Volume 126,number 4 PHYSICSLETFERSA 4 January1988

DIFFUSION OF ELECTRONS IN COHERENT LANGMUIR WAVE PACKETS

W. ROZMUS, J.C. SAMSONPhysicsDepartment,UniversityofAlberta,Edmonton,Alberta, CanadaT6G 2J1

and

A.A. OFFENBERGERElectricalEngineeringDepartment,UniversityofAlberta,Edmonton,Alberta, CanadaT6G2J1

Received3 August1987; revisedmanuscriptreceived8 October1987;acceptedfor publication6 November1987Communicatedby R.C.Davidson

A newdiffusioncoefficientfor electronsrepeatedlyinteractingwith coherentLangmuirwavepacketsis proposed.The modelaccountsfor regularislandsembeddedin a regionof connectedstochasticity,since theseislandshavea profoundeffecton thediffusionprocess.Thediffusioncoefficienthasa resonantformwhichhasbeenexplicitly calculatedincorporatingthefinite decaytimes of thecorrelationfunctions.Theoreticalpredictionscomparewell with numericalresultsderivedfromsolutionsof theequationof motion.

In this Letterwereporton numericalandanalyt-ical studiesof theinteractionbetweenelectronsand E(x, t) =E0 sech(Eox~ 1 1/2 Plocalized,cohetentLangmuirwave structures.The ‘~D (87tnk8T)existenceof suchnonlinear,soliton-likeobjectshas x cos(w I — kLx) , (I)beenwell establishedin theoryandexperiments(e.g.review [1]). In particular our work has been wherefi= 1/2~/~,kL is the electronplasmawavepromptedby recenttheoreticaldevelopmentsin the wave-number, w0= (4xe

2ne/me)112, and 2D =

strongLangmuir turbulencein which Doolenet al. (kB Te/4Ee2fle)~’2. ForkL = 0 expression(1) is asta-[2] proposedamechanismof localizationandcay- tionary solutionto theZakharovequations[1]. Theiton formation on preexistingion densitymodula- periodicity can be introducedinto eq. (1) by rep-tions. A coherentversion of this mechanismhas resentingthe field in termsof wave packetsof thefound application in the l-D models describing formlaser—plasmainteractionphysics [3,41.In this the-oryLangmuirwavesproducedby stimulatedRaman E(x, I) = ~ En cos(w

0t—k~x), (2)scattering(SRS)cannonresonantlyinteractwith ionwaves producedby simultaneousstimulatedBril- wherek~= kL + nkA, n= 0, ±1, ±2, ... andkA = 2x/lbum scattering(SBS) anddeveloplocalized,peri- definesperiodicity length 1. Eq. (2) hasthe form ofodic structuresin the fields of the electronplasma a Bboch function.Fig. I showsa particularchoiceofwaves, the wave packet for E~/8itnekB Te = 0.53, kA =2kL

Eachstructurein this periodicsequenceof local- =0.09k0.This choice couldbe relevantfor modelsizedfields canbe approximatedby thefollowing an- of SRSandSBSat ncr/4 (ncr is the critical densityatalytical expression which the laserfrequencyco0= w1,). In orderto ad-

equatelyrepresentthe localizedfields we haveused12 modeswith amplitudesE~.The insert in fig. 1

0375-9601/88/$03.50© ElsevierSciencePublishersB.V. 263(North-HollandPhysicsPublishingDivision)

Volume 126,number4 PHYSICSLETTERSA 4 January1988

E~

(87fl~lekBT)½

~!2n~0

° 1=70

-2 2

LI)0

-3 0 3

~-2.5 -3.2 -4.5 -7.5 -22.4 22.4 7.5 4.5 3.2 2.5 2.0 1.7

1k T \1/2Phase Velocity I~,—~.—)= 1

Fig. I. Thespectrumof theLangmuirwavepacket(2).The insertshowsthefield envelopeof a singleelement,overtheinterval 1, in theperiodicsequence.

showsthat the electric field amplitudeis well local- havea small numberof modes,quasi-lineartheoryized within theperiodicity length. providesonly a crude approximationfor the diffu-

