where Ω
!
B ¼ Ω B B
!
=B whereas C q
! and C Π
$ are the corresponding moments of the
collision operators [17, 22, 23]. We note, however, that for the calculation of the
collision-driven energy and momentum flux with sufficient accuracy, the evolution
equation of the higher-order Laguerre polynomials, corresponding to q
Ã
!
r
! , t
and
Π
Ã
$
r
! , t
, should be considered [13].
6.2 Collisionless Cross-Field Components of Energy
and Momentum Fluxes
For strongly magnetized plasmas, Ω B ) ν C , the leading terms in the fluid equations
(6.6), (6.7) and (6.8) and (6.20) and (6.21) are those proportional to the magnetic
field strength B. This allows solving the momentum balance Eq. (6.7) for the crossfield component of V
!
by expanding V
!
⊥ in the powers of 1/B and neglecting the
impact of the collision operators. Observing Eq. (6.7), one finds that in the first order
in 1/B, there is a contribution from both the electric field and the pressure gradient:
V
! 1
ð Þ
⊥ ¼ V
!
E þ V
!
p ,
ð6:22Þ
where
V
!
E ¼ c E
! Â B
!
=B
2 and V
!
p ¼ c B
! Â ∇p
=enB
2
ð6:23Þ
describe, respectively, the E
! Â B
!
and diamagnetic drift velocities. In the second
order, we find the components of V
!
⊥ related to inertial and collisionless viscosity
polarization
V
! 2
ð Þ
⊥ ¼ V
!
I þ V
!
Π ,
ð6:24Þ
where
V
!
I ¼
1
Ω B
B
!
B
Â
∂
∂t
þ V
!
E þ V
!
p
Á ∇
!
V
!
E þ V
!
p
&
'
,
ð6:25Þ
and
122
6 Fluid Description of Edge Plasma Transport