Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.001.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
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 ...
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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, the book begins by offering a step-by-step guide on how to design codes for simulating nonlinear time-dependent thermal convection in a 2D box using Fourier expansions in the horizontal direction and finite differences in the vertical direction. It then describes how to implement more efficient a nd accurate numerical methods and more realistic geometries in two and three dimensions. The third part of the book demonstrates how to incorporate more sophisticated physics, including the effects of magnetic field, density stratification, and rotation. The book features numerous exercises throughout, and is an ideal textbook for students and an essential resource for researchers. It explains how to create codes that simulate the internal dynamics of planets and stars, and builds on basic concepts and simple methods. The book shows how to improve the efficiency and accuracy of the numerical methods. It considers more relevant geometries and boundary conditions.Less
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, the book begins by offering a step-by-step guide on how to design codes for simulating nonlinear time-dependent thermal convection in a 2D box using Fourier expansions in the horizontal direction and finite differences in the vertical direction. It then describes how to implement more efficient a nd accurate numerical methods and more realistic geometries in two and three dimensions. The third part of the book demonstrates how to incorporate more sophisticated physics, including the effects of magnetic field, density stratification, and rotation. The book features numerous exercises throughout, and is an ideal textbook for students and an essential resource for researchers. It explains how to create codes that simulate the internal dynamics of planets and stars, and builds on basic concepts and simple methods. The book shows how to improve the efficiency and accuracy of the numerical methods. It considers more relevant geometries and boundary conditions.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, ...
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This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, focusing on thermal convection and internal gravity waves. Thermal convection and internal gravity waves are the two basic types of fluid flows within planets and stars that are driven by thermally produced buoyancy forces. The chapter then reviews the equations that govern fluid dynamics based on conservation of mass, momentum, and energy. It also examines the conditions under which the Boussinesq approximation simplifies conservation equations to a form very similar to that of an incompressible fluid. Finally, it discusses the key characteristics of the model of Rayleigh–Bénard convection.Less
This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, focusing on thermal convection and internal gravity waves. Thermal convection and internal gravity waves are the two basic types of fluid flows within planets and stars that are driven by thermally produced buoyancy forces. The chapter then reviews the equations that govern fluid dynamics based on conservation of mass, momentum, and energy. It also examines the conditions under which the Boussinesq approximation simplifies conservation equations to a form very similar to that of an incompressible fluid. Finally, it discusses the key characteristics of the model of Rayleigh–Bénard convection.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0002
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter describes a numerical method for solving equations of thermal convection on a computer. It begins by introducing the vorticity-streamfunction formulation as a means of conserving mass. ...
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This chapter describes a numerical method for solving equations of thermal convection on a computer. It begins by introducing the vorticity-streamfunction formulation as a means of conserving mass. The approach involves updating for the vorticity first and then solving for the fluid velocity each time step. The chapter continues with a discussion of two very different spatial discretizations, whereby the vertical derivatives are approximated with a finite-difference method and the horizontal derivatives with a spectral method. The nonlinear terms are computed in spectral space. The chapter also considers the Adams-Bashforth time integration scheme and explains how the Poisson equation can be solved at each time step for the updated streamfunction given the updated vorticity.Less
This chapter describes a numerical method for solving equations of thermal convection on a computer. It begins by introducing the vorticity-streamfunction formulation as a means of conserving mass. The approach involves updating for the vorticity first and then solving for the fluid velocity each time step. The chapter continues with a discussion of two very different spatial discretizations, whereby the vertical derivatives are approximated with a finite-difference method and the horizontal derivatives with a spectral method. The nonlinear terms are computed in spectral space. The chapter also considers the Adams-Bashforth time integration scheme and explains how the Poisson equation can be solved at each time step for the updated streamfunction given the updated vorticity.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0005
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and ...
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This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and store results in a file during the computer simulation, assuming the Fourier transforms to x-space are done within the main computational code during the simulation. It then describes the postprocessing code for reading these files and displaying the various fields, along with the use of graphics software packages that provide additional, more sophisticated visualizations of the scalar and vector data. It also discusses the computer analysis of several additional properties of the solution, focusing on measurements of nonlinear convection such as Rayleigh number, Nusselt number, Reynolds number, and kinetic energy spectrum.Less
This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and store results in a file during the computer simulation, assuming the Fourier transforms to x-space are done within the main computational code during the simulation. It then describes the postprocessing code for reading these files and displaying the various fields, along with the use of graphics software packages that provide additional, more sophisticated visualizations of the scalar and vector data. It also discusses the computer analysis of several additional properties of the solution, focusing on measurements of nonlinear convection such as Rayleigh number, Nusselt number, Reynolds number, and kinetic energy spectrum.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0006
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter focuses on internal gravity waves in a stable thermal stratification. When the amplitude of the fluid velocity is small relative to the amplitude of the phase velocity, a linear ...
