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Kinetic-simulation study of propagation of Langmuir-like ionic waves in dusty

plasma

S. M. Hosseini Jenab∗ and F. Spanier †

Centre for Space Research, North-West University, Potchefstroom Campus,

Private Bag X6001, Potchefstroom 2520, South Africa

(Dated: July 19, 2018)

The propagation of ionic perturbations in a dusty plasma is considered through a three-specieskinetic simulation approach, in which the temporal evolution of all three elements i.e. electrons, ionsand dust particles are followed based on the Vlasov equation coupled with the Poisson equation. Twocases are focused upon: firstly a fully electron depleted dusty plasma, i.e., a plasma consisting of ionsand dust-particles. The second case includes dusty plasmas with large electron-to-ion temperatureratios. The main features of the ionic waves in these two settings including the dispersion relationand the Landau damping rate are studied. It is shown that the dispersion relation of the ionicwaves perfectly matches the dispersion relation of Langmuir waves and hence are called Langmuir-like ionic waves and can be considered as Langmuir-like ionic waves. These waves can be theoreticallypredicted by the dispersion relation of the dust-ion-acoustic waves. The transition of ionic wavesfrom dust-ion-acoustic to Langmuir-like waves are shown to be sharp/smooth in first/second case.The Landau damping rates based on simulation results are presented and compared with theoreticalpredictions wherever possible.

I. INTRODUCTION

Langmuir waves have been the first kind of wavesdiscovered in plasma physics by Langmuir1, in whichthe electrons’ oscillations propagate through the

plasma media with a frequency ωpe =√

ne0e2

meǫ0≪ ω, is

the plasma frequency for electrons. It also possessesa dispersion relation in a restricted approximation ofsmall wavenumber kλDe ≪ 1 (electron Deybe length

λDe =√

ǫ0KBTe

ne0e2) known as the Bohm-Gross equation:

ω2 = ω2pe + 3

KBTe

me

k2, (1)

in which KB, Te and me are Boltzman constant, theelectron temperature and its mass respectively. How-ever more precise dispersion relations were presentedover the years with less restriction on kλDe ≪ 1 andhere the empirical equation suggested by Swanson2

(see Eq.10) is considered as well. The dispersion re-lation is achieved by considering the fact that ionsdue to their large mass react too slowly to fast oscil-lations of the electric field and thus can be regardedas stationary and immobile. Ions play the importantrole of a positive background which maintains the lo-cal quasineutrality condition throughout the plasmamedium. Electrons on the other hand are respond-ing to the electric field oscillations providing the in-ertia necessary for wave propagation by their masses.So the mechanism of Langmuir waves consists of res-onating charged particles, neutralized by oppositelycharged particles in the background, oscillating and

∗Email: Mehdi.Jenab@nwu.ac.za†Email: Felix@fspanier.de

resonating to electric field oscillation while the back-ground charged particles stay passive to the electricfield oscillation -in case of an ion-electron plasma, dueto large mass ratio of background particles to resonat-ing particles. Such a mechanism introduces a clearcut in dispersion relation called the cut-off frequency,waves with the frequency below the plasma frequencyof the resonating charged particles can’t prorogate inthe plasma media, e.g. in an ion-electron plasma,waves with angular frequency ω < ωpe are dampedheavily.As for ionic waves, the fundamental waves include

ion acoustic waves (IAWs)3 which consist of electro-static waves propagating with the frequency ωpi ≪ω ≪ ωpe in which, contrary to the Langmuir wavemechanism, both electrons and ions respond to theelectric field oscillation. This kind of wave can befound in dusty plasmas as well i.e. dust-ion-acousticwaves (DIAWs) kvTd ≪ kvTi ≪ ω ≪ kvTe althoughwith a modified version of the ion-acoustic dispersionrelation of an ion-electron plasma4:

ω2 = k2(ni0

ne0

kBTe

mi

1

1 + k2λ2De

+ 3kBTi

mi

). (2)

Dusty plasmas as an active frontier of the plasmaresearch5–8 consist of three constitutes including elec-trons, ions and dust particles. The dust particles in-clude heavy particles appearing in different plasmasranging from laboratory to astrophysics and comingfrom widely different sources such as scrap-off parti-cles of the wall in a tokamak or big molecules in theatmosphere. Dust particles due to their large massin comparison to both electrons and ions don’t ac-tively participate in DIA waves propagation i.e theycan’t resonate with the electric field oscillation. Dustparticles absorb mostly electrons and play the role ofneutralizing background charged particles and modifythe dispersion relation. In fact, due to the collisional

2

process of charging, the dusty plasma should be con-sidered as thermodynamically open systems9–11.

