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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

A Roadmap for a Lightning Modeling Grand Challenge

This document is a roadmap for building an interconnected model of the physical processes that produce a lightning discharge, and its observable optical and radio signals. We call this a Lightning Modeling Grand Challenge, recognizing that significant effort and coordination of human and financial resources is required to realize the capability. The roadmap serves to outline the coordination of resources necessary to enable stitching together existing knowledge and model components to make a lightning prediction, and to test these predictions with observations. Such a capability does not currently exist. The roadmap is motivated not only by a spirit of scientific inquiry, but by practical challenges faced by US Federal and societal stakeholders. Advancements in lightning observations have outpaced our tests of integrated understanding, leaving many stakeholders unsure how to design their missions to properly detect and discriminate lightning, and unsure how to apply the sometimes-disagreeing lightning signals from diverse instruments. The time is right to connect existing theories and models to support stakeholders in understanding the signals they observe, for needs as diverse as climate monitoring, national security, weather forecasting, public safety, and protection of natural and built environments. The roadmap’s two main technical sections describe the components of a linked physical model, followed by a description of models of lightning signals and sensors that are driven by outputs from the physical model. The goal is to predict the time-varying physical properties of lightning that are self-consistent with the thunderstorm’s structure and dynamics. These lightning signals then propagate through the storm, with realistic dispersion and attenuation, to receivers on the ground or in space. At a high level, the model begins with weather (cloud) model output, including explicit prediction of the electrification of cloud particles. The cloud’s electrical structure drives a model of lightning physics, from initiation, through channel development, and discharges along those channels. Key lightning parameters, such as the temperature and currents in the channel, and their space and time distribution, are then used to produce optical and electromagnetic signal sources that propagate to modeled receivers. This architecture therefore generates a dataset suitable for comparison to existing and envisioned observing systems. The need for additional measurements and field campaigns to support model development is described. In each model sub-component, inputs, outputs, uncertainties, evaluation methods, and next steps are summarized, interleaved with references to the scientific literature. Identifying boundaries between the model sub-components aids in segmenting an integrated, complex model into practical work packages and system sub-components, allowing a diverse team to contribute and maintain the system. We estimate that at least five years of effort and a $\$$10M initial investment is necessary to make a significant step forward. Mechanisms to facilitate community coordination, including annual workshops and open-source code repositories, are described.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Investigating spatial variability of aerosol, cloud condensation nuclei, and ice nucleating particles in mountainous terrain

The ASR-supported Surface Atmosphere Integrated field Laboratory (SAIL) in the East River Watershed (ERW) of the Upper Colorado River Basin in southwestern Colorado ran from fall 2021 to spring 2023. Two monitoring sites were deployed in the East River Watershed as part of SAIL. The two sites were the Aerosol Observation System (AOS) located on Crested Butte Ski Mountain, and the ARM Mobile Facility (AMF-2), located at the Rocky Mountain Biological Laboratory in Gothic, Colorado. To gain a more comprehensive understanding of aerosols in complex, mountainous terrain, Handix Scientific deployed SAIL-Net, a distributed network of six measurement nodes spanning the domain of the SAIL research area from October 2021 to July 2023. Each node measured aerosol particles between 140 nm and 3.4 μm in diameter using a small particle counter (POPS, (Gao et al., 2016)), CNN using a miniature CCN counter (CloudPuck), and INP using the Time-Resolved Aerosol Filter Sampler (TRAPS, Creamean et al. (2018)). Our approach was similar to other studies that aimed to better characterize and understand aerosols and gas-phase pollutants using networks of lower-cost sensors (Caubel et al., 2019; Kelly et al., 2021; Asher et al., 2022). Such studies have identified neighborhood-level variations in pollutant concentrations (Schneider et al., 2017; Popoola et al., 2018; Caubel et al., 2019). Small-scale variations such as this are poorly represented in models and poorly measured by a single monitoring system (Caubel et al., 2019). Previous work has shown the representation error (the ability of measurements to represent a larger area) increases with complex orography, leading to decreases in model accuracy (Schutgens et al., 2017). The overall goal of SAIL-Net was to improve our understanding of the variability of aerosol in ERW, thus increasing our knowledge of aerosol-cloud interactions in this region and informing the usefulness of distributed networks of measurements for future studies.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

ARM FY2026 Aerosol Operations Plan

The U.S. Department of Energy’s Atmospheric Radiation Measurement User Facility (ARM) deploys at each of ARM’s observatories a suite of aerosol and trace gas (further mention of aerosols will assume inclusion of trace gases) instrumentation that constitute the Aerosol Observing System (AOS; Uin et al. 2019).

