The Adiabatic 1D Kinetic Equilibrium of the Electron Diffusion Region During Anti‐Parallel Magnetic Reconnection

An earlier model of the electron distribution function accounts for the pressure anisotropy that develops in the inflow regions during anti‐parallel magnetic reconnection. However, the model is not applicable to the electron diffusion region (EDR), where the electron magnetic moments break as an adiabatic invariant. The earlier model is here generalized through the use of the current‐sheet adiabatic invariant of a 1D current layer, to become applicable also to the EDR. The new generalized model reproduces the main features of the EDR and its vicinity observed in a fully kinetic simulation, formally proving the 1D equilibrium relationship between the upstream electron pressure anisotropy and the EDR electron jets.


Introduction
The structure of the electron diffusion region (EDR) during anti-parallel magnetic reconnection is a subject of significant interest in part because of its relevance to magnetic reconnection in the Earth's magnetotail (Dungey, 1953).The common two-dimensional reconnection scenario is centered on an EDR characterized by strong electron current densities, for which resistive fluid models have been widely applied.In particular, given the results of the influential Sweet-Parker reconnection model (Parker, 1957), it is commonly assumed that the EDR electron current is driven by the reconnection electric field, E rec (by Faraday's law the inductive reconnection electric field in the direction aligned with EDR current layer, E rec , is the rate by which magnetic flux convects through the EDR).Meanwhile, through the study of reconnection using fully kinetic simulations, it has been suggested that the EDR current is more strongly coupled to electron pressure anisotropy that forms upstream of the EDR (Egedal et al., 2013;Le et al., 2010).However, because an analytical description has not been available that fully captures the important transition from the reconnection inflow into the EDR, the connection between the upstream pressure anisotropy and the EDR current has sofar not been explicitly proven.
In this present letter, a previous model (Egedal et al., 2008) (describing well-magnetized electrons within the reconnection inflows) is modified using results from a recent semi-analytical study of 1D current layers sustained by electron pressure anisotropy (Egedal, 2023).The resulting generalized model provides a reduced kinetic description that (for the first time) can account for the main features of the electron scale reconnection region observed in fully kinetic simulations (Karimabadi et al., 2007;Shay et al., 2007).Such 2D kinetic simulations in turn reproduce detailed spacecraft observations of reconnection regions in the Earth's magnetotail (Egedal et al., 2019;Schroeder et al., 2022;Torbert et al., 2018).

Generalized Model for the Electron Distribution Function, f e
The study in Ref. Egedal (2023) can account for static 1D current sheets separating regions of opposite magnetic fields.As illustrated in Figure 1a, in contrast to a Harris sheet (Harris, 1962), the considered geometry includes a small normal magnetic field, B z , which yields streaming of electrons across the current layer.iser, 1965) streaming in along the magnetic field lines as a regular cyclotron orbit before transitioning into the meandering motion across z = 0.As the electron leaves the current layer, its trajectory transitions back into the regular cyclotron motion.
The description of the geometry in Ref. Egedal (2023) utilizes the current sheet adiabatic invariant, J z ∝∮ v z dz, which is well conserved for a single transit of a Speiser orbit through this geometry (Buchner & Zelenyi, 1989;Janicke, 1965;Sonnerup, 1971).The transition to the meandering part of the orbit can be considered a merger of two cyclotron orbits, which in principle doubles the value of J z at the transition point.However, through the normalization J z remains an invariant as the particles pass in and out of the current sheet.Furthermore, for cyclotron orbits (with gyrophase ϕ) we have v z = v ⊥ sin ϕ and dz = (mv ⊥ /eB) sin ϕ dϕ, such that ∮ v z dz = πmv 2 ⊥ / (eB) = 2πμ/e.Thus, the factor of e/(2π) in Equation 1 ensures that in the asymptotic regions (large |z|) we simply have J z = μ.Note that a slightly modified normalization must be applied for the meandering electrons in the case where the electrostatic potential Φ(z) is non-monotonic.
Consistent with the results for 1D current layer driven by electron pressure anisotropy (Egedal, 2023), previous work has suggested that the EDR current layers of anti-parallel reconnection are similarly linked to pressure anisotropy (Egedal et al., 2013(Egedal et al., , 2023;;Le et al., 2010).Illustrated in Figure 2a, the electron anisotropy with T e‖ ≫ T e⊥ develops self-consistently within the inflow regions.The formation of the anisotropy is closely tied to the parallel bounce motion of trapped electrons with trajectories similar to that indicated by the magenta line in Figure 2a, yielding the type of electron distribution shown in Figures 2b and 2c.The trapping effectively reduces the electron heat-transport such that their equations of state (Le et al., 2009) approximately follow the Chew, Goldberger and Low (CGL) scalings, where p e‖ ∝ n 3 /B 2 and p e⊥ ∝ nB (Chew et al., 1956).Consequently, p e‖ / p e⊥ ∝ n 2 /B 3 , and strong electron anisotropy is observed as the EDR with low values of B is approached.The described effect is accurately captured by the model for the electron distributions first derived in Ref. Egedal et al. (2008).However, as emphasized with Figures 2d-2g, within the EDR the electron distributions are more complicated (Ng et al., 2011), and the model of Ref. Egedal et al. (2008) fails in this region where the magnetic moment, μ, breaks as an adiabatic invariant.
The main new insight of the present Letter is that the model of Ref. Egedal et al. (2008) can be generalized simply by replacing μ with J z .The electron distributions are then characterized by , trapped where E = m|v| 2 / 2. In addition to the change in the form for the trapped electrons, replacing μ by J z also influences the trapped passing boundary now given by With these changes the expression in Equation 2 becomes applicable to both the reconnection inflows as well as the EDR.While the methods for computing J z z,v y ,v z ) in Ref. (Egedal, 2023) only considers 1D configurations, we note that J z z,v y ,v z ) is independent of the imposed value of B z .Thus, by using an arbitrary fixed value for B z , the methods still provide accurate values of J z z,v y ,v z ) applicable to the EDR region (where B z varies in strength along the length of the current layer, including B z = 0 right at the X-line).

