Kinetic features and non-stationary electron trapping in paraxial magnetic nozzles

Abstract

The paraxial expansion of a collisionless plasma jet into vacuum, guided by a magnetic nozzle, is studied with an Eulerian and non-stationary Vlasov-Poisson solver. Parametric analyzes varying the magnetic field expansion rate, the size of the simulation box, and the electrostatic potential fall are presented. After choosing the potential fall leading to a zero net current beam, the steady states of the simulations exhibit a quasi-neutral region followed by a downstream sheath. The latter, an unavoidable consequence of the finite size of the computational domain, does not affect the quasi-neutral region if the box size is chosen appropriately. The steady state presents a strong decay of the perpendicular temperature of the electrons, whose profile versus the inverse of the magnetic field does not depend on the expansion rate within the quasi-neutral region. As a consequence, the electron distribution function is highly anisotropic downstream. The simulations revealed that the ions reach a higher velocity during the transient than in the steady state and their distribution functions are not far from mono-energetic. The density percentage of the population of electrons trapped during the transient, which is computed self-consistently by the code, is up to 25% of the total electron density in the quasi-neutral region. It is demonstrated that the exact amount depends on the history of the system and the steady state is not unique. Nevertheless, the amount of trapped electrons is smaller than the one assumed heuristically by kinetic stationary theories.

Full text

Kinetic features and non-stationary electron
trapping in paraxial magnetic nozzles
G. S´anchez-Arriaga1, J. Zhou1, E. Ahedo1, M.
Mart´ınez-S´anchez2, J.J. Ramos1,3
1Equipo de Propulsion Espacial y Plasmas (EP2), Universidad Carlos III de Madrid,
28911 Legan´es, Spain
2Department of Aeronautics and Astronautics, Massachusetts Institute of
Technology, Cambridge, 02139, Massachusetts, USA
3Plasma Science and Fusion Center, Massachusetts Institute of Technology,
Cambridge, 02139, Massachusetts, USA
October 2017
Abstract. The paraxial expansion of a collisionless plasma jet into vacuum, guided
by a magnetic nozzle, is studied with an Eulerian and non-stationary Vlasov-Poisson
solver. Parametric analyses varying the magnetic field expansion rate, the size of the
simulation box, and the electrostatic potential fall are presented. After choosing the
potential fall leading to a zero net current beam, the steady states of the simulations
exhibit a quasi-neutral region followed by a downstream sheath. The latter, an
unavoidable consequence of the finite size of the computational domain, does not
affect the quasi-neutral region if the box size is chosen appropriately. The steady
state presents a strong decay of the perpendicular temperature of the electrons, whose
profile versus the inverse of the magnetic field does not depend on the expansion rate
within the quasi-neutral region. As a consequence, the electron distribution function is
highly anisotropic downstream. The simulations revealed that the ions reach a higher
velocity during the transient than in the steady state and their distribution functions
are not far from mono-energetic. The density percentage of the population of electrons
trapped during the transient, which is computed self-consistently by the code, is up to
25% of the total electron density in the quasi-neutral region. It is demonstrated that
the exact amount depends on the history of the system and the steady state is not
unique. Nevertheless, the amount of trapped electrons is smaller than the one assumed
heuristically by kinetic stationary theories.
PACS numbers: 52.75.Di,52.25.Xz, 52.65.Ff
Keywords: electric propulsion, magnetic nozzles, electron trapping Submitted to: Plasma
Sources Sci. Technol.

2
1. Introduction
Plasma expansions in the presence of magnetic fields appear in astrophysical scenarios,
such us pulsars [1], stellar winds [2] and supernova remmants [3], and in laser-produced
laboratory plasmas [4, 5]. They have also engineering applications in electric propulsion.
One of the most relevant is a magnetic nozzle [6], where a magnetic field generated
by coils is used to guide, expand, and accelerate the plasma without using any
physical wall. Several types of thrusters under development, including the helicon
plasma thruster [7], the magnetoplasmadynamic thruster [8], and the Variable Specific
Impulse Magnetoplasma Rocket (VASIMR) [9] involve magnetic nozzles. Plasma flows
in magnetic nozzles have been characterized in the laboratory by using laser-induced
fluorescence techniques [10], Langmuir and Mach probes [11] and spectroscopic methods
[12] among others.
The modeling of magnetic nozzles has also attracted great attention. One [13],
two [14, 15, 16, 17] and three [18] dimensional fluid models have been developed, and
the transformation of the internal energy of the plasma into directed kinetic energy,
the plasma detachment, and the role played by the plasma-induced magnetic field have
been discussed. However, since plasma flows are generally weakly collisional, simple
closures of the fluid equation hierarchy for the pressure tensor and the heat fluxes are
doubtful. A self-consistent determination of these magnitudes needs inevitably a kinetic
description of the plasma.
Stationary solutions of the Vlasov equation in a magnetized plasma expansion
have been obtained recently [19]. After assuming steady conditions, a slender nozzle
geometry, and a fully magnetized plasma, the conservation of the total energy and the
magnetic moment were used to write rigorously the densities of the particles connecting
with the source as functions of the electrostatic potential and to compute the latter. It
was then found that there exist regions in phase space not connected with either the
source or the downstream region where doubly-trapped bouncing particles can exist.
Since in collisionsless plasmas the filling of that regions happens during the transient,
a stationary model cannot characterize rigorously the (doubly) trapped particles. The
plasma spatial solution and its numerical convergence turned out to be very sensitive to
the distribution of trapped particles on the divergent side of the nozzle. After adding an
heuristic population of trapped electrons, the authors found numerical solutions with
the electron density dominated by the confined electrons over most of the divergent jet.
Discussions on whether or not trapped populations of electrons are an essential
component of the solutions and how they are determined also arise in other areas of
plasma physics. For instance, for an electron-attracting Langmuir probe in flowing
plasma, it was argued that a population of electrons should exist at the ram side
of the probe [20]. The formation of such a trapped population during the transient
phase, which is an adiabatic process, has been observed recently in non-stationary direct
(eulerian) Vlasov simulations [21]. Adiabatic trapping in slowly varying time-dependent
electric fields [22] has been considered in analytical studies of magnetized plasma

