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NASA NTRS1978

Inflight optical measurement of antenna surfaces

A technology base was developed for a wide variety of applications oriented sensors to meet requirements for the fabrication, assembly, test, surface figure monitoring, and ultimately surface figure active control of large space antennas. An optical sensor technique is described which establishes an ideal centerline at each beam during fabrication or later during assembly. Deviations from the centerline, either in lateral deformation or in twist, are measured to produce limit warnings or to evoke active control at the building machine.

Neiswander, R. S.↗

NASA NTRS2004

The Impact of Subsampling on MODIS Level-3 Statistics of Cloud Optical Thickness and Effective Radius

The MODIS Level-3 optical thickness and effective radius cloud product is a gridded l deg. x 1 deg. dataset that is derived from aggregation and subsampling at 5 km of 1 km, resolution Level-2 orbital swath data (Level-2 granules). This study examines the impact of the 5 km subsampling on the mean, standard deviation and inhomogeneity parameter statistics of optical thickness and effective radius. The methodology is simple and consists of estimating mean errors for a large collection of Terra and Aqua Level-2 granules by taking the difference of the statistics at the original and subsampled resolutions. It is shown that the Level-3 sampling does not affect the various quantities investigated to the same degree, with second order moments suffering greater subsampling errors, as expected. Mean errors drop dramatically when averages over a sufficient number of regions (e.g., monthly and/or latitudinal averages) are taken, pointing to a dominance of errors that are of random nature. When histograms built from subsampled data with the same binning rules as in the Level-3 dataset are used to reconstruct the quantities of interest, the mean errors do not deteriorate significantly. The results in this paper provide guidance to users of MODIS Level-3 optical thickness and effective radius cloud products on the range of errors due to subsampling they should expect and perhaps account for, in scientific work with this dataset. In general, subsampling errors should not be a serious concern when moderate temporal and/or spatial averaging is performed.

Oreopoulos, Lazaros↗

DOE OSTI2017

Multi-resolution anisotropy studies of ultrahigh-energy cosmic rays detected at the Pierre Auger Observatory

Here, we report a multi-resolution search for anisotropies in the arrival directions of cosmic rays detected at the Pierre Auger Observatory with local zenith angles up to 80(o) and energies in excess of 4 EeV (4 × 10 18 eV). This search is conducted by measuring the angular power spectrum and performing a needlet wavelet analysis in two independent energy ranges. Both analyses are complementary since the angular power spectrum achieves a better performance in identifying large-scale patterns while the needlet wavelet analysis, considering the parameters used in this work, presents a higher efficiency in detecting smaller-scale anisotropies, potentially providing directional information on any observed anisotropies. No deviation from isotropy is observed on any angular scale in the energy range between 4 and 8 EeV. Above 8 EeV, an indication for a dipole moment is captured, while no other deviation from isotropy is observed for moments beyond the dipole one. The corresponding p-values obtained after accounting for searches blindly performed at several angular scales, are 1.3 × 10 -5 in the case of the angular power spectrum, and 2.5 × 10 -3 in the case of the needlet analysis. While these results are consistent with previous reports making use of the same data set, they provide extensions of the previous works through the thorough scans of the angular scales.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1991

Mesoscale and large-scale variability of the Antarctic circumpolar current

This investigation of the physical oceanography of the Southern Ocean will carry out two parallel efforts during the years preceding the launch of TOPEX/POSEIDON. First, the Geosat data will be used to develop a preliminary descriptive picture of the mesoscale and large-scale, low-frequency surface circulation of the Southern Ocean. Some of this analysis of Geosat data has already begun. For example, as a measure of the geographical distribution of mesoscale variability, a color-coded map of the standard deviation of sea level from two years of Geosat data is shown. Efforts are presently under way to investigate the seasonal and year-to-year variability of this mesoscale energy. The data are also being used to generate low-pass filtered fields of sea level from which the temporal evolution of large-scale variability in the Southern Ocean may be investigated. The second parallel effort is the development and test of modeling and data assimilation techniques that will later be applied to TOPEX/POSEIDON data during the postlaunch phase. One objective of the modeling and data assimilation is to investigate the relation between mesoscale sea level variations and eddy flux in the Southern Ocean. Uncertainties in present estimates of the various components of meridional oceanic heat transport are large. The evidence presented indicates that the very energetic mesoscale variability in the ACC apparently accounts for much of the estimated 0.45x10(exp 15) watts of poleward heat transport across the ACC required to balance the heat budget. Eddy variability is strongly coherent vertically in the ACC, at least in the vicinity of Drake Passage where nearly all of the historical in situ data have been collected.

