A Fnt Volu Sch fo th Two Flud Plasa Sst J. Lovch,. Shula Aospac and ngtcs Rsach Poga, nvst of Washngton ICOPS 26-3 Ma 22 Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Abstact A Fnt Volu Sch fo th Two Flud Plasa Sst J. Lovch,. Shula nvst of Washngton Aospac and ngtcs Rsach Poga o 35225 Sattl, WA 9895-225 In ths pap w psnt ou wo on a nucal on-dnsonal two flud plasa solv. W ta th collsonlss, non-latvstc sst of quatons consstng of on contnut, on ontu, on ng, cobnd wth lcton contnut, lcton ontu and lcton ng, coupld wth th full lctodnac Mawll s quatons. Th algoth s a fstod t, scond-od spac fnt volu foulaton of a Ro-tp appoat Rann solv and uss flu lts fo good shoc soluton wthout spuous oscllatons. W addss th ssu of stffnss ntoducd b th spd of lght and th stffnss assocatd wth th stong couplng of th souc ts n th hpbolc sst. oth plct and plct schs a dvlopd and th advantags of th plct sch a dscussd. It s shown how th algoth a b tndd to ultpl dnsons. Th algoth s tstd on vaous nucal and phscal pobls ncludng lctostatc and lctoagntc two flud plasa wavs and shoc pobls copang th two flud sults to th MHD sults. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Intoducton Nucal thods fo solvng th MHD quatons hav bn studd tnsvl []. Ths sst of quatons s usd to odl pntal fuson dvcs, spac populson concpts, spac plasa phscs and oth pobls nvolvng plasa phscs. Th MHD quatons, howv, do not odl potntall potant plasa ffcts l spac chag dstbutons, fnt Db lngth and fnt go adus ffcts, lcton nta, on cunt o gad- dfts. Futho, nucal solutons nvolvng th Hall t usng th MHD quatons a dffcult. Th MHD sst of quatons s dvd fo th two-flud sst and b solvng th two flud sst all thos ffcts just dscbd can b odld. Ou tchnqu s basd off of wo n coputatonal flud dnacs[3][5] and coputatonal lctoagntcs[4]. ltatl w would l to b abl to odl a wd g spannng th ang fo gas dnac to wal ond gas to an MHD-l flud, n ts cunt ncanaton th solv solvs th full Mawll s quatons ncludng dsplacnt cunt and s thfo abl to odl lctoagntc wav ntactons wth plasas as wll. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup Flud quatons Fo ou flud w ta th nvscd, non-latvstc quatons of flud dnacs wth th addton of th Lont foc. Two spcs a usd, lctons and ons, both of whch a dscbd b th own st of flud quatons. Th two flud quatons allow fo both lcton and on tpatus as wll as lcton and on nta. Th flud quatons a wttn n noald consvatv dvgnc fo as follows.. Spcs Contnut 2. Spcs Montu 3. Spcs ng ( ) q P t 2 γ p ( ) q P t t.4 γ
Mawll s quatons Th full Mawll s quatons a solvd ncludng dsplacnt cunt. Th Posson constant s satsfd ntall and th Magntc flu constant s satsfd autoatcall bcaus th pobl s on dnsonal. Faada s law and th Ap s law a usd to updat th lctc and agntc flds. oth lcton and on cunts contbut to Ap s law. Th quatons a noald such that th spd of lght s.. Ap s Law 2. Faada s Law 3. Posson s quaton 4. Magntc Flu t t q q q q Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup On Dnsonal Fo Th two flud quatons a solvd n on dnson and Rann solvs qu th soluton to on dnsonal pobls acoss ach cll ntfac. In th followng th on dnsonal consvaton fo of th two flud sst s wttn. q q q q q q t In on dnson dos not chang wth t, howv, t s lft as a consvd vaabl so that th algoth can b o asl gnald to 3 dnsons. On dnsonal Mawll s quatons ( ) q p p t 2 ach of th flud quatons has 4 souc ts whch a algbac functons of th consvd vaabls of th flud quatons and of Mawll s quatons. On dnsonal flud quatons
