The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory / Edition 1

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Overview

Magnetism is one of the most pervasive features of the Universe, with planets, stars and entire galaxies all having associated magnetic fields. All of these fields are generated by the motion of electrically conducting fluids, the so-called dynamo effect. The precise details of what drives the motion, and indeed what the fluid consists of, differ widely though. In this work the authors draw upon their expertise in geophysical and astrophysical MHD to explore some of these phenomena, and describe the similarities and differences between different magnetized objects. They also explain why magnetic fields are crucial in the formation of the stars, and discuss promising experiments currently being designed to study some of the relevant physics in the laboratory. This interdisciplinary approach makes the book appealing to a wide audience in physics, astrophysics and geophysics.

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Editorial Reviews

From the Publisher
"…a valuable book to have and study…an encyclopedic guide to some of the most interesting problems in astrophysics…should be savored over many sittings." (Physics Today, November 2005)

“…does contain many deep insights and there is valuable material…that is not easily accessible elsewhere…” (Geophysical and Astrophysical Fluid Dynamics, August 2005, Vol 99 (4))

"...the authors do a good job of conveying the excitement of the subject and of bringing together much of the research of important topics...a welcome addition to the bookshelves..." (The Observatory, Vol.25, No.1186, June 2005)

"Ruder and Hollerbach have brought together the significant research on this topic…will be of interest to both the geophysicist and the astrophysicist, and perhaps the physicist or engineer…" (E-STREAMS, April 2005)

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Product Details

  • ISBN-13: 9783527404094
  • Publisher: Wiley
  • Publication date: 8/1/2004
  • Edition number: 1
  • Pages: 343
  • Product dimensions: 6.93 (w) x 9.78 (h) x 0.87 (d)

Meet the Author

Güenther Ruediger received his PhD from the University of Jena, Germany. He is Professor at the Astrophysical Institute Potsdam, and lectures at the University of Potsdam. He worked at the University of Goettingen, and the High Altitude Observatory in Boulder, Colorado. He is also a former visiting professor at the University of Newcastle upon Tyne, England.

Rainer Hollerbach is Reader in Applied Mathematics at the University of Glasgow, Scotland. He has a PhD in Geophysics from the University of California, San Diego. He recently spent a year in Germany as a Research Fellow of the Alexander von Humboldt Foundation.

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Table of Contents

Preface.

1 Introduction.

2 Earth and Planets.

2.1 Observational Overview.

2.1.1 Reversals.

2.1.2 Other Time-Variability.

2.2 Basic Equations and Parameters.

2.2.1 Anelastic and Boussinesq Equations.

2.2.2 Nondimensionalization.

2.3 Magnetoconvection.

2.3.1 Rotationor Magnetism Alone.

2.3.2 Rotation and Magnetism Together.

2.3.3 Weakversus Strong Fields.

2.3.4 Oscillatory Convection Modes.

2.4 Taylor’s Constraint.

2.4.1 Taylor’s Original Analysis.

2.4.2 Relaxation of Ro=E=0.

2.4.3 Taylor States versus Ekman States.

2.4.4 From Ekman States to Taylor States.

2.4.5 Torsional Oscillations.

2.4.6 αΩ-Dynamos.

