#### Transcript Self-consistent mean field forces in turbulent plasmas

Self-consistent mean field forces in two-fluid models of turbulent plasmas C. C. Hegna University of Wisconsin Madison, WI Hall Dynamo Get-together PPPL via UW-Madison June 11, 2004 Theses • The properties of turbulent plasmas are described using the two-fluid equations. • Global constraints are derived for the fluctuation induced mean field forces that act on the ion and electron fluids. • Relationship between relaxation of parallel momentum flows and parallel currents C. C. Hegna, “Self-consistent mean-field forces in turbulent plasmas: current and momentum relaxation,” Physics of Plasmas 5, 2257 (1998); 3480 (1998). --- RFP physics was largely the motivation Outline • Brief review of mean field resistive MHD theory relevant to magnetized plasmas - applications to RFPs • Two-fluid theory – Constraints on the fluctuation induced mean-field forces – Heuristic derivations of local forms for the mean-field forces – A simple quasilinear theory Subsequent work - Mirnov, et al ‘03 - is much more complete – Relation to relaxation Steinhauer, Ishida, ‘98-’03; Mahajan and co-workers ‘01 In resistive MHD dynamo theory, a mean field force is identified • Fluctuations affects mean field dynamics in resistive MHD through a dynamo electric field E v B J Write all quantities as mean field and fluctuations ˜ Q Q Q The bracket <> notation denotes either an ensemble average or an average over the “small” spatial scales or “fast” time scales of the fluctuations Mean field Ohm’s Law E v B F J ˜ F v˜ B Global conservation laws have motivated local forms for the mean field force of resistive MHD • In resistive MHD, fluctuations do not dissipate helicity, but do dissipate energy. (Boozer, J. Plasma Physics, 1986; Bhattacharjee and Hameiri, PRL 1986; Phys. Fluids 1987; Strauss, Phys. Fluids 1985). d d 3 x F B 0, 3 x F J 0. These condititions are used to motivate a “local” form for the mean-field force in toroidal confinement devices --- fluctuations generate an additional electron viscosity or hyper-resitivity, not a “dynamo.” F|| Bo Jo Bo 2 (K ) Bo2 Bo2 K2 is a profile dependent positive function satisfying boundary conditions. Consistent with the Taylor state, F gets large, ---> J||/B = constant Two-fluid equations can be written in a concise form • The exact two-fluid momentum balance equations vs vs vs ) nsqs (E vs B) ps s Rs t – These equations can be written more concisely with the identification of the canonical momentum. msns ( m s v s ( v s v s ) E v s B qs t ms msv s2 As A v s, s qs 2qs Momentum balance equations A ps R s s s v s Bs s t nsqs nsqs nsqs Plasma flow for each species vs u J m e mi nsqsms me mi ms m v2 m A v s s s v s ( s v s ) v s B qs t 2qs qs t m m v2 m (A s v s ) ( s s ) v s (A s v s ) t qs 2qs qs As s v s Bs t A pressure equation is also used for each species • The pressure evolution equations ps vs ps ps vs ( 1)(Qs qs s : vs ) t – Q = collision energy transfer and Ohmic heating, last term represents viscous heating. – In general, compressibility is allowed. This modifies the usual definition of the mean field force and allows for anomalous particle transport. – In what follows, the effects of heat flux q are simplified. A weakness in the theory and a potential new area of investigation. Fluctuations induce mean field forces on both the ion and electron species. • For simplicity, we consider a cylindrical plasmas with all the usual boundary conditions. Quantities are split into equilibrium and fluctuating quantities, Q Qo Q˜ Nonlinearities produce fluctuation induced mean field