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NASA NTRS1994

Performance Measurement, Visualization and Modeling of Parallel and Distributed Programs

This paper presents a methodology for debugging the performance of message-passing programs on both tightly coupled and loosely coupled distributed-memory machines. The AIMS (Automated Instrumentation and Monitoring System) toolkit, a suite of software tools for measurement and analysis of performance, is introduced and its application illustrated using several benchmark programs drawn from the field of computational fluid dynamics. AIMS includes (i) Xinstrument, a powerful source-code instrumentor, which supports both Fortran77 and C as well as a number of different message-passing libraries including Intel's NX Thinking Machines' CMMD, and PVM; (ii) Monitor, a library of timestamping and trace -collection routines that run on supercomputers (such as Intel's iPSC/860, Delta, and Paragon and Thinking Machines' CM5) as well as on networks of workstations (including Convex Cluster and SparcStations connected by a LAN); (iii) Visualization Kernel, a trace-animation facility that supports source-code clickback, simultaneous visualization of computation and communication patterns, as well as analysis of data movements; (iv) Statistics Kernel, an advanced profiling facility, that associates a variety of performance data with various syntactic components of a parallel program; (v) Index Kernel, a diagnostic tool that helps pinpoint performance bottlenecks through the use of abstract indices; (vi) Modeling Kernel, a facility for automated modeling of message-passing programs that supports both simulation -based and analytical approaches to performance prediction and scalability analysis; (vii) Intrusion Compensator, a utility for recovering true performance from observed performance by removing the overheads of monitoring and their effects on the communication pattern of the program; and (viii) Compatibility Tools, that convert AIMS-generated traces into formats used by other performance-visualization tools, such as ParaGraph, Pablo, and certain AVS/Explorer modules.

Yan, Jerry C.↗
NASA NTRS2004

Over-Luminous Elliptical Galaxies

We have completed a first draft of a paper on the galaxy group ES03060170, the hottest known fossil group. We have submitted a first draft of the paper but the final completion is delayed due to several issues mentioned by the referee that we wish to revisit and discuss in more detail. The XMM data was combined with Chandra data which allowed a rich set of projects. The paper discusses the north-south elongation which is similar to that of the central dominant galaxy as well as the galaxy distribution. We detect an X-ray "finger" or small tail emanating from the central galaxy to the north, suggesting motion of the galaxy within the elongated gravitational potential. The overall agreement between XMM and Chandra data are excellent (although the XMM data extend to larger radii). Both data sets show a cool core centered on the dominant galaxy. Surprisingly, the temperature maps and detailed spectra indicate that the finger of gas is NOT cool, but has the same temperature as the ambient gas. We extracted surface brightness profiles, deprojected gas density profiles, cooling time profiles, and entropy profiles. There is a sharp discontinuity in gas temperature where the surface brightness profile starts to rise rapidly at 10 kpc. This produces a decrease in the cooling time and the gas entropy within 10 kpc. The central cooling time (within 10 kpc) is less than l0(exp 9) years and falls to almost half that value in the inner 5 kpc. Despite the very short cooling time, we find no evidence (even with the excellent statistics from XMM-Newton) for multi-phased gas, i.e., a cooling flow. We find two "edges" associated with the gas distribution (common in peaked X-ray groups and galaxies). On large scales, the temperature profile is flat and disagrees with the profile predicted by Loken et al. (2003) from detailed numerical simulations. We studied the galaxy distribution within one virial radius. The galaxy concentration associated with the group is detectable only within 0.3 of the virial radius (450 kpc) given the available depth of the optical galaxy catalogs at present. We have derived total mass and gas mass distributions (from the X- ray data) and find the gas fraction approaches a constant 8% (for Ho = 70).