2.4.7 Taylor’s Constraint in the Anelastic Approximation.

2.5 Hydromagnetic Waves.

2.6 The Inner Core.

2.6.1 Stewartson Layers on C.

2.6.2 Nonaxisymmetric Shear Layers on C.

2.6.3 Finite Conductivity of the Inner Core.

2.6.4 Rotation of the Inner Core.

2.7 Numerical Simulations.

2.8 Magnetic Instabilities.

2.9 Other Planets.

2.9.1 Mercury, Venus and Mars.

2.9.2 Jupiter’s Moons.

2.9.3 Jupiter and Saturn.

2.9.4 Uranus and Neptune.

3 Differential Rotation Theory.

3.1 The Solar Rotation.

3.1.1 Torsional Oscillations.

3.1.2 Meridional Flow.

3.1.3 Ward’s Correlation.

3.1.4 Stellar Observations.

3.2 Angular Momentum Transport in Convection Zones.

3.2.1 The Taylor Number Puzzle.

3.2.2 The Λ-Effect.

3.2.3 The Eddy Viscosity Tensor.

3.2.4 Mean-Field Thermodynamics.

3.3 Differential Rotation and Meridional Circulation for Solar-Type Stars.

3.4 Kinetic Helicity and the DIV-CURL-Correlation.

3.5 Overshoot Region and the Tachocline.

3.5.1 The NIRVANA Code.

3.5.2 Penetration into the Stable Layer.

3.5.3 A Magnetic Theory of the Solar Tachocline.

4 The Stellar Dynamo.

4.1 The Solar-Stellar Connection.

4.1.1 The Phase Relation.

4.1.2 The Nonlinear Cycle.

4.1.3 Parity.

4.1.4 Dynamo-related Stellar Observations.

4.1.5 The Flip-Flop Phenomenon.

4.1.6 More Cyclicities.

4.2 The α-Tensor.

4.2.1 The Magnetic-Field Advection.

4.2.2 The Highly Anisotropic α-Effect.

4.2.3 The Magnetic Quenching of the α-Effect.

4.2.4 Weak-Compressible Turbulence.

4.3 Magnetic-Diffusivity Tensor and η-Quenching.

4.3.1 The Eddy Diffusivity Tensor.

4.3.2 Sunspot Decay.

4.4 Mean-Field Stellar Dynamo Models.

4.4.1 The α2-Dynamo.

4.4.2 The αΩ-Dynamo for Slow Rotation.

4.4.3 Meridional Flow Influence.

4.5 The Solar Dynamo.

4.5.1 The Overshoot Dynamo.

4.5.2 The Advection-Dominated Dynamo.

4.6 Dynamos with Random α.

4.6.1 Aturbulence Model.

4.6.2 Dynamo Models with Fluctuating α-Effect.

4.7 Nonlinear Dynamo Models.

4.7.1 Malkus-Proctor Mechanism.

4.7.2 α-Quenching.

4.7.3 Magnetic Saturation by Turbulent Pumping.

4.7.4 η-Quenching.

4.8 Λ-Quenching and Maunder Minimum.

5 The Magnetorotational Instability (MRI).

5.1 Star Formation.

5.1.1 Molecular Clouds.

5.1.2 The Angular Momentum Problem.

5.1.3 Turbulence and Planet Formation.

5.2 Stability of Differential Rotation in Hydrodynamics.

5.2.1 Combined Stability Conditions.

5.2.2 Sufficient Condition for Stability.

5.2.3 Numerical Simulations.

5.2.4 Vertical Shear.

5.3 Stability of Differential Rotation in Hydromagnetics.

5.3.1 Ideal MHD.

5.3.2 Baroclinic Instability.

5.4 Stability of Differential Rotation with Strong Hall Effect.

5.4.1 Criteria of Instability of Protostellar Disks.

5.4.2 Growth Rates.

5.5 Global Models.

5.5.1 A Spherical Model with Shear.

5.5.2 A Global Disk Model.

5.6 MRI of Differential Stellar Rotation.

5.6.1 T Tauri Stars (TTS).

5.6.2 The Ap-Star Magnetism.

5.6.3 Decay of Differential Rotation.

5.7 Circulation-Driven Stellar Dynamos.

5.7.1 The Gailitis Dynamo.

5.7.2 Meridional Circulation plus Shear.

5.8 MRI in Kepler Disks.

5.8.1 The Shearing Box Model.

5.8.2 A Global Disk Dynamo?

5.9 Accretion-Disk Dynamo and Jet-Launching Theory.

5.9.1 Accretion-Disk Dynamo Models.

5.9.2 Jet-Launching.

5.9.3 Accretion-Disk Outflows.

5.9.4 Disk-Dynamo Interaction.

6 The Galactic Dynamo.

6.1 Magnetic Fields of Galaxies.

6.1.1 Field Strength.

6.1.2 Pitch Angles.

6.1.3 Axisymmetry.

6.1.4 Two Exceptions: Magnetic Torus and Vertical Halo Fields.

6.1.5 The Disk Geometry.

6.2 Nonlinear Winding and the Seed Field Problem.

6.2.1 Uniform Initial Field.

6.2.2 Seed Field Amplitude and Geometry.

6.3 Interstellar Turbulence.

6.3.1 The Advection Problem.

6.3.2 Hydrostatic Equilibrium and Interstellar Turbulence.

6.4 From Spheres to Disks.

6.4.1 1DdynamoWaves.

