In succession to our paper dedicated to Ed Spiegel, we proceed to establish a proportionality relation between the solar-cycle variation of the sky-brightness and that of the global warming. The increase of the optical depth appearing in the sky brightness may cause the solar-cycle global warming of a few degrees from the minimum to the maximum.

We wish to dedicate this paper to Douglas, in celebration of his 60th birthday anniversary.

Introduction

Solar magnetism not only controls the solar activity but also influences significantly the structure of the convection zone (Gough, 2001). On the other hand, the influence of solar activity on terrestrial meteorology such as found in tree rings, etc., has long been the subject of discussion (Eddy, 1976) but without finding the definitive causal relation explaining the physics involved. Recently, however, Sakurai (2002) analysed data of the sky background brightness observed with the Norikura coronagraph over 47 years (1951–1997) and found a clear 11.8-year periodicity as well as the marked annual variation, both exceeding the 95 per cent confidence level.

The annual variation is apparently meteorological, e.g., the famous Chinese yellow soil particles (rising up to 100 thousand feet high! – old Chinese sayings). The solar-cycle variation is also considered to be caused by increased aerosol formation (Sakurai, 2002); but if the solar activity changes the chemistry in the upper atmosphere; the observed time lag of 2 to 4 years of the sky-brightness variation relative to sunspot maximum is somewhat enigmatic.

Reduced partial differential equations valid for convection in a strong imposed magnetic field (vertical or oblique) are derived and discussed. These equations filter out fast, small-scale Alfvén waves, and are valid outside of passive horizontal boundary layers. In the regime in which the convective velocities are not strong enough to distort substantially the field, exact, fully nonlinear, single-mode solutions exist. These are determined from the reduced PDEs reformulated as a nonlinear eigenvalue problem whose solution also gives, for each Rayleigh number, the time-averaged Nusselt number and oscillation frequency together with the mean vertical temperature profile. In the oblique case a hysteretic transition between two distinct convection regimes is identified. Possible applications to sunspots are discussed.

Introduction

The study of convection in an imposed magnetic field is motivated primarily by astrophysical applications, particularly by the observed magnetic field dynamics in the solar convection zone (Hughes & Proctor 1988). Applications to sunspots (Thomas & Weiss 1992) have leds everal authors to investigate the suppression of convection by strong “vertical” or “horizontal” magnetic fields. However, the magnetic field in sunspots is neither vertical nor horizontal, and this has led to recent nonlinear investigation of convection in an oblique magnetic field (Matthews et al. 1992, Julien et al. 2000). Numerical simulations of magnetoconvection are unable to reach the parameter values, both in terms of field strengths and Reynolds number (Re), characteristic of convection in sunspots.

We review the effects of rotation on the oscillation spectrum of rapidly rotating stars. We particularly stress the novelties introduced by rotation: for instance, the disappearance of modes in the low frequency band due to the ill-posed natured of the underlying mathematical problem. This is mainly an effect of the Coriolis acceleration. The centrifugal effect changes the shape of the star in the first place. The possible consequences of this deformation on the oscillation spectrum are briefly analyzed. We also describe other possibly important effects of the centrifugal acceleration which come about on the time scale of star evolution.

A short introduction to rapidly rotating stars

All stars are affected by rotation but some of them, the rapid rotators, are more affected than the others! Astronomers usually qualify as rapid rotators all the stars with v sin i ≥ 50 kms−1, i.e. those with an equatorial velocity larger than 50kms−1. Such a value should be compared to the Keplerian limiting velocity which is

Vkep ∼ 440kms−1 (M/M⊙)0.1

for stars on the main sequence (we used the mass-radius relation given by Hansen and Kawaler 1994). Thus, for these stars the limiting velocity is weakly mass-dependent and rapid rotators appear as stars whose centrifugal acceleration exceeds 10% of the surface gravity; since this ratio measures the impact of rotation on the star structure, rapid rotators are those stars whose shape is significantly distorted by rotation.