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This chapter focuses on internal gravity waves in a stable thermal stratification. When the amplitude of the fluid velocity is small relative to the amplitude of the phase velocity, a linear analysis, which neglects advection, provides insight to the relation between the wavelength and frequency of internal gravity waves. Furthermore, when thermal and viscous diffusion play relatively minor roles the system can be further simplified by neglecting diffusion. The chapter first describes the linear dispersion relation before discussing the computer code modifications and simulations. In particular, it explains what modifications would be needed to convert one's thermal convection code to a code that simulates internal gravity waves, including the nonlinear and diffusive terms. Finally, it considers the computer analysis of wave energy.Less
This chapter focuses on internal gravity waves in a stable thermal stratification. When the amplitude of the fluid velocity is small relative to the amplitude of the phase velocity, a linear analysis, which neglects advection, provides insight to the relation between the wavelength and frequency of internal gravity waves. Furthermore, when thermal and viscous diffusion play relatively minor roles the system can be further simplified by neglecting diffusion. The chapter first describes the linear dispersion relation before discussing the computer code modifications and simulations. In particular, it explains what modifications would be needed to convert one's thermal convection code to a code that simulates internal gravity waves, including the nonlinear and diffusive terms. Finally, it considers the computer analysis of wave energy.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0007
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter discusses double-diffusive convection, with a particular focus on the initial instability and eventual nonlinear evolution. It first considers the “salt-fingering” instability and then ...
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This chapter discusses double-diffusive convection, with a particular focus on the initial instability and eventual nonlinear evolution. It first considers the “salt-fingering” instability and then the “semiconvection” instability before discussing the possibility that the onsets of these instabilities at marginal stability have an amplitude that oscillates in time. The goal is to find the conditions that would result in a zero growth rate of the oscillation amplitude in order to determine the marginal stability constraint on the Rayleigh numbers for the onset of an oscillating instability. The chapter also shows how, after evolving beyond the onset of the instability, thermal diffusion between the moving parcel and the surroundings can alter the initial linear vertical profile of the horizontal-mean temperature into a “staircase” profile. This evolution of the temperature profile is investigated via nonlinear simulations.Less
This chapter discusses double-diffusive convection, with a particular focus on the initial instability and eventual nonlinear evolution. It first considers the “salt-fingering” instability and then the “semiconvection” instability before discussing the possibility that the onsets of these instabilities at marginal stability have an amplitude that oscillates in time. The goal is to find the conditions that would result in a zero growth rate of the oscillation amplitude in order to determine the marginal stability constraint on the Rayleigh numbers for the onset of an oscillating instability. The chapter also shows how, after evolving beyond the onset of the instability, thermal diffusion between the moving parcel and the surroundings can alter the initial linear vertical profile of the horizontal-mean temperature into a “staircase” profile. This evolution of the temperature profile is investigated via nonlinear simulations.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0008
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter focuses on time integration schemes, including fourth-order accurate Runge–Kutta and predictor-corrector schemes as well as schemes that allow larger time steps (and therefore fewer ...
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This chapter focuses on time integration schemes, including fourth-order accurate Runge–Kutta and predictor-corrector schemes as well as schemes that allow larger time steps (and therefore fewer steps for a given amount of simulated time) by treating the linear diffusion terms implicitly. The nonlinear terms, however, couple all the modes and so would be extremely expensive to treat implicitly; therefore they are usually treated explicitly. Such “semi-implicit” schemes considerably improve the efficiency of the computer code. The chapter also describes the Crank–Nicolson scheme and concludes by showing how the current numerical model can easily be modified to study mantle convection (also called “geodynamics”) using the vorticity equation in the limit of an infinite Prandtl number.Less
This chapter focuses on time integration schemes, including fourth-order accurate Runge–Kutta and predictor-corrector schemes as well as schemes that allow larger time steps (and therefore fewer steps for a given amount of simulated time) by treating the linear diffusion terms implicitly. The nonlinear terms, however, couple all the modes and so would be extremely expensive to treat implicitly; therefore they are usually treated explicitly. Such “semi-implicit” schemes considerably improve the efficiency of the computer code. The chapter also describes the Crank–Nicolson scheme and concludes by showing how the current numerical model can easily be modified to study mantle convection (also called “geodynamics”) using the vorticity equation in the limit of an infinite Prandtl number.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0009
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter considers two ways of employing a spatial resolution that varies with position within a finite-difference method: using a nonuniform grid and mapping to a new coordinate variable. It ...
