An Introduction to Astrophysical Fluid Dynamics

by Thompson, Michael J.

• ISBN13: 9781860946332
• ISBN10: 186094633X
• Format: Paperback
• Copyright: 2006-01-30
• Publisher: World Scientific Pub Co Inc

Summary

This book provides an introduction for graduate students and advanced undergraduate students to the field of astrophysical fluid dynamics. Although sometimes ignored, fluid dynamical processes play a central role in virtually all areas of astrophysics. No previous knowledge of fluid dynamics is assumed. After establishing the basic equations of fluid dynamics and the physics relevant to an astrophysical application, a variety of topics in the field are addressed. There is also a chapter introducing the reader to numerical methods. Appendices list useful physical constants and astronomical quantities, and provide handy reference material on Cartesian tensors, vector calculus in polar coordinates, self-adjoint eigenvalue problems and JWKB theory. Contents: Basic Fluid Equations; Simple Models of Astrophysical Fluids and Their Motions; Theory of Rotating Bodies; Fluid Dynamical Instabilities; Magnetohydrodynamics; Numerical Computations; Planetary Atmosphere Dynamics; Accretion, Winds and Shocks; Viscous Accretion Disks; Jeans Instability and Star Formation; Radial Oscillations of Stars; Nonradial Oscillations and Helioseismology. Key Features Graduate and advanced undergraduate text for students interested in the important role of fluid dynamical processes in astrophysics Assumes no prior knowledge of fluid dynamics Includes useful appendices delving deeper into the mathematics required Readership: Graduate students and advanced-level undergraduates in astronomy, astrophysics, applied mathematics and physics.

