Amplitude equations for coupled electrostatic wave

arXiv:patt-sol/9708004v121Aug1997AmplitudeequationsforcoupledelectrostaticwavesinthelimitofweakinstabilityJohnDavidCrawfordDepartmentofPhysicsandAstronomyUniversityofPittsburghPittsburgh,PA15260EdgarKnoblochDepartmentofPhysicsUniversityofCaliforniaBerkeley,CA94720ABSTRACTWeconsiderthesimplestinstabilitiesinvolvingmultipleunstableelectrostaticplasmawavescorrespondingtofour-dimensionalsystemsofmodeamplitudeequations.Ineachcasethecoupledamplitudeequationsarederiveduptothirdorderterms.Thenonlinearcoefficientsaresingularinthelimitinwhichthelineargrowthratesvanishtogether.Thesesingularitiesareanalyzedusingtech-niquesdevelopedinpreviousstudiesofasingleunstablewave.Inadditiontothesingularitiesfamiliarfromtheonemodeproblem,therearenewsingularitiesincoefficientscouplingthemodes.Thenewsingularitiesaremostseverewhenthetwowaveshavethesamelinearphasevelocityandsatisfythespatialresonanceconditionk2=2k1.Asaresulttheshortwavemodesaturatesatadramaticallysmalleramplitudethanthatpredictedfortheweakgrowthrateregimeonthebasisofsinglemodetheory.Incontrastthelongwavemoderetainsthesinglemodescaling.Iftheseresonanceconditionsarenotsatisfiedbothmodesretaintheirsinglemodescalingandsaturateatcomparableamplitudes.August21,1997Contents1Introduction11.1Notation......................................41.2Summaryoflineartheory............................52Amplitudeequations:generalfeatures62.1Instabilitywithoutreflectionsymmetry:F0(v)6=F0(−v)...........92.2Instabilitywithreflectionsymmetry:F0(v)=F0(−v).............102.2.1Realeigenvalues..............................102.2.2Complexeigenvalues...........................112.3Amplitudeexpansions..............................122.4Previousresultsforthesinglemodeinstabilities................153Singularitiesinthemode-modecouplings163.1Theuniversalcouplingsp1(0)andr2(0).....................173.1.1F0(v)6=F0(−v),complexeigenvalues..................183.1.2F0(v)=F0(−v),realeigenvalues.....................213.1.3F0(v)=F0(−v),complexeigenvalues..................223.2ThecouplingsQ1(0),P2(0),Q3(0)andQ1(0)+P3(0).............233.3Thespatialresonances:q(0)ands(0)......................243.4Speciallimits:couplingsingularitieswithfixedions..............254Nonlinearscalings255Discussion306Acknowledgements321IntroductionRecentlywepresentedadetailedanalysisoftheamplitudeequationforasingleunstableelectrostaticmodeinanunmagnetizedVlasovplasma(henceforth(I)).[1]Theanalysisrevealsafundamentaldifficultywiththederivationofamplitudeequationsforthisclassofproblems:thecoefficientsintheamplitudeequationsbecomesingularinthelimitinwhichthegrowthrateγoftheunstablewaveisallowedtovanish.Althoughthesesingularitiescanberemovedbyanappropriateγ-dependentrescaling(seebelow)theanalysisshowsthatamplitudeequationsofthistypecannotbetruncatedatanyfiniteorder.[2,3]Nonethelessthescaling1identifiedbythetheorypredictstheamplitudeatwhichweaklygrowingwaveswillsaturate,andhenceisoffundamentalimportancebothinplasmaphysicsandinthecloselyrelatedproblemofshearflowinstabilityofidealfluids.Inviewoftheimportanceofthepredictedscalingforapplicationsweinvestigateheretheeffectsofincludingadditionalunstablemodes.Weconsideronlythesimplestpossibilities,thoserequiringafour-dimensionalsystemofamplitudeequations.Therearethreesuchinstabilitiesdistinguishedbythesymmetryoftheequilibriumandwhethertheunstablemodeshaverealorcomplexeigenvalues.Wefindthatunlessthetwomodessatisfyastrongresonanceconditionthepresenceofthesecondmodedoesnotalterthesaturationamplitudeoftheoriginalmode.However,intheimportantresonantcaseinwhichthephasevelocitiesofthetwomodesarethesameandtheirwavenumbersk1,k2satisfyk2=2k1thelongwavemodesuppressesdramaticallythesaturationamplitudeoftheshortwavemode.Theamplitudeequationdefines,inthelimitγ→0+,akindofsingularperturbationproblemwhosedetailedfeaturesrevealasymptoticscalingbehaviorofthenonlinearwave.Thisisakeyideabehindourapproachandthereaderisreferredto(I)foramoredetaileddiscussion.ForthesinglemodeinstabilitiesthereisonlyoneamplitudeA(t)andoneseeksascalingA(t)=γβa(γt)suchthattheevolutionequationfora(τ),τ≡γt,hasanon-singularlimitasγ→0+.Indissipativeproblems,thecriticaleigenvaluesareisolatedontheimaginaryaxis,andβ=1/2isthegenericallyexpectedexponent.Asaresult,inthegenericcase,theamplitudeequationcanbetruncatedatthirdorder.Thisisnotsoforanunstableelectrostaticwave.HerethesituationisquitedifferentbecausetheVlasovequationisHamiltonianandtheeigenvaluesofthemodemergewithacontinuousspectrumontheimaginaryaxisatcriticality,i.e.asγ→0+.Asaconsequencethenonlinearcoefficientsintheamplitudeequationaresingularasγ→0+signallingstrongnonlineareffectsthatsaturatetheunstablelineargrowthatexceptionallysmallwaveamplitudes.Aquantitative2signatureofthisreductionofthenonlinearwaveamplitudeisalargerexponent:β=5/2forplasmaswithmultiplemobilespeciesandβ=2inthelimitingcaseofinfinitelymassive(fixed)ionsandmobileelectrons.Infact,theanalysispresentedin(I)showedthatsettingβ=5/2yieldedatheorythatwasfinitetoallordersintheamplitudeexpansionasγ→0+.Inthispaperweinvestigatethecoupledamplitudeequationsfortwounstablemodesw