The element settling which occurs inside stars, due to the combined effect of gravity and thermal gradient (both downwards), radiative transfer (upwards) and concentration gradients, leads to abundance variations which cannot be neglected in computations of stellar structure. This process is now generally introduced as a “standard process” in stellar evolution codes. The new difficulty is to explain why, in some cases, element settling does not proceed at all as expected. Macroscopic motions, like rotation-induced mixing, may increase the settling time scales, but then it introduces in radiative regions extra mixing with consequences which are not always observed as predicted. We have recently developed a new approach for treating rotation-induced mixing in which we include the feedback effect of the settling-induced μ-gradients (Vauclair 1999, Théado & Vauclair 2001, 2002). This effect, which was not included in previous computations, leads to first order terms in the meridional circulation velocity. It results in a mixing process, just below the convective zone, quite different from that induced by normal circulation. For the first time, we have evidence of a mixing region which is precisely confined and directly modulated by the settling itself. This will have interesting consequences for the computations of abundance variations in stars.

Introduction

Although element settling inside stars was already recognized as a fundamental process at the very beginning of the computation of stellar structure and evolution (Eddington 1926), it has long been forgotten by stellar astrophysicists.

This volume, “Stellar Astrophysical Fluid Dynamics”, arises from a meeting held 25–29 June 2001 to celebrate the sixtieth birthday earlier that year of Douglas Gough. Douglas has been and continues to be an inspiring and enthusiastic teacher and colleague to many, as well as a highly original and influential researcher in astrophysical fluid dynamics. Many colleagues and former research students (the categories are far from mutually exclusive) came together to celebrate, of course, but also for scientific discussions of the highest quality. The meeting fully lived up to its title of “New Developments in Astrophysical Fluid Dynamics”, and although the title of the present volume has been specialiseda little to emphasise the dominant stellar aspect, the full breadth of the meeting's science is retained.

The choice of venue at the Chateau de Mons, an armagnac-producing chateau in the Gers region of south-west France, was inspired and highly appropriate given Douglas's love of the region and its spirit. The food, wine and armagnac blended with the science, celebration and personal interactions to make a truly memorable week. One particular high spot occurred during a banquet after the first day of the meeting when Douglas was initiated as a Mousquetaire d'Armagnac, a brotherhood dedicated to promoting the enjoyment of armagnac throughout the World.

Stars undergo some mild mixing in their radiation zones, which is due to a thermally driven large scale circulation, and presumably to turbulence caused by shear instabilities. It is the rotation of the star which is responsible for these motions, and therefore the transport of angular momentum must be described in time and space when modeling stellar evolution. We review the present state of the problem and discuss briefly the open questions.

The observational evidence

At first sight, there seems to be no mixing at all in stellar radiation zones, since a thoroughly mixed and therefore homogeneous star would not evolve to the red giant stage. This is why such mixing is ignored in the standard modeling of stellar evolution. However there are several signs that at least some partial mixing occurs in radiative interiors, and that this may have an impact on the later phases of stellar evolution.

Let us start by reviewing briefly the observational evidence pointing to such mixing.

• Models of built by pretending that there is no mixing in the radiation zones do not agree well with the observed global properties of stars, such as their luminosity and radius (or effective temperature). This is apparent when comparing theoretical isochrones with their observed counterpart in the Hertzsprung-Russel diagram, for stars with more than about 2 solar masses. The situation improves if one allows for some extra mixing beyond the convective core.

This chapter reviews recent research on the interaction of magnetic fields with MHD turbulence, with particular application to the question of the influence of Lorentz forces on the efficiency of large-scale field generation.

Scales for solar magnetic fields

The solar magnetic field outside the radiative core exists on a great range of length and time scales; these embrace all sizes from that of the disc itself to that of the diffusion length scales of a few km, well below present observational resolution. While it is the largest scales that force themselves on our attention, due to the visibility of sunspots and associated coronal structures, and the coherence of the solar cycle, it is not clear whether these large-scale fields control, or are controlled by, the small-scale fields that have much greater total energy. While the cycle is clearly global in nature, the “magnetic carpet” of small-scale field structures that appear in quiet regions would seem to be a local manifestation of dynamo action due to turbulent stretching.

Linear dynamo theory, in particular the “mean-field” or “α-effect” models, has proved amazingly successful in predicting aspects of the solar cycle such as the butterfly diagram. In fact some of this ‘success’ has nothing to do with the physics employed, but derives from the symmetry of the underlying geometry.

