 / 2} exp [\frac{- \frac{1}{2} m_{i} v^{2} - e ϕ}{T_{i}}],

(5)

f_{e} = f_{e 0} exp [\frac{- c_{e 1}}{T (c_{e 2})}] = n_{1} {(\frac{m_{e}}{2 π T_{e 1}})}^{3 / 2} exp [\frac{- \frac{1}{2} m_{e} v^{2} + e ϕ}{T (c_{e 2})}],

(6)

where the $T (c_{e 2})$ function is given by

T (c_{e 2}) = T_{e 1} + (T_{e 2} - T_{e 1}) \cdot [\frac{1 + \tanh (- S \cdot c_{e 2})}{2}] .

(7)

With $c_{e 2}$ given by Eq. (2), the electron distribution satisfies the electron Vlasov equation and takes the following form:

f_{e} = n_{1} {(\frac{m_{e}}{2 π T_{e 1}})}^{3 / 2} exp [\frac{- \frac{1}{2} m_{e} v^{2} + e ϕ}{T_{e 1} + (T_{e 2} - T_{e 1}) \cdot [\frac{1 + \tanh (- S (m_{e} v_{y} - e A_{y}))}{2}]}] .

(8)

The free parameter S (in units of ${[e B_{1} r_{L}]}^{- 1}$ ⁠, or equivalently inverse momentum) influences the thickness of the current layer and the shape of the electron distribution function by determining how sharply the temperature transitions from $T_{e 1}$ to $T_{e 2}$ in velocity space. On the MSP of the layer, f_e is a Maxwellian population with temperature $T_{e 1}$ ⁠, while on the MSH side of the layer, f_e is a hotter Maxwellian with temperature $T_{e 2}$ ⁠. In the center of the current layer, f_e exhibits a mixture of these two populations, which we discuss in detail below. All quantities are normalized by reference values taken on the MSP side of the layer; for example, the magnetic field, density, temperature, mass, velocity, and length are normalized by B₁, n₁, $T_{e 1}$ ⁠, m_e, v_th, and r_L, respectively, where $v_{t h} = \sqrt{T_{e 1} / m_{e}}$ is taken as the electron thermal speed, and $r_{L} = m_{e} v_{t h} / e B_{1}$ is the thermal electron gyroradius on the MSP side.

The model coordinate system {x, y, z} corresponds to typical GSE coordinates at the subsolar magnetopause: $+ x$ points sunward along the sun-Earth line, $+ z$ is directed northward along the magnetic field direction in the magnetosphere, and $+ y$ completes a right-handed system and is oriented along the current (opposite to the electron flow). For comparison to the two MMS events presented in Figs. 2–4, the model quantities are plotted with respect to –x. Additionally, the MMS data are plotted in field-aligned coordinates where the $⊥_{1}$ direction points along $(- U_{e} \times B) \times B$ (roughly aligned with the $E \times B$ direction), and the $⊥_{2}$ direction points along $- U_{e} \times B$ (roughly along E and $\nabla n_{e}$ ⁠). Since the $⊥_{1}$ direction is approximately oriented along the model –y direction, the figures are adjusted accordingly to facilitate visual comparisons.

A. Imposing quasineutrality

The electron distribution function f_e in Eq. (8) depends on both $ϕ$ and A_y. Because of the chosen form for the $T (c_{e 2})$ function, f_e cannot be written in a factorized form that conveniently separates the $ϕ$ and A_y dependences. Nevertheless, the quasineutrality requirement enables determination of $ϕ$ as a function of A_y. Integrating Eq. (5), the ion density is readily obtained as a function of $ϕ$ ⁠, which we then equate to the integral expression for the electron density

n_{i} (ϕ) = n_{e} (ϕ, A_{y}),

(9)

n_{1} exp [\frac{- e ϕ}{T_{i}}] = \int_{- \infty}^{+ \infty} f_{e} (ϕ, A_{y}) d^{3} v .

(10)

We numerically obtain a discretized representation of $ϕ (A_{y})$ by first constructing an array with k elements of A_y values, $A_{y}^{k}$ ⁠, and then implementing a root-finding algorithm to determine the value $ϕ = ϕ_{k}$ that satisfies Eq. (10) for each $A_{y}^{k}$ ⁠. Using this $ϕ_{k} (A_{y}^{k})$ ⁠, we numerically integrate f_e to acquire the electron moments as functions of A_y and $U_{e y} (A_{y}), T_{e} (A_{y})$ ⁠. Since A_y is assumed to be monotonically increasing, we have $B_{z} > 0$ ⁠, and so the magnetic field $B_{z} (A_{y})$ may be obtained directly from these particle moments using the pressure balance condition derived from integrating the force balance equation

\int \nabla \cdot P d x = \int J \times B d x \to B_{z} = \sqrt{2 μ_{0} [P_{1} - n (T_{i} + T_{exx})]},

(11)

where P₁ is the total pressure (a constant) given by $P_{1} = \frac{B_{1}^{2}}{2 μ_{0}} + n_{1} (T_{i} + T_{e 1})$ ⁠. Finally, utilizing the definition of the magnetic vector potential A_y, we obtain the spatial parameter x explicitly as follows:

\int_{x_{0}}^{x} d x = \int_{x_{0}}^{x} \frac{1}{B_{z}} \frac{d A_{y}}{d x} d x = \int_{A_{y} (x_{0})}^{A_{y} (x)} \frac{1}{B_{z}} d A_{y} \to x - x_{0} = \int_{A_{y 0}}^{A_{y}} \frac{1}{B_{z}} d A_{y},

(12)

from which we find the moments and fields quantities as functions of x, such as $U_{e y} (x), T_{e} (x), B_{z} (x)$ ⁠, etc.

Although Poisson's equation is not strictly satisfied, an electric field E_x consistent with the potential $ϕ (A_{y})$ acquired from quasineutrality is given by $E_{x} = - \partial ϕ / \partial x = - (\partial ϕ / \partial A_{y}) (\partial A_{y} / \partial x) = - (\partial ϕ / \partial A_{y}) B_{z}$ ⁠. From a dimensionless Gauss' law, the actual charge difference can be assessed via

\frac{e (n_{i} - n_{e})}{ϵ_{0}} = \frac{\partial E_{x}}{\partial x} \to n'_{i} - n'_{e} = (\frac{ϵ_{0} v_{t h} B_{1}}{e n_{1} r_{L}}) \frac{\partial E'_{x}}{\partial x'} = {(\frac{v_{A, e}}{c})}^{2} \frac{\partial E'_{x}}{\partial x'},

(13)

where $n'_{i} - n'_{e} = (n_{i} - n_{e}) / n_{1}$ is the normalized charge difference, the electric field $E'_{x} = E_{x} / (v_{t h} B_{1})$ and spatial coordinate $x' = x / r_{L}$ are dimensionless quantities, $v_{A, e} = B_{1} / \sqrt{μ_{0} m_{e} n_{1}}$ is the electron Alfvén speed based on B₁ and n₁, and $c = 1 / \sqrt{μ_{0} ϵ_{0}}$ is the speed of light. For the magnetopause conditions encountered by MMS in the events considered (⁠ $B_{1} \approx 30$ nT, $n_{1} \approx 10$ cm⁻³, $T_{e 1} \approx 70$ eV, $v_{A, e} \approx$ 9000 km/s), the ratio $v_{A, e} / c$ is on the order of 0.03. The peak electric field ranges from about 0.1 to 0.2 $v_{t h} B_{1}$ ⁠, and $\partial E_{x} / \partial x$ is on the order of 0.01 to 0.1 $v_{t h} B_{1} / r_{L}$ ⁠. With these parameters, we estimate the actual charge imbalance in Eq. (13) to be roughly 10^–4 or 10^–5. Thus, quasineutrality is a reasonable approximation for these events.

A visualization of various properties of the Vlasov–Maxwell solution described above is featured in Fig. 1, which displays results from twelve different model runs obtained by systematically varying each of the independent parameters β_e, $T_{i} / T_{e 1}, T_{e 2} / T_{e 1}$ ⁠, and S. Figures 1(a)–1(d) illustrate how the spatial dependence of the magnetic field $B_{z} (x)$ ⁠, density n(x), electron bulk velocity $U_{e y} (x)$ ⁠, and current density $J_{y} (x)$ changes with three different values (black, blue, and green traces) of the initial electron beta β_e, ion to electron temperature ratio $T_{i} / T_{e 1}$ ⁠, electron temperature ratio $T_{e 2} / T_{e 1}$ ⁠, and the distribution steepness parameter S, respectively. Figure 1(e) presents an example of the force balance between $J \times B$ (blue) and $\nabla \cdot P$ (green points), where $P = P_{i} + P_{e}$ ⁠, along with decompositions of $\nabla \cdot P$ into $\nabla \cdot P_{i}$ (red) and $\nabla \cdot P_{e}$ (black), and $\nabla \cdot P_{e}$ into $n_{e} \nabla \cdot T_{e}$ (orange) and $T_{e} \cdot \nabla n_{e}$ (aqua) for a model solution with the following parameters: S = 10, $T_{e 2} / T_{e 1} = 2, T_{i} / T_{e 1} = 10$ ⁠, and $β_{e} = 0.2$ ⁠. Scaling relationships with black, blue, and green points corresponding to the twelve model runs in Figs. 1(a)–1(d) (three runs per row) are provided in Figs. 1(f)–1(i) and will be described with more detail in Secs. II B–II E. The parameters displayed in the dashed boxes adjacent to Figs. 1(f)–1(i) were held constant for each set of three runs shown in Figs. 1(a)–1(d). The effect of the free parameter S on the half-thickness of the current layer and the shape of 2D electron distribution slices in v_x–v_y velocity space (taken along v_z = 0) is shown in Figs. 1(i) and Fig. 1(j–1), respectively, with 1D cuts along v_x = 0 included below each distribution slice. We performed two additional runs for the S = 2 and S = 5 cases shown in Fig. 1(j) and Fig. 1(k), respectively. The effect that increasing the electron temperature ratio $T_{e 2} / T_{e 1}$ has on the distribution's structure is presented in Figs. 1(m)–1(o), again with 1D cuts along v_x = 0 beneath the distribution panels.

FIG. 1.

