Accuracy requirements to test the applicability of the random cascade model to supersonic turbulence
Résumé
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 Zp of two-point velocity statistics as a function of two parameters, β and Δ. The model relates them to the dimension D of the most dissipative structures, D = 3 - Δ/(1 - β). While this description has proved most successful for incompressible turbulence (β = Δ = 2/3, and D = 1), its applicability in the highly compressible regime remains debated. For this regime, theoretical arguments suggest D = 2 and Δ = 2/3, or Δ = 1. Best estimates based on 3D periodic box simulations of supersonic isothermal turbulence yield Δ = 0.71 and D = 1.9, with uncertainty ranges of Δ ∈ [0.67,0.78] and D ∈ [2.04,1.60]. With these 5-10% uncertainty ranges just marginally including the theoretical values of Δ = 2/3 and D = 2, doubts remain whether the model indeed applies and, if it applies, for what values of β and Δ. 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 Zp, with p = 1,...,5, should allow for 2% (1%) accurate estimates of β and Δ in the highly compressible regime, but not in the mildly compressible regime. We argue that simulation-based Zp 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 β and Δ, 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.
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