In extreme cases dust particles actually absorb al-most all electrons and a kind of dust-ion plasmaemerges. In this case the electrons are so scarce andfully absorbed by dust particles that they can’t playany effective role in the dynamics of the plasma andhence in the wave propagation. It is theoretically aninteresting question whether the ionic waves can prop-agate in this situation and, if so, what dynamics theyfollow.

These kind of plasma has been observed in bothlaboratory12 and astrophysical plasmas. Severalplasma environments in the solar system, such as Ti-tan’s deep ionosphere13–17, the D-region of Earthsionosphere18,19, the E-ring plasma disk of Saturn? ? ,and the plume of Enceladus20,21. It is also sug-gested through a detailed modeling t hat strong elec-tron depletion can be found in comet’s environmentsuch as 67P/Churyumov-Gerasimenko, newly visitedby Rosetta22. Evidences from different flybys (espe-cially T70 (2004) and other 47 deep flybies -2004 un-til 2012-) of Cassini have monitored the deep iono-sphere of Titan. These astronomical observations bythe LP (Langmuir probe) and Cassini-RPWS (Radioand Plasma Wave Science)16 have shown deep de-cline in electron number density and it is concludedthat dust/aerosol-ion plasma15 exists especially in thenight side of the Titan. In case of Saturn’s rings, theCassini measurements during its mission (since 2004)have shown the existence of a decline in electron num-ber density in E-ring and the plume of Enceladus23

which is an active part of the ionosphere of the thismoon. These two environments are related since Ence-ladus plume is the main source of ionized gas for E-ring24.

However, even in the existence of electrons, whenthe electron to ion temperature ratio increases to alarge value, then the electrons’ movement is so fastthat they can be considered as the neutralizing back-ground and in this case again the same pattern of theionic propagation appears with the ions oscillating andthe other constituents, electrons and dust particles,providing the neutralizing background.

Langmuir waves along with ion acoustic and dustion acoustic waves are all subject to a well-known ki-netic effect named after its discoverer, Lev Landau,as the Landau damping25 which is basically damp-ing of electric energy and converting it into kineticenergy without collisions involved. After its purelytheoretical discovery, it has been one of the main fo-cuses in plasma physics, although its physical natureand even mathematical proof still remains challengingand controversial26. However the Landau dampinghas been proven repeatedly in countless experimentaland computational efforts. As main characteristics ofwaves in plasmas, the Landau damping along with dis-persion relation are focused upon in this paper, anddue to the pure kinetic origin of the Landau damping,

the kinetic simulation method is employed to studythese waves here.The kinetic simulation is considered as the most

comprehensive method to study many-body systemsand is based on following the temporal evolution ofdistribution functions, which basically is the densityof particles in the phase space. The main theory gov-erning the dynamics of distribution function is basedon Liouville’s theorem and the consistency of phasespace, which dictates the trajectories of the particlesremain unique for each of them. By ignoring colli-sions, the Vlasov equation can be derived form theformer-mentioned principle theory. Here by followingthe unique trajectories of the particles in the phasespace based on Leap-Frog scheme a high-precisionmethod for the kinetic simulation is constructed. Inthis method by using a randomized sampling in theinitial step the recurrence effect is removed from thesimulation results27. Here all the three elements ofdusty plasma are treated based on Vlasov equationsand hence a fully kinetic technique is adopted.