3
expansion [23]. Particles can also be trapped due to collisional effects. Particle collisions,
which can also produce trapped particles, have been included in non-stationary particle-
in-cell (PIC) simulations [24].
This work studies plasma expansions in magnetic nozzles by using a non-stationary
direct Vlasov code. As compared to stationary fluid models, this technique computes the
pressure tensor, the heat fluxes and the population of trapped electrons self-consistently.
PIC codes do also exhibit these two features. However, due to the numerical noise, they
do not give an accurate description if the number of macroparticles per cell is small, a
circumstance that is unavoidable in non-stationary simulations of a plasma expansion
into vacuum. Direct Vlasov codes, which are more demanding from a computational
point of view because they discretize the distribution function in real and velocity
spaces instead of using macroparticles, provide a better accuracy and degree of detail of
the distribution functions. Section 2presents the mathematical model and describes
briefly the numerical algorithm. The effects of fixing in the code the size of the
nozzle (finite simulation domain) and the electrostatic potential value at the exit, two
parameters that do not appear or are not externally imposed in a real infinite plasma
expansion, are shown in Sec. 3. The correct selection of these two parameters allows to
reproduce with the code the conditions of a real nozzle with zero net current. Section
4shows some kinetic features of the expansion such as the distribution functions, and
particle densities, temperature, and heat fluxes. The trapped electron population is
computed and the results are compared with previous studies. Section 5summarizes
the conclusions of the work.
2. Magnetic nozzle model based on guiding center theory
2.1. Plasma model
Let us consider a tank placed at z < z0<0 and filled with an electron-ion plasma
(see Fig. 1). We are interested in the time-dependent, magnetically-channeled plasma
expansion that is produced when a hole of radius R0at z=z0is opened at the plasma-
vacuum wall. For the sake of illustration, the geometry of the magnetic nozzle is the
one corresponding to a current loop of radius RL(RL> R0) placed at the plane z= 0.
It generates a stationary and non-uniform magnetic field in the vacuum region that
reaches its maximum value BTat z= 0 (the nozzle throat T). The forward distribution
functions of ions and electrons entering the nozzle are assumed semi-Maxwellian,
fα(t, z =z0, vk>0, v⊥) = N∗ mα
2πkBT∗
α!3/2
exp −mαv2
2kBT∗
α!, α =i, e, (1)
while the backward distribution functions will be determined self-consistently by the
expansion characteristics. Here N∗and T∗
αare reference parameters (not the actual
densities and temperatures at z=z0that also involve the backward distribution
function), v=qv2
k+v2
⊥is the velocity, and vkand v⊥the velocity components
parallel and normal to the magnetic field lines. For convenience, hereafter the axial

4
coordinate, time, velocities, magnetic field, electrostatic potential, particle distribution
functions, and densities, are all normalized and we will write z/λ∗
De →z,tω∗
pe →t,
vk,⊥/λ∗
Deω∗
pe →vk,⊥,B/BT→B,eφ/kBT∗
e→φ,fα/N∗(me/kBT∗
e)3/2→fα, where
λ∗
De =q0kBT∗
e/N∗e2is the Debye length, ω∗
pe =qN∗e2/me0the electron plasma
frequency, kBthe Boltzmann constant, methe electron mass, ethe elementary charge,
and 0the vacuum permittivity. As shown below, the plasma dynamics depends on the
following dimensionless parameters in our model
rL≡RL
λ∗
De
, δα≡T∗
α
T∗
e
, βα≡mα
me
, Zα,(2)
where the subscript α=e, i denotes electrons and ions, and mαand Zαare the mass
and the charge number of the α-species.
- z00zM
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
B
N*, Te
*, Ti
*
Figure 1. Geometry of the convergent-divergent magnetic nozzle.
In this work we follow a paraxial approximation and assume a slender and slowly-
varying magnetic field, i.e. we take RL/R0>> 1, and just look at the center line of the
magnetic nozzle. Under this hypothesis, the parameter R0does not appear anymore in
the model and one just needs the normalized magnetic field at the center line. It reads
B(z) = r3
L
(r2
L+z2)3/21z,(3)
where 1zis an unit vector along the z-axis. Therefore, we are having a convergent-
divergent nozzle with the maximum of the magnetic field B= 1 at z= 0 and B→0
as z→ ±∞; the analysis here will be focussed at the divergent side of the nozzle. The
model also assumes that the magnetic field is very strong and the normalized Larmor
radii satisfies ρLα ≡βv⊥/|Zα|B << rL. In the limit ρLα/rL→0, the slow drift motion of
the particles across the field lines can be ignored and the normalized magnetic moment
µα=βαv2
⊥
2B(4)
is conserved (for brevity, we will generally write µα→µ). Hereafter, we will work with
the gyrocenter variables (z, vk, µ, γ), and will also average the distribution functions of

5
the particles fαover the fast gyrophase γ
¯
fα(t, z, vk;µ) = 1
2πZ2π
0
fα(t, z, vk, µ, γ)dγ. (5)
The evolution of the gyrocenter distribution function ¯
fαis governed by the Vlasov
equation
∂¯
fα
∂t +vk
∂¯
fα
∂z +aα
∂¯
fα
∂vk
= 0,(6)
where we ignored the induced magnetic field and introduced the parallel dimensionless
acceleration
aα=−1
βα Zα
∂φ (t, z)
∂z +µdB (z)
dz !(7)
The normalized electric field E=EkB/B =−∂φ/∂z is given by the paraxial Poisson’s
equation
B∂
∂z Ek
B!=X
α=e,i
Zαnα(8)
with the particle densities computed from the distribution functions as
nα(z) = Zfαdv=2πB
βαZ+∞
−∞ Z+∞
0
¯
fαdvkdµ. (9)
Therefore, the dynamics of the electrons and the ions governed by the two Vlasov
equations in Eq. (6) are nonlinearly coupled through the electrostatic potential. This
set of equations must be integrated with appropriate boundary and initial conditions,
discussed in Subsec. 2.3.
2.2. Evolution of macroscopic quantities
The evolution equations of the main macroscopic quantities are helpful in the analysis
of the simulations. The average or mean value of any quantity ψ, is computed as
hψiα=1
nαZψfαdv=2πB
βαnαZ+∞
−∞ Z+∞
0
ψ¯
fαdvkdµ (10)
Interesting quantities are: densities given by Eq. (9); macroscopic velocities parallel
to the magnetic field uα=DvkEα; current densities jα=Zαnαuα; temperatures
Tkα=βαDc2
kαEαand T⊥α=Bhµiα, where we introduced the peculiar velocities
ckα=vk−uα; pressures Pkα=nαTkαand P⊥α=nαT⊥α; and (parallel) heat fluxes
of parallel and perpendicular energy, Qkα=1
2βαnαDc3
kαEαand Q⊥α=BnαDµckαEα,
respectively. According to Sec. 2, the normalization has been done with characteristic
variables involving the electron mass.
The evolution equations of these quantities are obtained straightforwardly by taking
velocity moments in Eq. (6). In the paraxial case, the equations for continuity, axial
momentum, total energy, and perpendicular energy are, respectively,
∂nα
∂t +B∂
∂z nαuα
B= 0,(11)

