Equlbrum and stablty of torodal plasmas wth flow n hgh-beta reduced MHD Atsush Ito and Noryosh Nakajma Natonal Insttute for Fuson Scence
Equlbrum wth flow n extended MHD models of fuson plasmas Equlbrum wth flow n fuson plasmas - In mproved confnement modes of magnetcally confned plasmas, equlbrum torodal and polodal flows play mportant roles lke the suppresson of nstablty and turbulent transport. Pressure ansotropy n plasma flow - Plasma flows drven by neutral beam njecton ndcate strong pressure ansotropy. Small scale effects n MHD equlbra - Equlbrum models wth small scale effects may be sutable for modelng steady states of mproved confnement modes that have steep plasma profles and for ntal states of mult-scale smulaton. - Two-flud equlbra wth flow and pressure ansotropy was studed for the case of cold ons [Ito, Ramos and Nakajma, PoP 4, 65 (7)]. - However, fnte on Larmor radus effects may be relevant for hghtemperature plasmas n magnetc confnement fuson devces. ( ) ρ = d β, ρ : on Larmor radus, d : on skn depth, β p B µ
Equlbrum wth flow n reduced MHD models - Flud moments n collsonless, magnetzed plasmas are smplfed - Grad-Shafranov type equlbrum equatons can be easly derved even n the presence of flow and several small scale effects. - Basc physcs of flow and non-deal effects can be nvestgated. - Reduced equlbrum models Two-flud MHD, FLR, polodal Alfvenc flow [Ito, Ramos and Nakajma, PFR 3, 34 (8)] MHD, polodal sonc flow [Ito, Ramos and Nakajma, PFR 3, 34 (8)] [Ito and Nakajma, PPCF 5, 357 (9)] Two-flud MHD, FLR, polodal sonc flow, sotropc pressure [Ito and Nakajma, AIP Conf. Proc. 69, (8)] [Raburn and Fukuyama, PoP 7, 54 ()] assumed adabatc pressure and parallel heat flux was neglected Two-flud MHD, FLR, polodal sonc flow, ansotropc pressure [Ito and Nakajma, NF 5, 36 ()] Parallel heat flux wth flud closure s taken nto account.
Reduced two-flud equlbra wth polodal-sonc flow Compressble hgh-β tokamak and slow dynamcs orderngs - Large aspect rato and hgh-β tokamak - Weak compressblty ( ) ε ar, B εb, pε B µ, p ar, B, p B : mnor and major rad of a torus : polodal and torodal magnetc felds p :pressure ( ) v εv a, v v + v MHD MHD MHD d The fast magnetosonc wave s elmnated. - Flow velocty for slow dynamcs v ~ v v δv, Π δ p δ m nv q pv δ pv gv MHD d th th th δ ρ / a, ρ : on Larmor radus R Flow velocty comparable to the polodal sound velocty ( p ) ( ) /, / a 3 ρv B B γ pε B µ δ ε - Transton between sub- and super-polodal-sonc flow appears. - Hgher-order terms should be taken nto account. Strong pressure ansotropy: p p p Parallel heat flux can not be neglected: q q pv
Equlbrum equatons n extended-mhd - Flud-moment equatons for magnetzed collsonless plasmas [Ramos, PoP 5 (5)] - Electron nerta s neglected: me - Two-flud equlbrum equatons wth on FLR ( λh, λ ) = (,) = (, ) = (,) Sngle - flud MHD Hall MHD FLR two - flud MHD ( nv) =, E=, µ j= B, E Φ, p p mnv v = j B p + B B λ Π s s gv s s= e, B λh pe p e E= v B+ j B pe + B B, ne B v= vb+ v, b B/ B, FLR effect (λ ): on gyrovscosty Π ε p gv Two-flud effects (λ H ): Hall current and electron pressure
- Strong pressure ansotropy: - Parallel and perpendcular heat flux: p p p q q pv - Equatons for ansotropc on and electron pressures ( ) λ ( ) v p + p v p b b v + q b + λ q, T T v p + p v+ pb ( b v) + λ ( qbb) + λ qb λqb ( b b), - Perpendcular (damagnetc) heat flux: q q q se, s sb st m s 3 ps qst ( v v ) ( ) fd ps v v v b eb s n ( ) ( ) m 3 s p ps p s s ps sb ( v v ) ( ) fd p q s v v v b + eb s n b b n - Parallel heat flux: λ = adabatc on pressure = on pressure wth parallel heat flux q q q s e, s sb st q m s 3 m 3 sb ( v vs ) ( s ) fd, st ( v vs ) ( s ) fd v v v q v v v,
- Parallel heat flux equatons for ons [Ramos, PoP 5 86 (8)] q q q B T λ λ v b neb v b neb + p qt + qt + p ( ) p p p p p + b b B, m n m nb λ 3 p p v+ p b qb +, neb b m n Knetc effects n the fourth-order moments are neglected - Parallel heat flux equatons for mass-less electrons, me ( ) e B ( e e ) B p n =, p p B = Electron pressure s obtaned from the parallel heat flux equatons. Electron parallel heat flux s obtaned from the pressure equatons.
