不透水河道中随机对流扩散方程概率分布函数的持续不对称性
Persisting asymmetry in the probability distribution function for a random advection-diffusion equation in impermeable channels
摘要：
研究了不渗透边界对无规流平流作用下无源标量场对称性的影响。我们专注于广泛的一类非线性剪切流乘以一个固定的，Ornstein-Uhlenbeck（OUU）的时变过程，包括他们的一些极限情况下，如高斯白噪声或活塞流。对于前一种情况下，线性剪切，最近的研究\cite{camassa2019symmetry}数值表明，衰减的被动标量的长时间限制概率分布函数（PDF）可能是负偏的存在下，不可渗透的通道边界，在严格的结果，在自由空间中建立的限制PDF是正偏的\cite{mclaughlin1996explicit}。在这里，在设置长时间限制偏度的PDF的边界的作用是严格建立上述类使用长时间渐近展开的$N$点相关的随机场从基态本征值微扰方法提出的\cite{bronski1997scalar}。我们的分析结果验证了在\cite{camassa2019symmetry}中数值模拟得到的线性剪切流的结论。此外，我们证明了在足够低的Péclet数的任何剪切流的极限分布是负偏态的。我们表明，前三个时刻的长时间限制明确地依赖于$γ$的值，这是相反的结论在\cite{vanden2001non}的限制PDF在自由空间。本文推导了无空间依赖性和高斯时间涨落流动的$N$点相关器的精确公式，推广了\cite{bronski2007explicit}的结果。该公式的长时间分析与我们的一般剪切流理论是一致的。所有结果都通过蒙特卡罗模拟进行了验证。
英文摘要：
We study the effect of impermeable boundaries on the symmetry properties of a random passive scalar field advected by random flows. We focus on a broad class of nonlinear shear flows multiplied by a stationary, Ornstein-Uhlenbeck (OU) time varying process, including some of their limiting cases, such as Gaussian white noise or plug flows. For the former case with linear shear, recent studies \cite{camassa2019symmetry} numerically demonstrated that the decaying passive scalar's long time limiting probability distribution function (PDF) could be negatively skewed in the presence of impermeable channel boundaries, in contrast to rigorous results in free space which established the limiting PDF is positively skewed \cite{mclaughlin1996explicit}. Here, the role of boundaries in setting the long time limiting skewness of the PDF is established rigorously for the above class using the long time asymptotic expansion of the $N$-point correlator of the random field obtained from the ground state eigenvalue perturbation approach proposed in \cite{bronski1997scalar}. Our analytical result verifies the conclusion for the linear shear flow obtained from numerical simulations in \cite{camassa2019symmetry}. Moreover, we demonstrate that the limiting distribution is negatively skewed for any shear flow at sufficiently low Péclet number. We show that the long time limit of the first three moments depends explicitly on the value of $γ$, which is in contrast to the conclusion in \cite{vanden2001non} for the limiting PDF in free space. We derive the exact formula of the $N$-point correlator for a flow with no spatial dependence and Gaussian temporal fluctuation, generalizing the results of \cite{bronski2007explicit}. The long time analysis of this formula is consistent with our theory for a general shear flow. All results are verified by Monte-Carlo simulations.
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