Solv An Appoat Rann Solv s usd to solv ach of th 3 ssts of quatons, lcton flud, on flud and Mawll s quatons. Th doan s bon nto sval fnt volus o clls and a Rann pobl s solvd acoss ach cll ntfac. Th thod chosn follows closl that outlnd b Hatn[2] and s foulatd n th ann dscbd b LVqu[3]. Th thod s fst od accuat n t and scond od accuat n spac cpt at shocs wh th algoth ducs to fst od accuat n spac. Flu lts a usd so that no spuous oscllatons occu na shocs. A Ro at[5] s usd fo th flud quatons. Th Ro at was dvlopd n th al 8 s and s coonl usd n fnt volu solvs. Th Ro at allows th us of a splfd consvatv algoth fo th flud quatons. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Intgal quatons and Dsctaton Th flud quatons and Mawll s quatons a spcfc apls of th gnal consvaton law whch s wttn. Th consvaton law s ntgatd to obtan th ntgal fo Q t Q t F dv ( Q) Ψ( Q) F ( Q) dv Ψ( Q) dv sng th dvgnc tho th volu ntgal s tansfod nto an aa ntgal fo whch th dsctaton s obtand. Th plct dsctaton s Th plct dsctaton s Q Q Q t dv F ( Q) da Ψ( Q) dv A V n n ~ ~ Q Δt / 2 / 2 A V ( ) n n n F F Δt Ψ n n ~ ~ Q Δt / 2 / 2 ( ) n n n F F Δt Ψ Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Iplct Souc Whn th sst s stff an plct tchnqu s ncssa. Th plct soluton s updatd usng a Nwton taton Δt Ψ Q ΔQ Δt ( ) ( ) n A ~ n ~ n Q Q F F Ψ V / 2 / 2 Th plct updat s v fast bcaus th Jacoban s a spl algbac functon of th consvd vaabls. As a sult, no nucal Jacoban nds to b calculatd. Futho th plct stp onl qus a sall at soluton fo ach cll. At v stp th lctoagntc fld and th lcton flud a updatd plctl whl th ons a updatd v 2 stps o so. Th sult s that at v stp a 7X7 sst s solvd fo ach cll to updat th lctoagntc fld and th lcton flud and v 2 stps an X sst s solvd fo ach cll whn th on flud s updatd as wll. Faada s law and th contnut quatons a solvd plctl bcaus th a hoognous quatons. Th soluton to th lna sst fo ach cll s pfod usng about 3 stc Gauss Sdl tatons fo v Jacoban stp. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup Iplct Stup Th Jacoban fo th full X plct updat follows. Rcall that all hoognous quatons a updatd plctl. Th plct stp s pfod fo ach cll. In stff pobls ost of th t s spnt dong th plct updat so a quc plct stp s potant. Ψ Q Q Ψ q q wh and
plct Souc Whn th souc ts a not stff th souc can b updatd plctl. Q A V n n ~ ~ Q Δt / 2 / 2 ( ) n n n F F Δt Ψ Th plct updat s slghtl fast than th plct updat whn th sst s not stff. Fo non-stff pobls, ost of th calculaton t s spnt calculatng th nucal flus. Fo stff pobls th plct soluton qucl blows up whl th plct soluton ans stabl. In od to spd up th algoth w hav td updatng th lctoagntc fld plctl fo sval stps and thn dong an plct updat wth th flud quatons, howv, w hav found that fo vn odatl stff pobls ths tchnqu can poducs un-phscal oscllatons and fquntl lads to aplfng solutons whch vntuall blow up. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Ro Mat A ~ A consvaton law can b wttn wth A dsct Jacoban can b calculatd fo an consvaton laws and s oftn usful n dsgnng ffcnt consvatv algoths. Fo th Mawll sst th dsct Jacoban s th sa as th act Jacoban. A Ro at s usd fo th dsct Jacoban of ach of th flud ssts. Th Ro at satsfs th followng popts. [ F F ] A[ Q Q ] L F Q A R ~ wh L Q Q A t F A Is th Jacoban. Q R A ~ aps Q F QL QR, A ~ A ~ As Th gnvctos of A( Q) a lnal ndpndnt Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Nucal Flu Th Nucal Flu s calculatd accodng to LVqu[3] n th followng wa. λ / 2 ~ Δt n F / 2 λ / 2 / 2 w / 2 / 2 2 λ Δ Is th th gnvalu of th dsct Jacoban calculatd at I/2 w / 2 Is th th gnwav of th dsct Jacoban calculatd at I/2 / 2 Is th th ght gnvcto of th dsct Jacoban calculatd at I/2 ~ sng th ltd gnwav And th nod flu lt w~ w / 2 / 2 S( θ ) S( θ ) n od(,θ ) w Wh θ and / 2 f / 2 > o w I / 2 I λ I / 2 f 3 λ / 2 < Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Langu Wavs As pat of th cod valdaton w hav tstd th algoth on a spl lctostatc wav. Th oscllaton fqunc and apltud ag wth that pdctd b lna tho. Th plct solv poducs slowl aplfng wavs whl th plct solv poducs a slowl dcang wav. In a shoc pobl v sla to th on dscbd on th nt pag wthout th fld, a Langu wav popagats fo th shoc a shot t nto th sulaton. Notc th wav stpnng that appas n th wavs n th ght half of th shoc. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup o and Wu MHD Shoc[] Idal MHD ntal condtons p p p 7.5 5 5 2 3 5 5 3 p p L 7.5 5.25 5.25 2 3 6 6 3 p p R Th ntal condtons a gvn n ts of ptv two flud vaabls. Ths condtons a quvalnt to th MHD ntal condtons gvn on th lft. Idal two flud ntal condtons L s th ntal condton on th lft half of th doan and R s th ntal condton on th ght half of th doan. Th ntal condtons a chosn so that all MHD wavs tavl at oughl th sa spd. 7.5 2 3 4 p L 7.5.25 2 3 5 p R
