plasma expansion into vacuum : hydrodynamic approach.pdf

PLASMA EXPANSION INTO VACUUM- A HYDRODYNAMIC APPROACH

Ch. SACK and H. SCHAMEL

Institut für TheoretischePhysik, Ruhr-Universikit Bochum,D-4630Bochum 1, West Germany

INORTH-HOLLAND - AMSTERDAM

PHYSICSREPORTS(Review Sectionof PhysicsLetters)156, No. 6 (1987) 311—395. North-Holland, Amsterdam

PLASMA EXPANSION INTO VACUUM - A HYDRODYNAMIC APPROACH

Ch. SACKt and H. SCHAMELtt

Institut für TheoretischePhysik,Ruhr-UniversitdtBochum,D-4630Bochum1, WestGermany

ReceivedJune1987

Contents:

1. Introduction 313 7. Comparison with previous hydrodynamic and kinetic2. The plasmaexpansionmodel 316 models 360

2.1. Basicequations,initial- and boundaryconditions 316 8. Navier—Stokesviscosity and implicit Lagrangianscheme 3642.2. Constraintson the electronthermodynamics 319 9. The early stageof viscid plasmaexpansion 367

3. Plasmaexpansionwithin the frameworkof gas dynamics 322 10. The long-time behaviour 3713.1. Self-similar theory 322 10.1. The debunchingprocess 3713.2. Simple wavesand Riemanninvariants 327 10.2. The asymptoticvelocity of fast ions 373

4. Numerical methodsand analysis 332 10.3. Intermediateasymptotics 3744.1. Lagrangianequationsof ion motion 332 11. Comparisonwith experiments 3774.2. Two iterative methods for the nonlinear Poisson’s 12. Summaryandconclusions 380

equation 334 AppendixA: Bunching and wave breaking in nonlinear4.3. Boundaryconditions 336 Langmuir oscillations 3814.4. Conservationlaws 339 AppendixB: Spatial discretizationof Poisson’sequation 3854.5. Variationalprinciple 342 AppendixC: Implicit differenceschemefor theequationsof

5. Numericalresultsfor thedissipationlessplasmaexpansion 343 viscid ion motion 3886. Theory of bunchingand wavebreakingin ion dynamics 351 AppendixD: Conventionalstability analysisof the discret-

6.1. Scalarwave equation 352 ized setof equations 3906.2. Perturbativesolutionof thescalarwaveequationfor References 393

isothermalelectrons 354

Abstract:Basedon thehydrodynamicdescription,the planarexpansionof a plasmainto vacuum is investigatednumerically and analytically. Several

dynamicalstructuresarefoundandexplained.In the inviscid case,ion wavecollapseis themoststrikingfeature.Viscosity preventsthecollapseandallows long-termcalculations.Threephasesin theevolution canbe distinguished.In thetime asymptoticregimetheself-similarstateis approached.This approachis precededby a stateof intermediateasymptoticsand applies to chargeseparationas well.

Now at JETJoint Undertaking,Culham Laboratory,Abingdon, Oxon0X143EA, England.~ Now at PhysikalischesInstitut, UniversitätBayreuth,D-8580Bayreuth,Postfach3008, WestGermany.

Singleordersfor thi.s issue

PHYSICSREPORTS(Review Sectionof PhysicsLetters)156, No. 6 (1987) 311—395.

Copies of this issue may be obtainedat theprice given below. All ordersshouldbe sentdirectly to the Publisher.Ordersmust beaccompaniedby check.

Single issueprice Dfl. 61.00,postageincluded.

0 370-1573/87 / $29.75 © Elsevier SciencePublishersB.V. (North-HollandPhysicsPublishingDivision)

Ch. SackandH. Schamel,Plasmaexpansioninto vacuum— A hydrodynamicapproach 313

1. Introduction

The investigationof a spontaneouslyreleasedor producedplasma expandinginto vacuum is ofconsiderableinterestto the generalunderstandingof unsteady,nonlinearbehaviourof ionized gases.The specialrole plasmaexpansionis playing as a basicprocessin astrophysicalflow problemshasbeenrepeatedlyemphasized[1, 2]. In fusion researchexpansionprocessesare just as fundamentalif onethinksof phenomenasuch as the arcing on surfacesof limiters anddivertors [31,the ablationof solidhydrogenpellets in refuelling of tokamaks[4, 5], and the laser—pelletinteraction in inertial confine-ment. In the latter case,e.g., adetailedknowledgeof the dynamicsof the coronalplasmaproducedbythe laserbeamis requiredin orderto estimateits effects on the implosionefficiency [6—12].As it is wellknown, the existenceof a componentof high energeticions deterioratesthe pellet compression.Although thesefast ions representonly a relativelysmall fraction of the total ablatedmass,theycarryoff up to 50 per cent of the absorbedlaser energy [8, 131. The feasibility of laser-inducedfusion,therefore,is essentiallycoupledto the minimizationof the numberof fast ions which presumesa basicunderstandingof their generationmechanism.

Acceleratedions in connectionwith expandingplasmasareobservedin other areas,too, suchas inthepolar wind [14,151, in cathodeflares[161,in vacuumarcs[17] as well asin explodingwires[18, 19].As earlyas 1930 Tanberg[201foundplasmajets of high speedin a pulsedgas dischargebeingejectedfrom the plasmaregion in front of the cathode.An explanationfor this effectbasedon the mechanismof ambipolarion accelerationby electronswasnot given until the earlysixties by Plyutto [211,andbyHendel and Reboul [22]. Due to the higher mobility of the electrons,an electric (ambipolar)field isgeneratedwhich preventsthe electronsfrom escapingand, at the sametime, acceleratesthe ionstowardsthe vacuumor the less densemedium.Experimental[23—31] andtheoretical[32—521investiga-tions carried out during the following years have further contributed to a clarification of the ionaccelerationprocess,andhavediscoverednonlinearphenomenabeinginvolved in the plasmaexpansionsuch as shockwaves, wavebreakingand anomalousdissipativeeffects.

According to the general plasma theory, expandingplasmas are describedby both kinetic andhydrodynamicmodels. Completedby Maxwell’s equations,quite complicatedsetsof equationsareobtainedwhich cannotbe solved analytically. Eventhe numericalsolutionof thesesetscausesa lot ofdifficulties. However, in order to comprehendbasicprocessesin plasmaexpansion,it is often sufficientto considersimplified modelsbeing suggestedby experimentalobservations.

Such a simplified model is the one-dimensionalplanarexpansionof a rarefied magneticfield-freeplasma into vacuum. The space-timeevolution of the rarefied plasma is describedby a systemofcollisionless kinetic equationsfor electronsand ions in connectionwith Poisson’sequationfor theelectrostaticpotential. Assumingthat the electronsalways stay in equilibrium with the electrostaticpotential and that the plasma behavesquasi-neutral,the set of equationsis reducedto one singleequation, the Ion—Vlasov equationwhich is, however, nonlinear. Becauseof the quasi-neutralityassumptionthis equationcan be transformedto self-similar variableswith referenceto usualhydro-dynamics (see, e.g. [53,54]). This meansthat in the solution of the problem the time t and thespace-coordinatex appearonly in the combinationx/t.

In the caseof Boltzmann-distributedisothermalelectronssuch solutionsof the plasmaexpansionproblem were firstly presentedby Gurevich et al. [33]. They indicate a strong, continuous ionaccelerationdriven by the ambipolarelectric field which has beenfound later on in other extendedinvestigations,too [37,38, 42, 46, 49, 50]. Coincidentwith theaccelerationtheeffectiveion temperaturerapidly cools down, and finally the motion of the acceleratedions can be describedby the hydro-

314 Ch. Sackand H. Schamel,Plasma expansioninto vacuum— A hydrodynamicapproach

dynamicequationsof an ideal, isothermalgas[33]. Assumingcold ions from the outset,the self-similarsolution of the hydrodynamic equationsyields at a fixed time t a density profile exponentiallydecreasing,and a velocity profile linearly increasingin x [36,43]. An unphysicalfeatureof this solutionis that the electric field is singularat t = 0 anddoesnot fall off for x —~ +~ at anyothertime. Becauseoftheseshortcomingsandbecauseof the fact that a characteristicscalelengthis missing,the self-similartheory is only conditionally suitable to describe the plasma expansionfor an arbitrary initial- andboundaryvalue problem. In which sensethe quasi-neutralself-similar solution is approachedin anexpandingplasma,will be studiedamongothersin this review-kind article.

A more generaldescriptionof the nonlinearmotion of a collisionlessquasi-neutralplasma,whichincludesthe self-similar solutions as a specialcase, is basedon the Riemannsolutionsfor so-calledsimple wavesof arbitraryamplitudein usual hydrodynamics(Riemann(1860),comparealso [53]). Themost noteworthyproperty of this solution is that any compressionalwave progressivelysteepens,andfinally breaksafter a finite time. In this casethe flow profile becomesmultiple-valuedin space,andtheequationsloose their global validity [53]. Whereasin usual hydrodynamicsthe occurrenceof singu-larities in the flow profile is preventedby dissipation,dispersiveeffects comeinto play in the plasmacase.They arebelievedto limit the steepeningof finite amplitudewavesandinhomogeneities,leadingto the developmentof solitary and/orperiodic waves [55].

Dispersiveeffects resultfrom the deviationfrom quasi-neutrality[32,34,35]. In this casethe Debyelength is the characteristicscale length of the problem, and the potential has to be calculatedfromPoisson’s equation. By this means, the singularity of the electric field at t = 0 existing in thequasi-neutralself-similar solution is removed.Therehasbeenseveralattemptsto simplify the systemofhydrodynamic ion equationsand Poisson’sequationby appropriateself-similar transformationsandthus, to renderthem accessibleto an at least approximateanalyticaltreatment[44, 48]. However, thegeneralinitial- and boundaryvalue problemfor an expandingplasmaincluding chargeseparationcanonly be solved numerically.

For the numerical investigationof plasmaflow on a hydrodynamicbasis finite differenceschemes,suchas,e.g.,Eulerian-andLagrangianschemes,havebeendeveloped.In thiscontextrepeatedinteresthas beendevotedto the isothermalexpansionof a cold ion plasmainto vacuum[37,42,49, 50]. Besidesa lot of according resultson this subjectbeingconfirmed by particle simulationsof the problem[46],and being discussedin detail in this review, quite contradictorystatementshavebeen recordedwithrespectto the concretespace-timebehaviourof the fastestions. Especiallyit wasnot clearwhethertheoccurrenceof a densityhump at the ion front shown,e.g.,by Crow et al. [42], andby Gurevichetal.[49], is dueto a numericaleffect or whetherthereis a physicalphenomenonhiddenbehindit. Trueetal. [50],who solvedthe samesetof equations,simply statetheyhadnot observedsucha densityhump.

In an earlier paper of the authors [56], the one-dimensionalplanar expansionof a collisionlessplasmawas studiedunderthe influenceof anobliquely incidentelectromagneticwave.Using an explicitLagrangianschemefor the ion hydrodynamics,a local, explosiveincreasewas foundin the ion densityleadingto a suddenbreakdownof the numericalsolution.A similarbreakdownwas alsoseenwhentheradiationfield was switchedoff.

In subsequentpapers[57,58], the presentauthorsinvestigatedthis pureplasmaexpansionproblemin moredetail. It was foundthat the densityhump andthe breakdownof the hydrodynamicsolutionareof the sameorigin being connectedwith the appearanceof a simple wave structurein the expansionproblemwhich developsfrom an extremely inhomogeneousinitial density.

The main topic of this reviewis to evaluatethe nonlinearstructuresarisingin the plasmaexpansionmodelledby the hydrodynamicequationsof motion. The paperis organizedas follows:

Ch. Sackand H. Schamel,Plasmaexpansioninto vacuum— A hydrodynamicapproach 315

In section2 the mathematicalformulation of the problemandits physical foundationis presented.Section 2.1 contains the basic equationsincluding the general initial- and boundaryconditions. Insection2.2 it is shown that in the caseof a polytropic electronequationof statethe specificationof theboundaryconditions in Poisson’sequationimposes constraintson the electron thermodynamicbe-haviour.

The analogyof the plasmaexpansionmodelto the usual hydrodynamicsof an ideal neutralgas isdiscussedin section3. Section3.1 dealswith the analyticaltreatmentof the plasmaexpansionwithin theframework of the self-similar theory. Section 3.2 contains,the more general formulation of theexpansionproblem by meansof the theory of simple waves of finite amplitude and of Riemanninvariants.

In section4 we presentthe foundationsof the numericalsolution methodsfor the dissipationlesshydrodynamic equations and for Poisson’s equation as well as some corresponding analyticalpeculiarities. Section 4.1 contains the transformation of the Eulerian equationsof motion to theLagrangianpicture, whereassection4.2 is devotedto the iterative solution methodsfor the nonlinearPoisson’sequation. In section 4.3 the boundaryconditions for the ion equationsand for Poisson’sequationareadaptedto the numericalsolutionmethods.The conservationlaws resultingfrom the basicequationsarededucedin section4.4. In section4.5 it is shownthat the ion hydrodynamicequationscanbe completelyderivedfrom a variational principle under the constraintof Poisson’sequation.

In section5 the numericalresultsof the dissipationlessplasmaexpansionarepresented.The effectsof chargeseparationand of quasi-neutralityon the global behaviourof the solution are discussed,aswell as the influenceof electron thermodynamics.In those caseswherethe solution breaksdown, asharpspike in the ion densityemerges,increasingexplosively. At the sametime the electric field andthe ion velocity steepenwithout limits.

These propertiesof the solution being analytically analysed in detail in section 6, suggesttheoccurrenceof wavebreakingin connectionwith densitybunchingas it is knownfor Langmuirwavesandfor ion wavesof finite amplitude.In section6.1, a single scalarwave equationis deriveddescribingthewhole ion dynamicsincludingchargeseparation.Its solutionby perturbativemeansin the vicinity of thebreakingpoint given in section6.2 reproducesall the propertiesfound numerically. Thus,it is proventhat the singularity in the ion dynamicsis not a numericaleffect but an expressionof the extremenonlinearity of the plasmaexpansioninto vacuum.

In section 7 comparisonsare madewith previoushydrodynamic and kinetic investigations.Thekinetic calculationsfor an equivalentproblemgive a hint which types of phasespacestructurescan beexpectedafter wavebreaking.

Within the hydrodynamicdescriptionwavebreakingis preventedby includingdissipativeeffects. Forthis purposea viscosity term of the Navier—Stokestype is introduced in the ion momentumequationrenderingpossiblethe implicit formulationof the differencescheme.In section8 the propertiesof sucha scheme,especiallyits numericalstability, are discussed.The full-implicit schemeturns out to be infavourof the stability behaviourof the codeallowing long-termcalculationsof the expandingplasmawithoutshowing anynumericalinstability. The resultsof thesecalculationsarepresentedin sections9and 10.

Section9 dealswith the earlystageof viscid plasmaexpansionwherethe initial accelerationandtheion bunchingoccur. The ion bunchingprocessis now stabilizedby theviscosity termandmanifestsitselfin a pronounceddensity hump propagatingat a constantspeedof several times of the ion soundvelocity. The quantitativepropertiesof the accelerationphaseandof thebunchingphasedependon theviscosity, on the electronthermodynamics,andon the initial conditions.

316 Ch. Sack and H. Schamel.Plasmaexpansioninto vacuum— A hydrodynamicapproach

In section10 the long-termbehaviourof the expandingplasmais discussed.Section10.1 showsthatthe two phasesexplainedin section9 entera third phasewhich is initiated by the debunchingprocess.The debunchingleadsto a rapid decreaseof the densityhump andgivesrise to a furtheraccelerationoftheion front. As presentedin section10.2 the fast ionsat the front achievea velocity which is given bythe self-similar theory.The self-similar behaviourof the plasmaexpansionas a state of intermediateasymptoticsis exemplarily provenin section10.3 by a perturbativeanalysisof the single scalarwaveequationfor the isothermalquasi-neutralcase.

In section11 the resultsof our numericalcalculationsare usedto interpret the measurementsofsomeselectedplasmaexpansionexperimentsbeingespeciallycontradictorywith respectto the velocityof the fastestions.

Section 12 containsthe summaryof this review and the conclusions.In the Appendix the nonlinearbehaviourof Langmuirwavesand the details of the finite difference

schemeand of the stability analysisare treatedseparately.

2. The plasma expansion model

2.1. Basic equations,initial- and boundaryconditions

As mentionedin the introduction,weinvestigatethe one-dimensional,planarexpansionof astronglyinhomogeneous,magnetic field-free plasma into vacuum. It is supposedthat the plasma consistsofsingle chargedions with mass m~and charge+e, and of electronswith mass me and charge —e; erepresentsthe elementarycharge.Assumingthat the ion temperatureT1 is negligible comparedto theelectrontemperatureT~,e.g. T1/T~—~0,we havethe following two-fluid equations:

+ ~i~(n1v)= 0 (2.1)

+ v~a~v~= —-f-- (2.2)

+ ax(neve)= 0 (2.3)

e Ic31V~+ Ve i3xVe = — — — (2.4)

(2.5)

The equations(2.1)—(2.4)describethe space-timedevelopmentof thedensityn andof thefluid velocityo of the two-componentplasma,where the indices i and e distinguishion and electronquantities.Equation(2.5) for the electronpressureis obtainedby insertingthe equations(2.3) and(2.4) into theelectronenergyequation;We representsthe electronheatconduction.The setof equations(2.1)—(2.5)is supplementedby Poisson’sequationfor the electrostaticpotential4, which couplesthe ions to theelectrons:

= 4~re(n~— ni). (2.6)


The equations(2.1)—(2.5) are not yet completebecausean equationfor We is missing. Closure is

achievedby assuminga polytropic equationof statefor the electrons:Pe1~T~~’~k=const., (2.7)

wherey is the polytropic exponent.The polytropic equationof statecorrespondsto a nonvanishingdivergenceof the heatconduction

We~given by

axwe=—(3—y)kaxve=—(3—y)peaxve. (2.8)

Inserting (2.8) into (2.5), one easilygets

(P5\_ d (Pe~

~ (2.9)

from which follows by integrating(2.9)

Pe~12~ (2.10)n~0

wherePeo andn~0are the correspondingquantitiesof the electronpressureand of the density in theunperturbedhomogeneousplasma.

Assumingthat the electronfluid behaveslike an ideal gas, i.e.

PehleKTe (2.11)

whereK is the Boltzmann constant,we get from (2.10) and (2.11) a relation betweenthe electrontemperatureand the electrondensity:

Ten~~~const. (2.12)

In the isothermalcase,y = 1, it follows from (2.12) Te = const.as it should be. Owing to the highthermalconductivity of the plasma,the electrontemperaturewill stay constantduring the expansionprocess. This standard case is describedby an isothermal equationof state, ‘y = 1. In certaincircumstances,however,the heattransportcan be reducedespeciallywhenregionsof stronggradientsor extremelydilute plasmasare involved. We, therefore,allow in our general descriptionalso fornonisothermalequationsof states,y � 1. In fact, equationsof statesdeviatingfrom isothermalityhavebeenidentified in particle simulations(seee.g. Denavit[46]) and in laboratoryexperiments(seee.g.Hendeland Reboul[22] or Eselevichand Fainshtein[30]).

Being interested in processestaking place on the ionic time scale, we normalize the system(2.1)—(2.4) and (2.6) as follows:

a~n+ 9~(nv)= 0 (2.13)

+ u 9~v= —a~4 (2.14)


atne + 8x(neve)= 0 (2.15)

m 1

—s (a,v~+ Ve axve) = — — a~n~ (2.16)

(2.17)

wheret, x, o, n and 4’ arenormalizedby w ~‘, AD, c~,n0 and K Teo/e. The latter quantitiesare the ionplasmafrequency~ = (4~rn0e

2/m1)

1~,the electronDebyelength AD = (K Teo/4lTnoe2)”2and the ionsound speedc~= (K Ten/mi)112,where the index 0 refers to the unperturbed,homogeneousplasma(nen =

As long as the ion fluid velocity is much lessthanthe electronthermalvelocity °the= (K Teo/me)’ / 2

noting that Vthe/CS = (m1/m~)

112~‘ 1 in theseunits, we can neglect the left-handside in the electronmomentumequation(2.16),which then reads

(2.18)

Integratingeq. (2.18) yields the electrondensityne as a function of the electrostaticpotential4:—1 l/(y-l)

= (~+7 . (2.19)

Fromthe requirementthat the electrondensityfalls off from unity in the unperturbedplasmato zerointhevacuum,it follows that the potentialis restrictedto the interval0 and —‘y/(y — 1). In the isothermalcase, ‘y = 1, we obtainfrom (2.19) the well-known Boltzmannlaw:

ne(4)= e’~. (2.20)

For y = 2 we get a linear relation betweenthe electrondensityand the potential:

‘~e(4~)= 1 + cb/2. (2.21)

The set of equationswhich will be the basisfor the following investigations,therefore,reducesto:

a~n+a~(no)=0 (2.22)

+ v = —a,~ (2.23)

0~4= ne(4) — n (2.24)

— 1 ~

= (i + ~ 7 � 1 (2.25)e4, y=1

We note parentheticallythat this systemcan be derivedfrom a variationalprinciple which will be givenin section4.5.

Ch. Sackand H. Schamel,Plasmaexpansioninto vacuum— A hydrodvnamicapproach 319

The solution of the equations (2.22)—(2.25) is subject to the initial andboundaryconditions:a) initial conditions(t = 0):

n(x,0)=n0(x), ~(x,0)=4.0(x), v(x,0)=u0(x); (2.26)

b) boundary conditions (x —~ ±so):

v(—co,t)=0(2.27)

n(+cc,t)=0, ~(+so,t)=—7

1 , v(+co,t)=0.

The potential 4~i0(x)consistentwith the initial ion densityn0(x) is obtained by solving Poisson’s equation(2.24) at t = 0. In contrastto Widneret al. [37], Crow et al. [42] and Gurevichet al. [49], who haveuseda discretedensitystepfor the ions, we prefer a densityprofile with a finite transition length1:

n0(x) = ~ arctg[exp— (x7v0)], n0(x0)=0.5; (2.28)

for x—~±so,n0(x) reachesthe valuesgiven in (2.27).A diffuse densityprofile correspondsto amorerealisticinitial situationthana densitystep,havingin

mind, e.g.,the laser—pelletinteraction.In this case,the laserprepulseproducesa finite densitygradientthroughheating,ablationandionization.The parameter1 in (2.28) allows to control the steepnessandthe transition width of the density profile; for l—~0 (2.28) degeneratesto a density step. In ourcalculationswe havechosen1 = 4 and 1 = 1, respectively.This choice, especially1 = 1, guaranteesasharpdensity transition, in front of which the ion densityis negligible.

In most of the casestreatedin this review, the ions are assumedto rest initially, i.e. v0(x) 0. Insection9 alsotwo caseswith v0(x)~ 0 are considered.

Mathematicallythe equations(2.22)—(2.25)are a couplednonlinearsystem of partial differentialequations,the generalsolution of which can only be obtainedby numericalmeans.In specialcases,however,one can find analyticalsolutionsof the systemor one can at least studysomepropertiesofpossiblesolutionsanalytically. In section3 we shall comeback to this point.

In the following subsectionwe presenta specialanalyticalsolutionof Poisson’sequationanddiscussthe consequenceson the thermodynamicbehaviourof the electrons.

2.2. Constraintson the electron thermodynamics

From earlier numerical investigations of the plasma expansion into vacuum taking into accountcharge separation (see, e.g. [37,42, 46, 49, 50]), it is known that the ions acquire a well-defined front,beyond which there exists a pure electron cloud. Wemay suppose the same in our case, and introducex~, the time-dependent position of the ion front. In this case Poisson’s equation is given in the electroncloud region by

— 1 1/(y-l)= = (i + ~ , x > x~, (2.29)

wherewe assumefor the moment ‘y � 1.

320 Cli. Sackand H. Schamel,Plasma expansioninto vacuum— A hydrodynamicapproach

Multiplying eq. (2.29) with çb’ and integrating, we obtain:

—1 ~ E2~ =(i+~ ~) as~, (2.30)

where we use the boundary conditions 4’(x—* +so) = 0 and ç5(x—~+co) = —y/(y — 1). Taking thesquareroot of (2.30) and replacingthe electrostaticfield E by —~‘, we areable to performa furtherintegration:

1 — 1 — (2~’2~~1) ~[x — x~(t)] = ~ [(~+ ~ ~ ]~(() (2.31)

Solving (2.31) for 4, we get the potentialwithin the electroncloud x > x~:

2 — —2ey—1)/(2—y)

~(x, t) = — ~ fi — A2~’~[i + ~ A~2~(x— xf(t))] } (2.32)with

— 1A=[1+7 ~t(t)]

where 1 takesthe value4~at the ion front Xf. The time dependence of the potentialin the electroncloud region enters through xf(t) and 4

1(t) which are analyticallyunknown,andwhich will be providedby the numerical solution treating the full problem.

If one assumes a step profile for the ions

Ji, ~�~0nsi~(t=O)n(x,0)—i~0

one gets ~(x0, 0) 4~(t= 0) = —1. This result follows from the continuity of çb and E at the densityjump x = x0 and hasalreadybeenformulatedby Crow et al. [42] for the isothermalcasey = 1; it is,however,valid for any ‘y.

