ASTR/ATOC-5410: Fluid Instabilities, Waves, and Turbulence September 4, 2016, Axel Brandenburg

Handout 0: Basic equations

1 Continuity equation

\frac{∂ρ}{∂t} + ∇ · (ρu) = 0  (1)

It can be written in various other forms, such as D ln ρ/Dt = −∇ · u.

2 Navier-Stokes equation

ρ\frac{Du}{Dt} = −∇P − ρ∇Φ + J × B + ∇ · τ  (2)

where Φ is the gravitational potential, which obeys ∇^{2}Φ = 4πGρ. The solution for a sphere, for example, is Φ = −GM/r, while for a plane layer it is Φ = zg, up to an additive constant.

For a monatomic gas, the stress tensor is τ = 2ρνS with S_{ij} = \frac{1}{2}(u_{i,j} + u_{j,i}) − \frac{1}{3}δ_{ij}∇ · u. In that case, ∇ · τ = ρν(∇^{2}u + \frac{1}{3}∇∇ · u + 2S∇ ln ρν).

• Note that the ∇Φ term comes with a minus and an additional ρ factor.

• All the other terms come with no ρ factor.

• It is sometimes useful to note that ρ^{−1}∇P = ∇h − T ∇S, where h is the specific enthalpy. For an ideal gas we have h = c_{p}T.

3 Energy equation

ρT \frac{DS}{Dt} = −∇ · F_{rad} + τ : ∇u + J^{2}/σ + nucl. fusion  (3)

The τ : ∇u is supposed to mean τ_{ij}u_{i,j}. This is the viscous heating term and it is positive definite. In the homework, we are supposed to find that for τ = 2ρνS we have τ : ∇u = 2ρνS^{2}, which is manifestly positive definite. In the optically thick case, the radiative flux is F_{rad} = −K∇T. The electric conductivity σ can also be written as 1/σ = μ_{0}η, where μ_{0} is the vacuum permeability and η the magnetic diffusivity.

• In the strictly isentropic case (S = const), we have P ∝ ρ^{γ}. However, viscous heating can usually not be neglected!

• The lhs of Equation (3) can also be written as ρDe/Dt + P ∇ · u.

• For an ideal gas, we have e = c_{v}T and can write the lhs also as ρc_{v}DT/Dt + P ∇ · u.

• Again for an ideal gas, we can write the lhs as ρc_{p}DT/Dt − DP/Dt.

Recall that R/μ = c_{p} − c_{v}, where R = k_{B}/m_{u} is the universal gas constant^{1} and μ is the mean molecular weight (=1 for neutral hydrogen, ≈ 1.2 for a neutral hydrogen-helium mixture, 0.6 for an ionized hydrogen-helium mixture).

4 Induction equation

\frac{∂B}{∂t} = ∇ × (u × B − J/σ)  (4)

1Here, k_{B} = 1.3806505×10^{−16} erg K^{−1} is the Boltzmann constant, m_{u} = 1.66053886^{−24} g is the atomic mass unit [0.993 times the proton mass], and so R = 8.31447 × 10^{7} erg g^{−1} K^{−1}. In astrophysics, μ is dimensionless rather than in g/mol.