Since initially the plasma density was not homogeneous in the x-direction, such a
protrusion will cause a weak plasma inhomogeneity along both z- and y-directions.
As a result, the electrons, which are much lighter and, therefore, much faster than the
ions, will try to escape from the plasma density bump, leaving the bulky and slow
ions alone. However, this charge separation along the magnetic field can only go
until the electric field that emerges from such plasma polarization stops further
electron escape, see Fig. 7.1a. We notice that due to the inclination of the shifted
plasma slab, the resulting electric field has a component in the y-direction (see
Fig. 7.1b), which will cause the E
! Â B
!
plasma drift in the x-direction. The direction
of this E
! Â B
!
plasma drift will be different on the different slopes of the density
protrusion: on one side it will move the plasma protrusion toward its initial position
(and decrease the protrusion), whereas on the other one it will move the plasma out
from its initial position (and increase the protrusion) as shown in Fig. 7.1b. As a
result, the plasma protrusion will be “advected” in the y-direction, exhibiting a wavelike motion. However, we note that even though such a wave of plasma density
perturbation propagates in the y-direction, the displacement of the plasma density
per se occurs only in the x-direction.
This simplest physical picture of collisionless drift waves can be easily
supplemented by a quantitative description (e.g. see [1–4]). For this, we will assume
that the perturbations of the plasma density, e n , are small je n j ( n
ð
Þ and all
nonlinear effects can be ignored. In addition, we will consider the characteristic
wavelength of the perturbations much larger than the Debye length, and the plasma
can be considered quasi-neutral, e n e ffi e n i e n, even though some charge polarization
effects exist. We will ignore the ion motion along the magnetic field lines and
assume that parallel dynamics of fast electrons reaches equilibrium virtually instantaneously so the gradient of the electron pressure along the magnetic field is balanced
by the electric force, which gives the Boltzmann relation for the perturbed electron
density and electrostatic potential e
φ:
e n e
n
¼
ee φ
T e
e
ϕ:
ð7:2Þ
Finally, we assume that the cross-field plasma (both electron and ion) velocity is
determined by E
! Â B
!
drift:
V
!
E
! ÂB
! ¼ À
c
B
2
∇φ Â B
!
:
ð7:3Þ
Then, taking into account that for a straight constant magnetic field ∇ Á V
!
E
! ÂB
! ¼ 0,
we have the following form of the plasma continuity equation
142
7 Anomalous Cross-Field Transport in Edge Plasma
protrusion will cause a weak plasma inhomogeneity along both z- and y-directions.
As a result, the electrons, which are much lighter and, therefore, much faster than the
ions, will try to escape from the plasma density bump, leaving the bulky and slow
ions alone. However, this charge separation along the magnetic field can only go
until the electric field that emerges from such plasma polarization stops further
electron escape, see Fig. 7.1a. We notice that due to the inclination of the shifted
plasma slab, the resulting electric field has a component in the y-direction (see
Fig. 7.1b), which will cause the E
! Â B
!
plasma drift in the x-direction. The direction
of this E
! Â B
!
plasma drift will be different on the different slopes of the density
protrusion: on one side it will move the plasma protrusion toward its initial position
(and decrease the protrusion), whereas on the other one it will move the plasma out
from its initial position (and increase the protrusion) as shown in Fig. 7.1b. As a
result, the plasma protrusion will be “advected” in the y-direction, exhibiting a wavelike motion. However, we note that even though such a wave of plasma density
perturbation propagates in the y-direction, the displacement of the plasma density
per se occurs only in the x-direction.
This simplest physical picture of collisionless drift waves can be easily
supplemented by a quantitative description (e.g. see [1–4]). For this, we will assume
that the perturbations of the plasma density, e n , are small je n j ( n
ð
Þ and all
nonlinear effects can be ignored. In addition, we will consider the characteristic
wavelength of the perturbations much larger than the Debye length, and the plasma
can be considered quasi-neutral, e n e ffi e n i e n, even though some charge polarization
effects exist. We will ignore the ion motion along the magnetic field lines and
assume that parallel dynamics of fast electrons reaches equilibrium virtually instantaneously so the gradient of the electron pressure along the magnetic field is balanced
by the electric force, which gives the Boltzmann relation for the perturbed electron
density and electrostatic potential e
φ:
e n e
n
¼
ee φ
T e
e
ϕ:
ð7:2Þ
Finally, we assume that the cross-field plasma (both electron and ion) velocity is
determined by E
! Â B
!
drift:
V
!
E
! ÂB
! ¼ À
c
B
2
∇φ Â B
!
:
ð7:3Þ
Then, taking into account that for a straight constant magnetic field ∇ Á V
!
E
! ÂB
! ¼ 0,
we have the following form of the plasma continuity equation
142
7 Anomalous Cross-Field Transport in Edge Plasma