Solar partially ionizd plasma Fluid dscription of multi-componnt solar partially ionizd plasma E. Khomnko, 1, 2, a M. Collados, 1, 2 A. Díaz, 3 and N. Vitas 1, 2 1 Instituto d Astrofísica d Canarias, 3825 La Laguna, Tnrif, Spain 2 Dpartamnto d Astrofísica, Univrsidad d La Laguna, 3825, La Laguna, Tnrif, Spain 3 Dpartamnt d Física, Univrsitat d ls Ills Balars, E-7122 Palma d Mallorca, Spain Datd: 8 August 214 W driv slf-consistnt formalism for th dscription of multi-componnt partially ionizd solar plasma, by mans of th coupld quations for th chargd and nutral componnts for an arbitrary numbr of chmical spcis, and th radiation fild. All approximations and assumptions ar carfully considrd. Gnralizd Ohm s law is drivd for th singl-fluid and two-fluid formalism. Our approach is analytical with som ordrof-magnitud support calculations. Aftr gnral quations ar dvlopd w particulariz to som frquntly considrd cass as for th intraction of mattr and radiation. I. INTRODUCTION Th plasma in th lowr atmosphr of th Sun photosphr and chromosphr is only partially ionizd. Th prsnc of nutrals is xtrmly high in th photosphr, whr th stimations basd on th standard atmosphric modls provid about on ion vry tn thousand nutrals s,.g. Vrnazza, Avrtt, and Losr, 1981. Highr up in th chromosphr th ionization fraction incrass, but it rmains blow unity up to th transition rgion and corona. This fact alon dos not ncssarily imply that th magntohydrodynamic MHD approximation with som modifications can not b safly applid. Thr ar numrous xampls of th succss of MHD modls dscribing basic procsss in th photosphr and blow, such as convction, magnto-convction, formation of magntic structurs, flux mrgnc, wav propagation, tc. Asplund t al., 2; Khomnko and Collados, 26; Chung, Schüsslr, and Morno-Insrtis, 27; Morno-Insrtis, Galsgaard, and Ugart-Urra, 28; Nordlund, Stin, and Asplund, 29. Such dscription is possibl bcaus, dspit th prsnc of a significant amount of nutrals, th plasma is strongly isionally coupld in th photosphr and blow. Howvr, whn daling with strongly magntizd rgions, such as sunspots, on has to b carful sinc th cyclotron frquncis of lctrons and ions may bcom larg nough to ovrcom th isional frquncis lading to a brak of assumptions of idal MHD vn in th photosphr. Th situation is diffrnt in th chromosphr bcaus th isional coupling of th plasma wakns with hight. Th physics of this rgion is lss undrstood and th modls of chromosphric phnomna ar not so wll dvlopd Carlsson, 27; Lnaarts, 21. In th chromosphr, th plasma changs from prssur- to magntic fild-dominatd and, du to th wakning of isional coupling, th lctrons and ions bcom maga Elctronic mail: khomnko@iac.s ntizd starting from som hight, vn for rgions with rlativly wak magntic fild. All ths factors togthr man that th assumptions of MHD and th approach basd on th thrmodynamic quilibrium conditions bcom invalid in th chromosphr. It has long bn known that th chromosphric plasma can not b tratd in th local thrmodynamic quilibrium LTE approach for procsss rlatd to th transfr of radiativ nrgy and intraction of plasma and radiation. In rcnt yars it has also bcom accptd that, for th dscription of th plasma procsss thmslvs, th MHD approach, basd on th total isional coupling of th plasma, fails and altrnativ modls must b applid. On th xtrm dg of modlling, thr is th kintic dscription of th plasma. This, howvr, can not b of practical us in th photosphr and th chromosphr bcaus ths rgions ar far too dns. Th intrmdiat approach consists in using th fluid-lik dscription. Diffrnt modifications of such dscription includ singl-fluid quasi-mhd quations, or multi-fluid quations. In th multi-fluid approach, th quations for diffrnt plasma componnts - ions, nutrals and lctrons - ar solvd sparatly, coupld by mans of th isional trms. In th quasi-mhd approach, th quations for diffrnt componnts ar addd up togthr lading to apparanc of dissipativ trms in th induction and nrgy consrvation quations. In both cass, ths quations nd to b coupld to th quations dscribing th radiation fild. Thr is an incrasing numbr of works in th litratur whr multi-fluid or singl-fluid approachs hav bn applid for th dscription of various phnomna in th solar atmosphr. Dviations from classical MHD ar found to b important for: i propagation of diffrnt typs of high-frquncy wavs Kumar and Robrts, 23; Khodachnko t al., 24, 26; Fortza, Olivr, and Ballstr, 27; Pandy, Vranjs, and Krishan, 28; Vranjs t al., 28; Solr, Olivr, and Ballstr, 29, 21; Zaqarashvili, Khodachnko, and Ruckr, 211b; Zaqarashvili t al., 212, as th rlativ motion btwn th nutral and chargd spcis incrass th isional damping of ths wavs in th photosphr, chromo-
Solar partially ionizd plasma 2 sphr and prominnc plasmas, modifis thir xcitation rats and producs cut-off frquncis; ii rconnction procsss Zwibl, 1989; Brandnburg and Zwibl, 1994, 1995; Lak t al., 212 sinc th isional damping du to nutrals modifis th rconnction rats; rconnction studis in th two-fluid approach show that ions and nutrals can hav rathr diffrnt vlocitis and tmpraturs Sakai, Tsuchimoto, and Sokolov, 26; Smith and Sakai, 28; Sakai and Smith, 29; iii quilibrium balanc of magntic structurs by facilitating th cration of potntial forc-fr structurs in th chromosphr Arbr, Botha, and Brady, 29, and offring nw mchanisms for cration of intns photosphric flux tubs Khodachnko and Zaitsv, 22; iv magntic flux mrgnc Lak and Arbr, 26; Arbr, Hayns, and Lak, 27 by incrasing th amount of mrgd flux du to th prsnc of a diffusiv layr of partially ionizd plasma in th photosphr; v plasma instabilitis in prominncs Solr t al., 212; Díaz, Solr, and Ballstr, 212; Díaz, Khomnko, and Collados, 213; Khomnko t al., 214a,b by rmoving th critical wavlngth and making th plasma unstabl at all spatial scals; vi hating of chromosphric plasma D Pontiu and Harndl, 1998; Judg, 28; Krasnoslskikh t al., 21; Khomnko and Collados, 212; Martínz-Sykora, D Pontiu, and Hanstn, 212 sinc th Joul dissipation of lctric currnts is nhancd by ordrs of magnitud du to th prsnc of nutrals and wakning of th isional coupling. Bcaus this subjct is gaining so much attntion, it is important to rvis th drivation of th basic quations for th dscription of th multi-componnt solar plasma using a common fram of assumptions and simplifications. Th purpos of th prsnt articl is to provid such drivation. With no criticism to any of th abov mntiond studis, th quations ar oftn takn from diffrnt sourcs, lading to a diffrnt formulation of various trms. Mir and Shumlak 212 discussd rcntly th sam problm and providd a formalism for th dscription of multi-componnt solar plasma, including ionization and rcombination procsss, aiming at gnral plasma physics applications. In our approach, th macroscopic quations of motion for th plasma componnts ar slf-consistntly drivd starting from Boltzmann quation. W includ th intraction with radiation from th vry bginning, via xcitation/dxcitation and ionization/rcombination procsss and trat photons and yt anothr typ of particls intracting with th rst of th mixtur. Th gnralizd Ohm s law is drivd for th singl-fluid and two-fluid cass. Unlik prvious works, w considr not only hydrogn or hydrognhlium plasma, but gnraliz to th cas of multipl spcis in diffrnt ionization and xcitation stats. W thn mak ordr of magnitud stimats of th diffrnt trms in th gnralizd Ohm s law for th typical solar conditions. II. MACROSCOPIC EQUATIONS W considr a plasma composd of a mixtur of atoms of diffrnt atomic spcis, nglcting molculs. Ths atoms can b xcitd to xcitation lvls and/or to diffrnt ionization stags. Each particl is dtrmind by thr numbrs: its atomic spcis, its ionization stag I and its xcitation lvl E, dfining its micro-stat, {IE} 1. Th cas of lctrons and photons will b considrd apart. Th following rlation is applid: n = I n I = I,E n IE 1 whr n IE is th numbr dnsity of particls of lmnt with ionization stag I and xcitation lvl E; n I is th numbr dnsity of particls of lmnt and ionization stag I, and n is th numbr dnsity of particls of lmnt in all ionization and xcitation stags. Th numbr of lctrons is qual to th numbr of ions of all spcis and ionization stats, taking into account that th I s ionization stat givs I lctrons: n =,I I n I 2 Th gnral transport quation of a scalar quantity χ v is drivd from th Boltzmann quation, writtn for an nsmbl of particls of a givn micro-stat {IE}: f IE + v f IE + a v f IE = fie 3 whr f IE r, v, t is th on-particl distribution function, a is acclration and v is th particl vlocity. Th trm on th right hand sid of Eq. 3 dscribs th variation of th distribution function of our nsmbl of particls du to isions with othr particls, not blonging to {IE}. Th isional trm must satisfy th usual conditions of th consrvation of particls, thir momntum and nrgy, s.g. th txtbooks by Godblod and Podts 24 and Balscu 1988. Th gnral transport quation for a scalar quantity χ v can b obtaind by multiplying th Boltzmann quation by χ v and intgrating it ovr th whol vlocity spac Braginskii, 1965; Bittncourt, 1986; Balscu, 1988. W assum that th variabl χ v dos not xplicitly dpnd on tim nor spac variabls χ/ =, χ =. Th transport quation thn taks th form: n IE χ IE + n IE χ v IE n IE a v χ IE fie = χ d 3 v 4 V 1 Through th papr w will us I = for nutral atoms, I = 1 for singly-ionizd ions, tc., so that ionization stag I givs I lctrons. Similarly, w us E = to indicat th ground stat, and E > for subsqunt xcitation stats.
