An Introduction To Astrophysical Fluid Dynamics

by Michael John Thompson

Overview

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.

Product Details

ISBN-13:	9781860946158
Publisher:	Imperial College Press
Publication date:	01/18/2006
Pages:	240
Product dimensions:	6.10(w) x 9.00(h) x 1.00(d)

Table of Contents

Preface v

1 Basic Fluid Equations 1

1.1 The Material Derivative 2

1.2 The Continuity Equation 3

1.3 The Momentum Equation 3

1.4 Newtonian Gravity 6

1.5 The Mechanical and Thermal Energy Equations 7

1.6 A Little More Thermodynamics 9

1.7 Perfect Gases 11

1.8 The Virial Theorem 12

1.9 Vorticity 14

2 Simple Models of Astrophysical Fluids and Their Motions 19

2.1 Hydrostatic Equilibrium for a Self-gravitating Body 20

2.1.1 Spherically symmetric case 20

2.1.2 Plane-parallel layer under constant gravity 22

2.2 Equations of Stellar Structure 23

2.3 Small Perturbations about Equilibrium 25

2.3.1 Isothermal fluctuations 26

2.3.2 Adiabatic fluctuations 27

2.4 Lagrangian Perturbations 28

2.5 Sound Waves 28

2.6 Surface Gravity Waves 30

2.7 Phase Speed and Group Velocity 32

2.8 Order-of-magnitude Estimates for Astrophysical Fluids 33

2.8.1 Typical scales 33

2.8.2 Importance of viscosity 33

2.8.3 The adiabatic approximation 35

2.8.4 The approximation of incompressibility 35

3 Theory of Rotating Bodies 37

3.1 Equation of Motion in a Rotating Frame 38

3.2 Equilibrium Equations for a Slowly Rotating Body 38

3.3 The Roche Model 40

3.4 Chandrasekhar-Milne Expansion 41

3.5 Dynamics of Rotating Stellar Models 45

3.6 Solar Rotation 46

3.7 Binary Stars 50

4 Fluid Dynamical Instabilities 55

4.1 Convective Instability 55

4.1.1 The Schwarzschild criterion 55

4.1.2 Effects of dissipation 60

4.1.3 Modelling convection: the Boussinesq approximation 61

4.2 The Rayleigh-Taylor Instability 62

4.3 Rotational Instability 63

4.4 Shear and the Kelvin-Helmholtz Instability 64

4.4.1 The Kelvin-Helmholtz instability 64

4.4.2 CriticalRichardson and Reynolds numbers 67

4.4.3 Turbulence and the Kolmogorov spectrum 68

5 Magnetohydro dynamics 71

5.1 Maxwell's Equations and the MHD Approximation 71

5.2 MHD Waves 74

5.3 Some MHD Applications 76

5.3.1 Solar prominences 76

5.3.2 Dynamo theory 78

5.3.3 Coronal heating 81

5.4 MHD Instabilities 82

6 Numerical Computations 85

6.1 The Formulation of Finite Differences 86

6.2 The von Neumann Stability Analysis 87

6.3 Various Finite-difference Schemes 89

6.3.1 The Lax method 89

6.3.2 Upwind differencing 89

6.3.3 The staggered leapfrog method 90

6.3.4 The Lax-Wendroff method 90

6.3.5 Implicit schemes: the Crank-Nicholson method 91

6.4 Considerations for More Complex Systems 92

6.5 Operator Splitting 93

6.6 Examples of Implementations 95

6.6.1 1-D Lagrangian scheme with artificial viscosity 95

6.6.2 2-D scheme using operator splitting 98

6.6.3 Codes for computing astrophysical flows 100

6.7 Smoothed Particle Hydrodynamics 101

7 Planetary Atmosphere Dynamics 107

7.1 The Importance of Rotation: the Rossby Number 107

7.2 Relative and Absolute Vorticity 108

7.3 Potential Vorticity 110

7.4 Baroclinicity and the Thermal Wind Equation 110

7.5 Geostrophic Motion 112

7.6 Some Approximate Models 116

7.6.1 The shallow-water model 117

7.6.2 f-plane and ?-plane models 119

7.7 Waves 120

7.8 Ekman Layers 122

8 Accretion, Winds and Shocks 127

8.1 Bernoulli's Theorem 128

8.2 The de Laval Nozzle 129

8.3 The Bondi Problem 130

8.4 The Parker Solar-Wind Solution 134

8.5 Nonlinear Acoustic Waves 134

8.6 Shock Waves 139

8.7 Blast Wave from a Supernova 140

9 Viscous Accretion Disks 143

9.1 Role of Angular Momentum and Energetics of Accretion 143

9.2 Thin Accretion Disks 145

9.3 Diffusion Equation for Surface Density 147

9.4 Steady Disks 150

9.5 The Need for Anomalous Viscosity 153

10 Jeans Instability and Star Formation 155

10.1 Links to Observations 156

10.2 Jeans Instability 156

10.3 Jeans Instability with Rotation 158

10.3.1 Jeans instability for a rotating system 159

10.4 Ambipolar Diffusion 161

10.5 Fragmentation 162

10.6 Some Comments on Star Formation 163

11 Radial Oscillations of Stars 165

11.1 Linear Adiabatic Wave Equation for Radial Oscillations 165

11.1.1 Boundary conditions 168

11.1.2 Eigenvalue nature of the problem 169

11.1.3 Self-adjointness of the problem 170

11.1.4 A lower bound on the fundamental frequency 172

11.1.5 Homology scaling for the fundamental frequency of stars 172

11.2 Non-adiabatic Radial Oscillations 173

11.2.1 Physical discussion of driving and damping 176

11.3 The Quasi-adiabatic Approximation 178

12 Nonradial Oscillations and Helioseismology 181

12.1 Nonradial Modes of Oscillation of a Star 181

12.2 Mode Classification 185

12.3 The Cowling Approximation 186

12.4 A Simplified Discussion of Nonradial Oscillations 187

12.5 A More General Asymptotic Expression 190

12.6 Helioseismology: The Duvall Law 193

12.7 Tassoul's Formula 196

12.8 Asymptotics of g Modes 198

12.9 Probing the Sun's Internal Rotation 199

Appendix A Useful Constants and Quantities 205

A.l Fundamental Physical Constants 205

A.2 Astronomical Quantities 205

A.3 Cartesian Tensors: Index Notation and Summation Convention 206

Appendix B Vector Calculus in Spherical and Cylindrical Polar Coordinates 207

B.l Cylindrical Polar Coordinates (ω,ø,z) 207

B.2 Spherical Polar Coordinates (r,&theda;,ø) 208

Appendix C Self-adjoint Eigenvalue Problems 209

C.l Reality of Eigenvalues 209

C.2 Orthogonality of Eigenfunctions 210

C.3 Eigenhmction Expansions 210

C.4 Variational Principle 211

Appendix D The JWKB Method 213

Bibliography 217

Index 223