The Concept of Geochemical Classification

The Norwegian geochemist Victor Goldschmidt is the father of the notion of geochemical classifications of the chemical elements. Goldschmidt's (1954) posthumous work Geochemistry is still of great value. His basic aim was to divide the elements into groups which might be identified with the major divisions of the earth during its cooling history. He thought there might be three separate liquid phases, one metal, one silicate, and one primarily iron sulfide. These would be surrounded by a gaseous phase. He classified the elements from their association with, or preference for one or the other of these phases.

Let us begin with a consideration of the chemistry of meteorites and the earth. The earth may be divided into a core, a mantle and a crust. The chemistry of the core must be largely inferred, and this is essentially true for most of the mantle (Ringwood 1975, Pasteris 1984). The upper continental crust is relatively well sampled (Taylor and McLennan 1985), but it is not representative. Meteorites, on the other hand, have been repeatedly and thoroughly analyzed in the laboratory. Moreover, they are thought to be pieces of a broken-up planet, not unlike the earth (see, e.g., McSween 1987). Because of this they have been used to infer the chemistry of the earth as well as of much of the cosmos.

Introduction

The isotopic abundances of cosmic materials may change for a number of reasons. If a substance contains radioactive nuclei, there will be a continual decrease in the parent and a buildup of the daughter isotopes. Bombardment of materials by cosmic rays or other high-energy particles can also alter the isotopic complement of a sample. During radioactive decays or nuclear fission, particles are emitted which can affect the surrounding nuclei. Fission fragments remain in the neighborhood of the parent nuclei. A third possibility is fractionation, by either diffusion or small mass-dependent effects in chemical reactions. All three of these contingencies have been mentioned or intimated previously. We shall now take up certain aspects of these processes in detail.

It will not be possible for us to discuss most of the dating techniques. The interested reader may consult the textbooks of Faure (1986) or Durrance (1986). Richardson and McSween (1989) have an excellent chapter on radioactive dating.

Rubidium–Strontium Dating; Sample and Model Ages

One of the most straightforward methods of age determination makes use of the decay of 87Rb to 87Sr. We shall discuss this particular method here in detail, because of its pedagogical advantages. We shall have time to mention only briefly other methods, some of which are now more actively pursued than rubidium–strontium.

Both rubidium and strontium are geochemically dispersed, that is they occur primarily as impurities in major minerals.

My career as a professional astronomer was some 15 years old when it first became necessary for me to learn something of the new developments in the solar system. At that time, in the mid-1970's, I was about as ignorant of the solar system as one trained in astronomy could possibly be. Worse than that, I had an attitude typical of many astronomers today. Because the field was old, I thought it was dull and uninteresting! Nevertheless, when it became necessary for me to give an introductory course in solar system astronomy, I thought I must try to understand what all the fuss over moon rocks was all about.

Moon rocks are not so terribly different from terrestrial rocks, and so I began to read an introductory geology text. Soon, I was making trips to the building next door to visit the Geology Department. I became an amateur geologist, and a rockhound. On automobile trips I would stop at various rock formations, and bash off samples with a rock hammer. These samples were typically shown to a geologist, sometimes in a nearby university or college, sometimes back at Michigan.

The experience of becoming an amateur geologist was immensely broadening. Not only did I become a great fan of planetary science, but I began to be interested in other areas of astronomy that had never particularly appealed to me. Eventually, I began to realize that there was a single theme behind all of these endeavors – the history of matter.

Introduction

Most stars are converting hydrogen to helium by either the proton–proton chain, or the CNO cycle. We will discuss these processes in §10.2. Nuclear energy is available by the combination or fusion of nuclei less massive than A = 56 (Fe). This nucleus has the maximum binding energy per nucleon. By far the largest fraction of energy in this process is released in the first step, the formation of helium from hydrogen. A rough figure for the ratio of the time that a star spends during hydrogen burning (its main-sequence lifetime) to all other phases of its lifetime is ten to one. The stellar lifetimes are terminated by supernovae explosions, or by quiescent deaths that may involve mass loss followed by the formation of a white dwarf or neutron star.

