Advanced Plasma

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Advanced Plasma Physics Notes 2008 Spike April 6, 2008 Contents 1 Lecture 1: Kinetic Theory 3 1.1 Liouville’s Theorem ....................................... 3 1.2 μ-space .............................................. 3 1.3 μ-space vs. Γ-space ........................................ 3 1.4 The BBGKY Hierarchy ..................................... 4 2 Lecture 2: Derivation of BBGKY Hierarchy 4 2.1 BBGKY Hierarchy ........................................ 5 3 Lecture 3 6 3.1 Dilute Systems .......................................... 6 3.2 Boltzmann Collision Integral from the BBGKY Viewpoint .................. 7 3.3 Time’s Arrow ........................................... 8 4 Lecture 4 8 4.1 Boltzmann Collision Operator .................................. 8 4.1.1 Boltzmann Collision Operator .............................. 9 4.2 Plasma ............................................... 9 5 Lecture 5 10 5.1 Electron Scattering by Ions ................................... 10 5.2 Fokker-Planck Equation ..................................... 11 6 Lecture 6 12 6.1 More on the collision operator: BGK (Bahatnagar-Gross-Krook) Equation ......... 12 6.2 Vlasov Equation ......................................... 12 6.3 Debye Length ........................................... 13 7 Lecture 7 15 7.1 Weakly and Strongly Coupled Plasmas ............................. 15 7.2 Dielectric Properties of a Plasma ................................ 15 7.3 Charge and Current Density ................................... 16 8 Lecture 8 17 8.1 Landau Prescription ....................................... 17 8.2 Charge Density .......................................... 17 8.3 Plasma Dispersion function ................................... 18 9 Lecture 9 20 9.1 Plasma Dielectric Constant ................................... 20 9.2 Plasma Dispersion Relation ................................... 20 9.3 Dispersion Relation for Longitudinal Waves .......................... 21 9.4 Dispersion Relation for Transverse waves ............................ 21 9.5 Transverse Waves ......................................... 22 1

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AdvancedPlasmaPhysicsNotes2008SpikeApril6,2008Contents1 Lecture1: KineticTheory 31.1 LiouvillesTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 -space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 -spacevs. -space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 TheBBGKYHierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Lecture2: DerivationofBBGKYHierarchy 42.1 BBGKYHierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Lecture3 63.1 DiluteSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 BoltzmannCollisionIntegralfromtheBBGKYViewpoint . . . . . . . . . . . . . . . . . . 73.3 TimesArrow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Lecture4 84.1 BoltzmannCollisionOperator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.1.1 BoltzmannCollisionOperator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Lecture5 105.1 ElectronScatteringbyIons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Lecture6 126.1 Moreonthecollisionoperator: BGK(Bahatnagar-Gross-Krook) Equation. . . . . . . . . 126.2 VlasovEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.3 DebyeLength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Lecture7 157.1 WeaklyandStronglyCoupledPlasmas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.2 DielectricPropertiesofaPlasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3 ChargeandCurrentDensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Lecture8 178.1 LandauPrescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 ChargeDensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.3 PlasmaDispersionfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Lecture9 209.1 PlasmaDielectricConstant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.2 PlasmaDispersionRelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.3 DispersionRelationforLongitudinalWaves . . . . . . . . . . . . . . . . . . . . . . . . . . 219.4 DispersionRelationforTransverse waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.5 TransverseWaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2219.6 LongitudinalWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.7 PhysicalOriginofLandauDamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310Lecture10: MHD 2310.1 Hydrodynamiclimit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.2 ContinuityEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.3 MomentumEquation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.4 Multi-ElementFluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.5 BulkFluidEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611Lecture11 2711.1 OhmsLaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.2 EnergyEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.3 TransportCoecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.4 Hydro-MagneticApproximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912Lecture12: Ideal MHDApproximation 3012.1 PressureBalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.2 IncompressibleApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.3 BernoullisEquation-SteadyFlow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.4 KelvinsTheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.5 AlfvenWaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3213Lecture13: MagnetosonicWaves 3314Lecture14: Magneto-HydrodynamicWaves 3514.1 M.H.D.Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3715Lecture15: M.H.D.Shocks 3815.0.1 AlfvenWave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3915.1 BoltzmannsEquationasaFokker-Planck (LandausEquation) . . . . . . . . . . . . . . . 3916Lecture16 4116.1 Rosenbluth,MacdonaldandJuddForm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4216.1.1 RelationshipwithLandausFormula . . . . . . . . . . . . . . . . . . . . . . . . . . 4517Lecture17: CalculationoftheTransportCoecients 4621 Lecture1: KineticTheoryThemotionof individual particlescanbeconsideredasanensemble. WetakeasourstartingpointHamiltonsequationforthedynamicsof particles. Considerasystemof Nparticles, whereNisverylarge. Each particle has qi, pi( r, v). Wecan now consider a space, ,whichconsists of theco-ordinateaxis of qi, piof all of the particles. (A 6Ndimensional space). The particle state (as determined by qi, pi)appears as a point,and the particle trajectory appears as a line in the space. The probability functiongivestheprobabilitythataparticlewill befoundintheboxformedbydq1dq2. . . dqNatq1p1. . . qNpNisgivenbyProb = F(q1p1q2p2. . . qNpN)dq1dp1. . . dqNdpN.Iftheparticlepositionisinitiallyatsomeknownpoint,wecansaythatF(q1p1q2p2. . . qNpN, t) = Nt=1(qi1Qi1(t))(pi1Pi1(t)) . . . (1)where Qi1(t), Pi1(t) are the integrals of the Hamiltonian equation in time, Qi1(t), Pi1(t) is the classicaltrajectoryofparticlei.Infact, althoughwecannotdescribetheinitial conditionsexactly, wedoknowaprobabilitydistri-butionfortheparticles. Wecanusethistoformanensembleofpathsinphasespace(6N, qi, pi). Wefollowthemotionoftheensembleinphasespace.1.1 LiouvillesTheoremAswefollowthemotionoftheprobabilitycloud,Liouvillestheoremstatesthat:DFDT= 0 =Ft+ Ni=1 qi_Fqi+ piFpi_(2)ThisfollowsdirectlyfromHamiltonsequationforaconservativesystem: pi= Hqi; qi=Hpi(3)This onlyworks if thesystemis conservative. If weaddamagneticeld, sothat thesystemisnon-conservative, Hamiltonsequationsarestillapplicable,andLiouvillestheoremisvalid.N.B. Sofor, themotionofthesystemiscompletelythermodynamicallyreversible. However, arealplasmaisnotreversible. Howisthis?1.2 -spaceThisisasix-dimensionalphasespace: (r, v, t).Theprobabilitydistributionfunction, f (r, v, t)givestheprobabilityof ndinganyparticleinthevolumeelement(dr, dv)at(r, v)asf (r, v, t) dr dv.1.3 -spacevs. -spaceConsider theprobability in-space of ndingparticle1in r1, v1, dr1, dv1tobeAdr1, dv1andtheotherparticlesanywhereinphasespace.A =

2

NF(v1, r1, v2, r2, . . . , vN, rN, t)dr2, dv2, . . . , drN, dvN(4)N.B.Thereisnodr1, dv1inthisequation.But: theparticlesareindistinguishable;allparticleshavethesameprobabilityofbeinginthesamespaceelement.f(1)(r1, v1, t) = f (r1, v1, t) = NA = N

2

NF(v1, r1, v2, r2, . . . , vN, rN, t)dr2, dv2, . . . , drN, dvN(5)But: the integration from r2 rNremoves all of the correlation information, for example f (r1, v1, t)contains noenhancedprobabilityfor ndingparticle1intheneighbourhoodof 2. Inthis case, theprobabilitydistributionsoftheparticlesaretotallyindependentofeachother.3Considertheprobabilityof ndingaparticleinr1, dr1, v1, dv1andasecondinr2, dr2, v2, dv2si-multaneously. Thisisthetwo-particledistributionfunction: f(2)(r1, v1, r2, v2, t)dr1, dv1, dr2, dv2. Thisdistributionincludes correlation, i.etheprobabilityof ndingoneparticleis inuencedbythat of asecond,thedistributionsoftheparticlesarenot independent,onedependsontheother.Correlation isveryimportant. Therearetwotypesofcorrelationinplasmas. Collisionalcorrelationisshort-range, andthereisalsoalong-rangecorrelation. Indenseplasmastheremayalsobeathird,short-range correlation.Intermsof-spacedistribution:f(2)(r1, r2, v1, v2) =N!(N 2)!2!

3

NF(v1, r1, v2, r2, . . . , vN, rN, t)dr3, dv3, . . . , drN, dvN(6)whereN!(N 2)!2!=N(N 1)2(7)Thiscanbegeneralisedtoanynumberofparticles.Thegeneralparticledistributionfunction: r1, v1, dr1, dv1, . . . , rx, vx, drx, dvxgives:f(q)(r1, v1, . . . , rq, vq) =N!(N q)!q!

q+1

NF(r1, v1, . . . , rq, vq, rq+1, vq+1, . . . , rN, vN)drq+1, dvq+1, . . . , drN, dvN(8)gives asetof N-folddistributionfunctionsin-space. Thecompletesetcontainsalltheinformationinthe-spacedistribution. Calculatingthisisverydicult.Iftheparticlesareindependent(i.e. thereisnointeractionbetweentheparticles)thenf(2)(r1, v1, r2, v2) = f(1)(r1, v1)f(1)(r2, v2) (9)If the particle-particle interactions are weak, the motions are nearly independent, and the one-particledistributionparticleisnearlyadequate. Wecanthereforworkwithatruncatedsetofdistributions.1.4 TheBBGKYHierarchyAsetofequationswhichenablethecalculationoff(q)fromf(q1), f(q+1). Eventuallythissetmustbetruncatedwithsomef(q+1)assumedtobesmall. Fordilutegasplasmaswecantruncateatf(2),whichremovescorrelations. However, thefull setof-spacedistributionsarestill reversible. Thetruncationmustalsointroduceirreversibility.2 Lecture2: DerivationofBBGKYHierarchyForcesonaparticle:Externallyappliedforce: Fi. e.g. gravity. (N.B.F=distributionfunction. Notelackofindices)Internal forces: forcesfrommutual interactions: Fij, ingeneralafunctionoftheseparationoftheparticles: Fij([rirj[).Ft+ vi

Fri+ vi

Fvi= Ni=1Nj=1,j=iFijm

Fvi(10)Remember: Fisasymmetricfunctionw.r.t. theinterchangeofparticles.Multiply(10)byN!(Nn)!Integrateoverdrn+1dvn+1. . . drNdvNtoyieldf(n)(r1, v1, . . . , rn, vn)N!(N n)!

