  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~3.4 for
compressive forcing and beta~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~2.3 for compressive and D~2.6 for solenoidal
forcing, which is within the range of fractal dimension estimates inferred from
observations (D~2.0-2.7). The velocity dispersion to size relations for both
solenoidal and compressive forcing obtained from velocity spectra follow a
power law with exponents in the range 0.4-0.5, in good agreement with previous
studies.
