fields:
~
E
! ¼ iω e
!
z
~
A k =c and
~
B
! ¼ i
k
! Â e
!
z
~
A k . In addition, we have the following
expression for the fluctuating electric current associated with the vector potential
~
J
!
k ¼
c
4π
∇ Â
~
B
!
¼
ck
2
⊥
4π
~
A k e
!
z :
ð7:68Þ
The component of the fluctuating electric current perpendicular to B
!
is described
by Eq. (7.55). As a result, using expressions (7.55) and (7.68), from the condition
∇ Á
e
J
! ¼ 0 we find the relation between e
φ and ~
A k :
ω ~
A k =k k c ¼ ðω=k k V A Þ
2 ~
φ:
ð7:69Þ
For ω < k k V Te , the electron density perturbation can be found from the stationary
parallel electron momentum balance equation where electron temperature can be
assumed constant. However, now we need to allow for small bending of the
magnetic field lines, which is caused by electromagnetic effects. As a result, we have
À
T e
e
k k
~ n e
n 0
þ
ω Ã
ω
À 1
ω ~
A k
c
þ k k ~
φ ¼ 0:
ð7:70Þ
Then, using Eqs. (7.69) and (7.70), taking the ion density perturbation from
Eq. (7.11) and assuming the quasi-neutrality condition, we arrive at the following
dispersion equation
ðω Ã =ω À 1Þ
n
1 À ðω=k k V A Þ
2
o
À ρ
2
s k
2
⊥ ¼ 0:
ð7:71Þ
Equation (7.71) describes the so-called drift-Alfven wave. In particular, for the
large and small values of the ω Ã /k k V A ratio, it gives, respectively, the drift wave
Eq. (7.14) and the shear Alfven wave:
( ω ¼ ω Ã =ð1 þ ρ
2
s k
2
⊥ Þ, for ω Ã =k k V A ( 1
ω
2
¼ ðk k V A Þ
2 ð1 þ ρ
2
s k
2
⊥ Þ, for ω Ã =k k V A ) 1
:
ð7:72Þ
In our evaluation of electromagnetic effects, we assumed so far that the magnetic
field is constant and straight, B
! ¼ B e
!
z . As a result, the non-divergence-free crossfield plasma current still appears only due to ion inertia, recall Eq. (7.55). However,
in a tokamak magnetic configuration, we should allow for the contribution of the
diamagnetic current (7.54). Considering low β plasma and being interested in the
waves with the characteristic frequency ω ~
> k k V A , using Eq. (7.12) for ion density
perturbation, we can neglect the ion dynamics along the magnetic field lines (which
7.2 Linear Theory of Edge Plasma Instabilities
163
~
E
! ¼ iω e
!
z
~
A k =c and
~
B
! ¼ i
k
! Â e
!
z
~
A k . In addition, we have the following
expression for the fluctuating electric current associated with the vector potential
~
J
!
k ¼
c
4π
∇ Â
~
B
!
¼
ck
2
⊥
4π
~
A k e
!
z :
ð7:68Þ
The component of the fluctuating electric current perpendicular to B
!
is described
by Eq. (7.55). As a result, using expressions (7.55) and (7.68), from the condition
∇ Á
e
J
! ¼ 0 we find the relation between e
φ and ~
A k :
ω ~
A k =k k c ¼ ðω=k k V A Þ
2 ~
φ:
ð7:69Þ
For ω < k k V Te , the electron density perturbation can be found from the stationary
parallel electron momentum balance equation where electron temperature can be
assumed constant. However, now we need to allow for small bending of the
magnetic field lines, which is caused by electromagnetic effects. As a result, we have
À
T e
e
k k
~ n e
n 0
þ
ω Ã
ω
À 1
ω ~
A k
c
þ k k ~
φ ¼ 0:
ð7:70Þ
Then, using Eqs. (7.69) and (7.70), taking the ion density perturbation from
Eq. (7.11) and assuming the quasi-neutrality condition, we arrive at the following
dispersion equation
ðω Ã =ω À 1Þ
n
1 À ðω=k k V A Þ
2
o
À ρ
2
s k
2
⊥ ¼ 0:
ð7:71Þ
Equation (7.71) describes the so-called drift-Alfven wave. In particular, for the
large and small values of the ω Ã /k k V A ratio, it gives, respectively, the drift wave
Eq. (7.14) and the shear Alfven wave:
( ω ¼ ω Ã =ð1 þ ρ
2
s k
2
⊥ Þ, for ω Ã =k k V A ( 1
ω
2
¼ ðk k V A Þ
2 ð1 þ ρ
2
s k
2
⊥ Þ, for ω Ã =k k V A ) 1
:
ð7:72Þ
In our evaluation of electromagnetic effects, we assumed so far that the magnetic
field is constant and straight, B
! ¼ B e
!
z . As a result, the non-divergence-free crossfield plasma current still appears only due to ion inertia, recall Eq. (7.55). However,
in a tokamak magnetic configuration, we should allow for the contribution of the
diamagnetic current (7.54). Considering low β plasma and being interested in the
waves with the characteristic frequency ω ~
> k k V A , using Eq. (7.12) for ion density
perturbation, we can neglect the ion dynamics along the magnetic field lines (which
7.2 Linear Theory of Edge Plasma Instabilities
163