AI 中文总结
本研究通过欧拉式伪谱方法模拟粗糙二维Kraichnan流中的被动标量平流，验证相关标度律，明确模拟参数随速度粗糙度指数的调整规则，为Kraichnan模型扩展形式的模拟提供数值基准。
AI 中文摘要
我们对粗糙二维Kraichnan流中的被动标量平流开展了系统的欧拉式研究，覆盖了速度粗糙度指数h∈(0,1)的全范围。采用伪谱方法直接求解平流-扩散方程，网格分辨率最高达2048²个点，结合与载体流的时间白噪声构造一致的冻结噪声龙格-库塔格式进行积分。模拟结果重现了平流标量与载体流之间的对偶关系——载体越平滑，标量越粗糙——以及二阶统计量的预测标度律。一旦正确识别标度范围，便可在整个h范围内测量四阶峭度异常，其结果与Frisch等人（1999）的拉格朗日估计、Bernard等人（1998）以及Pumir等人（1997）的微扰预测一致。得益于欧拉式模拟可直接获取完整标量场的优势，我们还研究了标量增量的概率密度函数及对应的高阶统计量，结果显示其系统性偏离高斯分布和对数正态分布，在h取中间值时偏差最为显著。本研究的核心成果是系统说明了模拟参数（尤其是分子扩散率）需如何随h调整，为可靠模拟提供了实用指南；我们进一步表明，与理论标度律的残余偏差可由载体流的有限谱表示定量解释。我们的分析也为模拟Kraichnan模型更现实的扩展形式提供了数值基准，在这些扩展形式中，载体流与高斯乘性混沌耦合，而理论发展尚有限。
英文摘要
We present a systematic Eulerian study of passive scalar advection in rough two-dimensional Kraichnan flows, covering the full range of velocity roughness exponent $h\in(0,1)$. The advection--diffusion equation is integrated directly using a pseudo-spectral method at resolutions up to $2048^2$ grid points, with a frozen-noise Runge--Kutta scheme consistent with the white-in-time construction of the carrier flow. The simulations recover the duality between advected scalar and advecting flow---the smoother the carrier, the rougher the scalar---together with the predicted scaling laws for the second-order statistics. Once the scaling range is properly identified, the fourth-order flatness anomaly is measured across the whole range of $h$, in agreement with the Lagrangian estimates of Frisch \textit{et al.} (1999) and with the perturbative predictions of Bernard \textit{et al.} (1998) and Pumir \textit{et al.} (1997). Benefiting from the fact that our Eulerian simulations give direct access to the full scalar field, we also examine the probability density functions of scalar increments and the corresponding higher-order statistics, which show systematic departures from Gaussianity and from log-normality, with the strongest deviations manifesting for intermediate values of $h$. A central outcome of this work is a systematic account of how the simulation parameters, in particular the molecular diffusivity, must be adjusted with $h$, providing practical guidelines for reliable simulations; we further show that the residual deviations from the theoretical scaling laws are quantitatively accounted for by the finite spectral representation of the carrier flow. Our analysis also serve as a numerical baseline for simulating more realistic extensions of the Kraichnan model, where the carrier flow is coupled with a Gaussian multiplicative chaos and theoretical developments are limited.