Ftdrn+1dvn+1. . . drNdvN=f(n)t(r1, v1, . . . , rN, vN) (11)4Integralsfori > nwhichcontainderivativeswithrespecttooneofthevariablesofintegration,maybeintegratedbypartstogiveasurfaceintegralattheboundaryandadierentvolumeintegral,whosevaluesarezero,forexample

vi

Fridri=_

dSi viF

driF virr_ = 0since limrF= 0,andviandriareindependentvariables.N!(N n)!

vi

Fridrn+1dvn+1. . . drNdvN=___vi

f(n)ri(r1, v1, . . . , rN, vN) ifi n0 ifi > n___Notingthat theonlyexternal forcewhichdepends onthevelocityis theLorentzforceduetoanexternalmagneticeldv B,thereisnocomponentparalleltothevelocityandthereforeFi/vi= 0N!(N n)!1m

Fi

Fvidrn+1dvn+1. . . drNdvN=___1mFi

f(n)vi(r1, v1, . . . , rN, vN) ifi n0 ifi > n___ThemutualinteractionbetweenparticlesistreatedbybreakingthesumintothreepartsNi=1Nj=1,j=i= ni=1nj=1,j=i + ni=1Nj=n+1,j=i + Ni=n+1Nj=1,j=i(12)N!(N n)!

Fij(rirj)m

Fvidrn+1, dvn+1, . . . , drN, dvN=Fij(rirj)m

f(n)vi(13)wherei n and j n.N!(N n)!

Fijm

Fvidrn+1dvn+1, . . . , drN, dvN=1(N n)

Fij(rirj)m

f(n+1)vidrj dvj(14)wherei n and j> n.Totalcontributiontothesum: multiplyby(N-n)totakeintoaccountidenticalparticles.N!(N n)!

Fij(rirj)m=Fvidrn+1 dvn+1. . . , drN dvN(15)Now,substitutetheseintotheLiouvilleequation:f(n)t+ni=1vi

f(n)ri+ni=1Fim

f(n)vi+ni=1j=i,j=iFijm f(n)vi= ni=1Nj=n+1

rjvjFij(rirj)m

f(n+1)vi(16)2.1 BBGKYHierarchyToobtainf(n)youneedf(n+1). Thecompletesetf1). . . f(N)areequivalenttotheLiouvilleequation.Weneedtointroduceclosure: Limitthenumberof n, usuallyton=1(orn=2fordilutesystems).Usingn = 1weget:f(1)t+v f(1)r1+Fm f(1)v= 1m

F12(r1r2) r1_f(2)(r1, v1, r2, v2)_dr2dv2. .(17)wherethethirdtermisthecollisionterm,givingtheinteractionforce. IfitisassumedthatthereisnoexternalforcetheBoltzmannequationisrecovered.5f(2)t+ v1

f(2)r1+ v2

f(2)r2+1m_F12(r1r2) f(1)v1+F21(r2r1) f(2)v2_ =

_F13(r1r3) f(n+1)r1+F23(r2r3) f(n+1)r2_dv3dr3(18)Findanapproximationbasedonphysical considerationsforthecollisionterm. Whatarethecon-straintsonthephysicstogetasensiblecollisionterm?3 Lecture3ConsiderthestreamingderivativefortheoneparticledistributionDf(1)Dtf(1)t+v1

f(1)r1+F1m f(1)v1. .(19)Inthe uiddynamic analogy, this is the Lagrangiantime derivativefor the uid owinginphasespace.Driving term for the velocity, namely the acceleration, is determined by the external forces alone.In addition the time variation of the one particle distribution depends on an additional term determinedbytheinteractionforces,whichitselfdependsofthetwoparticlederivative.n = 1canalsobewrittenas:Df(1)Dt=f(1)tcoll(20)3.1 DiluteSystemsIfthesystemisdilutetherearethreewellseparatedtimescales:1. Collisiontime: 0 avwherea =rangeofintermolecularforce.2. Collisionfrequency(timebetweencollisions): t0 vwhere =themeanfreepath.3. Macro. time: 0 LvswhereL =Labscale,andVs=speedofsound.Since Therangeoftheinterparticleforceaisverysmall Themeanfreepathisrelativelylarge aislarge Thescaleofthelaboratory apparatusisextremelylargeL Consequently0 t0 0We prepare thesystem insome state at t = 0, not in equilibrium. Rapid changes occur over times0dueto theinterparticle force. Themultipleparticle distributionfunctionsf(s)s 2 change rapidly overthesetimescales,andestablishanequilibriumamongst themselves. Ontheotherhandtheoneparticledistributionf(1)onchangesmoreslowlyduetothestreamingderivativeandchangesonlyoverastimet0asinteractionstakeplace.Weareusuallynotinterestedinchangesover0,asitistoosmallandgivesalevelofdetail,whichis not required. Howeverchangesovert0determiningtheoneparticledistributiondirectlyrelatetoexperimentallymeasurablequantitiessuchastemperature/densityetc. Wecanthereforeaveragethedetaileductuationsovertimes 0withoutlosinganyimportantinformation.Whenthis averagingis carriedout, the behaviour of the multiple particle distributionfunctionsrapidlyachievetheequilibriumformsassociatedwiththeinstantaneousoneparticledistribution. Since6thisequilibrationtimeisveryshort,weimaginethatmultipleparticledistributions,onaveraging, haveatimedepending,whosetimedependenceisdeterminedbythatoftheoneparticledistributionf(s)(r1, v1, . . . , rs, vs, t) f(s)_r1, v1, . . . , rs, vs)f(1)(t)_(21)wheres 2.Asaresulttheequationforn = 1becomesDf(1)Dt= A_r, v,f(1)_ =f(1)tcoll(22)We still have reversibility problems. There is no arrowof time. A loss of reversibility can be introducedbyintroducingadirectionoftimebywhichbeforeandafter areclearlydistinguished. Thisassumesmolecularchaos:1. Beforeacollision(2-body)the2bodiesarestatisticallyindependent.2. Afterthecollisiontheyarecorrelated.3. Intimethecorrelationislostbetweencollisions. .Asaresult,theparticlesinvolvedinthecollisionsarealwaysuncorrelatedbeforethecollision.3.2 BoltzmannCollisionIntegralfromtheBBGKYViewpointFordilutegasescollisionsarebinaryandcompletedoververyshort timescales.Assumptions:i 3-body(andhigher)termsarenegligible.ii Timevariationduringcollisionisnegligible.iii Spatialvariationoverthecollisionrange(a)isnegligible.ivChangeinvelocityduetoexternalforcesduringcollisionsisnegligible.Taken = 2:

(ii)f(2)t+v f(2)r+v

f(2)r+Fm f(2)v+Fm f(2)v+$$$$$$$$X(iv)F(r rm

f(2)v+$$$$$$$$$X(iv)F(r r)m

f(2)v1= 1m

__$$$$$$$$$X(i)F(r r) f(3)r+$$$$$$$$$X(i)F(rr) f(3)r__drdr(23)Since there is no spatial inhomogeneity (iii), a small displacement leaves the two particle distributionunchanged.f(2)(r, v, r, v, t) = f(2)((r +), v, (r +), v, t) (24)Wecannowdierentiatewithrespectto:f(2)r+f(2)r= 0 (25)So(23)becomes:(v v)f(2)r+F(r r)mf(2)v+F(rr)mf(2)v= 0 (26)Forn = 1:7f(1)t+v f(1)r+Fmf(1)v= .. _F(rr)mf(2)v_drdv. .(27)wherethelefthandsideisJ, andtheintegrandoftherighthandsidecanbesubstitutedfrom(26)togive:J= _(v v) f(2)rF(rr)m

f(2)v_drdv(28)integratethesecondtermoverv,usingGaussstheorem.

F(rr)m

f(2)vdv= 0 (29)Integratethersttermoverr,canuseGausstogive:J=

dv

dS (v v)f(2)

dr$$$$$$$$X0f(2)r(v v) (30)Itispossibletocancelthesecondtermonthelefthandsideasv&varenotfunctionsofr. Thesurfaceintegralisover thecollisionvolume.Consider the collision volume as a sphere of radius a, centred on the centre of mass. Vis therelativevelocity, where V= vv,which is unchanged in magnitude in an elastic collision. We consider (dierent)pairsofparticleswiththesamevelocitiesafterthecollisionasthosebefore.r istheradiusvectoroftheoutgoingparticleatthesurfaceontop(exit),ristheradiusvectorof theincomingparticleatthesurfaceonthebottom(entry). TheelementdSofthesurfacecorrespondingtotheentryandexitparticles is projectedintodontheequatorial plane. Thus dis theelement of thecollisioncrosssection.3.3 TimesArrowAssumption(v)Molecularchaos. Incomingparticlesarestatisticallyindependent.f(2)(r, v, r, v) = f(1)(r, v) f(1)(r, v) (31)Thisappliestotheentry,butnottheexit,particles.For an elastic collision: Conservation of total momentum and energy, therefore centre-of-mass velocityisconstantandrelativevelocityisrotatedthroughascatteringangle. canbespeciedfromthecross-section as= (v, )d. Hencethevelocitiesafterthecollisionvandvcanbecalculated.FromLiouvillestheoremthetwoparticledistributionfunctionafterthecollisionmustbeequal toitsvaluebeforethecollisionforthesametwoparticles. Thusif r, vandr, varetheinitial particlepositionsandvelocitieswhichgiverisetovelocitiesr, vandr, vrespectivelyaftercollision:f(2)(r, v, r, v) = f(1)(r, v) f(1)(r, v) (32)4 Lecture44.1 BoltzmannCollisionOperatorFromtheclosingoftheBBGKYHierarchy:J=

dvV = f(2)(r, v,r, v, t) f(2)(r, v,r, v, t)dw (33)wherev =relativevelocity,dw =dierentialcross-sectionforcollisionthroughanangle .Wemake theadditional assumption of molecular chaos, where two particles have such along historybeforethecollisionthattheyareessentiallyuncorrelatedbeforethecollision.8BeforeCollision:f(2)(r, v,r, v) = f(1)(r, v) f(1)(r, v) (34)LiouvillesTheorem: Distributionfunctionf(2)isunchangedthroughthecollision.AftercollisionassumingMolecularChaos:f(2)(r, v,r, v, t) = f(1)(r, v, t)f(1)(r, v, t) (35)r, vandr, vaftercollisionresultfromr, v, r, vbeforecollision.Sincethecollisionvolumeisverysmall,r = r= r = r= r= rJ=

dvdw_f(1)(r, v, t) f(1)(r, v, t) f(1)(r, v, t) f(1)(r, v, t)_(36)wherevandvleadtovandvafterthecollision.J=

dvd(V, )_f(1)(r, v) f(1)(r, v) f(1)(r, v) f(1)(r, v)_(37)where dw = (v, )disthedierentialcross section forscattering throughsolid angle dfor arelativevelocityV andangleofscatter.4.1.1 BoltzmannCollisionOperatorBoltzmannCollisionOperator(fora1particledistributionfunction):ft+vfr+Fmfv=ftcoll(38)where f f(1)andftcoll= J. [Note that henceforward we shall omit distribution function superscriptsunlessnecessary].Therateof changeof f duetocollisions=therateof collisionsbringingparticlesintothephasespacevolumedrdv-rateofcollisionsexitingdrdv.Thenumberofcollisionsexitingdrdvperunittime=f (r, v)drdv

f (r, v)V dvdrNumberofcollisionsenteringthevolumeelementr v rv r v r v

f (r, v)f (r, v) Vdr dv drdvThevolumeelementisgivenbytheJacobian, forwhichdrdvdrdv=drdvdrdvfollowsfromthedynamics.Henceweobtainthesameresultasbeforeftcoll=

