SP2018 00332

Collisional Effects in Non-stationary Plasma Expansions alongConvergent-Divergent Magnetic Nozzles

J. Zhou1,3, G. Sanchez-Arriaga1, E. Ahedo1, M. Martınez-Sanchez2 and J. Ramos1

1Equipo de Propulsion Espacial y Plasmas, Universidad Carlos III de Madrid, 28911 Leganes, Spain, Email3:jiewei.zhou@uc3m.es

2Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, 02139,Massachusetts, USA

KEYWORDS:

Magnetic nozzles, Boltzmann equation, weakly col-lisional plasmas

ABSTRACT:

The electron-electron collisional effect on the non-stationary expansion of a plasma in a convergent-divergent magnetic nozzle is studied. Under parax-ial and fully magnetized plasmas approximations,an Eulerian code has been adapted to solve Pois-son’s equation coupled with the kinetic transportequations for plasma species, i.e. a Vlasov equa-tion for singly-charged ions and a Boltzmann equa-tion with a Bhatnagar-Gross-Kook operator for elec-trons. The study is focused on weakly collisionalplasma plumes, which have a collisional time scalelarger than the transit time in the nozzle of typi-cal electrons. A kinetic analysis shows that phase-space regions of isolated, doubly-trapped electronsthat are nearly empty in the collisionless case areprogressively populated due to the electron-electroncollisions. Such a higher density of trapped elec-trons modifies the profile of the electrostatic poten-tial, which keeps almost unaltered the density of freeelectrons and decreases the density of the reflectedones. As compared with the collisionless case, thecollisions decrease the length of the downstreamsheath and the parallel electron temperature whileincreasing the normal one. Therefore the steadyplasma state is more isotropic. The simulationsshow that collisions erase the time history of thesystem and, unlike the collisionless case, the steadystate is unique.

1. INTRODUCTION

Many astrophysical scenarios, and also some en-gineering devices such as magnetic nozzles, in-volve the expansion of a plasma from a sourceinto the vacuum and in the presence of a magneticfield. In many of them, the collision frequency isso low as compared with other characteristic timesof the system, that the plasma expansion can betaken as collisionless. However, in the very longterm, the effect in the steady state of such rare

collisions is unclear. This is particularly interest-ing for the phase-space regions occupied by parti-cles that are doubly-trapped between two axial co-ordinates. In a collisionless plasma, these specialregions are populated during the transient phaseand plasma species injected from the source can-not access them in the steady state. Collisions,even if they are rare, can slowly populate them andchange the final steady state. Fluid models can-not describe correctly this trapping phenomenon inweakly collisional plasmas and a kinetic descriptionis necessary. Self-consistent solutions of the sta-tionary Vlasov-Poisson system has been obtainedfor a magnetic nozzle with slender geometry [1].Since a stationary model cannot provide rigorouslythe amount of particles trapped during the tran-sient, the authors added an heuristic population andfound that the steady state solution was sensitiveto the particular choice. In a recent work, the non-stationary expansion of a collisionsless plasma ina paraxial magnetic nozzle was studied numerically[2]. Vlasov simulations showed that both the elec-tron trapping and the steady state of the plasma de-pend on the transient phase. Therefore, it was con-cluded that the final state of a collisionless plasmaexpansion depends on the history of the system.In this work, we add collisional effects and study theplasma expansion though numerical simulations ofthe Boltzmann equation. The purpose of the study istwofold. In the first place, it reveals the impact of thecollisions on macroscopic quantities, such as elec-tric potential, plasma densities and temperatures,and also some kinetic features such as trapped par-ticle population. In addition, non-stationary Boltz-mann simulations can give insight into an interestingtheoretical question: do collisional effects make col-lapse to the same state all the steady states of col-lisionless plasma expansions in magnetic nozzles ?The answer to this question clarifies whether or not,in the long term, collisions erase completely the his-tory of the plasma expansion and make the steadystate independent on the transient phase.