forces (actually forces per unit charge) T˜s ln(1 n˜ s /nso ) msv˜ s2 ˜ ˜ Fs v s Bs qs 2qs Note, the first term contains both the MHD and Hall dynamo terms. For the electrons, ve=u - J/ne + O(me/mi). ˜ J ˜ ˜ Fe u˜ B ( ) B ... ne Three global properties of the mean-field forces can be shown • Mean field momentum balance equations Aso pso R s so v so Bso Fs ( s )o ( )o t nso qs nsqs nsqs • Three global constraints to be shown dV Bo Fe 0 to O() dV B F 0 to O() dV(n ev F nev F ) 0 o i o io i eo e The last condition can also be written using F||M = F||i -F||e, F||O = (miF|||e+meF||i)/(mi+me) dV(n eu F o M Jo FO ) 0 A number of assumptions are used to prove the three global constraints • Simplifying assumptions used in the constraint derivations: – Fluctuation amplitudes are small compared to the mean magnetic field, typically valid in all MFE devices 2 – 1 1 ˜2 B msn sv˜ s2, p˜ , B o 2 2o 2o The equilibrium quantities evolve on a slow diffusive time scale Q ~ Q t o a 2 Viscosities and radial mean flow are ordered with resistivity. Parallel heat flux is ordered small to be consistent with the neglect of heat flux, (again, probably a weak point) B T ~ O( ) B oVA a2 A number of assumptions are used to prove the three global constraints • Assumptions (continued) – The viscous force is dissipative for both species : v s 0 – All other equilibrium flows are ordered small - probably not a crucial assumption,may be generalized to equilibrium with flow Jo Bo po – Ion and electron skin depths are small. With b~ 1, r ~ d s s c ds ps a velocity and magnetic field fluctuations are small, gradients of While fluctuating quantities may be large, in general J˜ au˜ ~ ~ O(1) Jo VA The first two conditions can be shown from the generalized helicity evolution equations • Two separate ways to generate the evolution of the mean generalized helicity – The first from the total momentum balance – The last from the mean momentum balance • Subtracting the average of the first equation from the last equation. 2Fs Bso C1 ˜ ˜ ˜ ˜ ˜ s As Bs 2 J B 2 Bs t n sqs 2 2 (1 ln n so )Bso Tso (1 ln n s )Bs Ts q qs With the assumed orderings and appropriate boundary conditions, the first two conditions are derived • The previously derived condition 2Fs Bso C1 ˜ ˜ ˜ ˜ ˜ s As Bs 2 J B 2 Bs t n sqs 2 2 (1 ln n so )Bso Tso (1 ln n s )Bs Ts q qs – All the terms on the right hand side are smaller than O() dV 2F B s so dS C 0 1 to O() – C1 is the fluctuation induced generalized helicity flux. – In the me = 0 limit, the electron condition corresponds to the same as that derived for resistive MHD. In two-fluid theory, there two constraints, one for each fluid. Energy balance relations are used to prove the third condition • Total energy conservation B2 nsmsv s2 p nsmsv s2 p v ( ) [ v s E B ( s s v s s )] 0 t 2o s 2 1 2 1 s s • Construct mean magnetic energy evolution from pressureequations nsqsvso Momentumbalances 0 s s – Subtract this from O() average of the top equation p˜ sv˜ s ˜ : v˜ noev io Fi noev eo Fe pso C2 J˜ 2 s s p os s s – C2 is the leading-order energy flux caused by the fluctuations – Third term denotes anomalous cross-field transport --- similar bits show up in resistive MHD - Hameiri and Bhattacharjee, ‘87. By accounting for the cross-field diffusion in our definition of F, the fluctuations are shown to dissipate energy • One can redefine the mean field force to account for turbulence induced cross field heat and particle transport – This redefinition doesn’t affect The first two conditions – The final condition is derived dV(n ev o io F F F , f p˜ ev˜ e (1 f ) p˜ iv˜i F Bo [ ] peo poi ˜ : v˜ ) 