_f (r, v)f (r, v) f (r, v)f (r, v)V dvd (39)All occur at the same place in space (due to the collision volume being small) so ris thesame for all.4.2 PlasmaLongrangeforcesbetweenparticles(CoulombForce):CoulombForce: (r, r) 1|rr|violatestheshort-range assumption.ImposeclosureontheBBGKY: OnlyatshortrangeistheCoulombforcestrong. Assumeadiluteplasma-canseparate astrong eldregion andconsidertwo bodycollisions onlyinthisregion. Outsidethisregionweconsidertheeldasweak, wecanthereforetreattheeldasaperturbation, followingmanybodycollisions.Strongcollisionregion: collisionsareessentiallybinary. Impactparameterlessthanoforderof90oscatteringbyaCoulombeld,b bmin,Landaulength.Weakcollisionregion: Multi-particle, cooperativeregionImpactparameterbmin b D(DebyeLength).(e.g. plasmawaves).9Collectiveregion: Impactparameterb D. Interactioncharacterisedbycollectivebehaviourre-sultinginplasmawavesduetonon-localself-generatedelds.For small collisions: The particle wanders through the plasma with numerous small deections - manybodycollisions: (interactionthroughthecoulombeld.ElectronIonscattering: Momentumchangeoftheparticleisthesumofmanyterms. p=p1 +p2 ++ pN. Interactingparticlesarerandomlydistributed(molecularchaos)givingpx 0butp2x ,= 0. The statistical distribution of the particle deections. These are randomly distributed in angle,Gaussianwithvariance proportionaltolengthof pathandproportionaltotime. TheR.M.S.Deectionisproportionaltothesquarerootoftime.Inthe remainder of this course we shall consider dilute plasmainwhichthe particle densityissucientlysmallthatcorrelations arealmost unimportant. However wecannot neglectthemaltogetherastheyformthebasisforbothcollisionsandcollectivebehaviour. Fordilateplasmawemayconsideronlytheoneparticledistributionfunction(f)(1)greatlysimplifyingcalculations.5 Lecture55.1 ElectronScatteringbyIonsWeconsiderthescatteringofelectronsbyrandomlydistributedstationaryions.Theimpulseofeachionontheelectron: piTotalimpulseduetomanyionsi= ipiIf the electron is reasonably well separated from the ions (i.e. the ions are far apart) the impulse fromeachionissmall, andcanbetreatedasaperturbationonthemotionof theelectron. Themotionistherefore notmuchdisturbedbyeach impulse. Sinceeachperturbationissmallitactsindependentlyoftheothers despitebeingsimultaneous. (Thisconditionisonlytruefordistant collisions). Sincetheionsare randomlydistributed,thenetimpulseontheelectronwillbezero,aslong asthepathissucientlylong. Howeveri[pi[2,=0. (Inaparticulardirectionj, i[pij[2yieldsthemeansquareimpulseinthedirectionj.)p2j=(ipij)2N(40)whereNisthenumberofionsencounteredHencethemeansquaredimpulseintimet:p2x=i(pix)2+$$$$$$$$$X02i=j(pixpjx)N(41)= [px(b)]2Nvbdbddt (42)wherethetaistheazimuthalangleandbtheimpactparameter.Themeansquareddeectionangleisgivenby:2x=p2x(mv)2(43)where(mv)2isthemomentumoftheelectron.Sincetheelectron-ioninteractionisweak,theelectronpathisapproximatelyastraightlinez= vt.Thetransverse impulseatimpactparameterbis:pxi=qQ40

bdz(b2+z2)32=2qQ40bv(44)pi=2qQ40bmv2(45)whereQistheionchargeandqtheelectroncharge.102x=

bdb

d_ 2qQcos 40bmv2_2Nvdt (46)=4q2Q2N(40)2m2v3

dbbdt (47)where

dbbisalogarithmicsingularity,givingproblemsforbbothlargeandsmall.Wenowlookatthecasewherebislarge. Herewestarttogetsomescreeningof ionpotential byother electrons, up tob > D(see later). (If b > Dthere is no impulse,this gives anupper limit on theintegralofD.)Turning our attention to the case where b is small, the deection will be very large, and the assumptionof smallangle deectionisviolated. Wecantreat theproblem asaseries of two particlecollisions usingthe Rutherford scattering formula (equivalent to a Boltzmann collision term). It is found that this resultis equivalent to a cut-o at bmin(the Landau parameter/distance). This is the distance that correspondsto90oscattering.bmin=qQ40mv2(48)2x=4q2Q2N(40)2m2v3

Dbmindbbdt (49)=4q2Q2N(40)2m2v3ln_Dbmin_(50)Letln = ln_Dbmin_knownastheSpitzerparameter/Coulomb logarithm. ( 5 10). Thephysicalsignicance of is that it is the ratio of the number of small-angle large-impact collisions to the numberoflarge-anglesmall-impactcollisions. Post-hocjusticationoftheuseofperturbationisprovidedbybeinglarge.p2x, 2xleaddirectlytothe Fokker-Planckequation, amethodfor calculatingthe distributionfunction.5.2 Fokker-PlanckEquationMultiplesmall interactionsthatchangethedistribution. Weareinterestedintheprobabilityfunctionthat anelectronwithvelocityvundergoesacollisionthat changesits velocityby(v, dv) namely(v)dvinatimedt.f (r, v, t) =

_f (r, (v v), (t dt)(v, v)dv (51)Ifvissmallwecanexpect,viatheTaylorseries:f (r, v, t) =

d(v)f (r, v, (t dt))(v, v)v_fv + fv_+12vivj_2fvivj+ 2fvivj+ f2vivj_

(v, v)d(v) = 1 (52)ftcoll= vi_f vi+122vivj_f vivj(53)wheretherst terminsquarebracketsis thecoecient of dynamical frictionandthesecondis thecoecientofdiusion._vivivj_ =1t

(v, v)_vivivj_d(v)11Referring to the mean change in thevelocity on the top line,and the mean change in theproduct of twovelocitiesonthebottom.Forelectronscatteringfromstationaryions,wemayusetheexpressionsobtainedearlier:vx= y 0 (54)v2x= v2y, 0 (55)vxvy 0 (56)wherexandyareperpendiculartotheelectronmotion. Writing forthetotalperpendicularvelocitycomponentand |fortheparallel wenotethat becausethecollisionsareelastic:12m(v2

+ 2v2) =isunchanged. Hencev2=8q2Q2N ln (40)2m2vv

= 121vv2=(4q2Q2N ln 40)2m2v2v2

0(57)Ifweaddthesebackinto(53)wegettheFokker-Planck equation.Wewilldiscusselectron-electroncollisionslater.Iftheplasmaisnon-isotropicvxmaynotbezero,andvxvymayalsonotbezero.Inequilibrium(i.e. thermodynamicbalance)(53):ftcoll= 0 (58)whichxesthenumericalconstants.6 Lecture66.1 Moreonthecollisionoperator: BGK(Bahatnagar-Gross-Krook)Equa-tionDeningtherelaxationtimethatthedistributiontakestogointoaMaxwellian:ftcoll= 1 (f f0) (59)Wheref0isMaxwellian,andistherelaxationtime.6.2 VlasovEquationThus far we have considered collisional eects, where the electron is scattered by short range eects. (i.e.byrandomlydistributedscatteringelds,l < D).Now, welookatco-operativeeects: scatteringbyself-generatedelds. If theinteractionisoverlongerdistances(l >D)thesystemactsmorelikeauid, andwelosegraininess. ThisgivesrisetotheVlasovequationdescribingthedistributionfunction(f)(1).Imaginethattheplasmaissubdivided.emisconstant, whereeistheelectronicchargeandmtheelectronmass. Therearenochangesonthemacroscopicscale, andweholdthetotal chargeandmassconstant. Now let thenumber of particles, N,tendtoinnity,but withtotal charge andmass constant.e, m 0. Considerthestaticelds,whichwillproduceanacceleration:a(0)i= em_E(0)+viB(0)_(60)whereemisaconstant.Nowconsidertheinteractionsbetweentwoparticles,iandj:12a(j)i= e240m(rirj)[rirj[3 e2m 0 (61)Inthislimittwobodyeectsaresmall, soweassumethattheparticlesareweaklycorrelated, i.e.independentofoneanother.f(2)(r1, v1, r2, v2) = f(1)(r1, v1)f(1)(r2, v2) (62)Putthisintothe1sttermoftheBBGKY:f(1)t+v1

f(1)r1+_a(0)+

a(2)1 f(1)(r2, v2, t)dr2dv2_

f(1)v1= 0 (63)wherea(0)isthestaticeldterm,andtheintegralisovertheself-consistenteldterm.Substitutingfora(2)1weintroducetheself-consistentelectriceld:Eself(r,t)= e40

