Docstoc

ramos

Document Sample
ramos Powered By Docstoc
					 CEMM Meeting, Atlanta GA, November 2009.




 LONG-PARALLEL-MEAN-FREE-PATH KINETIC CLOSURES
                     FOR SLOW EXTENDED-MHD∗


                                     J.J. Ramos
                       M.I.T. Plasma Science and Fusion Center




*Work supported by the U.S. Department of Energy
.                                      INTRODUCTION
A theoretical model of the electron dynamics for slow, macroscopic plasma processes (such as
the ”neoclassical” tearing instabilities) in a long-parallel-mean-free-path collisionality regime
will be presented.


The model is a hybrid one, with fluid conservation equations for particle number, momen-
tum and energy, and drift-kinetic closures.


Key to this work is a careful choice of the orderings relating fundamental parameters, aimed at
describing as realistically as possible the low-collisionality, fusion-relevant plasmas of interest.
The conventional ordering of the collisionality in neoclassical theory is deemed too high for
the ions, even in the banana regime. Instead, the orderings ρι/L ∼ L/λcoll ∼ (me/mι)1/2           1
are adopted, which still yield a theory equivalent to the one based on the neoclassical banana
orderings for the electrons.
.              BASIC FRAMEWORK AND ORDERING ASSUMPTIONS


Quasineutral plasma with one ion species of unit charge:
                                      ∂n               ∂n
              nι = ne = n ,              + ∇ · (nuι) =    + ∇ · (nue) = 0 ,
                                      ∂t               ∂t
                  ∂B
                     = − ∇×E ,                  j = en(uι − ue) = ∇ × B .
                  ∂t


Small ion Larmor radius fundamental expansion parameter:

                                      δ ∼ ρι/L ∼ kρι    1.



Small mass ratio and low collisionality orderings, linked to δ:

                         (me/mι)1/2 ∼ δ ,     hence    ρe/L ∼ kρe ∼ δ 2

and

                νι ∼ δνe ∼ δ 2Ωcι ,   hence    λcoll ∼ vthι/νι ∼ vthe/νe ∼ δ −1L .
Macroscopic flows of the order of the diamagnetic drifts:

                                   uι ∼ ue ∼ u∗ι,e ∼ δvthι ∼ δ 2vthe .

Close to Maxwellian distribution functions with comparable ion and electron temperatures and
small parallel temperature gradients:
                                                             
                                                |v − us | 
                                                          2
                                   n
                                          3 exp − 2 v 2                            vths ≡ Ts/ms ,
                                                                                    2
       fs = fM s + fN M s =                                 + fN M s     with
                              (2π)3/2    vths         ths

           b · ∇Ts ∼ δ 2 Ts/L,            Te ∼ Tι,      fN M ι ∼ δ fM ι,    fN M e ∼ δ 2 fM e .



Using Ωcι as reference, we have the following hierarchy of time scales:
                              O(δ −2) : Ωce = vthe/ρe
                              O(1) :       Ωcι = vthι/ρι ∼ vthe/L
                              O(δ) :       νe ∼ ωA = kcA ∼ ωS = kcS ∼ vthι/L
                              O(δ 2) :     νι ∼ kuι,e ∼ ω∗ι,e = ku∗ι,e
                              O(δ 3) :     collisional dynamics
                         FLUID AND DRIFT-KINETIC APPROACH



Non-Maxwellian parts of the distribution functions, fN M s, evaluated in the moving reference
frames of their macroscopic flows, like the Maxwellian parts.


1, v − us and |v − us|2 velocity moments of fN M s equal to zero.


Density, flow velocities and temperatures determined by fluid moment equations.


