3D BGK model for electron phase-space holes including polarization drift [^1]

Nonlinear kinetic structures are regularly observed in space and experimental magnetized plasmas, called electron phase-space holes (EH). The existence of EH is conditioned and varies according to the ambient magnetic field and the parameters of the electron beam that may generate them. The objective of this paper is to extend the 3D Bernstein-Greene-Kruskal (BGK) model with cylindrical geometry to include finite effects due to (i) the strength of the ambient magnetic field $\mathbf{B}_0$, by modifying the Poisson equation with a term derived from the electron polarization current, and (ii) the drift velocity $u_e$ of the background plasma 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 interacting with the EH, as well as the width-amplitude relationships for their existence.

\[sec:level1\]Introduction
==========================

Electron phase-space holes (EH) are non-linear kinetic structures observed in the Earth's magnetosphere by spacecraft missions (@matsumoto1994electrostatic [@ergun1998fast; @bale1998bipolar; @cattell2005cluster; @norgren2015slow; @le2017lower; @tong2018simultaneous; @holmes2018electron; @steinvall2019observations; @andersson2009new; @tao2011model; @shustov2021dynamics] and the reference therein) in various regions (*e.g.* the auroral region, the bow-shock, the dayside magnetopause and the geomagnetic tail), as well as in the Saturn's magnetosphere by Cassini mission (@williams2006electrostatic [@pickett2015electrostatic]), and very recently in the induced Venus magnetosphere by Parker Solar Probe and Solar Orbiter missions (@malaspina2020plasma [@haddid2021Solar]). First observations have evidenced electrostatic structures (@ergun1998fast [@williams2006electrostatic; @krasovsky1997bernstein]). Recently, observations in the tail of the Earth's magnetosphere have reported structures with an electromagnetic signature, characterized by a magnetic field perturbation in the direction parallel to the ambient magnetic field (@andersson2009new [@tao2011model; @holmes2018electron; @steinvall2019observations]). Moreover, this type of structures can be generated by laboratory experiments such as intense laser (*e.g.* @montgomery2001observation), magnetic reconnection facilities (*e.g.* @fox2008laboratory) and by beam injection (*e.g.* @lefebvre2010laboratory). Some theories have been developed for solitary waves in quantum plasma (*e.g.* @mamun2010solitary) based on a fluid model or a Bernstein-Greene-Kruskal (hereafter, BGK) approach for quantum Vlasov equation (*e.g.* @haas2020bernstein). More generally, coherent structures appear in many fields of physics such as biophysics, condensed matter or fluid dynamics (@dauxois2006physics). EH can typically be generated during the nonlinear phase of beam instabilities caused by the interaction of two plasma populations (@omura1996electron [@mottez2001instabilities; @umeda2004two]). These structures of a dozen Debye lengths move at a velocity $v_{\mathrm{EH}}$ close to the beam velocity, and are commonly characterize in two types: slow (*e.g.* @norgren2015slow) or fast (*e.g.* @le2017lower). They are characterized by a bipolar electric field in the direction parallel to the ambient magnetic field, with a positive electric potential, and an associated depletion of electrons. In the examples cited above, the plasma conditions vary and, in particular, the magnitude of the ambient magnetic field. How does the ambient magnetic field strength impact the existence and characteristics of EHs ?

After a presentation of the BGK model and the assumptions for including the polarization effects (section \[sa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7BGK\]), the characteristics (distribution functions and densities) of the passing and trapped particles are theoretically determined in section \[sa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7Particles\]. Section \[sa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7Existence\] aims at deriving the conditions of existence of these structures.

BGK Model \[sa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7BGK\]
======================