r r[r r[3. .f(1)(r, v, t)drdv(64)wherethetermsoverthebraceareequaltoa(2).Similarly we may include a self-generated magnetic eld arising from the current in the plasma Bself,whichwilladdanadditionalacceleration v Bselftoa(2).ft+v fr+a fv= 0 (65)wherewehaveomittedthe(unnecessary) superscriptonthedistributionfunction,anda = em_E0+ Eself+v (B0+Bself)_(66)and Eselfand Bselfare found by integrating over the local charge density and current density throughouttheplasma.ThisistheVlasovEquation.The Vlasovequationis collisionfree, uid-like longrange interactions. The interactiontermiscollective,butnon-collisional.EmpiricallythecollisionscanbeaddedbyaddinganextratermtotheVlasovequation:ft+v fr+a fv=ftcoll(67)ThecollisionfreeVlasovequationgiveswavesolutions. Thecollisiontermaddsdamping.6.3 DebyeLengthLengthoverwhichindividual particlegraininessexists. ConsideraplasmaoftemperatureTifortheionsandTefortheelectrons. TheionchargeisZeandtheelectrondensityisn0e. Theplasmawill bequasi-neutral: ni neZ . Theelectronsmovingaroundtheionsexperienceanattractiveelectro-staticeld. We would therefore expect the density of electrons close to the ions to be greater than the ambientdensity:ne= n0e exp_ekTe_; ni= n0iexp_ZekTi_(68)Consider the value around a single ion: charge Ze at r = 0. Assume that the interparticle ion distanceissmallcomparedtothepenetrationdistanceoftheeld,i.e. thatelectronsarenotxedtoparticularions. Overlongtimesandlargedistanceswewill losethegraininessof theindividual particles. Thestatistical distributionof theelectronswill giveaslowlyvaryingpotential, , describedbyPoissonsequation:2 = 10[Znie nee] (69)13Sphericallysymmetricnearanion:1r2ddr_r2ddr_ = 10n0e_exp_ZekTi_exp_ekTe__(70)Boundarycondition1: Neartheion: =Ze40rasr 0Boundarycondition2: Farfromtheioneldmustvanish: 0asr .1r2ddr_r2ddr_ =n0ee20_1kTe+ZkTi_. . =2D(71)Substitute: =1xf (72)x =rD(73)toobtaind2fdx2= f (74)subjecttof (0) = 1 (75)f (x) 0 as x (76)(77)whosesolutionisf (x) exp x (78)andthus =Ze40r2exp_rD_(79)where2D=n0ee20_1kTe+ZkTi_istheDebyelength.Wemaywritethisexpressioninalternativeconvenientforms:2D=0kTenee2_1 +ZkTekTi_(80)DividethisintoElectronandIonparts:2e=0kTenee22i=0kTinie2_12D=12e+12iTheeldfallsorapidlyasrD,screening theCoulombeld,andreducingtherange oftheeldfrom to D.TheelectronscreeningofeldsismoregeneraloverD. Imagineawall attheedgeoftheplasma.Theelectronratetothewall is14nv, wheretheelectronsarelost. Thisproducesanelectron-decientareaadjacenttothewall, whichwill createaeld. Thiseldwill retardelectronsandaccelerateions,andtheareaoverwhichthisviolationofchargeneutralityoccursisknownasthesheath. ItextendsadistanceDfromthewall.TheDebyelengthmeasuresthepenetrationof electriceldsintotheplasma. However, theDebyelengthonlywellformedifthenumberof particlesintheDebyeSphereislarge, oralternatively that theDebyelengthismuchlarger thantheinterparticleseparation.147 Lecture77.1 WeaklyandStronglyCoupledPlasmasThe analysis leading to the Debye shielding concept is only applicable to a weakly coupled plasma wherethethermalenergiesarelargecomparedtoelectro-staticenergiesTheconditionsforDebyeLengthanalysis:1. DilutePlasma2. OveraDebyeLengththeparticlesshouldformastatisticallyaveragedcontinuum. (i.e. n13i 1thentheplasmaisstronglycorrelated. _Z2e2_0n1/3i_kTeElectrostaticpotentialenergybetweenparticlesThermalEnergyperparticle(83)Therefore the plasma is strongly correlated whenthe electrostatic energy isgreater than thethermalenergy. Stronglycoupledplasmasareunusual,andveryhardtomakeplasmaswhere 1 100ish.In this case the distributing can no longer be written in terms of a single particle distribution function asthehigherordertermsarenotsmall. Calculationsofthesesystemsareverydicult, andMonte-Carlomethodsmustoftenbeused. (Notethatanioniccrystalisessentiallyastronglycoupledplasma.)7.2 DielectricPropertiesofaPlasma Unmagnetised,Warm,Dilute LongWavelengths(>> D) Coherent/Cooperative motion VlasovEquationConsideranimposedE.M.waveeldwithwavevectorkandfrequency.E0 exp [i(t kr)] (84)Theresponseoftheplasmatothiseldisgivenby:ft+v fr+qmE fv= 0_ftcoll_(85)Assume that the applied eld is weak: f = f0+f1(perturbation) where f0is the ambient distribution(Maxwellian) and f1is the perturbation. This gives equations for the electrons and ions. As the ion massislarge,theeectonionsismuchweakerthanthatonelectrons. Wethereforeneglectionmotion:f1t+v f1r emE0 exp i(t kr) f0v= 0 (86)15Writingf1(v, r, t)asaFouriertransformf1(v, k, )f1(v, r, t) =f1(v, k, ) expi(t kr) (87)i( rv)f1=emE0

f0v(88)f1=emE0

f0vi( kv)(89)Denominatorsingularwhenv

(vcomponent |tok)satises:kv

= (90)Wecantreatthisinoneofthreewaystomaintainphysicalreality:1. Aninitialvalueproblem: Switcheldonatt = 0 LaplaceTransforms.2. Switchon: slowlyincreaseeldfromt = . Introduceatermexp (t)intotheamplitude.3. AddincollisionsviaBGKtermwithacollisiontime(whichcanbeverylarge).All threecaseschangethedenominatorof(89)to[i( kv) + ] where=forcase2, or1forcase 3, but is a very small quantity (0). This explicitly introduces causality, and removes reversibility.7.3 ChargeandCurrentDensityWecalculatetheFouriercomponentsofthechargeandcurrentdensity. Thesearewavetermswiththesamefrequencyandwavenumberastheappliedeld.(r, t) = (k, )j(r, t) =j(k, )_exp [i(t kr)] (91) = e

f1dv overallvelocityspace (92)= e2mlim0

E0

f0v dv[i( kv) +](93)= e2mlim0

E0

f0v (i( kv) +)( kv)2+2dv (94)= e2mlim0

E0

f0v [i( kv)]( kv)2+2dve2mlim0

E0

f0v( kv)2+2dv (95)= e2mP

E0

f0v[i( kv)]dv em2k

E0f0v dvv

=k(96)FortherstintegralweintroducethePrincipalValueoftheintegralas 0P

f (x)xdx =lim0

xf (x)(x2+2) dx =lim0_

f (x)xdx +

f (x)xdx_(97)providedthefunctionf (x)iscontinuousatx=0. Forthesecondintegral wenotethatfor( kv

)notsmall,theintegrandissmall,thereforethedominantcontributionoccurswhenkv

= . Hencelim0

E0

f0v( kv

)2+2dv

=lim0E0

f0vv

=k

dv

( kv

)2+2= E0

f0Vv

=k #1k(98)16usingthestandardintegral

1a2+x2dx =aThisyieldstheLandauPrescription(standard):

1x= P

1x+i(x) (99)where(x)istheDiracdeltafunction.8 Lecture88.1 LandauPrescription

L1x= P

1x+i(x) (100)1. Oscillating part of thewaves: exp [i(t kr)]. Ifthisisexp [i(kr t)]theplusintheLandauPrescription becomesaminus.2. (x)termwith+iisadirectconsequenceofcausality: expressionoftimesarrow.3. Singularityiswhenv

=k,whenparticlevelocitymatchesthephasevelocity-resonance.8.2 ChargeDensity = e

f dv = e2mlim0

E0

f0v dvi( kv)= e2m

LE0

f0V dVi( kv)(101)f0issymmetricinv . Thisisanevenfunctionofv.f0v

,f0v1,f0v2arealloddinv

,v1andv2respectively. (v1, v2) = v.v

f0v

iseven.Integrating:E0

f0V1dv1(i( kv))= 0 = E0

f0V1dv1(i( kv

))(102) = e2mE0

Lf0v

dv

(i( kv

))(103)jisthecharge velocity.j = e2m

v E0

f0v dvi( kv

)(104)j

= e2mE0 v

f0v

dvi( kv

)(105)j1,2= e2mE01,2

v1,2f0v1,2dvi( kv

)(106)The charge density and current densities are not independent,but must satisfy an equation of continuityt+ j = 0 (Currentcontinuityequation) (107)i( k j) = e2mE0

B1i( kv)i( kv)

U0f0v

dv (108)17Wehavetwodecoupledterms: |longitudinalparalleltotheeldpropagation direction,k transverse, perpendicularparalleltotheeldpropagationdirection,k,twocomponents(1,2)Introducethedielectricconstant: (k, ).Fromelectrostatics:D = E= 0E +P (109)whereP=dipolemomentperunitvolume.j =Pt(Rateofchangeofpolarisation),P isanoscillatingterm. (110)= iP0(111)Dt= 0Et+j (112)= Et(113)i = i0 +jE0(114)where2p=nee20m, andf0=f0ne. f0isthenumberofelectronsperunitvolumeofphasespace, neistheelectrondensity,andf0istheprobabilityofndingtheelectronperunitvolume.

0= 1 +2p2 v

f0v

dv(1 kv

)(115)= 1 +2p2___2k2_ v

f0v

dv(k v

)__(116)0= 1 +2p2

vf0vdv(1 kv

)(117)= 1 2p2k

f0dv(k v

)(118)

0= 1 +2p1_k_(119)0= 1 2p2_k_(120)1(x) = x2

Lf0v

dv(x v

)(121)2(x) = x

Lf0(x v

)dv (122)1, 2arecomponentsoftheplasmadispersionfunction:8.3 PlasmaDispersionfunction182 4 6 8z-2-1.5-1-0.500.511.52G G G G G0Figure1: PlotoftherealandimaginarycomponentsoftheplasmadispersionfunctionG(z)anditsderivativeG(z)forrealvaluesoftheargumentz.19G(z) = 12

Le2(z )d (123)I(z) = P_12

e2(z )d_(124)= I(z) (125)= 2ez2

z0et2dt (DawsonsIntegral) (126)G(z) = 2ez2 z0et2dt +it12ez2(127)G(z) = 212

e2i(z )= 2(1 zG(z)) (128)

0= 1 +2p2z2G(z)0= 1 2p2zG(z)___z=k_m2kT_12TistheelectrontemperatureForsmallz:G(z) i12ez2+ 2z(1 2z23+4z415. . . ) (129)Forlargez:G(z) =$$$$$Xverysmalli12ez2+z1(1 +12z2+34z4+. . . ) (130)where = 0, 1, or 2 if z< 0, = 0, or > 09 Lecture99.1 PlasmaDielectricConstantTherearetwo. Oneparalleltothewave:

0= 1 +2p2z2G(z) (131)andoneperpendiculartothewave:0= 1 2p2zG(z) (132)where:z =k_m2kT_12(133)G(z) = 2(1 zG(z)) (134)9.2 PlasmaDispersionRelationThisistherelationshipbetweenfrequencyandwavelength. WrittenasD(, k).StartfromMaxwellsequations:2010B = j +0Et(135)E = Bt(136) B = 0 (137) E =0(138)(E) = 0jt+00Et(139)2E 002Et2= 0jt+10p (140)Addtheplanewaves:Ejtilde___exp [i(t kr]_k22c2_

E= 10_ 1c2j k_(141)9.3 DispersionRelationforLongitudinal WavesTakethescalarproductk E:_k22c2_k E= i0_ 1c2k j k2 _(142)Applyequationofcontinuityofcurrent:t+ j = 0 (143)i( k j) = 0 (144)c k j k2 =1_2c2 k2_k j (145)_k22c2_k E = 1i0_k22c2_k j (146)Nowintroducethedielectricfunction:

