3D cylindrical BGK model of electron phase-space holes with finite velocity and polarization drift

Nonlinear electron kinetic structures are regularly observed in space and experimental magnetized plasmas, called electron phase-space holes (EHs). The existence of EHs is conditioned and varies according to the ambient magnetic field and the parameters of the electron beam(s) that may generate them. The objective of this paper is to extend the 3D Bernstein-Greene-Kruskal (BGK) model with cylindrical geometry developed by Chen et al. (2004,2004) to include simultaneously finite effects due to (i) the strength of the ambient magnetic field $\vec{B}_0$, by modifying the Poisson equation with a term derived from the electron polarization current, and (ii) the drift velocity $\vec{u}_e$ of the background plasma electrons with respect to the EH, by considering velocity-shifted Maxwellian distributions for the boundary conditions. This allows us to more realistically determine the distributions of trapped and passing particles forming the EHs, as well as the width-amplitude relationships for their existence.


I. INTRODUCTION
Electron phase-space holes (EHs) are non-linear kinetic structures observed in the Earth's magnetosphere by spacecraft missions in various regions (e.g, the auroral region, the bow-shock, the dayside magnetopause and the geomagnetic tail), [3][4][5][6][7][8][9][10][11][12][13][14][15][16] as well as in the Saturn's magnetosphere by Cassini mission, 17,18 and very recently in the induced Venus magnetosphere by Parker Solar Probe and Solar Orbiter missions. 19,20oreover, this type of structures can be generated by laboratory experiments such as intense laser, 21 magnetic reconnection facilities 22 and by beam injection. 23Some theories have been developed for solitary waves in quantum plasma 24 based on a fluid model or a Bernstein-Greene-Kruskal (hereafter, BGK) approach 25 for quantum Vlasov equation. 26More generally, coherent structures appear in many fields of physics such as biophysics, condensed matter or fluid dynamics. 27rst observations in magnetized space plasmas have evidenced electrostatic structures propagating along the ambient magnetic field B 0 , 3,17,28 although, for fast moving structures, a perpendicular magnetic perturbation to B 0 (δ B ⊥ ) corresponding to the Lorentz field was detectable. 4 Recently, observations in the tail of the Earth's magnetosphere have also reported structures with another electromagnetic signature, characterized by a magnetic field perturbation parallel to B 0 (δ B ∥ ) [12][13][14][15] .These structures associated with electron trapping and a hole in phase space, are characterized in real space by a positive electric potential and an electron density depletion in their center.They therefore have a divergent electric field and their crossing in the direction parallel to B 0 shows a bipolar (monopolar) spike of the parallel (perpendicular) electric field component.Their parallel width is generally in the order of a few to ten Debye lengths, while their perpendicular width can be much larger.The determination of these lengthscales and their ratio (shape of the structure) are most probably related to the generation mechanism, and to some existence criteria to which this article tries to bring new elements of under-standing.0][31][32] In a magnetized plasma, they move along the field line at a velocity v EH close to the average velocity v b of the electron beam(s), and are commonly characterized into two types: slow 8 or fast 4 , depending on whether |u e | ≪ v T or |u e | ≥ v T , with u e the drift velocity of the background plasma electrons with respect to the EH and v T the parallel thermal velocity of electrons.As demonstrated numerically by Muschietti et al. 33,34 and experimentally by Fox et al., 22 one of the criteria for the existence of such a nonlinear structure is that the electrons remain sufficiently magnetized to be stable.
In the examples cited above, the plasma conditions vary and, in particular, the magnitude of the ambient magnetic field.Historically, BGK models were developed in 1D 25,35 for unmagnetized plasma and then theoretical additions or suggestions were made: shifted distribution 2,36 , non-maxwellian distribution (e.g, flap-top, 37 Lorentzian 2,38 ), effects of electron polarization current, 1,39 and 2D 40 or 3D 1,2 extension.However, these previous studies did not investigate dimensionality and finite velocity effects at the same time, 2,25,36,37,41 and did not give a quantified estimate of electron polarization effects. 1,39In this paper, we develop a BGK theoretical model that includes the effects of dimensionality, finite ambient magnetic field, and a drift velocity of the background plasma electrons with respect to the EH (i.e, 3D axisymmetric cylindrical coordinates, shifted electron distribution function and electron polarization current).It aims at discussing the previous studies, modeling slow and fast EH observations for relatively weak and strong magnetized plasma, and relating the model to the PIC simulations. 3,29,31,32In particular, the questions addressed in this paper are: how do the ambient magnetic field strength and the finite electron drift velocity influence the existence conditions and characteristics of such 3D EHs?
After a presentation of the 3D BGK model of EHs with cylindrical geometry, and its underlying assumptions, such as those that allow the inclusion of electron polarization effects (section II), the characteristics (distribution functions and densities) of the passing and trapped particles are theoretically determined in section III.Section IV aims at deriving the conditions of existence of these EHs.Our model is then discussed in section V in the light of space and laboratory measurements, as well as simulation results.