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Spatial evolution of the Vlasov–Maxwell solution and associated scaling relationships for a variety of initial parameter selections. (a) Magnetic field strength B_z, (b) electron density n_e, (c) electron bulk velocity $- U_{e y}$ ⁠, (d) current density J_y, and (e) decomposition of the force balance within the layer: divergence of the total pressure ${(\nabla \cdot P)}_{x}$ (green points), ${(J \times B)}_{x}$ force (blue), ion pressure divergence ${(\nabla \cdot P_{i})}_{x}$ (red), electron pressure divergence ${(\nabla \cdot P_{e})}_{x}$ (black), electron temperature gradient term $n_{e} {(\nabla \cdot T_{e})}_{x}$ (orange), and electron density gradient term ${(T_{e} \cdot \nabla n_{e})}_{x}$ (aqua). (f) Dependence of $B_{2} / B_{1}$ on the initial β_e. (g) Dependence of $n_{2} / n_{1}$ on the ion to initial electron temperature ratio $T_{i} / T_{e 1}$ ⁠. (h) Maximum U_ey as a function of the electron temperature ratio $T_{e 2} / T_{e 1}$ with several approximate scaling curves (described in Sec. II). (i) Effect that increasing S has on the half-thickness of the current layer $L / 2$ (in units of r_L). The black, blue, and green points in panels (f)–(i) correspond to the three solutions shown in (a)–(d), respectively. (j)–(l) Effect that increasing S has on the crescent-shaped electron velocity distribution's structure near the location of maximum current density. (m)–(o) Electron distribution structures as a function of $T_{e 2} / T_{e 1}$ across the layer. The panels directly below the 2D distribution panels in (j)–(o) show 1D cuts of the distributions taken along v_x = 0.

The Vlasov–Maxwell solutions displayed in Fig. 1 are asymmetric (⁠ $n_{2} > n_{1}$ and $B_{2} < B_{1}$ ⁠) and have thicknesses on the order of a few electron thermal gyroradii. The structure of the 2D electron distribution slices significantly depends on the electron temperature ratio $T_{e 2} / T_{e 1}$ and the steepness parameter S. For small values $S ≲ 2$ ⁠, there is a relatively smooth transition in velocity-space along v_y between the hotter electron population with $T_{e 2}$ and the colder population with $T_{e 1}$ [Fig. 1(j)]. For larger values $S ≳ 10$ ⁠, the velocity-space transition is sharp, resulting in pronounced semi-circular (crescent-like) contours of the phase space density [Fig. 1(l)]. For a given S, increasing the electron temperature ratio $T_{e 2} / T_{e 1}$ exaggerates the transition between the hot and cold populations as seen in Figs. 1(m)–1(o). When the temperature ratio is small $T_{e 2} / T_{e 1} ≲ 1.2$ ⁠, there is only a slight change in the distribution's structure [Fig. 1(m)], whereas increasing the temperature ratio to $T_{e 2} / T_{e 1} ≳ 1.5$ visibly amplifies the velocity-space transition [Fig. 1(o)].

B. Scaling properties across the asymmetric current layer

In this section, we derive several useful scaling relationships between characteristic parameter ratios for the Vlasov–Maxwell solution.

1. Density ratio: $n_{2} / n_{1}$

From quasineutrality in this model, the density ratio $n_{2} / n_{1}$ is set by the ion and electron temperature ratios. On the high-density (MSH) side of the layer as $x \to \infty$ ⁠, we have $ϕ (x) \to ϕ_{2}$ and $A_{y} (x) \to \infty$ ⁠, which means that $T (c_{e 2}) \to T_{e 2}$ ⁠. In this limit, the electron distribution function becomes

f_{e 2} = n_{1} {(\frac{m_{e}}{2 π T_{e 1}})}^{3 / 2} exp [\frac{- \frac{m_{e}}{2} v^{2} + e ϕ_{2}}{T_{e 2}}] .

(14)

Thus, integrating over velocity space, the MSH electron density n₂ is given by

n_{2} = n_{1} exp [\frac{e ϕ_{2}}{T_{e 2}}] {(\frac{T_{e 2}}{T_{e 1}})}^{3 / 2} .

(15)

Equating this expression to the MSH ion density, we solve for $ϕ_{2}$ as follows:

n_{2} = n_{i} (ϕ_{2}) = n_{e} (ϕ_{2}),

(16)

n_{2} = n_{1} exp [\frac{- e ϕ_{2}}{T_{i}}] = n_{1} exp [\frac{e ϕ_{2}}{T_{e 2}}] {(\frac{T_{e 2}}{T_{e 1}})}^{3 / 2},

(17)

ϕ_{2} = - \frac{T_{i}}{e} ln [{(\frac{T_{e 2}}{T_{e 1}})}^{a}],

(18)

where the exponent a is given by

a = \frac{3 / 2}{(\frac{T_{i}}{T_{e 2}}) + 1} = \frac{3 / 2}{(\frac{T_{i}}{T_{e 1}}) (\frac{T_{e 1}}{T_{e 2}}) + 1} .

(19)

Substituting Eq. (18) into the expression for the ion density and dividing by n₁, we obtain the desired density ratio

\frac{n_{2}}{n_{1}} = {(\frac{T_{e 2}}{T_{e 1}})}^{a},

(20)

which is plotted as a function of T_i in Fig. 1(g) (black trace). Considering the theoretical limits $T_{i} / T_{e} ≫ 1$ and $T_{i} / T_{e} ≪ 1$ ⁠, which correspond to either infinite or zero ion temperature, the range for a is $0 \leq a \leq 3 / 2$ ⁠. In the limit where $T_{i} = T_{e}$ ⁠, then a = 3/4. For a strong temperature gradient (⁠ $T_{e 2} / T_{e 1} = 2$ ⁠) near the magnetopause (⁠ $T_{i} / T_{e 1} = 10$ ⁠), then a = 1/4 and $n_{2} / n_{1} \approx 1.2$ ⁠.

Despite the relatively large ion temperature $T_{i} / T_{e 1} = 10$ in Fig. 1(e), the electrons' contribution to the overall force balance is roughly equal to that of the ion $T_{i} \nabla n$ term (red trace), mainly due to the large $n_{e} \nabla \cdot T_{e}$ term (orange trace) associated with the electron temperature divergence. Even in the limit of infinite ion temperature (⁠ $T_{i} / T_{e 1} \to \infty$ ⁠), the finite $n_{e} \nabla \cdot T_{e}$ term will still contribute significantly to the force balance. This is because increasing T_i reduces $n_{2} / n_{1}$ as shown in Eq. (20), which in turn decreases the magnitude of $\nabla n$ and thus limits the overall effect of the ion $T_{i} \nabla n$ term.

2. Magnetic field ratio: $B_{2} / B_{1}$

Pressure balance across the current layer requires

\frac{B_{1}^{2}}{2 μ_{0}} + n_{1} (T_{i} + T_{e 1}) = \frac{B_{2}^{2}}{2 μ_{0}} + n_{2} (T_{i} + T_{e 2}),

(21)

which is readily obtained by integrating the force balance equation $J \times B = \nabla \cdot P \to - B_{z} (\partial B_{z} / \partial x) / μ_{0} = \partial P_{x x} / \partial x$ ⁠. Dividing both sides of Eq. (21) by $n_{1} T_{e 1}$ ⁠, we have

\frac{1}{β_{e}} + \frac{T_{i}}{T_{e 1}} + 1 = \frac{1}{β_{e}} {(\frac{B_{2}}{B_{1}})}^{2} + \frac{n_{2}}{n_{1}} (\frac{T_{i}}{T_{e 1}} + \frac{T_{e 2}}{T_{e 1}}),

(22)

where $β_{e} = 2 μ_{0} n_{1} T_{e 1} / B_{1}^{2}$ ⁠. Using Eq. (20), we can solve for $B_{2} / B_{1}$ in terms of the ion and electron temperatures

\frac{B_{2}}{B_{1}} = \sqrt{1 + β_{e} [1 + \frac{T_{i}}{T_{e 1}} - {(\frac{T_{e 2}}{T_{e 1}})}^{a} (\frac{T_{i}}{T_{e 1}} + \frac{T_{e 2}}{T_{e 1}})]} = \sqrt{1 - \frac{β_{e}}{β_{e}^{*}}},

(23)

where $β_{e}^{*}$ is given by

β_{e}^{*} = - {[1 + \frac{T_{i}}{T_{e 1}} - {(\frac{T_{e 2}}{T_{e 1}})}^{a} (\frac{T_{i}}{T_{e 1}} + \frac{T_{e 2}}{T_{e 1}})]}^{- 1},

(24)

and where β_e is chosen such that $0 < β_{e} < β_{e}^{*}$ ⁠. Equation (23) is plotted as a function of β_e in Fig. 1(f) (black trace). For $T_{i} > T_{e 1}$ and $T_{e 2} > T_{e 1}$ ⁠, the expression in brackets under the square root in Eq. (23) is negative. This implies that for a given set of temperature parameters, the ratio $B_{2} / B_{1}$ in Eq. (23) would become imaginary if $β_{e} > β_{e}^{*}$ ⁠. As $β_{e} \to β_{e}^{*}$ ⁠, we have $B_{2} / B_{1} \to 0$ ⁠, and the structure of the Vlasov–Maxwell solution undergoes a bifurcation: for $β_{e} > β_{e}^{*}$ ⁠, the current layer becomes symmetric with a B_z reversal and a corresponding A_y profile that is no longer a monotonically increasing function of x. Since the focus of this paper is on asymmetric equilibria, we consider only cases for which $β_{e} < β_{e}^{*}$ ⁠.

3. Approximation for the maximum $U_{e y} / v_{t h}$

From Ampère's law in Eq. (4), we can obtain an approximation for the maximum $U_{e y} / v_{t h}$ as a function of the previously derived parameter ratios. Near the maximum velocity $U_{e y}^{*}$ at the center of the current layer, we estimate $d B / d x \approx (B_{2} - B_{1}) / L r_{L}$ ⁠, where the dimensionless quantity $L = Δ x / r_{L}$ is a number specifying the width of the layer in units of r_L, and we can approximate the density as $n_{e} \approx (n_{2} + n_{1}) / 2$ ⁠. Inputting these estimations into Ampère's law, we have

\frac{d B}{d x} = μ_{0} e n_{e} U_{e y} \to \frac{B_{2} - B_{1}}{L r_{L}} \approx μ_{0} e (\frac{n_{2} + n_{1}}{2}) U_{e y}^{*} .