Reduced equlbrum equatons - Axsymmetrc equlbra: ϕ =, B = ψ ϕ + I ϕ R,, Z cylndrcal geometry - Asymptotc expansons: f = f + f + f + f +, f ε f, f ε f, f ε f, - Lowest order quanttes are functons of ψ Φ =Φ ( ψ ), n = n ( ψ ), p = p ( ψ ), p = p ( ψ ), p = p ( ψ ), p = p ( ψ ), I = I ( ψ ) e e e e 3 3 3 - Hgher-order quanttes are determned by ψ and ψ p p xr C P, s{, } = s{, }ψ + s{, } ψ + s{, } ψ ( ) ( ) ( ) ( xr ) C Φ( ) ( ) v ( xr) C ( ψ ) + v ( ψ ) Φ =Φ + +Φ ψ ψ ψ,, v ( ) ( ) ( ) n= n + xr C + n ψ, n ψ ψ ( ) ( ψ) ( ψ ), + q ( xr) C ( ψ ) + q ( ψ ) q xr C q s sq s Shft from magnetc surfaces, sb sqb sb P, v, Φ, n, q, q are arbtrary functons of s{, } s sb ψ C,,,,,, s{, } Cv CΦ Cn Csq CsqB C are obtaned from I the equatons for p, v, Φ, n, q, q, I {, } {, } B{, } s s
- Polodal force balance - Radal force balance yelds the Grad-Shafranov (GS) type equatons x I + ψ = µ R ( p s + p s ) + g *, R Z R s= e, ( s s ) ( s s s + ) * R s= e, R s= e, ( x R R x a ) x I ( s s ) ψ ψ + ψ + µ R p + p + µ Rg * + ψ = + F + ψ + F R Z R s= e, R R R Z x x µ R E + P + P + p + p + C V E ψ B p + p + I = const., µ R e s s µ R R s= e, x x B ps 3 ( ps ps ) + ( Cs + Cs ps + ps ) + I3 s= e, R R µ R I ψ + ++ v + q µ R µ ( I I ψ ) g ψ F λ ( χ χ ) E ( ψ ) C s. - GS equaton for ncludes the effect of flow, FLR and pressure ansotropy: p F( ψ ) V ( λ λ ) V ( V λ V ) + ( ) V ( ) p s s E H d E H d s= e, B µ Gyrovscous cancellaton (FLR effect) B x p + p + I ( p p ) g, ψ, d ψ : Polodal Alfven Mach numbers of the E B drft and the on damagnetc drft veloctes, Pressure ansotropy, ( ψ )
Analytc soluton for sngle-flud, sotropc and adabatc pressure case [A. Ito and N. Nakajma, Plasma Phys. Control. Fuson 5, 357 (9] Magnetc structure s modfed to yeld a forbdden regon and the pressure surface departs from magnetc surface due to polodal-sonc flow. Shft of the magnetc axs from the geometrc axs Shft of the pressure maxmum Beyond beta lmt
Magnetc surfaces (gray: statc equlbrum) Pressure surfaces (gray: magnetc surfaces) Sub-polodal -sonc flow p c MApc.5 Super-polodal -sonc flow p c MApc.5 - The pressure maxmum s shfted outwards for sub-polodalsonc flow and nwards for super-polodal-sonc flow
Analytc soluton for sngle-flud equlbrum wth flow and pressure ansotropy: ( λh, λ ) = (,) [A. Ito and N. Nakajma, J. Phys. Soc. Jpn 8 (3) 645.] Ansotropc, double adabatc on pressure, Sngularty M = 3 p + p (slow magnetosonc wave) Apc c e c Qualtatvely same as the sotropc case λ = Shft of the magnetc axs due to flow Ansotropc on pressure n the presence of the parallel heat flux, λ = Sngularty ( ) M Apc= 6 p c+ pe c± 4 p c+ pe c (slow magnetosonc and on acoustc waves) M = p Apc c (on acoustc wave)
Complcated characterstcs n the regon around the polodal sound velocty due to pressure ansotropy and the parallel heat flux have been found Dependences of polodal sonc sngularty on the pressure ansotropy for ons α p p c c, p p p c sc s c se, s fxed. In the shaded regon,the equlbrum s regular and the polodal Mach number les between the two sngular ponts for all values of α
Profles of pressures and the magnetc flux n the mdplane (Left) and the radal component of the damagnetc current n a polodal cross secton (rght) for the polodal Mach number.45, α =. (top) and.75 (bottom). - Perpendcular (damagnetc) current j ψ Cs ( ps ps ) x, ψ B s= e, { } Decreasng a wth the polodal Mach number fxed n the shaded regon moves the equlbrum from the super polodal sonc regon to the sub polodal sonc one smlarly to how ncreasng the polodal Mach number at a fxed a does.
