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DOE OSTI2020

A Computationally Efficient Linear Semi-Lagrangian Scheme for the Advection of Microphysical Variables in Cloud-Resolving Models

An efficient semi-Lagrangian (SL) numerical scheme for the advection of positive-definite microphysical variables in cloud-resolving models is proposed. The scheme is a linear combination of SL schemes of the first and second order. The proposed scheme is compared with two high-order, monotonic, nonlinear Eulerian schemes, in two 2D tests. In the first test, advection of a positive scalar field with large gradients is performed within complicated idealized steering flows. In the second test, a hail storm is simulated using the 2D Hebrew University Cloud Model, describing the interactions between 15 size distributions of different microphysical quantities (each distribution is described by 43 mass bins). The proposed scheme is computationally efficient, has a low numerical diffusion and a high level of mass conservation accuracy, and preserves the sum of multiple advected variables as well as the linear relationships between the advected scalar variables. Although the proposed SL scheme is not exactly positive definite, the negative values and their number, which may appear only in the case of very strong gradients, are negligible. The scheme produces results similar to those of nonlinear Eulerian advection schemes while reducing computation time by approximately a factor of 10.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS2005

Analysis of Multiple Precipitation Products and Preliminary Assessment of Their Impact on Global Land Data Assimilation System (GLDAS) Land Surface States

Land surface models (LSMs) are computer programs, similar to weather and climate prediction models, which simulate the stocks and fluxes of water (including soil moisture, snow, evaporation, and runoff) and energy (including the temperature of and sensible heat released from the soil) after they arrive on the land surface as precipitation and sunlight. It is not currently possible to measure all of the variables of interest everywhere on Earth with sufficient accuracy and space-time resolution. Hence LSMs have been developed to integrate the available observations with our understanding of the physical processes involved, using powerful computers, in order to map these stocks and fluxes as they change in time. The maps are used to improve weather forecasts, support water resources and agricultural applications, and study the Earth's water cycle and climate variability. NASA's Global Land Data Assimilation System (GLDAS) project facilitates testing of several different LSMs with a variety of input datasets (e.g., precipitation, plant type). Precipitation is arguably the most important input to LSMs. Many precipitation datasets have been produced using satellite and rain gauge observations and weather forecast models. In this study, seven different global precipitation datasets were evaluated over the United States, where dense rain gauge networks contribute to reliable precipitation maps. We then used the seven datasets as inputs to GLDAS simulations, so that we could diagnose their impacts on output stocks and fluxes of water. In terms of totals, the Climate Prediction Center (CPC) Merged Analysis of Precipitation (CMAP) had the closest agreement with the US rain gauge dataset for all seasons except winter. The CMAP precipitation was also the most closely correlated in time with the rain gauge data during spring, fall, and winter, while the satellitebased estimates performed best in summer. The GLDAS simulations revealed that modeled soil moisture is highly sensitive to precipitation, with differences in spring and summer as large as 45% depending on the choice of precipitation input.

Gottschalck, Jon↗

NASA NTRS2005

Evaluation of Ten Methods for Initializing a Land Surface Model

Rodell, M.↗

NASA NTRS1991

A geometric approach to regulator and tracker design for an aerospace plane

The paper presents a nonlinear design approach drawing from singular perturbations, feedback linearization, and variable structure control, that leads to regulators with automatic gain scheduling which exhibit similar dynamic behavior over the entire flight envelope of the aerospace plane. Additionally, design approach provides for a systematic way to counter disturbance effects as well as modeling uncertainties. The unifying feature of the three nonlinear feedback control methodologies is that they all have a geometric interpretation. First, the translational dynamics are decomposed into reduced-order slow and fast dynamics by way of a formal singular perturbation analysis. After feedback linearization the fast dynamics are robustly stabilized via a variable structure control approach. The slow dynamics are stabilized using conventional proportional-integral compensation based on the nominal slow dynamics. A number of sample command and disturbance responses at opposite ends of the flight envelope are presented for a nonlinear aerospace plane model.