(25)

Solving for $U_{e y}^{*} / v_{t h}$ ⁠, we find

\frac{U_{e y}^{*}}{v_{t h}} \approx (\frac{2}{L}) (\frac{1}{μ_{0} e r_{L} v_{t h}}) (\frac{B_{2} - B_{1}}{n_{2} + n_{1}}) = (\frac{4}{L}) (\frac{1}{β_{e}}) (\frac{\frac{B_{2}}{B_{1}} - 1}{\frac{n_{2}}{n_{1}} + 1}) .

(26)

For the runs presented in Fig. 1(c), choosing a thickness in the range L = 2–3 provides a reasonable estimate of how the peak U_ey scales with density, temperature, and magnetic field strength [see the magenta and purple traces in Fig. 1(h)]. Section II C describes a more sophisticated approach for obtaining an explicit representation of U_ey and hence a more accurate approximation for $U_{e y}^{*}$ ⁠.

C. Piecewise Maxwellian limit for f_e

In the limit as $S \to \infty$ ⁠, the $T (c_{e 2})$ function in Eq. (7) approaches a step function in velocity-space along the v_y direction. Thus, in this limit f_e approaches a piecewise Maxwellian

f_{e} = {\begin{matrix} f_{e 1}, & v_{y} \geq v_{y}^{*}, \\ f_{e 2}, & v_{y} < v_{y}^{*}, \end{matrix}

(27)

where $f_{e 1} = f_{e 0} exp [- c_{e 1} / T_{e 1}], f_{e 2} = f_{e 0} exp [- c_{e 1} / T_{e 2}]$ ⁠, and $v_{y}^{*} = e A_{y} (x) / m_{e}$ is the spatially dependent velocity marking the velocity-space transition between $f_{e 1}$ and $f_{e 2}$ ⁠. This conveniently allows the moments of f_e to be computed analytically in a piecewise fashion.

The exact expression for the electron density n_e as a function of $ϕ$ and A_y in this limit is given by

n_{e} (ϕ, A_{y}) = \frac{n_{1}}{2} ([1 - erf (\frac{A_{y}}{B_{1} r_{L} \sqrt{2}})] exp (\frac{e ϕ}{T_{e 1}}) + {(\frac{T_{e 2}}{T_{e 1}})}^{3 / 2} [1 + erf (\frac{A_{y}}{B_{1} r_{L}} \sqrt{\frac{T_{e 1}}{2 T_{e 2}}})] exp (\frac{e ϕ}{T_{e 2}})),

(28)

= n_{1} [Φ (\frac{- A_{y}}{B_{1} r_{L}}) exp (\frac{e ϕ}{T_{e 1}}) + {(\frac{T_{e 2}}{T_{e 1}})}^{3 / 2} Φ (\frac{A_{y}}{B_{1} r_{L}} \sqrt{\frac{T_{e 1}}{T_{e 2}}}) exp (\frac{e ϕ}{T_{e 2}})],

(29)

where $erf (x)$ is the error function, and $Φ (x)$ is the cumulative distribution function for the standard normal distribution. As $A_{y} \to - \infty$ ⁠, we have that $n_{e} \to n_{1}$ ⁠, while for $A_{y} \to + \infty$ ⁠, Eq. (29) reduces to Eq. (15). To compute the v_y integrals, we make use of the symmetry of the Gaussian functions to conveniently adjust the limits of integration, and we utilize the relationship between the cumulative distribution function $Φ (x)$ and the error function $erf (x)$ (see the Appendix for full derivations).

For the electron bulk velocity U_ey, we find

\begin{matrix} U_{e y} (ϕ, A_{y}) = (\frac{v_{t h}}{\sqrt{2 π}}) {exp [\frac{- 1}{2} {(\frac{A_{y}}{B_{1} r_{L}})}^{2}] exp [(1 + \frac{T_{e 1}}{T_{i}}) \frac{e ϕ}{T_{e 1}}] \\ - {(\frac{T_{e 2}}{T_{e 1}})}^{2} exp [\frac{- 1}{2} (\frac{T_{e 1}}{T_{e 2}}) {(\frac{A_{y}}{B_{1} r_{L}})}^{2}] exp [(1 + \frac{T_{e 2}}{T_{i}}) \frac{e ϕ}{T_{e 2}}]} . \end{matrix}

(30)

The black trace in Fig. 1(h) is a plot of Eq. (30) evaluated at A_y = 0, which corresponds to a spatial location close to where the maximum U_ey occurs, and hence represents a more accurate scaling relationship for the ratio $U_{e y}^{*} / v_{t h}$ ⁠. This can be understood by considering how the electron distribution in Eq. (8) depends on A_y. Starting on the MSP side with $A_{y} = - \infty$ ⁠, the electron distribution $f_{e} = f_{e 1}$ is composed entirely of colder electrons with temperature $T_{e 1}$ ⁠. As A_y increases, electrons from the hotter $T_{e 2}$ population begin to be added to the $- v_{y}$ side of the distribution. The contribution to U_ey from these hotter $T_{e 2}$ electrons on the $- v_{y}$ side surpasses the colder $T_{e 1}$ electrons' contribution on the $+ v_{y}$ side, resulting in the net U_ey < 0 signature throughout the layer. As more $T_{e 2}$ electrons are added to the distribution, the magnitude of U_ey increases up until roughly A_y = 0, where the electron nongyrotropy is large and the distribution takes the form of Eq. (27) with $v_{y}^{*} = 0$ ⁠. Increasing A_y beyond A_y = 0 adds additional hot $T_{e 2}$ electrons to the $+ v_{y}$ side of the distribution, which makes the electron distribution more gyrotropic and decreases $| U_{e y} |$ up until the MSH side where $A_{y} = + \infty$ and $f_{e} = f_{e 2}$ ⁠. Thus, the location of maximum U_ey occurs close to where the most $T_{e 2}$ electrons appear on the $- v_{y}$ side of the distribution, which is near A_y = 0 (⁠ $v_{y}^{*} = 0$ ⁠).

D. Phase-space structures and electron orbits

Figures 2(a)–2(c) show how the phase space structure of the modeled current layer in the x-v_y plane depends on various choices of the initial electron beta β_e with a comparison to the phase space structure of an electron-scale layer observed by MMS on December 23, 2016, in Fig. 2(e) (more details about this event are presented in Sec. III). Figure 2(d) shows a characteristic phase space trajectory in three-dimensional x-v_x-v_y space for an energetic electron with initial parameters {x₀, $v_{x 0}, v_{y 0}$ } = { $12 r_{L}, 3 v_{t h}$ ⁠, 0} taken from the solution with $β_{e} = 0.468$ ⁠, which is close to the maximum $β_{e}^{*}$ given by Eq. (24) for the initial parameters and ratios used in Figs. 2(a)–2(d): S = 10, $T_{i} / T_{e 1} = 10$ ⁠, and $T_{e 2} / T_{e 1} = 1.7$ ⁠.

FIG. 2.

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Phase space structure of the electron-scale current layer in the Vlasov–Maxwell model and MMS event on December 23, 2016. (a)–(c) Slices of the modeled electron phase space density $f (x, v_{y})$ at v_x = 0 for three different choices of the initial electron beta β_e. The colorbar and axis labels for (a) and (b) are the same as those shown for (c). (d) Characteristic orbit (solid black trace) and projections (dashed traces) of an energetic electron starting from ${x_{0}, v_{x 0}, v_{y 0}} = {12 r_{L}, 3 v_{t h}, 0}$ ⁠. Projections of the orbit in the x–v_y plane for the test electron are shown as magenta traces in (a)–(d) and are aligned with the curve given by $v_{y}^{*} = e A_{y} (x) / m_{e}$ (black and white dashes), which defines the phase space transition plotted in (a)–(c). (e) Reduced phase space density in t- $v_{⊥ 1}$ observed by MMS 1 on December 23, 2016 (see Fig. 4 for an overview of this event). Using the structure speed of 60 km/s determined from four spacecraft timing analysis, the time axis can be converted to distance, which is shown normalized by an upstream electron thermal gyroradius taken near 02:53:10.5 UT. Integrating the observed magnetic field over this distance yields an estimate of $A_{⊥ 1} (x_{⊥ 2})$ ⁠, which is shown (black and white dashed trace) for comparison with the Vlasov–Maxwell solution. We note that the x axis in (a)–(c) here has been relabeled since it shows a larger interval of the model solution than what is shown in the other figures.

Electrons with large $- v_{y}$ (+ $v_{⊥ 1}$ ⁠) penetrate deepest into the MSP side of the layer (up to several r_L in the –x direction), which produces the visible protrusion of phase space density toward the low-density (MSP) side. Prior to this region of enhanced phase space density in $- v_{y}$ ⁠, there is also a depletion of electrons with $+ v_{y}$ (⁠ $- v_{⊥ 1}$ ⁠) that, depending on the initial β_e, can extend up to tens of r_L toward the higher density (MSH) side of the layer. The overall shift of phase space density toward $- v_{y}$ corresponds to the bulk $E \times B$ = $- E_{x} B_{z} \hat{y}$ drift in this region. Just as $v_{y}^{*}$ denotes the velocity-space transition between $f_{e 1}$ and $f_{e 2}$ [see Eq. (27)], here the curve defining the edge in phase space is given by $v_{y}^{*} (x) = e A_{y} (x) / m_{e}$ ⁠. This curve is found by setting the argument of the hyperbolic tangent function equal to 0 in Eq. (8). Similarly, any electron orbit's projection in the x-v_y plane [magenta traces in Figs. 2(a)–2(d)] is governed by $A_{y} (x)$ from the expression for $c_{e 2}$ in Eq. (2): $v_{y} (x) = (c_{e 2} + e A_{y} (x)) / m_{e}$ ⁠. The MMS observations in Fig. 2(e) have $β_{e} = 0.33$ and are qualitatively consistent with these Vlasov–Maxwell model solutions. Figure 2(e) shows that MMS detected a slanted enhancement of phase space density in $+ v_{⊥ 1}$ preceded by a tilted depletion of phase space density in $- v_{⊥ 1}$ that roughly aligns with the $A_{⊥ 1} (x_{⊥ 2})$ curve obtained from integrating the magnetic field over the interval.