Solution of drift-kinetic equations for fN M s to provide the fluid closure terms. Since fN M s
are obtained in the reference frames of their macroscopic flows, the evaluation of the stress
and heat flux tensors is direct without the need of substracting the mean flows.
                     FLUID MOMENTUM AND TEMPERATURE EQUATIONS



       ∂uι
mι n       = en(E + uι × B) − ∇(nTι)−mιn(uι · ∇)uι − ∇ · (pι − pι⊥)(bb − I/3) + PGV −∇ · Pcoll + Fcoll
                                                                                 ι        ι⊥      ι
       ∂t


                ∂ue
         me n       = − en(E + ue × B) − ∇(nTe) − ∇ · (pe − pe⊥)(bb − I/3) + Fcoll .
                                                                              e
                 ∂t
                             2
                       O(nmevthe/L)                             2
                                                       O(δ 2nmevthe/L)                       2
                                                                                    O(δ 3nmevthe/L)




                                                                            
           3n ∂Tι     3n                                    5nTι
                  = −    uι · ∇Tι − nTι∇ · uι − ∇ · qι b +      b × ∇Tι + Gcoll .
                                                                             ι
            2 ∂t       2                                    2eB

                                                                             
           3n ∂Te     3n                                    5nTe
                  = −    ue · ∇Te − nTe∇ · ue − ∇ · qe b −      b × ∇Te + Gcoll .
                                                                             e
            2 ∂t       2                                    2eB
                                                 3
                                        O(δ 2nmevthe/L)                                    3
                                                                                  O(δ 3nmevthe/L)
               KINETIC EQUATION FOR THE NON-MAXWELLIAN PART
                    OF THE ELECTRON DISTRIBUTION FUNCTION
In terms of velocity-space coordinates (v , χ, α) in the reference frame of the electron macro-
scopic flow:

           v = ue(x, t) + v cos χ b(x, t) + v sin χ [cos α e1(x, t) + sin α e2(x, t)] ,

the non-Maxwellian part of the electron distribution function can be represented as

                                           ¯                     ˜
                  fN M e(v , χ, α, x, t) = fN M e(v , χ, x, t) + fN M e(v , χ, α, x, t)

with

                                  ˜
                                  fN M e   α   ≡ (2π)−1 dα fN M e = 0 .
                                                           ˜



Then, keeping the accuracy of O(δ 2fM e) + O(δ 3fM e):
                                                     
                                                 2
               ˜              me v  me v
               fN M e = fM e             − 5 sin χ (cos α e2 − sin α e1) · ∇Te
                                             
                             2eBTe    Te
    ¯
and fN M e obeys the following drift-kinetic equation:

                                                                                                       
  ¯
∂ fN M e                  ¯
                        ∂ fN M e Te               ¯
                                                ∂ fN M e    sin χ  Te             v2              ¯
                                                                                                ∂ fN M e
         + cos χ v b ·         +    b · ∇ ln n           −           b · ∇ ln n − b · ∇ ln B           =
   ∂t                     ∂x      me              ∂v           v     me             2               ∂χ
                                                                                                           
         v      me v 2                   v       2
= cos χ     5 −         b · ∇Te + cos χ     b ·  ∇(pe − pe⊥) − pe − pe⊥ ∇ ln B − Fcoll  +
                                                                                     e
        2Te       Te                      nTe      3
                                                                             
                 me v 2                                       1  me v 2
     + P2(cos χ)        ∇ · ue − 3b · [(b · ∇)ue]         +             − 3 ∇ · (qe b) − Gcoll +
                                                                            
                                                                                            e
                  3Te                                       3nTe   Te
                                                                               
            1              me v 2  me v 2       m2 v 4      me v 2
         +     2P2 (cos χ)                − 5 +  e
                                                          − 10        + 15 (b × κ) · ∇Te +
                                                                          
           6eB               Te       Te            Te2         Te
                                                                           
         1              me v 2  me v 2       m2 v 4      me v 2
      +     −P2 (cos χ)                − 5 +  e
                                                   2
                                                       − 10        + 15 (b × ∇ ln B) · ∇Te +
                                                                       
        6eB               Te       Te            Te          Te
                                          me v 2
                              + P2(cos χ)        (b × ∇ ln n) · ∇Te fM e +
                                          3eBTe
             (3)                  (3)                        (3)              (3)
          + Cee (fM e, fN M e) + Cee (fN M e, fM e)   α   + Ceι (fM e, fι) + Ceι (fN M e, fM ι)   α   .
                                       ¯
With the 1, v cos χ and v 2 moments of fN M e equal to zero, the 1, v cos χ and v 2 moments of
this drift-kinetic equation are satisfied identically.
                                        COLLISION OPERATORS


