Accuracy requirements to test the applicability of the random cascade
  model to supersonic turbulence
  A model, which is widely used for inertial rang statistics of supersonic
turbulence in the context of molecular clouds and star formation, expresses
(measurable) relative scaling exponents Z_p of two-point velocity statistics as
a function of two parameters, beta and Delta. The model relates them to the
dimension D of the most dissipative structures, D=3-Delta/(1-beta). While this
description has proved most successful for incompressible turbulence
(beta=Delta=2/3, and D=1), its applicability in the highly compressible regime
remains debated. For this regime, theoretical arguments suggest D=2 and
Delta=2/3, or Delta=1. Best estimates based on 3D periodic box simulations of
supersonic isothermal turbulence yield Delta=0.71 and D=1.9, with uncertainty
ranges of Delta in [0.67, 0.78] and D in [2.04,1.60]. With these 5-10\%
uncertainty ranges just marginally including the theoretical values of
Delta=2/3 and D=2, doubts remain whether the model indeed applies and, if it
applies, for what values of beta and Delta. We use a Monte Carlo approach to
mimic actual simulation data and examine what factors are most relevant for the
fit quality. We estimate that 0.1% (0.05%) accurate Z_p, with p=1...5, should
allow for 2% (1%) accurate estimates of beta and Delta in the highly
compressible regime, but not in the mildly compressible regime. We argue that
simulation-based Z_p with such accuracy are within reach of today's computer
resources. If this kind of data does not allow for the expected high quality
fit of beta and Delta, then this may indicate the inapplicability of the model
for the simulation data. In fact, other models than the one we examine here
have been suggested.