At a given spatial location x, velocity distributions in v_x-v_y (⁠ $v_{⊥ 1}$ - $v_{⊥ 2}$ ⁠) space are vertical slices of the phase space density structures shown in Figs. 2(a)–(c) and 2(e). Thus, velocity space distributions in v_x-v_y represent a mixture of both the low-temperature electron population and the energetic penetrating electrons. The projection of the particle's orbit in the v_x-v_y plane is approximately circular [rightmost projection in Fig. 2(d)]. The direction of an electron's net drift in the y-direction depends on both the electron's initial energy and its proximity to the current layer, which determine whether the $\nabla B$ drift (pointing in the $+ y$ direction for electrons) overpowers the $E \times B$ drift (directed along –y). Since there are no forces acting parallel to the magnetic field in the z-direction, the particle's initial velocity $v_{z 0}$ remains unchanged.

The visible elongation of the electron orbit in $+ x$ toward the high density (low magnetic field) side in Fig. 2(d) is responsible for the skewed spatial structure of the layer with its tail extending toward the $+ x$ direction (e.g., Fig. 1). The orbit's shape is understood by considering the combined effects of $E_{x} (x)$ and $B_{z} (x)$ on the gyrating electron. The particle's orbit in the x-v_x-v_y phase space [Fig. 2(d)] is closed as a result of the solution's symmetries. As the electron gyrates toward the center of the current sheet (toward –x), the increasing electric force $- e E_{x}$ accelerates the electron faster toward the central layer. The electron's $| v_{x} |$ increases up until it encounters the maximum E_x near $x \approx 10 r_{L}$ ⁠. At this stage, the electron's gyrofrequency is higher due to the larger magnetic field strength, so B_z works to quickly reverse the electron's velocity from $- v_{x}$ to $+ v_{x}$ ⁠. The rapidly gyrating electron with large acquired $+ v_{x}$ promptly exits the central current layer region and travels far (up to tens of r_L depending on β_e) into the high temperature side of the layer back to its original x location. During this stage, the electric force $- e E_{x}$ acts to decelerate the electron. On the high temperature (low magnetic field) side of the layer (toward +x), the electron's gyrofrequency is smaller, but eventually B_z rotates the electron's velocity back toward $- v_{x}$ ⁠, and the cyclic orbit repeats. The acceleration by E_x in this model is analogous to the acceleration from the normal electric field reported by Bessho et al. (2016) in the context of asymmetric reconnection simulations, except here electrons do not perform meandering, crescent-shaped orbits since there is no magnetic field reversal in our model.

E. Thickness of the current layer

Figures 1 and 2 show that the thickness of the current layer, L, visibly increases in any of the following limits: $β_{e} \to β_{e}^{*}$ ⁠, $T_{e 2} / T_{e 1} \to \infty$ ⁠, and $S \to 0$ ⁠. When we let $S \to \infty$ (discussed in Sec. II C above), the velocity-space boundary between the $T_{e 1}$ and $T_{e 2}$ electron populations becomes infinitely sharp, yet L cannot become smaller than the scale of a thermal electron gyroradius, r_L. In the $S \to \infty$ limit, the velocity space gradient term $| (F / m_{e}) \cdot \nabla_{v} f_{e} | \approx | (e B_{z} / m_{e}) v_{x} \partial f_{e} / \partial v_{y} | \to \infty$ ⁠, which must balance with a corresponding structure in the spatial gradient term $v \cdot \nabla f_{e} = v_{x} \partial f_{e} / \partial x \to \infty$ along the curve $v_{y}^{*} (x)$ ⁠.

In the limit as $β_{e} \to 0$ ⁠, the magnetic field ratio $B_{2} / B_{1} \to 1$ ⁠. In this case, the magnetic pressure dominates the plasma pressure so that the particles' dynamics do not appreciably alter the magnetic field profile. In this situation, in order for β_e to vanish, either $B_{1} = B_{2} = B_{0} \to \infty$ ⁠, or $P_{e 1} = n_{1} T_{e 1} \to 0$ ⁠. Alternatively, in the limit as $β_{e} \to β_{e}^{*}$ given in Eq. (24), we have $B_{2} / B_{1} \to 0$ from Eq. (23), which is shown by the scaling relationship between $B_{2} / B_{1}$ and β_e in Fig. 1(f). In this limit, there are two possibilities: either (1) $B_{1} \to \infty$ (corresponding to an infinitely sharp magnetic wall), or (2) $B_{2} \to 0$ ⁠. If $B_{1} \to \infty$ ⁠, then for a finite electron temperature we have $r_{L} \to 0$ ⁠, and the layer becomes arbitrarily small to sustain the infinite magnetic gradient. If $B_{2} \to 0$ with a finite B₁ [similar to Fig. 1(a) or Fig. 2(c)], then the slope of $A_{y} (x)$ on the MSH side of the layer approaches 0. Thus, as $A_{y} (x)$ flattens, the skew of the layer toward the MSH side extends infinitely since the vanishing B₂ is unable to rotate the electron's velocity back toward the MSP side of the current layer.

III. SPATIAL EVOLUTION OF THE MMS OBSERVATIONS AND THE VLASOV–MAXWELL MODEL

In this section, we compare the Vlasov–Maxwell equilibrium solution directly to MMS observations of two thin current sheets exhibiting electron-scale temperature gradients recently found in the vicinity of Earth's magnetopause (Shuster et al., 2019; 2021). The Fast Plasma Investigation (FPI) onboard MMS measures a full 3D distribution function for ions and electrons at unprecedented time cadences of 150 and 30 ms, respectively (Pollock et al., 2016), which is fast enough to resolve electron-scale current layers with thicknesses on the order of a few electron gyroradii. Table I lists relevant plasma parameter ratios observed by MMS for the two events featured in Figs. 3 and 4 alongside ratios from the Vlasov–Maxwell (VM) model runs chosen for comparison.

TABLE I.

Comparison of observed MMS and modeled Vlasov–Maxwell (VM) plasma parameter ratios for the events on November 28, 2016, and December 23, 2016, shown in Figs. 3 and 4, respectively.

Event	November 28, 2016		December 23, 2016
Ratio	MMS	VM	MMS	VM
$\frac{n_{2}}{n_{1}}$	1.15	1.03	1.27	1.06
$\frac{B_{2}}{B_{1}}$	0.48	0.48	0.38	0.38
$\frac{T_{i}}{T_{e 1}}$	11	11	13	13
$\frac{T_{e 2}}{T_{e 1}}$	1.15	1.20	1.18	1.50
${(\frac{T_{e 2}}{T_{e 1}})}_{⊥}$	1.19	1.20	1.48	1.50
β_e	0.37	1.44	0.33	0.59
$β_{e}^{*}$	0.48	1.86	0.39	0.69
$\frac{β_{e}}{β_{e}^{*}}$	0.77	0.77	0.85	0.85
${\| \frac{U_{e y}}{v_{t h}} \|}_{\max}$	0.22	0.16	0.39	0.41

Event	November 28, 2016		December 23, 2016
Ratio	MMS	VM	MMS	VM
$\frac{n_{2}}{n_{1}}$	1.15	1.03	1.27	1.06
$\frac{B_{2}}{B_{1}}$	0.48	0.48	0.38	0.38
$\frac{T_{i}}{T_{e 1}}$	11	11	13	13
$\frac{T_{e 2}}{T_{e 1}}$	1.15	1.20	1.18	1.50
${(\frac{T_{e 2}}{T_{e 1}})}_{⊥}$	1.19	1.20	1.48	1.50
β_e	0.37	1.44	0.33	0.59
$β_{e}^{*}$	0.48	1.86	0.39	0.69
$\frac{β_{e}}{β_{e}^{*}}$	0.77	0.77	0.85	0.85
${\| \frac{U_{e y}}{v_{t h}} \|}_{\max}$	0.22	0.16	0.39	0.41

FIG. 3.

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Comparison of MMS observations from November 28, 2016, to the Vlasov–Maxwell solution with similar initial parameter ratios. (a)–(g) MMS data: (a) Total magnetic field strength $| B |$ ⁠. (b) Electron density n_e (black) with two measures of the density gradient ${(\nabla n_{e})}_{⊥ 2}$ (aqua, dashed) and $V^{- 1} \partial n_{e} / \partial t$ (aqua, solid). (c) Electron perpendicular bulk velocity $U_{e ⊥ 1}$ with two measures of the velocity gradient ${(\nabla U_{e ⊥ 1})}_{⊥ 2}$ (magenta, dashed) and $V^{- 1} \partial U_{e ⊥ 1} / \partial t$ (magenta, solid). The gradient terms based on $\partial / \partial t$ in (c) and (d) were computed using V = 30 km/s determined from four-spacecraft timing analysis. (d) Total $T_{e, tot}$ (black), parallel $T_{e ∥}$ (blue), and perpendicular $T_{e ⊥}$ (red) electron temperature profiles with the $T_{e, ⊥ 1, ⊥ 1}$ component of the temperature tensor (green). (e) Decomposition of the electron pressure divergence ${(\nabla \cdot P_{e})}_{⊥ 2}$ (black) into the temperature gradient $n_{e} {(\nabla \cdot T_{e})}_{⊥ 2}$ (orange) and density gradient ${(T_{e} \cdot \nabla n_{e})}_{⊥ 2}$ (aqua) terms. (f) Electric field $E_{⊥ 2}$ (black) and $U_{e ⊥ 1} B$ (green) terms. (g) Pressure balance P_tot (green) showing the sum of magnetic pressure $B^{2} / 2 μ_{0}$ (blue), ion thermal pressure $n_{i} T_{i}$ (red), and electron thermal pressure $n_{e} T_{e}$ (black). (h)–(n) Vlasov–Maxwell results: (h) magnetic field B_z with simulation parameters displayed to the right of the panel. (i) Electron density n_e (black) and density gradient ${(\nabla n_{e})}_{x}$ (aqua, dashed). (j) Electron bulk velocity $- U_{e y}$ and velocity gradient ${(\nabla U_{e y})}_{x}$ (magenta, dashed). (k) Total $T_{e, tot}$ (black), parallel $T_{e ∥}$ (blue), and perpendicular $T_{e ⊥}$ (red) electron temperature profiles with the T_eyy component of the temperature tensor (green). (l) Decomposition of the electron pressure divergence ${(\nabla \cdot P_{e})}_{x}$ (black) into the temperature gradient $n_{e} {(\nabla \cdot T_{e})}_{x}$ (orange) and density gradient ${(T_{e} \cdot \nabla n_{e})}_{x}$ (aqua) terms. (m) Electric field E_x (black) and $- U_{e y} B$ (green) terms. (n) Pressure balance P_tot (green) showing the sum of magnetic pressure $B^{2} / 2 μ_{0}$ (blue), ion thermal pressure $n_{i} T_{i}$ (red), and electron thermal pressure $n_{e} T_{exx}$ (black). (t₁–t₇) Electron distribution function 2D $v_{⊥ 1}$ – $v_{⊥ 2}$ slices with corresponding 1D cuts along $v_{⊥ 2} = 0$ directly below each 2D slice for the seven consecutive times indicated by vertical gray bars in the MMS panels (a)–(g). The 1D cuts show $f_{e} (v_{⊥ 1})$ (black diamonds) with gray error bars and light yellow background indicating the measurement uncertainty. The blue and red dotted lines indicate cold and hot reference Maxwellians with temperatures of 40 and 70 eV, respectively. (x₁–x₇) Electron distribution function 2D v_x–v_y slices with corresponding 1D cuts along v_x = 0 directly below each 2D slice for the seven consecutive locations indicated by vertical gray bars in the Vlasov–Maxwell panels (h)–(n). The 1D cuts show $f_{e} (- v_{y})$ (black) with blue and red dotted lines corresponding to the cold and hot reference Maxwellians $f_{e 1} (- v_{y})$ and $f_{e 2} (- v_{y})$ ⁠, respectively.

