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At least 199 records · Page 11
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Diurnal evolution of non-precipitating marine stratocumuli in a large-eddy simulation ensemble

Abstract. We explore the cloud system evolution of non-precipitating marine stratocumuli with a focus on the impacts of the diurnal cycle and free-tropospheric (FT) humidity based on an ensemble of 244 large-eddy simulations generated by perturbing initial thermodynamic profiles and aerosol conditions. Cases are categorized based on their degree of decoupling and the cloud liquid water path (LWPc, based on model columns with cloud optical depths greater than one). A budget analysis method is proposed to analyze the evolution of cloud water in both coupled and decoupled boundary layers. More coupled clouds start with a relatively low LWPc and cloud fraction (fc) but experience the least decrease in LWPc and fc during the daytime. More decoupled clouds undergo greater daytime reduction in LWPc and fc, especially those with higher LWPc at sunrise because they suffer from faster weakening of net radiative cooling. During the nighttime, a positive correlation between FT humidity and the LWPc emerges, consistent with higher FT humidity reducing both radiative cooling and the humidity jump, both of which reduce entrainment and increase LWPc. The LWPc is more likely to decrease during the nighttime for a larger LWPc and greater inversion base height (zi), conditions under which entrainment dominates as turbulence develops. In the morning, the rate of the LWPc reduction depends on the LWPc at sunrise, zi, and the degree of decoupling, with distinct contributions from subsidence and radiation.

Chen, Yao-Sheng (ORCID:0000000208354132)↗
DOE OSTI2023

Heterogeneity or illusion? Track the carbon Kuznets curve of global residential building operations

Residential buildings, the “last mile” sector in the global decarbonization, have become the most significant uncertain factor hindering carbon neutrality with increasing household energy demand. To track the operational carbon in buildings, this study investigates the carbon Kuznets curve (CKC) and the corresponding decoupling status of residential building operations at four emission scales by using the data of 30 countries from 2000 to 2019. The results show that (1) the CKC model can fit more than half of the samples. Most curves have an inverted U-shape, with 76% of emission per household and 82% of total emissions. (2) In the presence of the CKC, over four-fifths of global residential buildings peak regardless of any emission scale. The analysis denotes that the carbon emissions of developed countries reach their peaks earlier. In the total emissions, the samples’ peaking proportion is 20% and 25% with income per capita < 20,000 United States dollars (USD) and 20,000–40,000 USD, respectively. (3) The Tapio decoupling analysis and the threshold regression effectively verify the robustness and the heterogeneity of CKCs, respectively. Strong decoupling effects of CKCs in most countries are demonstrated at the scales of emission per floor space and the total emissions, and the heterogeneity proves the classic inverted U-shaped relationship between economy and emissions doesn’t exist in all emitters. Overall, this study tracks the historical carbon emission trajectories of residential building operations at a global scale, providing reference for different economies to simulate the dynamic of building carbon emissions along with the economic booming.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2025

The gravitational index of 5d black holes and black strings

The supersymmetric index of 5d black strings and spinning black holes in M-theory is related to that of 4d black holes in type IIA supergravity when both theories are compactified on the same Calabi-Yau threefold. We find the finite-temperature saddles for the 5d gravitational supersymmetric index by uplifting the recently found attractor saddles of the corresponding 4d index. We study uplifts for two types of geometries: 5d black holes and 5d black strings. For 5d black holes, the uplift guarantees that the index of 4d and 5d black holes match. For 5d black strings, the saddle reproduces the microscopic index at leading order in G N , even without the conventional decoupling limit taken in AdS/CFT. In particular, when the temperature is set to be finite in the 5d flat space region, the black string index is computed from an asymptotically flat solution where the AdS throat is absent. Further, as the temperature is lowered and eventually becomes infinitesimally small in the flat space region, the solution admits a novel decoupling limit in which the AdS 3 throat takes the form of a finite-temperature BTZ black hole that is known to compute the index in AdS 3 /CFT 2 . This represents the first step towards understanding holography for supersymmetric observables in flat space, away from the decoupling limit.

