COLLISIONS AND TRANSPORT

Transcription

1 COLLISIONS AND TRANSPORT Temperature are in ev the correponding value of Boltzmann contant i k = erg/ev mae µ, µ are in unit of the proton ma e α = Z α e i the charge of pecie α. All other unit are cg except where noted. Relaxation Rate Rate are aociated with four relaxation procee ariing from the interaction of tet particle labeled α treaming with velocity v α through a background of field particle labeled β: lowing down tranvere diffuion parallel diffuion energy lo dv α = ν α\β v α d v α v α = να\β v α d v α v α = να\β d v α = ν α\β v α, v α where v α = v α and the average are performed over an enemble of tet particle and a Maxwellian field particle ditribution. The exact formula may be written 19 ν α\β = 1 + m α /m β ψx α\β ν α\β ν 1 α\β = 1/x α\β ψx α\β + ψ x α\β ν α\β ν α\β = ψx α\β /x α\β ν α\β ν α\β = m α /m β ψx α\β ψ x α\β ν α\β, where ν α\β = 4πe α e β λ αβ n β /m α v α 3 x α\β = m β v α /kt β ψx = x t 1/ e t π ψ x = dψ dx, and λ αβ = ln Λ αβ i the Coulomb logarithm ee below. Limiting form of ν, ν and ν are given in the following table. All the expreion hown 3

2 have unit cm 3 ec 1. Tet particle energy and field particle temperature T are both in ev µ = m i /m p where m p i the proton ma Z i ion charge tate in electron electron and ion ion encounter, field particle quantitie are ditinguihed by a prime. The two expreion given below for each rate hold for very low x α\β 1 and very fat x α\β 1 tet particle, repectively. Slow Fat Electron electron ν e e /n eλ ee T 3/ / ν e e /n eλ ee T 1/ / ν e e /n eλ ee T 1/ T 5/ Electron ion ν e i /n iz λ ei.3µ 3/ T 3/ / ν e i /n iz λ ei µ 1/ T 1/ / ν e i /n iz λ ei µ 1/ T 1/ µ 1 T 5/ Ion electron ν i e /n ez λ ie µ 1 T 3/ µ 1/ 3/ ν i e /n ez λ ie µ 1 T 1/ µ 1/ 3/ ν i e /n ez λ ie µ 1 T 1/ µ 1/ T 5/ Ion ion ν i i µ 1/ 1/ 1 + µ T 3/ n i Z Z λ ii µ µ ν i i n i Z Z λ ii ν i i n i Z Z λ ii µ 1/ µ 1 T 1/ µ 1/ µ 1 T 1/ µ + 1 µ µ 1/ µ 1/ 3/ 3/ µ 1/ µ 1 T 5/ In the ame limit, the energy tranfer rate follow from the identity ν = ν ν ν, except for the cae of fat electron or fat ion cattered by ion, where the leading term cancel. Then the appropriate form are ν e i n i Z λ ei 3/ µ µ/t 1/ 1 exp 1836µ/T ec 1 33

3 and ν i i n i Z Z λ ii 3/ µ 1/ /µ 1.1µ + µ /µµ µ /T 1/ 1 exp µ /µt ec 1. In general, the energy tranfer rate ν α\β i poitive for > α * and negative for < α *, where x* = m β /m α α */T β i the olution of ψ x* = m α m β ψx*. The ratio α */T β i given for a number of pecific α, β in the following table: α\β i e e e, i i e p e D e T, e He 3 e He 4 α * T β When both pecie are near Maxwellian, with T i < T e, there are jut two characteritic colliion rate. For Z = 1, ν e = nλt e 3/ ec 1 ν i = nλt i 3/ µ 1/ ec 1. Temperature Iotropization Iotropization i decribed by dt = 1 dt = ν α T T T, where, if A T /T 1 >, ν α T = πe α e β n α λ αβ A 3 + A + 3 tan 1 A 1/. m 1/ α kt 3/ A 1/ If A <, tan 1 A 1/ /A 1/ i replaced by tanh 1 A 1/ / A 1/. For T T T, ν e T = nλt 3/ ec 1 ν i T = nλz µ 1/ T 3/ ec 1. 34

