Plasma notes: Ohm s Law and Hall MHD by E. Alec Johnson, October, 2007.

Transcription

1 1 Plama not: Ohm Law and Hall MHD by E. Alc Johnon, Octobr, Boltzmann quation Th Boltzmann quation for pci i an volution quation for particl dnity in pha pac. Pha pac i a collction of coordinat which bijctivly pcify th tat of a particl. For point particl, uch a lctron and proton, point in pha pac ar rprntd by x, ṽ, th poition and propr vlocity of th particl. Ignor th blu txt and wid tild if you do not car about th pcial rlativitic rgim. If you do car about rlativitic ffct th blu txt i incomplt and nd to b fixd; rad Crcignani 2002 book on th Rlativitic Boltzmann Equation intad. So jut ignor th blu txt. For molcul with othr dgr of frdom, pha pac mut alo includ othr variabl which pcify not only tranlational mod, but uch tat quantiti a rotational and vibrational mod. Th rat of motion of a particl through pha pac i ẋ, ṽ : v, a. Th Boltzmann quation pcifi particl balanc in pha pac: t f + vf + v a f C. Hr i th pci indx, x i poition in pac, v i particl vlocity, ṽ γv i propr vlocity, f f p t, x, ṽ i th particl dnity of pci i.., f p t, x, ṽ dx dṽ i th numbr of particl in th infinitimal box dx dṽ, a q m E + v B + g d t v i th rat of chang of th propr vlocity of a particl whr q i charg pr particl, m i ma pr particl, E i lctric fild, B i magntic fild, g i gravitational fild, and C C p, known a th colliion oprator, i th rat of chang in particl dnity du purly to colliion within pci or with othr pci. Not that C i an oprator which dpnd on th ditribution function of ach pci rgardd a a function of vlocity but not of poition or tim: C [v f β t, x, v β Σ ], whr Σ i th t of all pci indic. 1.1 Not on th rlatitic Boltzmann quation W u dτ to dnot th lap of propr tim. By th invarianc of th Lorntz mtric undr inrtial tranformation, dτ 2 dt 2 dx/c 2. Dividing altrnativly by dτ 2 and dt 2 how that th Lorntz factor γ : dt dτ a a function of vlocity and a a function of propr vlocity i γv 1 v 2 c 1/2 γṽ 1 + v 2 c 1/ Diffrntation of th Lorntz factor To xpr drivativ of th Lorntz factor in trm of drivativ of th propr vlocity, w diffrntiat th rlation γ ṽ 2. W gt γ dγ dṽ ṽ, i.., d γṽ dṽ ṽ γṽ.

2 Drivation of th Boltzmann quation t f + x d t xf + v d t ṽ f C. t f + x d τ x f + v d τ ṽ f C. γ γ t f + vf + v a f C Convntion of intrprtation Not that by multiplying by m or q and, rpctivly, by making th rdfinition f : f m : m f p and C : C m : m C p, or f : f q : q f p and C : C q : q C p, w can alo rgard th Boltzmann quation a a tatmnt of conrvation of ma or charg. Hncforth, w drop th dfault pci indx xcpt a a rmindr, particularly whn making dfinition until w conidr multipl pci. So thr i an implicit pci indx on mot variabl xcpt for th indpndnt variabl t, x, ṽ and χṽ blow and th fild variabl E, B, g. 2 Spci balanc law Hncforth w viw th Boltzmann quation by dfault a ma-conrvation in pha pac. Taking momnt of th Boltzmann quation yild balanc law for dnity, momntum, and nrgy. Dfin v : v R. Dfin 3 : R v f v, M : Mf, th avrag vlocity u v, and th thrmal vlocity c : v u. Dfin th avrag propr vlocity ũ ṽ and th thrmal propr vlocity c : ṽ ũ. Lt χ b a momnt function powr of v.g., 1, v, v 2, or vv. To comput momnt w multiply th Boltzmann quation by χ and intgrat ovr ṽ: χ χ: v χ tf + v χ x vf + v χ v af v χc t v χf + x v vχf + v v aχf v fa vχ v χc t χ + x vχ + a v χ v χc Dfining : x, w hav th momnt quation: χ χ: t χ + vχ a v χ + v C χ, χ 1: t + u 0 + v C : S, χ ṽ: t ũ + vṽ a + v C ṽ : A, χ c γṽ ṽ 0 : t cγ + vcγ a v/c + v C cγ, χ ṽ 2 /2: t ṽ 2 /2 + vṽ 2 /2 a ṽ + v C ṽ 2 /2 : Q, χ ṽṽ: t ṽṽ + vṽṽ aṽ + ṽa + v C ṽṽ. Hr c i th pd of light, S i th rat of production of ma of pci du to colliion, and A ũs + R : c cc i th rat of production of momntum du to colliion, whr R i th ritiv forc on pci du to colliion. Q S ũ 2 /2 + R ũ + H : c c2 C /2 i th rat of nrgy production du to colliion with othr pci, whr R ũ i th rat of work du to th ritiv forc and H i th rat of production of hat in pci du to colliion.

