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 is an element of [0.67, 0.78] and D is an element of [ 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.