Licon-Salaiz, Jose 
ORCID: 0000-0002-8733-2256
(2020).
Topological Aspects of Turbulence in the Planetary Boundary Layer.
PhD thesis, Universität zu Köln.
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Abstract
The second decade of the 21st century has seen significant advances in the nascent field of Topological Data Analysis (TDA), both from the theoretical and applied standpoints. The subject of this dissertation is the investigation, by topological means, of the spatial structure of turbulent convection in the planetary boundary layer (PBL) of the Earth. It thus serves a dual purpose: first, to introduce atmospheric science as a source of rich and challenging problems to the applied topology community; second, to introduce topological ideas and methods to the atmospheric science community, as a complement to commonly applied methods such as global bulk measurements and spectral transforms. The first problem considered here is the interaction of the atmosphere with complex land surface patterns, where the limitations of classical approaches are known. Low-order topological invariants, the Betti numbers, are computed for the vertical wind velocity fields of Large-Eddy Simulation (LES) models, and used to quantify the structural changes in convective flow induced by different surface patterns. These invariants are also shown to capture structural properties of the boundary layer and its temporal evolution, as they induce a physically meaningful partition of the model domain into the corresponding boundary layer subregions. Here, the results obtained from LES model data are compared with those from a direct numerical simulation (DNS). Next, the connectivity of updrafts in an LES is determined algorithmically, and used in the quantitative analysis of the hierarchical organization of convective flow. An empirical law of updraft scaling is derived, which agrees with the expected Kolmogorov scaling. A tree-like representation is used to quantify the rate of plume coalescence, and its dependence on land surface heterogeneity. This representation is then used to assess the relative efficiency of different land surface types in sustaining convection. Finally, the spatial distribution of shallow cumulus clouds is analyzed. To this end, the stable rank invariant from persistent homology is interpreted as a homological density function, which justifies its use as a spatially descriptive statistical measure. This allows direct comparison of the spatial pattern of a cloud field with different spatial point processes. An index for spatial organization is introduced based on persistent homology, and comparison with the classical \(I_\text{org}\) index shows it is more resilient to spatial noise. Finally, a morphological classification of cloud fields based on the persistence contour formalism is described.
| Item Type: | Thesis (PhD thesis) | ||||||||
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| URN: | urn:nbn:de:hbz:38-121644 | ||||||||
| Date: | 23 September 2020 | ||||||||
| Language: | English | ||||||||
| Faculty: | Faculty of Mathematics and Natural Sciences | ||||||||
| Divisions: | Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute | ||||||||
| Subjects: | Natural sciences and mathematics Mathematics |
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| Date of oral exam: | 4 February 2020 | ||||||||
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| Refereed: | Yes | ||||||||
| URI: | http://kups.ub.uni-koeln.de/id/eprint/12164 |
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