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

Optical Particle Measurements during EPCAPE Field Campaign Report

This campaign requested the deployment of the U.S. Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) User Facility optical particle counter (OPC) at the first ARM Mobile Facility (AMF1) located at the Scripps Pier in La Jolla, California during the Eastern Pacific Cloud Aerosol Precipitation Experiment (EPCAPE). The addition of the OPC was requested for two reasons. (1) Close the gap between the scanning mobility particle sizer (SMPS) and aerodynamic particle sizer (APS) size distribution from the Aerosol Observing System (AOS) measurements. (2) Principal investigator Petters has been working with Tracking Aerosol Convection Interaction Experiment (TRACER) data to compute particle fluxes from Doppler lidar (Petters et al. 2024). Briefly, backscatter flux is obtained using the eddy covariance technique using the Doppler vertical velocity and attenuated backscatter. Building upon prior studies, we were able to relate backscatter to particle number concentration by calibrating the lidar retrievals against optical particle counter-measured ground-based aerosol size distribution and radiosonde-interpolated relative humidity at lidar sample height. Performing similar analysis was of interest to EPCAPE to better understand the emissions and vertical transport of large particles into the overlying stratus clouds. However, as stated above, this analysis requires an optical size distribution that covers the 0.3-30-μm-diameter size range. The OPC was deployed between 2023-04-14 and 2024-02-14. The deployment, data quality analysis, and data archiving was handled by the DOE ARM instrument mentor team without additional involvement by the principal investigator. Data quality was marked as “routine” for the majority of the campaign.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

Arctic Observing Experiment (AOX) Field Campaign Report

Our ability to understand and predict weather and climate requires an accurate observing network. One of the pillars of this network is the observation of the fundamental meteorological parameters: temperature, air pressure, and wind. We plan to assess our ability to measure these parameters for the polar regions during the Arctic Observing Experiment (AOX, Figure 1) to support the International Arctic Buoy Programme (IABP), the Arctic Observing Network (AON), the International Program for Antarctic Buoys (IPAB), and the Southern Ocean Observing System (SOOS). Accurate temperature measurements are also necessary to validate and improve satellite measurements of surface temperature across the Arctic.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Accuracy of Vaisala RS41 and RS92 Upper Tropospheric Humidity Compared to Satellite Hyperspectral Infrared Measurements

Radiosondes are important for calibrating satellite sensors and assessing sounding retrievals. Vaisala RS41 radiosondes have mostly replaced RS92 in the Global Climate Observing System (GCOS) Reference Upper Air Network (GRUAN) and the conventional network. This study assesses RS41 and RS92 upper tropospheric humidity (UTH) accuracy by comparing with Infrared Atmospheric Sounding Interferometer (IASI) upper tropospheric water vapor absorption spectrum measurements. Using single RS41 and RS92 soundings at three GRUAN and DOE Atmospheric Radiation Measurement (ARM) sites and dual RS92/RS41 launches at three additional GRUAN sites, collocated with cloud-free IASI radiances (OBS), we compute Line-by-Line Radiative Transfer Model radiances for radiosonde profiles (CAL). We analyze OBS-CAL differences from 2015 to 2020, for daytime, nighttime, and dusk/dawn separately if data is available, for standard (STD) RS92 and RS41 processing, and RS92 GRUAN Data Processing (GDP; RS41 GDP is in development). We find that daytime RS41 (even without GDP) has ~1% smaller UTH errors than GDP RS92. RS41 may still have a dry bias of 1–1.5% for both daytime and nighttime, and a similar error for nighttime RS92 GDP, while standard RS92 may have a dry bias of 3–4%. These sonde humidity biases are probably upper limits since “cloud-free” scenes could still be cloud contaminated. Radiances computed from European Centre for Medium-Range Weather Forecasts (ECMWF) analyses match better than radiosondes with IASI measurements, perhaps because ECMWF assimilates IASI measurements. Relative differences between RS41 STD and RS92 GDP, or between radiosondes and ECMWF humidity profiles obtained from the radiance analysis, are consistent with their differences obtained directly from the RH measurements.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Time-domain Modulation of HD 189733 Activities by Its Planet