Themain purposeof this studyis to investigatea sion coefficient.statisticaldescriptionof electronsinteractingwith Webeginour analysiswith numericalsolutionsofthe localized fields (2). We describea diffusion the equationof motionmodel which allows for the existenceof regular is- d2 elandswithin theregionofstochasticityinphasespace. x( t)= — — ~ E~cos(w~t—k~x). (3)Thismodel uses,as oneof its parameters,the finite me

decaytimesof the correlationfunctions. The resultsare presentedin fig. 2 which showstheTheproblemof electronsinteractingwith sharply phaseplaneconstructedby mappingnumericalso-

localizedLangmuir fields wasanalyzedin ref. [5]. lutions every T~=2x/cv1,. The initial velocities

It wasfound that if the localizedfields hadrandom spannedthe region betweenthe phasevelocitiesofphasesthe quasi-lineardiffusion gavea good de- the fastermodesin the wave packet(2). Ourchoicescriptionof theevolutionof theparticledistribution of plasmaparametersillustratesquite well all pos-function.A similarconclusionwasreachedby Fuchs sible kindsof particletrajectoriesincluding regionsetal. [6] for coherentwavepacketswith a largenun- of connectedstochasticity, and trappedand un-berof overlappingmodes.Our resultsshow that for trappedorbits.realisticwavepacketsof Langmuirwaves(2) which Exceptfor the modesn =0, — 1, all the modesin

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Volume126, number4 PHYSICSLETTERSA 4 January1988

30.0 in the velocity range vI ~l.SVth. Secondly,thoseelectronswhich start at low velocities in the sto-

20.0 ~ chasticregionanddiffusetohighervelocitieswill not

~-. 10 0 ~ get trappedin the potentialwells formedby modes.+) .~ : —~ - n = 1, ±2, —3. However,the existenceof theseis-

o.o . . lands still influencesthe nonadiabatictrajectories,

10 0 .‘ ,.~ . particularlyin the vicinity of thevelocity boundsofseparatriceswhen particlesare rapidly accelerated

—20.0 and havehigh diffusion rates.This featurewill beclarified in the calculations of the diffusion

30.0 71. coefficient.

Pha In orderto introducea statisticaldescriptionintoe the dynamicalsystemdescribedabovewe will con-

Fig. 2. The Poincarésurfaceof sectionplot, basedon numerical centrateon the region of mixing as shownin fig. 2solutionsto theequationsof motion(3). Pointsaremappedevery andtry toconstructa diffusion coefficient.DiffusionT~=2ir/o.i~.Velocity is in unitsof thethermalvelocity andphaseis givenin units of

27t/kL. The length of thecell is 21 (notethat of electronsby coherentwavepacketshasbeendis-2kL= kA). cussedrecentlyin greatdetailby Fuchset al. [6]. In

theweak-fieldregime,i.e. whenthetrappingtimeforthe wave packet (3) satisfy the overlappingcrite- the wavesis greaterthanautocorrelationtime, andnon. If particlesaregiveninitial velocitieswithin the whentheoverlappingcriterionis satisfied,thequasi-interval definedby the overlappingresonances,a linear theorygives an adequatedescriptionof thelargeregionofconnectedstochasticityis evident.The diffusion process.Thereare certainsimilaritiesbe-low velocity boundingthe regionof stochasticityis tweenour problemandthe model analyzedin ref.determinedby the regular trajectoriesdue to the [6]. Unfortunately,only the slowest modesin theponderomotivepotential.In fig. 2 thetwo orbitsseen wavepacket(2) (I n I ~ 4) satisfythe conditionsre-in thevicinity of v= 0 are due to particleswhich are quiredfor the validity of the quasi-lineartheory, asreflectedby the ponderomotivepotentialof the two discussedin ref. [6]. Theremainderof thespectrumlocalizedstructures,one in thecenterof thebox and (cf. fig. 1) correspondsto large amplitudemodes,the otheron the edges.The high velocity boundary for which the trappingtime becomesless than theis determinedby the regular trajectoriesof un- time of onepassof a resonantelectronthroughthetrappedparticles.The blank areaaroundzero ye- localizedfield. Themodesstill overlap(exceptn = 0,