AdS-CFT correspondence↗
DOE OSTI2024

Solid-State, Single-Ion Conducting, Polymer Blend Electrolytes with Enhanced Li + Conduction, Electrochemical Stability, and Limiting Current Density

The development of solid-state polymer electrolytes with high lithium conductivity is crucial to improve lithium-ion battery performance and ameliorate the safety challenges associated with current solvent-based electrolytes. Unfortunately, sluggish polymer segmental dynamics are known to constrain conductivity enhancements in solid-state polymer electrolyte systems, limiting overall performance. In this work, a glassy single-ion-conducting polymer, poly[lithium sulfonyl(trifluoromethane sulfonyl)imide methacrylate] (PLiMTFSI), was blended with a flexible polymer, poly(oligo-oxyethylene methyl ether methacrylate) (POEM), and the impact of PLiMTFSI molecular weight and ion concentration on the thermal and ion-conducting behavior of blend electrolytes was investigated. High ionic conductivities approaching 1 × 10 -2 S/cm at 150 °C were realized in this polymer blend electrolyte system as a result of decoupling Li + transport from polymer segmental dynamics. The decoupled ion transport was attributed to the packing frustration of the glassy PLiMTFSI – sufficient percolating free volume was generated to produce effective ion diffusion pathways. This decoupling was tunable as the ion transport could be altered from being closely coupled to the polymer segmental dynamics (Vogel–Tammann–Fulcher-like) to hopping (Arrhenius-like) by increasing the PLiMTFSI molecular weight and ion concentration. Moreover, the immobilized TFSI anion resulted in high Li + selectivity (Li + transference number = 0.9), high electrochemical stability (up to 4.7 V against Li + / Li), and limiting current density of 1.8 mA/cm 2 (electrolyte thickness = 0.05 cm). These features suggest that this single-ion-conducting, polymer blend electrolyte might be a promising alternative to a benchmark system – salt-doped polyethylene oxide. Moreover, the above characteristics can support the battery operation at higher voltages using energy-dense Li metal anodes, with faster charging rates and enhanced energy/power densities. Altogether, the results suggest that polymer chain packing frustration can be exploited to overcome the constraints of slow polymer segmental relaxations to achieve rapid and highly selective ion transport and enhanced performance in solid-state polymer electrolytes.

36 MATERIALS SCIENCE↗
DOE OSTI2021

A second order yield-temperature relation for accurate inference of burn-averaged quantities in multi-species plasmas

Measured yields and ion temperatures inferred from the fusion product energy spectra can be used as metrics for the performance of an ICF implosion. This can be used to infer species separation, thermal decoupling, flows, or other effects that can cause the inferred ion temperatures to deviate from the true underlying thermal temperature and the yield ratio to deviate from the expected value. Direct inference of the impact of these effects on observed temperatures and yields can be difficult to uncover due to the underlying dependence on the shape and time evolution of the temperature and density profiles of the fusing plasma. Due to differences in the temperature dependence of the reactivities, different fusion products are emitted from different regions and times within the implosion. In order to properly account for this, a second-order analytical expression relating the apparent temperatures and yield ratios is developed. This expression can be coupled to models of yield and/or temperature altering effects to infer their burn-averaged impact on an implosion. The second-order expression shows significant improvement over lower-order expressions in synthetic data studies. Demonstrations of its applications to synthetic data coupled with models of ion thermal decoupling and radial flows are presented. In the case of thermal decoupling, both first and second-order expressions show reasonable levels of accuracy. To consistently infer the amplitude of radial flow with a <10% error, the second-order equation is required.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2026

Molecular dynamics based study on the effects of cation size on the local structure and diffusion in polymerized ionic liquids

We have used coarse-grained molecular dynamics simulations to understand the effect of cation size on ion diffusion in polymerized ionic liquids at temperatures well above the glass transition temperature of the polymers. We investigated dependencies of the diffusion constant on the cation radius and static dielectric constant and interpreted these results in terms of underlying structural changes, and decoupling phenomenon. We have found non-monotonic effects of the cation radius on the diffusion constant, with a maximum at an intermediate radius resulting from two different size dependent effects. Changes in the radial distribution functions characterizing spatial distribution of cations with respect to anions result in a monotonic increase in the coordination number. This yields an increase in the effective interaction energy as a function of cation radius, which can explain the decrease in the diffusion constant. This increase is counteracted at small cation radii by a corresponding increase in decoupling of cation and anion dynamics. In addition, the diffusion constant is found to increase on increasing the static dielectric constant for all radii of the cations. These results highlight the significant role of decoupling in designing polymerized ionic liquids with an enhanced diffusion constant of cations.

