An Exposition of Discrete Morse Theory and Applications

The classical Morse theory is a powerful tool to study topological properties of a smooth manifold by examining critical points of some differentiable functions on that manifold. Robin Forman developed a discrete variant of Morse theory by adapting it on abstract simplicial complexes that resulted in a new theory with wide applications in other fields of mathematics, computer science, data science, and others. In this thesis, we present Forman’s construction of discrete Morse theory, as well as its main theorems such as the Collapse theorem, discrete Morse inequalities, the theorem for cancelling critical simplices, and some additional topics. We also discuss some applications of discrete Morse theory with a major focus on the concept of persistence in topological data analysis. While many results exist in the literature, we wrote our own proofs, added more details and explanations, and adapted it to our own settings with a strong topological flavor whenever possible

An Exposition of Discrete Morse Theory and Applications

The classical Morse theory is a powerful tool to study topological properties of a smooth manifold by examining critical points of some differentiable functions on that manifold. Robin Forman developed a discrete variant of Morse theory by adapting it on abstract simplicial complexes that resulted in a new theory with wide applications in other fields of mathematics, computer science, data science, and others. In this thesis, we present Forman’s construction of discrete Morse theory, as well as its main theorems such as the Collapse theorem, discrete Morse inequalities, the theorem for cancelling critical simplices, and some additional topics. We also discuss some applications of discrete Morse theory with a major focus on the concept of persistence in topological data analysis. While many results exist in the literature, we wrote our own proofs, added more details and explanations, and adapted it to our own settings with a strong topological flavor whenever possible

Full wave propagation modelling in view to integrated ICRH wave coupling/RF sheaths modelling

RF sheaths rectification can be the reason for operational limits for Ion Cyclotron Range of Frequencies (ICRF) heating systems via impurity production or excessive heat loads. To simulate this process in realistic geometry, the Self-consistent Sheaths and Waves for Ion Cyclotron Heating (SSWICH) code is a minimal set of coupled equations that computes self-consistently wave propagation and DC plasma biasing. The present version of its wave propagation module only deals with the Slow Wave assumed to be the source of RF sheath oscillations. However the ICRF power coupling to the plasma is due to the fast wave (FW). This paper proposes to replace this one wave equation module by a full wave module in either 2D or 3D as a first step towards integrated modelling of RF sheaths and wave coupling. Since the FW is propagative in the main plasma, Perfectly Matched Layers (PMLs) adapted for plasmas were implemented at the inner side of the simulation domain to absorb outgoing waves and tested numerically with tilted BD in Cartesian geometry, by either rotating the cold magnetized plasma dielectric tensors in 2D or rotating the coordinate vector basis in 3D. The PML was further formulated in cylindrical coordinates to account for for the toroidal curvature of the plasma. Toroidal curvature itself does not seem to change much the coupling. A detailed 3D geometrical description of Tore Supra and ASDEX Upgrade (AUG) antennas was included in the coupling code. The full antenna structure was introduced, since its toroidal symmetry with respect to the septum plane is broken (FS bars, toroidal phasing, non-symmetrical structure). Reliable convergence has been obtained with the density profile up to the leading edge of antenna limiters. Parallel electric field maps have been obtained as an input for the present version of SSWICH

Mechanisms of global ocean ventilation age change during the last deglaciation

Marine radiocarbon (14C) is widely used to trace deep-ocean circulation, providing insight into the atmosphere–ocean exchange of CO2 during the last deglaciation. Evidence shows a significantly depleted Δ14C in the glacial deep ocean, suggesting an increased ventilation age at the Last Glacial Maximum (LGM). In this study, using two transient simulations with tracers of 14C and ideal age (IAGE), we found that the oldest ventilation age is not observed at the LGM. In contrast, the models show a modestly younger ventilation age during the LGM compared to the present day. The global mean ventilation ages averaged below 1 km are approximately 800 (630) years and 930 (2000) years at the LGM and in the present day, respectively, in two simulations. This younger glacial ventilation age is mainly caused by the stronger glacial Antarctic Bottom Water (AABW) transport associated with sea ice expansion. Notably, the ocean ventilation age is significantly older predominantly in the deep Pacific during deglaciation compared to the age at the LGM, with global mean ventilation ages peaking at 1900 and 2200 years around 14–12 ka in two simulations, primarily due to the weakening of AABW transport.</p

Plasma noise in TianQin time delay interferometry

TianQin is a proposed geocentric space-based gravitational wave observatory mission, which requires time-delay interferometry (TDI) to cancel laser frequency noise. With high demands for precision, solar-wind plasma environment at $\sim 10^5$ km above the Earth may constitute a non-negligible noise source to laser interferometric measurements between satellites, as charged particles perturb the refractivity along light paths. In this paper, we first assess the plasma noises along single links from space-weather models and numerical orbits, and analyze the time and frequency domain characteristics. Particularly, to capture the plasma noise in the entire measurement band of $10^{-4} - 1$ Hz, we have performed additional space-weather magnetohydrodynamic simulations in finer spatial and temporal resolutions and utilized Kolmogorov spectra in high-frequency data generation. Then we evaluate the residual plasma noises of the first- and second-generation TDI combinations. Both analytical and numerical estimations have shown that under normal solar conditions the plasma noise after TDI is less than the secondary noise requirement. Moreover, TDI is shown to exhibit moderate suppression on the plasma noise below $\sim 10^{-2}$ Hz due to noise correlation between different arms, when compared with the secondary noise before and after TDI.Comment: 12 pages, 15 figures, accepted by Phys. Rev.

SOL RF physics modelling in Europe, in support of ICRF experiments

A European project was undertaken to improve the available SOL ICRF physics simulation tools and confront them with measurements. This paper first reviews code upgrades within the project. Using the multi-physics finite element solver COMSOL, the SSWICH code couples RF full-wave propagation with DC plasma biasing over “antenna-scale” 2D (toroidal/radial) domains, via non-linear RF and DC sheath boundary conditions (SBCs) applied at shaped plasma-facing boundaries. For the different modules and associated SBCs, more elaborate basic research in RF-sheath physics, SOL turbulent transport and applied mathematics, generally over smaller spatial scales, guides code improvement. The available simulation tools were applied to interpret experimental observations on various tokamaks. We focus on robust qualitative results common to several devices: the spatial distribution of RF-induced DC bias; left-right asymmetries over strap power unbalance; parametric dependence and antenna electrical tuning; DC SOL biasing far from the antennas, and RF-induced density modifications. From these results we try to identify the relevant physical ingredients necessary to reproduce the measurements, e.g. accurate radiated field maps from 3D antenna codes, spatial proximity effects from wave evanescence in the near RF field, or DC current transport. Pending issues towards quantitative predictions are also outlined