Annu. Rev. Astron. Astrophys. 2004. 42: 275-316
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6. SUMMARY AND REFLECTIONS

The turbulence that is observed directly on resolvable scales in the ISM also has important effects on very small scales, down to the level of atomic diffusion, mixing, and viscosity, and continuing far below to the thermal ion gyroradius. The resolvable turbulence in the neutral medium was reviewed in the first part of this series (Interstellar Turbulence I), along with general theory and simulations. The smaller scale implications were reviewed here.

Turbulence helps to disperse the elements made in supernovae and other stellar sources by stretching and folding the contaminated gas until the gradient length approaches the collision mean free path. Then atomic diffusion does the final step of interatomic mixing. This mixing and homogenization may occur partly at the source, in the shock fronts and contact discontinuities around the expanding flow following dynamical instabilities, and it may occur partly in the ambient ISM after the expansion subsides.

Observations suggest that the dispersion in elemental abundances among stars inside clusters and field stars of the same age, and the abundance differences between HII regions and diffuse clouds, are only a few percent up to perhaps 30%. Observations of clusters also suggest that most of the mixing occurs within several times 108 years. This homogenization has to occur between the injection scale of supernovae and superbubble explosions and the star formation scale containing on the order of a solar mass. The corresponding range of spatial scales is ~ 100 for gas at the ambient density. Homogenization cannot be significant on scales much larger than a superbubble, because then it would diminish the galactic radial abundance gradient over a Hubble time. These gradients enter the problem in another way too because turbulence mixes the gas over galactic radius to increase the elemental dispersion locally at the same time as it mixes this gas locally and leads to homogenization.

While the distribution of elemental abundances appears to be fairly narrow, with a dispersion on the order of 10%, it may also have a fat tail caused by occasional odd stars and diffuse clouds with very different abundances. This tail is reminiscent of other fat tails in the distribution functions for turbulent media, and these other tails seem to originate with intermittency. In the case of elemental dispersion, this means that the homogenization process is spotty so some regions survive for long periods of time with very little mixing.

Several methods for studying turbulent mixing have been employed. Because of the wide range of scales involved, these studies often employ simplifying assumptions that appear to capture the essential physics. They include artificial stochastic velocity fields and closure methods using moment equations. Direct numerical simulations of ISM elemental dispersion have only just begun. The theory suggests that a turbulent medium mixes passive scalars like elemental abundances faster than a Gaussian velocity field because of the long-range correlations that are associated with turbulence.

Turbulence affects interstellar chemistry by mixing regions with different properties, heating the gas intermittently on the viscous scale, and enhancing ion-neutral collisions in regions with strong magnetic field gradients. Turbulent mixing is more far-reaching than diffusive mixing because of long-range correlated motions in a turbulent flow. Such mixing spreads out each chemical species over a large radial range inside a cloud. Turbulent heating promotes temperature-sensitive reactions inside otherwise cold clouds. Applications to the formation of CH+ and OH in diffuse clouds look promising. Chemical reaction networks that include these turbulent processes are only beginning to understand some of the implications, and direct simulations of chemistry in a turbulent medium are limited to only a few studies so far.

The scattering and acceleration of cosmic rays depends strongly on the presence of turbulence in the ISM. The scale for this turbulence is the gyroradius, which is less than the collisional mean free path for most cosmic rays, which have energies less than 1 GeV. Thus, scattering relies entirely on magnetic irregularities in collisionless plasma turbulence. Cosmic ray acceleration is by two processes, both of which involve turbulence: The first-order Fermi mechanism accelerates cosmic rays by cycling them through shock fronts where they pick up a relative velocity kick comparable to the shock speed at each passage. This cycling occurs because magnetic irregularities on each side of the front scatter the out-streaming cosmic rays back into the front. The second-order Fermi mechanism accelerates cosmic rays through turbulent diffusion. Each collision with a randomly moving magnetic irregularity turns the cosmic ray around with a reflection speed in the moving frame that is comparable to the incident speed. Over time, the result is a transfer of energy from the turbulence to the cosmic rays.

Cosmic ray scattering occurs in several ways. Fast moving particles can interact weakly but resonantly with numerous magnetic irregularities, and successive interactions of this type can randomly change the pitch angles of their helical motions along the field. This eventually leads to a complete reversal in the direction of motion. Cosmic rays can also interact strongly with large magnetic irregularities, as would occur in shock fronts and at the edges of dense cloud complexes. These interactions change the particle directions significantly each time. The nature of cosmic ray diffusion in both momentum and space is not understood well because the structure and strength of the important magnetic irregularities are not observed directly. If MHD turbulence is highly anisotropic on the scale of cosmic ray scattering, with transverse irregularities much stronger than parallel as suggested by theory, then resonant scattering processes during motions along the mean field can be very weak. The structure of magnetic waves below the mean free path is also unclear, as the usual fast and slow magnetosonic modes do not exist.

