COLLISIONLESS SLOWING DOWN OF NOVA AND SUPERNOVA SHELLS IN MAGNETIZED INTERSTELLAR MEDIUM

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1 Astrphysics, Vl., N., 3 COLLISIONLESS SLOWING DOWN OF NOVA AND SUPERNOVA SHELLS IN MAGNETIZED INTERSTELLAR MEDIUM D. A. Osipyan, H. B. Nrsisyan, and H. H. Matvsyan UDC:5.35 Th cllisinlss intractin f an xpanding plasma clud with a magntizd backgrund plasma is xamind in th framwrk f a 3D kintic-hydrdynamic mdl. Th slwing dwn f a hydrgn clud is studid fr high Alfvn-Mach numbrs and magnt-laminar intractin paramtrs. A particl-in-cll mthd is usd t study th dynamics f th magntic fild, plasma clud, backgrund plasma, and cllisinlss shck wav gnratd by th intns particl flux. A numrical simulatin is cnsistnt with th nnstatinary intractins btwn th plasma shlls frmd during nva and suprnva xplsins and th intrstllar plasma mdium. Kywrds: (stars:) nva, suprnva -ISM: magntic filds. Intrductin Th prblm f th intractin rarfid plasma flws with a surrunding backgrund magntizd plasma ariss during rsarch n th dynamics f slar flars and n th flw f th slar wind arund th arth s magntsphr, in activ xprimnts with plasma cluds in spac, and in th curs f intrprting a numbr f astrphysical bsrvatins [-]. Rsarch n this prblm is f cnsidrabl intrst in cnnctin with xprimnts n cntrlld thrmnuclar fusin [7,8] and labratry simulatins f lasr plasmas in xtrnal magntic filds [3,9]. Phnmna f this srt, f curs n substantially highr nrgy and spatial-tmpral scals, hav rcntly bn undr intns discussin in cnnctin with xplsiv astrphysical prcsss, which ar charactrizd by th rlas f immns nrgy and ar accmpanid by th frmatin f pwrful high-vlcity plasma structurs xpanding sphrical and annular shlls, jts, tc. Accrding t astrphysical bsrvatins and thrtical studis [], th rathr massiv shlls jctd during xplsins intract strngly with th intrstllar mdium and with th intrgalactic magntic fild. Thus, fr xpansin f th jctd shll f a suprnva is pssibl nly in th arly stags f its vlutin. Thn it is slwd dwn with th transfr f nrgy-mmntum t th surrunding plasma backgrund and magntic fild. Th prblm Institut f Radi Physics and Elctrnics, Natinal Acadmy f Scincs f Armnia. -mail: hrachya@irph.am Translatd frm Astrfizika, Vl., N., pp , Octbr-Dcmbr, 3. Original articl submittd July, 3; accptd fr publicatin August 7, /3/-3$5. 3 Plnum Publishing Crpratin

2 f th cllisinlss slwing dwn f th rsidu f suprnva was first frmulatd by Ort [] and subsquntly analyzd in dtail by Shklvskii []. Th slwing dwn prcss is charactrizd by th radius fr slwing dwn f th clud by th magntic fild, R H, and th gas dynamic slwing dwn radius R. An xprssin fr R H can b btaind by quating th initial kintic nrgy W f a sphrical clud t th nrgy f th magntic fild that it pushs ut in xpanding t radius R H [3], i.., ( W H ) 3 R H =. Hr H is th intnsity f th unprturbd magntic fild. As th clud xpands, it draws th backgrund plasma int a cmbind mtin. Alng with this, th mass f xplld plasma incrass. Th radius f th sphr within which th mass f th clud and that f th backgrund plasma drawn int th cmbind mtin bcm qual is rfrrd t as th gas dynamic slwing dwn radius: R = ( 3 M πn ) 3 * m*, whr n * and m * ar th dnsity and mass f