∂n
∂t
þ ∇ Á V
!
E
! ÂB
! n
∂n
∂t
þ ∇n Á V
!
E
! ÂB
! ¼ 0:
ð7:4Þ
As a result, from expressions (7.2 and 7.4) we find
∂e n
∂t
þ U dw
∂ e
ϕ
∂y
¼
∂e n
∂t
þ U dw
∂e n
∂y
¼ 0,
ð7:5Þ
where
U dw x
ð Þ ¼ À
cT e
eB
dℓn n
ð Þ
dx
cT e
eB
Λ n x
ð Þ:
ð7:6Þ
It is easy to see that the general solution of Eq. (7.5) can be written as e n r
!
, t
¼
f y À U dw t
ð
Þ and describes, in agreement with our physical picture, the wave
propagating along the y coordinate (here f(y) is an arbitrary function, which can
also depend on both the x and z coordinates).
We notice that U
!
dw is equal to the electron diamagnetic velocity, V
! dia
e , which
occurs due to inhomogeneity of the electron Larmor circles and can be found from
the cross-field electron momentum balance equation where the electron pressure
gradient (the density gradient, for the case of constant temperature) is balanced by
the Lorentz force:
ÀT e
dn x
ð Þ
dx
e
!
x À
eB 0
c
V
! dia
e  e
!
z
n x
ð Þ ¼ 0:
ð7:7Þ
However, the expression for the evolution of the perturbed plasma density (7.5)
does not take into account some crucially important effects. First, in the derivation of
Eq. (7.5) we assumed that the cross-field motion of the charged particles is only due
to E
! Â B
!
drift, which is the same for both electrons and ions. However, the mass
difference causes an important disparity in the dynamics of the electrons and ions. It
results in a more complex governing equation for the perturbed plasma density,
which becomes important for both the linear stability and nonlinear interactions of
the drift waves.
Secondly, and more importantly, Eq. (6.5) shows that the amplitudes of both the
perturbed plasma density and electrostatic potential, linked to the density perturbation through the Boltzmann relation (7.2), remain the same and do not grow in time.
Therefore, it cannot describe instabilities resulting in large plasma density/potential
fluctuations and strong cross-field anomalous transport observed in experiments.
To address the first issue we consider the equation for the ion velocity, V
!
i , which
follows from the ion momentum balance equation and in the cold ion approximation
reads:
7.2 Linear Theory of Edge Plasma Instabilities
143
∂t
þ ∇ Á V
!
E
! ÂB
! n
∂n
∂t
þ ∇n Á V
!
E
! ÂB
! ¼ 0:
ð7:4Þ
As a result, from expressions (7.2 and 7.4) we find
∂e n
∂t
þ U dw
∂ e
ϕ
∂y
¼
∂e n
∂t
þ U dw
∂e n
∂y
¼ 0,
ð7:5Þ
where
U dw x
ð Þ ¼ À
cT e
eB
dℓn n
ð Þ
dx
cT e
eB
Λ n x
ð Þ:
ð7:6Þ
It is easy to see that the general solution of Eq. (7.5) can be written as e n r
!
, t
¼
f y À U dw t
ð
Þ and describes, in agreement with our physical picture, the wave
propagating along the y coordinate (here f(y) is an arbitrary function, which can
also depend on both the x and z coordinates).
We notice that U
!
dw is equal to the electron diamagnetic velocity, V
! dia
e , which
occurs due to inhomogeneity of the electron Larmor circles and can be found from
the cross-field electron momentum balance equation where the electron pressure
gradient (the density gradient, for the case of constant temperature) is balanced by
the Lorentz force:
ÀT e
dn x
ð Þ
dx
e
!
x À
eB 0
c
V
! dia
e  e
!
z
n x
ð Þ ¼ 0:
ð7:7Þ
However, the expression for the evolution of the perturbed plasma density (7.5)
does not take into account some crucially important effects. First, in the derivation of
Eq. (7.5) we assumed that the cross-field motion of the charged particles is only due
to E
! Â B
!
drift, which is the same for both electrons and ions. However, the mass
difference causes an important disparity in the dynamics of the electrons and ions. It
results in a more complex governing equation for the perturbed plasma density,
which becomes important for both the linear stability and nonlinear interactions of
the drift waves.
Secondly, and more importantly, Eq. (6.5) shows that the amplitudes of both the
perturbed plasma density and electrostatic potential, linked to the density perturbation through the Boltzmann relation (7.2), remain the same and do not grow in time.
Therefore, it cannot describe instabilities resulting in large plasma density/potential
fluctuations and strong cross-field anomalous transport observed in experiments.
To address the first issue we consider the equation for the ion velocity, V
!
i , which
follows from the ion momentum balance equation and in the cold ion approximation
reads:
7.2 Linear Theory of Edge Plasma Instabilities
143