  Compressible turbulence shapes the structure of the interstellar medium of
our Galaxy and likely plays an important role also during structure formation
in the early Universe. The density PDF and the power spectrum of such
compressible, supersonic turbulence are the key ingredients for theories of
star formation. However, both the PDF and the spectrum are still a matter of
debate, because theoretical predictions are limited and simulations of
supersonic turbulence require enormous resolutions to capture the
inertial-range scaling. To advance our limited knowledge of compressible
turbulence, we here present and analyse the world's largest simulations of
supersonic turbulence. We compare hydrodynamic models with numerical
resolutions of 256^3-4096^3 mesh points and with two distinct driving
mechanisms, solenoidal (divergence-free) driving and compressive (curl-free)
driving. We find convergence of the density PDF, with compressive driving
exhibiting a much wider and more intermittent density distribution than
solenoidal driving. Analysing the power spectrum of the turbulence, we find a
pure velocity scaling close to Burgers turbulence with P(v) k^(-2) for both
driving modes in our hydrodynamical simulations with Mach = 17. The spectrum of
the density-weighted velocity rho^(1/3)v, however, does not provide the
previously suggested universal scaling for supersonic turbulence. We find that
the power spectrum P(rho^(1/3)v) scales with wavenumber as k^(-1.74) for
solenoidal driving, close to incompressible Kolmogorov turbulence, k^(-5/3),
but is significantly steeper with k^(-2.10) for compressive driving. We show
that this is consistent with a recent theoretical model for compressible
turbulence that predicts P(rho^(1/3)v) k^(-19/9) in the presence of a strong
div(v) component as is produced by compressive driving and remains remarkably
constant throughout the supersonic turbulent cascade.
