Content uploaded by Gonzalo Sanchez-Arriaga

Author content

All content in this area was uploaded by Gonzalo Sanchez-Arriaga on Oct 13, 2014

Content may be subject to copyright.

Direct Vlasov simulations of electron-attracting cylindrical Langmuir probes in flowing

plasmas

G. Sánchez-Arriaga and D. Pastor-Moreno

Citation: Physics of Plasmas (1994-present) 21, 073504 (2014); doi: 10.1063/1.4889732

View online: http://dx.doi.org/10.1063/1.4889732

View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/21/7?ver=pdfcov

Published by the AIP Publishing

Articles you may be interested in

A direct Vlasov code to study the non-stationary current collection by a cylindrical Langmuir probe

Phys. Plasmas 20, 013504 (2013); 10.1063/1.4774398

Relativistic current collection by a cylindrical Langmuir probe

Phys. Plasmas 19, 063506 (2012); 10.1063/1.4729662

Comment on “On higher order corrections to gyrokinetic Vlasov–Poisson equations in the long wavelength limit”

[Phys. Plasmas16, 044506 (2009)]

Phys. Plasmas 16, 124701 (2009); 10.1063/1.3272151

One-dimensional Vlasov simulation of parallel electric fields in two-electron population plasma

Phys. Plasmas 14, 092302 (2007); 10.1063/1.2770002

Numerical study of a direct current plasma sheath based on kinetic theory

Phys. Plasmas 9, 691 (2002); 10.1063/1.1432316

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

2.138.102.237 On: Wed, 23 Jul 2014 11:23:12

Direct Vlasov simulations of electron-attracting cylindrical Langmuir probes

in flowing plasmas

G. S

anchez-Arriaga and D. Pastor-Moreno

Departamento de F

ısica Aplicada, Escuela T

ecnica Superior de Ingenieros Aeron

auticos,

Universidad Polit

ecnica de Madrid, Plaza de Cardenal Cisneros 3, 28040 Madrid, Spain

(Received 15 April 2014; accepted 24 June 2014; published online 10 July 2014)

Current collection by positively polarized cylindrical Langmuir probes immersed in ﬂowing

plasmas is analyzed using a non-stationary direct Vlasov-Poisson code. A detailed description of

plasma density spatial structure as a function of the probe-to-plasma relative velocity Uis pre-

sented. Within the considered parametric domain, the well-known electron density maximum close

to the probe is weakly affected by U. However, in the probe wake side, the electron density mini-

mum becomes deeper as Uincreases and a rariﬁed plasma region appears. Sheath radius is larger at

the wake than at the front side. Electron and ion distribution functions show speciﬁc features that

are the signature of probe motion. In particular, the ion distribution function at the probe front side

exhibits a ﬁlament with positive radial velocity. It corresponds to a population of rammed ions that

were reﬂected by the electric ﬁeld close to the positively biased probe. Numerical simulations

reveal that two populations of trapped electrons exist: one orbiting around the probe and the other

with trajectories conﬁned at the probe front side. The latter helps to neutralize the reﬂected ions,

thus explaining a paradox in past probe theory. V

C2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4889732]

I. INTRODUCTION

Plasma structure in the neighborhood of material walls

is a fundamental problem in plasma physics. Theoretical

models were developed during almost one century and used

in relevant technological applications. Two examples are

Langmuir probes,

1

which are routinely used for plasma

diagnostic and bare electrodynamic tethers,

2

that can be

used to deorbit satellites at the end of life and mitigate the

space debris problem. The steady-state properties of plasma

sheaths in non-ﬂowing plasma are now well understood.

1,3–7

In ﬂowing plasmas, both analytical works in particular

regimes

1,8–10

and numerical simulations of the Vlasov-

Poisson system

11–14

were carried out. Some issues, however,

remain open.

An interesting paradox arises for cylindrical Langmuir

probes positively polarized within the so called mesother-

mal regime;

15

i.e., probe velocity small (large) compared

with electron (ion) thermal velocity. The faraway electron

population would still be (nearly) isotropic and an important

and very general result by Laframboise and Parker

applies:

16

electron density N

e

is less that the unperturbed

plasma density N

0

everywhere. This result is valid even if

the electric potential depends on the azimuthal angle h,asit

happens in the case of ﬂowing plasmas. On the other hand,

since probe bias is positive and ions are repelled, the hyper-

sonic ion ﬂow will result in an ion density N

i

exceeding N

0

in a broad region at the probe front side. This would break

the quasineutrality in a region much larger than the Debye

length.

The explanation of the paradox may come from the vio-

lation of (at least) one of the hypothesis used in Ref. 16 to

ﬁnd the condition N

e

/N

0

<1. Besides the isotropic character

of the electron distribution function, Ref. 16 assumed that (i)

plasma reached an steady state and (ii) that there are no

trapped particles. Violation of hypothesis (i) was invoked in

Ref. 17, where plasma oscillations were detected in labora-

tory experiments. Possible electron trapping in energy

troughs due to collisional effects was discussed in Ref. 3.

