Introduction to Modeling Convection in Planets and Stars

Book Description

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

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Dedication Page
  5. Contents
  6. Preface
  7. Part I. The Fundamentals
    1. Chapter 1 A Model of Rayleigh-Bénard Convection
      1. 1.1 Basic Theory
      2. 1.2 Boussinesq Equations
      3. 1.3 Model Description
      4. Supplemental Reading
      5. Exercises
    2. Chapter 2 Numerical Method
      1. 2.1 Vorticity-Streamfunction Formulation
      2. 2.2 Horizontal Spectral Decomposition
      3. 2.3 Vertical Finite-Difference Method
      4. 2.4 Time Integration Scheme
      5. 2.5 Poisson Solver
      6. Supplemental Reading
      7. Exercises
    3. Chapter 3 Linear Stability Analysis
      1. 3.1 Linear Equations
      2. 3.2 Linear Code
      3. 3.3 Critical Rayleigh Number
      4. 3.4 Analytic Solutions
      5. Supplemental Reading
      6. Exercises
      7. Computational Projects
    4. Chapter 4 Nonlinear Finite-Amplitude Dynamics
      1. 4.1 Modifications to the Linear Model
      2. 4.2 A Galerkin Method
      3. 4.3 Nonlinear Code
      4. 4.4 Nonlinear Simulations
      5. Supplemental Reading
      6. Exercises
      7. Computational Projects
    5. Chapter 5 Postprocessing
      1. 5.1 Computing and Storing Results
      2. 5.2 Displaying Results
      3. 5.3 Analyzing Results
      4. Supplemental Reading
      5. Exercises
      6. Computational Projects
    6. Chapter 6 Internal Gravity Waves
      1. 6.1 Linear Dispersion Relation
      2. 6.2 Code Modifications and Simulations
      3. 6.3 Wave Energy Analysis
      4. Supplemental Reading
      5. Exercises
      6. Computational Projects
    7. Chapter 7 Double-Diffusive Convection
      1. 7.1 Salt-Fingering Instability
      2. 7.2 Semiconvection Instability
      3. 7.3 Oscillating Instabilities
      4. 7.4 Staircase Profiles
      5. 7.5 Double-Diffusive Nonlinear Simulations
      6. Supplemental Reading
      7. Exercises
      8. Computational Projects
  8. Color insert
  9. Part II. Additional Numerical Methods
    1. Chapter 8 Time Integration Schemes
      1. 8.1 Fourth-Order Runge-Kutta Scheme
      2. 8.2 Semi-Implicit Scheme
      3. 8.3 Predictor-Corrector Schemes
      4. 8.4 Infinite Prandtl Number: Mantle Convection
      5. Supplemental Reading
      6. Exercises
      7. Computational Projects
    2. Chapter 9 Spatial Discretizations
      1. 9.1 Nonuniform Grid
      2. 9.2 Coordinate Mapping
      3. 9.3 Fully Finite–Difference
      4. 9.4 Fully Spectral: Chebyshev-Fourier
      5. 9.5 Parallel Processing
      6. Supplemental Reading
      7. Exercises
      8. Computational Projects
    3. Chapter 10 Boundaries and Geometries
      1. 10.1 Absorbing Top and Bottom Boundaries
      2. 10.2 Permeable Periodic Side Boundaries
      3. 10.3 2D Annulus Geometry
      4. 10.4 Spectral-Transform Method
      5. 10.5 3D and 2.5D Cartesian Box Geometry
      6. 10.6 3D and 2.5D Spherical-Shell Geometry
      7. Supplemental Reading
      8. Exercises
      9. Computational Projects
  10. Part III. Additional Physics
    1. Chapter 11 Magnetic Field
      1. 11.1 Magnetohydrodynamics
      2. 11.2 Magnetoconvection with a Vertical Background Field
      3. 11.3 Linear Analyses: Magnetic
      4. 11.4 Nonlinear Simulations: Magnetic
      5. 11.5 Magnetoconvection with a Horizontal Background Field
      6. 11.6 Magnetoconvection with an Arbitrary Background Field
      7. Supplemental Reading
      8. Exercises
      9. Computational Projects
    2. Chapter 12 Density Stratification
      1. 12.1 Anelastic Approximation
      2. 12.2 Reference State: Polytropes
      3. 12.3 Numerical Method: Anelastic
      4. 12.4 Linear Analyses: Anelastic
      5. 12.5 Nonlinear Simulations: Anelastic
      6. Supplemental Reading
      7. Exercises
      8. Computational Projects
    3. Chapter 13 Rotation
      1. 13.1 Coriolis, Centrifugal, and Poincaré Forces
      2. 13.2 2D Rotating Equatorial Box
      3. 13.3 2D Rotating Equatorial Annulus: Differential Rotation
      4. 13.4 2.5D Rotating Spherical Shell: Inertial Oscillations
      5. 13.5 3D Rotating Spherical Shell: Dynamo Benchmarks
      6. 13.6 3D Rotating Spherical Shell: Dynamo Simulations
      7. 13.7 Concluding Remarks
      8. Supplemental Reading
      9. Exercises
      10. Computational Projects
  11. Appendix A A Tridiagonal Matrix Solver
  12. Appendix B Making Computer-Graphical Movies
  13. Appendix C Legendre Functions and Gaussian Quadrature
  14. Appendix D Parallel Processing: OpenMP
  15. Appendix E Parallel Processing: MPI
  16. Bibliography
  17. Index

Product Information

  • Title: Introduction to Modeling Convection in Planets and Stars
  • Author(s): Gary A. Glatzmaier
  • Release date: November 2013
  • Publisher(s): Princeton University Press
  • ISBN: 9781400848904