th backgrund plasma ins ( snwplw mdl []) and M is th mass f th jctd shll. Taking n *.cm, M = M ( M = g is th mass f th sun), and m * = m H (m H is th mass f a hydrgn in) as typical valus f th paramtrs, w btain R =. pc. Th smallr f th radii, R H r R, dtrmins th prdminant mchanism fr slwing dwn f th clud 3 magntic r gas dynamic. Th rlatinship R R =, whr M A = u VA is th Alfvn-Mach numbr (u is th 3 H M A initial xpansin vlcity f th clud) and VA = H πn* m* is th Alfvn vlcity in th backgrund plasma, implis that fr M A << th clud lss nrgy as a rsult f th dfrmatin and displacmnt f th magntic fild, whil fr M A >> th slwing dwn is causd by th intractin with th backgrund plasma [-8]. Sinc th charactristic vlcitis f suprnva shlls ar f rdr u cm/s fr typical magntic filds 8 9 H 3 G, w find that M A 5 5. Thus, th jctd shlls f suprnva can nly b slwd dwn as a rsult f thir intractin with th backgrund plasma mdium; that is, th slwing dwn is gas dynamic in charactr. In this papr, w study this prblm using a hybrid mdl fr th plasma with a numrical simulatin mplying a particl-in-cll mthd.. Statmnt f th prblm An analysis shws that th gas dynamic slwing dwn can nly b nsurd by a cllisinlss laminar (r turbulnt) mchanism [3] assciatd with th gnratin f vrtical lctric filds E i in th lading dg f th clud r by a cllisinal mchanism wing t pairwis cllisins f ins frm th shll with ins, nutral atms, and lctrns in th intrstllar mdium. W bgin by brifly xamining th ffcts f pairwis cllisins, basd n an analysis f Rfs. 9 and. Th ins in th clud ls thir nrgy and transfr it t th ins in th backgrund plasma in multipl Culmb in-in r in-lctrn scattring vnts with a man fr path givn by λ ii mu = π( + m m* ) ZZ* n*, lnλ () r i = ( m m)( + m m * ) λ ii λ, rspctivly, and in scrnd Culmb rpulsins with a man fr path givn by 35

3 λ mu 3 3 in = S + S* ass* n*. () Hr Z and Z * ar th chargs f th backgrund and clud ins, S and S * ar th chargs f th nucli in th clud and backgrund (in a hydrgn plasma Z=S=), a is th Bhr radius, ln Λ is th Culmb lgarithm f th backgrund plasma, and m is th mass f th clud ins. As fr scrnd plarizatin nuclar attractin prcsss, thy ar f n imprtanc hr sinc th man fr path fr that prcss undr astrphysical cnditins is nrmusly gratr than and λ in [9,]. Nt als, that Eqs. () and () hav bn btaind fr cnditins such that th magntic fild ds nt influnc th Culmb scattring prcss. In fact, fr H 3 G, n *. cm -3, and m * = mh (hydrgn plasma), w btain th fllwing stimats fr th lctrn cycltrn and plasma frquncis: ω =. 8 p λii s - and ωc ω p 3 3 <<, which is th cnditin fr applicability f Eqs. () and (). W nw stimat th man fr paths fr ths cllisinal prcsss during prpagatin f suprnva rsidus at a charactristic vlcity u = 9 cm/s thrugh th intrstllar mdium in th cas f hydrgn shll and backgrund plasmas with an lctrn tmpratur n th rdr f T = V ( lnλ 5 ) and with th paramtr valus givn abv. Th gas dynamic slwing dwn radius fr a typical suprnva is...pc.., whil Eqs. () and () imply that R =. pc, 5 3 λ ii =. 75 pc, and λ in = 3. 