II. 3D BGK MODEL
We consider the case of an uniform magnetized plasma with the assumption of an unperturbed neutralizing ion background.The dynamics of ions is thus ignored for reasons of simplicity and because of their large mass ratio with electrons.The parallel dynamics of electrons (of charge −e and mass m) is analyzed from the evolution of their distribution function f e , while their perpendicular dynamics is described by a fluid representation of their polarization drift.

A. Parallel Vlasov dynamics
The original BGK model 25 of EH is a one dimensional, stationary (∂ t f e = 0) nonlinear exact solution of the Vlasov-Poisson system of equations for a given shape of potential.Based on the observations [11][12][13][14][15] , the EH potential can be represented in cylindrical coordinates (r, θ , z) independent by rotation around the ambient magnetic field axis (B 0 = B 0 ẑ), and can be written in the EH frame in the following double-Gaussian form: with φ 0 the amplitude of the potential structure, ℓ ⊥ and ℓ ∥ its half-width in the perpendicular and parallel directions to B 0 , respectively.It is also reasonable to consider the magnetic field perturbation to be much smaller than the ambient magnetic field (δ B ≪ B 0 ) and the perturbed Lorentz term to be negligible compared to the electric field perturbation The Larmor radius of the electrons being generally much smaller than the perpendicular size of the EH (ρ L ≪ ℓ ⊥ ) and their travel time in the structure sufficiently slow compared to their gyroperiod ((|u e | + v T )/ℓ ∥ ≪ ω c ), 15,22,34,39 their motion can be approximated by their guiding center motion.Due to the cylindrical symmetry, azimuthal drift displacements (δ E × B 0 ) do not contribute to equilibrium in phase space and will not be considered 15,42 .Indeed, as a first approximation for describing the kinetic dynamics along the ambient magnetic field, we will neglect the radial and azimuthal motions of the electrons (i.e, assume no dependence on perpendicular velocities), and we will solve the corresponding simplified Vlasov equation, just along the z-axis for a given r value, the electrons being closely tied to a cylindrical magnetic field surface of radius r 34,42 : This equation represents a simplified Vlasov equation, in the context of azimuthal symmetry and uniform magnetic field, where the dependence of θ and perpendicular velocities are neglected.Its conditions of validity deserve some additional comments, which we now address.The condition to neglect the finite Larmor radius effects on the electrons can be rewritten using the ratio between the electron cyclotron frequency and the electron plasma frequency, and considering the anisotropy ratio of their temperature: with λ D the (parallel) Debye length.This implies for ω c /ω p ≤ 1 that ℓ ⊥ ≫ λ D (unless T e⊥ ≪ T e∥ , as in the FAST observation case 4 ).The present model has therefore a limit and cannot correctly describe all the structures of small perpendicular size when the plasma is weakly magnetized.In particular, as stressed by Hutchinson 43 , such a model is not valid for EHs with a scale of the order of one Debye length (ℓ ⊥ ∼ λ D ) in the case of ω p /ω c = ρ L /λ D ≫ 1.However, even in the ℓ ⊥ ∼ ρ L limit, the guiding center approximation should still give some qualitatively acceptable results.
The finite frequency effects on the electrons are negligible when, in the reference frame moving at their parallel velocity, the time scale of the variations they undergo is much larger than their gyroperiod.Similarly, this second condition for the electrons to remain magnetized can be rewritten to give a second condition on the electron cyclotron to plasma frequency ratio: Therefore, for weak magnetic field conditions, small parallel and/or fast velocity structures could hardly be described by our model.Note that the previous reasoning also applies to trapped electrons 35,37 and, insofar as eφ 0 ≤ T ∥ , implies that their bounce frequency has to be much lower than their cyclotron frequency for the guiding center approximation to be valid: 33,34 Note that this condition is similar to that etablished by Chen et al., 1 as given by their Eq.(10).However, while we agree that this is a condition for preventing trapped electrons from escaping, i.e. their demagnetization during their bounce motion, we disagree that this condition implies that the effects of polarization drift are negligible.Indeed these latter can be present without causing the running away of electrons.In view of these limitations, there are however space observations and laboratory measurements for which the conditions developed in this paper are valid and which will be discussed in section V (see Table I and references therein).