Mushotzky, Richard↗
NASA NTRS2004

Over-Luminous Elliptical Galaxies

We have completed a first draft of a paper on the galaxy group ESO3060170, the hottest known fossil group. We have submitted a first draft of the paper but the final completion is delayed due to several issues mentioned by the referee that we wish to revisit and discuss in more detail. The XMM data was combined with Chandra data which allowed a rich set of projects. The paper discusses the north-south elongation which is similar to that of the central dominant galaxy as well as the galaxy distribution. We detect an X-ray 'finger' or small tail emanating from the central galaxy to the north, suggesting motion of the galaxy within the elongated gravitational potential. The overall agreement between XMM and Chandra data are excellent (although the XMM data extend to larger radii). Both data sets show a cool core centered on the dominant galaxy. Surprisingly, - the temperature maps and detailed spectra indicate that the finger of gas is NOT cool, but has the same temperature as the ambient gas. We extracted surface brightness profiles, deprojected gas density profiles, cooling time profiles, and entropy profiles. There is a sharp discontinuity in gas temperature where the surface brightness profile starts to rise rapidly at 10 kpc. This produces a decrease in the cooling time and the gas entropy within 10 kpc. The central cooling time (within 10 kpc) is less than 109 years and falls to almost half that value in the inner 5 kpc. Despite the very short cooling time, we find no evidence (even with the excellent statistics from XMM-Newton) for multi-phased gas, i.e., a cooling flow. We find two 'edges' associated with the gas distribution (common in peaked X-ray groups and galaxies). On large scales, the temperature profile is flat and disagrees with the profile predicted by Loken et al. (2003) from detailed numerical simulations. We studied the galaxy distribution within one virial radius. The galaxy concentration associated with the group is detectable only within 0.3 of the virial radius (450 kpc) given the available depth of the optical galaxy catalogs at present. We have derived total mass and gas mass distributions (from the X-ray data) and find the gas fraction approaches a constant 8% (for H0 = 70).

Mushotsky, Richard↗
NASA NTRS2008

High-Resolution Specification of the Land and Ocean Surface for Improving Regional Mesoscale Model Predictions

The exchange of energy and moisture between the Earth's surface and the atmospheric boundary layer plays a critical role in many meteorological processes. High-resolution, accurate representations of surface properties such as sea-surface temperature (SST), soil temperature and moisture content, ground fluxes, and vegetation are necessary to better understand the Earth-atmosphere interactions and improve numerical predictions of sensible weather. The NASA Short-term Prediction Research and Transition (SPoRT) Center has been conducting separate studies to examine the impacts of high-resolution land-surface initialization data from the Goddard Space Flight Center Land Information System (LIS) on subsequent WRF forecasts, as well as the influence of initializing WRF with SST composites derived from the MODIS instrument. This current project addresses the combined impacts of using high-resolution lower boundary data over both land (LIS data) and water (MODIS SSTs) on the subsequent daily WRF forecasts over Florida during May 2004. For this experiment, the WRF model is configured to run on a nested domain with 9- km and 3-kin grid spacing, centered on the Florida peninsula and adjacent coastal waters of the Gulf of Mexico and Atlantic Ocean. A control configuration of WRF is established to take all initial condition data from the NCEP Eta model. Meanwhile, two WRF experimental runs are configured to use high-resolution initialization data from (1) LIS land-surface data only, and (2) a combination of LIS data and high-resolution MODIS SST composites. The experiment involves running 24-hour simulations of the control WRF configuration, the MS-initialized WRF, and the LIS+MODIS-initialized WRF daily for the entire month of May 2004. All atmospheric data for initial and boundary conditions for the Control, LIS, and LIS+MODIS runs come from the NCEP Eta model on a 40-km grid. Verification statistics are generated at land surface observation sites and buoys, and the impacts of the high-resolution lower boundary data on the development and evolution of mesoscale circulations such as sea and land breezes are examined, This paper will present the results of these WRF modeling experiments using LIS and MODIS lower boundary datasets over the Florida peninsula during May 2004.

Case, Jonathan L.↗
NASA NTRS2008

A New Concurrent Multiscale Methodology for Coupling Molecular Dynamics and Finite Element Analyses

The coupling of molecular dynamics (MD) simulations with finite element methods (FEM) yields computationally efficient models that link fundamental material processes at the atomistic level with continuum field responses at higher length scales. The theoretical challenge involves developing a seamless connection along an interface between two inherently different simulation frameworks. Various specialized methods have been developed to solve particular classes of problems. Many of these methods link the kinematics of individual MD atoms with FEM nodes at their common interface, necessarily requiring that the finite element mesh be refined to atomic resolution. Some of these coupling approaches also require simulations to be carried out at 0 K and restrict modeling to two-dimensional material domains due to difficulties in simulating full three-dimensional material processes. In the present work, a new approach to MD-FEM coupling is developed based on a restatement of the standard boundary value problem used to define a coupled domain. The method replaces a direct linkage of individual MD atoms and finite element (FE) nodes with a statistical averaging of atomistic displacements in local atomic volumes associated with each FE node in an interface region. The FEM and MD computational systems are effectively independent and communicate only through an iterative update of their boundary conditions. With the use of statistical averages of the atomistic quantities to couple the two computational schemes, the developed approach is referred to as an embedded statistical coupling method (ESCM). ESCM provides an enhanced coupling methodology that is inherently applicable to three-dimensional domains, avoids discretization of the continuum model to atomic scale resolution, and permits finite temperature states to be applied.

