AI 中文总结
为随机介质中平均傍轴波传播开发基于维格纳的相空间框架，从随机抛物波方程推导相关分布演化，经多层建模得出分析结果并验证求解器，与非局部动力学模型比较，还引入大气特化。
AI 中文摘要
我们为随机介质中平均傍轴波传播开发了一个基于维格纳的相空间框架。从随机抛物波方程出发，我们推导了与实现相关的维格纳分布的精确演化，并将系综平均维格纳函数确定为自然的二阶状态变量。平均方程包含一个由混合场 - 介质相关性给出的闭合缺陷，这使得在没有额外假设的情况下无法获得封闭的输运方程。因此，我们将建模组织成一个层次结构，从随机波方程到精确的维格纳公式，再到非局部动力学闭合，最后到小角度区域的局部福克 - 普朗克简化。对于最小的均匀各向同性福克 - 普朗克模型，我们推导了二次矩的闭合演化定律，展示了距离的三次方对光束扩展的贡献，并获得了明确的高斯和高斯 - 谢尔传播公式。这些分析结果用于在一维横向基准中验证相空间分裂求解器。与非局部动力学模型的比较表明，扩散近似对于窄动量转移核是准确的，并且随着有限跳跃效应变得显著，以可控的方式失去有效性。最后，我们基于正则化湍流谱引入了第一个大气特化，得到了一个用标准大气参数表示的有效扩散系数。
英文摘要
We develop a Wigner-based phase-space framework for mean paraxial wave propagation in random media. Starting from the random parabolic wave equation, we derive the exact evolution of the realization-dependent Wigner distribution and identify the ensemble-averaged Wigner function as the natural second-order state variable. The averaged equation contains a closure defect, given by a mixed field--medium correlation, which prevents a closed transport equation from being obtained without additional assumptions. We therefore organize the modelling as a hierarchy from the random wave equation to an exact Wigner formulation, then to a nonlocal kinetic closure, and finally to a local Fokker--Planck reduction in the small-angle regime. For the minimal homogeneous isotropic Fokker--Planck model, we derive closed evolution laws for the quadratic moments, exhibit the cubic-in-distance contribution to beam spreading, and obtain explicit Gaussian and Gauss--Schell propagation formulas. These analytical results are used to validate a phase-space splitting solver in one-dimensional transverse benchmarks. Comparisons with nonlocal kinetic models show that the diffusive approximation is accurate for narrow momentum-transfer kernels and loses validity in a controlled way as finite-jump effects become significant. Finally, we introduce a first atmospheric specialization based on a regularized turbulence spectrum, yielding an effective diffusion coefficient expressed in terms of standard atmospheric parameters.
Comments27 pages, 4 figures
