Φ 0 r
!
, v
0 2
¼
X 1
j¼2
a
j
ð Þ
0
r
!
L
1=2
ð
Þ
j
v
0 2 =v
2
T
,
ð6:13Þ
Φ i r
! , v
0 2
¼
X 1
j¼1
a
j
ð Þ
i
r
!
L
3=2
ð
Þ
j
v
0 2 =v
2
T
,
ð6:14Þ
Φ ik r
!
, v
0 2
¼
X 1
j¼0
a
j
ð Þ
ik r
!
L
5=2
ð
Þ
j
v
0 2 =v
2
T
,
ð6:15Þ
where L
α
ð Þ
j
x
ð Þ are the generalized Laguerre (or Sonin-Laguerre) polynomials and
v
2
T ¼ 2T=m (e.g. see [13, 16, 17] and the references therein). We notice that all the
terms in the expansion (6.12), (6.13), (6.14) and (6.15) are orthogonal to each other
(with the weight function proportional to F Mxw
n, T, v
0
!
, r
!
, t
). Moreover, taking
into account such orthogonality, we find that Φ does not contribute to n r
! , t
and
T r
!
, t
whereas implementing Eq. (6.12) into Eq. (6.9) and (6.10) it is easy to show
that the components of the energy flux q i and tensor Π ik can be expressed through the
vector a
1
ð Þ
i and tensor a
0
ð Þ
ik :
q i r
!
, t
¼
Z
v
0
i
m
2
v
0 2 L
3=2
ð
Þ
1
v
0 2 =v
2
T
f v
0
!
, r
!
, t
d v
0
! ¼ À
2
5
pT
m
a
1
ð Þ
i ,
ð6:16Þ
and
Π ik ¼
pT
m
a
0
ð Þ
ik :
ð6:17Þ
However, the orthogonal functions used for the decomposition of Φ v
0
!
, r
!
, t
in
(6.12), (6.13), (6.14) and (6.15) are not eigenfunctions of the linearized collision
operators. Therefore, the components of Φ i r
!
, v
0 2
contribute not only to the heat
flux q
! r
!
, t
but also to C
!
V which results in the so-called thermal forces depending
on ∇T (see [13] for the discussion of the physical nature of the thermal force). Thus,
we arrive at the set of infinite number of equations for the coefficients a
j
ð Þ
i and a
j
ð Þ
ik .
The solutions of these equations can only be found by keeping only a finite number
of these coefficients. But the accuracy of physically meaningful parameters a
1
ð Þ
i and
a
0
ð Þ
ik , determining the energy and momentum fluxes (6.16) and (6.17) and contributing to C n , C
!
V , and C p depends on the number of the terms kept in these sets of
equations. In [13, 18] it was found that sufficient accuracy of a
1
ð Þ
i
and a
0
ð Þ
ik can be
120
6 Fluid Description of Edge Plasma Transport
!
, v
0 2
¼
X 1
j¼2
a
j
ð Þ
0
r
!
L
1=2
ð
Þ
j
v
0 2 =v
2
T
,
ð6:13Þ
Φ i r
! , v
0 2
¼
X 1
j¼1
a
j
ð Þ
i
r
!
L
3=2
ð
Þ
j
v
0 2 =v
2
T
,
ð6:14Þ
Φ ik r
!
, v
0 2
¼
X 1
j¼0
a
j
ð Þ
ik r
!
L
5=2
ð
Þ
j
v
0 2 =v
2
T
,
ð6:15Þ
where L
α
ð Þ
j
x
ð Þ are the generalized Laguerre (or Sonin-Laguerre) polynomials and
v
2
T ¼ 2T=m (e.g. see [13, 16, 17] and the references therein). We notice that all the
terms in the expansion (6.12), (6.13), (6.14) and (6.15) are orthogonal to each other
(with the weight function proportional to F Mxw
n, T, v
0
!
, r
!
, t
). Moreover, taking
into account such orthogonality, we find that Φ does not contribute to n r
! , t
and
T r
!
, t
whereas implementing Eq. (6.12) into Eq. (6.9) and (6.10) it is easy to show
that the components of the energy flux q i and tensor Π ik can be expressed through the
vector a
1
ð Þ
i and tensor a
0
ð Þ
ik :
q i r
!
, t
¼
Z
v
0
i
m
2
v
0 2 L
3=2
ð
Þ
1
v
0 2 =v
2
T
f v
0
!
, r
!
, t
d v
0
! ¼ À
2
5
pT
m
a
1
ð Þ
i ,
ð6:16Þ
and
Π ik ¼
pT
m
a
0
ð Þ
ik :
ð6:17Þ
However, the orthogonal functions used for the decomposition of Φ v
0
!
, r
!
, t
in
(6.12), (6.13), (6.14) and (6.15) are not eigenfunctions of the linearized collision
operators. Therefore, the components of Φ i r
!
, v
0 2
contribute not only to the heat
flux q
! r
!
, t
but also to C
!
V which results in the so-called thermal forces depending
on ∇T (see [13] for the discussion of the physical nature of the thermal force). Thus,
we arrive at the set of infinite number of equations for the coefficients a
j
ð Þ
i and a
j
ð Þ
ik .
The solutions of these equations can only be found by keeping only a finite number
of these coefficients. But the accuracy of physically meaningful parameters a
1
ð Þ
i and
a
0
ð Þ
ik , determining the energy and momentum fluxes (6.16) and (6.17) and contributing to C n , C
!
V , and C p depends on the number of the terms kept in these sets of
equations. In [13, 18] it was found that sufficient accuracy of a
1
ð Þ
i
and a
0
ð Þ
ik can be
120
6 Fluid Description of Edge Plasma Transport