Hybrid Simulations

							      Hybrid Simulations
on Kinetically Excited Alfvenic
     Instabilities: Phase I

             S. Hu
     College of Science, GZU
            Liu Chen
     Dept of Phys & Astr, UCI

        Supported by NSFC
              Outline
•   Motivation
•   Theoretical model
•   Numerical scheme
•   Alfven waves in toroidal plasmas
•   Numerical demonstration
•   Summary
               Motivation
• Alfven waves and energetic-particle
  physics are important in fusion plasmas.
• Wave-particle interactions play important
  roles in kinetic destabilizations of Alfven
  waves by energetic/thermal particles.
• Gyrokinetic-MHD hybrid simulations, with
  the help of theoretical studies, provide a
  powerful way to demonstrate the kinetic
  excitation mechanisms for Alfvenic
  instabilities.
             Objective
• To focus on basic physical pictures
• To apply simplified equation system
• To clarify kinetic mechanisms of
  Alfvenic instabilities
• To collaborate the simplified numerical
  studies with more sophisticated
  simulations
• Education: Understanding/training as
  a bridge to massive simulations
     Coupled GKE-MHD Equations
 Gyrokinetic equation :
                                ~     q  f k  e f  
      v//
      X        ik   v D  i  g  i   0   //  0   J 0  
                                  
                                                                       ~
            //                         m      Ω    X  

                J0    D J 0    v J1   B//  i v// J1  
                        ~              ~              ~                       ~ 
                                                                                 
                                                 kc                      X // 

 Generalized parallel Ampere's law (voticity equation) :
                ~
      k  k   4
           2
                                              k  2 ~ 4 k 
                                                2             2
B0                   2  qJ 0D g S j  2   
                                      ~
                                                                            m n             ~
                                                                                        Pj
   X //  B0 X //  c j
                                                                                 j 0j
                                           VA           B02                j


    4                            
   2 k   e //    k   e //  
                                         P P
                                        total
                                       0
                                                 total
                                                0 //  ~     4
                                                        2 k   e //     
                                                                                      2
                                                                            P0total  ~
    B0                                      X              B0             X  
    4                   P0total 4          qv f 0 g
                                                   2
                                                                   4                  P0total ~
       k   e //     2   
                         X B                                  2 k   e //    B//
                                                                 ~
    cB0                            0      j    2 B0  j        cB0                  X 
Coupled GKE-MHD Equations (cont.)
 Generalized perpendicular Ampere's law :
   2
  kc ~                               k  c 2 k   e // P0total ~
                                        2
                                                                      q 2 v J12 f 0 g
                                                                           2
                                                                                                      ~
      B//   qv k  J1 g
                         ~            2                    
                                                                                                   B//
  4        j
                                 j
                                       B0                X       j    mc B0                  j

                                               q 2 v k  f 0 g
                                                               J 0 J1         
                                                                               ~ ~
                                           j       m B0                 j


 Quasi - neutrality condition :
        q 2 f 0 g 2                                        2         2
                                                           kc 2 ~ kc 2
                 J0          
                            ~ ~
                                               ~
                                           qJ 0 g                                        ~
                                                                                 m j n0 jPj
    j   m            j               j
                                                    j
                                                          4 VA 2
                                                                     B02        j


                                                                 q 2 f 0 g v J 0 J1       ~
                                                                                          B//
                                                             j   m B0  k  c          j


   Closed equation set for  ~       ~
                           ~,  , and B with g by the gyrokinetic equation
                                       //
                                             ~                                                        
                            Chen and Hasegawa, JGR,1991
           Hybrid Simulations

• Fluid components (MHD description)
     by finite difference algorithm
• Particle components (Gyrokinetic description)
    byδf simulation method
• Grid-particle coupling
     by particle-in-cell (PIC) technique
       Numerical Scheme
• The coupled gyrokinetic-MHD system
     Time-advanced for a given
  toroidal/azimuthal wavenumber
• Particle loading
    Markers with equilibrium distribution
• Boundary condition
    Vanishing perturbations applied
         Theoretical Model
       Frieman and Chen, PoF, 1982 
      Chen and Hasegawa, JGR, 1991
                                   