3 3 2.1 χ 1: Balanc of particl, ma, and charg In th momnt quation w can rgard a th particl numbr dnity n : v f p, th ma dnity : v f m, or th charg dnity σ : v f q. Dfin th currnt J : σ u. Dfin δ t : a t a + u a, th tranport or bulk drivativ for pci. Thn w can writ particl, ma, and charg balanc a δ tn t n + nu S p : C p, v δ t t + u S : S m : C m ms p, δ tσ t σ + J S q : qs p. v 2.2 χ ṽ: Balanc of momntum i.. ma flux and currnt i.. charg flux Rcall that c : v u i th thrmal vlocity of pci. Obrv that c v u v u 0. Likwi, c : ṽ ũ i th propr thrmal vlocity of pci, and obrv that c ṽ ũ ṽ ũ 0. U vṽ u + cũ + c uũ + c c. U a q m E + u B + g. Dfin th prur tnor P : c c, i.. th flux of thrmal momntum acro a boundary convctd with th pci vlocity u, and dfin th lctrokintic prur tnor P q : σ c c, i.. th flux of thrmal charg flux acro a boundary convctd with th pci vlocity u. Th firt momnt quation bcom t ũ + uũ + c c nqe + u B P σe+j B q 2 + g + A, i.., t σũ + σuũ + σ c c n E + u B + σg + A q : q/ma. } m {{} J P q q m σe+j B 2.3 χ ṽ 2 /2: Claical balanc of nrgy U vṽ 2 u + cũ + c ũ + c uũ 2 + u c 2 + 2ũ c c + c c 2. U ṽ 2 ũ 2 + c 2. U a ṽ q m E + v B + g ṽ mṽ q E + ṽ g mũ q E + ũ g a ṽ. Th cond momnt quation bcom ũ2 t 2 E k + c2 2 E t } {{ } E + u ũ c2 + ũ c c }{{ 2 } P E + c c2 2 q nqũ E + nm ũ g + Q, J whr E k : ũ 2 /2 i th claical kintic nrgy, E t : c 2 /2 i th claical thrmal nrgy, E : E k + E t u c2 2 i th ga nrgy, q : c c 2 /2 i th hat flux, i.. th flux of hat nrgy through a boundary convctd with th pci vlocity u, and Q : v Cm ṽ2 /2 i th rat of nrgy production du to colliion with othr pci. That i, t E + ue + ũ P + q J E + ũ g + Q.

4 4 2.4 Convctiv drivativ Dfin d t : t + u, th convctiv drivativ for pci. Vrify th following Libnitz rul for tranport drivativ: δ t ab δ t ab + ad t b. So by ma balanc δ t S, w can writ for any quantity b: δ t b d t b + S b. Sinc δ t ũ d t ũ + S a ũ and A ũs + R, w can writ momntum balanc a: d tũ + P σ E + J B + g + R. 2.5 Claical kintic nrgy balanc To gt an quation for kintic nrgy balanc, w imply dot ũ with momntum balanc. Sinc ũ d t ũ d t ũ 2 /2 δ t ũ 2 /2 S ũ 2 /2, w gt δ t ũ 2 /2 + ũ P J E + ũ g + S a ũ 2 /2 + ũ R. 2.6 Claical thrmal nrgy balanc Rcall nrgy balanc: δ t ũ 2 /2 + c 2 /2 + ũ P + q J E + ũ g + S ũ 2 /2 + ũ R + H. To obtain thrmal nrgy balanc, w ubtract kintic nrgy balanc from nrgy balanc. W gt δ t c 2 /2 + ũ : P + q H. Obrv that all macrocopic forc hav diappard, a on would xpct. 2.7 χ c 2 γ: rlativitic balanc of nrgy Rcall rlativitic balanc of nrgy: t c 2 γ + c 2 vγ a v + v C c 2 γ. Rlativitic kintic nrgy i dfind to b th nrgy minu th rt nrgy. Rcall balanc of rt nrgy: t c 2 + c 2 u C c 2 c 2 S. v Subtracting th rt nrgy from th nrgy giv microcopic kintic nrgy balanc. Thi i not th balanc of macrocopic kintic nrgy.

