Transcript Discrete Alfven Eigenmodes in Tokamaks

Summer School 2007, Chengdu
Alfven Waves
in Toroidal Plasmas
S. Hu
College of Science, GZU
Supported by NSFC
Outline
• Introduction to Alfven waves
• Alfven waves in tokamaks
• Toroidicity-induced Alfven Eigenmodes
(TAE)
• Energetic-particle modes (EPM)
• Discrete Alfven eigenmodes ( TAE)
• Summary
Introduction to Alfven Waves
•
•
•
•
Basic pictures of Alfven waves
Importance of Alfven waves
Alfven waves in nonuniform plasmas
Shear modes vs. compressional modes
Alfven Waves (Shear Modes)
Field line B 0 

~
B
String

~
x


~
2~
 B
2  B
 VA
2
2
t
x//

2
B0
2
VA 
2
0 0

2~
 2~
x

x
2
 VP
2
2
t
x//

TF
2
VP 
m
Alfven Waves &Energetic Particles
• Importance in Fusion Studies:
The Alfven frequencies are comparable
to the characteristic frequencies of
energetic / alpha particles in heating /
ignition experiments.
• Basic Waves in Space Investigations:
The Alfven waves widely exist in space,
e.g., the Earth’s magnetosphere, the solarterrestrial region, and so on. The
interactions between the Alfven waves and
the energetic particles also play important
roles in physical understandings.
Alfven Waves
Linearizat ion :

~
Q

Q

Q

  u
0




u


u

J

B

~
  t
2~
2~
2
u

0
,
E

0

B//
 u
VA
0

 0
2  u
 

V


 2
A

2
B  const.

t

l
B
t
 B

0
 
//
0









E

 ~

t

 B//
~


B


u
0


  B   0 J
 t

E  u  B  0
B 0  B0e z ,  0   0  x   VA  x 
 2

2
~ ~



k
z
2
~




Q

Q
x
exp
ik
y

ik
z

i

t


u x    2
  VA
A
y
z
 2  k z2 u~x  0
             2

 V
x 
2
2 x

 2  kz  k y
  A
 VA

Alfven Waves
(Compressional Modes)
  2
2 ~
VA


 0  const. 
 0  V 2  k z u x  0, shear mode 

x  A
     
 2~
2
  u x     k 2  k 2 u~  0,  compressio nal 
z
y  x


 x 2  V 2
mode


A



~ ~
Shear component : B  u

 
~
Compressio nal component : B//
~
 shear mode
coupled
to
B
//
 












2
2
2


~

~




V
~

~
2
u



A

u  - eq and B// - eq

V



B

A
 //
2

t  t 2






l
B


//
0
Fluid displaceme nts  

  compressio nal 
~
B

  B0     , 
 //
 component 
Alfven Waves in Tokamaks
•
•
•
•
Basic equations
Ballooning formalism
Shear Alfven equation
The s-  diagram
[ Lee and Van Dam, 1977
Connor, Hastie, Taylor, 1978 ]
Basic Equations
 



u


 u  J  B  P
  t

 
 B
A E//  0 A//

B  0



E






E











 t
t
t
l//

E  u  B  0

  A
2
2
  B   J B










A


A


A  0 J
0



0

 q

  J  t  0



 
 P

u


P


0

 P
 t

 u  P   P P  u  0


t
 

 t     u   0



Ballooning Formalism
 Toroidal geometry r ,  ,   :     q P  ,
  a R 
A   P  B    A   P  
  B  0  B  f  g    

