Collisionless Hall-MHD Modeling Near a Magnetic Null. D. J. Strozzi J. J. Ramos MIT Plasma Science and Fusion Center

Transcription

1 Collisionlss Hall-MHD Modling Na a Magntic Null D. J. Stoi J. J. Ramos MIT Plasma Scinc and Fusion Cnt

2 Collisionlss Magntic Rconnction Magntic connction fs to changs in th stuctu of magntic filds, bought about by ocsss outsid of idal MHD. A siml xaml is a chang in magntic fild toology du to sistivity: a cunt sht foms, and fild lins diffus within it (g, th Swt-Pak modl). Many astohysical and laboatoy instancs of connction occu in basically collisionlss gims. W nd a mchanism to xlain connction without sistivity o oth collisional ffcts, and ultimatly dict th connction at. W us a fluid modl with a collisionlss Ohm s law that includs an lcton ssu and Hall tm. W show that this systm s quilibium can b dscibd by two could Poisson-lik quations. W solv this systm numically fo an imosd oloidal cus fild. This systm has nontivial lasma otis, including th ossibility to bak th fon-in law. Ou modl alis in aamt gims wh d i ~systm si L and d <<L, such as thos of th Vsatil Tooidal Facility (VTF) ximnt at MIT lad by A. Fasoli.

3 Govning Equations Continuity : Momntum : Ohm's Law : Stat : ρ v ( ρ ) t dv ρ dt E v nm Ti, T cold ions, total ssu - n i m nt Dv Dt isothmal along fild lins In quilibium, / t Nondimnsionali quations using systm si L, tyical magntic fild, dnsity n, and Alfvén sd. Continuity : Momntum: Ohm's : v ( ρ ) v v β E v di L ( ) n d L *advctiv Dv Dt tms d intial skin dth c ω, ω n q ε m

4 -D Equilibium, / ˆ ψ ˆ ˆ ψˆ Lt v u v ˆ Intgat Continuity : ( ρu ) ρu ˆ ϕ Tmatu a flux function : T T T[ψ ] W do tms of od (d /L), as thy a small in many gims of intst (g, VTF). Sinc d i ~L in VTF, w tain ths tms non-idal Ohm s Law. Tak comonnt and cul of Ohm s Law to liminat lctic fild, which is - φ in quilibium.

5 Rduction to Scala PDEs On can constuct 4 quations of th fom ( ψ o ϕ) f f f [ ψ o ϕ]. This givs : di F[ψ ] ϕ div G[ ϕ] ψ v F'[ ψ ] ρ U '[ ψ ] di ( ρ ρ log[ ρt '[ ψ ]]) v U[ ψ ] d i T[ ψ ]( log ρ) W[ ϕ] noulli - lik Systm ducs to PDEs: ψ ρt '[ ψ ]log ρ d ( F'[ ψ ] i vg'[ ϕ] W '[ ϕ] ( ϕ) ρ d d ρ ρ algbaic noulli qn.fo i i ρ ρv ρu '[ ψ ])

6 Solving fo ρ log ρ a b ρ, a T ( [ ]) v W ϕ, b ϕ T Lt ρ a, Q b a log Q. Q.6 susonic. subsonic (w us this oot) Q At Q Existnc of /, Q Q solutions : Q < Q Q A( Q oloidal Mach numb M / Q) α u c s maximum Q ϕ ρ T

7 T [ ψ ] β Σψ F[ ψ] ϕ G[ ϕ] atanh σ F U[ ψ] W[ ϕ] T[ F [ ϕ]]{ T K[ x] log F [ ψ ],, F Functions G K[ F / [ AT [ x] ] F [ ψ] w choos β f [ ϕ]]} G tanh[ ψ/σ] /,. Ths choics nsu solutions of ρ quation xist on th bounday, wh Q is lagst. Also choos ϕ, ounday Conditions W want to simulat an xtnally imosd cus fild bnd bnd F [xy]. v bnd F [xy] Ths choics giv ux x const sch [σy] imosd nomal flow dos off away fom saatix ψ xy bnd aidly

8 Numical Solution W hav two could Poisson-lik PDEs fo ϕ and ψ. W scify bounday conditions on a ctangl, and this dtmins th solution insid. W us finit diffncs, and tat th oblm as a could, nonlina systm fo ϕ and ψ at th disct gid oints. Nwton s mthod: quadatic convgnc Solv f ( u), u ψ o ϕ. f ( u) f ( u ) f '( u )( u u ) O(( u u Sinc f ( u ), * f ( u ) f ( u ) f '( u )( u u ) * * Taylo xand about a guss, u, na solution u * ) ). : u * u f '( u ) f ( u ) Guss an initial u Comut th Jacobian f'(u) - Calculat nw u itat until avag valu of f(u) is ~, givn oundoff o. Imlmntd in MATLA. Woks blindingly fast: 3 to 5 itations fo convgnc. W st d i L and us a x domain in units of L. Disctid to x gid oints.

9 Equilibium Dnsity: confind, almost flux function Tmatu T[Ψ]

10 Poloidal : clos to cus x

11 Poloidal u (magnitud~.v A ) v

12 Poloidal : magnitud~ ( ψ, fo ψ xy)

13 > Poloidal ( 5) Dnsity, tmatu, and oloidal vy clos to cas : shiftd by 5, no < idgs

14 5 Poloidal u V (no ngativ idgs)

15 5 Poloidal (magnitud~e-4) guid fild baks symmty among quadants

16 Possibility of aking Fon-in Law ρ ψ ρ ψ

17 Elctic Fild Fom Ohm s Law

18 Rlativ Imotanc of tms, Intgatd ov domain Poloidal Ohm ( ) - n v u E E-6.64 E-7 > u u Poloidal Momntum 4.5E-3.6E-7 > 4.6E u to oundoff, Ohm : of - com. E u u to oundoff Mom.: - com.of u

19 Summay Hall-MHD quations simlifid to scala PDEs Solutions xist with nontivial lasma cunts, flows, magntic filds, and lctic filds Possibility of baking fon-in law na saatics sinc dnsity not a flux function nono flows ndd fo baking fon-in Futu Poscts Tim-dndnt oblm Alid lctic fild (as in VTF) Estimat of connction at