Solar partially ionizd plasma 3 Hr, th triangular brackts,, man avraging of a quantity ovr th distribution function in vlocity spac. Similar quation can b writtn for a vctor quantity χ v: n IE χ IE + n IE χ v IE n IE a v χ IE fie = χ d 3 v 5 V whr rprsnts th xtrnal or tnsorial vctor product. Ths two quations will b usd blow to driv th quations of consrvations of mass, momntum and nrgy for particls in th micro-stat {IE}. Particl dnsity, momntum and nrgy ar dfind as usual as th momnts of th distribution function and thir corrsponding consrvation quations ar drivd as th momnts of th Boltzmann quation, s blow. Th avrag vlocity u IE of ach spcis in a givn microstat {IE} is obtaind from th momntum, dividing by th corrsponding particl s mass. Th mass of particls is xactly th sam for all xcitation stats E, and diffrs btwn th ionization stats by th mass of corrsponding lctrons. This givs only a small diffrnc btwn th masss of particls in diffrnt ionization stags, but w will prsrv it hr for consistncy. W will assum that th macroscopic vlocitis of particls of a givn xcitation stat ar th sam as of thir corrsponding ground stat ithr ionizd or nutral sinc thr is no rason to xpct that particls in xcitd stat xprinc diffrnt forcs than particl in th ground stat. Howvr, macroscopic vlocitis of th nutral and ionizd stats can b diffrnt, as th lattr xprinc th action of th Lorntz forc and th formr do not: m IE = m I 6 u IE = u I u I if I I It is also usful to dcompos th vlocity of a particl as th sum of th macroscopic u and random vlocitis c: v = u IE + c IE = u I + c I 7 Photons can b considrd anothr typ of particls and, as such, a similar Boltzmann quation can b writtn also for thir distribution function f R Mihalas, 1986: f R + v f R + F p f R = fr 8 whr th function f R = f R r, p, t givs th numbr dnsity of photons at th location r, r + d r and with momntum p, p + d p. Equivalntly, on can writ f R = f R r, n, ν, t to rprsnt th numbr dnsity of photons at r, r + d r, with frquncy ν, ν + dν propagating with vlocity c in th dirction n. Sinc photons do not hav mass at rst, if no rlativistic ffcts ar prsnt, no forc is acting on thm, F =, and photon propagation will b along straight lins with v = c n. Sinc ach photon has nrgy hν bing h Planck constant, th amount of nrgy crossing a surfac ds in th dirction n in th intrval dt is de = chνf R ds cosθdωdνdt 9 whr θ is th angl btwn th dirction of n and th normal to th surfac ds, and Ω is th solid angl. This givs us th rlation btwn th photon distribution function and th spcific intnsity: I ν r, n, t = chνf R r, n, ν, t 1 Substituting this rlation into th Boltzmann quation Eq. 8 on obtains th usual form of th radiativ transfr quation: 1 chν [ ] Iν + c n I ν = fr 11 Th right-hand sid of th quation is rlatd to th gnration/rmoval of photons in th radiation fild du to xcitation/d-xcitation and ionization/rcombination procsss s Eqs. 17 and 18. This isional trm is usually xprssd as th diffrnc btwn photon sourcs j ν and losss which ar proportional to th radiation fild and ar writtn as k ν I ν. Th lattr notation is th standard on in th litratur on radiativ transfr.g., Mihalas, 1986, and usually rads as fr = j ν k ν I ν hν 12 Aftr nglcting th tmporal drivativ of th radiation fild in Eq. 11 on gts th usual transfr quation: A. Mass consrvation d I ν d s = j ν k ν I ν 13 Th quation of mass consrvation for particls in a micro-stat {IE} is drivd from Eq. 4 by stting χ = m IE = m I. This way χ IE = m I, χ v IE = m I v I = m I u I and v χ = and on obtains: ρ IE + ρ IE u I = m I V fie d 3 v = S IE 14 whr ρ IE = m I n IE and Eqs. 6 and 7 ar takn into account for vlocitis. Th ision trm on th right hand sid S IE accounts for isions with particls of anothr kind that lad to cration or dstruction of particls of th microstat {IE}, including th intraction with photons i.. radiation fild as anothr typ of particls. If particl idntity at th micro-stat {IE} is maintaind during th ision, such ision is calld lastic. Th trm S IE is zro for lastic isions. Th isions that lad to cration/dstruction of particls ar calld inlastic. Inlastic procsss most rlvant for th solar
Solar partially ionizd plasma 4 atmosphr ar ionization, rcombination, xcitation and d-xcitation. Scattring can b tratd as a particular cas of photon absorption and r-mission by any of th abov procsss. Thus, in a gnral cas, th isional trm can b writtn in th following form: S IE = m I P I IE n IE P IEI 15 I,E n I whr th summation gos ovr all ionization stats I and thir corrsponding xcitation stats E I lading to cration/dstruction of particls of micro-stat {IE} of th lmnt. Th first trm dscribs transitions from othr xcitation/ionization stats of lmnt to th stat IE, and th scond trm dscribs th opposit procsss. Th rat cofficints P IEI ar th sum of isional C IEI and radiativ F IEI rat cofficints: P IEI = F IEI + C IEI 16 Th radiativ rats F IEI hav to b spcifid sparatly for th upward initial microstat with a lowr nrgy than th final on and for th downward procsss. Th formr ons ar photoxcitation E < E and photoionization I < I, whil th lattr ons ar photodxcitation E > E and photorcombination I > I. Th unifid radiativ rats F IEI can b found lswhr in standard radiativ transfr tutorials,.g. Ruttn 23; Carlsson 1986. Thy dpnd on th radiation fild and on th atomic paramtrs of th transitions. Th unifid radiativ rats F IEI for th upward procsss xcitation, ionization, IE < I can b writtn as: σiei F IEI = hν I νdωdν 17 Th cofficint σ IEI is th xtinction cofficint of intraction of an atom with a photon to giv ris to th transition from stat IE to I. Th intgral ovr angl and frquncy coms th fact that photons coming from diffrnt dirctions or frquncis may b absorbd to produc a giv ris to an xcitation or ionization procss. For th downward radiativ procsss dxcitation, rcombination, IE > I th unifid rats F IEI ar: σiei 2hν 3 F IEI = hν G IEI c 2 + I ν dωdν 18 It is important to not that ths xprssions rprsnt th important link btwn th radiation fild and th atomic xcitation/dxcitation/ionization/rcombination procsss. Through ths radiativ procsss, th photon absorption/mission has to b takn into account in addition to th corrsponding chang in th population of th {IE} microstat. Th xtinction cofficints σ IEI and th cofficint G I IE hav diffrnt forms for bound-bound and boundfr procsss and can b found in standard tutorials on radiativ transfr. Th rats for th inlastic isions btwn a particl in microstat {IE} and anothr particl idr β that turn {IE} into {I } may b writtn in a gnral form as: C IEI,β = n β v σ IEI v β fv β v β dv β, 19 whr fv β is th vlocity distribution of th idrs and σ IEI,β is th isional cross-sction of spcis and β. th total isional rat cofficint, σ IEI, is obtaind aftr adding up th contribution off all idrs, i.., C IEI = β C IEI,β 2 Th most common idrs in th solar atmosphr ar th lctrons. Th isional rats for th boundbound and bound-fr procsss including fr lctrons ar usually spcifid by th approximation of van Rgmortr 1962 s Ruttn, 23, drivd assuming local thrmodynamical quilibrium. If w sum Eq. 14 ovr all possibl microstats IE of th atom, th right hand sid bcoms idntical to zro as no nuclar ractions ar considrd and no transformation btwn th atoms is allowd S IE IE = S =. Howvr, th right hand sid of Eq. 14 writtn for a spcific microstat S IE may also bcom zro undr crtain approximations. It is crtainly idntical to zro if th plasma is in thrmodynamical quilibrium TE sinc in that cas all procsss ar in dtaild balanc, so n I P I IE = n IE P IEI. Th sam applis for th approximation of local thrmodynamical quilibrium LTE whr TE is assumd to b valid locally. Furthrmor, it is still idntical to zro whn th LTE approximation is rlaxd to th approximation of th statistical quilibrium SE that is commonly solvd within th instantanous non-lte radiativ transfr problm. All ths approximations rquir that th macroscopic changs of th atmosphr happn on tmporal and spatial scals such that th atmosphr may b considrd stationary ρ IE / = and static ρ IE u I =. Th condition of SE may formally b writtn as S IE = m I P I IE n IE P IEI = 21 I,E n I In a gnral cas, th nt sourc trms S IE cannot b nglctd in th solar chromosphr whn th SE fails to dscrib th plasma conditions Carlsson and Stin, 22; Lnaarts t al., 27 du to th imbalanc btwn th ionization and rcombination rats. Eq. 14 can b summd ovr th xcitation stats E to giv: ρ I + ρ I u I = S I 22 whr w dfin S I = m I P I IE n IE P IEI 23 E I I,E n I