FIG. 3.

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Comparison of MMS observations from November 28, 2016, to the Vlasov–Maxwell solution with similar initial parameter ratios. (a)–(g) MMS data: (a) Total magnetic field strength $| B |$ ⁠. (b) Electron density n_e (black) with two measures of the density gradient ${(\nabla n_{e})}_{⊥ 2}$ (aqua, dashed) and $V^{- 1} \partial n_{e} / \partial t$ (aqua, solid). (c) Electron perpendicular bulk velocity $U_{e ⊥ 1}$ with two measures of the velocity gradient ${(\nabla U_{e ⊥ 1})}_{⊥ 2}$ (magenta, dashed) and $V^{- 1} \partial U_{e ⊥ 1} / \partial t$ (magenta, solid). The gradient terms based on $\partial / \partial t$ in (c) and (d) were computed using V = 30 km/s determined from four-spacecraft timing analysis. (d) Total $T_{e, tot}$ (black), parallel $T_{e ∥}$ (blue), and perpendicular $T_{e ⊥}$ (red) electron temperature profiles with the $T_{e, ⊥ 1, ⊥ 1}$ component of the temperature tensor (green). (e) Decomposition of the electron pressure divergence ${(\nabla \cdot P_{e})}_{⊥ 2}$ (black) into the temperature gradient $n_{e} {(\nabla \cdot T_{e})}_{⊥ 2}$ (orange) and density gradient ${(T_{e} \cdot \nabla n_{e})}_{⊥ 2}$ (aqua) terms. (f) Electric field $E_{⊥ 2}$ (black) and $U_{e ⊥ 1} B$ (green) terms. (g) Pressure balance P_tot (green) showing the sum of magnetic pressure $B^{2} / 2 μ_{0}$ (blue), ion thermal pressure $n_{i} T_{i}$ (red), and electron thermal pressure $n_{e} T_{e}$ (black). (h)–(n) Vlasov–Maxwell results: (h) magnetic field B_z with simulation parameters displayed to the right of the panel. (i) Electron density n_e (black) and density gradient ${(\nabla n_{e})}_{x}$ (aqua, dashed). (j) Electron bulk velocity $- U_{e y}$ and velocity gradient ${(\nabla U_{e y})}_{x}$ (magenta, dashed). (k) Total $T_{e, tot}$ (black), parallel $T_{e ∥}$ (blue), and perpendicular $T_{e ⊥}$ (red) electron temperature profiles with the T_eyy component of the temperature tensor (green). (l) Decomposition of the electron pressure divergence ${(\nabla \cdot P_{e})}_{x}$ (black) into the temperature gradient $n_{e} {(\nabla \cdot T_{e})}_{x}$ (orange) and density gradient ${(T_{e} \cdot \nabla n_{e})}_{x}$ (aqua) terms. (m) Electric field E_x (black) and $- U_{e y} B$ (green) terms. (n) Pressure balance P_tot (green) showing the sum of magnetic pressure $B^{2} / 2 μ_{0}$ (blue), ion thermal pressure $n_{i} T_{i}$ (red), and electron thermal pressure $n_{e} T_{exx}$ (black). (t₁–t₇) Electron distribution function 2D $v_{⊥ 1}$ – $v_{⊥ 2}$ slices with corresponding 1D cuts along $v_{⊥ 2} = 0$ directly below each 2D slice for the seven consecutive times indicated by vertical gray bars in the MMS panels (a)–(g). The 1D cuts show $f_{e} (v_{⊥ 1})$ (black diamonds) with gray error bars and light yellow background indicating the measurement uncertainty. The blue and red dotted lines indicate cold and hot reference Maxwellians with temperatures of 40 and 70 eV, respectively. (x₁–x₇) Electron distribution function 2D v_x–v_y slices with corresponding 1D cuts along v_x = 0 directly below each 2D slice for the seven consecutive locations indicated by vertical gray bars in the Vlasov–Maxwell panels (h)–(n). The 1D cuts show $f_{e} (- v_{y})$ (black) with blue and red dotted lines corresponding to the cold and hot reference Maxwellians $f_{e 1} (- v_{y})$ and $f_{e 2} (- v_{y})$ ⁠, respectively.

Not all of the parameter ratios can be exactly matched due to the relative simplicity of the model, which assumes the distribution function is isotropic on either side of the current layer and varies only along one spatial dimension, the implications of which will be discussed in more detail below. For the model, the isotropy assumption requires that ratios of the total, perpendicular, and parallel temperatures across the layer are the same: $T_{e 2} / T_{e 1} = {(T_{e 2} / T_{e 1})}_{⊥} = {(T_{e 2} / T_{e 1})}_{∥}$ ⁠, whereas for the MMS observations this may not necessarily be the case. For instance, in the second MMS event we consider, there is a mechanism producing an electron temperature anisotropy $T_{e ∥} / T_{e ⊥} \approx 1.3$ on the MSP side [Fig. 4(d)] that is not included in the model. Thus, we choose these model run parameters by first matching the electron perpendicular temperature ratio ${(T_{e 2} / T_{e 1})}_{⊥}$ ⁠, which is the parameter that most influences whether or not an electron crescent distribution structure will be evident in the $v_{⊥ 1}$ - $v_{⊥ 2}$ velocity space [see Figs. 1(m)–1(o)]. Additionally, we matched the ion to electron temperature ratio $T_{i} / T_{e 1}$ ⁠, and we adjusted β_e so that the magnetic field ratio $B_{2} / B_{1}$ in the model would match the MMS observations. As a consequence of these parameter choices, the model density ratios $n_{2} / n_{1}$ are smaller than what was observed by MMS. Although the chosen β_e parameters do not match exactly with the MMS observations, they are still on the order of about unity (β_e for the model runs and MMS observations all fall between 0.5 and 1.5), and we note that matching $B_{2} / B_{1}$ results in the modeled ratio $β_{e} / β_{e}^{*}$ matching exactly with the MMS observations due to the relationship derived in Eq. (23). For these run parameters, the modeled maximum electron velocities of the current layers are close to the observed MMS values, which were ${(U_{e y})}_{\max}$ $\approx$ 0.2 v_th for the first event, and ${(U_{e y})}_{\max}$ $\approx$ 0.4 v_th for the second.

FIG. 4.

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Comparison of MMS observations from December 23, 2016, to the Vlasov–Maxwell solution with similar initial parameter ratios, following the same layout as in Fig. 3. Panel (a) shows the magnetic field B_z, and the gradient terms based on $\partial / \partial t$ in (c) and (d) were computed using V = 60 km/s determined from four-spacecraft timing analysis.

We chose the velocity-space steepness parameter S so that the modeled v_x-v_y electron distribution structures would best resemble the observed $v_{⊥ 1}$ - $v_{⊥ 2}$ distribution signatures observed by MMS. As we showed in Figs. 1(j)–1(l), increasing S sharpens the velocity space boundary of the crescent-shaped population. Although the perpendicular electron temperature ratio has the largest effect on the distribution's structure, for sufficiently low values of this steepness parameter $S ≲ 1$ corresponding to broader current layers, the crescent structure would not be discernible even if a large temperature gradient was present. Since the electron-scale layers passed by MMS in only a few tenths of a second, it is difficult to precisely determine the current sheet thicknesses, and hence the exact value of S, to use for the corresponding model runs. Using multi-spacecraft timing analysis, we estimated the structure velocity to be V $\approx$ 30 km/s for the event on November 28, 2016 (Fig. 3), and V $\approx$ 60 km/s for December 23, 2016 (Fig. 4). Thus, we estimate the thickness L of these layers to be roughly 5 or 6 km, with corresponding half-thicknesses $L / 2$ of about 2.5 to 3 r_L (since r_L $\approx$ 1 km for these events), which is consistent with the model scaling results [Fig. 1(i)]. This range of thicknesses corresponds to values of S between roughly 1 and 10.

Figure 3 features an electron-scale current layer observed on November 28, 2016 (Shuster et al., 2019), with electron distribution slices that exhibit a gradual temperature transition with ${(T_{e 2} / T_{e 1})}_{⊥} = 1.19$ ⁠, and hence, they lack a pronounced crescent-shaped distribution. Nevertheless, there is still a strong bulk electron flow signature with $U_{e ⊥ 1} \approx$ 700 km/s supported by a visible shift in the electron distribution function in the $v_{⊥ 1}$ -direction (see Fig. 3, times t₄ and t₅). For the corresponding Vlasov–Maxwell solution, a lower distribution steepness parameter S = 3 is chosen to model the gradual velocity-space transition. Figure 4 presents a second electron-scale current layer observed on December 23, 2016 (Shuster et al., 2019; 2021), which features an electron bulk flow of $U_{e ⊥ 1} \approx$ 1300 km/s (almost twice that of the first event) and a pronounced crescent-shaped electron distribution function corresponding to the substantially larger perpendicular electron temperature gradient with ${(T_{e 2} / T_{e 1})}_{⊥} = 1.48$ ⁠. A higher steepness parameter S = 10 is chosen for the Vlasov–Maxwell model to capture the sharper velocity-space transition, which results in a crescent-like distribution structure comparable to the MMS measurements for this event. In particular, comparing the 1D cuts of the 2D MMS distributions t₄-t₆ to the modeled distribution locations x₄-x₆ in Fig. 4, there is a notable bump in phase space density occurring within 4000 to 8000 km/s in $v_{⊥ 1}$ (1 to 2 v_th in $- v_{y}$ for the model distributions). This signature corresponds to the more pronounced crescent-shaped distribution that emerges due to the combined effects of the higher perpendicular temperature ratio and larger S parameter.