II. MODEL AND NUMERICAL PROCEDURE

To follow the trajectories of the particles of differentspecies in a collisionless plasma, the Vlasov equationas partial differential equations for each of them

∂fs(x, v, t)

∂t+ v

∂fs(x, v, t)

∂x

+qsE(x, t)

ms

∂fs(x, v, t)

∂v= 0 . (3)

has been broken into to two ordinary differential equa-tion using the method of characteristics as follows:

dxs(t)

dt= vs

dvs(t)

dt=

qsE(x, t)

ms

in which s = e, i, d stands for electrons, ions anddust particles respectively. qs and ms are represent-ing charge and mass for each of the species. It isworth mentioning that since the trajectories are be-ing followed in the phase space hence velocity andconfiguration space are independent from each other.The above two equations are solved numerically basedon a modified version of the Leap-Frog scheme28. Tofollow the temporal evolution of the electric field, thePoisson equation is employed

∂2φ(x, t)

∂x2= −

1

ǫ0

∑

s=e,i,d

ρs . (4)

Finally the set of aforementioned equations are closed

3

via density integrals over the velocity direction

ρs(x, t) = qsns0

∫fs(x, v, t)dv. (5)

Simulations are carried out in a so called 1D1Vphase space which indicates that there are one spa-tial direction x and one velocity direction vx. Atthe initial step, the distribution function, which is as-sumed to be a Maxwellian distribution function, issampled randomly by four points in each grid cell i.e.for each phase points which has random x and vx avalue of f will be associated based on the distributionfunction. Then at each time step, the distributionfunction over grid points are constructed using the fvalues of the phase points and interpolating them tothe grid points. Knowing the distribution functionallows the code to calculate densities -by integratingthe distribution function over velocity direction- andultimately the electric field over the spatial directionusing density integrals and the Poisson equation re-spectively. In return the electric field is fed into theVlasov equation and the new coordinates (x, vx) ofeach phase point for the next time step are calcu-lated. In this method the value f , the phase pointdistribution function, remains untouched throughoutthe simulation which avoid some numerical errors aris-ing from constructing them.Since the focus here is on ionic propagation, for

computational reasons all the parameters are normal-ized to ionic parameters (the normalized quantitiesare marked by a acute accent) such as follows: Spaceand time are scaled by λDi and ω−1

pi respectively,

where ωpi =√ni0e2/(miǫ0) is the ion plasma fre-

quency and λDi =√ǫ0KBTi/(ni0e2) is the ion Debye

length. The velocity variable v has been scaled by theion thermal speed vthi

=√KBTi/mi, while the elec-

tric field and the electric potential have been scaledby KBTi/(eλDi) and KBTi/e, respectively (here KB

is Boltzmann’s constant). The densities of the threespecies are normalized by ni0

29. The quasineutralitycondition connects the three zeroth-order densities:

σ = 1− δ . (6)

in which σ = ne0/ni0 , δ = Zdnd0/ni0 are the nor-malized zero-order charge densities of electrons anddust particles. In order to excite an ionic propaga-tion, the small periodic perturbation is imposed oninitial distribution function i.e. Maxwellian distribu-tion function (fm) of ions.

fi(x, vx, t) = fm[1 + α cos

(2π

λx

)]. (7)

Note that quantities normalized to ionic scale are

marked by acute accent ω(= ω/ωpi

)and k

(= kλDi

),

however for the sake of comparison some quantitiesscaled by electron parameters will be mentioned and

they are marked by tilde ω(

= ω/ωpe

)and k

(=

kλDe

). Worth mentioning that the parameters with-

out acute accent or tilde are parameters without anynormalization.The normalized Bohm-Gross dispersion relation of

Langmuir waves (normalized to electron parameters,ω and k by ωpe and 1/λDe respectively

ω2 = 1 + 3k2, (8)

and dust ion acoustic waves (normalized to ion pa-rameters)

ω2 = k2( θ

1− δ + θk2+ 3

), (9)

are presented for the sake of comparison.Main parameters used in this simulations are as

follows: md/mi = 105, mi/me = 1836, Ti/Td =400, Zd = 1000. Other parameters, like δ, L = λ(= the length of the simulation box), α and Te/Ti

(hereafter θ) are varied for different sets of simula-tions and will be addressed where appropriate. Thegrid applied for phase space discretization (Mx pointsin x-direction and Mvx points in the x-component ofthe velocity) is set to (Mx,Mvx) = (25, 3000). As aconsequence, the phase space is covered by 7.5 × 104

cells, and in each of the cells, at the initial step 4phase points are scattered randomly inside the cell.Consequently, the distribution function of each of theplasma elements is sampled by 3 × 105 points in thephase space. The number of phase points during thetemporal evolution should stay unchanged since theperiodic boundary condition is adopted into the code.