Star–planet interaction can be studied by analyzing exoplanetary orbit and stellar rotation modulations of the transit emissions from exoplanetary systems. Here we carry out a time-domain analysis of the multiband activities of the HD 189733 star–planet system observed in X-ray, atomic lines, and broadband spectra reported in the literature. We find that the number of activities occuring at beat angles (of the planetary orbit and the stellar rotation) concentrating on a sector with a width of 108° are significantly larger than expected from a uniform distribution of activities, which indicates the existence of a major stellar active magnetic field region. The concentration of activities in this active magnetic field region shows their possible origins from the magnetic field interaction between HD 189733 and its close planet HD 189733 b.

47 OTHER INSTRUMENTATION↗
DOE OSTI2024

LIGHT-bgcArgo-1.0: using synthetic float capabilities in E3SMv2 to assess spatiotemporal variability in ocean physics and biogeochemistry

Since their advent over 2 decades ago, autonomous Argo floats have revolutionized the field of oceanography, and, more recently, the addition of biogeochemical and biological sensors to these floats has greatly improved our understanding of carbon, nutrient, and oxygen cycling in the ocean. While Argo floats offer unprecedented horizontal, vertical, and temporal coverage of the global ocean, uncertainties remain about whether Argo sampling frequency and density capture the true spatiotemporal variability in physical, biogeochemical, and biological properties. As the true distributions of, e.g., temperature or oxygen are unknown, these uncertainties remain difficult to address with Argo floats alone. Numerical models with synthetic observing systems offer one potential avenue to address these uncertainties. Here, we implement synthetic biogeochemical Argo floats into the Energy Exascale Earth System Model version 2 (E3SMv2), which build on the Lagrangian In Situ Global High-Performance Particle Tracking (LIGHT) module in E3SMv2 (E3SMv2-LIGHT-bgcArgo-1.0). Since the synthetic floats sample the model fields at model run time, the end user defines the sampling protocol ahead of any model simulation, including the number and distribution of synthetic floats to be deployed, their sampling frequency, and the prognostic or diagnostic model fields to be sampled. Using a 6-year proof-of-concept simulation, we illustrate the utility of the synthetic floats in different case studies. In particular, we quantify the impact of (i) sampling density on the float-derived detection of deep-ocean change in temperature or oxygen and on float-derived estimates of phytoplankton phenology, (ii) sampling frequency and sea-ice cover on float trajectory lengths and hence float-derived estimates of current velocities, and (iii) short-term variability in ecosystem stressors on estimates of their seasonal variability.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

Prognostic simulations of mixed-phase clouds with model AC-1D v1.0: the impact of aerosol types and freezing parameterizations on ice crystal budgets

Mixed-phase clouds at high latitudes contribute to the uncertainty in predicting cloud feedbacks and climate sensitivity, mainly due to the complexity of microphysical processes that influence the partitioning between the supercooled liquid and ice phases, and hence, cloud radiative effects on regional scales. Particularly in Arctic mixed-phase clouds, the activation of ice-nucleating particles (INPs) from various aerosol populations remains a leading source of uncertainty. We developed an aerosol-cloud one-dimensional (AC-1D) model, which provides a novel framework to prognostically treat INP and ice crystal budgets while explicitly accounting for polydisperse and multicomponent aerosol that activate INPs following different freezing parameterizations. The AC-1D model is informed by large-eddy simulations to probe the impact of INP representation on predicted ice crystal number concentrations (N i ) and ice crystal budgets in mixed-phase Arctic stratus. We apply three immersion freezing (IMF) parameterizations, two time-independent (singular) and one time-dependent (classical nucleation theory), to predict the evolution of the INP reservoir and resulting ice crystal budget from polydisperse mineral dust, organic (humic-like substances), and sea spray aerosol particle size distributions. Our analysis focuses on how variations in aerosol number concentration and cloud system parameters such as cloud cooling rate, cloud-top entrainment rate, and ice crystal fall speed influence the INP reservoir and ice crystal budgets. Furthermore, this study investigates the competitive ice nucleation dynamics in mixed aerosol environments and provides a process-level quantification of the INP budget terms, which directly controls ice crystal budgets. For all studied case scenarios, the aerosol types and associated particle size distributions significantly impact INP and N i , and the choice between a time-dependent and a singular freezing description yields orders-of-magnitude differences in the predicted INP and N i over the 10 h simulation time, reflecting typical cloud lifetimes. Our results show that the influence of cloud cooling, INP entrainment, and sedimentation varies significantly depending on the chosen freezing parameterization. These findings underscore the critical need for robust IMF parameterizations and precise cloud system observations to enhance the accuracy of models in predicting mixed-phase cloud structure and evolution.