However, since collisional trapping rate can be very small,

6

a non-stationary process called adiabatic trapping

18

has been

proposed in Ref. 15 to explain the paradox. As electron

potential wells develop in time, some electrons become

trapped adiabatically. The adiabatic term comes from the

fact that time variations in the electric potential, which are

controlled by the ions, are very slow as compared with the

motion of the electrons.

The conﬁrmation of electron trapping at the front side

cannot be carried out with stationary Vlasov-Poisson solv-

ers

11,12,14

because they explicitly neglect this particle popu-

lation. The computation of trapped particles requires a non-

stationary Vlasov-Poisson solver, like the particle-in-cell

(PIC) codes used to study Langmuir probes in the

past.

13,19,20

However, PIC codes do not give an accurate rep-

resentation when the number of particles per cell is small

and they also introduce numerical noise. An alternative is

the implementation of Eulerian or direct Vlasov codes.

Unlike PIC codes, which use macro-particles to discretize

Vlasov equation, direct Vlasov codes make a discretization

of the distribution functions in both real and velocity space.

The result is a code free of numerical noise but expensive

from a computational point of view.

This work used numerical simulations to conﬁrm that

electron trapping at the probe front side occurs. Section II

shows the modiﬁcation carried out in the direct Vlasov code

named Kilaps (Kinetic Langmuir probe software)

21

to

1070-664X/2014/21(7)/073504/8/$30.00 V

C2014 AIP Publishing LLC21, 073504-1

PHYSICS OF PLASMAS 21, 073504 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

2.138.102.237 On: Wed, 23 Jul 2014 11:23:12

incorporate the probe-to-plasma relative motion. In Sec. III,

the time evolution of the macroscopic variables, including

collected current, is discussed. Sections IV and Vdescribe

the structure of the particle densities and distribution func-

tions. The signatures of the probe motion in plasma variables

are highlighted. In Sec. VI, two populations of trapped elec-

trons, one of them at the probe front side, are shown.

Discussion of results and comparisons with previous works

are presented in Sec. VII.

II. MATHEMATICAL MODEL AND NUMERICAL

ALGORITHM

A perfectly absorbing cylindrical probe of radius Rand

bias U

p

(t) is moving with velocity U(t) inside a collisionless,

non-magnetized, Maxwellian plasma made of electrons an

single-charged ions. Faraway from the probe, plasma density

is equal to N

0

, and electrons’ and ions’ temperatures are T

e

and T

i

, respectively. Cylindrical coordinates (r,h), with ori-

gin at the center of the probe and angle h¼0(h¼p) corre-

sponding to the wake (front or ram) side, are used. Time,

position, and velocity vectors, electric potential, particles

distribution functions, and densities are normalized as

xpet!t;r

kDe !r;v

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

kTe=me

p!v;

eU

kTe!U;kTefa

N0me!fa;Na

N0!Na;(1)

where the subscript a¼eand idenotes electrons and ions.

We also introduce the following dimensionless parameters

and variables:

qR

kDe

;lama

me

;daTa

Te

;

/pt

ðÞeUpt

ðÞ

kTe

;St

ðÞUt

ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

kTi=mi

p:(2)

Using this normalization, Vlasov-Poisson system in cy-

lindrical coordinates reads

@fa

@tþvr

@fa

@rþvh

r

@fa

@hþv2

h

ra

la

@U

@r

!

@fa

@vr

1

rvrvhþa

la

@U

@h

@fa

@vh¼0;(3a)

@2U

@r2þ1

r

@U

@rþ1

r2

@2U

@h2¼NeNi;(3b)

where

e

¼1,

i

¼1 and

Naðt;r;hÞ¼ððfaðt;r;h;vr;vhÞdvrdvh:(4)

The boundary conditions of the problem are

faðt;r¼q;h;vr>0;vhÞ¼0;(5a)

fat;r!1;h;vr;vh

ðÞ

¼la

2pda

exp la

2da

v2

rþv2

hþdi

li

S22Sﬃﬃﬃﬃ

di

li

svrcos hvhsin h

ðÞ

2

43

5

8

<

:9

=

;

;(5b)

Uðt;r¼q;hÞ¼/pðtÞ;Uðt;r!1;hÞ!0:(5c)

Boundary condition (5a) imposes that no particle is emitted

by the probe. Faraway from the probe, a drifting Maxwellian

is assumed (see Eq. (5b)).

