INTRODUCTION

Starting with the paper of Parker (1955), one of the main purposes of scientists working in Solar physics was directed towards the understanding of the mechanisms governing the generation and the maintenance of the magnetic field that plays a central role in the Solar activity process. Many characteristic features, for example the well-known ‘butterfly’ diagram, are well described by kinematic dynamo theories. Great efforts are now directed towards the derivation of dynamo models that could agree with recent helioseismological data, and there is accumulating evidence that the seat of the dynamo is in the overshoot layer at the base of the convection zone (see for example, De Luca & Gilman 1991).

However, even the most sophisticated theories based mainly on the magnetic field are not able to explain topological aspects of Solar activity structures like active longitudes.

The screw dynamo is one of the simplest examples of a conducting fluid flow in which magnetic field can be self-excited provided the magnetic Reynolds number is sufficiently large (see, e.g., Roberts 1993). Such a flow can be encountered in some astrophysical objects and also in such technological devices as breeder reactors. For example, jet outflows in active galaxies and near young stars can be swirling. A flow of this type is used for modelling the dynamo effects in laboratory conditions (Gailitis 1993). The generation of magnetic fields by a laminar flow with helical streamlines was discussed by Lortz (1968), Ponomarenko (1973), Gailitis & Freiberg (1976), Gilbert (1988), Ruzmaikin et al. (1988) and other authors as an eigenvalue problem. Below we use the results of the asymptotic analysis of this problem for large Rm by Ruzmaikin et al.

We introduce an axisymmetric velocity field whose cylindrical polar components are (0, rω(r), v2(r)), with (r, φ, z) the cylindrical coordinates. We

consider smooth functions v2(r) and ω(r) vanishing as r → ∞. Both v2 (0) and ω(0) are assumed to be of order unity.

For Rm ≫ 1, an eigenmode of the screw dynamo represents a dynamo wave concentrated in a cylindrical shell of thickness ≃Rm−1/4 a certain radius r0.

INTRODUCTION

Stretching of line elements in fluids with or without conductivity is studied from various points of view. Firstly, simultaneous concentration of vortex and magnetic tubes is considered in section 2 by presenting an exact solution of the axisymmetric MHD equation for a viscous, incompressible, conducting fluid. This solution (Kambe 1985) tends to a stationary state that results from complete balance of convection, diffusion and stretching. Sections 3 and 4 are concerned with mathematical formulation based on Riemannian differential geometry, and global (mean) stretching of line elements is considered.

The general form of the Riemannian curvature tensor for any Lie group was derived by Arnold (1966), where explicit formulae for T2 (two-torus) were given. Explicit expressions for diffeomorphism curvatures on Tn (and even on any locally flat manifold) were described by Lukatskii (1981). Recently, Nakamura et al. (1992) considered the curvature form on T3 corresponding to three-dimensional motion of an ideal fluid with periodic boundary conditions in a cubic space. An immediate consequence is the property that the section curvature in the space of ABC diffeomorphisms is a negative constant for all the sections (Kambe et al. 1992).

INTRODUCTION

Chaotic dynamical systems can be grouped into two classes according to the characteristics of their behavior. One class is characterized by aperiodic modulations of already periodic signals while the other class is characterized by signals which exhibit apparently random switching between qualitatively different kinds of behavior. The latter behavior is called intermittency. Examples of intermittency are abundant in nature. They include intermittent bursts of turbulence in otherwise laminar pipe flow in fluid dynamics, sunspot activity in astrophysics, and stock market crashes in economics. A model of intermittency in terms of dynamical systems as well as a partial classification of some types of intermittency was given by Pomeau & Manneville (1980). In general, signals produced by this scenario are periodic oscillations interrupted from time to time by some aperiodic bursts of activity. Another model of intermittency, crisis-induced intermittency, was introduced by Grebogi, Ott, Romeiras & Yorke (1987). This intermittency involves a collision in phase- space of two chaotic attractors as some parameter is varied, and it is again characterized by random switching between different aperiodic oscillations.

ABSTRACT

We present infrared images and infrared spectroscopy of the suspected very young supernova remnant G25.5+0.2 first detected at radio wavelengths. The 2.2 μm image exhibits a similar nebular structure to that seen in the radio. Spectroscopic measurements at 2.17 μm show a Brackett gamma line that has the line-to-continuum ratio expected from ionized hydrogen at a temperature near 10,000 K. At 50 km/s resolution, the line is resolved with a FWHM of 200 km/s. Ten micron photometry clearly establishes a connection between G25.5+0.2 and the IRAS source 18344–0632. Most important, the infrared image reveals a point source in the center of the nebula that has properties of a blue luminous star that could excite an ionized ring nebula with the observed radio properties. The supernova hypothesis is ruled out and G25.5+0.2 is almost certainly a ring nebula around a mass losing luminous blue star ˜13 kpc distant and reddened by 20 magnitudes of visual extinction.