Based on the complete form of the linearized Fokker-Planck operators and using the elec-
tron collision frequency definition
                                                   c4e4n ln Λe
                                              νe ≡         3   ,
                                                    4πm2vthe
                                                        e

the gyrophase averaged collision operators that enter in the drift-kinetic equation are as follows:


                                                                                        
                                                                              3
                                  νeme  Ti              vthe   v      4πvthe
            (3)
           Ceι (fM e, fι)   α   =           − 1 M e(v ) 
                                               f              φ      −        fM ι(v ) +
                                                                                         
                                   mι Te                   v      vthι      n
                                                                             
                                                              3
                      νej vthe           vthι   v      4πvthι
                    +      3   fM e(v )       ξ      −        fM ι(v ) v cos χ
                                                                         
                       envthι              v      vthι      n


where
                                                                                                
                    2           x                                            1           dφ(x) 
          φ(x) =                    dt exp(−t2/2)           and       ξ(x) = 2 φ(x) − x
                                                                                               .
                 (2π)1/2        0                                            x            dx
 (3)                  (3)
Cee (fM e, fN M e) + Cee (fN M e, fM e)         α
                                                       (3)
                                                    + Ceι (fN M e, fM ι)            α
                                                                                                ¯
                                                                                            ≡ C[fN M e] is Legendre diagonal:
                                                              
                                       ∞                                   ∞
                               C          fl (v )Pl (cos χ)          =         Pl (cos χ) Cl [fl (v )]
                                   l=0                                     l=0

with
                               νe                                              −1
              Cl [fl (v )] =      fM e(v ) 4πvthefl (v ) − vtheΦR [fl (v )] + vtheΞR [fl (v )] +
                                              3
                                                                l                  l
                               n
                                                                                                 
                                3
                             νevthe ∂           
                                                
                                                   v   ∂fl (v )   v2       
                                                                               
                                                                               
                           +                    
                                                
                                                 ξ       v      + 2 fl (v ) −
                                                                               
                              v 2 ∂v                vthe      ∂v     vthe
                                                                                                   
                                3
                     νel(l + 1)vthe   v        v         v       v 
                   −                φ       −ξ       + φ      −ξ       fl (v )
                          2v 3           vthe      vthe        vthι      vthι


and
                                                        
                                     R           
                       1 ∂     
                                  2 ∂Φl [fl (v )]
                                                     l(l + 1) R
                               
                               
                                v                 
                                                  
                                                    −         Φl [fl (v )] = −4πfl (v ) ,
                      v 2 ∂v           ∂v                v2
                                                                            2
                                                                       2∂       ΨR [fl (v )]
                                                                                 l
                                             ΞR [fl (v
                                              l          )] = v                              ,
                                                                                 ∂v 2
                                                        
                                     R           
                       1 ∂     
                                  2 ∂Ψl [fl (v )]
                                                     l(l + 1) R
                               
                               
                                v                 
                                                  
                                                    −         Ψl [fl (v )] = ΦR [fl (v )] .
                                                                              l
                      v 2 ∂v           ∂v                v2
                                      ELECTRON CLOSURE VARIABLES

The kinetically defined closure terms in the electron fluid equations are:


                      ∞                    π
(pe − pe⊥) = 2πme          dv v 4                                 ¯
                                               dχ sin χ P2(cos χ) fN M e =                2               2
                                                                                 O(δ 2nmevthe) + O(δ 3nmevthe) ,
                      0                   0




                  ∞                   π
       qe = πme       dv v 5                             ¯                 3               3
                                          dχ sin χ cos χ fN M e = O(δ 2nmevthe) + O(δ 3nmevthe) ,
                  0               0




                                               2meνe          meνen
                          Fcoll =
                           e                            j −           (b × ∇Te) −
                                              3(2π)1/2e     (2π)1/2eB
                                                                                              