The 1D Structure of the EDR
During anti-parallel reconnection scalar quantities such as n e and |B| (as well as vector components evaluated in a coordinate system rotating with the magnetic field, see below) have largely 1D structures within the region of the EDR; we will document how these structures can be accounted for by Equation 2. In Figure 3 a selection of profiles are shown from a 2D fully kinetic simulation for anti-parallel Harris sheet reconnection using the code VPIC (Bowers et al., 2009) with open boundary conditions (Daughton et al., 2006).The simulation is implemented with m i /m e = 1,836, T i∞ /T e∞ = 5, ω pe /ω ce = 2, 400 particles per species per cell, and an upstream normalized electron pressure, β e∞ = 2 5 .More technical details on the run can be found in Ref. Egedal To visualize how the structures of the EDR roughly rotate like a solid 1D body, Figures 3e and 3f provide profiles of the reconnecting magnetic field B x , and the Hall magnetic field B y , respectively.From these profiles we obtain ∠B xy = arctan(B y /B x ) displayed in Figure 3g, corresponding to the angle of the B xy -field relative to the x-direction.
Between the two black lines where the magnetic field magnitude is small, |B| < 0.05, this angle is not well defined, and we here interpolate ∠B xy to the values observed along the EDR edge (the black lines where |B| = 0.05).Similar to Ref. Hesse (2006), we introduce coordinates rotating with the reconnection geometry, x′ = x cos ∠B xy + ŷ sin ∠B xy , ŷ′ = x sin ∠B xy + ŷ cos ∠B xy , and ẑ′ = ẑ.Figure 3h displays the electron current J ey′ in the rotated frame, with a structure nearly independent of x accounting for nearly all the current within the EDR.In Figures 3i-3k, for visualizing the near 1D form of the anisotropic electron heating, we display the temperature components in the rotating frame defined by T ejj = m ∫ (v j 〈v j 〉) 2 f e (v)d 3 v/ n e , where j ∈ {x′, y′, z′}.
In Figure 3l, the current layer rotation as a function of x for z = 0 is shown, from which the length scale of rotation can be defined, 1/l u = ∂∠B xy /∂x ≃ 1/(18d e∞ ).In Ref. Egedal et al. (2023), l u is shown to regulate the magnitude of the off-diagonal pressure tensor elements near the center of the EDR; for example, P exy ≃ (x/l u )(P exx P eyy ).
Meanwhile, for the 1D model l u = ∞, such that here P exy = 0.
To further document the structure of the reconnection inflows and EDR, in Figures 4a-4d cuts at x = 0 are provided for B x , eΦ ‖ /T e∞ , T ex′x′ , T ey′y′ , T ez′z′ , and J ey′ .In Figure 4a, the blue line is B x observed in the simulation whereas we denote the red line as B x,i (z), obtained as B x,i (z) = B x (z) μ 0 ∫ z 0 J ey z′) dz′.Thus, B x,i (z) has contributions from the electrons subtracted off, and can be considered the magnetic field caused by the ions; this profile is used as input for the 1D modeling, to be detailed below.Given the symmetry of the simulation, B is purely in the x-direction for x = 0, such that in Figure 4c the profile of T ex′x′ coincides with T e‖ , whereas T ey′y′ and T ez′z′ are representative of T e⊥ outside the EDR.Within the EDR the electron orbits are no longer circular in the plane perpendicular to B, such that T ey′y′ and T ez′z′ here have separate values.

