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