E

= 0 E

+1ij

(147)_k22c2_

k E = 0 (148)Ifk2=2c2thisisanelectro-magneticwaveinfreespace. Nottherightsolution.Ifk E= 0 thereisnoappliedeld: Alsoawrongsolution.Finally,if

(k, ) = 0 thisisthedispersionrelationshipforlongitudinalwaves.9.4 DispersionRelationforTransversewavesTakethevectorproductwithk21_k22c2_k E = i0c2k j=2c21i0k j (149) E= 0 E +1ij Usethistoeliminatejtermabove (150)_k22c20_k E = 0 (151)Ifk E= 0thereisnoappliedwave.Thedispersionrelationfortransverse wavesisthereforegivenby:k2=2c20(152)9.5 TransverseWavesk=c_0(153)Ifthephasevelocityk>>ThermalVelocity__2kTm_,then:z=k_2kTm1 (154)andwecanusethelargezapproximationfortheplasmadispersionrelation.Thelargezapproximation:G(z) i12ez2+z1_1 +12z2+34z4+. . ._(155)Theimaginarypartofthisapproximation isnegligible,andthereisnodamping.0= 1 2p2zG(z) (156)= 1 2p2_1 +12z2+34z4+. . ._(157) 1 2p2(Duetozbeingverylarge) (158)k=c__1 2p2_(159)> c (160)Thephasevelocitygreaterthanthespeedof light(>p). Thephasevelocityisreal, otherwisethe light is strongly attenuated. Therefore light will only propagate if the density is less than the criticaldensity. (p= ).Thegroupvelocityisgivenby:ddk= c_1 2p2_(< c) (161)Theinformationisthereforetransferredslowerthanthespeedoflight.Transverse waves electro-magneticwaves inplasmas.Collisional damping does occur,resulting ininverse bremsstrahling or collisional absorption. (Inthisanalysisthedampingtermiszero).229.6 Longitudinal Waves

= 0 (162)12G(z) +k22D= 0 (163)zG(z) = 1 +k22D(164)Hereziscomplex,andiscomplexduetothedamping:= r + i..(165)wheretheindicatedtermisthedampingterm.Assume a small kD(i.e. the wavelength is long compared to the Debye length, D). This givesthatzislarge.G(z) = i12ez2+z1_1 +12z2+34z4+. . ._(166)Theimaginarytermissmall. TheasymptoticformofG(z)gives2= 2p +32v2k2= 2p(1 + 3k22D) (167)wherev=_2kTem. ThisresultistheBohm-Grossfrequency. Inacoldplasma2= 2p coldplasmawave. 3k22Disthewarm plasma correction. Warm plasma changes theresonant frequency,introducingawavelengthdependence.Landaudampingisgivenby:= () =_8_12 p(k33D) exp_12k22D32_(168)Thisisweakforlongwavelengths. WegetstrongdampingwhenkD1. Nowaveisformedforveryshortwavelengths(ifkD1)asitisdampedbeforeoscillation.9.7 PhysicalOriginofLandauDampingThe imaginary part is due to the resonance where v

=k(phase velocity). This depends on the gradientofthedensitydistributionatv

=k_f0v

v

=k_.Thewavetends toaccelerate(pull along...) particleswithv |k. Theslowparticlesareaccelerated,thefastdecelerate. However,therearemoreslowparticlesthanfast,whichleechesenergyfromthewave,givingwavedamping.Intheframeof thewave, theeldhasawell, inwhichparticlescanbecometrapped. Thewavechanges phase slightlydue tothe trappedelectrons, whichintroduces phaseincoherence. This alsodampsthewave.10 Lecture10: MHD10.1 HydrodynamiclimitConsiderthesysteminanaveraged sense-averaged overthedistribution:n =

f (v)dv (169)wherenisthenumberdensity. Thisgivestheparticlenumbersperunitdensity.nmu =

mvf (v)dv (170)23wherenmuisthemomentumperunitvolume.nE=

12mv2f (v)dv (171)wherenEistheenergyofaparticle. Thevelocityofaparticlehas2parts. Theaveragedowvelocity(u),andtherandom(thermal)velocity(w).v = u +w (172)Thethermalenergyisgivenby:n =

12mw2f (v)dv (173)Sincethemeanofwiszero:nmw =

wf (v)dv= 0 (174)Thereforeenergyisgivenby:E= +12mu2(175)where =12mw2=32nkTifthedistributionisisotropic.UsetheBoltzmannequationasastartingpointtoformaverages. FirstdeneaquantityQ(v):Q(r, t) =

Q(v)f (r, v, t)dvn(r, t)(176)Theforceontheparticles:F= ma = q(E +v B) m (177)Whereisthegravitationalpotential. ConsiderthemomentsofvBabouttermsintheBoltzmannequation:

Qftdv =t

Qf dv =(Qn)t(178)

Qv frv =r

Qvf (v)dv =r(nQv) (179)

Qa fvdv =

Qaifvidv =

fvi(Qai)dv (180)afterinteractionbyparts, notingthatboundarytermsatv :f 0. Sincetheonlyforcesontheparticledependingonvelocityarethoseduetothemagneticeld,whichhavenocomponentinthedirectionxidependingonvi. Thereforeaixi= 0 (181)

Qa fvdv = na Qv(182)

Qftcolldv =t(nQ)coll(183)since the particle number, momentum and energy are conserved in the collision. Inserting the momentumintotheBoltzmannequationgives:t(nQ) +r(nQv) na Qv=t(nQ)coll(184)2410.2 ContinuityEquationQ(v) = 1nt+r(nv) = 0 (185)This excludes ionisation events, and is the equation for both electrons and ions (dierent number density).10.3 MomentumEquationQ(v) = mv (186)Forasingleelementuid(e.g. agas):t(nmv) +rj(nmvivj) nFimvj(mvj) = 0 (187)inCartesiantensornotation. Fisthetotalforceoneachparticle,mitsmass.vivj= uiuj +wiwj+B0uiwj+B0wiuj(188)wherenmuiujisthemomentumux,nmwiwjisthemomentumtransfer(stresstensor),andtheothertermscanceltozerobecausetheaverage ofwiszero.nmwiwj= ij(189)wherei,jisthetotalstresstensor.Iftheuidisassumedtobeisotropic: wiwj=0wherei ,= j. andtheonlystressisthehydrodynamicpressure,actinginwards:ij= pij= 13nmw2ij(190)wherei,j=_1 if (i = j)0 otherwiseistheKronekerdeltaThereisanotherstressterm, associatedwithviscosityinavelocitygradient,whentheuidisnon-isotropic:ij= 2( ij13 iiij) + ijij(191) ij=12_uirj+ujri_(192)andhencethetotalstressi,j= i,j pi,j(193)Makinguseoftheequationofcontinuityweobtainthefamiliarequationfromuidmechanicsnmut+nm(u)u xii,j nF= 0 (194)2510.4 Multi-ElementFluidMomentumisexchangedbetweenparticlesinacollision:t_nQcoll t[nmv]coll ,= 0 (195)Introduce anexchange term: Pij=averagemomentumexchange fromparticle i to particle j.(sec/unitvolume). Fromparticlei:nimi_uit+_ui

r_ui_&&&&b+rpirl[il] niFi= Pij(196)Inplasmaviscosityisgenerallynegligible:rlr(pi) = pi(197)Replacingtheexternalforcebythatduetotheelectricandmagneticeldsnimi_uit+ (ui )ui_+pi= zeni[E +uiB] +Pil(198)neme_uet+ (ue )ud_+pd= ene[E +viB] +Pil(199)wheretherstequationisforions, andthesecondisforelectrons. Theviscoustermsandgravityareneglected.10.5 BulkFluidEquationne zni quasi-neutrality Debyelengthisshort.Equationofcontinuityt(nimi) +t(neme) t+ (u) = 0 (200)u =(miui+meue)mi+meMeanowvelocity (201) = nimi +nemeDensity (202)MomentumEquation_ut+ (u)u_ = p + .. qE +j B (203)isEulersEquation,withaddedE.M.eldforce.q = zeniene 0 ChargeDensity (204)j = zeniuieneui CurrentDensity (205)me mi(206)u ui(207)Iftheplasmaisquasi-neutral: ui ue(208)and uiand uedier only by a rotational vector, i.e. the dierence in velocity vector forms closed loops intheplasma. Sincethebulkvelocity isapproximately equaltotheionvelocity issmall,theseare currentloopsofelectronsindependentofbulkmotion,governedbyOhmsLaw:26me mi(209)menee2_zeniuiteneuet_=menee2jt(210)= E +u B j Bnee+ penee+Piene 0 (211)ue= u jnee(212)ue ui(213)Thejttermisusuallynegligible. ThegeneralisedeldE= E +u B j Bnee+ penee(214)Inanisotropic(non-magnetisedplasma),themomentumexchangePienee (uiue) (215) j (216)= j(whereistheconductivity) (217)= E(eqn.213) (218)11 Lecture1111.1 OhmsLawThisderivesfromthecurrentequation. Thecurrent iscarriedbyelectrons,astheyarelighterthantheions,andiscausedbythedierencebetweentheelectronvelocityandmeanspeed:j= ene(u ue) (219)menee2_zeeniniteueuet_=menee2! 0(almostalways)jt(220)=eectiveelectricaleld(withelectronpressure) .. E..E-Field+ u B. .Lorentzforce(induced)j Bnee. .Hallterm+Pienee+penee..electronpressure(221)Ohmslaw:E=j(222)whereistheconductivity(scalarinanunmagnetisedplasma,secondordertensorwhenmagnetised).WeexpectPie j(currentdensity),leadingto,assumingthatthemediumisisotropic,Ohmslaw:1= Pienee(223)Thisisnegativebecausewearedealingwithnegativecharges.27Thermo-electricity,theThompsoneect,iscurrentdrivenbyatemperaturegradient.Pienee= j+T (224)where is the thermo-electric coecient. In a magnetic eld , are both tensors with three independentcomponents. (Again,assumeanon-isotropicmediuminthepresenceofamagneticeld): =__

