Introduction

Although sometimes ignored, there is no doubt that hydrodynamical processes play a central role in virtually all areas of astrophysics. If they are neglected in the analyses of observations and the modelling, the results for any object must become questionable; the same is therefore true of the understanding of basic astrophysical phenomena and processes that result from such investigations.

Investigations of astrophysical fluid dynamics are hampered by both theoretical and observational problems. On the theoretical side it is evident that the systems being studied are so complex that realistic analytical investigations are not possible. Furthermore, the range of scale, extending in the case of stars from the stellar radius to scales of order 100m or less, entirely prevents a complete numerical solution. Observationally, the difficulty is to find data that are sensitive to the relevant processes, without being overwhelmed by other, similarly uncertain, effects. Progress in this field therefore requires a combination of physical intuition combined with analysis of simple model systems, possibly also experiments analogous to astrophysical systems, detailed numerical simulations to the extent that they are feasible, together with a judicious choice of observations and development and application of analysis techniques that can isolate the relevant features. Douglas Gough has excelled in all these areas.

In this brief introduction we make no pretense of reviewing the whole vast field of hydrodynamical processes in astrophysics, or even in stars.

The discovery of extrasolar planets and the determination of their orbital properties have provided golden opportunities for new advancements in the quest to understand the origin and evolution of planets and planetary systems. While their bizarre variety presents a challenge for the existing theories, their ubiquity suggests that planetary formation is a robust process. Combining data obtained from solar system exploration, star formation studies and the searches for extra solar planets, we address some outstanding issues concerning critical processes of grain condensation, planetesimal coagulation, and gas accretion. Some implications of these investigations are: 1) the amount of heavy elements available for planetary formation in protostellar disks is retained at a similar level as that empirically inferred for the primordial solar nebula, through self regulated processes and 2) the critical stages of planet formation, from grain condensation, planetesimal coagulation, to gas accretion, proceed on the timescale of a few million years.

Observations

Ongoing searches of extra solar planets (ESPs) have led to their discovery around ten per cent of the solar-type stars on various target lists (Marcy & Butler 1998). The dynamical properties of many ESPs are very different from those of planets in the solar system. The first ESP discovered, while having a mass (Mp) similar to that of Jupiter (MJ), is located 100 times closer to its host star 51 Peg than Jupiter is to the Sun (Mayor & Queloz 1995). The period (P) distribution of ESPs has a noticeable concentration between 3–7 days.

Pulsation is a common phenomenon in stars. It occurs in a wide range of their masses and in all evolutionary phases, exhibiting large variety of forms. Stochastic driving and just two distinct instability mechanisms are the cause of the widespread phenomenon. The diversity of pulsation properties in stars across the H-R diagram is partially explained in terms of differences in the ranges of unstable modes and in terms nonlinear mechanisms of amplitude limitation. Still a great deal remains to be explained.

Introduction

Excitation of the fundamental radial mode was the essence of the pulsation hypothesis when it was first proposed by Ritter in 1879, as an explanation of periodic variability in stars. Radial symmetry of the motion was confirmed for a number of objects by means of observational tests. Excitation of the same, presumably fundamental, mode in all δ Cephei type stars got support in the discovery of the period-luminosity relation, which at some point seemed unique. Soon, the hypothesis that only the fundamental radial mode may be excited became a dogma like the earlier one that stars do not vary.

Referring to Schwarzschild's (1942) suggestion that RRc stars might be first overtone pulsators, Rosseland (1949) wrote: This hypothesis involves the very difficult problem of how to excite a higher mode to pulsation while leaving the fundamental mode unexcited.

Douglas Gough & Michael McIntyre proposed, in 1998, the first global and self-consistent model of the solar tachocline. Their model is however far more complex than analytical methods can deal with. In order to validate their work and show how well it can indeed represent the tachocline dynamics, I report on progress in the construction of a fully nonlinear numerical model of the tachocline based on their idea. Two separate and complementary approaches of this study are presented: the study of shear propagation into a rotating stratified radiative zone, and the study of the nonlinear interaction between shear and large-scale magnetic fields in an incompressible, rotating sphere. The combination of these two approaches provides good insight into the dynamics of the tachocline.

Introduction

The tachocline was discovered in 1989 by Brown et al.; it is a thin shear layer located at the interface of the uniformly rotating radiative zone and differentially rotating convective zone of the sun. Several issues about these observations remain unclear. Why is the radiative zone rotating uniformly despite the latitudinal shear imposed by the convection zone, and why is the tachocline so thin? How can the tachocline operate the dynamical transition between the magnetically spun-down convection zone and the interior? The first model of the tachocline was presented by Spiegel & Zahn (1992).

