We use the standard, adiabatic shell evolution to predict the size distribution N(R) for populations of SN-driven superbubbles in a uniform ISM. We derive N(R) for simple cases of superbubble creation rate and mechanical luminosity function. We then compare our predictions for N(R) with the largely complete HI hole catalogue for the SMC, with a view toward the global structure of the ISM in that galaxy. We also present a preliminary derivation for N(v), the distribution of shell expansion velocities.

Introduction

Core-collapse supernovae (SNe) tend to be correlated in both space and time because of the clustering of the massive (≳ 8M⊙) star progenitors. These clustered SNe, along with stellar winds of the most massive stars, produce superbubble structures in both the warm ionized (104 K) and atomic H I components of the interstellar medium (ISM) in star-forming galaxies. The hot, coronal component of the ISM is thought to originate largely from the shock heating of material interior to shells of superbubbles and supernova remnants (SNRs). Total kinetic energies deposited into the interstellar environment are in the range 1051 − 1054 erg for OB associations, and ≳ 1055 erg for starburst phenomena. Hence, the large-scale structure and kinematics of the multi-phase ISM could be largely determined by this superbubble activity. Likewise, this effect should influence turbulence on global, macroscopic scales, which then cascades to smaller scales.

H2O masers near young stars show turbulent motions of many times sound speed. These motions appear on scales of 1 to 300 AU, much smaller than the 104 AU sizes of H2O maser clusters. Turbulent velocity differences between the masers are typically 10 to 100 km s−1, much larger than typical sound and Alfven speeds of ∼ 0.8 km s−1. These velocity differences show the powerlaw correlation functions characteristic of fluid turbulence, over several orders of magnitude in separation. The index is close to that predicted by the Kolmogorov theory. Maser features also show internal turbulence, on scales of < 1 AU, consistent with Alfvenic turbulence.

Introduction

H2O masers are among the most spectacular astrophysical masers; they are found near late-type stars, around galactic nuclei, and near young stars. Those near young stars are among the brightest and most numerous. As Strelnitski & Sunyaev (1973) first proposed, the strong winds from these stars accelerate and power the masers. The population inversion required for maser action arises at shocks, in the outflowing wind and where it meets ambient material (Litvak 1969, Strelnitski 1984, Elitzur, Hollenbach, & McKee 1989, Kaufman & Neufeld 1996). Each masing region contains between one and several hundred individual masing cloudlets, known as features.

The kinematics of clusters of H2O masers have been studied in detail with very-long baseline interferometry (VLBI), in part because comparison of proper motions with Doppler shifts can yield trigonometric distances to these objects (Genzel et al. 1981, Reid et al. 1988, Gwinn, Moran, & Reid 1992).

Introduction

The Probability Density Function of mass density is an important statistical property of the ISM that relates, for example, to gravitational collapse and star formation. Log-Normal PDFs have been discussed occasionally in both the cosmological and interstellar contexts (Hubble 1934, Peebles 1980, Ostriker 1984, Zinnecker 1984, Coles & Jones 1991). Vázquez-Semadeni (1994) noticed that the density PDFs in his 2-D numerical simulations of turbulence were consistent with a Log-Normal, and discussed possible reasons for the lognormality. Padoan et al. (1997) showed that the standard deviation of the Log-Normal PDFs in their isothermal 3-D simulations was approximately equal to half the rms Mach number. Scalo et al. (1998) raised the questions of how a polytropic equation of state, and more generally a realistic ISM cooling function, might influence the PDF.

In this contribution we investigate the question of the shape of the PDF for isothermal and polytropic equations of state, using analytical methods and by looking at results from 3-D simulations of supersonic turbulence.

Introduction

Is the core difficulty in the so-called convective term of the equation, i.e. in the simple fact that moving matter is self-advected? In which case, the understanding of threedimensional (3D) incompressible turbulence would be the key to our own case of compressible MHD turbulence (CMT). Or will the many added features – such as Alfvén and magnetosonic waves and a preferred direction in the presence of a strong uniform magnetic field – change the behavior of CMT flows altogether? How far does the concept of universality carry out? This is one of the questions that the detailed observations of astrophysical flows can help settle. In this short review, I shall begin by giving the humongus list of parameters that have to be considered a priori, the rule of the game being to determine which parameters are relevant and which can be ignored. In the next Section, some of the key features of the temporal and spatial development of compressible flows and MHD flows will be recalled. Section 4 is devoted to the formation of large scales (as opposed to small scales), the agent being here the magnetic helicity and Section 5 deals with intermittency and its measure through high-order structure functions.