From eq. (2.32) we obtain the electric field E = —4.’ and the electron density ‘~‘e= [1+ ((‘y—1)/y) ~]1/(Y-1):

2 — -y/(

2-y)E = V’~A [i + ç~—~A~2~(x— xi)] (2.33)

2 —2/(2—y)

= A21~[i + ç~-~A~2~(x— xi)] (2.34)

with y ~ 1 and y ~ 2. The casey = 1 was obtainedby Crow et al. [42]:

~=~t21n[i+~ (x_x~)] (2.35)

Ch. Sackand H. Schamel,Plasma expansioninto vacuum— A hydrodynamicapproach 321

E = ~exp(~) [i + exp(~/2) (x — xi)] (2.36)

~,1 exp(q5~/2) 12nee Li+ ,~ (x—xf)] . (2.37)

For ‘y = 2 we get an exponential dependence of ~, E and ne:

= —2{1 — (1 + çbfl2) exp[—(x — x~)!V~]} (2.38)

E \/‘~ (1 + ~~/2)exp[—(x — x~)/\/~] (2.39)

= (1 + ~~/2)exp[—(x — xf)/\/~]. (2.40)

In the following we investigate the compatibility of the equations (2.32)—(2.34) with the boundaryconditions for x—~~so (see eq. (2.27)). All equations (2.32)—(2.34) contain the term

r ~, —

Ii ~ ~‘ A(2Y)17( I

i’ r— ‘~ ~ X~IL V 7 J

which must go to zero for x—~+so, in order to satisfy the boundary conditions

4(x—*+co)=—y/(y—i), E(x—*+cc)=0 and n~(x_~~+so)=0 (2.42)

for all times t � 0. This is the case when the polytropic exponent y lies in the interval

l<y<2, (2.43)

because the exponent — 11(2 — y) in (2.41) remainsnegative,and the expression(2.41) vanishes forx—* ~ In the special cases ‘y = 1 and y = 2, the solutions (2.35)—(2.37) and (2.38)—(2.40), respective-ly, also satisfy the boundaryconditions(2.42), so that the inequality (2.43) can be extended to

l~y~2. (2.44)

For y >2, however, the exponent in (2.41) becomespositive,and the boundaryconditionscannotbesatisfiedanymore,becausein the equations(2.32)—(2.34)4, E and tie divergeor becomeimaginaryforx—> +so. This unphysicalfeatureof the solutioncan beavoidedby assumingan electronfront at finite x.To prove this statement, we consider (2.41) for y >2:

rIi — ~ — 2 45L \,/~ ~,X X~ .

The conceptof the plasmaexpansioninto vacuum,which implies the monotonicvanishingof ne andE,can only be maintainedas long as the expressionin the squarebracketsis non-negative.From

1— ~

322 Ch. Sackand H. Schamel.Plasmaexpansioninto vacuum— A hydrodynamicapproach

we obtain the position of the electronfront x~1:

Xet = Xf + 2 ~ y >2. (2.46)

For y >2, the boundaryconditions(2.42) mustbe modified as follows:

7—1’ E(x=xCf)=O and fle(X~et)O (2.47)

We thus arrive at the conclusionthat thereexists a uniquerelation betweenthe thermodynamicbehaviourof electrons,being describedby the polytropic exponenty, and the initial and boundaryconditions. The restrictionof y to the interval 1 s y ~ 2 implies that the electronfront is at infinityalreadyat t = 0. For y > 2, the electronfront is in the finite region,which is alsotrue if one usesanelectronequationof statedescribingmore sophisticatedtrappedelectrons(see [59]). The numericalmethodsused in this reviewrequirethe restrictionof the polytropic exponentto the interval describedby inequality (2.44) (seesection4.3). Someresultsin this subsection,especiallyeq. (2.30), turnout tobe usefulin establishingthe boundaryconditionfor the numericalschemeas alsoshownin section4.3.

In the nextsectionwe investigatethe system(2.22)—(2.25)with respectto its analyticalsolubility inthe quasi-neutralregime.The frameworkof ordinaryhydrodynamicswill providea convenientbasisforthe derivationof somepropertiesbeing recoveredin the numericalsolution.

3. Plasma expansion within the framework of gas dynamics

3.1. Self-similartheory

The formulation of the plasma expansion on a hydrodynamic basis suggests a relation to ordinary gasdynamics. This relation is most evident if one first assumes quasi-neutrality,

— 1 l/(y-i)tie=~=(i+~ ~) , (3.1)

and, thus, expressesthe potential 4 in (2.23) in terms of the ion density n:

3,n + a~(nv)= 0 (3.2)

+ o = —yn~2a~n. (3.3)

The equations (3.2) and (3.3) are identical with those describing the dynamics of an ideal neutral gas.In order to solve theseequationsonemay, therefore,apply the knownmethodsderivedfor Eulerianhydrodynamics,such as dimensionalanalysis or similarity treatment,which can be found, e.g., inLandau—Lifschitz’s text book [53]. However, we are not solving the equations(3.2) and (3.3) by adimensionalanalysis,but shall use a generalmathematicalformalismbasedon the theory of transfor-mation groupsfor partial differential equations(Ames [60], Lonngren [61]). Within this theory it ispossibleto derive self-similar variablesandto investigatethe self-similar behaviourof partial differen-


tial equations.The independentvariablesx and t are combinedto a new variable reducingthe set ofpartial differential equations to an ordinary one. The solutions of the latter are called self-similar,becauseby rescalingthe solution in x-spacethe solutionfor different timescan bemappedonto itself.The connectionof the similarity transformationto the theory of Lie groupsis given as follows (seeLonngren[61]):

Similarity variables are identical with the invariants of a certain one parameter or more parametergroupof transformations.The mostgeneralgroupis called“infinitesimal group”; it containsall possiblesimilarity variables.

The derivation of similarity variables from the infinitesimal group, however, is algebraicallyverycumbersome(Lonngren [61]). We restrict ourselveshenceforthto the definition of a one parameterlinear group G:

I t=aa~t,G:=~_ ~ - , (3.4)

~n=a1n, v=a2v

wherea is a positive, real constant.The exponentsa, andf3~~i = 1, 2, haveto be determinedsuchthatthe equations(3.2) and (3.3) are “constantconformallyinvariant” under the groupG [60,61].

Definition: A function F(y) is said to be “constantconformallyinvariant” (CCI) under G, when

F(y)=f(a) F(~), (3.5)

where f(a) is any function of the parametera. If f(a) 1, F(y) is said to be “absolute constantconformallyinvariant” (ACCI).

We insert(3.4) and the transformedderivatives

= aa, a = a~3~ (3.6)

into the hydrodynamicequations(3.2) and (3.3):

aa1~,a~+ aa2~,~2a~(n~Y)= 0 (3.7)

a~1I323~s~+ a2~’~2~2

1j ~j _ai(~

2)~2Pi y ñ~’~a~. (3.8)

In order that the equations(3.7)and(3.8) areCCI (inclusivelyACCI) accordingto the definitiongivenabove, it must hold:

(3.9)

a1 f~2 a2 —2/32 a2—(y—1)13k. (3.10)

The nontrivial solutionsof the equations(3.9) and(3.10) generatethe similarity variableswhichareatthe sametime invariantsof the group G [60,61].

Firstly we determinetheseinvanantsand eliminatethe time variable t. From the first equationin

324 Cli. Sack and H. Schamel,Plasma expansioninto vacuum— A hydrodynamicapproach

(3.4) it follows:

a = (~/t)’~’ and x (1/t)a2/~1x, a = const . (3.11)

with a1 � 0. For a, = 0, the casewe are not consideringherefurthermore,we obtain anotherset ofinvariants(Ames [60]). With (3.11) we obtainthe first invariant

= x/ta2~1 r = r(x, t). (3.12)

In a similar way the secondset of equationsin (3.4) yields the further invariants

n(x, t) /tP1~a1=: t~(r), o(x, t) /~P2~1=: i~(r). (3.13)

Fromthe equations(3.9) and (3.10) we get the following relationsfor the exponentsa2/a1,/31/a1 and

f321a1:

~21a1a2~11~~1, A—a2/a, (3.14)

f31/a, = 2(A — 1)/(y —1), (3.15)

wherey � 1 is assumed.In order to determinetheseexponentsuniquely,we requirethat the boundaryconditionsof the expansionproblemformulatedoriginally in x andt, aretransformedwithout changeonto the T-dependentvariables.For x —~ —so the boundaryconditionfor the ion densityreads

n(x—*—so,t)=1. (3.16)

Becausen(x—~—so, t) transformsthrough(3.13) to i~(r—*—so), (3.16) demands,that /31/a1 = 0.From (3.14) and (3.15) we thusget f32/a1 = 0 andA = 1. With theseresultsthe self-similar variables

areobtainedfrom (3.12) and (3.13) as follows

rx/t, n(x,t)=n(r), v(x,t)v’T(r). (3.17)

(For the sakeof simplicity we write n andv insteadof ñ and 5’!)With (3.17) and the derivativesformulatedin T

= —~ a~, a~= a~ (3.18)

the systemof partial differentialequations(3.2) and (3.3) becomesan ordinaryone:

(v—T)n’+nv’=O (3.19)

yn~2n’+(o—r)v’=0, (3.20)

where the prime indicates the derivative with respectto T. The equations(3.19) and (3.20) are


nontrivially solvable,providedthat

(v — r)2 = ‘y n~1 (3.21)

wherey n~’representsthe squareof the soundvelocity of the system:

2 y—1 y—l

c = dpe/dne= 7 tie 7 ii = dPe/dfl• (3.22)Becauseof the quasi-neutralityassumptiontie = n, one hasPe = n~.For the velocity v in (3.21) oneobtains:

V = T ±\/~ ne’l~= T ± C. (3.23)

In (3.23) the positive sign hasto be chosento give ararefactionwavepropagatinginto theunperturbedplasma(negativex-direction).In the unperturbed,homogeneousplasmait shouldhold n = 1 andv = 0.Therefore,it follows from (3.23), thatwe only haveto considerthe solutionsfor r � ~ The pointr = ~ dividing the regionsof stationaryandmoving plasma,is a point of weakdiscontinuity.Thefirst derivativesof n and v are discontinuousthere(see[43]).

Eliminating v andv’ in (3.19) by meansof (3.23) andintegrating,we get the densityn, the velocityv, the potential4 andthe electric field E as functionsof r:

— 1 2/(y—1)~(y+1) (r+~)] (3.24)

v(r)=~i(r+\~y) (3.25)

~(r) = y ~ - 1) ~ + - (3.26)

~ (3.27)

The solutions (3.24)—(3.27) makesenseonly if the sound velocity c(T) is non-negative.We expressc = ~ ~ in termsof the velocity u [seeeqs. (3.24) and (3.25)]:

c=V~(1—~ ~), v�0. (3.28)

From c � 0 it follows, that the velocity u hasto satisfythe inequality

v ~ 2V~/(y— 1). (3.29)

If v reachesthe upper limit, the density and the electric field are zero, and the potentialadoptsthevalue —y/(y — 1). This result is consistentwith the boundaryconditions in the expansionregion.


Altogether T hasto be restrictedby the following inequalities:

T~2\1~/(yl). (3.30)

The equations(3.24)—(3.27)differ from thoseof ref. [53] throughthe choice of the sign in eq. (3.23)only, which simply accountsfor the opposite direction of the expansion. In our caseat fixed t thevelocity shouldincreasewith increasingx.

In the limit y = 1, the set of equations(3.24)—(3.27)reducesto the well-known self-similarsolutionsfor the isothermalexpansion(Gurevichet al. [33,43], Allen et al. [36]):

n(r)=eT~ (3.31)

v(T)=T+1 (3.32)

4(r) = —(T + 1) (3.33)

with T=x/t and —1=<r<so.From (3.33) one gets for the electric field:

E=—a~4=1/t. (3.34)

In the isothermalcase,‘y = 1, the electric field is spaceindependentand doesnot vanishfor x —* + so; inaddition, it hasa singularity at t = 0. If one initially assumesa densitystep, ions beinglocatedat thediscontinuity x = are acceleratedto infinity by the infinite electric field as soonas t> 0. The iondensitydecreasesexponentiallyfor t> 0 andextendsto infinity. The ion velocity increaseslinearly in xfor a given t and, hence,reachesvery largevalues for largex (v —÷ +so for x—~+so). The rarefactionfront, on the other hand,propagateswith ion sound speedtowards the unperturbedplasma.In theisothermalcasethe self-similar solution doesnot exhibitan ion front, andions initially locatednearthedensitystepare acceleratedaccordingto [36,46]:

= ln(t/t~)+ 00 , o,~= const . (3.35)

Numericalinvestigationsof the plasmaexpansionproblemwith chargeseparation,whereindeedanion front occurs,claim that the time behaviourof the ion front velocity can be describedby eq. (3.35)(e.g. [46]). We shall comeback to this point in section10.

We terminatethis subsectionwith someremarksconcerningthevalidity of the self-similartheory andthe correspondingsolutions (3.24)—(3.27), respectively (3.31)—(3.34). In general, the existenceofself-similar variablesimplies the lack of characteristiclengths and times [53,54]. In the plasma acorrespondingcharacteristiclength is the Debyelength. Thequasi-neutralityassumption,Tie = n, beingimposed at the beginning of this subsection,takes careof the fact that the Debyelength looses itsimportance as a characteristiclength. By this way, it was possible to investigatethe self-similarbehaviourof the plasmaexpansionproblem. On the otherhand, the group theoreticalansatzfor thegenerationof self-similar variables can be consideredas a mathematicaltool to reduce partialdifferential equationsto ordinary oneswithout takingcare of characteristiclengthsandtimes. This hasbeendoneby Shen andLonngren[44] (seealso [61])for the isothermalplasmaexpansionwith charge


separationwherethe Debyelength comesinto play. In contrastto the quasi-neutralcasethe resultingsystemof equationscan no longer be solved analytically; solutions are obtainedonly under veryrestrictive assumptions[48].

In any case, the existenceof self-similar variables has to be checked by meansof the grouptheoreticalformalism. In caseswherephysically relevantself-similar solutionsare found, they usuallydescribethe asymptoticbehaviourof an extendedproblem(see[54,61]). In the nextsubsectionandinsection10 it will be shown that this statementholds for the plasmaexpansionproblem,too.

Next we turnto the theoryof simple wavesandof Riemanninvariants,anddiscussthe propertiesofsolutionswhich contain the self-similar behaviourof the plasmaflow as a specialcase.We no longerinsist on the quasi-neutralityassumptionwhich was necessaryto get the resultsin this subsection.

3.2. Simplewavesand Riemanninvariants

In order to formulatetheplasmaexpansionproblemin analogyto thedescriptionof one-dimensionalpropagatingwavesof arbitraryamplitude in ordinaryhydrodynamics(compare,e.g. [53]), we requirethat the velocity, the densityand the potentialarefunctionsof eachother. It is, furthermore,assumedthat all quantitiesare piecewisemonotone.Undertheseassumptionsthe equations(2.22) and (2.23)

a,n + a~(nv)= 0,

a1v + o a5v =

transform to

an d(nv) an—+ —=0 (3.36)at dn ax

av / d4\au(3.37)

We do not haveto considerthat the potential t~ follows from Poisson’sequation.In view of

an/at ax au/at axand

an/ax at au/ax at

it follows from (3.36) and (3.37):

ax/atI~= d(nu)/dn= u + n dvldn (3.38)

ax/at~0= u + d4/du. (3.39)

The densityn uniquely determines u. It does,therefore,not matterwhetherthe derivativeis takenatconstant n or u, so that

ax/atI~ ax/at~~. (3.40)

328 Cli. Sackand H. Schamel,Plasmaexpansioninto vacuum— A hydrodynamicapproach

From the equations(3.38) and (3.39)we get

n dv/dn= d4/dv = (d4ildn) (dn/dv) (3.41)

and

dv/dn= ±(1/n)\/~~7’. (3.42)

Analogousto ordinaryhydrodynamics[53]we identify the expression\/~d4.ildn in (3.42)with a soundvelocity c, so that

cas\/~d4/dn. (3.43)

In the following we shall usethe term “pseudo-soundvelocity” for c in (3.43) to distinguishit to thesoundvelocity in an ideal neutralgas. The relation to this sound velocity follows from (3.43) as aspecialcaseby assumingquasi-neutrality,ne = n, andby differentiatingeq. (2.25) which is resolvedfor4 with respectto n (see also eq. (3.22) and ref. [53]):

c~:=~j’~ ~(~‘)‘2~ (3.44)

In the caseof isothermality, y = 1, CON becomesindependentof the density,e.g. CON 1. In theunperturbed,homogeneousplasma,wherethe ion densityequalsunity, it holds:

CQN(n—1)V~ . (3.45)

Assumingthat the potential~ is dependenton x only throughtheion densityn, oneobtainsfrom (3.43)the actual local soundvelocity of the system:

2 a,b/ax nEC (x, t) = n an/ax = — an/ax (3.46)

Equivalentto the above mentionedformulation of the plasmaexpansionis the replacementof theelectrostaticfield E by a pseudo-pressureJ3’ in the ion momentumequation(2.23)

E=—1a~p=—a~~. (3.47)

Using the well-known definition for the soundvelocity c2 asdp/dn (see [53]), we obtainfrom

C2=j~= ak/ax (348)dn an/ax .

andfrom (3.47) again the equation (3.46) (seealso [56]). Returningto eq. (3.42), its integrationwithrespectto n orp

o=±J~dn=±J ~ - , (3.49)n n(p)c(p)

Cli. Sackand H. Schamel,Plasmaexpansioninto vacuum— A hydrodynamicapproach 329

yields a relation betweenthe velocity and the ion densityor the pseudo-pressure,respectively.From(3.47) andeq. (3.49) which is differentiatedwith respectto ~, it follows:

(3.50)

By meansof (3.50) we substituted4i/du in eq. (3.39) andintegrateover t:

x = [v ±c(v)] t + x0(u). (3.51)

The function x0(v) is an arbitrary function of the velocity being determinedby the initial conditions;c(u) is given by eq. (3.49).

The equations(3.49) and (3.51) representthe general solution of the problem and correspondformally to the solutionsfor propagatingsimplewavesof arbitraryamplitudein anideal gas(Riemann(1860), see also [53]). Both signs in (3.51) describewaves, moving relatively to the medium in thenegativeor in the positive x-direction. In contrastto the gas dynamiccase,the u-dependenceof thesoundvelocity can no longer be describedanalytically in the caseof chargeseparation(seeeq. (3.49)and (3.50), respectively).

In the gas dynamic(quasi-neutral)case,we obtain for the u-dependenceof c (see [53]):

c(v)~(1±~v). (3.52)

In the contraryto section3.1, wherewe haveinvestigatedthe self-similar behaviourof a quasi-neutralplasma,the xt-dependenceof the velocity v and, therefore,of the otherquantities,too, is only givenimplicitly. The self-similar solution is obtainedas a specialcaseof a simple wave, in which x0(u) in(3.51) vanishes(see(3.51) in comparisonto (3.23) and (3.25)).

The velocity of the pointsof the waveprofile describedby (3.49) and (3.51)

u=u±c (3.53)

hasto be understoodas the superpositionof the propagationof a disturbancewith soundvelocityrelative to the medium and of the motion of the medium itself with the velocity v. Becauseof thedifferent local velocities, the profile can change its shape in the courseof time. If there is acompressionalwave,i.e., a certainregion in which the velocity v, the densityn andthe pressurep aredecreasingin the direction of propagation,the waveprofile will progressivelysteepen.Finally, thewaveprofile can bendout of shapein sucha way that the dependentflow variables(e.g. n, u) areno longerdefineduniquely in spacefor fixed t. Threedifferent valuesof n and v, respectively,arebelonging tothe same point x. Such a processis called “wave breaking” and opposesthe foundationsof thehydrodynamicdescription.In fact, therearestrongdiscontinuitiesandshockwaves,respectively,in theflow which are characterizedby singularitiesin the derivatives and by jumps in the flow variables.Exceptthesepoints and surfacesof discontinuity, the spacedependenceof the flow profile is uniqueand can be describedfurtheron by the equationsfor an ideal gas (see [53]).

In the casewhere surfacesof discontinuityare formed in a flow, the wave is reflectedfrom thesesurfaces.Therefore, the wave is no longer propagatingin one direction, and the uniquedependence


betweenn, u andj3’ in the equations(3.49) and(3.51) is lost. This meansthatgenerallythesolution inthe form of a simple wave is only valid for a finite time interval. In the specialcaseof a rarefactionwave,however,in which the density increasesmonotonicallyin the direction of wavepropagation,thesolution is valid for all times. Such a waveis obtained,e.g.,whenapistonis drawnout of an infinitelylong tube filled with gas [53].

The occurrenceof the discontinuity is connectedwith a singularity in the first derivativeof thevelocity. In the casewherethe systemdoesnot contain anyadditionalinternalboundaries,the secondderivativealso becomessingular. Therefore,it is possibleto determinethe instant andthe location atwhich the waveceasesto exist. For the inversefunction x(v, t) it must hold [53]:

ax/av~,= 0, a2x/av2j,= 0. (3.54)

For the gasdynamic(quasi-neutral)case,wherec(v) is given by (3.52),one obtainsby differentiation

of (3.51) with respectto o from the conditions (3.54):

2 ~ (3.55)~ y+1 av ‘ av2

The steepeningtime t5 dependson the polytropic exponenty, and on the slopeof the function x0(v)

obtainedfrom the generalsolutionof (3.51) whereax0/av<0 representsa compressionalwave.Eachcompressionalwavesteepensin the courseof time anddevelopsdue to (3.55) at the time t~averticaltangentin the velocity profile v(x).

In ordinaryhydrodynamicsthe presenceof suchdiscontinuitiesin an otherwiseideal flow producesan energydissipationwhichcausesa strongdampingof the wave.Thestrengthof thedissipationcanbedeterminedby applying the conservationlaws of mass,energyand momentumto both sides of thediscontinuity yielding the Rankine—Hugoniotequations[53]. Furthermore,the appearanceof dissipa-tive effectscausesan entropyincreaseand,therefore,irreversibility. In the plasma,however,dispersiveeffects arisingfrom chargeseparationmay counteractthe progressivesteepeningwithout affecting thereversibility. Montgomery [34], who investigatednonlinear ion acoustic waves, suggestedthat thesteepeningof the wavecan be balancedby dispersion.Chargeseparationeffects comeinto play whenthe scale length of the electrostaticpotential is comparableto the Debyelength. This suggestioniscertainly true for small amplitudeperturbationsand weak inhomogeneity,respectively (seeSagdeev[55]). However, if there is an extremelyinhomogeneousand nonlinearexpansionof a plasmaintovacuum,which is typical for the initial stage,unlimited steepeningandthe resultingwavebreakingwilloccurdespitedispersion.This will be shownin sections5 and6. An indication for this statementis thegeneralformulation of the expansionproblemwith chargeseparationas given by (3.49) and (3.51). Incontrastto the quasi-neutralcase[see(3.55)], the steepeningtime t5 cannotbe calculatedin a simplemanner,since c(v) is not given analytically.

A furtherrelationbetweenthe plasmaexpansionproblemandordinary hydrodynamicsresultsfromthe constructionof Riemanninvariants (see [531).For this purposewe use the following trick andreplacethe electrostaticfield in the ion momentum equationby —(c

2/n)a~n,where c2 is given by(3.46). For the hydrodynamicion equations(2.22) and (2.23) we obtain:

a,n+oa~n+na~o=0 (3.56)


a1u+ua~u+Ea~n=o. (3.57)

By multiplying eq. (3.56) with ±cln,and by addingit to (3.57), one obtains:

[a1+(u±c)a5]R+=0 (3.58)

with

R := v ± f dn’ c(n’) = ~± f dp’ (3.59)± J n’ J c(p’) n(p’)

R~andR are the Riemanninvariants. The differentialoperatorsactingon R~andR turn out to bethe derivativesin the direction of the characteristicsC~and C in the xt-plain, given by

(dx/dt)~= u + c, (dxldt)_= u — c (3.60)

(seealso (3.53)).On eachcharacteristicC.. andC, respectively,the correspondingRiemanninvariantremainsconstant.

This kind of formulation representsan alternativeto the numerical treatment of the expansionproblem by finite difference methods. By meansof the Riemanninvariants the hydrodynamicionequations(3.56) and (3.57) can be formulatedas a Cauchyinitial valueproblem,andcan be solvedbythe methodof characteristics.In the quasi-neutral,isothermalcase(y = 1), whereR.,. is given by

R~=u±lnn (3.61)

this was done by Felberand Decoste[9].The comparisonwith eq. (3.49) showsthe correspondencebetweenRiemanninvariants(3.59) and

simple waves.With R~asO we obtain from (3.59)

— f dn’ c(n’)n’

from which follows dueto (3.46)

a~u=~a~n=±~E. (3.62)

The replacementof E in the ion momentumequation(2.23) by ±ca~vyields finally:

[a1+(u~c)a~]u=0. (3.63)

With (3.63) wethushavesucceededin formulatingthe ion momentumequation(a1 + u a~)u= E in theform of a simple waveequation.The pseudo-soundvelocity c(x, t) definedby (3.46) is introducedbythe analysisof simple wavesof arbitrary amplitude [see(3.49) and (3.51)] or by the constructionof

332 Cli, Sackand H. Scharnel,Plasma expansioninto vacuum— A hydrodynamicapproach

Riemanninvariants[see(3.58) and(3.59)]. The pseudo-soundvelocity canbe obtainedas a function ofx and t by evaluatingthe numericaldatafor the ion densityn and the electrostaticfield E, the latterfollowing from Poisson’sequation.It can alsobeusedto characterizethe stateof chargeseparation.Weshall come back to this point in section5. Furthermore,c(x, t) from (3.46) allows a conventionalstability analysisof the numericaldifference scheme(seesection8 and appendixD).

In the next section we discussthe foundationsof the numerical methods for solving the hy-drodynamicion equations(2.22), (2.23)andPoisson’sequation(2.24).Of specialimportancewill bethederivationof the boundaryconditionsadoptedto the finite spaceinterval.

4. Numerical methods and analysis

4.1. Lagrangianequationsof ion motion

In many hydrodynamicproblems,especially in one-dimensionalsystems,it is useful to transformEuler’sequationsof motion to a coordinatesystemmoving with the local flow velocity. In this so-calledLagrangianpicture the equationsof motion get a very simple form, which can be handledmoreeasily,analytically and numerically. Often the analytical investigation of hydrodynamic problemsis notpossible until Lagrangiancoordinatesare introduced [62,63,64]. In section 6 we shall discusstheconsequencesof such a formulation in the concretecaseof the plasmaexpansion.

