Fundamentals of Plasma Physics

234 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma

Using this identity, substitution of Eq.(8.24) into Eq.(8.5) gives

nσ 1 = −

qσ

κTσ

φ 1 nσ 0

     

1 −

ω

π^3 /^2 v^3 Tσ

∑

n

∫∞

−∞

dv‖

∫ 2 π

0

dφ

∫∞

0

v⊥dv⊥

e−i

k⊥v⊥

ωcσ sinφ+inφJn(k⊥v⊥

ωcσ )e

−(v^2 ‖+v^2 ⊥)/v^2 Tσ

(

ω−k‖v‖−nωcσ

)

     

= −

qσφ 1 nσ 0

κTσ











1 −

2 ω

√

πv^3 Tσ

∑

n

∫∞

−∞

dv‖

∫∞

0

v⊥dv⊥

Jn^2 (

k⊥v⊥

ωcσ

)e

−

v‖^2 +v⊥^2

v^2 Tσ

(

ω−k‖v‖−nωcσ

)











= −

qσ

κTσ

φ 1 nσ 0

[

1+α 0 σe−k

(^2) ⊥r (^2) Lσ∑
n
In

(

k^2 ⊥r^2 Lσ

)

π^1 /^2

∫∞

−∞dξ

e−ξ

2

ξ−αnσ

]

= −

qσ

κTσ

φ 1 nσ 0

[

1+α 0 σe−k

2

⊥r

2

Lσ

∑

n

In

(

k^2 ⊥r^2 Lσ

)

Z(αnσ)

]

(8.26)

where

αnσ=

ω−nωcσ

k‖vTσ

, (8.27)

the Larmor radius is

rLσ=

√

κTσ/mσ

ωcσ

, (8.28)

andZis the plasma dispersion function. Finally, Eq.(8.26) is substituted into Eq.(8.4) to
obtain the warm magnetized plasma electrostatic dispersion relation

D(ω,k)=1+

∑

σ

1

k^2 λ^2 Dσ

[

1+α 0 σe−k

(^2) ⊥rLσ 2

∑∞

n=−∞

In

(

k^2 ⊥rLσ^2

)

Z(αnσ)

]

=0. (8.29)

Note thatD(ω,k)refers to the dispersion relation and should not be confused withD
the off-diagonal term of the cold-plasma dielectric tensor, nor withDthe displacement
vector. A similar, but more involved calculation using unperturbed orbit phase integrals
for the perturbed current density gives the hot plasma version of the full electromagnetic
dispersion, i.e., the finite temperature generalization of the cold plasmadielectric tensor
←→
K. Although the calculation is essentially similar, it is considerably more tedious, and
the interested reader is referred to specialized texts on plasmawaves such as those by Stix
(1992) or Swanson (2003).

8.2 Analysis of the warm plasma electrostatic dispersion relation

Equation (8.29) generalizes the unmagnetized warm plasma electrostatic dispersion rela-