Parallel Vlasov dynamics
------------------------

We consider the case of an uniform magnetized plasma composed of ions and electrons and we represent these species by a distribution function $f_s$ where $s=e,i$ for electrons (of charge $-e$ and mass $m$) and ions (of charge $Ze$ and mass $M$), respectively. The BGK model (@bernstein1957exact) of EH is a one dimensional, stationary ($\partial_{t}f_{s}=0$) nonlinear exact solution of the Vlasov equation for a given shape of potential. Based on the observations (@andersson2009new [@tao2011model; @holmes2018electron; @steinvall2019observations; @tong2018simultaneous]), the EH potential can be represented in cylindrical geometry ($r,\theta, z$) independent by rotation around the ambient magnetic field axis ($\mathbf{B}_0 = B_0\hat{\mathbf{z}}$), and can be written in the EH frame in the following double-Gaussian form: $$\phi(r,z) = \phi_0\,\exp\left(-\frac{r^2}{2\ell_{\perp}^2} - \frac{z^2}{2\ell_{\parallel}^2}\right) \label{eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7}$$ with $\phi_0$ the amplitude of the potential structure, $\ell_{\perp}$ and $\ell_{\parallel}$ its width in the perpendicular and parallel directions with respect to $\mathbf{B}_0$, respectively. We assume therefore that the magnetic perturbation is much smaller than the ambient magnetic field ($\delta B\ll B_0$). Also, due to the cylindrical symmetry, azimuthal displacements ($\mathbf{E}\times \mathbf{B}_0$, with $\mathbf{E}=-\mathbf{\nabla}\phi$) do not contribute to equilibrium in phase space and will not be considered (@tao2011model). In order to simplify, we will also consider the perturbed Lorentz term negligible with regard to the electric perturbation ($\mathbf{v}\times \delta \mathbf{B} \ll \mathbf{E}$). In addition, the electron Larmor radius being much smaller than the size of the EH ($\rho_{Le}\ll \ell_{\perp}$) (@muschietti2002modeling [@franz2000perpendicular; @tao2011model]), the electron motion can be approximated by their guiding center motion. Indeed, as a first approximation for describing the kinetic dynamics along the ambient magnetic field, we will neglect the perpendicular motion of the particles, and we will solve the corresponding simplified Vlasov equation, just along the $z$ axis for a given $r$ value, as follows: $$v_z\cdot\frac{\partial f_{s}(r,z,v_z)}{\partial z}-\frac{q_s}{m_s}\frac{\partial \phi(r,z)}{\partial z}\cdot\frac{\partial f_{s}(r,z,v_z)}{\partial v_z}=0 \label{eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7}$$

Polarization effects
--------------------

![image](ChargeDensity.pdf){width="\linewidth"}

Since the typical time scale of the EH is much smaller than the ions typical time scale, we assume that the ion provide neutrality (*i.e.* $Zn_i = n_0$, where $n_0$ is the background electron plasma density). We decompose the electron charge density as a sum of two terms: $\rho_e = \rho_{e\parallel}+ \rho_{e\perp}$, where $\rho_{e\parallel}$ is the contribution to the electron charge density due to their parallel motion, which will be determined from equation (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\]); and where $\rho_{e\perp}$ is an additional contribution due to their perpendicular motion. Inside the EH, the only perpendicular motion we will then consider is the polarization drift of electrons. We try to take into account this effect in the form of an equivalent charge, $\rho_{\mathrm{pol}}=\rho_{e\perp}$, and an additional current in the Maxwell equations. This electron current is written to zeroth order as $\mathbf{J}_{\mathrm{pol}} = n_0m\, \partial_t \mathbf{E}_{\perp}/B_0^2$. Taking the divergence: $$\mathbf{\nabla}\cdot \mathbf{J}_{\mathrm{pol}}= \frac{\partial }{\partial t}\left[ \mathbf{\nabla} \cdot \left(-\varepsilon_0\frac{\omega_{p}^2}{\omega_{c}^2} \mathbf{\nabla}_{\perp}\phi\right)\right]= - \frac{\partial \rho_{\mathrm{pol}}}{\partial t}$$ Hence, the Poisson equation for electrons and ions, $$\varepsilon_0\mathbf{\nabla}^2\phi = -\rho_{\parallel}-\rho_{\mathrm{pol}}$$ where $\rho_{\parallel} = en_0+\rho_{e\parallel}$, can take the following form (@franz2000perpendicular [@lee1983gyrokinetic; @jovanovic2002stability; @vasko2017electron; @hutchinson2021oblate]): $$\mathbf{\nabla}^{2}\phi+\frac{\omega_p^{2}}{\omega_c^{2}}\mathbf{\nabla}_{\perp}^{2}\phi=-\frac{\rho_{\parallel}(r,z)}{\varepsilon_{0}} \label{eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7}$$ where $\omega_p$, $\omega_c$ are the electron plasma and cyclotron frequencies respectively. The second term on the left side includes thus the perpendicular displacement of electrons due to their finite polarization drift. Then using the form of the potential as given by equation (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\]), we obtain the following result for the charge density: $$\frac{\rho_{\parallel}}{\varepsilon_{0}}  = \left\{ \frac{1}{\ell_{\parallel}^{2}}\left[\frac{r^{2}}{\ell_{\perp}^{2}}+2\ln\left(\frac{\phi}{\phi_{0}}\right)+1\right]  - \frac{\Lambda}{\ell_{\perp}^{2}}\left(\frac{r^{2}}{\ell_{\perp}^{2}}-2\right) \right\} \phi \label{eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7}$$ where $\Lambda = 1+\omega_p^{2}/\omega_c^{2}$. Figure \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\] represents the charge densities $\rho_{\parallel}$ and $\rho_{\mathrm{pol}}$ along $z$ and $r$ directions as calculated by equation (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\]). Then, taking into account the polarization current shows that the contribution of $\rho_{\mathrm{pol}}$ (see Figs. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](b)–(d)) to the total charge density $\rho_{\mathrm{tot}}=-\varepsilon_0\mathbf{\nabla}^2\phi$ can be of the same order when $\omega_c/\omega_p\leq 1$. Hence, this in turn changes the values of $\rho_{\parallel}$, see Figs. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](a)–(c). We notice that electron drift current for small frequency ratio $\omega_p/\omega_c$ ($\Lambda \simeq 1$), is consistent to Chen criteria (@chen2004bernstein) for neglecting polarization drift effect. Moreover, for consistency, one can easily verify that the total charge of the electron hole is zero (@hutchinson2017electron).