5 5 2.8 Rlativitic kintic nrgy balanc To gt an quation for balanc of bulk macrocopic kintic nrgy, w bgin with it dfinition: Ẽ k : γũ 1. So δ t Ẽk δ t γũ 1 d t γũ 1 + S γũ 1 d t ũ u γũ + S γũ 1 d t ũ ũ/ γũ + S γũ 1. So w can gt an quation for kintic nrgy balanc by taking th dot product of momntum conrvation with not that in gnral ũ γũu, i.., ṽ γvv γ v v : u γu d t ũ + P σe + J B + g + R ũ/ γũ. d t γũ 1 + ũ/ γũ P σe + J B + g + R ũ/ γũ. δ t γũ 1 + ũ/ γũ P σe + J B + g + R ũ/ γũ + S γũ 1. 3 Spci contitutiv quation [From hr unl othrwi indicatd rult hold only for th non-rlativitic domain.] To clo th two-fluid ytm, it i ncary to upply contitutiv rlation for th prur tnor, hat flux, and drag forc in trm of th ma, momntum, and nrgy of ach pci. 3.1 Contitutiv rlation for prur Dfin th calar prur p to b on-third th trac of th prur tnor. Th tranlational thrmal nrgy i half th trac of th prur tnor, i.., tranlational thrmal nrgy 3/2p. Th implt contitutiv rlation for th prur tnor i to aum th thrmodynamic quilibrium condition that th thrmal nrgy i qually ditributd among all dgr of frdom of th ytm. Aum that thr ar α dgr of frdom. Thn thrmal nrgy α/2p. Th tranlational thrmal nrgy dnity along a particular axi i half th prur along that axi. So in quilibrium th prur in all dirction i qual, i.., th prur tnor qual th calar prur tim th idntity tnor: P pi. In ummary, E α/2p + 1/2u 2.

6 6 3.2 Contitutiv quation for drag forc W driv a contitutiv quation for th intrpci drag forc, which w will nd to upply a conitutiv quation in Ohm law for th ritanc du to colliion. Currnt dnity i th flux rat of charg dnity. Momntum i th flux rat of ma dnity. Colliion conrv momntum, but thy do not conrv currnt dnity. Thi impli that a contitutiv rlation for intr-pci drag forc will b ndd in th balanc law for nt currnt. W k a contitutiv rlation for R, th ritiv drag on pci du to colliion with othr pci that do not produc or dtroy pci. Auming that colliion do not dtroy pci if and only if thy do not dtroy pci p, Nwton third law giv R p R p. W aum that drag forc i jointly proportional to th dniti of th intracting pci and thir diffrnc in vlocity: R p : ν p n n p u p u, whr n i th numbr dnity of pci. So ν p ν p. 4 Entropy invarianc In th adiabatic ca P pi, q 0, H 0, S 0, α 2 δ tp + up 0, i.., α 2 d tp + α + 2 p u. 2 But ma conrvation ay that u d t ln, o d t ln p α + 2 α d t ln, i.., d t lnp γ 0, whr γ : α + 2 α. 5 Nt fluid balanc law [W rum noting rlativitic corrction.] Th quation of magntohydrodynamic MHD ar a t of balanc law for th nt ma, momntum, and nrgy ummd ovr all pci and for th magntic fild. A balanc law for currnt, calld Ohm law, provid a contitutiv rlation that allow u to liminat th lctric fild. 5.1 Balanc of ma Balanc of rt ma pr pci i t + u S, whr i th pci indx, i ma dnity, u i fluid vlocity, and S i th rat of production of pci du to colliion.