B. Polarization effects
We decompose the electron charge density as a sum of two terms: ρ e = ρ e∥ + ρ e⊥ , where ρ e∥ is the contribution to the electron charge density due to their parallel motion, which will be determined from Eq. ( 2); and where ρ e⊥ is an additional contribution due to their perpendicular motion.As a direct consequence of azimuthal symmetry, the only perpendicular motion to be considered is the polarization drift.We take into account this effect in the form of an equivalent charge, ρ pol = ρ e⊥ , and an additional current in the Maxwell equations.In the reference frame moving at their mean parallel velocity, this electron current is written to lowest order as , where n 0 is the electron background plasma density.Taking the divergence: leads to an expression for the contribution of the electron polarization displacement in the density balance that depends on time.Integrating then with respect to time, this gives us an expression for the electron polarization charge density valid in the EH frame.Hence, the Poisson equation for mobile electrons and infinitely heavy ions, where ρ ∥ = en 0 + ρ e∥ , can take the following form: 39,43,44 The second term on the left-hand side includes thus the perpendicular displacement of electrons due to their finite polarization drift.As Hutchinson 43 pointed out, this term is in principle negligible in the setting where the guiding center approximation applies with structures at Debye lengthscales (as shown in Eqs. ( 3)-( 4) for ℓ ⊥ , ℓ ∥ ∼ λ D ), and should not be invoked for explaining the ℓ ⊥ /ℓ ∥ ≃ (1 + ρ 2 L /λ 2 D ) 1/2 scaling observed by Franz. 39Nevertheless, with the limitations highlighted in the previous section on the validity conditions of our model, for sufficiently large EHs (a few to a few tens of Debye lengths as estimated from observations), the electron polarization current can play a role in the charge balance, which is evaluated in the next sections.
Using the form of the potential as given by Eq. ( 1), we obtain the following result for the charge densities: where Λ = 1 + ω 2 p /ω 2 c . Figure 1 represents the charge densities ρ ∥ and ρ pol along z and r directions as calculated by Eqs. ( 9) - (10).We then observe that the contribution of the polarization current ρ pol (see Figures 1(b) and 1(d)) to the total charge density ρ tot = −ε 0 ∇ 2 φ can be of the same order when ω c /ω p ≤ 1, which consequently changes significantly the values of ρ ∥ (see Figures 1(a) and 1(c)).For consistency, one can easily verify that the total charge of the electron hole is zero.

C. Integral equation
We introduce the total energy ε(r, z, v z ) = mv 2 z /2 − eφ , which is a constant of the electron motion along a cylindrical magnetic field surface parameterized by r.We consider two types of electrons with respect to the potential φ : trapped electrons if their total energy ε do not exceed the potential barrier, i.e, if −eφ 0 ≤ ε ≤ 0, and passing electrons such as ε > 0. This allows us to write the total electron distribution function by introducing two distribution functions f t and f p for trapped and passing electrons, respectively, such as: Far away from the potential influence (i.e, z → ±∞), the passing electron distribution must match the boundary conditions: We can then decompose the electron density using f t and f p as in the BGK approach: 25,37 0 with V = eφ < eφ 0 the potential energy and, The set of Eqs. ( 12)-( 13) has the form of an Abel integral equation that can be solved using Laplace techniques 25,37 or another method from §12 Landau & Lifshitz, 45 and considering physical distributions, i.e, g(r, 0) = 0 : where f are the integrals corresponding to the two terms of g as given by Eq. ( 13).The term f