5 pc. Ths stimats imply that cllisinal prcsss cannt slw th clud dwn, sinc R << λi, λ in, and λ ii. Thus, it is imprtant t xamin cllisinlss mchanisms fr this intractin. Th first grup f cllisinlss intractin mchanisms ar cllctiv turbulnt mchanisms (anmalus viscsity, anmalus rsistivity) during th dvlpmnt f in-in r lctrn-in bam instabilitis []. Th cnditin fr xcitatin f th in-in instability has th frm [] V A c s u + r cs M A +, (3) V whr c s = T m* is th in sund spd in th plasma. Fr typical valus f th paramtrs, cs cm/s andva cm/s. Thus, accrding t Eq. (3), M A < and th bsrvd slwing dwn f th xpanding cluds cannt b causd by turbulnt anmalus viscsity, sinc th charactristic Alfvn-Mach numbr M A 5 5. Th scnd grup is th cllisinlss laminar slwing dwn mchanism assciatd with th gnratin f vrtical lctric filds. It is knwn that th rl f vrtical lctric filds bcms prdminant as M A incrass, sinc A E E M [3,], whr E p is th plarizatin lctric fild that ariss as a rsult f th drp in th gas dynamic and magntic prssur at th bundary f th clud. A mdl fr nrgy xchang btwn th clud and th backgrund plasma wing t th cmbind ffct f th gyrrtatin f th ins and th gnratin f vrtical lctric filds whn M A > (th magntic laminar mchanism (MLM) fr slwing dwn) has bn prpsd [,5]. Analytic slutins fr th initial xpansin phas, whn nly a vrtical lctric fild E i dvlps, shwd that th fractin f nrgy givn up by th clud is prprtinal t δ = ( R R ) L (th MLM intractin paramtr), whr R L is th Larmr radius f th clud ins. Thus, th intnsity f th cllisinlss intractin btwn th clud and th backgrund plasma is dtrmind by th paramtr δ, as wll as by th Alfvn-Mach numbr M A. i p A 3

4 3. Numrical simulatin basd n a hybrid plasma mdl W shall carry ut a numrical simulatin f th cllisinlss intractin f a suprnva shll with th intrstllar mdium using a hybrid plasma mdl, in which th in cmpnnt is dscribd by a Vlasv kintic quatin and th lctrn cmpnnt, by th quatins f gas dynamics. Nt that this kind f mthd has bn usd xtnsivly lswhr. (S, fr xampl, Rfs. -8). This hybrid mdl is justifid by th fact that, bcaus f th slwing dwn f th shll, a cllisinlss shck wav can b gnratd in th backgrund plasma with hydrdynamic braking f th lading dg and frmatin f a multiflux flw. Thus, th structur f this srt f suprcritical cllisinlss shck wav n spatial scals f R R can nly b dscribd adquatly in trms f a hybrid apprximatin []. Nt als, that, althugh th apprximatin f cllisinlss magnthydrdynamics is valid fr δ >>, it is unsuitabl fr dscribing th frmatin and dynamics f a cllisinlss shck wav. Th initial systm f quatins cnsists f a Vlasv kintic quatin fr th ins, th quatins f mtin and intrnal nrgy fr th lctrn cmpnnt, and Maxwll s quatins fr th lctrmagntic fild. In th cas f a hydrgn clud bing slwd dwn in a hydrgn backgrund plasma this systm f quatins has th fllwing frm: fi t + v fi r + F fi v =, F = E+ m c H [ v H], () v t + c m n ( v ) v = E + [ v H] ( nt ), m (5) T t + ( v ) T+ T v =, () 3 πn H = c n + t n v i (7) t ( n ) = + ( nv ) =, H i (8) c t ( v v ), E =, H = E = n = n = ni = fi, i = i v n ( r v, t) dv, v ( r, t) = v v f ( r, v, t) d. (9) Hr E and H ar th lctric and magntic fild intnsitis, m H is th mass f th hydrgn in, v, and v i ar th mass avragd lctrn and in vlcitis, and T is th lctrn tmpratur. Nt that in Eqs. ()-() w hav mittd all th dissipativ trms assciatd with th finit cnductivity, viscsity, and thrmal