Table of Contents

Preface	v
Basic Fluid Equations	1	(18)
The Material Derivative	2	(1)
The Continuity Equation	3	(1)
The Momentum Equation	3	(3)
Newtonian Gravity	6	(1)
The Mechanical and Thermal Energy Equations	7	(2)
A Little More Thermodynamics	9	(2)
Perfect Gases	11	(1)
The Virial Theorem	12	(2)
Vorticity	14	(5)
Simple Models of Astrophysical Fluids and Their Motions	19	(18)
Hydrostatic Equilibrium for a Self-gravitating Body	20	(3)
Spherically symmetric case	20	(2)
Plane-parallel layer under constant gravity	22	(1)
Equations of Stellar Structure	23	(2)
Small Perturbations about Equilibrium	25	(3)
Isothermal fluctuations	26	(1)
Adiabatic fluctuations	27	(1)
Lagrangian Perturbations	28	(1)
Sound Waves	28	(2)
Surface Gravity Waves	30	(2)
Phase Speed and Group Velocity	32	(1)
Order-of-magnitude Estimates for Astrophysical Fluids	33	(4)
Typical scales	33	(1)
Importance of viscosity	33	(2)
The adiabatic approximation	35	(1)
The approximation of incompressibility	35	(2)
Theory of Rotating Bodies	37	(18)
Equation of Motion in a Rotating Frame	38	(1)
Equilibrium Equations for a Slowly Rotating Body	38	(2)
The Roche Model	40	(1)
Chandrasekhar-Milne Expansion	41	(4)
Dynamics of Rotating Stellar Models	45	(1)
Solar Rotation	46	(4)
Binary Stars	50	(5)
Fluid Dynamical Instabilities	55	(16)
Convective Instability	55	(7)
The Schwarzschild criterion	55	(5)
Effects of dissipation	60	(1)
Modelling convection: the Boussinesq approximation	61	(1)
The Rayleigh-Taylor Instability	62	(1)
Rotational Instability	63	(1)
Shear and the Kelvin-Helmholtz Instability	64	(7)
The Kelvin-Helmholtz Instability	64	(3)
Critical Richardson and Reynolds numbers	67	(1)
Turbulence and the Kolmogorov spectrum	68	(3)
Magnetohydrodynamics	71	(14)
Maxwell's Equations and the MHD Approximation	71	(3)
MHD Waves	74	(2)
Some MHD Applications	76	(6)
Solar prominences	76	(2)
Dynamo theory	78	(3)
Coronal heating	81	(1)
MHD Instabilities	82	(3)
Numerical Computations	85	(22)
The Formulation of Finite Differences	86	(1)
The von Neumann Stability Analysis	87	(2)
Various Finite-difference Schemes	89	(3)
The Lax method	89	(1)
Upwind differencing	89	(1)
The staggered leapfrog method	90	(1)
The Lax-Wendroff method	90	(1)
Implicit schemes: the Crank-Nicholson method	91	(1)
Considerations for More Complex Systems	92	(1)
Operator Splitting	93	(2)
Examples of Implementations	95	(6)
1-D Lagrangian scheme with artificial viscosity	95	(3)
2-D scheme using operator splitting	98	(2)
Codes for computing astrophysical flows	100	(1)
Smoothed Particle Hydrodynamics	101	(6)
Planetary Atmosphere Dynamics	107	(20)
The Importance of Rotation: the Rossby Number	107	(1)
Relative and Absolute Vorticity	108	(2)
Potential Vorticity	110	(1)
Baroclinicity and the Thermal Wind Equation	110	(2)
Geostrophic Motion	112	(4)
Some Approximate Models	116	(4)
The shallow-water model	117	(2)
f-plane and β-plane models	119	(1)
Waves	120	(2)
Ekman Layers	122	(5)
Accretion, Winds and Shocks	127	(16)
Bernoulli's Theorem	128	(1)
The de Laval Nozzle	129	(1)
The Bondi Problem	130	(4)
The Parker Solar-Wind Solution	134	(1)
Nonlinear Acoustic Waves	134	(5)
Shock Waves	139	(1)
Blast Wave from a Supernova	140	(3)
Viscous Accretion Disks	143	(12)
Role of Angular Momentum and Energetics of Accretion	143	(2)
Thin Accretion Disks	145	(2)
Diffusion Equation for Surface Density	147	(3)
Steady Disks	150	(3)
The Need for Anomalous Viscosity	153	(2)
Jeans Instability and Star Formation	155	(10)
Links to Observations	156	(1)
Jeans Instability	156	(2)
Jeans Instability with Rotation	158	(3)
Jeans instability for a rotating system	159	(2)
Ambipolar Diffusion	161	(1)
Fragmentation	162	(1)
Some Comments on Star Formation	163	(2)
Radial Oscillations of Stars	165	(16)
Linear Adiabatic Wave Equation for Radial Oscillations	165	(8)
Boundary conditions	168	(1)
Eigenvalue nature of the problem	169	(1)
Self-adjointness of the problem	170	(2)
A lower bound on the fundamental frequency	172	(1)
Homology scaling for the fundamental frequency of stars	172	(1)
Non-adiabatic Radial Oscillations	173	(5)
Physical discussion of driving and damping	176	(2)
The Quasi-adiabatic Approximation	178	(3)
Nonradial Oscillations and Helioseismology	181	(24)
Nonradial Modes of Oscillation of a Star	181	(4)
Mode Classification	185	(1)
The Cowling Approximation	186	(1)
A Simplified Discussion of Nonradial Oscillations	187	(3)
A More General Asymptotic Expression	190	(3)
Helioseismology: The Duvall Law	193	(3)
Tassoul's Formula	196	(2)
Asymptotics of g Modes	198	(1)
Probing the Sun's Internal Rotation	199	(6)
Appendix A Useful Constants and Quantities	205	(2)
Fundamental Physical Constants	205	(1)
Astronomical Quantities	205	(1)
Cartesian Tensors: Index Notation and Summation Convention	206	(1)
Appendix B Vector Calculus in Spherical and Cylindrical Polar Coordinates	207	(2)
Cylindrical Polar Coordinates (w,φ,z)	207	(1)
Spherical Polar Coordinates (r,θ,φ)	208	(1)
Appendix C Self-adjoint Eigenvalue Problems	209	(4)
Reality of Eigenvalues	209	(1)
Orthogonality of Eigenfunctions	210	(1)
Eigenfunction Expansions	210	(1)
Variational Principle	211	(2)
Appendix D The JWKB Method	213	(4)
Bibliography	217	(6)
Index	223