drift wave, the so-called ion temperature gradient (ITG) drift mode [22] can become
unstable.
As an example, we consider plasma with homogeneous both density and electron
temperature embedded into a constant magnetic field B
!
0 ¼ B 0 e
!
z but having the ion
temperature depending on the x-coordinate, ∂T i /∂x 6 ¼ 0.
Then, the ion E
! Â B
!
drift will result in a perturbation of the ion temperature, e
T i ,
similar to that of the plasma density described by Eq. (7.4):
∂
∂t
e
T i
T i
þ U dw,i
∂ e
ϕ
∂y
¼ 0,
ð7:36Þ
where
U dw,i ¼ À
cT e
eB 0
dℓn T i
ð Þ
dx
:
ð7:37Þ
For the case of k k 6 ¼ 0, the perturbation of the ion temperature e
T i results in the
perturbation of the plasma pressure along the magnetic field lines, which is somewhat similar to the plasma density inhomogeneity along the magnetic field shown in
Fig. 7.1.
Standard wisdom would suggest that the perturbation of plasma pressure should
result in some sort of a “sound” wave, similar to the ion sound waves, Eq. (7.17),
with ω ~ C p k k , where C p is the plasma sound speed. Although this option exists, we
consider a very different regime of the ion dynamics, which will finally bring us to
the unstable ITG mode. Following Eq. (7.1) we assume that ω ) V T i k k . We will see
that such an inequality becomes possible because in inhomogeneous plasma, among
different characteristic frequencies of the waves, there are drift frequencies such as
ω Ã ¼ U dw k y (Eq. (7.12)) and, as we will see,
ω Ã,i ¼ U dw,i k y ,
ð7:38Þ
which do not depend on k k .
Then, the ion dynamics can be considered as a motion under a “fast”
ðω ) V T i k k Þ applied force caused by the ion temperature perturbation. As a result
of such forced ion motion, the perturbations of both the plasma density and electrostatic potential will be relatively small but vital for the instability:
ee φ
T i
∼
e n i
n
(
e
T i
T i
:
ð7:39Þ
We assume here that T e ~ T i . Then, from ion momentum balance along the magnetic
field and the continuity equations we have
152
7 Anomalous Cross-Field Transport in Edge Plasma
unstable.
As an example, we consider plasma with homogeneous both density and electron
temperature embedded into a constant magnetic field B
!
0 ¼ B 0 e
!
z but having the ion
temperature depending on the x-coordinate, ∂T i /∂x 6 ¼ 0.
Then, the ion E
! Â B
!
drift will result in a perturbation of the ion temperature, e
T i ,
similar to that of the plasma density described by Eq. (7.4):
∂
∂t
e
T i
T i
þ U dw,i
∂ e
ϕ
∂y
¼ 0,
ð7:36Þ
where
U dw,i ¼ À
cT e
eB 0
dℓn T i
ð Þ
dx
:
ð7:37Þ
For the case of k k 6 ¼ 0, the perturbation of the ion temperature e
T i results in the
perturbation of the plasma pressure along the magnetic field lines, which is somewhat similar to the plasma density inhomogeneity along the magnetic field shown in
Fig. 7.1.
Standard wisdom would suggest that the perturbation of plasma pressure should
result in some sort of a “sound” wave, similar to the ion sound waves, Eq. (7.17),
with ω ~ C p k k , where C p is the plasma sound speed. Although this option exists, we
consider a very different regime of the ion dynamics, which will finally bring us to
the unstable ITG mode. Following Eq. (7.1) we assume that ω ) V T i k k . We will see
that such an inequality becomes possible because in inhomogeneous plasma, among
different characteristic frequencies of the waves, there are drift frequencies such as
ω Ã ¼ U dw k y (Eq. (7.12)) and, as we will see,
ω Ã,i ¼ U dw,i k y ,
ð7:38Þ
which do not depend on k k .
Then, the ion dynamics can be considered as a motion under a “fast”
ðω ) V T i k k Þ applied force caused by the ion temperature perturbation. As a result
of such forced ion motion, the perturbations of both the plasma density and electrostatic potential will be relatively small but vital for the instability:
ee φ
T i
∼
e n i
n
(
e
T i
T i
:
ð7:39Þ
We assume here that T e ~ T i . Then, from ion momentum balance along the magnetic
field and the continuity equations we have
152
7 Anomalous Cross-Field Transport in Edge Plasma