locity vi <1.SvLhis boundedby regularorbitsandis — 1) but give fairly large adiabaticislandsin thethereforeinaccessibletoparticlesfromthestochastic phasespace(cf. fig. 2).Westressagainthatourpar-region. The same is true for the regular islands ticular choiceof conditionsfor this exampleillus-embeddedin the regionof mixing.Theseislandsare tratesmostof the featuresfor the particleorbits. Aparticularlylargeformodesn= 1, —2 (VPh = ±l.SVth). morelocalizedsolitonwill give largermixing regionWehaveintroducedparticlesin theseislandsin or- (andvice versa),but will still havethe samesetofderto show thetrappedorbits. Finally, fig. 2 shows orbitswhich are representedin fig. 2.two large resonances(n= 0, — 1) at velocities In thederivationof thediffusion coefficient [7,8]v= ±22.4Vthseparatedby regularorbitsfrom thesto- onearrivesat the formulachasticregion. / e ~2

Basedonthemapinfig.2onecanmaketwoim- D(v)= J dt~>.(~—)PE~I2portant predictionsfor the evolution of an initial o me

maxwellian(v1,, = 1) distributionfunction.First, the

bulkof theparticlesmove in the ponderomotivep0- X <exp{1[9~k~(t)—~k~(O)]}> (4)tentialwell. Theyaretrappedbetweenlocalizedfields wheretheaverages< > aretakenwithrespecttomi-andundergototal reflection. In effect the distribu- tial phasesand ~,k,,(t)=k~x(t)—w0t is evaluatedtionfunction,averagedoverphases,doesnotchange alonga particletrajectory.In quasi-lineartheory [7],

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Volume 126, number4 PHYSICSLETTERSA 4 January1988

x( t) is approximatedby a free-streamingtrajectory 1.0

leadingto thewell known singularresult for the dif- 0.8

fusion coefficient 0.6

2

DQL=~E(~’) ~ lEk~I2ö(wp—kflv). ~ 0.4\me, n 0.2

-..

S

Onemight arguethat in the presenceof mixing and 0.0

in the caseof very short autocorrelationtimes for — .2

particleswithin the localizedfields, the quasi-lineardescriptionis a valid model. The presenceof sto- — .4

0.0 1.0 2.0 3.0 4.0 5.0 6.0chasticinstability in the particlemotion andthe ex-istenceof a densepopulationof overlappingmodes, Time (passes)makesit possibleto useatransitionto a continuousrepresentation~— (l/~k).f dk, for which 1.0

2 0.8

DQL=(I_) ~l~E2 1 (5)I k=w/v —

lvi 06

The aboveargumentsprovedto work well for the ~ 0.4casesdiscussedin ref. [6]. In our model,which rep- 0.2

resentstypical Langmuir wave packets,the transit ~ 0.0

timeof electronsis comparableto the trappingtime.Overlappingmodesleave large regularislandsand — .2

consequentlywe need to examineeq. (4) more ______________________________carefully. 0.0 1.0 2.0 3.0 4.0 5.0 6.0

Westartby abandoningtheassumptionof freeIra- Time (passes)jectoriesand introducea discrete map as an ap- Fig. 3. CorrelationfunctionsC (eq. (8)) calculatedfrom thenu-

proximationfor the motion of particles mericalsolutionsof theequationsof motion(3) (solidcurve).

The circlesareobtainedfrom the approximatemap (9). The

V ± = v” — —~—E,,, cos~ “ , (6) dashedline showsanestimateof theenvelopeof thecorrelationm~ function.(a) correspondsto v=7.18,(b) v=5.81.

