Astrophysical Fluid Dynamics Mathematical Tripos, Part III 1 Ideal gas dynamics and MHD Dr Gordon Ogilvie Michaelmas Term 2008 1.1 Review of ideal gas dynamics Provisional synopsis 1.1.1 Fluid variables Equations of ideal gas dynamics and MHD, including compress- A fluid is characterized by a velocity field u(x, t) and two independent • ibility, thermodynamic relations and self-gravitation. Microphys- thermodynamic properties. Most useful are the dynamical variables: ical basis and validity of a fluid description. the pressure p(x, t) and the mass density ρ(x, t). Other properties, e.g. temperature T , can be regarded as functions of p and ρ. The specific Physical interpretation of MHD, with examples of basic phenom- • volume (volume per unit mass) is v = 1/ρ. ena. We neglect the possible complications of variable chemical composition Conservation laws, symmetries and hyperbolic structure. Stress • associated with ionization and recombination, or chemical or nuclear tensor and virial theorem. reactions. Linear waves in homogeneous media. • 1.1.2 Eulerian and Lagrangian viewpoints Nonlinear waves, shocks and other discontinuities. • Spherical blast waves: supernovae. In the Eulerian viewpoint we consider how fluid properties vary in time • at a point that is fixed in space, i.e. attached to the (usually inertial) Spherically symmetric steady flows: stellar winds and accretion. • coordinate system. The Eulerian time-derivative is simply ∂ Axisymmetric rotating magnetized flows: astrophysical jets. • ∂t Waves and instabilities in stratified rotating astrophysical bodies. • In the Lagrangian viewpoint we consider how fluid properties vary in time at a point that moves with the fluid. The Lagrangian time- derivative is then Please send any comments and corrections to [email protected] D ∂ = + u · ∇. Dt ∂t 1.1.3 Material points and structures A material point is an idealized fluid element, a point that moves with the bulk velocity u(x, t) of the fluid. (Note that the true particles of which the fluid is composed have an additional random thermal motion.) 1 Material curves, surfaces and volumes are geometrical structures com- If δm = ρδV is a material mass element, it can be seen that mass is posed of fluid elements and moving (and distorting) with the fluid flow. conserved in the form An infinitesimal material line element δx evolves according to Dδm = 0. Dδx Dt = δu = δx · ∇u. Dt 1.1.5 Equation of motion It changes its length and/or orientation in the presence of a velocity gradient. The equation of motion, Infinitesimal material surface and volume elements can be defined from Du material line elements according to ρ = ρ∇Φ ∇p, Dt − − δS = δx(1) δx(2), × derives from Newton’s second law with gravitational and pressure forces. (1) (2) (3) δV = δx · δx δx . Φ(x, t) is the gravitational potential. Viscous forces are neglected in × They therefore evolve according to (exercise) ideal gas dynamics. DδS =(∇ · u)δS (∇u) · δS, Dt − 1.1.6 Poisson’s equation DδV =(∇ · u)δV. Dt The gravitational potential is related to the mass density by Poisson’s equation, (See, e.g., Batchelor, An Introduction to Fluid Dynamics, chapter 3.) S In Cartesian coordinates and suffix notation the equation for δ reads 2Φ = 4πGρ, ∇ DδSi ∂uj ∂uj = δSi δSj where G is Newton’s constant. The solution Dt ∂xj − ∂xi ′ ′ ρ(x , t) 3 ′ ρ(x , t) 3 ′ Φ(x, t)= G ′ d x G ′ d x 1.1.4 Equation of mass conservation − x x − ˆ x x ZV | − | ZV | − | =Φint +Φext The equation of mass conservation, generally involves contributions from both inside and outside the fluid ∂ρ + ∇ · (ρu) = 0, region V under consideration. ∂t has the typical form of a conservation law: ρ is the mass density and Non-self-gravitating means that (variations in) Φint can be neglected. ρu is the mass flux density. An alternative form is Then Φ(x, t) is known in advance and Poisson’s equation is not coupled to the other equations. Dρ = ρ∇ · u. Dt − 2 3 1.1.7 Thermal energy equation can be found from the equation of state. For an ideal gas with negligible radiation pressure, χρ = 1 and so Γ1 = γ. In the absence of non-adiabatic heating (e.g. by viscous dissipation or We often rewrite the thermal energy equation as nuclear reactions) and cooling (e.g. by radiation or conduction), Dp Γ p Dρ Ds = 1 , = 0, Dt ρ Dt Dt where s is the specific entropy (entropy per unit mass). Fluid elements and generally write γ for Γ1. undergo reversible thermodynamic changes and preserve their entropy. This condition is violated in shocks (see later). 