Yamakov, Vesselin↗
NASA NTRS2015

R2U2: Monitoring and Diagnosis of Security Threats for Unmanned Aerial Systems

We present R2U2, a novel framework for runtime monitoring of security properties and diagnosing of security threats on-board Unmanned Aerial Systems (UAS). R2U2, implemented in FPGA hardware, is a real-time, REALIZABLE, RESPONSIVE, UNOBTRUSIVE Unit for security threat detection. R2U2 is designed to continuously monitor inputs from the GPS and the ground control station, sensor readings, actuator outputs, and flight software status. By simultaneously monitoring and performing statistical reasoning, attack patterns and post-attack discrepancies in the UAS behavior can be detected. R2U2 uses runtime observer pairs for linear and metric temporal logics for property monitoring and Bayesian networks for diagnosis of security threats. We discuss the design and implementation that now enables R2U2 to handle security threats and present simulation results of several attack scenarios on the NASA DragonEye UAS.

Formal Methods↗
NASA NTRS2018

CALIPSO IIR Version 2 Level 1b Calibrated Radiances: Analysis and Reduction of Residual Biases in the Northern Hemisphere

Version 2 of the Level 1b calibrated radiances of the Imaging Infrared Radiometer (IIR) on board the Cloud-Aerosol Lidar and Infrared Satellite Observation (CALIPSO) satellite has been released recently. This new version incorporates corrections of small but systematic seasonal calibration biases previously revealed in Version 1 data products mostly north of 30° N. These biases – of different amplitudes in the three IIR channels 8.65 µm (IIR1), 10.6 µm (IIR2), and 12.05 µm (IIR3) – were made apparent by a striping effect in images of IIR inter-channel brightness temperature differences (BTDs) and through seasonal warm biases of nighttime IIR brightness temperatures in the 30–60° N latitude range. The latter were highlighted through observed and simulated comparisons with similar channels of the Moderate Resolution Imaging Spectroradiometer (MODIS) on board the Aqua spacecraft. To characterize the calibration biases affecting Version 1 data, a semi-empirical approach is developed, which is based on the in-depth analysis of the IIR internal calibration procedure in conjunction with observations such as statistical comparisons with similar MODIS/Aqua channels. Two types of calibration biases are revealed: an equalization bias affecting part of the individual IIR images and a global bias affecting the radiometric level of each image. These biases are observed only when the temperature of the instrument increases, and they are found to be functions of elapsed time since night-to-day transition, regardless of the season. Correction coefficients of Version 1 radiances could thus be defined and implemented in the Version 2 code. As a result, the striping effect seen in Version 1 is significantly attenuated in Version 2. Systematic discrepancies between nighttime and daytime IIR–MODIS BTDs in the 30–60° N latitude range in summer are reduced from 0.2 K in Version 1 to 0.1 K in Version 2 for IIR1–MODIS29. For IIR2–MODIS31 and IIR3–MODIS32, they are reduced from 0.4 K to close to zero, except for IIR3–MODIS32 in June, where the night-minus-day difference is around −0.1 K.

Garnier, Anne↗
NASA NTRS2024

Azimuthal Size Scales of Solar Wind Periodic Density Structures

Periodic Density Structures (PDSs) are quasi-periodic variations of solar wind density ranging from a few minutes to a few hours. PDSs advect with the solar wind and have radial length scales (Lx) of tens to several thousand Mm, thus belonging to the class of “mesoscale structures”. Current interplanetary multi-spacecraft observations are not at spatial separations capable of directly measuring the 3D size scale of PDSs or other mesoscale structures. Instead, previous investigations estimated characteristic spatial scales in solar wind parameters using cross-correlation and/or coherence analysis applied to multi-spacecraft observations. For solar wind density and interplanetary magnetic field (IMF) intensity, the reported size scales perpendicular to the Sun-Earth line (Ly) ranged between ≈30 and ≈200 Earth Radii (RE). Here, we implemented a similar approach on the same parameters but focused on high density slow solar wind intervals with PDSs observed by the Wind and ARTEMIS-P1 spacecraft. Additionally, this is the first statistical study of the IMF intensity periodicities in relation to PDSs. We identified intervals in which the two spacecraft observed the same periodicity obtaining two PDS groups based on their radial length scale: Lx1 ≈86RE and Lx2 ≈35RE. Then, we classified the events based on the periodic variations’ coherence level. Reproducing the results with simulations of the PDSs’ transit, we inferred the Ly order of magnitudes for the two PDS groups: Ly1 ≈340RE and Ly2 ≈187RE. Knowing the PDSs’ size scales is fundamental to constrain models aimed at reproducing these structures and is critical to better understand the PDS-magnetosphere coupling.