6
∂
∂t (βαnαuα) + B∂
∂z βαnαu2
α
B!=−Zαnα
∂φ
∂z
+"Pkα−P⊥α∂ln B
∂z −∂Pkα
∂z #,(12)
∂
∂t "nα βα
2u2
α+Tkα
2+T⊥α!#+
B∂
∂z "nαuα
B βα
2u2
α+3
2Tkα+T⊥α!+Qkα+Q⊥α
B#+jα
∂φ
∂z = 0.(13)
∂
∂t (nαT⊥α) + B2∂
∂z 1
B2(nαuαT⊥α+Q⊥α)= 0,(14)
This set of macroscopic equations is incomplete, unless equations for the parallel heat
fluxes Qkαand Q⊥αare added, which will introduce higher order magnitudes. A closure
of the set of the fluid equations is not simple in a collisionless plasma.
Here, the consistent kinetic solution is obtained directly, so the fluid equations are
used to interpret the results, mainly the steady-state ones in Section 3. In this respect,
in the above equations, 1/B plays the role of the effective beam area. Furthermore, it is
in fact the natural spatial variable (instead of z) in the divergent paraxial nozzle. Thus,
in (11), nαuα/B is the species flow (i.e. the flux area integrated), which is constant
spatially in steady-state. The species current, jα/B, and the total plasma current,
I= (je+ji)/B, are constant in steady-state too. In (12), the two last terms on the
right-hand side are the contribution of the divergence of the pressure tensor (i.e. the
net pressure force). Then, the steady-state limit of (13) yields that the total enthalpy
flow, ˙
Hα, is constant spatially,
˙
Hα≡nαuα
B βα
2u2
α+3
2Tkα+T⊥α+Zαφ!+Qkα+Q⊥α
B=const (15)
Here, (Qkα+Q⊥α)/B is the total heat conduction flow. The steady state limit of
(14) yields that the (convection plus conduction) flow of perpendicular energy evolves
proportional to B−1
nαuαT⊥α+Q⊥α
B2=const, (16)
which is the direct consequence of the conservation of the magnetic moment of the
species. These conservation laws were already used in Ref. [25] to analyze the plasma
response in a convergent magnetic field. Finally, if the mean kinetic energy is eliminated
from (13) by using equations (11) and (12), the evolution equation for the internal energy
is obtained,
∂
∂t Pkα
2+P⊥α!+B∂
∂z 1
Buα3
2Pkα+P⊥α+ (Qkα+Q⊥α)
−uα"∂Pkα
∂z +P⊥α−Pkα∂ln B
∂z #= 0,(17)
which can susbtitute for (13).

7
2.3. Simulation domain and boundary conditions
We are interested in the time-dependent plasma expansion along the (semi-infinite)
divergent nozzle, extending from z= 0 (the throat T) to z=∞. However, since the
numerical simulation requires to work with a finite domain, the downstream end of the
domain (point M) will be placed at a certain zM1, with BM<< 1. A parametric
analysis of the combined influence of zMand rLon the solution is carried out below.
Furthermore, it turns out that, in spite of applying quasineutrality at the upstream
end of the simulation domain, a non-desirable Debye sheath, extending a few Debye
lengths develops there. In order to eliminate its spurious influence, the usptream end
of the domain has been placed at the convergent side of the nozzle, in particular, at
z0=−rL/2.
At the domain entrance, we set Maxwellian functions for the injected particles,
¯
fα(t, z =z0, vk>0; µ) = χα(t)¯
fMα (18)
with
¯
fMα = βα
2πδα!3/2
exp −βαv2
k
2δα
−Bµ
δα!(19)
and χe= 1. The parameter χi(t) is dynamically varied to accomplish quasineutrality
at entrance section z=z0, once reflected-back particles are taken into account there.
(For instance, if no ions are reflected back and all electrons are, χi= 2.) At the domain
downstream end, in order to simulate the vacuum at infinity, we impose no incoming
particles into the domain,
¯
fα(t, z =zM, vk<0; µ)=0.(20)
Regarding initial conditions, one would initially set ¯
fα(t= 0, z > z0, vk;µ) = 0.
However, since such a hard transition can lead to numerical issues, our simulations used
¯
fα(t= 0, z > z0, vk;µ) = ¯
fMα ×exp −z−z0
L0(21)
with L0a dimensionless parameter that controls the density gradient of the initial plasma
profile. A value L0= 2, which yields a profile with width about a few Debye lengths, is
enough to provide a smooth transition at t= 0 in the simulations.
Finally, Poisson’s equation requires two boundary conditions on the electrostatic
potential. Clearly, there is the freedom to take φ(z=z0) = 0. With respect to the
boundary condition at the downstream end, the studies of the semi-infinite, stationary
nozzle with a simple plasma have shown two things. First, the potential decays
monotonically to an asymptotic value φ=φ∞<0 (i.e. yielding dφ/dz|∞→0). Second,
the net electric current of the plasma beam Iis not a parameter independent of φ∞:
a parametric current-voltage curve I(φM) with ∂I/∂|φM|>0 exists. This behavior of
the current-voltage curve is simple to explain: for normalized distribution functions at
injection, the more negative is φ∞, a larger fraction of electrons injected into the nozzle
is reflected back to the reservoir while (near) all ions cross freely the nozzle, and thus
the more positive becomes I. Therefore, in the downstream end of our finite simulation