1.1.8 Simplified models The thermal variables (T, s) can be related to the dynamical variables A polytropic gas is an ideal gas with constant cv, cp, γ and µ. The (p, ρ) via an equation of state and standard thermodynamic identities. polytropic index n (not generally an integer) is defined by γ =1+1/n. The most important case is that of an ideal gas together with black-body Equipartition of energy for a classical gas with N degrees of freedom radiation, per particle gives γ = 1+2/N. For a classical monatomic gas with kρT 4σT 4 N = 3 translational degrees of freedom, γ = 5/3 and n = 3/2. In p = pg + pr = + , reality Γ is variable when the gas undergoes ionization or when the µmp 3c 1 gas and radiation pressure are comparable. The specific internal energy where k is Boltzmann’s constant, mp is the mass of the proton and c of a polytropic gas is the speed of light. µ is the mean molecular weight (the average mass of the particles in units of mp), equal to 2.0 for molecular hydrogen, p N 1 e = = 2 kT . 1.0 for atomic hydrogen, 0.5 for fully ionized hydrogen and about 0.6 (γ 1)ρ µmp − for ionized matter of typical cosmic abundances. Radiation pressure is usually negligible except in the centres of high-mass stars and in the A barotropic fluid is an idealized situation in which the relation p(ρ) immediate environments of neutron stars and black holes. is known in advance. We can then dispense with the thermal energy equation. e.g. if the gas is strictly isothermal and ideal, then p = c2ρ We define the first adiabatic exponent s with cs = constant being the isothermal sound speed. Alternatively, ∂ ln p if the gas is strictly isentropic and polytropic, then p = Kργ with Γ = , 1 ∂ ln ρ K = constant. s related to the ratio of specific heats γ = cp/cv by (exercise) An incompressible fluid is an idealized situation in which Dρ/Dt = 0, implying ∇ · u = 0. This can be achieved formally by taking the limit Γ1 = χργ, γ . The approximation of incompressibility eliminates acoustic → ∞ where phenomena from the dynamics. ∂ ln p The ideal gas law itself is not valid at very high densities or where χ = ρ ∂ ln ρ quantum degeneracy is important. T 4 5 1.2 Elementary derivation of the MHD equations In the ideal MHD approximation we regard the fluid as a perfect electri- cal conductor. The electric field in the rest frame of the fluid vanishes, Magnetohydrodynamics (MHD) is the dynamics of an electrically con- implying that ducting fluid (ionized plasma or liquid metal) containing a magnetic E = u B field. It is a fusion of fluid dynamics and electromagnetism. − × in a frame in which the fluid velocity is u(x, t). 1.2.1 Induction equation This condition can be regarded as the limit of a constitutive relation such as Ohm’s law, in which the effects of resistivity (i.e. finite conduc- We consider a non-relativistic theory in which the fluid motions are slow tivity) are neglected. compared to the speed of light. The electromagnetic fields E and B are governed by Maxwell’s equations without the displacement current, From Maxwell’s equations, we then obtain the ideal induction equation ∂B ∂B ∇ u B = ∇ E, = ( ). ∂t − × ∂t × × This is an evolutionary equation for B alone, and E and J have been ∇ · B = 0, eliminated. The divergence of the induction equation ∇ B = µ J, × 0 ∂ (∇ · B) = 0 where µ0 is the permeability of free space and J is the electric current ∂t density. The fourth Maxwell equation, involving ∇ · E, is not required ensures that the solenoidal character of B is preserved. in a non-relativistic theory. These are sometimes called the pre-Maxwell equations. 1.2.2 The Lorentz force Exercise: Show that these equations are invariant under the Galilean transformation to a frame of reference moving with uniform relative velocity v, A fluid carrying a current density J in a magnetic field B experiences a bulk Lorentz force ′ x = x vt, − 1 Fm = J B = (∇ B) B ′ t = t, × µ0 × × ′ E = E + v B, per unit volume. This can be understood as the sum of the Lorentz × forces on individual particles, ′ B = B, ′ qv B = qv B. J = J, × × X X as required by a ‘non-relativistic’ theory. (In fact, this is simply Galilean, (The electrostatic force can be shown to be negligible in the non-relativistic rather than Einsteinian, relativity.) theory.) 6 7 In Cartesian coordinates is often referred to as the total pressure. The ratio ∂B p (µ F ) = ǫ ǫ m B β = 0 m i ijk jlm ∂x k B2/2µ l 0 ∂B ∂B = i k B is known as the plasma beta. ∂x − ∂x k k i ∂B ∂ B2 = B i . k ∂x − ∂x 2 1.2.3 Summary of the MHD equations k i Thus A full set of ideal MHD equations might read 1 B2 F = B · ∇B ∇ .