                                                                                     3     2
                                  ∞              π
                                                                    ¯              δ nme vthe 
             − 2πmeνevthe
                      3
                                  0
                                          dv     0
                                                     dχ sin χ cos χ fN M e   b = O             ,
                                                                                       L

                                                                                       
                                                               3     3
                                     2meνen                  δ nme vthe 
                          Gcoll
                           e      =           (Tι − Te) = O              .
                                    (2π)1/2mι                    L
     APPLICATION: STATIONARY AXISYMMETRIC SYSTEM




    ∇·B=0 ,               j=∇×B ,              E = −∇Φ − V0∇ϕ

                                                           1
          ∇ · (nuι) = ∇ · (nue) = 0 ,         ue = uι −      j
                                                          en

b · ∇Te = O(δ 2 Te/L) ,      b · ∇Tι = O(δ 2 Tι/L) ,        Te − Tι   Te


            −en(E + uι × B) + ∇(nTι) = O(δ 2 Tι/L)


en(E + ue × B) + ∇(nTe) + ∇ · (pe − pe⊥)(bb − I/3) − Fcoll = 0
                                                      e

                                                            
     3n                                    5nTe
        ue · ∇Te + nTe∇ · ue + ∇ · qe b −      b × ∇Te = 0
      2                                    2eB
                      LOWEST-ORDER STATIONARY FLUID RELATIONS

The axisymmetric magnetic field is

                                   B = ∇ψ × ∇ϕ + RBϕ∇ϕ

and the lowest-order stationary fluid system (valid on the MHD time scale t < δ −1Ω−1) yields
                                                                           ∼      cι

the well known relations:

      n = N (0)(ψ),   Ts = Ts(0)(ψ),   Φ = Φ(1)(ψ) = O(Ts/e),         RBϕ = (RBϕ)(0)(ψ) ≡ I(ψ)
                                                                           
                                                (1)             (0)
                                              dΦ      1 d(N           Ts(0)) 
            us = u(1) = Us(ψ)B + R2 
                  s                              +                             ∇ϕ = O(δvthι)
                                               dψ   esN (0) dψ


From these, it follows that:

                      ∇ψ · (b × κ) = ∇ψ · (b × ∇ ln B) = I(ψ) b · ∇ ln B

                                              ∇ · us = 0

                                b · [(b · ∇)us] = Us(ψ) b · ∇ ln B
             HIGHER-ORDER STATIONARY ELECTRON FLUID RELATIONS



                                         2
Keeping the highest accuracy of O(δ 3nmevthe/L), the parallel component of the stationary
electron momentum equation yields
                                                                               
     2                                                      eΦ    n     Te     eV0N (0)I
b ·  ∇(pe − pe⊥) − pe − pe⊥ ∇ ln B − Fe
                                       coll 
                                              = N Te b · ∇  (0) − (0) − (0)  +
                                                 (0) (0)
     3                                                      Te    N     Te         BR2




                     3
and keeping O(δ 3nmevthe/L), the stationary electron temperature equation yields
                                                
                               (0) (0)
                           5N    Te         (0)  5N (0)Te(0)I dTe(0)
           ∇ · qe b   = ∇·            b × ∇Te  =                     b · ∇ ln B .
                              2eB                      eB        dψ




Notice that, within this highest available accuracy, the stationary electron temperature equa-
tion does not provide any information on the higher-order correction Te(x) − Te(0)(ψ)!
                     STATIONARY ELECTRON DRIFT-KINETIC EQUATION

                                                           ¯       (0)                                          ∞
Using the previous stationary fluid results and calling g ≡ fN M e/fM e =                                        l=0   gl Pl (cos χ), the
stationary electron drift-kinetic equation can be written as:


                                                                                               
                                              ∂g   1                ∂g 
                                 v cos χ b ·    + b · ∇ ln B sin χ      =
                                              ∂x   2                ∂χ
                                                                                                           
                                                                                 2
                                      eΦ             n                 3 me v         Te        eV0I 
             = v cos χ b · ∇ 
                             
