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).

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.

Diffraction and refraction of radio waves by irregularities in the interstellar electron density produce a wide range of phenomena that allow inferences of the wavenumber spectrum of the irregularities, anisotropies of the fluctuations, the strength of the variations as a function of location in the Galaxy, and characteristic velocities. I summarize the empirical constraints on these aspects of density microstructure on length scales from about 100 km up to ∼ 1 pc.

Introduction

In this paper I discuss turbulence in the diffuse, ionized component (DIG) of the interstellar medium (ISM). The DIG has been probed on scales that are phenomenally small, by astronomical standards (≳100 km) using innovative radio astronomical techniques. I discuss the propagation effects underlying those techniques and summarize the current state of knowledge about turbulence in the DIG. The information gleaned includes the galactic distribution of the free-electron density, ne, and its fluctuations; the wavenumber spectrum of δne, including the level, shape and extent in wavenumber; and the conclusion that, in addition to the Kolmogorov-like fluctuations that pervade the ISM, there are additional structures on ∼ 1 AU scales but which appear to be independent of the Kolmogorov fluctuations.

Radio-Wave Propagation Effects & Parameters

Radio waves are strongly influenced by the free electron density along the line of sight. Dispersive and scattering effects are determined by the cold-plasma refractive index, nr(v) = (1 − vp2/v2)1/2.

Introduction

The properties of the interstellar medium strongly suggest that it is turbulent. Here turbulence is understood as unpredictable spatial and temporal behavior of nonlinear systems as preached by Scalo (1985, 1987).

The importance of turbulence in molecular clouds and its relation to star formation has long been appreciated (Dickman 1985). Recent progress in numerical simulations of molecular cloud dynamics (see Ostriker, this volume) indicates the intrinsic connection between the turbulence in different phases of the interstellar medium (McKee & Ostriker 1977).

Introduction

Most of the talks at this meeting have focused on the nature of turbulence on scales where the eddies are fully developed, which conventionally means sizes less than the outer scale, typically in the range of 10−2 − 10 pc (see Cordes, this meeting).

Introduction

In 1981, Larson found the size-line width relation toward various molecular clouds, so called “Larsons's Laws” (Larson 1981). After his investigation, many investigators reported the same relation from large scales (molecular clouds) to small scales (dense cores).

The nature of turbulence in the warm ionized component of the interstellar medium (WIM) can be investigated using Fabry-Perot spectroscopy of optical emission lines. The Hα intensity provides the emission measure (EM) along a line of sight, which is used in conjunction with the scattering measure, rotation measure, and dispersion measure to study interstellar turbulence. Observations at high spectral resolution (∼ 10 km s−1) allow measurements of the bulk radial velocity structure of the emitting gas and investigations of thermal and non-thermal (turbulent) broadening mechanisms through the line widths. By measuring the widths of the Hα line and an emission line from a heavier atom (e.g. the [S II] λ6716 line), one can separate the thermal and non-thermal contributions to the line width. Preliminary studies using this method have shown that the broad range of Hα line widths (typically 15 – 50 km s−1) is mostly due to differences in the non-thermal component of the width and that along many lines of sight this component dominates. The Wisconsin Hα Mapper (WHAM) is in the process of producing a very sensitive kinematic map of the northern sky in Hα at 1° angular resolution and 12 km s−1 spectral resolution. WHAM is also mapping emission lines from heavier atoms such as sulfur and nitrogen for selected regions of the sky. This data set will provide unique new information concerning turbulence in the WIM.

Introduction

Molecular cloud lifetimes are of order 3 × 107 yr (Blitz & Shu 1980), while free-fall gravitational collapse times are only tff = (1.4 × 106 yr) (n/103 cm−3)−½. In the absence of non-thermal support, these clouds should collapse and form stars in a small fraction of their observed lifetime. Supersonic hydrodynamical (HD) turbulence is suggested as a support mechanism by the observed broad lines, but was dismissed because it would decay in times of order tff. A popular alternative has been sub- or trans-Alfvénic magnetohydrodynamical (MHD) turbulence, which was first suggested by Arons & Max (1975) to decay an order of magnitude more slowly. (Also see Gammie & Ostriker 1996).

However, analytic estimates and computational models suggest that incompressible MHD turbulence decays as t−η, with a decay rate 2/3 < η < 1.0 (Biskamp 1994; Hossain et al. 1995; Politano, Pouquet, & Sulem 1995; Galtier, Politano, & Pouquet 1997), while incompressible HD turbulence has been experimentally measured to decay with 1.2 < η < 2 (Comte-Bellot & Corrsin 1966; Smith et al. 1993; Warhaft & Lumley 1978).

