Partially Ionized Plasma - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 9

Leaving aside details of reducing of 3-fluid hydrodynamics to the one-fluid
magnetohydrodynamics, let us write the MHD equations [5]
ρ ∂ u ) u = P + 1

∂t +( u

∇ − ∇

c [ j

× B ] , (1.33)

∂ρ
∂t +

∇ · ( ρ u )=0 , (1.34)

and Maxwell's equations are

1
c

∂ B
∂t ,

∇ × E = − (1.35)

B = 4 π ∇ ×

c j ,

(1.36)

j = σ 0 E + 1 B ] .

c [ u

× (1.37)

Here σ 0 is electrical conductivity which can be expressed as

σ 0 = Ne 2
m e ν
The mass density ρ , total pressure P and macroscopic velocity u are given by

ρ = N e m e + N i m i + N n m n , (1.38)

P = P e + P i + P n , (1.39)

1
ρ ( N e m e v e + N i m i v i + N n m n v n ) .

u = (1.40)

Equations (1.33), (1.34) are obtained from the general condition for the
hydromagnetic approximation: the collision frequencies should be larger than
the ion cyclotron frequencies

ω ω ci ν. (1.41)

Then all plasma components are involved into the motion and from (1.26)-
(1.27) follows

v e ≈ v i ≈ v n .

The conditions of applicability of MHD-approximation in the strong magnetic
field can be essentially weaker than the inequality (1.32). In particular, MHD-
approximation in the strong magnetic field can provide correct results even in
a collisionless plasma. In this case, instead of small parameter l/L controlling
the applicability of hydrodynamics, a new small parameter r ci /L is introduced,
where r ci is the Larmour radius r ci = v Ti /ω ci . For the cold plasma it is
necessary for the frequency ω to be small in comparison with the ion cyclotron
frequency ω ci .