There has been a long-standing discrepancy between the number of neutrinos expected from the sun and the number we actually detect. One possible interpretation for this was that our theoretical solar model was wrong. However, recent progress of helioseismology has shown that the real sun is very close to the latest solar models. On the other hand, very recent experiments of neutrino detection provided us evidence for neutrino oscillation. I discuss what we should do and what we can do in this situation for the neutrino physics from the astrophysical side.

Historical review: the solar neutrino problem

The energy source of sunshine (and shining of stars in general) is now thought to be nuclear fusion. To get direct evidence that nuclear reactions are really occurring in the sun is, however, a very challenging task. It takes ∼ 104 years for photons generated by nuclear fusion near the solar centre to reach the solar surface, because the photons interact so frequently with matter in the sun. Hence, the photons by which we can see the sun right now do not tell us the physical state of the present solar core. On the other hand, since neutrinos interact little with matter, unlike photons, and travel at the speed of light, the neutrinos generated by nuclear reactions in the sun reach the earth only eight minutes after they are generated.

The last decade has seen an impressive improvement in the quality and quantity of helioseismic data. While much of the progress has come from a new generation of instruments, such as GONG and MDI, data analysis has also played a major role. In this review I will start with a brief discussion of how the basic analysis of helioseismic data is done. I will then discuss some of the data analysis problems, their influence on our inferences about the Sun and speculate on what improvements may be expected in the near future. Finally I will show a selection of recent results.

Introduction

Until recently most research in helioseismology has used modes in the low (l ≤ 3) and medium (3 < l ≤ 200) degree (l) ranges. Here I will concentrate on the methods and problems in the study of medium-degree modes as well as show selected results. Most studies of modes of high degree (l > 200) have used entirely different analysis methods, such as time-distance analysis, which is discussed elsewhere in this volume (Kosovichev 2003). However, I will touch on some of the issues regarding the analysis of the high-degree modes by methods similar to those used for the medium-degree modes. The reader is also referred to Haber et al. (2002) for results from a technique known as ring diagrams which also uses high-degree modes.

I will start by providing some background material on solar oscillations in Section 17.2.

In this contribution I discuss how recent advances in numerical techniques and computational power can be applied to problems in astrophysical fluid mechanics. As a case in point some results of simulations of radio relics are presented which have provided strong support for a model that explains the origin of these peculiar objects. Radio relics are extended radio sources which do not appear to be associated with any radio galaxy. Here a model is presented which explains the origin of these relics in terms of old plasma that has been compressed by a shock wave. Having taken into account synchrotron, inverse Compton and adiabatic energy losses and gains, the relativistic electron population was evolved in time and synthetic radio maps were made which reproduce the observations remarkably well. Finally, some other examples are discussed where hydrodynamical simulations have proven very useful for astrophysical problems.

Introduction

With the advent of powerful computers and more accurate algorithms, simulations of astrophysical fluids have become increasingly useful. Most fields of astrophysics, such as solar physics, star formation, stellar evolution and cosmology have benefitted greatly from hydrodynamical simulations and hopes for further advances are high.

Essentially, there are two main approaches to the numerical solution of the equations of hydrodynamics: Finite-grid simulations and Smoothed Particle Hydrodynamics (SPH). In the former approach the equations are discretised on a computational mesh before they are solved. The latter method avoids the notion of a mesh and employs particles to track the fluid.

Oscillations and waves in the quiet and active solar atmosphere constitute a zoo of distinct and overlapping phenomena: internetwork oscillations, K-grains, running penumbral waves, umbral oscillations, umbral flashes etc. The distinctive oscillation spectra associated with the network, the internetwork, and sunspots and pores are a strong indicator that the magnetic field has a significant dynamical effect on wave motions. This immediately raises two questions i) Can waves be used as diagnostic indicators of the magnetic field? and ii) Do the different properties of wave motions in various field geometries have consequences for the efficiency of wave-heating in the atmosphere and corona? I will discuss some new numerical calculations of wave propagation in a variety of model atmospheres, which throw some light on these questions.

Introduction

The field of helioseismology has shown how waves which propagate through the deep solar interior can be used to determine the internal properties of the Sun – including its stratification, differential rotation, and sub-surface flow fields. Given the wide variety of waves and oscillations observed in the atmosphere of the Sun, in both Quiet and Active Regions, it is natural to ask whether the structures of these regions can also be determined from a wave analysis.

However, a brief consideration of the problem indicates that there are a number of critical differences between the atmospheric-wave problem and the p-mode problem which make the former vastly more difficult to study.

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.