随机流体动力学的几何学
The Geometry of Stochastic Fluid Dynamics
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中文总结 AI 辅助
本文是一篇教学综述，梳理了从李群不变随机变分原理出发的随机几何力学框架进展，将其应用于上层海洋动力学建模，推导了SALT、LA-SALT方程，构建了SOAM模型回归Hasselmann范式。
中文摘要 AI 辅助
随机几何力学（SGM）具有量化地球海洋和大气全球气候建模中不确定性的潜力，同时能保留理想流体流动的基础平流输运特性。本文是一篇教学综述，梳理了近期从李群不变随机变分原理出发、针对上层海洋动力学建模的随机几何力学数学框架的进展，分为五个部分：第一部分讨论几何力学在确定性流体动力学中应用的起源；第二部分以确定性三维欧拉布辛涅斯克（EB）方程为例；第三部分为三维欧拉布辛涅斯克（EB）方程添加随机输运，推导得到SALT方程（SALT是随机李输运平流的缩写）；第四部分聚焦拉格朗日平均随机李输运，缩写为LA-SALT，LA-SALT将大气“气候”视为系综期望，大气“天气”则视为路径依赖波动场，如Ed Lorenz在1995年著名讲座中所讨论的；第五部分将SALT和LA-SALT应用于构建随机海气模型，缩写为SOAM。SOAM方法回归了Hasselmann 1976年的范式，该范式将通用气候模型分解为确定性和随机两部分。
英文摘要
Stochastic geometric mechanics (SGM) is known for its potential utility in quantifying uncertainty in global climate modelling of the Earth's ocean and atmosphere while also preserving the fundamental advective transport properties of ideal fluid flow. This paper is a pedagogical review of the recent developments of the mathematical framework of stochastic geometric mechanics obtained from Lie group-invariant stochastic variational principles in the context of model building for upper ocean dynamics, The paper is divided into the following five parts. Part I discusses the origins of geometric mechanics applications in deterministic fluid dynamics. Part II focuses on the example of the deterministic 3D Euler Boussinesq (EB) equations. Part III adds stochastic transport to the 3D Euler Boussinesq (EB) and derives its SALT equations. (SALT is the abbreviation of Stochastic Advection by Lie Transport.) Part IV focuses on Lagrangian Averaged Stochastic Lie Transport, abbreviated as LA-SALT. LA-SALT treats atmospheric `climate' as the ensemble expectation, while the atmospheric `weather' is treated as a field of pathwise fluctuations, as discussed in Ed Lorenz's famous 1995 lecture. Part V applies SALT and LA-SALT to create stochastic Ocean--Atmosphere Models, abbreviated as SOAM.. The SOAM approach brings us back to Hasselmann's 1976 paradigm, which decomposes a general climate model into its deterministic and stochastic parts.