Van Buren, Mark A.↗

DOE OSTI2023

A high‐order discontinuous Galerkin approach for physics‐based thermospheric modeling

Abstract The accurate prediction of aerodynamic drag on satellites orbiting in the upper atmosphere is critical to the operational success of modern space technologies, such as satellite‐based communication or navigation systems, which have become increasingly popular in the last few years due to the deployment of constellations of satellites in low‐Earth orbit. As a result, physics‐based models of the ionosphere and thermosphere have emerged as a necessary tool for the prediction of atmospheric outputs under highly variable space weather conditions. This paper proposes a high‐fidelity approach for physics‐based space weather modeling based on the solution of the Navier–Stokes equations using a high‐order discontinuous Galerkin method, combined with a matrix‐free strategy suitable for high‐performance computing on GPU architectures. The approach consists of a thermospheric model that describes a chemically frozen neutral atmosphere in nonhydrostatic equilibrium driven by the external excitation of the Sun. A novel set of variables is considered to treat the low densities present in the upper atmosphere and to accommodate the wide range of scales present in the problem. At the same time, and unlike most existing approaches, radial and angular directions are treated in a nonsegregated approach. The study presents a set of numerical examples that demonstrate the accuracy of the approximation and validate the current approach against observational data along a satellite orbit, including estimates of established empirical and physics‐based models of the ionosphere‐thermosphere system. Finally, a one‐dimensional radial derivation of the physics‐based model is presented and utilized for conducting a parametric study of the main thermal quantities under various solar conditions.

Engineering↗

DOE OSTI2019

The Tropical Atlantic Observing System

The tropical Atlantic is home to multiple coupled climate variations covering a wide range of timescales and impacting societally relevant phenomena such as continental rainfall, Atlantic hurricane activity, oceanic biological productivity, and atmospheric circulation in the equatorial Pacific. The tropical Atlantic also connects the southern and northern branches of the Atlantic meridional overturning circulation and receives freshwater input from some of the world's largest rivers. To address these diverse, unique, and interconnected research challenges, a rich network of ocean observations has developed, building on the backbone of the Prediction and Research Moored Array in the Tropical Atlantic (PIRATA). This network has evolved naturally over time and out of necessity in order to address the most important outstanding scientific questions and to improve predictions of tropical Atlantic severe weather and global climate variability and change. The tropical Atlantic observing system is motivated by goals to understand and better predict phenomena such as tropical Atlantic interannual to decadal variability and climate change; multidecadal variability and its links to the meridional overturning circulation; air-sea fluxes of CO 2 and their implications for the fate of anthropogenic CO 2 ; the Amazon River plume and its interactions with biogeochemistry, vertical mixing, and hurricanes; the highly productive eastern boundary and equatorial upwelling systems; and oceanic oxygen minimum zones, their impacts on biogeochemical cycles and marine ecosystems, and their feedbacks to climate. Past success of the tropical Atlantic observing system is the result of an international commitment to sustained observations and scientific cooperation, a willingness to evolve with changing research and monitoring needs, and a desire to share data openly with the scientific community and operational centers. The observing system must continue to evolve in order to meet an expanding set of research priorities and operational challenges. This paper discusses the tropical Atlantic observing system, including emerging scientific questions that demand sustained ocean observations, the potential for further integration of the observing system, and the requirements for sustaining and enhancing the tropical Atlantic observing system.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Analytic amplitudes for a pair of Higgs bosons in association with three partons

The pair production of Higgs bosons at the LHC can give information about the triple Higgs boson coupling. We perform an analytic one-loop calculation of the amplitudes for a pair of Higgs bosons in association with three partons, retaining the exact dependence on the quark mass circulating in the loop. These amplitudes constitute the real radiation corrections in the calculation of Higgs boson pair production at next-to-leading order in the strong coupling. The results of an analytic generalised-unitarity computation are simplified via analytic reconstruction in spinor variables. Compact ansätze for kinematic pole residues are iteratively fitted via p-adic evaluations near said poles and subtracted until no pole remains. A new ansatz construction is introduced to minimally parametrise coefficients of amplitudes with multiple massive external legs. The simplified expressions are faster to evaluate than automatic codes and can lead to more stable results near singular regions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2016

Measurement of total and differential W + W - production cross sections in proton-proton collisions at √s=8 TeV with the ATLAS detector and limits on anomalous triple-gauge-boson couplings

