Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation: 24 (Princeton Series in Astrophysics) - Softcover

Glatzmaier, Gary A.

 
9780691141732: Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation: 24 (Princeton Series in Astrophysics)
Softcover
ISBN 10: 0691141738 ISBN 13: 9780691141732
Verlag: Princeton University Press, 2013

Inhaltsangabe

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

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Über die Autorin bzw. den Autor

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.

Von der hinteren Coverseite

"Glatzmaier's work is synonymous with the cutting edge of research in this field, and his tried-and-true presentation has been perfected over many years of teaching. I know of no other book that focuses on computer modeling of convection in planets and stars as this one does. It is an ideal tutorial for graduate students, and will also be of great interest to senior researchers."--James M. Stone, Princeton University

"The computational methods Glatzmaier describes can be applied to a huge range of nonlinear problems, with a variety of physical effects. There is a great deal of potential here for new investigations. In fact, our generation has barely scratched the surface! This is an important message for young scientists, who will find in this book some of the tools they will need to make future advances in astrophysics and geophysics."--Chris A. Jones, University of Leeds

"I am certain that this book will prove to be extremely useful to students and professionals alike. It is engagingly written, timely, comprehensive, and perhaps most importantly, graduated in its approach. Gary Glatzmaier is internationally recognized as one of the best computational scientists in geophysics and astrophysics."--Peter L. Olson, Johns Hopkins University

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Introduction to Modeling Convection in Planets and Stars

Magnetic Field, Density Stratification, Rotation

By Gary A. Glatzmaier

PRINCETON UNIVERSITY PRESS

Copyright © 2014 Princeton University Press
All rights reserved.
ISBN: 978-0-691-14173-2

Contents

Preface....................................................................xi
PART I. THE FUNDAMENTALS...................................................1
Chapter 1 A Model of Rayleigh-Bénard Convection............................3
Chapter 2 Numerical Method.................................................17
Chapter 3 Linear Stability Analysis........................................27
Chapter 4 Nonlinear Finite-Amplitude Dynamics..............................35
Chapter 5 Postprocessing...................................................51
Chapter 6 Internal Gravity Waves...........................................59
Chapter 7 Double-Diffusive Convection......................................68
PART II. ADDITIONAL NUMERICAL METHODS......................................83
Chapter 8 Time Integration Schemes.........................................85
Chapter 9 Spatial Discretizations..........................................95
Chapter 10 Boundaries and Geometries.......................................115
PART III. ADDITIONAL PHYSICS...............................................167
Chapter 11 Magnetic Field..................................................169
Chapter 12 Density Stratification..........................................193
Chapter 13 Rotation........................................................229
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


CHAPTER 1

A Model of Rayleigh-Bénard Convection


There are two basic types of fluid flows within planets and stars that are drivenby thermally produced buoyancy forces: thermal convection and internal gravitywaves. The type depends on the thermal stratification within the fluid region. TheEarth's atmosphere and ocean, for example, are in most places convectively stable,which means that they support internal gravity waves, not (usually) convection (butsee Chapter 7). On warm afternoons, however, the sun can heat the ground surface,which changes the vertical temperature gradient in the troposphere and makes theatmosphere convectively unstable; the appearance of cumulus clouds is an indicationof the resulting convective heat (and moisture) flux. Thermal convection likelyalso occurs in the Earth's liquid outer core, which generates the geomagnetic field,and, on a much longer time scale, in the Earth's mantle, which drives plate tectonicsand, on a much shorter time scale, initiates earthquakes and volcanic eruptions.Thermal convection is seen on the surface of the sun and likely occurs in the outer30% of the solar radius, where solar magnetic field is generated. Below this depthbuoyancy likely drives internal gravity waves. Rotation strongly influences the styleof the convection and waves in all of these examples except the mantle, which isdominated by viscous forces.

Computer simulation studies, over the past few decades, have significantly improvedour understanding of these phenomena. Some studies, like those for theatmospheres of the Earth and sun, have provided physical explanations and predictionsof the observations. Others, like those for the deep interiors of the Earth andsun, have provided detailed theories and predictions of the dynamics that cannot bedirectly observed. As computers continue to improve in speed and memory, computerprograms are able to run at greater spatial and temporal resolutions, whichimproves the quality of and confidence in the simulations. Numerical and programmingmethods have also improved and need to continue to improve to take fulladvantage of the improvements in computer hardware.