For the derivationof the Lagrangianequationsfrom the Eulerianpicturewe assigna certainlabel ~to each fluid element; ~ is the Lagrangianspace-coordinateand representsthe position of a fluidelementat the time t = 0. The relationbetweenthe space-coordinatex andthe local flow velocity u isgiven by the equationof motion

dx(t)/dt = u(x(t), t) . (4.1)

The integrationof eq. (4.1) yields implicitly the coordinatex of the fluid elementfor t >0:

x(t) = ~ + v(x(T), T) dT (4.2)

with

v(x(r),r)=v(~+Jo(x(r’),r’)dr’,r)as5’(~,r). (4.3)

Fromthe equations(4.2) and(4.3) we obtainthe transformationof the Eulerianvariables(x, t) to theLagrangian ones (~,T) [42,50,56, 62]:

x=x(~,r)=~+Jff(~,r’)dr’ (4.4)

t=t(~,r)=r. (4.5)

Ch. Sack and H. Schamel,Plasmaexpansioninto vacuum— A hydrodynamicapproach 333

Accordingto (4.4) and (4.5), respectively,the partial derivatives a1 and a~transformas follows:

a a a d

(4.6)

= (i + a5’(~,r’)dr’) ~. (4.7)

Equation(4.6) definesthe convectivetime derivative.The transformation(4.7) is uniquely determinedonly if

J a5’(~’)dr’ � -1. (4.8)

The equalitysign meansthat in the xt-spacetwo particletrajectoriescrosseachother. In this casethedependentvariablesareno longer unique,andthe plasmacannotbedescribedby ideal hydrodynamics(seesections5 and6).

Inserting the equations(4.6) and (4.7) into the ion continuity equation (2.22) and the ionmomentumequation(2.23),we obtaintogetherwith eq. (4.1) the descriptionof the ion motion in theLagrangianformulation [42,50,56]:

x(~,t) = u(x(~,t), t) (4.9)

n(x(~,t), t) = n~(~)~ (4.10)

v(x(~, t), t) = E(x( ~, t), t) (4.11)

with the initial conditions given by

n(x(~,0),0) = n0(~) and x(~,0)=

For solving the continuumproblemnumerically,the equations(4.9)—(4.11)arediscretizedin spaceandtime [65,66]. An essentialadvantageof the Lagrangianformulationis the absenceof thenonlinearconvectiveterm u a~u,which causesmost trouble in the Eulerian picture. Theequationsof motion caneasily be discretizedvia an explicit Lagrangiandifferencescheme.This procedureexcludesnumericaldissipationanddispersionwhichusuallyleadto an undesirablestrongsmoothingof the solution.Suchafalsifying influencemustbe expectedif the Eulerian equationsof motion arediscretizedby an explicitLax scheme[65,66]. The Lagrangianmethodalsoguaranteesthat the differencemeshis coupledto thelocal flow velocity, so that stronginhomogeneitiesin the velocity and in the densityprofile areresolved’in an optimummanner.Furtherdetailsof the discretizationprocedureareto be found in appendixC.The explicit differenceschemeturns out to be a specialcaseof the more generalformulation.

For solving the equations(4.9)—(4.11)the knowledgeof the potential~ and of the electric field E,respectively,is required. In the next subsectionwe presenttwo solution methodsfor the nonlinearPoisson’sequation.


4.2. Two iterative methods for the nonlinear Poisson’s equation

After replacingthe electron density by the electrostaticpotential, Poisson’sequationbecomesanonlineardifferentialequation[excepty = 2, seeeqs. (2.24) and (2.25)1. Its numericalsolutioncausessome problems with respect to the numerical procedureand to the formulation of the boundaryconditions.Becauseof the nonlinearterm [1 + ((y — 1)Iy)~]’~’~’~,y = 1 and ed’, y = 1, respectively,the electrostaticpotentialis not availableby a direct spatialdiscretizationof Poisson’sequation(2.24),evenif the ion density is known.To getrid of this difficulty, it is necessaryto developsuitableiterationprocedures.

In the first iteration procedureusedby Mason[38], the potential 4 is replacedby the electric fieldE = —4’. Differentiating Poisson’s equation (2.24) with respectto x, and expressingthe electrondensitytie by n — E’ yields a differentialequationof secondorder for the electric field:

E”—~(n—E’)2~E—n’=0. (4.12)

In the isothermalcase,y = 1, eq. (4.12)simplifies to that equationsolvedby Mason[38].Insofar (4.12)representsan extensionof the model to more generalpolytropic equationsof state.

In implementing the iteration procedure,we assumethat the electric field E is known for theiterationstepp ateachgrid point i. In orderto obtainE for the nextiterationstepp + 1, we insertE ofthe pth iteration stepat the grid points i + 1 and i — 1, andsolvewith respectto E,. The advantageofthis procedureis that we get E at eachgrid point i by solving asimple algebraicequation.The detailsofthis iteration scheme,especially the discretizationof (4.12) on a non-equidistantgrid, have beendescribedin ref. [56] for the isothermalcase,y = 1.

The seconditerative procedure,which yields the potentialas the solution, usedthe Newtonmethodand was developedamongothersby Dewaretal. [67] andForslundet al. [68]. It is basedon a Taylorexpansionof the nonlinearelectrondensityunderthe conditionthat 4 can bedescribedby 4 = +

~. Linearizing Poisson’sequation(2.24) with respectto ~, gives

dnc~’~ t~~——4~+n(4~)—n (4.13)

‘I’D

with

— 1 lI(y-l)

Tie(~o)=(i+ ~ y~1 (4.14)y1

and

1 — 1 (2—y)/(y—l) 1dn = - (i + 7 = - [ne(~o)]2~, ~ ~ 1~ (4.15)

d~0 ~

y1

A version which can be handled numerically more easily, is obtained by subtracting the term

Cli. Sackand H. Schamel,Plasma expansioninto vacuum— A hydrodynamicapproach 335

(dneldq5o) q50 from both sides of (4.13):

dn dn

(4.16)

In contrastto (4.13), eq. (4.16) containsonly one spatialderivativeof secondorder. In addition, eq.(4.16) represents the more suitable form to implementtheboundaryconditionswhich are formulatedinq~instead in 4’,.

The spatialdiscretizationof eq. (4.16) yields a tridiagonalcoefficientmatrixfor the potential4’ at thegrid point i, the first andlast row of which arefilled up by the boundaryconditions(seesection4.3). Adetailed description of the discretization procedureon the non-equidistantgrid will be given inappendixB.

Prescribingthe initial value4~andthe boundaryconditions,weobtain a preliminarysolutionfor 4’,which is insertedinsteadof 4~in the next iterationstep. In the caseof convergence,i.e., 4’ — 4.~—>0,oneobtainsfrom eq. (4.16) the original Poisson’sequation(2.24).For t = 0, the quasi-neutralsolutionis used,given by

~‘ (n~—1), y~14’0(x,t=0)= y—1 . (4.17)

lnn, y1

It is found, however, that the iteration procedureis rather insensitive to the choice of the initialfunction. Testcalculationswith 4’0(x, t = 0) as0 havegiven identical solutionscomparedto those with(4.17) as initial function. However, more iterations have been necessaryto gain convergence.Therefore, eq. (4.1) has been used as initial function in all of the calculationswithin the seconditerationprocedure.For t > 0, 4~is identifiedwith 4’ from the precedingtime stepasthe initial functionfor the iteration.

An essential advantage of the second iterative procedure is the fact, that besides 4” further spatialderivatives as, e.g., n’in eq. (4.12) are not involved. Thus, the convergence is not deteriorated by aninaccurate calculation of possibly steep gradients and of small scale structures. In section 5 we shall seethat this property of the iterationprocedureplays an important role for the resolution of nonlinearstructures occurring in the expansion problem. Moreover, the seconditeration procedureconvergesconsiderablyfaster than the first one. After ten iterations, e.g., the maximum relative error (seeappendixB) amountsto about 108. Using the first iteration procedure,many more iterationsareneededto reachthis degreeof accuracy[56], especiallyin the region of low density.Becauseof theseproperties we prefer the second iteration procedure, henceforth called “Poisson solver”.

We notethat the seconditerationprocedurecan beapplied to anyequationof statefor the electrons[68]. In the presentcase,only thoseequationsof stateareconsideredwhichimply anelectronfront atinfinity. In section2.2 wehaveshownthat this is the casefor 1 ~ y �2. For y >2 (andfor an equationof state describing trapped electrons, respectively), dfle/d4’0 in eq. (4.16) becomes singular when theelectron density ne goes to zero [seeeq. (4.15)]. In this case,the Newton procedure cannot be usedappropriately.A way out of this dilemmais offeredby the fitting procedureof Crow et al. [42],wherethe ions are given as a stepfunction. In this casea numericalsolution of Poisson’sequationin theregion of the pure electron cloud is not necessary since there exists already an exact analytical solution(seesection2.2). Therefore,Poisson’sequationhasto be solvednumericallyonly in the region where


the ion density is non-vanishing,for which the seconditeration procedurecan be usedagain. Theresulting solution is thenmatchedto the analyticalsolution from the pure electronregion.

In thisand in the last subsectionwe havediscussedthe basicpropertiesof the proceduresfor solvingPoisson’s equation and the hydrodynamic ion equations. A complete solution of the equations(4.9)—(4.11)and(4.16),however,is not possibleuntil the generalboundaryconditions,eqs. (2.27),areadaptedto the numericalsolution method. In the next subsectionwe shall derive such adjustedboundaryconditions.

4.3. Boundaryconditions

It is clear that numericalsolutionscan only be obtainedin a finite integrationinterval. In order tosolvesystemswith boundariesat infinity, onehasto transformthemto a finite interval. In the presentcase,the length of the integration interval is 2L, — L ~ x c + L, at the beginning. The boundaryconditionsfor this finite intervalat x = ±L haveto beformulatedsuchthat the boundaryconditionsofthe continuousproblemaresatisfiedautomatically,whenx—> ±so~This conceptleadsto so-calledopenboundaryconditions,which are representedby differential equations.Thereby it is not the functionitself which is prescribedat the boundaries,but its asymptoticbehaviourin differential form.

Firstly, we investigatethe asymptoticbehaviourof the system(2.22)—(2.25)formulatedin Eulerianform for the undisturbed,homogeneousplasma,x—~—so andx = —L, respectively.It is assumedthatthe density, the velocity and the potentialvary only weakly in this region so that linearizationof thesystemwith respectto smallperturbationsis justified. We write

n=n0+5’, v=u0+5’, 4’=4’~+4, ne=neo+ne, (4.18)

wheren0 = neo = 1, 00 = 0, and 4’~= 0 aregiven by their valuesat x = —so [seeboundaryconditionsinsection2.1, eqs. (2.27)]; the perturbedquantitiesaremarkedby tildes.With (4.18) wederive from theequations(2.22)—(2.25) the linearizedsystemof equations

a1i~+a~5’=0 (4.19)

a,5’= —a~4’ (4.20)

a~=1-cb—n. (4.21)

Making a Fourieransatz

f=~f~exp[i(kx—wt)]

with (4.22)

Jss(,~,5’, ~) and fk as(ilk’ Uk, ~k)’

andinsertingit into (4.19)—(4.21),we obtainthe well-knowndispersionrelationfor ion acousticwaves:

2 yk ±_÷ ~f~k(0 = resp. ~k —— wkE’~. (4.23)k 1+

7k

2 \/1+

7k

2


As seenfrom the definition, w~and Wk differ in the sign in front of k, i.e., w(w~)representswaves

propagatingto the right (left),(4.24)

Demandingthat no wavesshouldcomein from x = —so atx = —L, we obtainfor the perturbation4’ ofthe electrostaticpotential

4’ = ~ ~ exp{i(kx + wkt)} + ~ ~kexp{i(kx + w~’t)} (4.25)k>0 k<0

with w~>0 for k>0, and Wk <0 for k<0. From the reality property of 4’, it follows

= 4’k~ (4.26)

whereuse is madeof Wi~ik= ~ = ~ and asteriskmeanscomplexconjugation.Equation(4.25) thenbecomes

= ~ ~k exp[i(kx + ~ ~k2 ~)] (4.27)

which in the long wavelengthlimit (k ~ 1) reducesto

~—~4’kexp[i(kx+v5~ kt)]. (4.28)

Becauseof k~ 1 the dispersiveterm beingproportionalto k3 hasvanished.(It is well-known that thisterm representsdispersion in the Korteweg—de Vries equation [62] which is derived for weaklynonlinearwaves.)

From (4.28) we immediately get by differentiationthe correspondingdifferential equationwe arelooking for:

a,çb— \/~ a~cb= 0, (4.29)

where4’ is replacedby 4’ [see(4.18)]. With E = —4” we also get

a1E—\/3a~E=0, (4.30)

andthe asymptoticbehaviourof the velocity in the unperturbed,homogeneousplasmais described,ofcourse,by a differential equationof the samekind:

a1u —\/~ a~v=0. (4.31)

The space-timediscretizedversion of (4.31) is used to determineu at the left boundaryx= — L in thenumericalsolution (seeappendixC). Thus,it is not the valueof v that is prescribedbut its variation, indifferential form. Onehasnot to distinguishbetweentheEulerianandthe Lagrangianpicture,sincetheterm u a~vis lost after linearization.

338 Cli. Sackand H. Scliamel,Plasma expansioninto vacuum— A hydrodynamicapproach

In principle it is possible to use (4.29) as a boundarycondition for Poisson’sequationat x= — L.Since in Poisson’sequationtime appearsonly as a dummyvariable,we havechosenanotherboundaryconditioncontainingonly spatial derivatives.This so-called“gradientboundarycondition” hasalreadybeenderivedby Crow et a!. [42] for the caseof the isothermalplasmaexpansion.

We demandagain that the density and the potential at the left boundary,x = — L, vary weakly.Furthermore,we assumethat the variation of the ion densitycan be describedby a power seriesin 4’with time dependentcoefficients a,,, v >0, ii E ~J:

n=1+E, e=~a,,4’~. (4.32)

Poisson’sequationlinearizedwith respectto 4’ (2.24) then reads:

— 1 l/(y—l) 1 1~ 4’) —n1+—4’+...—(1+ai4’+...)~(——a1)4’. (4.33)

After multiplication of (4.33) with 4” and integrationusing the boundaryconditions4’ = 4” = 0 forx—>—so we obtain:

(4.34)

Since the right-handside of (4.34) is constantin x, we can write

= 4”4’~-L+~~, (4.35)

wherez~xdenotes the spatial step. In a similar mannerwe obtainthe gradientboundaryconditionforthe E”-equation (4.12) (seealso ref. [56]):

E’/ELL = E’/ELL÷~X . (4.36)

From the spatial discretizationof (4.35) and of (4.36) we get for 4’ and E the left-handboundarycondition. It dependson the choiceof the iterationproceduresdescribedin section4.2. Note, that inthe contrary to (4.29) and (4.30) the relations (4.35) and (4.36) are nonlinear. Consistentwith thecorrespondingiteration procedures,the boundary conditions (4.35) and (4.36), respectively, areiteratedas well. Hence,the behaviourof the plasmaquantitiesat the left-handboundaryis completelydetermined by (4.31), (4.35) and (4.36), respectively.

In the vacuum region, x—> + so and x= +L, respectively, the boundary conditions must bedeterminedseparately.Again it is our intention to derive differential equationsof the first order,describingthe asymptoticbehaviourof thesolutionfor x—> + so correctly.A differentialequationof thiskind for the potential has already been derived in section 2.2. This equationresulted from theintegrationof Poisson’sequationby neglectingthe ion density:

4’F2 — 1 y/(y—l) E2

~=(i+~ 4’) =~=n~ (4.37)


with Tie from (2.25). For 4” it follows from (4.37):

= —“/~ n,,~’2. (4.38)

A further integrationof (4.38) to determinethe boundaryconditions is not meaningfulbecausetwointegrationconstants,xf and4’~,are introduced,which arenot availableuntil the solution of the entireset of equationsis known (seesection2.2).

According to the preparationof the seconditeration procedure,eqs. (4.13)—(4.16),the nonlinearterm n~2in (4.38) is linearizedby a Taylor expansion:

n~2asF(4’)~F(4’0)+~4’i; 4’=4’~+4’~, 4’1HI4’0~ (4.39)

with

F(4’0) = [n,,(4’0)]~”2 and dF/d4’

0=

We insertn~2from (4.39) into (4.38) andadd on both sides the term (1/V’~)[ne(4’o)]1’~”24’o:

4” + ~ [(4’~)]~4’ = ~ {[n~(4’~)]~ - 1[ (4’)]1~2 4’o}. (4.40)

The spatial discretization of (4.40) determinesthe boundary condition for Poisson’s equation atx= +L. Consistentwith the differenceschemefor (4.16),eqs. (4.35)and (4.40) fill up thefirst and thelast row of the tridiagonalmatrix, the solution of which yields the potential 4’ at the grid point i [seeappendixB].

Usingthe first iterationprocedure,we obtainthe correspondingboundaryconditionforx = + L fromthe integrationof the E”-equation(4.12) with n = n’ = 0:

E’ + (E2I2)1~= 0. (4.41)

The velocity v at the right-handboundary,x = +L, is determinedby thespatial averageof the equation[seeappendixC]:

du/dt=E. (4.42)

The boundaryconditionsatx = ±L, are,therefore,given completelyby eqs. (4.31), (4.35)and(4.40),(4.42), respectively.

In the next sectionwe formulatethe conservationlaws derivedfrom the set of equations(2.22)—(2.25), which turn out to be important tools for testing the reliability and the accuracyof numericalsolutions.

4.4. Conservationlaws

Multiplying the continuity equation(2.22) with u212, and the ion momentumequation(2.23) withno, andaddingthe resulting equations,one getsin the Euleriandescription

a,(nu212) + a~(nu3I2)= nuE. (4.43)

340 Cli. Sackand H. Schamel.Plasma expansioninto vacuum— A hydrodynamicapproach

In contrastto anoppositestatementmadeby Mora et a!. [47]andby Trulsen[69],it will beproventhateq. (4.43)can be formulatedin conservativeform. Eliminatingno through tieVe — a~Eby Ampere’slaw,andusing the electroncontinuityequation,one gets

a1(nv2/2+ E2/2) + a~(nv3/2+ 4’neoe) = 4~atne . (4.44)

It is worthnoticingthat thisequationholdstrueevenif theelectroninertia is included.In theinertialesslimit the right-handside of (4.44) can be expressedas a partial time derivative

—4’ a,n~= —atFe(4’), (4.45)

where

Fe(4’) = ne(4’) 4’ — Pe(4’), (4.46)

noting thatp~(4’)= tie(4’) and ne(4’) is given by (2.25). We therefore,arrive at the following energy

conservationlaw

a,[nv2/2 + E212+ Fe(4’)] + a~[nv3/2+ (neve)4’] = 0, (4.47)

which is in conservativeform.The quantity ~‘,, definedby

= J dx [nv2/2+ E2/2+ Fe(4’)], (4.48)

doesnot changein time. Anotherconservativeformulationof the energylaw can be found. First, wenotice that Fe expressed in terms of Pc becomesin the limit of inertialesselectrons

Pc 7 1/y

Fe(Pe)= y1 — y—1 Pc ‘ y1 , (4.49)p~(lnp~~l), 7=1

whereuse is madeof p~= n~and of eq. (2.25).With this form of Fe, eq. (4.47) transformsto

+ ç + y~1) + a~[~+ Tie Ve (4’~&)] =0 (4.50)

with 4’~= 4’(x = so, t) = —y/(y — 1), y ~ 1 (see(2.27)).The quantity ~‘2

~2 Jdx[no2/2+E2/2+ ~], (4.51)


is constantin time as well. It representsthe total energyand reflects the balancebetweenion kineticenergy,electrostaticfield energyand internal energyof the electrons.

On the other hand, taking into account electron inertia effects, it follows from the electronmomentumequation(seeeq. (2.16))

= fle[1 + 6E’ (arve + Oe axve)], (4.52)

where6 = me/mi. Equation(4.44) thenbecomes

a[ç + + Fe] + a5[~ + fleVe4’] + 6[TieE1 a4’ (a

1v~+ o~axue)] = 0. (4.53)

In the inertialesslimit, 6—*0, we recoverthe old expression(4.47).A straightforwardcalculation shows that (4.53) is equivalentto the following conservativeformu-

lation:

a,[ç+6 ~.~+ç+‘~1]+a1[~+o ~-~+ ~‘1 VePe]reO, (4.54)

which is the well-known expressionfor the energyconservationusinga polytropic electronequationofstate.This equationreducesto (4.50) in the limit 6—*0 usingPc n~and

7 7 / y—1\

y—l VePe= )‘l neuel~1+ y

Similarly, the momentum can be expressed in conservative form,

a,(nv) + a~[nv2 + Pc — E2/2] = 0, (4.55)

whereE = —~~4’and Poisson’sequationhasbeen usedto replace—na~4’by a~[E2/2— PeL

The total momentum

p= fnudx (4.56)

increaseslinearly in t:

P(t) = t, (4.57)

which can be seenby calculatingP(t) andusing the boundaryconditionp~—~1 for x—> —so, noting thatthe otherquantitiesvanishat the boundaries.It is the electronpressureat minus infinity whichpushesthe ions and gives rise to a monotonicincreaseof the total momentum.

The conservationlaws (4.47) and (4.50), respectively,and (4.55) are in the appropriateform forimplementationin the numericalcode.

342 Cli. Sackand H. Scliamel, Plasmaexpansion into vacuum— A hydrodynamicapproach

4.5. Variational principle

The energyconservationlaw (4.47) suggeststhe following Hamiltonian

H = J dx [no2/2+ (a~4’)2/2+ Fe(4’)] (4.58)

for the formulationof a Hamiltonian principle.Introducing the potential representationof the curl-free velocity field (Clebsch[70], Elsässerand

Schamel[71])

o = —asp , (4.59)

we get by variation with respectto ~, n and4’ underthe constraintthatthe 4’ variation is dependentonbn and satisfiesPoisson’sequation,the following Hamiltonian set of equations

a,cp=~H/~n=v2/2+ 4’, (4.60a)

a1n = —~HIF~p= —a~(nv). (4.60b)

The first one is the well-known Bernoulli equation,that is the integratedform of the momentumequation,and the secondone is the continuity equation.We note that the Hamiltonian density~1Cin(4.58)or the equivalentformula in which Fe(4’) is replacedby (4.49)can be written up to a constantas

= [n 02/2+ E2/2+ ne W(ne)], (4.61)

where

W(n~):= —J (Pc — 1) d(1/fle), (4.62)

is the compressionalenergyof the electronsper unit fluid mass(Elsàsserand Schamel[71]).The Lagrangiandensity from which the Hamiltonian densitycan be derivedreads

~t= n[~ - ~(a~~)2]- ~(a~4’)2- F~(çb), (4.63)

from which immediately follows that ~ andn are canonicallyconjugatedvariables

pas6L/&~b=n, (4.64)

being alreadyimplied by (4.60).The Euler—Lagrangianequationsminimizing the functional j’ dt J dx ~ areagaingivenby the set of

equations(4.60a,b) and(2.22), (2.23), respectively,wherethe~4’variation is, as before,relatedto ~nunder the constraintof Poisson’sequation.

Ch. Sack and H. Schamel,Plasma expansioninto vacuum— A hydrodynamicapproach 343

Hence,we seethat our basicequations(2.22)—(2.25)can be completelyderivedfrom a variationalprinciple.

In the next sectionwe presentthe numericalsolutionsobtainedby the dissipationlessnumericalschemedescribedin this section.Our emphasisis to clarify the role of chargeseparation,andof chargeneutrality, respectively,andof the electronthermodynamicson the dynamicalevolutionof the system.In view of the intimate relationshipto ordinaryhydrodynamicsas discussedin section3, it will not besurprising that almost all solutionsexhibit discontinuities.We shall describethesediscontinuitiesandshall offer ansätzeto understandthe singularbehaviourof the plasmaflow.

5. Numerical resultsfor the dissipationlessplasmaexpansion

Figure 1 showsthe space-timedependenceof the ion densityin the isothermalcase,y = 1, takinginto account chargeseparation.The global behaviourof the ion density for threedifferent times isshown in fig. la. The framedpart in fig. la is drawnon an enlargedscalein fig. lb for five successivetime steps.Startingwith a diffuse ion densityprofile (1 = 4 in (2.28)), a sharpion front quickly developsin front of whichapureelectroncloud exists.Passingaplateau-likestate,aspike is formedat thefrontgrowing explosively. Finally, at t = t~, 17.94, the numerical solution breaks down without anyindication of a numericalinstability (e.g., oscillationsfrom grid point to grid point). In section8 weshall describesucha caseof numericalinstability. Neitherthe increaseof the numberof grid pointsnorthe introductionof an ion pressureandof variousartificial viscositiescould avoidthe singularbehaviourof the plasmaflow within an explicit discretizationscheme(seealso ref. [56]).

Figure 2 gives a first informationaboutthe origin of the singularity. In fig. 2a the ion densityn, theelectric field E and the ion velocity v of the initial state (v as0 for t = 0) are comparedwith the

n(xt) ri(x,t)tO 0.15

t~130.1 \\ ,‘~

~ \\z-Z~\‘\—‘ - Z~.A7r._ ,,~—

:: a 0 ~ ~~~•-~•ó ~

Fig. 1. Ion densityn asa functionof space and time for y = 1 in the case of charge separation;(a) gIooal behaviour of the densityfor three differenttime steps; (b) expansion front on an enlarged scale, —20~ x < 10.


