22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 2: Derivation of Ideal MHD Equation

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1 .65, MHD Thory of Fuson Systms Prof. Frdbrg Lctur : Drvton of Idl MHD Equton Rvw of th Drvton of th Momnt Equton. Strtng Pont: Boltzmnn Equton for lctrons, ons nd Mxwll Equtons. Momnts of Boltzmnn Equton: consrvton of mss, momntum nd nrgy. mss df F m v dv momntum c mv nrgy 3. Accountng: ( ) v = u,t + v, u = flud vlocty, v = rndom vlocty n u = F dv dnsty = vf dv = n v flud vlocty P = n m vv prssur tnsor p = mn v 3 sclr prssur h nm = v v ht flux R = m vc d β v frcton du to collsons Q m v = C βdv ht gnrtd du to collsons Gnrl Flud Equtons B E = B = 0 E σ B =μ J+ E = c 0 n ( ) + n u = 0.65, MHD Thory of Fuson Systms Lctur Prof. Frdbrg Pg of

2 du mn = qn ( E+ u B) P + R 3 dt n + P : u Q = h, ( n) σ= n ( ) J = nu n u Physcl Assumptons Ldng to Idl MHD. Momnt qutons s thy now stnd r xct, but not closd.. Crtn ssumptons ld to closur - flud MHD modl Asymptotc Assumptons. MHD s concrnd wth low frquncy - long wvlngth mcroscopc bhvor. Th frst smplfcton of th flud qutons lmnts short wvlngth, fst tm scl phnomn: wll stsfd ssumptons xprmntlly 3. Asymptotc ssumptons chng bsc mthmtcl structur of th tm voluton. spd of lght lctron nrt 0 Frst Asymptotc Assumpton c. Mxwll qutons low frquncy Mxwll qutons. Formlly lt 0 0 E B = u0j+ μ J 0 c nglct dsplcmnt currnt 0 n n = E 0 qusnutrlty 3. Equtons r now Gllln nvrnt 4. Conons for vly: v no plsm osclltons T ω ωp λd ω p.65, MHD Thory of Fuson Systms Lctur Prof. Frdbrg Pg of

3 ω v T v c T k no hgh frquncy wvs 5. Not: n = n n dos not mply E or E = 0. Only tht 0 En Scond Asymptotc Assumpton m 0. Th lctron rspons tm s ssntlly nstntnous bcus m m. W thn nglct lctron nrt n th momntum quton 0 n E+ u B P + R ( ) 3. Conons for vly ω ωp λd no lctron plsm osclltons to B ω ω r no lctrons cyclotron osclltons c c 4. Both c, m 0 ssumptons r wll stsfd for MHD bhvor Subtl Effct. Nglct of lctron nrt long B cn b trcky. For long wvlngths, lctrons cn stll rqur fnt rspons tm vn though m s smll. Ths s rgon of th drft wv 3. W shll s tht MHD consstntly trts moton poorly, but for MHD bhvor, rmrkbly ths dos not mttr!! 4. To trt such bhvor mor sophstctd modls r rqurd. Th rsultng nstblts r much wkr, (nd stll mportnt) thn for MHD. Th two Flud Equtons wth Asymptotc Assumptons B n E = B = 0 + nu = 0 ( ) n + nu = 0 B =μ0n u u n = n =n.65, MHD Thory of Fuson Systms Lctur Prof. Frdbrg Pg 3 of

4 du mn n( E u B) P R + + = n E u B P R ( + ) + + = Sngl Flud Equtons. Introduc sngl flud vrbl 3 dt n + P : u + J h = Q v = u th momntum of flud s crrd by ons snc m = 0 p = p + p totl prssur ρ= mn mss dnsty ( ) J = n u u currnt dnsty. Us ll nformton!!. Ths s not trvl!! Intlly th unknowns r E, B, J, V, n, p (9 vrbls). Th fnlly unknowns r E, B, J, V, n, p (4 vrbls) 3. Mxwll qutons OK s s n low frquncy form 4. Mss consrvton. M on ρ + ρ v = 0 b. (on-lctron) n ( u u ) = J = 0 Ths s utomtc from th low frquncy Mxwll qutons B =μ0 J J =0 5. Momntum Equton (on + lctron) dv. ρ n ( u u ) B + ( P + P ) = R + R J B ( p + p) Ι+Π +Π dv m v c mv c + = 0.65, MHD Thory of Fuson Systms Lctur Prof. Frdbrg Pg 4 of

5 dv b. ρ J B + p = ( Π + Π ) 6. Elctron Momntum quton. R E + u B = P n J J u = u = v n n E + v B = R P J B n + b. 7. Enrgy Equton (ons) P: u 3 d p. n p u Q h : n + = Π u 3 dp 3 p dn b. : n c. : n n dn + nv = 0 = + v n + n v = n v p dn p v = n d. +: 3 dp 5 p dn 3 d p n n 53 = n 53. d p = Q h : r r Π v ρ 3ρ r=5/3 8. Enrgy Equton (lctrons). d p J p d = Q r r h Π : v + + Π r : ρ 3ρ n ρ n from d from Π : u.65, MHD Thory of Fuson Systms Lctur Prof. Frdbrg Pg 5 of