III. SIGNATURE OF TRAPPED AND PASSING ELECTRONS A. Distribution functions
As a first step, using the form of the density as determined by Eq. ( 10), the first integral becomes: where v T = (T e∥ /m) 1/2 is the parallel thermal velocity of electrons far from the EH, and λ D = v T /ω p .This term is negative at the bottom of the well, i.e, when ε → −eφ 0 , then becomes positive and cancels for ε → 0. For the case of a negligible electron polarization current, i.e. for Λ = 1, we obtain the same result as Chen et al. 1 The passing electrons for |z| ≫ ℓ ∥ are all corresponding to the background plasma.Thus, if at infinity their distribution f ∞ is a velocity shifted Maxwellian distribution (independent of r), everywhere else it must be of the following form: with ε > 0 and u e the drift velocity of the background plasma with respect to the EH.The contribution of passing electrons to the distribution of trapped electrons can then be written as: with and where we have defined the quantities β = (−ε/T e∥ ) 1/2 and ζ = u e / √ 2v T .As an integral function of a positive integrand, we get a positive function.The general expression (19)  does not allow an analytical calculation, and must be evaluated numerically, except when the integral takes the form of the Dawson's integral, for b = 0.In that case of zero drift velocity (u e = 0), corresponding to the EH immobile with respect to the background plasma, we get: where the erf function is defined as √ π erf(x)/2 = x 0 exp(−t 2 )dt.For the case of zero drift velocity, we then well find the result obtained by previous authors, whether in the 1D 35,36 or 3D 1,42 BGK model.For the case of a finite drift velocity, the solution ( 18) is in accordance with Turikov's result. 36The continuity property of this general solution when ε → 0 can be easily verified.First, we obtain straightforwardly f , which is consistent with the populations present outside the hole and generation mechanism.t .As a result, in that case with ω c /ω p = 0.5, the distribution of trapped electrons f t is found to be slightly negative, which is unphysical.As we shall see later, the conditions for the existence of EHs will be defined on the basis of this limitation.In addition, the pseudo-Maxwellian part of f p is moved at higher velocities, allowing for particle acceleration.Comparing Figures 2(b)-(d) shows the impact of the drift polarization effect on f t .As Λ increases, there is a decrease in the density of trapped electrons f t at the hole center (v = v EH ).
Figure 2(a) with u e ≪ v T represents the case of a slow EH (e.g, as observed in the magnetotail 8,9 ) that may result from counterstreaming instability. 30,46,47Conversely, Fig- ures 2(c)-(d) with u e ≥ v T represent the case of fast EH and show a plateau-like structure as observed in space plasmas, 12 or in Particle-In-Cell (PIC) simulations of bumpon-tail type. 29,31,32e choice of a drift velocity u e as high as 2v T may raise questions for a reader accustomed to Schamel's results, showing that values greater than about 1.3 are impossible under physical conditions. 48Unlike the BGK integral equation method we used to solve the Vlasov-Poison equation sys-tem, Schamel's approach, which used a differential equation method, leads to a limited number of possible solutions.Indeed, keen to find "preferred" BGK states, Schamel introduced a particular shape of the trapped distribution function, which is smoother at the separatrix ε = 0 than most BGK solutions, and in particular ours as can be seen on Figure 2. Justifying what might be an acceptable trapped distribution function remains an open question 49 and is beyond the scope of our work.However, we can acknowledge that such structures have long been observed in space and laboratory plasmas.For instance, Andersson et al. 14 reported fast EHs in the Earth's magnetotail with velocities u e ≥ 2.5, whereas histogram given by Holmes et al. 12 show velocities between 1 and 1.5.In addition, Fox et al. 22 reported EHs with velocities u e ∼ 2 under laboratory conditions, as did Lefebvre et al., 23 who found 80% of structures with velocities between 1.3 and 2.3, with 2.1 as the median velocity.