III. RESULTS AND DISCUSSION

Firstly, we employed our numerical simulation ap-proach to study Langmuir waves in an ion-electronplasma in order to use the outcomes as a benchmarkfor our results related to the ionic perturbation. Herea small periodic perturbation is imposed on the initialdistribution function of electrons. The results of sim-ulation are shown in Fig.1, here the normalization iscarried out based on electron parameters. In Fig.1 twoanalytical dispersion relations are sketched, first theBohm-Gross dispersion relation (see Eq.8). The otherone is an empirical formula proposed by Swanson2:

ω =[1 +

1.37k2 + 10.4k4

1 + 11.1k3

](10)

which is more accurate than the Bohm-Gross disper-

sion relation up to 0.2 percent for k < 0.6. As it isshown in the Fig.1 the simulation results match both

Eq.10 and Eq.8 for k < 1.0. However for 1.0 ≤ k < 2.5a divergence occurs between the two analytical predic-tion, and the Swanson formula (Eq.10) claimed to bemore precise, and expectedly the Bohm-Gross disper-sion relation fails since it is derived for a restricted

4

FIG. 1: Comparison between analytical and simulationresults -black solid curve- of dispersion relation of Lang-muir waves as the benchmark. The normalization is basedon electron parameters and the wavenumber axis is de-picted in logarithmic scale. For the analytical results thediagrams of two equations are shown, The Bohm-Grossdispersion relation -blue dotted curve- and an empiricalformula suggested by Swanson -red dashed curve.

FIG. 2: Normalized angular frequency is shown versusnormalized wavenumber -in logarithmic scale- for δ = 1for ionic propagation. The normalization is based on ionparameters and the trends of curves can be compared toFig. 1 which is for Langmuir waves. The simulation re-sults -black solid curve- are sketched versus two analyticalrelation i.e. the Bohm-Groass equation -blue dotted curve-and empirical formula suggested by Swanson -red dashedcurve.

approximation of k ≪ 1. As it is shown in Fig.1, oursimulation results agrees with the Swanson formula

(Eq.10) for the regime of 1.0 ≤ k < 2.5 perfectly.As the first step in studying the ionic propagation,

we focus on the extremely electron depleted case inwhich electrons are mostly absorbed by dust parti-cles so that the plasma actually consists of ions anddust particles. The results of this case δ = 1.0 is de-

FIG. 3: Normalized Landau damping rate is sketched forfully electron depleted dusty plasma δ = 1.0 -black solidline- and compared versus two analytical prediction onefor small wavenumbers (k ≪ 1.0) -dashed red line- and for

large wavenumbers (k ≫ 1.0) -dotted blue line-.

picted in Fig.2 (the normalization is based on ionicparameters and the results can be compared by Fig.1in terms of the trend but not in the scale since the scaleof two figures are different, electrons/ions parametersfor Fig.1/Fig.2). Therefore in the case of completeelectron-depleted dusty plasma i.e ion and dust par-ticle plasma, Langmuir-like ionic waves arise whichshow the same dispersion relation as usual Langmuirwaves i.e electron Langmuir waves and attributes likethe cut-off frequency ,for waves with the frequencysmaller than the ion plasma frequency ω ≪ ωpi,clearly manifests in the dispersion relation of thiswave. The transition from dust-ion-acoustic waveswith no cut-off frequency to Langmuir-like ionic waveswith cut-off is quite sharp, although it depends onthe electron-to-ion temperature ratio as well (cf. nextsection). Theoretically this can be justified by settingδ = 1 in Eq.9 which results in the Bohm-Gross disper-sion relation and predicts the existence of Langmuir-like ionic waves.The Landau damping rate as one the main features

of any waves including the Langmuir-like ionic wavesare presented in Fig.3. Simulation results are com-pared to two different analytical predictions30, first

for k ≪ 1:

γ =

√π

8

1

k3exp

(−1

2k2− 1.5

), (11)

and secondly for k ≫ 1:

γ = k

√2 ln

( 1√2π

k2). (12)

The simulation results display a good agreement withboth the analytical predictions in their limit of valid-

5

FIG. 4: Dispersion relation of ionic propagation issketched for different electron to ion temperature ratioθ = 5, 10, 50, 500 and 5000. As θ increases the dispersionrelation converge to Langmuir waves dispersion relationwhich approaches zero as wavenumber drops.

ity, for first one not beyond k ≫ 0.5 and for the second

one for above k ≫ 2.0.

