发表机构
University of California, Los Angeles; Weizmann Institute of Science（加州大学洛杉矶分校; 魏茨曼科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出辛几何闭合（SGC），基于多层随机模型几何约化构建，保留二维湍流双级串所需几何与哈密顿结构，重现Kraichnan理论算子架构，为几何约束的稳定数据驱动闭合提供基础。
AI 中文摘要
二维湍流的次网格参数化必须保留支持双级串的几何结构。经典闭合方法与数据驱动闭合方法通常缺乏在线稳定性所需的对称性。我们引入辛几何闭合（Symplectic Geometric Closure, SGC），该方法源自多层随机模型的几何约化。其核心是一个未解析辛自由度的隐藏储层，通过约束哈密顿交换对解析流产生反馈。稳定性由几何方式保证。该相互作用由辛泛函$\n\naught$产生，它保留增广拟涡能，为Navier–Stokes–$\n\nbeta$核心产生拉回有界性，并在超粘性实现中产生紧致随机拉回吸引子。同一生成器产生一个涌现的哈密顿次网格速度，其强迫恰好是解析涡量的李输运。因此，未解析的随机活动在保持不可压缩二维运动哈密顿结构的同时，对平流几何进行了重正化。消除储层会得到有限记忆的欧拉理论。在一圈线重正化阶，储层收缩重现了Kraichnan型直接相互近似理论的算子结构。关键是，嵌套雅可比顶点与随机伽利略变换兼容，并在欧拉相互作用中对均匀扫掠模式施加了本征的四阶红外抑制$\n\nmathcal{O}(p^4)$。由于去相关由应变诱导变形而非扫掠控制，修饰后的欧拉输运理论自洽地承认经典的$k^{-5/3}$逆能量级串和$k^{-3}$正向拟涡能级串。因此，SGC为Kraichnan纲领提供了欧拉几何实现，并为几何约束、保持稳定性的数据驱动闭合提供了原则性基础。
英文摘要
The equations of fluid motion are known; retaining their physics after removing most degrees of freedom remains a fundamental problem. Closure must preserve the geometry of the dynamics it replaces. We introduce Symplectic Geometric Closure (SGC), a prognostic stochastic field theory carrying Euler's Hamiltonian, area-preserving transport into two-dimensional and $β$-plane turbulence. SGC couples resolved vorticity to a hidden stochastic reservoir. The coupling conserves augmented enstrophy while permitting bidirectional energy transfer. This yields a compact random attractor in the hyperviscous Navier--Stokes--$β$ realization. Numerically, SGC sustains jets, vortices and filaments in high-Reynolds-number turbulence with high fidelity to filtered DNS at an inertial-range cutoff, preserving geometric identities to machine precision. The induced transport folds, stretches and rearranges vorticity: cross-gradient coupling selects interactions through resolved--reservoir gradient misalignment, retaining geometric selectivity and memory. The same geometry addresses spurious sweeping decorrelation, a longstanding obstacle to Eulerian closure. The interaction vertex excludes uniform translation and is Random Galilean compatible. Eliminating the reservoir generates finite memory, stochastic backscatter and a Dyson--Volterra equation for the dressed propagator. Its one-loop, line-renormalized self-energy reproduces Kraichnan-type Direct-Interaction Approximation architecture with fourth-order infrared suppression of sweeping. Deformation-controlled memory and conservative triad constraints yield the $k^{-5/3}$ inverse-energy and $k^{-3}$ forward-enstrophy cascades under standard assumptions. SGC thus realizes Kraichnan's program through Eulerian field dynamics, opening a route to data-driven closures that learn admissible geometric interactions rather than unconstrained forces.
Comments66 pages, 3 Figures