Given initial conditions, U(0, r,h) and f

a

(0, r,h,v

r

,v

h

),

and laws /pðtÞand S(t), Sys. 3 is integrated with Kilaps,

which is a modiﬁcation of the code presented in Ref. 21.It

implements a ﬁnite-difference method combined with an

explicit Runge-Kutta algorithm to carry out the time integra-

tion. Electrons and ions share the same spatial grid (r,h), but

they have different grids in velocity space (v

r

,v

h

). The semi-

inﬁnite spatial domain [qr<1] is truncated up to a maxi-

mum radius r

max

and discretized with N

r

points. These points

are not uniformly distributed, i.e., a ﬁne grid is taken close to

the probe. Thanks to the symmetry of the problem, Kilaps

just simulates the range 0 hpusing a grid of N

h

points

uniformly distributed. Regarding velocity space, the inﬁnite

domains are truncated to the intervals vrmax <vr<vrmax and

vhmax <vr<vhmax .Nvrand Nvhequispaced points are used

to created a grid in velocity space. For convenience, we take

Nvrand Nvhodd numbers. Due to the disparate masses, limits

of the velocity intervals are different for electrons and ions,

and we set velectrons

max ¼ﬃﬃﬃﬃ

li

pvions

max.

The total number of grid points for each specie is

N¼NhNvhNrNvrNvr1

2

;(6)

where we took into account that no grid point is necessary at

r¼qand v

r

>0, because the distribution function is known

(see Eq. (5a)). The values of the distribution function of the

specie aat the grid points are organized in two vectors F

a

of

dimension N(a¼e,i).

After substituting the differential operator by appropri-

ate ﬁnite-differences formulae, Vlasov equation becomes a

system of 2Nordinary differential equation

dFa

dt ¼

Ma

Lþa

la

Ma

NL Fe;Fi

FaþCa(7)

that we integrate with a third order Runge-Kutta method.

Matrices

Ma

L, which have a sparse structure and only depend

on the grid parameters, are computed at the beginning of the

073504-2 G. S

anchez-Arriaga and D. Pastor-Moreno Phys. Plasmas 21, 073504 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

2.138.102.237 On: Wed, 23 Jul 2014 11:23:12

simulation. Vectors C

a

, which take into account the bound-

ary conditions and the law S(t), and matrices

Ma

NL, involving

the non-linear term in Vlasov equation, are calculated each

time step. Potential Uis also found each time step by solving

Poisson equation with ﬁnite-difference methods. Boundary

conditions Uðq;h;tÞ¼/pðtÞand U(r

max

,h,t)1/rare

imposed.

The above scheme has been parallelized in Kilaps. Each

processor computes the distribution function evolution

within a subdomain at certain hvalue and within a radial

range. In order to compute the radial and azimuthal deriva-

tives at the boundaries of the subdomains, communication of

adjacent processors each time step is needed. This strategy

ensures the scalability of the code and a similar computa-

tional load to each processor.

III. EVOLUTION OF MACROSCOPIC VARIABLES

Parameter values of the simulation were R/k

De

¼0.5,

T

e

/T

i

¼1, and the unrealistic mass ratio m

i

/m

e

¼100. This

value, which guarantees that ion motion is much slower

than electron motion, helps to save computational resour-

ces. From Ref. 6, we conclude that probe radius is small

enough to make the probe operate within the orbital-

motion-limited (OML) regime if no probe motion would

exist. The numerical parameters were N

r

¼153, N

h

¼18,

N

vr

¼65, N

vh

¼65 and r

max

¼120 (normalized units). The

limits of the simulation box in velocity space for the elec-

trons were set to 7.5 <v

r

<7.5 and 7.5 <v

h

<7.5. The

ion box in velocity space was scaled by a factor ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

me=mi

p,

and the equations were integrated with a ﬁxed time step

equal to Dt¼0:001x1

pe .AsshownbySec.IV–VI, the nu-

merical box and the resolution were large enough to impose

the boundary conditions appropriately and capture the main

physics of the problem. The simulation ran in 145 processor

during one month, approximately. The external laws /pðtÞ

and S(t) were varied as shown in Fig. 1(note the logarith-

mic scale in the horizontal axis).

Initially, the probe bias is zero (/pð0Þ¼0), the probe is

at rest (S(0) ¼0), and the electron and ions distribution

functions are given by Eqs. (11) and (12) in Ref. 21. The nor-

malized potential was increased until the value /p0¼10,

which is reached at x

pe

t20. This part of the simulation is

similar to the one performed in Ref. 21, but with a lower

probe potential value. The current had an overshoot, and it

approached to the OML current (per unit length) I

OML

after-

wards. We recall that I

OML

is given by

IOML ¼2eN0Rﬃﬃﬃﬃﬃﬃ

2e

me

rﬃﬃﬃﬃﬃﬃ

/p

pþ1

2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

pkTe

e

rexp e/p

kTe

erfc e/p

kTe

"#

(8)

and corresponds to the maximum current in steady condi-

tions of a long cylindrical Langmuir probe of radius Rand

bias /poperating at rest in unmagnetized, collisionless,

Maxwellian plasmas of electron temperature T

e

.