Integral equation
-----------------

We introduce the total energy $\epsilon(r)=mv_{z}^2/2-e\phi$, which is a constant of particle motion along a magnetic field line parameterized by $r$. We consider two types of electrons with respect to the potential $\phi$: trapped electrons if their total energy $\epsilon$, do not exceed the potential barrier, *i.e.* if $-e\phi_{0}\leq\epsilon\leq0$ and, passing electrons such as $\epsilon>0$. 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: $$f_{e}(r,\epsilon)=
    \left\{
  \begin{array}{ c l }
f_{p}(r,\epsilon) & \mathrm{if}\quad\epsilon>0\\
f_{t}(r,\epsilon) &\mathrm{if}\quad-e\phi_{0}\leq\epsilon\leq0
  \end{array} \right.$$ Far away from the potential influence (*i.e.* $z\to \pm \infty$), the passing electron distribution must match the boundary conditions $f_p(r,\epsilon) = f_{\infty}(r,\epsilon)$. We can then decompose the electron density using $f_t$ and $f_p$ as in (@bernstein1957exact [@muschietti1999phase]): $$\int_{- {V}}^{0}\frac{f_{t}(r,\epsilon)\,\mathrm{d}\epsilon}{\sqrt{2m(\epsilon+ {V})}}= \mathfrak{g}(r, {V})\label{eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7}$$ with $V= e\phi<e\phi_0$ the potential energy and, $$\mathfrak{g} =  -\frac{\rho_{\parallel}}{e}- \int_{0}^{+\infty}     \frac{f_{p}(r, { {\epsilon}})\,\mathrm{d} { {\epsilon}}}{\sqrt{2m( { {\epsilon}}+ {V})}}+ \int_{ {V}}^{+\infty}     \frac{Zf_{i}(r, { {\epsilon}})\,\mathrm{d} { {\epsilon}}}{\sqrt{2M( { {\epsilon}}-Z {V})}} \label{eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7}$$ The set of equations (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\])–(\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\]) has the form of an Abel integral equation that can be solved using Laplace techniques (@bernstein1957exact [@muschietti1999phase]) or another method from §12 Landau & Lifshitz (vol. 1) (@landau1976mechanics) and considering physical distributions, *i.e.* $\mathfrak{g}(r,0)=0$ : $$\begin{aligned}
f_{t}(r,\epsilon) &=& \frac{\sqrt{2m}}{\pi}\int_{0}^{-\epsilon}\frac{\mathrm{d}\mathfrak{g}(r, {V})}{\mathrm{d} {V}}\frac{\mathrm{d} {V}}{\sqrt{-\epsilon- {V}}}\\
&=& f_{t}^{(\mathrm{a})}+ f_{t}^{(\mathrm{b})}+ f_{t}^{(\mathrm{c})}
    \label{eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7-bis}\end{aligned}$$ where $f_{t}^{(\mathrm{a})}$, $f_{t}^{(\mathrm{b})}$ and $f_{t}^{(\mathrm{c})}$ are the integrals corresponding to the three terms of $\mathfrak{g}$ as given by equation (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\]). The term $f_{t}^{(\mathrm{a})}$ stands for the total charge density, $f_{t}^{(\mathrm{b})}$ for the contribution of the passing electrons, and $f_{t}^{(\mathrm{c})}$ for the ion contribution.