6.4.2 Oscillatory vs. Steady Solutions.

6.5 Linear 3DModels.

6.6 The Nonlinear Galactic Dynamo with Uniform Density.

6.6.1 The Model.

6.6.2 The Influences of Geometry and Turbulence Field.

6.7 Density Wave Theory and Swing Excitation.

6.7.1 Density Wav Theory.

6.7.2 The Short-Wave Approximation.

6.7.3 Swing Excitation in Magnetic Spirals.

6.7.4 Nonlocal Density Wave Theory in Kepler Disks.

6.8 Mean-Field Dynamos with Strong Halo Turbulence.

6.8.1 Nonlinear 2D Dynamo Model with Magnetic Supported Vertical Stratification.

6.8.2 Nonlinear 3D Dynamo Models for Spiral Galaxies.

6.9 New Simulations: Macroscale and Microscale.

6.9.1 Particle-Hydrodynamics for the Macroscale.

6.9.2 MHD for the Microscale.

6.10 MRI in Galaxies.

7 Neutron Star Magnetism.

7.1 Introduction.

7.2 Equations.

7.3 Without Stratification.

7.4 With Stratification.

7.5 Magnetic-Dominated Heat Transport.

7.6 White Dwarfs.

8 The Magnetic Taylor–Couette Flow.

8.1 History.

8.2 The Equations.

8.3 Results without Hall Effect.

8.3.1 Subcritical Excitation for Large Pm.

8.3.2 The Rayleigh Line (a = 0) and Beyond.

8.3.3 Excitation of Nonaxisymmetric or Oscillatory Modes.

8.3.4 Wave Number and Drift Frequencies.

8.4 Results with Hall Effect.

8.4.1 Hall Effect with Positive Shear.

8.4.2 Hall Effect with Negative Shear.

8.4.3 A Hall-Driven Disk-Dynamo?

8.5 Endplate effects.

8.6 Water Experiments.

8.7 Taylor–Couette Flow as Kinematic Dynamo.

9 Bibliography.

Index.

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First Chapter

The Magnetic Universe

Geophysical and Astrophysical Dynamo Theory
By Günther Rüdiger Rainer Hollerbach

John Wiley & Sons

Copyright © 2004 Wiley-VCH Verlag GmbH & Co. KGaA
All right reserved.

ISBN: 3-527-40409-0


Chapter One

Introduction

Magnetism is one of the most pervasive features of the Universe, with planets, stars and entire galaxies all having associated magnetic fields. All of these fields are generated by the motion of electrically conducting fluids, via the so-called dynamo effect. The basics of this effect are almost trivial to explain: moving an electrical conductor through a magnetic field induces an emf (Faraday's law), which generates electric currents (Ohm's law), which have associated magnetic fields (Ampere's law). The hope is then that with the right combination of flows and fields the induced field will reinforce the original field, leading to (exponential) field amplification.

Of course, the details are rather more complicated than that. The basic physical principles may date back to the 19th century, but it was not until the middle of the 20th century that Backus (1958) and Herzenberg (1958) rigorously proved that this process can actually work, that is, that it is possible to find 'the right combination of flows and fields.' And even then their flows were carefully chosen to make the problem mathematically tractable, rather than physicallyrealistic. For most of these magnetized objects mentioned above it is thus only now, at the start of the 21st century, that we are beginning to unravel the details of how their fields are generated.

The purpose of this book is to examine some of this work. We will not discuss the basics of dynamo theory as such; for that we refer to the books by Roberts (1967), Moffatt (1978) and Krause & Rädler (1980), which are still highly relevant today. Instead, we wish to focus on some of the details specific to each particular application, and explore some of the similarities and differences.