7 7 Thi i a Lorntz-invariant quation which art that th four-divrgnc of th calar γu tim th four-vctor ũ i th rat of production of propr ma, S : v C: ct γu cγu + Summing ovr all pci giv t + u 0, γu u S. γu whr : i th total ma dnity, u : u, th total ma flux, dfin th nt fluid vlocity u, and S : S, th nt rat of ma production, i aumd to b zro. Dfin δ t : a t a + ua x µ a γu γuu µ, th tranport or bulk drivativ. Thn th ma balanc law bcom δ t 0. [From hr on rult hold only for th non-rlativitic domain.] 5.2 Balanc of momntum Rcall balanc of momntum for ach pci: t ũ + u ũ + P σ E + J B + g + A, whr P i th prur tnor th flux of momntum acro a boundary convctd with th pci, σ i th charg dnity, J i th currnt dnity, E i th lctric fild, B i th magntic fild, g i th gravitational fild, and A i th ritiv drag du to colliion with othr pci rgardl of whthr th colliion do or do not crat or dtroy pci. Summing ovr all pci giv nt balanc of momntum: t ũ + ũũ + P σe + J B + g, whr : i th total ma dnity, ũ : ũ dfin th ovrall fluid vlocity, w : u u i th diffuion or drift vlocity of pci, w : ũ ũ i th propr diffuion or drift vlocity of pci, P d : a Pd : w w i th total diffuion prur tnor, P t : P i th total thrmal prur tnor, P : P t + P d i th total prur tnor i.., th total flux of momntum du to thrmal and pci diffuion acro a boundary convctd by global man vlocity, σ : a σ i th nt charg dnity, and J : J i th nt currnt. Not that A 0, inc momntum i conrvd in all colliion whthr particl ar convrtd to othr typ of pci or not Convctiv drivativ Obrv that th firt two trm of th momntum balanc quation ar th tranport drivativ of th momntum, δ t u. A bfor, dfin d t : t + u, th convctiv drivativ for th bulk fluid. Vrify/rcall th Libnitz rul for tranport drivativ: δ t ab δ t ab + ad t b. So by th conrvation of ma quation, δ t 0, w can pull out of a tranport drivativ, thrby turning it into a convctiv drivativ: for xampl, δ t u d t u, i.., t u + uu t u + u u.

8 Conrvation of momntum To rcat momntum balanc a a conrvation law, w u Maxwll quation to rcat th ourc trm a th tim drivativ of th momntum of th magntic fild plu th divrgnc of a tr tnor: σe + J B ɛ 0 EE + µ 1 0 B ɛ 0 t E B ɛ 0 EE + µ 1 0 B B ɛ 0 t E B ɛ 0 EE + µ 1 0 B B B 2 /2 [ ɛ 0 t E B E t B ] ɛ 0 t E B + µ 1 0 B B B 2 /2 ɛ 0 E E + ɛ 0 EE ɛ 0 t E B + µ 1 0 B B B 2 /2 +ɛ 0 E E E E + EE ɛ 0 t E B + µ 1 0 BB IB 2 /2 +ɛ 0 EE E 2 /2 t ɛ 0 E B + BB IB 2 /2 +ɛ 0 EE IE 2 /2 t S/c 2 + T EB, µ 1 0 whr S : µ 1 0 E B i th Poynting vctor, S/c2 i th momntum of th lctromagntic fild, and T EB : µ 1 0 BB IB 2 /2 +ɛ 0 EE IE 2 /2 i th lctromagntic tr tnor. Writing g χ, conrvation of momntum rad δ t u+ t ɛ 0 E B + P + χ T EB. Th on-fluid thory liminat th lctric fild by nglcting th diplacmnt currnt t E and condordr trm in E. Thi ffctivly t E 0 in th momntum quation, o you can dlt all th colord txt, and on-fluid conrvation of momntum rad δ t u + P + µ 1 0 IB 2 /2 BB + χ Balanc of currnt Ohm law Th vrion of Ohm law ud i what dtrmin th modl of MHD: idal, ritiv, Hall, or xtndd. Ohm law pcifi th volution of lctrical currnt in a plama in rpon to lctromagntic fild in trm of th quantiti of th on-fluid plama modl MHD. Rcall Faraday law: t B + E 0. Ohm law i a balanc law for th currnt. Ohm law i a contitutiv rlation that w u to liminat E from Faraday law. Th primary aumption ud to driv th gnralizd Ohm law ar quainutrality and that thr ar only two pci, ion and lctron. Each momntum balanc quation bcom a currnt balanc law if w multiply it by th ratio of charg to ma: t σ u + σ u u + P q q m σ E + J B + σ g + A q, whr for pci P q : q m P i th lctrokintic prur tnor, and A q : q m A i th production of currnt du to colliion. Summing ovr all pci giv nt currnt balanc. Th nt currnt i σ u J : J. Dfin alo J : J : σ w, th currnt dnity in th rfrnc fram of th fluid. Obrv that