0 00 0 __where

= electricaleldalongmagneticeld= electricaleldperpendiculartothemagneticeld= cross-product componentarisingwhenEandBarenormaltoeachother11.2 EnergyEquationsSetQ =12mv2,andions,electronsandE.M.eld,wegettheforcebalanceequation:Ut+ W= 0 (225)WhereUistheenergydensity,andwistheenergyux:U =32nikTi +32nekTe +12u2+0E22+B220(226)W =_32nikTi +pi_. .Enthalpyof theionsui +_32nekTe + pe_. .Enthalpyof theelectronsue +_12miniu2i_. .ionK.E. densityui +_12meneu2e_. .electronK.E. densityue(227)+10E B. .Poynting vector ijvj. .viscousworkdone+ q..thermalux(228)Where qis the energy ux due to thermal transport, and is the heat transfer term in the above equation.q= T E(229)WhereEistheneteectiveelectriceld. Inamagnetised, non-isotropicplasmaandaretensors,where= Te +52kTee (230)whereisthesameasthatinequation(224)asrequiredbytheOnsagerrelations(reciprocal kineticprocess).11.3 TransportCoecientsFor a small perturbation, the ux is proportional to the appropriate driving force: Conductivity, ThermalConductivity, Thermo-electricCoecient, Viscosity1. Theseareall collisional terms, and, if therearemagneticeldsaswell,thesearealsoalltensors.If youapplya force to aparticle distribution, the distributionwill be distortedawayfromtheMaxwellian. Provided theforceisweak,thedistortionwillbesmall(perturbation),andtheuxwillbelinearlyproportionaltotheforce. Iftheforceisstrong,thislinearitywillbreakdown.Distributionfunctioncanbewrittenwithaperturbationf1onasteadystateMaxwellianf01Onlyassociatedwithions,usuallyneglected28f = f0 + f1(231)Appliedtoelectronsfor, , , andtoionsfor. Ifweassumethatf1issmall,theconditionsonfaregivenby:0 =f0tc(232)ThisgivesaMaxwellianforf0.f0t+v f0r+a f0v. .Force= C(f0[f1))..Collisionterm(233)where C(f0[f1) is the rst order perturbationtermintroducedby f1. Hence knowing that (f)0isMaxwellianyieldsthel.h.s. fromwhichtheperturbeddistributionfunctionf1canbecalculated. Thesolutionoff1givestheperturbeddistribution,andf1givesyoutheux.11.4 Hydro-MagneticApproximationWeassumethat theplasmahasinniteconductivity, (or, morecorrectly, that themagneticReynoldsnumberRm(=Lu) , whereListhecharacteristiclength, anduisthecharacteristicvelocity). Sincej isnite, thisgivesE=0. WeassumethatthereisnoHall term, andnoelectronpressureterm E +u B= 0. ThisleadstoFaradays Law(bytakingthecurl)fortheux:Bt= (u B) (234)WenowconsiderthroughanarbitraryloopSwhichisxedintheuid, i.e. madeupofthesameuidparticlesforalltime: =

ABdA (235)whereAistheareavector, whichchangesintimeas theuidmoves. Theareachangesduetothemotion:dAdt=

Au dl (236)wheretheintegralisaroundtheloopHenceddt=

ABt.._1dA +

Bu dl. ._2(237)Therstterm _1 isduetotimevariationinthemagneticeld,thesecond _2 theresultofuxsweptoutbythemotion.ThuschangingtheorderoftermsinthescalartripleproductandusingStokesTheorem:ddt= 0 (238)andtheuxremainsconstantthroughtheloop.Consider a tubeof force withwalls parallel toB. Put aloopinto awall of thetube. istherefore 0sincethenormal component of Biszero . Thisremains constant (=0) as theplasma moves, asthereisno Bthrough the loop. The loop is arbitrary, moves with theuid along a stream line,and

B dA = 0for all such loops, so the uid,which ties the loop, also follows a ux line in addition to the stream lines.Itmayalsobesaidthattheuxlinesaretiedtothemotion,thefrozenincondition.29(u B) = (B )u (u)B +$$$$X0u B B u (239)t+ (u) = u (240)ddt_B_=ddt_B_+u_B_(241)=_B_u (242)Considerthemovementofanelementlxedintheeld:ddt[l] =t[l] +u[l] = (l)u (243)Iflinitiallyliesonaeldline,thenitwillcontinuetodoso.B l (244)12 Lecture12: IdealMHDApproximation12.1 PressureBalanceIftheplasmaisatrest,theaccelerationtermcanberewrittenas:dudt= 0 (245)= p +j B (246)p = j B (247)IfweuseMaxwellslaws:j =10(B) (248)p = j B (249)=10(B) B (250)= 10__12(B2). .Mag. Pressure Term (B)B. .LongitudinalStress__(251)wherethelongitudinalstressactsalongtheeldlines.Formanyapplications(B)B= 0,asweareonlyinterestedintheabilityofthemagneticeldtoconstraintheplasma. Thepressurebalanceequationistherefore:p +120B2= Constant (252)Theplasma,which,formagneticconnementdevices,relatestothestabilityetc,isdenedas:=Pmax_B2max20_ (253)Forallconnedplasmas,< 1,andshouldbesignicantlylessthanone.30Inthetheadiabaticlimitwhen: , and q 0,i.enodissipationorentropy increasetheenergyequationreducestotheadiabaticgaslaw:32dpdt+52pddt= 0 (254)whereddt=t+ucdot (255)isthetimederivativemovingwiththeplasma.Hence_p_53=constant12.2 IncompressibleApproximationTheequationofmotion:__ut+ ( u) u +12(u2). .(u)u__= _p +120B2_+10(B)B (256)andthevorticity= v. Ifthelasttermiszero,i.e. jBcanbewrittenasapotential, theowmaybedescribedintermsoftheusualuidmechanicsrelations.12.3 BernoullisEquation-SteadyFlowForauid:12u2+p= Constant alongastreamline (257)Ifwetakethescalarproductofuwiththeequationofmotion:u_p+B220+12u2_ =10u(B)B. .Mag. StressTerm(258)wherethemagneticstresstermmust bemuchlessthan1forBernoullitoapply. Ifweintegratealong astreamline,wendthat:p+B220+12u2= Constant (259)Wenowre-writethestresstermintermsofacomponentalongtheeldlines, andanothernormaltotheeldlines. Letbbetheunitvectorintheelddirection,and ubetheunitvectornormaltotheeldline:10u(B)B =10___u uB2R. .NormaltotheFieldline+(u b)(b(12B2). .Alongtheeldline___(260)WhereR =radiusofcurvatureoftheeldline.3112.4 KelvinsTheoremThecirculationaroundaloopxedintheeldisconstant,andgivenby: =

udl =

u. .Vorticityds (261)Wenowtakethecurlofoutequationofmotion( = 0):_ut+ u_ =10[(B)B] (262)Ifweintegratethelefthandsideover theloop:ddt

ds = 0 recover Kelvinstheorem (263)Buttherighthandsidegivesanon-zerocontribution(Kelvinstheoremfailsintheplasma). Theplasmageneratesvorticity. Incontrasttoauid, amagnetisedplasmacansupportshear,i.e. theeldimposesadegreeofstiness totheplasma. Inconsequencealthoughuidsonlysupportlongitudinalwaves(onlysoundwaves,notransversewaves),aplasmaallowstransversewaves(viavorticity)whichhavevorticityassociatedwiththem,inadditiontothelongitudinalwaves.Transverse:Bxz,= 0 Non-zerocurl = Vorticity (264)Longitudinal:uxz= 0 : ux= 0 (265)Solidsalsosupportbothlongitudinalandtransversewaves (e.g. earthquakes).12.5 AlfvenWavesThesimplest possiblecaseof transversewaves: uniformincompressibleinniteconductivityplasma,movingataconstantvelocity, u, inauniformeld,B. Wemovetotherestframeoftheplasma. TheplasmaisperturbedbysmallchangesinvelocityuandeldB. Thelinearisedequationsare:ut= p +10(B) B (266)AddingMaxwellgives:Bt= E= (B )u(267)whereE= u B.Wenowdierentiate(266)withrespecttotime,andsubstitutefor(267):2ut2= _pt_+10[(B)u] B (268)Nowtakethecurl:2t2=10(B)[(B)] (269)N.B.Bisaconstant.Introduceaplanewave:= 0 exp [i(t kr)] (270)whereisthevorticity,andistheangularfrequency.Thephasevelocityisgivenby:32V=k= CA cos (271)WhereCAistheAlfvenvelocity, andistheanglebetweenkandB, theangleof propagationwithrespecttotheeldlines.CA=B0(272)Stringtension:B20,Stringmass: .2t2=10(B)[(B)] = Vorticity (273)TheonlydirectioninwhichvariesisalongB. Since= u,thisimpliesthatparticlevelocityuisperpendiculartothemagneticeldsuperimposedonthesteadybackgroundmotion.Fortransversewaves: u= 0,astheseareincompressiblewavesanyway,andthereisnoneedtomaketheincompressibleapproximation.13 Lecture13: MagnetosonicWavesAnincompressibleplasmacanonlysupporttransversewaves(Alfvenwaves). Plasmasare, however,compressible. This adds longitudinal (sonic) waves. We expect that the newwaves will be mixedlongitudinal and transverse. Consider a perturbation on an ambient condition assuming that the plasmaisinitiallyatrest: Density: 0 + ( istheperturbation, 0istheambientcondition Pressure: p0 +p Magneticeld: B0 +B Velocity: v(v0= 0afterchangingtheframetotheplasmarestframe) Displacement: ,wherev=ddt.LinearisingtheM.H.D.equations,andassumingadiabaticow:t+0 v = 0 (274)0

vt= p +10(B) B0(275)pt= p000t(276)E +v B = 0 (277)Bt= E (278)v =ddt=tTheothertermisneglectedhere) (279)Here,(274)becomes: +0 = 0 (280)Ifwecombine(276)and(280)weget:p = p00 = p0 (281)33Combining(277),(278)and(279) gives:Bt= 0_t B0_(282)B = ( B0) (283)2t2 p00( ) 100(B) B0= 0 (284)Let:(B) B0=__B20 (B0 )(B0 )+ (B0 2) B0_(B0 )( )_(285)Therefore:02t2+p0( ) 10_B20 (B0 )(B0 10(B0 )2 +10B0(B0 )( ) = 0(286)WenowintroducetheAlfvenspeed:C2A=B2000(287)Thesoundspeed:C2s=p00(288)Andtheratio: =C2sC2A=_2_ (289)where=p0B220Thisgives:1C2A2t2 _(1 +) (b)(b)(b)2 + (b)( )b = 0 (290)whereb =B0|B0|,andistheunitvectorintheB-direction.N.B.2t2is not parallel to , so the wave is not longitudinal. Indeed, it must include an Alfven wave,whichistransverse.Wenowintroduceaplanewave: =0 exp[i(t kr)]. Theequationwill contain3vectorcom-ponents. WecouldgotothethreeCartesiancoordinatesareobtainequationsoftheform:V2C2Ai +