Solar partially ionizd plasma 5 In this summation th contributions from th transitions btwn diffrnt xcitation stats EI of th sam ionization stat I cancl out. Th rmaining contributions includ transitions that chang th ionization stat whatvr th xcitation stat and thrfor gnrat or rmov a fr lctron. In th cas of lctrons, by substituting χ = m into quation Eq. 4 w obtain: ρ + ρ f u = m d 3 v = S 24 V Th isional trm for lctrons taks into account all th procsss lading to th apparanc/disapparanc of lctrons ionization and rcombination for all atomic spcis of th systm. S = m I In IEP IEI n I P I IE I,E I >I,E 25 This trm only bcoms strictly null in th cas of TE/LTE/SE. Sinc photons do not hav mass, thr is no quivalnt mass consrvation quation for thm. B. Momntum consrvation By stting χ = m I v in Eq. 5, th quation of momntum consrvation for particls of a micro-stat {IE} is drivd. According to Eq. 7, w hav v = u IE + c IE = u I + c I and m IE = m I. Using th continuity quation for th micro-stat {IE}, Eq. 14, and dfining th kintic prssur tnsor, ˆp IE, according to Eq. A1 s Appndix A, th quation of momntum consrvation is obtaind: [ ] ui ρ IE + u I ui + ˆp IE n IE F IE = 26 fie m I v d 3 fie v m I u I d 3 v V V Introducing th total drivativ and assuming that th avrag forc F has an lctromagntic and gravitational natur, th quation of motion is writtn as: whr ρ IE D u I Dt = n IE q I E + u I B + ρ IE g ˆp IE + R IE u I S IE. 27 R IE = m I V v fie d 3 v 28 Th trm R IE provids th momntum xchang du to isions of particls in th micro-stat {IE} with othr particls, including photons. Th xprssions for lastic isions btwn nutrals, ions and lctrons xcluding photons ar givn in Appndix B. Th abov quation can b summd up for all xcitation stats E. In this summation it is important to tak into account that w assumd that macroscopic vlocitis and masss of all xcitation stats ar th sam. This maks th summation straightforward. Th following momntum quation is thn obtaind: ρ I D u I Dt = n I q I E + u I B + ρ I g ˆp I + R I u I S I 29 whr E R IE = R I Appndix B, and othr dfinitions ar givn in Appndix A. For lctrons, an quation similar to Eq. 27 can b obtaind introducing χ = m v: ρ D u Dt = n E+ u B+ρ g ˆp + R u S 3 whr w hav usd q =. To gt th momntum consrvation for photons, w can multiply Eq. 11 by n, th propagation dirction of photons, and intgrat ovr all solid angls and all frquncis. With this, on gts th quation 1 F R c 2 + ˆP R = 1 c j ν k ν I ν ndωdν 31 whr th radiativ nrgy flux, F R, is dfind as F R r, t = ni ν r, n, ν, tdωdν 32 and th radiation prssur, ˆP R, as ˆP R r, t = 1 n ni ν r, n, ν, tdωdν 33 c Nglcting th tim variations of th radiation fild, on gts th momntun consrvation quation for photons, ˆP R = 1 j ν k ν I ν ndωdν 34 c Th rlation btwn th isional momntum trm R IE and th radiation fild dpnds on particular transition, and it is of no us hr to introduc th complxity to dscrib all possibl transitions, that can b found lswhr. For th purpos of illustration, w provid a simpl xampl of a radiativ xcitation procss, whrby an atom passs from micro-stat {IE } to microstat {IE} aftr th absorption of a photon. With this procss, th momntum of microstat {IE} is incrasd with th initial momntum of th particls in micro-stat {IE } plus th momntum of th absorbd photon. For this particular xampl, th contribution R rad IE to th trm R IE by this procss would b R rad IE = n IE m I u I F IE IE + 1 c σ IE IE I ν dωdν 35
Solar partially ionizd plasma 6 Th intgral in solid angl taks into account that photons coming from diffrnt dirctions can b absorbd, whil th intgral in frquncy includs all possibl photons that may lad to th transition through th crosssction factor σ IE IE. Similar xprssions can b writtn for th othr radiativ procsss lading to changs of th momntum of ach microstat {IE}. Th total valu for R IE is obtaind aftr adding up th contributions of all procsss that populat/dpopulat microstat {IE}. C. Enrgy consrvation In a gnral cas, th gas intrnal nrgy of a givn micro-stat maks up of kintic nrgy of particl motion, potntial nrgy of thir intraction xcitation stats and ionization nrgy Mihalas and Mihalas, 1986. For an idal gas, th potntial and ionization nrgis of th particls ar nglctd and th intrnal nrgy of a gas is only du to random thrmal motions: IE = ρ IE c 2 I /2 36 Taking into account th dfinition of th scalar prssur, p IE = 1 3 ρ IE c 2 I, th intrnal nrgy can b xprssd as: IE = 3 2 p IE 37 In a mor gnral situation, th intrnal nrgy will mak up of kintic nrgy and potntial nrgy of th xcitation-ionization lvl IE, E IE, with rspct to th ground lvl of th nutral stat, E : IE = 3 2 p IE + n IE E IE 38 To driv th consrvation law for th quantity givn by Eq. 38, w hr tak: χ = χ 1 + χ 2 = m I v 2 /2 + E IE 39 Th potntial nrgy associatd to a givn microstat is a scalar constant. W start by stting χ = χ 1 = m I v 2 /2 in Eq. 4. Following th standard stps, th nrgy quation is writtn: 3 2 p IE + 1 2 ρ IEu 2 I + 1 2 ρ IE v 2 v IE n IE F v IE = 1 2 m I v 2 fie d 3 v 4 V whr F is a gnral forc. W rwrit th scond trm on th lft hand sid using th dfinition of th man and random vlocitis Eq. 7, prssur tnsor ˆp IE, Eq. A1 and hat flow vctor q IE, Eq. A2. Undr th additional assumption that any xtrnal forc is indpndnt of vlocity or dos not hav a componnt paralll to th vlocity which includs both, lctromagntic and gravitational, forcs, combining th abov quation with thos of mass Eq. 14 and momntum consrvation Eq. 27 givs: whr D 3p IE + 3 Dt 2 2 p IE u I + ˆp IE ui + q IE = M IE = 1 2 m I M IE u I RIE + 1 2 u2 IS IE 41 V v 2 fie d 3 v 42 This trm is zro in Braginskii 1965. Howvr, in our point of viw, it should b rtaind, givn that th trm M IE includs kintic nrgy losss/gains of th particls of micro-stat {IE} du to isions with othr particls, which in gnral will not vanish. Now w considr th scond contribution to th intrnal nrgy and st χ = χ 2 = E IE in quation Eq. 4. W obtain for a givn micro-stat: Dn IE E IE Dt + n IE E IE ui = E IE S IE /m I 43 Th trm E IE S IE taks into account th chang of potntial nrgy of a micro-stat du to inlastic isions with othr particls, including photons, i.. potntial nrgy chang during radiativ and isional ionization, rcombination, xcitation and dxcitation. Adding up th nrgy quations Eqs. 41 and 43 and using th dfinition of th intrnal nrgy of a micro-stat {IE} from Eq. 38 w rwrit th consrvation quation for intrnal nrgy lik follows: D IE Dt + IE ui + ˆp IE ui + q IE = Q IE 44 whr w dfind th intrnal nrgy losss/gains trm as: Q IE = M IE u IRIE 1 + 2 u2 I + E IE /m I S IE 45 In this dfinition, th Q IE trm includs nrgy losss/gains du to: lastic isions of particls of th micro-stat {IE} with othr particls M IE trm, and part of th u IRIE trm, Eq. B4, and inlastic isions with photons and othr particls u 2 I/2 + E IE /m I S IE, and th corrsponding part of th u IRIE trm, s Eq. 35 for an xampl. Adding up Eq. 44 for all xcitation stats on obtains: with and D I Dt + I ui + ˆp I ui + q I = Q I 46 Q I = M I u I RI + 1 2 u2 IS I + Φ I 47 I = 3p I /2 + χ I 48