MHD Shoc Scalng Th ntal condtons of th pvous pag suggst that th s on unqu two flud soluton to th o and Wu MHD shoc. H w phas that t s not th ntal condtons whch a potant, but th MHD wav spds. On can agn changng th plasa s o ncasng th agntc fld and th nub dnst whl png th MHD wav spds th sa. Ths aounts to nothng o than changng th Db lngth and go adus whl png th ato th sa. Th stuctu of th o and Wu MHD shoc s copltl dtnd b th Db lngth and go adus povdd th spcs ass ato s th sa and th spcs chag to ass ato s th sa. In th followng, th non-dnsonalaton fo th two flud shoc wth vaabl Db lngth and go adus s psntd. Th Db lngth s d and th on go adus s. Spcs Montu Spcs ng Ap s Law Posson s quaton t P t t d ( P ) d 2 2 ( ) ( ) ( ) Th lcton chag to ass ato s 3 and on chag to ass ato s and on to lcton ass ato s. It should b cla now that f th Db lngth and go adus a sall th pobl bcos stff and w sa th flud-lctoagntc couplng s stong. Whn th th Db lngth and go adus a lag th pobl bcos lss stff and th flud-lctoagntc couplng s wa. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Two Flud Shoc, MHD lt In th followng w hav usd a sall Db lngth and go adus to t and atch th MHD soluton to th o and Wu shoc. As can b sn blow uch of th shoc stuctu s captud dspt havng fnt Db lngth and go adus. Th t stp fo ths sulaton s oughl.5 ts th plasa fqunc. Fo stff pobls th t soluton s donatd b th plasa fqunc not th spd of lght. Notc that ach gd cll s oughl 8 Db lngths and.8 on go ad. Th a 4 gd clls. In th dnst plot th bup on th ght s th alfvn wav, and th afacton wav on th lft s th fast agntosonc wav. oth plots atch th MHD soluton wll. Rducng th gd soluton to stll gvs dcnt sults as can b sn n th shoc copason lat on. Ths ss to suggst that fo ths pobls t s not ncssa to solv th Db lngth. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Two Flud Shoc, MHD lt Th MHD quatons do not pdct a bul vloct n th Z dcton. In th two flud cas, th lctons hav nta and th lcton cunt suppots ost of th agntc fld. Consquntl a bul Z vloct s pdctd b th two flud quatons. Th MHD quatons do not pdct a Z agntc fld. Howv, n th two flud sst, th s nducd b a J whch s causd b th J cossd wth th. Includng th hall t and th at of chang of cunt allows th agntc fld to b solvd n th two flud sst. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Two Flud Shoc, MHD lt vang gd H w hav changd th gd soluton on th sa pobl as on th pvous pag, d3.6-5, g3.6-3. At a gd soluton of 5 ost of th stuctu that appas at a gd soluton of 4 s solvd. Fo Db lngth alon, on ght pct that th soluton would not b wll solvd untl th doan s splt nto about 3, gd clls, but t appas that w can avod solvng th Db lngth scals and stll gan ost of th MHD stuctu wth addd two flud phscs. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Two Flud Shoc, MHD lt vang gd Cunt sps whch suppot th agntc shocs a uch btt solvd at hgh solutons. Th low solutons do a good job of atchng th lag wavlngth stuctu. In th two flud sst, fo v stff pobls, not onl do w nd to b abl to solv shoc dscontnuts, w also nd to b abl to solv cunt shts whch tavl ppndcula to th shoc. At hgh soluton, a cunt sht s clal solvd at.6 unts n th plot of v. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Two Flud Shoc, MHD lt vang couplng Th two flud sst adts a contnuous ang of solutons fo th o and Wu MHD shoc bcaus th couplng can b adjustd whl png th MHD wav spds th sa. Ths plot shows a squnc of two flud solutons to th MHD shoc pobl. It s appant that ncasng th couplng povs th agnt btwn th MHD soluton and th two flud soluton. As statd pvousl th v dffs substantall bcaus n th dal MHD soluton th cunt s psntd b th cul of th agntc fld and n th two flud sst th cunt s th cobnaton of th lcton and on cunts. Th two flud sst pdcts stong dpndnc of v on th couplng. gd clls w usd n ths copason sulatons. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Gas Dnac Shoc Th Db lngth and go adus a ncasd to th pont that th lctoagntc fld s dcoupld fo th flud and th two fluds ov ndpndntl. Th lcton shoc popagats at th lcton acoustc spd and th on shoc at th on acoustc spd, th plot blow plots th lcton flud dnst at /4 th t that th on dnst s plottd. oth spcs tavl slow than th spd of lght wth th lctons tavlng at a au of.3c. Th shoc n poducs a shoc n and th lctoagntc wav qucl popagats out of th doan at th spd of lght wthout an ntacton wth th flud. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Two Flud Shoc, Gas Dnac lt In ths sulatons w dcas th couplng n th o and Wu shoc pobl to th pont that w gt gas dnac shocs. Th MHD dnst pofl whch s calculatd as th wghtd avag of th lcton dnst and th on dnst s plottd blow. Th dnst pofl vas wldl as Db lngth and go adus a changd. As th couplng s ncasd, lss and lss of th shoc s suppotd b th dsplacnt cunt and o s suppotd b th flud cunt. In patcula onl hgh fqunc wavs a suppotd b th dsplacnt cunt. Ths s th assupton that s ad n MHD, and th valdt of ths assupton s donstatd n th two flud soluton blow and on th nt pag. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Two Flud Shoc, Gas Dnac lt As th couplng s ncasd hgh and hgh fquncs of th shoc a fon nto th flud. In th plot blow t s appant that fo wa couplng, all th lctoagntc wavs tavl na th spd of lght. As th couplng s ncasd th low fqunc wavs tavl slow than th hgh fqunc wavs untl th pont that th shoc s fon nto th flud. Notc that on th hgh dnst sd of th shoc on both plots th couplng s stong and lctoagntc wavs a hgh fqunc than on th low dnst sd. In both cass th hgh fqunc lctoagntc wavs tavl fast than th low fqunc wavs. Th plot blow also donstats that as th couplng s ncasd th potanc of th dsplacnt cunt s ducd. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
tnson to 3 Dnsons and Futh Wo tnson to 3 Dnsons. 3 dnsonal appoat Rann solvs fo flud quatons a wll undstood and a coonl usd n ndust and acada. Th tnson to 3 dnsons s spl bcaus th sa solv can b usd cpt now a on dnsonal pobl ust b solvd acoss ach cll fac. 2. Appoat Rann Solvs fo Mawll s quatons wth souc ts a latvl nw[4]. Anoth gaduat studnt, Chs Abl, has shown that spl fst od appoat Rann solvs fo Mawll s quatons wth souc ts wo on tangula shs. It s blvd that hgh od thods wll wo on ctangula shs. 3. In 3 dnsons th souc ts can stll b updatd plctl, cll b cll, on plct updat p cll. As a sult th plct coputaton t wll not ncas whn th sst s tndd to 3 dnsons. Futh wo. Th stff pobls a t consung and ths wll bco o potant n 3 dnsons. Rsolvng th spd of lght as th sst coputatonal ntnsv, but solvng th plasa fqunc can a th pobl vn had to solv. 2. plctl satsfng Posson s quaton wll b potant; w a cuntl wong on thods to dal wth ths constant. 3. Th agntc flu constant wll nd so consdaton n 3 dnsons. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Sua & Conclusons Iplct and plct schs of a on-dnsonal appoat Rann solv fo th non-latvstc dal two flud plasa sst hav bn dvlopd. Tadtonal tchnqus of flud dnacs ncludng th Ro at and flu lts hav bn appld to th two flud sst. Th two flud quvalnt of th o and Wu MHD shoc has bn dscbd and tstd allowng fo vang flud-lctoagntc couplng. Th sa algoth has bn shown to poduc coct solutons to lctostatc, lctodnac, gas dnac and MHD-l pobls. W hav dscussd how ths solv can b tndd to ultpl dnsons. Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup
Rfncs [] M. o and C.C. Wu, An pwnd Dffncng Sch fo th quatons of Idal Magntohdodnacs, JCP 75 998, pg 4-422 [2] A Hatn, Hgh Rsoluton Schs fo Hpbolc Consvaton Laws, JCP 49, 983 pg 357-393 [3] Randall J LVqu, Fnt Dffnc Mthods fo Dffntal quatons, Lctu Nots fo AMATH 585-586, nvst of Washngton [4] C.D. Mun, R. Schnd, and. Vos, A Fnt-Volu Mthod fo th Mawll quatons n th T Doan, Sa Jounal on Scntfc Coputng 22 (2), no. 2, 449-475. [5] P.L. Ro, Appoat Rann Solvs, Paat Vctos, and Dffnc Schs JCP 35 (pntd 997) pg 25-258 Aospac & ngtcs Rsach Poga - nvst of Washngton Plasa Dnacs Goup