The layouts of Figs. 3 and 4 are the same: panels (a)–(g) show an overview of the MMS plasma moments and fields data, and panels (t₁-t₇) show a 2D slice of the electron distribution structure in the $v_{⊥ 1}$ - $v_{⊥ 2}$ plane observed by MMS for seven times indicated by the vertical gray bars in panels (a-g). Below each 2D slice, 1D cuts of the distribution taken along $v_{⊥ 2} = 0$ are included with uncertainties for the electron phase space density measurements shown as gray error bars with a light yellow background (e.g., Gershman et al., 2015). The same moment and field quantities from the Vlasov–Maxwell model are shown in panels (h)–(n), with panels (x₁-x₇) showing 2D slices in the v_x-v_y plane of the modeled electron distribution structure at seven spatial locations indicated by the vertical gray bars in panels (h)–(n), with 1D cuts taken along v_x = 0 included below. Figure 3(a) and Fig. 4(a) show MMS 1 magnetic field data from the fluxgate magnetometer (FGM) (Russell et al., 2016), and Figs. 3(f) and 4(f) include electric field data from the electric field double probes (EDP) (Lindqvist et al., 2016).

Going from the MSP to MSH side of the layer [right to left in Figs. 3(a)–3(g) and 4(a)–4(g)], both of the MMS electron-scale temperature gradient events maintain approximate pressure balance and exhibit qualitatively similar plasma properties, each of which are captured by the Vlasov–Maxwell solutions [Figs. 3(h)–3(n) and 4(h)–4(n)]: (1) the magnetic field strength decreases by about a factor of 2 or 3, (2) the density increases on the order of 10%, (3) the ion temperature is about an order of magnitude larger than the electron temperature, (4) there is a strong electron bulk velocity signature on the order of 1000 km/s supported by enhanced electron phase space density in the $v_{⊥ 1}$ (⁠ $- v_{y}$ ⁠) direction, (5) the electron pressure divergence $\nabla \cdot P_{e}$ is dominated by the $n_{e} \nabla \cdot T_{e}$ term, and (6) there is an enhanced DC electric field $E$ that deviates from $U_{e} \times B$ due to the nonzero $\nabla \cdot P_{e}$ ⁠. The structure of each current layer is also skewed slightly toward the MSP (low density) side of the layer, which is evident for instance in the $U_{e ⊥ 1}$ and $E_{⊥ 2}$ profiles. The gradient structure of the layers observed by MMS also agrees with the Vlasov–Maxwell models based on comparing the density gradient (aqua trace) and bulk velocity gradient (magenta trace) terms in Figs. 3(b), 3(c), 4(b), and 4(c) to Figs. 3(i), 3(j), 4(i), and 4(j). Since these layers are approximately steady-state structures moving past the spacecraft, single spacecraft MMS gradients can be inferred via $\partial / \partial x \approx V^{- 1} \partial / \partial t$ [solid aqua and magenta traces in Figs. 3(b), 3(c), 4(b), and 4(c)], which are more consistent with the model results [in Figs. 3(i), 3(j), 4(i), 4(j)].

There are also aspects of the current layers observed by MMS that are not captured by the simplified 1D kinetic model. The electron distribution function f_e in Eq. (8) is based on a varying temperature function $T (c_{e 2})$ that only depends on v_y (⁠ $v_{⊥ 1}$ ⁠); thus, the modeled f_e is symmetric about the other two velocity-space directions v_x and v_z (⁠ $v_{⊥ 2}$ and $v_{∥}$ ⁠). However, the MMS distributions do not reflect such symmetry. Because the model does not account for this structure of the electron distribution in the $v_{∥}$ (v_z) direction, the model incorrectly represents the parallel electron temperature $T_{e ∥}$ profile [blue traces in Figs. 3(k) and 4(k)]. Consequently, despite the matched perpendicular electron temperature ratio, the modeled total electron temperature ratio $T_{e 2} / T_{e 1}$ is larger than what was observed by MMS, which leads to an artificially large deviation between the E_x and $U_{e y} B_{z}$ terms, and an exaggerated variation in the modeled T_eyy-component of the electron temperature tensor [green trace in Figs. 3(k) and 4(k)] when compared to the MMS observations [green trace for the $T_{e, ⊥ 1, ⊥ 1}$ -component in Figs. 3(d) and 4(d)]. These limitations are consequences of the relative simplicity and assumed symmetry of the 1D model. Nevertheless, the model successfully reproduces many qualitative and quantitative properties of the bulk plasma parameters and produces electric and magnetic fields that are self-consistent with the electron distribution's crescent-like structure observed by MMS.

IV DISCUSSION AND CONCLUSIONS

We formulated an exact Vlasov–Maxwell equilibrium solution suitable for modeling thin (electron gyroradius scale) asymmetric current layers observed by MMS at the magnetopause. Our solution self-consistently produces a spatially evolving, nongyrotropic electron crescent-type distribution structure in velocity space similar to the kinetic electron structures detected by MMS. Nongyrotropic, crescent-shaped electron velocity distribution functions can be indicative of non-ideal energy conversion processes associated with the electron diffusion region of magnetic reconnection (Burch et al., 2016; Bessho et al., 2017; Torbert et al., 2018; Webster et al., 2018). While electron crescents may be a necessary consequence of asymmetric reconnection dynamics, they alone do not constitute a sufficient condition for reconnection to occur. Electron crescent distributions appear more generally as the manifestation of diamagnetic drifts and finite Larmor radius effects occurring within well-magnetized plasmas (Egedal et al., 2016; Rager et al., 2018), which is consistent with the crescent-like electron distributions produced in our 1D solution where the associated particle orbits remain closed in phase space throughout the current layer.

Qualitatively similar semi-circular, crescent-resembling electron distributions were reported by (Camporeale and Lapenta, 2005) [see their Fig. 3, cases (a) and (b)] in their study of symmetric, bifurcated current layers applicable to magnetotail geometries. The similar structures develop primarily due to the dependence on the constant of motion associated with the electron's y-momentum [ $c_{e 2}$ in our Eq. (2)], but there are also several distinctions to note. In their study, the distribution function's dependences on $ϕ$ and A_y could be separated for the forms of f_e that were chosen based on the separable forms considered by Schindler and Birn (2002). Additionally, the temperature parameter T_e is kept constant in their solutions, and thus, the relative magnitude between the two electron populations on either side of the cutoff velocity $v_{y}^{*}$ is different than in our solution. Several forms for the ion distribution f_i were also considered in their study, whereas here for simplicity we model the ions with a non-drifting Maxwellian population in the electron-scale layer (i.e., we view the current layer from the frame of reference moving with the ion bulk flow, where the ion diamagnetic drift balances the $E \times B$ drift so that U_i = 0). We note that for particular choices of $β_{e}^{*}$ [see Eq. (24)], our solution also permits symmetric, bifurcated-type current layers, although our focus in this paper is on the asymmetric case applicable to the MMS magnetopause observations.

The two parameters in our model which most influence the crescent-shaped distribution structure are (1) the perpendicular electron temperature ratio across the layer ${(T_{e 2} / T_{e 1})}_{⊥}$ ⁠, and (2) the velocity space steepness parameter S, which affects the thickness and strength of the current layer by controlling the sharpness of the velocity-space transition between the two electron populations of differing temperatures. These parameters may be adjusted for facilitating comparison to MMS encounters of electron-scale current layers with varying temperature ratios and thicknesses, some of which may support a strong current density without exhibiting a visibly apparent crescent-shaped distribution signature (Shuster et al., 2019). An electron crescent distribution in $v_{⊥ 1}$ - $v_{⊥ 2}$ is likely to be detected in sufficiently thin (⁠ $S ≳ 3$ ⁠) electron-scale current layers that exhibit a strong perpendicular electron temperature gradient ${(T_{e 2} / T_{e 1})}_{⊥} ≳ 1.5$ ⁠. However, for broader current layers (⁠ $S ≲ 1$ ⁠) with a weaker electron temperature gradient ${(T_{e 2} / T_{e 1})}_{⊥} ≲ 1.2$ ⁠, a crescent-shaped distribution is less likely to be observed. In the limit as $S \to 0$ ⁠, the gradient vanishes as the electron distribution reduces to a uniform Maxwellian. On the other hand, in the limit as $S \to \infty$ ⁠, the electron distribution function becomes piecewise Maxwellian in velocity-space, which conveniently permits analytic solutions to the self-consistent plasma moments and fields.

Having an exact Vlasov–Maxwell solution for the case of an asymmetric gradient in the electron density, bulk velocity, and temperature is relevant to recent MMS magnetopause observations of terms in the Vlasov equation $\nabla f_{e}$ and $v \cdot \nabla f_{e}$ (Shuster et al., 2021). The form of the electron distribution we chose takes two electron temperatures $T_{e 1}$ and $T_{e 2}$ as initial parameters to specify the size of the overall temperature gradient across the layer. The self-consistent solution also yields gradients in the electron density and bulk velocity, consistent with the MMS events presented here, and relevant to recent observations of similar electron-scale gradients found in association with magnetopause current layers. Our solution may be used to further validate MMS observations of the spatial gradient term $v \cdot \nabla f_{e}$ and future measurements of the velocity-space gradient term $(F / m_{e}) \cdot \nabla_{v} f_{e}$ ⁠. For the spatial gradient term, one could fly several virtual spacecraft probes through our 1D layer at various $Δ x$ spacings to simulate how the $\nabla f_{e}$ gradient distribution structures seen by MMS vary in velocity-space, and assess the accuracy of the spatial gradient computation for a given MMS spacecraft separation (e.g., $Δ x \approx$ 10 km).

ACKNOWLEDGMENTS

We especially thank the MMS instrument teams for their dedication and commitment to providing unprecedented, high-quality data sets. This research was supported in part by NASA grants to the Fast Plasma Investigation, FIELDS team, and Theory and Modeling program of the MMS mission. J.R.S. was supported by NASA under Grant Nos. 80NSSC19K1092 and 80NSSC21K0732. S.W. was supported by DOE under Grant No. DE-SC0020058. J.N. was supported by NASA under Grant No. 80NSSC21K1046.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

All MMS data are made publicly available through the Science Data Center (SDC) at the Laboratory for Atmospheric and Space Physics (LASP) via at https://lasp.colorado.edu/mms/sdc/public/, LASP (2015). The codes used to solve the one-dimensional, steady-state Vlasov-Maxwell system of equations and generate the results shown in the figures will be made available upon reasonable request.