Introduction

Numerical tools are likely to play an important role in the investigation of MHD turbulence in cold molecular clouds if for no other reason than because the observed linewidths are highly supersonic, and as of yet there does not exist a comprehensive analytic theory of compressible MHD turbulence. Our group (C. Gammie, E. Ostriker, and myself) has begun a project to study systematically the internal dynamics of magnetized, self-gravitating molecular clouds in two- and three-dimensions. Our motivations are two-fold: not only do we wish to understand the dynamics of compressible MHD turbulence as a well-defined physics problem, but also we would like to use the dynamical models as a basis with which to interpret the enormous collection of astronomical observations of molecular clouds that have been collected over the past several decades.

The viscous dissipation process in supersonic flows is known to be highly intermittent in space and time. It means that the volume over which the turbulent energy is released is as small as the character of intermittency is pronounced. In the interstellar medium, this phenomenon is predicted to induce large fluctuations at small scales of the gas properties. Since the interstellar viscous dissipation scale is out of reach of current observational capabilities, indirect signatures of the presence of this process in the interstellar medium would be valuable. A few possible indirect signatures are presented here.

Introduction

On average in our Galaxy, about 10−3 L⊙/M⊙ are permanently driven into heat due to the viscous dissipation of supersonic turbulence. On average again, this is negligible compared to the heating due to the UV stellar radiation which is about 1 L⊙/M⊙. But it is now known that a large fraction of the energy released by the dissipation of turbulence is not distributed evenly but is concentrated in localized regions of space and time because most of the dissipation occurs in bursts. This phenomenon is known as the intermittency of turbulent dissipation. In those regions of interstellar space where viscous dissipation occurs, the corresponding heating term becomes temporarily dominant, even in regions poorly shielded from the ambient interstellar radiation field. This significantly modifies the subsequent evolution of the gas which has received this energy, in comparison to that of gas of its neighborhood which has not received it.

In the previous section we showed that gravity, as described by the curvature tensor, exerts a converging influence on a uniform flow of inertial particles, given that energy is locally positive. This is in accord with everyday experience, in that gravity is always observed to be an attractive force. Indeed, as we have seen, this effect provides the necessary link between general relativity and Newtonian gravity. We now turn our attention to the effect gravity has on a uniform flow of massless particles, that is, a geodesic null congruence. We shall show that, even in this situation, gravity still has a converging influence. This is important, since it highlights the difference between general relativity and Newtonian gravitational theory, where light rays are unaffected by gravity. Indeed, the converging influence of gravity on light rays has led to some of the more remarkable predictions of general relativity, such as black holes and spacetime singularities.

Throughout this chapter we take la to be an affine tangent vector to a geodesic null congruence and take r to be an affine parameter such that Dr = 1, where D = la∇a.

Surface-Forming Null Congruences

A null congruence is surface-forming if its tangent vector la has the form la = ∇au for some null function u. An important example of such a congruence is that formed by future-directed null rays emanating from some timelike world line. Here the level surfaces of u will be future null cones with vertices on the world line.

Introduction

The focus of this meeting is on Interstellar turbulence, but the scientfic program reflects the importance of examining this in a general context, including earlier studies of incompressible turbulence in terrestrial fluids, as described in the classical theory of Kolmogorov and its modern extensions (Frisch 1995).

A model for the initial stellar mass function based on random sampling in a hierarchical cloud is reviewed. The Salpeter function is readily obtained, with a flattening at low mass where cloud pieces cannot become self-gravitating. Fluctuations around the IMF are considered.

Introduction

The initial stellar mass function (IMF) shares two properties with turbulence: it is partly scale-free, with nearly a power law distribution for a factor of ∼ 100 in mass, and it is ubiquitous. The scale-free behavior is also like turbulence in the sense that the power law appears beyond a physical boundary, which in this case is set by the inability of gas to form stars at very low mass (at a given temperature and pressure). There is probably an upper boundary for stellar mass too, but this has not been observed yet because high mass stars are rare.

The IMF is ubiquitous as well, having about the same power law slope for the mass distribution function in a wide variety of environments, from old globular clusters to OB associations and young clusters. There are clear deviations from this average slope, and there are sometimes gaps and bumps in the IMF for particular clusters, but it is possible that these deviations and features are within the range of statistical fluctuations, as in the model discussed here. It is also possible that really significant differences in the IMF occur as a result of differences in the one physical parameter that enters this distribution, the lower mass limit.

Introduction

I first heard about the Kolmogorov law (Kolmogorov 1941) for the velocity structure of turbulent fluids, in a lecture given by Chandrasekhar on the theory of the origin of the solar system proposed by C.F. von Weizsäcker (1943), when I was at the Yerkes Observatory as a Junior staff member. I could not imagine then that 50 years later I still would be talking about the matter. Shortly afterwards, when I joined the Mt. Wilson and Palomar Observatories, the work of von Hörner (1951) on the gas motions in the Orion Nebula, based on sixty measured radial velocities (Campbell & Moore 1918), became known and led to observations of the nebula using the coudé spectrograph of the 5 meter Hale telescope, with the highest angular and spectral resolution then possible. A few years later Wilson et al (1959) published about 10,000 radial velocities of the nebula, in the [OIII], [OII] and Hβ lines, besides a sample of line profiles from photographic plates. Their study (Münch 1958) essentially confirmed von Hörner's result, in the sense that the rms difference in radial velocity of two points on the nebula separated in the sky by a distance r, varies nearly as r0.4, somewhat more steeply than the r1/3 predicted by Kolmogorov law. This rather unexpected agreement was difficult to explain at the time, but it clearly implied that the formation path of the [OIII] line analyzed, determined by extinction and scattering by dust along the line of sight, is not large enough to smooth out the velocity variations of mass motions along the line of sight.