Arctic clouds↗
DOE OSTI2026

Evaluation of a high-resolution regional climate simulation for surface and hub-height wind climatology over North America

Assessing the availability of key wind resources requires augmenting observations to support the implementation of wind energy infrastructure. However, observations are limited, necessitating the development of high-resolution, long-term gridded datasets. This study presents a robust, dynamically downscaled climatological dataset, offering 20 years of hourly wind data at a 4 km spatial resolution across North America, and evaluates its performance against observations, including meteorological towers and automated surface-observing system (ASOS) stations, as well as coarse-resolution reanalysis data (the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis version 5 (ERA5)). Results demonstrate that the downscaled high-resolution wind data outperform ERA5 in regions of complex terrain and coastal areas, with improved overlap coefficients for wind data distributions and reduced root mean square errors (RMSEs) for hub-height and near-surface diurnal wind patterns. The downscaled simulation also captures the synoptic drivers of seasonal wind direction patterns reasonably well, indicated by high wind rose similarity indices. This study also provides an analysis of interannual variability, utilizing the dataset's full 20-year period, and model uncertainty, generated by varying model initial conditions and physics parameterizations across 1-year ensemble members, which are key considerations for wind resource assessment in wind farm development.

17 WIND ENERGY↗
DOE OSTI2021

CACTI ARM Mobile Facility (AMF) Measurements of Ice Nucleating Particles

The dataset comprises measures, using Colorado State University's Ice Spectrometer (IS, an immersion freezing device with a range from 0°C down to -26 to -29°C) of atmosperic ice nucleating particle (INP) concentrations taken on ARM's Mobile Facility (AMF-1) Aerosol Observing System, near Villa Yacanto in central Argentina, during the Cloud, Aerosol, and Complex Terrain Interactions (CACTI) Experiment. A filter sampler was mounted on the AOS trailer (2 m below the AOS inlet). Single-use filter units open to the atmosphere fitted with pre-cleaned and pre-sterilized, 47-mm dia. Nuclepore polycarbonate filters were used. Filters were typically drawn for an 8-hour period, totaling 6000 liters sampled on average. After collection, filters were stored at -20°C until processed. Initial processing to obtain spectra of INP number concentration active via the immersion freezing mechanism versus temperature was conducted using CSU's IS instrument (McCluskey et al., 2018). For measurment of INPs, collected aerosol particles were re-suspended in 7-8 mL of 0.1 µm-filtered deionized (DI) water. Aliquots of each suspension, and serial dilutions, were dispensed into trays which were fit into aluminum blocks in the IS. Samples are cooled at 0.33°C min-1 and the freezing temperatures of wells recorded automatically. Cumulative INP concentrations were determined by first calculating the INPs per mL of suspension based on Vali (1971) and then converting to concentration per standard liter of air using the proportion of the total liquid sample dispensed and the air sample volumes. An aliquot of suspension fro selected samples were also heat treated (95°C for 20 min) to denature and deactivate biological INPs, and another aliquot digested in 10% H2O2 at 95°C under UV-B to remove all organic carbon INPs. McCluskey, C. S., J. Ovadnevaite, M. Rinaldi, J. Atkinson, F. Belosi, D. Ceburnis, S. Marullo, T. C. J. Hill, U. Lohmann, Z. A. Kanji, C. O’Dowd, S. M. Kreidenweis, P. J. DeMott, 2018: Marine and Terrestrial Organic Ice Nucleating Particles in Pristine Marine to Continentally-Influenced Northeast Atlantic Air Masses, Journal of Geophysical Research: Atmospheres, 123, 6196–6212, https://doi.org/10.1029/2017JD028033. Vali, G., 1971: Quantitative evaluation of experimental results on the heterogeneous freezing nucleation of supercooled liquids. J. Atmos. Sci., 28, 402–409.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

