4.1. Decaying hydrodynamic turbulence
We have computed the -variance spectra for two models of decaying hydrodynamic turbulence, one with initial rms Mach number , noted as Model D in Mac Low et al. (1998), and one otherwise identical model with initial , not published before. As noted above, these models were excited with a flat-spectrum pattern of velocity perturbations, which would correspond to a rather steep spectrum in 2D or in 3D, respectively. Both were run at a resolution of .
The first time steps in Fig. 6 show that only hypersonic turbulence provides a self-similar behavior, indicated by a power-law -variance spectrum. In this case, there appears to be structure corresponding to a power-law spectrum of , somewhat steeper than the that would be expected from a simple box full of step-function shocks, but approaching the steepness observed for real interstellar clouds at large scales. When the turbulence decays to supersonic rms velocities at later times, or in the model having only supersonic initial velocities, the spectrum indicates no self-similar structure but a distinctive physical scale that evolves with time to larger sizes.
We speculate that a physical explanation for this observation might be drawn from the nature of dissipation in supersonic turbulence. Energy does not cascade from scale to scale in a smooth flow through wavenumber space as is assumed by analyses following Kolmogorov (1941) for subsonic turbulence. Rather, energy on large scales is directly transferred to scales of the shock thickness by shock fronts, and there dissipated. As a result, energy is not added to small and intermediate scale structures at the same rate that it is dissipated. Combined with a fairly steep power spectrum, this means that the smaller scale structures will be lost to viscous dissipation first, moving the typical size to larger and larger scales.
We can quantify the change in typical scale by simply fitting a power-law to the lag at which reaches a peak. This was done for the 3D -variance where the peak appears more prominent than in Fig. 6 and represents the true length scale without projection effects. For the model starting at Mach 5 we find a variation in time of with .
These decaying turbulence models were found by Mac Low et al. (1998) to lose kinetic energy at a rate , with . However, Mac Low (1999) showed that driven hydrodynamic turbulence dissipates energy , corresponding to a kinetic energy decay rate of if the effective decay length scale were independent of time. From this observation, a time dependence of was deduced. Mac Low (1999) also showed that the characteristic driving length-scale was the most likely identification for . Identifying for decaying turbulence with the length scale containing the most power in the -variance spectrum seems natural, and yields excellent agreement in the time-dependent behavior of the length scale, since .
4.2. Driven hydrodynamic turbulence
In Fig. 7 we show the -variance spectra for models of supersonic hydrodynamic turbulence driven with a fixed pattern of Gaussian random perturbations having only a narrow range of wavelengths and two different energy input rates. The driving wavelengths are 1/2, 1/4, and 1/8 of the cube size, corresponding to driving wavenumbers of , 4, and 8. In the upper graph (models HE2, HE4, and HE8 from Mac Low 1999), the driving power is by a factor 10 higher than in the lower graph (models HC2, HC4, and HC8). The equilibrium rms Mach numbers here are 15, 12, and 8.7, for the high energy models driven with , 4, and 8 respectively, and 7.4, 5.3, and 4.1 for the low energy simulations. All of these models were run at resolution.
All spectra show a prominent peak characterizing the dominant structure length. It is obviously related to the scale on which the turbulence is driven but the exact position depends on the energy input rate. Whereas all peak positions in the strongly driven case are at about 0.5 times the driving wavelength (correcting the scales from Fig. 7 by the projection factor ), they change in the lower graph from 0.8 times the driving wavelength for to 0.6 for . Thus, only the strongly hypersonic models provide a constant relation between the driving scale and the dominant scale of the density structure.
Below the peak scale, a power-law distribution of structure is observed, while above this scale, the spectrum drops off very quickly. The power-law section of the spectrum has a slope between 0.45 for the high Mach number models and 0.75 for the lower Mach numbers, corresponding to a power spectrum power law of . This agrees with the slope observed in the case of hypersonic decaying turbulence and lies within the range of power law exponents observed in real molecular clouds. However, the only peaks found in -variance spectra of the observations correspond to the total size of the molecular clouds, strongly suggesting that processes injecting energy at distinct intermediate scales do not provide a major contribution to the cloud structuring.
Further simulations should systematically study the transition from supersonic to hypersonic velocities in driven models to find the critical parameters for the onset of a self-similar behaviour and the exact relation between the peak position, the driving scale, and the viscous dissipation length in this case.
4.3. MHD models
Now we can examine what happens when magnetic fields are introduced to models of both decaying and driven turbulence. In Fig. 8 we begin by examining the -variance spectra of a decaying model with and initial rms Alfvén number , equivalent to a ratio of thermal to magnetic pressure . This model was described as Model Q in Mac Low et al. (1998).