B. Electron densities in the hole
From the passing distribution function f p , we can explicitly define the density n p of passing electrons as: with and where ψ = eφ /T e∥ , the ratio between the potential and thermal energy of electrons.In particular for zero drift velocity (u e = 0), For finite drift velocity, the integral ( 22) must be evaluated numerically.Using the definition of the charge density ρ ∥ , the trapped electron density n t writes: and can be determined from Eqs. ( 10) and ( 21).In the case where u e = 0 and Λ = 1 (negligible effects of EH velocity and electron polarization drift), the trapped density n t calculated by Chen et al. 1,2 is retrieved.We can also define the density of the trapped electrons from their distribution as: Figure 3 displays the different electron densities in the hole (n t , n p and their sum n t + n p ) as a function of parallel axis z.As indicated by the Poisson Eq. ( 8) and Eq. ( 24), the quantity n p + n t represents the electron density −ρ e∥ /e induced by the potential of the structure and the perpendicular polarization drift effect.It can be observed that despite the variations of the quantities n p and n t , the overal neutrality remains preserved and the total load of the structure is zero.The impact of the polarization effect can be seen by comparing Figures 3(a) and 3(b).We observe that this leads to a reduction of the density of electrons trapped in the potential well, and more importantly at the center than at the edges (dashed line in Figure 3(a)).Thus, for ω c /ω p < 1, the distribution of trapped electrons consists of two humps.This is because the polarization current brings additional electrons to the center of the structure (see Figures 1(b) and 1(d)), independently of the parallel electron dynamics, so fewer trapped electrons are needed to satisfy Poisson's equation.Figures 3(b) and 3(d), or 3(a) and 3(c), illustrate the impact of the drift velocity.As it results in more passing electrons, there is also less need for trapped electrons.For ω c /ω p = 0.5, u e = −2v T and assuming a spherical shape (ℓ ∥ = ℓ ⊥ = 5λ D ), as shown in Figure 3(c), the case is even impossible, as it would require a non-physical negative density of the trapped electrons.In this exemple, theoretical determination of the distribution of trapped electrons ( f t ) in fact reveal negative values around its center, similar to what is observed in Figure 2(b).We can note that the lowest order approximation of J pol we have used, is in principle strictly consistent for small parallel perturbations of the electron density, i.e, n t + n p ∼ n 0 .For this reason, all our evaluations carried out with strong disturbances are to be considered with caution and aim at identifying trends.
This presentation of the electron density composition in the EH allows us to recall an important observation made by Chen and Parks 35,42 on the nature of the charge density "shielding" of the core of a BGK structure.As Figure 3 shows, the passing electrons in the middle of the hole are in deficit (relative to the ions), simply because they are accelerated there.The positive core is then "shielded" by the trapped electrons that oscillate in the potential structure.In fact, the trapped electrons must distribute themselves in such a way as to counterbalance the positive charge density produced by the depletion of passing electrons inside the potential well, thereby producing a total charge density consistent with the specified potential profile.A BGK EH is a self-consistent and self-sustaining object with zero total charge and does not require any thermal screening by the surrounding plasma (Debye shielding).This result contradicts the idea that the positive core of the EH is due to a deficit of trapped electrons, and that this positive core is screened by the passing electrons, as for example recently described by Hutchinson. 43

