∇ Á j
! dia
q ¼ À
2c
qB
b
! Á e
!
R Â ∇P q
R
:
ð7:45Þ
A similar result can be found from the divergence of the flux n q V
!
B,q .
Another implication of the spatial variation of the magnetic field is that the E
! Â B
!
drift velocity is also no longer divergence-free even for the potential electric field
E
! ¼ À∇φ. Indeed, assuming that B / 1/R, for this case from Eq. (7.3) we find
∇ Á V
!
E
! ÂB
! ¼
2c
B
b
! Á ∇ℓn B
ð Þ Â ∇φ
ð
Þ ¼ À
2c
B
b
! Á e
!
R Â ∇φ
R
:
ð7:46Þ
Although in large aspect ratio (R ) a) tokamaks j V B, q j is relatively small
j V B,q j/
1
R
( j U dw j , j U dw,i j/
1
a
,
ð7:47Þ
the cross-field drift velocity V
!
B can cause plasma polarization (recall Fig. 7.2),
which, as we will see, results in a new class of plasma instabilities. In what follows,
along with the frequencies ω Ã and ω Ã, q , we will use frequencies ω B,q ¼ V
!
B,q Á k
!
,
which, according to Eq. (7.47), have the following ordering
j ω B,q j( ω Ã , ω Ã,q
ð7:48Þ
As an example, we consider the impact of a weakly toroidal magnetic field on the
ITG mode. We introduce a local coordinate system (x,y,z), where x, y, and z are in
the radial, poloidal, and B
!
directions respectively and, as we did before, take constant
plasma density and electron temperature and assume that the ion temperature is
varying in the “radial” direction: ∂T i /∂x 6 ¼ 0.
We recall that for the case of a constant magnetic field, the only reason for
establishing the perturbations of the plasma density and corresponding electrostatic
potential was the ion motion along the magnetic field caused by the ion temperature
(pressure) perturbation. However, in the presence of toroidal effects, perturbation of
the ion temperature (pressure) results in a finite divergence of the diamagnetic ion
flux (7.45) and, therefore, in the ion density perturbation. To emphasize the toroidal
effects, we completely neglect here the impact of the ion dynamics along the
magnetic field lines and the effects of the ion inertia. We will be interested in plasma
fluctuations with the frequency ω such that
j ω B,q j< ω < ω Ã , ω Ã,q :
ð7:49Þ
7.2 Linear Theory of Edge Plasma Instabilities
155
! dia
q ¼ À
2c
qB
b
! Á e
!
R Â ∇P q
R
:
ð7:45Þ
A similar result can be found from the divergence of the flux n q V
!
B,q .
Another implication of the spatial variation of the magnetic field is that the E
! Â B
!
drift velocity is also no longer divergence-free even for the potential electric field
E
! ¼ À∇φ. Indeed, assuming that B / 1/R, for this case from Eq. (7.3) we find
∇ Á V
!
E
! ÂB
! ¼
2c
B
b
! Á ∇ℓn B
ð Þ Â ∇φ
ð
Þ ¼ À
2c
B
b
! Á e
!
R Â ∇φ
R
:
ð7:46Þ
Although in large aspect ratio (R ) a) tokamaks j V B, q j is relatively small
j V B,q j/
1
R
( j U dw j , j U dw,i j/
1
a
,
ð7:47Þ
the cross-field drift velocity V
!
B can cause plasma polarization (recall Fig. 7.2),
which, as we will see, results in a new class of plasma instabilities. In what follows,
along with the frequencies ω Ã and ω Ã, q , we will use frequencies ω B,q ¼ V
!
B,q Á k
!
,
which, according to Eq. (7.47), have the following ordering
j ω B,q j( ω Ã , ω Ã,q
ð7:48Þ
As an example, we consider the impact of a weakly toroidal magnetic field on the
ITG mode. We introduce a local coordinate system (x,y,z), where x, y, and z are in
the radial, poloidal, and B
!
directions respectively and, as we did before, take constant
plasma density and electron temperature and assume that the ion temperature is
varying in the “radial” direction: ∂T i /∂x 6 ¼ 0.
We recall that for the case of a constant magnetic field, the only reason for
establishing the perturbations of the plasma density and corresponding electrostatic
potential was the ion motion along the magnetic field caused by the ion temperature
(pressure) perturbation. However, in the presence of toroidal effects, perturbation of
the ion temperature (pressure) results in a finite divergence of the diamagnetic ion
flux (7.45) and, therefore, in the ion density perturbation. To emphasize the toroidal
effects, we completely neglect here the impact of the ion dynamics along the
magnetic field lines and the effects of the ion inertia. We will be interested in plasma
fluctuations with the frequency ω such that
j ω B,q j< ω < ω Ã , ω Ã,q :
ð7:49Þ
7.2 Linear Theory of Edge Plasma Instabilities
155