APPENDIX: DERIVATION OF ELECTRON MOMENTS IN THE PIECEWISE MAXWELLIAN LIMIT

Here, we compute electron moments in the limit $S \to \infty$ where f_e becomes the piecewise Maxwellian distribution defined in Eq. (27) as discussed in Sec. II C.

The electron density n_e is derived as follows:

n_{e} = \int_{- \infty}^{+ \infty} f_{e} (ϕ, A_{y}) d^{3} v,

(A1)

= \int_{- \infty}^{+ \infty} d v_{x} \int_{- \infty}^{+ \infty} d v_{z} [\int_{- \infty}^{v_{y}^{*}} f_{e 2} d v_{y} + \int_{v_{y}^{*}}^{+ \infty} f_{e 1} d v_{y}],

(A2)

\begin{matrix} = n_{1} {(\frac{m_{e}}{2 π T_{e 1}})}^{3 / 2} [(\frac{2 π T_{e 1}}{m_{e}}) exp (\frac{e ϕ}{T_{e 1}}) \int_{- \infty}^{- v_{y}^{*}} exp (\frac{- m_{e} v_{y}^{2}}{2 T_{e 1}}) d v_{y} \\ + (\frac{2 π T_{e 2}}{m_{e}}) exp (\frac{e ϕ}{T_{e 2}}) \int_{- \infty}^{v_{y}^{*}} exp (\frac{- m_{e} v_{y}^{2}}{2 T_{e 2}}) d v_{y}] \end{matrix}

\begin{matrix} = n_{1} [exp (\frac{e ϕ}{T_{e 1}}) (\frac{1}{v_{t h} \sqrt{2 π}}) \int_{- \infty}^{- v_{y}^{*}} exp (\frac{- v_{y}^{2}}{2 v_{t h}^{2}}) d v_{y} + {(\frac{T_{e 2}}{T_{e 1}})}^{3 / 2} exp (\frac{e ϕ}{T_{e 2}}) (\frac{1}{v_{t h, 2} \sqrt{2 π}}) \int_{- \infty}^{v_{y}^{*}} exp (\frac{- v_{y}^{2}}{2 v_{t h, 2}^{2}}) d v_{y}] \end{matrix}

= \frac{n_{1}}{2} ([1 + erf (\frac{- v_{y}^{*}}{v_{t h} \sqrt{2}})] exp (\frac{e ϕ}{T_{e 1}}) + {(\frac{T_{e 2}}{T_{e 1}})}^{3 / 2} [1 + erf (\frac{v_{y}^{*}}{v_{t h, 2} \sqrt{2}})] exp (\frac{e ϕ}{T_{e 2}}))

= \frac{n_{1}}{2} ([1 - erf (\frac{A_{y}}{B_{1} r_{L} \sqrt{2}})] exp (\frac{e ϕ}{T_{e 1}}) + {(\frac{T_{e 2}}{T_{e 1}})}^{3 / 2} [1 + erf (\frac{A_{y}}{B_{1} r_{L}} \sqrt{\frac{T_{e 1}}{2 T_{e 2}}})] exp (\frac{e ϕ}{T_{e 2}})),

(A3)

n_{e} (ϕ, A_{y}) = n_{1} [Φ (\frac{- A_{y}}{B_{1} r_{L}}) exp (\frac{e ϕ}{T_{e 1}}) + {(\frac{T_{e 2}}{T_{e 1}})}^{3 / 2} Φ (\frac{A_{y}}{B_{1} r_{L}} \sqrt{\frac{T_{e 1}}{T_{e 2}}}) exp (\frac{e ϕ}{T_{e 2}})],

(A4)

where $v_{t h, 2} = \sqrt{T_{e 2} / m_{e}}$ ⁠. As $A_{y} \to - \infty$ ⁠, we have that $n_{e} \to n_{1}$ ⁠, while for $A_{y} \to + \infty$ ⁠, Eq. (28) reduces to Eq. (15). To compute the v_y integrals, we make use of the symmetry of the Gaussian functions to conveniently adjust the limits of integration, and we utilize the relationship between the cumulative distribution function for the standard normal distribution $Φ (x)$ and the error function $erf (x)$

Φ (\frac{v_{y}^{*}}{v_{t h}}) = \frac{1}{v_{t h} \sqrt{2 π}} \int_{- \infty}^{v_{y}^{*}} exp (\frac{- v_{y}^{2}}{2 v_{t h}^{2}}) d v_{y} = \frac{1}{2} [1 + erf (\frac{v_{y}^{*}}{v_{t h} \sqrt{2}})],

(A5)

where the error function is defined as

erf (x) \equiv \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- t^{2}} d t .

(A6)

Invoking quasineutrality, the expression for n_e in Eq. (28) can be equated to $n_{i} = n_{1} exp [- e ϕ / T_{i}]$ ⁠, and the resulting equation can be numerically solved for $ϕ (A_{y})$ via

2 = [1 - erf (\frac{A_{y}}{B_{1} r_{L} \sqrt{2}})] exp [(1 + \frac{T_{e 1}}{T_{i}}) \frac{e ϕ}{T_{e 1}}] + {(\frac{T_{e 2}}{T_{e 1}})}^{3 / 2} [1 + erf (\frac{A_{y}}{B_{1} r_{L}} \sqrt{\frac{T_{e 1}}{2 T_{e 2}}})] exp [(1 + \frac{T_{e 2}}{T_{i}}) \frac{e ϕ}{T_{e 2}}],

(A7)

1 = Φ (\frac{- A_{y}}{B_{1} r_{L}}) exp [(1 + \frac{T_{e 1}}{T_{i}}) \frac{e ϕ}{T_{e 1}}] + {(\frac{T_{e 2}}{T_{e 1}})}^{3 / 2} Φ (\frac{A_{y}}{B_{1} r_{L}} \sqrt{\frac{T_{e 1}}{T_{e 2}}}) exp [(1 + \frac{T_{e 2}}{T_{i}}) \frac{e ϕ}{T_{e 2}}] .

(A8)

For the electron bulk velocity U_ey, we have

U_{e y} = \frac{1}{n_{e}} \int_{- \infty}^{+ \infty} v_{y} f_{e} d^{3} v,

(A9)

= \frac{1}{n_{e}} \int_{- \infty}^{+ \infty} d v_{x} \int_{- \infty}^{+ \infty} d v_{z} [\int_{- \infty}^{v_{y}^{*}} v_{y} f_{e 2} d v_{y} + \int_{v_{y}^{*}}^{+ \infty} v_{y} f_{e 1} d v_{y}],

(A10)

\begin{matrix} = \frac{n_{1}}{n_{e} (ϕ, A_{y})} {(\frac{m_{e}}{2 π T_{e 1}})}^{3 / 2} [(\frac{2 π T_{e 1}}{m_{e}}) exp (\frac{e ϕ}{T_{e 1}}) \int_{v_{y}^{*}}^{+ \infty} v_{y} exp (\frac{- m_{e} v_{y}^{2}}{2 T_{e 1}}) d v_{y} \\ + (\frac{2 π T_{e 2}}{m_{e}}) exp (\frac{e ϕ}{T_{e 2}}) \int_{- \infty}^{v_{y}^{*}} v_{y} exp (\frac{- m_{e} v_{y}^{2}}{2 T_{e 2}}) d v_{y}] \end{matrix}

= \frac{n_{1}}{n_{1} exp (\frac{- e ϕ}{T_{i}})} (\frac{1}{v_{t h} \sqrt{2 π}}) [v_{t h}^{2} exp (\frac{- v_{y}^{*}^{2}}{2 v_{t h}^{2}}) exp (\frac{e ϕ}{T_{e 1}}) - v_{t h, 2}^{2} (\frac{T_{e 2}}{T_{e 1}}) exp (\frac{- v_{y}^{*}^{2}}{2 v_{t h, 2}^{2}}) exp (\frac{e ϕ}{T_{e 2}})]

\begin{matrix} U_{e y} = v_{t h} (\frac{1}{\sqrt{2 π}}) [exp [\frac{- 1}{2} {(\frac{A_{y}}{B_{1} r_{L}})}^{2}] exp [(1 + \frac{T_{e 1}}{T_{i}}) \frac{e ϕ}{T_{e 1}}] \\ - {(\frac{T_{e 2}}{T_{e 1}})}^{2} exp [\frac{- 1}{2} (\frac{T_{e 1}}{T_{e 2}}) {(\frac{A_{y}}{B_{1} r_{L}})}^{2}] exp [(1 + \frac{T_{e 2}}{T_{i}}) \frac{e ϕ}{T_{e 2}}]] . \end{matrix}

(A11)

To arrive at equation (A), we made use of the result

\int_{a}^{b} v_{y} exp (\frac{- v_{y}^{2}}{2 v_{t h}^{2}}) d v_{y} = - v_{t h}^{2} exp (\frac{- v_{y}^{2}}{2 v_{t h}^{2}}) |_{a}^{b} .

