Accuracy requirements to test the applicability of the random cascade model to supersonic turbulence

TL;DR

This paper investigates the applicability of a widely used turbulence model to supersonic regimes, analyzing the accuracy needed in velocity statistics to reliably estimate model parameters and assess its validity.

Contribution

It introduces a Monte Carlo approach to determine the precision required in simulation data to validate the turbulence model in highly compressible flows.

Findings

0.1% accuracy in velocity scaling exponents allows reliable parameter estimation

Simulation data with such accuracy is feasible with current computational resources

Failure to fit parameters accurately may indicate the model's inapplicability

Abstract

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.