Chelton, Dudley B.↗

NASA NTRS2019

The Simons Observatory: Science Goals and Forecasts

The Simons Observatory (SO) is a new cosmic microwave background experiment being built on CerroToco in Chile, due to begin observations in the early 2020s. We describe the scientific goals of the experiment, motivate the design, and forecast its performance. SO will measure the temperature and polarization anisotropy of the cosmic microwave background in six frequency bands centered at: 27,39, 93, 145, 225 and 280 GHz. The initial configuration of SO will have three small-aperture 0.5-mtelescopes and one large-aperture 6-m telescope, with a total of 60,000 cryogenic bolometers. Our key science goals are to characterize the primordial perturbations, measure the number of relativistic species and the mass of neutrinos, test for deviations from a cosmological constant, improve our understanding of galaxy evolution, and constrain the duration of reionization. The small aperture telescopes will target the largest angular scales observable from Chile, mapping≈10% of the sky to a white noise level of 2μK-arcmin in combined 93 and 145 GHz bands, to measure the primordial tensor-to-scalar ratio, r, at a target level ofσ(r) = 0.003. The large aperture telescope will map ≈40% of the sky at arcminute angular resolution to an expected white noise level of 6μK-arcminin combined 93 and 145 GHz bands, overlapping with the majority of the Large Synoptic Survey Telescope sky region and partially with the Dark Energy Spectroscopic Instrument. With up to an order of magnitude lower polarization noise than maps from the Planck satellite, the high-resolution sky maps will constrain cosmological parameters derived from the damping tail, gravitational lensing of the microwave background, the primordial bispectrum, and the thermal and kinematic Sunyaev–Zel’dovich effects, and will aid in delensing the large-angle polarization signal to measure the tensor-to-scalar ratio. The survey will also provide a legacy catalog of 16,000 galaxy clusters and more than20,000 extragalactic sources.

Peter Ade↗

DOE OSTI2015

The scatter in the hot gas content of early-type galaxies

Optically similar early-type galaxies are observed to have a large and poorly understood range in the amount of hot, X-ray-emitting gas they contain. To investigate the origin of this diversity, we studied the hot gas properties of all 42 early-type galaxies in the multiwavelength ATLAS{sup 3D} survey that have sufficiently deep Chandra X-ray observations. We related their hot gas properties to a number of internal and external physical quantities. To characterize the amount of hot gas relative to the stellar light, we use the ratio of the gaseous X-ray luminosity to the stellar K-band luminosity, L{sub X{sub g{sub a{sub s}}}}/L{sub K}; we also use the deviations of L{sub X{sub g{sub a{sub s}}}} from the best-fit L{sub X{sub g{sub a{sub s}}}}–L {sub K} relation (denoted ΔL{sub X{sub g{sub a{sub s}}}}). We quantitatively confirm previous suggestions that various effects conspire to produce the large scatter in the observed L{sub X}/L{sub K} relation. In particular, we find that the deviations ΔL{sub X{sub g{sub a{sub s}}}} are most strongly positively correlated with the (low rates of) star formation and the hot gas temperatures in the sample galaxies. This suggests that mild stellar feedback may energize the gas without pushing it out of the host galaxies. We also find that galaxies in high galaxy density environments tend to be massive slow rotators, while galaxies in low galaxy density environments tend to be low mass, fast rotators. Moreover, cold gas in clusters and fields may have different origins. The star formation rate increases with cold gas mass for field galaxies but it appears to be uncorrelated with cold gas for cluster galaxies.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRSDate not supplied