2. MAGNETIC NOZZLE MODEL

Besides collisions, the physical configuration of theplasma expansion and the numerical method usedin this work are similar to the ones introduced in

1

Ref. [2]. For this reason, this section presents brieflythe most important features of the model and paysspecial attention to the differences introduced by thecollisions. Interested readers can find more detailsin Appendix A and Ref. [2].

2.1. Plasma Model

An infinitely large tank placed at z < z0 < 0 is filledwith an electron-ion plasma. At z > z0 there is vac-uum and an external magnetic field generated by acurrent loop of radius RL placed at z = 0. Such amagnetic field is stationary and non-uniform and itreaches its maximum value BT at z = 0 (the noz-zle throat T). Suddenly, the nozzle is started andthe plasma of the tank expands into the vacuum.There are several possibilities for the forward distri-bution functions of ions and electrons entering thenozzle. For simplicity, we will assume here that theyare semi-Maxwellian,

fα(t, z =z0, v‖ > 0, v⊥) = χα(t)N∗(mα

2πT ∗α

)3/2

exp

(−mαv

2

2T ∗α

), α = i, e (Eq. 1)

with subscript α = e and α = i referring to elec-trons and ions. Therefore, we assume a fully ion-ized plasma without neutrals. Parameters with a stardenote reference parameters such as particle den-sity N∗ and temperature T ∗α. However, these arenot the actual densities and temperatures at z = z0,because they also involve the backward distributionfunctions that are computed self-consistently by thecode. For convenience, we write the velocity asv =

√v2‖ + v2⊥ with v‖ and v⊥ the velocity compo-

nent parallel and normal to the magnetic field lines.Finally the paramenter χα(t) is χe = 1 for elec-trons and χi(t) is adjusted dynamically for ions toaccomplish quasineutrality at the entrance z = z0.We will follow the nomenclature of Ref. [2] and ax-ial coordinate, time, velocities, magnetic field, elec-trostatic potential, particle distribution functions, anddensities, are normalized according to z/λ∗De → z,tω∗pe → t, v‖,⊥/λ

∗Deω

∗pe → v‖,⊥, B/BT → B,

eφ/kBT∗e → φ, fα/N∗ (me/kBT

∗e )

3/2 → fα, whereλ∗De =

√ε0kBT ∗e /N

∗e2 is the Debye length, ω∗pe =√N∗e2/meε0 the electron plasma frequency, kB the

Boltzmann constant, me the electron mass, e the el-ementary charge, and ε0 the vacuum permitivity.The models assumes a slender and slowly-varyingmagnetic field and the analysis is focus at the cen-ter line of the nozzle. The normalized magnetic fieldas a function of the axial distance at the axis of thenozzle then reads

B(z) =r3L

(r2L + z2)3/2

1z, (Eq. 2)

where 1z is an unit vector along the z-axis andrL ≡ RL/λ

∗De. We will also assume that the mag-

netic field is very strong, i.e. the Larmor radius sat-isfies ρLα ≡ βv⊥/|Zα|B << rL, and the plasmaparticles are fully attached to the magnetic fieldlines. Therefore, the normalized magnetic momentµα = βαv

2⊥/2B is conserved. Here βα ≡ mα/me

and Zα are the mass and the charge number of theα-species. The effect of the induced magnetic fieldwill be also ignored. Under this set of hypotheses,the gyrocenter distribution functions fα, i.e.the distri-bution functions averaged over the fast gyrophase,are governed by the Boltzmann equations

∂fα∂t

+ v‖∂fα∂z

+ aα∂fα∂v‖

= Qα(fe, fi

), (Eq. 3)

where Qα is a collisional operator, and we intro-duced the normalized acceleration

aα = − 1

βα

(Zα

∂φ (t, z)

∂z+ µ

dB (z)

dz

)(Eq. 4)

with Zα the charge number. Such acceleration in-volves the electric field E = E‖B/B = −∂φ/∂z,which is given by the paraxial Poisson’s equation

B∂

∂z

(E‖

B

)=∑α=e,i

Zαnα (Eq. 5)

and the densities computed from the distributionfunctions as

nα(z) =

∫fαdv =

2πB

βα

∫ +∞

−∞

∫ +∞

0

fαdv‖dµ.