Solar partially ionizd plasma 7 whr χ I and Φ I ar dfind according to Eq. A3 and Eq. A4, s Appndix A. For lctrons, th abov quation simplifis vn furthr sinc thr ar no ionization-xcitation stats, nithr potntial nrgy corrsponding to such stats. D Dt + u + ˆp u + q = M u R + 1 2 u2 S 49 with = 3p /2. Th nrgy quation for photons is obtaind aftr intgrating Eq. 11 for all solid angls and for all frquncis E R + F R = j ν k ν I ν dωdν 5 whr th radiativ nrgy is dfind as E R r, t = 1 I ν r, n, ν, tdωdν 51 c Th standard quation is obtaind aftr nglcting th tmporal variation of th radiation fild, to lad to th xprssion F R = j ν k ν I ν dωdν 52 Again, w can rinforc th concpt of th coupling btwn th plasma particls and th radiation fild. To that aim, w can us th sam xampl as in Sct. II B, xcitation procsss whrby particls in microstat IE chang to microstat IE aftr absorbing a photon. In this cas, w hav and M IE = 1 2 n IE m Iu 2 IF IE IE 53 E IE E IE S IE /m I = D. Assumptions of macroscopic quations σ IE IE I ν dωdν 54 Th systm of quations Eqs. 14, 27, 44, and 13 is so far rathr gnral, but not closd, sinc th quations for th third momntum of th distribution functions ar not prsnt. Th ffct of thrmal motions, of random phas cyclotron motions ar avragd in ths quations and ar prsnt in th form of kintic prssur tnsor also calld strss tnsor, ˆp IE, hat flow vctor q IE, and lastic and inlastic isional trms S IE, R IE, and M IE. To clos th systm, on has to provid th xprssions for ths quantitis. Th prssur tnsor can b approximatd by a scalar prssur only th lmnts of th diagonal ar non-zro FIG. 1. Plasma scals for th solar photosphr and chromosphr. Uppr panl: isions btwn nutral hydrogn with lctrons rd dashd, Eq. 6 and with ions rd dottd, Eq. 59; isions btwn hydrogn ions with lctrons blu dashd, Eq. 61 and with thmslvs blu dottd, Eq. 62; cyclotron frquncis of gyration black dottd and dashd, for ions and lctrons, Eq. 57 for th fild B = 1 xp z/6 G; and Langmuir frquncy black solid, Eq. 55. Lowr panl: Dby radius black solid, Eq. 56; Larmor radius of gyration for lctrons and ions black dottd and dashd, Eq. 58. in ssntially two cass: 1 whn th plasma is isotropic and ision-dominatd, implying small man fr paths, and 2 for strongly magntizd plasmas whn th cyclotron motion dominats ovr th isions and th plasma bcoms anisotropic in th dirction prpndicular to th magntic fild, implying small gyration radius, much smallr that any gradint in any variabl s,.g. Spitzr, 1956. In th first cas all thr lmnts on th diagonal ar th sam. In th scond cas, thr ar two indpndnt componnts of prssur, paralll and prpndicular to th magntic fild. Th non-diagonal componnts of th prssur tnsor ar th origin of viscosity. Radiativ procsss ar considrd to happn instantanously, whil othr procsss hav charactristic tmporal or spatial scals. Typical valus usd to dtrmin th scal of th diffrnt physical mchanisms ar th
Solar partially ionizd plasma 8 following: Frquncy of Langmuir oscillations arising whn th local charg nutrality of plasma is violatd: ω p = 2 n /m ɛ 1/2 55 Spatial scal of Langmuir oscillations is th Dby radius r p = k B T ɛ / 2 n 1/2 56 Frquncy of cyclotron rotation of particls of ach spcis ω ci = q I B/m I, I 57 Spatial scal of cyclotron rotation is Larmor radius r ci = 2k B T I m I 1/2 / q I B. 58 Ths frquncis and scals for typical paramtrs of th solar atmosphr ar givn in Fig. 1, calculatd aftr th VAL-C atmosphric modl Vrnazza, Avrtt, and Losr, 1981 and assuming a magntic fild varying with hight as B = 1 xp z/6 G, approximating quit solar rgions. Th largst spatial scal is th Larmor radius of ions raching about 1 mtr in th uppr chromosphr. Th lowst frquncy is th ion cyclotron on raching 1 4 Hz in th chromosphr corrsponding to a priod of 2π 1 4 sc. Th plasma frquncis and spatial scals dfind abov should b compard with th frquncy of isions btwn th chargd particls and isions btwn th nutral and chargd particls. Th frquncy of isions btwn nutral atoms of spci β with ions of spci, and lctrons ar givn in,.g., Spitzr 1956, although som nwr dvlopmnts can b found in Vranjs and Krstic 213: 8k B T ν in β = n βn Σ in 59 πm in β ν nβ 8k B T = n βn Σ n 6 πm in β whr m in β = m i m nβ /m i + m nβ and Σ in = 5 1 19 m 2, Σ n = 1 19 m 2 ar th ion-nutral and th lctron-nutral cross sctions, rspctivly. Th isional frquncis of lctrons with ions of spci, and of ions of diffrnt spcis ar providd by Braginskii 1965, his Appndix A1 s also Lifschitz 1989; Rozhansky and Tsdin 21 and Bittncourt 1986, chaptr 22: ν i = 4 n i Λ 3ɛ 2 m2 3/2 m = 3.7 1 6 n iλ 61 2πk B T T 3/2 miβ ν iβ i = 4 n i ΛZi 4 3/2 3 = 6 1 8 n iλzi 4 2ɛ 2 m2 i β 2πk B T T 3/2 62 whr Z i is th charg of th ion Z i = 1, 2,..., and th mass of th ion is assumd to b approximatly qual to th proton mass in valuating th cofficint of th last quality. Th Coulomb logarithm Λ for T < 5 V i.. solar tmpraturs approximatly blow 6. K is Λ = 23.4 1.15 log 1 n + 3.45 log 1 T 63 with n givn in cm 3 and T in K. Ths frquncis for th solar atmosphr ar plottd in rd in Fig. 1. Th thrmal vlocitis of ions ar takn to b of th ordr of 1 1 km s 1 and thos of lctrons of th ordr of 1 3 km s 1. Th quantitis ˆp IE, q IE, S IE, RIE, and M IE can b xprssd in trms of ρ IE, p IE, u I and th magntic fild dirction b = B/B if th plasma tmporal scal, τ, and th spatial scals, L and L paralll and prpndicular to th magntic fild, satisfy th following conditions Lifschitz, 1989: τ ωc 1 64 τ ν 1, ν 1 ii L ν 1 c L ν 1 u L ωc 1 c L ωc 1 u Th tmporal scals should b much largr than th cyclotron priod and th tim btwn isions. Th spatial scals paralll to th magntic fild should b much largr than thos dfind by th ision frquncis and th thrmal and plasma vlocitis i.. thrmal fr path scal. In th cas of scals prpndicular to th magntic fild, thos should b largr than th dimnsions dfind by th cyclotron frquncy and th vlocitis i.. radius of cyclotron gyration around th magntic fild lins. Fig. 1 shows that all ths conditions ar satisfid in th solar atmosphr, for th typical obsrvd spatial and tmporal scals. It is important to not that th motion dtrmind from macroscopic quations dos not agr with th microscopic particl drifts, dscribd by th quation of motion of individual particls. Only th drift du to lctric fild rmains in th macroscopic quations. Th drifts du to th gradint of th magntic fild is not prsnt in th macroscopic dscription. Th lattr drift is rsponsibl for,.g., th motion of chargd particls in th Earth s magntosphr magntic mirrors, and is usually dscribd in th microscopic approach. This apparnt paradox in th dscription of plasma drifts and its origin is discussd in Spitzr 1956 and has to do with th diffrnt way of avraging in th macroscopic and microscopic quations.
