stands for the Geodesic Acoustic Mode, which is a specific, toroidally symmetric
mode in a tokamak, see [34, 35] and the references therein).
Another example can be found in [21], where the resistive drift waves and the
resistive interchange mode limit of the resistive drift mode (driven by magnetic
drifts) were observed correspondingly at the High Field Side (HFS) inboard and Low
Field Side (LFS) outboard sides of the Helimak toroidal device, see Fig. 7.7.
7.2.7 Electromagnetic Effects
As we can notice, all plasma waves we have considered so far were accompanied by
electric currents. These currents generate fluctuating electromagnetic fields, which in
plasma are manifested as the Alfven waves. However, we neglected the electromagnetic effects in all preceding considerations. This can be justified for the case where
the characteristic frequencies of the waves considered are much lower than the
frequency of the Alfven waves or, to be more precise, the frequency of the shear
Alfven wave, k k V A , where V A ¼ B=
ffiffiffiffiffiffiffiffiffiffiffi ffi
4πMn
p
is the Alfven speed.
As an example, we consider the impact of electromagnetic effects on the
collisionless drift wave described by Eq. (7.14). Recall that to avoid the Landau
damping on electrons when deriving Eq. (7.14), we assumed ω < k k V Te . Therefore,
the inequality ω < k k V A is “automatically” satisfied for the case where V Te < V A
(β e nT e /(B
2 /4π) < m/M), which gives ω < k k V Te < k k V A . However, for larger
β e (β e > m/M), the impact of electromagnetic effects can be important. To address
this issue, in addition to the electrostatic potential φ we introduce a vector potential
A
! ¼ e
!
z A k
r
! , t
and will assume that both of them are small. Adopting the eikonal
approximation for the perturbed quantities, we find the following contributions from
the electromagnetic term to the amplitudes of fluctuating electric and magnetic
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
0
1
2
3
4
5
(a)
(b)
LFS
HFS
1.2 kHz
f (kHz)
L c (m)
Fluctuation spectrum (a.u.)
6
20
20
40
40
60
60
80
80
100
100
120
120
140
140
δn/n
Fig. 7.7 (a) Fluctuation frequency spectra at HFS (of resistive drift wave, blue curve) and LFS
(of resistive interchange mode, red and black curves) of the Helimak device; and (b) Relative
density fluctuation at both LFS and HFS versus the connection length along helical magnetic field
between the end plates. (Reproduced with permission from [21], © AIP Publishing 2006)
162
7 Anomalous Cross-Field Transport in Edge Plasma
mode in a tokamak, see [34, 35] and the references therein).
Another example can be found in [21], where the resistive drift waves and the
resistive interchange mode limit of the resistive drift mode (driven by magnetic
drifts) were observed correspondingly at the High Field Side (HFS) inboard and Low
Field Side (LFS) outboard sides of the Helimak toroidal device, see Fig. 7.7.
7.2.7 Electromagnetic Effects
As we can notice, all plasma waves we have considered so far were accompanied by
electric currents. These currents generate fluctuating electromagnetic fields, which in
plasma are manifested as the Alfven waves. However, we neglected the electromagnetic effects in all preceding considerations. This can be justified for the case where
the characteristic frequencies of the waves considered are much lower than the
frequency of the Alfven waves or, to be more precise, the frequency of the shear
Alfven wave, k k V A , where V A ¼ B=
ffiffiffiffiffiffiffiffiffiffiffi ffi
4πMn
p
is the Alfven speed.
As an example, we consider the impact of electromagnetic effects on the
collisionless drift wave described by Eq. (7.14). Recall that to avoid the Landau
damping on electrons when deriving Eq. (7.14), we assumed ω < k k V Te . Therefore,
the inequality ω < k k V A is “automatically” satisfied for the case where V Te < V A
(β e nT e /(B
2 /4π) < m/M), which gives ω < k k V Te < k k V A . However, for larger
β e (β e > m/M), the impact of electromagnetic effects can be important. To address
this issue, in addition to the electrostatic potential φ we introduce a vector potential
A
! ¼ e
!
z A k
r
! , t
and will assume that both of them are small. Adopting the eikonal
approximation for the perturbed quantities, we find the following contributions from
the electromagnetic term to the amplitudes of fluctuating electric and magnetic
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
0
1
2
3
4
5
(a)
(b)
LFS
HFS
1.2 kHz
f (kHz)
L c (m)
Fluctuation spectrum (a.u.)
6
20
20
40
40
60
60
80
80
100
100
120
120
140
140
δn/n
Fig. 7.7 (a) Fluctuation frequency spectra at HFS (of resistive drift wave, blue curve) and LFS
(of resistive interchange mode, red and black curves) of the Helimak device; and (b) Relative
density fluctuation at both LFS and HFS versus the connection length along helical magnetic field
between the end plates. (Reproduced with permission from [21], © AIP Publishing 2006)
162
7 Anomalous Cross-Field Transport in Edge Plasma