Gillespie, Colin [ORNL] (ORCID:0000000320287080)↗
DOE OSTI2026

MATEY: multiscale adaptive transformer models for spatiotemporal physical systems

Accurate representation of the multiscale features in spatiotemporal physical systems using vision transformer architectures requires extremely long, computationally prohibitive token sequences. To address this issue, we propose two novel adaptive tokenization schemes that dynamically adjust patch sizes based on local features: one ensures convergent behavior to uniform patch refinement, while the other offers better computational efficiency. Moreover, we present a set of spatiotemporal attention schemes, where the temporal or axial spatial dimensions are decoupled, to evaluate their baseline computational and data efficiencies and to determine whether adaptive tokenization can improve this performance. We assess the performance of the proposed multiscale adaptive model, MATEY, in a sequence of experiments. Compared to a full spatiotemporal attention scheme or a scheme that decouples only the temporal dimension, we find that fully decoupled axial attention is less efficient and expressive, requiring more training time and model parameters to achieve the same accuracy. The experiments on the adaptive tokenization schemes show that, compared to a uniformly refined model, the proposed schemes achieve comparable or improved accuracy at a much lower cost in the tested two-dimensional settings. While the asymptotic analysis suggests the potential for favorable scaling, empirical validation at substantially longer sequence lengths remains to be performed in future work. Finally, we demonstrate in two fine-tuning tasks featuring different physics that models pretrained on PDEBench data outperform the ones trained from scratch, especially in the low data regime with frozen attention.

adaptive tokenization↗
DOE OSTI2023

Confronting interacting dark radiation scenarios with cosmological data

Dark radiation (DR) is generally predicted in new physics scenarios that address fundamental puzzles of the Standard Model or tensions in the cosmological data. Cosmological data have the sensitivity to constrain not only the energy density of DR but also whether it is interacting. In this paper, we present a systematic study of five types of interacting DR (free-streaming, fluid, decoupling, instantaneous decoupling, and recoupling DR) and their impact on cosmological observables. We modify the Boltzmann hierarchy to describe all these types of interacting DR under the relaxation time approximation. We, for the first time, robustly calculate the collision terms for recoupling scalar DR and provide a better estimation of the recoupling transition redshift. We demonstrate the distinct features of each type of DR on the cosmic microwave background and matter power spectra. We perform Markov-chain Monte Carlo scans using the Planck 2018 data and baryon acoustic oscillation data. Assuming no new physics in the standard model neutrino sector, we find no statistically significant constraints on the couplings of DR, although there is a slight preference for the fluidlike limit of all the cases. In the case of instantaneous decoupling DR, this limit corresponds to a late transition redshift around recombination. Furthermore, the ΔN eff constraint varies marginally depending on the type of DR.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Developing a Reduced 240-Bus WECC Dynamic Model for Frequency Response Study of High Renewable Integration

The ongoing changes in the generation resource mix, driven by the rapid adoption of inverter-based resources (IBR) as well as the early retirement of synchronous generators, are bringing new challenges to the planning and operation of bulk electric power systems. Consequently, there is an increasing need to understand, design, and quantify the reliability service provision from IBRs by performing integrated scheduling and dynamic simulations. However, test systems that have consistent scheduling and dynamic models rarely exist largely because of the decoupled nature of the two simulations on a synchronous generator-dominated system. This paper develops the dynamic model of a reduced Western Electricity Coordinating Council (WECC) system. In conjunction with the existing scheduling model, it is suitable for integrating scheduling and dynamic simulations. The reduced 240-bus WECC model reflects the generation resource mix of the Western Interconnection as of 2018. Moreover, the developed dynamic model is validated against field frequency events measured by FNET/GridEye and preserves the dominant inter-area oscillation mode in WECC.