The production of W boson pairs in proton-proton collisions at √s=8 TeV is studied using data corresponding to 20.3 fb -1 of integrated luminosity collected by the ATLAS detector during 2012 at the CERN Large Hadron Collider. The W bosons are reconstructed using their leptonic decays into electrons or muons and neutrinos. Events with reconstructed jets are not included in the candidate event sample. A total of 6636 WW candidate events are observed. Measurements are performed in fiducial regions closely approximating the detector acceptance. The integrated measurement is corrected for all acceptance effects and for the W branching fractions to leptons in order to obtain the total WW production cross section, which is found to be 71.1 ± 1.1(stat) -5.0+5.7(syst)±1.4(lumi) pb. This agrees with the next-to-next-to-leading-order Standard Model prediction of 63.2-1.4+1.6(scale)±1.2(PDF) pb. Fiducial differential cross sections are measured as a function of each of six kinematic variables. The distribution of the transverse momentum of the leading lepton is used to set limits on anomalous triple-gauge-boson couplings.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRSDate not supplied

TPSAS-NF1676L-10726-DND

The principal component radiative transfer model PCRTM (Xu 2006) simulates spectral longwave radiance at a 0.5cm-1 resolution over the wavelengths (50-2760cm-1) using 280 principal components. The CERES product SSF containing CERES field-of-view (FOV) cloud properties based on MODIS pixel information are used as inputs. NASA GMAO(Global Modeling and Assimilation Office)Geos-4 temperature and humidity profiles, as well as SMOBA (Stratosphere Monitoring Ozone Blended Analysis) ozone profiles allow computations for five years (Jan 2001- Dec 2006) of FOV nadir spectral radiance. The approximately twice daily FOV PCRTM simulations along with CERES broadband measurements are time-space averaged to 10deg zonal monthly data. Analysis of monthly seasonal and deseasonalized interannual variability in terms of brightness temperature are presented. Comparison of interannual variability in the portions of the spectrum seen by the AIRS instrument are shown. Comparisons of spectral integrated PCRTM simulations to CERES broadband longwave radiance measurements are made. In order to study the sensitivity of the spatially and temporally averaged TOA spectrum to various atmospheric properties, we perturbed atmospheric and surface temperatures, humidity, cloud height, cloud fraction, and particle size. The sensitivity of the spectrum to these properties depends on region and seasons. Month to month variability of the longwave spectra to these perturbations is 10 to 20% of the global monthly mean perturbation response. In addition, it depends on time period and size of the area used for averaging spectra. In this poster, the magnitude of sensitivity variability and how the sensitivity changes depending on temporal and spatial scale of averaging will be discussed.

Fred G. Rose↗

DOE OSTI2017

An étude on global vacuum energy sequester

Recently two of the authors proposed a mechanism of vacuum energy sequester as a means of protecting the observable cosmological constant from quantum radiative corrections. The original proposal was based on using global Lagrange multipliers, but later a local formulation was provided. Subsequently other interesting claims of a different non-local approach to the cosmological constant problem were made, based again on global Lagrange multipliers. We examine some of these proposals and find their mutual relationship. We explain that the proposals which do not treat the cosmological constant counterterm as a dynamical variable require fine tunings to have acceptable solutions. Furthermore, the counterterm often needs to be retuned at every order in the loop expansion to cancel the radiative corrections to the cosmological constant, just like in standard GR. These observations are an important reminder of just how the proposal of vacuum energy sequester avoids such problems.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2021

Order-disorder in room-temperature ionic liquids probed via methyl quantum tunneling

Room-temperature ionic liquids are promising candidates for applications ranging from electrolytes for energy storage devices to lubricants for food and cellulose processing to compounds for pharmaceutics, biotransformation, and biopreservation. Due to the ion complexity, many room-temperature ionic liquids readily form amorphous phases upon cooling, even at modest rates. Here, we investigate two commonly studied imidazolium-based room-temperature ionic liquids, 1-ethyl-3-methylimidazolium tetrafluoroborate and 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide, as well as their mixtures, to demonstrate how the complex interplay between the crystalline and amorphous phases is affected by the processing conditions, such as thermal history, liquid mixing, and applied pressure. We show that quantum tunneling in the cation methyl groups, measured by high-resolution inelastic neutron scattering, can be used to probe the order-disorder in room-temperature ionic liquids (crystalline vs amorphous state) that develops as a result of variable processing conditions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2019