8
box we can impose either φM=φ(zM) or I. The first choice is the natural one for the
numerical scheme. The case of most practical interest, I= 0, which corresponds to a
current-free plasma beam, requires to iterate on φM.
2.4. Direct Vlasov solver
This section discusses briefly the main features of the novel direct Vlasov code
VLASMAN (VLAsov Simulator for MAgnetic Nozzle), that has been developed for
the numerical integration of (6). A mesh of points ziwith i= 1,...Nzis defined within
the interval z0≤z≤zM. These points are distributed non-uniformly in order to keep
constant the ratio between the resolution of the mesh and the local Debye length, which
is expected to vary as λDe ∼n−1/2∼B−1/2. The velocity space, involving vkand µ,
was truncated as −vα
max ≤vk≤vα
max and 0 ≤µ≤µα
max and discretized with Nvk×Nµ
points. Unlike the spatial mesh, which is common for both species, different maximum
velocities and magnetic moments are chosen for electrons and ions. For both species, the
velocity mesh is uniform. The unknowns of the code are the values of the distribution
functions at the points of the mesh and at discrete times tm,¯
fα(tm, zi, vkj, µk).
Since µappears as a parameter in (6), the algorithm just needs to solve a one
dimensional equation Nµtimes. Given the distribution function ¯
fα(tm, zi, vkj, µk), the
value at tm+ ∆tis found by using a splitting algorithm that treats the convective terms
in the zand vkdirections separately, and gives a scheme of second order in ∆t[26]. A
short summary of the splitting algorithm and a description of the numerical schemes
implemented for the interpolation and the numerical integration are given in Appendix
A.
In the simulations, we took the physical parameters δi= 1, βi= 100, Zi= 1
and considered several values of rL. The value of βiis not realistic for an electron-
ion plasma but it still separates significantly the electron and ion response times and
helped us save computational resources. Regarding the geometry of the nozzle, we set
z0=−rL/2, i.e. the divergent and convergent segments have lengths equal to rL/2 and
zM, respectively. The most relevant numerical parameters are Nvk= 77, Nµ= 101,
ve
max = 5, vi
max = 0.5, µe
max =µi
max = 12.5, and ∆t= 0.03. Tradeoffs analysis varying
the numerical parameters zMand φMare shown in Sec. 3.
3. Stationary solution and parametric analysis
The effect of the truncation of the computational box up to a length zMand the
setting of the electrostatic potential value φMat that position have been investigated
by running a large number of simulations. After taking an expansion rate of rL= 50,
the physical and numerical parameters explained in Sec. 2.4, and several values of
zMand φM, we integrate the Vlasov-Poisson system forward in time until the plasma
reached the corresponding stationary state. For each simulation, the latter was verified
by monitoring the time evolution of the most important variables, such as density and

9
potential, and the z-profiles of nαuα/B, which becomes uniform at stationary conditions.
-0.04 -0.02 0 0.02 0.04 0.06
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
Figure 2. Net current-to-magnetic field ratio j/B versus the total potential drop
(φM) and the one between the throat and zM(φT M ) for several expansion rates and
box sizes. The curves practically overlap, also for the case rL= 100 (not shown)
First of all, Figure 2shows the results of investigating the influence of the total
potential fall φMon the electric current Iacross the nozzle in steady-state. As expected
and known from previous models, the current I(abcissa) is positive for large, negative
values of φM(ordinate), and negative otherwise. The most relevant result here is the
universal character of the curve φM(I): it is practically independent of rLand zM(as
long as BM1), which allows to infer that this curve reproduces the behavior of
the semi-infinite nozzle too. Furthermore, Fig. 2shows that the curve φT M (I) for
the potential fall along the divergent part of the nozzle is (near) universal too. The
principal, current-free beam case has potential falls of φM≈ −2.75 and φT M ≈ −2.34.
Such potential drops are consistent with previous calculations from stationary kinetic
models (extrapolate to mi/me= 100 the results in Fig. 4c of Ref. [19]). Hereafter, we
focus the analysis at discussing the (approximate) ’current-free case’ φM=−2.75. The
steady states values of the normalized species current are Ii=−Ie'0.074
Figure 3shows (near) stationary axial profiles for different rLand zM. Panel (a)
plots the relative space charge for rL= 50 and several lengths of the simulation box;
panels (b) and (c) plot the electric potential profile versus zand B−1, and panel (d) the
ion density versus B−1. Three different spatial regions can be distinguished in panels
(a)-(c). First, a small sheath (with a relative space charge <5% and extending a few
Debye lengths) forms at the entrance of the simulation domain, in spite of having forced
quasineutrality locally at z=z0. This ’numerical’ sheath is caused by the need of the
electric field to adapt the entrance distribution functions of ions and electrons and it
was the reason to include a small convergent part of the nozzle, even though the work
is focused on the divergent nozzle behavior. Second, there is the large quasineutral
region, with a decreasing electric field as we move downstream. Third, there is a second

10
0 200 400 600 800
-3
-2
-1
0
1
100101102103104105
-3
-2
-1
0
0 200 400 600 800
0
0.2
0.4
0.6
100101102103104
10-5
10-4
10-3
10-2
10-1
100
Figure 3. Panels (a), (b) and (d) show, respectively, the normalized space charge,
φversus zand niversus 1/B for rL= 50 and several zM. Panel (c) displays φversus
1/B for zM= 800 and several expansion rates.
Debye sheath at the downstream end of the simulation. Since the Debye length is
proportional to n−1/2
α[and thus nearly proportional to B−1/2, according to panel (d)], it
increases by 1-2 orders of magnitude along the discharge, thus giving the impression that
the downstream sheath is thick. Just for reporting, the relative space-charge and the
potential fall in the downstream sheath are nearly constant, because of the low electric
field at the sheath entrance and the need to adjust the total potential fall to φM. Panel
(c) shows that φdepends more naturally on B−1in the quasineutral region. In panel
(d), we see that niis near proportional to B, indicating a much gentler dependence of
uion B(to have niui/B=const). The electron density nebehaves as niexcept at the
downstream sheath where it decreases more sharply. Normalized plasma densities at the
entrance and the throat are ne0= 0.97 and neT = 0.62, the difference indicating the jet
acceleration in the small convergent region of the nozzle. Finally, at the simulation box
entrance, the ratio of ion-to-electron densities for forward moving particles, that is χiin
(18) is 1.53: this ratio would be close to 2 and 1 if the entrance would be, respectively,
at the throat and further upstream in the convergent magnetic nozzle.
The stationary momentum equation of ions and electrons, (12), states the balance
FI
α=FP
α+FE
α,(22)
between the inertial force (or flow of species momentum) FI
α, the pressure force FP
α, and
the electric force FE
α. Both FI
αare positive, both FP
αare expected to be positive, while
FE
i>0 and FE
e<0. Furthermore, one has: FE
e'FE
iin the quasineutral region; and

