7.1. ASTROPHYSICAL FLUID DYNAMICS 409
whereP= (Pr,Pθ,Pφ)is the momentum density field,Tis the temperature,pis the pressure,
ρis the energy density,νandμis the viscosity coefficient,βis the coefficient of thermal
expansion,κis the thermal diffusivity,αis as in (7.1.70),∆P,(P·∇)P, ̃∆T,divPare as in
(7.1.57)-(7.1.60), and
(7.1.72) (P·∇)T=
Pθ
r∂T
∂ θ+
Pφ
rsinθ∂T
∂ φ+Pr∂T
∂r.
The equations (7.1.71) are supplemented with the boundary conditions:(7.1.73)
Pr= 0 ,
∂Pθ
∂r= 0 ,
∂Pφ
∂r=0 atr=r 0 ,r 1 ,T=T 0 atr=r 0 ,
T=T 1 atr=r 1 ,whereT 0 andT 1 are approximatively taken as constants and satisfy the physical condition
T 0 >T 1.A few remarks are now in order:Remark 7.6. First, there are three important parameters: the Rayleigh number Re, the
Prandtl number Pr and theδ-factorδ, which play an important role in astrophysical fluid
dynamics:
(7.1.74) Re=
mGr 0 r 1 β
κ νT 0 −T 1
h, Pr=ν
κ, δ=2 mG
c^2 r 0.
Theδ-factorδreflects the relativistic effect contained the Laplacian operator.
Remark 7.7.Astronomic observations show that the Sun has three layers of atmospheres:
the photosphere, the chromosphere, and the solar corona, where the solar atmospheric con-
vections occur. It manifests that the thermal convection isa universal phenomenon for stellar
atmospheres. In the classical fluid dynamics, the Rayleigh number dictates the Rayleign-
B ́enard convection. Here, however, both the Rayleigh number Re and theδ-factor defined by
(7.1.74) play an important role in stellar atmospheric convections.
Remark 7.8.For rotating stars with angular velocity~Ω, we need add to the right hand side
of (7.1.71) the Coriolis term:
− 2 ~Ω×P= 2 Ω(sinθPr−cosθPθ,cosθPφ,−sinθPφ),whereΩis the magnitude of~Ω.
Fluid dynamical equations inside open balls
As the fluid density in a stellar atmosphere is small, the equations (7.1.71) can be regarded
as a precise model governing the stellar atmospheric motion. However, for a fluid sphere with
high density, the fluid dynamic equations have to couple the gravitational field equations.