0 Fi noev eo Fe ) dV( J˜ 2 s s s - This condition can also be written using FM = Fi - Fe, FO = (miFe + meFi)/(me + mi) dV(n eu F o M Jo FO ) 0 • Fluctuations dissipate energy. Energy delivered to electrons and ions via Ohmic and viscous heating. The global constraints imply a particular choice for the local form of the mean field forces • In analogy with the hyper-resistivity form implied by the resistive MHD global constraints, local forms are implied by the two-fluid global constraints. – The first two conditions imply dV B F o s|| 0 Bo Fs|| xs where xs vanishes on the boundary – The third condition can then be written dV qsnsvso Fs|| dV xs ( s s – A solution that guarantees the above is n v B xs f st ( to to2 o ) Bo t with conditions on the coefficients fst. qsnsv so Bo )0 2 Bo The implied local forms suggest a coupling between current and flow evolution • The parallel components of the turbulent mean field force F|| e Bo n v B n v B [ke2( o eo2 o ) Le ( o io 2 o )], 2 Bo Bo Bo F|| i Bo n ov io Bo n ov eo Bo 2 [k ( ) L ( )], i ei Bo2 Bo2 Bo2 – Can rewrite these equations as Those that appear in Ohm’s law And the total momentum balance F||O Bo Jo Bo eno uo Bo 2 [ ( ) ( )], e e Bo2 Bo2 Bo2 F|| M Bo eno uo Bo Jo Bo 2 [ ( ) ( )], i i Bo2 Bo2 Bo2 Coefficients are spatially dependent functions that vanish on the boundary and satisfy ke2 > 0, ki2 > 0, (Le + Li)2 < ke2ki2/4 e2 > 0, i2 > 0, (e + i)2 < e2i2/4 The local forms are also derivable from quasilinear tearing mode theory • Simplified, incompressible, low-b, me = 0, s = 0 limit – Mean field forces are derived from quasilinear expression for the tearing mode resonant on some surface. < > = average over layer width ˜ Fs B o v˜ s (Bo As ) , ˜ ˜ ˜ J FO Bo (Bo A)(u ) ne m 1 ˜ ˜ FM Bo i u˜ (Bo u˜ ) J (Bo A) e ne • In the Ohm’s law, the resistive MHD and Hall dynamos • In the total momentum balance, the Reynolds and Maxwell stresses (to within a factor of ne). A simple model is used in the linear layer analysis • A four field model B bˆ B bˆ , u bˆ V bˆ • Equilibrium near the rational surface x2 Bo Bo (ro ),o o '' , 2 J o J o (ro ) J o '(ro )x Vo Vo (ro ) Vo '(ro )x The mean field forces derived from quasilinear theory exhibit the same structure as that implied from the global constraints • After linear theory dJo dV n oeN e o ), dx dx dV dJ FM Bo (n oeMi2 o N i o ), dx dx FO Bo (M e2 – In the resistive MHD limit, Me2 is the largest coefficient - hyperresistivity dominates and FM is small. – In the two-fluid limit, the dominant drive comes in the combination dVe/dx=dVo/dx - dJo/dx (ne)-1 • Highly simplified theory --- A much more complete job calculation of two-fluid tearing mode growth rates has been calculated by Mirnov, et al. Relationship to relaxation theory • Constraints imply a relaxation theory via the minimization of W - eKe - iKi where Ks = dV As.Bs – To lowest order in c/(pia), the minimizing solutions yield J o 1Bo n oeuo 2 Bo A state discussed by many - Sudan, ‘79; Finn and Antonsen, ‘83; Avinash and Taylor ‘91; Steinhauer and Ishida, ‘98; Mahajan et al ‘01. Summary • Using a modest number of assumptions, three global constraints are derived for turbulence induced mean field forces in two-fluid models of plasmas. dV Bo Fe 0 to O() dV B F 0 to O() dV(n ev F nev F ) 0 o i o io i eo e • These constrains imply functional forms for the parallel mean-field forces in the Ohm’s law and the total momentum balance equations suggesting the fluctuations relax the plasma to states with field aligned current and bulk plasma momentum. • Applications to flow profile evolution during discrete dynamo events on MST?