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

 Gyrokineti c equation :  

v

//  

X

// 

i

k

 

v

D 

i

   

J

0    D

J

0 

q i m

 

f

  0 

v



J

1

k



c



k

 

e

//

Ω

 

f

0 

X

~

B

// 

i v

// 

J

1  

J

0   

X

//  ~    Generalize d parallel Ampere' s law (voticity equation) :

B

0  

X

//  4 

B

0 2  

k

2  

k B

0 

k

 

e

//  

X

//     

k



c

 4  2

e

// 

j

 

qJ

0  D 

S

 

P

0 total   

X



P

0 total //

j



k

2    ~

V A

2     2  4 

B

0 2 

k

  4  

B

0 2

k

2 

e

//   

j m j



P

0 total  

X



n

0   2

j

 

Pj

   4 

cB

0 

k

 

e

//     

P

0 total  

X

 4 

B

0 2      

j qv

2  2 

f

0 g

B

0  

j

  ~  4 

cB

0 2 

k

 

e

//   

P

0 total  

X

  ~

B

//

Coupled GKE-MHD Equations (cont.)

 Generalize d perpendicu lar Ampere' s law :

k

2  4 

c

~

B

//  

j qv



k



J

1

j



k

2 

c

2

B

0 2

k

  

e

//  

P

0 total  

X

  ~  

j q

2

v

2 

J

1 2

mc



f

0 g

B

0  

j

~

B

//  

j q

2

v



k



m



f

0 g

B

0  

J

0

J

1

j

   ~   Quasi neutrality condition 

j q

2

m



f

0 g  

J

0 2

j

    : 

j qJ

0

j



k

4  2 

c

2

V A

2 

k

 2 

c B

0 2 2 

j m j n

0

j

 

Pj

 ~  

j q

2

m



f

0 g

B

0  

v



J

0

J

1

k



c

~

B

//  Closed equation set for  ~  Chen , and , and ~

B

// with Hasegawa, JGR, by the 1991  gyrokineti

j

c equation 

Hybrid Simulations

• Fluid components (MHD description) by finite difference algorithm • Particle components (Gyrokinetic description) by

δf

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 Chen and and Chen, Hasegawa, PoF, JGR, 1982 1991    Two component plasmas (core, energetic)    

C

 

E

,

T C T E

~  2 ,

k

 

E

~  1 2  Ideal MHD : 

E

//  0  Gyrokineti c formalism

For Toroidal Plasmas

   Chen Chen, and PoP, 1994 Hasegawa, JGR, 1991     

C

~  1 , 

E

~  2 ,  

a R

 

B

//  4 

q E

 ˆ *

P C

  

Bm E

  Ballooning mode representa tion

Equations for Toloidal Plasmas

 Vorticity equation :  1  2  0 cos      2   2

A

0 

t

2 

i

  

A P

0   

A

0 

t

    2    2 

V

  4 

q E q S

2

R

2

f

1 2

c

2

J

0

Ω d



G

 Gyrokineti c equation :    

t



q S v

// 

R

  

i



d

  

G



i q E m E QF

0

f

1 2 

Ω



J

0 

Ω P J

2    Parameters

V

 

s

  : cos   2 

d

  

k



Ω d



k



Ω

 

f

2

k

 

e

//

Ω



k

 

e

//

Ω

  cos    

v

2  2

B

0     

v

2  2 

B

0 

X



f

,

f



v

// 2   , 

P

 1  

v

// 2        

s



k



Ω P

   

k

    

P

,  

Pj

  4 

B

0 2

v

2  2   ˆ sin   2  ˆ 

j P

0 

j m j n

0

j



P

0 total  ,    , 

k

  

k r n dq S

 

C

 

E

 

q S

2

R d



dr dr

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]

  Gyro -

Wave-Particle Resonances

kinetic equation  ~ : 

X

// 

i



v

//     D  

i



v

//   ~

S

1  ~

S i

  ~

S

2    Resonances

a



b

, ,             

G I a l



C a s

   

G



l a

  exp 2

ds v b

cos  //

I

cot

a s



i

   ,

I



a b a



S a s

 ~

S

1

C

D    exp

a s

 ,

Q

sin

I i

  ~

S

2

a s



I S



a a

2

s b

 

l a b a

  , 

b ds v

// ~

S

1

dl v

//  ~

S S Q

,  2 

a s

exp  

b

~

S

2

b i

 

s C a s

 

a



b



dl v

// : cot

I a b

 cot    2

b

    D     1 

K

        

b

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.