(A12)

For P_exx, we have

P_{exx} = m_{e} \int_{- \infty}^{+ \infty} v_{x}^{2} f_{e} d^{3} v,

(A13)

= m_{e} \int_{- \infty}^{+ \infty} d v_{z} \int_{- \infty}^{+ \infty} d v_{x} v_{x}^{2} [\int_{- \infty}^{v_{y}^{*}} f_{e 2} d v_{y} + \int_{v_{y}^{*}}^{+ \infty} f_{e 1} d v_{y}],

(A14)

\begin{matrix} = m_{e} n_{1} {(\frac{m_{e}}{2 π T_{e 1}})}^{3 / 2} [\sqrt{\frac{2 π T_{e 1}}{m_{e}}} exp (\frac{e ϕ}{T_{e 1}}) (\int_{- \infty}^{+ \infty} v_{x}^{2} exp (\frac{- v_{x}^{2}}{2 v_{t h}^{2}}) d v_{x}) (\int_{- \infty}^{- v_{y}^{*}} exp (\frac{- v_{y}^{2}}{2 v_{t h}^{2}}) d v_{y}) \\ + \sqrt{\frac{2 π T_{e 2}}{m_{e}}} exp (\frac{e ϕ}{T_{e 2}}) (\int_{- \infty}^{+ \infty} v_{x}^{2} exp (\frac{- v_{x}^{2}}{2 v_{t h, 2}^{2}}) d v_{x}) (\int_{- \infty}^{v_{y}^{*}} exp (\frac{- v_{y}^{2}}{2 v_{t h, 2}^{2}}) d v_{y})], \end{matrix}

(A15)

\begin{matrix} = m_{e} n_{1} {(\frac{1}{v_{t h} \sqrt{2 π}})}^{3} [v_{t h} \sqrt{2 π} exp (\frac{e ϕ}{T_{e 1}}) (v_{t h}^{3} \sqrt{2 π}) (\frac{v_{t h} \sqrt{2 π}}{v_{t h} \sqrt{2 π}}) (\int_{- \infty}^{- v_{y}^{*}} exp (\frac{- v_{y}^{2}}{2 v_{t h}^{2}}) d v_{y}) \\ + v_{t h, 2} \sqrt{2 π} exp (\frac{e ϕ}{T_{e 2}}) (v_{t h, 2}^{3} \sqrt{2 π}) (\frac{v_{t h, 2} \sqrt{2 π}}{v_{t h, 2} \sqrt{2 π}}) (\int_{- \infty}^{v_{y}^{*}} exp (\frac{- v_{y}^{2}}{2 v_{t h, 2}^{2}}) d v_{y})], \end{matrix}

(A16)

= \frac{n_{1} m_{e} v_{t h}^{2}}{2} ([1 + erf (\frac{- v_{y}^{*}}{v_{t h} \sqrt{2}})] exp (\frac{e ϕ}{T_{e 1}}) + {(\frac{v_{t h, 2}}{v_{t h}})}^{5} [1 + erf (\frac{v_{y}^{*}}{v_{t h, 2} \sqrt{2}})] exp (\frac{e ϕ}{T_{e 2}})),

(A17)

P_{exx} = \frac{n_{1} T_{e 1}}{2} [Φ (\frac{- A_{y}}{B_{1} r_{L}}) exp (\frac{e ϕ}{T_{e 1}}) + {(\frac{T_{e 2}}{T_{e 1}})}^{5 / 2} Φ (\frac{A_{y}}{B_{1} r_{L}} \sqrt{\frac{T_{e 1}}{T_{e 2}}}) exp (\frac{e ϕ}{T_{e 2}})],

(A18)

where we have made use of the following Gaussian integral:

\int_{- \infty}^{\infty} v_{x}^{2} exp (- \frac{v_{x}^{2}}{2 v_{t h}^{2}}) d v_{x} = v_{t h}^{3} \sqrt{2 π} .

(A19)

For P_eyy, we have

P_{eyy} = m_{e} \int_{- \infty}^{+ \infty} {(v_{y} - U_{e y})}^{2} f_{e} d^{3} v,

(A20)

= m_{e} [\int_{- \infty}^{+ \infty} v_{y}^{2} f_{e} d^{3} v - 2 U_{e y} \int_{- \infty}^{+ \infty} v_{y} f_{e} d^{3} v + U_{e y}^{2} \int_{- \infty}^{+ \infty} f_{e} d^{3} v],

(A21)

= m_{e} \int_{- \infty}^{+ \infty} v_{y}^{2} f_{e} d^{3} v - m_{e} n_{e} U_{e y}^{2} = I - m_{e} n_{e} U_{e y}^{2} .

(A22)

Evaluating the integral I in Eq. (A22), we find

I = m_{e} \int_{- \infty}^{+ \infty} v_{y}^{2} f_{e} d^{3} v,

(A23)

= m_{e} \int_{- \infty}^{+ \infty} d v_{x} \int_{- \infty}^{+ \infty} d v_{z} [\int_{- \infty}^{v_{y}^{*}} v_{y}^{2} f_{e 2} d v_{y} + \int_{v_{y}^{*}}^{+ \infty} v_{y}^{2} f_{e 1} d v_{y}],

(A24)

\begin{matrix} = m_{e} n_{1} {(\frac{m_{e}}{2 π T_{e 1}})}^{3 / 2} [(\frac{2 π T_{e 1}}{m_{e}}) exp (\frac{e ϕ}{T_{e 1}}) \int_{- \infty}^{- v_{y}^{*}} v_{y}^{2} exp (\frac{- v_{y}^{2}}{2 v_{t h}^{2}}) d v_{y} \\ + (\frac{2 π T_{e 2}}{m_{e}}) exp (\frac{e ϕ}{T_{e 2}}) \int_{- \infty}^{v_{y}^{*}} v_{y}^{2} exp (\frac{- v_{y}^{2}}{2 v_{t h, 2}^{2}}) d v_{y}] \end{matrix},

(A25)

\begin{matrix} = m_{e} n_{1} {(\frac{1}{v_{t h} \sqrt{2 π}})}^{3} [(2 π v_{t h}^{2}) exp (\frac{e ϕ}{T_{e 1}}) v_{t h}^{3} \sqrt{2 π} [(\frac{1}{2}) erf (\frac{- v_{y}^{*}}{v_{t h} \sqrt{2}}) + (\frac{v_{y}^{*}}{v_{t h} \sqrt{2 π}}) exp (\frac{- v_{y}^{*}^{2}}{2 v_{t h}^{2}}) + \frac{1}{2}] \\ + (2 π v_{t h, 2}^{2}) exp (\frac{e ϕ}{T_{e 2}}) v_{t h, 2}^{3} \sqrt{2 π} [(\frac{1}{2}) erf (\frac{v_{y}^{*}}{v_{t h, 2} \sqrt{2}}) - (\frac{v_{y}^{*}}{v_{t h, 2} \sqrt{2 π}}) exp (\frac{- v_{y}^{*}^{2}}{2 v_{t h, 2}^{2}}) + \frac{1}{2}]] \end{matrix},

(A26)

\begin{matrix} = \frac{n_{1} m_{e} v_{t h}^{2}}{2} [exp (\frac{e ϕ}{T_{e 1}}) [1 + erf (\frac{- v_{y}^{*}}{v_{t h} \sqrt{2}}) + (\frac{2 v_{y}^{*}}{v_{t h} \sqrt{2 π}}) exp (\frac{- v_{y}^{*}^{2}}{2 v_{t h}^{2}})] \\ + {(\frac{v_{t h, 2}}{v_{t h}})}^{5} exp (\frac{e ϕ}{T_{e 2}}) [1 + erf (\frac{v_{y}^{*}}{v_{t h, 2} \sqrt{2}}) - (\frac{2 v_{y}^{*}}{v_{t h, 2} \sqrt{2 π}}) exp (\frac{- v_{y}^{*}^{2}}{2 v_{t h, 2}^{2}})]] . \end{matrix}

(A27)

Therefore,

\begin{matrix} P_{eyy} = \frac{n_{1} T_{e 1}}{2} [exp (\frac{e ϕ}{T_{e 1}}) {Φ (\frac{- A_{y}}{B_{1} r_{L}}) + (\frac{A_{y}}{B_{1} r_{L}} \sqrt{\frac{2}{π}}) exp (\frac{- 1}{2}) {\frac{A_{y}}{B_{1} r_{L}})}^{2})} \\ + {(\frac{T_{e 2}}{T_{e 1}})}^{5 / 2} exp (\frac{e ϕ}{T_{e 2}}) {Φ (\frac{A_{y}}{B_{1} r_{L}} \sqrt{\frac{T_{e 1}}{T_{e 2}}}) - (\frac{A_{y}}{B_{1} r_{L}} \sqrt{\frac{2 T_{e 1}}{π T_{e 2}}}) exp (\frac{- 1}{2} (\frac{T_{e 1}}{T_{e 2}}) {(\frac{A_{y}}{B_{1} r_{L}})}^{2})}] - m_{e} n_{e} U_{e y}^{2}, \end{matrix}

where n_e and U_ey are derived above, and we used the result that

\int_{a}^{b} v_{y}^{2} exp (\frac{- v_{y}^{2}}{2 v_{t h}^{2}}) d v_{y} = v_{t h}^{3} \sqrt{2 π} [(\frac{1}{2}) erf (\frac{v_{y}}{v_{t h} \sqrt{2}}) - (\frac{v_{y}}{v_{t h} \sqrt{2 π}}) exp (\frac{- v_{y}^{2}}{2 v_{t h}^{2}})] |_{a}^{b} .

(A28)

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Author(s)

Electron-scale temperature gradients in kinetic equilibrium: MMS observations and Vlasov–Maxwell solutions

I. INTRODUCTION

II. VLASOV–MAXWELL EQUILIBRIUM FOR AN ELECTRON-SCALE TEMPERATURE GRADIENT

A. Imposing quasineutrality

B. Scaling properties across the asymmetric current layer

1. Density ratio: $n_{2} / n_{1}$

2. Magnetic field ratio: $B_{2} / B_{1}$

3. Approximation for the maximum $U_{e y} / v_{t h}$

C. Piecewise Maxwellian limit for f_e

D. Phase-space structures and electron orbits

E. Thickness of the current layer

III. SPATIAL EVOLUTION OF THE MMS OBSERVATIONS AND THE VLASOV–MAXWELL MODEL

IV DISCUSSION AND CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

APPENDIX: DERIVATION OF ELECTRON MOMENTS IN THE PIECEWISE MAXWELLIAN LIMIT

References

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

Electron-scale temperature gradients in kinetic equilibrium: MMS observations and Vlasov–Maxwell solutions

I. INTRODUCTION

II. VLASOV–MAXWELL EQUILIBRIUM FOR AN ELECTRON-SCALE TEMPERATURE GRADIENT

A. Imposing quasineutrality

B. Scaling properties across the asymmetric current layer

1. Density ratio: n2/n1

2. Magnetic field ratio: B2/B1

3. Approximation for the maximum Uey/vth

C. Piecewise Maxwellian limit for fe

D. Phase-space structures and electron orbits

E. Thickness of the current layer

III. SPATIAL EVOLUTION OF THE MMS OBSERVATIONS AND THE VLASOV–MAXWELL MODEL

IV DISCUSSION AND CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

APPENDIX: DERIVATION OF ELECTRON MOMENTS IN THE PIECEWISE MAXWELLIAN LIMIT

References

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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1. Density ratio: $n_{2} / n_{1}$

2. Magnetic field ratio: $B_{2} / B_{1}$

3. Approximation for the maximum $U_{e y} / v_{t h}$

C. Piecewise Maxwellian limit for f_e