Solar partially ionizd plasma 9 III. TWO-FLUID DESCRIPTION A. Multi-spcis continuity quations Th solar plasma is composd by particls of diffrnt atomic lmnts in diffrnt ionization stags. On can considr that, onc th ision coupling wakns, all ths spcis bhav in a diffrnt way, moving with diffrnt vlocitis. Howvr, as a first approximation, it is rasonabl to assum that th diffrnc in bhavior btwn nutrals and ions is largr than btwn th nutrals/ions of diffrnt kind thmslvs, sinc th lattr fl th prsnc of th magntic fild and th formr do not. This assumption allows to dcras th numbr of quations for diffrnt spcis Eq. 14, 27 and 44 to just two, for an avrag nutral particl and an avrag chargd particl photons considrd apart. Th two-fluid dscription has th limitation that it rquirs a strongr coupling btwn chargd particls than btwn chargd and nutral particls s, Zaqarashvili, Khodachnko, and Ruckr, 211a,b and thus it is only valid for z>1 km according to Fig. 1. Also, in som circumstancs, if th isional coupling btwn th nutral spcis is wakr than btwn ionizd and nutral ons, th nutral spcis can dcoupl as wll on from anothr, s,.g. Zaqarashvili, Khodachnko, and Ruckr 211a,b. Th approach dscribd in this sction can b asily xtndd to dscrib this cas. Blow w driv th two-fluid quations for a plasma composd by an arbitrary numbr of nutral and ion spcis not only hydrogn. Th volution of th chargd spcis will b dscribd by quations that combin jointly lctrons and all ions. Th volution of nutral spcis will b dscribd by th combind quations for all nutral componnts. Plasma quasi-nutrality is assumd and th tmpratur of lctrons and ions is assumd th sam though th nutral componnt can hav a diffrnt tmpratur. W assum that thr ar N spcis prsnt in th plasma, i.. w hav N diffrnt ions and N diffrnt nutrals 2N + 1 spcis in total, accounting for lctrons as wll. W will only considr singly-ionizd ions sinc th abundanc of multiply-ionizd ions is not larg in th rgions of intrst of th solar atmosphr i.., rgions whr th prsnc of nutrals is important, photosphr and chromosphr. Th conditions of th solar photosphr ar such that th most abundant nutral atoms ar hydrogn, and th donors of lctrons ar ionizd iron atoms. Th summation don in this sction allows to obtain quation for an avrag nutral/ion atoms taking th lattr fatur into account. In th nxt subsctions, w us indics n and i, instad of I = or I = 1, to rfr to nutral or ions, following th standard notation of prvious works. Whr appropriat, w also us th indx c to account for th combind ffct of all chargs, i., ions and lctrons. Using Eq. 22 and th dfinitions from Appndix C on trivially obtains: In a similar way, for chargs ρ n + ρ n u n = S n 65 ρ c + ρ c u c = S i + S = S c = S n 66 whr ρ c = ρ i+ρ, s Appndix C. Th dfinitions of th inlastic isional trms S n, S i and S ar givn in Appndix B. Th last quality coms from th fact that isions can not chang th total dnsity. B. Photon momntum and nrgy quations for procsss involving nutrals and chargd spcis Bfor procding to th drivation of th momntum and nrgy quations for nutrals and chargs, it is convnint to split th photon momntum and nrgy quations, Eqs. 34 and 52, taking into account th natur of th intraction of particls and photons. In that quation, w can dfin th missivity j ν and absorption k ν cofficints as th sum of sparat contributions from nutrals suprindx n and chargs suprindx c: j ν = j n ν + j c ν 67 k ν = k n ν + k c ν 68 Exampls of this sparation ar: xcitationdxcitation procsss of nutral atoms will b includd in kν n jν n, xcitation/dxcitation of ionizd spcis contribut to kν/j c ν; c th ionization of a nutral spcis is takn into account in kν n a photon disappars from th radiation fild du to th absorption by th nutral, th rcombination of an ionizd atom with an lctron to giv a nutral atom is includd in jν c a photon appars as a consqunc of th ion-lctron intraction, tc. With this, on can writ ˆP R = ˆP n R + ˆP c R 69 whr ˆP n R and ˆP c R ar dfind as ˆP n R = 1 jν n kν n I ν ndωdν 7 c and ˆP c R = 1 c Analogously, on can dfin jν c kνi c ν ndωdν 71 F R = F n R + F c R 72
Solar partially ionizd plasma 1 whr and F n R = F c R = jν n kν n I ν dωdν 73 jν c kνi c ν dωdν 74 and intnsitis I ν ar obtaind from complt radiativ transfr quation, Eq. 13 with total cofficints j ν and k ν. C. Multi-spcis momntum quation for nutrals First w start with nutrals, adding Eq. 29 for nutral spcis: D u n ρ n = ρ n g ˆp n Dt + R n u n S n 75 Using th continuity quation, w xpand th drivativ part as usual: ρ n D u n Dt + u n S n = ρ n u n Th prssur tnsor trm givs: ˆp n = ˆp n + + ρ n u n u n ρ n w n w n ρ n w n w n 76 whr th drift vlocity w n is dfind according to Eq. C7. Th trms containing w n cancl out aftr th summation of th abov two xprssions. Th isional trm taks into account th lastic isions btwn all nutral spcis with ions and lctrons dnotd by R n, as wll as inlastic isions through th momntum carrid by photons givn by th intgral blow. Th isions btwn diffrnt pairs of nutrals go away aftr summation s Appndix, Eq. B8: R n = R n + 1 kν n I ν jν n ndωdν 77 c In a photon absorption/mission procss, th nutral atoms gain/los th momntum givn by th scond trm on th right-hand sid of th quation. Finally, taking into account Eq. 7, th momntum quation for th nutral componnt bcoms: ρ n u n + ρ n u n u n = ρ n g ˆp n ˆP n R + R n 78 or, altrnativly, using th continuity quation this can b rwrittn as: D. Multi-spcis momntum quation for chargs Th chargd componnt momntum quations ar Eq. 29 and 3: ρ D u Dt = n E + u B + ρ g ˆp + R u S ρ i D u i Dt = q i n i E + u i B + ˆp i + R i u i S i ρ i g Th sum of th drivativ trms givs, sam as bfor ρ i D u i Dt D u + ρ Dt + u i S i + u S = ρ c u c + ρ c u c u c + ρ i w i w i + ρ w w Th Lorntz forc trm q i n i E+ u i B n E+ u B = [ J B] 8 according to th dfinition of th currnt dnsity, J givn by Eq. C6. Th prssur trm ˆp i + ˆp = ˆp i ρ i w i w i ρ w w 81 according to Eq. C9. Th ision trm writs as R i + R = R i + kνi c ν jνdωdν c = R n + kνi c ν jνdωdν c 82 whr th lastic isions btwn ions and lctrons with nutrals ar takn into account trm R and th isions btwn diffrnt pairs of ions or ions-lctrons go away s Appndix, Eq. B9 and B6. Th intgral trm givs th momntum xchang du to inslatic isions. Th last quality is drivd from th consrvation of th total momntum by lastic isions. With all this, th ion-lctron momntum quation bcoms: ρ c u c + ρ c u c u c = [ J B]+ρ c g ˆp i ˆp c R R n 83 Using th continuity quation, Eq. 66, it can b rwrittn as ρ n D u n = ρ n g ˆp n ˆP n R + R n + u n S n 79 ρ c D u c = [ J B] + ρ c g ˆp i ˆp c R R n + u c S n 84
