466 Chapter 16. Non-neutral plasmas
Thelthazimuthal mode is assumed to have a time dependenceexp(−iωlt)so that
φ 1 (r,θ,t)=
∑
l
φ ̃l(r)eilθ−iωlt (16.32)
and the Fourier coefficient is given by
̃φl(r)=^1
2 π
∫ 2 π
0
dθφ 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
rB
d
dr
∇^2 φ 0 =0. (16.35)
Using the relations
∇^2 ̃φl =
1
r
d
dr
(
r
d ̃φl
dr
)
−
l^2
r^2
φ ̃l (16.36)
∇^2 φ 0 =
1
r
d
dr
(
r
dφ 0
dr
)
(16.37)
this temporal-azimuthal Fourier transform of Eq.(16.29) can be expanded as
(
ωl−
luθ 0 (r)
r
)(
1
r
d
dr
(
r
d ̃φl
dr
)
−
l^2
r^2
̃φl
)
+
l ̃φl
rB
d
dr
(
1
r
d
dr
(
r
dφ 0
dr
))
=0 (16.38)
The equilibrium angular velocity is
ω 0 (r)=
uθ 0 (r)
r
=
1
rB
dφ 0
dr
(16.39)
and so Eq.(16.38) can be expressed as
(ω−lω 0 (r))
(
1
r
d
dr
(
r
dφ ̃l
dr
)
−
l^2
r^2
φ ̃l
)
+
lφ ̃l
r
d
dr
(
1
r
d
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
∫r
0
n 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
∫r
0
n 0 (r′)r′dr′. (16.42)