  We use large-scale three-dimensional simulations of supersonic Euler
turbulence to study the physics of a highly compressible cascade. Our numerical
experiments describe non-magnetized driven turbulent flows with an isothermal
equation of state and an rms Mach number of 6. We find that the inertial range
velocity scaling deviates strongly from the incompressible Kolmogorov laws. We
propose an extension of Kolmogorov's K41 phenomenology that takes into account
compressibility by mixing the velocity and density statistics and preserves the
K41 scaling of the density-weighted velocity v=rho^{1/3}u. We show that
low-order statistics of 'v' are invariant with respect to changes in the Mach
number. For instance, at Mach 6 the slope of the power spectrum of 'v' is -1.69
and the third-order structure function of 'v' scales linearly with separation.
We directly measure the mass dimension of the "fractal" density distribution in
the inertial subrange, D_m=2.4, which is similar to the observed fractal
dimension of molecular clouds and agrees well with the cascade phenomenology.