Cosmic rays can also generate turbulence by streaming instabilities following particle-wave resonances or by fire-hose and mirror instabilities that operate even without resonances. The expected anisotropy of Alfvén wave turbulence diminishes the first of these mechanisms significantly, however. The others are problematic because they require strong cosmic ray pressure anisotropies. As a result, the impact of cosmic rays on turbulence is currently not understood.

Radio wave scintillation is indirect evidence for interstellar plasma turbulence. These radio observations span a very wide range of spatial scales through a combination of diffraction and refraction effects. The scales are mostly below the collision mean free path, and they are far below the limits of angular resolution. The first of these limits makes it difficult to understand the origin of the density irregularities. Unlike the turbulence that scatters cosmic rays, which requires only magnetic irregularities and no density structure, the turbulence that causes scintillation requires small-scale density irregularities in the ionized medium. The associated magnetic irregularities are not observed, and the connection to cosmic ray scattering is unclear, even though the length scales are about the same. The origin of density structures below the collision mean free path is unknown. Atomic diffusion should smooth them out on a sound crossing time unless magnetic field irregularities hold them in place. In that case they could be the result of slight temperature variations, with the cooler regions having higher electron densities, all divided up and mixed together by transverse magnetic motions. The nature of these motions below the mean free path is unclear, because, as mentioned above, the usual fast and slow compressional modes do not exist, nor do the usual thermal and pressure-regulated processes.

The second of the two limits on spatial scale imply that the geometrical properties of scintillation turbulence are difficult to observe. The scintillation is clearly anisotropic, but whether it is in sheets or filaments, for example, is unknown.

The importance of scintillation observations for studies of ISM turbulence is that they give the power spectrum of electron density fluctuations fairly accurately. This is usually close to the Kolmogorov spectrum of incompressible turbulence. Rarely are the spectra so steep that the medium can be interpreted as a superposition of sharp edges, like shock fronts. One recent observation with fairly high precision obtained the relatively shallow Iroshnikov-Kraichnan power spectrum, leading to the conjecture that the slope varies from region to region. As discussed in Interstellar Turbulence I, this variation may arise from a variation in the relative strength of the magnetic field compared to the turbulent motions, with the Iroshnikov-Kraichnan spectrum present in regions of relatively strong fields.

Observations suggest that scintillation arises in a distributed fashion from the ambient ionized medium and also from discrete high-density places like the edges of local bubbles, ionized molecular clouds, and HII regions. These discrete regions are likely to be highly turbulent and they also have a juxtaposition of hot and cool gas, which is necessary for isentropic mixing and electron density structure.

There are evidently many uncertainties in the nature of ISM turbulence on small scales, even though the evidence for this turbulence is pervasive. Part of the problem is that none of the features of this turbulence have been observed directly: not the densities, magnetic fields, temperatures, or motions. Still, the density irregularities are revealed indirectly through scintillation, the magnetic field irregularities through cosmic ray scattering, the temperature fluctuations through chemistry, and the motions through elemental and chemical mixing. An additional problem is that many small scale effects of ISM turbulence rely on details of the theory that are independent of the usual scaling relations, such as viscous heating and elemental diffusion, which arise at the bottom of the cascade in the neutral medium. Turbulence in the ionized medium is also below the collisional mean free path, where pressure and thermal effects are relatively unimportant. Moreover, the small scale ISM processes are often strongly dependent on the large scale processes, such as turbulent shock formation, energy and metal injection, and galactic-scale gradients. This means that direct simulations of small scale turbulence are impossible without simplifying assumptions about the large-scale medium - assumptions that require more knowledge about ISM turbulence on the large scale than is presently available.

While the observations and theory of ISM turbulence have come a long way from the first efforts in the 1950s, the details of this new information have led to a growing awareness that the complete problem is far too large to solve any time soon. We rely on future generations of astronomers and physicists to continue this work, and hope that they find this field as intriguing and challenging as we do today.


Acknowledgments

We are grateful to A. Brandenburg, D. Lambert, F. Matteucci, S. Oey, and G. Tenorio-Tagle for helpful comments on Section 2; E. Falgarone, D. Hollenbach and J. Le Bourlot for helpful comments on Section 3; B. Chandran, R. Jokipii, A. Lazarian, V.S. Ptuskin, and R. Schlickeiser for helpful comments on Section 4; B. Rickett, S. Spangler, and D. Stinebring for helpful comments on Section 5 and to A. Hill for Figure 2.

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