We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection–diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or heat equation. By this coordinate transformation, the reversible effects of advection are separated from the irreversible joint effects of advection and diffusion. In this framework, LCSs express themselves as (boundaries of) metastable sets under the Lagrangian diffusion process. In the case of spatially homogeneous isotropic diffusion, averaging the time-dependent family of Lagrangian diffusion operators yields Froyland’s dynamic Laplacian. In the associated geometric heat equation, the distribution of heat is governed by the dynamically induced intrinsic geometry on the material manifold, to which we refer as the geometry of mixing. We study and visualize this geometry in detail, and discuss connections between geometric features and LCSs viewed as diffusion barriers in two numerical examples. Our approach facilitates the discovery of connections between some prominent methods for coherent structure detection: the dynamic isoperimetry methodology, the variational geometric approaches to elliptic LCSs, a class of graph Laplacian-based methods and the effective diffusivity framework used in physical oceanography.

中文翻译：

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拉格朗日相干结构的几何热流理论

我们将拉格朗日相干结构（LCSs）视为物质子集的边界，这些子集的对流演化在弱扩散下是亚稳态的。为了进行检测，我们首先将欧拉对流扩散方程式转换为拉格朗日坐标，其中采用时间依赖性扩散或热方程式。通过这种坐标变换，对流的可逆作用与对流和扩散的不可逆联合作用分开。在此框架中，LCS在拉格朗日扩散过程中表示为亚稳态集（的边界）。在空间均质各向同性扩散的情况下，平均时间相关的拉格朗日扩散算子族可得出Froyland的动态拉普拉斯算子。在相关的几何热方程中混合几何。我们将详细研究和可视化此几何，并在两个数值示例中讨论被视为扩散障碍的LCS与几何特征之间的联系。我们的方法有助于发现一些用于相干结构检测的著名方法之间的联系：动态等距方法，椭圆LCS的变分几何方法，基于拉普拉斯图的方法以及物理海洋学中使用的有效扩散率框架。