Probe potential was kept constant and equal to 10 in the

rest of the simulation. At about x

pe

t1000, the probe-to-

plasma relative velocity was increased until it reached S¼0.5

at x

pe

t1200. We remark that parameter Saffects the

boundary conditions 5b. In the interval 1200 <x

pe

t<3500,

the (normalized) ram velocity is kept ﬁxed and equal to 0.5.

This is enough time to reach an state close to equilibrium.

During the transient, the current dropped below I

OML

,and

then it reached an asymptotic value slightly higher than I

OML

.

The black solid line in Fig. 1then looks thicker because the

current exhibits small oscillations with frequency close to x

pe

(see inset in Fig. 1).

Within the interval 3500 <x

pe

t<4000, Swas increased

from 0.5 to 1 and then kept ﬁxed until the end of the simula-

tion at x

pe

t7000. Plasma response was similar to the one

exhibited in the interval 1000 <x

pe

t<3500; collected cur-

rent decreased for a moment and it then increased up to a

value slightly higher than I

OML

. Very small oscillations in

the collected current are present at the end of the simulation.

A close look to the simulated variables at x

pe

t3500 and

x

pe

t7000, including collected current, electric potential,

and plasma distribution functions, indicates that the simula-

tions were long enough to reach an state close to equilibrium.

The small difference between the asymptotic collected

FIG. 1. Evolution of the normalized current I/I

OML

, probe potential

/pðtÞ=/p0and probe-to-plasma relative velocity S(t). Here /p0¼10. The

inset shows the time evolution of I/I

OML

in the interval 3450 <x

pe

t<3480.

FIG. 2. Normalized electron density N

e

/N

0

in log-polar coordinates at the

end of a simulation (/p¼10 and S¼1).

073504-3 G. S

anchez-Arriaga and D. Pastor-Moreno Phys. Plasmas 21, 073504 (2014)

2.138.102.237 On: Wed, 23 Jul 2014 11:23:12

current value and I

OML

is attributed to the slightly anisotropic

character of the electron distribution function as r!1

(note that in our simulation, we just have ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

mi=me

p¼10 in

Eq. (5b)).

IV. ELECTRON AND ION DENSITIES

For ﬁnite probe motion, electron density has a complex

spatial structure. Figure 2shows a map of the electron den-

sity at the end of the simulation. In order to magnify the

region close to the probe, a log-polar coordinate system,

with ð~

x;~

yÞ¼log10ðr=RÞðcos h;sin hÞ, is used. Very close to

the probe, there is a rariﬁed plasma with normalized electron

density close to 0.5 (this is the value predicted by OML

theory if no probe motion exists). As radial distance is

increased, electron plasma density grows monotonically until

a maximum value is reached. The maximum, about

N

e

/N

0

0.94, does not depend on the azimuthal coordinate

value (see Fig. 3). Its position, however, is closer to the

probe in the wake (r/k

De

1.17) than in the front side

(r/k

De

1.6). For higher radial distances, the electron density

decreases until a minimum value, which depends strongly on

the probe side; at the front (wake), one has N

e

/N

0

0.87 (N

e

/

N

0

0.57) at r/k

De

4.3 (r/k

De

4.8). The low electron

plasma region is especially important at the wake side,

within the range 2 <r/k

De

<10 and 0 <h<p/2. After the

minimum, the electron density grows monotonically and it

approaches to the faraway value (N

e

/N

0

!1). These features

are in agreement with the results given by stationary-like

codes.

22

Figure 3shows electron density proﬁles at certain h

angles. We recall that h¼0(h¼180) corresponds to the

wake (front) side. There is a h-range in the front side where

the electron density is slightly above the background density

within certain radial domain. For h¼159, this radial range

extends from 40k

De

to the border of the simulation box. This

behavior, in apparent contradiction with Ref. 16, is due to a

population of trapped particles and it will be analyzed in

Sec. VI.

Probe-to-plasma velocity effect is illustrated in Fig. 4,

which shows density proﬁles at the front (top panel) and

wake (bottom panel) probe sides for different Svalues. The

thick solid black lines correspond to numerical results of a

stationary code (see for instance Ref. 5or Ref. 22). The three

thin lines are Kilaps results for S¼0, 0.5, and 1 at

x

pe

t¼1000, 3500, and 7000, i.e., once the plasma reached

an equilibrium state. Clearly, the maximum in the electron

density is higher for Kilaps. This is due to particle trapping

(see Sec. VI), an effect not included in stationary codes. As S

is increased, the maximum in the electron density decreases,

and it approaches (moves away from) the probe in the wake

(front) sides. Regarding the density minimum, we observe

that it is less pronounced in the front side as Sis enhanced

but it becomes very deep in the wake side.