 Two- component plasmas (core, energetic)

    C   E , TC TE ~  , k  E ~ 
                           2               12



 Ideal MHD :  E//  0

 Gyrokinetic formalism
 For Toroidal Plasmas
       Chen, PoP,1994       
 Chen and Hasegawa, JGR,1991
                            
 C ~  1,  E ~  2 ,   a R

  B//  4 qE* PC BmE 
               ˆ

 Ballooning - mode representation
    Equations for Toloidal Plasmas
 Vorticity equation :
                    2   P    2            4 qE qS R 2
                                                               2
  1  2 0 cos  2 2  i
                    t                 
                                           2  V  f 1 2 c 2   J 0 Ωd  G
                   A0       A0  A0 t 

 Gyrokinetic equation :
      v         
              id  G  i E 1 2 Ω J 0  ΩP J 2 
                           q QF0
    //
   t q R       
       S                 mE f

 Parameters:
V  s   cos      f 2   cos f ,    f  1  s   k    sin  
                  2                                                      2


            k e          v B0
                             2
                                        2 
                                                                     j P0 j
                                                                    ˆ                    kr
d  k Ωd   //       2 B X     v//       P , Pj 
                                                                              , k 
              Ω           0                                       m j n0 j         n dqS dr
             k   e //  v  2
                                  2                      4 v2
                                                                                 C   E
  k Ω                v// ,  P  k ΩP   2
                            2                                   P0 , 
                                                                  ˆ   total

                Ω                                       B0 2                  qS R d dr
                                                                                      2
Alfven Waves in Toroidal Plasmas
 • TAE: Frequencies located inside the
   toroidal Alfven frequency gap
 • EPM: Frequencies determined by typical
   frequencies of particles via wave-particle
   resonance conditions
 • alpha-TAE: Bound states in the
   potential wells due to the ballooning drive
 • Low-frequency Alfven continuum:
   Physics to be understood
Alfven Continuum with Gap




    [Chen and Zonca, 1995]
         Wave-Particle Resonances
                                                           ~
                                                       S 
                                                           
                                                                               
                              ~ i
 Gyro - kinetic equation :
                             g
                                       D g  i S1  i S 2
                                               ~       ~      ~
                            X // v//              v//
                        ~
                                                                                    
                                                               l
                                                                 ds ~
                         g  G exp i I a  exp i I a 
                                           l                l
                                                                      S exp i I sl
    ~ a,   g a,  
               ~                                               a
                                                                 v//
  g


  ~
      b   
   g , g  
                 
                        
                                 2
                                         a
                                             ~
                                               
               ~ b,  G    b cot I b S C s  S S s  S S s  S C s
                                               1 a
                                                      ~
                                                         2 a    ~
                                                                     1 a
                                                                          ~
                                                                           2 a      
                              l                               b                b
                        I a  
                           l     ds
                                       D , Q   Q,  b  
                                                            2 dl                    dl
                              a
                                 v//                        b a v//            a
                                                                                    v//
                         s
                         Ca  cos I as , S as  sin I as , b  2  b
                                 b             1                 b
 Resonances: cot I a  cot   D   
                        b

                                2                K     D  Kb
Discrete Alfven
eigenmodes
trapped in the
potential wells
Quasi-
marginal
stability
 Discrete
 Alfven
 eigenmodes
 excited by
 energetic
 particles
   d  K b
                Summary
• A gyrokinetic-MHD hybrid simulation code
  is developed to study Alfvenic instabilities
  excited by energetic/thermal particles via
  wave-particle interactions.
• It is to be applied to study instabilities
  associated with toroidal Alfven frequency
  gap modes, energetic-particle continuum
  modes, discrete Alfven eigenmodes, as
  well as the low-frequency Alfven
  continuum modes.