= ~ + k1 (0 LIt” (7) linearizingthe mapwith respectto the strengthofp

whereLIt”= ~l(l/v”+ l/v”~’) istheapproximatetime the field, we canrewrite eq. (8) asof onepassthroughthelocalizedfield duringthenth 2,t

collision. In writing eq. (6) we haveassumed(cf. Cm ~_ $ dq,~exp(— mLIt°w~+ mk~1ref. [6]) that in the zero order approximationelec-tronsinteractwithonemodeforwhich i/~ WrJkandthat during the accelerationthe phasein eq. (6) ~ [1 +2(m—r)] cos

9~r_i) (9)changesalongthefree-streamingorbit. With eqs.(6) r= i

and(7) describingmotionof electrons,onecanwrite where Sk~= ~(e/me)Ek, 2 and &o = 1/v0 is the

for the correlationfunction time of one pass,basedon the initial velocity.

C= <exp{i[cok~(t)—qk~(O)]}> Fig. 3 presents comparisonbetweenresultsof theapproximatemap (9) andthe numericalresultsfor

/ / ,~i(‘exp~,i~ (~— ~ ))). (8) eq.(8) basedonsolutionsto theequationofmotion.In the numericalstudiesthe initial conditionsput\ r=l

particleson stochasticorbitsandaveragesweretakenSimplifyingevenfurtherthetimeevolutionof~~by with respectto time, ratherthan phases.The cor-

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Volume 126, number4 PHYSICSLE1’TERS A 4 January1988

relation function for v0= 7.16 in fig. 3 corresponds

2.OE+07to ratherlong decay times in termsof numberofpasses.Theinitial velocity ofthis trajectoryis closer 1.6E+07

to the phasevelocity of the wave with a regularis-land. This particle experiencessmaller transferof ~, 1.2E+07

momentumandlongerdecay time for the correla- ~tidewith velocity v0= 5.81 (fig. 3b),which ismixing ]~‘tion functionsas comparedto, for example,a par- ~ 8.OE+06on much fastertime scale. 4 OE+06

The decaytimesfor the calculationof the cone- 0.0

lation function canbe approximatedby rc=LIt°Ia, —12.0 —8.0 —4.0 0.0 4.0 8.0 12.0whereLIt°= i/v andthe coefficient a variesslightly

Velocitydependingon the mode.Due to the exponentialde- (k T/m ) 1/2 = 1cay of a correlation function in time one can cal- B

culate the integral in eq. (4) and obtain a time Fig. 4. Diffusion coefficients.Solid curve— theoreticalpredic-

independentdiffusion coefficient.As one might ex- tionsbasedon eq.(11), longdash— numericalresults(12),shortdash— quasi-lineartheory(5).pect the modification of the trajectorieswill alsochangethe characteristicfrequencyof C( t),

C(t, v) ~eI~t_h/tc (10) D(v°)= < [v(LIt°) —v°12>, (12)

where for the free orbit approximation one has whereLIt°= 1/v0 is the time of onepass.The ensem-(2= kvt— cvi,. We proposethat the nth modein the ble of particlesfor a given velocity v°is constructedwave packet(2), which givesa regularislandin the by introducingparticlesat different timesinto theregion of mixing contributestwo characteristicfre- field. By this procedureandby appropriatechoicesquenciesto the correlationfunction.Thesefrequen- of the initial phaseswe ensurethat only stochasticciesare givenby (2,,= k,,vt—w~,±6w,, where 8cv,, is trajectoriescontributeto the valuesof D(v°).Theofthe orderofthe halfwidth of the undisturbedres- resultsareshownin fig. 4 togetherwith quasi-linearonancei.e. 8w,, ~ 2~/k,,(e/m~)E,,. This approxima- results (5) andvaluesgivenby ourmodel (11). Thetion leadsto a diffusion coefficientof the following resonantstructureis clearlyvisible with minima atform the phasevelocitiesof the modes.Thetheoreticalre-

2 sult (11) reproducesthe resultsfrom the numericalD( v) = ~ I E,.,, 2 solutionof eq. (3) reasonablywell with the coeffi-

~ \m~ cienta,, 2 estimated from numerical and analytical

< ( va,,/l studiesof the correlationfunction (8) for mostof(va,,!!)2+(w

0 +8w,, —k,,v)2 the modesexceptn= 1, —2. The latter components

of the spectrum correspond to the largest islands andva,,/i ) (11) have a,,=l.5.

