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1 !"#$%'()%"*#%*+,-./-*+01.2(.* *!"#$%"'()'*+,-."/0.%+1'23"45'46'7.89:89'/' ;8-,"$ ,8:+#9' Dr. Ptr T. Gallaghr Astrphyscs Rsarch Grup Trnty Cllg Dubln :2;$-$(01*%<*=,-./-*=0;"%/;"-*!"#$%#!'()*%()# +,)-"(./#012"( 3*1*) 4,$50,671*) 87)21*) 9*"5:%#)) ;<7#1*) 3*%()5$*=9*"5:%#)) ;<7#1*) +,)-"(>7,? 37"1/"( >7,?$

2 Undr crtan crcumstancs, apprprat t cnsdr ntr plasma as a sngl flud. D nt dffrntat btwn ns and lctrns. Apprach s calld magnthydrdynamcs (MHD). Gnral mthd fr mdllng hghly cnductv fluds, ncludng salt watr, crnal lps, ISM, tkamaks, tc. Sngl-flud apprach apprprat whn dalng wth slwly varyng cndtns. MHD s usful whn plasma s hghly nsd and lctrns and ns ar frcd t act n unsn, thr bcaus f frqunt cllsns r by th actn f a strng xtrnal magntc fld. >2"?,;@A'2*;F'-)%"*<%$*<',,1*2%"2.;*=,-./-* Can cmbn multpl-flud quatns nt a st f quatns fr a sngl flud. Assumng tw-spcals plasma f lctrns and ns (j = r ): "n j + # $(n j v j ) = 0 %"v m j n j ( j ' +(v j #$)v j * = +$ #P j + q j n j (E + v j, B)+ P j ) (9.1a) (9.1b) Fr a fully nsd tw-spcs plasma, ttal mmntum must b cnsrvd: P = -P As >> th tm-scals n cntnuty and mmntuquatns fr ns and lctrns ar vry dffrnt. Th charactrstc frquncs f a plasma, such as plasma frquncy r cycltrn frquncy ar much largr fr lctrns.

3 Whn plasma phnmna ar larg-scal (L >>! D ) and hav rlatvly lw frquncs (" << " plasma and " << " cycltrn ), plasma s n avrag lctrcally nutral (n # n ). Indpndnt mtn f lctrns and ns can thn b nglctd. Can thrfr trat plasma as sngl cnductng flud, whs nrta s prvdd by mass f ns. Gvrnng quatns btand by cmbnng Eqns Frst dfn macrscpc paramtrs f plasma flud: " m = n + n J = n q v + n q v v = n v + n v n + n P = P +P Mass dnsty Elctrc currnt Mass vlcty Ttal prssur tnsr D:E*/-..*-"*(0-$?;*(%".;$G-)%"* Usng Eqn 9.1a: "n j + # $(n j v j ) = 0 Multply by q and q and add cntnuty quatns t gt: "# + $ %(J) = 0 Cntnuty f lctrc currnt whr J s th lctrc currnt dnsty dnsty s " = n q + n q J = n q v + n q v and th lctrc charg Nw multpl Eqn. 9.1a by and, "# m + # m $ %(v) = 0 Mass cnsrvatn whr " m = n + n vlcty s th sngl-flud mass dnsty and v s th flud mass v = n v + n v n + n

4 D:E*HF'-)%"*%<*/%)%"* Equatn f mtn fr bulk plasma can b btand by addng ndvdual mmntum transprt quatns fr ns and lctrns (Eqns. 9.1b). (n + n ) "v = #$ %(P +P )+(n q + n q )E + J B (9.2) Nt that w hav nglctd (v "#)v as w ar dalng wth small prturbatns fr whch th gradnts ar nglgbl. Scnd trn Eqn. 9.2 s zr as plasma s nural. Thrfr, #v " m #t = $% P+ J ' B Equatn f mtn Fr an strpc plasma, "#P = "p whr th ttal prssur s p = p + p and " m #v #t = $%p + J B Equatn f mtn I;";$-,2J;*K0/L.*M-N* Th fnal sngl-flud quatn dscrbs th varatn f currnt dnsty J. Cnsdr th mmntuquatns fr lctrn and ns (Eqn. 9.1b): %"v m j n j ( j ' +(v j #$)v j * = +$ #P j + q j n j (E + v j, B)+ P j ) Multpl lctrn quatn by q / and n quatn by q / and add: = # q $ %P # q $ %P + n q 2 + n q 2 ) ( + E ' * + n q 2 ( ' v + n q 2 + q P + q P v ) +, B *

5 I;";$-,2J;*K0/L.*M-N* Fr an lctrcally nutral plasma q n # q n and usng J = n q v + n q v and v = n v + n v n + n w can wrt = # q $ %P # q $ %P + n q 2 + n q 2 ) ( + (E + v, B) ' * As << => q / >>q / and n q 2 / >>n q 2 /. In thrmal qulbrum, kntc prssurs f lctrns s smlar t n prssur (P! P ) + q + q ) ( + (J, B) ' * + q # q ) ( + P ' * = # q $ %P + n q 2 (E + v B)+ q (J B)+ q P (9.3) I;";$-,2J;*K0/L.*M-N* Th cllsnal trm can b wrttn: P = "q 2 n 2 (v # v ) whr $ s th spcfc rsstvty, q 2 rlats t fact that cllsns rsult frm Culmb frc btwn ns (q ) and lctrns (q ) and ttal mmntum transfrrd t lctrns n an lastc cllsn wth an n s v v. Nw q = -q and n = n and J = n q (v v ) => P = "n q #J Can thrfr wrt Eqn. 9.4 as = # q $ %P + n 2 q (E + v B)+ q (J B)# n 2 q ˆ '% J (9.5) whr $ s a tnsr. Ths s gnralsd Ohm s law.

6 I;";$-,2J;*K0/L.*M-N* Fr a stady currnt n a unfrm B, / = 0,# $P = 0 and B = 0 s that E = $ J => J = 1 / $ E Th lctrc fld E can b fund frm Eqn. 9.5: E = "v # B " J # B + $ %P + ˆ % J + 'J n q n q n q 't Cnsdr rght hand sd f ths quatn: Trm 1: E asscatd wth plasma mtn. Trm 2: Hall ffct. Trm 3: Ambplar dffusn frm E-fld gnratd by dnsty gradnts. Trm 4: Ohmc lsss/jul hatng Trm 5: Elctrn nrta >2/=,2O;*D:E*HF'-)%".* W hav nw drvd th fllwng: "# m = # q $ %P + n q 2 + # m $ %(v) = 0 " m #v #t = $% P+ J ' B (E + v B)+ q (J B)# n q 2 ˆ '% J Nw assum plasma s strpc, s that "#P = "p. Als nglct Hall ffct and ambplar dffusn n gnralsd Ohm s law snc nt mprtant n mst cass. Fr slw varatns, J = cnstant, s can wrt gnralsd Ohm s law as: Rarrangng gvs, 0 = n q 2 (E + v " B)# n 2 q $J J = "(E + v # B)

7 >2/=,2O;*D:E*HF'-)%".* Th smplfd MHD quatns can thrfr b wrttn: "# m + # m $ %v = 0 "v # m m = $p + J ' B J = ((E + v ' B) Flud quatns must b slvd wth rducd Maxwll s quatns fr flds: " # B = µ 0 J " # E = $ %B %t " E = 0 " B = 0 Hr w hav assumd that thr s n accumulatn f spac charg (.., % = 0). Cmplt st f quatns nly whn quatn f stat fr rlatnshp btwn p and n s spcfd.