7.2. STARS 419
- Dynamic transitions of (7.2.24) take place at(u,λ) = ( 0 ,λ 1 )provided thatλ 1 satisfies
the following principle of exchange of stability (PES):
(7.2.25)
Reβ 1
< 0 forλ<λ 1 (orλ>λ 1 ),
= 0 forλ=λ 1 ,
> 0 forλ>λ 1 (orλ<λ 1 ),
Reβk(λ 1 )< 0 ∀k≥ 2.- Dynamic transitions of all dissipative systems described by (7.2.24) can be classified
into three categories: continuous, catastrophic, and random. Thanks to the universal-
ity, this classification is postulated in citeptd as a general principle called principle of
dynamic transitions. - Letuλbe the first transition state. Then we can also use the same stratege outlined
above to study the second transition by considering PES for the following linearized
eigenvalue problem
Lλφ+DG(uλ,λ)φ=β(^2 )(λ)φ.
Also we know that successive transitions can lead to chaos.
7.2.3 Stellar interior circulation
The governing fluid component equations are (7.1.85). We first make the nondimensional.
Let
(r,τ) = (r 0 r′,r^20 τ′/κ),(P,T,p,ρ) =(
κ ρ 0 P′/r 0 ,−dT ̃
drr 0 T′,ρ 0 κ^2 p′/r^20 ,ρ 0 ρ)
.
Then the equations (7.1.85) are rewritten as (drop the primes):
(7.2.26)
∂P
∂ τ+
1
ρ(P·∇)P=Pr∆P+1
κFGP+σ(r)T~k−∇p,∂T
∂ τ+
1
ρ(P·∇)T= ̃∆T+Pr,divP= 0 ,whereP= (Pθ,Pφ,Pr),~k= ( 0 , 0 , 1 ),σ(r)andFGPare as in (7.2.11) and (7.2.14), Pr=ν/κ
is the Prandtl number, and the∆is given by
(7.2.27)
∆Pθ= ̃∆Pθ+2
r^2∂Pr
∂ θ−
2cosθ
r^2 sin^2 θ∂Pφ
∂ φ−
Pθ
r^2 sin^2 θ,
∆Pφ= ̃∆Pφ+2
r^2 sinθ∂Pr
∂ φ+
2cosθ
r^2 sinθ∂Pθ
∂ φ−
Pφ
r^2 sin^2 θ,
∆Pr= ̃∆Pr−2
αr^2(
Pr+
∂Pθ
∂ θ+
cosθ
sinθPθ+1
sinθ∂Pφ
∂ φ