Comparison of two Calibrations of NASA's MC60 Five-component Semi-span Balance

Two separate manual calibrations of NASA’s MC60 five–component semi–span balance were compared. The calibrations were performed in 1999 and 2019using CALSPAN Force Measurement System’s Large Load Rig. The Iterative Method was chosen as the load prediction algorithm and data reduction matrices were generated from both calibration data sets. Then, the applied calibration load schedules, the standard deviations of the calibration load residuals, the prime sensitivities of the five gages, and the maximum outputs at capacity of the five gages were compared. In addition, the calibration data of 1999 was used as check load data for the data reduction matrix that was obtained from the 2019 data. Overall, the agreement between the data analysis results is very good considering the facts that two different load schedules were used for the calibrations, different people performed the calibrations, different instrumentation was used to measure the loads and the outputs, the Large Load Rig was moved to a different site in between the calibrations, and no tare load corrections could be applied to the 1999 data set. Nevertheless, the calibration data of 2019 is clearly the better data set. This observation is no surprise considering that CALSPAN Force Measurement Systems made significant investments in calibration hardware and process improvements during the last 20 years.

wind tunnel balance↗

NASA NTRS2022

Predicting Antarctic Net Snow Accumulation at the Kilometer Scale and Its Impact on Observed Height Changes

Sub-grid-scale processes occurring at or near the surface of an ice sheet have a potentially large impact on local and integrated net accumulation of snow via redistribution and sublimation. Given observational complexity, they are either ignored or parameterized over large-length scales. Here, we train random forest (RF) models to predict variability in net accumulation over the Antarctic Ice Sheet using atmospheric variables and topographic characteristics as predictors at 1 km resolution. Observations of net snow accumulation from both in situ and airborne radar data provide the input observable targets needed to train the RF models. We find that local net accumulation deviates by as much as 172% of the atmospheric model mean. The correlation in space between the predicted net accumulation variability and satellite-derived surface-height change indicates that surface processes operate differently through time, driven largely by the seasonal anomalies in snow accumulation.

Antarctic↗

DOE OSTI2018

Central and Eastern North America Ground-Motion Characterization (NGA-East)

This document is the final project report of the Next Generation Attenuation for Central and Eastern North America (CENA) project (NGA-East). The NGA-East objective was to develop a new ground-motion characterization (GMC) model for the CENA region. The GMC model consists of a set of new ground-motion models (GMMs) for median and standard deviation of ground motions and their associated weights to be used with logic-trees in probabilistic seismic hazard analyses (PSHA). NGA-East is a large multidisciplinary project coordinated by the Pacific Earthquake Engineering Research Center (PEER), at the University of California. The project has two components: (1) a set of scientific research tasks, and (2) a model-building component following the framework of the “Seismic Senior Hazard Analysis Committee (SSHAC) Level 3” (Budnitz et al. 1997; NRC 2012). Component (2) is built on the scientific results of component (1) of the NGA-East project. This report documents the tasks under component (2) of the project. Under component (1) of NGA-East, several scientific issues were addressed, including: (a) development of a new database of ground motion data recorded in CENA; (b) development of a regionalized ground-motion map for CENA, (c) definition of the reference site condition; (d) simulations of ground motions based on different methodologies; and (e) development of numerous GMMs for CENA. The scientific tasks of NGA-East were all documented as a series of PEER reports. The scope of component (2) of NGA-East was to develop the complete GMC. This component was designed as a SSHAC Level 3 study with the goal of capturing the ground motions’ center, body, and range of the technically defensible interpretations in light of the available data and models. The SSHAC process involves four key tasks: evaluation, integration, formal review by the Participatory Peer Review Panel (PPRP), and documentation (this report). Key tasks documented in this report include review and evaluation of the empirical groundmotion database, the regionalization of ground motions, and screening sets of candidate GMMs. These are followed by the development of new median and standard deviation GMMs, the development of new analyses tools for quantifying the epistemic uncertainty in ground motions, and the documentation of implementation guidelines of the complete GMC for PSHA computations. Appendices include further documentation of the relevant SSHAC process and additional supporting technical documentation of numerous sensitivity analyses results. The PEER reports documenting component (1) of NGA-East are also considered “attachments” to the current report and are all available online on the PEER website (https://peer.berkeley.edu/). The final NGA-East GMC model includes a set of 17 GMMs defined for 24 ground-motion intensity measures, applicable to CENA in the moment magnitude range of 4.0 to 8.2 and covering distances up to 1500 km. Standard deviation models are also provided for site-specific analysis (single-station standard deviation) and for general PSHA applications (ergodic standard deviation). Adjustment factors are provided for consideration of source-depth effects and hanging-wall effects, as well as for hazard computations at sites in the Gulf Coast region.

