2.4 Two-fluid equations 37(i) “pulling” both the time and space derivatives out of the velocity integral,
(ii) writingv=v′(x,t) +u(x,t)wherev′(x,t)is therandompart of a given velocity,
i.e., that part of the velocity which differs from the mean (note thatvis independent
of bothxandtbutv′is not;alsodv=dv′),
(iii) integrating by parts in 3-D velocity space on the acceleration term and using
(
∂v
∂v)
ij=δij.After performing these manipulations, the first moment of the Vlasovequation be-
comes
∂(nσuσ)
∂t
+
∂
∂x·
∫
(v′v′ +v′uσ+uσv′+uσuσ)fσdv′−
qσ
mσ∫
(E+v×B)fσdv′=−1
mσ
Rσα(2.21)
whereRσαis the net frictional drag force due to collisions of speciesσwith speciesα.
Note thatRσσ = 0since a species cannot exert net drag on itself (e.g., the totality of
electrons cannot cause frictional drag on the totality of electrons). The frictional terms
have the form
Rei=νeimene(ue−ui) (2.22)
Rie=νiemini(ui−ue) (2.23)
so that in the ion frame the drag on electrons is simply the total electronmomentum
meneuemeasured in this frame multiplied by the rateνeiat which this momentum is
destroyed by collisions with ions. This form for frictional drag has the following proper-
ties: (i)Rei+Rie= 0showing that the plasma cannot have a frictional drag on itself, (ii)
friction causes the faster species to be slowed down by the slower species, and (iii) there is
no friction between species if both have the same mean velocity.
Equation (2.21) can be further simplified by factoringuout of the velocity integrals
and recalling that by definition
∫
v′fσdv′ =0. Thus, Eq. (2.21) reduces tomσ[
∂(nσuσ)
∂t+
∂
∂x
·(nσuσuσ)]
=nσqσ(E+uσ×B)−∂
∂x·
←→
Pσ−Rσα (2.24)where thepressure tensor
←→
Pis defined by←→
Pσ=mσ∫
v′v′fσdv′.Iffσis an isotropic function ofv′,then the off-diagonal terms in
←→
Pσvanish and the three
diagonal terms are identical. In this case, it is useful to define the diagonal terms to be the
scalar pressurePσ, i.e.,
Pσ =mσ∫
vx′vx′fσdv′=mσ∫
vy′v′yfσdv′=mσ∫
v′zv′zfσdv′=
mσ
3∫
v′·v′fσdv′.