Plasma Simulation Algorithm for the Two-Fluid Plasma Model

Transcription

1 ELIGIBLE Plasma Smulaton Algorthm for th Two-Flud Plasma Modl U. Shumlak, C. Abrl, A. Hakm, and J. Lovrch Arospac & Enrgtcs Rsarch Program Unvrsty of Washngton, Sattl, USA Confrnc on Computatonal Physcs Sptmbr 2004

2 Abstract Many currnt plasma smulaton cods ar basd on th magntohydrodynamc (MHD) modl whos drvaton nvolvs svral asymptotc and smplfyng assumptons that svrly lmt ts applcablty, partcularly for Hall ffct physcs. Th two-flud plasma modl only assums local thrmodynamc qulbrum wthn ach flud (on and lctron flud). Thrfor, th twoflud plasma modl mor accuratly rprsnts th approprat physcal procsss. An algorthm s prsntd for th smulaton of plasma dynamcs usng th two-flud plasma modl. Th two-flud quatons ar drvd n consrvatv/dvrgnc form and an approxmat Rmann solvr s dvlopd. Th algorthm uss th approxmat Rmann solvr to comput th fluxs of th lctron and on fluds at th computatonal cll ntrfacs and an upwnd charactrstc-basd solvr to comput th lctromagntc flds. Th sourc trms that coupl th fluds and flds ar tratd mplctly to rlax th stffnss. Th algorthm s valdatd wth th coplanar Rmann problm, Langmur plasma oscllatons, and th lctromagntc shock problm. Th lctromagntc shock problm has bn smulatd wth th MHD plasma modl. In nondmnsonal form th two-flud plasma modl has thr physcally rlvant paramtrs that dfn th mportanc of th two-flud ffcts, th lctron to on mass rato m /m, th Dby lngth λ D, and th on Larmor radus r L. Th MHD modl assums m /m = 0, λ D = 0, and r L = 0. By adjustng th valu of r L, th two-flud modl can approach thr th unmagntzd gas dynamc lmt r L or th MHD lmt r L 0. Comparsons ar prsntd that show ths lmts for th lctromagntc shock problm. Furthrmor, a physcal analyss of th ntrmdat valus of r L ar prsntd.

3 Background and Motvaton Plasmas may b most accuratly modld usng kntc thory. Th plasma s dscrbd by dstrbuton functons n physcal spac, vlocty spac, and tm, f(x, v, t). Th voluton of th plasma s thn modld by th Boltzmann quaton. f f q f f + vα + ( E+ vα B) = t x m v t α α α α α α α collsons for ach plasma spcs α = ons, lctrons. Coupld wth Maxwll's quatons ths provds a complt dscrpton of th plasma dynamcs.

4 Kntc Modl Th Boltzmann quaton s svn dmnsonal. As a consqunc plasma smulatons usng th Boltzmann quaton ar only usd n vry lmtd applcatons wth narrow dstrbutons, small spatal xtnt, and short tm duratons. Th svn dmnsonal spac s furthr xacrbatd by th hgh vlocty spac that s unusd xcpt for th tal of th dstrbuton or nrgtc bams. Boundary condtons ar dffcult to mplmnt n kntc smulatons.

5 Partcl n Cll (PIC) Modl Partcl n cll (PIC) plasma modl appls th Boltzmann quaton to rprsntatv suprpartcls whch ar far fwr (10 7 ) than th numbr of partcls n th actual plasma (10 20 ). Th partcls fll th sx dmnsonal phas spac and ar trackd n tm. PIC smulatons hav smlar lmtatons as smulatons usng kntc thory du to statstcal rrors causd by th rlatvly fw suprpartcls. Boundary condtons can also b dffcult to mplmnt n PIC smulatons.

6 Sngl-Flud (MHD) Modl Smplr plasma modls ar gnratd by takng momnts of th Boltzmann quaton and avragng ovr vlocty spac for ach spcs. Two-Flud Plasma Modl Th two-flud quatons ar combnd usng smplfyng assumptons to form th sngl-flud MHD modl. low frquncy, zro lctron mass, zro Larmor rad, zro Dby lngth Plasma smulaton algorthms basd on th MHD modl hav bn succssful n modlng plasma dynamcs and othr phnomna.