58 GEOSCIENCES↗

DOE OSTI2023

A forecast for large-scale structure constraints on Horndeski gravity with CO line intensity mapping

Here we consider the potential for line intensity mapping (LIM) of the rotational CO(1-0), CO(2-1), and CO(3-2) transitions to detect deviations from General Relativity from 0 < z < 3 within the framework of a very general class of modified gravity models, called Horndeski’s theories. Our forecast assumes a multitracer analysis separately obtaining information from the matter power spectrum and the first two multipoles of the redshift space distortion power spectrum. To achieve ±0.1 level constraints on the slope of the kinetic gravity braiding and Planck mass evolution parameters, a mm-wave LIM experiment would need to accumulate ≈10 8 –10 9 spectrometre-hours, feasible with instruments that could be deployed in the 2030s. Such a measurement would constrain the parameters of Horndeski’s theory at a level at worst competitive to and at best an order of magnitude tighter than existing constraints from the CMB and LSS. Our modelling code is publicly available.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2016

Red, Straight, no bends: primordial power spectrum reconstruction from CMB and large-scale structure

We present a minimally parametric, model independent reconstruction of the shape of the primordial power spectrum. Our smoothing spline technique is well-suited to search for smooth features such as deviations from scale invariance, and deviations from a power law such as running of the spectral index or small-scale power suppression. We use a comprehensive set of the state-of the art cosmological data: Planck observations of the temperature and polarisation anisotropies of the cosmic microwave background, WiggleZ and Sloan Digital Sky Survey Data Release 7 galaxy power spectra and the Canada-France-Hawaii Lensing Survey correlation function. This reconstruction strongly supports the evidence for a power law primordial power spectrum with a red tilt and disfavours deviations from a power law power spectrum including small-scale power suppression such as that induced by significantly massive neutrinos. This offers a powerful confirmation of the inflationary paradigm, justifying the adoption of the inflationary prior in cosmological analyses.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2019

The Simons Observatory: science goals and forecasts

The Simons Observatory (SO) is a new cosmic microwave background experiment being built on Cerro Toco in Chile, due to begin observations in the early 2020s. We describe the scientific goals of the experiment, motivate the design, and forecast its performance. SO will measure the temperature and polarization anisotropy of the cosmic microwave background in six frequency bands centered at: 27, 39, 93, 145, 225 and 280 GHz. The initial configuration of SO will have three small-aperture 0.5-m telescopes and one large-aperture 6-m telescope, with a total of 60,000 cryogenic bolometers. Our key science goals are to characterize the primordial perturbations, measure the number of relativistic species and the mass of neutrinos, test for deviations from a cosmological constant, improve our understanding of galaxy evolution, and constrain the duration of reionization. The small aperture telescopes will target the largest angular scales observable from Chile, mapping ≈ 10% of the sky to a white noise level of 2 μK-arcmin in combined 93 and 145 GHz bands, to measure the primordial tensor-to-scalar ratio, r, at a target level of σ(r)=0.003. The large aperture telescope will map ≈ 40% of the sky at arcminute angular resolution to an expected white noise level of 6 μK-arcmin in combined 93 and 145 GHz bands, overlapping with the majority of the Large Synoptic Survey Telescope sky region and partially with the Dark Energy Spectroscopic Instrument. With up to an order of magnitude lower polarization noise than maps from the Planck satellite, the high-resolution sky maps will constrain cosmological parameters derived from the damping tail, gravitational lensing of the microwave background, the primordial bispectrum, and the thermal and kinematic Sunyaev-Zel'dovich effects, and will aid in delensing the large-angle polarization signal to measure the tensor-to-scalar ratio. In conclusion, the survey will also provide a legacy catalog of 16,000 galaxy clusters and more than 20,000 extragalactic sources.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS2019

