Annu. Rev. Astron. Astrophys. 2004. 42: 211-273
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6. SUMMARY AND REFLECTIONS

Interstellar turbulence has been studied using power spectra and structure functions of the distributions of radial velocity, emission, and absorption, using statistical properties of line profiles, unsharp masks, and wavelet transforms, one-point probability distribution functions of column densities and velocity centroids, fractal dimensions and multifractal spectra, and various other techniques including the Spectral Correlation Function and Principal Component Analysis, which are applied to spectral line data cubes.

The results are often ambiguous and difficult to interpret. The density of the neutral medium seems to possess spatial correlations over a wide range of scales, possibly from the sub-parsec limit of resolution to hundreds of parsecs. Such correlations probably reflect the hierarchical nature of turbulence in a medium with a very large Reynolds number. The power spectrum of the associated intensity fluctuations is a robust power law with a slope between -1.8 and -2.3 for the Milky Way, LMC, and SMC. A steeper slope has been obtained for Galactic CO using Principle Component Analysis. However, no clear power spectrum or autocorrelation function has been found yet for centroid velocities as a function of spatial lag inside individual clouds, even though they might be expected if the clouds are turbulent.

Statistical properties of the ISM are difficult to recover with only line-of-sight motions in projected cloud maps that have limited spatial extent and substantial noise. Linewidths often seem correlated with region size in an average sense, but there are large variations between different regions and surveys. At the moment, this relation seems to be dominated by scatter. Because the linewidth-size correlation is analogous to a second order structure function, which is well-defined for incompressible turbulence, the regional variations are difficult to understand in the context of conventional turbulence theory. Certainly a case could be made that ISM turbulence is not conventional: it is not statistically homogeneous, stationary, or isotropic.

Considering the difficulty of observing three-dimensional structure and motion in space, the value of statistical descriptors lies primarily in their ability to make detailed comparisons between observations and simulations. Many such comparisons have been made, but a comprehensive simulation of a particular region including many of the descriptors listed above is not yet available.

Among the many differences between interstellar turbulence and classical incompressible turbulence is the broad spatial scale for energy input in space. In spite of the wide ranging spatial correlations in emission and absorption maps, there is no analogy with classical turbulence in which energy is injected on the largest scale and then cascades in a self-similar fashion to the very small scale of dissipation. In the ISM, energy is injected over a wide range of scales, from kiloparsec disturbances in spiral density waves, shear instabilities, and superbubbles, to parsec-sized explosions in supernovae, massive-star winds and H II regions, to sub-parsec motions in low-mass stellar winds and stellar gravitational wakes, to AU-sized motions powered by cosmic ray streaming instabilities. Estimates of power input indicate that stellar explosions are the largest contributor numerically, but this does not mean that other sources are unimportant on other scales.

Energy is also dissipated over a wide range of scales. Shock fronts, ambipolar and Ohmic diffusion, Landau wave damping, and viscosity in vortex tubes should all play a role. The geometrical structures of the dissipating regions are not well constrained by observations. Energy is also dissipated in regions with little density sub-structure, as in the ionized gas that produces scintillation or along perturbed magnetic field lines that scatter some cosmic rays. Viscosity and Landau damping can produce heat from the tiny wave-like motions that occur in these regions (see Interstellar Turbulence, Part II), although the nature of this dissipation below the collision mean free path is not understood yet.

Self-gravity makes interstellar turbulence more difficult to understand than terrestrial turbulence. The contribution to ISM motions from self-gravity appears to increase with cloud mass until it dominates above 104 - 105 Modot. Many selection effects may contribute to this correlation, however. Self-gravity is also important in the smaller clumps that form stars. What happens between these scales remains a mystery. We would like to know how the distribution function of the ratio of virial mass to luminous mass for ISM clouds and clumps varies with the spatial scales of these structures. What fraction of clouds or clumps are self-gravitating for each mass range? Does the apparent trend toward diminished self-gravity on small scales turn around on the scale of individual protostars?

The balance between solenoidal and compressible energy density in the ISM varies with time. Compressibility transfers energy between kinetic and thermal modes, short-circuiting the cascade of energy in wavenumber space that occurs in a self-similar fashion for incompressible turbulence. Consequently the ISM turbulent power spectra for kinetic and magnetic energy are not known from terrestrial analogies, and there is not even a theoretical or heuristic justification for expecting self-similar or power-law behavior. Numerical simulations make predictions of these quantities, but these simulations are usually idealized and they are always limited by resolution and other numerical effects.

