Lagrangian averaging theories, most notably the generalized Lagrangian mean (GLM) theory of Andrews and McIntyre, have been primarily developed in Euclidean space and Cartesian coordinates. We reinterpret these theories using a geometric, coordinate-free formulation. This gives central roles to the flow map, its decomposition into mean and perturbation maps, and the momentum 1-form dual to the velocity vector. In this interpretation, the Lagrangian mean of any tensorial quantity is obtained by averaging its pull-back to the mean configuration. Crucially, the mean velocity is not a Lagrangian mean in this sense. It can be defined in a variety of ways, leading to alternative Lagrangian mean formulations that include GLM and Soward and Roberts's volume-preserving version. These formulations share key features that the geometric approach uncovers. We derive governing equations both for the mean flow and for wave activities constraining the dynamics of the perturbations. The presentation focuses on the Boussinesq model for inviscid rotating stratified flows and reviews the necessary tools of differential geometry.

中文翻译：

---

拉格朗日平均的几何方法


拉格朗日平均理论，最著名的是安德鲁斯和麦金太尔的广义拉格朗日均值 （GLM） 理论，主要在欧几里得空间和笛卡尔坐标中发展起来。我们使用几何、无坐标的公式重新解释这些理论。这为流图提供了核心作用，将其分解为均值图和扰动图，以及动量 1 形式对偶到速度矢量。在这种解释中，任何张量的拉格朗日平均值都是通过将其回调到平均值配置来获得的。至关重要的是，从这个意义上说，平均速度不是拉格朗日平均值。它可以以多种方式定义，从而产生替代的拉格朗日均值公式，包括 GLM 和 Soward 和 Roberts 的体积保持版本。这些公式具有几何方法揭示的关键特征。我们推导出了平均流和限制扰动动力学的波浪活动的控制方程。本演示文稿重点介绍了用于无粘性旋转分层流的 Boussinesq 模型，并回顾了微分几何的必要工具。