Signature of trapped and passing particles \[sa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7Particles\]
=============================================================

![image](Distrib.pdf){width="\linewidth"}

Distribution functions
----------------------

As a first step, using the form of the density as determined by the equation (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\]), the first integral becomes: $$\begin{aligned}
    f_{t}^{(\mathrm{a})}  =&& \frac{2n_0\lambda_D^2}{\pi\,v_{T}}  \sqrt{\frac{-2\epsilon}{k_BT_e}} \left[\kappa+\frac{1}{\ell_{\parallel}^{2}} \left( 1-2\ln \left(\frac{-4\epsilon}{e\phi_{0}}\right) \right) \right]\\ \nonumber
    \mathrm{with}\:&&\quad\kappa = -\frac{2\Lambda}{\ell_{\perp}^{2}}+r^{2}\left(\frac{\Lambda}{\ell_{\perp}^{4}}-\frac{1}{\ell_{\perp}^{2}\ell_{\parallel}^{2}}\right)\end{aligned}$$ where $v_{T}$ is the thermal velocity of electrons (at temperature $T_e$) far from the EH, and $\lambda_D(=v_{T}/\omega_p)$ is the electron Debye length. This term is negative at the bottom of the well, *i.e.* when $\epsilon\to-e\phi_{0}$, then becomes positive and cancels for $\epsilon\to0$. The passing electrons for $|z|\gg \ell_{\parallel}$ are all corresponding to the background plasma. Thus, if at infinity their distribution $f_{\infty}$ is a velocity shifted Maxwellian distribution, everywhere else it must be of the following form: $$f_{p}(r,\epsilon) = \frac{n_{0}}{\sqrt{2\pi}v_{T}} \sum_{\sigma=\pm1} \exp \left[-\frac{(\sigma\sqrt{2\epsilon/m}-u_{e})^{2}}{2v_{T}^{2}}\right]$$ with $\epsilon>0$ and $u_{e}$ the drift velocity of the background plasma with respect to EH. The contribution of passing electrons to the distribution of trapped electrons can then be written as: $$\begin{aligned}
    f_{t}^{(\mathrm{b})} 
    &=&\frac{2n_{0}}{\pi v_{T}\sqrt{2\pi}}\left[ {I}(\beta,\zeta)+ {I}(-\beta,\zeta)\right] \\
    \mathrm{with}\:&\quad& {I}(\beta,\zeta)=\int_{0}^{+\infty}\frac{\mathrm{e}^{-(\beta x -\zeta)^{2}}}{1+x^{2}}\mathrm{d}x \label{eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7}\end{aligned}$$ and where we have defined the quantities $\beta = \sqrt{-\epsilon/k_{B}T_{e}}$ and $\zeta=u_{e}/\sqrt{2}v_{T}$. As an integral function of a positive integrand, we get a positive function. The general expression (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\]) does not allow an analytical calculation except if the integral takes the form of Dawson's integral: $ {I}(\beta,0)$ (@turikov1984electron). In that case, the limit $\zeta\to 0$ gives $$f_{t}^{(\mathrm{b})} = \frac{n_0}{v_T}\,\sqrt{\frac{2}{\pi}} \exp(-\beta)\left[1-\mathrm{erf }(\sqrt{-\beta})\right]$$ as calculated (@chen2005width) with $\mathrm{erf}$ function defined as $\sqrt{\pi}\mathrm{erf}(x)/2= \int_0^x\exp(-t^2)\mathrm{d}t$. Ions are considered in the rest frame of the structure to have a kinetic energy well above $Ze\phi_0$. Indeed, the ion contribution term is proportional to the mass ratio $\sqrt{m/M}$. It is worth noting that if we consider large ion thermal velocities ($v_{T_i} >v_{\mathrm{EH}}$), the mass ratio effect could be counterbalanced. However, we will not consider such a case and the ion contribution will be neglected in the rest of the calculation; we will make $f_{t}^{(\mathrm{c})}=0$ in equation (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7-bis\]). For the restricted case of $\zeta=0$ and $\Lambda\simeq1$ (or $\omega_p\ll \omega_c$), we find the same results as with the 3D BGK models developed by the authors (@schamel1979theory [@chen2005width; @turikov1984electron]) in the following, we generalized studies for other $\zeta$ values. We can study the limit of this distribution when $\epsilon\to0$. First, we obtain $f_t (r,\epsilon\to0^-)=f_t^{\mathrm{(b)}}(r,\epsilon\to0^-)$ since $f_t^{\mathrm{(a)}}(r,\epsilon\to 0^-)=0$. Secondly, from the continuity relation for $f_t^{\mathrm{(b)}}$, we can deduced $f_t^{\mathrm{(b)}}(r,\epsilon\to0^- ) = f_{p}(r,\epsilon\to0^+ )$. In conclusion, we get $f_e(r,\epsilon\to0^-)=f_e(r,\epsilon\to0^+)$, which is consistent with the populations present outside the hole and generation mechanism.