A considerable effort has been put into modelling compressible MHD turbulence under idealized conditions, i.e., without self-gravity and without dispersed and realistic energy sources (e.g., explosions). Some models predict analytically and confirm numerically a Kolmogorov energy spectrum transverse to the mean field when the magnetic energy density is not much larger than the kinetic energy density. These models also predict intermittency properties identical to hydrodynamic turbulence in this trans-field direction. Other models find steeper power spectra, however. One has mostly solenoidal motions on large scales and sheet-like dissipation regions on small scales. Another has more realistic heating and cooling. At very strong fields, the turbulence becomes more restricted to the two transverse dimensions and then the energy spectrum seems to become flatter, as in the Iroshnikov-Kraichnan model. Overall the situation regarding the energy spectrum of MHD ISM turbulence is unsettled.

Our understanding of the ISM has benefited greatly from numerical simulations. In the 10 or so years since the first simulations of ISM turbulence, numerical models have reproduced most of the observed correlations and scaling relations. They have demonstrated that supersonic turbulence always decays quickly and concluded that star formation is equally fast, forcing a link between cold clouds and their energy-rich environments. They have also predicted a somewhat universal probability function for density in an isothermal gas, implying that only a small fraction of the gas mass can ever be in a dense enough state to collapse gravitationally - thereby explaining the low efficiency of star formation.

Simulations have demonstrated that magnetism does not support clouds, prevent collapse, systematically align small clumps, or act like an effective pressure. Magnetism can slow collapsing cores and remove angular momentum on large and small scales, and it may contribute to filamentary structures that control the accretion rate of gas onto protostars. Simulations have also shown that thermal instabilities do not lead to bimodal density distributions as previously believed, although they probably enhance the compressibility of the ISM and might contribute to the turbulent motions. The high abundance of atomic gas in what would be the thermally unstable regime of static models is easily explained in a turbulent medium.

Simulations have a long way to go before solving some of the most important problems, however. Models are needed that include realistic cooling, ionization balance, chemistry, radiative transfer, ambipolar diffusion, magnetic reconnection, and especially realistic forcing because it is not clear that any of these effects are negligible. Detailed models should be able to form star clusters in a realistic fashion, continuing the progress that has already been made with restricted resolution and input physics. Complete models need to include energy sources from young stellar winds, ionization, and heating, and they should be able to follow the orbital dynamics of binaries. Successful models should eventually explain the stellar initial mass function, mass segregation, the binary fraction and distribution of binary orbital periods, the mean magnetic flux in a star, and the overall star-formation efficiency at the time of cloud disruption.

Realistic simulations of galactic-scale processes are also challenging because they should include a large range of scales and fundamentally different physical processes in the radial and vertical directions. Background galactic dynamics and the possibility of energy and mass exchange with a three-dimensional halo are also important.

The turbulent model of the ISM also raises substantial new questions. Why is the power spectrum of ISM density structure a power law when direct observation shows the ISM to be a collection of shells, bubbles, comets, spiral wave shocks and other pressurized structures spanning a wide range of scales? Do directed pressures and turbulence combine to produce the observed power spectrum or does one dominate to other? Does the input of energy on one scale prevent the turbulent cascade of energy that is put in on a larger scale? In this respect, is ISM turbulence more similar to turbulent combustion than incompressible turbulence because of the way in which energy is injected?

Our current embrace of turbulence theory as an explanation for ISM structures and motions may be partly based on an overly-simplification of available models and a limitation of observational techniques. This state of the field guarantees more surprises in the coming decade.

Acknowledgments: We are grateful to J. Ballesteros-Paredes, J. Dickey, N. Evans, E. Falgarone, A. Goodman, A. Lazarian, P. Padoan, and E. Vázquez-Semadeni for helpful comments on Section 2; to C. Norman for a careful reading of Section 3; to A. Brandenberg, A. Lazarian, P. Padoan, T. Passot, E. Vázquez-Semadeni, and E. Vishniac for helpful comments on Section 4; and to A.G. Kritsuk, M-M. Mac Low, E. Ostriker, P. Padoan, and E. Vázquez-Semadeni for helpful comments on Section 5. We also thank J. Kormendy for helpful comments on the manuscript.

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