Numerical experiments on three-dimensional convection in the presence of an externally imposed magnetic field reveal a range of behaviour that can be compared with that observed at the surface of the Sun (and therefore expected to be present in other similar stars). In a strongly stratified compressible layer small-scale convection gives way to a regime with flux separation as the field strength is reduced; with a weak mean field magnetic flux is concentrated into narrow lanes enclosing vigorously convecting plumes. Small-scale dynamos, generating disordered magnetic fields, have been found in Boussinesq calculations with very high magnetic Reynolds numbers; there is a gradual transition from dynamo action to magnetoconvection as the strength of the imposed field is increased.

Introduction

Thirty-seven years ago, when I was a postdoc at Culham, Roger Tayler told me that he was sending a very bright young research student to spend the summer there – and so I first met Douglas. When I moved to Cambridge a year later he was finishing his Ph.D. and then he and Rosanne went off to the States for a few years. We've been in close contact ever since they returned to Cambridge and it has been a great pleasure having Douglas as a colleague and a friend – always stimulating and often argumentative, but never causing any serious disagreement. So I am very glad to have a chance of saying ‘Thank you’ here.

As we have already been reminded, Douglas's third paper (Gough & Tayler 1966) was on magnetoconvection.

Deep convection occurs in the outer one-third of the solar interior and transports energy generated by nuclear reactions to the surface. It leads to a characteristic pattern of time-averaged differential rotation, with the poles rotating approximately 20% slower than the equator. A particularly notable feature of the solar differential rotation is that it departs significantly from the Taylor-Proudman state of rotation constant on cylinders aligned with the rotation axis. Although this observation contrasts with results from early numerical simulations, such simulations provide the best hope of understanding the observations. Many studies have adopted the DNS (Direct Numerical Simulation) approach and justified the artificially large viscosities and thermal diffusivities used as modelling transport by unresolved eddies. LES (Large Eddy Simulation) techniques, which use a suitable turbulence closure model, offer a superior alternative but face the problem of choosing an appropriate turbulence closure; this can be difficult in the face of complicating factors such as stratification and rotation. An alternative approach is to shift responsibility for truncating the turbulent cascade to the numerical scheme itself. Since this approach abandons the rigorous notions of the LES approach, we refer to it as a VLES (Very Large Eddy Simulation). This paper compares results of DNS simulations carried out with a spherical harmonic code, and preliminary results obtained using a VLES-type code. Both make the anelastic approximation.

The tachocline has values of the stratification or buoyancy frequency N two or more orders of magnitude greater than the Coriolis frequency. In this and other respects it is very like the Earth's atmosphere, viewed globally, except that the Earth's solid surface is replaced by an abrupt, magnetically-constrained ‘tachopause’ (Gough & McIntyre 1998). The tachocline is helium-poor through fast ventilation from above, down to the tachopause, on timescales of only a few million years. The corresponding sound-speed anomaly fits helioseismic data with a tachocline thickness (0.019 ± 0.001) R⊙, about 0.13 × 105 km (Elliott & Gough 1999), implying large values of the gradient Richardson number such that stratification dominates vertical shear even more strongly than in the Earth's stratosphere, as earlier postulated by Spiegel & Zahn (1992). Therefore the tachocline ventilation circulation cannot be driven by vertically-transmitted frictional torques, any more than the ozone-transporting circulation and differential rotation of the Earth's stratosphere can thus be driven. Rather, the tachocline circulation must be driven mainly by the Reynolds and Maxwell stresses interior to the convection zone, through a gyroscopic pumping action and the downward-burrowing response to it. If layerwise-two-dimensional turbulence is important, then because of its potential-vorticity-transporting properties the effect will be anti-frictional rather than eddy-viscosity-like. In order to correctly predict the differential rotation of the Sun's convection zone, even qualitatively, a convection-zone model must be fully coupled to a tachocline model.

The sun is a magnetic star whose variable activity has a profound effect on our technological society. The high speed solar wind and its energetic particles, mass ejections and flares that affect the solar-terrestrial interaction all stem from the variability of the underlying solar magnetic fields. We are in an era of fundamental discovery about the overall dynamics of the solar interior and its ability to generate magnetic fields through dynamo action. This has come about partly through guidance and challenges to theory from helioseismology as we now observationally probe the interior of this star. It also rests on our increasing ability to conduct simulations of the crucial solar turbulent processes using the latest generation of supercomputers.