dynamic simulation↗
DOE OSTI2022

The Roles of Westward-Propagating Waves and the QBO in Limiting MJO Propagation

Abstract A recent study categorized the Madden–Julian oscillation (MJO) during boreal winter season into four types called stand, jump, slow, and fast MJO. This study focuses on the stand and jump MJO. Based on whether their convection penetrates the Maritime Continent (MC), stand and jump MJOs are seen as non-penetrating (NP) MJOs, while the other two are seen as eastward-penetrating (EP) MJOs. Results reveal the relative roles of the westward-propagating wave (WPW), as well as the QBO and ENSO, in limiting MJO propagation. Lack of the premoistening over the southern sea surface of the MC stops NP MJO from penetrating the MC. The active convection of the WPWs hinders the descending branch of the NP MJO circulation and therefore leads to the insufficient meridional advective moistening over the southern sea surface of the MC. The independent convection over the Pacific for jump MJOs is influenced by a combined effect of the QBO and ENSO. The tropopause instability induced by the MJO is found to significantly decouple from its convection over the Pacific in westerly QBO (QBOW) winters more than in easterly QBO (QBOE) winters. For jump MJOs, the independent convection over the central Pacific comes from local WPWs whose amplification and further development into deep convection are correlated to jump the MJOs’ decoupled tropopause instability. For stand MJOs, however, the seasonal-mean La Niña–like cool SST anomalies weaken the WPW activity over the central Pacific and confine WPWs within the western Pacific. Therefore, the decoupled tropopause instability of stand MJOs is out phase of WPWs and fails to induce an independent convection over the central Pacific.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

A Multidisciplinary Approach to Integrated Energy Systems: Advanced Nuclear Plants with Thermal Storage for Dynamic and Flexible Operation in Diverse Markets

Energy supply, distribution, and demand are continuously evolving in accordance with the integration of new power generation sources. In the United States, there is a progressive shift toward incorporating fluctuating energy sources such as wind and solar into the energy mix, alongside the distributed generation. Simultaneously, we are seeing a revolution in consumer technologies such as electric vehicles, energy devices for both residential and commercial use, and industrial processes—all contributing to a fundamental change in energy consumption dynamics. These scenarios have led to huge demand for enhanced flexibility from nuclear power plants (NPPs), requiring them to operate with unprecedented adaptability. Furthermore, advanced NPPs (A-NPPs) could potentially apply to scenarios in which power generation flexibility is prioritized over its current and conventional baseload generation capacity for traditional demand patterns. The present work covers several configurations for coupling nuclear energy production with thermal energy storage (TES). Previous work evaluated systems entailing a simple nuclear-TES configuration that used steam as a heat source for the TES of three different advanced reactors (ARs), but its heat diversion ratio (HDR) for charging was limited. The present work proposes a new configuration that features full decoupling of the nuclear reactor and the power cycle, using reactor coolant (i.e., gas) as a heat source for the TES and offering a theoretically unlimited HDR. This configuration is, however, more complex, as both high- and low-temperature TES (HT-TES and LT-TES) systems are required in order to cover the extended temperature range. Although this configuration imposes certain limitations on the discharge system configuration, it also presents the opportunity to design a more efficient system. For this case in particular, a steam reheat cycle is advantageous. Conventional TES systems take advantage of nominal efficiency when discharging, whereas the reheat cycle in the fully decoupled TES system affords a significant efficiency increase over conventional cases. Material selection, component/equipment sizing, and costing analyses were performed, followed by economic dispatch and size optimization. Dynamic models were developed for the decoupled system in order to generate insights into aspects of off design operation (e.g., drops in cycle efficiency), and these models enable exploration of potential mitigation strategies for addressing such drops. The models also highlight the need to design fail-safes that ensure continuous operational stability and minimize the impact of power ramping on the reactor parameters.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2021

Scaling of Turbulent Viscosity and Resistivity: Extracting a Scale-dependent Turbulent Magnetic Prandtl Number