1.0 _‘_\ v.25

t0 020.75 E

n Q15

0.5 :i0.1

025 ‘~

Q05

a :O ..,.,. . ~•1~- ,‘‘.. 0-200 -100 0 100

to Q25\ /1 3°

t=17 I iL 02

v 11 EI \/i~.\ / t’t~. 20 0.15

0.5 \ I ~\/ ~\ 0.1

S025 ~~ DOS

0 .,,~,....,..,. 0-200 -100 0 100

Fig. 2. Ion density n (solid line), electric field E (dotted line), and ion velocity u (broken line) as functions of x for y = 1 and charge separation attwo different times: t = 0 (a) and t = 17(b) for u(x,0) 0.

correspondingquantitiesat a latertime justbeforethe break-down.At the beginningthe electric field ishump-like. During the elapseof time the velocity profile exhibitsthis structure,too. In the early stageof the evolution, the velocity maximumstays a little bit behindthe ion front causingthe narrowingofthe meshpoint distances~ as x~.÷1— x1. in the region close to the ion front. Due to the conservationofmass given by ni.~1 = const. in the Lagrangianpicture (see eq. (4.10)), the density increasesin thisregion. At the sametime the velocity andthe electric field strongly steepen(seefig. 2b). Themain partof theplasmain the rearbehindthe front approachesthe stateof quasi-neutralityandof self-similarity;this is expressedin fig. 2b by the electric field forming a plateau,andby the linearvelocity increase[seealso eqs. (3.31)—(3.34)].The quasi-neutralself-similar region broadensin the courseof time. A moredetailedpicture of the solution nearthe ion front will be presentedin fig. 7.

Assumingquasi-neutralityfrom the outsetwe obtain a completelydifferent solution, althoughtheinitial densityprofile is the same.Figure3 shownn, E andu for t = 0 (fig 3a) andfor t = 30 (fig. 3b). Atanytime the ion densitydecreasesmonotonically,whereasthe velocity andthe electric field increasesmonotonically.In contrastto the casediscussedabove,y = 1 and chargeseparation,the solution doesnot break down. This behaviouris due to the alteredinitial condition for the electric field. In the


to0.3

t~O E E

02: ~ :‘

—200 -100 0 100

n ~ V~Et 0.3

0.5 \ V\/! 602

02: b ~.‘ .

-200 -100 0 100x

Fig. 3. y= I and quasi.neutrality: t = 0 (a) and t = 30 (b).

quasi-neutralcasewith y = 1, the electric field is given by

E=—a~lnn=—1a~n. (5.1)

Takinginto accountthe prescribedion densityprofile, eq. (2.28), the evaluationof the term —(1/n) a5n

for x —* + so yields

E=1/l=0.25, (5.2)

i.e., the electric field doesnot vanishin the regionof low density.The velocity assumesthevalue v = Et

(seefig. 3b). The smaller value of the plateaucorrespondsto the self-similar solution, eqs. (3.31)—(3.34). In contrastto the caseof ‘y = 1 andof chargeseparation,the spatially increasingion velocitycausesthe fluid particles to move apartin the low densityregion and,therefore,the solutiondoesnotbreakdown. One can get a hump-likestructureof E and, therefore,of v, too, if one usesa differentdensityprofile decreasing,e.g., algebraically,n ‘-~ x~,a >0 for x—+ +so. In this casea break-downofthe solution even for y = 1 and quasi-neutralityoccurs. Here, therefore, the rather unphysical

346 Cli. Sackand H. Schamel,Plasmaexpansioninto vacuum— A liydrodynamicapproach

behaviourof E, namely E(x—* +so) = 1/1 = const., which is due to the specialchoice of the initialdensityprofile, does not allow the collapse.This is, of course,an exception.In general, one has toexpectthat a collisionlessexpandingplasmais exposedto this typeof collapse,as it will be shownin thefollowing.

Maintaining quasi-neutrality,sucha situationis found if onechoosesa differentpolytropic exponentkeepingthe initial densityprofile. In fig. 4a andfig. 4b, respectively,n, u andE areplottedfor t = 0 andt = 10 in the casey = 2. The initial profile for the electricfield shownin fig. 4a follows from [see(2.21)and (2.28)]:

E=_2a~~)=~_sech(~-_1-~°),1=4. (5.3)

This initial situationis very similar to that in fig. 2a, and, thereforeit is not surprisingthatthe solutionalsobreaksdown. The collapse,however,takesplaceat an earlier time. As in the precedingcases,thevelocity assumesthe structureof the electric field from the initial state(seefig. 4a). Due to (5.3) the

1.0’n

‘0.15t=O

0200 -100 0 x-_::

1.0 to 1.0V E

n0~75 ~ 08 08

0.6 ~6

Q:5~\Q2 02

Fig. 4. y = 2 and quasi-neutrality: t = 0 (a) and t = 10 (b).

Ch. Sack and H. Schamel,Plasmaexpansioninto vacuum— A hydrodynamicapproach 347

field variation is enhanced(e.g., the electric field decreasesmorequickly for x—* +so than in the casey = 1 and chargeseparation).This affects the velocity and, as a result, the collapseis speedingup.

Figure 4b is of greatinterestbecauseit undoubtedlyprovesthat wavebreakingassociatedwith ionbunchingdoesnot rely on the assumptionof chargeseparation.It occursin the quasi-neutralplasmaaswell. Sincethe quasi-neutraldescriptioncoversat thesametimealsothe expansionof a neutralgas,weconcludethatbunchingis also to be seenin the expansioninto vacuumof an inviscid ordinary gas. Ananalogousdiscontinuity,as mentionedalreadyin subsection3.2, developsin the gas flow driven by anadvancingpiston. The straightcharacteristicsC~of a simple compressionwave(sometimesalso calledcondensationwave)originatingat the piston form a cusp-shapedenvelopein the t—x plane.The tip ofthis cusp marks the break down of the flow description through a simple wave, resulting in adiscontinuityandsingularity,respectively,of the flow pattern[53,72]. A phenomenonof this typewasperhapsfirst noticed by Stokesin 1848. To realize this breakingphenomenonnumerically, a codeisnecessaryin which dissipationis moreor less absent.Dissipationusually incorporatedintrinsically inthe numericalcodes(e.g. LAX codes)would smooth out the density peak,giving rise to the knownshock-typestructures.We shall comebackto this point in section9 in the descriptionof fig. 10.

For the sakeof comparison,the casey = 2 andchargeseparationis shownin fig. 5a, t = 0, andfig.Sb, t = 12. The collapse takes place at a higher value of the density similar to the casey = 2 and

1.0

I 0.15

t~00.75 E

:~tivE . 01

05 i\I005

a :‘\~ a

—200 -100 0 100

1.0 ___-_______~....~,,~ 1.25 0.25

t~12 tO 0.20.75 \ ,Ii~E v E

\j 0.75 0.150.5 0.5 ‘0.1

0.25

/ 0.25 0.05

0 ‘4.— 0-100 -50 0

Fig. 5. y = 2 and charge separation, t = 0 (a) and t = 12 (b).

348 Ch. Sack and H. Schamel,Plasma expansioninto vacuum— A hydrodynamicapproach

quasi-neutrality.This is due to the smaller width andthe strongerdecreaseof the electric field at thetime t = 0. Rememberthat the electric field falls off exponentially[see(2.39)] in the region wheretheion density is negligible. In the casey = 1 and chargeseparation,however, the electric field falls offonly algebraically[see(2.36)]. The collapsetime is comparableto that for y = 2 and quasi-neutrality.Behindthe ion front againa self-similar region forms. For y = 2, the electric field from the self-similarsolution,however,falls off linearly in x [see(3.27)].This decreasecanbe seenin fig. 5b behindthe ionpeak.

For valuesof y within the interval 1 ~ y ~ 2, e.g. y = 1.2 (seeref. [57]), we obtain underthe sameinitial andboundaryconditionssimilar solutions.

Figures 3 and4 point out the limits of the quasi-neutralityassumptionin theplasmaexpansion.Theuseof quasi-neutralityis justified only if the inhomogeneitylengthsarelarge comparedto the Debyelength. This is no longer true if the density, the electric field andthe velocity steepenand vary on theDebye length scale (see, e.g. fig. 4b). Through this the physical assumptionof the quasi-neutraldescriptionlosesits validity alreadybeforethe numericalsolutionbreaksdown. The Riemannsolutionsfor simple waves of arbitrary amplitude derived in section 3.2, express the break-downof thequasi-neutrality assumptionanalytically (see also [34]). Hence, it is necessaryto include chargeseparationeffects in caseswhere profile steepeningtakes place (see section 3.2). In general, thesolutionof the completemodel includingchargeseparationeffectshasto approvewhetherandin whichway the plasmaapproachesthe quasi-neutralstate.In somecases(seefig. 2 andfig. 5), this approachcan be investigatedby comparisonwith the self-similar solutions.An alternativeprocedureis given bythe pseudo-soundvelocity c(x, t), the squareof which [see(3.46)] is shownin fig. 6a for threedifferenttimes as a function of spacefor —60~ x~ +20, y = 1 andchargeseparation(seefig. 1 andfig. 2). Theplot of c2 given by

c2(x, t) = —nEIa,~n (5.4)

hasbeencalculatedfrom the numericaldata for the ion densityn andthe electric field E. Figure 6bshowsn and E for t = 9, and t = 15 in the interval —40 ~ x � 0. In fig. 6a we can clearlydistinguishtworegions.The left-handpartin fig. 6 representsthe quasi-neutralregion which expandsin the courseoftime andwhich in the isothermalcase,y = 1, is describedby c2 = 1 [see(3.44) and(5.4), respectivelywith E from (5.1)]. As it is expected,thereare significant deviationsfrom quasi-neutralitywithin andaheadof the ion front region whichcan be seenon the right-handsideof fig. 6a (seefor comparisonfig.6b). The strong increaseof c2 is causedby the formation of a plateauin the ion density.For a~n—+0and E >0, E � 0, c2 from (5.4) becomessingular. When the ion hump appears,thereexists a regionwith c2(x, t) < 0 behindthe ion front (seealsofig. 1 andfig. 2). In this casethe pseudo-soundvelocityc(x, t) is imaginary and, therefore,cannotbe interpretedanymore as a velocity. Consequentlythetransformationof the plasmaexpansionproblemto a hydrodynamicone (seesection 3.2) using theconceptof pseudo-soundvelocity is not applicableglobally. In particularthe constructionof Riemanninvariants,eqs. (3.62) and (3.63), becomesmeaningless.In thoseregions,however,wherec2(x, t) ispositive, the pseudo-soundvelocity can be used to distinguishbetweenregionsof quasi-neutralityandof chargeseparationin the expandingplasma.It is, hence,a keenindicator for the breakdownof thequasi-neutralityassumption.

In section 6 we introduce a further formulation of the plasma expansionmodel, allowing theanalytical investigationof the breakdown discussedin this section.We shall treatthe specialcaseofisothermality, y = 1, and chargeseparation(seefig. 1 andfig. 2). Before doing this, it is necessaryto


6

a

5

~t=15

3

2

~t~9

t~O

0 I I I

-60 -50 -40 -30 -20 -10 0 .10 .20

Fig. 6. Square of the pseudo-sound velocity c as a function of x for three different time steps: (a) in comparison to the space-time evolution of theion density n and the electric field E; (b) in the isothermal case, y = 1, with charge separation.

analysethe numericalresultsfor this casein moredetail. For thispurposethe space-timebehaviourofthe solution nearthe collapseis shownon an enlargedscalein fig. 7.

Figure 7a shows the gradualsteepeningof the ion velocity. If t approachesthe collapse time= 17.94, the gradient of v becomesminus infinity. Figure 7b shows the electric field which also

steepensand whosegradientgoesto plus infinity for t—+ t,. The specific volumeV= 1/n illustratedinfig. 7c yields someinterestingdetailsbeingof importancefor the analyticalinvestigationof the collapse.At any fixed time t the inversedensityis V-shapedconsistingof two brancheswith differentslopewhichremainsfinite for t—* ti,. The minimum of V decreaseslinearly and becomeszero, i.e. n—* +so, at’x= x~,= 4.112. At t = 17.935, e.g.,the maximumvalueof the densityequals11.35. In thefollowing wecall thepoint (x~,t~)“critical point”. The electrondensityandthe electrostaticpotentialwhich arenotshownhere,remainsmooth andmonotonicallydecreasingat the critical point.

It is remarkablethat thenumericalprocedure,consistingof thousandsof iterations,is able to resolvethesefine structuresup to the breakingpoint. Furthermore,we mention that by using the (second)


2.6 4.00 4.04 4.p8 x—’.

0.16

Fig. 7. Spatial dependence of the ion velocity v, (a), of the electric field E (b), and of the specific volume V~1/n (c) for four equidistant time stepsjust before wave breaking (see also fig. 1 and fig. 2); the minimum of the specific volume approaches zero linearly (broken line in (c)).

Poissonsolverwithin the Lagrangianscheme,additionalspatial derivativeswhich maybecomesingular,areavoided(seesection4.2). In the (second)Poissonsolveronly the potential4’ is differentiated,whichis the smoothestquantity in the system.Therefore,the Lagrangianschemetogetherwith PoissonsolverII not containinga1n, is advantagouscomparedto the Eulerian schemeand Mason’sPoissonsolver[38].

The structureswe are treatinghere appearon a smaller scalethan the Debye length. Thus thequestionmayarisewhetherthe basicmodel is valid globally. At the moment,however,we deal withthe mathematicalaspectsof the dissipationlessplasmaexpansion;its physical aspectswill be discussedin section7.

The numericalresultspresentedin detail in fig. 7 andthe characteristicfeaturesdeducedtherefrom

av/ax~—so, aE/ax~+so, V~0 (n~so) for (x, t)~(x~, t~), (5.5)

Cli. Sackand H. Scliamel,Plasma expansioninto vacuum— A hydrodynamicapproach 351

as well as the linear behaviourof V in the critical region are the starting point for the analyticalinvestigationof the collapse.

The solution shown in fig. 7c can be describedin good approximationby the following empiricalformulas:

V(x, t) = ‘1(t) t — X(t) [x — xmjn(t)] + . . . (5.6)

with

Xmin(t) = ~(t) t. (5.7)

Equation (5.6) and (5.7) hold in the region x � xmin(t), wherexmjn representsthe minimum of theV-profile. A similar ansatzwith another.~K(t)can be madefor the secondbranchx ~ xmjn(t). In (5.6)and (5.7) x and t representtransformedvariables,correspondingto the original variablesvia thetransformation

x—*x~—x and t—*t~—t. (5.8)

This transformationrepresentsa reflection of the V-profile on the critical point (x~, t~)and asubsequentshift of the latter to the origin. The critical point is approachedfrom the right, x, t—* 0~.Insertingthe transformation(5.8) into Poisson’sequation(2.24)andinto the equationsof motionin theLagrangianpicture, eqs. (4.9)—(4.11), the transformationof the dependentvariablesreads

v-*v, V—*V, 4’—*4’, E—*-E. (5.9)

To lowestorder in the smallness-parametert [see(5.8)] the quantities‘1(t), ~‘{(t)and~‘(t) in (5.6) and(5.7) are non-vanishingconstants.

In the next sectionwe shall justify the empirical formulas (5.6) and (5.7) and clarify the collapsemechanism.

6. Theory of bunching and wave breaking in ion dynamics

The characteristicfeaturesof the numerical break-down [see (5.5)] describedin the precedingsection indicate the occurrenceof wave breaking. The densitybecomesstrongly peakedand finallydiverges.As mentionedin the Introduction, this unboundedincreaseof the local ion density is called“bunching”. The relationshipbetweenthis behaviourof the ion systemandthe dynamicsof nonlinearLangmuir oscillations,which will be treatedin appendixA, enablesus to investigateanalytically thecollapsein the ion dynamics.In analogyto the analysisof nonlinearLangmuir oscillations(Davidson[62] and Kalman [73], see appendixA) we deducein section 6.1 a single scalarwave equation. Itdescribesthe ion systemincludingchargeseparation.Comparedto the caseof electrons,whereonegetsa simple oscillatorequationfor theelectronvelocity, this wave equationis a morecomplicatedoneandin general cannotbe solvedanalytically. In section6.2 a perturbativesolution of this equationin thevicinity of the critical point (x~,t~)is presentedfor the isothermalcase,y = 1. It reproducesall thepropertiesof the numericalsolution (seefig. 7) proving the existenceof wavebreaking.


6.1. Scalarwaveequation

In order to deducethe scalarwave equationfor the ionic system,two importantstepshaveto beperformed.The first stepconsistsof the transformationto the Lagrangianmassvariable[see(A27) inappendixA]:

(6.1)

With referenceto thermodynamics,in the secondstepthe specific volume V= 1/n is introduced.It isusefulto replacethe densityn by the specific volume,sincein contrastto n, V is a limited function inthe caseof unlimited bunching(V—* 0 for n—~so). In differential formulation (6.1) reads:

(6.2)

wheret is held fixed. With regard to the transformationmadein section5, eq. (5.8), —~ is replacedby~. From (6.1) it follows that i~—*0’~’for x—*0”. With (6.2) eqs. (4.9)—(4.11) transformto:

x(~,t) = v(~, t) (6.3)

(6.4)

o(,~,t) = E(~,t) = - V(71, t) ~ 4’(m t). (6.5)

For Poisson’sequationwe obtain:

~- [~ ~- 4’(~,t)] = V(~, t) Tie(4’) -1. (6.6)

Equations(6.3)—(6.5) can be summarizedto give

a~VasV=_a71[~a~4’]. (6.7)

The addition of equations(6.6) and (6.7) yields:

ne(4’) = (1- ~)/V. (6.8)

Inserting for ne(4’) in eq. (6.8) the generalizedequationof state,eq. (2.25), the potential 4’ can beexpressedin terms of the specific volume V:

- ~ [(1~ti1 69y-1 t~v i i’


where,at the moment,y ~ 1 is assumed.By meansof (6.9), 4’ is eliminatedfrom (6.7):

+ a~[~(1 ~)~2 a~ ~)] = 0, 1 ~ y ~2. (6.10)

Equation(6.10) representsthe scalarwave equationwe arelooking for. Its solution in connectionwitheqs. (6.3)—(6.5)yields the nt-dependenceof x, v, E and4’. Finally the solutionin the Eulerianpictureis obtainedby invertingx(q, t)—* ~(x, t).

In contrastto the electrondynamicswhereone obtainsa simple oscillatorequation[see,e.g. eq.(A30)], which can completelybe solved analytically, the waveequation(6.10) for the ion dynamicsismoreintricate. It representsa nonlinear,partial differential equationin t and ~, which can be solvedonly by perturbativemeans.Moreover, the appearanceof ~j-derivativesin (6.10) gives rise to awave-like motion in the a-spaceso that in the stationarycoordinatesystemalways new particles,respectivelyfluid elements,startparticipatingin the dynamics.This is, however,not the casefor thenonlinearmotion of electrons(Kalman[73]).There,awave-likepropagationalong the 7)-coordinateismissing (a,1 = 0), and only thoseparticleswhich are originally disturbed,participatein the bunchingprocess.

For y = 1 and y = 2, we get the following specialversionsof eq. (6.10):

y=1: v+a,1[~a,1(~/)]o (6.11)

y=2: V+ a,1[.~ a,1( _V)] =0. (6.12)

Equation (6.11) is used in section6.2 to investigateexemplarily the collapsein the ion dynamicsbyperturbation theory.

In the quasi-neutralcase,Tie(4’) from (6.8) mustbe replacedby ~e(4’) = 1 /V(asn). We obtain for 4’:

4’ y~i [(1)7] (6.13)

By inserting 4’ from (6.13) in (6.7) we get the quasi-neutralform of the scalarwaveequation:

V+a~(1/V)~=0. (6.14)

Even this much simpler scalar wave equationcan generallynot be solved analytically. Assumingespeciallya self-similar behaviourof the system,the solutionof (6.14) can be written in a very simpleform by meansof the group theoreticalformalism in section3.1:

V(~)= ( /~2)1/(Y+1) ~as ~/t , (6.15)

where~ is given by (6.1). In the isothermalcase,y = 1, the solution reads:

V(fl=1/~~~, ~ (6.16)

354 Cli. Sackand H. Scliamel,Plasma expansioninto vacuum— A hydrodynamicapproach

It is a characteristicfeatureof the solution (6.16) that the secondorder t- and i~-derivativesin (6.14)vanish simultaneously.Becauseof V= 1/n we obtainfor the ion density n( ~):

n(fl=~~~, —1~~0, (6.17)

i.e., in the c-spacethe ion densitydecreaseslinearly with the slope —1.Differentiating (6.17) via ~= ij/t = (1 /t) $~n(x’, t) dx’ with respectto x we get

lan a 1——=—lnn=——. (6.18)nax ax t

In accordanceto (3.31) the x-integrationof (6.18) yields the functionaldependenceof the ion densityon the self-similar variable T = x/t.

The next subsectionis devotedto the perturbativesolution of (6.11) in the vicinity of the criticalpoint, i.e., for small values of ij and t.

6.2. Perturbativesolution of the scalar wave equationfor isothermalelectrons

We investigatethe solubility of the scalarwaveequation(6.10)by perturbationtheory in the specialcasey = 1 [see(6.11)]. With somemodificationsthe formalism usedis applicableto othervaluesof y,too. It is our aim to determinethe functions ‘1(t), ~1{(t)and~t) in (5.6) and (5.7),and to prove thebehaviourof the dependentvariablesv, E and V, as it is shown in fig. 7.

For the solution of (6.11) we makethe following ansatz:

V(~,t) = at[1 — ht — (b0 — b1t)z + (c1 + c2t)z

2+ d1z

3+ 0(r4)] (6.19)

with z = — lilt and 0< ij ~ lilt or — lilt < z ~ 0, respectively;~ = fit representsthe position of theminimumof the V-profile in the ijt-space.The quantities(1, a, h, b

0, b1,. . . areconstants.Accordingto the transformation(5.8),eq. (6.19) representsthe left-handbranchof the V-profile. A similaransatzwith other constantsb1, c1,... holds true for the right-handbranch. We assumethat t and z arequantitiesof the orderr and determinethe constantsin (6.19) by comparingthe coefficients in eachpowerof r. It is shown that

i) within eachorder of r the wave equation(6.11) is satisfiedby the ansatz(6.19),ii) the solution representsexactly all propertiesof the numericalsolution in the vicinity of the

critical point (x~,t~),

iii) to lowestorder the solution is describedby the equation

a~v+ v a1v = 0. (6.20)

In the following, the limit t—*0, ij—~0,is made under the constraint ~q—*lit, i.e. we are at theminimumof V. A moregeneralapproachto the critical point, e.g., ‘q = KIlt, 0< K S 1, however,doesnot affect the essentialresults.

At first it follows from the ansatz (6.19) that V—*0 for t—*0, guaranteeingautomaticallythedivergenceof the ion densityat the critical point, ~ = t = 0 [seetransformation(5.8)]. Since the ion


densityne remainsfinite at the critical point, it follows from (6.8), that for t—+ 0 and ~j = flt—~ 0, V

must go to unity. In the ansatz(6.19) we perform the first andthe secondtime differentiation:V’= a[1 + (b012 — 2h)t — b0z — b111t

2+ 2(b1 — c111)zt+ c1z

2

+(2c2 — 3d111)z

2t— 2c2Qzt

2+ d1z

3+ 0(e4)] (6.21)

12=a[2(b0uil—h)—a1t—a2z+/31t

2—/32zt—/33z

2+O(e3)], (6.22)

where

a1 = 2fl(2b1 — c1Q), (6.23a)

(6.23b)

2c21’12, (6.23c)

1~2 = 211(4c2 — 3d111), (6.23d)

/33 = 2(3d1Q — c2). (6.23e)

With V(~—+0~,t—*0~)—-*1we obtain from eq. (6.22) the coefficient h:

h = b0ul — 1/2a. (6.24)

Thus 1 — V= 0(e), andit follows from (6.22)

1— V= a[a1t+ a2z— 131t2 + /3

2zt+ /33z2 + 0(e3)]. (6.25)

Furthermore,we get from the boundednessof ne the coefficient a1 it holds [see(6.8)]:

ne=(1~V)/V—+ne*; t—+0~, ~=Qt—~~0~ . (6.26)

By meansof (~.25)and (6.19) we get for ~, t—+0:

a1 = 211(2b1— c1fl) fle* (6.27)

ne* representsthe finite value of the electrondensityat the critical point.In order to determinethe other coefficients, we insert the ansatz(6.19) into the wave equation

(6.11). By using (6.19) and (6.25) and expanding1/V up to the order 0(1), we obtain explicitly theterm:

1 VV = a[a1t + a2z — $1t2 + $

2zt+ $3z2+ 0(r3)] 1 [1+ ht + b

0z+ 0(62)]. (6.28)


For the ij-derivation of (6.28) one gets:

a,1 (1 V)=t [a2+g1t+g2z+0(r2)], (6.29)

where

= /32 + ha2 + b0a1 (6.30a)

g2 = 2(f33 + b0a2). (6.30b)

Next we determinefrom (6.25) the term 1/(1 — V) throughTaylor expansion:

1 1 [ R(~jt) 21

1— = a S(~, t) L1 — S(~,t) + 0(r)] (6.31)

with

S(ij, t) = a1t+ a2z = 0(e) (6.32a)

R(-q, t) = —f31t2+ /3

2zt + f33z2= 0(62). (6.32b)

The multiplication of (6.29) and (6.31) yields up to 0(1):

l_~,1(V)atS(fl,t)[a21t2z_a2S(~,t)+0(6)] (6.33)The n-derivative of (6.33) then yields the secondterm in the waveequation(6.11):

a,1[1 1 ~ a,1(

1 ~)] = ~ {—a~S + S {g2S — a2 [(/32 + g1)t + (2/33 + g2) z]} + 2a~R

+ 0(r~)}. (6.34)

For the sakeof simplicity the argumentsof S and R are droppedin (6.34) [compare(6.32a,b)].Finally we insert (6.34) into the wave equation(6.11) and multiply with atS

3‘-~ 0(r~).Includingtermsof order0(62) we get:

—cx~S+ S {g2S— a2 [(1~2 + g1)t+ (2f33 + g2) z} + 2a~R+ 0(r~) 0. (6.35)

The demandthat thewaveequation(6.11)is satisfiedto anyorderof r allows the determinationof thecoefficients.Comparingthe coefficientswith respectto r~and ~2 in (6.35),we obtaina2 = 0 andg2 = 0from which by meansof (6.23a—e), (6.27) and (6.30a,b)follows

b1 = ~ c1 = ~~eJ6Q2, d

1 = c2/3h2, 1~2= 6c2u1. (6.36)

For the evaluationof the termproportionalto 0(r~)the order~ in (6.19) mustbe takeninto account,


which will not be pursuedfurther.Therefore,we obtain by meansof (6.36)and (6.24) for V(-q, t) in(6.19):

V(~,t) = at[1 + ~— — b~i~+ ~ (~2— 112t2) + (~— lit)2 (ij + 211t) +

~ (6.37)

To determinec, in (6.37) we usethe fact that the derivative of the electrondensity

1 1 /1—1~\

axne= ~ anne= ~ a,1~\ ~ ) (6.38)

at the critical point is finite, too. Therefore,we obtain

c2 = ~bofle*/6hl• (6.39)

The final solution V(’q, t) dependsonly on four constantsa, b0, 11 andne.,which can be obtainedfromthe numericalsolution. To the lowest order,a representsthe slope of the line V= Vmjn in fig. 7c; b0

denotesthe finite gradientat the minimumof V.