jAijj= 0 (291)whereV isthephasevelocity,andAij(b, k, )iswhatweneedtosolveforthewave.Formingthecomponentsofthewave inthreedirections: k, (tokinplane(k, b)),and(tokandb). Formk: givescomponentsparalleltok. Formb: givescomponentsparalleltob. Form(bk): givescomponentsperpendiculartobothbandk.341st:k _V2C2A(1 +)_k + (bk)(b) = 0 (292)2nd:b V2C2Ab (bk)(k) = 0 (293)3rd:(bk) _V2C2A(bk)2_bk = 0 (294)Thesethreeequationsmustbesatisedbyanygeneral wave. If wehaveawavewithparallel tob k,i.e. perpendiculartobothbandk,weget:_VCA_ = bk (295)The phase velocity V= CA cos where () is the angle between b and k. These are the Alfven waves.Theyarecharacterisedbyhavingdisplacement, perpendicular tobandk. Twocoupledequationswhereisintheplaneofkandb:_VCA_4(1 +)_VCA_2+ cos2 = 0 (296)Obtaining consistence for theequations for(k) and(b). Thisis a quadratic equation for V2. Therearetwodistinctwaves, eachofwhichcanmoveforwardsorbackwards(fastorslowwaves).V2=12___(C2A +C2s) _(C2A +C2s) 4C2AC2s cos2. .Twowaves, neitherall longitudinal norall transverse___(297)If = +thisisafastwave,if = thisisaslowwave,withtheconditionthatk, b, allcouple.14 Lecture14: Magneto-HydrodynamicWaves1C2A2t2 [(1 +) (b)(b)] (b)2 + (b)( )b = 0 (298)Introduceaplasmawave: = 0 exp_i2(V t kr)wherevisthephasevelocity,andkistheunitvectorinthewavedirection.Formthetermsk, b, (bk))._V2C2A(1 +)_k + (bk)(b) = 0 (299)V2C2Ab (bk)(k) = 0 (300)_V 62C2A(bk)2_bk = 0 (301)Case1: perpendiculartobandk:k= b = 0 (302)bk ,= 0 (303)V2= (bk)2C2A(304)35Letbetheanglebetweenbandk.V2= C2A cos2 Alfvenwaves (305)Case2: liesintheplaneofbandk.bk = 0 (306)Theremainingtwoequationsarecoupled.There are two waves, since there are two solutions. We require consistency between the two equations._V2C2A (1 +)_V2C2A+(bk)2= 0 (307)V4C4A(1 +)V2C2A+cos2 = 0 (308)(309)Solutions:V2=12_(C2A +C2s) _[(C2A +C2s) 4C2AC2s cos2]_(310)Twowaves,fast(+)andslow(). Thesehavebothlongitudinalandtransversecomponents.If = 0 kisparalleltob.V+=12_C2A +C2s+[C2AC2s[ = LargerofCAorCs(311)V=12_C2A +C2s [C2AC2s[ = SmallerofCAorCs(312)If =2 bisperpendiculartok.V+=_(C2A +C2s) (313)V= 0 (314)Thisisapurelylongitudinalwave,withnosecondwave. Themagneticpressureaddstothekineticpressure,likeasoundwave.The longitudinal and transverse component (to the magnetic eld) are given by: k

for longitudinal,andkfortransverse.k = _V2C2A__V2C2A(1 +)_k

(315)k k

=(1 +) _[(1 +)24cos2](1 +) _[(1 +)24cos2](316)Thiswillbegreaterthanzeroforafastwave,andlessthanzeroforaslowwave.Themajordierencebetweenthewavesisthatinfastwavesthelongitudinal andtransversepartsare in phase,but inslow waves theyare out of phase. Givenan arbitrary initialcondition,displacementandvelocity,therelativephasesxtheinitialfastandslowcomponents.3614.1 M.H.D.ShocksInuiddynamicsalargeamplitudesoundwavegoesintoashock:c2=p(317)p (1)(318)c2 (1)(319)Thusasthedensityincreases, thesoundspeedincreasesandwavesfromthebackbuildupintoasharplyrisingfront-shockwave.Large amplitude magneto-sonic waves can develop into shock waves. We consider shocks with normalinthexdirection. Thecomplexityoftheshockwaveismuchincreasedduetothemagneticeld. i.e.thewaveisnotonlylongitudinal, whichmeansthatthetangential symmetryofthethegasdynamicsislost. Wegetowsalongthesurfaceoftheshock(y,zdirections). Letuslookatthejumpconditionsintherestframeoftheshock,rememberingthatoneithersideoftheshockfrontwehavep1, 1, B1, u1andp2, 2, B2, u2,whereuistheowvelocity,u = (u, v, w).Conservationofmass1u1= 2u2(A)Conservationofmomentumnormaltotheshockfrontp11u21 +B2y1 +B2z120= p22u22 +B2y220+B2z220(B)Conservationofmomentumtransversetotheshock1u1v1Bx1By10= 2u2v2Bx2By20y-direction1u1w1Bx1Bz10= 2u2w2Bx2Bz20z-direction___(C)Energy:1u1_h1 +12(u21 +v21 +w21)_+E1B10x= 2u2_h2 +12(u22 +v22 +w22)_+E2B20x(D)whereh= +p=enthalpyperunitmass,orspecicenthalpy.Thetransverseelectricandnormalmagneticeldsarebothcontinuousacross theshock:Et1= Et2and Bx1= Bx2(E)We can simplify these equations. The velocity components may be set by an appropriate transforma-tion to lie in the(x, y) plane only. A further transformation in yallows either thetransverse momentumuxorthetransverseelectriceldtobezeroed(normallythetransverseelectriceld). Wecanassumeinnite conductivity - Ideal M.H.D.- so that E+v B 0. The transverse eld equations now become:u1By1v1Bx1= u2By2v2Bx2u1Bz1w1Bx1= u2Bz2w2Bx2_(F)Asaresulttheenergyequation(E)becomes:1u1_h1 +12(u21 +v21 +

w21)_= 2u2_h2 +12(u22 +v22 +

w22)_+u10_B2y1 +&&B2z1_+u20_B2y2 +&&B2z2_v1Bx1By10w10Bx1Bz1v2Bx2By20w20Bx2Bz2(D)37followingtheframetransformations.Example1: FlownormaltoshockonEntry:v1= 0 (320)By1= 0 (321)Bx1= Bx2= Bx(322)By2= 02u2v2Bxviamomentum (323)=v2Bxu2viatransverse electriceld (324)Eitherv2= By2= 0,anormalgasdynamicshock,oru22=B2x202By2 ,= v2 ,= 0,switchonshock.15 Lecture15: M.H.D.Shocks1u1= 2u2= j (325)p11u21 +B2y120+B2z120= p2 +2u22 +B2y220+B2z220(326)1u1v1Bx1By10= 2u2v2Bx2By20(327)1u1w1Bx1Bz10= 2u2w2Bx2Bz20(328)1u1_h1 +12(u21 +v21 +w21)_+E1B10x= 2u2_h2 +12(u22 +v22 +w22)_+E2B20x(329)Duetotheassumptionthattheplasmahasinniteconductivity:Et1= Et2(330)Bx1= Bx2(331)E +v B = 0 (332)u1By1v1Bx1= u2By2v2Bx2(333)u1Bz1 w1Bx1= u2Bz2w2Bx2(334)Ifv1= 0, By1 = 0Thereisnoupstreamvariationineitheroworeld.By2=02u2v2Bx(335)=v2Bxu2(336)u22=B2xo2(337)or v2= By2= 0 (338)Where(337)and(338) representtwopossiblesolutions:v2= By2= 0 Notransverseeld,sameasanordinarygasdynamicshock.u2 =Bx02Transverse elds,switched-onshock.Inpractice, thesolutionisdeterminedbythestabilityandevolutionof theshock. 1stgetthelesslikelyshock,andthengetthesecondshock. Theshockinvolvingthelargestentropychangedominatesthe interaction (by experiment). For weak shocks, this is the ordinary gas dynamics shock. Past a criticalpoint,increasedstrength=switched-onshock.The three types of sonic waves, Alfven, Fast and Slow, can all produce shocks, and all have their owntypeofshock. Thisproducesverycomplexsituations.3815.0.1 AlfvenWaveConsiderv, w By, Bz: thereforev/By/v = w/Bz= . AssumethatB2y1 +B2z1= B2y2 +B2z2andthereforev2y1 +v2z1= v2y2 +v2z2Deningthechangeacrosstheshockby,thusBy= By2By1Henceweobtainfromthejumpcondition(C)By By= Bz Bzv =1ojBxByw =10jBx BzMakinguseoftheinitialassumptionsandconditions(F)weobtainu1By1v1Bx= 1By1= u2By2v2Bx= 2By2u1Bz1w1Bx= 1Bz1= u2Bz2w2Bx= 2Bz2wherej= 1u1 = 2u2; 1,2= u1,2 andvw=ByBzThustheeldtermsin(D)maybeequatedto_u(B2x +B2y) v(BxBy) w(BxBz) = 0Itis easytoshowthat thereexistsolutionswheretheowvariablesnormal totheshockremainunchanged,i.e. thedisturbanceistransversealongtheshock,1= 2u1= u21= 2p1= p2Hencethewavespeedisgivenbyu By= Bx v =B2xou By(339)Thewavespeed,_B2x0, istheAlfvenspeed. ThisresultsinadiscontinuousjumpwavemovingattheAlfvenspeed, aniteamplitudeAlfvenwave. Thusfar, wehavenotincludedcollisions, andareconsidering only collision free shocks. Where does the entropy change originate?Microscopic turbulence.15.1 BoltzmannsEquationasaFokker-Planck(LandausEquation)Boltzmannsequationisabinarycollisionequation. TheFokker-Planck equationisamanybodyequa-tion. Our route here will be to start with weak collisions and small impulses, and add velocity slowly andlinearly. Alargenumberofsmallbinariesisamanybodycollision. WeuseBoltzmann,butwithweakcollisionsonly, astherearemanymoreweakcollisionsthanstrongcollisions. Weassumetwoidentical39particles,hereelectron-electron scattering,withvelocityv1, v2,relativevelocityu = v2v1,andcentreofmassvelocityV=v1+v22. Thecross-section forcoulombscatteringthroughis: =_e22mr_2_u sin_2__4(340)wheremr=12m-electronreducedmass.TheBoltzmanncollisionintegral:I=_e2m_2

dv2

sin ddu_u sin_2_4_f_v112u_f_v2 +12u_f (v1)f (v2)_(341)uf= Ou Oisarotation (342)u = uf u (343)[uf[ = [u[ (344)v1_v112u_(345)v2_v212u_(346)Unitvectorlisparalleltou. Normaltolareorthogonal unitvectorslarem.n.u = 2u sin_2__l sin_2_+mcos_2_cos () +nsin_2_sin ()_(347)sincemagnitudeofuisunchangedmagnitudeonscatter(elasticcollision).Ifthescatterisweak,uissmall,wecanexpandtermsviatheTaylorseries:f_v112u_f_v2 +12u_f (v1)f (v2)=$$$$$f (v1)f (v2) $$$$$f (v1)f (v2) +_f (v1)fvv2+ f (v2)fvv1_ 12u+12_f (v1)2fvivjv22fviv1fvjv2+ f (v2)2fvivjv2_ 12ui12uj(348)Integrationovertheazimuthalangle eliminatestermsincos , sin , cos sin since