The Simons Observatory: science goals andforecasts

The Simons Observatory (SO) is a new cosmic microwave background experiment being built on Cerro Toco in Chile, due to begin observations in the early 2020s. We describe the scientific goals of the experiment, motivate the design, and forecast its performance. SO will measure the temperature and polarization anisotropy of the cosmic microwave background in six frequency bands centered at: 27, 39, 93, 145, 225 and 280 GHz. The initial configuration of SO will have three small-aperture 0.5-m telescopes and one large-aperture 6-m telescope, with a total of 60,000 cryogenic bolometers. Our key science goals are to characterize the primordial perturbations, measure the number of relativistic species and the mass of neutrinos, test for deviations from a cosmological constant, improve our understanding of galaxy evolution, and constrain the duration of reionization. The small aperture telescopes will target the largest angular scales observable from Chile, mapping ≈ 10% of the sky to a white noise level of 2 μK-arcmin in combined 93 and 145 GHz bands, to measure the primordial tensor-to-scalar ratio, r, at a target level of σ(r)=0.003. The large aperture telescope will map ≈ 40% of the sky at arcminute angular resolution to an expected white noise level of 6 μK-arcmin in combined 93 and 145 GHz bands, overlapping with the majority of the Large Synoptic Survey Telescope sky region and partially with the Dark Energy Spectroscopic Instrument. With up to an order of magnitude lower polarization noise than maps from the Planck satellite, the high-resolution sky maps will constrain cosmological parameters derived from the damping tail, gravitational lensing of the microwave background, the primordial bispectrum, and the thermal and kinematic Sunyaev-Zel'dovich effects, and will aid in delensing the large-angle polarization signal to measure the tensor-to-scalar ratio. The survey will also provide a legacy catalog of 16,000 galaxy clusters and more than 20,000 extragalactic sources.

Peter Ade↗

DOE OSTI2024

Large-Volume Injection and Assessment of Reference Standards for n -Alkane δD and δ 13 C Analysis via Gas Chromatography Isotope Ratio Mass Spectrometry

Compound-specific stable isotope analysis of hydrogen (δD) and carbon (δ 13 C) in organic compounds is a valuable tool in biogeochemical research. A key limitation of this method is the relatively large amount of sample required to achieve desirable precision. We developed a large-volume (20 μL) injection method that allows for high throughput analysis of less concentrated samples and tested it for δ 13 C and δD measurements of n-alkanes. We also conducted a comparison of reference standards and assessed several methods to normalize and correct n-alkane δD and δ13C measurements. The mean precision of the δD method based on 233 environmental n-alkane samples (two to three replications per sample) is 4.0‰ (1σ, estimated from the weighted mean of the pooled unbiased standard deviations) and 0.46‰ (1σ) for δ 13 C from 37 environmental samples (two to three replications per sample). The evaluation of reference standards shows that the use of n-alkane standards with large offsets in δD values in adjacent n-alkane chains can lead to biases in measurement correction. The large-volume injection method shows good reproducibility of δ 13 C and δD measurements of n-alkanes and reduces the required sample concentration by about 80%. We propose that for δD measurements, a reference standard set should be used in which each reference standard has a limited range of δD values and no adjacent n-alkane chains, to minimize memory effects.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1990

Observations of the earth's polar cleft at large radial distances with the Hawkeye 1 magnetometer

Based on 364 spacecraft passes through the dayside region, the position of the polar cleft at large radial distances was determined with the magnetometer flown on Hawkeye 1. This data set is ideal for the study of the cusp and cleft region. Identification of the cleft depended on noting strong negative deviations of the magnetic field strength in the region from that of the dipole field. In solar magnetic coordinates, cleft observations were found between 40 deg and 70 deg latitude and + or - 75 deg latitude and + or - 75 deg longitude, while in geocentricmagnetospheric coordinates, these observations were found between 20 deg and 75 deg latitude and + or - 75 deg longitude. The extreme longitudinal extent of 150 deg is larger than those reported in some previous studies. Large magnetic depressions associated with the cleft extend out to 12 earth radii.