MOSAiC-Colorado State University Ice Spectrometer

This data set contains atmospheric ice nucleating particle (INP) measurements, using Colorado State University&rsquo;s (CSU) Ice Spectrometer (IS), of filter collections taken at the U.S. DOE ARM AMF2 site onboard the R/V Polarstern P-deck during the Multidisciplinary Drifting Observatory for the Study of Arctic Climate (MOSAiC) field campaign. Samples were collected from October 27, 2019 to September 24, 2020. A filter sampler was mounted approximately 15 m above ground level on a railing in proximity to (and approximately 3 m below) the Aerosol Observation System (AOS) inlet. Single-use filter units open to the atmosphere were pre-cleaned and pre-loaded with 47-mm diameter Nuclepore polycarbonate (0.2 &micro;m pore-diameter) filters. Filters were typically drawn for a three-day period, with an average volume of air filtered of 87,000 standard liters. Total volumes were calculated through recorded daily flow rates using a mass flow meter (TSI). After collection, filters were stored and transported frozen until analysis using CSU&rsquo;s IS instrument (McCluskey et al., 2018). Aerosol particles were first re-suspended in 8 mL of 0.1 &micro;m-filtered deionized (DI) water. Aliquots of each suspension, and corresponding 11-fold dilutions, were dispensed into polymerase chain reaction (PCR) trays and placed into the aluminum blocks of the IS. Samples were cooled at approximately 0.33 &deg;C min -1 and freezing detected optically with corresponding temperatures recorded. Cumulative INP concentrations were determined through calculating the number of INPs per mL of suspension (Vali, 1971) and converting to concentration per standard L of air by accounting for the proportion of liquid used and volume of air collected. All samples were corrected for the number of INPs on the average of four field blanks (cleaned, handled, transported, and analyzed in the same way without air flow). Two-tailed, 95% confidence intervals for binomial sampling are provided (Agresti and Coull, 1998). Select samples were also heat treated (95 &deg;C for 20 min) to denature and deactivate biological INPs present and digested in 10% H 2 O 2 at 95 &deg;C under UV-B for 20 min to remove any organic carbon INPs. Agresti, A, and BA Coull. 1998. "Approximate is better than &ldquo;exact&rdquo; for interval estimation of binomial proportions." American Statistics 52: 119&ndash;126. https://doi.org/10.2307/2685469 McCluskey, CS, J Ovadnevaite, M Rinaldi, J Atkinson, F Belosi, D Ceburnis, &hellip; and PJ DeMott. 2018. "Marine and Terrestrial Organic Ice-Nucleating Particles in Pristine Marine to Continentally Influenced Northeast Atlantic Air Masses." Journal of Geophysical Research: Atmospheres 123 (11): 6196&ndash;6212, https://doi.org/10.1029/2017JD028033 Vali, G. 1971. "Quantitative Evaluation of Experimental Results and the Heterogeneous Freezing Nucleation of Supercooled Liquids." Journal of the Atmospheric Sciences 28: 402-209. https://doi.org/10.1175/1520-0469(1971)028<0402:QEOERA>2.0.CO;2

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Gradient-Descent-Based Learning Gain for Backstepping Controller and Disturbance Observer of Nonlinear Systems

This paper proposes a gradient-descent-based learning (GL) gain for backstepping controller and disturbance observer (DOB) of nonlinear system. The proposed method consists of the GL gain update law, controller, and DOB. The GL gain update law is proposed to adapt the control gain and DOB gain according to the direction that minimizes the cost function. The mathematical analysis reveals that the GL gain always has a positive sign and upper bound. The controller is designed via a backstepping procedure to track the desired output with GL control gain. The DOB is designed to estimate the unknown external disturbance with the GL DOB gain. Because the control and DOB gains are simultaneously tuned to achieve improved performance, the time consumption for tuning can be reduced. In addition, the peaking phenomenon can be avoided initially by a small initial value of GL gains. The stability of the closed-loop system is guaranteed using the input-to-state stability property. The performance of the proposed method was validated via simulations and experiments using a DC motor.