11
0 100 200 300 400 500 600 700 800
z
10-8
10-6
10-4
10-2
10-1
0 100 200 300 400 500 600 700 800
z
10-10
10-8
10-6
10-4
10-2
10-1
Figure 4. Stationary, spatial profiles of inertial, pressure, and electric forces of ions
and electrons for rL= 50 and zM= 800.
FI
e/F I
i∼me/mi1 for the current-free and ’small current’ cases. The combination
of these two facts implies that the electron inertia is negligible, and the electric and
electron pressure forces balance each other, i.e. FP
e' −FE
eFI
e. For ions, the inertial
force is dominant, and the relevance of the ion pressure depends on upstream conditions
for Ti/Te. These trends are confirmed by Figure 4(a), which plots FE
e,FP
e, and FI
i;
the two other forces are obtained by just applying (22). In this case the ion flow is
supersonic and thus accelerated freely by the electric force.
Figure 4(b) assesses the different contribution to the net electron pressure force,
that is the pressure tensor divergence. This can be expressed in two different ways:
∇ · ¯
¯
Pα=∂Pkα
∂z +P⊥α−Pkα∂ln B
∂z ≡B∂
∂z (Pkα
B) + P⊥α
∂ln B
∂z .(23)
The first division is based on the parallel pressure gradient and the magnetic mirror
effect, and the second is based on the Pkαand P⊥αcontributions. The panel
shows, interestingly, that the parallel pressure gradient and the magnetic mirror effect
are individually much larger than their difference (i.e. they compensate practically
each other). This makes them not very suitable to characterize the total pressure
contribution. On the contrary the Pkαand P⊥αcontributions are of the same order
as their difference. For this particular case, the P⊥α-contribution dominates mildly
upstream, while the Pkα-contribution dominates totally downstream. The milder
behavior of the z-derivative of Pkα/B compared to that of Pkα, and the drop of the
P⊥α-contribution, are due to nα∝Bapproximately and the behaviors of Tkαand T⊥α
shown below. Although not shown, the ion pressure contributions behave in the same
way as the electron ones.
Figure 5(a)-(c) displays the stationary spatial profiles of the parallel and
perpendicular temperatures of ions and electrons for zM= 800 and rL= 25,50, and 100.
In this collisionless plasma, these kinetic temperatures simply express the dispersion of
particle velocities. The first interesting feature is that both perpendicular temperatures
decrease with B−1, a behavior related to the conservation of magnetic moment. On the
contrary, the parallel temperatures are rather constant spatially, except for the decrease

12
100101102103104105
0
0.25
0.5
0.75
1
0 200 400 600 800
0.9
1
1.1
1.2
1.3
0 200 400 600 800
0.8
1
1.2
1.4
1.6
100101102103104105
0
1
2
3
4
Figure 5. Panels (a) and (b) show the perpendicular temperature of the electron and
the ions, respectively.Panel (c) shows their parallel temperatures and (d) the mean
kinetic energy of the ions. zM= 800 and solid, dashed, and dot-dash lines correspond
to rL= 100,50, and 25, respectively.
of Tkeat the downstream sheath. Therefore, the plasma expansion along the divergent
nozzle implies both anisotropy and cooling [the average temperature is (Tkα+ 2T⊥α)/3].
This behavior agrees qualitatively with stationary, kinetic, fully-quasineutral models (see
figure 7 in [19]). Apparently, the main difference of the non-stationary model, which
computes the population of trapped electrons self-consistently, is a softer decay of the
parallel temperature. Values of normalized temperatures at the throat are TkiT '0.30,
T⊥iT '1.20, TkeT '0.86, T⊥eT '0.90, and they are explained by the analysis of Ref.[25]
for a convergent magnetic geometry: there the ion distribution function is determined
by the combination of the magnetic mirror and the free extraction at the throat, while
electrons remain near Maxwellian. Figure 5(d) displays the ion axial kinetic energy,
which increases downstream thanks to the electric potential energy. For this current-
free case, the electron axial energy behaves exactly the same in the quasineutral region
(but it is me/mitimes lower).
Figure 6analyzes the z-profiles of the heat (or internal energy conduction) flows.
We recall that Qkand Q⊥represent parallel fluxes of the parallel and perpendicular
thermal energies (perpendicular fluxes are zero in our model). Notice that, since this
is a collisionless plasma, no Fourier-type law is expected to apply for these heat flows.
Panel (a) shows that: the area-integrated parallel heat flows, Qkα/B, are approximately
constant (except, as often, near the ends of the simulations) while the parallel flow
of perpendicular thermal energy, Q⊥α/B, decrease proportional to B2. The same
dependence with Bwas found for the internal energy convection flows, nαuα(3Tkα/2)

13
0 100 200 300 400 500 600 700 800
z
10-4
10-2
0 100 200 300 400 500 600 700 800
z
0
0.2
0.4
0.6
a)
b)
Figure 6. Axial profiles of the heat fluxes [panel (a)] and their relative rates versus
the pressure fluxes [panel (b)] rL= 50 and zM= 800.
and nαuαT⊥α. To complete this, panel (b) determines the relative rates of internal
energy conduction versus the pressure flux for ions and electrons. Focusing on the
main quasineutral region, conduction of parallel and perpendicular internal energies of
electrons is about 50-52% of its convection; for ions conduction represents 4% and 11% in
the parallel and perpendicular cases; more parametric analyses are needed to ascertain
how these values depend on the upstream plasma conditions. In spite of this, it seems
clear that, both for ions and electrons, the flow of the perpendicular internal energy
decreases in proportion to B, Eq. (16), and becomes negligible downstream. Then,
since the total enthalpy flow is conserved, Eq. law (15), that loss of perpendicular
energy goes mainly into increasing the axial kinetic energy, which is the macroscopic
equivalence of the classical particle mirror effect, for ions, and the electric potential
energy for electrons.
4. Kinetic features of the expansion
There are several ways of presenting the four-dimensional distribution functions
fα(t, z, vk, µ) and each of them highlights a particular physical feature of the plasma
expansion. For instance, as shown in Figs. 7and 8, the development of the plasma
plume structure is evident when fαis plotted versus the axial distance and the parallel
velocity for a given value of µand t. In panels (a)-(c) of Fig. 7, which corresponds to
that representation with µe= 1 and t= 500, 2000, and 7200 for rL= 50 and zM= 800,
one can observe how the simulation box is filled progressively by the plasma. Note that
our simulations used ve
max = 5 but we only showed the range −3≤vke≤3 for clarity.
As shown by the red line, the distribution feis not symmetric with respect to the axis
vke= 0; for a given z, there are more outgoing electrons than reflected ones. As shown
in panel (c), the depletion of particles with negative velocity as zincreases gives rise to

14
a lower dispersion in vke, thus producing the moderate drop in Tkedisplayed in figure 5.
Figure 7. Panels (a), (b), and (c) show the electron distribution function for µ= 1
and times t= 500, 2000 and 7200, respectively, for rL= 50 and zM= 800. The red
line corresponds to vke= 0.
The distribution functions of ions with magnetic moment µi= 1 and times t= 500,
2000 and 7200 are shown in figure 8. For convenience, we did not plot the full dynamic
range of the simulation (−0.5≤vki≤0.5) but only the region of interest. Panels
(a)-(c) reveal that ions reach higher velocities during the transient than in stationary
conditions. Once the plume reached its equilibrium state [panel (c)], most of the
acceleration happens within the first sixty Debye lengths from the throat [consistently
with the electrostatic potential profile of panel (c) in Fig. 3]. Although there is a certain
amount of dispersion, the ion distribution function is not far from mono-energetic.
Figure 8. Panels (a), (b), and (c) show the ion distribution function for µ= 1 and
times t= 500, 2000 and 7200, respectively, for rL= 50 and zM= 800. The red line
corresponds to vki= 0.

