Partially Ionized Plasma - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 10

Equations for Disturbances

Suppose that in the equilibrium state, the velocities v a and electric field E
vanish and the plasma is embedded into external magnetic field B 0 . Unper-
turbed values of concentration denote as N 0 = N e 0 = N i 0 , N n 0 , and pressure
P α 0 . Denote small perturbations of magnetic and electric fields, particle num-
ber densities, velocities and pressure by b , E ,n α , v ( α ) and p α .
Substituting into (1.23)-(1.27)

B 0 + b , E ,N α 0 + n α , v ( α ) ,P α 0 + p α ,

(1.42)

and neglecting all quadratic terms of the perturbations, we obtain linearized
equations of 3-fluid hydrodynamics. From now on, if it is not specified, we
drop 0 when there was no risk of ambiguity and write N α instead of N α 0 ,
for easy of notation. Linearized continuity equations (1.23)-(1.25) allow us to
find perturbations of concentrations of plasma components from divergencies
of mean velocities:

∂n e
∂t

∂n i
∂t

∂n n
∂t

= − N ∇ · v e , = − N ∇ · v i , = − N n ∇ · v n . (1.43)

From the momentum equations (1.26)-(1.27) we get

E + 1

B 0 ]

∂ v e
∂t

e
m e

− ∇

p e
m e N e −

= −

c [ v e ×

ν ei ( v e − v i ) − ν en ( v e − v n ) ,

(1.44)

E + 1

B 0 ]

∂ v i
∂t

e
m i

− ∇

m i N i + m e

p i

=

c [ v i × m i ν ei ( v e −

v i ) − ν in ( v i − v n ) ,

(1.45)

∂ v n
∂t

− ∇

p n
m n N n −

m e
m n

N e
N n ν en ( v n −

m i
m n

N i
N n ν in ( v n −

= v e ) − v i ) . (1.46)

A linear approximation to (1.33)-(1.34) is given in Chapter 4, where wave
disturbances in the linear magnetohydrodynamics are considered.

1.3 Electromagnetic Field Equations

Let us introduce the electric induction vector D ( r ,t )as

t

j ( r ,t )dt

D ( r ,t )= E ( r ,t )+4 π

(1.47)

−∞

making it possible to combine density of induced charges and currents with
displacement currents in the field equations. Thus, Maxwell's equations (1.9)