Solar partially ionizd plasma 11 E. Multi-spcis nrgy quation for nutrals Adding up nrgy quations for th nutral componnts Eq. 41 and 43, summd ovr E ionization stats, w gt: 3 Dp n + 3 2 Dt 2 p n u n + ˆp n un + q n = M n u nrn + 1 2 u2 ns n 85 and Dχ n Dt + χ n un = Φ n 86 whr χ n and Φ n ar dfind by Eqs. A3 and A4. Using th momntum and continuity quation, as wll as dfinitions from Appndix C, aftr som lngthy but straightforward calculations, on obtains th total nrgy quation for nutrals: whr n + 1 2 ρ nu 2 n + + q n = ρ n u n g + n = 3p n /2 + u n n + 1 2 ρ nu 2 n + ˆp n u n M n + Φ n 87 χ n = 3p n /2 + χ n 88 whr q n is dfind according to Eq. C12. As bfor w can us momntum and continuity quations for nutrals to rmov th kintic nrgy part from this quation, gtting: D n Dt + n u n + ˆp n un + q n + F n R = M n + 1 2 u2 ns n u n Rn 89 whr w hav usd th dfinition M n + Φ n = 9 M n + kν n I ν jν n ndωdν = M n F R n Th trm M n accounts for th nrgy xchang of nutrals through th lastic isions with chargs and th intgral rprsnts th nrgy gain/losss of nutral spcis aftr absorbing/mitting photons. F. Multi-spcis nrgy quation for chargs Using th nrgy quations for ions Eq. 41 and 43 and Eq. 49 for lctrons, w gt: 3 Dp i + 3 p i ui + ˆp i ui + 2 Dt 2 q i = M i u iri + 1 2 u2 is i 91 and D 3p Dt 2 + 3 2 p u + ˆp u + q = Dχ i Dt M u R + 1 2 u2 S 92 + χ i ui = Φ i 93 with χ i and Φ i ar dfind by Eqs. A3 and A4. In a similar way as for nutrals, aftr som lngthy manipulations th nrgy quation for chargs is obtaind: whr i + 1 2 ρ cu 2 c + q i = ρ c u c g + J E + i = 3p i /2 + u c i + 1 2 ρ cu 2 c M i + M + + ˆp i u c + Φ i 94 χ i = 3p i /2 + χ i 95 and q i is dfind according to Eq. C12. W us th continuity and momntum quations for chargs Eq. 66 and 84 to rmov th kintic nrgy part from this quation: D i Dt + i uc + ˆp i uc + q i + F c R = J E + [ u c B] M n 1 2 u2 cs n + u c Rn 96 whr w hav usd th rlation M i + Φ i + M = M n F R, c 97 du to th nrgy consrvation in lastic isions and th nrgy xchang with photons. Th drivd systm of quations Eqs. 65, 66, 79, 84, 89 and 96 looks formally th sam as in th cas of purly hydrogn plasma. Howvr th variabls ar dfind in th diffrnt way and rprsnt avrag quantitis ovr all N componnts of th plasma. It is also important to not that only th isions with or by nutrals nd to b considrd. G. Collisional R trms In ordr to mak practical us of th isional trms R in th momntum quations on has to xprss thm via th avrag vlocity of chargs and nutrals instad of individual vlocitis as in Eqs. B6, B8 and B9. To that aim, w us th particular form of th isional frquncis, Eqs. 59, 6 and 61. Ths frquncis ssntially dpnd on th numbr of iding particls, thir masss, and a givn function of tmpratur. Assuming
Solar partially ionizd plasma 12 In a similar way w gt for th nutral trm, R n, Eq. B8: R n ρ u n u ρ i u n u i N β=1 N ν nβ β=1 and for th lctron trm, R, Eq. B9: R N ν i m J + N ν in β 13 N ν nβ ρ u n u 14 β=1 FIG. 2. Map of th isional frquncis computd according to Eqs. 6 and 59 at h = 515 km of th VAL-C modl atmosphr. Th x-axis is th atomic numbr of th first idr, th y-axis th atomic numbr of th scond idr. w hav isions btwn lctrons, ions of typ and nutrals of typ β, on can writ: ν i = n i m 1/2 ft ν in β = n βn m 1/2 i f1 n β T = n βn m 1/2 p ν nβ = n βn m 1/2 f 1 T A 1/2 β f 2 T 98 whr m p is th proton mass, A is th atomic mass of spcis, and A β = A A β /A + A β is th rducd mass. Collisional frquncis satisfy th following proprty: ρ i ν in β = ρ βn ν nβ i 99 ρ ν nβ = ρ βn ν nβ 1 ρ i ν i = ρ ν i 11 Using ths proprtis w can rwrit th ion isional trm R i, Eq. B6, as: R i = ρ m 1/2 ft m 1/2 p f 1 T N ρ i N n i u i u N β=1 ρ m 1/2 ft J m 1/2 p m J n βn A 1/2 β u i u βn f 1 T n n ρ i u i u n N N N ν i ρ i u i u n ν in β 12 β=1 whr w hav nglctd th trm β n βn w βn, sinc w do not xpct larg diffrncs btwn th individual vlocitis of nutrals of typ β with rspct to thir cntr of mass vlocity. Thrfor, w assum th magnitud of this trm to b of scond ordr. Additionally, w hav assumd that th factor A 1/2 β dos not diffr significantly from its valu for hydrogn atoms. This numbr is only larg for isions btwn pairs of havy particls, with larg atomic numbr, as.g., F. Sinc th abundanc of ths atoms in th solar atmosphr is significantly smallr that of th lightst atoms H, H, w can safly nglct thir contribution in th summation. To prov th lattr ida w hav calculatd th frquncy of isions according to Eqs. 59 and 6, multiplid by th numbr dnsity of both idrs, i.., ν in β 8k B T = n i n βn Σ in 15 πm p A β W hav computd ν in β for th first 3 lmnts of th priodic systm and thir first ions. For that, w hav usd thir solar rlativ abundancs and hav computd th Saha distribution of population dnsitis for th firstionizd stats of thos 3 lmnts. Thn w hav solvd th quations of th instantanous chmical quilibrium for th 3 lmnts and 14 molculs that ar common in th solar atmosphr CH, NH, OH, CN, CO, NO, TiO, H 2, H + 2, C 2, N 2, LiH, SiH and HF. Finally, w hav computd th isional frquncis. Th computation has bn prformd for th lctron dnsity, tmpratur and total hydrogn dnsity of th VALC modl Vrnazza, Avrtt, and Losr, 1981. Th rsult of ths calculation is givn in Fig. 2. W hav chosn as rprsntativ an arbitrary hight in th uppr photosphr, 515 km. Th rlativ wight btwn th isional frquncis dos not significantly dpnd on hight. It appars that th most frqunt isions ar btwn th most abundant lmnts, H and H, with ions of othr lmnts. Th rst of idrs ar much lss abundant and thrfor, th isional transfr of momntum btwn thm can b safly nglctd. Thrfor, w can safly assum A 1/2 β only wakly varying, and prform th summation abov.
Solar partially ionizd plasma 13 H. Ohm s law for two-fluid dscription Ohm s law is frquntly drivd from th lctron momntum quation, by nglcting th lctron inrtia trms ρ u /, ρ u u and gravity acting on lctrons ρ g, s,.g. Zaqarashvili, Khodachnko, and Ruckr 211a. Howvr, w now dal with th 2N + 1 componnt plasma N ionizd, N nutral and 1 lctron componnts and w find mor convnint to us th sam stratgy as in Bittncourt 1986. W driv an volution quation for th lctric currnt by multiplying th momntum quations Eq. 29 and 3 of ach spcis by q i /m I and add thm up. Sinc nutral spcis hav zro charg, summatoris only xtnd to ions = 1,.., N and lctrons = N + 1. This lads to N+1 N+1 N+1 u i n i q i + n i q u i ui + q i u i S i = m i ni qi 2 m i E + u i B + n i q i g q i ˆp i + m i q i m i Ri 16 Using th continuity quation for spcis, on gts: N+1 u i n i q i + n i q i u i ui + q i u i S i = m i J + N+1 n i q i u i u i 17 Hr w hav usd th currnt dnsity dfinition J = n iq i u i according to Eq. C6. Expanding th convctiv trm in Eq. 17, and substituting th vlocity u i using th dfinition of th rlativ vlocitis of ions and lctrons Eq. C7 with rspct to th cntr of mass vlocity of chargs u c u i = w i + u c and u = w + u c w hav: n i q i u i u i = J u c + u c J + N+1 N n i q i w i w i n w w Hr w hav xplicitly rplacd spcis = N + 1 by lctrons. Th currnt dnsity J is dfind as J = N n i q i w i n w = J 18 th latr quality holds bcaus th charg nutrality is assumd. All four trms in th abov xpansion ar of scondordr, givn that thy involv doubl products of vlocitis. Gnrally, th Ohm s law is usd only up to first ordr, and thy ar nglctd. W will do th sam approximation latr. W kp thm for th momnt, in cas thy nd to b rtaind partially or fully for som rason. Th gravity trm in Eq. 16 also cancls out bcaus of th charg nutrality. This givs th Ohm s law th following form: J + J u c + u c J + n w w = N+1 N+1 N n i q i w i w i q i m i ˆp i + q i m i Ri 2 ni q i E m + u i B i 19 Up to now it is gnral and no assumptions hav bn mad xcpt for th charg nutrality and th nglcting of multiply ionizd atoms. To mak a practical us of this Ohm s law w will particulariz it to our cas. Stting q = for ions and putting xplicitly th lctron contribution, th Lorntz trm sums to: N+1 n i qi 2 E m + u B = i 2 N n n i m + 1 [ E m n m + u c B] [ J i m B] + 2 m N n i w i 1 + m m i B 11 W procd by nglcting all th trms proportional to m /m i in th Lorntz forc trm. Th drift vlocity trm is xpandd as: N n i w i = N n i u i n i u c 111 Th rsult of th summation can b rasonably assumd small in th cas of multipl ions, sinc w do not xpct strong dviations btwn th individual vlocitis of ions u i and th cntr of mass vlocity of chargs u c. Aftr nglcting this trm, w gt N+1 n i qi 2 E m + u i B 2 n [ E i m + u c B] m [ J B] 112 Th sum of th prssur trms givs: [ N+1 ] q i ˆp i = N m ˆp i ˆp m i m m i ˆp m 113