While the fusion of hydrogen into helium is undoubtedly one phase of nucleosynthesis, it is difficult to know the extent to which the observable helium throughout the universe was made in stars. It is relatively straightforward to demonstrate that the present luminosity of the galaxy, if constant over a lifetime of some 10 billion years, is insufficient to account for the present hydrogen-to-helium ratio. We shall return to this question in §10.7.

Helium, as we shall see shortly, is burned to carbon and oxygen, which can be returned to the interstellar medium in either quiescent mass loss or supernovae explosions. While the helium abundance in the Galaxy is arguably constant, this is surely not the case for carbon.

The General Framework

In this chapter we consider the results of the analytical methods discussed in Chapter 11 insofar as they apply to stars, and to the integrated starlight of some star clusters. Many of these results were obtained with methods that are directly applicable to the analysis of extragalactic systems. We shall postpone a general discussion of the chemical evolution of our own and external galaxies until Chapter 16.

The main pillars of any description of stars and stellar systems are spectral classification, and the Hertzsprung–Russell (H–R) diagram or some variation of it. Classification itself provides mostly information about the temperatures and pressures in the photospheres of stars. Since the majority of stars have rather similar compositions, classification mainly discriminates between the broad categories of normal and peculiar chemistry. The chemically peculiar stars are in many ways the most interesting. The position of a star on the Hertzsprung–Russell diagram (§13.3) indicates its state of evolution. Usually, this tells us something of the chemistry of the star's interior, but occasionally, what has happened in the interior of a star is manifested on its surface. Only a very brief summary of these concepts can be given here.

Most stars belong to double or multiple systems. Double stars were discovered by Sir William Herschel in the late eighteenth century, and a century later, their study was well developed. The marvelous Father Angelo Secchi (1878, p. 228) suggested that perhaps half of the visible stars had physically bound companions.

INTRODUCTION

In this chapter I attempt a review of theories of convection in a spherical geometry in the presence of magnetic fields and rotation. The understanding of such motion is essential to a proper theory of the geodynamo. Even though, as discussed by Malkus and Braginsky (chapters 5 and 9), the nature of the driving mechanism for the convection is not certain, and is likely to be compositional in nature, we shall generally, following other authors, look only at thermal convection, which is the simplest to study. In addition, it will be assumed (incorrectly) that the core fluid has essentially constant viscosity, density, etc., allowing the Boussinesq approximation to be employed.

The excuse for these simplifications is readily to hand: the dynamical complexities induced by the interaction of Coriolis and Lorentz forces are still not fully resolved, and transcend the details of the forcing or of compressibility effects. The effects of this interaction on global fields are discussed by Fearn (chapter 7) but here we shall confine ourselves to a small part of the complete picture: the non-axisymmetric instabilities of an imposed (and prescribed) axisymmetric magnetic field and differential rotation in a rotating sphere. This task is the mirror-image of the ‘intermediate’ models of Braginsky (chapter 9) and the non-linear ‘macrodynamic’ dynamos driven by the a-effect, described by Fearn (chapter 7), in that these works parametrize the small, rather than the global fields.

In what follows, we shall begin by defining a geometry and non-dimensionalization for the system. We shall mainly be working in a spherical geometry, but use for illustration simplified (e.g., planar, cylindrical) geometry where appropriate.

Magnetic fields are observed to exist wherever there is matter in the visible universe; they exist on planetary, stellar and galactic length-scales, indeed wherever there is a sufficiently large mass of rotating conducting fluid. Dynamo theory is that branch of fluid mechanics that seeks to explain both the origin of these magnetic fields, and the manner of their variation in space and time. The subject has exerted a profound challenge, and great advances have been made over the last few decades. Nevertheless, acute problems remain in relation to both planetary and stellar magnetism. The NATO Advanced Study Institute on Stellar and Planetary Dynamos, and the six-month Dynamo Theory Programme of the Isaac Newton Institute of which it formed part, set out both to review the present state of knowledge in this broad field, and to define the critical problems that now demand attention.