Decentralized Voltage and Power Control of Multi-Machine Power Systems With Global Asymptotic Stability

Maintaining power system stability is becoming urgent due to the large-scale interconnection of power grids and the high penetration of uncertain renewable energy sources. The excitation control and governor control of synchronous generators have been considered as two crucial measures for enhancing the power system stability. However, a major challenge is to simultaneously achieve global asymptotic stability (GAS), voltage regulation (VR), and power regulation (PR) in the excitation and governor control. In this paper, a Lyapunov-based decentralized control (LBC) is proposed to address this challenge. The time-derivative of the Lyapunov function is designed by the feedback control of synchronous generators in order to guarantee GAS. VR and PR are ensured by considering voltage and power deviations as the feedback variables. The simulation results on the New-England ten-machine power system validate the effectiveness of the proposed LBC in improving power system transient stability and simultaneously achieving VR and PR. Although the proportional-integral- and power system stabilizer-based control can also perform VR and PR, the proposed control has much better dynamic performance and can more significantly improve the system transient stability.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2018

Hurricane Maria Puerto Rico Landsat Analysis

Hurricane Maria made landfall as a strong Category 4 storm in southeast Puerto Rico on September 20th, 2018. The powerful storm traversed the island in a northwesterly direction causing widespread destruction. This study focused on a rapid assessment of Hurricane Marias impact to Puerto Ricos forests. Calibrated and corrected Landsat 8 image composites for the entire island were generated using Google Earth Engine for a comparable pre-Maria and post-Maria time period that accounted for phenology. Spectral mixture analysis (SMA) using image-derived end members was carried out on both composites to calculate the change in the non-photosynthetic vegetation (Delta-NPV) spectral response, a metric that quantifies the increased fraction of exposed wood and surface litter associated with tree mortality and crown damage from the storm. Hurricane simulations were also conducted using the Weather Research and Forecasting (WRF) regional climate model to estimate wind speeds associated with forest disturbance. Dramatic changes in forest structure across the entire island were evident from pre- and post-Maria composited Landsat 8 images. A Delta-NPV map for only the forested pixels illustrated significant spatial variability in disturbance, with patterns that associated with factors such as slope, aspect and elevation. An initial order-of-magnitude impact estimate based on previous work indicated that Hurricane Maria may have caused mortality and severe damage to 23-31 million trees. Additional field work and image analyses are required to further detail the impact of Hurricane Maria to Puerto Rico forests. A minor update to this dataset was posted on April 20, 2018. The previous version is being retired. If you need access to the prior version of the data, email ngee-tropics-archive@lbl.gov.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2023

Quantification of LEU Holdup using gamma ray imaging and inverse transport solver

