AI 中文总结
针对湍流随机壳模型的核心问题，本文基于第一性原理微扰方法推导任意阶结构函数的反常标度指数，经数值模拟验证，将隐藏对称微扰框架扩展至非线性级联系统。
AI 中文摘要
从运动方程推导反常标度指数仍是湍流统计理论的核心问题。本文针对非线性随机二进壳模型，获得了第一性原理微扰解。该模型保留了确定性动力学的保守级联结构和精确标度对称性，而随机输运涨落提供了微扰环境，其中领头阶重标度动力学为高斯型。利用惯性区方程经统计恢复的隐藏标度对称性，确定了重标度变量的平稳统计特性。随后将反常标度表述为乘子统计的Perron-Frobenius本征值问题。所得微扰展开在弱噪声区给出了任意阶结构函数标度指数的显式解析表达式。直接数值模拟对理论预测进行了独立验证。结果表明，该隐藏对称微扰框架可从线性随机模型扩展至真正的非线性级联系统。
英文摘要
Deriving anomalous scaling exponents from the equations of motion remains a central problem in the statistical theory of turbulence. Here we obtain a first-principles perturbative solution for a nonlinear stochastic dyadic shell model. The model preserves the conservative cascade structure and exact scaling symmetry of the deterministic dynamics, while stochastic transfer fluctuations provide a perturbative setting in which the leading-order rescaled dynamics is Gaussian. Using the statistically restored hidden scaling symmetry of the inertial-range equations, we determine the stationary statistics of the rescaled variables. We then formulate anomalous scaling as a Perron--Frobenius eigenvalue problem for the multiplier statistics. The resulting perturbative expansion yields explicit analytical expressions for the scaling exponents of structure functions of arbitrary order in the weak-noise regime. Direct numerical simulations provide an independent verification of the theoretical predictions. The results demonstrate that the hidden-symmetry perturbation framework extends from linear random models to a genuinely nonlinear cascade system.
Comments37 pages, 8 figures