15
4.1. Trapped electrons
One of the main features that distinguishes the non-stationary code VLASMAN from
other codes is the self-consistent computation of the doubly-trapped particles (electrons,
here). A detailed analysis of this population is carried out by noting first that the
characteristic equations of the Vlasov equation
dz
dt =vk,(24)
dvk
dt =aα(z, t, µ),(25)
give the following evolution law for the particle total energy E=βαv2
k/2 + Zαφ+µB
dE
dt =Zα
∂φ
∂t .(26)
Therefore, in stationary conditions (∂φ/∂t = 0), the energy is also conserved. In the
steady state and for a given energy and zposition, the maximum value of the magnetic
moment
µmax (z, E ) = E−Zαφ
B(27)
is found by setting vk= 0 in the definition of the energy. A particle of energy Eis
trapped between two axial coordinates zmin ≤z≤zmax if its magnetic moment µ
intersects the curve of µmax at those points, i.e. µmax (zmin, E) = µmax (zmax , E) = µ.
For this reason, the analysis of the trapped particles is easier with the parametrization
fe(t, z, E, µ), instead of fe(t, z, vk, µ). Since for a given energy there are two possible
velocities vk=±q2 (E−Zαφ−µB)/βα, the total distribution function fe(t, z, E, µ)
involves outgoing f+
e(vk>0) and ingoing f−
e(vk<0) distribution functions. In the
analysis below we will always refer to the total one.
Panels (a)-(c) in Fig. 9show the electron distribution function versus zand µfor
E= 2.1 and times t= 500, 2000 and 7200. At large t[panel (c), considered as steady
state], the curve µmax (upper red solid line) exhibits a minimum, say µ∗, at a position z∗
close to the throat. For this energy level, two populations of electrons exist in the steady
state. First, reflected electrons, with µ<µ∗or µ>µ∗and z < z∗: they are injected
from the left of the simulation box, propagate to the right until the location where
µ=µmax, where vk= 0, and they are reflected back to the nozzle entrance and the
reservoir. Second, trapped electrons, with µ > µ∗and z > z∗, which bounce between
two axial coordinates satisfying µmax (zmin, E) = µmax (zmax, E) = µ. Physically, in
the divergent nozzle, electrons injected upstream gain parallel velocity from the (anti)
mirror effect in the decaying Band lose it from the electrostatic force. Therefore, their
reflection is always due to the dominance of the electrostatic force. The same type of
arguments apply to the bouncing motion of the double-trapped electrons.
Similarly to panel (c), panel (d) shows the steady-state distribution function for
electrons of a higher energy (E= 3), µmax(zmax )>0. In this case, there is a population
of free electrons with µ < µ∗that leave the computational box together with the ions,

16
there is a small region of reflected electrons, there are no doubly trapped electrons, and
the region µ>µ∗and z > z∗will be totally void of electrons at t→ ∞. Hereafter we
will denote the doubly-trapped, reflected and free electrons with the subscripts et,er,
and ef, respectively.
The filling of the regions of the phase space with trapped particles, which is not
connected with the injection zone (z= 0) at stationary conditions, is explained in terms
of two transient mechanisms. Since the curve µmax and its minimum µ∗depend on time,
an electron moving to the right with µ<µ∗at a given time could be trapped if it meets
the condition µ>µ∗when moving backwards after electrostatic reflection. The other
mechanism cannot be visualized in the µ−zplane because it is related to the transport
of particles along a third dimension (the energy is not conserved during the transient).
A particle, with energy Ejat instant tjand non-trapped in the µ−zdiagram with
E=Ej, could be trapped at a latter instant tkif its energy decreases to a value Ekthat
traps the particle at the µ−zdiagram with E=Ek< Ej. Both mechanisms, which
can trap or untrap the particles, act simultaneously during the transient period of the
simulation.
Figure 9. Electron distribution functions in the µe−zplane for several values of
Eand t. Simulations results for rL= 50 and zM= 800. The red line corresponds to
µmax in equation 27 and µ=µ∗.
In order to assess the relative importance of the three populations of electrons we
analyzed the distribution function at the end of the simulations with rL= 25,50, and
100. From the z−µdiagrams for all the energies involved in the simulations, the densities
of the trapped (net), free (nef ), and reflected (ner ) electrons were found. The results
are displayed in the three panels of Fig. 10, which show the densities versus the inverse
of the magnetic field. In the extensive quasineutral region the three densities are of the

17
same order and the trapped electrons represent about 20% of the total. Such a value
shows that the population of trapped electrons is neither dominant nor negligible. In
any case, it is smaller than the one assumed heuristically by a recent stationary kinetic
model, which filled completely the phase-space region where doubly-trapped electrons
could potentially exist [19]. At the downstream sheath, where only the most energetic
electrons can arrive, the density of the free electrons is clearly dominant but the density
of trapped electrons still has a finite value.
100101102103104105
1/B
0
0.2
0.4
net/ne
100101102103104105
1/B
0
0.5
1
nef/ne
100101102103104105
1/B
0
0.5
1
ner/ne
rL=100 rL=50
rL=100
rL=25rL=50
rL=100
rL=25
rL=50 rL=25
(a)
(c)
(b)
Figure 10. Panels (a), (b), and (c) show the electron densities of the trapped, free,
and reflected electrons for zM= 800. Solid, dashed, and dot-dashed lines correspond
to rL= 100,50, and 25, respectively.
The results of the code Vlasman can be post-processed to investigate more advanced
features related with the trapping of the particles. For instance, once a simulation is
finished and the history of the electrostatic potential φ(z, t) and the distribution function
of the electrons in the steady state Fe(z, vk, µ, tF) are known, the trajectories of the
trapped particles can be computed as follows. First, by using the z−µdiagrams of
Fe(z, vk, µ, tF) for all the energies, we computed the distribution function Fet(z, vk, µ, tF)
of the trapped electrons at the end of the simulation. Then, taking the values of zand
vkof the trapped electrons as initial conditions, we integrated numerically Eqs. 24-25
backward in time with the corresponding value of µas a parameter and taking into
account the computed time history of the accelerations. Panel (a) in Fig. 11 shows two
examples of trapped orbits for rL= 50 and zM= 800. When analyzed from t=tF
to t= 0, they exhibit periodic motions that correspond to the trapped phases in the
steady state, and then a segment that connects them to the entrance of the nozzle at
the z=z0. For one of the orbits, panel (a) shows the time at different points of the
trajectory. We observe that this particular trapped electron is injected at t= 1287
and the characteristic time to complete a periodic motion is ∼1500. This bouncing
time depends on the energy and the magnetic moment of the electrons and, for some of
them, it can be affected by the size of the computational box. We also mention that,
as t→ ∞, the orbits of the trapped particles are periodic because, once the stationary