9 9 J σ u u J σu. Thn th currnt flux um a: σ u u σ u + w u + w σ uu + σ w u + σ uw + σ w w σuu + J u + uj + Ju + uj σuu + σ w w σ w w. Dfin P qd, and P qd : Pqd : σ w w J J /σ q m P d, th diffuion lctrokintic prur tnor for pci σ w w, th total diffuion lctrokintic prur tnor. W alo dfin P q : Pq, th total thrmal lctrokintic prur tnor. So total currnt balanc rad t J + Ju + uj σuu + whr A q : Aq. J J /σ + P q q m σ E + J B + σg + A q, Ohm law i currnt balanc olvd for E. For thi to giv ytm clour, all two-fluid quantiti in Ohm law nd to b xprd in trm of on-fluid quantiti, and it i ncary to upply a contitutiv rlation for A q. W will do o for th ca of a quainutral two-pci plama with ngligibl diplacmnt currnt and linar colliional drag forc. 5.4 Balanc of nrgy Rcall balanc of nrgy for ach pci: t E + u E + u P + q J E + u g + Q. Summing ovr all pci giv total nrgy balanc. Rcall th dcompoition of th pci vlocity a u : u + w, th um of th nt fluid vlocity and th pci diffuion vlocity, whr w 0. Dfin E k : w/2, 2 th kintic nrgy of pci in th rfrnc fram of th nt fluid flow, and dfin E : E t + E k, th ga-dynamic nrgy of pci in th rfrnc fram of th nt fluid flow. W now how that th nonlinar trm giv ri to highr-ordr trm which ar naturally aborbd into th highr-ordr momnt, th hat flux. Th total ga-dynamic nrgy i E : E. For th nonlinar flux trm, dfin q : q. W comput th nonlinar flux trm: u E + u P + q ue + w E + u P + w P + q ue + ue + w E t + u 2 /2 + u P + w E t + w/2 2 +u w w + u P + E P d w P + q w P + q.

10 10 Rordring th trm to aborb th bad tuff into th prur tnor and hat flux, thi i ue + u P + w w + q + w P + w E t + w 2 /2. E }{{ } P }{{ d } q P So w hav formulatd nt nrgy balanc in th familiar form, t E + ue + u P + q J E + u g, whr E : E i th total ga-dynamic nrgy, P : P + P d : P + w w i th total thrmal and diffuion prur, and q : q + q d + W d : q + w E + w P i th total hat flux du to thrmal diffuion, pci diffuion, and work pr pci Conrvation of nrgy To rcat thi a a conrvation law, w u Ampr and Faraday law to rcat th oppoit of th ourc trm a th tim drivativ of omthing w will call th nrgy of th magntic fild plu th divrgnc of omthing w will call th lctromagntic nrgy flux. J E ɛ 0 t E µ 1 0 B E ɛ 0 t 1 2 E2 µ 1 0 E B ɛ 0 t 1 2 E2 µ 1 0 B E E B ɛ 0 t 1 2 E2 + µ 1 0 t 1 2 B2 + µ 1 0 E B t ɛ0 1 2 E2 + µ B2 + E B. µ } 0 {{} Call E f Call S Writing g χ, and obrving that u χ uχ uχ uχ + t χ uχ + t χ t χ, conrvation of nrgy rad t E + χ + t ɛ0 E 2 /2 + B2 2µ 0 + ue + χ + u P + q + 1 µ 0 E B In MHD w nglct ɛ 0 E 2 /2. W will alo nglct χ. 6 Nt fluid balanc law for two pci 6.1 Expring two-fluid quantiti in trm of on-fluid quantiti W aum that thr ar only two pci, ion i and lctron. Aum ingly chargd ion. W can xpr J i and J in trm of on-fluid quantiti uing th dfinition of th nt firt momnt, th charg and ma, J and u: { } J Ji + J,, i.., u i u i + u [ ] J 1 1 Ji u m i m J,