(Eq. 6)Particle collisions are modeled by the Bhatnagar-Gross-Kook operator [3]

QBGKα =1

τα

[fMα (nα, uα, Tα)− fα

](Eq. 7)

with τα a normalized collision time. Such a opera-tor, which is not rigorous but useful to gain insightinto the physics of the expansion, pushes the actualdistribution function fα towards the function

fMα (nα, uα, Tα) =nα

(βα

2πTα

)3/2

exp

[−βα(v‖ − uα

)22Tα

− Bµ

Tα

],

(Eq. 8)

which is a Maxwellian distribution function with thesame density nα, mean velocity along the mag-netic field line uα, and temperature Tα, as the ac-tual distributions fα. The density is given by Eq.6, and the mean velocity uα =

⟨v‖⟩α

and tempera-ture Tα = (T‖α+2T⊥α)/3 are found straightforwardlyafter introducing the average or mean value of any

2

quantity ψ as

〈ψ〉α =1

nα

∫ψfαdv

=2πB

βαnα

∫ +∞

−∞

∫ +∞

0

ψfαdv‖dµ, (Eq. 9)

with T‖α = βα

⟨c2‖α

⟩α

, c‖α = v‖ − uα the peculiar

velocity, and T⊥α = B 〈µ〉α.

2.2. Setup of the Simulations

The VLAsov Simulator for MAgnetic Nozzles (VLAS-MAN), explained in detail in Ref. [2], has been ex-tended to include collisions and solve (Eq. 3) and(Eq. 5). Unlike particle-in-cell simulators, which usemacroparticles, VLASMAN is an Eulerian code thatsolves the kinetic equation on a finite grid of phasespace, i.e. z0 ≤ z ≤ zM , −vαmax ≤ v‖ ≤ vαmax and0 ≤ µ ≤ µαmax. Consequently, it is computationallydemanding but provides a high precision in regionswhere the distribution function is small, as it natu-rally happens in a plasma expansion to the vacuum.The main differences of the extended version of thecode include the implementation of the collision op-erator (see details in Appendix A) and the µ-grid.Since the magnetic moment grid is constructed ina finite domain 0 ≤ µ ≤ µαmax, the truncationvalue µmax should be high enough to ensure thatthe distribution function is small at the boundaries.Otherwise, the boundary condition f(t, z, v‖, µ >µmax) = 0 imposed by the code would yield wrongresults. Our collision operator in (Eq. 7) involves theMaxwellian distribution function with the exponen-tial of −Bµ/Te [see (Eq. 8)]. For large z, the mag-netic field is very small [see (Eq. 2)] and a high valueof µ is needed to capture correctly the Maxwellian.For this reason, we set the computational box lim-its to z0 = −10 and zM = 100 (smaller than theone used in Ref. [2]), increased the value of µemaxup to 105, and implemented a nonuniform grid dis-tribution in µ. For electrons, a total of 40 grid pointswere distributed uniformly from 0 ≤ µ ≤ 5.6 and41 non-equispaced points were used in the interval5.6 < µ ≤ µemax. Ions are not problematic becausewe ignored their collisions. Such strategy providesenough resolution in the region of interest, wheremost of the trapping process occurs, and also cap-tures correctly the behavior of the Maxwellian distri-bution function appearing in the collision operator.In the simulations we set the following physical pa-rameters T ∗e = T ∗i , βi = 100, rL = 20, Zi = 1,.Such unphysical mass ratio helps us to save com-putational resources. The boundary conditions ofPoisson’s equation are φ(z0) = 0 and

φM (t) = φ0 + (φF − φ0)

{1

2+

1

πarctan [ω (t− t0)]

}(Eq. 10)

Two cases were considered: (i) constant potentialwith φ0 = φF = −2.75 and (ii) time-dependentpotential with φ0 = −5, φF = −2.75, t0 = 500,and ω = 5. We remark that they present differenttime histories but provide exactly the same bound-ary condition in the long term, i.e. a potential valueof −2.75 that gives zero net current in the stationarystate according to Ref. [2]. The number of points inz- and v‖-grids were Nz = 451 and Nv = 77, respec-tively and we used vemax = 5 and vimax = 0.5.