Turbulent viscosity νt and resistivity ηt are perhaps the simplest models for turbulent transport of angular momentum and magnetic fields, respectively. The associated turbulent magnetic Prandtl number Pr t ≡ ν t /η t has been well recognized to determine the final magnetic configuration of accretion disks. Here, we present an approach to determining these "effective transport" coefficients acting at different length scales using coarse-graining and recent results on decoupled kinetic and magnetic energy cascades. By analyzing the kinetic and magnetic energy cascades from a suite of high-resolution simulations, we show that our definitions of ν t , η t , and Pr t have power-law scalings in the "decoupled range." We observe that Pr t ≈ 1–2 at the smallest inertial-inductive scales, increasing to ≈5 at the largest scales. Furthermore, based on physical considerations, our analysis suggests that Pr t has to become scale independent and of order unity in the decoupled range at sufficiently high Reynolds numbers (or grid resolution) and that the power-law scaling exponents of velocity and magnetic spectra become equal. In addition to implications for astrophysical systems, the scale-dependent turbulent transport coefficients offer a guide for large-eddy simulation modeling.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Vertical profiles of trace gas and aerosol properties over the eastern North Atlantic: variations with season and synoptic condition

Abstract. Because of their extensive coverage, marine low clouds greatly impact the global climate. Presently, the response of marine low clouds to the changes in atmospheric aerosols remains a major source of uncertainty in climate simulations. One key contribution to this large uncertainty derives from the poor understanding of the properties and processes of marine aerosols under natural conditions and the perturbation by anthropogenic emissions. The eastern North Atlantic (ENA) is a region of persistent but diverse subtropical marine boundary layer (MBL) clouds, where cloud albedo and precipitation are highly susceptible to perturbations in aerosol properties. Here we examine the key processes that drive the cloud condensation nuclei (CCN) population in the MBL using comprehensive characterizations of aerosol and trace gas vertical profiles during the Aerosol and Cloud Experiments in the Eastern North Atlantic (ACE-ENA) field campaign. During ACE-ENA, a total of 39 research flights were conducted in the Azores: 20 during summer 2017 and 19 during winter 2018. During summer, long-range-transported aerosol layers were periodically observed in the lower free troposphere (FT), leading to elevated FT CCN concentrations (NCCN). Both biomass burning and pollution from North America contribute to submicron aerosol mass in these layers, with pollution likely the dominant contributor. In contrast, long-range transported continental emissions have a much weaker influence on the aerosol properties in the ENA during the winter season. While the entrainment of FT air is a major source of particle number in the MBL for both seasons, on average it does not serve as a direct source of CCN in the MBL because the average FT NCCN is the same or even lower than that in the MBL. The particle number flux due to FT entrainment is dominated by pre-CCN (particles that are too small to form cloud droplets under typical conditions, i.e., particles with sizes below the Hoppel minimum) due to the elevated Npre-CCN in the lower FT. Once these pre-CCN are entrained into the MBL, they can grow and reach CCN size range through condensational growth, representing an indirect and major source of MBL CCN in the ENA. The impact of synoptic conditions on the aerosol properties is examined. Under pre-front and post-front conditions, shallow convective activity often leads to a deep and decoupled boundary layer. Coalescence scavenging and evaporation of drizzle below clouds lead to reduced NCCN and larger accumulation-mode particle sizes in the upper cloud-containing decoupled layer, indicating that surface measurements overestimate the NCCN relevant to the formation of MBL clouds under decoupled conditions.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

A second order yield-temperature relation for accurate inference of burn-averaged quantities in multi-species plasmas

Measured yields and ion temperatures inferred from the fusion product energy spectra can be used as metrics for the performance of an ICF implosion. This can be to infer species separation, thermal decoupling, flows or other effects that can cause the inferred ion temperatures to deviate from the true underlying thermal temperature and the yield ratio to deviate from the expected value. Direct inference of the impact of these effects on observed temperatures and yields can be difficult to uncover due to underlying dependence on the shape and time evolution of the temperature and density proles of the fusing plasma. Due to differences in the temperature dependence of the reactivities, different fusion products are emitted from different regions and times within the implosion. In order to properly account for this, a second order analytic expression relating the apparent temperatures and yield ratios is developed. This expression can be coupled to models of yield and/or temperature altering effects to infer their burn-averaged impact on an implosion. The second order expression shows significant improvement over lower order expressions in synthetic data studies. Demonstrations of its applications to synthetic data coupled with models of ion thermal decoupling and radial flows are presented. In the case of thermal decoupling both first and second order expressions show reasonable levels of accuracy. To consistently infer the amplitude of radial flow with <10% error the second order equation is required.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