1.1 BASIC THEORY

We begin with a simple description of the fundamental dynamics expected in afluid that is convectively stable and in one that is convectively unstable. Then wereview the equations that govern fluid dynamics based on conservation of mass,momentum, and energy.


1.1.1 Thermal Convection and Internal Gravity Waves

The thermal stability of a fluid within a gravitational field is determined by itshorizontal-mean (i.e., ambient) vertical temperature gradient. The classic way ofdescribing this is to consider a fluid in hydrostatic equilibrium, i.e., the weightof the fluid above a given height (per cross-sectional area) is supported by thepressure at that height. Therefore, the vertical pressure gradient is negative. (Asusual, "vertical" here and throughout this book refers to the direction of increasingheight or radius, opposite to that of the gravitational acceleration.) In the interiorsof planets and stars the horizontal-mean density and temperature also decreasewith height. The question is how does the vertical temperature gradient of this fluid(atmosphere) compare with what an adiabatic temperature gradient would be.

Consider a small (test) parcel of fluid (Fig. 1.1) that, at its initial position (1), hasthe same pressure, density, and temperature as the surrounding atmosphere at thatposition. Imagine raising the parcel to a new height (2), fast enough so there is noheat transfer between it and the surrounding atmosphere but slowly enough that itremains in pressure equilibrium with its surroundings; that is, its upward velocity ismuch less than the local sound speed. Assuming this process is reversible and alsoadiabatic since there is no heat transfer, the parcel's entropy remains constant whilerising; that is, this is an isentropic process. However, since it remains in pressureequilibrium with the surroundings, its density and temperature both decrease as itrises because the decrease in pressure causes it to expand. If, when reaching its newhigher position (2), its temperature has decreased more than the temperature of thesurrounding atmosphere has decreased over that change in height, its density therewill be greater than the density of the surrounding atmosphere there (assuming atypical coefficient of thermal expansion). Therefore, the parcel will be antibuoyantand, when no longer externally supported, will fall. As the parcel falls its temperatureincreases faster than the surrounding temperature and when it passes the initialposition its temperature exceeds the temperature of surrounding atmosphere, causingthe now buoyant parcel to eventually stop falling and then to start rising. Thisprocess of accelerating downward when it is above the initial position and acceleratingupward when below the initial position is called an internal gravity waveand the surrounding atmosphere is said to be convectively stable. Recall that thisoccurs when the surrounding temperature decreases less rapidly with height than anadiabatic temperature profile would, since the test parcel moves adiabatically. Thatis, the surrounding temperature gradient is subadiabatic. The temperature stratificationwould be extremely stable if the surrounding temperature increased withheight. In reality, thermal and viscous diffusion cause internal gravity waves todecay with time unless they are continually being excited.

Now consider the case for which the changes in the parcel's temperature as itmoves up and down are less than that of the surrounding atmosphere (Fig. 1.2). Thatis, consider a surrounding atmosphere with a superadiabatic temperature gradient.In this case, when reaching its new higher position (2), the parcel's temperaturewill be higher than the temperature of the surroundings there and so its density willbe less than that of the surroundings. This makes the parcel buoyant and so, whenno longer externally supported, it continues to rise. Likewise, if the parcel wereinitially lowered, it would continue to sink. Typically, a rising parcel will eventuallyencounter a cold impermeable top boundary where it gives up heat by conduction,contracts, and becomes antibuoyant. This causes it to fall until it encounters a hotimpermeable bottom boundary where it absorbs heat by conduction, expands, becomesbuoyant, and rises. This process is called thermal convection and in this casethe surrounding atmosphere is said to be convectively unstable. In reality, thermaland viscous diffusion would cause this process to decay with time unless the excesstemperature drop maintained across the region (compared to the adiabatic temperaturedrop) is larger than a critical value (Section 3.3). That is, if thermal diffusion(i.e., conduction) is not efficient enough at transferring heat upward through theatmosphere, thermal convection will occur, which will transfer heat as convectiveheat flux.

