The interstellar medium (ISM), which had formerly been better known for the allegoric shapes of its nebulae and dark clouds than for its physical properties, has developed into a fascinating area of astrophyscical research during the past few decades. An important aspect is the turbulent flows observed in many regions of the ISM, with velocities that, at least in the cooler parts, far exceed the speed of sound. These flows seem to play a decisive role in the cloud dynamics, slowing down gravitational contraction and star formation. In contrast to the objects studied in the two previous chapters, which had well-defined physical and geometrical properties and thus allowed a detailed analysis, the ISM is a rather diffuse system, whose modeling is far more uncertain and arbitrary. While the various atomic processes, such as transition probabilities and excitation rates, are well known and also the thermodynamic properties are fairly well understood, we have only a coarse picture of their hydrodynamics, including the effect of magnetic fields. Thus it is, for instance, difficult to apply hydrodynamic-stability theory without very special assumptions regarding geometry and flows. Hence the results we discuss in this chapter are mainly of qualitative nature, in which general arguments, such as equipartition and virialization, play an important role. The precise numerical factors, which are often found in the astrophysical literature, imply special choices of geometry and profiles and should not be taken too literally. In Section 12.1 we give a brief overview of the characteristic properties of the ISM.

Turbulence is usually associated with the idea of self-similarity, which means that the spatial distribution of the turbulent eddies looks the same on any scale level in the inertial range. This is a basic assumption in the Kolmogorov phenomenology K41 and, on the same lines, the IK phenomenology introduced in Section 5.3.2. It is, however, well known that this picture is not exactly true, since it ignores the existence of small-scale structures, which cannot be distributed in a uniform space-filling way. In fact, in a real turbulence field experiments as well as numerical simulations show that smaller eddies, or higher frequencies, become increasingly sparse, or intermittent, which apparently violates self-similarity. This chapter deals with the various aspects of intermittency.

Section 7.1 gives a brief introduction. We illustrate the concept of self-similarity by some simple examples and clarify the notion of intermittency, distinguishing between dissipation-range and inertial-range intermittency. Section 7.2 deals with structure functions, in particular the set of inertial-range scaling exponents, which are convenient parameters for a quantitative description of the statistical distribution of the turbulence scales. We discuss the important constraints on these exponents imposed by basic probabilistic requirements. Since experiments and, even more so, numerical simulations deal with turbulence of finite, often rather low, Reynolds number, the scaling range may be quite short, or even hardly discernable, especially for higher-order structure functions, which makes determination of the scaling exponents difficult. The scaling properties can, however, be substantially improved by making use of the extended self-similarity (ESS), which often provides surprisingly accurate values of the relative scaling exponents.

Until now we have viewed the turbulent motions and fields mainly in space, or configuration space. Though spatial structures are important in MHD turbulence, for instance as final states of selective decay processes, the most characteristic property of fully developed turbulence is the presence of a wide spectrum of different scales. Turbulence scales have already been used in a very loose way in Section 4.2.2 in discussing the mechanism of dynamic alignment. In this chapter these ideas will be given a more precise meaning. In Section 5.1 we introduce the concept of homogeneous turbulence, a very useful idealization of a turbulence field far away from boundary layers. Here the Fourier components of the field play the role of the amplitudes at a certain scale l ∼ k−1. Section 5.2 considers an approximation that, at first sight, has little resemblance to real turbulence, namely a nondissipative system of Fourier modes truncated at a finite wavenumber and its relaxed states, which are called absolute equilibrium distributions. In spite of their artificial character such states can provide valuable information about the tendencies of the spectral evolution in dissipative systems, in particular the direction of the spectral fluxes called cascades. In Section 5.3 we then switch on dissipation in order to study the spectral properties of MHD turbulence.

In the previous chapters turbulence was assumed incompressible. As discussed in Section 2.3, this assumption is valid if either the sonic Mach number of the flow is small, Ms = υ/cs « 1, or the Alfvén Mach number is small, MA = υ/υA « 1. The former condition applies to a weakly magnetized plasma, in which υA « cs, or to motions along the magnetic field, while the latter applies to motions perpendicular to the field. If the flow is turbulent, there is some arbitrariness in the definition of the Mach numbers, since one may choose (a) the mean flow velocity, (b) the r.m.s. velocity fluctuation υ = 〈ῦ2〉1/2 = (Ek)1/2, or (c) the local velocity. Following convention in turbulence theory, we refer to the Mach number in terms of the r.m.s. velocity, noting that local Mach numbers may be considerably higher.