B   P  0   P   P r 
q  r B R B 
 B   P    q P    e B  e B  
 s  r q q
~ t , r ,  ,    exp i n  i t  ˆ nqr   mexp  im  

m
 
ˆ nqr   m  ˆ nqr   m  1  ˆ nqr   m  1 2
rm : nqrm   m  0, k // nm  nq  m  qR 

 n  1 : rm ~ O1 m   1
  l  
//


Shear Alfven Equation
~  f 1 2~


 2~  2
~  V  ~  0,




1

2

cos



2
2

A
 B  B0 1   cos  
   ,  : the extended coordinate along the magnetic field line

2
2
2






V



cos

f

s


cos

f
,
f

1

s



sin


  q 2 R ,   V qR ,   2  P B 2
A
A
0
0

 First term : field - line bending
Second term : inertial contributi on
Third term : ballooning /interchan ge drive
The s-  Diagram
• First ballooning-mode stable regime
(with the low pressure-gradient)
• Ballooning-mode unstable regime
(with pressure-gradient inbetween)
• Second ballooning-mode stable regime
(with the high pressure-gradient)
TAE
•
•
•
•
Localized and extended potentials
Alfven continuum and frequency gap
Toroidicity-induced Alfven eigenmodes
TAE features
[ Cheng, Chen, Chance, AoP, 1985 ]
Localized and Extended Potentials
 2~
2
~  V~  0,




Ω
1

2

cos


 2
V   cos  f  s   cos  2 f 2

 f  1  s   sin  2 , Ω    A
 Extended potential     : Mathieu' s equation
 2~
2
~  0  Alfven frequency spectrum



Ω
1

2

cos


 2
 Localized potential  ~ O1 :
 2~
2~
~  0,   1  possible Alfven eigenmodes



Ω


V


 2
Alfven Frequency Spectrum
 2~
2
~  0,   1


 Mathieu' s equation :

Ω
1

2

cos


 2
 Eigenvalue s of Ω : Continuum spectrum of Alfven frequencie s




Ω 2  Ω 2  0, ; Ω 2  ΩLj2 , ΩUj2 , j  1,2, ,
2
 j
Alfven frequency gaps around Ω  Ω    : ΩLj2  Ω 2j  ΩUj2
2
2
2
j
 
Ω j  ΩUj  ΩLj ~ O  j
1
 The case of j  1 : Ω 
41   
2
U
L


Ω 2  ΩL2 , ΩU2  No coupling with the continumm spectrum
Toroidal Alfven Eigenmodes
 Contributi on of the localized potential with   1 :
2
2
2~



V


cos

f

s


cos

f


2~
~
 Ω   V    0, 
2

 f  1  s   sin  2
 Dispersion relation
s  1,   1 :
Ω 2  ΩL2
 
 C s  
2
2
ΩU  Ω
s

2
2 2
Ω

Ω
1
  1
2
2
2
2
L
U C s   s 
Ω 





Ω
~

Ω
,
Ω
L
U
2
2
4
1  C s   s 
2


TAE Features
• Existence of the Alfven frequency gap due
to the finite-toroidicity coupling between the
neighboring poloidal harmonics.
• Existence of eigenmodes with their
frequencies located inside the Alfven
frequency gap.
• These modes experience negligible
damping due to their frequencies
decoupled from the continuum spectrum.
EPM
•
•
•
•
Gyro-kinetic equation
Vorticity equation
Wave-particle resonances
EPM features
[ Chen, PoP, 1994 ]
Gyro-Kinetic Equation
f
f q 
v
 f
 Vlasov equation :
 v   E  B
0
t
x m 
c
 v
ˆ

 q v
 
ˆ
~  L0 f 0  0, L0   v     B 0  
t
x m  c
Q  Q0  Q 
 v
   

q  ~ v ~  f 0
 Lˆ ~
E  B
0 f  

m
c
 v
 Transforma tion to guiding - center space : x, v   X, V 
  v 2 2 ,   v2 2 B0 ,   sign v// 

 v   v e1 cos   e 2 sin  , C  qB0 mc
 C ~ k //  C ~  C L ~ O 
 Gyro - ordering : 
12


k

~
O

,   C  v C 
 C

X  x  v  e // C
,

V   ,  ,  ,  
 
Gyro-Kinetic Equation (cont.)
QF0






D




q  f 0 g k   e // f 0 g  
~
~
 v



i
k

v

i

g

i



J






D
0
 // X //
m

C
X  





ˆ * F0



D
v J1   ~
J1   
~
~
~

 J 0   
J 0   
B//  i
v// 
 

kc
 X //


~ ~
q  f 0 g ~ QF0

J 0  ~ exp iLk 
 f g  g exp iLk   
m  



 
f 0 g  ~ v// ~ 
v J1   ~

   i
1  J 0   exp iLk  

B// exp iLk  


B0  
 X // 
kc
 

 X
  Lk  k  e //  v C 

with perturbati on form exp i  k   dX   i t ,  ~

 iA//  c ~ X //


Vorticity Equation

 B0
2
B0
Q    d  d  d
Q
v//
 0
0
0
Gyro - kinetic equation           
 Voticity equation
k