Figure \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\] displays the different contributions to the distribution of trapped electrons ($f_t$, with the virtual terms $
f_t^{(\mathrm{a})}$ and $f_t^{(\mathrm{b})}$ determined above), the distribution of passing electrons ($f_p$), and the electron distribution at infinity ($f_{\infty}$), as a function of velocity in the EH reference frame. The drift-free case ($u_e=0$) is represented by the figure \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](a) as reference. The impact of $u_e$ is visualized by comparing to the Fig. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](b), which represents a case with finite drift ($u_e = -2v_T$). We observe an asymmetric distribution, a decrease in the virtual value of $f_t^{\mathrm{(b)}}$ without modifying the term from the potential $f_t^{\mathrm{(a)}}$. In addition, the pseudo-Maxwellian part of $f_p$ is moved at higher velocities, allowing for particle acceleration. Figures \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\] (b) and (c) show the impact of the drift polarization effect (which increases with $\Lambda$, *i.e.* decreases with $\omega_c/\omega_p$) on $f_t^{\mathrm{(a)}}$, without modifying $f_t^{\mathrm{(b)}}$. We obtain a decrease in the density of trapped electrons $f_t$ through the center of the hole ($v=v_{\mathrm{EH}}$). Fig. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](a) with $u_e \ll v_T$ represents the case of a slow EH (*e.g.* as observed in the magnetotail (@norgren2015slow)) that may result from counterstreaming instability (@mottez2001instabilities). Conversely, Figs. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](b)–(d) with $u_e > v_T$ represent the case of a fast EH and show a plateau as observed in space plasmas (@holmes2018electron), as well as in Particle-In-Cell (PIC) simulations of bump-on-tail type (@umeda2004two).

