This book provides readers with the skills they need to write computer codes that simulate convection, internal gravity waves, and magnetic field generation in the interiors and atmospheres of rotating planets and stars. Using a teaching method perfected in the classroom, Gary Glatzmaier begins by offering a step-by-step guide on how to design codes for simulating nonlinear time-dependent thermal convection in a two-dimensional box using Fourier expansions in the horizontal direction and finite differences in the vertical direction. He then describes how to implement more efficient and accurate numerical methods and more realistic geometries in two and three dimensions. In the third part of the book, Glatzmaier demonstrates how to incorporate more sophisticated physics, including the effects of magnetic field, density stratification, and rotation.
Featuring numerous exercises throughout, this is an ideal textbook for students and an essential resource for researchers.
- Describes how to create codes that simulate the internal dynamics of planets and stars
- Builds on basic concepts and simple methods
- Shows how to improve the efficiency and accuracy of the numerical methods
- Describes more relevant geometries and boundary conditions
- Demonstrates how to incorporate more sophisticated physics
Recenzijos
"This book provides readers with the skills they need to write computer codes that simulate convection, internal gravity waves and magnetic field generation in the interiors and atmospheres of rotating planets and stars. It is very useful for readers having a basic understanding of classical physics, vector calculus, partial differential equations, and simple computer programming."--Claudia-Veronika Meister, Zentralblatt MATH
Preface xi PART I. THE FUNDAMENTALS 1
Chapter 1 A Model of
Rayleigh-Benard Convection 3 1.1 Basic Theory 3 1.2 Boussinesq Equations 10
1.3 Model Description 13 Supplemental Reading 15 Exercises 15
Chapter 2
Numerical Method 17 2.1 Vorticity-Streamfunction Formulation 17 2.2
Horizontal Spectral Decomposition 19 2.3 Vertical Finite-Difference Method 21
2.4 Time Integration Scheme 22 2.5 Poisson Solver 24 Supplemental Reading 25
Exercises 25
Chapter 3 Linear Stability Analysis 27 3.1 Linear Equations 27
3.2 Linear Code 29 3.3 Critical Rayleigh Number 30 3.4 Analytic Solutions 31
Supplemental Reading 34 Exercises 34 Computational Projects 34
Chapter 4
Nonlinear Finite-Amplitude Dynamics 35 4.1 Modifications to the Linear Model
35 4.2 A Galerkin Method 36 4.3 Nonlinear Code 38 4.4 Nonlinear Simulations
43 Supplemental Reading 48 Exercises 49 Computational Projects 49
Chapter 5
Postprocessing 51 5.1 Computing and Storing Results 51 5.2 Displaying Results
51 5.3 Analyzing Results 54 Supplemental Reading 57 Exercises 57
Computational Projects 57
Chapter 6 Internal Gravity Waves 59 6.1 Linear
Dispersion Relation 59 6.2 Code Modifications and Simulations 62 6.3 Wave
Energy Analysis 66 Supplemental Reading 66 Exercises 67 Computational
Projects 67
Chapter 7 Double-Diffusive Convection 68 7.1 Salt-Fingering
Instability 69 7.2 Semiconvection Instability 72 7.3 Oscillating
Instabilities 74 7.4 Staircase Profiles 76 7.5 Double-Diffusive Nonlinear
Simulations 79 Supplemental Reading 80 Exercises 80 Computational Projects 80
PART II. ADDITIONAL NUMERICAL METHODS 83
Chapter 8 Time Integration Schemes
85 8.1 Fourth-Order Runge-Kutta Scheme 85 8.2 Semi-Implicit Scheme 87 8.3
Predictor-Corrector Schemes 89 8.4 Infinite Prandtl Number: Mantle Convection
91 Supplemental Reading 92 Exercises 93 Computational Projects 93
Chapter 9
Spatial Discretizations 95 9.1 Nonuniform Grid 95 9.2 Coordinate Mapping 97
9.3 Fully Finite Difference 98 9.4 Fully Spectral: Chebyshev-Fourier 102 9.5
Parallel Processing 108 Supplemental Reading 112 Exercises 112 Computational
Projects 112
Chapter 10 Boundaries and Geometries 115 10.1 Absorbing Top and
Bottom Boundaries 115 10.2 Permeable Periodic Side Boundaries 117 10.3 2D
Annulus Geometry 122 10.4 Spectral-Transform Method 130 10.5 3D and 2.5D
Cartesian Box Geometry 133 10.6 3D and 2.5D Spherical-Shell Geometry 135
Supplemental Reading 162 Exercises 162 Computational Projects 164 PART III.
ADDITIONAL PHYSICS 167
Chapter 11 Magnetic Field 169 11.1
Magnetohydrodynamics 170 11.2 Magnetoconvection with a Vertical Background
Field 173 11.3 Linear Analyses: Magnetic 179 11.4 Nonlinear Simulations:
Magnetic 182 11.5 Magnetoconvection with a Horizontal Background Field 184
11.6 Magnetoconvection with an Arbitrary Background Field 187 Supplemental
Reading 189 Exercises 190 Computational Projects 191
Chapter 12 Density
Stratification 193 12.1 Anelastic Approximation 194 12.2 Reference State:
Polytropes 207 12.3 Numerical Method: Anelastic 214 12.4 Linear Analyses:
Anelastic 219 12.5 Nonlinear Simulations: Anelastic 222 Supplemental Reading
227 Exercises 227 Computational Projects 228
Chapter 13 Rotation 229 13.1
Coriolis, Centrifugal, and Poincare Forces 229 13.2 2D Rotating Equatorial
Box 233 13.3 2D Rotating Equatorial Annulus: Differential Rotation 241 13.4
2.5D Rotating Spherical Shell: Inertial Oscillations 247 13.5 3D Rotating
Spherical Shell: Dynamo Benchmarks 259 13.6 3D Rotating Spherical Shell:
Dynamo Simulations 264 13.7 Concluding Remarks 275 Supplemental Reading 277
Exercises 278 Computational Projects 279 Appendix A A Tridiagonal Matrix
Solver 283 Appendix B Making Computer-Graphical Movies 284 Appendix C
Legendre Functions and Gaussian Quadrature 288 Appendix D Parallel
Processing: OpenMP 291 Appendix E Parallel Processing: MPI 292 Bibliography
295 Index 307
Gary A. Glatzmaier is professor of earth and planetary sciences at the University of California, Santa Cruz. He is a fellow of the American Academy of Arts and Sciences and a member of the National Academy of Sciences.
