Partially Ionized Plasma - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 7

Three-Fluid Magnetohydrodynamics

The simplest and often used approximation is an assumption about isotropy
of the pressure tensor P ( α ) jk and reducing it to

P ( α ) jk = P α δ jk . (1.21)

Henceforth in this topic we put P α = kN α T α .
The friction R ( α ) acting on particles α is connected with their collisions
with all other particles. Let R αβ be the change of momentum at collisions of
particles α with particles β. Then
R ( α ) =
β

R αβ .

It follows from the law of momentum conservation at collisions that

R αβ = − R βα .

Consider, for instance, the force between electrons and one of the ionic
species. Electrons within the time of

∼

1 /ν ei lose their relative velocity v e −

v i

by collisions. The ions in the process acquire the momentum m e ( v e −

v i )per

each electron. This means that the friction v i ) acts on electrons.

Opposite and equal force acts on ions. Thus, R ( α ) can be approximated by
expressions proportional to relative mean velocities of the collide particles

− m e N e ( v e −

N α
β

− µ αβ ν aβ ( v α −

v β ) ,

R ( α ) = (1.22)

= α

where ν αβ is the collision frequency of the particle α with all the particles β ;
the reduced masses

m α m β
m α + m β ,

µ αβ =

where m α , m β are the mass of particles α , β , respectively. The reduce mass
µ eβ ≈ m e for collisions of electrons with heavy particles.

Consider a three-component plasma containing electrons, ions and neu-
trals. Then continuity equation (1.17) gives

∂N e
∂t

+ ∇ · ( N e v e )=0 , (1.23)

∂N i
∂t

+ ∇ · ( N i v i )=0 , (1.24)

∂N n
∂t

+ ∇ · ( N n v n )=0 , (1.25)