cnductivity f th plasma. It is asy t shw that this apprximatin is satisfid with wid margins undr th cnditins f xpanding plasma shlls in th rarfid intrstllar mdium. In fact, fr xampl, w can stimat th charactristic diffusin tim fr th magntic fild: τ m = π R L σ c, whr σ is th cnductivity f th plasma. This implis that τ = λ R m T M A( ii ) >>. Hr T = R u is th charactristic slwing dwn tim and λ ii is givn by Eq. (). Similar stimats can als b btaind fr th thr transprt cfficints. In additin, in th Maxwll quatins w nglct th displacmnt currnt bcaus f th xpansin f th clud is nnrlativistic. Ths apprximatins frm th basis f th hybrid cllisinlss plasma mdl mplyd in this papr. It shuld, 37

5 hwvr, b mphasizd that, althugh arlir calculatins and simulatins hav dmnstratd th practical ralizability f this laminar intractin mchanism [3], thy wr carrid ut nly vr a fairly narrw rang f paramtrs δ < and R cm and, thrfr, ( ) M A, whil typical suprnva hav L δ = R RL fr M A 5 5. Althugh numrical simulatins with ths paramtrs ar nt rprducibl at th prsnt tim and will scarcly b pssibl vn in th distant futur, th numrical simulatins in th arlir paprs and in this papr d shw that, fr sufficintly high valus f δ > and M A >, th nrgy lss curvs acquir a univrsal charactr. Thus, thr is vry rasn t supps that this cllisinlss intractin mchanism can prvid fr th slwing dwn f th jctd shll undr th cnditins f a typical suprnva. This univrsality can asily b graspd if w cnvrt Eqs. ()-(9) fr th hybrid mdl t dimnsinlss frm using th fllwing variabls: f i ω r c pi H H H r, t ωci t, i i V E ( v, v ) ( v, v ),, E, ( VA c) H mhva V n 3 ( A r, v, t ) f ( r, v,t), n, F, * i n n * T A T A, F V ω ci () () () whr ω pi and ωci ar th in plasma and cycltrn frquncis. Thn it is asily sn that slutins f Eqs. ()-(9) dpnd nly n th paramtr m mh. Lt us cnsidr a tw-dimnsinal, axially symmtric mdl fr th dynamics f a pint xplsin that frms a clud f dns plasma xpanding int a magntizd backgrund. At th initial tim t =, an xplsin ccurs at th pint r =, z = in a cylindrical rgin r Lr, Lz z Lz with a unifrm magntic fild = (,, H ) H and filld with a plasma f dnsity n. Th xplsin frms a clud f dns plasma f radius R cntaining N particls with a ttal kintic nrgy W. At th initial tim, th vlcity f th ins in th clud is distributd linarly alng th radius, i.., u ( R, ) u =, m R R, R R R > R. (3) Hr R = r + z is th magnitud f th radius vctr in sphrical crdinats and u m is th maximum vlcity f th ins in th clud and is dtrmind by th initial nrgy W f th clud, with ( ) ( ) Th quatin fr th cnsrvatin f nrgy in this systm has th frm u =. m W 3NmH = u 5 3 W = π Lz L r 3 nm v nt + H + rdr + W 8π kin. () Th ttal nrgy f th systm cnsists f th thrmal and kintic nrgis f th lctrn gas (th first tw trms), th nrgy f th magntic fild, and th kintic nrgy W kin f th ins. 38