18
state has been reached and φdoes not depend on time, Eqs. (24)-(25) are autonomous
and integrable. In our analysis, we computed the 105trapped trajectories with the
highest values of the distribution function in the simulation for rL= 50 and zM= 800.
From them, the values of the parallel and perpendicular velocities of the particles when
they were injected at z=z0and the injection time tinj were found. The color in panel
(b) of Fig. 11 shows the value of the electron distribution function at z= 0 given by
Eq. (18) in the vke−v⊥eplane, and the red dots the region of the phase that yields
trapped trajectories in the interval tinj ≤t < tF. The figure shows that there is a
single phase space region for trapped particles in the injection and it is centered about
(vke, v⊥e)≈(2,2). Almost the same region was found in the simulation with rL= 25
and zM= 800.
Each red point in panel (b) produces a set of trapped orbits during a certain
interval of time. Beyond a certain time, particles cannot be trapped anymore because
the system reaches the steady state and the orbits cannot connect with the trapped
regions showed in Fig. 9. In order to investigate the injection time distribution of the
trapped trajectories we constructed a histogram. As shown in panel (c) of Fig. 11, we
summed the values of ¯
feof the particles trajectories that were injected within a certain
temporal interval of width ∆t= 200. The histogram shows that particles injected at the
very beginning of the simulation are not trapped, and the contribution of the trapped
particles injected after t≥4000 is negligible. It reveals a complex behavior with several
maxima and minima and highlights the non-linear character of the transient phase.
Figure 11. Trapped particles results for rL= 50, zM= 800, and φM=−2.75. Panel
(a), (b) and (c) shows two trapped trajectories, the region of the phase space at z= 0
that yields trapped electrons within the time Tinj ≤t≤TF, and the histogram of the
injection time, respectively.
Since the trapping happens during the transient phase and there is an infinitude of
possible scenarios describing the turn on of the nozzle, the natural question about the
robustness of the previous results arises. This topic has been investigated by running
a simulation with rL= 25, zM= 800, and the following boundary condition for the

19
electrostatic potential
φM(t) = φ0+ (φF−φ0)1
2+1
πarctan [ω(t−t0)],(28)
where we took the parameters φ0=−5, φF=−2.75, ω= 0.1, and t0= 2000.
Such a law keeps the potential at the exit of the simulation box roughly equal to
φ0=−5 before t0= 2000, then makes a smooth transition to the current-free condition
φM=−2.75, and keeps this value constant afterwards. Interestingly, for this simulation,
the maximum percentage of the density of trapped particles (∼27%) is greater than the
one obtained when φMis constant [about a 20% as shown by panel (a) in Fig. 12]. As
shown in panels (b) and (c), the regions of the phase space at z=z0that are trapped at
later times and the histogram are also different from the ones obtained for constant φM.
We then conclude that the steady state of a collisionless plasma expansion in a magnetic
nozzle is not universal and depends on the particle history of the system. On the one
hand, this justifies the uncertainties related to the trapped population in steady-sate
models. On the other hand, this enhances the relevance of taking into account collisions,
even if they are very rare, since they could be the mechanism leading to a final, unique
distribution for the doubly-trapped population and a unique steady-state solution.
0123
0
1
2
3(b)
0.01
0.02
0.03
0.04
0.05
0.06
0 2000 4000 6000 8000
0
20
40
60 (c)
100101102103104105
0
0.1
0.2 (a)
Figure 12. Trapped particles results for rL= 25, zM= 800, and φMgiven by Eq.
28. Panel (a), (b), and (c) show the relative density of the trapped population, the
region of the phase space at z= 0 that are trapped within the time tinj ≤t≤tF,
and the histogram of the injection time, respectively. For comparison panel (a) also
reproduces the relative density of the trapped population for the case φM=const.
5. Conclusions
The Vlasov-Poisson solver revealed interesting kinetic features of the unsteady paraxial
expansion of collisionless plasma jets in magnetic nozzles. Its most relevant properties,
i.e. its non-stationary character and the discretization of the Vlasov equation on a mesh
in phase space, provided a self-consistent description of the particles trapped during the

20
transient with a high accuracy. The parametric analysis showed how the electrostatic
potential drop and the size of the simulation box should be selected to reproduce relevant
physical conditions in the simulations. In general, the results are aligned with previous
work based on stationary models, which considered a heuristic population of trapped
particles. Such results include the electrostatic potential drop that yields a zero net
current, the main features of the electron cooling (temperatures and heat fluxes), and
the shape of the ion distribution function that is close to monoenergetic. Interestingly,
the profiles within the quasineutral region of several quantities, such as the normal
electron temperature and the particle densities, do not depend on the expansion rate if
presented versus the inverse of the magnetic field.
The analysis provided quantitative information about the relative importance of the
different electron populations. Reflected electrons are the dominant population in the
quasineutral region, followed by the free and the trapped electrons. The latter represent
about 20% of the total. This figure is much smaller than the one needed to make
stationary Vlasov-Poisson solver converge in previous studies. Therefore, a population
of trapped electrons seems to be a fundamental component of the expansion but it is
not as large as considered earlier. This is one of the most important conclusions of this
work. Moreover, it has been shown that the exact amount of trapped electrons depends
on the particular history of the system and several steady states are possible.
The model of our work is restricted to the the paraxial and fully magnetized limit
and therefore describes entirely the beam behavior at the axis of the nozzle only. These
and other hypotheses are now revisited. Once the external magnetic field is fixed, the
paraxial approximation adopted in this model, i.e. RL>> R0, imposes a constraint
to the radius R0of the hole at the plasma-vacuum wall. Due to the divergence of the
nozzle, it also limits the maximum axial distance that is meaningful in the numerical
simulations. For a given magnetic field geometry, an estimation of such a distance can be
found by computing the axial position where a magnetic line turns back. The paraxial
hypothesis could be removed from the model by modifying the code slightly: Eq. 6
could be solved for many field lines and Eq. 8should be extended to two-dimensional
(rand z) geometries. On the other hand, at some distance downstream, the local ion
Larmor radius will become comparable to the jet radius and the ions will no longer be
magnetized. Unlike its extension to two-dimensional geometries, the incorporation of
de-magnetization effects to the code, which would involve transport of particles across
different field lines, would not be straightforward.
As presently formulated, the code captures the adiabatic trapping of some electrons
during the initial transient that leads to the development of a fully magnetized expansion
to vacuum. Collisions are entirely neglected in this version, which is appropriate for the
short time scale considered. However, this leaves open the question as to the ultimate
steady state of the trapped electron population over the time scale of many collision
times. The extension of the code to such scenario would require the computation at
each time and location of the multiple integrals over velocities involved in the Boltzmann
collision operator. Since for the usual weak collisionality case the new term will only be

