In this paper, 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 (Camassa et al., 2019) 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 (McLaughlin and Majda, 1996). 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 Bronski and McLaughlin (1997). Our analytical result verifies the conclusion for the linear shear flow obtained from numerical simulations in Camassa et al. (2019). Moreover, we demonstrate that the limiting distribution is negatively skewed for any shear flow at sufficiently low Péclet number. We demonstrate the convergence of the Ornstein–Uhlenbeck case to the white noise case in the limit $\gamma \to \infty$ of the OU damping parameter, which generalizes the results for free space in Resnick (1996) to the channel domain problem. We show that the long time limit of the first three moments depends explicitly on the value of $\gamma$, which is in contrast to the conclusion in Vanden Eijnden (2001) for the limiting PDF in free space. To find a benchmark for theoretical analysis, 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 Bronski et al. (2007). 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.

在本文中，我们研究了不可渗透边界对随机流平流的随机被动标量场对称性的影响。我们专注于一大类非线性剪切流乘以平稳的 Ornstein-Uhlenbeck (OU) 时变过程，包括它们的一些极限情况，例如高斯白噪声或塞流。对于具有线性剪切的前一种情况，最近的研究（Camassa 等人，2019 年）从数值上证明，与严格的通道边界相比，衰减的被动标量的长时间限制概率分布函数 (PDF) 可能在不渗透通道边界的情况下呈负偏斜。导致建立限制 PDF 的自由空间正偏斜（McLaughlin 和 Majda，1996 年）。这里，$N$从 Bronski 和 McLaughlin (1997) 提出的基态特征值扰动方法获得的随机场的点相关器。我们的分析结果验证了从 Camassa 等人的数值模拟中获得的线性剪切流的结论。(2019)。此外，我们证明了在足够低的 Péclet 数下，对于任何剪切流，极限分布都是负偏斜的。我们证明了 Ornstein-Uhlenbeck 情况在极限情况下与白噪声情况的收敛$\gamma \to \infty$OU 阻尼参数，它将 Resnick (1996) 中自由空间的结果推广到通道域问题。我们表明前三个时刻的长时间限制明确取决于$\gamma$，这与 Vanden Eijnden (2001) 中关于自由空间中有限 PDF 的结论形成对比。为了找到理论分析的基准，我们推导出精确的公式$N$无空间依赖性和高斯时间波动的流的点相关器，概括了 Bronski 等人的结果。(2007)。这个公式的长时间分析与我们对于一般剪切流的理论是一致的。所有结果均通过蒙特卡罗模拟验证。