INTRODUCTION

Cowan et al. (1989) reported extensive observations of the galactic radio source G25.5+0.2. These included radio continuum images at four frequencies, 21 cm absorption measurements, and a search for radio recombination line emission. Based primarily on the absence of the H76α recombination line, Cowan et al. concluded that G25.5+0.2 is a very young galactic supernova remnant, perhaps only 25 years old. Based on 21 cm HI absorption lines, Cowan et al. give a minimum distance to G25.5+0.2 of 7.2 kpc. White and Becker (1990) and Green (1990) suggest that identification of G25.5+0.2 as a planetary nebula is more likely, based on the radio observations and IRAS fluxes.

INTRODUCTION

The nature of turbulent magnetic diffusion and the α-effect has been a puzzle for several decades. Until recently, virtually all the work in this subject has been based on analytical theory, but the advent of ready access to supercomputers now allows us to address the question of turbulent magnetic diffusion and the turbulent α-effect from the perspective of numerical experiments. In this paper, we shall describe numerical simulations of an idealized model of mean field dynamos.

In order to understand how fields are generated, the mean field theoretical approach is widely used (see Moffatt 1978). This two-scaled approach conveniently parametrizes the effects of small scale turbulence on large scale fields into two coefficients, α and β. The central problem of mean field electrodynamics is to calculate these transport coefficients from the statistical properties of the flow and the magnetic diffusivity, η. Explicit in these calculations is that the fluid flow is not affected by the presence of magnetic fields.

In typical magnetofluid circumstances, this is assumed to be the case, unless the magnetic energy of the large scale component is comparable to the energy in the flow. Recent two-dimensional simulations suggest that this is not the case and that turbulent diffusivity can be severely suppressed even when the mean field is less than the equipartition field value (see Cattaneo & Vainshtein 1991).

INTRODUCTION

An interesting asymptotic limit in the theory of dynamical systems enforces a highly-developed chaotic structure by the assumption of large-amplitude particle excursion in flows and maps. An example of such a method applied to diffusion of a scalar is given by Rechester & White (1980). This important idea has been developed by Soward (1992) in the context of fast dynamo theory and in particular for the case of pulsed helical waves. Our purpose in this note is to apply the large-amplitude method of Soward (1992), to the simpler SFS map (Bayly & Childress 1987, 1988). In the SFS (stretchfold- shear) map, a simple baker's map in the xy-plane is supplemented by a lateral shear in the z-direction. Numerical calculations indicate that, when the map operates in a perfectly conducting fluid on a magnetic field of the form (B(y)eikz, 0, 0), the average of the field over planes z=constant can be made to grow exponentially for sufficiently large shear. This property of ‘perfect’ fast dynamo action has never been proved in the SFS problem, however, despite the existence of an especially simple adjoint eigenvalue problem, where the growing eigenfunctions, if they exist, are known to be smooth (Bayly & Childress 1988). Moreover, numerical studies show clearly the existence of these eigenfunctions for the perfect fast dynamo problem.

The fourth “Canary Islands Winter School of Astrophysics” was held in Playa de las Américas (Adeje, Tenerife) from the 7th to 18th December 1992, organized by the Instituto de Astrofísica de Canarias and the Universidad Internacional Menéndez-Pelayo. A total of 85 participants from 21 countries world-wide attended the meeting.

This volume contains a series of nine courses which were delivered during the School. The aim of the lectures was to portray a thorough, up-dated view on the field of Infrared Astronomy.

The last School dedicated to Infrared Astronomy before this one (the legendary ISA Course held in Erice, Italy) took place in 1977, fifteen years ago. Since then, dramatic changes in our understanding of the Infrared Universe -pushed forward by the corresponding advances in telescopes, instruments and detector capabilities- have strongly influenced all branches of Astrophysics. One of the primary goals at the present School was to put the new generation of astrophysicists into contact with the extremely important new findings which have surfaced from recent infrared research, and to present and discuss the fundamental physical ideas emerging from those results.

We wish to express our gratitude to the Commission of the European Community (Human Capital and Mobility Programme -Euroconferences), the Dirección General de Investigación Científica y Técnica (Spanish Ministry of Education and Science) and the Government of the Canary Islands for helping us to fund the School. They, together with HOTESA and their “Hotel Gran Tinerfe”, made it possible to allocate grants to over 75% of attendees.

INTRODUCTION

The mathematical difficulties in deriving solutions for growing magnetic fields in spherical geometries have long puzzled dynamo theoreticians. In contrast to the solutions of the kinematic dynamo problem found by Roberts (1970, 1972) and others in the case of periodic velocity fields in infinitely extended electrically conducting fluids, dynamo action often seems to disappear as soon as insulating boundaries are introduced. Motivated by this observation Bullard & Gubbins (1977) have investigated kinematic dynamos in a spherical domain of constant conductivity with insulating exterior for velocity fields with alternating signs as a function of radius. As expected the critical magnetic Reynolds number decreases significantly as the number of sign changes increases and the limit of a periodic velocity field is approached. When the radial velocity component does not change sign, a dynamo solution was not obtained unless an outer shellular region of finite conductivity was introduced.