+ (va,,/i)2+(w~—8w,,—k,,v)2‘ In fact onecanusethe numericalresults ofD(v)

(11) to estimateproperdecaytimes; aswell asthewherewe haveused the explicit form of the cone- frequency shifts 6w,,, for which 6w,,=lation decay time, r~= i/va,,. Theresonantstructure 2\/k,,(e/me)Ek~is only a first order approximation.in eq. (11) indicates an enhancement of the diffu- We know that the true extent of the islands is dif-sioncoefficientforparticlesin the regionswherethe ferent due to nonlinear coupling between reso-separatricesof adjacentmodesare closest. nances.Ourmodel of the diffusion coefficientgives

Thenumericalresultsfor thediffusion coefficient a good approximationto the numericalresultsex-calculatedfor onepassof electronsthroughthe lo- ceptperhapsnearthe regularorbitswhich boundthecalizedfield were obtainedfrom regionof connectedstochasticity.In theseregionsthe

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Volume 126,number4 PHYSICSLETTERSA 4 January1988

model (11) givesresultswhich aretoohigh.Wehave leadsto an enhancementof thediffusion coefficientfound thatin orderto usethe diffusion modelto ad- at velocities(w~±6w,,)/k,,,where 8w,, is defined bycurately predictthe temporalevolutionof the dis- thewidth of a resonance.In order to construct antribution function, the diffusion coefficient must be accurate model the decay times of the correlationtruncated at these regular orbits. functions were estimated using approximatemap-

The quasi-lineartheory gives the right order of ping and numerical solutionsto the equationsofmagnitudefor thediffusion coefficient,butdoesnot motion.reproducethe resonantstructuresandalsoextendsbeyondthe stochasticregion. A modified quasi-lin- Weacknowledgeveryusefuldiscussionswith Drs.ear theorywith a cutoffof the diffusion coefficient D. DuBois,H. Roseandhelpof Mr. R. Teshimainor velocitiescorrespondingto the first invariant tori the computercalculations.The researchwas sup-seenin fig. 2, gives good agreementfor the heating portedby theNaturalSciencesandEngineeringRe-ratesandan evolutionof the distribution function. searchCouncil of Canada.We will reporton calculationsof thetime evolutionof the distribution in a laterpaper.Finally we alsoobserve(cf. ref. [6]) thatwhenthe particlesfill the Referencesaccessiblevelocityinterval definedby theregularor- [1] M.V. Goldman,Rev.Mod. Phys.56 (1984) 709.bits,thediffusioncoefficientshouldchangeandcon- [2] G.D. Doolen,D.F. DuBois andH.A. Rose,Phys.Rev.Lett.

sequently a second diffusion coefficient will be 54 (1985)804.requiredto adequatelydescribethe longtime evo- [3] C.H. Aldrich, B. Bezzerides,D.F. DuBois andH.A. Rose,

lution of the distribution function. PlasmaPhys.Contr.Fusion10 (1986) 1.In conclusionwehavefounda generalform of the [4] W. Rozmus,R.P.Sharma,J.C. SamsonandW. Tighe, Phys.

Fluids30(1987)2181.diffusion coefficient for electronsrepeatedlyinter- [5] G.J.MoralesandY.C. Lee,Phys.Rev.Lett. 33 (1974) 1534.

actingwith coherentlocalizedLangmuirwaves.This [6] V. Fuchs,V. Krapchev,A. Ramand A. Bers,PhysicaD 14

modelgives satisfactoryresultsevenwhenthewave (1985) 141.packethasonly a few modesandthe regionof con- [7] J. Krommes,in: Handbookof plasmaphysics, Vol. 2, eds.

M.N. Rosenbluthand R.Z. Sagdeev(North-Holland,Am-nectedstochasticity,defmedby thewaveoverlap,hassterdam,1984)p. 183.largesizeregularislandswhich areinaccessibleto the [81 G.M.Zaslavsky,Chaosin dynamicsystems(Harwood,New

stochastictrajectories.Thepresenceof theseislands York, 1985).

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