3. SIMULATION RESULTS

3.1. Macroscopic Quantities

We here present four simulations. Two of themare collisionless and have ΦM constant and vari-able given by (Eq. 10), respectively. The other twohave exactly the same conditions, except that thereare electron-electron collisions with a collision timeequal to τe = 1000. This collision time is larger thanthe typical residence time of reflected and free elec-trons in the simulation box. The simulations wererun up to a time large enough to guarantee that thesystem reached a stationary state. Such a condi-tion was monitored by plotting the z-profile of certainquantities that are z-independent in stationary con-ditions, like for instance the ratio j/B, and also thetemporal evolution of the density of trapped particlesand the temperature at several axial coordinates.

Top panel in Fig. 1 shows the electrostatic potentialat the end of the simulation versus the inverse of themagnetic field, which is proportional to the cross-sectional area of the nozzle. The two collision-less simulations with constant and time-dependentφM , which correspond to solid red and dashedgreen lines, converge to different steady states. ForφM given by law (Eq. 10) the amount of particlestrapped during the transient is larger and it yieldsa lower value of the electrostatic potential. Interest-ingly, the two simulations with collisions have exactlythe same steady state (dark dashed-dot and bluedotted lines in Fig. 1 overlap). The electron-electroncollisions yield a lower value of the electrostatic po-tential because more particles are trapped (see be-low) and they also shorten the downstream sheathas compared with the collisionless simulations (seecharge density in the bottom panel).

3

100 101 102-3

-2

-1

0

100 101 102

1/B

-0.2

0

0.2

0.4

0.6(b)

(a)

1/B

Figure 1: Electrostatic potential (top) and chargedensity (bottom) versus the inverse of the magneticfield.

The electron parallel and perpendicular tempera-tures versus 1/B for the four simulations are shownin the top and bottom panels of Fig. 2. As expected,the collisions make the distribution function moreisotropic because they decrease the parallel tem-perature and increase the perpendicular one. In-terestingly, the temperatures at zM are almost thesame with and without collisions but the z-profilesare totally different. In the collisionless case theparallel temperature is almost constant and dropsabruptly in the downstream sheath. For weakly colli-sional plasmas the parallel electron temperature de-creases almost exponentially with the inverse of themagnetic field (note the logarithmic scale in the x-axis).

100

101

102

1/B

0

0.5

1

100

101

102

1/B

0

0.5

1

Figure 2: Parallel (top) and perpendicular (bottom)electron temperatures versus the axial coordinate.See legend in Fig. 1.

There are three populations of electrons in the mag-netic nozzle: (i) free electrons that have enough en-ergy to leave the simulation box at zM , (ii) reflected

electrons that are injected and leave the simulationbox at z0, and (iii) doubly trapped electrons. The lat-ter population is injected at z0 and during the tran-sient or due to collisions gets trapped between twoaxial positions. These three electron population aredenoted by subscripts F , R, and T . As shown in Fig.3, collisions influence considerably the relative com-position of the plasma. The ratio of the free electrondensity to the total density is practically the same forthe four simulations (see middle panel). Therefore,it is not affected by the collisions and the time his-tory of φM . Such a density ratio, and the net current,seems to be controlled by just the final value of ΦM .The major effect of the collisions is to increase therelative composition of the trapped electron popu-lation and decrease the reflected one (see top andbottom panels). The understanding of this featureneeds a kinetic analysis made in the next section.

100

101

102

0

0.5

ne

T/n

e

100

101

102

0

0.5

1

ne

F/n

e

100

101

102

1/B

0

0.5

1

ne

R/n

e

Figure 3: Trapped (top), free (middle), and reflected(bottom) electron densities versus the axial coordi-nate. See legend in Fig. 1

Figure. 4 shows the evolution of the electron den-sities (top) and temperatures (bottom) at axial posi-tion z = 90. As mentioned at the beginning of thissection, the system reaches a steady state after thetransient phase. The figure also confirms that thesimulation time was long enough, and the resultspresented in this work correspond effectively withsteady states.

