检测有限时间相干集的诞生与消亡
Detecting the birth and death of finite-time coherent sets
AI总结：
针对现有方法无法确定相干集完整寿命及生灭时间的问题，提出基于膨胀动态拉普拉斯算子与半物质FTCS的新构造，建立其与标准动态拉普拉斯的数学联系，可直接获取相干集的数量、寿命与演化信息。
摘要：
有限时间相干集（FTCSs）是相空间中在有限时间段内抵抗与周围空间混合的特殊区域；其物理表现形式分别包括海洋中的涡旋和大气中的涡旋。有限时间相干集的边界是拉格朗日相干结构（LCSs）的典型例子。实际计算FTCS和LCS时，时长的选择对计算成功至关重要。如果该时长超过单个对象的相干寿命，现有方法将无法检测到寿命更短的相干性。确定相干对象的完整寿命具有明确的实际意义，但在复杂的实际场景中，例如一片寿命各异的海洋涡旋场，现有方法无法实现这一点。此外，确定相干集的出现和消亡时间具有重要的科学意义。在本研究中，我们提出了新的构造方法来解决这些问题。核心组成部分是膨胀动态拉普拉斯算子和半物质有限时间相干集（semi-material FTCSs）的概念。我们建立了膨胀动态拉普拉斯算子与标准动态拉普拉斯算子[Froyland 2015]之间的紧密数学联系，证明后者是前者的极限情况。膨胀动态拉普拉斯算子的谱和本征函数可直接提供相干集的数量、寿命和演化信息。
英文摘要：
Finite-time coherent sets (FTCSs) are distinguished regions of phase space that resist mixing with the surrounding space for some finite period of time; physical manifestations include eddies and vortices in the ocean and atmosphere, respectively. The boundaries of finite-time coherent sets are examples of Lagrangian coherent structures (LCSs). The selection of the time duration over which FTCS and LCS computations are made in practice is crucial to their success. If this time is longer than the lifetime of coherence of individual objects then existing methods will fail to detect the shorter-lived coherence. It is of clear practical interest to determine the full lifetime of coherent objects, but in complicated practical situations, for example a field of ocean eddies with varying lifetimes, this is impossible with existing approaches. Moreover, determining the timing of emergence and destruction of coherent sets is of significant scientific interest. In this work we introduce new constructions to address these issues. The key components are an inflated dynamic Laplace operator and the concept of semi-material FTCSs. We make strong mathematical connections between the inflated dynamic Laplacian and the standard dynamic Laplacian [Froyland 2015], showing that the latter arises as a limit of the former. The spectrum and eigenfunctions of the inflated dynamic Laplacian directly provide information on the number, lifetimes, and evolution of coherent~sets.
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