11 11 whr m i : m i and m : m. Solving thi linar ytm for J i and J giv [ ] Ji 1 m 1 J m/ mi J m i + m m i 1 u m/ m whr th rducd ma µ, dfind by µ 1 : m 1 i + m 1 1 J + 1 u m i + m m 1, i.., µ m im m i + m, i lightly mallr than th ma of an lctron, and whr m : m o m m µ m. m/ mi J + u/ m, m/ m J u/ m i Tranforming to th rfrnc fram of th fluid i.. xpring in trm of rlativ vlociti via u u + w and multiplying th momntum quation by, thi bcom { J J i + } [ ] J, J 1 1 J 0 m i J i m J, i.., i, 0 m i m J which ha olution J i 1 m i + m J [ m 1 m i 1 ] J 0 1 m i + m m m i J m/ mi J. m/ m W could hav obtaind th currnt in th rt fram from th currnt in th fluid fram or vic vra, uing th fact that J J + σ u, a follow. W can xpr σ i and σ in trm of on-fluid quantiti uing th dfinition of th nt zroth momnt of th charg and ma, σ and : { σ σi + σ, i + }, i.., σ [ ] 1 1 σi. m i m Solving thi linar ytm for σ i and σ giv [ ] σi 1 m 1 σ m/ mi σ m i + m m i 1 m/ m σ 1 σ + 1 m i + m m/ mi σ + / m. m/ m σ / m i In ummary, n i w i J i m m i J σu, n w J m m J σu, n i u i J i m m i J + u m, n u J m m J u m i, n i σ i m m i σ + m, n σ m m σ m i. Dividing th top two pair of quation by th bottom pair giv u formula for th pci vlociti: w i J i J σu σ i σ + m m J if σ 0, w J J σu σ σ m m ij i u i J i J + u m σ i σ + m u + w i, u J J + u m i σ σ + m u + w. i if σ 0, So w hav formula for th zroth and firt momnt of ach pci in trm of two-fluid variabl.

12 12 W now ubtitut to obtain xprion for th cofficint of th Lorntz forc law in trm of on-fluid variabl. q σ a σ i + σ 1 m m σ + m + 1 m m σ m m m i m i m i m i 1 m i 1 m σ + 2 n m i m µ if σ 0. Similarly by making th rplacmnt σ J and u, q J a m 1 m 1 m J + i u 1 1 m i m m i m J + 2 n µ u if σ 0. Th gnral xprion for th diffuion prur tnor in trm of 1-fluid quantiti don t m to implify to anything nicr, o w comput it hr only in th quainutral ca. Sinc J J σu, σ 0, ay that J J. So P qd σ w w J w m2 m 2 i JJ m i + m m2 m 2 i JJ m i + m m m i JJ, JJ n if m m i. Finally, w driv a contitutiv rlation for th ritanc, i.., th rat of production of currnt du to colliion. Aum that thr ar no colliion which annihilat ithr pci. So S 0, and A R. So A q Rq q p m ν p n n p u p u n i n ν i q i m i q m u i u. Auming ingly chargd ion, A q n in ν i u i u µ 1 ν i n J i + n i J. 6.1 µ So in th ca of quainutrality n i n n i + n /2 : n, A q nν i µ J. Th ngativ ign indicat that th ritiv drag act to dcra th magnitud on th currnt, a xpctd. 6.2 Ohm law: Currnt balanc xprd in trm of on-fluid quantiti So th aumption of quainutrality yild th gnralizd Ohm law, t J + Ju + uj + m m i JJ m i + m n whr rcall that P q : Pi m i + P m. Solving for E giv + P q 2 n 1 E + u B + 1 J B nν i µ m i m µ J. E B u + ν 2 J + µ 1 1 J B + µ [ n m m }{{ i } 2 P q + t J + Ju + uj + m m i JJ ]. 6.2 n m i + m n Hall trm