Holdup is the residual amount of special nuclear material (SNM) remaining in a processing facility after the bulk materials have been cleaned out. In commercial uranium processing facilities, quantification of holdup is a major challenge because of the highly variable shapes and sizes of the deposits. Any method that attempts to generalize and calibrate deposit shapes in order to quantify holdup will be prone to high uncertainties. Uncertainties on the order of ±50% are typical in holdup results. In international safeguards applications, a ±50% uncertainty can result in a large amount of material unaccounted for (MUF) thereby increasing the difficulty of detecting material diversion and facility misuse. An imaging-based methodology has been developed with the objective of significantly reducing this uncertainty by using the true deposit shape, instead of relying on oversimplified geometric assumptions. The project is a collaboration between ORNL, Y-12, and the University of Tennessee, Knoxville, TN. Uranium sources of known masses were measured using the Germanium Gamma-ray Imager (GeGI), a high-resolution imaging spectrometer, creating a pixelated map for each spectral bin. Two different gamma imaging methods are employed in this work: coded aperture imaging and Compton imaging. A validated MonteCarlo model of the detector has been developed using the GEANT4 code for determining the intrinsic response of the detector, its enclosure, and the coded aperture mask. An inverse transport solver based on the Markov Chain Monte-Carlo approach known as Differential Evolution Adaptive Metropolis (DREAM) is employed to use the measurement data from the image pixels (coded aperture or Compton) to solve for the mass of 235 U in the deposit. A reliable method based on the DREAM solver has been developed to flag the infinite thickness condition of a uranium deposit. The project team is working towards improving the image reconstruction for Compton imaging so that a better localization of the source can be achieved. Besides treating the coded aperture and Compton imaging methods independently, the project is also evaluating a combined method that uses the Compton scatter data from a coded aperture measurement. GEANT4 simulations are being performed to evaluate the combined approach. The impact on the DREAM optimization as the source thickness progressively approaches infinite thickness is being evaluated. A number of uranium sources available at ORNL have been measured, and the DREAM results have been tested and validated for the coded aperture imaging. A similar effort will be carried out to validate the Compton based method once the development of algorithms for better localization are complete. The imaging based quantification is very amenable to unattended monitoring of holdup accumulation at key measurement points. A proof of concept measurement has been completed to demonstrate this capability The current work used the high energy resolution imager GeGI. However, the approach and methodologies are applicable to other imagers such as the cadmium zin telluride (CZT) based imager manufactured by H3D, Inc.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2023

Spectroscopic Analysis of Milky Way Outer Halo Satellites: Aquarius II and Boötes II

In this paper, we present a chemical and kinematic analysis of two ultrafaint dwarf galaxies (UFDs), Aquarius II (Aqu II) and Boötes II (Boo II), using Magellan/IMACS spectroscopy. We present the largest sample of member stars for Boo II (12), and the largest sample of red giant branch members with metallicity measurements for Aqu II (eight). In both UFDs, over 80% of targets selected based on Gaia proper motions turned out to be spectroscopic members. In order to maximize the accuracy of stellar kinematic measurements, we remove the identified binary stars and RR Lyrae variables. For Aqu II, we measure a systemic velocity of -65.3 ± 1.8 km s-1 and a metallicity of [Fe/H] = $-2.57^{+0.17}_{-0.17}$. When compared with previous measurements, these values display a ~6 km s -1 difference in radial velocity and a decrease of 0.27 dex in metallicity. Similarly for Boo II, we measure a systemic velocity of $-130.4^{+1.4}_{-1.1}$ km s -1 , more than 10 km s -1 different from the literature, a metallicity almost 1 dex smaller at [Fe/H] = $-2.71^{+0.11}_{-0.10}$, and a velocity dispersion 3 times smaller at σ v hel = $2.9^{+1.6}_{-1.2}$km s -1 . Additionally, we derive systemic proper-motion parameters and model the orbits of both UFDs. Finally, we highlight the extremely dark-matter-dominated nature of Aqu II and compute the J-factor for both galaxies to aid searches of dark matter annihilation. Despite the small size and close proximity of Boo II, it is an intermediate target for the indirect detection of dark matter annihilation due to its low-velocity dispersion and corresponding low dark matter density.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2020

Constraining the Twomey effect from satellite observations: issues and perspectives

