particular, the characteristic length of spatial variation of these parameters, L, (in the
absence of the magnetic field or in the direction parallel to B
!
) should be larger than
the mean free path of the thermal particles, λ C . Then, as we have seen above, the
distribution function f v
!
, r
!
can be expanded in the series of the integer powers of
the parameter γ λ C /L < 1 (here for simplicity we consider a stationary process) and
the closed system of fluid equations is derived. In most cases, only linear, / γ, terms
are held in this expansion. However, such an approach poses some questions. First, a
practical one: how small the parameter γ should be to ensure that the linear
approximation describes the transport properties of the gas/plasma well? And the
second, somewhat more academic question: are we sure that the distribution function
does not have some terms (e.g. / exp (ÀCγ
Àr
), where C and r are some positive
constants) which cannot be expanded in the series of any powers of γ, but which can
still be important for some range of γ (see also [16, 28, 29]).
We start our consideration with the simpler, first issue. Historically, in plasmarelated applications, this issue was first raised with respect to the validity range of the
Spitzer-Härm expression [30] for the electron conductive heat flux. According to
Eq. (6.16), the heat flux q
! is determined by the function Φ i (v
0 ) ¼ γ b
Φ(v
0 ) / γ, which
can be found either from expansion (6.14) or from [30], where Φ i (v
0
) was obtained
from the numerical solution of linearized electron kinetic equation. In [31, 32] it was
pointed out that the integral expression (6.16) for the heat flux contains high powers
of the electron velocity. As a result, the main contribution to this integral comes from
the velocities v
0
∼ v cond ∼ 2 Ä 3 Â v T e . However, the magnitude of Φ i (v
0 ) increases
with increasing v
0
> v T e (e.g. see both the expression (6.14) and the results from
[30]) and for v
0 ~ v cond we have b
Φ(v cond ) % 10
2 [30]. Thus, the applicability of the
linearized solution for electron heat conduction, which requires Φ v cond
ð
Þ
γ b
Φ v cond
ð
Þ< 1 gives the limitation for the validity of the Spitzer-Härm expression
for the electron conductive heat flux: γ e
< 10
À2 . Such a severe limitation can be
understood by recalling that the Coulomb mean-free path, λ v , of a particle with
velocity v scales as λ v ¼ λ T Â v=v T e
ð
Þ
4 . Therefore, the linear approximation for the
electron conductive heat flux, which assumes that the electron distribution function
is close to the Maxwellian, can only be valid if the electrons with velocities
v
0
∼ v cond > v T e are collisional, λ v cond =L e
< 1, which, finally, results in γ e
< 10
À2 .
However, in the edge plasmas, the inequality γ e
< 10
À2 usually does not hold and
the classical expression for the electron heat flux is not applicable. A similar problem
often occurs in the plasmas related to inertial confinement experiments (e.g. see
[32]). For the case of γ > 10
À2 , the electron distribution function starts to deviate
significantly from the Maxwellian distribution at v
0 ~ v cond . To describe this
“nonlocal” effect for heat conduction along the magnetic field (in the z-direction),
the expansion of the distribution function in the powers of the parameter γ becomes
impractical and an integral expression for the heat flux
6.4 The Electron Heat Transport in a Weakly Collisional Regime
127
absence of the magnetic field or in the direction parallel to B
!
) should be larger than
the mean free path of the thermal particles, λ C . Then, as we have seen above, the
distribution function f v
!
, r
!
can be expanded in the series of the integer powers of
the parameter γ λ C /L < 1 (here for simplicity we consider a stationary process) and
the closed system of fluid equations is derived. In most cases, only linear, / γ, terms
are held in this expansion. However, such an approach poses some questions. First, a
practical one: how small the parameter γ should be to ensure that the linear
approximation describes the transport properties of the gas/plasma well? And the
second, somewhat more academic question: are we sure that the distribution function
does not have some terms (e.g. / exp (ÀCγ
Àr
), where C and r are some positive
constants) which cannot be expanded in the series of any powers of γ, but which can
still be important for some range of γ (see also [16, 28, 29]).
We start our consideration with the simpler, first issue. Historically, in plasmarelated applications, this issue was first raised with respect to the validity range of the
Spitzer-Härm expression [30] for the electron conductive heat flux. According to
Eq. (6.16), the heat flux q
! is determined by the function Φ i (v
0 ) ¼ γ b
Φ(v
0 ) / γ, which
can be found either from expansion (6.14) or from [30], where Φ i (v
0
) was obtained
from the numerical solution of linearized electron kinetic equation. In [31, 32] it was
pointed out that the integral expression (6.16) for the heat flux contains high powers
of the electron velocity. As a result, the main contribution to this integral comes from
the velocities v
0
∼ v cond ∼ 2 Ä 3 Â v T e . However, the magnitude of Φ i (v
0 ) increases
with increasing v
0
> v T e (e.g. see both the expression (6.14) and the results from
[30]) and for v
0 ~ v cond we have b
Φ(v cond ) % 10
2 [30]. Thus, the applicability of the
linearized solution for electron heat conduction, which requires Φ v cond
ð
Þ
γ b
Φ v cond
ð
Þ< 1 gives the limitation for the validity of the Spitzer-Härm expression
for the electron conductive heat flux: γ e
< 10
À2 . Such a severe limitation can be
understood by recalling that the Coulomb mean-free path, λ v , of a particle with
velocity v scales as λ v ¼ λ T Â v=v T e
ð
Þ
4 . Therefore, the linear approximation for the
electron conductive heat flux, which assumes that the electron distribution function
is close to the Maxwellian, can only be valid if the electrons with velocities
v
0
∼ v cond > v T e are collisional, λ v cond =L e
< 1, which, finally, results in γ e
< 10
À2 .
However, in the edge plasmas, the inequality γ e
< 10
À2 usually does not hold and
the classical expression for the electron heat flux is not applicable. A similar problem
often occurs in the plasmas related to inertial confinement experiments (e.g. see
[32]). For the case of γ > 10
À2 , the electron distribution function starts to deviate
significantly from the Maxwellian distribution at v
0 ~ v cond . To describe this
“nonlocal” effect for heat conduction along the magnetic field (in the z-direction),
the expansion of the distribution function in the powers of the parameter γ becomes
impractical and an integral expression for the heat flux
6.4 The Electron Heat Transport in a Weakly Collisional Regime
127