20dcos =

20dsin =

20dsin cos = 0HencesubstitutingI = 8_e2m_2

dv2

0d_2_cos_2__uiu3sin_2__f (v1)fviv2f (v2)fviv1++12_f (v1)2fvivjv22fviv1fvjv2+ f (v2)2fvivjv2_

1u_u2u1u2sin_2_+12(mimj +ninj)_1sin_2_ sin_2_____(349)40But

0cos (2)d(2)sin (2)diverges at small and large . Physically this is due to the neglect of correlationsat small 0 (Debye shielding). As before we cut-o at the integral for small impact parameters at theDebyelengthcorrespondingtomin Dbmin(350)Henceln__sin_max2_sin_min2___ = ln (351)ln islarge.16 Lecture16I = 8_e2m_2

dv2

0d_2_cos_2__uiu3sin_2__f (v1)fviv2f (v2)fviv1++12_f (v1)2fvivjv22fviv1fvjv2+ f (v2)2fvivjv2_

1u_u2u1u2sin_2_+12(mimj +ninj)_1sin_2_ sin_2_____(352)

0cos_2_d_2_sin_2_ ln__sin_max2_sin_min2___ = ln Spitzer/CoulombLog. (353)

0d_2_cos_2_sin_2_=12(354)Assumeln islarge.mimj +ninjistheprojectionoperator. Itprojectsavector,a,ontotheplaneperpendiculartog.[mimj +ninj]a = mi(m a) +ni(na) Componentof alyinginm, nplane. (355)= a l(la) (356)Intensornotation (357)mimj +ninj= ij liljN.B.lisparalleltou. (358)mimj +ninj= ij uiuju2(359)NewTensor: (360)ij=1u_ij uiuju2_(361)ijui=2uiu3N.B.Repeatedindex=summation (362)I = 2_e2m_2ln

dv2_2_f (v1)fviv2f (v2)fviv1_ ijui+vjv1_f (v2)fviv1f (v1)fviv2]ij+_f (v1)2fvivjv2fviv1fvjv2_ij_(363)41Notingijui= ijvij=ijv2jandintegratingbypartsasnecessary,weobtainLandausequation:I = 2_e2m_2ln vij

dv2_f (v2)fviv1f (v1)fviv2_ij(364)LandausEquationvj

dv2_f (v2)fviv1 f (v1)fviv2_ij= vj_f (v1)_

dv2fviv2_ij_+2vjvi_f (v1)_dv2f (v2)ij__vij_f (v1)

dv2f (v2)ijvij_= 2vij___f (vi)__

dv2fviv2ij. .

_____+2vivj___f (v1)_dv2f (v2)ij_. .

___(365)(366)Sinceijv1i=ijv2j16.1 Rosenbluth,MacdonaldandJuddFormThe detailedcalculationof the collisionterms foll wingCoulombscatteringis straightforward, butcomplicated(Rosenbluth,MacdonaldandJudd,PhysRev107,11957). Itleadstoanequivalentresultasthatof Landauforelectron-electron collisions,butismoregeneral astheparticlesmaybedissimilar.Weconsider thecase of twoparticles of masses mand mmoving withvelocitiesvand v. Asiswellknown thecollision occurs with scattering about thecentre of mass withrotation of therelative velocityvector VThevelocity of each particleintermsof thecentre of mass velocity Uandtherelative velocityV arev = U+mm+mVv= U+mm+mVHencethechangeinthevelocityvasaresultofthecollisionv =mm+mVThescatteringgivesrisetovelocitychangesintheplaneperpendicular,, andparallel, |, totheinitialrelativevelocityV asabove. Thecrosssectionforscatterthroughanangleismodiedbytheinclusionofthereducedmassmm =mmm+m42but is otherwise unchanged. Therefore we may use the averages we have already calculated but with themassreplacedbythereducedmass.However therelativevelocityV changes indirectionandthe centreof mass velocityUmust beremovedbeforewecanrelatetheparticlevelocitiestotheirvaluesinthelaboratoryframe. TodothisweidentifyasetofdirectionswithrespecttoV , namelyparalleltoV andandperpendiculartoV sothatV= V(1 cos ) = 2Vsin2(/2)V= Vsin cos = 2Vsin(/2)cos(/2)cos V= Vsinsin = 2Vsin(/2)cos(/2)sin (367)beingtherotationangleintheplane, perpendiculartoV .Usingeither thesmall angleapproachasinsection5.1ordirectlyfromthefull Rutherfordcrosssectionasinsection15.1() d =Z2e44 m2V4csc4(/2) d (368)forparticlesofchargeeandZe. Theresultantaverages arecarriedoutasdescribedinsection5.1,butinthecentreofmassframeV) = V

) = 4Z2e4m2V2ln V) = V) = 12V) = 0V2) = V

2) = 0V2) = V2) = 12V2) =4Z2e4m2Vln (369)Theunitvectorsinthedirections,andare =VV =k VVx2+Vy2and = where(,j,k)aretheunitvectorsinthe(x,y,z)directionsThusV inthelaboratorysetofCartesianco-ordinates(x,y,z)maywrittenVx= ( )V + ( )V + ( )VwithsimilarexpressionsforVyandVz. ThustakingtheaverageVx) = ( )V) + ( )$$$$X0V) + ( )$$$$X0V)=VxV V

) (370)withsimilarexpressionsfor Vy)and Vz).The product terms Vx Vy) etc. are obtained in a similar manner, noting that several terms canceltozeroVx Vy) =__( )V + ( )V + ( )V_ _(j )V + (j )V + (j )V_ _=___ _ _j __V2+__ __j _+_ __j __$$$$$V V+__ _ _j __V2+_( )_j _+_ __j __$$$$$V V+_( )_j __V2+__ __j _+ ( )_j __$$$$$V V_=__ __j _+ ( )_j __12V2_43HencegeneralisingVi Vj_= [ij +ij]12V2_= [ij ij]12V2_(371)sincethecomponentsof theunit vectors, , ontheorthogonal setof unit vectors,j,knamely(i,j,k), (i,j,k)and(i,j,k)respectivelysatisfytheconditionij+ij + ij= ijReturningtothelaboratoryframe,thechangeintheparticleaverages aregivenbyvi_=mm+mVi_= 4 Z2e4ln mmViV3=4 Z2e4ln mmVi_1V_(372)sinceV=_(vivi) (vivi)andvi vj_=4 Z2e4ln m2_ij ViVjV2_=4 Z2e4ln m22VViVj(373)Thesetermsarewrittenintermsofthefunctionsg(v) =

dvf(v)[v v[ (374)h(v) =

m+mm

dvf(v)[v v[1(375)thesumsbeingtakenoveralltheperturbingspecies. Thederivativesofg(v)andh(v)yieldgvi=

dvf(v)ViV2gvivj=

dvf(v)_ijViVjV2_(376)hvi=

m+mm

dvf(v) ViV3(377)SubstitutingweobtaintheMacDonald, RosenbluthandJuddformof theFokker-Planckcollisionoperator.vi_=4 Z2e4ln m2hvi(378)vi vj_=4 Z2e4ln m22gvivj(379)Inpolarco-ordinates,withoutsymmetry,asimilaranalysisgivesbasicallyasimilar,butmorecom-plicated,result.4416.1.1 RelationshipwithLandausFormulaThisresultiseasilyrelatedtotheLandauformulaforidentical particlesm=m, section15.1. Usingthesamesymbolsasbeforeu V , andv1 vandv2 vweobtainI, theintegral involvedthenalsolution.ij=1u_ijuiuju2_(380)ijuij=ijvij(381)=ijv2j(382)= 2uiu3(383)Dene:g(v1) =

dv2 f (v2) u (384)h(v1) = 2

dv2f(v2)u (385)gv1i=

dv2 f (v2)uiu ] (386)2gv1iv1j=

dv2 f (v2)_ij uiuju2_(387)=

dv2 f (v2) ij(388)hv1i= 2

dv2 f (v2)uiu(389)=

dv2 f (v2)ijv1j(390)=

dv2 f (v2)ijv2j(391)3gv1iv1jv1j=

dv2f (v2) ijv1j(392)=hv1i(393)

dv2_f (v0)fv1jf (v1)fv2j_ij=

dv2_f (v2)fv1j+ f (v1)fv2j_ij+2

dv2 f (v1) f (v2) ijv2j(394)=fv1j2gv1iv1j+ f (v1)3gv1jv1jv1j2f (v1)hv1j(395)HenceI= 4e4m2ln ___v1i_f (v1)hv1i_. .

+122v1iv1j_f (v1)2gv1iv1j_. .

___(396)givesthedynamicalfrictionandthediusionSphericallysymmetricdistributionforf (v):45I= (4)2e4m2ln 1v2ddv_f +dfdv_(397)where: =

v0f (v)v2dv(398) =13_1v

v0f (v)v4dv +v2

vf (v)vdv_(399)Fordissimilarparticleswithmassmandm,usethereducedmass:mmm+m. Theconstantinh(v)ischangedtobym+mm. i.e. replacethe2bym+mminthefrontofthedenition.)17 Lecture17: CalculationoftheTransportCoecientsThetransportcoecients are: Electrical Conductivity, Thermal Conductivity, andViscosity. Let usimagine that the plasma is weakly perturbed by an external agency/force, such as a temperature gradient.Wenow have theambient thermaldistribution,andtheperturbationtothedistributioninducedbytheforce. Weassumethatthereisnonetuxassociatedwiththebackground(isotropicdistribution). Wewishtondtheuxfromtheperturbation, thisgivesthepropertiesof theplasma. Wewouldliketobeabletocalculatetheperturbationduetoaforceonthedistributionfunction, andcalculatetheuxfromthisperturbation. Weassumesteadystatebeforetheperturbation:f (v)..FromDistribution= f0(v). .Initial Maxwellian+ f1(v). .Perturbation(400)Boltzmannequation:!0duetosteadystateft+v fr a0

fv=ftcI (401)Apply(400),neglecttermsofsecondorderandhigher Torstorder:f0_mv22kt52_1T viTxif0eZkT Eivi= I(f1, f 0) +I(f0, f 1) (402)Solveforf1. Ingeneral, thesystemhascylindrical symmetryif thereisnomagneticeldpresent.Thismeansthatf1 cos ,whereisthepoloidalanglewithrespecttotheelddirection.f1 = D(v) cos (403)whereDissomefunctionof v. FormoredetailsseeSpitzerandH arm(PhysRev89, 977,1952), andCohen,SpitzerandRoutly(PhysRev80,230,1950).46