Electron densities in the hole
------------------------------

From the passing distribution function $f_p$, we can define separately the passing electron density $n_p$ as: $$\begin{aligned}
 \nonumber
    n_p(r,z)&=& \frac{n_0}{\sqrt{2\pi}v_T}\left( \int_{+v_T\sqrt{2\psi}}^{+\infty}
    \exp{\left[-\frac{(\sqrt{v^2-\frac{2e\phi}{m}}-u_e)^2}{2v_T^2}\right]}\mathrm{d}v\right.\\ \nonumber &&+ \left. \int^{-v_T\sqrt{2\psi}}_{-\infty}
    \exp{\left[-\frac{(-\sqrt{v^2-\frac{2e\phi}{m}}-u_e)^2}{2v_T^2}\right]}\mathrm{d}v \right)\\
    &=& n_0\left[{J}_{\sigma=+1}(\sqrt{2\psi}, \zeta)+{J}_{\sigma=-1}(\sqrt{2\psi}, \zeta) \right] \label{eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7pdensity} \\\nonumber
        \mathrm{with}\:&&  {J}_{\sigma}(\theta,\zeta) = \frac{1}{\sqrt{2\pi}}\int_{\theta}^{+\infty} \mathrm{e}^{-(\sigma \sqrt{x^2-\theta^2}- \sqrt{2}\zeta)^2/2}\mathrm{d}x\end{aligned}$$ for $\theta =\sqrt{2\psi}$ and, $\psi = e\phi/k_BT_e$ the ratio of potential energy and thermal energy. In particular for $u_e =0$, $ {J}_{\sigma=+1}(\theta,0) =  {J}_{\sigma=-1}(\theta,0)= \exp(\theta^2/2)\left[1-\mathrm{erf}(\theta/\sqrt{2})\right]/2$, whence $$n_p(r,z) = n_0 \exp(\psi)\left[1-\mathrm{erf}(\sqrt{\psi})\right]\leq n_0$$ Using the definition of the charge density $\rho_{\parallel}$, the trapped electron density $n_t$ writes: $$n_t(r,z) = n_0 - n_p(r,z) - \frac{\rho_{\parallel}(r,z)}{e} \label{eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7}$$ and can be determined from equations (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\]) and (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7pdensity\]). In the case where $u_e = 0$, the trapped density $n_t$ calculated by Chen (@chen2001trapped [@chen2004bernstein; @chen2005width]) is retrieved. We can also define the density of the trapped electrons from their distribution as: $$n_t(r,z) = \int_{-v_T\sqrt{2\psi}}^{v_T\sqrt{2\psi}} f_t(r,z,v) \mathrm{d}v$$ Figure \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7-bis\] displays the different densities in the hole ($n_t$ trapped, $n_p$ passing and their sum $n_t+n_p$) as a function of parallel axis $z$. As indicated by the Poisson equation (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\]) and equation (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\]), the quantity $n_p+n_t$ represents the electron density $-\rho_{e\parallel}/e$ 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 Figs \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7-bis\](a) and \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7-bis\](b). We observe that it reduces the density of electrons trapped in the well and generates an accumulation of electrons on the edges. Furthermore, the hole is deeper for $\omega_c/\omega_p < 1$. Hence, the polarization effect creates the running away of trapped electrons and can lead to the disintegration of the EH (@chen2004bernstein). In the case of the Fig. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7-bis\](c), we can note that the case is impossible because of a non-physical negative density of the trapped electrons. Moreover, the case with finite drift ($u_e=-2v_T$) decreases the trapped particle density (see Fig. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7-bis\](d)) due to a larger amount of passing electrons. We can note that our zeroth order approximation of $\mathbf{J}_{\mathrm{pol}}$ is in principle strictly consistent only for small perturbations of the electron density, *i.e.* $n_t+n_p\sim  n_0$.

![Passing ($n_p$), trapped ($n_t$) electron densities at $r=0$, for different values of cyclotron to plasma frequency ratio $\omega_c/\omega_p$ and Maxwellian mean velocity $u_e$ (with $e\phi_0=k_BT$ and $\ell_{\parallel} = \ell_{\perp} = 5\lambda_D$).<span data-label="figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7-bis"></span>](Densities.pdf){width="\linewidth"}

Existence criteria \[sa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7Existence\]
=====================================

![image](Lpara.pdf){width="\linewidth"}

We define and analyse the criteria that allow these structures to exist to obtain information on EH size. The trapped distribution function (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7-bis\]) has to be physical, and thus $f_{t}$ must be non-negative. Writing $f_t(\epsilon)\geq 0$ for $-e \phi_0 \leq \epsilon <0$, we obtain: $$\frac{\ell_{\parallel}^2}{\lambda_D^2} \geq \frac{2\ln4-1}{ {G}(\psi_0,\zeta)-2\Lambda \lambda_D^2/\ell_{\perp}^2} \label{eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7}$$ where $\psi_0 = e\phi_0/k_BT_e$ is the ratio $\psi$ at $r=z=0$ corresponding to the maximum potential and, where we have introduced the function $${G}(\psi_0,\zeta) = \frac{ {I}(\sqrt{\psi_0},\zeta)+ {I}(-\sqrt{\psi_0},\zeta)}{2\sqrt{\pi \psi_0}}$$ This function has the following limits, for $\psi_0 \to 0^+$: $ {G}(\psi_0,\zeta)\sim \sqrt{\pi/4\psi_0}\exp(-\zeta^2)$, and for $\psi_0 \to +\infty$: $ {G}(\psi_0,\zeta)\sim 1/2\psi_0$. Drift-free case ($u_e=0$) of $ {G}$ function gives $${G}(\psi_0,0) =\frac{\sqrt{\pi}}{2\sqrt{\psi_0}} \exp(\psi_0)[1-\mathrm{erf}(\sqrt{\psi_0})]$$ (see Figs. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\] (a)–(b)). The sign of the denominator in equation (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\]) imposes a second condition on the perpendicular scale, which must be respected and can be expressed as: $$\frac{\ell_{\perp}^2}{\lambda_D^2}\geq \frac{2\Lambda}{ {G}(\psi_0,\zeta)}
\label{eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7}$$ Equations (\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\])–(\[eqa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\]) give amplitude-width criteria in both perpendicular and parallel directions. Figure \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\] represents the minimum parallel sizes of the hole as a function of the amplitude of the well for several values of the perpendicular size, and plasma parameters.