4 Thermal Equilibration If the component of a plama have different temperature, but no relative drift, equilibration i decribed by where ν α\β dt α = β ν α\β T β T α, = m α m β 1/ Z α Z β n β λ αβ ec 1. m α T β + m β T α 3/ For electron and ion with T e T i T, thi implie ν e i /n i = ν i e /n e = Z λ/µt 3/ cm 3 ec 1. Coulomb Logarithm For tet particle of ma m α and charge e α = Z α e cattering off field particle of ma m β and charge e β = Z β e, the Coulomb logarithm i defined a λ = ln Λ lnr max /r min. Here r min i the larger of e α e β /m αβ ū and h/m αβ ū, averaged over both particle velocity ditribution, where m αβ = m α m β /m α + m β and u = v α v β r max = 4π n γ e γ /kt γ 1/, where the ummation extend over all pecie γ for which ū < v T γ = kt γ /m γ. If thi inequality cannot be atified, or if either ūω cα 1 < r max or ūω cβ 1 < r max, the theory break down. Typically λ 1. Correction to the tranport coefficient are Oλ 1 hence the theory i good only to 1% and fail when λ 1. The following cae are of particular interet: a Thermal electron electron colliion λ ee = 3.5 lnn e 1/ T e 5/ ln T e /16 1/ b Electron ion colliion λ ei = λ ie = 3 ln = 4 ln = 3 ln c Mixed ion ion colliion n e 1/ ZT 3/ e n e 1/ T 1 e λ ii = λ i i = 3 ln, T i m e /m i < T e < 1Z ev, T i m e /m i < 1Z ev < T e n i 1/ T i 3/ Z µ 1, T e < T i Zm e /m i. ZZ µ + µ µt i + µ T i ni Z T i + n i Z T i 1/. 35

5 d Countertreaming ion relative velocity v D = β D c in the preence of warm electron, kt i /m i, kt i /m i < v D < kt e /m e λ ii = λ i i = 35 ln ZZ µ + µ µµ β D ne T e 1/. Fokker-Planck Equation Df α fα f + v f α + F v f α α = Dt t t where F i an external force field. The general form of the colliion integral i f α / t coll = β v J α\β, with J α\β = πλ αβ e α e β m α d 3 v u I uuu 3 coll { } 1 f α v m v f β v 1 f β v v f α v β m α Landau form where u = v v and I i the unit dyad, or alternatively, J α\β = 4πλ αβ e α e β m α { f α v v Hv 1 v f α v v v Gv },, where the Roenbluth potential are Gv = f β v ud 3 v Hv = 1 + m α m β f β v u 1 d 3 v. If pecie α i a weak beam number and energy denity mall compared with background treaming through a Maxwellian plama, then J α\β = m α ν α\β m α + m β v I vv 1 4 να\β 36 vf α 1 να\β vv v f α v f α.

6 B-G-K Colliion Operator For ditribution function with no large gradient in velocity pace, the Fokker-Planck colliion term can be approximated according to Df e Dt = ν eef e f e + ν ei F e f e Df i Dt = ν ie F i f i + ν ii F i f i. The repective lowing-down rate ν α\β given in the Relaxation Rate ection above can be ued for ν αβ, auming low ion and fat electron, with replaced by T α. For ν ee and ν ii, one can equally well ue ν, and the reult i inenitive to whether the low- or fat-tet-particle limit i employed. The Maxwellian F α and F α are given by 3/ { } mα mα v v α F α = n α exp πkt α kt α 3/ { } mα mα v v α F α = n α exp, πk T α k T α where n α, v α and T α are the number denity, mean drift velocity, and effective temperature obtained by taking moment of f α. Some latitude in the definition of T α and v α i poible one choice i T e = T i, T i = T e, v e = v i, v i = v e. Tranport Coefficient Tranport equation for a multipecie plama: m α n α d α v α d α n α + n α v α = = p α P α + Z α en α E + 1 c v α B + R α 3 n d α kt α α + p α v α = q α P α : v α + Q α. Here d α / / t + v α p α = n α kt α, where k i Boltzmann contant R α = β R αβ and Q α = β Q αβ, where R αβ and Q αβ are repectively the momentum and energy gained by the αth pecie through colliion with the βthp α i the tre tenor and q α i the heat flow. 37