Introduction

The intensely turbulent convection zone of the sun, occupying the outer 30% by radius or 200Mm in depth, exhibits some remarkable dynamical properties that have largely defied theoretical explanation. The most central issues concern the difierential rotation with radius and latitude that is established by the convection redistributing angular momentum, and the manner in which the sun achieves its 22-year cycles of magnetic activity. These dynamical issues are closely linked: the global dynamo action is most likely very sensitive to the angular velocity Ω profiles realized within the sun. It is striking that the underlying solar turbulence can be both highly intermittent and chaotic on the smaller spatial and temporal scales, and yet achieve a large-scale order that is robust in character (e.g. Brummell, Cattaneo & Toomre 1995).

Introduction

When the phrase solar–terrestrial activity is used, the intent is to describe those changes of energetic particles and electromagnetic fields that originate at the Sun, travel to the Earth's magnetosphere, and have drastic effects upon the Earth's atmosphere and geomagnetic field. The activity is on time scales that are short in the human perception of events. The Sun is said to be “active” when the magnitude of such changes is distinguishably large with respect to the average behavior over tens of years. A specific region or process on the Sun is said to be an active source region when a particle or field disturbance in the Earth's magnetosphere can be traced to some special change in that region of the Sun. The vagueness in these definitions should disappear as we become more specific in the description of such phenomena as sunspots, flares, coronal holes, coronal mass ejections, solar wind, geomagnetic storms, ionospheric disturbances, auroras, and substorm processes.

We call the moving plasma of ionized particles and associated magnetic fields that are expanding outward from the Sun the solar wind. Its associated field is the interplanetary magnetic field (IMF). The wind exists out past 150 times the Sun–Earth distance because the pressure of the interstellar medium is insufficient to confine the energetic particles coming from the hot solar corona. We call this solar-wind dominated region the heliosphere.

Outer space is filled with particles and fields originating from the formation of the universe and from stars.

International Unions and Programs

The following quotation was taken, with permission, from The National Geomagnetic Initiative copyright 1993 by the National Academy of Sciences, courtesy of the National Academy Press, Washington, D.C. Revisions of this quoted material have been provided by J.H. Allen in order to modernize the statement to year 2002.

The study of the Earth is intrinsically global. This was recognized by geologists, geodeticists, and geophysicists in the nineteenth century. During the past hundred years, the need for global collaboration in geosciences has become axiomatic; many mechanisms have been developed to encourage international cooperation in Earth sciences. Much international cooperation in science takes place under the non-governmental International Council for Science (ICSU).

By the latter part of the nineteenth century, international expeditions and exchange of datawere common in the geosciences. This led to the development of international mechanisms for ongoing cooperation in geophysical and geological sciences. Seismic and magnetic observatories were being established worldwide. These de facto global networks of magnetic and seismic observatories led to international agreements on measurement standards and data exchange. These international activities led to the formation of an international organization that was the predecessor to the modern International Union of Geodesy and Geophysics (IUGG). The objectives of IUGG are the promotion and coordination of physical, chemical, and mathematical studies of the Earth and geospace environment. IUGG now consists of seven essentially autonomous associations: one of these, the International Association of Geomagnetism and Aeronomy (IAGA), is principally concerned with geomagnetism.

This second edition of Introduction to Geomagnetic Fields has been redesigned as a classroom textbook for a semester course in geomagnetism. Student exercises have been added at the end of each chapter. Outdated figures and tables are replaced with more modern equivalents. Recent discoveries, field information, and references have been added along with special websites and computer programs. The basic structure of the original edition remains, providing a condensed and more readable coverage of geomagnetic topics than is afforded by existing textbooks.

My intention has been to focus upon the basic concepts and physical processes necessary for understanding the Earth's natural magnetic fields. When mathematical presentation is required, I have tried to remove the mystery of the scientists' special jargon and to emphasize the meanings of important equations, rather than obscure the relationships with complex formulas. Because some formulas are needed to appreciate geomagnetism, I have included, in an appendix, a succinct review of the required mathematical definitions and facts. For those readers who are approaching the subject of Earth magnetic fields for the very first time it may be helpful to start with the small layman's presentation, devoid of all mathematical equations, that I provided as Earth Magnetism: A Guided Tour Through Magnetic Fields, Academic Press, San Diego, 151 pp, 2001.

The student reader is expected to have a familiarity with the elementary scientific concepts identified by words of specific meaning, such as “force, velocity, energy, temperature, heat, charge, light waves, and fields of electric, magnetic, and gravitational nature”.