IV. EXISTENCE CRITERIA
We define and analyse the criteria that allow these EH structures to exist in order to obtain information on their size.The trapped distribution function (15) has to be physical, and thus f t must be non-negative.Writing f t (ε) ≥ 0 for −eφ 0 ≤ ε < 0, we obtain: where ψ 0 = eφ 0 /T e∥ is the ratio ψ at r = z = 0 corresponding to the potential maximum, and where we have introduced the function This function has the following limits, for ), and for ψ 0 → +∞: G(ψ 0 , ζ ) ∼ 1/2ψ 0 .The case without drift velocity (u e = 0) gives: The sign of the denominator in Eq. ( 26) imposes a second condition on the perpendicular scale, which must be respected and can be expressed as: Eqs. ( 26) and ( 29) give amplitude-width criteria in both parallel and perpendicular directions.In the case where u e = 0 and Λ = 1, we obtain the same criteria as those found by Chen et al. 1,2 Figure 4 represents the minimum parallel sizes of the hole (from Eq. ( 26)) as a function of the amplitude of the potential well for several values of the perpendicular size, and plasma parameters.While the observations indicate that the ratio eφ 0 /T e∥ does not seem to exceed 1 by much, we have extended its range to 10 in order to show the common trend for the different values of ℓ ⊥ .The area under these curves represents forbidden zones, see for example the shaded area of Figure 4(a) corresponding to the case ℓ ⊥ = ∞, i.e, the 1D limit.We note that for a finite perpendicular length the weaker the magnetic field strength or the higher the electron drift velocity, the larger the forbidden zone.Indeed, polarization drift effects tend to reduce the possibility of perpendicular small scale, large amplitude structures (compare Figures 4(a) and  4(b) where u e = 0).When a shifted Maxwellian distribution is taken into account, which should be more realistic regarding to a beam instability generation mechanism, we find that the allowed zone is still more reduced, making structures below the Debye lengthscale in the parallel direction hardly possible (compare Figures 4(b) and 4(d)).This last result is consistent with Goldmann's. 38Furthermore, when the finite effects of electron polarization current and electron drift velocity are both considered, perpendicular small scale structures can only exist for very small amplitudes (see Figure 4(c)).
Figure 5 represents the minimum perpendicular sizes of the hole (from Eq. ( 29)) as a function of amplitude of the potential well for several ω c /ω p ratio, in case of u e = 0 (Figure 5(a)) and u e = −2v T (Figure 5(b)).For both cases, the polarization drift effects lead to an increase in the slope of ℓ ⊥ (ψ 0 ) and of the forbidden zone for small ω c /ω p values.For a given ω c /ω p ratio, the velocity drift effects also contribute significantly to the expansion of the forbidden zone.In addition, the width-amplitude relations ℓ ⊥ (ψ 0 ) shown in Figure 5 can also represent the maximum well amplitudes ψ 0,max as a function of ℓ ⊥ .The vertical asymptotes in Figure 4 correspond to the limit of ℓ ∥ (ψ 0 → ψ 0,max ), precisely.Consistently, from Figure 4(c), we observe that the conditions used to make Figure 3(c) (i.e, eφ 0 = T e∥ and ℓ ∥ = ℓ ⊥ = 5λ D , with u e = −2v T , and ω c /ω p = 0.5) are in the forbidden zone.

V. DISCUSSION
Our study allows to precise the 3D EH existence conditions depending on the electron drift velocity (u e ) and the plasma magnetization (ω c /ω p ).In this section, we compare our results with various measurements from space and laboratory summarized in Table I.For instance, the first statistical results of EHs observed by the Fast mission 4 in highly magnetized auroral plasma (ω c /ω p > 5) and moving with velocities u e < 0.6v T , have interpreted the observational relationship between eφ 0 /T e and ℓ ∥ /λ D from a 1D BGK model. 37Most of these observations (eφ 0 /T e ≈ 0.05 − 1.1, ℓ ∥ /λ D ≈ 0.5 − 4) belong to the allowed region for EH having ℓ ⊥ /λ D ≤ 3 (Figure 4(b)).As the same manner, a statistical study of EHs detected in the cusp and in the Plasma Sheet (PS)/Plasma Sheet Boundary Layer (PSBL) regions by the Polar satellite, showed a relationship between their potential amplitude and parallel size. 6In the cusp region, ω c /ω p < 1 (resp.PS/PSBL region, ω c /ω p ∼ 2), the EH velocity was found in the range 0.1-2 (resp.0.3-1) of the thermal velocity.The potential eφ 0 /T e was found between 10 −3 and 10 −1 (resp.between 10 −5 and 10 −2 ).Considering these different EH velocities and ω c /ω p ratios, the cusp (resp.PS/PSBL) observations can be related to our results in Figures 4(a  is consistent with the observed scale ratio (ℓ ∥ /ℓ ⊥ < 0.5), see Figure 4(a).From other observations provided by MMS 12,13 in weakly magnetized plasma (ω c /ω p ∼ 0.5), fast and largeamplitude EHs (u e /v T ≥ 1, eφ 0 /T e ∼ 1) have been also reported.In such conditions and in accordance with observations, these structures are allowed if ℓ ⊥ ≥ 10λ D with a minimum allowed value of ℓ ∥ ∼ 4λ D , see Figure 4(c).

