Then we will see that in our case, the ion temperature has the largest relative
perturbation and, similarly to Eq. (7.41), e
T i =T i ) e
ϕ ∼ e n i =n 0 (we assume here
T i ~ T e ). As a result, the ion temperature perturbation can still be described by
Eq. (7.36), which gives
e
T i
T i
¼
ω dw,i
ω
e
ϕ:
ð7:50Þ
Finding e n i we can allow for the ion temperature perturbation in the ion diamagnetic
flux (7.44) only and neglect the compressibility of the E
! Â B
!
drift flow since
e
T i =T i ) e
ϕ. So we have
e n i
n 0
¼
ω B,i
ω
e
T i
T i
:
ð7:51Þ
Assuming the Boltzmann relation for the electron density perturbation and electrostatic potential together with the plasma quasi-neutrality, from Eqs. (7.50 and 7.51)
we find
ω
2
¼ ω Ã,i ω B,i 2
T i
T e
cT e
eB 0
2 k
2
y
R
dℓn T i
ð Þ
dx
,
ð7:52Þ
which satisfies the inequality (7.49) and all other assumptions made in the course
of our derivation of Eq. (7.52) and shows the instability of the toroidal ITG for
∂T i /∂x < 0.
We notice that x is the local coordinate and for the “standard” temperature
distribution over the minor radius, ∂T i /∂x < 0 region corresponds to the outboard
side of the torus (or to the so-called “bad” curvature region), whereas at the inboard
side (where the curvature is “good”) ∂T i /∂x > 0 and the mode is neutrally stable
(within our simplified treatment).
Equation (7.52) gives the growth rate of the ITG instability in the eikonal
approximation. More complex numerical simulations go beyond the eikonal approximation and treat the solution of the corresponding differential equations as an
eigenfunction-eigenvalue problem. Such solutions provide information about the
mode structure (somewhat similar to what we did for a nonlocal solution of the drift
wave arriving at Eqs. (7.20) and (7.21). As an example, in Fig. 7.4 one can find the
spatial structure of the eigenfunction of the electrostatic potential at some toroidal
angle found for the ITG mode. We notice that in agreement with our simplified
consideration, the amplitude of the electrostatic potential shows a strong enhancement at the outer side of the torus, where, according to Eq. (7.52), the driving
mechanism is localized. However, at relatively high radial gradients of the plasma
parameters, the spatial structure of the most unstable ITG eigenmode can be very
different (e.g. see [24, 25] and the references therein).
156
7 Anomalous Cross-Field Transport in Edge Plasma
perturbation and, similarly to Eq. (7.41), e
T i =T i ) e
ϕ ∼ e n i =n 0 (we assume here
T i ~ T e ). As a result, the ion temperature perturbation can still be described by
Eq. (7.36), which gives
e
T i
T i
¼
ω dw,i
ω
e
ϕ:
ð7:50Þ
Finding e n i we can allow for the ion temperature perturbation in the ion diamagnetic
flux (7.44) only and neglect the compressibility of the E
! Â B
!
drift flow since
e
T i =T i ) e
ϕ. So we have
e n i
n 0
¼
ω B,i
ω
e
T i
T i
:
ð7:51Þ
Assuming the Boltzmann relation for the electron density perturbation and electrostatic potential together with the plasma quasi-neutrality, from Eqs. (7.50 and 7.51)
we find
ω
2
¼ ω Ã,i ω B,i 2
T i
T e
cT e
eB 0
2 k
2
y
R
dℓn T i
ð Þ
dx
,
ð7:52Þ
which satisfies the inequality (7.49) and all other assumptions made in the course
of our derivation of Eq. (7.52) and shows the instability of the toroidal ITG for
∂T i /∂x < 0.
We notice that x is the local coordinate and for the “standard” temperature
distribution over the minor radius, ∂T i /∂x < 0 region corresponds to the outboard
side of the torus (or to the so-called “bad” curvature region), whereas at the inboard
side (where the curvature is “good”) ∂T i /∂x > 0 and the mode is neutrally stable
(within our simplified treatment).
Equation (7.52) gives the growth rate of the ITG instability in the eikonal
approximation. More complex numerical simulations go beyond the eikonal approximation and treat the solution of the corresponding differential equations as an
eigenfunction-eigenvalue problem. Such solutions provide information about the
mode structure (somewhat similar to what we did for a nonlocal solution of the drift
wave arriving at Eqs. (7.20) and (7.21). As an example, in Fig. 7.4 one can find the
spatial structure of the eigenfunction of the electrostatic potential at some toroidal
angle found for the ITG mode. We notice that in agreement with our simplified
consideration, the amplitude of the electrostatic potential shows a strong enhancement at the outer side of the torus, where, according to Eq. (7.52), the driving
mechanism is localized. However, at relatively high radial gradients of the plasma
parameters, the spatial structure of the most unstable ITG eigenmode can be very
different (e.g. see [24, 25] and the references therein).
156
7 Anomalous Cross-Field Transport in Edge Plasma