Figure 5shows an ion density map at x

pe

t¼7000. Ion

density practically vanishes within the sheath, and it

increases monotonically until the background plasma density

N

0

. The sheath has a radius about 10k

De

, and, unlike the elec-

tron density, it exhibits a quasi-isotropic behavior. However,

there is small h-dependence, and the sheath radius is smaller

in the front than in the wake side. An analysis similar to

FIG. 3. Normalized electron density proﬁles at the end of the simulation

(/p¼10 and S ¼1).

FIG. 4. Electron density at the front (top panel) and the wake (bottom panel)

sides for different probe-to-plasma relative velocity. Results from Ref. 5,

which ignores particle trapping, are also shown.

FIG. 5. Normalized ion density N

i

/N

0

in log-polar coordinates at the end of

the simulation.

073504-4 G. S

anchez-Arriaga and D. Pastor-Moreno Phys. Plasmas 21, 073504 (2014)

2.138.102.237 On: Wed, 23 Jul 2014 11:23:12

Fig. 4applied to the ions (not shown) reveals that the sheath

radius decreases (increases) in the front (wake) side as the

parameter Sin enhanced. As compared with stationary simu-

lations, Kilaps shows that ion density in ﬂowing plasmas is

higher at the front and lower at the wake; as expected,

rammed ions reach positions closer to the probe.

V. DISTRIBUTION FUNCTIONS

Electron and ion distribution functions in ﬂowing plas-

mas exhibit a complex structure in real and velocity spaces.

Besides a dependence on the radial and azimuthal coordi-

nates, the distribution functions are not symmetric in velocity

space (except at the speciﬁc angles h¼0 and 180). Close to

the probe, they are far from Maxwellian functions and present

different features depending on the specie under considera-

tion. Some examples are given in Figs. 6and 7, which also

show that both the computational box and the resolution in

velocity space were large enough in the simulations.

Figure 6shows the normalized electron distribution

function at several positions. Panels (a) and (b) correspond

to r¼qat h¼0 and h¼p, respectively. Similarly, panels (c)

and (d) display f

e

at the same azimuthal angles but at

r¼20k

De

. Panels (a) and (b) show that, at the probe radius,

f

e

has a shape of half a ring with a radius in velocity space of

the order of ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2eUp=me

p. As compared with the no-ﬂowing

case (see Fig. 3 in Ref. 21), there is a lack of electrons arriv-

ing at the probe with vanishing azimuthal velocity. This

effect is important in the wake [see panel (a)] and practically

negligible (but still present) in the front [panel (b)], where

the electron distribution function reaches its maximum value

for a normalized velocity magnitude of approximately 2. For

higher radial coordinate (bottom panels), the distribution

function resembles to a Maxwellian, but with vanishing

number of particles at a cone that extends for positive radial

velocity and has its axis at zero azimuthal velocity. The aper-

ture angle of this cone, which is a signature of the no-

emissive character of the probe, decreases as r!r

max

.At

other azimuthal angles, the electron distribution functions is

not symmetric with respect to the line v

h

¼0; for instance, at

r¼Rand h¼32, there are much more particles with nega-

tive azimuthal velocity (not shown).

Panels (a)–(d) in Fig. 7display the ion distribution func-

tion at the same positions as in Fig. 6. Close to the probe,

inside the sheath, the ion distribution function almost van-

ishes. However, due to the probe velocity, there are more

ions in the front than in the wake side [see panels (a) and

(b)]. As rincreases, interesting and different features appear

in the wake and the front sides. Panel (c) in Fig. 7shows the

ions distribution function in velocity space at r¼20k

De

and

h¼0 (wake). There are two peaks with centers at vr

0:1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

kTe=me

pand vh60:1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

kTe=me

p. These peaks do not

appear for S¼0 simulations, where the ion distribution func-

tion has an isotropic behavior in velocity space (except for a

cone-like region with no particles due to the no emissive

character of the probe).

21

Panel (d) shows the ion distribution

function at the same radial distance but at the front side.

Even though the probe does not emit particles, there is a

prominent ﬁlament with positive radial velocity. Its azi-

muthal velocity range is very narrow, and it extends until

vr0:3ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

kTe=me

p. This population corresponds to rammed

ions that are reﬂected by the electric ﬁeld (note that the probe

is biased positively).

Figure 8shows the structure of the ﬁlament in more

detail. For convenience, we plot with a thin solid line the

shifted Maxwellian distribution function given by Eq. (5b)

with S¼1. The thick lines are ion distribution sections at

v

h

¼0 and h¼180at the end of the simulation

(x

pe

t¼7000). For r¼2k

De

and 8k

De

, the ion distribution

function has a maximum at v

r

¼0; i.e., inside the sheath, the

inﬂuence of the ram motion in the ion distribution function is

weak. At r¼11k

De

we observe the appearance of two max-

ima that, as rincreases, separate apart. The left peak, with

v

r

<0, perfectly ﬁts with the shifted Maxwellian, and it rep-

resents the incoming population of ions. The right peak,

which is the ﬁlament with v

r

>0 in Panel (d) of Fig. 7,isa

population of ions that have been reﬂected by the (positively

polarized) probe. Although, here, we just showed the ion dis-

tribution function at the speciﬁc value h¼p, ions are

reﬂected in a broad h-range. The consequence is a large

FIG. 6. Electron distribution function in velocity space at x

pe

t¼7000.