7 Lmtatons of th MHD Modl Th MHD modl has lmtatons that ar ntroducd n ts drvaton. Th gnraton of non-solnodal magntc flds. Th Hall ffct and damagntc trms ar oftn nglctd. Ths trms rprsnt th sparat motons of th ons and lctrons and account for on currnt and th fnt Larmor radus of th plasma consttunts. Hall thrustrs, MPD thrustrs, Lorntz forc thrustrs Anod and cathod fall Plasma currnt drv Magntc rlaxaton Spac plasma ffcts

8 Th Hall Effct In gnral th Hall trms ar dffcult to stablz bcaus thy lad to th whstlr wav branch of th dsprson rlaton whch can lad to larg propagaton spds. 10 ωτ A ω c τ = 0 (MHD) ω c τ = 1 ω c τ = 2 ω c τ = k z

9 Hall Effct Physcs Sm-mplct tchnqus hav bn mplmntd to trat th Hall ffct trms.* Th mthod corrcts th dal MHD voluton of th magntc fld and nrgy to account for th Hall ffct. Th oprator uss a 45 pont stncl n 3D. Th mthod works for small Hall paramtrs but bcoms slow to convrg or unstabl for larg Hall paramtrs. * Harnd and Mkc, JCP 83, 1 (1989).

10 Two-Flud Plasma Modl Th two-flud modl s drvd by takng momnts of th Boltzmann quaton for ach spcs. Th modl has th sam dmnsonalty as th MHD modl xcpt thr ar two fluds. Th only approxmaton mad s local thrmodynamc qulbrum of ach flud but not wth th othr flud. Th modl conssts of govrnng quatons for th contnuty, momntum, and nrgy of th lctrons and ons. Maxwll s quatons complt th modl.

11 Consrvaton Form ( ) ( ) ( ) ( ) = n n n m n m n p n p p m n p m n n n t R E j R E j R B j E R B j E j j I j j I j j j j j j 0 0 ε ε ε ε

12 Charactrstcs n 1D Q F Q Q + = + A = t x t x 0 Hyprbolc quaton st, so th flux Jacoban A hav complt sts of gnvalus jx jx γt jx jx γt λ =, ±,, ± n n m n n m Frst ordr fluxs ar splt and upwnd dffrncd to form a Rotyp,* approxmat Rmann solvr 1 1 F = ( F + F) l ( Q Q ) λ r k + 1 k k k * Ro, JCP 135, 250 (1997).

13 Approxmat Rmann Solvr for Two-Flud Plasma Modl Ovrall soluton s composd of th solutons to th Rmann problm dfnd at th grd ntrfacs. t v ntrfac -c v +c -v -c -v -v +c v Stat 1 Stat 2 x Informaton propagats along charactrstcs.

14 Elctromagntc Fld Modl Maxwll s quatons for th lctromagntc flds ar rqurd to comput th RHS and complt th plasma modl. E = n ( n)/ ε B = 0 o B = E t ε E = B/ µ t j j o o φ t A t 2 2 φ µ oεo = n ( )/ 2 n εo 2 2 A µ oε o = µ 2 o j+ j ( ) whr th magntc vctor potntal A and th lctrc scalar potntal φ ar ntroducd.

15 Elctromagntc Fld Modl For th lctromagntc fld modl, Maxwll s quatons ar solvd B + E = 0 t E ε o B µ t o = j j subjct to th ntal condtons ( ) / 0 / E = n n ε and B = 0 Th fld quatons ar solvd wth a fnt-volum, upwnd schm* whr th quatons ar wrttn as Q + F = S t * Shang and Fthn, JCP 125, 378 (1996).

16 Multdmnsonal Smulatons Satsfyng th dvrgnc constrant condtons on th lctromagntc flds bcoms mor dffcult du to n-plan flds. Corrcton potntals* ar addd to th flds algorthm. B = E ψ t E ε o = B/ µ o j j εo φ t φ E = n ( n) / εo t ψ + B = 0 t * Munz, Schndr, and Vos, SIAM JSC 22, 449 (2000).