The problem of the origin of the Earth's magnetic field has challenged the imagination of great scientists of past centuries. Edmund Halley showed extraordinary prescience three hundred years ago when, in considering the possible causes of the secular variation of the geomagnetic field, he wrote:

This view of the inner structure of the Earth was not confirmed till Jeffreys' discovery in 1926 of the liquid outer core and Bullen's discovery in 1946 of the solid inner core.

INTRODUCTION

This chapter is devoted to understanding the nature of the transitions that are possible in rotating systems. Rotation is implicated in most instabilities of astrophysical and geophysical interest. These include, for example, the baroclinic instability responsible for the formation of weather fronts in the earth's atmosphere, the instability that forms the spiral arms of galaxies, and of course the dynamo instability. The approach we take emphasizes generic, i.e., model-independent, behaviour. As a result the discussion that follows focuses on the symmetries of the system which are often responsible for much of the observed behaviour. As such we do not address specific physical mechanisms that give rise to the instabilities, or even specific model equations that might be used to describe them. Nonetheless we find that the approach used provides a number of new insights into the type of dynamics that are characteristic of rotating systems. In addition it points out the shortcomings of local studies of rotating systems that have been used to simplify the analysis. Moreover, since the results are model-independent, they apply to any system sharing the same symmetry properties. Thus our results shed light not only on the possible transitions in dynamo theory, but also on those occurring in baroclinic and other rotating flows.

We begin by pointing out that a rotating cylinder and a rotating sphere have the same symmetry: both are invariant under proper rotations about the rotation axis. In fact any figure of revolution rotating about its axis has this symmetry. For a solid body the meaning of this statement is quite intuitive.

INTRODUCTION

The picture of the solar dynamo has evolved continuously over the past twenty years. Both observations and theoretical considerations have introduced new aspects to the problem. Meanwhile, there has been a growing recognition of the importance of the overshoot layer beneath the solar convection zone (CZ) as the place where magnetic flux tubes can be ‘stored’ over time scales comparable with the solar cycle period (e.g., Spiegel & Weiss 1980). Less clear, however, is the question of whether the field is also generated there, or whether the magnetic field is actually generated in the convection zone, and only then transported into the overshoot layer where it accumulates (Brandenburg et al 1991, Nordlund et al 1992).

Numerical simulations of hydromagnetic convection show generation of magnetic fields in the entire CZ, but there is a strong turbulent downward pumping, that causes the magnetic field to build up at the interface between the CZ and the radiative interior. Magnetic buoyancy causes magnetic flux tubes to float upwards, but at the same time convective motions push them down again. In numerical simulations it is seen that under these conditions the magnetic field plays an active role and can still be amplified. It is questionable whether the interface can be considered in isolation. Consequently, throughout this chapter we consider the evolution of the magnetic field in the entire CZ and allow for the interaction between the CZ and the radiative interior in most of the cases.

The magnetic fields generated by turbulence are so intermittent that it is difficult to understand how the solar magnetic field has such a systematic orientation, as demonstrated by Hale's polarity law (cf. Schiissler 1987).

INTRODUCTION

Cosmic Dynamos

Many stars, planets and galaxies possess magnetic fields whose origins are not easily explained. Even the ‘solid’ planets cannot be sufficiently ferromagnetic to account for their magnetism, for the bulk of their interiors are above the Curie temperature at which permanent magnetism disappears; obviously the stars and galaxies cannot be ferromagnetic at all. Nor are the magnetic fields transient phenomena that just happen to be present today. Palaeomagnetism, the study of magnetic fields ‘fossilized’ in rocks at the time of their formation in the remote geological past, shows that Earth's magnetic field has existed at much its present strength for the past 3 x 109 years, at least. But, unless they contain sources of electric current, conducting bodies of spatial dimension, L, can retain their magnetic fields only for times of the order of the electromagnetic diffusion time τη = L2/η, where η is the magnetic diffusivity of their constituents; η = l/μ0σ, where σ is their electrical conductivity and μ0 is the permeability of free space. (SI units are used throughout.) Being proportional to L2, this time may be very considerable, but is as nothing compared with the ages of the bodies concerned. For example, Earth contains a highly conducting region, its core, of radius about L = 3.48 x 100 m, and its conductivity is about 4 x 105 S/m. This gives η ≍ 2 m2/s and τη ≍ 200,000 years. Similarly, it is thought that the magnetic fields of other planets cannot be fossil relicts of their birth. A mechanism is required to maintain them.