 ~ 
q 2 v//2 f 0 g
c2
  k 2 
4
2
B0
1  J 0   
 1  2 2 

2
4
X //  B0  k  c j
m B0 

X
j

 // 

  qJ 0  
j

j
D ~
g

 i  q v//
j
j
q 2  f 0 g ~ QF0


m  

J1    ~
g
 X //

j
 2 ~ D J 02   ~ v J 0  J1   ~ 

B// 
J0 

kc


f 0 g 
v J 0  J1   ~  
2
~

B//  
 1  J 0    
B0  
kc



j
Vorticity Equation (cont.)
~
f 0 g   0, B//  0
~
 E//  0  ~  ~
        Shear Alfven equation :
c2
  k 2 ~ 
D ~


B0
  qJ 0  
g
2


4
X //  B0 X //  j

j
2
ˆ


F
QF

J
q 2  f 0 g
2
2
* 0
0
D 0   ~
1  J 0   
J 0   




m   


 
2
 2


j

j
- line bending Inertial
Ballooning drive
Field















TC TE ~ 

2
2
2
~ 



k


k
4
π
q
2
 B
2  ~
~

nE nC ~ 


ˆ




ω
F
ω

* 0 D
    0 X //  B0 X // 
VA2
c 2m
C

4 qE

~  Kinetic compressio n




J

g
D 0
E

c2
2
Wave-Particle Resonances
~
S 

~
g i
  D g~  i S~1  i S~2
 Gyro - kinetic equation :

X // v//
v//


l
~
ds ~
l
l
l





g

G
exp
i

I

exp
i

I
S
exp
i

I

a
a 
s
v//
a

~
~
 g a,   g a,  
b
~ s
~ s ~ s
~
b ~
s
~
g b,   g b,  G   cot I a S1Ca  S 2 S a  S1S a  S 2Ca

     
2
l
b
b

ds
2
dl
dl
l
I a  
  D , Q   Q,  b  
v//
 b a v//
v//

a
a
 s
s
s
s
C

cos
I
,
S

sin
I
a
a
a , b  2  b
 a
b
 b
 1 
b
 Resonances : cot I a  cot    D   
2
  K     D  Kb


 

EPM Features
• The Alfven modes gain energy by resonant
interactions between Alfven waves and
energetic particles.
• The mode frequencies are characterized by
the typical frequencies of energetic
particles via the wave-particle resonance
conditions.
• The gained energy can overcome the
continuum damping.
 TAE
• Theoretical model
• Bound states in the second
ballooning-mode stable regime
• Basic features
• Kinetic excitations
[ Hu and Chen, PoP, 2004 ]
Theoretical Model
Basic Equations
Some Definitions
 TAE Features
• Existence of potential wells due to
ballooning curvature drive.
• Bound states of Alfven modes trapped in
the MHD potential wells.
• The trapped feature decouples the
discrete Alfven eigenmodes from the
continuum spectrum.
Summary
• Introduction to shear Alfven waves in
tokamaks and their interaction with
energetic particles.
• Discussions on the toroidicity-induced
Alfven eigenmode (TAE), the
energetic-particle continuum mode
(EPM), as well as the discrete Alfven
eigenmode ( TAE).
Alpha-TAE vs. EPM/TAE
• alpha-TAE: Bound states in the potential
wells due to the ballooning drive.
• EPM: Frequencies determined by the
wave-particle resonance conditions.
• TAE: Frequencies located inside the
toroidal Alfven frequency gap.