∇ Á J
! dia
⊥ ¼ À
2c
B
b
! Á e
!
R Â ∇ e P tot
R
,
ð7:54Þ
where e P tot is the perturbation of the total plasma pressure.
Quite often Eq. (7.53) is considered in the Boussinesq approximation where the
expression (7.53) is simplified as follows:
∇ Á J
! inert
⊥ ¼ Àen
∂
∂t
þ V
!
E
! ÂB
! Á ∇
c∇
2 φ
BΩ Bi
&
'
:
ð7:55Þ
For linear theory, such an approximation is equivalent to the omission of the inertial
part of the cross-field ion velocity in the term V
!
i⊥ Á ∇n in Eq. (7.12). Then, the linear
vorticity equation reads
∇ Á
~
J
! Àen
∂
∂t
c∇
2
~
φ
BΩ Bi
À
2c
B
b
! Á
e
!
R Â ∇ ~
P tot
R
þ ∇ k Á ~ J k ¼ 0:
ð7:56Þ
To simplify our algebra, we will assume that the electron temperature is constant
(which gives an order of unity correction for the growth rate, recall Eq. (7.32)) so that
for the perturbed electron pressure we take e P e ¼ e nT e .
Finding the parallel component of the electric current, ~ J k , we will ignore the ion
dynamics along the magnetic field lines. Then, assuming ω/ν ei < 1, from the parallel
electron momentum balance equation in linear approximation we have
∇ k Á ~ J k ¼ enν k
~
ɸ À
~ n
n
:
ð7:57Þ
First, we consider the case of ν k ! 0, which corresponds to reduced electron
mobility along the magnetic field lines and, in the absence of toroidal effects, results
in a relatively slow instability (7.35). However, toroidal effects, causing magnetic
drifts, provide plasma polarization which is not related to electron mobility along the
magnetic field lines and, therefore, is not bounded by the small magnitude of ν k .
Therefore, in the simplest case, we can take J k ¼ 0 and relax Eq. (7.56) to
ρ
2
s k
2
⊥
∂
∂t
ee φ
T e
À
2cT e
eBR
∂
∂y
e n
n
¼ 0:
ð7:58Þ
The variation of plasma density can be found from the ion continuity equation.
Considering the case ρ
2
s k
2
⊥ < 1 and recalling the inequality (7.48), we can neglect
the compressibility in both the inertial and E
! Â B
!
ion drift flows and use Eq. (7.4) for
the plasma density perturbation. As a result, we arrive at the following equation
158
7 Anomalous Cross-Field Transport in Edge Plasma
! dia
⊥ ¼ À
2c
B
b
! Á e
!
R Â ∇ e P tot
R
,
ð7:54Þ
where e P tot is the perturbation of the total plasma pressure.
Quite often Eq. (7.53) is considered in the Boussinesq approximation where the
expression (7.53) is simplified as follows:
∇ Á J
! inert
⊥ ¼ Àen
∂
∂t
þ V
!
E
! ÂB
! Á ∇
c∇
2 φ
BΩ Bi
&
'
:
ð7:55Þ
For linear theory, such an approximation is equivalent to the omission of the inertial
part of the cross-field ion velocity in the term V
!
i⊥ Á ∇n in Eq. (7.12). Then, the linear
vorticity equation reads
∇ Á
~
J
! Àen
∂
∂t
c∇
2
~
φ
BΩ Bi
À
2c
B
b
! Á
e
!
R Â ∇ ~
P tot
R
þ ∇ k Á ~ J k ¼ 0:
ð7:56Þ
To simplify our algebra, we will assume that the electron temperature is constant
(which gives an order of unity correction for the growth rate, recall Eq. (7.32)) so that
for the perturbed electron pressure we take e P e ¼ e nT e .
Finding the parallel component of the electric current, ~ J k , we will ignore the ion
dynamics along the magnetic field lines. Then, assuming ω/ν ei < 1, from the parallel
electron momentum balance equation in linear approximation we have
∇ k Á ~ J k ¼ enν k
~
ɸ À
~ n
n
:
ð7:57Þ
First, we consider the case of ν k ! 0, which corresponds to reduced electron
mobility along the magnetic field lines and, in the absence of toroidal effects, results
in a relatively slow instability (7.35). However, toroidal effects, causing magnetic
drifts, provide plasma polarization which is not related to electron mobility along the
magnetic field lines and, therefore, is not bounded by the small magnitude of ν k .
Therefore, in the simplest case, we can take J k ¼ 0 and relax Eq. (7.56) to
ρ
2
s k
2
⊥
∂
∂t
ee φ
T e
À
2cT e
eBR
∂
∂y
e n
n
¼ 0:
ð7:58Þ
The variation of plasma density can be found from the ion continuity equation.
Considering the case ρ
2
s k
2
⊥ < 1 and recalling the inequality (7.48), we can neglect
the compressibility in both the inertial and E
! Â B
!
ion drift flows and use Eq. (7.4) for
the plasma density perturbation. As a result, we arrive at the following equation
158
7 Anomalous Cross-Field Transport in Edge Plasma