6 Th clud radius R is cnsidrably smallr than th stp siz h f th cmputatinal grid; that is, w can assum that at th initial tim th clud is cncntratd at a pint. At th initial tim t = th backgrund particls ar distributd unifrmly vr th ntir rgin and th particls in th clud ar distributd unifrmly in th clud. Th unprturbd valus f all quantitis ar spcifid at th bundaris f th rgin r = L r and z = ±L z n th axis r = w hav H r z Ez = r vz = r n T = = r r =, v = v = E = E = H = H =. With ths bundary cnditins th calculatin can b cntinud until th tim th prturbatin rachs th bundary f th rgin. Th quatins f mtin fr th ins ar th quatins f th charactristics f th kintic quatin, i.., r r r (5) d ri dt d vi = vi, = F. () dt W hav usd a particl-in-cll tchniqu t slv ths quatins. Th Maxwll quatins and th hat cnductin quatin ar slvd using finit diffrnc rslutin schms. A mr dtaild dscriptin f th mathmatical mdl fr th prcss and th numrical simulatin algrithm can b fund, fr xampl, in Rfs Rsults f th numrical simulatins Numrical simulatins wr dn using a particl-in-cll mthd and a study was mad f th nrgy and dynamical charactristics f th xpansin f a hydrgn plasma int a unifrm magntizd backgrund hydrgn plasma at larg Alfvn-Mach numbrs M A = 5. fr a magntic-laminar intractin paramtr δ = 8.. W mphasiz that, as ntd abv, th fllwing rsults rflct ral astrphysical prcsss nly with an accuracy crrspnding t th scals f th rsulting curvs a clud backgrund b.8. T W H W t t Fig.. Th tim variatin in th nrgy f th clud and backgrund plasma fr M A = 5. and δ=8.. (a) Th variatin in th ttal nrgy f th backgrund plasma (5) and clud (n). (b) Th variatin in th ttal thrmal nrgy f th lctrns (n) and f th magntic fild (l), and in th kintic nrgy f th lctrns (5). 39

7 TABLE. Valus f th Paramtrs fr th Numrical Simulatin (M A = 5., δ = 8. ) R L =.88 cm R =. cm RH = 337 cm T = 9.5 µs H = G W =.. rg u =.7. cm/s T =. V () n * =.3. 7 cm -3 V A =.5. cm/s 5 ci = 9 58 s - ω =. 7 pi s - ω a θ = E r E E z b θ = 9 E r E E z c θ = -5.5 d θ = 9.8. H H r H z..9 H H r H z θ = Clud Backgrund f θ = 9 Clud Backgrund Fig.. Th distributin vr R (in cm) f th lctrmagntic filds and dnsitis f th clud and backgrund plasma fr t =.5T, M A = 5., and δ = 8. and fr tw angls θ. (a) and (b) shw th distributins f th r (smth curv), (dashd), and z (fin dts) cmpnnts f th lctric fild fr θ = and θ = 9, rspctivly. (c) and (d) shw th sam fr th magntic fild. () and (f) shw th dnsitis f th clud (smth curvs) and backgrund plasma (dttd) fr θ = and θ = 9, rspctivly.

8 Figur shws plts f th clud and backgrund nrgis, and f th variatin in magntic fild nrgy and lctrn nrgy, as functins f tim. Th clud and backgrund nrgis ar nrmalizd t th initial clud nrgy W and th tim is xprssd in units f th gas dynamic slwing dwn tim T = R u. Th valus f th charactristic quantitis ar listd in Tabl, whr () T is th initial lctrn tmpratur. Fr t < T, th nrgy f th plasma is cncntratd mainly in th kintic nrgy f th clud (Fig. a). With th passag f tim, this nrgy dcrass and at t = T, abut 5% f th initial clud nrgy has bn cnvrtd t kintic nrgy f th backgrund plasma, which rprsnts slwing dwn by th MLM (magntic-laminar mchanism). Th thrmal nrgy f th lctrns has risn by % and th magntic nrgy has incrasd by 5-% (Fig. b). Figurs - shw plts f th lctric and magntic filds, clud and backgrund dnsitis, and lctrn tmpratur as functins f R at tims t =.5T (Figs. and a) and t =.5T (Figs. 3 and b) at angls θ = and θ = 9 t th a c θ = θ = θ = EE r E z H r H H z Clud Backgrund b d f θ = 9 E r E E z θ = 9 H r H H z θ = 9 Clud Backgrund Fig. 3. Sam as in Fig., but fr t =.5T.