17 Tratmnt of th Two-Flud Sourc Trms Th two-flud quaton st contans sourc trms whch rsult from th lctromagntc flds: th Lorntz forc and Joul hatng (fluds) and currnt dnsts (flds). Whn th sourc trms ar larg (larg Lorntz forc), thy ar calculatd mplctly wth an tratv mthod. Q n+ 1 Q t n ( ) = F F + S n n n Th sourc trms ar lnarzd, and a local sourc Jacoban S Q s dfnd. 1 S Q Q = + t Q t k k n n+ 1 k ( ) ( ) n n k k Q Q F+ 1 F 1 S 2 2

18 Normalzaton of th Two-Flud Modl In th normalzaton procss svral mportant physcal paramtrs ar dntfd. Ionzaton stat, Z (st to 1). Ion / Elctron mass rato, m /m (st to 1836, hydrogn plasma). r = v ω = v m B Ion Larmor radus, r L, whr. L Th c Th λ = v ω = ε kt n Dby lngth, λ D, whr. D Th p 0 / 2

19 Bnchmarks and Valdaton Tsts Th algorthm has bn appld to th classcal 1-D MHD shock problm and rvals an ffct of two-flud physcs and th transton btwn a gas dynamc shock and an MHD shock. 1-D Elctromagntc Plasma Shock* Th algorthm has also bn appld to svral 2-D problms and compard to publshd rsults. 2-D Cylndrcal Elctromagntc Plasma Shock Collsonlss Magntc Rconncton * Bro and Wu, JCP 75, 400 (1988).

20 Elctromagntc Plasma Shock

21 Elctromagntc Plasma Shock

22 Elctromagntc Plasma Shock

23 Elctromagntc Plasma Shock

24 Elctromagntc Plasma Shock

25 Elctromagntc Plasma Shock

26 Elctromagntc Plasma Shock

27 Fast Wav Analyss n th Plasma Shock Th wav vctor s prpndcular to th dscontnuty and paralll to th longtudnal magntc fld,. Th plasma dsprson rlaton for ths cas ylds th lft and rght crcularly polarzd wavs (L mod and R mod), n addton to th slowr Alfvn wavs. Th low frquncy porton of th lowr branch of th R mod s th whstlr wav. k " B ck ωp = 1 2 ω ω ω ω ( ± ) c

28 Elctromagntc Plasma Shock 5 0RGH / 0RGH ω 5 0RGH $OIYHQ 0RGH N

29 Cylndrcal Plasma Shock Th lctromagntc plasma shock smulaton s rpatd n a cylndrcal gomtry. No longtudnal magntc fld s appld. \ An azmuthal currnt sht supports th dscontnuous jump n th magntc fld. Th currnt sht braks du to a Klvn-Hlmholtz nstablty whn th sht thcknss dcrass blow th lctron skn dpth (c/ωp). [

30 Collsonlss Magntc Rconncton Magntc rconncton n a currnt sht tars and rconncts magntc flux through th orgnal currnt layr. Rconncton usually rqurs rsstvty, but t can also occur wthout collsons f two-flud ffcts ar ncludd.* Shown s th magntud of B x rvalng rconnctd flux through th currnt layr. * Brn t al., JGR 106, 3715 (2001).

31 Collsonlss Magntc Rconncton Brn t al. tracks th tm voluton of th rconnctd magntc flux for dffrnt plasma modls. Two-flud modl rsults ar also shown.

32 Summary Th two-flud plasma modl rlaxs som of th constranng assumptons usd n MHD and modls mor complt physcs. W hav dvlopd a nw algorthm basd on a Ro-typ approxmat Rmann solvr for th two-flud plasma modl. W hav mplmntd an lctromagntc fld solvr and an mplct tratmnt of stff sourc trms. Th algorthm has bn bnchmarkd to publshd problms and producs accurat rsults. Smulatons hav rvald mportant physcal nsght and dmonstrat th mportanc of two-flud ffcts. MHD Plasma Shock (1-D and 2-D) Collsonlss Magntc Rconncton