![Perpendicular width-amplitude relations for different values of cyclotron to plasma frequency ratio $\omega_c/\omega_p$ and Maxwellian mean velocity $u_e$.<span data-label="figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7"></span>](Lperp.pdf){width="\linewidth"}

We note that the weaker the magnetic field strength, the larger the forbidden zone, *i.e.* the area under the curves (*e.g.* see Fig. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](a) case $\ell_{\perp}=\infty$). Note that the case $\ell_{\perp}= \infty$, in Fig. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\], corresponds to the 1D limit. We notice that for 1D structures, taking into account Maxwellian mean velocity $u_e$ increases the forbidden zone, especially for small structures (see Figs \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](c)–(d)). Hence, polarization drift effects tend to reduce the generation of very small scale and large amplitude structures. 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 also reduced (see Figs. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](c)–(d)). Figure \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\] represents the minimum perpendicular sizes of the hole as a function of amplitude of the well for several $\omega_c/\omega_p$ ratio, in cases of $u_e=0$ (Fig. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](a)) and $u_e = -2v_T$ (Fig. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](b)). The polarization effects lead to an increase in the slope of $\ell_{\perp}(\psi)$ and of the forbidden zone for small $\omega_c/\omega_p$ values. From Fig.\[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](c), we observe that the conditions used to make the Fig. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7-bis\] (*i.e.* $e\phi_0=k_BT$ and $\ell_{\parallel}= 5\lambda_D$) are in the forbidden zone. Figure \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\] can also show the maximum well amplitude $\psi_{0,\max}$ as a function of $\ell_{\perp}$. The vertical asymptotes in Fig. \[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\] correspond to the limit of $\ell_{\parallel}(\psi_0\to \psi_{0,\max})$ precisely. We note that observed fast structures ($u_e > v_T$) where $\ell_{\perp}\geq 10 \lambda_D$, $\psi\simeq 1-2$ and $\omega_c \simeq 0.5\,\omega_p$ (*e.g.* recent observations (@le2017lower [@holmes2018electron; @steinvall2019observations])) correspond to a minimum value for $\ell_{\parallel}\simeq 5-6 \lambda_D$ (see Fig.\[figa67d:ce30:8b00:4856:c02f:75af:3b8:dfd7\](c)), which is close to the observed perpendicular to parallel width ratio (@steinvall2019observations): $\ell_{\perp}=1.6\ell_{\parallel}$ . The reason for which minimum value for $\ell_{\parallel}$ is observed while larger $\ell_{\parallel}$ can exist (see *e.g.* @andersson2009new) still need to be investigated.

Conclusions
===========

To conclude, this model describes the criteria of existence of the EH including the polarization drift of electrons ($\omega_c/\omega_p \lesssim 1$). Moreover, we have used more realistic boundary conditions on the electron distribution functions which include a global drift consistent to the observations (*e.g* @holmes2018electron [@steinvall2019observations]). These theoretical improvements have shown an important impact on the shape and position of the distribution functions and the densities of the passing and trapped electrons in EH structures, as well as on their condition of existence. Particle measurements by the MMS mission even provided at 30 ms time resolution (@burch2016electron) do not allow to fully resolve the electron distribution function in the EH which have 1 ms time resolution. Depending on the number of EH detected in 30 ms, measured electron distribution functions can be representative to EH or to the ambient plasma. Therefore, theoretical developments are crucial to better understand the self consistent structure of EH. This will be verified by PIC simulations and observations in weakly magnetized media $\omega_c/\omega_p \lesssim 1$, which will be presented in a forthcoming study.

[^1]: Released on

{'source': 'arXiv_out/2105'}