9 lins f frc f th magntic fild. All f ths quantitis ar nrmalizd in accrdanc with Eqs. () and (). As xpctd, fr θ = th lctric fild is mainly dirctd alng z ( E r = E =, E z ) and fr θ = 9 its azimuthal cmpnnt prdminats ( E z =, E r << E ). Fr arbitrary angls θ, th magntic fild is mainly dirctd alng H ( H r, H, H z ). Th slwing dwn f th clud is accmpanid by th frmatin f an axially symmtric layr f cmprssd plasma in th backgrund plasma that mvs alng with th cmprssd magntic fild; i.., a cllisinlss shck wav dvlps. Th amplitud f th magntic fild at th shck frnt is. tims th unprturbd valu. Th thicknss f th layr is n th rdr f th in Larmr radius R L =.88 cm, in agrmnt with an stimat f th thicknss f th cllisinlss shck wav, RL [], fr th cas f braking and frmatin f a multiflux mtin. Th plasma clud acquirs th shap f an almst sphrical shll in which almst all th kintic nrgy and mass f th initially sphrical clud is cncntratd (Figs. and 3), with th lattr bing dfrmd mainly alng th dirctin f th unprturbd magntic fild H. Backgrund particls ar transprtd ut f th xpansin rgin; this lads t th frmatin f a plasma cavity. It crrlats with th magntic cavity, a rgin f radius R, in which th magntic fild is lwr than th unprturbd fild bcaus it is squzd ut (Figs. and 3). Thus, thr is ssntially n lctric fild in th cavity. In th initial phas f xpansin (Fig. ), th amplitud f th azimuthal cmpnnt... is cnsidrably gratr than that f th radial max max cmpnnt E ; i.., E r, E Er 5. At tim t =.5T, th psitins f thir maxima ssntially cincid with th pak dnsity f th clud. With th passag f tim, th utr bundary f th rgin transfrs nrgy t th backgrund plasma thrugh lctrmagntic intractins. Fr this rasn th distanc btwn th utr bundary f th clud and th innr bundary f th cllisinlss shck wav incrass. At tim t =.5T th rati E E 7.. Hr th psitin f ssntially th sam as th utr bundary f th cllisinlss shck, and E r is a maximum in th shck frnt. max max r max E is T ( - ) 8 t =.5T t =.5T θ = θ = θ = 9 θ = 9 3 a 3 5 b Fig.. Distributin vr R (in cm) f th lctrn tmpratur fr M A = 5., δ = 8., and t =.5T (a) and t =.5T (b) fr tw angls è= (smth curvs) and è= 9 (dashd curvs).

10 An lctrn thrmal wav is gnratd in th plasma alng with th cllisinlss shck. Th spatial distributin f that wav dpnds significantly n θ (Fig. ). A layr f lctrns hatd by th cllisinlss shck accmpanis th shck wav (cmpar Figs. -) and is distributd in a way s as t cmpnsat th in charg f th plasma (s Eq. (9)). This mdl fr th xpansin f plasma cluds is valid fr t < T. At latr tims th xpansin vlcity f th clud and, thrfr, th Alfvn-Mach numbr, dcras and ur mdl fails. An analysis shws that undr ths cnditins, th backgrund plasma ins undrg a cmplicatd multiflux mtin and quasipridic lp structurs shw up distinctly in th phas plans. (Th distanc btwn thm is R L.) Ths ar vidnc f th braking f th wav. In