INTRODUCTION

Although kinematic dynamo models reveal some of the basic features of Solar and stellar magnetic fields, a fully satisfactory model must allow the dynamics to emerge as part of the solution of the governing system of equations. This has been attempted in a number of mean-field studies starting with Proctor (1977). It has been shown that solutions exist in which axisymmetric fields become saturated at a finite energy by the action of the macroscopic Lorentz force acting on the fluid.

The mean-field formalism is used in turbulent convection zones to parametrise the effects of the small scale dynamics on the magnetic field (Steenbeck et al. 1966). The influence of the small-scale turbulence on the macroscopic motions can be similarly modelled by the ‘Λ-effect’, representing the Reynolds stresses of anisotropic turbulence induced in a rotating, stratified medium (Rudiger 1989). The resulting mean-field equations describe the evolution of quantities averaged over time or length scales greater than those of the turbulence.

INTRODUCTION

Our galaxy and others are permeated by magnetic fields. They play an important role in star formation, in the support of molecular clouds against collapse, and in cosmic ray confinement. With a field strength of a few microgauss, they are comparable in in energy density to thermal energy, radiation, and cosmic rays. These fields are widely assumed to be the result of a dynamo operating on an initial seed field.

Dynamos work by folding magnetic field lines back on themselves constructively more often than destructively. Mean field theory assumes that the many folds in the field with no net contribution are destroyed, usually by resistivity. What would happen if these small disordered fields were not destroyed? They would obscure the growing large-scale field and might dominate the total magnetic energy. This is indeed a concern for galactic dynamo theory as magnetic loops 0.1 pc across need 1022 years to decay ohmically.

THE DYNAMO IN THE CONVECTION ZONE

In 1969, Steenbeck & Krause presented results of the first hydrodynamic dynamo model acting in the turbulent convection zone (CZ) and based on the idea of mean field electrodynamics. They introduced two spherical shells for the induction effects: in the inner, there is the differential rotation (Ω ∼ r) and in the outer, one has the turbulent rotating matter (α ∼ Cos v). This simple model is in agreement with most of the observed magnetic patterns, such as the butterfly diagram (Figure 1), H ale's polarity rule and the 22 year period of the Solar cycle.

During half a cycle, i.e. eleven years, the activity belts, as a measure of the toroidal field, move from about ±30° latitude towards the equator. In the vicinity of the pole, there are no active regions. But observations of torsional oscillations (Howard & LaBonte 1982) and Solar wind (Legrand & Simon 1991) suggest that the toroidal field starts the reversal of its polarity there (Schussler 1981).

INTRODUCTION

This paper is concerned with the effect of an imposed magnetic field on thermal convection. Although this is not directly relevant to dynamo theory, the question of the interaction of convection with magnetic fields is important for a full understanding of a convectively driven dynamo. The motivation for this work is to understand some aspects of convection in the Sun, particularly in regions of high magnetic field, such as sunspots, where the strong, predominantly vertical field inhibits thermal convection, causing the spot to appear dark. This work is part of an ongoing collaboration with Michael Proctor and Nigel Weiss. This brief report summarises some of the main results of this work; further details will be published in a future paper (Matthews, Proctor & Weiss 1993).

Linear theory for the onset of magnetoconvection in an incompressible fluid was discussed extensively by Chandrasekhar (1961).

INTRODUCTION

My objective in these five lectures was to introduce the fundamental physical ideas underlying cosmological models of an isotropic hot Big-Bang and the development of large-scale structure in such an expanding universe. My instructions from the Organizing Committee were to assume that the audience had not studied cosmology before, and that is what I have tried to do. My final lecture reviewed a number of results in infrared observational cosmology, but constraints of time and space have not permitted inclusion of that material here.

Much of the material on isotropic cosmological models is developed very well in any number of existing sources (although this is less so for the theory of galaxy formation) and it is with some diffidence that I offer my version. The references list a number of other developments of these topics, and I have borrowed from most of them. Perhaps the best recent text which covers these and many other topics in modern cosmology very thoroughly, and yet very readably, is that by Kolb and Turner (1990). My own research in infrared observational cosmology received an important stimulus from a series of similar lectures presented by Malcolm Longair (1977) over 15 years ago. I hope these lectures might, in a similar way, give young astronomers an introduction which will generate enthusiasm for inventing new infrared observing programs bearing on fundamental cosmological problems.

THE ISOTROPIC UNIVERSE

Introduction

We begin by taking a very large-scale view of the universe, and make a simplicity approximation that the universe is smooth, with no structure. In this approximation we can think of the universe as a fluid (of galaxies) of density ρ, pressure p.

A hotly debated topic is the origin of the large scale galactic magnetic field. Originally, it was supposed by Fermi and others that the field had a primordial origin and was maintained against Ohmic decay by the large inductance of the galactic disk. (The time scale for Ohmic decay by ordinary Spitzer resistivity is extremely long, of order 1026 years.) However, there have been several objections to the primordial theory (Parker 1979). One objection is that turbulent resistivity is sufficiently large to destroy the field in a Hubble time. A second objection is that if it is not destroyed by turbulent resistivity, it can escape from the galactic disc by ambipolar diffusion. Probably the strongest objection has been that there seems no known way to produce a magnetic field in the early universe on a large enough scale and of sufficient strength to provide a primordial origin.