By meansof (6.2)—(6.5)we now prove the singularbehaviourof a~vand ~ at the critical point,= t = 0; from (6.37) we obtain to lowestorder:

1 1 Vi(6.40)

1 1 ..V 1a~E= ~ a~E= ~ a,~x= -~ -~ —~+so. (6.41)

Reversingthe transformation(5.8) and(5.9),we get a1u—-+ —so, and~,~E—++so in accordancewith thebehaviourshown in fig. 7a and fig. 7b. Equation (6.4), relating the Eulerian and the Lagrangianvariables is usedto transform from the ~jt-variablesto the xt-variables. The 7)-integration of (6.4)yields:

x(~,t) = Xmjn(t) — fd~’V(~’,t)

= Xmin(t) + at {(i + (,~— fit) — ~~0(2 — Q2t2)

+ 2 [~(n3— f13t3) — 112t2(~— lit)] + 0(r~)), 0 ~ ~ lit; (6.42)

Xmin(t) describesthe time dependenceof the minimumof V.

By defining

~:= [x — xmjn(t)]Iat, (6.43)

358 Ch. Sackand H. Schamel.Plasmaexpansioninto vacuum— A hydrodynamicapproach

we obtain iteratively from (6.42) to eachorder the desiredinverse transformation~ = ij(x, t). Thisiteration proceduredemandsthat ~= 0(r) andx — xmjn(t) = 0(62), respectively.The smallnessof ~meansthat the expansionis done in the vicinity of xmjn(t). The region of validity of the expansion,therefore,shrinksto zero for t—+0.

Up to secondorder in e we obtainfrom (6,42) and (6.43):

— flt= ~[i + (~ ~ + ht)] +0(~~), (6.44)

whereh is given by (6.24). With ~j from (6.44) it follows for thex-dependenceof the specific volume in(6.37):

V(x, t) = at{i — ht— b0 ~+ ~ (~+ 2ult) — b0 (~-~ + ht)] + o(r3)}. (6.45)

In a similar way we obtainby integrationof the equationsa~u= V anda~E= V with respectto ij andbyreplacingi~by eq. (6.44) the velocity v. and the electric field E dependingon x andt:

v(x, t) = Vmjn(t) + a~(1 + t/2a) + 0(~~) (6.46)

E(x, t) = Emin(t) + ~ + 0(r~), (6.47)

whereVmin(t) andEmin(t) areintegrationconstantswith respectto ~ or x. The latterdeterminethetimebehaviour of the velocity and of the electric field at the minimum of the V-profile, x = xmjn(t),

correspondingto i~= lit in the n-space.By meansof eq. (6.42) and of the relations~ t) = v and~(vj, t) = v(rj, t) = E(~, t), evaluatedat the point i~= lilt, we calculateXmjn(t) andvmjn(t):

~min(t) — alIt (1 — ht) = Vmin(t) (6.48)

t3min(t) — all + lit (aboli — ~+ an~) = Emjn(t) (6.49)

with Emin(t) = —E~+ 0(t), and Vm10(t) = v~.+ 0(t); v~,and E~representthe finite values of thevelocity and of the electric field at the critical point, respectively. In the two lowest orders thet-integrationof eq. (6.49) resolvedwith respectto limin yields:

Vmin(t) = v~+ (—E~+ all) t + 0(t2), Vmin(0)= v~. (6.50)

From (6.50) we determinexmjn(t) by a further t-integrationof (6.48):

xmjn(t) = v~t+ ~(—E~+ 2ali) t2 + 0(t3), xmjn(0) = 0. (6.51)

By comparisonof (6.45) and (6.51) with the empirical formulas (5.6) and (5.7) we get

‘1(t) = a(1 — ht +...) (6.52)

~(t) = b0[1 + (h — n~./3lib~)t +...] (6.53)

C/s. Sackani”H. Schamel,Plasma expansioninto vacuum— A hydrodynamicapproach 359

= v~.+ ~(—E~+ 2afl) t (6.54)

with h from (6.24).Taking into account the third order in r for v(x, t) in (6.46), we get a term proportional to= [x — xmin(t)]

2/a2t.Therefromit follows that for t—+ 0 the secondderivativeof v with respectto xdiverges,a2vlax2—+so, i.e., a2xlav2—+0. Hence,for the inversefunction x(v) the critical point is aturningpoint. This resultexactlycorrespondsto the characteristicbehaviourof the velocity in the caseof wavebreakingof the hydrodynamicsystemintroducedin section3.2 [see(3.54)]. In a similarway weobtainfor the electric field a2E/ax2—+ so and a2x/aE2—*0, respectively.

Fromeqs. (6.46) and (6.47) and from the Eulerianequationof motion

a~v+ v a~v= E (6.55)

we get a furtherproof for the existenceof wavebreaking.At first we verify that eq. (6.55) is satisfiedup to 0(r). Inserting (6.50) into (6.46),we getx/t as the leadingterm,which is of order0(1) becauseof x, t—0(e):

v(x, t) = x/t + 0(r). (6.56)

To lowestorder a~vand v a~vare 0(r~)such that eq. (6.55) reads

a1v + v a~v= 0. (6.57)

In the sameway we showto lowest order by meansof

n(x, t)= 1/V(x, t) = 1/at+ 0(1), (6.58)

and

E(x, t) = —E~+ 0(r), (6.59)

that the equationof continuity and Poisson’s equationare satisfied. The leading term in (6.58) is0(6_i) and reflectsion bunching.

Equation(6.57) is the most simple version of an equationdescribingwave steepeningand provesdefinitely that the ion dynamicsis subject to wave breakingin the vicinity of the critical point. Justbefore wavebreakingthe electric field becomesunimportant,andthe systembehaveslike an ordinaryfluid. The dynamicsof thesystemis therebycontrolledby the convectiveterm v a~v,which diverges.Ina compressionalwave, in which a region of high densitypropagatesin the direction of low density,steepeningand wave breakingare driven by a decreasingvelocity profile. This is in accord with thenumericalresultsshown in fig. 7 which were the startingpoint of our analyticalinvestigation.Similarconclusionscan be drawn from figs. 4 and5.

The case y = 1 and quasi-neutrality(see fig. 3) is an exception. Here, the velocity increasesmonotonically, and the conditions for wave breakingare not met. As mentionedin section5, this,however, is due to the specialchoiceof the initial conditions.If we had started in this casewith asufficiently decreasinginitial velocity v(x,0) or with anotherdensity profile, we also could haveobtainedwavebreaking.

360 Cli. Sack and H. Scliamel,Plasma expansioninto vacuum— A liydrodynamicapproach

In the following considerationit is shown,thatthe specialcasey = 2 andquasi-neutrality(seefig. 4)is equivalentto other physical systemswhich are known to exhibit wave breaking.With tte(4’) = =

1 + 4’/2 [seeeq. (2.21)], andthe definition h(x, t) 2 n(x, t), the hydrodynamicion equations(2.22)and (2.23) turn into

a~h+ a~(vh)= 0, (6.60)

a~v+va~v+a~h=0. (6.61)

In the ordinary theory of fluids the equations(6.60) and (6.61) describethe rotationlessdynamicsoflong wave length, shallow waterwavesin a channelwhereh standsfor the depth of the channel[55].The solutionsof theseequationsarethe Riemannsimple waveswhich, as known, showwavebreaking(comparesection 3.2). Therefore, it is not surprising that in our systemwavebreakingexists, too,especiallywhenwe startwith a moregeneralset of ion equations.The equations(6.60) and(6.61) are,furthermore,identicalto thosedescribingthe wave propagationperpendicularto a magneticfield in asmall /3 plasma(/3 plasma (f3 = nT/(B2/8ir) with wave lengths exceedingthe collisionlessskindepthcIw,~[55]. In this case,the depth of the channelh has to be replacedby the magneticfield B and thesoundspeedby the fastmagneto-acousticspeed.

Summarizing,the collapsein the ion dynamicscan be understoodby the following simplepicture.The ambipolarelectric field, beingfinite atthe front of the plasmaandhump-like,acceleratesthe ionsand generatesa similarly shapedvelocity profile. This profile continuouslysteepensdue to the ionnonlinearityand,in a furtherstage,whenthe ion flow velocity becomessupersonic,theconvectivetermin the ion momentum equation overrulesthe electric field. Accompaniedby an unlimited densitybunchingthis leadsto a simple wavestructureanalogousto an ordinaryhydrodynamiccompressionalwave.

The main conclusiondrawn from this section,therefore,is that the dissipationlessexpansionof aplasmainto vacuum is subject to the phenomenonof wavebreaking,a phenomenonwhich cameintoattentiononly recently.

It is knownfrom ordinaryhydrodynamicsthatafter the adventof wave breaking,the wave doesnotremain in a simple wave structure. In the configurationspacethe dependentquantitiesas e.g., thevelocity andthe density,are no longer uniquelydefined[53]. The descriptionof the plasmaexpansionwithin the framework of a dissipationlesshydrodynamic model itself, breaks down. In order toinvestigatethe ion dynamicsfurther, thereareessentiallytwo concepts.In the first onethe dissipation-less regime is kept. This requiresthe transition to the microscopicVlasov description. In the nextsectionwe shall discussamongothers the kinetic structuresarising within this descriptionafter wavebreaking.It will be shownthat in the caseof strongnonlinearityour solutioncoincideswith the kineticone. In the secondconceptone keepsthe hydrodynamicdescriptionandintroducesa viscosity term inthe ion momentumequation which preventsthe unlimited steepeningof the velocity profile and,therefore,the wavebreaking.This procedureis well known in hydrodynamics[65,66]. Our numericalinvestigationsusing the secondconceptare presentedin sections8, 9 and 10.

7. Comparison with previous hydrodynamic and kinetic models

In the quasi-neutralcase the breaking of ion acoustic waves with finite amplitude has beeninvestigatedby Akhiezer [32] and by Gurevichet al. [35,43]. Akhiezer’s model was basedon the


kinetic descriptionof the electronsusingthe Vlasov equationand on the hydrodynamicequationsforcold ions. Gurevichet a!. used the ion-Vlasovequationat finite ion temperatureandinvestigatedtheirnonlinearself-similar behaviour;therebythe electronswere assumedto be isothermal.Ignoringfor amomentthe fact that quasi-neutralityis violatedat the steepenedfront, we can get analogousresultsbymaking use of eq. (6.14). By solving eq. (6.11) explicitly we haveshownthatthe structureof a simplewave doesnot rely on the assumptionof quasi-neutralitywhichlosesits validity prior to wave breaking.Simple wavescan be found in the caseof chargeseparation,too. In somepublications[34,35,43], itwas conjecturedthat the inclusionof dispersion,which in the descriptionof ion wavescomesinto playthroughthe terma~4’~ 0, can preventwave breaking.Our resultsdo not supportthis conjecturein theextremely nonlinear case of plasma expansioninto vacuum. In addition the collapse cannot bepreventedeither by finite ion temperaturein the realm of a collisionlessmodel [56].

Fromthe numericalpoint of view the resolutionof wave breakingin connectionwith ion bunchingisonly possiblewithin a numericallydissipationlessdiscretizationprocedure,e.g., the explicit Lagrangianscheme[65,66]. Explicit algorithmsfixed in spacesuch as Lax schemesarelesssuitable.The associatednumerical dispersionand dissipation[66] causean unacceptablestrong smoothingof the nonlinearstructures. Such a procedurewas used by Widner et al. [37], who were the first to investigatenumerically the isothermalhydrodynamicplasmaexpansioninto vacuum, taking into accountchargeseparation(seefig. 8a). The limitation of the ion front velocity to about3c~found by them togetherwith the ion peak,is an indication for the onsetof wavebreaking. However,as mentionedby Crow eta!. [42], the influence of inaccurateboundaryconditions for Poisson’sequation (4’ = const. at theboundaries)cannotbe excluded.

An improveddescriptionof the plasmaexpansionwith respectto the boundaryconditionswasgivenby Crow et al. [42].The hydrodynamicion equationsweresolvedby the explicit Lagrangianproceduresimilar to our method.The main differenceto the expansionmodeldiscussedin this reviewis the choiceof the initial ion densityprofile which was a stepfunction. Thus 4” is discontinuousat the stepwhichsurvivesfor all times. Using the continuity of the potential and of the electric field at the front, thesolution in the electroncloud region is matchedto the numerically obtainedsolutionin the ion region.In the early stageof the expansionone observesa well-pronouncedhump in the ion densitywhichdisappearsfor largetimes.The densityprofiles in thisstagearesimilar to thoseshownin fig. 1. Crow etal., however,do not makeanystatementsaboutthe appearanceand disappearanceof the hump.Thespace-timehistory of their solution for the ion density is illustrated in fig. 8b. It is possible that thedisappearanceof the ion peakin their solution is dueto the fitting procedure.Fromhydrodynamicsit isknown that the applicationof jump and continuity conditions,respectively,in the region of discon-tinuities introducesdissipation [53]. This smoothessmall scalestructuresand reducesamplitudes.

Shock fitting, in general, hassomeundeniableshortcomings[65]: Firstly, shock fitting can only beusedif the solution is knownexactlyin front of the discontinuityas in themodel of Crow et a!. Usuallythis is not the caseif in an initially diffuseprofile discontinuitiesappear[65]. In gasesor fluids, shockscan developspontaneously,and thereis no way to find out when and whereshock fitting should beapplied. Secondly,the finite grid size does not allow to localize the shock front precisely. Thisintroducesinaccuraciesof the dependentvariables,falsifying thenumericalsolution.Crow et a!. do notspecify how they havetried to circumventthis problem.

In a morerecentpublication,Gurevichand Meshcherkin[49] working out the model of Crow et a!.oncemore, presentedidentical solutions.Theirion hump shownin fig. 8c, was not commented.

A similar modelwas usedby True et al. [50]. The ion hydrodynamicswas solvedin the Lagrangiandescriptionby meansof the conservativeleap-frog method[66]. The stability analysis of the lattermethodleadsto two uncoupleddifferencemesheswhich maydrift apartin the courseof time [66]. Such


n,n

a ::

I ~ o.:.

—80 —40 0 40 80 120 180 —50 0 50 100 150x x

.0,1-I 0

1.0 ~n

0.5 c 00 d

—2 —1 0 1 2 3 4 r= —~— 0 50 100 150 200 250 300 350cot x

Fig. 8. Space-time behaviour of ion(electron) density profiles taken from: (a) ref. [37],(b) ref. [421,(c) ref. [491and (d) ref. [50].Note the similarityof density profiles a, b and c with those shown in Figs. 1 and 2.

a decoupling can be avoided either by defining the dependentvariableson one net only or byintroducing diffusion that couplesboth meshes.The paper of True et al. does not explain whichpossibility was chosen.The striking smoothnessof their density profile shown in fig. 8d, one maysuspect,indicatesstrongnumericaldissipation.With respectto theion hump found by Crow eta!. [42],True et al. simply state: “We havenot observedsuch a bump with the given initial conditions”.

It is remarkablethat four articlessolving the sameset of equationsgive essentiallythreedifferentanswers.As discussedwe attribute thesediscrepanciesto the uncontrolledaction of dissipation.Tostudy nonlinear structures within dissipationlession hydrodynamics,a Lagrangian dissipationlessschemeis preferable,like the onepresentedandreviewedin this paper.A schemelike oursoffers thecorrect numerical answer to the plasma expansionproblem, at least for diffusive step-like initialprofiles.

Finally, we discussin this sectionthe phenomenonof wave breakingwithin the framework of thecollisionlessmicroscopicmodel. Using particlesimulations,the breaking of finite amplitude ion waves


hasbeeninvestigatedby Forslundetal. [74], andby BuchelnikovaandMatochkin[75].In both worksahomogeneousplasma is modulatedinitially by a sinusoidalve!ocity perturbationwith contro!ablestrengthv0. Both simulationsshow-the occurrenceof wavebreakingin connectionwith ion bunching.As a result, kinetic structuresarise, the propertiesof which are dependingon v0, T1ITe, and theelectronpolytropic exponenty [74,75]. For small values of v0 (v0 ~ c,), doublelayer structuresof theslow ion acoustictype [76,77] are seen. In this case,the region betweenthem is characterizedby alaminarincreaseof the ion temperature[78]. For largervalues of v0 (v0 ~2—5c~) a symmetricx-pointstructurein theion phasespaceemerges.For still largervalues,ofv0(v0 5—10 c5) amixed typeof wavebreakingstructureis obtained,in which ion reflectionandfree streamingoccurssimultaneously(seeref.[74] fig. 7 and ref. [75] fig. 18). For velocitiesfar beyondion acousticspeed (v0 ~ 20 c5) an invertedS-typestructurearisesexpressingmultiple ion free streaming.Figure9, takenfrom Forsiundet a!. [74],illustratesthis specific pattern.In this casechargeseparationlimits theelectrostaticpotential4’, so thatthe conditionsfor ion trappingand reflection cannotbe reached.In a simpleanalyticalmodel, Forslund

:~*I 200

~

Fig. 9. Ion wave behaviour taken from ref. [74]for v~= 30 c, and TI T, = i0~ (a) ion phase space v versus x; (b) ion density; (c) electrostaticpotential; (d) electron density; (e) ion phase space v versus x after wave breaking. The wave has broken by multiple ion free streaming.

364 Ch. Sackand H. Schamel.Plasma expansioninto vacuum— A hydrodynamicapproach

et a!. show that the electronscannoteffectively shield the ions over distancessmaller thanseveralDebyelengths.The electrondensitylagsbehingthe ion density.

As long asthe ion velocity is unique,the curvesof n, v and4’ obtainedin theseparticlesimulations,coincide with our hydrodynamicresults.By comparisonwith the kinetic results,we can decidewhichkind of phasespacestructurewould beapproachedafterwave breakingif a kinetictreatmenthadbeenused.

In the quasi-neutralcasethe potential grows with n, and ion reflection will be the saturationmechanism.Therefore,we expectin a kinetic descriptionthe transition to two doublelayers or to ax-point structurewhich maybe modified by free streaming.

If chargeseparationis takeninto account,4’ and ~e(4’) are moremoderateand do not follow theincreaseof the ion density. Consequently,ion free streamingwith triple velocity will comeup afterbreaking,resemblingthe caseof ultrasonicv0.

The expansionof a plasma into vacuum can be interpretedas the extremecase of a densitymodulatedplasmain which the densityat the minimumreacheszero.In thissense,the plasma—vacuumsystembelongsto the strongestnonlinearsystemsonecan imagine.Ion bunchingandion wave collapseare naturaleventsin such a system.

Appropriate kinetic calculations of the plasma—vacuumsystem are not yet available. Particlesimulationsarenot suitedif steepeningoccursin the !ow densityregion (y 1) becauseof the enlargednumericalnoiseat low densities.Codes,solving Vlasov equationby direct discretizationdo not containsuch kind of noise and seemto be more advantageousto investigatenonlinearphenomenaassociatedwith plasmaexpansioninto vacuum.Onehas,however,to be sure that numericaldissipativeeffectsareminimized by the choiceof the differencemethod.Oneshouldnot referto LAX schemes[66,page191ff.]. In any case, the entropy production f dx dv f In f has to be calculated to check the rate ofdissipationbeing awarethat dissipationcannotbe eradicatedcompletely.

In the hydrodynamicdescription,one way to get rid of the collapsephenomenonin a controlablemanner, is to introduce a Navier—Stockesviscosity. Its physical foundation and its impact on thenumericalprocedurewill be discussedin the next section.

8. Navier—Stokesviscosity and the implicit Lagrangian scheme

The particle simulationsconsultedin section7 indicatea thermalizationof the ion distributionaftersteepening,pointing towardsmicro-instabilities. These anomalouscollision effects are modelledbyaddinga Navier—Stokesterm to the ion momentumequation(2.23), (4.11) respectively:

a,v + ~ a~v dv/dt= —a~~+ a~v, = const. (8.1)

beingawareof theshortcomingsof this approximation.A further justification resultsfrom the fact thatthe energyof the directedplasmaflow is transferredto smallerscale-lengthbringing viscouseffectsintoplay [55]. This assumesthat T. is not completelynegligible [79].

The numericalresultsin sections9 and 10 will show that the behaviourof the expandingplasmaisrather insensitiveto the strengthof v.

Introducing the Navier—Stokesviscosity, the ion momentumequationbecomesparabolicin spaceallowing the implicit formulation of the discretizedset of equations(see,e.g. [66]). Accordingly, the


discretizationof p a~vin (8.1) reads

2 2 n+1/2 2 2 n—i/2

v[a(a v/ax ). + (1 — a) (~v/ax ). ], (8.2)

involving both, the new andthe old time step. In (8.2) j andn representdiscretespaceandtime !abels;n+1/2

adenotesthe implicitnessparameter,0 � a ~ 1. The explicit schemeis givenby a = 0, in whichis determineda!gebraica!lyby the knownquantitiesat the previoustime step.For a � 0 the actualtimeis introduced,and the schemeis called implicit. The special,choicea = ~ yields the half-implicit orCrank—Nicholsonscheme[66]. Of specialimportanceis the full-implicit scheme,a = 1, which is usedtoget the resultspresentedin sections9 and 10.

With a� 0, eq. (8.1) is transformedto a set of linear equationsfor v’~~2 characterizedby a

tn-diagonalcoefficient matrix. The first and the last row of the latter haveto be completedby theboundaryconditions. For this purposewe use the discretizedversion of eqs. (4.33) and (4.44),respectivelyfrom section4.3. Furtherdetailsof the implicit differenceschemearediscussedin appendixC.

An essentialpoint in the implementationof evolutionequationsby finite differencesis the stabilityproperty of the algorithm. Suchan algorithm is stableif smallcomputationalerrorsdo not accumulatein the courseof time [66]. The stability propertycaneasilybe establishedfor linear systemsof partialdifferential equations.Basicdifficulties arisein non-linearsystems.In hydrodynamicsor magnetohy-drodynamics,e.g., the pressuregradient term —(1/n) a~pis replacedby —(c2In)a~nwhere c(x, t)dp/dn~,denotesthe characteristicsoundvelocity. The closureof the systemis achievedby assumingapolytropic equation of statep — n7 correspondingto isentropy, s — ln p/n7 = const. Keeping c2/nconstantwith respectto x and t, the pressureterm becomeslinear allowing a stability analysis(see[65,66]).

Our system is similar. The electrostaticpotential4’, replacingp, is known by solving an additionalequation(Poisson’sequation)andcanbe consideredas a functionalof the ion density n. Basedon thisconcept we have introduced in section 3.2 the pseudo-soundvelocity c, eq. (3.46), which in thequasi-neutralcasebecomesc = V5~~ In the homogeneousplasma(n 1) c correspondsto theusualion acousticvelocity, andthe linearizationmethodis applicablehere,too.

In the chargeseparationcase,however, c2 can becomenegativeand c imaginary (see section5).There exist obviously regionswhere the concept of pseudo-soundvelocity and the relatedstabilityanalysis lose their validity. In which sensethe stability conditions describe the behaviour of ournumericalmodel will be discussednext.

The ion momentumequation(8.1) is first formulatedin the Lagrangiancoordinate~:

dv/dt= C a~V+ A a~v+ B a~v, (8.3)

wherethe ~- and t-dependentcoefficientsare definedby

c2V0 V V V

2

C:=—~-, A:=V-~af-~ and B:=p—~. (8.4)In (8.4) c2 representsthe pseudo-soundvelocity in termsof ~

c2 = EV2/V

0 a~V (8.5)

366 Cli. Sack and H. Scliamel.Plasmaexpansioninto vacuum— A hydrodynarnicapproach

Linearizationof (8.3) is achievedby assuminglocally constantcoefficients (see,ref. [65, page297]).The discretizedversionof eqs. (4.9), (4.10) and(8.3) gives rise to aclosedset of linearizedequations,the stability of which can now be studied. Details of the stability analysisaretreatedin appendixD.Here we discussthe resultsonly.

Accordingto thesign of c2, two caseshaveto bedistinguished.First we considerthe generalstabilitycondition, inequality (D18) in appendixD, for the casec2 > 0, and three different values of theimplicitness parametera (see(Di9)). For a = onegets

p2sin2kh~4 resp. p2~4, (8.6)

if onechoosesthe maximumvalueof the sinus(kh = ir/2) wherep2 C2(V~5.~t/Vh)

2andh as~ Sincein thishalf-implicit case(8.6) is independentof ~‘, the viscosity doesnot contributeto the stability of thescheme.This propertycan beverified numerically.In fig. 10 the ion front is plottedasa function of x atdifferent timesfor v = 1. Ion bunchingis weak,but in the front region oscillationsfrom grid point togrid point developincreasingin amplitude.Shortlyafter this time, the solutionbreaksdown. Obviouslythis break-downis of numericalorigin in contrastto the collapsedescribedin sections5 and6. Insertingtypical valuesof the front region,the stability condition (8.6) is violated.We notethat inequality (8.6)coincideswith the stability conditionfor the explicit, inviscid Lagrangianscheme.