4

0 2000 4000 6000 8000 100000

0.5

1

0 2000 4000 6000 8000 100000

0.2

0.4

0.6(b)

(a)

Figure 4: Evolution of the electron densities (top)and temperature (bottom) at z = 90. Simulation withcollision and constant φM

3.2. Electron Trapping

The characteristic equations of (Eq. 3) are

dz

dt=v‖ (Eq. 11)

dv‖

dt=aα (Eq. 12)

dfαdt

=QBGKα (Eq. 13)

From (Eq. 11) and (Eq. 12) one finds that the energyE = βαv

2‖/2 + Zαφ+ µB is governed by

dE

dt= Zα

∂φ

∂t(Eq. 14)

Therefore, in the steady state with ∂φ/∂t = 0, theenergy is conserved (dE/dt = 0). For this rea-son, certain analyses are easier by changing fromv‖ to E by using v‖ = ±

√2 (E − Zαφ− µB) /βα,

and using the parametrization f±α (t, z, E, µ) insteadof fα(t, z, v‖, µ). The ± signs in the superscript ofthe distribution function indicate positive and nega-tive axial velocities. The positive and negative dis-tribution functions in the steady state should thensatisfy the equation

±

√2 (E − Zαφ− µB)

βα

∂f±α∂z

=f±Mα − f±α

τα(Eq. 15)

where f±Mα are the Maxwellian distribution functionwith positive and negative axial velocity in terms ofz, µ, and E and we used dE/dt = dµ/dt = 0. Ac-cording to (Eq. 15) and for a given energy level inthe steady state, f±α are z-independent piecewisefunctions in the collisionless case. For weakly col-lisional plasma (τα >> 1), f±α are almost constantwith z.The population of trapped particles in the steadystate can be identified by plotting the distribution

function in a z − µ plane for a given value of theenergy at the end of the simulations. We remarkthat, at a certain position and energy, the magneticmoment should be smaller than

µmax(z, E) =E − Zαφ(z)

B(z)(Eq. 16)

and a particle is doubly-trapped in the region zmin ≤z ≤ zmax if its magnetic moment µ satisfies µ =µmax(zmin, E) = µmax(zmax, E). The curve µmaxversus z can exhibit a minimum at a certain value ofthe magnetic moment, say µ∗.Figure 5 shows the sum of the electron distribu-tion functions f±e at the steady state in the z − µplane for a given energy level. For convenience,the µmax − z curve is also presented. Top panelscorrespond to collisionless simulations and energylevels E = 2.2 [panel (a)] and E = 2.4 [panel (b)].It is evident that there are trapped electrons, butthey just occupy a relatively narrow µ-band abovethe minimum of the µmax-curve. Therefore, most ofthe phase-space region that trapped electrons couldpotentially occupy is empty. As shown in panels (c)and (d), which present the same results but for asimulation with collisions, the electron-electron col-lisions change this picture considerably. We firstnote that the µmax-curves for a given energy levelare not the same, because the collisions changethe densities and, therefore, the electrostatic poten-tials in the steady state are different. Moreover, thecollisions fill the phase-space region with doubly-trapped electrons, thus explaining the higher densityof the trapped electron population presented in Sec.3.1.. The panels also corroborate a fact already an-ticipated by (Eq. 15): f± are z-independent and al-most z-independent in the collisionless and weaklycollisional cases, respectively.

Figure 5: Steady state electron distribution functionin the z − µ plane for a given energy. The top andthe low row correspond to collisionless and weaklycollisional plasmas, respectively, and left and rightcolumns to E = 2.2 and E = 2.4.