Application of the Generalized Model for f e
We next examine the extent to which the kinetic simulation results can be accounted for by a 1D model based on Equation 2. Because the model imposes B y = 0, the in-plane potential, Φ, coincides with the acceleration potential of Ref. Egedal et al. (2009), Φ‖ = ∫ ∞ x E ⋅ dl.The model is implemented as follows: For given profiles of B x (z) and Φ ‖ (z) we compute J z as a function of (z, v y , v z ) using the methods developed for a 1D current sheet in Ref. Egedal (2023).In turn, this permits an evaluation of f e (z, v) in Equation 2, where the upstream distribution f ∞ (E) is taken as an isotropic Maxwellian with density and temperature matching the kinetic simulation in regions far outside the reconnection layer.Self-consistent profiles are then obtained through an iterative scheme, where at step k the electron distribution f k e (z,v) is computed based on profiles B k x (z) and Φ‖ k (z).From f k e (z,v) the electron current, J k ey (z), and its contribution to the magnetic field is evaluated, B k x,e (z) = μ 0 ∫ z 0 J k ey z′) dz′.In the following iteration step we then use x,e (z))/2.The acceleration potential is updated using ) . (4) Here n i (z) is the profile of the ion density in the VPIC simulation, n k e (z) is the electron density obtained from f k e (z,v), while λ defines the dimensionless "step-size" used in the iterations.Equation 4 is derived such that for the fictional case of Boltzmann electrons, n e ∝ exp(eΦ ‖ /T e ), the profile of Φ ‖ is found in a single iteration when using λ = 1.In reality, the electrons do not follow the Boltzmann scaling, but for λ = 0.15 stable conversion toward quasi-neutrality n e ≃ n i is observed in about 20 iteration when initialized by Φ‖ 1 (z) = 0 and B 1 x (z) = B x,i (z).
Results based on the 1D model are shown in Figures 4e-4h, reproducing many of the details from the VPIC run in Figures 4a-4d.In Figure 4e the red line is B x,i (z) obtained from the VPIC run and applied as an input to the 1D model.Meanwhile, the blue line includes a sharp jump centered on z = 0 corresponding to the EDR electron current layer predicted by the model.Likewise, the profile of Φ ‖ in Figure 4f provides a detailed match to the VPIC profile in Figure 4b.However, for the central portion, |z|/d e∞ < 0.8 the inversion of Φ ‖ yields an electronhole structure (Chen et al., 2011;Egedal, 2023) for which the relationship between n e and Φ ‖ is not consistent

Geophysical Research Letters
10.1029/2024GL108895 EGEDAL with ∂n e /∂Φ ‖ > 0 implied by Equation 4. Thus, as encircled in red, in this small region we impose "by hand" the local enhancement in Φ ‖ observed in VPIC.The enhancement causes the current layer to bifurcate and is associated with localized T ezz -heating (Egedal, 2023).
The agreement between the VPIC profiles in Figures 4c and 4d and the 1D model in Figures 4g and 4h may appear surprising.For example, the enhancement in T ex′x′ is solely due to the acceleration potential Φ ‖ that in the 1D model develops independent of E rec .Similarly, because the 1D model is derived in the adiabatic limit where E rec vanishes, the current profiles in Figure 4h are solely caused by the orbit motion and its relationship with the pressure anisotropy that forms within the reconnection inflow regions.The current integrated across the 1D model profile is given by K = 2 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ (P e‖ P e⊥ )/μ 0 √ , where P e‖ P e⊥ is the value observed at the edge of the EDR (Egedal, 2023).In the VPIC run a small fraction of the current can be attributed E rec , such that for the run slightly less anisotropy is required for sustaining the EDR current.Thus T ex′x′ -VPIC reach an amplitude about 25% smaller than T ex′x′ -1D.
For the same three positions as considered in Figures 2b-2f, in Figure 5 we show similar electron distributions predicted by Equation 2. The distribution in Figures 5a and 5b is obtained at z/d e∞ = 3 just outside the edge of the EDR, and has the morphology previously explored in Refs.Egedal et al. (2008Egedal et al. ( , 2013)).Meanwhile, for the points at z/d e∞ = 1.5 considered in Figures 5c and 5d the effect of using J z in place of μ in Equation 2becomes  (Ng et al., 2011).In contrast, the 1D adiabatic model is obtained in the limit of infinitely fast bounce motion in the z direction, eliminating the striations.
At the X-line in Figure 5e, the distribution of the 1D model has a near perfect semicircular shape in its v x v yprojection, where the cutoff velocity is set by the v x -extent of the distribution at the edge of the EDR.In comparison, the distribution in Figure 2f has a triangular shape where the slight tip at v x = 0 is a result of energization by the reconnection electric field, E rec (Ng et al., 2011).Meanwhile, the splitting of the distributions into two parts in the v y v z projections, observed both in Figures 2g and 5f, is caused by the non-monotonic profile of Φ ‖ (z) discussed in Refs.Chen et al. (2011) andEgedal (2023).

