6.3 21-Moment Grad Approximation
As we have already mentioned above, for sufficient accuracy of the calculations of
the energy and momentum fluxes q
! r
!
, t
and Π
$
r
!
, t
, as well as C
!
V and C p , one
needs also to consider the q
! Ã
r
! , t
and Π
$ Ã
r
!
, t
moments determined by the
expressions (6.18) and (6.19). This increases the number of independent moments
that must be found. However, from the definition of Π
$
, recall Eq. (6.10), it is easy to
see that Π ik ¼ Π ki and Π ik δ ki ¼ 0, so Π
$
is determined by five independent moments.
The same is applicable for Π
$ Ã
. Then, allowing also for n, p (or T), V
!
, q
!
, and q
! Ã
, we
find that we need to determine 21 moments.
To find these moments, we consider Eqs. (6.20) and (6.21), keeping only the
highest order terms. Then, for the multi-component plasma, we come to the following equations for q
!
α and Π
$
α :
5
2
p α
m α
∇T α À q
!
α Â Ω
!
Bα ¼
X
β
C
α,β
q
! ,
ð6:34Þ
Π
$
α Â Ω
!
Bα À Ω
!
Bα Â Π
$
α
¼ p α ∇V
!
α þ ∇V
!
α
Tr À
2
3
I
$
∇ Á V
!
α
þ
2
5
∇ q
!
α þ ∇ q
!
α
Tr À
2
3
I
$ ∇ Á q
!
α
À
X
β
C
α,β
Π
$ ,
ð6:35Þ
where C
α,β
q
! depends on V
!
α , V
!
β , q
!
α , q
! Ã
α , q
!
β , and q
! Ã
β , whereas C
α,β
Π
$ depends on Π
$
α , Π
$ Ã
α ,
Π
$
β and Π
$ Ã
β . Somewhat similar equations can be found from the evolution equations
for q
! Ã
r
! , t
and Π
$ Ã
r
!
, t
with the collisional terms corresponding to C
α,β
q
!Ã and C
α,β
Π
$Ã .
Having found q
!
α , q
! Ã
α , q
!
β , and q
! Ã
β , one can calculate C
!
Vα that, together with Π
$
α ,
closes the balance equations (6.6), (6.7) and (6.8) completely.
However, such calculations for multi-component plasma are extremely cumbersome and go beyond the scope of our consideration. The detail of such derivation can
be found in [17]. Nonetheless, just for illustration, we consider here the derivation of
the electron energy flux parallel to the magnetic field, q ke , assuming that the plasma
has one kind of ions with charge Z.
6.3 21-Moment Grad Approximation
125