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.

INTRODUCTION

In the belief that only unkind gods would arrange two energy sources for planetary dynamos as equally important, this re-exploration of plausible sources seeks to eliminate rotational energy in favor of convection. Recent experiments and theory of the ‘elliptical’ instabilities in a rotating fluid due to precessional and tidal strains provide quantitative results for velocity fields and energy production. The adequacy of these flows to produce a. dynamo on both terrestrial and giant planets is assessed in the context of ‘strong field’ scaling. With little ambiguity it is concluded that Mercury, Venus, and Mars can not have a dynamo of tidal or precessional origin. The case for today's Earth is marginal. Here precessional strains (accidentally comparable to tidal strains) also are potential sources of inertial instabilities. The ancient Earth with its closer Moon, as well as all the giant planets, have tides well in excess of those needed to critically maintain dynamos. Hence the project proposed here proves to be successful only in part – an Earth in the distant future will not be able to sustain the geodynamo with its rotational energy. On the other hand, convection remains a possible dynamo energy source, with such a large number of undetermined processes and parameters that it is unfairly easy to establish conditions for its inadequacy. A large literature explores its adequacy. A brief review of this literature, in both a ‘strong-field’ and ‘weak-field’ context, advances several cautionary restraints to be employed on that day when the limits of validity of a quantitative dynamo-convection theory are to be determined.

INTRODUCTION

Many astrophysical bodies possess magnetic fields that arise from dynamo action. The case of the Earth is a unique one because the observational data available are much more detailed for the Earth than for any other astrophysical body, making possible a rather detailed comparison of geodynamo theory with observations. To meet this unique opportunity we therefore need a geodynamo theory that is very detailed. To develop the fully-fledged theory of such a complicated system as the geodynamo, even with the help of modern computers, it is however necessary to possess a qualitative understanding of its structure. This can be achieved by preliminary ‘scouting’ calculations of some artificially simplified models that are much simpler than the full geodynamo model but nevertheless help to understand it. A kinematic dynamo theory is the first step towards this goal. Kinematic models provide us with an understanding of its electrodynamics (the magnetic field generation process). The next necessary step is an understanding of its mechanics. The model-Z geodynamo emerges as a result of this step of scouting calculations. It may be considered as a specific case of a more general model that we call the nonlinear (pseudo-) axisymmetric dynamo model. This is a natural generalisation of the linear, nearly axisymmetric, kinematic dynamo model (Braginsky 1964a, b, c, d), and it is ‘intermediate’ between the kinematic and the complete theories of the geodynamo.

The nonlinear axisymmetric dynamo model aims at understanding the specific features of the main convective flow and the production of axisymmetric field in the core while the field generation due to the non-axisymmetric motion (a-effect) is considered as given. Another direction for an essential ‘intermediate’ investigation is to explore non-axisymmetric magnetoconvection.

Quasars, which can be a thousand times brighter than an ordinary galaxy, are the most distant objects observable in the Universe. How quasars produce the luminosity of 1013 suns in a volume the size of the solar system continues to be a major question in astronomy. Distant quasars are very rare objects whose study has been blocked by their scarcity. Recent technical advances, however, have opened new paths for their discovery. Forty quasars with redshifts greater than 4 have been found since 1986. Redshift 4 corresponds to a light travel time of more than 10 billion years. As a result, we are now able to probe the epoch shortly after the Big Bang when quasars may have first formed and to study the universe when it was less than a tenth its present age.

Quasars were one of the main discoveries thirty years ago that revolutionized astronomy. While they and the black holes thought to occur in their centers have become household words today, quasars are as enigmatic in many ways as they were when first discovered. Whatever their nature, they offer us views of the Universe never before seen, especially at distances far beyond what astronomers of the previous generation expected to see. In this chapter I wish to review briefly their history, how extraordinary their properties are, and how they serve as probes of the Universe to nearly as far as the visible horizon.

This book originated as a symposium at the American Association for the Advancement of Science annual meeting in San Francisco in 1989. The topic, The Farthest Things in the Universe, suggested itself to me as the most interesting and significant topic that people could hear about. An earlier AAAS Symposium had led to a book, The Redshift Controversy, that was still in use, and we hope that this volume will prove itself of similarly lasting interest.

Two of the original speakers, Hyron Spinrad of the University of California at Berkeley, and Patrick Osmer, then of the National Optical Astronomy Observatories, revised their pieces to bring them up-to-date for inclusion in this book. Further, Ed Cheng of the COBE Science Team and NASA's Goddard Space Flight Center agreed to write a new piece for inclusion in the book. We appreciate his taking time during the period of his duties as Chief Scientist for the Hubble Space Telescope's repair mission to complete his piece. During the interval from the time of the symposium to the present, the Cosmic Background Explorer spacecraft was launched and has had its tremendous successes in showing that the Universe has a blackbody spectrum and in finding ripples in space that may be the seeds from which galaxy-formation began. Thus this book appears at an optimum time.

The technical ability of astronomers to obtain images and spectra of very faint galaxies has improved greatly over the last decade. Since galaxies are vast collections of gas and stars, they must physically evolve with time. We should be able to directly observe the time-evolution of galaxies by studying very distant systems; the look-back internal corresponding to the mostdistant galaxies known in 1992 now approaches 15 billion years (80% of the total expansion age of the Universe)!

The line spectra of these faint galaxies are invaluable for redshift determination and physical study. The realization that Ly α (121.6 nm), formed in neutral hydrogen gas, is a strong emission line in most active galaxies and perhaps normal star-forming galaxies also, has helped us measure much larger redshifts in 1987–92 than was previously possible. Recall that this wavelength is in the ultraviolet; it can be observed only by satellites. But when galaxies are very far away, their Doppler effect shifts this spectral line into the region of the spectrum that we can observe with large telescopes on Earth. The largest redshifts for radio galaxies now approach z=3.8. Differing selection effects control which galaxies can be seen/isolated that far away. At least some red galaxies must form at redshift zf>5 (where the subscript f stands for the epoch of star formation).

When we look out into space at night, we see the Moon, the planets, and the stars. The Moon is so close, only about 380000 kilometers (240000 miles) that we can send humans out to walk on it, as we did in the brief glorious period from 1969 to 1972. Even the planets are close enough that we can send spacecraft out to them, notably the Voyager spacecraft, one of which has passed Neptune. Whereas light and radio signals from spacecraft take only about a second to reach us from the Moon, the radio signals from Voyager 2 at Neptune took several hours to travel to waiting radio telescopes on Earth. We say that the distance to the Moon is 1 light-second and the distance to Neptune is several light-hours.

Aside from our Sun, the nearest star at 8 light-minutes away, the distances to the stars are measured in light-years. The nearest star system is Alpha Centauri, visible only in the southern sky, and the single nearest star is known as Proxima Centauri, about 4.2 light-years away. We know so little about the stars that new evidence in 1993 indicates that Proxima Centauri might not be a member of a triple-star system along with the other parts of alpha Centauri, as has long been thought. The speeds at which those stars are moving through space may be sufficiently different that Proxima is only temporarily near Alpha's components.