                                          (0)
                                                −     (0)
                                                           +
                                                                    
                                                                        −   (0)
                                                                                 b · ∇
                                                                                          (0)
                                                                                               +
                                                                                                   (0)
                                                                                                            −
                                      Te            N                   2 2Te            Te       Te BR  2

                                                                                                         
         
                     me v 2                         mev 2  5 mev 2  I dTe(0) 
                                                                                
     −   P2(cos χ)
         
                          (0)
                                 UeB + 2 + P2(cos χ)        −
                                                       (0) 2
                                                                     
                                                                 (0) eB dψ     
                                                                                  b · ∇ ln B +
                       Te                            3Te       2Te

                                                                      ˆ
                                                       + v cos χ D1 + C[g]

where
                                                             
                                       3                                                                       ∞
     νej vthe     vthι   v      4πvthι                                                       1       (0)
D1 ≡      3
                       ξ      −        fM ι(v )
                                                                          and           ˆ
                                                                                         C[g] ≡ (0) C gfM e =                ˆ
                                                                                                                  Pl (cos χ) Cl [gl ]
      envthι        v      vthι      n                                                         fM e           l=0
The stationary electron drift-kinetic equation can be solved with the methods of neoclassical
theory in the banana regime. Thus, using the variable

λ(x, χ) = sin2 χ Bmax(ψ)/B(x),             0 ≤ λ ≤ Bmax/B,          v (x, v , λ) = ±v (1 − λB/Bmax)1/2 ,

one gets
                                                                                                    
                                                   e Φ − Φ(1)     n−N      (0)
                                                                                         3 mev  Te − Te(0)
                                                                                                 2
 g = gp0 + (v /v ) gp1 + h     (2)
                                     +h  (3)
                                               =        (0)
                                                                −                +        −                +
                                                       Te          N (0)                  2 2Te(0)    (0)
                                                                                                     Te
                                                         
           meUeB        me I5  mev 2  dTe(0)                   ˆ
 +v    −      (0)
                     +        −
                          (0) 2    (0)
                                                + σ(v ) H(1 − λ) K(ψ, v , λ) + h
                                                                                  (3)
                                                                                      (x, v , λ) ,
            Te         eBTe      2Te      dψ
where gp1 ∼ h(2) ∼ δ 2 and gp0 ∼ h(3) ∼ δ 3.


The function h(3)(x, v , λ) = O(δ 3) satisfies
                                                           
     ∂h(3)(x, v , λ)         eV0I
v b·                 = v  (0)
                                   + D1(x, v ) + C σ(v ) H(1 − λ) K(ψ, v , λ) + (v /v ) gp1
                                                  ˆ                ˆ
          ∂x               Te BR  2

which has the integrability constraint
                                                                                                         
        dl ˆ                                                                         eV0I
                            ˆ
           C σ(v ) H(1 − λ) K(ψ, v , λ) + (v /v ) gp1 = −                  dl    
                                                                                      (0)
                                                                                               + D1(x, v )
                                                                                                          
        v                                                                            Te BR2
                                                                 ˆ
and the solution of this Spitzer problem determines the function K(ψ, v , λ).
                                          ODD PARALLEL CLOSURES



                                         ˆ
Once the solution of the Spitzer problem K(ψ, v , λ) is known, it specifies the poloidal flow
stream function Ue(ψ) and the odd parallel closures qe (x) and Fecoll (x):


                                         2π          ∞           (0)           1
                                                                                      ˆ
                    Ue(ψ) =                              dv v 3fM e(ψ, v )         dλ K(ψ, v , λ) ,
                                  N (0)(ψ) Bmax(ψ)   0                        0



                                               
                  5N (0)Te(0)  I dTe(0)            πmeB          ∞            (0)             1
         qe   = −                       + Ue B  +
                                                
                                                                 0
                                                                       dv v 5fM e(ψ, v )      0
                                                                                                       ˆ
                                                                                                    dλ K(ψ, v , λ) ,
                      2         eB dψ               Bmax

