Content uploaded by Gonzalo Sanchez-Arriaga

Author content

All content in this area was uploaded by Gonzalo Sanchez-Arriaga on Feb 15, 2018

Content may be subject to copyright.

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 =q0kBT∗

e/N∗e2is the Debye length, ω∗

pe =qN∗e2/me0the 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

Buα3

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 zM1, 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 BM1), 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/mi1 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

eFI

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∗

αz, 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.

[1] Alice K. Harding. The neutron star zoo. Frontiers of Physics, 8(6):679–692, 2013.

[2] Brian E. Wood. Astrospheres and solar-like stellar winds. Living Reviews in Solar Physics, 1(1):2,

2004.

[3] Stephen P. Reynolds. Supernova remnants at high energy. Annual Review of Astronomy and

Astrophysics, 46:89–126, 2008.

[4] X. L. Zhang, R. S. Fletcher, S. L. Rolston, P. N. Guzdar, and M. Swisdak. Ultracold plasma

expansion in a magnetic ﬁeld. Phys. Rev. Lett., 100:235002, Jun 2008.

[5] W.-M. Wang, P. Gibbon, Z.-M. Sheng, and Y.-T. Li. Magnetically assisted fast ignition. Physical

Review Letters, 114(1), 1 2015.

[6] S. A. Andersen, V. O. Jensen, P. Nielsen, and N. D’Angelo. Continuous supersonic plasma wind

tunnel. The Physics of Fluids, 12(3):557–560, 1969.

[7] C. Charles, R. W. Boswell, and M. A. Lieberman. Xenon ion beam characterization in a helicon

double layer thruster. Applied Physics Letters, 89(26):261503, 2006.

[8] Mariano Andrenucci. Magnetoplasmadynamic Thrusters. John Wiley & Sons, Ltd, 2010.

[9] Alexey V. Areﬁev and Boris N. Breizman. Theoretical components of the vasimr plasma propulsion

concept. Physics of Plasmas, 11(5):2942–2949, 2004.

[10] S. A. Cohen, N. S. Siefert, S. Stange, R. F. Boivin, E. E. Scime, and F. M. Levinton. Ion

acceleration in plasmas emerging from a helicon-heated magnetic-mirror device. Physics of

Plasmas, 10(6):2593–2598, 2003.

[11] Kazunori Takahashi, Aiki Chiba, Atsushi Komuro, and Akira Ando. Experimental identiﬁcation of

an azimuthal current in a magnetic nozzle of a radiofrequency plasma thruster. Plasma Sources

Science and Technology, 25(5):055011, 2016.

[12] Y Takama and K Suzuki. Experimental studies on nonequilibrium plasma ﬂow in a

convergentdivergent magnetic ﬁeld. Plasma Sources Science and Technology, 17(1):015005, 2008.

[13] Donal L. Chubb. Fully ionized quasi-one-dimensional magnetic nozzle ﬂow. AIAA Journal,

10(2):113–114, 1972.

[14] Alexey V. Areﬁev and Boris N. Breizman. Magnetohydrodynamic scenario of plasma detachment

in a magnetic nozzle. Physics of Plasmas, 12(4):043504, 2005.

[15] E. Ahedo and M. Merino. Two-dimensional supersonic plasma acceleration in a magnetic nozzle.

Physics of Plasmas, 17(7):073501, 2010.

24

[16] Mario Merino and Eduardo Ahedo. Plasma detachment in a propulsive magnetic nozzle via ion

demagnetization. Plasma Sources Science and Technology, 23(3):032001, 2014.

[17] Mario Merino and Eduardo Ahedo. Eﬀect of the plasma-induced magnetic ﬁeld on a magnetic

nozzle. Plasma Sources Science and Technology, 25(4):045012, 2016.

[18] Heath Lorzel and Pavlos G. Mikellides. Three-dimensional modeling of magnetic nozzle processes.

AIAA Journal, 48(7):1494–1503, 2010.

[19] M. Martinez-Sanchez, J. Navarro-Cavall´e, and E. Ahedo. Electron cooling and ﬁnite potential

drop in a magnetized plasma expansion. Physics of Plasmas, 22(5):053501, 2015.

[20] Juan Ram´on Sanmart´ın. Active charging control and tethers. In C.P. Catani, editor, Space

environment: prevention of risk related to spacecraft charging. Space technology course, pages

515–533. C´epadues, Tolouse, 2002.

[21] G. S´anchez-Arriaga and D. Pastor-Moreno. Direct vlasov simulations of electron-attracting

cylindrical langmuir probes in ﬂowing plasmas. Physics of Plasmas, 21(7):073504, 2014.

[22] A. V. Gurevich. Distribution of Captured Particles in a Potential Well in the Absence of Collisions.

Soviet Journal of Experimental and Theoretical Physics, 26:575, March 1968.

[23] M A Raadu. Expansion of a plasma injected from an electrodeless gun along a magnetic ﬁeld.

Plasma Physics, 21(4):331, 1979.

[24] Albert Meige, Rod W. Boswell, Christine Charles, and Miles M. Turner. One-dimensional particle-

in-cell simulation of a current-free double layer in an expanding plasma. Physics of Plasmas,

12(5):052317, 2005.

[25] M. Martinez-Sanchez and E. Ahedo. Magnetic mirror eﬀects on a collisionless plasma in a

convergent geometry. Physics of Plasmas, 18(3):033509, 2011.

[26] C.Z Cheng and Georg Knorr. The integration of the vlasov equation in conﬁguration space.

Journal of Computational Physics, 22(3):330 – 351, 1976.

[27] Yuan Hu and Joseph Wang. Fully kinetic simulations of collisionless, mesothermal plasma emission:

Macroscopic plume structure and microscopic electron characteristics. Physics of Plasmas,

24(3):033510, 2017.

[28] Min Li, Mario Merino, Eduardo Ahedo, Junxue Ren, and Haibin Tang. Full-pic code validation and

comparison against ﬂuidmodels on plasma plume expansions. In Proceedings of International

Electric Propulsion Conference, pages IEPC–2017–230. Georgia Institute of Technology, Oct

2017.

A preview of this full-text is provided by IOP Publishing.

Content available from

**Plasma Sources Science and Technology**This content is subject to copyright. Terms and conditions apply.