Since laboratory plasmas are usually confined by a strong magnetic field, they can be considered incompressible, the dynamics consisting mainly of cross-field motions. Also the motions in the liquid core of the Earth, which drive the Earth's dynamo, are incompressible, since Ms « 1 (here inertial effects are often neglected altogether, which is called the magnetostrophic approximation). By constrast, most astrophysical plasmas are compressible, for instance the interstellar medium, which is rather cold, such that, in the turbulent motions observed, Ms, and possibly also MA, tend to be large (see Chapter 12), or the turbulence in the interplanetary plasma, which is riding on the supersonic and super-Alfvénic solar wind (Chapter 10).

In the preceding chapters we considered the dynamics of an individual system. Starting from a smooth state, fine structures develop, which, in general, become unstable at some point. After the onset of instability the structure of the flow is very complex and irregular and, most importantly, the further behavior is unpredictable in the sense that minimal changes would soon lead to a completely different state. Such a behavior is commonly called turbulent. Though a direct view of the continuously changing patterns is certainly most eyecatching and fascinating, a pictorial description of these structures is not very suitable for a quantitative analysis. On the other hand, it is just this chaotic behavior which makes turbulence accessible to a theoretical treatment involving statistical methods. While individual shapes and motions are intricate and volatile, the average properties of the turbulence described by the various correlation functions are, in general, smooth and follow rather simple laws. A well-known paradigm is the turbulent behavior in our atmosphere. We try to predict the short-term changes, called weather, in a deterministic way for as long as is feasible, which, as daily experience shows, is not very long, while predictions of the long-term behavior, called climate, can be made only on a statistical basis.

Dividing the fields into mean and fluctuating parts, we derive equations for the average quantities, the generalized Reynolds equations, which contain second-order moments of the fluctuating parts, the turbulent stresses.

Helioseismology provides us with means to investigate the otherwise invisible solar interior. The seismic approach is indispensable for the study of internal structure and evolution of the sun. It is even more so, however, for the study of dynamical aspects of the sun, because of the lack of other reliable means. The current status of seismology of solar rotation is reviewed and outstanding problems are discussed.

Introduction

In 1984, Douglas Gough started his paper, entitled ‘On the rotation of the Sun’, by pointing out our lack of understanding of the dynamical history of the sun (Gough 1984). The question of how the sun has evolved dynamically, since its arrival on the main sequence, still stands as one of the big questions in astronomy. With an increased level of interest attracted by the issue of how our solar system (and other ‘solar’ systems) formed and evolved, it may be a problem of even greater importance today.

Another big problem regarding the solar rotation is what is behind the solar cycle, and if a dynamo mechanism is responsible, as is generally believed, how it works. Here, too, the problem seems to be recognized in a wider community because of the great interest currently shown towards the solar-terrestrial study.

In tackling both problems, an important key is the dynamical structure of the sun today, and in particular how it rotates. Observational clues are not many.

Telechronohelioseismology (or time-distance helioseismology) is a new diagnostic tool for three-dimensional structures and flows in the solar interior. Along with the other methods of local-area helioseismology, the ring diagram analysis, acoustic holography and acoustic imaging, it provides unique data for understanding turbulent dynamics of magnetized solar plasma. The technique is based on measurements of travel time delays or wave-form perturbations of wave packets extracted from the stochastic field of solar oscillations. It is complementary to the standard normal mode approach which is limited to diagnostics of two-dimensional axisymmetrical structures and flows. I discuss theoretical and observational principles of the new method, and present some current results on large-scale flows around active regions, the internal structure of sunspots and the dynamics of emerging magnetic flux.

Introduction

Telechronohelioseismology (or telechronoseismology) is defined as a subdiscipline of helioseismology by Gough (1996) in his reply to criticism of the term ‘asteroseismology’ (Trimble 1995). Gough argued that, being derived from all classical Greek words, ‘thoroughbred’ telechronohelioseismology should be preferred to ‘oedipal combinations’ of Greek and Latin words. Telechronohelioseismology belongs to a new class of helioseismic measurements, broadly defined as epichorioseismology (also calledlocal-area helioseismology), which provides three-dimensional diagnostics of the solar interior.

Helioseismology is originally basedon interpretation of the frequencies of normal modes of solar oscillation.

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.