This volume contains the texts of the invited lectures presented at the NATO Advanced Study Institute ‘Theory of Solar and Planetary Dynamos’ held at the Isaac Newton Institute for Mathematical Sciences in Cambridge from September 20 to October 2 1992. Its companion volume ‘Solar and Planetary Dynamos’, containing the texts of the contributed papers, has recently been published in the same series as the present one, and contains a full list of participants and their addresses. It is a measure of the recent growth of the subject that one volume has proved insufficient to contain all the material presented at the meeting: indeed, dynamo theory now acts as an interface between such diverse areas of mathematical interest as bifurcation theory, Hamiltonian mechanics, turbulence theory, large-scale computational fluid dynamics and asymptotic methods, as well as providing a forum for the interchange of ideas between astrophysicists, geophysicists and those concerned with the industrial applications of magnetohydrodynamics.

The topics of the lectures cover almost all the principal parts of the subject. Authors were asked to give reviews of a pedagogical nature. Earlier chapters cover relatively fundamental aspects of the subject; later chapters treat more specialised topics. Although each chapter is self-contained, there are cross-references to other lectures where appropriate; in addition, the Editors have striven to maintain uniformity of notation and style, in the hope that the resulting complete text will find favour as a unified work of reference, rather than as a disparate set of reviews.

INTRODUCTION

The aim of this chapter is to provide a link between observations of magnetic fields in the Sun and other active stars, and the theory that is presented elsewhere in this volume. I shall begin therefore by considering the observational background and the phenomenological picture that emerges from it. Then I shall go on to discuss a hierarchy of idealized dynamo models that help to explain different aspects of these observations. This material has been the subject of several recent reviews (Weiss 1989, Belvedere 1990, Brandenburg & Tuominen 1991, Stix 1991, DeLuca & Gilman 1991, Rosner & Weiss 1992, Schmitt 1993).

This treatment relies heavily on what has become known as the solarstellar connection. Figure 2.1 shows a Hertzsprung-Russell diagram, with the relative luminosity of the stars plotted as a function of their effective surface temperature (or, equivalently, their spectral type). The stars on the main sequence form a one-parameter family, with their positions determined by their masses. Some of the hot stars to the left of the vertical line have strong magnetic fields, which vary only as a consequence of the star's rotation. The fields in these magnetic stars (the Ap stars) are apparently fossil relics and will not be considered here. Stars to the right of the vertical line are sufficiently cool that hydrogen only becomes ionized beneath their visible surfaces; as a result, they have deep convective envelopes. The combination of convection and rotation is associated with magnetic activity in these cool stars. Their behaviour is similar to that found in the Sun, and it is with them that we are concerned.

PREFACE

Almost 400 hundred years ago Galileo noticed that the period of a pendulum is the same for all small amplitudes. Not long afterwards, Galileo and his contemporaries (see figure 8.1) proved that sunspots really were on the sun. So the same person was involved in discovering the paradigm of periodicity and establishing an exemplar of irregularity. But just how irregularly do sunspots behave? In modern terms, this question comes down to asking how many degrees of freedom are involved in the phenomenon. If the mechanism I am going to describe here, on/off intermittency, is operative, this question cannot be answered soon (Platt, Spiegel & Tresser 1993a). That I should begin this discussion by mentioning aperiodicity is a sign of where we are in the long saga of sunspot studies. Shortly after Galileo's discoveries, serious work on sunspots got under way. This was somewhat disappointing for a time because sunspots had become quite scarce, with only a few per year being detected. This intermission in solar activity lasted approximately throughout the life of Newton, being most extreme when he was in his prime and ending about a decade before his death (Eddy 1978). So the question of the changing level of solar activity must have been much on astronomers' minds at that time. By the time this puzzle was fadinga from memory, a new issue was raised in the middle of the nineteenth century, when it was noticed that the level of solar activity (as judged mainly by sunspots) was found to vary with some regularity.