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Astron. Astrophys. 353, 339-348 (2000)

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5. Conclusions and outlook

5.1. Conclusions

We have examined the structure of supersonic, MHD turbulence using the [FORMULA]-variance, a wavelet transform method introduced by Stutzki et al. (1998) to characterize observed structure in molecular clouds. The [FORMULA]-variance spectrum can be analytically related to the more commonly used Fourier power spectrum, but has distinct advantages: it explicitly reveals finite map size and finite resolution effects; it works in the absence of periodic boundary conditions; and it reveals characteristic structure scale even in the presence of shocks and other sharp discontinuities. We find that one note of caution is called for in its use, however: 2D spectra are proportional to the 3D spectra multiplied by the lag, and this can introduce apparent power-law behavior even in cases where the 3D spectra do not appear to have any such behavior intrinsically.

We computed [FORMULA]-variance spectra for the numerical simulations of compressible, hydrodynamical and MHD turbulence described by Mac Low et al. (1998) in the freely decaying case, and by Mac Low (1999) in the case of driven turbulence, along with a few extra models run to expand the parameter space in interesting directions. Resolution studies reveal that the [FORMULA]-variance spectra cleanly pick out the scale on which artificial viscosity operates, which appears as a steeply dropping section of the spectrum at small lags. Examination of spectra from widely different times for driven models in equilibrium shows that the [FORMULA]-variance spectrum offers a stable characterization of the dynamically varying structure.

Decaying hydrodynamical turbulence excited initially with a range of length scales only appears to have self-similar, power-law behavior in the hypersonic regime. Once the rms Mach number drops below [FORMULA] or so, a distinct length scale appears that grows as the square root of time. This appears to confirm the prediction made by Mac Low (1999) that the effective driving scale must increase to explain the inverse linear dependence of the kinetic energy dissipation rate on the time in this regime.

Driven, supersonic, hydrodynamical turbulence can maintain self-similar, power-law behavior at scales less than the driving scale, while decaying hydrodynamical turbulence does not show power-law spectra indicative of self-similarity, but rather shows characteristic scales. Strongly driven turbulence has a power-law slope that lies directly in the range of slopes observed for real molecular clouds (Bensch et al. 1999). The observations show power-laws extending to the largest scales in the map that can be analyzed, suggesting that driving mechanisms may be acting that add power on scales larger than those of the individual clouds and clumps that are mapped. Ballesteros-Paredes et al. (1999) suggest that molecular clouds are transient density enhancements formed in supernova-driven interstellar turbulence. The resulting large-scale driving would be consistent with our results.

Molecular clouds are observed to have magnetic fields strong enough for the Alfvén velocities to be of the same order of magnitude as the observed rms velocities (e.g. Crutcher 1999). We have therefore examined the effects of magnetic fields on our results from the hydrodynamic models. We find that even strong magnetic fields often have fairly small effects, but that they do tend to transfer power from large to small scales, with implications for the support of small Jeans unstable regions by large-scale driving mechanisms. (Note, however, that Mac Low et al. 1999 find this insufficient to support against collapse.) Contrary to some expectations, we find that magnetic fields do not tend to create self-similar behavior, but rather tend to destroy it, and that stronger fields tend to do so more. Hypersonic turbulence with Alfvén numbers of a few appears to be consistent with the observations of both power-law behavior and relatively strong magnetic fields in molecular clouds.

5.2. Outlook

We expect that the combined analysis of the velocity and density structure in molecular clouds will be able to help distinguish between the possible mechanisms driving interstellar turbulence and to provide information on the internal relaxation or virialization of the clouds on different scales.

The next step in our work is to move from the general characterization of supersonic turbulence presented here to attempts to fit observations of specific real interstellar clouds, using what we have learned so far to guide our search. This should yield constraints on the effective Mach and Alfvén numbers in these clouds, and begin to show whether supersonic, super-Alfvénic turbulence can indeed give a good description of the structure of molecular clouds.

To get a detailed comparison between observations and simulations we have to solve the full radiative transfer problem relating the simulated structure to maps in common lines such as the lower CO transitions, which are often optically thick. Having the full radiative transfer computations also allows the fit to include not just the observed map scaling relations but also the peak intensities, line ratios, and line shapes, placing significant additional constraints on the models.

Furthermore the structure analysis must be extended beyond the investigation of isotropic scaling behaviour. Appropriate measures for anisotropy or filamentarity, and the relationship between the density and the velocity structure have to be found. Our first results presented here have only scratched the surface of the possibilities for systematic comparison between cloud observations and direct turbulence simulations.

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© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999