Solar partially ionizd plasma 14 Nxt w dal with th friction forcs trms. Not that bcaus of th multiplication by q, th friction forc of nutrals has no ffct. In ordr to oprat ths trms w hav to tak into account th form of xprssion for isional frquncis, Eqs. 6, 59, Eq. 61, and Eq. 98. N m I Ri Ri J m N N ν i + β=1 N N N +n u c u n ν nβ β=1 β=1 ν in β ν nβ 114 whr w hav nglctd trms such as n i w i and assumd wak variations of th factor A 1/2 β, similar to Sction III G. Aftr assuming stationary currnts and nglcting scond-ordr trms all trms on th lft hand sid of Eq. 19, th two-fluid Ohm s law for multi-componnt plasma bcoms: E = [ E + u c B] = 1 [ J n ˆp B] n + ρ n 2 ν i + β ρ u c u n n β ν nβ ν nβ J 115 β ν in β This quation closs th systm. From lft to right th trms in th Ohm s law ar: Hall trm; battry trm, Ohmic trm, and ambipolar trm. Th Ohm quation has similar form as th on for hydrogn plasma drivd in.g., Zaqarashvili, Khodachnko, and Ruckr 211b xcpt that w us u c instad of u i. A. Mass, momntum and nrgy consrvation Th mass, momntum and nrgy quations for all spcis, including photons, ar addd up lading to th singl-fluid quations in th following form: whr ρ + 2N+1 ρ u = S = 116 ρ D u Dt = J B + ρ g ˆp ˆP R 117 + 12 ρu2 + u + 1 2 ρu2 + ˆp u + + q + F R = J E + ρ u g 118 = 3 2N 2 p + χ 119 and q dfind by quation Eq. D8 and χ by Eq. A3 with th indx running ovr N nutrons plus N ions. In ths quations w hav not nglctd th lctron inrtia. Th individual inlastic isional S -trms hav disappard sinc thr can not xist any mass dnsity variation du to isions. Th momntum transfr btwn particls also vanishs th lastic ision trms btwn particls sum to zro according to Eqs. B9, B6 and B8 and only th momntum xchang with photons rmains through th trm ˆP R. Th sam happns with th nrgy transfr btwn particls and only th nrgy xchang with photons is lft with th trm F R. W can rmov th kintic nrgy trms from th nrgy quation using th momntum Eq. 117 and continuity Eq. 116 quations, obtaining: D Dt + u + ˆp u + q + F R = J[ E + u B] 12 IV. SINGLE-FLUID DESCRIPTION B. Ohm s law for singl-fluid dscription Whn th isional coupling of th plasma is strong nough, it is convnint to us a singl-fluid quasi- MHD approach i., including rsistiv trms that ar not takn into account by th idal MHD approximation. Th drivation of th singl-fluid quations for a multicomponnt plasma N nutrals, N ions and on lctron componnt gos ssntially through th sam stps as for two-fluids abov, xcpt for slightly diffrnt dfinitions of th macroscopic variabls. Ths dfinitions ar givn in Appndix D. In this sction w provid th final singl-fluid quations of consrvation of mass, momntum and nrgy, and driv th gnralizd Ohm s law for th cas of singl-fluid dscription. Blow w us singl indx to rfr to ach of 2N +1 componnts, without xplicit indication of nutrals and ions. Using th sam stratgy as for th two-fluid dscription, w multiply th individual momntum quations of ach spcis by q /m and add thm up. This lads to an quation similar to Eq. 19, xcpt that avrag cntr of mass vlocity u appars aftr th summation, instad of th vlocity of chargs u c and a diffrnt dfinition of th drift vlocitis, w i, now masurd with rspct to th avrag vlocity, u, rathr than with rspct to th avrag vlocity of th chargs, u c : J + J u + u J + N+1 n q w w = N+1 n q 2 m E + u B q ˆp + q R 121 m m
Solar partially ionizd plasma 15 FIG. 3. Magntic fild as a function of hight assumd for th quit Sun modl lft and for th sunspot umbra modl right. FIG. 4. Ionization fraction ξ i as a function of hight in th th VALC modl lft and in th sunspot umbra modl right. whr th summation is don only ovr th chargd componnts ions and lctrons sinc q n =. Th summation of th Lorntz forc trm lads to th following N+1 + 2 m n q 2 E m + u B 2 n [ E m + u B] + N n w B J B 122 m whr w hav nglctd th trm proportional to m /m. Th summation in th abov quation gos only ovr N ions. W furthr assum that th drift vlocity of individual ions is not diffrnt from thir avrag drift vlocity, i.., w u i u 123 with u i bing cntr of mass vlocity of ions. Using th dfinition from Eq. D2, th rlation btwn th cntr of mass vlocitis of chargs u c, nutrals u n, and th avrag on ovr th whol fluid u can b approximatly xprssd as: u = ρ c ρ u c + ρ n ρ u n = ξ c u c + ξ n u n 124 Th lattr quation provids that ξ n +ξ c = 1. Combining th two rlations abov on obtains: u i u = ξ n u c u n + u i u c ξ n w 125 Th cntr of mass vlocity of ions u i and of chargs u c th lattr including th contribution from lctrons, s Eq. C5 can b assumd approximatly th sam if on nglcts th lctron mass compard to th ion mass. This assumption has alrady bn usd in this sction, as wll as for th drivation of th Ohm s law for th twofluid formalism. In th quation abov w hav dfind th rlativ ion-nutral vlocity w as: w = u c u n u i u n 126 Th summation of th Lorntz forc trm finally givs: N+1 n q 2 E m + u B 2 n [ E m + u B] + 2 n m ξ n w B m J B 127 Th summation of th prssur trms, and th lastic ision R trms, ar both th sam as in Eq. 113 and Eq. 114, corrspondingly. Assuming that currnts ar stationary in Eq. 121, th partial drivativ vanishs. For situations not far from quilibrium, vlocitis can b assumd to b small and scond-ordr trms including doubl products can b nglctd. Thus, all th trms on th lft-hand sid of Eq. 121 can b rmovd. Th quation for th lctric
Solar partially ionizd plasma 16 FIG. 5. Diffusion paramtrs from th induction quation for th quit Sun modl lft and th sunspot umbra modl right. Not th diffrnc in scals at th vrtical axis. fild E thn rads as: E = [ E + u B] = ξ n w B J + B ˆp n n + ρ N N n 2 ν i + ν nβ J β=1 ρ N N N ν nβ ν in n β w 128 β=1 β=1 Unlik th two-fluid approach, w can not dirctly us this quation, bcaus thr ar still trms dpnding on w and w nd furthr assumptions to procd from hr. W nd th quation for th rlativ ion-nutral vlocity w and w obtain it following Braginskii 1965. W add up th ion-lctron and th nutral momntum quations Eq. 83 and Eq. 78, multiplid by ξ n and ξ c, corrspondingly. Th rsult is: ρc u c ξ n + ρ c u c u c ρn u n ξ c + ρ n u n u n = [ ] ξ n J B G N J + ρ ν nβ n n w 129 β=1 whr G and n ar givn by G = ξ n ˆpi ξ i ˆpn 13 N N N n = ρ ν nβ + ρ i ν in β 131 β=1 β=1 Th gravity trm cancls out sinc gρ c ξ n ξ c ρ n = W can xprss th total drivativ in trms of th drift vlocity ρc u c ξ n ρn u n ξ c + ρ c u c u c + ρ n u n u n 132 ρξ n ξ c w and this trm is nglctd whn compard to th friction trms, th lattr assumd to b w/τ col, whr τ col 1/ν β is a typical isional tim scal, s Díaz, Khomnko, and Collados 213. Thn th xprssion for w bcoms: w = ξ n n [ J B ] G n + N J ρ ν nβ 133 n n β=1 Introducing this rsult in Eq. 128 and rarranging trms, on can obtain an xprssion for th lctric fild E : E = c j J + cjb [ J B] + c jbb [ J B B] + c p ˆp + c pt G + cptb [ G B] 134 whr th cofficints of th Ohm s law ar givn by: c j = 1 N N n 2 ρ ν i + o ρ ν nβ n 2 c jb = 1 n 1 2ξ n ɛ 1 + ξ n ɛ 2 c jbb = ξ2 n n c p = 1 n c pt = 1 n ɛ 1 ɛ 2 β=1 c ptb = ξ n n 135 In ths quations, w hav dfind th additional paramtrs: = N N ρ ν i + ρ ν nβ 136 β=1
