AI 中文总结
该研究针对广义拉格朗日平均理论的体积不守恒缺陷，提出基于极分解的保体积拉格朗日平均方法，开发数值算法并在二维不可压缩和浅水模型模拟中验证其有效性。
AI 中文摘要
Andrews与McIntyre提出的广义拉格朗日平均（GLM）理论为研究波与流的相互作用提供了强大框架。该理论的一个缺陷是，即使对于不可压缩流体，拉格朗日平均速度也会出现散度，因为将流体微团的拉格朗日标签映射到其平均位置的平均流映射不保持体积，例如会导致涡旋在拉格朗日平均下收缩。我们通过修改平均流映射的定义来克服这一缺陷，选择它为最接近“裸”GLM平均映射的保体积映射。最优传输理论的一个标准结果表明，新的平均映射是GLM平均映射极分解中的保体积因子。我们开发并实现了一种数值方法，用于从模拟数据计算对应的拉格朗日平均场，该实现基于近期开发的使用指数和巴特沃斯滤波器实时计算拉格朗日平均的算法。我们在二维不可压缩和浅水模型的模拟中证明了保体积拉格朗日平均的价值，并比较了有无保体积约束时得到的拉格朗日平均场。
英文摘要
The generalised Lagrangian mean (GLM) theory of Andrews & McIntyre provides a powerful framework to study the interactions between waves and flows. A drawback of this theory is that the Lagrangian mean velocity is divergent even for incompressible fluids because the mean flow map, which sends the Lagrangian labels of fluid parcels to their mean positions, does not preserve volume. This results, for instance, in vortices shrinking under Lagrangian averaging. We overcome this drawback by revising the definition of the mean flow map, choosing it as the volume-preserving map closest to the "bare" GLM mean map. A standard result of optimal-transport theory then shows that the new mean map is the volume-preserving factor in the polar factorization of the GLM mean map. We develop and implement a numerical method for the computation of the corresponding Lagrangian mean fields from simulation data. The implementation builds on recently developed algorithms for the on-the-fly computation of Lagrangian means using the exponential and Butterworth filters. We demonstrate the value of volume-preserving Lagrangian averaging in simulations of the two-dimensional incompressible and shallow-water models. We compare the Lagrangian-mean fields obtained with and without the volume-preservation constraint.