2020 International Conference on Smart Grids and Energy Systems (SGES)

This paper is concerned about phasor-based linear state estimation of three-phase power grids which contain non transposed lines and unsymmetrical branches. It builds upon our previous work where the linear measurement equations were decoupled into three independent sets via the symmetrical component transformation. Such a transformation may however not fully decouple the measurement equations if there are nontransposed transmission lines and unsymmetrical branches in the system. This paper addresses this limitation by deriving additive terms that compensate for the remaining coupling between sequence measurements after the application of the conventional transformation. It will be shown that general unbalanced threephase power grids can still be transformed into three decoupled subsystems and their state estimation solution can be obtained in parallel by the proposed approach.

Khalili, Ramtin↗
DOE OSTI2024

Zero-Strain Metal-Insulator Transition by the Local Fluctuation of Cation Dimerization

The coupled electronic and structural transitions in metal-insulator transition (MIT) hinder ultrafast switching and ultimate endurance. Decoupling these transitions and achieving a zero-strain electronic MIT can overcome the fundamental limitations of MIT in solid materials. Here, this study demonstrates that iso-valent Ti dopants in supercooled VO 2 epitaxial films cause MIT with minimal hysteresis without changing unit-cell volume and crystal symmetry. The Ti dopants in the VO 2 lattice locally alter the configuration of V-V pairs, where the long-range ordering in V-V pairs is disrupted, and the nano-domains of V-V dimers are formed. Strikingly, these local V-V dimers persist even above the electronic transition temperature (T MI ), facilitating the zero-strain electronic MIT with nanoscale structural heterogeneity. The geometrically compatible interface between insulating and metallic phases drastically enhances switching speed and endurance during electrically and optically driven zero-strain MIT. In conclusion, this discovery offers a fresh perspective on the scientific understanding of MIT and the improved functionality in terms of device speed and reliability by decoupling electronic and structural transitions.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Crossing the Oxo‐Peroxo Wall for Selective Electrochemical Epoxidation

Electrochemical oxidation in water requires the formation of reactive oxygen species to be able to oxidize unsaturated hydrocarbons to epoxides, aldehydes, and ketones. These reactions, broadly classified as alternative oxidation reactions (AOR), directly compete with the prevalent oxygen evolution reaction (OER). In molecular catalysis, the Oxo-Wall dictates a transition from a stable oxo intermediate (OER active) to a meta-stable metal-oxo (OER inactive) generally occurs. In this work on heterogeneous catalysis, the same Oxo-Wall applies, however, a meta-stable oxo preferentially coordinates with lattice oxygen to form a more stable surface peroxo intermediate. A universal free energy onset of this process is identified at 3.39 eV under electrochemical activation in water and show that it is completely decoupled from the OER oxo species. Such decoupling gives rise to a new region of oxygen reactivity relevant for AOR where a selective oxidation of the unsaturated C-C bonds is predicted to occur instead of OER. A distinct AOR overpotential volcano is constructed and identify recently reported electrocatalysts, including palladium-platinum for propylene epoxidation and silver-nickel for ethylene epoxidation, along with others such as TiO 2 and CuO. Broader implications and limitations of electrochemical AOR are discussed, highlighting their potential to enable electrochemically enhanced thermal catalysis.

Electrocatalysis↗
DOE OSTI2023

Large U(1) charges from flux breaking in 4D F-theory models

We study the massless charged spectrum of U(1) gauge fields in F-theory that arise from flux breaking of a nonabelian group. The U(1) charges that arise in this way can be very large. In particular, using vertical flux breaking, we construct an explicit 4D F-theory model with a U(1) decoupled from other gauge sectors, in which the massless/light fields have charges as large as 657. This result greatly exceeds prior results in the literature. We argue heuristically that this result may provide an upper bound on charges for light fields under decoupled U(1) factors in the F-theory landscape. We also show that the charges can be even larger when the U(1) is coupled to other gauge groups.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