13 13 Auming ngligibl lctron ma m m i yild: E B u + ν 2 J + 1 n J B 1 n P + m [ 2 t J + Ju + uj JJ ] n n. 6.3 But in th ca of an lctron-poitron pair plama m m i 2µ, E B u + ν 2 J + 1 2n P i P + µ [ ] 2 t J + Ju + uj. n 6.3 Balanc of momntum for two pci Auming two pci allow u to writ out th total diffuion prur tnor in trm of bulk quantiti. For implicity w aum quainutrality. Thn w i w J n. Alo from quainutrality, So w i µ J m i n, i.., iw i µ J, and w µ J m n, i.., w µ J. P d w w µ Jw i w µ 2 n JJ. So momntum quation 5.1 i δ t u + P + µ 2 n JJ Gnrally JJ i nglctd a a cond-ordr trm. + µ 1 0 IB 2 /2 BB + χ 0. 7 Mapping btwn MHD and 2-fluid variabl Th mapping from 2-fluid to 1-fluid variabl i traightforward. For conrvd variabl: Alo, i + u u i + u E E i + E B B P P i + P, J µ 1 0 B.

14 14 Sinc th mapping from 2-fluid to 1-fluid variabl i not on-to-on, th mapping from 1-fluid to 2-fluid variabl rquir auxiliary information or aumption. Th four pcific thing that w mut pcify ar 1 th ratio of particl numbr dniti i.. quainutrality and 2 th ratio of tmpratur btwn th two pci, and contitutiv rlation for th 3 currnt typically by nglct of diplacmnt currnt and 4 lctric fild uing om form of Ohm law. Th MHD aumption of quainutrality ay that th ratio of particl numbr dniti i on. So th contribution of ach pci to th dnity i: n /m i + m, m i i, m i + m m, m i + m Th MHD magntotatic aumption giv th currnt: J µ 1 0 B. Combining thi with quainutrality giv th currnt and thu th vlocity of ach pci: m J i J, m i + m m i J J, m i + m w i J i /n m J/, w J /n m i J/, u i u + w i, u u + w, Auming calar prur, combining th idal ga law p n kt with quainutrality n n, and uing p p i + p giv th contribution of ach pci to th prur: T i p i p, T i + T T p p. T i + T For conrvation of nrgy w nd E E i + E. Th contitutiv rlation E i α i 2 p i + i u 2 i /2, E α 2 p + u 2 /2, whr α i th numbr of nrgy mod of pci α 3 for purly tranlational mod,.g. for lctron and proton, add to giv th nt contitutiv rlation E α MHD p + u 2 J 2 /2 + m i m 2 2, whr α MHD, th ffctiv numbr of dgr of frdom of th MHD ytm, i dtrmind by th rquirmnt that p MHD p i + p and α MHD p MHD α i p i + α p

15 15 Dividing th quation, w can dduc that th ffctiv numbr of dgr of frdom of th MHD ytm i th avrag of th numbr of dgr of frdom of ach componnt, wightd according to th ratio of prur or thrmal nrgy dniti, or tmpratur in th quainutral ca: α MHD α ip i + α p p i + p α it i + α T T i + T In ummary, th mor corrct MHD contitutiv rlation E α MHD p + u 2 J 2 /2 + m i m 2 2, α MHD α it i + α T T i + T fail to agr with th contitutiv rlation that w u for MHD, E α MHD p MHD + u 2 /2, 2 unl w incorporat th diffuion kintic nrgy into th prur: p MHD : p + 2 α MHD m i m J 2 2. W u th following procdur to obtain th nrgi and prur: 1. Comput th kintic nrgy of ach pci and add to gt th total pci kintic nrgy. If thi xcd th total MHD kintic nrgy, thn it will not b poibl to atify nrgy conrvation without modifying th diffuion vlociti w computd on th aumption of th pr-maxwll Ampr law and quainutrality. E k pci : 1/2[ i u 2 i + u 2 ]; E MHD α MHD /2p MHD + u 2 /2. 2. Subtract th total pci kintic nrgy from th total ga nrgy to gt th total pci thrmal nrgy. E t pci E MHD E k pci 3. Split th total pci thrmal nrgy uing th ratio of tmpratur and, mor gnrally, of th ga contant, if thy diffr of th two pci. Ei t α i T i E α i T i + α T pci; t p i 2/α i Ei t ; E t α T E α i T i + α T pci; t p 2/α E. t Finally, th lctric fild i upplid by Ohm law 6.2: E ηj + B u + m i m J B + 1 m P i m i P + m i m t J + uj + Ju + m m i JJ.