42 ENGINEERING↗
DOE OSTI2021

Causal Evaluations for Identifying Differences between Observations and Earth System Models

We use a nascent data-driven causal discovery method to find and compare causal relationships in observed data and climate model output. We consider ten different features in the Arctic climate collected from public databases on observational and Energy Exascale Earth System Model (E3SM) data. In identifying and analyzing the resulting causal networks, we make meaningful comparisons between observed and climate model interdependencies. This work demonstrates our ability to apply the PCMCI causal discovery algorithm to Arctic climate data, that there are noticeable similarities between observed and simulated Arctic climate dynamics, and that further work is needed to identify specific areas for improvement to better align models with natural observations.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

Predicting Adaptively Chosen Observables in Quantum Systems

Recent advances have demonstrated that 𝒪⁡(log 𝑀) measurements suffice to predict 𝑀 properties of arbitrarily large quantum many-body systems. However, these remarkable findings assume that the properties to be predicted are chosen independently of the data. This assumption can be violated in practice, where scientists adaptively select properties after looking at previous predictions. This work investigates the adaptive setting for three classes of observables: local, Pauli, and bounded-Frobenius-norm observables. We prove that Ω⁡(√𝑀) samples of an arbitrarily large unknown quantum state are necessary to predict expectation values of 𝑀 adaptively chosen local and Pauli observables, where the system size scales exponentially and polynomially in 𝑀, respectively. We also present computationally efficient algorithms that achieve this information-theoretic lower bound. In contrast, for bounded-Frobenius-norm observables, we devise an algorithm requiring only 𝒪⁡(log 𝑀) samples, independent of system size. These results highlight the potential pitfalls of adaptivity in analyzing data from quantum experiments and provide algorithmic tools to safeguard against erroneous predictions in quantum experiments.

Machine learning↗
DOE OSTI2022

Optimal Voltage Control in Low-Observable Unbalanced Distribution Systems

The increased integration of distributed energy resources (DERs) in distribution systems brings both advantages and technical challenges. High levels of DERs can often cause over/under voltage problems. Classical voltage control algorithms are based on full knowledge of voltage states across all nodes in the system. However, this may not be a practical assumption since many locations in distribution systems are unobservable. Therefore, this paper proposes a new model predictive control (MPC) based control algorithm that accounts for system unobservability to efficiently eliminate voltage violations with as low as 50% fraction of observable nodes. Additionally, an analytical voltage sensitivity framework is employed to quickly determine the change in voltage states due to PV injections. The effectiveness of the proposed method is validated via simulations on the unbalanced IEEE 37 node test system.

Abujubbeh, Mohammad↗
DOE OSTI2020

Time-dependent relaxation of observables in complex quantum systems

Abstract We consider time-dependent relaxation of observables in quantum systems of chaotic and regular type. Using statistical arguments and exact numerical solutions we show that the spread of the initial wave function in the Hilbert space and the main characteristics of evolution of observables have certain generic features. The study compares examples of regular dynamics, a completely chaotic case of the Gaussian orthogonal ensemble, a bosonic system with random interactions, and a fully realistic case of the time evolution of various initial non-stationary states in the nuclear shell model. In the case of the Gaussian orthogonal ensemble we show that the survival probability obtained analytically also fully defines the relaxation timescale of observables. This is not the case in general. Using the realistic nuclear shell model and the quadrupole moment as an observable we demonstrate that the relaxation time is significantly longer than defined by the survival probability of the initial state. The full analysis does not show the presence of an analog of the Lyapunov exponent characteristic for examples of classical chaos.

Volya, Alexander (ORCID:0000000217656466)↗