High resolution 21 cm observations of the Ursa Major cirrus revealed highly filamentary structures down to the 0.03 pc resolution. These filaments, still present in the line centroid map, show multi-Gaussian components and seem to be associated with high vorticity regions. Probability density functions of line centroid increments and structure functions were computed on the line centroid field, providing strong evidences for the presence of turbulence in the atomic gas.

Introduction

Many statistical studies of the density and velocity structure of dense interstellar matter have been done on molecular clouds where turbulence is seen as a significant support against gravitational collapse that leads to star formation. Less attention has been devoted to turbulence in the Galactic atomic gas (HI). The cold atomic component (T ∼ 100 K, n ∼ 100 cm−3), alike molecular gas, is characterized by multiscale self-similar structures and non-thermal linewidths.

A detailed and quantitative study of the turbulence and kinematics of HI clouds has never been done. Here we present a preliminary analysis of this kind based on high resolution 21 cm observations of an HI cloud located in the Ursa Major constellation. To characterize the turbulent state of the atomic gas, a statistical analysis of the line centroid field has been done. We have computed probability density functions of line centroid increments and structure functions.

HI Observations

The Ursa Major cirrus (α(2000) = 9h36m, δ(2000) = 70°20′) has been observed with the Penticton interferometer.

Introduction

The interstellar medium (ISM) is in a state of a compressible, inhomogeneous and anisotropic turbulent flow. There are several energy sources for the interstellar turbulence. Stellar winds, supernova (SN) explosions and superbubbles heat, accelerate and compress the ISM driving shock waves (e.g. Ostriker & McKee 1988).

Events

Lightning striking a tree, a brief encounter between friends, the battle of Hastings, a supernova explosion, a birthday party – these are all examples of events. An event is simply an occurrence at some specific time and at some specific place. Events, as we shall see, form the basic elements of the spacetime description of the universe.

The world line of a particle is the sequence of events that it occupies during its lifetime. Birthday parties, for example, form a particularly important set of events on any person's world line. A brief encounter between two friends is an event common to both their world lines (Fig. 2.1).

Most real events are very fuzzy affairs with no definite beginning or end. A pointlike event, on the other hand, is one that appears to occur instantaneously to any observer capable of seeing it.† A collision between two pointlike particle, for example, is a pointlike event. It is, of course, possible to have a nonpointlike event that appears to be instantaneous to some observer, but, due to the finite velocity of the propagation of light, such an event will not in general appear to be instantaneous to some other observer. We say that two pointlike events occupy the same spacetime point if they appear to occur simultaneously to any observer capable of seeing them. If this is not the case, we say that they occupy distinct spacetime points. It is, of course, possible to have two events occupying distinct spacetime points that appear simultaneously to some observer, but, again because of the finite velocity of the propagation of light, they will not, in general, appear simultaneous to some other observer.

Introduction

The turbulence in Galactic molecular clouds has a rather different character from the other forms of interstellar turbulence discussed at this meeting. Strong molecular cooling brings the ambient temperatures to the range T = 10 – 30 K, which renders turbulence with velocities of a few km s−1 not just nonlinear, but hypersonic. Most observational evidence on magnetic field strengths suggest that Alfvén speeds are of same order than the turbulent speeds or a few times larger, and in any case unlikely to be smaller than the sound speed. Thus, the turbulence in molecular clouds is highly compressible, and strongly magnetic. In addition, although high-latitude unbound molecular clouds exist, most of the molecular material in the Galaxy resides in much more massive, self-gravitating, giant molecular clouds and cloud complexes (GMCs).

Introduction

Early spectroscopic observations of interstellar lines of HI, OH, and CO have revealed that observed line widths (or velocity dispersions) in interstellar clouds are larger than thermal line widths expected for these low-temperature regions (see e.g. Myers 1997 and references therein).

Introduction

The conference for which this paper is being written has been instrumental in opening the eyes of astronomers to the need for understanding turbulent processes in astrophysics. While several astrophysical environments where turbulent processes could be important were identified by numerous contributors in this conference, the pulsar scintillation measurements and the study of lines in molecular clouds provide two environments where the need for magnetohydrodynamic (MHD) turbulence is observationally well-founded. Since the MHD equations are highly non-linear analytical approaches sometimes prove to be of limited utility.