For example, what is the mechanism that drives the fluid flow in the first place, and hence ultimately supplies the energy for the field? In planets and stars it turns out to be convection, whereas in accretion disks it is the differential rotation in the underlying Keplerian motion. In galaxies it could be either the differential rotation, or supernova-induced turbulence, or some combination of the two.

Next, what is the mechanism that ultimately equilibrates the field, and at what amplitude? The basic physics is again quite straightforward; what equilibrates the field is the Lorentz force in the momentum equation, which alters the flow, at least just enough to stop it amplifying the field any further. But again, the details are considerably more complicated, and again differ widely between different objects.

Another interesting question to ask concerns the nature of the initial field. In particular, do we need to worry about this at all, or can we always count on some more or less arbitrarily small stray field to start this dynamo process off? And yet again, the answer is very different for different objects. For planets we do not need to consider the initial field, since both the advective and diffusive timescales are so short compared with the age that any memory of the precise initial conditions is lost very quickly. In contrast, in stars the advective timescale is still short, but the diffusive timescale is long, so so-called fossil fields may play a role in certain aspects of stellar magnetism. And finally, in galaxies even the advective timescale is relatively long compared with the age, so there we do need to consider the initial field.

Accretion disks provide another interesting twist to this question of whether we need to consider the initial condition. The issue here is not whether the dynamo acts on a timescale short or long compared with the age, but whether it can act at all if the field is too weak. In particular, this Keplerian differential rotation by itself cannot act as a dynamo, so something must be perturbing it. It is believed that this perturbation is due to the Lorentz force itself, via the so-called magnetorotational instability. In other words, the dynamo can only operate at finite field strengths, but cannot amplify an infinitesimal seed field. One must therefore consider whether sufficiently strong seed fields are available in these systems.

Accretion disks also illustrate the effect that an object's magnetic field may have on its entire structure and evolution. As we saw above, the magnetic field always affects the flow, and hence the internal structure, in some way, but in accretion disks the effect is particularly dramatic. It turns out that the transport of angular momentum outward - which of course determines the rate at which mass moves inward - is dominated by the Lorentz force. Something as fundamental as the collapse of a gas cloud into a proto-stellar disk and ultimately into a star is thus magnetically controlled. That is, magnetism is not only pervasive throughout the Universe, it is also a crucial ingredient in forming stars, the most common objects found within it.

We hope therefore that this book will be of interest not just to geophysicists and astrophysicists, but to general physicists as well. The general outline is as follows: Chapter 2 presents the theory of planetary dynamos. Chapters 3 and 4 deal with stellar dynamos. We consider only those aspects of stellar hydrodynamics and magnetohydrodynamics that are relevant to the basic dynamo process; see for example Mestel (1999) for other aspects such as magnetic braking. Chapter 5 discusses this magnetorotational instability in Keplerian disks. Chapter 6 considers galaxies, in which the magnetorotational instability may also play a role. Chapter 7, concerning neutron stars, is slightly different from the others. In particular, whereas the other chapters deal with the origin of the particular body's magnetic field, in Chapt. 7 we take the neutron star's initial field as given, and consider the details of its subsequent decay. We consider only the field in the neutron star itself though; see Mestel (1999) for the physics of pulsar magnetospheres. Lastly, Chapt. 8 discusses the magnetorotational instability in cylindrical Couette flow. This geometry is not only particularly amenable to theoretical analysis, it is also the basis of a planned experiment. However, we also point out some of the difficulties one would have to overcome in any real cylinder, which would necessarily be bounded in z. Where relevant, individual chapters of course refer to one another, to point out the various similarities and differences. However, most chapters can also be read more or less independently of the others. Most chapters also present both numerical as well as analytic/asymptotic results, and as much as possible we try to connect the two, showing how they mutually support each other. Finally, we discuss fields occurring on lengthscales from kilometers to megaparsecs, and ranging from [10.sup.-20] to [10.sup.15] G - truly the magnetic Universe.

(Continues...)



Excerpted from The Magnetic Universe by Günther Rüdiger Rainer Hollerbach Copyright © 2004 by Wiley-VCH Verlag GmbH & Co. KGaA. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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