The Physics of Fluids and Plasmas

Transcription

1 The Physics of Fluids and Plasmas An Introduction for Astrophysicists ARNAB RAI CHOUDHURI CAMBRIDGE UNIVERSITY PRESS

2 Preface Acknowledgements xiii xvii Introduction Fluids and plasmas in the astrophysical context Characteristics of dynamical theories Different levels of theory Ensembles in phase space. Liouville's theorem Collisionless Boltzmann equation 13 Part I Neutral fluids 17 2 Boltzmann equation Collisions in a dilute neutral gas The collision integral The Maxwellian distribution Boltzmann's H theorem The conservation equation 28 Exercises 30 3 March towards hydrodynamics The moment equations An excursion into stellar dynamics. Oort limit Zero-order approximation Transport phenomena Comparison with experiments Hydrodynamics at last Concluding remarks 48 Exercises 51 vn

3 viii Contents Properties of ideal fluids 4.1 Macroscopic derivation of hydrodynamic equations The equation of motion The equation of energy 4.2 The vorticity equation. Incompressible and barotropic fluids 4.3 Hydrodynamic equations in conservative forms 4.4 Hydrostatics. Modelling the solar corona 4.5 Bernoulli's principle for steady flows 4.6 Kelvin's vorticity theorem 4.7 Potential flows. Flow past a cylinder 4.8 Stream function Exercises Viscous flows 5.1 Tangential stress in a Newtonian fluid 5.2 Navier-Stokes equation 5.3 Flow through a circular pipe 5.4 Scaling and Reynolds number 5.5 Viscous flow past solid bodies. Boundary layers 5.6 Aerodynamic lift 5.7 Accretion disks in astrophysics The basic disk dynamics Steady disk Exercises Gas dynamics 6.1 Thermodynamic properties of a perfect gas 6.2 Acoustic waves 6.3 Emission of acoustic waves 6.4 Steepening into shock waves. The method of characteristics 6.5 The structure of shock waves 6.6 Spherical blast waves. Supernova explosions 6.7 One-dimensional gas flow. Extragalactic jets 6.8 Spherical accretion and winds Exercises Linear theory of waves and instabilities 7.1 The philosophy of perturbation analysis 7.2 Convective instability and internal gravity waves 7.3 Rayleigh-Benard convection 7.4 Perturbations at a two-fluid interface Surface gravity waves Rayleigh-Taylor instability

4 ix Kelvin-Helmholtz instability Jeans instability Stellar oscillations. Helioseismology A few general comments. Beyond the linear theory 155 Exercises Turbulence The need for a statistical theory Kinematics of homogeneous isotropic turbulence Kolmogorov's universal equilibrium theory Turbulent diffusion The mean equations Turbulence in astrophysics 173 Exercises Rotation and hydrodynamics Introduction Hydrodynamics in a rotating frame of reference The geostrophic approximation Vorticity in a rotating frame Taylor-Proudman theorem Self-gravitating rotating masses Maclaurin spheroids Jacobi ellipsoids Rotation in the world of stars Rotation in the world of galaxies 192 Exercises 193 Part 2 Plasmas Plasma orbit theory Introductory remarks The effect of a perpendicular force Gradient drift Curvature drift Magnetic mirrors Formation of the Van Allen belt Cosmic rays. Particle acceleration in astrophysics 209 Exercises, Dynamics of many charged particles Basic properties of plasmas Debye shielding. The plasma parameter Different types of plasmas BBGKY hierarchy From the Vlasov equation to the two-fluid model 228

5 11.6 Fokker-Planck equation Concluding remarks 234 Exercises Collisionless processes in plasmas Introduction Electromagnetic oscillations in cold plasmas Plasma oscillations Electromagnetic waves Warm plasma waves Vlasov theory of plasma waves Landau damping Two-stream instability Electromagnetic wave propagation parallel to magnetic field Faraday rotation Whistlers Pulsars as probes of the interstellar medium 254 Exercises Collisional processes and the one-fluid model Collisions and diffusion in weakly ionized plasmas Diffusion across magnetic fields Collisions in a fully ionized plasma Towards a one-fluid model Transport phenomena in fully ionized plasmas Lorentz transformation of electromagnetic fields. The non-relativistic approximation A brief note on pulsar magnetospheres. 272 Exercises Basic magnetohydrodynamics The fundamental equations Some consequences of the induction equation Magnetohydrostatics Pressure-balanced plasma column Stability of plasma columns Force-free fields A note on fusion research Hydromagnetic waves Magnetoconvection and sunspots Bipolar magnetic regions and magnetic buoyancy Parker instability Magnetic field as a transporter of angular momentum Magnetic braking during star formation 309

6 xi Magnetized winds Jets from accretion disks MHD applied to weakly ionized plasmas 314 Exercises Theory of magnetic topologies Introduction Magnetic reconnection Magnetic helicity. Woltjer's theorems Taylor's theory of plasma relaxation Parker's theory of coronal heating 334 Exercises Dynamo theory Magnetic fields in the Cosmos Origin of astronomical magnetic fields as an MHD problem Cowling's theorem Parker's turbulent dynamo. Qualitative idea Mean field magnetohydrodynamics A simple dynamo solution Concluding remarks 361 Exercises 362 Epilogue Virial theorem Application to gravitationally bound systems Some comments on global methods Relativistic hydrodynamics Radiation hydrodynamics A guided tour through the world of astrophysical fluids and plasmas Stellar convection, rotation and oscillations Magnetohydrodynamics of the Sun and the solar system Neutron stars and pulsars Interstellar medium Active galactic nuclei Stellar dynamics Final goodbye to the reader 385

7 xii Contents Appendix A Useful vector relations 389 Appendix B Integrals in kinetic theory 392 Appendix C Formulae and equations in cylindrical and spherical coordinates 393 Appendix D Values of various quantities 396 Appendix E Basic parameters pertaining to plasmas 398 Suggestions for further reading 399 References 408 Index 419