5

Figure 6 shows in detail the profiles of the forwardand backward distribution function of the electronsfor z = 50 and E = 2.4 in the collionless and weaklycollisional simulations [panels (a) and (b) respec-tively]. These two panels are obtained by makinga section at z = 50 of Fig. 5.b and 5.d. For conve-nience, we also plotted the value of µmax for z = 50,and the minimum of µmax curve for E = 2.4. Thedistribution functions vanish for µ > µmax and thetrapped electron region correspond to µ∗ ≤ µ ≤µmax. The panels include the forward and back-ward Maxwellian distribution functions with density,drift velocity, and temperature equal to the local val-ues of the distribution function obtained in the sim-ulation. These distributions f+Me and f−Me are ob-tained by writting (Eq. 8) as a function of the en-ergy (instead of v‖) and they are different becausethe local value of ue is not zero. In the collision-less simulation the distribution functions are far fromMaxwellian and they decreases abruptly to the rightof µ∗, i.e. only a narrow µ-range is filled with trappedelectrons [see panel (a)]. Even in the weakly colli-sional case the distribution functions do not followa Maxwellian, although they are much closer in thetrapped region. Clearly, the collisions increased theamount of trapped electrons.

0 1 2 3 4 5 6 70

2

4

6

810-3

(a)

0 1 2 3 40

2

4

6

810-3

(b)

Figure 6: Forward and backward electron distribu-tion functions versus mu for z = 50 and E = 2.4in the steady state. The top and the low panel cor-respond to collisionless and weakly collisional plas-mas, respectively.

Figure 7 shows the distribution function of the elec-trons in the v‖ − µ plane for z = 50 [panels (a) and(c)] and z = 75 [panels (b) and (d)]. Similarly toFig. 5, panels (a)-(b) and (c)-(d) correspond to colli-sionless and weakly collisional simulations, respec-tively. The presence of collisions yields a long tail ofthe distribution function for large µ values and nearlyzero axial velocity. Since B decreases with z, sucha tail is longer as we move from the throat of thenozzle to zM .

-4 -2 0 2 40

5

10

(a)

0

5

10

10-3

-4 -2 0 2 40

5

10

(b)

0

2

4

6

8

10-3

-4 -2 0 2 40

5

10

(c)

0

2

4

6

810-3

-4 -2 0 2 40

5

10

(d)

0

2

4

10-3

Figure 7: Steady state electron distribution functionin the v‖ − µ plane for a given axial coordinate. Thetop and the bottom row correspond to collisionlessand weakly collisional plasmas, respectively, andleft and right columns to z = 50 and z = 75, re-spectively.

4. CONCLUSIONS

The numerical simulations show that collisions, evenif they are rare, play a fundamental role in the steadystate of magnetized plasma expansions. The ax-ial profiles of the electrostatic potential and elec-tron temperature are modified considerably. For in-stance, collisions make the electron parallel temper-ature decay almost exponentially with the inverseof the magnetic field. Moreover, collisions keepinvariant the relative density of free electrons butmodify the relative amount of trapped and reflectedelectrons, respectively. The phase-space region ofdoubly-trapped electrons is progressively filled bythe collisions until a final and unique steady stateis reached. Therefore, unlike collisionless plas-mas, the simulations suggest that the steady stateis unique for weakly collisional plasmas. The equi-librium distribution function is far from Maxwellian,even though the Bhatnagar-Gross-Kook operator isproportional to the difference between the actualdistribution function and a Maxwellian with the samelocal macroscopic properties.

The conclusions of this work are based on a lim-ited number of simulations and may be consolidatedby a deeper analysis by varying important physi-cal and numerical parameters such as the collisionfrequency, the simulation box size, and the down-strean electrostatic potential. We also mention thatelectron-neutral collisions, ignored in this work, maybe dominant for not fully ionized plasma. More re-fined analysis will be presented in a forthcomingwork.

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ACKNOWLEDGMENTS

J.Z. was supported by Airbus DS (GrantCW240050). G.S-A was supported by the Min-isterio de Economıa y Competitividad of Spain(Grant RYC-2014-15357). E.A. was supportedby the MINOTOR project, that received fundingfrom the European Union’s Horizon 2020 researchand innovation programme, under grant agreement730028. J.R. and M.M-S stays at UC3M for this re-search were supported by a UC3M-Santander Chairof Excellence and by National R&D Plan (GrantESP2016-75887), respectively.

