magnetic field). Therefore, as an illustration, we consider the motion of a charged
particle in a constant magnetic field B
!
0 ¼ B 0 e
!
z and a 2D electrostatic potential,
φ(x, y, t), with a characteristic magnitude, φ 0 , and a spatial scale length, λ ⊥ , which
varies on the time-scale τ φ ~ ω
À1 . Assume that λ ⊥ is larger than the particle gyroradius whereas ω is smaller than the particle gyrofrequency. In this case, the charged
particle will mostly experience E
! Â B
!
drift along the equipotentials of φ(x, y, t) with
a characteristic speed V E
! ÂB
! ∼ cφ 0 = λ ⊥ B 0
ð
Þ. However, the equipotentials can be
considered “fixed” only for the time t e
< τ φ and in this time, the particle would
move by the distance δ ∼ V E
! ÂB
!τ φ along the equipotentials, which we assume to be
smaller than λ ⊥ . If the “landscape” of the φ(x, y, t) equipotentials is completely
changed by the time t e
> τ φ , then the particle at t e
> τ φ will drift along an equipotential completely different from what it was for t e
< τ φ . As a result, the distance δ can
be considered as a “jump” of the particle in the (x,y) plane, which occurs in random
directions within the time ~τ φ , and particle transport in the φ(x, y, t) potential has a
diffusive nature. Taking into account that we assumed δ e
< λ ⊥ , and estimating
eφ 0 ~ T (where T is the plasma temperature), we find that the particle diffusion
coefficient D φ e
< D B ¼ cT=eB 0 , where D B is the so-called Bohm diffusion
coefficient.
This physical picture shows that the presence of electric field fluctuations having
relatively small spatiotemporal scales can result in cross-field plasma transport
exceeding, in practice, the classical one which is governed by quasi-stationary and
large-spatial-scale processes.
However, in reality, anomalous cross-field plasma transport is much more complex. Nonlinear interactions of plasma fluctuations result, on the one hand, in some
sort of self-regulation and even suppression of anomalous transport. On the other
hand, they can change transport from relatively slow diffusive to very fast convective. Today we have no full understanding of all processes governing anomalous
plasma transport. Therefore, in what follows, we present just very basic ingredients
of plasma instabilities and features of anomalous plasma transport, with the emphasis on the processes more typical for the edge plasmas. Comprehensive review of
instabilites in inhomogeneous plasmas is given in [148].
In this chapter, we adopt the following notations: all parameters with “tilde” are
considered to be perturbations small in comparison with the stationary (or quasistationary) parameters having no “tilde” sign, or resulting in such small perturbations: e.g. a perturbation of the plasma density e n is much smaller than the background
plasma density n je n j=n ( 1
ð
Þ ;
e
V
!
is a small perturbation of the velocity, which
either is small in comparison with the background velocity V
!
, or produces a small
variation of such plasma parameters as the pressure, density, etc.
140
7 Anomalous Cross-Field Transport in Edge Plasma
particle in a constant magnetic field B
!
0 ¼ B 0 e
!
z and a 2D electrostatic potential,
φ(x, y, t), with a characteristic magnitude, φ 0 , and a spatial scale length, λ ⊥ , which
varies on the time-scale τ φ ~ ω
À1 . Assume that λ ⊥ is larger than the particle gyroradius whereas ω is smaller than the particle gyrofrequency. In this case, the charged
particle will mostly experience E
! Â B
!
drift along the equipotentials of φ(x, y, t) with
a characteristic speed V E
! ÂB
! ∼ cφ 0 = λ ⊥ B 0
ð
Þ. However, the equipotentials can be
considered “fixed” only for the time t e
< τ φ and in this time, the particle would
move by the distance δ ∼ V E
! ÂB
!τ φ along the equipotentials, which we assume to be
smaller than λ ⊥ . If the “landscape” of the φ(x, y, t) equipotentials is completely
changed by the time t e
> τ φ , then the particle at t e
> τ φ will drift along an equipotential completely different from what it was for t e
< τ φ . As a result, the distance δ can
be considered as a “jump” of the particle in the (x,y) plane, which occurs in random
directions within the time ~τ φ , and particle transport in the φ(x, y, t) potential has a
diffusive nature. Taking into account that we assumed δ e
< λ ⊥ , and estimating
eφ 0 ~ T (where T is the plasma temperature), we find that the particle diffusion
coefficient D φ e
< D B ¼ cT=eB 0 , where D B is the so-called Bohm diffusion
coefficient.
This physical picture shows that the presence of electric field fluctuations having
relatively small spatiotemporal scales can result in cross-field plasma transport
exceeding, in practice, the classical one which is governed by quasi-stationary and
large-spatial-scale processes.
However, in reality, anomalous cross-field plasma transport is much more complex. Nonlinear interactions of plasma fluctuations result, on the one hand, in some
sort of self-regulation and even suppression of anomalous transport. On the other
hand, they can change transport from relatively slow diffusive to very fast convective. Today we have no full understanding of all processes governing anomalous
plasma transport. Therefore, in what follows, we present just very basic ingredients
of plasma instabilities and features of anomalous plasma transport, with the emphasis on the processes more typical for the edge plasmas. Comprehensive review of
instabilites in inhomogeneous plasmas is given in [148].
In this chapter, we adopt the following notations: all parameters with “tilde” are
considered to be perturbations small in comparison with the stationary (or quasistationary) parameters having no “tilde” sign, or resulting in such small perturbations: e.g. a perturbation of the plasma density e n is much smaller than the background
plasma density n je n j=n ( 1
ð
Þ ;
e
V
!
is a small perturbation of the velocity, which
either is small in comparison with the background velocity V
!
, or produces a small
variation of such plasma parameters as the pressure, density, etc.
140
7 Anomalous Cross-Field Transport in Edge Plasma