The casea = 0 (a= 1) can be treatedcommonly (see (D19)):

~) [1 (a fl2 + (fl2] ~ 1, (8.7)

where the lower (upper) sign holds for a = 0 (a = 1). In (8.7) the stabilizing influenceof v for thefull-implicit scheme(a = 1) is visible. In the explicit scheme(a = 0), on the other hand, viscositydeterioratesthe stability. Within the explicit scheme,oscillations like thosefor a = 0.5 (see fig. 10)arise.The solutionbreaksdown the earlier, the larger i.’ is. Generally speaking,inequality (8.7) is inagreementwith the behaviourof our codein casewherethe densitydecreaseswith x (c2 >0).

0.12

Fig. 10. Spatial dependence of the ion density for four different times obtained from the semi-implicit code, a = 0.5, v = 1, y = I and chargeseparation. The fluctuations in the front region at t 21.9 are due to a numerical instability.

Ch. Sackand H. Sc/same!,Plasmaexpansioninto vacuum— A hydrodynamicapproach 367

In regionsof increasingdensity,i.e. c2 <0,one obtainsan expressionwhichformally implies that thenumericalschemeis unconditionally unstable(seeinequality (D20) in appendixD). This is accom-plishedeasiestif a = 1. Accordingly, the full-implicit schemeshould becomeunstableimmediatelyafterion-humpformation in the region of increasingdensity.This is, however,not in accordancewith ournumericalexperience.

Our essentialconclusionis that the stability analysis,basedon the conceptof pseudo-soundvelocity,yields reasonableresultsin thoseregionswherec is a real quantity and, thus,can be interpretedas avelocity. It must, however,fail if c is imaginaryand consequentlyloses its physical relevance.It isindeedquestionablewhetherstability conditionsderived in such a limited sense,describethe actualbehaviourof a numericalmodel for an extremelynonlinear and inhomogeneousproblem. Strictlyspeaking,the set of discretizedequationsis a multi-dimensional,highly nonlineardiscretemapping.Itspropertiesshouldbederivedwithin the frameworkof the methodsfor investigatingnonlineardynamicalsystems(compare,e.g. [80]). Especially, it is contradictoryto keepthe quantitiesA, B, C, in (8.4)constantif, at the same time, the a-dependenceof afV and afv is taken into account. Such aninconsistencyholds for any linearization procedureof nonlinear equations.Conventiona!stabilitymethodsincluding hydrodynamiconescan at bestyield a roughestimateof the dynamicalbehaviour;theybecomethe more unreliablethe more nonlinearand inhomogeneousthe problemis.

By meansof thefull-implicit Lagrangianscheme,a = 1, one is ableto preventwave breakingandtofollow the expandingplasmaover long timeswithout anynumericalinstability. In the next sectionthenumericalresultsfor the earlystageof viscid plasmaexpansionare presentedpayingspecialattentiontothe dependenceof the solution on the viscosity v, on the charge conditions (chargeseparation,-neutrality) and on the initial conditions.

9. The earlystageof viscid plasmaexpansion

Figure ha showsthe spatial dependenceof the ion densityin theisothermalcase(y = 1) with chargeseparationandwith u = 5 for threedifferent time steps.In fig. lib the densityn, the ve!ocity v, andtheelectric field E in the ion front region are plotted on an enlargedscalefor t= 60. Different to the

a __-200 -100 0 100 200

x

Fig. ii. Ion density as a function of x for (a) three different time steps, for v = 5, y = 1 and charge separation; (b) shows the spatial behaviour of n,E, v at the leading front for t = 60. The dots representing n are the actual grid points.

368 Cli. Sackand H. Scliamel.Plasma expansioninto vacuum— A hydrodynamicapproach

correspondinginviscid case(seefigs. 1, 2 and 7) where the solution broke down at t~,= 17.94, wavebreakingdoesnot occur here.The ion hump,which expressesthe densitybunching,increasesin thecourseof time, but the rate of increaseis not explosive. Thus, the implementationof a controllableviscosity takes care of stabilizing the ion bunching process.Within the full-implicit discretizationscheme,a = 1, no numericalinstability develops,althoughthe ion hump is very spiky and steepasshownin fig. lhb. Eachpoint in fig. hib correspondsto a grid point, i.e. the ion hump is resolvedby alarge number of grid points. For higher values of v, e.g. ii = 10, the ion bunchingis further delayed(compareref. [57]). The qualitative structure of the ion front appearsto be independentof thenumericalvalue of v.

The influenceof the chosenchargemodelon the viscid solution is depictedin fig. 12 for the caseofy = 1.2, a = 1 and x = 5. Figure 12a showsthe ion densityat t = 60 resulting from chargeseparation(solid line) andfrom chargeneutrality (brokenline). The densitycurvesdiffer significantly in the frontregion whereasin the rear they are indistinguishable.Figure 12a reflects the delaying influence ofchargeseparationon the ion front velocity, confirming suggestionsin this direction,which havebeenmadeby other authorstoo (seee.g. Denavit [46]). Behindthe front the plasmabehavesquasi-neutral.In this region the ion motion can be describedby the self-similar theory (seesection3.1).

1.0’

0.75

\ a

—200 -100 0 ~__._. 100

b

0.3 / 0.3 25V ~ —2.2 -2

0.224 ‘~~j\ 240.2 ~\~n v ~ n V

/ ~1’ 2.6 20 -42.35 ...

0.1 ‘~ 0.1

2.3 ~. .

84 ~ 88 90 107 109 111 113 115

Fig. 12. (a) Ion density as a function of x for y = 1.2, v = 5 at t = 60, comparing the effect of charge separation (solid line) with that of chargeneutrality (dashed line). (b) and (c) show the. spatial dependence of n, cb, v at the front of a larger scale.


A furthercomparisonof the expansionpatternwith and without chargeseparationis presentedinfigs. 12b and 12c, wheren, v and 4’ in the front region are plottedas functionsof x for t = 60. In thecase of chargeseparation,fig. h2b, the ion density is needle-like,whereasthe velocity v and thepotential 4’ remainrelatively smooth.Consistentwith the chargeseparationmodel the width 4 of theion front is comparableto the local Debyelengthgiven by AD(x, t) = fle(X, t)~~2~2.Sucha consistencyis missing in the quasi-neutralcase,fig. 12c, where the front width doesnot satisfy the necessarycondition4 ~‘ AD(x, t). In contrastto fig. 12b no pronouncedion hump appears.Insteada sharpdensitystep is formed. Disregardingthat the quasi-neutralityassumptionbreaksdown at the densitystep, thebehaviourof the dependentquantitiesin fig. h2c resemblesa usual hydrodynamicshock. This is,however,not surprising,becausein the quasi-neutralcasethe setof equationscoincideswith that of anideal neutralgas (seesection3).

In order to investigatethe dependenceof the solutionon the initial conditions,two calculationshavebeenperformed,in which the initial velocity of the ions was unequalzero. The correspondingdensityprofiles in comparisonto v (x, 0) = 0 (broken line) are given in fig. 13 for y = 1.2, i.’ = 5, and chargeseparationat t = 60. The solid curve resultsfrom the initial velocity

v(x, 0) = v0[1 — n(x, 0)] , (9.1)

where v0 = 1; n(x, 0) is the initial densityprofile from eq. (2.28). The dottedcurve shows a densityprofile following from

v(x, 0) = \/—2 4’(x, 0); (9.2)

4’(x,0) is the self-consistentpotential calculatedat t=0. Both expressions(9.1) and (9.2) try todescribea plasmastatein whichthe diffuse initial densityprofile, dueto somecomplicatedandin detailunknowngenerationmechanisms,is correlatedwith an initial momentumof the particles.Figure 13clearly reflects the dependenceof the front behaviouron the initial conditions.The numberand thepresentposition of the fast ions essentiallydependon the detailsof the initial conditions.

An importantquantity for the descriptionof the dynamicsof expandingplasmasbeingof interest,especiallyin experiments,is the velocity of the fastestions at a given stage.Figure14 presentsthe time

-200x

Fig. i3. Ion density as a function of x at t = 60 for three choices of the initial ion velocity (t = 0): v(x,0) = 0 (dashed line); v0(1 — n0(x,0)) (solidline); u(x, 0) = V—2 cb(x, 0) (dotted line), (y = 1.2, p = 5, charge separation).


8

7

6

5

4

~ ______

9

0 ~ I0 10 20 ~) 40 50 60 70 90

Fig.14. Maximumionvelocityasafunctionoftfor1:y1,QN,v=0;2:y’~1.2,CS,i’5,v(x,0)V—2~(x,0);3:yi,CS,v0;4:yi.CS, p1;5: y=i,CS, v=5;6: y=1, CS, v10;7: y=1.2, CS, r5, v(x,0)=v

0(1—n(x,0)),u0—1; 8: yi.2, QN, v=5;9: y1.2,C5,v = 5. QN and CS denote quasi-neutrality and charge separation, respectively; except for curves 1 and 3 a = i is used in the calculations(_* full-implicit scheme).

evolution of this maximum ion velocity Vmax(t) for a seriesof differentcases.(Noticethat vmax shouldnot be mixed up with the final velocity of the fast ions, which will be investigatedin subsection10.2.)Firstly, it is striking that the expansionelapsesmainly in two phases.The first phaseis characterizedbya continuousincreaseof Vmax~This increaseresultsfrom the initial accelerationof the plasma,which isparticularlystrong,if the plasmawas atrestat t = 0, i.e. v(x, 0) 0. If, dueto the initial conditions,theelectric field is hump-like,this behaviourtransfersto the velocity profile as well in that first phase(seecurves3—6, 8 and9). At the time wherethe solutionof the dissipationlessmodelbreaksdowndue tothe unlimited ion bunching,andthe following wave breaking,the maximumion velocity reachesin thedissipative case, v ~0, a constantvalue of several c5, dependingon i.’, on y, and on the chargeconditions. In this second phase the ion bunching is stabilized by the viscosity. For the sake ofcomparisoncurve 3 shows the maximum ion velocity in the inviscid case for ‘y = 1 with chargeseparation.The crossat the end of the curvemarksthe critical time t~,= 17.94, at which the solutionbreaksdown (seesections5 and 6).

If the plasmahasan initial velocity at t = 0, the accelerationis diminisheddueto the reducedelectricfield (seecurves2 and 7). In the casewhere the initial velocity is not adaptedself-consistently,eq.(9.1), a time evolution of the velocity similar to the caseswith v(x, 0) 0 can be seen(seecurve 7).Different to this, the useof a self-consistentinitial velocity profile, eq. (9.2), leads to a weak, but

C/s. Sackand H. Schamel,Plasmaexpansioninto vacuum— A hydrodynamicapproach 371

continuousincreaseof the maximumion velocity (seecurve2). Therefore,no densitybunchingtakesplace in the consideredtime interval (seealso dotted line in fig. 11).

In fig. 14 a specialcaseis the quasi-neutralonewith y = 1, v = 0 andv(x, 0) 0 whichshowsa strongcontinuousincreaseof the maximum ion velocity (seecurve 1). This behaviourcan be understoodbythe following argumentation.Using quasi-neutrality,ne = n, and an electronequationof statewith apolytropic exponenty> 1, one obtainsfrom eq. (2.25) the relation

~E=a~(n71). (9.3)

For x—* +so the right-handside of eq. (9.3) vanishesbecausen anda~nboth go to zero. As long asy ~ 1, the electric field also vanishes.A finite valueof E in the vacuumregion,which is necessaryfor astrongcontinuousaccelerationof the plasma,can only be achievedfor y = 1. In thisspecialcaseit holdsE = —a~ln n, and a constantvalue of E is necessarilycorrelatedwith an exponentiallydecreasingdensity (see also fig. 3 and section 5). This is exactlythe casehere. In all the othercasesthe electricfield vanishesfor largex within our simplified model, andconsequently,the conditionsfor ion bunchingandwavebreakingare met.

In this sectionthe accelerationand the bunchingphaseand its dependenceon various parametershavebeeninvestigatedwithin the dissipativemodel. An essentialresult is the stabilizationof the ionbunchingby the dissipativeterm beingintroducedin form of the Navier—Stokesviscosity.Furthermore,it hasbeen found that the stabilization of the bunching is correlatedwith a reducedmaximumionvelocity. Onecould believethat this is the final stateof the expandingplasma.In the nextsectionit isshown that the two phasesdiscussedhere are followed by a third phase in which the plasma isacceleratedagain.A simpleexplanationis givenfor thetransitionof thisthird phase.For long times theion front velocity approachesthe value given by theself-similar theory (seeeq. (3.29)).Despitethefactthat oscillationsof the dependentquantitiesemergein the rearof the ion front, the main part of theplasmabehavesquasi-neutralandcan be time-asymptoticallydescribedby the self-similar solution.Theexistenceof theseso-calledintermediateasymptoticsis provenby a perturbativeanalysisof the singlescalarwave equationfor the isothermalquasi-neutralcase(seeeq. (6.14)) usingthe self-similarsolutionas a starting point.

10. The long-time behaviour

10.1. Thedebunchingprocess

To get animpressionof the late phaseof plasmaexpansionwe madeseverallong-termcalculationswhich go beyondthe bunchingphase.Figure 15 exemplarilyshowsthe evolution of the ion densityinthe time interval0~t~140 for y = 1, v = 5 and 1= 1 (seeeq. (2.28)) at ninetime steps.Onerealizesthe stabilizing influenceof viscosity. The existenceof the finite peakandthe associatedfast ions is therelic of the nonlinearprocesstaking place in the early stage.At t = 95 (seedashedline in fig. 15) thepeakreachesmaximumheight. Shortly after this the peakdecaysrapidly, the decaytime beingshorterthan the time for the formation of the peak. The mechanismresponsiblefor this decay can beunderstoodby mass conservationand by the characteristicchangeof the velocity profile in the frontregion.This is illustratedin fig. 16 wherethe x-dependenceof the ion densityn, of the velocity v, and

372 Cli. Sackand H. Schamel.Plasma expansioninto vacuum— A hycirodynarnicapproach

14Q~

120100

40

20

1.00 \\\

118 \ /\

0.6\

0.4—

0.2

0 I I I 1

-200 0 2(X) 400 600—.x

Fig. 15. Space-time evolution of the ion density for v = S. y 1 in the time interval 0 ~ t~140; at r = 95 (dashed curve) ion’ bunching is maximum,whereas for t >95 debunehing takes place.

as

b C ~0.4 -A 0,4 A 0.4 ~ A

n ,~ E v E — . C va ~ 0.3 .3 03 3 03 E

.3 0.2 ‘ Q~ -2 0.2 \.~ 0.2 2

0.1 ~ .i as o.i 1 at at i

—r— 0 -0 ,~‘ ,-- 0 ‘0

268~ 256 260 254268 272 274 284 288 292 296V V

-0.1 -0.1 -0.1

1.90 1.95 1.100

Fig. 16. Spatial dependence of the ion density n (solid line), of the electric field E (dashed line), and of the ion velocity v (dotted line) in the frontregion for (a) t = 90, (b) t 95, and (c) t = 100; at t = 95 the velocity maximum overtakes the density hump initiating thereby the debunchingprocess.

of the electric field E is drawnon an enlargedscalefor the time stepst = 90 (fig. 16a), t = 95 (fig. 16b),and t = 100 (fig. 16c). For t = 90, the velocity maximum Umax lies behindthe density peak,causinganarrowingof the meshpoint distance4 in the ion front region. Due to massconservationn4 const.(sections4.1 and 5), the ion density is locally enhancedbut the collapse is preventedby viscousdissipation.At t = 95 the velocity maximumovertakesthe ion peak,beingin front of it at t = 100 (seefigs. 16b and c). This shift of Umax can be understoodas the desire of the plasmato approachthe

Ch. Sackand H. Schamel,Plasma expansioninto vacuum— A liydrodynamic approach 373

100

0 10 0

0 ~ Vf 00~i~

I 8 I

I. I10 • v = 0.2 I 0 6 i

o ~ 1 p °V • V V 0.2oy5 .~ ovrl

I ~. V V ~ ••~• .~‘‘~‘

V. 6~ II 2 0

It It • I I-2 a * ° b t. Ito

10 I 111114$ 1 .~ ,~— 0 ..... .1 5 10 20 50 100 t 200 300 1 5 10 20 50 100 ~200 300

Fig. 17. Time dependence of (a) the maximum electric field Ema.~and (b) of the ion front velocity, v1, on a double-, resp. single-logarithmic scale

for three different values of v: x: v = 5, 0: r’ = 1; •: v = 0.2; the three plasma expansion regimes are separated by t~and t0.self-similar state,being characterizedby a linearly increasingvelocity profile (seeeq. (3.32)). There-fore, it is not dissipationwhich causesthe suddendecreaseof the densitypeak but a processcalled“debunching”. In the latter, the bunching is reversed, the mesh point distancesbeing rectified.Consequently4 increasesand n decreases.After the passageof Umax through the ion front, thecurvatureof &‘, being always negativein this region, diminishesand the electric field can act moreeffectively on the dynamics.This instantmarksthe onset of a new, as it turns out final, accelerationphase.

In fig. 17athe time behaviourof the maximumelectric field Emax is plottedon a double-logarithmicscalefor threedifferentvaluesof the viscosity v. Figure 17b showsthe ion front velocity v~which is thevelocity at the positionof Emax (i.e. at the positionx~where~e equalsn) on a single-logarithmicscale.Three phasescan be distinguished.The first one is given by the initial accelerationof the plasmaresulting in steepgradients of fl, v and E, associatedwith ion bunching. At t= t~,,where in thedissipationlesscasethe collapsewould occur, the secondphasestarts. It is characterizedby an almostconstantfront velocity Vf of about3 c~(seefig. 14 for moredetails) and by a nearly constantelectricfield Emax. During this phasethe ion peakis present.Debunchingsets in at t = t~wherethe maximumion velocity overtakesthe densitypeak.Therefore,debunchingterminatesthe secondphasecalled forshort “plateauregime”. The last phaseis determinedby thedecreaseof the maximum electricfield andby the continuousincreaseof the ion front velocity. Small differencesin the behaviourof Emax and Uf

arevisible for differentv. With increasingv, debunchingis sloweddown, andthe electric field is slightlyenhanced.Neverthelessthe velocity Ut 15 reduceda little bit owing to the fact that the accelerationisdeterminedby E + v a~vsuchthat the increaseof E is compensatedby the negativeviscosity term. Thelarger electric field gives rise to a stronger increaseof Uf. Later on the influenceof t a~Ubecomesnegligible becausethe curvatureof v disappears.This is trueindependentof thepolytropic exponenty.

10.2. The asymptoticvelocity offast ions

Figure 18, takenfrom ref. [81],showson a semi-logarithmicplot the time-dependenceof the fastionvelocity Vf for variousvaluesof the electronpolytropicindex y (ii = 5). The scenarioof ion accelaration


— v~*(12)10-

v~(1,5)

.: ~ v,~’(1.7) VV

2 ~ (18)

0- ,,,,,,,,, ,,,l,,,,,,l,, -,1 5 10 20 50

Fig. 18. Fast ion velocity c4 as a function of time ton a single-logarithmic scale for several values of the polytropic exponent y; the approach of v4 to

the corresponding self-similar values u’ is evident.

presentedin section10.1 for y = 1 is negotiablefor y > 1, too. The moststriking featureof this figure isthe approachof v~to a constantvelocity for ~ x• This constantvelocity turns out to be identicalwiththe front velocity deducedfrom self-similar theory. The latter hasbeen derived in section 3, eqs.(3.24)—(3.30).Fromthe inequality (3.39) it follows that the self-similar front speedis given by

U~(y)=

2i/~/(y— 1) . (10.1)

It is exactly this speedwhich is asymptoticallyapproachedby the front of the expandingplasma.Withother words, for long times chargeseparationeffects at the front retreatand the deviationsto theself-similar statebecomesnegligible. The approachto the self-similar statewill be outlined further insubsection10.3. For ‘y = 1 thereis no limitation of the front velocity, a casewhich seemsto be inaccordancewith experiments(seesection11). For y > 1 the driving force of electronsis reducedandthe front acquiresa finite velocity.

As mentionedin ref. [81] the knowledgeof this final ion front velocity and o-f the front velocityduring the plateauphaseyields aninformation aboutthethermodynamicbehaviourof electronsduringtheexpansionprocessexpressedby y andaboutthe initial electrontemperatureTeoin theunperturbedplasma.

10.3. Intermediateasymptotics

This approachto the self-similar statesets in at an early stageof the evolution in the rear of theplasmaas mentionedalreadyin section5 (figs. 2, 5, 6). Figure 19 presentsthe space-timebehaviourofthe potential4’, correspondingto fig. 15 (y = 1, v = 5). All 4’ curveshaveanexactfix point, 4’ = —1, atx = —40, the position of the front at t = 0. The linear part in 4’ broadensin the courseof time. Itrepresentsthe self-similar stategiven by

(10.2)

and extendsalmost up to xf, the position of the ion front. At the front the curvatureof 4’ is


1~ 1~~200 2~0 490 6~

2

Fig. 19. Electrostatic potential as a function of x for five different time steps. The positions of the ion front x4 are indicated by arrows. For late

times, the bulk plasma behaves quasi-neutral (straight lines of the potential curves).

non-negligible and changesfrom a positive sign at the beginning to a negativesign at later times.Negativecurvaturecorrespondsto a surplus of ions indicating the presenceof the ion peak. Afterdebunchingwhich takesplaceat aboutt 95, anoscillatorybehaviourof 4’ appearsbehindthefront. Incontrastto the small scalelength at the front in the plateauregime (t ~ 95), the scalelengths of theoscillations behind the front have effectively widened. This is also seenin fig. 20 where both theself-similar quantities,labelledby the index ss and given by (3.24)—(3.27),and the actual numericalquantitiesareplottedas functionsof x for t = 175. Note the characteristicdeviationsbetweenthem.Theelectric field hasa small scale structureat the front wheren is appreciablydifferent from ~~e’ andexhibitsoscillationsof longer wave-lengthsin the rear wheren

600 700 ~ 800

Fig. 20. Ion density n, electron density n~(dotted line), electric field Eand ion velocity v for v = 5 at t = 175; only the leading front is drawn. Forthe sake of comparison, the corresponding self-similar curves, denoted by “ss”, are plotted too. The double arrow and Of represent the local Debyelength and the ion velocity, respectively, at the position of maximum electric field.

376 Cli. Sackand H. Schamel,Plasma expansioninto vacuum— A liydrodynamicapproach

In the following we shall prove that an intermediatestageexists indeedin which quasi-neutralityholds,but in whichthe self-similar behaviourhasnot yet beenachieved.This stagewill be characterizedby long wave-lengthoscillations.

The starting point for our analysisis the scalar wave-equationin the quasi-neutrallimit given by

V+a~(hIV)=0, (10.3)

which hasbeenderivedin section6.1 and assumes‘y = 1. V is the specific volume (V= 1/n), and~j theLagrangianmassvariabledefinedby eq. (6.1).

The self-similar solution of (10.3) is given by

VJ~’vj, t) = _t/*q —1 ~‘q/t~0, (10.4)

(seesection6.1, eq. (6.16)).We now look for a perturbativesolution of (10.3) aroundV

05 making the ansatz:

V=V~5(1+w), (10.5)

where wi =~1. Inserting (10.5) into (10.3) we get after linearization

a1(t2 a

1w) = a~(~2a~w). (10.6)

By meansof the transformation

u=—~7tw (10.7)

(10.6) becomes

t2u=~2u”, (10.8)

wheredot representsa1 and prime a~,respectively.

Equation (10.8) can be solvedby separationof variablesu(ij,t)=f(t)g(~). (10.9)

The symmetry in (10.8) implies that f(t) andg(~)are governedby the samedifferential equation

d2yldx2=(A1x2)y, (10.10)

which is called the Eulerdifferential equation(y(x) = f(t) or g(~),respectively);A is the separationconstant.For 4A + 1 <0 the solution reads[82]:

f(t) = ‘s/j [a1cos(sln t) + a2 sin(sln t)] , (10.lla)

g(ij) = -s/~i~[b1 cos(sln(—q)) + b2 sin(sln(—~))], (10.llb)

C/s. Sackand H. Schamel,Plasmaexpansioninto vacuum— A hydrodynamicapproach 377

wherea~,b~,i = 1,2 are constants,and s is given by

s=~Vi4A+1i.

To simplify the discussion,we set a2 = b2 = 0. With this, V in (10.5) becomes

V= + cos(sln t) cos(sln(_~))], (10.12)

wherec is constant.For fixed t the oscillatory patternin ~j is evident, a fact which transformsto theotherquantitiesand to the Eulerian spaceas well. Equation(10.12) is valid for largevaluesof t andIntl. Excludingthe front given by ~ 0, we see that the oscillationdampsout in the courseof time.Time-asymptoticallythe self-similar stateis reached.To our knowledge,eq. (10.12) representsthe firstconsistent,analyticexpressionin the expansionproblemwhich goesbeyondthe self-similar treatment.Sincechargeseparationis neglectedit appliesto the gasdynamicalexpansionas well.

The consequencesof this observationcan be statedas follows:1. Thereexists a stateof intermediateasymptotics[54] in which the solution no longer dependson

the details of the initial and/orboundaryconditionsbut is far from being in the limiting state.2. This limiting stateis self-similar and is stable.3. The final approachto the self-similar stateis not affectedby charge-separationin theplasmacase.

11. Comparison with experiments

Despitethe simplicity of the consideredion expansionmodel comparisonswith laboratoryexperi-mentsare possible.

