# Hybrid Simulations

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```							      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
toroidal/azimuthal wavenumber
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.

```
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