466 Chapter 16. Non-neutral plasmas
Thelthazimuthal mode is assumed to have a time dependenceexp(−iωlt)so that
φ 1 (r,θ,t)=∑
lφ ̃l(r)eilθ−iωlt (16.32)and the Fourier coefficient is given by
̃φl(r)=^1
2 π∫ 2 π0dθφ 1 (r,θ,t)e−ilθ+iωlt. (16.33)Using the equilibrium azimuthal velocity
uθ 0 (r)=−∇φ 0 ×B
B^2·ˆθ (16.34)the temporal-azimuthal Fourier transform of Eq.(16.29) can be written as
(
ωl−luθ 0 (r)
r)
∇^2 φ ̃l+l ̃φl
rBd
dr∇^2 φ 0 =0. (16.35)Using the relations∇^2 ̃φl =1
rd
dr(
rd ̃φl
dr)
−
l^2
r^2φ ̃l (16.36)∇^2 φ 0 =1
rd
dr(
rdφ 0
dr)
(16.37)
this temporal-azimuthal Fourier transform of Eq.(16.29) can be expanded as
(
ωl−luθ 0 (r)
r)(
1
rd
dr(
rd ̃φl
dr)
−
l^2
r^2̃φl)
+
l ̃φl
rBd
dr(
1
rd
dr(
rdφ 0
dr))
=0 (16.38)
The equilibrium angular velocity is
ω 0 (r)=uθ 0 (r)
r=
1
rBdφ 0
dr(16.39)
and so Eq.(16.38) can be expressed as
(ω−lω 0 (r))(
1
rd
dr(
rdφ ̃l
dr)
−
l^2
r^2
φ ̃l)
+
lφ ̃l
rd
dr(
1
rd
dr(
r^2 ω 0 (r))
)
=0 (16.40)
where thelsubscript has been dropped fromω.
Since Eq.(16.37) gives
dφ 0
dr=−
q
ε 0 r∫r0n 0 (r′)r′dr′, (16.41)the equilibrium angular velocity can be evaluated in terms of the equilibrium density to
obtain
ω 0 (r)=−q
ε 0 Br^2∫r0n 0 (r′)r′dr′. (16.42)