We study the convective boundary layer (CBL) through low-order topological properties of updrafts and downdrafts, that is, based solely on the sign of the vertical velocity. The geometric representation of the CBL as a pair of two-dimensional cubical complexes, one each for updrafts and downdrafts, is exemplarily obtained from two simulations of the CBL, a realistic daily cycle and an idealized quasi-steady CBL growing into linear stratification. Each cubical complex is defined as a set of grid cells that have the same sign of vertical velocity, either positive or negative. Low-order topological invariants, namely the Betti numbers of the cubical complexes, are found to capture key aspects of the boundary-layer organization and evolution over the diurnal cycle. An unsupervised-learning algorithm is trained using the topological invariants in order to classify the spatio–temporal evolution of convection over a whole day. The successful classification of the CBL by using this approach illustrates the potential of such simplified representation of turbulent flow for data reduction and boundary-layer parametrization approaches.

中文翻译：

---

从拓扑不变量推导出对流边界层的结构

我们通过上升气流和下降气流的低阶拓扑特性来研究对流边界层 (CBL)，即仅基于垂直速度的符号。作为一对二维立方复合体的 CBL 的几何表示，每个用于上升气流和下降气流，示例性地从 CBL 的两个模拟中获得，一个现实的日常周期和一个理想化的准稳态 CBL 成长为线性分层。每个立方复合体被定义为一组具有相同垂直速度符号（正或负）的网格单元。发现低阶拓扑不变量，即立方复合体的 Betti 数，可以捕捉到边界层组织和昼夜循环演化的关键方面。使用拓扑不变量训练无监督学习算法，以便对一整天对流的时空演变进行分类。使用这种方法对 CBL 的成功分类说明了湍流的这种简化表示在数据减少和边界层参数化方法中的潜力。