Numercal soluton for the FLR two-flud model Boundary condtons: Crcular cross-secton, Up-down symmetry ψ (,θ ),= ψ (,θ ). = Fnte element method GS Eq. for ψ: nonlnear, solved teratvely GS Eq. for ψ: lnear, solved by substtutng ψ Numercal results for two-flud equlbra Magnetc flux ψ Total pressure ( ps + ps ) 3 x x ( = ε., = VEc.4γ p= c, Vdc Ion stream functon s =,e Z - Isosurfaces of each quantty do not concde because of the flow, pressure ansotropy and the two-flud effects. x Ψ.γ pc )
Pressure profles n the mdplane ( p.75,.5,. c = pe c p c = p c pe c = pe c) pe Ansotropc pressures for ons and electrons are self-consstently obtaned. p pe p x - Solutons depend on the sgn of E B flow compared to that of on damagnetc flow ψ p Ψ x Red: same drecton x Blue: opposte drecton x
Reduced MHD equatons for stablty of polodal-sonc flow Reduced-MHD equatons wth - must nclude equlbra wth polodal-sonc flow when - requre the energy conservaton up to t, ϕ ( ) ε µ 3 O B t =, ϕ = - are needed for stablty of torodal equlbra wth strong polodal flow Dervaton of the reduced equatons for sngle-flud MHD We modfy the reduced equatons found by Strauss [Strauss, 983] to apply for hgh-beta plasmas wth polodal-sonc dynamcs and non-constant densty. ( ) ( ) ( ) v + xr U B B + v B B, B H ϕ + I ϕ, F F H ψ ξ( RZ, ) + ξ Θ( RZ, ). ξ ϕ Θ ϕξ R U + R U R U + R R p t ( ϕ) ( ρ ) ρ, {, } p jϕ p H + H ϕ + BR ϕ ( Rjϕ ) + µ + p =, B µ B ϕ BR ϕ ψ R U = ϕ ( U H) + B Φ, t R ϕ ϕ R p ρ + R( U ϕ) v ϕ BR ϕ p, t + + H + = BR B µ
p vr p + + ϕ BR H + ϕ p t BR B µ ( ) ( ) H ( H ) + R BR I U ϕ + U ϕ U ϕ p p vr p + γ p H ϕ + BR ϕ + B µ BR B µ ( ) ( ) H ( H ) + γ p R BR I U ϕ + U ϕ U ϕ γ pr + ( U ϕ) H+ H ( U ϕ) ϕ U + I U ϕ = BR H ( ) ( ), + ( U ϕ) + ( U ϕ) H + H ( U ϕ) I I I I R t RRB RRB RRB H U µ + ( H ϕ) + ( H U)( ϕ p) =, RRB RRB F F µ p ξ, J R ( ξ ) ϕ ξ ξ ξ + = Θ ξ Θ ξjb Φ Φ µ p U µ p U ξ + = ξ, + ξ ξ ξ ξ Θ ξ ξ B ξ ξ Θ B Θ These equatons contan shear Alfven and slow magnetosonc waves. - Energy conservaton ( ) ( H ) 3 p d ρ U + v + + I + =, t x µ R γ can be shown wth asymptotc expansons n terms of ε up to ε 3 ( µ ) O B.
Summary Reduced equatons for two-flud equlbra wth flow - Two-flud equlbra wth torodal and polodal flow, on FLR, pressure ansotropy and parallel heat flux have been derved. Analytc soluton for sngle-flud equlbra - The soluton ndcates the modfcaton of the magnetc flux and the departure of the pressure surfaces from the magnetc surfaces due to flow. - Complcated characterstcs n the regon around the polodal sound velocty due to pressure ansotropy and the parallel heat flux have been found. Numercal soluton for two-flud equlbra wth on FLR - The sosurfaces of the magnetc flux, the pressure and the on stream functon do not concde wth each other. - Pressure ansotropy assocated wth parallel heat flux has been ncluded n the numercal code. - Solutons depend on the drecton of E B flow compared to that of on damagnetc flow. Reduced MHD equatons for stablty of polodal-sonc Flow - We have derved reduced sngle-flud MHD equatons wth tme evoluton n order to study ther stablty of equlbra wth polodal-sonc flow - We have shown that the energy s conserved up to the order requred by the equlbra.