Discussion and Conclusions
It has recently been emphasized (Liu et al., 2017) how the rate of magnetic reconnection is often controlled by the inertia of the ions, with E rec ≃ 0.1v A B up , where v A and B up are the ion Alfvén speed and magnetic field observed at the edge of the EDR (Shay et al., 1999).Given the large inertia of the ions, from the perspective of the electrons, the reconnection electric field is small.To be more specific, it is natural to introduce Êrec = ed e∞ E rec / T e∞ as a measure of the relative energization an electron will experience when traveling 1 d e∞ in the direction of E rec .

Simple analysis then yields
, where β e∞ is the normalized electron pressure upstream of the reconnection region.For values of β e∞ considered here and typical of the Earth's magnetosphere it is found that Êrec ≪ 1, such that Êrec only marginally influences the rapid motion of the individual electrons through the EDR (Egedal et al., 2023) (for the particular run considered here we have Êrec ≃ 0.025, see Figure 10c in Ref. Egedal et al. (2023)).As a result, the structure of the inflow region and the EDR is well accounted for by the adiabatic electron equilibrium governed by Equation 2, where the low values of Êrec only yields small perturbations away from the 1D equilibrium geometry.
In summary, using the current sheet adiabatic invariant J z of a 1D current layer, a previous model for the anisotropic heating in the reconnection inflow is generalized.The new model reproduces the detailed inflow and EDR structure of anti-parallel reconnection previously only observed in fully kinetic simulation and spacecraft observations.The model is derived in the adiabatic limit of a vanishingly small reconnection electric field, emphasizing how the structure of the EDR in collisionless reconnection is mainly set by kinetic dynamics not tied to the rate of the reconnection process.In contrast, the overall anisotropic heating of the EDR is imposed by the constraint that J z be conserved for each electron.Given this constraint, the local electron density is regulated by the acceleration potential, Φ ‖ , which adjusts such that quasi-neutrality with the ions is maintained.The field aligned electron pressure anisotropy generated in the reconnection inflow couples to the motion of meandering Speiser-type orbits and is responsible for driving the current of the EDR, setting both the structure/width and magnitude of the EDR current layer.
The magnetic field lines are shown by the white lines, while the current density of the thin current layer (centered on z = 0) is represented by the color contours.The axes are normalized by the electron skin depth based on the density outside the current layer, d e∞ = ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ m e ⁄ (μ 0 n e∞ e 2 ) √ .The magenta line represents a typical electron Speiser orbit (Spe-

Figure 1 .
Figure 1.Projections of a typical electron Speiser orbit crossing a current layer including a small normal magnetic field.Supported by electron pressure anisotropy, the static 1D configuration is obtained by the methods in Ref. Egedal (2023), with T e‖0 /T e⊥0 = 4, T i /T e⊥0 = 10, and B Z = B 0 /10.

Figure 2 .
Figure 2. (a) Color contours of T e‖ /T e⊥ with black lines indicating the projection of the magnetic field lines.The magenta line represents a typical electron test orbit.This electron initially follows a trapped trajectory in the lower inflow region before reaching the EDR.Within the EDR, the electron trajectory transitions into the Speiser type shown in Figure 1.(b-g) Projections, ∫f(v x , v y , v z )dv z and ∫f(v x , v y , v z )dv x , of electron distributions for x/d e∞ = 0 and z/d e∞ ∈ { 3, 1.5, 0}.

Figure 3 .
Figure 3. Profiles for a range of quantities (see text) observed in a fully kinetic simulation of anti-parallel magnetic reconnection.The locations of the EDR are marked by magenta rectangles with sides z/d e∞ = ±2.5, x/d e∞ = ±15.

Figure 4 .
Figure 4. (a-d) Cuts of kinetic simulation profiles evaluated for x = 0 (data averaged over |x/d e∞ | < 2), where ∠B xy = 0 such that T ex′x′ = T exx , etc.In (a) the blue line is B x (z), while the red line represents the magnetic field of the ions, B x,i (z) = B x (z) μ 0 ∫ z 0 J ey z′) dz′, used as input for the 1D modeling, (b) shows the profile of Φ ‖ .In (c) temperature components T exx , T eyy , and T zz are shown, whereas (d) documents J ey .(e-h) Profiles obtained by the 1D model matching those of the kinetic simulation in (a-d).In the zoom-in plots the edges of the EDR, z/d e∞ = ±2.5, are marked by the dashed magenta lines.