Fast dynamo instabilities are the subject of intense research (reviewed in Childress 1992), through numerical simulations and analytical studies of simple models. However little is known about how a fast dynamo instability might saturate and what the resulting spatial structure and temporal behaviour of the field might be. Does a fast dynamo saturate by suppressing the flow field until the effective magnetic Reynolds number is reduced to a value of order unity or by modifying transport effects of the flow (Vainshtein et al. 1993)? Is the saturated magnetic energy in equipartition with the kinetic energy and how is the magnetic energy distributed; in particular how much energy is stored in large-scale field components (Vainshtein & Cattaneo 1992)? Does the field contain the fine structure typical of kinematic fast dynamo instabilities and is it intermittent in time? The difficulty in answering these questions is that dynamo action only allows growth of 3-d magnetic fields and through the Lorentz force this leads to all the complexities of 3-d MHD turbulence. Numerical studies are computationally expensive and only moderate values of Rm have been achieved (see, for example, Gilman 1983, Glatzmaier 1985, Meneguzzi & Pouquet 1989, Nordlund et al 1991 and Galanti et al 1992).

MOTIVATION

Solar activity manifests itself in many forms but perhaps most importantly through the presence of a magnetic field. The topic of this meeting is that of dynamos, in Solar and planetary contexts. In the case of the Sun the dynamo, which seems likely to be responsible for at least part of the Solar activity we observe, acts on a global scale. That is, the period of the dynamo is 22 years (a timescale distinct from those usually encountered on the Sun), sunspots appear within latitude bands and their numbers (in terms of monthly or yearly running means) increase and decrease over one cycle. There is however, a strong asymmetry of the Solar cycle in time, i.e. the growth phase is shorter (and dependent of the amount of activity) than the decay, or descending phase. In addition, the polar field of the Sun is observed to reverse around Solar maximum, again with a distinct asymmetry between hemispheres. Despite these global-scale features, the Solar magnetic field has many spatial components (Stenflo 1991) and the majority of the magnetic flux appears in small elements.

INTRODUCTION

The aim of this short paper is to attract attention to one feature in the Earth's fluid core. The feature is an internal shear layer induced by a relative rotation of the inner core. Large gradients of the velocity around this layer may be important for the geodynamo. Note, in particular, that in the geodynamo model-Z without an account of the inner core rotation one of the basic sources (the α-effect) is assumed to be concentrated near the core-mantle boundary (Braginsky 1993).

The inner core of the Earth can be considered as a hard iron ball of radius approximately 0.2R, where R is the Earth's radius. The rest of the planet is occupied by the outer liquid core and the rock mantle in the form of spherical shells of almost equal width, 0.4R. The other iron-rock planets (Mercury, Mars), except probably Venus, also have inner cores (Stevenson 1983). As the source of compositional convection (Loper & Roberts 1983) the inner core is apparently a necessary part of the planetary dynamo.

STATISTICAL PROPERTIES OF PLASMA TURBULENCE

Recent measurements have shown the existence of fine-scale density structures in Tokamak plasmas (Cripwell & Costley 1991). There is also experimental evidence that the anomalous (i.e. greater than collisional) particle and energy transport may be in some circumstances due to particle drift motion caused by microturbulence. These facts make the investigation of the turbulence in Tokamaks very important. To describe the phenomena, it is useful to know statistics of random magnetic and density fields.

In this paper we discuss the statistical properties of plasma which can be studied with the so-called refractometry technique probing plasma with a plane polarized electromagnetic laser or microwave beam (Gill & Magyar 1987; Weisen et al. 1988.

THEORETICAL EVOLUTION

In order to discuss the observations in a meaningful way, it is useful to first discuss the theory, because only then does one have a meaningful context in which to place the observations. One must note that this does not imply that the theory is well understood, because this is not necessarily the case. But this approach gives some insight as to what the key observations are, and by comparing the observations with the theory one can find the weakness in the theory which can be more thoroughly studied.

Review of the early evolution

The main sequence star transforms hydrogen into helium in the central regions of the star as its source of energy. This stage is usually referred to as hydrogen burning. As the hydrogen burns, the core gradually contracts and heats up, which produces an increase in the rate at which the hydrogen is burned. The increased burning rate closely offsets the diminished fuel supply, and the luminosity of the star does not change substantially. For the first 90% of its lifetime the star remains close to the main sequence, and may double its luminosity in this time. This is shown schematically in the H-R diagram, Fig. 1, as the motion from points A to B. The initial position on the main sequence is determined only by the mass of the star, as long as it consists mainly of hydrogen. Evolutionary tracks are shown for stars of 1.1 M⊙ and 5 M⊙. The former represent the ‘low’ mass stars and the latter represent ‘intermediate’ mass stars which will be presently more completely defined.

INTRODUCTION

Numerical computations comprise an increasingly important tool in the understanding of the Earth's dynamo, and, with the increased accessibility of supercomputers, direct, realistic simulations of the geodynamo are not far off. Any such simulation must solve the equations governing a three dimensional, rapidly rotating, dynamically consistent dynamo with Lorentz force J × B present in the dominant balance of forces. The simplest dynamo with these characteristics, first proposed by Childress & Soward (1972), uses the convective motions of rapidly rotating Benard convection to drive a dynamically consistent MHD dynamo. Computationally, the Childress–Soward dynamo has the advantage of permitting the expansion of the unknown fields in Fourier series in all directions, allowing three dimensional fast Fourier transforms (FFT's) to be used in calculating the nonlinear terms. Since no fast Legendre transform exists at the moment, the resulting programs will be faster than more realistic spherical dynamo simulations, while at the same time reflecting the important features of these models.

In this paper, the strong field branch of the Childress–Soward dynamo is investigated using direct numerical simulations.