                                                        
             2meνe  j            3N (0)I dTe(0)           3
                                                     2πmeνevtheB                     ∞        (0)           1
                                                                                                                   ˆ
Fecoll    =           + N Ue B −
                          (0)
                                                  −                                     dv fM e(ψ, v )         dλ K(ψ, v , λ) .
            3(2π)1/2 e             2eB dψ               Bmax                       0                        0
            CLOSURE PROBLEM FOR THE PRESSURE ANISOTROPY (pe − pe⊥)
                                                                                   ∞
 With a Legendre series expansion, h(2) + h(3) (x, v , χ) =                        l=0   hl (x, v ) Pl (cos χ), the l = 2
projection of the drift-kinetic equation can be expressed after algebraic elimination of the
electric potential using the fluid momentum equation as:
                                                                         
                             ∂ 
                                        2           2(pe − pe⊥)            
                                                                                        ∂g0(x, v )
                B 3/2 b ·       B −3/2  h2(x, v ) −
                                       
                                                                           
                                                                               + b·                =
                            ∂x 
                                        5                   (0)
                                                      3N (0)Te
                                                                            
                                                                                           ∂x
                                              
              5    me v 2       Te 
                                              Fecoll                  1 ˆ
          =  −            b · ∇        −            + D1(x, v ) +   C1[h1 + gp1] ,
               2 2Te(0)              (0)
                                   2Te       N
                                                    (0)
                                               (0) Te                 v
where g0 = h0 + gp0 is an unknown function that must satisfy
       ∞                       (0)                                    ∞                        (0)
      0
           dv v 2 g0(x, v ) fM e(ψ, v ) = 0              and          0
                                                                          dv v 4 g0(x, v ) fM e(ψ, v ) = 0 ,

and
                                                      3B(x)     1
                                                                       ˆ
                                   h1(x, v ) =                  0
                                                                    dλ K(ψ, v , λ) .
                                                     2Bmax(ψ)


                                                                            ∞                (0)          ∞           (0)
The pressure anisotropy is given by (pe − pe⊥) = (4πme/5)                   0 dv   v 4 h2 fM e, but the   0 dv   v 4 fM e
moment of the above equation results in an identity!
The conclusion is reached that, within the available accuracy of O(δ 3), the stationary and
axisymmetric drift-kinetic equation does not contain information on the pressure anisotropy
moment (pe − pe⊥)(x) = O(δ 3nTe).


This is consistent with the fluid moment equation for (pe − pe⊥) which, in the stationary
and axisymmetric case and within O(δ 3) accuracy, reduces to
                                                                             
                                                      (0)
                                                 N         Te(0)                    3N (0)Te(0)
∇ · (2qeB − qeT )b + 3qeT b · ∇ ln B + ∇ ·                         b×   ∇Te(0)
                                                                                  +             b × ∇Te(0) · κ = 0.
                                                      eB                                eB




So, the determination of (pe − pe⊥)(x) as well as Te(x) − Te(0)(ψ) in a stationary and ax-
ixymmetric system requires carrying the analysis at least to O(δ 4). This would necessitate a
drift-kinetic equation accurate to the second order of the electron gyroradius and including the
quadratic parts of the collision operators.
                                            SUMMARY

A closed fluid and drift-kinetic electron system for slow dynamics has been put forward. It is
accurate to the third order in a small ion gyroradius and large parallel collisional mean free
path expansion, and is compatible with the neoclassical theory for electrons in their banana
regime.


The stationary and axisymmetric limit of such electron system has been studied. Here, the
lowest significant order expressions for the poloidal flow and the odd parallel closures, qe and
Fecoll , have been derived. On the other hand, it has been shown that the lowest significant
order of the stationary pressure anisotropy, (p − p⊥)/(nTe) = O(δ 3), is not determined by the
available third order equilibrium system. Therefore, even though the parallel collisional friction
force is known with third order accuracy, Fecoll /(nTe) = O(δ 3/L), the equilibrium parallel electric
field can only be determined in its collisionless second order: b · ∇(eΦ/Te) = b · ∇ ln n = O(δ 2/L).

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:10
posted:12/19/2011
language:English
pages:22