Thermal convection exists in planets and stars and takes on a variety of forms.Thermal convection likely occurs throughout the mantles of terrestrial planets, likethe Earth, and also throughout most if not all of their liquid cores. However, becauseof their very different viscosities, the time scale for mantle convection is typicallya hundred million times longer than that for core convection and, whereas the styleof mantle convection is unaffected by planetary rotation (Coriolis forces), core convectionis dominated by rotation. The interactions between the Earth's mantle andits fluid core are discussed in Buffett (2007).Where, within the interior of stars andgiant planets, thermal convection occurs depends on the interior structure. Mostone-dimensional (1D) evolutionary models of giant gas planets predict convectionthroughout their liquid/gas interiors. Thermal conduction in stars is by radiativetransfer, which is less efficient at heat transfer in regions where atomic excitationand ionization occur, i.e., where the opacity is large and the adiabatic gradient isless steep, respectively. Convection typically occurs in these regions because thetemperature gradient would need to be steeper than the adiabatic gradient to conductall of the heat upward (i.e., outward). For example, the sun has a convectionzone in roughly the outer 30% of its radius; radiative transfer is sufficient to carrythe heat flux within the inner 70% of its radius, where the gas is fully ionized.Lower mass stars, which are cooler than the sun and therefore have a larger fractionof un-ionized gas, have much deeper convection zones; stars with less than30% of a solar mass are fully convective. Stars more massive (and so hotter) thanthe sun have very shallow outer convection zones, if any, and much larger starshave convection within their central cores where the strong temperature dependenceof the nuclear energy generation rate maintains a sufficiently steep temperaturegradient.

Computer modeling has made and will continue to make significant contributionsto our understanding of the interior dynamics of planets and stars. Our studyof computer modeling begins with simple models in this Part 1. In Chapters 1–5 wefocus on modeling thermal convection. In Chapter 6 we discuss the relatively simplechanges to the model needed to simulate internal gravity waves. A combinationof convection and gravity waves occurs in double-diffusive convection, which wedescribe in Chapter 7.


1.1.2 Equations of Motion

The fluid dynamics and thermal dynamics of these processes are governed by theclassical conservation laws for mass, momentum, and energy. However, since weare considering a continuous fluid, these laws are written in terms of the densitiesof mass, momentum, energy, and force.

The mass conservation equation,

[partial derivative]ρ/[partial derivative]t = [nabla] · ρv, (1.1a)


says that the local (Eulerian) time rate of change of mass density (ρ) is determinedby the convergence (i.e., negative divergence) of mass flux (ρv) at that locationand time (t); v is the fluid velocity. Note that the Lagrangian time derivative ofdensity (dρ/dt), which is the rate at which the density of a fluid parcel changes asit moves with the flow, is the sum of the Eulerian time derivative ([partial derivative]ρ/[partial derivative]t) and theadvection of density (v · [nabla])ρ. (The Lagrangian derivative is also called the materialderivative.) Therefore, Eq. 1.1a can also be written as

dρ/dt = -ρ[nabla] · v. (1.1b)


Newton's Second Law of motion applied to a fluid describes momentum conservation:mass density times acceleration equals the net force density on a fluidparcel as it moves. This equation,

ρ dv/dt = -[nabla]p + [nabla]·σ + gρ, (1.2)


is called the Navier-Stokes equation after Claude-Louis Navier and George GabrielStokes.

The first two terms on the right side are the macroscopic representation of theeffects due to molecules (or atoms). The first is the negative pressure gradient, aforce density from high to low pressure, p, which is due to static normal stress. Thesecond is the divergence of the viscous stress tensor, σ or "σij", to indicate that itis a tensor. Unless noted, we assume a Newtonian fluid; that is, viscous stress isproportional to the rate of strain of the fluid:

σij = 2ρν (eij - 1/3 ekk δi,j) (1.3)

where

eij = 1/2([partial derivative]vi/[partial derivative]xj + [partial derivative]vj/[partial derivative]xi) (1.4)


is the rate of strain tensor (for i = 1, 2,3 or x, y, z in cartesian coordinates), ν isthe viscous diffusivity (also called the kinematic shear viscosity), and the kroneckerdelta function δi,j is one for i = j and zero if not. Note, ekk = [nabla]·v. As is usuallydone for subsonic convection problems, we have neglected the small contributionto the viscous stress due to bulk viscosity: λekkδij, where λ is called the of kinematicbulk viscosity. If the dynamic shear viscosity ρν were constant in space, thedivergence of the viscous stress tensor (when neglecting the bulk viscosity) wouldreduce to

[nabla]·σ = ρν([nabla]2v + 1/3[nabla]([nabla]·v)). (1.5)


The last term on the right of Eq. 1.2 is the gravitational force density, g being thegravitational acceleration.