Panels (a) and (b) ((c) and (d)) correspond to sections at probe radius

(20k

De

) and h¼0and h¼180, respectively.

FIG. 7. Ion distribution function in velocity space at x

pe

t¼7000. Panels

(a)–(d) correspond to the same radial and azimuthal values as in Fig. 6.

073504-5 G. S

anchez-Arriaga and D. Pastor-Moreno Phys. Plasmas 21, 073504 (2014)

2.138.102.237 On: Wed, 23 Jul 2014 11:23:12

region at the probe front side with plasma density above N

0

.

As we will see, a trapped population of electrons is necessary

to recover quasi-neutrality faraway from the probe.

VI. ELECTROSTATIC POTENTIAL AND TRAPPED

PARTICLES

Figure 9shows a map of the normalized potential

(U=/p0) in log-polar coordinates at the end of the simulation.

The two insets correspond to potential proﬁles at h¼p(left)

and h¼0 (right). Clearly, the behavior of the potential is dif-

ferent at the front and wake sides of the probe. In particular,

the electrostatic potential reaches negative values at the

wake side, an effect already observed in stationary-like

codes.

22

As we will see, the potential asymmetry has conse-

quences in the electron trapped population, which happens at

the probe front side. We remark that the potential is not sta-

tionary at the end of the simulation but exhibits small oscilla-

tions at frequency close to x

pe

.

Figures 3and 4seem to contradict results from Ref. 16;

electron density close to the probe is higher in Kilaps

simulations, where one also has N

e

/N

0

>1 in the front. As

we will see, both differences are explained if trapped par-

ticles are considered. Trapped particles can be computed by

post-processing Kilaps simulations. At the end of the simula-

tion (x

pe

t¼7000), once the plasma is in equilibrium, the

(almost periodic) potential proﬁle U(t,r,h) is saved over one

electron period (2p/x

pe

). Particle trajectories can then be

computed by integrating Vlasov characteristic equations

dr

dt ¼vr;dvr

dt ¼a

la

@U

@rþv2

h

r;(9)

dh

dt ¼vh

r;dvh

dt ¼1

rvrvhþa

la

@U

@h

:(10)

Neither the energy nor the angular momentum is conserved,

because the potential depends on time and azimuthal

angle h.

Examining the electron distribution function (for

instance Fig. 6), we can select initial conditions (r

0

,h

0

,v

r0

,

v

h0

) where electrons exist at the end of the simulation.

Systems 9 and 10 are integrated for a long time (typically

x

pe

t50 000 in our calculations), and, if the particle does

not hit the probe and does not leave the simulation box, then

it is deemed to be trapped.

Figure 10 shows some examples of particle trajectories.

Trajectory (a) corresponds to a trapped electron orbiting

around the probe. This population of trapped particles is re-

sponsible for the high density (as compared with stationary

theory) exhibited by Kilaps simulations close to the probe

(see Fig. 4). A second population of trapped particles hap-

pens at the probe front side. An example is given by trajec-

tory (b) in Fig. 10. Neglecting this population yields to the

paradox explained in Sec. I. If it is included, one has

N

e

/N

0

>1 at the front, and plasma can reach quasineutrality

(note that N

i

/N

0

>1 at the front due to the reﬂected ions

showed in Fig. 8). Finally, electrons labeled by (c) and (d)

are captured by the probe and (e) goes around the probe and

leaves the simulation box.

FIG. 9. Normalized electric potential U/U

p0

in log-polar coordinates at the

end of the simulation. Left (right) inset corresponds to a section at h¼180

(h¼0).

FIG. 10. Electron trajectories: (a) trapped orbit around the probe, (b) trapped

orbit at the front side, (c) and (d) captured electrons and (e) trajectory leav-

ing the simulation box. Probe, with radius R/k

De

¼0.5, is plotted as a black

circle.

FIG. 8. The thick black line is the shifted Maxwellian given by Eq. (5b).

Thin lines correspond to sections of the ion distribution function (h¼180,

v

h

¼0) at x

pe

t¼7000.