Plasmaexpansionexperimentscan roughly be divided into two not necessarily’~omplementaryclasses,namelyinto

i) experimentsdealingwith vacuumarcs,exploding wiresetc. in which a metalplasmaproducedatthe cathodesurface expandstowardsthe anodeand into

ii) experimentsthat aremore or lessdesignedfor the studyof the quasione-dimensionalexpansionof an initially step-likeplasma(“diaphragmproblem”, “dam-breakproblem”).Ion accelerationexperimentsin laserproducedplasmaarepurposelyomittedhere.Their explanationshaveto take into accountamongothersthe irradiationfield, suprathermalelectrons,severalion speciesas well as possiblenonplanargeometries,effects thathavebeenignoredin our model.

A review on the first set of experimentshas beenrecently given by Jüttner[83]. It is generallybelievedthat a majormechanismfor theaccelerationof the ions in vacuumarcs is the electronpressuregradient [21,22,24]. The ions easily gain energiesexceedingthe applied voltage, a processwhichappearsto beenhancedby the Jouleheatingof the arccurrentor by anomalousheatingprocesses[84].Cathodicerosioncreatescratersat the cathodesurface,the cathodespots,giving rise to a continuousquasisteady-stateprocess.(Mathematicallyoneencountersherea somewhatdifferentboundaryvalueproblem. Instead of a rarefaction wave propagatingtowards the unperturbedplasma describedasymptoticallyby the time-dependentself-similar solution, thereis aquasi-stationarysolutionbeingdueto a plasmasourceat the cathodesurfacei.e. at a fixed boundary.)As pointedout by Mesyats[85],

378 Ch. Sack and H. Schamel,Plasma expansioninto vacuum— A hydrodynamicapproach

however, it cannotbe ruled out that the high velocity ions with U -= (1—2) X i0~rn/s are ejected byexplosivesurfaceprocesses,resulting from a processwhich seemsto be more relevantfor explodingwires.

The secondset of experimentsin which this later non-plasmadynamicalprocessis absentallows acloser contact.

Korn et al. [26] investigatedthe evolution of a large density discontinuity in a single-ended0machineand observedessentiallythe self-similar motion correctedby a finite initial ion drift velocityU(x, 0) and by a finite ion temperature,T1 = Te. When the electron temperaturewas raised bymicrowaveheatingat cyclotronresonance,the spatialprofile was observedto broaden,as expected.Apeakof fast ions was, however,not mentioned.Our numericalresultsindicatethatthe finite ion driftvelocity, a characteristicquantity in his experiment,suppressesparticlebunching(seesection9 andfig.13).

A groupof fast ions hasbeenrecordedby Tyulina [25]who measuredthe transverseexpansionof apulsed, nonmagnetizedarc. In this experimentthe existenceof a group of ions precedingthe plasmawas seenwith energieson the orderof ten timesthe estimatedelectronthermalenergy,correspondingto maximum ion velocities of about 3c5, c~= (kTeo/mi)”

2, ~ being the initial thermalenergyof theelectrons.Similar maximum ion velocitieswere found in [22—24].

A seriesof detailedmeasurementshavebeenpresentedmorerecently by EselevichandFainshtain[30] to which we drawthe attentionnow. In this experimentthe plasmawas producedin a prechamberat the entranceof a big vacuumchamberby ionizing pulse-fedgas. During plasmaexpansionthedensityand electrontemperaturewere nearly constantin this sectionof the chamber.In spiteof themodified boundaryconditions,the resultsbearresemblanceto ours in severalrespects,as follows.

The most remarkableonewas the measurementof fast ion velocitiesof 15 c0 at largedistancesfrom

the source,correspondingto an energyE 100kTeo.The accelerationtook place under isothermalconditionsinitially, later on Te decreasedin the front region. Thesehigh ion velocitiesare contrastedwith the relatively low ion front velocitiesin the earlier experiments[22—25].Oneinterpretationof thelatter is basedon the small dimensionsof the used vacuum chambers,which were not larger than0.2—0.4m. Consequently,in theseexperimentsonly a limited time interval was available for theexpansion;the fastestions could only be acceleratedto velocitiesof about3c~(see,e.g., the plateauregime in fig. 18, the curve for y = 1). In the experimentof Eselevichand Fainshtain,however,thevacuumchamberhad a length of 2.0m. By this it was possibleto follow the expansionof the plasmaover larger distancesand, thus, also over longer times. Therefore, the time evolution of the frontvelocity could reachthe third phasein fig. 18, the curvefor y = 1, in which the velocity monotonicallyincreasesandin which it assumesvaluesmuch largerthan 3c5.

Another reasonfor the reducedion front velocitiesin the experiments[22—25]maybe attributedtothe fact that the electronsdo not behaveisothermallyduring the expansion.In this casethe ion frontvelocity time-asymptoticallyapproachesthe valuesgiven by eq. (10.1)which arein accordancewith ournumericalresults (seefig. 18, the curvesfor y ~ 1).

More than a qualitative explanation of the expansionprocesscannot be expected within theframework of our simplified model. As an example, the plateau regime found in the numericalcalculations(seesection10.2, fig. 18) has not yet beendetectedexperimentally.Furthermore,detailedexperimentaldataconcerningthe thermodynamicbehaviourof the electronsin the initial statenearthegenerationpoint of the plasma and during the expansionare needed,to make estimatesfor theavailableion front velocities.

Eselevich and Fainshtain [30] comparetheir results with well-known kinetic [33,46] and hy-


drodynamic [42] models for the collisionless expansionof a plasma into vacuum. Especially, thehydrodynamicmodel appearsto be quitesuitablefor thedescriptionof the dynamicsof the acceleratedions, becausethe measuredeffective ion temperaturerapidly decreasesin the direction of theexpansion.Consequently,the thermalenergyof the ions can be neglectedcomparedto their kineticenergy(see also [33]).

The comparisonof the collision times and of the mean-freepathswith the characteristictime- andspace-scalesof the experimentshowsthat the effect of binary collisions andof collisonswith multiplesmall angledeviationson the expansionis negligible within the temperatureanddensityregionsgivenby Eselevichand Fainshtein:

Te = 2-10eV, T1 = 1-4eV, nei = 107_lOb cm3. (11.1)

For collisions of electronsandof ions with neutralparticlesEselevichand Fainshtain[30] (1980) give amean-freepath of more than 1.0m. The collision times Te and r~as well as the mean-freepaths forelectron—electronand ion—ion collisions are obtainedfrom (see[86]):

= 3.44 X 10~T~2/n~A, le = UtheTe (11.2)

7 3/2 2 1/2

= 2.09x 10 (T~/Z n1A)p. , ii = UIh1T , (11.3)

whereUthe = 4.19x iO7V~cm/s andUthi = 9.79 x i0~\/~7~cm/s denotethe thermalvelocitiesof the

electronsandions, respectively.Z, ~ andA arethe chargenumber,the fraction of ion to protonmass,= m

1Im~,,and the Coulomb logarithm, A 10—12, respectively.The temperaturesTe and T1 areexpressedin eV. For the single ionized argongas(Z = 1, ~ = 40) usedin the experimentonegets with,e.g., ~ei = i0

9 cm’3, Te = 6eV, T1 = 1 eV andA = 15 the following values:

Te sxs4 X 1O”~s, Uthe 10 cm/s,(11.4)

1.3x 102s, Uthi 1.5 X i05 cm/s

and

le~4x104cm, l1~=2X10

3cm.

The calculatedcollision times Tei in (11.4) are much larger than the flight times of the particlesthroughthe vacuumchamberbeingof the orderof (3—5) x i0~sfor the ions. The mean-freepathslei

exceedby far the maximumflight distanceof 2 x 102 cm.For the time evolution of the ion front velocity Eselevich and Fainshtain find a logarithmic

dependencebeing in good agreementwith the numericalresultsof the isothermalexpansionmodelofCrow et al. [42],at leastwith respectto the slope.However, ascan becheckedby meansof ref. [30]fig.4 (1981), the numericalvalue of the slope doesnot correspondto the given scalinglaw, ref. [30] eq.(11) (1981) (seealso eq. (3.35) in this review).

Measurementsof the spatialstructureof the ion front in the experimentof EselevichandFainshtainhaveyieldedtwo differentexpansionregimesin which the front width 4 can be either largeror smallerthan 10 AD! [30]: AD! is the local Debyelengthat the front. For4 ~ 10 AD! EselevichandFainshtainstate

380 Cli. Sackand H. Sc/same!,Plasmaexpansioninto vacuum— A hydrodynamicapproach

a quasi-neutralbehaviourof the front, wherethe densityfalls off almost monotonically.In the otherregime, where the condition 4 a 10 AD! is satisfied only at the plasmasource,the densitysteepensduring the expansion,and the front width 4 varies in the interval AD! ~ 4< 10 AD!; furthermore,adensityhump appearsin the ion densityprofile (seeref. [30] fig. 1 (1979) andref. [30] fig. 5 (1980)).This behaviouris typical for the presenceof chargeseparationeffectsat the expansionfront.

Our numericalresultsin section5 and in section9 offer a possibility to explain the front structuresfound in the experimentof ref. [30]. In section5 it hasbeenstated,that the steepeningof the densityprofile andthe subsequentbunchingaredriven by a spatially decreasingvelocity profile resultingfromthe hump-likeelectric field of theinitial state(seee.g. fig. 2). Steepeningandbunchingareparticularlypronounced,if the initial velocity of the plasma is low (seesection 9 andfigs. 13, 14), and thustheambipolar field dominatesin the expansionprocess.Such conditionscould havebeenpresentin theexperimentof Eselevichand Fainshtainin thosecaseswherethe decreaseof the front width frominitially 4 ~ 10 AD! to AD! ~ ~ < 10 AD! could be observed.In the othercasesin which always4 ~ 10 ADIwas measuredduring the expansionit can be concludedthat the electric field at the plasmasourceand/ or the initial velocity of the plasmawas spatially increasingandtherefore,no significantsteepeningcould take place.However, similar to the interpretationof the behaviourof the maximumion velocityat the beginningof this section,we areleft to a qualitativedescriptionof the occurringprocesses.For auniqueclarification of the space-timebehaviourof the front observedin the experimentof Eselevichand Fainshtaindetailedinformation about the plasma at its generationpoint are missing.This holdstrue for any other experiment,too.

12. Summary and conclusions

In this review we tried to summarizesome aspectsof the hydrodynamicevolution of a plasmaexpandinginto vacuum. A primitive form of the descriptionwas chosenneglectingimportanteffectssuch as the finite size of a plasma,magnetic fields, realistic collisions, two-electrontemperatures,severalion species,and so forth. On the other hand, the chosenmodel turns out to be by no meanstrivial. It allowsthe studyof the plasmaexpansionas awell-defined initial andboundaryvalueproblem.The study includesthe complete time evolution, starting from the very beginning wherethe plasmadistribution is step-likeup to the final self-similar state.

A variety of dynamicalstructureshavebeen found and explained.In the dissipationlesscasetheoccurrenceof wave breakingin connectionwith ion bunchingis oneof the most striking features.Ananalyticalconfirmationof this phenomenonwas given by deriving andsolving a scalarwaveequation.The latter describesthe whole ion dynamicsinclusively chargeseparation.The observationof wavebreakingwas renderedpossibleby using adissipationlessLagrangianschemewith moving boundaries.In addition, an effective Poissonsolver was developedinvolving the smoothestdependentquantity 4’only.

Long term calculationscould be performedby introducingaNavier—Stokesviscosity.It stabilizesionwavecollapse.As a relic of the nonlinearsteepeningprocess,a bunchof fast ion appears.In general,threedynamicalphaseshavebeenidentified. The first one is the initial accelerationphase.The secondoneis characterizedby the presenceof the stabilizedion bump,the velocity of which is approximatelyconstant(“plateauregime”) anddependson y, the electronpolytropic exponent.The third oneis thefinal accelerationphasestarting with the debunchingof the ion bump. It was shown that the whole


plasma,including the front behaviour,tendsto the self-similarstate.In addition,astateof intermediateasymptoticshas beendetected.

Although someof the featuresseemto be in accordancewith experiments,refinementsareneededto cope with realistic expansionprocesses.Ionization, recombination,Coulomb collisions, neutralparticles,electronheat conduction,andfinite energyreservoirare someof them.

Therehavebeenefforts to incorporatesomeof theseeffects in plasmaexpansionmodels[28,29, 40,41, 51, 52, 87, 88] inclusively the investigationsdealingwith the ion blowoff from laser-producedplasma[6—11,27—29, 45—52, 89—92]. A reasonableunderstandingof realistic plasmaexpansionis, however,still waiting for. Onereasonis the intricate natureof the involvedintrinsic processes,anotheronetheoccurrenceof nonlineardynamicalstructures,someof which having beenelucidatedin this report.

Acknowledgements

The authorswould like to thank Prof. Dr. K. Elsãsserfor fruitful discussionson the topic of thisreview. Furthermore,the authorsthankMrs. H. Schamelfor translatingthe text andMrs. B. Malik fortyping the manuscript.

Appendix A: Bunching and wave breaking in nonlinear Langmuir oscillations

The dynamics of nonlinear Langmuir oscillations can be describedwithin the framework of amacroscopicmodel. The hydrodynamic equations for the electronsare completedby Poisson’sequation.The ions are assumedto form a fixed neutralizingbackgroundof uniform constantdensity.Furthermore,the electronsareconsideredto be cold, so that the pressureis neglectedin the electronmomentumequation. From theseassumptionsthe following set of equationsis obtained(Davidson[62]):

atne+ ax(neUe)= 0 (Al)

atUe + Ue axUe = —(eIm~)E (A2)

= 4rre(ne — n0). (A3)

Equivalentto Poisson’sequation(A3) is the Ampere’s law in the electrostaticapproximation

— 41ren1,t.le= 0. (A4)

Within the context of eqs. (Al), (A2) and (A4) Poisson’s‘equationmay be consideredas an initialcondition. By virtue of the continuityequation(Al) and eq. (A4), eq. (A3) remainstrue for all time iftrue initially.

The linear analysisof eqs. (A1)—(A2) showsthat the only modesof oscillation are at the electronplasmafrequency ~

0pe= (41Tnoe2Ime)~”2.Assuming a sinusoidal initial perturbationfor the electrondensity, one obtainssolutions correspondingto standingwaves (wave length 21T/k, frequencyWpe)’

382 Ch. Sack and H. Schamel,Plasmaexpansioninto vacuum— A hydrodynamicapproach

which retain their sinusoidalwave form. In the caseof large wave amplitudes,however,wherethelinearizationprocedureis inapplicable,strongdistortionsof suchwave forms have to be expected.Thisis dueto the dependenceof the solutionon higher harmonicsin kx. Indeedthe conceptof distortionlessoscillationsis completelyuntenablein the investigationof eqs. (Al)—(A4), if large-amplitudeperturba-tions are admitted.

An exactsolutionof eqs. (Al)—(A4) is renderedpossibleby the introductionof Lagrangianvariables[62] correspondingto the ion dynamicalcase(seesection4.1). The setof equationstransformedin thismannerreads:

(AS)

(A6)

fle(~,T) ~X(~,T)=fle(~,0) (A7)

with mast and ~‘=X~j~dT’Ve(~,T’).

Equations(A3) and (A4) can be combinedto give

E(~,T) = 4~fl0~U~(~T). (A8)

Differentiating eq. (A6) with respectto m and using eq. (A8) one obtainsfor Ve(~, m) the harmonicoscillatorequation

~Ue(~,T)+W;eVe(~,T)=0. (A9)

The advantageresulting from the transformationof the set of equations(Al)—(A4) is evident.Equation(A9) is a linear scalarwave equation,which can be easilysolvedcomparedto the original setin the Eulerian picture. The Lagrangianposition variable 4 appearsonly as a parameter.From thegeneralsolution of eq. (A9)

Ue(~,r) = V(~) cos T + ~ X(~)sin Wpe r (AlO)

the remainingquantitiescan be derivedby integrationor differentiationof eq. (AlO):

x(~,m) = ~ + sin + X(~)(1 — cosWpe m) (All)

E(~,r) = Wpe [V(~) sin ~

0pe m — t0pe X(~)cosWpe T] (A12)

~e(~’ r) = ~ 0) [1 + V’(~)sin Wpe T + X’(~)(1 —cos Wpe T)] ; (Al3)

“ ‘ “ representsthe differentiationwith respectto ~.


The functional dependenceof V andX on ~ in eqs. (AlO)—(A13) is relatedto the initial conditionsfor the velocity and for the electric field as follows:

V(~,O)=Ue(~,O), X(~)=- 2 E(~,0). (Al4)

mewpe

In addition,X(~)is relatedto the initial densityTte(~, 0) through Poisson’sequationat m = 0, i.e.

X’(~)= fle(~,0)~’Tbo—1. (A15)

For the transformationfrom Lagrangianto Eulerian variables,i.e., the determinationof ~ as afunction of x andt from eq. (All), the initial conditionsV( ~)andX( ~)haveto be specifiedexplicitly.Generally,this transformationentails the solution of a transcendentalequationwhich can be estab-lishedby numericalmethods.

The condition,that theelectrondensityshouldbe positiveand finite for all time,restrictstheclassofinitial-value problems which may be treated by the procedureoutlined above. Additionally, thefollowing inequalitieshave to be fulfilled:

ne( ~, 0)> n0/2 (A16)

1 / (•~0~ \!/2

V’(~)i<(2 ~ S’ —1) . (A17)Wpe \ flo /

Mathematically,if inequalities(Al6) and(A17) areviolatedfor somerangesof ~, thetransformationfrom the Lagrangianto the Eulerian variablesloses its uniqueness(see eq. (All)). Physically, theviolation of (A16) and (Al7) leadsto the developmentof multi-streamflow. In particular,inequalities(Al6) and(Al7) excludeelectrontrappingwithin the contextof thepresentcold-plasmamodel [62].Toexplain this statement,considerthe initial conditions

X(~)=O, fle(~,0)fo, V(~)=U0sink~. (A18)

From eq. (A 10) and inequality (Al7) it follows that the maximum electronflow velocity is restrictedduring the evolution of the system, i.e.

k’eimax = VOl < iWpe/ki . (A19)

Since the plasma is cold, no electronsare moving exactly at the phase velocity Iw~~/kiof thefundamentalwave, i.e., thereexistsno resonancebetweenthe electronsandthe fundamentalwave,andthereforetrappingdoesnot occur.

Next we presentan example taken from ref. [62], which demonstratesthe essentialprocessesconnectedwith the transformationto Eulerianvariables.The initial conditionsarespecifiedas follows:

ne(~,0)no(1+4c05~), 4~<~ (A20)and

Ue(~,O)=V(~)=O. (A21)


Inequalities(Al6) and (Al7) are satisfiedfor this case,and one obtainsfrom eqs. (A10)—(Al3):

Ue(~, T) = —~ 4 sin k~sin m (A22)

m 2

E(~, m) = — e pe sin k~cos m (A23)

I + 4 cosk~

~ (A24)

with

kx = k~ + a(T) sin k~, a(r) = 24 sin2(wper/2). (A25)

In (A20) the condition ~I< ~ guaranteesthat a(r)i < 1 and that the inversion of (A25), i.e.x(~,m)—~~(x, t), is unique.

For small amplitudes,i.e. I~I~ 1, it is clear that the Eulerian and the Lagrangianvariablesareapproximatelythe same,x — 4. In this casethe resultscorrespondto those which follow from thelinearizationof the equationsof motion (Al)—(A4). For largervaluesof I~I,however,the solutionhasto be determinednumerically from (A25) at different times. ThenUe~E and~e’ in Eulerianvariables,arededucedfrom eqs. (A22)—(A24). Solutionsof this kind arepresentedby Davidson[62]for 4 = 0.45(seeref. [62] figs. 3.1 and3.2). The xt-dependenceof the solution is periodic with periods2ir/k and2~m/w~.The most interestingpropertyof the solution is that theelectrostaticforcescausea stronglocalincreaseof the electrondensity in the vicinity of

x(2n+l)ir/k, n0,±1,±2,.. . (A26)

This phenomenonis called“bunching” (compare[62]). The densitymaximum, lie = 5.5 n0 is reachedat

t = ir/w~. Later on the systemreturnsto its initial state.The correspondingelectric field exhibits asteepeningof the initial wave sin kx with any increasein the maximum amplitude. The electronvelocity, however,doesnot steepen;for t = (IT/Wpe) Ue is evenequalto zero.

The solutions presentedby Davidson [58] have a certain similarity to the solutionsfor the ionicsystemshownin fig. 7. However, in the caseof ion dynamics,the velocity steepens,too (seefig. 7).Such a behaviourcan also be obtainedfor nonlinearLangmuiroscillations,if oneusesa moregeneralformulationof the electronmotion. This formulation,allowing the separationinto a boundary-andaninitial valueproblem,hasbeenestablishedby Kalman [73]. Different to Davidson[62], Kalmandid notusethe initial coordinate~ as aLagrangianvariable,but the so-calledLagrangianmassvariablen(~I’inKalman’s work). However, thereis a uniquerelation between~ or x and n:

:= Jn5~’,0) d~’ fle(X’, t) dx’. (A27)

The transformation(A27) automaticallysatisfiesthe continuity equation

an/axl, = fle(x, t), an/ati~= ~eUe~ (A28)

so that atne+ ax(neUe)= 0 is valid.


Additionally, from (A28) it follows that thetotal time derivativeof n vanishes:

dn/dtO, (A29)

i.e. n = const.along the streamlines. Therefore,n is sometimescalled streamfunction.Inserting the transformation(A27) andeq. (A29) into the set of equations(A1)—(A4), one finally

gets again the scalarwave equationfor the velocity Ue:

a2Ue/at2+w~eUe=o. (A30)

Different to the waveequation(A9) Ve now dependson n and t and not on ~ andr(=t). In Kalman’swork [73], however, a scalar wave equation for the velocity is not deducedbut a correspondingequationfor the space coordinatex, which contains a free function of time. The solution of thisequationyields the velocity, the density, andthe electrostaticfields in dependenceon n andt. For anexplicit representationof the solution the boundaryand initial values have to be specified. Defining,e.g., a boundaryvalue problemof the kind

Ue = Ue0(l + ~.t sin tot) for x = 0 (A3l)

.1 = ~eü~’eü (A32)

E=0 (A33)

where~ is the relativedepthof the velocity modulationandto is its frequency,oneobtainssolutionsofthe problem,the structureof which is governedby the parameter

Co/COpe. (A34)

For a = 2, e.g.,the bunchingof the electronsis unlimited (the electrondensitybecomessingular!),andthe electric field andthevelocity steepenindefinitelywith a reversedsign of their gradients(seeref. [73]fig. 1). This propertyof the solution in the electroniccasecorrespondsexactlyto the propertiesof oursolution shown in fig. 7. It is, therefore, obvious to investigatethe nonlinear ion motion with aformalism analogousto that usedby Davidson[62]and Kalman [73]. The scalarwaveequationfor theionic systemdeducedin section6.1 is, however,partialandnonlinear.Therefore,it canonly betreatedby perturbativemeans(seesection6.2).

Appendix B: Spatial discretization of Poisson’sequation

According to Potter [66] a double mesh point net is used for the spatial discretization of thehydrodynamicion equationsin the Lagrangianpicture, eqs. (4.9)—(4.ll), respectively(8.1), and ofPoisson’sequation(2.24). The cell boundariesx1 andthe velocities U. aredefinedpointwiseon a net J,1 ~j ~ J+ 1; Jdenotesthe total numberof cells in the finite integrationinterval havingthe length 2L,

—L ~ x ~ +L, at t = 0. The propertiesof the densityandof the electrostaticpotential,respectivelyfieldare distributedwithin the cell. Therefore,the meanpoints of eachcell are definedon a secondnet I,

386 Ch. Sackand H. Sc/same!,Plasmaexpansioninto vacuum— A hydrodynamicapproach

1 � i ~ I(=J); eachcell i hasthe width

4, = x1÷1 — x. . (Bi)

The relation betweenthe f-net and the I-net is given by

x~= ~(x1~1+ x1). (B2)

At t = 0 the grid points have equal distance: 4, = ~.x= const. For our purposethe choice f = 500,

L = 200, and thus, ~x = 2L/J = 0.8 hasbeensufficient (see also [56]).SincePoisson’sequationis only spatiallydiscretized,the time variableis not explicitly consideredin

the following calculations.The discretizedversion of the linearized Poisson’sequationon the I-netreads

— (dfl~/d4’1~)~4’~= ~(dfle/d4’o)i4’~+ n~(4’~) — n,. (B3)

For the secondderivativeof 4’ in (B3) at the grid point i oneobtains

= - 4’~, k = i + (B4)xkxkl

with

—

4’j+1 — , — —

- ‘ 4’~-~-— (i+l x~

and

xk = ~(x1~1+ x~) , Xkl = ~(x~+ x~1) . (B6)

Inserting (B5) and (B6) into (B4) the resulting equation yields a three-point formula, i.e. 4~iscompletelydeterminedthroughthe discretevaluesof 4’ at the grid pointsx, ~, x, andx. ~ By this onegetsfor (B3) a setof linear equationswith a tn-diagonalcoefficientmatrixwhich can be formulatedasfollows

~ 2~i~I—l (B7)

with

2~ = (B7a)

— x~_1j~x~— x~1)

1 2 /dn\1a,, = —L + ~—~) ] (B7b)(x,÷1—x,)(x, —x~1) dçb0

2= / \1 \ (B7c)

— x1_1,~x~÷1— x~)


in (B7) o is identicalwith the right-handsideof (B3). The quantitiesfle(4’01) and(dne/dq5o)iaregiven

by (4.14) and (4.15), where 4’~denotesthe valuesof 4’, from the previousiterationstep(seesection4.2).