additin, w might xpct that at lng xpansin tims, turbulnt slwing dwn mchanisms will play an imprtant rl. Studis f ths qustins ar currntly undr way and th rsults will b publishd latr. Th authrs thank th dirctr f th Cmputatinal Physics Labratry f th Institut f High Tmpraturs f th Sibrian Branch f th Russian Acadmy f Scincs, Dr. G. I. Dudnikva fr a dtaild and critical analysis f ur rsults, as wll as fr usful cmmnts. REFERENCES. A. G. Pnmarnk, d., Physics f Csmic and Labratry Plasmas [in Russian], Nauka, Nvsibirsk, SO AN SSSR (989).. V. V. Adushkin, Yu. I. Ztsr, Yu. N. Kisilv t al., DAN 33, 8 (993). 3. Yu. P. Zakharv, A. M. Orishich, and A. G. Pnmarnk, Lasr Plasmas and Labratry Simulatin f Nnstatinary Prcsss in Outr Spac [in Russian], Nvsibirsk, ITPM SO AN SSSR (988).. R. Z. Sagdv, Cprativ prcsss and shck wavs in rarfid plasmas, in: M. A. Lntvich, d., Rviws f Plasma Physics, Vl., Cnsultants Burau, Nw Yrk (9), pp M. M. Lry, Phys. Fluids, 7 (983).. V. A. Vshivkv, G. I. Dudnikva, Yu. I. Mlrdv, and M. P. Fdruk, Vych. Tkhnlgii, 5 (997). 7. V. S. Imshnnik, in: K. V. Brushlinksii, d., Tw-dimnsinal Numrical Mdls f Plasmas [in Russian], IPM im. M. V. Kldysha AN SSSR, Mscw (979), p.. 8. A. G. Sgr and C. W. Nilsn, Phys. Fluids 9, (97). 9. B. A. Bryuntkin, U. Sh. Bgimkulv, V. M. Dyakin t al., Kvantvaya Elktrnika 9, (99).. T. A. Lzinskaya, Suprnva Stars and th Stllar Wind. Intractin with th Galactic Gas [in Russian] (98).. J. H. Ort, Mn. Ntic. Ry. Astrn. Sc., 59 (9).. I. S. Shklvskii, Suprnva Stars and Prcsss assciatd with thm [in Russian], Nauka, Mscw (97). 3. Yu. P. Raizr, PMTF, N., 9 (93).. A. I. Glubv, A. A. Slv v, and V. A. Trkhin, PMTF, N. 5, 33 (978). 5. V. P. Bashurin, A. I. Glubv, and V. A. Trkhin, PMTF, N. 5, (983).. V. A. Vshivkv, G. I. Dudnikva, Yu. P. Zakharv, A. M. Orishich, and A. G. Pnmarnk, A study f cllisinlss intractin prcsss btwn a plasma clud and a magntizd backgrund at high Alfvn-Mach numbrs. Physics f csmic and labratry plasmas [in Russian], Nvsibirsk (989). 7. Yu. A. Brzin, M. P. Fdruk, and P. V. Khnkin, Fizika Plazmy, 3 (988). 8. S. T. Surzhikv, Fizika Plazmy, 8 (). 9. D. W. Kpman, Phys. Fluids (959 (97).. Yu. A. Brzin, V. A. Vshivkv, Yu. P. Zakharv t al., Exprimntal and numrical study f a cllisinlss ambiplar mchanism fr th intractin f plasma flws in th absnc f a magntic fild [in Russian], ITPM SO AN SSSR, Prprint N. 7-8, Nvsibirsk (98). 3

11 . C. S. Wu, D. Winsk, Y. M. Zhu t al., Spac Sci. Rv. 3, 3 (983).. K. Papadpuls, J. Gphys. Rs., 38 (97). 3. Yu. A. Brzin, V. A. Vshivkv, G. I. Dudnikva, and M. P. Fdruk, Fizika Plazmy 8, 57 (99).. Yu. A. Brzin and V. A. Vshivkv, Particl-in-cll Mthds in Rarfid Plasma Dynamics [in Russian] Nauka, Nvsibirsk (98). 5. Yu. A. Brzin and M. P. Fdruk, Mathmatical Mdlling f Nnstatinary Plasma Prcsss [in Russian], Nauka, Nvsibirsk (993).