VI. CONCLUSIONS
To conclude, the model presented in this paper describes the criteria of existence of 3D cylindrical EHs including both the polarization drift of electrons (ω c /ω p ≲ 1) and a finite velocity of the EH with respect to the background electrons (|u e | ≳ v T ).For fast EHs, this allowed us to use more realistic boundary conditions on the electron distribution functions, which include a global drift consistent to the observations (e.g, MMS recent observations 12,13 ).These theoretical improvements have shown that they could have an important impact on the distribution functions and densities of electrons passing through and trapped in the EH structures, as well as on their conditions of existence.These two effects, the polarization drift of electrons and a finite parallel shift in their velocity distribution, indeed tend to restrict the possibilities of smallscale and large-amplitude EHs.But given the conditions of validity of the underlying gyrokinetic approach, our results can only show a modest effect for small scale EHs.
Since these existence conditions only determine the boundaries of a semi-open parameter space, our results (and previous ones, e.g., see discussion in Chen et al. 2 ) do not provide any strong constraints on the relationship between perpendicular and parallel lengthscales.Consequently, as suggested in the introduction, we believe that what determines the oblateness of these structures lies not so much in their self-consistent physics, but rather in the mechanisms by which they are generated.So, while we disagree with Hutchinson's approach 43 that the EH's positive core is shielded by passing electrons (rather than arising from their depletion due to their acceleration by the potential well; see discussion in section III B) and thus that the electron polarization effect would virtually be an anisotropic shielding mechanism, we can agree with his conclusion given in the abstract that "trapped electron charge distribution anisotropy must [...] underlie the oblate shape of electron holes".
[14][15] Particle measurements by the MMS mission, even provided at 30 ms time resolution, 51 do not allow to fully resolve the electron distribution function in fast EHs, which have characteristic times of 1 ms.Depending on the number of EHs detected in 30 ms, measured electron distribution functions can be statistically representative to the EH or to the ambient plasma.Therefore, theoretical developments are crucial to better understand in situ observations of such self consistent structures.This could be addressed by PIC simulations and observations in weakly magnetized media (ω c /ω p ≲ 1), in a forthcoming study.

DATA AVAILABILITY STATEMENT
(a) t stands for the electron charge density ρ e∥ , and f (b) t for the contribution of the passing electrons.

Figure 2
Figure 2 displays the different contributions to the distribution of trapped electrons ( f t , with the virtual terms f(a) t and f (b) t determined above), the distribution of passing electrons ( f p ), and the electron distribution at infinity ( f ∞ ), as a function of velocity in the EH reference frame.The drift-free case (u e = 0) is represented by the Figure 2(a) as reference.The impact of u e is visualized by comparing to the Figure 2(b), which represents a case with finite drift (u e = −2v T ).We observe an asymmetric distribution, an important decrease in the positive virtual value of f (b) t without any change in the equally with Λ, i.e, decreases with ω c /ω p ), without modifying f (b)

FIG. 4 .
FIG. 4. Parallel width-amplitude relations for different values of perpendicular width ℓ ⊥ , drift velocity u e and cyclotron to plasma frequency ratio ω c /ω p .

TABLE I .
Experimental data in different regions Article Context ω c /ω p ℓ ∥ /λ D ℓ ∥ /ℓ ⊥ eφ 0 /T e∥ |u e |/v T Perpendicular width-amplitude relations for different values of cyclotron to plasma frequency ratio ω c /ω p and drift velocity u e .