The representationof (B3) by (B7) cannot be transferredto the first and to the last row of thecoefficientmatrix becausein both rows only two insteadof threematrix elementsare available.Notethat a second derivative has to be resolved by three grid points. The missing matrix elementsaredefined by the boundary conditions being deducedin section 4.3, eqs. (4.37) and (4.42). Theseequationsaredifferentialequationsof first orderandareapproximatedin the discretecaseby two-pointformulae.The first row of the coefficientmatrix (i = 1) containsthe boundaryconditionsat x = — L (seeeq. (4.37)):

all4’l+a124’2=o-!=O (B8)

with

a11 = —4’~ (B8a)

— x3—x2a12 = 4’o2 + ô + 1 (4’~~— := — , - (B8b)

whereasthe last row (i = I) is filled by the boundaryconditionsat x= +L (seeeq. (4.42)):

+ a114’1 = 0~~ (B9)

with

a11..1 = —[1 —f11...1(x1 — x1.1)] (B9a)

a11 = 1 + f11 1(x1— x1..1) (B9b)

= [f1~...1(4’~1+ ~ — g1 ~ (x1 — x1...1) (B9c)

fj,1-1 ~ {[~e(4’oi)]1~~’2+ [n

0(4’01.1)]1~2} (B9d)

g1,1~1:= ~ {[n~(4’o1)]~

2+ [ne(4’oji)]~2. (B9e)

The numericalsolution of the setof linear equationsrepresentedby eqs. (B7)—(B9) is accomplishedthrougha Gaussianelimination procedureadaptedto the tn-diagonalband structureof the coefficientmatrix (see [93]). Since generally Poisson’s equation (2.24) is nonlinear (exception y = 2) and,therefore,can only be solvediteratively, the outlined procedurehas to berepeatedlyusedfor eachtimestep in order to determinethe new valuesof the potential. The maximumrelative error defined by

(BlO)

388 Ch. Sack and H. Sc/same!,Plasmaexpansion into vacuum— A hydrodynamicapproach

is usedas a convergencecriterion for the iteration; p numeratesthe iteration step. In the caseofconvergence6.,~hasto decreasemonotonicallyat least from a certainiterationstepon, so that ~ 6can be achieved.

Appendix C: Implicit difference schemefor the equations of viscid ion motion

For the time discretizationof the equationsof ion motion (4.9), (4.10) and (8.1), the time t isdivided into finite equidistantincrements~t, i.e., t = n~t=: t~,n = 0, 1,2 In order to guaranteean accuracyof the order O(~t)2it is necessaryto define eachcell boundaryat times n, whereasthevelocities U. of the cell boundariesare definedat the intermediatetime stepsn — ~ [66]. By this oneobtainsin the Lagrangianpicture the following differenceschemeon the double grid point net (seeappendixB):

— U~’2) /~t= ~(E~ + E~1)+ v[a(U”)~’

2 + (1 — a)(U”)~”2] (Cl)

XX~+~tU~~ (C2)

n+1 n n fl n+1 n+1n~ n~(x

1+~ —x1)/(x1+1 —x1 ), (C3)

whereE’ in (Cl) follows from

fl ! ~f fl ‘~ fl ~ fl ‘~

E1 = —~I(4’~+~— 4’1)/(x~+~—x1)+ (4’,- — 4’~1)/(x~ —x11)]; (C4)4’~resultsfrom the iterative solution of Poisson’sequation(seeappendixB), andx7 is given by (B2).Except the calculationsfor fig. 7, beingperformedwith z~t = 0.005, z~thas beenset equalto 0.025. Thespatially centredformulation of the secondderivativeof U in (Cl) is given by

m , m m rn(v )j ((U )J+1-2 — (v )~~1-2)/(x1+~12 — x1_117), m= n ± ~ , (CS)

where

m 1 ~n ~x1~12= ~(x1~1+ x1) (C6)

and

in m in in m in 1 fl n±1(U )J+1/2 = (U1÷1— U1 )/(x1~1 — x1 ), x1 = 1(x1 + x1 ). (C7)

In (Cl) the quantityadenotestheimplicitnessparameter,0~ a~ 1. For a = 0 the differenceschemeisexplicit, becausethe right-handsideof (Cl) only containsU of the previoustime stept”’~’

2. In the casea ~ 0 U of the actualtime stept~/2 comesinto play, andthe schemeis implicit. Specialcasesarea =

(half-implicit) anda = 1 (full-implicit) (seesection8). If the viscosity constanti.’ is equalto zero, thedissipationlessexplicit Lagrangianschemeis obtainedwhich has beenusedto get the resultspresentedin section5.

Similar to the discretizationof Poisson’sequationin appendixB the spatial discretizationof (Cl)

Cli. SackandH. Schamel,Plasma expansioninto vacuum— A hydrodynamicapproach 389

leadsto a set of linear equationswith a tn-diagonalcoefficient matrix:

b~1U~1+ b~V~+ b~~1U~~1= r’~2, 1 n + ~, 2~j~ J. (C8)

The coefficientsin (C8) andthe right-handside u~”~112areobtainedby inserting(C5)—(C7) into (Cl):

b~11= —s/4~1j1 ~ s = 2~Ata (C8a)

b~1= 1 + ~ ~ (C8b)

~ = ~‘~J+I,J! ~ (C8c)

r’~”2 = (E~+ E~

1)+ v~2 + p At (1— a) (U”)~2, (C8d)

= x~1— , = x~+~—x~, 4~-~= —x~1. (C8e)

The first andthe last row of the coefficientmatrix arespecifiedby the boundaryconditionsin the finiteintegrationinterval (seesection4.3). At the left-handboundary,x = —L(j = 1), the discretizedversionof eq. (4.33) yields:

b’11V~+ b~2t4= r~’ ; (C9)

b~1 1 +2Ata-’,h/(x’~—x~) (C9a)

b~2= 1 —2 Ata\/5~/(x’~—x~) (C9b)

~ = V~1+ U~~1+2At(l — a)-~~/y(Ut’ — v~’)I(x~~—xv’) (C9c)

with I from (C8). At the right-handboundary,x = +L(j = J + 1), the spatialmeanvalue of t~= E (seeeq. (4.44)) is used:

+ t)~,)= E

1. (ClO)

From (ClO) it follows:

I / I / n,n—1/2

~ + b~+1~+1U1+1= r~~1 , (Cll)

where

b1~1~= 1 = b1~11~, (Clla)and

r~11U2= V~1+ ~ + 2 AtE~. (Cllb)

390 C/s. Sackand H. Sc/same!,Plasma expansioninto vacuum— A hydrodynamicapproach

In the implicit case,a� 0, an iterative procedureis requiredfor the determinationof the velocityU~U

2in (Cl) by meansof (C8), (C9) and(Cll), which can be understoodas follows. Assumethat f,U”112, E~and n’ are known at the time step n. In order to calculate U’’~2 from (Cl), one needs,accordingto (CS)—(C7) x’ 1; however,x’ + is not known,sincefor its determinationthe knowledgeofUnL(2 is necessary(see(C2)). Consequently,one hasto iteratebetween(Cl) and (C2) at each timestep.As initial valuefor iterationthe valuesof x from the previoustime stepareused.Typically two orthreeiterationsare necessaryto reduce the maximum relative error 6~below i07 6~is definedby

:= maxi(U~— U~)/(U~’ + V~)I, (C12)

wherep denotesthe iteration step. An expressionsimilar to (C12) holds for xc.At each iteration step the set of linear equationsconsistingof (C8), (C9) and (Cil) is solved by

meansof the Gaussianeliminationprocedurementionedin appendixB.

Appendix D: Conventional stability analysis of the discretized set of equations

In section8 the assumptionsfor the derivationof a conventionalstability condition of the implicitdifferenceschemehavebeendiscussed,andits accordancewith the stability behaviourof the numericalsolutionhasbeeninvestigated.A crucialpoint for the derivationof the stabilityconditionis the useofthe pseudo-soundvelocity (see section 3.2 and eq. (8.5)) and the assumptionof an appropriatelinearizationof the ion equations(seee.g. eqs. (8.3) and (8.4)). Thus, the equations(4.9), (4.10) and(8.3) are discretizedwith respectto the Lagrangiancoordinate~ to give:

(x~ —x~)!At= U~2 (Dl)

V0(x~~1—xc)/h=Vc+112 (D2)n+1/2 n—1/2

U- U- C

At = ~ (V~÷3~2— V~1~2+ V7±112— Vc_312)

v’~--’2 V~~’~2 v”~2 v~112

+~ [a ~ h ‘‘ +(l—a) j+1 j-I ]n+1/2 n+1/2 n+l/2 n—1/2 n—!/2 n—1/2I V-~~—2V- +V-

1 U-~1 —2U- +U-1 1+BLa ‘2 +(l—a) ~2

(D3)

whereV0 = 1 /n0(fl denotesthe specific volume at t = 0, andh(=A~)is the equidistantspace-step;a isthe implicitness parameter(compare(8.2), resp. (Cl)). The quantitiesA, B, C in (D3) aredefinedby(8.4) andareheld locally constant,as well 1/~in (D2). By meansof the discretetimedifferentiationof(D2) and by using (Dl) the spacevariable x can be eliminated:

At n—! /2 n—1/2 n n—iV0 -~- (v~÷~— U1 ) = VJ÷1/2 — VJ+!/2. (D4)


(D3) and(D4) representa closedset of linearizedequations,thestability of which is investigatedin thefollowing.

Inserting the Fourier transformation

f~= j~exp(ik~1) (D5)

into (D3) and(D4), where denotesthe discretevaluesof ~ andf standsfor U andV respectively,oneobtains:

= y 12;1 + (w + ry) ~n-1/2 (D6)

= ~n-1 + rC~2. (D7)

In (D6) V~hasbeenexpressedby (D7). For the coefficients r, y, u and w, resp.,onegets:

r = 2i I’~~ sin(~) (D8a)

y = iC ~ cos(~)sin kh (D8b)

u=l+2(l—coskh) C2—2isin khC1 (D8c)

w = 1 —2(1 — coskh) ~2 + 2i sin kh ë~ (D8d)

with

/C1\ / a ~AAt /C2\ / a \BAtI 1=1 I— and I — I=I

\l—a/ 2h ~C2~ \l—a/ h

The eigenvaluesof the coefficient matrix of the linearizedset (D6, D7), i.e.

(~ (w+ry)Iu) (D9)

yield the stability condition. The characteristicequationfor the eigenvaluesof (D9) reads

A2— A(u + ~ ry) + =0. (Dl0)

In generalits complexsolutionsare

A!2 = (L ±VL

2 — 4uw) (Dll)

with

L=u+w+ry. (D12)

392 Cli. Sackand H. Sc/same!,Plasma expansioninto vacuum— A hydrodynamicapproach

The systemis stable, if A11 ~1, j = 1, 2 (compare[65,66]). L and the squareroot in (Dll) can be

formulatedas follows

L = 2[1 — Q2 — Y

1(l — 2a)+ iT(1 — 2a)] (D13)

\/L2—4uw=2\[X~iY (D14)

with

X = — 2Q2+ 2Q2 ~“1 (1 — 2a) + Y~— T2 (Dl5a)

Y = —2T [Q2(l — 2a) + Y1] (DlSb)

Q2 = ~p2sin2 kh (DiSc)

BAt (l—coskh) (D1Sd)

T= BAt af(ln ~) sin kh (DlSe)

p2 = c2 (1/a At/V/i)2, (DlSf)

where c2 is given by (8.5).From thesequantitiesone getsfor A!

2 in (Dll):

= ~ [L ±\/~(Vi~+ X + iV7~— X)] u~, (Dl6)

whereR = + Y2 >0; U is given by (D8c), and u’~ is the complexconjugateof u. From A

1 2I2 ~ 1

one deducesafter a lengthy calculationthe relation

0~R2~2Y~—X. (D17)

After a further quadratureit follows

O~Y2~4Y2(Y~ — X). (Dl8)

(Dl8) is the conditionfor numericalstability we havelooked for.Accordingto the sign of c2 two caseshaveto be distinguished.For c2 > 0 one obtainsfrom (Dl8)

and (DlSa):

Q2 [T2 (1 — 2a)2 + Y~]~ [2Y~ — 2Y1 (1 — 2a) (T

2 + Y~)]. (Dl9)

Q, Y1 and Tare defined by eqs. (l5c—e). From this the casesa = ~, inequality (8.6), anda=0,1,

inequality (8.7), are deduced,wherein the latter casecoskh = 0 (kh = ir/2) hasbeenused.


In regionsof increasingdensity, i.e. c2 <0, (D18) reads:

1Q21 [T2 (1 — 2a)2+ Y~]~ —[2Y~— 2Y1 (1— 2a)(T

2 + Y~)]. (D20)

In comparisonto (D19), (D20) has a negativesign on its right-handside. This meansthat the set ofdiscretizedequationsis numerically unconditionallyunstable,if the bracketon the right-handside of(D20) is positive. Since ~! is always greaterthanzero, this is the case,if a is setequalto one, i.e., inthe full-implicit scheme.However, such an instability hasnot been detectedin our calculations.

References

[1] U. Samir and N.H. Stone,Bericht der “XVI International Conference on Phenomena in Ionized Gases (ICPIG)” Düsseldorf, FRG, 1983; p.84;K.H. Wright Jr., N.H. Stone and U. Samir, J. Plasma Phys. 33(1985) 71.

[2] i.E. Borovsky, MB. Pongratz, R.A. Roussel-Dupréand T.-H. Tan, Astrophys. J. 280 (1984) 802;T.-H. Tan and J.E. Borovsky, Preprint LA-UR-84-754, Los Alamos National Laboratory, Los Alamos, New Mexico, USA, 1984.

[3] R. Behrisch, in: Proc. XIth Intern. Symp. on Discharges and Electrical Insulation in Vacuum, Vol. 1, Ostberlin, GDR, 1984 p. 123;L.M. Baskin, AL. Gromov and G.N. Fursey, ibid., p. 165.

[4] PB. Parksand R.J. Turnbull, Phys. Fluids 21(1978)1735.[5] CT. Chang, LW. Jorgensen,P. Nielsen and L.L. Lengyel, NucI. Fusion 20 (1980) 859.[6] K.A. Bruckner and S. Jorna,Rev.Mod. Phys. 46 (1974) 325.[7] E.J. Valeo and Ira B. Bernsteiii, Phys. Fluids 19 (1976) 1348.[8] R. Decosteand B.H. Ripin, Phys. Rev.Lett. 40 (1978) 34.[9] F.S. Felber and R. Decoste,Phys. Fluids 21(1978)520.

[10] A. Gurevich,D. Anderson and H. Wilhelmsson, Phys. Rev. Lett. 42 (1979) 769.[11] CE. Max, in: Laser-Plasma Interaction (North-Holland, Amsterdam, New York, Oxford. 1980) p. 301.[12] F. Begay and D.W. Forslund, Phys. Fluids 25 (1982) 1675.[13]P. Wagli and T.P. Donaldson, Phys. Rev. Lett. 40 (1978) 875.[14] P.M. Banks and T.E. Holzer, J. Geophys. Res. 73(1968)6846.[15] N. Singh and R.W. Schunk, J. Geophys. Res. 87 (1982) 9154.[16] ED. Korop, Zh. Tekh. Fiz. 46 (1976) 2187 [Soy.Phys. Tech. Phys. 21(1976)1284].[17] RB. Baksht, NA. Ratakin and BA. Kablamaev,Zh. Tekh. Fiz. 50 (1980) 487 [Soy.Phys. Tech. Phys. 25(1980) 294].[18] B. Eiselt, Z. Phys. 132 (1952) 54.[19] IF. Kvarkhtsava,A.A. Plyutto, A.A. Chernovand V.V. Bondarenko, Soy. Phys. — JETP 3 (1956) 40.[20] R. Tanberg, Phys. Rev. 35(1930)1080.[21] A.A. Plyutto, Zh. Eksp. Teor. Fiz. 39 (1960) 1589 [Soy.Phys. — JETP 12 (1961) 1106].[22] H.W. Hendel andT.T. Reboul, Phys. Fluids 5 (1962) 360.[23]P.F. Little, Journal of Nuclear Energy, Part C, 4 (1962) 15.[241A.A. Plyutto, V.N. Ryzhkov and AT. Kapin, Zh. Eksp. Teor. Fiz. 47 (1964) 494 [Soy.Phys. —JETP 20 (1965) 328].[25] M.A. Tyulina, Zh. Tekh. Fiz. 35 (1965) 511 [Soy.Phys. Tech. Phys. 10 (1965) 396].[26] P. Korn, T.C. Marshall and S.P. Schlesinger, Phys. Fluids 13 (1970) 517.[27] W. Demtröderand ‘A’. Jantz. PlasmaPhys. 12 (1970) 691.[28] G.J. Tallents, Plasma Phys. 22 (1980) 709.[29] P. Pitsch, K. Rohr andH. Weber,J. Phys. D: AppI. Phys. 14 (1981) L51.[30] V.G. Eselevichand V.G. Fainshtain, Doki. Akad. Nauk SSSR224 (1979) 1111 [Soy.Phys. Dokl. 24 (1979) 1141; Zh. Eksp. Teor. Fiz. 79

(1980) 870; [Soy.Phys. — JETP 52 (1980) 441]; Fiz. Plasmy7 (1981) 503; [Soy.J. PlasmaPhys. 7 (1981) 2711.[31] C. Chan, N. Hershkowitz,A. Ferreira,T. Intrator, B. Nelsonand K. Lonngren, Phys. Fluids 27 (1984) 266.[32] A. Akhiezer, Zh. Eksp. Teor. Fiz. 47 (1964) 952 [Soy.Phys. — JETP 20 (1964) 637].[33] A.V. Gurevich, LV. Pariiskaya and L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. 49 (1965) 647 [Soy.Phys. — JETP 22 (1966) 449].[34] D. Montgomery, Phys. Rev. Lett. 19 (1967) 1465.[35] A.V. Gurevich and L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. 56 (1969) 1778 [Soy.Phys — JETP 29 (1969) 954]; Zh. Eksp. Teor. Fiz. 60(1971) 2155

[Soy.Phys. —JETP33(1971)1159].[36] J.E. Allen and J.G. Andrews, J. Phasma Phys. 4 (1970) 187.

394 Cli. Sack and H. Schamel,Plasma expansioninto vacuum— A liydrodynamicapproach

[37] M. Widner, J. Alexeff and W.D. iones, Phys. Fluids 14 (1971) 795.[38] R.J. Mason, Phys. Fluids 14 (1971) 1943.[39] PD. Prewett and J.E. Allen, I. Plasma Phys. 10 (1973) 451.[40] G.S. Voronov and L.E. Chernyshev, Zh. Tekh. Fiz. 43(1973)1484 [Soy.Phys. Tech. Phys. 18 (1974) 940].[41]E.E. Lovetskii, AN. Polyanichev and VS. Fetisoy, Fiz. Plasmy 1 (1975) 773 [Soy.J. Plasma Phys. 1 (1975) 422].[42]J.E. Crow, P.L. Auer and i.E. Allen, J. Plasma Phys. 14 (1975) 65.

[431A.V. Gurevich and L.P. Pitaeyskii, Prog. Aerospace Sci. 16 (1975) 227.[441H. Shen and K.E. Lonngren, J. Engineering Math. 10 (1976) 135.[45] B. Bezzerides, D.W. Forslund and EL. Lindman, Phys. Fluids 21(1978) 2179.[46]J. Denavit, Phys. Fluids 22 (1979) 1384.[47] P. Mora and R. Pellat, Phys. Fluids 22 (1979) 2300.[48] K.E. Lonngren and N. Hershkowitz, IEEE Transactions on Plasma Science PS-7, No. 2 (1979) 107.[49]A.V. Gureyich and A.P. Meshcherkin, Zh. Eksp. Teor. Fiz. 53(1981)1810 [Soy.Phys. —JETP 53(1981) 937].[50] MA. True, JR. Albritton and E.A. Williams, Phys. Fluids 24 (1981) 1885.[51] A.V. Gureyich and A.P. Meshcherkin, Fiz. Plasmy 8(1982)502 [Soy.J. Plasma Phys. 8(1982) 283].[52] SI. Anisimoy, M.F. Ivanov, Yu.V. Medvedev and V.F. Shvets, Fiz. Plasmy 8 (1982) 1045 [Soy.I. Plasma Phys. 8 (1982) 591].[53] L.D. Landau and EM. Lifschitz, Lehrbuch der theoretischen Physik Vol. VI (Akademie-Verlag, Berlin, 1966).[54] G.J. Barenblatt, Similarity, Self.Similarity, and Intermediate Asymptotics (Consultants Bureau, New York and London, 1979).[55] R.Z. Sagdeev. Reviews of Plasma Physics Vol. 4 (Consultants Bureau, New York, 1966) p. 23;

J.E. Allen and L.M. Wickens. J. Physique, Colloque C7 (1979) 547.[56]Ch. Sack and H. Schamel, J. Comput. Phys. 53(1984) 395.[57]Ch. Sack and H. Schamel, Plasma Phys. Controlled Fusion 27 (1985) 717.[58] Ch. Sack and H. Schamel, Phys. Lctt. AllO (1985) 206.[59] R.L. Morse, Phys. Fluids 8 (1965) 308;

D.W. Forslund and CR. Shonk, Phys. Rev. Lett. 25 (1970) 1699;H. Schamel, Plasma Phys. 14 (1972) 905.

[60] W,F. Ames, Nonlinear Partial Differential Equations in Engineering, Mathematics in Science and Engineering Vol. 18 (Academic Press, NewYork, 1965).

[61] K.E. Lonngren, in: Recent Advances in Plasma Physics, Proc. Workshop held at Physical Research Laboratory, Ahmedabad, ed. B. Buti(1977) p. 39.

[62]R.C. Davidson, Methods in Nonlinear Plasma Theory (Academic Press, New York, 1972).[63] S.F. Johnson and K.E. Lonngren, Physica Scripta 25(1982)583.[64] H.L. Lewis, Preprint LA-UR-82-3100, Los Alamos National Laboratory, Los Alamos, New Mexico, USA, 1982.[65] RD. Richtmyer and K.W. Morton, Difference Methods for Initial Value Problems (Interscience, New York, 1967).[66] D. Potter, Computational Physics (Wiley, New York, 1973).[67] R.L. Dewar and E.J. Valeo, Proc. 6th Conf. on Num. Sim. of Plasmas, 1973.[68] D.W. Forsiund, J.M. Kindel, K. Lee and EL. Lindeman, Proc. 7th Conf. on Num. Sim. of Plasmas, Courant Institute NYU, 1975.[69] J. Trulsen, Tromsö Report, Tromsö, Norway, 1980.[70] A. Clebsch, J. Reine und Angew. Math. 56 (1859) 1.[71] K. Elsässer and H. Schamel, J. Plasma Phys. 15(1976) 409.[72] Ya. B. Zel’dovich and Yu.P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena (Academic, New York, 1967);

R. Courant and K.O. Friedrichs, Supersonic Flow and Shock Waves (Springer-Verlag, New York, Heidelberg, Berlin, 1976).[73] G. Kalman, Ann. of Physics 10 (1960) 1.[74] D.W. Forslund, i.M. Kindel, K. Lee and B.B. Godfrey, Phys. Fluids 22 (1979) 462.[75] N.S. Buchelnikova and E.P. Matochkin, Institute of Nuclear Physics, Report 83-88, Novosibirsk, USSR, 1983.[76] KY. Kim, Phys. Lett. A97 (1983) 45.[77] H. Schamel, Z. Naturforsch. 38a (1983) 1170.[78] P.K. Sakanaka, Phys. Fluids 15 (1972) 1323.[79] SI. Braginskii, Reviews of Plasma Physics Vol. 1 (Consultants Bureau, New York, 1965) p. 205.[80] A.J. Lichtenberg and MA. Lieberman, Regular and Stochastic Motion, Applied Mathematical Sciences Vol. 38 (Springer-Verlag, New York,

Heidelberg, Berlin, 1983).[81] Ch. Sack, H. Schamel and R. Schmalz, Phys. Fluids 29 (1986) 1337.[82] E. Kamke, Differentialgleichungen, Losungsmethoden und Ldsungen I (Teubner.Verlag, Stuttgart, 1977).[83] B. Jüttner, XVII Intern. Conf. on Phenomena in Ionized Gases, Invited papers, Budapest (1985) p. 262.[84] V.A. lvanov, B. Jüttner and H. Pursch, Preprint 84-6, Akademie der Wissenschaften der DDR, Zentralinstitut für Elektronenphysik (1984).[85] GA. Mesyats, Proc. 10th Intern. Symp. Discharges and Electrical Insulation in Vacuum, Columbia (1982) p. 37.[86] DL. Book, NRL Plasma Formulary, Laboratory of Computational Physics, Naval Research Laboratory, Washington, D.C. 20375, 1980.[87] Yu.I. Chutov. XVI Intern. Conf. on Phenomena in Ionized Gases (ICPIG), Invited Papers, Düsseldorf, FRG, 1983, p303.

Ch. Sackand H. Schamel,Plasma expansion into vacuum— A hydrodynainicapproach 395

[88] L.P. Gorbachev, Magnetohydrodyn. 20 (1985) 398.[89]A.W. Ehler, J. AppI. Phys. 46 (1975) 2464. —

[901P.M. Campbell, R.R. Johnson, F.J. Mayer, LV. Powers and D.C. Slater, Phys. Rev. Let!. 39 (1977) 274.[91]1.5. Pearlman, J.J. Thomson and CE. Max, Phys. Rev. Lett. 38 (1977) 1397.[92]L.M. Wickens and J.E. Allen, Phys. Fluids 24 (1981) 1894. —

[93] G. Jordan-Engeln and F. Reutter, Numerische Mathematik für Ingenieure, BI-Htb 104, Mannheim, Wien, Zurich, FRG, 1973.

plasma expansion into vacuum : hydrodynamic approach.pdf

Documents

pure electron cloud

simple oscillator equation

ion acoustic waves

velocity maximum overtakes

scalar wave equation

partial differential equations

ion momentum equation

group theoretical formalism