幂律搅拌下二维turbulence的标度区：理论与数值实验
Scaling regimes of 2d turbulence with power law stirring: theories versus numerical experiments
摘要：
本文研究了二维不可压速度场的纳维-斯托克斯方程和二维标量场的对流扩散方程对的统计性质。该系统是在一个制度的充分发展的湍流搅拌强迫场与高斯统计，白噪声在时间和自相似的空间。在这种设置中，如果搅拌集中在小的空间尺度，好像由于热波动，它是可能的，进行第一原理紫外重整化群分析的模型的标度行为。Kraichnan的二维湍流唯象理论支持惯性范围的存在，其特征在于在比搅拌更大的尺度上的逆能量传递。然而，对于我们的模型，Kraichnan的理论意味着标度预测与重整化群的结果根本不一致。我们进行精确的数值实验，以评估实际的统计特性的二维湍流与幂律搅拌。我们的研究结果清楚地表明，一个适应版本的Kraichnan的理论是一致的观察到的现象。我们还提供了一些理论方案来解释重整化群分析和观察到的现象之间的差异。
英文摘要：
We inquire the statistical properties of the pair formed by the Navier-Stokes equation for an incompressible velocity field and the advection-diffusion equation for a scalar field transported in the same flow in two dimensions (2d). The system is in a regime of fully developed turbulence stirred by forcing fields with Gaussian statistics, white-noise in time and self-similar in space. In this setting and if the stirring is concentrated at small spatial scales as if due to thermal fluctuations, it is possible to carry out a first-principle ultra-violet renormalization group analysis of the scaling behavior of the model. Kraichnan's phenomenological theory of two dimensional turbulence upholds the existence of an inertial range characterized by inverse energy transfer at scales larger than the stirring one. For our model Kraichnan's theory, however, implies scaling predictions radically discordant from the renormalization group results. We perform accurate numerical experiments to assess the actual statistical properties of 2d-turbulence with power-law stirring. Our results clearly indicate that an adapted version of Kraichnan's theory is consistent with the observed phenomenology. We also provide some theoretical scenarios to account for the discrepancy between renormalization group analysis and the observed phenomenology.
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