Solar partially ionizd plasma 17 ɛ 1 = ɛ 2 = o = N ρ ν nβ / n 1 137 β=1 N β=1 N β=1 N ρ ν in β / n 1 138 N ρ i ν in β / n 1 139 according to th dfinition of th isional frquncis Eq. 98. Th xprssion for th cofficint c jbb in Eq. 135 can b compard with thos givn in othr studis that includ th contribution of nutral hlium by Zaqarashvili, Khodachnko, and Solr 213 and Solr, Olivr, and Ballstr 21. Our xprssion for this cofficint is mor compact than givn in Zaqarashvili, Khodachnko, and Solr 213 and is mor similar to Solr, Olivr, and Ballstr 21. Zaqarashvili, Khodachnko, and Solr 213 showd that th xprssion by Solr, Olivr, and Ballstr 21 dos not hav a gnral validity and is only valid for wakly ionizd mdium, sinc qual vlocitis of nutral hydrogn and hlium wr assumd for its drivation. In our cas, w hav not mad any assumptions about th ionization dgr of th plasma, but w supposd that th trms n i w i and β n βn w βn rprsnting th sum of th individual drift vlocitis of ion and nutral spcis with rspct to th avrag ion and nutral vlocity ar small. Thrfor, our xprssion dos not suffr th sam rstrictions as th on by Solr, Olivr, and Ballstr 21. C. Total induction quation To gt th induction quation, w us Ohm s law, Eq. 134, and Faraday s law and Ampr s law nglcting Maxwll s displacmnt currnt: B = E; J = 1 µ B 14 Th gnralizd induction quation can b rwrittn in trms of currnt dnsity J as: B [ = u B ηµ J η Hµ B [ J B]+ + η Aµ B 2 [[ J B] B] + η pµ ˆp B c ptg cptb [ G B] ] 141 whr w hav dfind th diffusivity cofficints, making us of Eqs. 135. η = c j /µ = n 2 µ η H = c jb /µ B = B n µ 142 143 η A = c jbb /µ B 2 = n µ B 2 144 η p = c p /µ B = ξ2 n B n µ 145 In th induction quation, th trms on th right hand sid ar: convctiv trm, Ohmic trm, Hall trm, ambipolar trm, Birman battry trm, trms dpnding on th prssur gradints distribution and gravity trms. Th vctor G bcoms small if th nutral fraction is small, i.. in th wakly ionizd plasma. Not that in th abov dfinitions of diffusion cofficint w formally scald η p with B /µ to hav th sam dimnsions as othr diffusion cofficints η, η H, η A l 2 /t, i.. m 2 s 1 in th SI units. Th xprssion of th cofficint multiplying th Birmann battry prssur trm ˆp is xactly th sam as of th Hall trm [ J B]. Th rlativ importanc of th both trms is thus dtrmind by th strngth of th currnt dnsity J, magntic fild B and th valu of th lctron prssur that may b small for a wakly ionizd plasma, as wll as th amplitud of spatial variations of ths quantitis. Anothr oftn usd paramtr is Cowling conductivity Khodachnko t al., 24, 26; Lak and Arbr, 26; Fortza, Olivr, and Ballstr, 27; Arbr, Hayns, and Lak, 27; Arbr, Botha, and Brady, 29; Sakai and Smith, 29. It can b introducd combining th Ohm and Ambipolar trms: η A µ = ξ2 n B 2 n = µη c µη = 1/σ c 1/σ 146 σ c = σ σ = 1 + σ ξ2 n B 2 1 + ση A µ n 147 By introducing this conductivity, th ambipolar trm, togthr with th Ohmic trm can b rwrittn as: ηµ J η A µ J B B B 2 = ηµ J + η c µ J 148 so that th Cowling conductivity is rsponsibl for th prpndicular currnts to th magntic fild. Th Cowling conductivity dpnds on th magntic fild and on th ionization fraction. If thr ar no nutrals ξ n =, both Ohmic and Cowling conductivity ar th sam. V. OHM S LAW IN THE SOLAR ATMOSPHERE To stimat th rlativ importanc of th Ohmic, Hall, Ambipolar and othr trms in th induction quation,
Solar partially ionizd plasma 18 FIG. 6. Spatial and tmporal scals dfining th rang whn th Ohmic, Hall and Ambipolar trms bcom important rlativ to th convction trm in th induction quation, for th quit Sun modl fild. FIG. 7. Spatial and tmporal scals dfining th rang whn th Ohmic, Hall and Ambipolar trms bcom important rlativ to th convction trm in th induction quation, for th sunspot umbra modl. on has to compar th valus of η, η H, η A, η p, tc. for th typical conditions of th solar atmosphr. W compar two atmosphric modls. In th first modl w tak thrmodynamic paramtrs from th VALC modl atmosphr Vrnazza, Avrtt, and Losr, 1981, and th magntic fild is assumd to dpnd on hight as B = 1 xp z/6 G, roughly rprsnting a quit solar atmosphr outsid activ rgions. Th scond modl is xtractd from th non-lte invrsion of th IR Ca triplt by Socas-Navarro 25 and rprsnts a sunspot umbra. Ths two modls giv th two xtrm cass of th conditions that can b found in th solar photosphr and chromosphr. Figurs 3 and 4 show th magntic fild stratification and th ionization fraction ξ i in th both modls. Th magntic fild rachs th maximum of 2.5 kg in th sunspot modl, but only 1 G in th quit Sun modl. Th ionization fraction is always blow 1. In th quit Sun th minimum of 1 4 is rachd in th photosphr. In th sunspot modl, th ionization fraction is vn lowr, as xpctd for th coolr tmpraturs appropriat for th umbra. Th diffusion cofficints for th both modls ar shown in Figur 5. This figur givs th cofficints η, η H qual to η p in our dfinition and η A. Th comparison of th diffusion cofficints for th quit modl lft shows that th Hall trm bcoms mor important at hights abov 2, whr th lctrons dcoupl from th rst of th plasma, according to Fig. 1. At this hight th cyclotron frquncy of lctrons bcoms largr than thir isional frquncy. Th Ambipolar trm bcoms mor important abov 1 km whn th ions dcoupl from th rst, according to Fig. 1. As for th sunspot modl, th whol stratification sms to b shiftd downward in hight Willson dprssion. Th Ohmic diffusivity is of th sam ordr of magnitud as in th VALC modl. Howvr both Hall and Ambipolar trms ar two ordrs of magnitud largr du to th largr magntic fild. Th Ambipolar trm dominats ovr almost all hight rang starting from 3 km in th photosphr. To dfin th typical tmporal and spatial scals whr th Ohmic, Hall and Ambipolar trms bcom important, compard to th convctiv trm u B, w hav to rlat thm btwn ach othr. If u is of th ordr of Alfvén spd v A = B/ µρ, thn th spatial scal is givn by: L = η v A 149 Th tmporal scal is ν = v A /L making it as: τ = 1 ν = η v 2 A 15
Solar partially ionizd plasma 19 FIG. 8. Spatial distribution of th trms of th Ohm s law in a snapshot of magnto-convction with th avrag magntic fild strngth of 18 G. From lft to right, from top to bottom, th pans giv th continuum intnsity, and th y componnts of th following trms: th Ohmic trm ηµj y, Hall trm η H µ J B y/ B, Ambipolar trm η A µ[ J B] B y/ B 2, battry trm η pµdp /dy/ B, and th prssur trm c ptb G B y. Th units ar SI units mlq 1 t 2. Th diffrnt trms ar multiplid by a factor whn ncssary so that th scal is th sam in all th panls. Th Hall trm is takn as a rfrnc. Th largr is th multiplying factor, th smallr is th trm. Th quantitis ar givn at hight of 6 abov th photosphr. In th abov quations η stays for ithr Ohmic, Hall or Ambipolar diffusivity cofficint. Th scals go with B as: L ohm B 1 151 τ ohm B 2 152 L H = constb 153 τ H B 1 154 L A B 155 τ A = constb 156 so th importanc of th Ambipolar diffusion should incras in th strong-fild structurs, whil th importanc of th Hall ffct should incras in th wak-fild structurs. Anothr way of valuating th importanc of ths trms is by comparing thir scals with typical frquncis of th atmosphr, such as cyclotron frquncis and isional frquncis. Using th xprssions for th η cofficints abov, on radily obtains N τ ohm i + N β=1 ν n β ω ci ω c 157 τ H 1 ω ci 158 τ A ξ 2 n N N β=1 ν i n β. 159 Thrfor, th Hall trm bcoms important for tim scals clos to th ion cyclotron priod, th Ambipolar trm bcoms important on scals proportional to ionnutral ision tim s Zaqarashvili, Khodachnko, and Ruckr, 211b and th Ohmic trm is important on scals givn by th ratio of th lctron isional frquncy to th product of th ion and lctron cyclotron frquncis. Th typical scals obtaind from th abov stimation ar givn in Fig. 6 for th quit modl and in Fig. 7 for th sunspot umbra modl. In th quit modl th largst spatial scal for th Hall ffct rachs 1 m at 6 km; th largst tmporal scal is.1 sc, also for th Hall ffct in th uppr photosphr. For th Ambipolar trm, th largst scals ar at th uppr chromosphr, raching th sam valus. In th sunspot umbra modl, th Ohmic trm only