21
noticeable for times much longer than the initial transient time, it may be appropriate
to neglect collisions until a reasonable steady state is reached and only then to turn
collisions on and allow the gradual relaxation to the final steady state. This extension
of the code would be an important contribution. It could clarify whether or not collisions
erase completely the information about the transient, and if all the possible steady states
of collisionless expansions collapse to a unique state.
Acknowledgments
G.S-A was supported by the Ministerio de Econom´ıa y Competitividad of Spain (Grant
RYC-2014-15357). J.Z. was supported by Airbus DS (Grant CW240050). J.R. and
M.M-S stays at UC3M for this research were supported by a UC3M-Santander Chair of
Excellence and by National R&D Plan (Grant ESP2016-75887), respectively. E.A. was
supported by the MINOTOR project, that received funding from the European Unions
Horizon 2020 research and innovation programme, under grant agreement 730028.
Appendix A. Description of the numerical scheme
The discretization in velocity space was carried out with Nvk×Nµpoints uniformly
distributed as (vα
kj=−vα
max + 2(j−1)vα
max/(Nvk−1) and µα
k= (k−1)µα
max/(Nµ−1)
with j= 1 . . . Nvkand k= 1 . . . Nµ). The values of vα
max and µα
max, which control the
maximum ranges of the velocity mesh, depend on the specie and were chosen such that
βα
2(vα
kmax)2=Cvkδα, µα
max ˜
B(z0) = Cµδα,(A.1)
with Cvkand Cµtwo constants. In our simulations we set Cvk= 12.5 and Cµ= 8.95,
which was shown to be large enough (the value of the distribution functions were
negligible close to the boundaries of the velocity mesh).
The distribution function ¯
fα(tm, zi, vα
kj;µα
k) at time tmwas propagated to time
tm+1 =tm+ ∆twith the splitting-time algorithm introduced Ref. [26]. It involves
the following steps (the variable µkis omitted for brevity )
(i) Perform a half time step shift along the z-axis, ¯
f∗
α(z, vk) = ¯
fα(tm, z −vk∆t/2, vk).
(ii) Compute the acceleration aα(z, tm+ ∆t/2) by using ¯
f∗
αin 7and 8.
(iii) Perform a shift along the vk-axis, ¯
f∗∗
α(z, vk) = ¯
f∗
αz, vk−aα∆t
(iv) Perform a half time step shift along the z-axis, ¯
fα(t+∆t, z, vk) = ¯
f∗∗
α(z−vk∆t/2, vk).
A key component of the code is the interpolation scheme, which is needed to
make the shifts along zand vk. After several tests monitoring the conservation of
mass, momentum, and energy by using 11-13 and the results provided by the code, a
cubic interpolation scheme for the z-shifts was selected. For electrons, vk-shifts were
carried out with cubic splines, unless such operation is carried out at the three cells
that adjoin the entrance and the exit of the simulation domain. For these cases, where
the distribution function can be discontinuous due to the imposed boundary conditions,

22
we used linear interpolation. The filamentation of the electron distribution function
was avoided with the Fourier filter explained in [26]. For ions, spline interpolation in
vkdoes not produce very accurate results due to the discontinuous character of the
distribution function (note that no injected ions come back to the nozzle entrance).
Linear interpolation for the vkinterpolation of the ions gives more accurate results and
was finally used in the simulations. The integrals in velocity space of the distribution
functions, such us nαin 9, were carried out with a Sympson method. The two integrals
in real space appearing in 8were solved with a trapezoidal method.
The code was implemented with OpenMP (shared memory) in Fortran and the
parallel computation took advantage of the conservation of µ. The time propagation
of a piece of the distribution function with a certain µ-range was assigned to a specific
processor.
Appendix B. Effect of the convergent segment
Although this work is focused on the divergent segment of the nozzle, we here explain
briefly why it is convenient to add a small convergent segment in the simulation domain,
and the impact of its length, |z0|, in the results of the expansion in the divergent region.
A short parametric analysis for z0=−50,−25,−12.5 and rL= 50, is presented next.
Panel (a) of Fig. B1 displays the normalized charge density profiles in stationary
conditions. The sheath forms always at the left edge of the simulation box and its
strength mitigates as the length of the convergent segment is increased [see inset in
panel (a)]. Eventually the sheath would disappears if the whole plasma source were
included, as it is the case in the stationary model of Ref. [19]. Panels (a) and (b)
show that both the charge density and the electric potential (when referenced to the
throat) are almost unaltered by the segment length (if this is larger than a few Debye
lenghts). The same conclusion was reached for other plasma properties in the divergent
nozzle, like for instance the temperature profiles. In this respect, the results of our
work are robust. If the left edge of the simulation were placed at the throat, a non-
negligible sheath would appear next to the throat affecting significantly the potential
fall along the divergent nozzle. It is clear that there are distribution functions that avoid
the development of a sheath at the left edge, but we do not see a way to characterize
them. The presence of this spurious sheath has been reported also in full particle-in-cell
simulations of unmagnetized and magnetized plumes too [27, 28]

23
0 200 400 600 800
-0.2
0
0.2
0.4
-40 -20 0 15
-0.15
-0.1
-0.05
0
0 200 400 600 800
-3
-2
-1
0
1
(a)
(b)
Figure B1. Panels (a) and (b) show the charge densities and potential profiles for
simulations with lengths of the convergent segment equal to z0=−50 (solid black),
z0=−25 (dashed blue) and z0=−12.5 (red dashed-dotted). The curves practically
overlap.
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