7.4. GALAXIES 453
which is similar to the Taylor-Couette flow in a rotating cylinder. The rotating instability can
generate a circulation in the galactic nucleus, as the Taylor vortices in a rotating cylinder, as
shown in Figure7.9. The instability is caused by the forceF= (Fr,Fθ,Fφ,T)in the equations
of (7.4.26) given by
(7.4.30)
Fr=2 P ̃φ
ρ αrPφ−P ̃φ
ρrsinθ∂Pr
∂ φ,
Fθ=2cosθP ̃φ
ρrsinθPφ−P ̃φ
ρrsinθ∂Pθ
∂ φ,
Fφ=−1
ρ(
P ̃φ
r+
∂P ̃φ
∂r)
Pr−1
ρr(
cosθ
sinθP ̃φ+∂P ̃φ
∂ θ)
Pθ−P ̃φ
ρrsinθ∂Pφ
∂ φ,
T=−
P ̃φ
ρrsinθ∂T
∂ θ.
3.Rayleigh-B ́enard instability.Due to the nuclear reaction (fusion and fission) and the
large pressure gradient, the galactic nucleus possesses a very large temperature gradient in
(7.4.25) as
(7.4.31) DT=T 0 −T 1 ,
which yields the following thermal expansion force in (7.4.26), and gives rise to the Rayleigh-
B ́enard convection:
(7.4.32) Fr=β ρ
MbG
αr^2T, T=
1
ρdT ̃
drPr.4.Instability due to the gravitational effects.Similar to (7.2.7), there is a radial force in
the termν∆urof the third equation of (7.4.26):
(7.4.33) Fr=
ν
2 α∂
∂r(
1
αdα
drPr)
,
where
(7.4.34) α= ( 1 −Rs/r)−^1 , Rs<r<R 1.
In (7.4.33) and (7.4.34), we see the term
(7.4.35) fr=
ν
1 −Rs/rR^2 s
r^4
Pr,which has the property that
(7.4.36) fr=
{
+∞ forPr>0 atr=Rs,
−∞ forPr<0 atr=Rs.