In a systematic study , we compare the density statistics in high-resolution numerical experiments of supersonic isothermal turbulence , driven by the usually adopted solenoidal ( divergence-free ) forcing and by compressive ( curl-free ) forcing .
We find that for the same rms Mach number , compressive forcing produces much stronger density enhancements and larger voids compared to solenoidal forcing .
Consequently , the Fourier spectra of density fluctuations are significantly steeper .
This result is confirmed using the \Delta -variance analysis , which yields power-law exponents \beta \sim 3.4 for compressive forcing and \beta \sim 2.8 for solenoidal forcing .
We obtain fractal dimension estimates from the density spectra and \Delta -variance scaling , and by using the box counting , mass size and perimeter area methods applied to the volumetric data , projections and slices of our turbulent density fields .
Our results suggest that compressive forcing yields fractal dimensions significantly smaller compared to solenoidal forcing .
However , the actual values depend sensitively on the adopted method , with the most reliable estimates based on the \Delta -variance , or equivalently , on Fourier spectra .
Using these methods , we obtain D \sim 2.3 for compressive and D \sim 2.6 for solenoidal forcing , which is within the range of fractal dimension estimates inferred from observations ( D \sim 2.0 \dots 2.7 ) .
The velocity dispersion to size relations for both solenoidal and compressive forcings obtained from velocity spectra follow a power law with exponents in the range 0.4 \dots 0.5 , in good agreement with previous studies .