073504-6 G. S

anchez-Arriaga and D. Pastor-Moreno Phys. Plasmas 21, 073504 (2014)

2.138.102.237 On: Wed, 23 Jul 2014 11:23:12

Probe-to-plasma relative motion also produces a loss of

regularity in electrons trajectories. In non-ﬂowing plasmas

and once it reached equilibrium and electric potential is

time-independent, both angular momentum and energy are

conserved. Systems 9 and 10 are integrable, and particle dy-

namics is regular. However, in ﬂowing plasma, systems 9

and 10 are (in principle) non-integrable. This issue was

investigated by plotting Poincar

e sections of trapped elec-

trons orbiting around the probe and at the front; each time an

electron trajectory intersected the hypersurface h¼p,we

plotted v

r

versus r. The results are shown in panels (a) and

(b) of Fig. 11, where colors were used to denote trajectory

started with different initial conditions. Trapped electrons

trajectories are conﬁned to certain bounded region of the

phase space and exhibit a complex behavior. In the case of

trapped electrons at the probe front side [panel (b)], the tra-

jectories seem to ﬁll tori in phase space.

VII. CONCLUSIONS

This paper presented numerical simulations of current

collection by a positively polarized inﬁnite cylinder in ﬂow-

ing plasmas, a problem with applications to Langmuir probes

and electrodynamic tethers. The non-stationary Vlasov-

Poisson solver named Kilaps was extended to incorporate the

probe-to-plasma relative velocity. Since there is no longer az-

imuthal symmetry in ﬂowing plasmas, the problem is very

demanding from a computational point of view, and parallel

computing is required. Both normalized probe bias /pand

probe velocity Swere varied smoothly and then kept constant

during enough time to let the plasma reach an equilibrium

state. Repeating this strategy for several Svalues allowed to

make an analysis of plasma properties as a function of S.

The discretization algorithm implemented by Kilaps, an

Eulerian Vlasov code, provided results free of numerical

noise. However, due to the high computational cost, the

dimensionless parameters used in the simulations are far

from real space tether missions operating within the meso-

thermal regime. Since the code is parallelized, more realistic

dimensionless values will be achieved in the future thanks to

the development of computer clusters. This drawback did

not prevent to get insight into the paradox raised in Ref. 15.

The numerical simulations conﬁrmed that particle densities

exceed the background density at the probe front side (see

Fig. 3), where electron trapping also occurs. This population

is crucial to recover quasineutrality at the probe front and

should not be neglected. A second population of electrons or-

biting around the probe was also detected. Unlike the no-

ﬂowing plasma case, trapped particles trajectories do not

seem to be regular and exhibit a complex behavior. We

remark that neither particle energy nor angular momentum

are conserved because the potential depends (periodically)

on time and on the azimuthal coordinate. Another interesting

footprint of the probe motion is the presence of a ﬁlament in

the ion distribution function at the probe front side. This ﬁla-

ment corresponds to rammed ions that were reﬂected by the

probe.

Besides electron trapping, most of the feature of the sim-

ulations are in agreement with results from steady-state solv-

ers. Electron density proﬁles (Figs. 2–4) are similar to the

one found in Sec. 4.6.1 of Ref. 22; in the ram side, electron

density minimum is less pronounced as Sincreases, but it

becomes deeper in the wake side. A second common feature

with Ref. 22 is the potential depression (below 0) on the

wake side (see right inset in Fig. 9). The main difference

between Kilaps simulations and the one reported in Ref. 22

is the collected current. It is slightly above I

OML

in our simu-

lations but decays with Sin Ref. 22. Experimental measure-

ments indicated that plasma ﬂow leads to a current

enhancement over that predicted by the OML theory.

22

Particle trapping occurs in the simulations during the

transient. It suggests that the ﬁnal state of the plasma

depends on the speciﬁc S(t) and U

p

(t) temporal proﬁles. This

could be demonstrated by running two simulations with dif-

ferent S(t) and /pðtÞhistories but equal /pðt!1Þ and

S(t!1) values. Such a calculation is beyond the scope of

this paper due to computational limitations. However, the ex-

istence of multiple solutions for exactly de same value of the

parameters is not new. For instance, the solution found in

Ref. 21, which included particle trapping, was different to

the one found in Ref. 5, and they both correspond to the

same parameter values.

The parallelization scheme and the use of special sub-

routines to handle sparse matrices were essential to investi-

gate Langmuir probes with Eulerian Vlasov codes. Any

additional physical effect that would keep the problem

within a 2-dimensional geometry, could be incorporated to

the algorithm without increasing the computational cost

noticeably. Some examples are the analysis of emissive

Langmuir probes and the presence of an external magnetic

ﬁeld component along the probe axis. Extensions of Kilaps

in these directions are currently in progress.

ACKNOWLEDGMENTS

The authors wish to acknowledge very helpful

comments by the referee. D. Pastor-Moreno was supported

by a grant from the European Commission, FP-7 Space

FIG. 11. Poincare sections (with the hypersurface h¼p) of trapped elec-

trons. Top (bottom) panel corresponds to electron orbiting around the probe

(at the front).

