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.

There is no purely internal physical experiment which will demonstrate the relative motion of a Galilean reference system – this is the principle of relativity. It follows that all physical laws must necessarily have the same form in all inertial systems (see Chapter 2). To express this property in a simple analytic way, and to avoid the need to show explicitly in each case that the principle of relativity is satisfied, it is extremely useful to write the equations of physics in a form which is manifestly covariant under Lorentz transformations. Not only do we avoid the need for such proofs (often very delicate, and subject to error), but this also allows us to set physics in the context of space-time furnished with a pseudo-metric. Further, the need for manifest covariance of the analytic formulation of a physical phenomenon restricts its possible forms. For this reason we shall use various representations of the Lorentz group: spinors, tensors, etc.

Tensor formalism

(1) Unless otherwise stated, we shall only use cartesian coordinates which are orthogonal in the sense of the metric on M. They thus have a timelike axis Ox0 and three spacelike axes Oxi (i = 1, 2, 3) orthogonal in pairs, in the sense of the metric of R3; these three spatial axes are orthogonal to the axis Ox0, the time axis, in the sense of the metric of Minkowski space (Fig. 3.1).

We shall see at the end of this chapter that relativistic gravitation will require the introduction of curved space–time. We should ask what “curved space” actually is. Our intuition, based on surfaces in R3, can be extended to spaces of dimension larger than two. We can deduce the essentials from simple examples [like the sphere] of surfaces in R3. We shall do this, first by studying some geometric properties of known surfaces, and then comparing them with corresponding properties of the plane R2. We then define the Riemann curvature and finally give arguments leading to curved space–time.

Some manifestations of curvature

Here we consider only a sphere of radius R embedded in R3: clearly this is a curved surface. We shall try to construct elementary geometrical figures whose properties we compare with the analogous plane figure.

(1) Geodesic triangle (Fig. 6.1): In the plane, a triangle is formed by the intersection of three non-parallel straight lines. On a sphere, arcs of great circles play the role of straight lines: a straight line in the plane R2, is the shortest path (geodesic) between two points, while for the sphere S2 the geodesies are arcs of great circles. We thus can construct a triangle on the sphere S2; between two points A and B there is an arc of a great circle (exactly one, if the distance AB is to be a minimum and A, B are not at poles).

The usual equations of motion of a particle or a system of particles can be deduced from a variational principle [principle of least action leading either to Lagrange's or Hamilton's equations, depending on the variables used, see H. Goldstein (1980) or L. Landau and E. Lifschitz (1960)], in both the Newtonian and relativistic cases [see A.O. Barut (1965) or J.L. Anderson (1967)] with some subtleties concerning the constraint uµuµ = 1 in the latter case [G. Kalman (1961); A. Peres, N. Rosen (1960)].

In the same way, the equations satisfied by the fields (continuous systems with an infinite number of degrees of freedom), whatever tensor nature they may have, can often be deduced from variational principles. This is true of the equations of electromagnetism, for example, but not the equation for heat transfer.

There are many analytic procedures for describing the motion of a particle or the evolution of a field. There is no a priori reason to confine oneself to differential equations or second order partial differential equations.

The main advantage of a variational formalism is that it allows one to find the conserved quantities in the motion (i.e. the first integrals) and directly to exploit the symmetries of the physical problem considered; these two aspects are connected, as we shall see. We shall introduce such a formalism here only because it allows us to define the energy and momentum of a field very simply.

For classical physics, space and time provide the arena in which the phenomena of nature unfold. These phenomena do not change the space–time frame, which is inert and absolutely fixed for all time. Moreover, space and time are regarded as completely distinct and having no connection with each other. Relativity theory links space and time, and reaches its culmination in General Relativity, which connects the space–time properties with the dynamical processes occurring there.

Newtonian space–time

Physical space possesses the usual properties of continuity, homogeneity and isotropy which we attribute to the space R3 when equipped with its affine structure (parallelism, existence of straight lines) and its usual metric structure (Pythagoras' “theorem”). However, we must understand the physical significance of the mathematical concepts connected with R3. Thus, the existence of physical phenomena which can be represented by straight lines (mathematics) leads to the (experimental) notion of alignment: three points are (physically) aligned if we can find a viewing point from which they appear to coincide. From this it follows that light constitutes our standard of straightness; it is only by a further step (which may prove to be incorrect) that we can identify the trajectory of a light ray with a straight line in R3. Similarly, the mathematical concept of parallelism in R3 is directly related to the (physical) notion of rigid transport and of distance. Finally, we must recognise that the (mathematical) properties of homogeneity and isotropy of physical space only express our experience of mechanical systems: that these remain unaltered when placed in any position or place.