试验随机叶栅模型对超音速turbulence的适用性的精度要求
Accuracy requirements to test the applicability of the random cascade model to supersonic turbulence
摘要：
在分子云和恒星形成的背景下，一个被广泛应用于超声速湍流惯性范围统计的模型将两点速度统计的（可测量的）相对标度指数Z_p表示为两个参数β和δ的函数。该模型将它们与最耗散结构的维度D相关联，D=3-Delta/（1-beta）。虽然这种描述已被证明是最成功的不可压缩湍流（β =Delta=2/3，D=1），其适用性在高度可压缩的制度仍然存在争议。对于这种制度，理论上的论点建议D=2和Delta=2/3，或Delta=1。基于超声速等温湍流的3D周期性箱模拟的最佳估计结果为Delta=0.71和D=1.9，Delta的不确定性范围为[0.67，0.78]，D为[2.04，1.60]。由于这些5- 10%的不确定性范围仅略微包括Delta=2/3和D=2的理论值，因此仍然存在疑问，该模型是否确实适用，以及如果适用，β和Delta的值是多少。我们使用蒙特卡罗方法来模拟实际的模拟数据，并检查哪些因素与拟合质量最相关。我们估计，0.1%（0.05%）的准确Z_p，p=1... 5，应允许在高度可压缩状态下对β和δ进行2%（1%）的准确估计，但在轻度可压缩状态下则不然。我们认为，基于模拟的Z_p与这样的精度是在今天的计算机资源的范围内。如果此类数据不允许beta和Delta的预期高质量拟合，则这可能表明模型不适用于模拟数据。事实上，除了我们在这里研究的模型之外，还有其他模型被提出。
英文摘要：
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.
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