In the case of dusty plasma with all three ele-ments namely electrons, ions and dust particles, set-ting δ = 0.5 as an example to start with, we focuson the case of large electron to ion temperature ra-tio θ = Te

Ti

≫ 1. Fig.4 presents the effect of theion to electron temperature ratio on the behavior ofthe dispersion relation compared with the Langmuirwaves dispersion relation in the case of an ionic prop-agation. As the temperature ratio grows the moresimulation results resemble the Langmuir waves dis-persion relation which indicates the existence of theLangmuir-like ionic waves discussed above. However,in these circumstances despite the presence of elec-trons they don’t participate in the ionic propagationunlike the ion acoustic or dust ion acoustic waves.Here the collective movement of electrons can’t fol-low the collective displacement of ions since electronsare moving too fast due to the sharp temperature ratiobetween them. Fig.5 presents the temporal evolutionof number densities N(x, t) =

∫f dv of electrons and

ions in two different set of parameters, one for dust-ion-acoustic waves in which L = 11.34, θ = 5 andthe other one for the Langmuir-like ionic waves with

L = λ = 11.34, θ = 5000, L and λ are the simulationbox length and wavelength respectively. Fig.5 showshow differently electrons reacts in these two cases tothe ionic propagation, in case of dust-ion acousticwaves, electrons follow the movement of ions contraryto the case of Langmuir-like ionic waves in which elec-trons remain passive to the wave propagation and onlyprovide charged background like dust particles. As isthe case of complete electron depleted dusty plasma,the existence of Langmuir-like ionic waves can be pre-dicted from Eq.9. By setting θ ≫ 1 in Eq.9 the Bohm-

Time

Num

ber

Den

sity

0 5 10 15 20 250.99

0.995

1

1.005

1.01

1.015

IonsElectrons

(a)

Time

Num

ber

Den

sity

0 5 10 15 20 250.99

0.995

1

1.005

1.01

1.015

IonsElectrons

(b)

FIG. 5: The number density of electrons and ions areshown for a) the case of dust ion acoustic waves with pa-

rameters L = 11.34, θ = 5 and b) Langmuir-like ionic

propagation with parameters L = 11.34, θ = 5000 . Inlater case the electrons don’t participate in the wave prop-agation and their number density don’t oscillate with elec-tric field.

Gross dispersion relation Eq.8 can be achieved.Furthermore, another parameter which plays an im-

portant role is the wavenumber. Fig.6 shows the effectof the wavenumber on the dispersion relation and itsconnection with temperature ratio of electrons overions. It is worth mentioning that the conservation oftotal energy and entropy and total numbers of par-ticles of each species have been checked through outeach simulation run and the deviation has been keptunder 0.001 (cf Fig.7, Fig.8)Fig.10 presents the relative deviation of simulation

results from Langmuir waves dispersion relation

=ωA − ωS

ωA

(13)

in which ωA and ωS presents the wave frequency ofanalytical predictions and simulation results respec-tively. In Fig.10 areas with less than 5 percent de-viation are shown in black. A clear transition line

6

(�)

FIG. 6: Dispersion relation of ionic propagation -normalized frequency over normalized wavenumber- isshown while electron to ion temperature ratio acts as thethird parameter. Both normalized wavenumber and elec-tron to ion temperature ratio θ are represented in loga-rithmic scale.

Time

Rel

ativ

eT

ota

lEne

rgy

0 20 40 60 80 100-4.0E-05

-2.0E-05

0.0E+00

2.0E-05

4.0E-05

FIG. 7: The conservation of total energy has been testedthroughout simulation as validity criteria of simulation re-

sults. Relative total energy ε(t) = ε(0)−ε(t)ε(0)

of a few sim-

ulations is sketched versus normalized time and the de-viation of the initial value of the total energy is under4× 10−5.

can be identified in Fig.10 which depends on bothwavenumber and temperature ratio of electrons overions as it can be expected from Eq.9. This suggestswhat combination of two parameters namely δ and θcan produce Langmuir-like ionic waves and actuallywhen electrons role in ionic wave propagation can beignored. The Landau damping rate from simulationresults for different combination of δ and θ is shownin Fig.11.