No power law behavior is observed, with the spectra showing a uniformly curved shape remarkably devoid of distinguishing features. We emphasize that this behavior is preserved through a resolution study encompassing a factor of four in linear resolution, suggesting that it is not simply due to numerical diffusivity, but rather is a good characterization of the structure of a strongly magnetized plasma. Thus we conclude that self-similar, power-law behavior is not a universal feature of MHD turbulence, and that observations showing such curved -variance spectra may reflect the true underlying structure, rather than being imperfect observations of self-similar structure. The magnetic field tends to transfer power from larger to smaller scales quickly, overpowering the evolution of the characteristic driving scale seen in the hydrodynamical models.
A similar behavior is visible in the driven turbulence models shown in Fig. 9. In the upper part of the figure the -variance spectra for three models with driving wavenumber and ratios of thermal to magnetic pressure of , 0.08, and 2.0 are shown along with a hydrodynamical model () with identical driving. The MHD models all have equilibrium rms Mach number ; their equilibrium rms Alfvén numbers are about 0.8, 1.6, and 8 respectively. In the lower graph we have plotted the equivalent extreme cases of and for the driving.
We find again that the magnetic fields have some tendency to transfer energy from large to small scales, presumably through the interactions of non-linear MHD waves. The more energy that is transferred down to the dissipation scale, the less power is seen in the -variance spectra, suggesting that the strong field () is more efficient at energy transfer than the weaker, higher fields. The larger-scale driving admittedly shows much less drastic effects than the driving, emphasizing that the magnetic effects are secondary in comparison to the nature of the driving.
This transfer of energy to smaller scales has implications for the support of molecular clouds. There have been suggestions by Bonazzola et al. (1987) and Léorat et al. (1990) that turbulence can only support regions with Jeans length greater than the effective driving wavelength of the turbulence. The transfer of power to smaller scales might increase the ability of turbulence driven at large scales to support even small-scale regions against collapse. Computations including self-gravity that may confirm this are described by Mac Low et al. (1999).
4.4. The velocity space
The -variance measuring the density structure of the HD/MHD simulations can be compared directly to the analysis of astrophysical maps taken in optically thin tracers. However, there is much additional information in the velocity space which has to be addressed too.
Here, the -variance cannot be applied to the observations since they retrieve only the line-of-sight integrated one-dimensional velocity component convolved with the density. Nevertheless, we can apply it to analyze the characteristic quantities in the simulations where we have the full information on the spatial distribution of the velocity vectors. As the -variance measures the relative amount of structure on certain scales in the density cubes it can be applied in the same way to the velocity components or the energy density.
Fig. 10 shows the -variances for for the kinetic energy density, and the x-velocity component of the driven hydrodynamic model discussed in Sect. 4.2. The plots can be compared to the -variances of the corresponding density structures shown in the upper part of Fig. 7. We see a shift of the dominant structure size from the driving wavelength that is directly seen in the velocity structure to smaller scales for the density structure. The energy density structure shows an intermediate behavior as a combination of density and velocity structure.
The same comparison for the supersonic model shown in the lower part of Fig. 7 provides much smaller differences in the peak position of the -variance for the three quantities. This means that hypersonic velocities are not able to create density structures at the scale of injection but only on some smaller scales whereas smaller velocities produce void and compressed regions directly at the scale of their occurrence.
The slopes in the self-similar range at smaller scales are different for the density and velocity structure. The Gaussian perturbations in velocity space create a -variance slope of 2.1 in the projected velocities but do not translate into the same structural variations in the other quantities. In the density structure, perturbations are created more efficiently at smaller scales so that we obtain a slope of 0.45. The energy density structure turns out to be dominated by the density variations so that we find about the same slope there.
The difference in the -variances between the three quantities is however probably due to the special driving mechanism. If we apply the same analysis to the decaying turbulence models, we find that the peak position and slopes in all three quantities approach each other after some time, so that an equipartition of structure in density and velocity is produced. In the first steps of the decaying model from Fig. 6 we still find a difference in the slopes of the -variances between the density and velocity structure of a factor 1.5 to 2 whereas the slopes are almost identical at the latest step. Applying the same line of reasoning to the astrophysical observations, the comparison of density and velocity structure there might help to clarify the state of relaxation and the driving mechanism creating structure in interstellar clouds.
© European Southern Observatory (ESO) 2000
Online publication: December 8, 1999