Interplanetary physics↗
NASA NTRS2013

The Influence of Observation Errors on Analysis Error and Forecast Skill Investigated with an Observing System Simulation Experiment

The Global Modeling and Assimilation Office (GMAO) observing system simulation experiment (OSSE) framework is used to explore the response of analysis error and forecast skill to observation quality. In an OSSE, synthetic observations may be created that have much smaller error than real observations, and precisely quantified error may be applied to these synthetic observations. Three experiments are performed in which synthetic observations with magnitudes of applied observation error that vary from zero to twice the estimated realistic error are ingested into the Goddard Earth Observing System Model (GEOS-5) with Gridpoint Statistical Interpolation (GSI) data assimilation for a one-month period representing July. The analysis increment and observation innovation are strongly impacted by observation error, with much larger variances for increased observation error. The analysis quality is degraded by increased observation error, but the change in root-mean-square error of the analysis state is small relative to the total analysis error. Surprisingly, in the 120 hour forecast increased observation error only yields a slight decline in forecast skill in the extratropics, and no discernable degradation of forecast skill in the tropics.

Observation Error↗
NASA NTRS1991

Effect of short-term exposure to stereoscopic three-dimensional flight displays on real-world depth perception

High-fidelity color pictorial displays that incorporate depth cues in the display elements are currently available. Depth cuing applied to advanced head-down flight display concepts potentially enhances the pilot's situational awareness and improves task performance. Depth cues provided by stereopsis exhibit constraints that must be fully understood so depth cuing enhancements can be adequately realized and exploited. A fundamental issue (the goal of this investigation) is whether the use of head-down stereoscopic displays in flight applications degrade the real-world depth perception of pilots using such displays. Stereoacuity tests are used in this study as the measure of interest. Eight pilots flew repeated simulated landing approaches using both nonstereo and stereo 3-D head-down pathway-in-the-sky displays. At this decision height of each approach (where the pilot changes to an out-the-window view to obtain real-world visual references) the pilots changed to a stereoacuity test that used real objects. Statistical analysis of stereoacuity measures (data for a control condition of no exposure to any electronic flight display compared with data for changes from nonstereo and from stereo displays) reveals no significant differences for any of the conditions. Therefore, changing from short-term exposure to a head-down stereo display has no more effect on real-world relative depth perception than does changing from a nonstereo display. However, depth perception effects based on sized and distance judgements and on long-term exposure remain issues to be investigated.

Busquets, Anthony M.↗
NASA NTRS2000

Increase in the Random Dopant Induced Threshold Fluctuations and Lowering in Sub 100 nm MOSFETs Due to Quantum Effects: A 3-D Density-Gradient Simulation Study

In this paper we present a detailed simulation study of the influence of quantum mechanical effects in the inversion layer on random dopant induced threshold voltage fluctuations and lowering in sub 100 nm MOSFETs. The simulations have been performed using a 3-D implementation of the density gradient (DG) formalism incorporated in our established 3-D atomistic simulation approach. This results in a self-consistent 3-D quantum mechanical picture, which implies not only the vertical inversion layer quantisation but also the lateral confinement effects related to current filamentation in the 'valleys' of the random potential fluctuations. We have shown that the net result of including quantum mechanical effects, while considering statistical dopant fluctuations, is an increase in both threshold voltage fluctuations and lowering. At the same time, the random dopant induced threshold voltage lowering partially compensates for the quantum mechanical threshold voltage shift in aggressively scaled MOSFETs with ultrathin gate oxides.

Asenov, Asen↗
NASA NTRS1992

Hit rates and radiation doses to nuclei of bone lining cells from alpha-particle-emitting radionuclides

Factors relating the local concentration of a bone-seeking alpha-particle emitter to the mean hit rate have been determined for nuclei of bone lining cells using a Monte Carlo procedure. Cell nuclei were approximated by oblate spheroids with dimensions and location taken from a previous histomorphometric study. The Monte Carlo simulation is applicable for planar and diffuse labels at plane or cylindrical bone surfaces. Additionally, the mean nuclear dose per hit, the dose mean per hit, the mean track segment length and its second moment, the percentage of stoppers, and the frequency distribution of the dose have been determined. Some basic features of the hit statistics for bone lining cells have been outlined, and the consequences of existing standards of radiation protection with regard to the hit frequency to cell nuclei are discussed.

