Partially Ionized Plasma - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 12

approximation). The colder is the plasma, the weaker is spatial dispersion.
For example, at low temperatures the terms with pressure can be omitted in
(1.44)-(1.46). Then, the electron and ion velocities, the current density j (1.31)
and the induction D (1.47), depend only on the electric field E at the point r .
Let us apply the Fourier transform to (1.52)-(1.53). Suppose that
E ( r ,ω )= E ( r ,t )exp( iωt ) dt

and, similarly, for D ( r ,t )and j ( r ,t ) . The same notation E is used for Fourier
transforms E ( r ,ω ) and images E ( r ,t ) . This does not lead to misunderstand-
ing because the arguments are given explicitly, the same for other variables.
Then, (1.52), (1.53) and (1.47) become

D ( r ,ω )= ε ( r ,ω ) E ( r ,ω ) , (1.54)

j ( r ,ω )= σ ( r ,ω ) E ( r ,ω ) ,

(1.55)

ε ( r ,ω )=1+ i 4 π

ω σ ( r ,ω ) ,

(1.56)

where

t t

dt e iωt ε ( r ,t ) , dt e iωt σ ( r ,t ) .

ε ( r ,ω )= σ ( r ,ω )= (1.57)

−∞ −∞

ε ( r ,ω ) is called the tensor of complex dielectric permeability. It can be present
as a sum of Hermitian and anti-Hermitian parts

ε ( r ,ω )= ε ( r ,ω )+ i 4 π

ω σ ( r ,ω ) ,

(1.58)

where ε ( r ,ω )and σ ( r ,ω ) are Hermitian ones. σ ( r ,ω ) is the tensor of com-
plex conductivity. It can be also presented as a sum of Hermitian and anti-
Hermitian parts

i ω σ ( r ,ω )= σ ( r ,ω )

4 π ε ( r ,ω ) .

− (1.59)

Specific expressions for ε ( r ,ω )and σ ( r ,ω ) for the cold three-component
plasma containing electrons, ions and neutrals are obtained in the next section.

1.4 Dielectric Permeability and Conductivity

Drag of the Neutral Gas with Ions

ε ( r, ω )and σ ( r, ω ) can be found from (1.44)-(1.46). From (1.46) for the mole-
cule velocity we have

v n = a v e + b v i (1.60)