ω þ
2i
ρ 2
s k
2
⊥
C s
L cl
1 À
1
2
þ Λ sh
ω Ã,e
ω
n
o
¼ 0,
ð7:83Þ
where
ω Ã,e ¼ À
cT e
eB 0
dℓn T e
ð Þ
dx
k y :
ð7:84Þ
One can see that the growth rate of the instability described by Eq. (7.83) can be of
the order of ω Ã, e .
So far we consider the plasma waves and instabilities related to plasma polarization caused by the electron dynamics along the magnetic field and the cross-field
magnetic drift of the charged particles. However, plasma polarization can also be
related to the interplay of the electric current along the magnetic field lines and crossfield inhomogeneity of the plasma conductivity (e.g. see [27, 42]). Indeed, for the
case where the plasma current flows only along the magnetic field lines, it should be
maintained constant. However, displacement of a plasma slab inclined to the magnetic field lines, similar to that shown in Fig. 7.9, causes a perturbation of the plasma
conductivity along the magnetic field lines. Therefore, to keep the electric current
constant, some additional electric field appears inside the slab, caused by charge
accumulation at the boundaries of the slab. But due to the inclination of the slab, a
cross-field electric field and corresponding E
! Â B
!
plasma drift emerge, which can
displace this fluid element even more.
Indeed, from the conservation of parallel current, j ¼ σ(x)E ¼ const., where
σ(x) is the plasma conductivity and E is the electric field, we find
E
dσ x
ð Þ
dx
e x þ ik z σe φ ¼ 0:
ð7:85Þ
Here e x is the displacement and e
φ the perturbation of the electrostatic potential.
However, on the other hand, we have
Fig. 7.9 Plasma
polarization due to electric
current along the magnetic
field and cross-field
inhomogeneity of plasma
conductivity
7.2 Linear Theory of Edge Plasma Instabilities
167
2i
ρ 2
s k
2
⊥
C s
L cl
1 À
1
2
þ Λ sh
ω Ã,e
ω
n
o
¼ 0,
ð7:83Þ
where
ω Ã,e ¼ À
cT e
eB 0
dℓn T e
ð Þ
dx
k y :
ð7:84Þ
One can see that the growth rate of the instability described by Eq. (7.83) can be of
the order of ω Ã, e .
So far we consider the plasma waves and instabilities related to plasma polarization caused by the electron dynamics along the magnetic field and the cross-field
magnetic drift of the charged particles. However, plasma polarization can also be
related to the interplay of the electric current along the magnetic field lines and crossfield inhomogeneity of the plasma conductivity (e.g. see [27, 42]). Indeed, for the
case where the plasma current flows only along the magnetic field lines, it should be
maintained constant. However, displacement of a plasma slab inclined to the magnetic field lines, similar to that shown in Fig. 7.9, causes a perturbation of the plasma
conductivity along the magnetic field lines. Therefore, to keep the electric current
constant, some additional electric field appears inside the slab, caused by charge
accumulation at the boundaries of the slab. But due to the inclination of the slab, a
cross-field electric field and corresponding E
! Â B
!
plasma drift emerge, which can
displace this fluid element even more.
Indeed, from the conservation of parallel current, j ¼ σ(x)E ¼ const., where
σ(x) is the plasma conductivity and E is the electric field, we find
E
dσ x
ð Þ
dx
e x þ ik z σe φ ¼ 0:
ð7:85Þ
Here e x is the displacement and e
φ the perturbation of the electrostatic potential.
However, on the other hand, we have
Fig. 7.9 Plasma
polarization due to electric
current along the magnetic
field and cross-field
inhomogeneity of plasma
conductivity
7.2 Linear Theory of Edge Plasma Instabilities
167