随机线性剪切模型概率分布函数尾部的严格估计
Rigorous estimates of the tails of the probability distribution function for the random linear shear model
摘要：
在以前的工作中，Majda和McLaughlin计算了由线性剪切流以积分形式在${\bf R}^N$上平流的被动标量的$2N$阶矩的显式表达式。在本文中，我们首先计算这些矩的渐近大矩数。我们能够使用这些关于矩的大N行为的信息，以及关于有限阶整函数的一些基本事实，来计算概率分布函数尾部的渐近性。我们发现当能量集中在最大尺度时，概率分布具有高斯尾。随着初始能量向越来越小的尺度移动，我们发现分布的尾部变长，并且分布从高斯平稳地移动到指数和“拉伸指数”。我们还表明，标量的衍生物越来越间歇性，与实验观察一致，并与标量的指数的标量衍生物的指数。
英文摘要：
In previous work Majda and McLaughlin computed explicit expressions for the $2N$th moments of a passive scalar advected by a linear shear flow in the form of an integral over ${\bf R}^N$. In this paper we first compute the asymptotics of these moments for large moment number. We are able to use this information about the large N behavior of the moments, along with some basic facts about entire functions of finite order, to compute the asymptotics of the tails of the probability distribution function. We find that the probability distribution has Gaussian tails when the energy is concentrated in the largest scales. As the initial energy is moved to smaller and smaller scales we find that the tails of the distribution grow longer, and the distribution moves smoothly from Gaussian through exponential and ``stretched exponential''. We also show that the derivatives of the scalar are increasingly intermittent, in agreement with experimental observations, and relate the exponents of the scalar derivative to the exponents of the scalar.
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