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This chapter considers two ways of employing a spatial resolution that varies with position within a finite-difference method: using a nonuniform grid and mapping to a new coordinate variable. It first provides an overview of nonuniform grids before discussing coordinate mapping as an alternative way of achieving spatial discretization. It then describes an approach for treating both the vertical and horizontal directions with simple finite-difference methods: defining a streamfunction, which automatically satisfies mass conservation, and solving for vorticity via the curl of the momentum conservation equation. It also explains the use of the Chebyshev–Fourier method to simulate the convection or gravity wave problem by employing spectral methods in both the horizontal and vertical directions. Finally, it looks at the basic ideas and some issues that need to be addressed with respect to parallel processing as well as choices that need to be made when designing a parallel code.Less
This chapter considers two ways of employing a spatial resolution that varies with position within a finite-difference method: using a nonuniform grid and mapping to a new coordinate variable. It first provides an overview of nonuniform grids before discussing coordinate mapping as an alternative way of achieving spatial discretization. It then describes an approach for treating both the vertical and horizontal directions with simple finite-difference methods: defining a streamfunction, which automatically satisfies mass conservation, and solving for vorticity via the curl of the momentum conservation equation. It also explains the use of the Chebyshev–Fourier method to simulate the convection or gravity wave problem by employing spectral methods in both the horizontal and vertical directions. Finally, it looks at the basic ideas and some issues that need to be addressed with respect to parallel processing as well as choices that need to be made when designing a parallel code.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0010
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter examines how boundary and geometry affect convection. It begins with a discussion of how one can implement “absorbing” top and bottom boundaries, which reduce the large-amplitude ...
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This chapter examines how boundary and geometry affect convection. It begins with a discussion of how one can implement “absorbing” top and bottom boundaries, which reduce the large-amplitude convectively driven flows within shallow boundary layers or the reflection of internal gravity waves off these boundaries in a stable stratification. It then considers how to replace the impermeable side boundary conditions with permeable periodic side boundary conditions to allow fluid flow through these boundaries and nonzero mean flow. It also introduces “two and a half dimensional” geometry within a cartesian box geometry and describes how a fully 3D cartesian box model could be constructed. Finally, it presents a model of convection in a fully 3D spherical-shell and shows how it can be easily reduced to a 2.5D spherical-shell model. The horizontal structures are represented in terms of spherical harmonic expansions.Less
This chapter examines how boundary and geometry affect convection. It begins with a discussion of how one can implement “absorbing” top and bottom boundaries, which reduce the large-amplitude convectively driven flows within shallow boundary layers or the reflection of internal gravity waves off these boundaries in a stable stratification. It then considers how to replace the impermeable side boundary conditions with permeable periodic side boundary conditions to allow fluid flow through these boundaries and nonzero mean flow. It also introduces “two and a half dimensional” geometry within a cartesian box geometry and describes how a fully 3D cartesian box model could be constructed. Finally, it presents a model of convection in a fully 3D spherical-shell and shows how it can be easily reduced to a 2.5D spherical-shell model. The horizontal structures are represented in terms of spherical harmonic expansions.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0011
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter focuses on magnetoconvection, which refers to thermal convection of an electrically conducting fluid within a background magnetic field maintained by some external mechanism. It first ...
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This chapter focuses on magnetoconvection, which refers to thermal convection of an electrically conducting fluid within a background magnetic field maintained by some external mechanism. It first provides a brief overview of magnetohydrodynamics and the magnetohydrodynamic equations before explaining how to make a 2D model of magnetic field. In this approach, the case of a uniform vertical background field and the case of a uniform horizontal background field are both considered. The chapter then describes how one could simulate a case of a uniform background field that is tilted relative to both the vertical and horizontal axes. It also considers what can be learned about the stability and structure of magnetoconvection and the dispersion relation for magneto-gravity waves from analytical analyses without the nonlinear terms. Finally, it discusses nonlinear simulations of magnetoconvection in a box with impermeable side boundaries, along with magnetoconvection with a horizontal background field and an arbitrary background field.Less
This chapter focuses on magnetoconvection, which refers to thermal convection of an electrically conducting fluid within a background magnetic field maintained by some external mechanism. It first provides a brief overview of magnetohydrodynamics and the magnetohydrodynamic equations before explaining how to make a 2D model of magnetic field. In this approach, the case of a uniform vertical background field and the case of a uniform horizontal background field are both considered. The chapter then describes how one could simulate a case of a uniform background field that is tilted relative to both the vertical and horizontal axes. It also considers what can be learned about the stability and structure of magnetoconvection and the dispersion relation for magneto-gravity waves from analytical analyses without the nonlinear terms. Finally, it discusses nonlinear simulations of magnetoconvection in a box with impermeable side boundaries, along with magnetoconvection with a horizontal background field and an arbitrary background field.
