Significant advances in our understanding of the geodynamo have been made over the last ten years. In this review, we consider the extent to which this knowledge can be used to understand the origin of the magnetic fields in other planets. Since there is much less observational data available, this requires a ‘first principles’ understanding of the physics of convection driven dynamos.

Introduction

The basic structure of the interior of the Earth has been worked out by seismologists. The iron core is divided at ricb = 1220 km, the inner core boundary (ICB), into the solid, mainly iron, inner core below and the fluid outer core above. The exact composition of the outer core is not known, but the most plausible models suggest it is a mixture of liquid iron and various impurities, probably sulphur and oxygen (Alfè et al., 2000). The whole core is electrically conducting. Above the core-mantle boundary (CMB), at rcmb = 3485 km, lies the rocky mantle. The electrical conductivity of the mantle is very small, except possibly very close to the CMB itself, where iron may have leaked into the mantle. The basic structure of the other terrestrial planets, in which we include the larger satellites, is believed to be similar to that of the Earth, but the size of the iron core varies considerably, and the division between the fluid outer core and the solid inner core, if it exists, has to be computed from theoretical models.

Rapidly oscillating Ap stars have proved to be extremely interesting objects, for they combine in a unique way different physical properties, like stellar magnetism and abnormal chemical abundances, with important physical phenomena, like acoustic oscillations. In this paper we will discuss how the indirect effect of the magnetic field and the presence of chemical peculiarities may influence different aspects of the pulsations in roAp stars and will try to discuss their implications to our understanding of the latter.

Introduction

Rapidly oscillating Ap stars (hereafter roAp stars) have now been known for a couple of decades (Kurtz 1982). They are located in the main-sequence part of the classical instability strip, close to the δ Scuti stars, but unlike the latter, roAp stars are small-period pulsators, oscillating with periods that vary typically from 5 to 15 minutes. They are found among the coolest subgroup of classical Ap stars, and, thus, not only are they chemically peculiar, but also they have strong large scale magnetic fields, with typical intensities of a few kG. This combination of properties makes roAp stars extremely interesting targets for asteroseismology. Moreover, the oscillations they exhibit are interpreted as high-order, low-degree modes, opening the possibility of applying asymptotic techniques.

The magnetic fields present in roAp stars influence the oscillations both directly and indirectly.

3He transport in the solar core has been suggested as a solution to the solar neutrino problem. I investigate the consequences of imposing a flow on the solar core and show that it is unlikely that a flow could exist that would reproduce the best-fit astrophysical solution to the experimental neutrino fluxes from Homestake, SAGE, GALLEX and SuperKamiokande.

Introduction

Before the announcement of the results from the Sudbury Neutrino Observatory (Ahmad et al. 2001), the measurements of the fluxes of neutrinos coming from nuclear reactions in the core of the sun were inconsistent with solar models. It has been argued that a so-called standard solar model can never be consistent with the experimental fluxes, and this has been used as an argument for the necessity of flavour transitions. However, non-standard solar models where 3He is burnt out of equilibrium have been suggested as astrophysical solutions to the neutrino problem (e.g. Dilke & Gough, 1972 and Gough, 1991), and Cumming & Haxton (1996) showed how a solar model with a redistribution of 3He in the core could overcome the problems of a standard solar model.

It was argued (Bahcall et al., 1997), using simple one-dimensional models with a mixed core, that Cumming and Haxton's model was inconsistent with helioseismology. As helioseismology had measured only a horizontal average of quantities in the solar interior, and as the suggested mechanism is fundamentally at least two-dimensional, the mechanism cannot be ruled out until a more realistic, two-dimensional model has been produced.

For many years the principle source of excitation of oscillations in solar-like stars was under considerable debate. This was related to the fact that mode stability in such stars is governed not only by the perturbations in the radiative fluxes (via the κ-mechanism) but also by the perturbations in the turbulent fluxes (heat and momentum). The study of mode stability therefore demands a theory for convection that includes the interaction of the turbulent velocity field with the pulsation. It is now widely believed that the observed low-amplitude oscillations in the Sun are determined by the balance between the stochastic driving due to the acoustic radiation by turbulent convection and the damping of the intrinsically stable p modes. Acoustic radiation by turbulence also effects the stratification of the equilibrium model by reducing the estimated convective velocities and consequently influencing oscillation properties. In this contribution I review the mechanisms responsible for mode damping in solar-type stars and for stochastic driving by turbulent convection. Amplitude predictions for models of the Sun and β Hydri are compared with observations. Finally I discuss the effect of acoustic radiation by turbulence on the retardation of the convective velocities in solar-type stars with masses 1.0 – 1.9M⊙ and on mode stability in a Delta Scuti star of 1.65M⊙.

Helioseismology has become a very successful diagnosis of the equation of state of the plasma of the solar interior. Although the gas in the solar interior is only weakly coupled and weakly degenerate, the great observational accuracy of the helioseismological measurements puts strong constraints on the nonideal part of the equation of state. The helioseismic verification of major nonideal effects in the equation of state of solar matter has become well established. The dominant contribution is the Coulomb pressure, conventionally described in the Debye-Hückel approximation. However, in the last years, the increased precision of the helioseismic diagnosis has brought significant observational progress beyond the Debye-Hückel approximation. The helioseismic detection of a signature of relativistic electrons was a striking example. Very recently, effects of the excited states of the atoms and ions of heavy elements were discovered, which have a promising potential both for statistical mechanics and solar physics, in particular, the helioseismic determination of the heavy-element abundance.

Introduction

Precise measurements of solar oscillation frequencies provide data for accurate inversions for the sound speed in the solar interior. Except in the very outer layers, the stratification of the convection zone is almost adiabatic and the Reynolds stresses are negligible. The sound-speed profile is governed principally by the specific entropy, the chemical composition and the equation of state, and it is therefore essentially independent of the uncertainties in the radiative opacities.

The description of a stellar system as a continuous fluid represents a convenient first approximation to stellar dynamics, and its derivation from the kinetic theory is standard. The challenge lies in providing adequate closure approximations for the higher-order moments of the phase-space density function that appear in the fluid dynamical equations. Such closure approximations may be found using representations of the phase-space density as embodied in the kinetic theory. In the classic approach of Chapman and Enskog, one is led to the Navier–Stokes equations, which are known to be inaccurate when the mean free paths of particles are long, as they are in many stellar systems. To improve on the fluid description, we derive here a modified closure relation using a Fokker–Planck collision operator. To illustrate the nature of our approximation, we apply it to the study of gravitational instability. The instability proceeds in a qualitative manner as given by the Navier–Stokes equations but, in our description, the damped modes are considerably closer to marginality, especially at small scales.

A kinetic equation

If we have a system of N stars, with N very large, and wish to study its large-scale dynamics, we have to choose the level of detail we can profitably treat. Even if we could know the positions and velocities of all N stars for all times, we would be mainly interested in the global properties that are implied by this information.

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.