By doing a Taylor expansion about both position and time, the left side of Eq. 1.2can be written in the Eulerian form as ρ([partial derivative]v/[partial derivative]t + (v·[nabla])v), which by using Eq. 1.1also equals [partial derivative]ρv/[partial derivative]t + [nabla]·(ρvv). Above we called ρv mass flux; here we call itmomentum density. The Reynolds stress tensor, ρvv, is the momentum flux due tothe flow. That is, it states how each of the three components of momentum is beingtransported in each of the three directions. For example, ρvxvz is the rate that thex-component of momentum is being transported in the z-direction, which is alsothe rate that the z-component of momentum is being transported in the x-direction.The divergence of this tensor is a vector equal to the net rate that each of the threecomponents of momentum is diverging at the given position and time.

A few more words may be appropriate about the Eulerian and Lagrangian timederivatives. In an Eulerian representation we ask how the properties of the fluidare changing in time on, for example, a set of grid points in space, without keepingtrack of where the current fluid parcels at these locations originated. In a Lagrangianrepresentation, on the other hand, there are no set grid points in space. Instead, weask how the properties and the coordinate locations of a given set of fluid parcelschange with time. The Eulerian approach is preferred for a continuous fluid that fillsa defined volume. The Lagrangian approach is preferred for a discontinuous set ofparticles interacting within an otherwise empty volume of space. We are adoptingthe Eulerian approach.

The first law of thermodynamics describes internal energy conservation: the rateof change of the internal energy of a fluid parcel plus the rate the fluid parceldoes work equals the rate it absorbs heat. Note that "work" and "heat transfer"are process functions, not properties of the fluid. However, internal energy conservationcan also be described in terms of state functions. The rate the fluid doeswork per mass is pressure times the rate of change of the volume per mass (i.e.,specific volume, which is 1/ρ); therefore, using Eq. 1.1b, the rate fluid does workper volume is p]nabla]·v. The rate the fluid absorbs heat per volume can also be writtenin terms of state variables as ρT dS/dt, where S is specific entropy (i.e., entropyper mass). Therefore, conservation of internal energy density is

ρ de/dt + p]nabla]·v = ρT dS/dt = [nabla]·(k]nabla]T) + Q, (1.6)


where e is internal energy per mass (i.e., specific internal energy). The first heatingterm on the far right side of Eq. 1.6 is the convergence of diffusive heat flux,-k]nabla]T, where T is temperature, k = cpρκ is thermal conductivity, cp is specificheat capacity at constant pressure, and κ is thermal diffusivity. The remainingterm, Q, represents viscous and ohmic heating and any other heating or cooling,e.g., nuclear.

This relationship between process and state functions is valid within the verygood approximation of local thermodynamic equilibrium (LTE). That is, since"fluid dynamics" is a macroscopic description of the state and evolution of a fluidaveraged over length and time scales large compared to the molecular structureand processes, state variables like temperature, pressure, density, specific internalenergy, and specific entropy are defined as continuous functions of space and timethat usually vary slowly enough on macroscopic length and time scales that thermodynamicequilibrium can be assumed in small neighborhoods around every locationand time within the domain of study. For example, although temperature can varyin space (and therefore drive a diffusive heat flux), around any point within thefluid there is a small neighborhood in which the velocities of the particles have awell-defined Maxwellian distribution defined by the local temperature.


(Continues...)
Excerpted from Introduction to Modeling Convection in Planets and Stars by Gary A. Glatzmaier. Copyright © 2014 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Verlag
Princeton University Press
Erscheinungsdatum
2013
Sprache
Englisch
ISBN 10 
0691141738
ISBN 13 
9780691141732
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Tapa blanda
Anzahl der Seiten
328
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