Farrell, W. M.↗

DOE OSTI2022

Universal correlation for falling film evaporation on a horizontal plain tube

Falling film evaporators are widely used in many industrial sectors, such as desalination, refrigeration & air-conditioning, and chemical processes. The heat transfer correlations suitable for wider operating conditions and various refrigerants are useful for designing falling film evaporators. Although many researchers developed the correlation for their own experimental data, there is a limited universal correlation for falling film evaporators. In addition, the dry-out effects were not considered in the available universal correlations. Here, therefore, this study proposes a new correlation for falling film evaporation on a single horizontal plain tube, which includes nucleate boiling, film evaporation, and dry-out effects. The proposed correlation covers a wider range of operating conditions (Heat flux is of 2.5–168kWm -2 , and film Reynolds number is up to 3500), various refrigerants (R134a, R245fa, R1234ze(E), R123, R32, R410A, R290, and R600a), and the large number of experimental data (1101 data in total). The proposed correlation is validated against the available standard correlation with regard to mean absolute deviation (MAD) and R 2 values. The MAD of the new correlation is about 10%, respectively, which is the lowest among other correlations. Similarly, the R 2 showed a value of 0.92, which is the highest compared to the existing correlations. A simplified correlation is also proposed.

42 ENGINEERING↗

DOE OSTI2019

Comparing smoothing techniques for extracting MOSFET threshold voltage

Measurement noise acts as a barrier to the accurate calculation of threshold voltage by derivative-based extraction methods. We examined several smoothing techniques and their effects on measurement noise in threshold voltage extracted by the linear extrapolation method and the g m /I d method. We applied these techniques on a set of SiC power MOSFET devices before and after they had undergone accelerated thermal testing. The smoothing methods examined are least-squares polynomial fitting, low-pass filtering using Butterworth and Chebyshev fillers, and differential smoothing. These methods are compared by their shift in threshold voltage compared to unsmoothed data and their standard deviation among the measurements of each device. All methods performed poorly at smoothing when extracting via the g m /I d due to the large amounts of high-magnitude subthreshold noise, mostly tending to extract threshold voltages far below the expected value. Differential smoothing provided results closest to the expected value with an absolute shift of 1.2 V on average. The smoothing methods performed better with the LE method. On a failed device, all smoothing methods except polynomial smoothing preserved the large threshold voltage anomaly for four out of five measurements. These results show that choice of smoothing method, if used must be evaluated based on the device and extraction method.

42 ENGINEERING↗

DOE OSTI2022

Finite orbit width effects in large aspect ratio stellarators

New orbit-averaged equations for low collisionality neoclassical fluxes in large aspect ratio stellarators with mirror ratios close to unity are derived. The equations retain finite orbit width effects by employing the second adiabatic invariant $J$ as a velocity-space coordinate and they have been implemented in the orbit-averaged neoclassical code KNOSOS (Velasco et al. , J. Comput. Phys. , vol. 418, 2020, 109512; Velasco et al. , Nucl. Fusion , vol. 61, 2021, 116013). The equations are used to study the $1/\nu$ regime and the lower collisionality regimes. For generic large aspect ratio stellarators with mirror ratios close to unity, as the collision frequency decreases, the $1/\nu$ regime transitions directly into the $\nu$ regime, without passing through a $\sqrt {\nu }$ regime. An explicit formula for the neoclassical fluxes in the $\nu$ regime is obtained. The formula includes the effect of particles that transition between different types of wells. While these transitions produce stochastic scattering independent of the value of the collision frequency in velocity space, the diffusion in real space remains proportional to the collision frequency. The $\sqrt {\nu }$ regime is only recovered in large aspect ratio stellarators close to omnigeneity: large aspect ratio stellarators with large mirror ratios and optimized large aspect ratio stellarators with mirror ratios close to unity. Neoclassical transport in large aspect ratio stellarators with large mirror ratios can be calculated with the orbit-averaged equations derived by Calvo et al. ( Plasma Phys. Control. Fusion , vol. 59, 2017, 055014). In these stellarators, the $\sqrt {\nu }$ regime exists in the collisionality interval $(a/R) \ln (R/a) \ll \nu _{ii} R a/\rho _i v_{ti} \ll R/a$ . In optimized large aspect ratio stellarators with mirror ratios close to unity, the $\sqrt {\nu }$ regime occurs in an interval of collisionality that depends on the deviation from omnigeneity $\delta$ : $\delta ^{2} |\ln \delta | \ll \nu _{ii} R a/\rho _i v_{ti} \ll 1$ . Here, $\nu _{ii}$ is the ion–ion collision frequency, $\rho _i$ and $v_{ti}$ are the ion gyroradius and thermal speed, and $a$ and $R$ are the minor and major radii.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