Several authors have now suggested that some interstellar clouds above the plane of the Galaxy are interacting with the Reynolds' layer, the warm ionized gas extending well above (H ≅ 910 pc) the Galactic plane (Reynolds 1993). Characterizing the interaction between these clouds and their surroundings should be useful in understanding one source of interstellar turbulence: vertical shear flows. This paper discusses how studies of the morphology and drag coefficient of falling clouds might be used to constrain the Reynolds number for the flow, and hence the effective viscosity of the warm ionized medium. If arguments based on morphology are correct, the effective viscosity of the warm ionized medium is significantly higher than the classical values. Possible resolutions to this problem are suggested.

Turbulence from Vertical Flows

The spectrum of density and velocity fluctuation in the ionized interstellar medium (ISM) measured by scintillation of pulsars suggests that on small scales much of the structure of the diffuse ionized ISM may arise as the result of turbulent processes. Turbulence arises in regions of viscous shear flows. In the Galaxy, such flows have a large range of outer length scales, and include galactic rotational shear in both the radial and vertical (c.f. Walterbos 1998, this volume) directions, spiral density waves, stellar mass outflows (jets, winds, and explosions), and photoionization-driven flows. The structures formed contain energy over a range of length scales which is ultimately dissipated via viscous (hydrodynamical) and resistive (magneto-hydrodynamical) processes.

Faraday rotation measures for extragalactic sources were determined in a ∼ 12° by 10° area of the sky. The Hα emission from this region of the sky was also measured. These measurements allowed the unambiguous detection of turbulent magnetic field fluctuations in the diffuse interstellar medium. We compare these observations with the predictions of several ISM turbulence models. We find that the observed turbulence cannot be explained by an ensemble of magnetosonic waves propagating at large angles with respect to the mean magnetic field lines. The measurement of the turbulent magnetic field fluctuations allows us to quantify the energy contained in the turbulence which gives us an estimate of the turbulent dissipation rate. The effects of this turbulent dissipation on the heating of the diffuse ISM are investigated. It is found that the turbulent heating can explain the differences in observed line intensity ratios (such as [S II]/Hα and [N II]/Hα) between H II regions and the diffuse ionized gas (DIG) in our galaxy.

Observations

The Faraday rotation measures of 38 extragalactic sources, many of which are double lobed radio sources, were measured in a ∼ 12° by 10° region of the sky (RA 2h–3h, DEC 33°–43°) (Minter & Spangler 1996). This region of the sky was chosen due to the Hα emission from the diffuse ionized gas (DIG ≡ WIM ≡ Reynolds layer) in our galaxy having been previously mapped by Reynolds (1980).

Linear Polarization Observations of Water Masers in W51 M

High-resolution polarization observations of water masers provide a powerful tool for studying Alfvenic turbulence and magnetic fields in dense circumstellar regions. Here we present some main results of the first 22 GHz linear polarization observations of water masers in the central low-velocity range of W51M, 54 < Vlsr < 68 km s−1, obtained with VLBA (Leppänen, Liljeström, & Diamond 1998). The principal difference of polarimetric VLBI from total intensity VLBI is the need to calibrate the instrumental polarization parameters, which have been solved by Leppänen (1995) with a feed self-calibration algorithm (see also Leppänen, Zensus, & Diamond 1995). The uniformly weighted restoring (CLEAN) beam obtained was 0.71 × 0.26 mas; the velocity resolution was 0.2 km s−1.

Figure la shows the spatial distribution of the maser spots. Superimposed on the spots are the linear polarization vectors with their lengths proportional to the degrees of polarization. The inset of Figure la is an enlargement of the compact maser concentration near the reference position (0,0) of W51 M. The dotted line in the inset separates blueshifted (west of the dotted line) and redshifted (east of the dotted line) maser spots with respect to the velocity centroid, 61.5 km s−1, of this maser concentration, hereafter called the protostellar cocoon.

Introduction

This paper will deal with turbulence in the ionized portion of the interstellar medium. The theoretical ideas invoked will therefore be from the field of plasma turbulence, which in some respects differs from hydrodynamic turbulence. My primary interest, in keeping with the title of this paper, will be in the fluctuations which occur on spatial scales much less than the outer scales of the turbulence, scales which may be termed part of the inertial subrange.