6 b. d = + v =on convctv drvton Assumptons Ldng to Idl MHD. Phlosophy: Idl MHD s concrnd wth phnomn occurrng on crtn lngth nd tm scls.. Ordrng: Usng ths, w cn ordr ll th trms n th on flud qutons. Aftr gnorng smll trms, w obtn dl MHD. 3. Sttus: At ths pont only th ssumptons c, m 0 hv bn usd n th quton Chrctrstc Lngth nd Tm Scls for Idl MHD v ω. T. x k mcroscopc MHD phnomn 3. v vt 4. mcroscopc lngth 5. v mcroscopc on vlocty T 6. vt corrspondng mcroscopc tm scl Two Approchs to Idl MHD A. Collson domntd plsm: rgons lmt to dl MHD B. Collson fr lmt: lso works but for subtl rsons Collson Domntd Lmt. Th lctrons nd ons r ssumd collson domntd. Ths s th bsc rqurmnt to kp th prssur sotropc. Mny collsons kp prtcl clos togthr. Ths llows us to dvd th plsm nto smll flud lmnt nd provds good physcl dscrpton. 3. Thr r conons for collson domntd plsm. on th tm scl of ntrnl thr r mny collsons, so th plsm s nr mxwllon ons: on-on coulomb collsons domnt.65, MHD Thory of Fuson Systms Lctur Prof. Frdbrg Pg 6 of

7 lctrons: lctron-on, lctron-lctron collsons r comprbl vtτ ons: ωτ lctrons: vt m vtτ ωτ ωτ τ m Rcll: τ τ ( m m ) τ nd τ ( m m ) Th on conon s most svr τ EQ v T τ MHD Lmt b. Th mcroscopc lngth scl must b much lrgr thn th mn fr pth for collsons. λ = v τ T λ vtτ ons = (sm s bfor) λ vtτ vtτ lctrons (sm s ons). Us th collson domntd ssumpton to obtn dl MHD. Svrl donl ssumptons wll lso b rqurd 3. Vrous momnts n th qutons r pproxmtd by clsscl trnsport thory of Brgnsk. 4. Trnsport coffcnts cn lso b drvd n th homwork problms Rducton of Flud Equton. Mxwll Equtons OK. Mss consrvton OK 3. Momntum Equton u. ons: Π μ u u μ 3 vscosty.65, MHD Thory of Fuson Systms Lctur Prof. Frdbrg Pg 7 of

8 u b. lctrons: Π μ J c. Not: u = v n J p T r nv Bnv Bv T ssum smll gyro rdus d. u u : smll dffrnc n th flow vlocts gnrt mcroscopc currnt dnsty J, but u u v. Ordrng: u T Π μ μ v T μ m μ Π m Π Π T P μv μ ntτ vscosty coffcnt p Π τ v collson domntd ssumpton p T Both Π trms r nglgbl n momntum quton f. dv ρ = J B p momntum quton 4. Ohms Lw 4 3 E v B R P J B n + = + Hll ffct Elctron dmgntsm ω r Rsstvty.65, MHD Thory of Fuson Systms Lctur Prof. Frdbrg Pg 8 of

9 . rl / 4 J nv smll gyro rdus ssumptons b. R rsstvty momntum trnsfr du to collsons m R = nηd, η = n τ 3 / 4 J r r τ v TB τb m VTτ m m m n c. r m m v τ T d. Th plsm must b lrgr nough so tht rsstv dffuson dos not ply n mportnt rol. 5. Enrgy quton v. ons: Π p p, J n p p, J n vp b. lctrons: Π ( ) ( Π ) d p c. r r Q h = 3 ρ ρ d p d. r r Q h = 3 ρ ρ. = κ κ T h T f. h = κ T κ T domnnt contrbuton s from thrml conducton g. In gnrl κ κ h. Q ( T ) n T = τ q qulbrton. Q ( ) n T T J R = + qulbrton plus ohmc htng τ n q.65, MHD Thory of Fuson Systms Lctur Prof. Frdbrg Pg 9 of

10 j. Not: cons. of nrgy Q + Q J R n = 0 k. Compr JR m r ω p = n m vtτ smll ohmc htng n MHD tm scl d p l. r r n ( T ) = κ + n 3 ( T T ) ρ ρ EQ d p m. r r n ( T ) = κ + n 3 τ ( T T ) ρ ρ EQ n. But MHD s sngl flud modl - prssur, tmprtur τ T T o. Ths occurs f s vry smll, forcng EQ τ p. Smll τ EQ rqur nt τ EQ ωp or ωτeq q. m v τ m T Ths s mor svr thn th collson domntd momntum conon nrgy qulbrton τ momntum qulbrton τ. r. If ths s tru thn st nformton T T T nd nformton (dd qutons) d p ( ) r r ρ 3ρ = κ + κ T But ( ) κ m m κ, κ nt τ m Thus T m vt ωp m κ τ.65, MHD Thory of Fuson Systms Lctur Prof. Frdbrg Pg 0 of

11 Ths gvs Idl MHD Equton B E = B = 0 B = u0j n = n = n p + ρ v = 0 dv ρ = J B p E+ V B = 0 d p r ρ = 0.65, MHD Thory of Fuson Systms Lctur Prof. Frdbrg Pg of

Fundamentals of Continuum Mechanics. Seoul National University Graphics & Media Lab

Fundamentals of Continuum Mechanics. Seoul National University Graphics & Media Lab Fndmntls of Contnm Mchncs Sol Ntonl Unvrsty Grphcs & Md Lb Th Rodmp of Contnm Mchncs Strss Trnsformton Strn Trnsformton Strss Tnsor Strn T + T ++ T Strss-Strn Rltonshp Strn Enrgy FEM Formlton Lt s Stdy