INTRODUCTION

It is widely though erroneously believed that one can see the Milky Way Galaxy. In fact, one's image of the Milky Way depends more on how one looks at it than on what is available to be seen. For reasons which are related to population biology more than to astrophysics, our eyes are optimised to detect the peak energy output from thermal sources with a surface temperature near 6000K. Thus, unless such an object is typical of the entire contents of the Galaxy, there is no reason why we should be able to see by eye a representative part of whatever may be out there. If we had X-ray or UV sensitive eyes we would ‘see’ only hotter objects, if infrared or microwave eyes only cooler objects.

No single section of the electro-magnetic spectrum provides the ‘best’ view of the Galaxy. Rather, all views are complementary. However, some views are certainly more representative than are others. The most fundamental must be a view of the entire contents of the Galaxy. Such a view would require access to a universal property of matter, which was independent of the state of that matter. This is provided by gravity, since all matter, by definition, has mass. Mass generates the gravitational potential, which in turns defines the size and the shape of the Galaxy. While the most reliable and comprehensive, such a view is also the hardest to derive. Nonetheless, we will repeatedly return to the gravitational picture of the Galaxy in these lectures.

INTRODUCTION

The solution of model Z has been found in many cases with account taken of both viscous and electromagnetic core-mantle coupling (Braginsky 1978; Braginsky & Roberts 1987; Braginsky 1988; Braginsky 1989; Cupal & Hejda 1989). Apart from Braginsky (1989), the time evolution of the solution was used simply as an aid to obtain the steady-state solution. Cupal & Hejda (1992) found numerically a transient solution of model Z having the form of a decaying oscillation. The accuracy of such solutions depends on the numerical method used, on the density of space and time discretization, and for that matter, on the character of the solution itself. An important question is which characteristics of the time behaviour of the solution reflect the real (physical) behaviour of the system and which follow from the limitations of the numerical method.

MOTIVATION

In the absence, so far, of fully hydrodynamic dynamo models representative of the Earth or the planets, the main focus of planetary dynamo theory remains with (axisymmetric) mean-field dynamo models in which the contribution from the nonlinear interaction of the non-axisymmetric components of the problem are parameterized through a prescribed α-effect (see for example Roberts 1993).

MOTIVATION

The interaction between convection and magnetic fields plays a central role in the theory of stellar dynamos. In order to investigate this interaction in detail, we consider a simplified problem: two-dimensional convection in a vertical magnetic field. To represent the astrophysical situation, in which there are no sidewalls, we consider a box with periodic boundary conditions in the horizontal direction, allowing horizontal flows. It is found that convection can be unstable to a horizontal shearing motion.

INTRODUCTION

The study of passive scalar advection provides fundamental understanding of various phenomena such as convection and mixing that are ubiquitous in nature. In particular, the probability distribution of amplitude and its spatial gradients are of vital importance in relation to recent active studies of non-Gaussian probability density functions (PDFs) endemic in turbulence.

Since Castaing et al. (1989) reported exponential-like tails on the PDF of temperature fluctuations in thermal convection at very high Rayleigh numbers, there has been increasing interest in the mechanism of the non-Gaussian tails on PDFs of amplitudes. In a recent paper, Pumir, Shraiman & Siggia (1991) have suggested that the non-Gaussian tails for an advected passive temperature field may be induced by the presence of a mean-temperature profile. A simple physical mechanism for this is proposed in the present paper. The resultant non-Gaussian statistics will be shown by numerical simulations and theoretical analysis for a transport equation without molecular diffusion. In this paper, the result on PDFs is summarized; other details will be presented elsewhere (Kimura & Kraichnan 1993).

This volume contains papers contributed to 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 ‘Lectures on Solar and Planetary Dynamos’, containing the texts of the invited lectures presented at the meeting, will appear almost contemporaneously. 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 papers included have all been refereed as though for publication in a scientific journal, and the Editors are most grateful to the referees for helping to get all the papers ready in such a short time. They also wish on behalf of the Scientific Organising Committee to record their appreciation of the dedication of the staff of the Isaac Newton Institute, who coped cheerfully with many bureaucratic complexities, and to give special thanks to the Deputy Director, Peter Goddard, for making the whole meeting possible.

INTRODUCTION

Given the existence of a naturally occurring magnetic field, be it astrophysical or geophysical, it is natural to ask whether the field is generated by dynamo action or if instead it is a fossil field, trapped in the body since its formation. In certain contexts it is possible to give a definitive answer. For example, the Ohmic diffusion time of the Earth's core is of the order of 10 years whereas paleomagnetic records show that the magnetic field of the Earth has existed for 109 years. Consequently, since the field has been maintained for so many Ohmic decay times it must be generated by some sort of dynamo process. For astrophysical bodies on the other hand, for which typically the Ohmic time is comparable to the lifetime of the body itself, it is not so straightforward to assert that a field is dynamo-generated. Of course, there may be other factors suggesting the origin of the field, but simply on the basis of the Ohmic decay time the issue often cannot be decided. What we would like therefore is a test to distinguish between these two possibilities.