Abstract. The Twomey effect describes the radiative forcing associated with a change in cloud albedo due to an increase in anthropogenic aerosol emissions. It is driven by the perturbation in cloud droplet number concentration (ΔNd, ant) in liquid-water clouds and is currently understood to exert a cooling effect on climate. The Twomey effect is the key driver in the effective radiative forcing due to aerosol–cloud interactions, but rapid adjustments also contribute. These adjustments are essentially the responses of cloud fraction and liquid water path to ΔNd, ant and thus scale approximately with it. While the fundamental physics of the influence of added aerosol particles on the droplet concentration (Nd) is well described by established theory at the particle scale (micrometres), how this relationship is expressed at the large-scale (hundreds of kilometres) perturbation, ΔNd, ant, remains uncertain. The discrepancy between process understanding at particle scale and insufficient quantification at the climate-relevant large scale is caused by co-variability of aerosol particles and updraught velocity and by droplet sink processes. These operate at scales on the order of tens of metres at which only localised observations are available and at which no approach yet exists to quantify the anthropogenic perturbation. Different atmospheric models suggest diverse magnitudes of the Twomey effect even when applying the same anthropogenic aerosol emission perturbation. Thus, observational data are needed to quantify and constrain the Twomey effect. At the global scale, this means satellite data. There are four key uncertainties in determining ΔNd, ant, namely the quantification of (i) the cloud-active aerosol – the cloud condensation nuclei (CCN) concentrations at or above cloud base, (ii) Nd, (iii) the statistical approach for inferring the sensitivity of Nd to aerosol particles from the satellite data and (iv) uncertainty in the anthropogenic perturbation to CCN concentrations, which is not easily accessible from observational data. This review discusses deficiencies of current approaches for the different aspects of the problem and proposes several ways forward: in terms of CCN, retrievals of optical quantities such as aerosol optical depth suffer from a lack of vertical resolution, size and hygroscopicity information, non-direct relation to the concentration of aerosols, difficulty to quantify it within or below clouds, and the problem of insufficient sensitivity at low concentrations, in addition to retrieval errors. A future path forward can include utilising co-located polarimeter and lidar instruments, ideally including high-spectral-resolution lidar capability at two wavelengths to maximise vertically resolved size distribution information content. In terms of Nd, a key problem is the lack of operational retrievals of this quantity and the inaccuracy of the retrieval especially in broken-cloud regimes. As for the Nd-to-CCN sensitivity, key issues are the updraught distributions and the role of Nd sink processes, for which empirical assessments for specific cloud regimes are currently the best solutions. These considerations point to the conclusion that past studies using existing approaches have likely underestimated the true sensitivity and, thus, the radiative forcing due to the Twomey effect.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Influence of simple terrain on the spatial variability of a low-level jet and wind farm performance in the AWAKEN field campaign

In wind energy research, scientific challenges are often associated with complex terrain sites, where orography, vegetation, and buildings disrupt flow uniformity. However, even sites characterized as simple terrain can exhibit significant spatial variability in wind speed, particularly during stable boundary layers (SBLs) and low-level jets (LLJs). This study investigates these terrain interactions using both simulations and observations from the American WAKe ExperimeNt (AWAKEN). We employ a multiscale Weather Research and Forecasting (WRF) model simulation, integrating mesoscale forcing in the coarse domains and representing three rows of turbines from the King Plains wind farm as generalized actuator disks (GAD) in the large-eddy simulation (LES) domains. During a nocturnal LLJ event on 3 April 2023, the downstream, wake-affected turbine rows outperformed the upstream, unwaked row by 25 %–51 %. This counterintuitive result arises from terrain-induced streamwise variations in hub-height wind speed of approximately 4 m s −1 over 5 km – equivalent to ∼50 % of the upstream reference speed. This enhancement outweighs the wake-induced reduction in mean wind speed (∼12 %) and global blockage effects reported in the literature (∼1 %–3.4 %). The multiscale simulations capture the intra-farm spatial variability in power performance observed in SCADA data. Terrain-induced vertical displacement of the LLJ, coupled with large wind shear below the jet maximum, drives the substantial streamwise acceleration within the wind farm. These findings underscore the importance of accounting for spatial variability related to terrain, even in simple landscapes, particularly during LLJ conditions. Incorporating such effects into reduced-order modeling frameworks for wind farm design and control could significantly enhance their effectiveness.

17 WIND ENERGY↗

NASA NTRS1971

The three-dimensional steady radial expansion of a viscous gas from a sonic source into a vacuum.

The three-dimensional steady radial expansion of a viscous, heat-conducting, compressible fluid from a spherical sonic source into a vacuum is analyzed using the Navier-Stokes equations as a basis. It is assumed that the model fluid is a perfect gas having constant specific heats, a constant Prandtl number of order unity, and viscosity coefficients varying as a power of the absolute temperature. Limiting forms for the flow variable solutions are studied for the Reynolds number based on the sonic source conditions, going to infinity and the Newtonian parameter both fixed and going to zero. For the case of the viscosity-temperature exponent between .5 and 1, it is shown that the velocity as well as the pressure approach zero as the radial distance goes to infinity. The formulations of the distinct regions that span the domain extending from the sonic source to the vacuum are presented.

Bush, W. B.↗