A NUMERICAL SCHEME

The simulations have been carried out with anupdated version of the code named VLASMAN(VLAsov Simulator for MAgnnetic Nozzle). A de-tailed description of this code, including mesh in realand velocity space and filtering algorithm to avoidfilamentation, can be found in Ref. [2]. We now ex-plain the numerical implementation of the electroncollisions, which is the novel feature of this work.For convenience, we write the Boltzmann equationas

∂fα∂t

= (S + F + C) fα (Eq. 17)

where the operators for streaming S, forces F ,and collisions C are S fα = −v‖∂fα/∂z, F fα =

−aα∂fα/∂v‖, and Cfα =(fMα − fα

)/τα. Our

numerical scheme implements a splitting method.Given the distribution functions at time t, thescheme computes it at t + ∆t by using the follow-ing formula [4]

fα (tm + ∆t) = C1/2S1/2FS1/2C1/2fα (tm)(Eq. 18)

where for brevity we omitted the dependence withz, v‖, and µ of the distribution functions, and C1/2,S1/2, and F , are operators that propagate the dis-tribution function, according to collisions, streamingand forces, for time steps equal to ∆t/2, ∆t/2, and∆t, respectively.The collisional part in the splitting method, ∂fα/∂t =Cfα, is discretized with a second-order accuracy in∆t Crank-Nicolson approach [5]

C1/2fα(tm, z, v‖) =fα(tm, z, v‖)+

1

2

∆t

τα + ∆t/2

[fMα − fα(tm, z, v‖

](Eq. 19)

where we omitted again the dependence with µ be-cause it enters as a parameter in the algorithm. Thedistribution function fMα is computed from Eq. 8and its arguments, ne, ue, and Te, are the macro-scopic values of fα(tm, z, v‖). For the inner part of

Eq. 18, i.e. streaming and force (S1/2FS1/2), weused the well-known second order scheme [6]

S1/2fα(tm, z, v‖) = fα(tm, z − v‖∆t/2, v‖) (Eq. 20)

F fα(tm, z, v‖) = fα(z, v‖ − aα∆t

)(Eq. 21)

At each time step, the electric field appearing inaα is found by solving Poisson’s equation with thedensities computed from the distribution functionS1/2C1/2f .

REFERENCES

[1] Martinez-Sanchez, M., Navarro-Cavalle, J., andAhedo, E., “Electron cooling and finite poten-tial drop in a magnetized plasma expansion,”Physics of Plasmas, Vol. 22, No. 5, 2015,pp. 053501.

[2] Sanchez-Arriaga, G., Zhou, J., Ahedo, E.,Martınez-Sanchez, M., and Ramos, J., “Kineticfeatures and non-stationary electron trappingin paraxial magnetic nozzles,” Plasma SourcesScience and Technology , Vol. 27, No. 3, 2018,pp. 035002.

[3] Bhatnagar, P. L., Gross, E. P., and Krook, M., “AModel for Collision Processes in Gases. I. SmallAmplitude Processes in Charged and NeutralOne-Component Systems,” Phys. Rev., Vol. 94,May 1954, pp. 511–525.

[4] Schiller, U. D., “A unified operator splittingapproach for multi-scale fluid–particle couplingin the lattice Boltzmann method,” ComputerPhysics Communications, Vol. 185, No. 10,2014, pp. 2586 – 2597.

[5] Dellar, P. J., “An interpretation and derivationof the lattice Boltzmann method using Strangsplitting,” Computers & Mathematics with Appli-cations, Vol. 65, No. 2, 2013, pp. 129 – 141,Special Issue on Mesoscopic Methods in En-gineering and Science (ICMMES-2010, Edmon-ton, Canada).

[6] Cheng, C. and Knorr, G., “The integration of thevlasov equation in configuration space,” Journalof Computational Physics, Vol. 22, No. 3, 1976,pp. 330 – 351.

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