However there is no sharp transition fromLangmuir-like ionic waves to dust ion acoustic wavesand the transition happens rather smoothly. There is

Time

Rel

ativ

eT

otal

Num

ber

0 20 40 60 80 100.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

FIG. 8: The conservation of total number of parti-cles Π(t) =

∫fs(x, v, t)dvdx has been considered as an

essential and minimum measurement of validity of thesimulation results. The relative deviation of total num-bers of electrons in comparison to the initial state (t=0):

Π(t) = Π(0)−Π(t)Π(0)

has been presented here for of a few simu-

lations. The deviation remains under 10−4 for whole timeof the simulations.

Time

Rel

ativ

eE

ntr

opy

0 20 40 60 80 100-1.0E-05

-5.0E-06

0.0E+00

5.0E-06

1.0E-05

FIG. 9: The conservation of entropy of electrons S(t) =∫fe(x, v, t)lnfe(x, v, t)dvdx has been checked throughout

the simulation as one of the yardstick of reliability of the

simulation results. The relative deviation S(t) = S(0)−S(t)S(0)

has been shown here versus time and it remains under10−5.

no point in which two modes occur at the same timeand temporal evolution of electric field gets distorted.The cut-off frequency effect doesn’t exist and for eachtemperature there exists a threshold in wavenumberbelow which the dispersion relation goes below ω = 1contrary to the case of fully electron depleted dustyplasma in which the cut-off frequency exists.

7

FIG. 10: The relative deviation of wave frequency ω ofsimulation results from analytical prediction of Langmuirwaves are sketched. The area with less than 5 percentdifference are shown in black. A clear transition bordercan be indemnified. As wavenumber and electron to iontemperature ratio grows, the dispersion relation from thesimulation converges more and more into Langmuir wavesdispersion relation.

(θ)

FIG. 11: Normalized Landau damping rates γ is sketchedfor different values of wavenumber k and θ electron toion temperature ratio -both in logarithmic scale - in threedimensional diagram.

IV. CONCLUSIONS

Propagation of ionic waves in a dusty plasma hasbeen studied for two cases i.e fully electron depleted

dusty plasma and in the case of high temperatureratios of electrons over ions via fully kinetic simula-tion technique. The simulation results are comparedto the Langmuir waves’ dispersion relation and theconditions under which the ionic propagation followsLangmuir dispersion relation is shown. The existenceof cut-off frequency is shown for fully electron de-pleted although for the second case the cut-off fre-quency doesn’t exist. The transition from non cut-offregime of ionic waves i.e. dust ion acoustic waves tocut-off frequency regime i.e. Langmuir-like waves arestudied for both cases and smooth transition is pre-sented for the second case. The transition for the firstcase is sharp although it depends on temperature ratioof electrons over ions as well. The Landau dampingrate as the other main feature of waves in plasmas isstudied for the vast range of variables and are com-

pared to the analytical predictions for both k ≪ 1 and

k ≫ 1.

In the studied circumstances, dust particles acquirethe role of heavy immobile particles delivering thequasineutrality condition by providing the negativecharge background for ions which are acting as the in-ertia provider for the propagation by responding theelectric field oscillations. For the second case the hotelectrons - in comparison to ions- play the role of op-positely charged particles in background and don’t ac-tively participate in the wave propagation. It is sug-gested that in these two cases ionic propagation fol-lows the dynamics similar to Langmuir waves and itcan be called Langmuir-like ionic waves.

Acknowledgments

M. Jenab is thankful to Prof. I. Kourakis and Prof.M. A. Hellberg for their fruitful discussions. This workis based upon research supported by the National Re-search Foundation through MWL grant no. 87616 andDepartment of Science and Technology. Any opin-ion,findings and conclusions or recommendations ex-pressed in this material are those of the authors andtherefore the NRF and DST do not accept any liabil-ity in regard thereto.

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