7 The tranport coefficient in a imple two-component plama electron and ingly charged ion are tabulated below. Here and refer to the direction of the magnetic field B = bb u = v e v i i the relative treaming velocity n e = n i n j = neu i the current ω ce = B ec 1 and ω ci = m e /m i ω ce are the electron and ion gyrofrequencie, repectively and the baic colliional time are taken to be τ e = 3 m e kt e 3/ 4 = / T e π nλe 4 nλ ec, where λ i the Coulomb logarithm, and τ i = 3 m i kt i 3/ 4 = / T i πn λe 4 nλ µ1/ ec. In the limit of large field ω cα τ α 1, α = i, e the tranport procee may be ummarized a follow: 1 momentum tranfer R ei = R ie R = R u + R T frictional force R u = nej /σ + j /σ electrical σ = 1.96σ σ = ne τ e /m e conductivitie thermal force R T =.71n kt e 3n b kt e ω ce τ e ion heating Q i = 3m e nk T e T i m i τ e electron heating Q e = Q i R u ion heat flux q i = κ i kt i κ i kt i + κ i b kt i ion thermal κ i = 3.9 nkt iτ i conductivitie m i electron heat flux q e = q e u + qe T frictional heat flux κ i = nkt i m i ωci τ κ i = 5nkT i i m i ω ci q e u =.71nkT eu + 3nkT e ω ce τ e b u thermal gradient q e T = κe kt e κ e kt e κ e b kt e heat flux electron thermal κ e = 3. nkt eτ e conductivitie m e κ e = 4.7 nkt e m e ωce τ κ e = 5nkT e e m e ω ce tre tenor either P xx = η W xx + W yy η 1 W xx W yy η 3 W xy pecie 38

8 P yy = η W xx + W yy + η 1 W xx W yy + η 3 W xy P xy = P yx = η 1 W xy + η 3 W xx W yy P xz = P zx = η W xz η 4 W yz P yz = P zy = η W yz + η 4 W xz P zz = η W zz here the z axi i defined parallel to B ion vicoity electron vicoity η i =.96nkT iτ i η i 1 = 3nkT i 1ω ci τ i η i 3 = nkt i ω ci η i 4 = nkt i ω ci η e =.73nkT eτ e η e 1 =.51 nkt e ω ce τ e η e 3 = nkt e ω ce η e 4 = nkt e ω ce. For both pecie the rate-of-train tenor i defined a η i = 6nkT i 5ωci τ i η e =. nkt e ωce τ e W jk = v j x k + v k x j 3 δ jk v. When B = the following implification occur: R u = nej/σ R T =.71n kt e q i = κ i kt i q e u =.71nkT eu q e T = κe kt e P jk = η W jk. For ω ce τ e 1 ω ci τ i, the electron obey the high-field expreion and the ion obey the zero-field expreion. Colliional tranport theory i applicable when 1 macrocopic time rate of change atify d/ 1/τ, where τ i the longet colliional time cale, and in the abence of a magnetic field macrocopic length cale L atify L l, where l = vτ i the mean free path. In a trong field, ω ce τ 1, condition i replaced by L l and L lr e L r e in a uniform field, where L i a macrocopic cale parallel to the field B and L i the maller of B/ B and the tranvere plama dimenion. In addition, the tandard tranport coefficient are valid only when 3 the Coulomb logarithm atifie λ 1 4 the electron gyroradiu atifie r e λ D, or 8πn e m e c B 5 relative drift u = v α v β between two pecie are mall compared with the thermal velocitie, i.e., u kt α /m α, kt β /m β and 6 anomalou tranport procee owing to microintabilitie are negligible. 39

9 Weakly Ionized Plama Colliion frequency for cattering of charged particle of pecie α by neutral i ν α = n σ α kt α /m α 1/, where n i the neutral denity and σ α\ i the cro ection, typically cm and weakly dependent on temperature. When the ytem i mall compared with a Debye length, L λ D, the charged particle diffuion coefficient are D α = kt α /m α ν α, In the oppoite limit, both pecie diffue at the ambipolar rate D A = µ id e µ e D i µ i µ e = T i + T e D i D e T i D e + T e D i, where µ α = e α /m α ν α i the mobility. The conductivity σ α atifie σ α = n α e α µ α. In the preence of a magnetic field B the calar µ and σ become tenor, J α = σ α E = σ α E + σ α E + σ α E b, where b = B/B and σ α = n αe α /m α ν α σ α = σα ν α /ν α + ω cα σ α = σα ν αω cα /ν α + ω cα. Here σ and σ are the Pederen and Hall conductivitie, repectively. 4