073504-7 G. S

anchez-Arriaga and D. Pastor-Moreno Phys. Plasmas 21, 073504 (2014)

2.138.102.237 On: Wed, 23 Jul 2014 11:23:12

project BETs (No. 262972). The authors thankfully

acknowledge the computer resources, technical expertise,

and assistance provided by the Supercomputing and

Visualization Center of Madrid (CeSViMa).

1

H. M. Mott-Smith and I. Langmuir, “The theory of collectors in gaseous

discharges,” Phys. Rev. 28, 727–763 (1926).

2

J. R. Sanmartin, M. Martinez-Sanchez, and E. Ahedo, “Bare wire anodes

for electrodynamic tethers,” J. Propul. Power 9, 353–360 (1993).

3

I. B. Bernstein and I. N. Rabinowitz, “Theory of electrostatic probes in a

low-density plasma,” Phys. Fluids 2, 112–121 (1959).

4

S. H. Lam, “Uniﬁed theory for the Langmuir probe in a collisionless

plasma,” Phys. Fluids 8, 73–87 (1965).

5

J. G. Laframboise, “Theory of spherical and cylindrical Langmuir probes

in a collisionless, Maxwellian plasma at rest,” Ph.D. dissertation

(University of Toronto, Canada, 1966).

6

J. R. Sanmart

ın and R. D. Estes, “The orbital-motion-limited regime of cy-

lindrical Langmuir probes,” Phys. Plasmas 6, 395–405 (1999).

7

G. S

anchez-Arriaga and J. R. Sanmart

ın, “Relativistic current collection

by a cylindrical Langmuir probe,” Phys. Plasmas 19, 063506 (2012).

8

M. Kanal, “Theory of current collection of moving cylindrical probes,”

J. Appl. Phys. 35, 1697–1703 (1964).

9

W. R. Hoegy and L. E. Wharton, “Current to a moving cylindrical electro-

static probe,” J. Appl. Phys. 44(12), 5365 (1973).

10

R. Godard and J. G. Laframboise, “Total current to cylindrical collectors

in collisionless plasma ﬂow,” Planet. Space Sci. 31, 275–283 (1983).

11

G. Z. Xu, “The interaction of a moving spacecraft with the ionosphere:

Current collection and wake structure,” Ph.D. dissertation (York

University, 1992).

12

J. C. McMahon, “The interaction of inﬁnite and ﬁnite cylindrical probes

with a drifting collisionless Maxwellian plasma,” Ph.D. dissertation (York

University, Canada, 2000).

13

T. Onishi, “Numerical study of current collection by an orbiting bare teth-

er,” Ph.D. dissertation (MIT, Cambridge, 2002).

14

E. Choiniere and B. E. Gilchrist, “Self-consistent 2-D kinetic simulations

of high-voltage plasma sheaths surrounding ion-attracting conductive cyl-

inders in ﬂowing plasmas,” IEEE Trans. Plasma Sci. 35, 7–22 (2007).

15

J. R. Sanmart

ın, “Active charging control and tethers,” in CNES-Space

Technology Course: Prevention of Risks Related to Spacecraft Charging,

edited by J. P. Catani (Cepadus, Toulouse, France, 2002), pp. 515–533.

16

J. G. Laframboise and L. W. Parker, “Probe design for orbit-limited cur-

rent collection,” Phys. Fluids 16, 629–636 (1973).

17

J.-M. Siguier, P. Sarrailh, J.-F. Roussel, V. Inguimbert, G. Murat, and J.

SanMartin, “Drifting plasma collection by a positive biased tether wire in

LEO-like plasma conditions: Current measurement and plasma diag-

nostic,” IEEE Trans. Plasma Sci. 41, 3380–3386 (2013).

18

A. V. Gurevich, “Distribution of captured particles in a potential well in

the absence of collisions,” Sov. J. Exp. Theor. Phys. 26, 575 (1968).

19

A. C. Calder and J. G. Laframboise, “Time-dependent sheath response to

abrupt electrode voltage changes,” Phys. Fluids B 2, 655–666 (1990).

20

F. Iza and J. K. Lee, “Particle-in-cell simulations of planar and cylindrical

Langmuir probes: Floating potential and ion saturation current,” J. Vac.

Sci. Technol. A 24, 1366–1372 (2006).

21

G. S

anchez-Arriaga, “A direct Vlasov code to study the non-stationary

current collection by a cylindrical Langmuir probe,” Phys. Plasmas 20(1),

013504 (2013).

22

E. Choiniere, “Theory and experimental evaluation of a consistent steady-

state kinetic model for two-dimensional conductive structures in iono-

spheric plasmas with application to bare electrodynamic tethers in space,”

Ph.D. dissertation (University of Michigan, 2004).

073504-8 G. S

anchez-Arriaga and D. Pastor-Moreno Phys. Plasmas 21, 073504 (2014)

2.138.102.237 On: Wed, 23 Jul 2014 11:23:12