NASA Discipline Musculoskeletal↗
NASA NTRS2018

Melt Pond Conditions on Declining Arctic Sea Ice Over 1979-2016: Model Development, Validation, and Results

A melt pond (MP) distribution equation has been developed and incorporated into the Marginal Ice-Zone Modeling and Assimilation System to simulate Arctic MPs and sea ice over 1979-2016. The equation differs from previous MP models and yet benefits from previous studies for MP parameterizations as well as a range of observations for model calibration. Model results show higher magnitude of MP volume per unit ice area and area fraction in most of the Canada Basin and the East Siberian Sea and lower magnitude in the central Arctic. This is consistent with Moderate Resolution Imaging Spectroradiometer observations, evaluated with Measurements of Earth Data for Environmental Analysis (MEDEA) data, and closely related to top ice melt per unit ice area. The model simulates a decrease in the total Arctic sea ice volume and area, owing to a strong increase in bottom and lateral ice melt. The sea ice decline leads to a strong decrease in the total MP volume and area. However, the Arctic-averaged MP volume per unit ice area and area fraction show weak, statistically insignificant downward trends, which is linked to the fact that MP water drainage per unit ice area is increasing. It is also linked to the fact that MP volume and area decrease relatively faster than ice area. This suggests that overall the actual MP conditions on ice have changed little in the past decades as the ice cover is retreating in response to Arctic warming, thus consistent with the Moderate Resolution Imaging Spectroradiometer observations that show no clear trend in MP area fraction over 2000-2011.

sea ice↗
NASA NTRS2018

Global Changes in the Diurnal Cycle of Surface Ozone

Changing emissions of NOx and other ozone precursors drive trends in both production and loss of surface ozone, leading to surface ozone trends that differ according to the time of day. Consequently, the magnitude of the diurnal cycle in surface ozone is changing in several regions of the world. Changes in the diurnal cycle of ozone have implications for the metrics used to assess the impact of ozone on human health and vegetation, since different metrics are sensitive to different portions of the diurnal cycle. We use a high resolution model simulation to examine global changes in the magnitude of the diurnal cycle of O3 between 1980 and 2015. The simulation reproduces the negative trends in the tropospheric NO2 column over the eastern United States and Europe, and the positive trends over East Asia, seen by the Ozone Monitoring Instrument (OMI). It also gives a reasonable reproduction of the change in the diurnal cycle of surface ozone seen at rural sites in the eastern United States between the 1990s and 2000s. The simulation shows that the magnitude of the surface O3 diurnal cycle is increasing in regions with positive changes in NOx emissions, such as South and East Asia, and decreasing in regions with reductions in NOx emissions. It also shows changes in the diurnal cycle of the tropospheric ozone column, although these have fewer regions with statistically significant trends. These changes suggest that daily mean ozone is responding less than the mid-day ozone measured by the Total Ozone Mapping Spectrometer (TOMS) and OMI.

Trends↗
NASA NTRS2002

Decoupling the Role of Particle Inertia and Gravity on Particle Dispersion

Particle dispersion and the influence that particle momentum exchange has on the properties of a turbulent carrier flow in micro-gravity environments challenge present understanding and predictive schemes. The objective of this effort has been to develop and assess high-fidelity simulation tools for predicting particle transport within micro-gravity environments suspended in turbulent flows. The computational technique is based on Direct Numerical Simulation (DNS) of the incompressible Navier-Stokes equations. The particular focus of the present work is on the class of dilute flows in which particle volume fractions and inter-particle collisions are negligible. Particle motion is assumed to be governed by drag with particle relaxation times ranging from the Kolmogorov scale to the Eulerian timescale of the turbulence and particle mass loadings up to one. The velocity field was made statistically stationary by forcing the low wavenumbers of the flow. The calculations were performed using 96(exp 3) collocation points and the Taylor-scale Reynolds number for the stationary flow was 62. The effect of particles on the turbulence was included in the Navier-Stokes equations using the point-force approximation in which 96(exp 3) particles were used in the calculations. DNS results show that particles increasingly dissipate fluid kinetic energy with increased loading, with the reduction in kinetic energy being relatively independent of the particle relaxation time. Viscous dissipation in the fluid decreases with increased loading and is larger for particles with smaller relaxation times. Fluid energy spectra show that there is a non-uniform distortion of the turbulence with a relative increase in small-scale energy. The non-uniform distortion significantly affects the transport of the dissipation rate, with the production and destruction of dissipation exhibiting completely different behaviors. The spectrum of the fluid-particle energy exchange rate shows that the fluid drags particles at low wavenumbers while the converse is true at high wavenumbers for small particles. A spectral analysis shows that the increase of the high wavenumber portion of the fluid energy spectrum can be attributed to transfer of the fluid-particle covariance by the fluid turbulence. This in turn explains the relative increase of small-scale energy caused by small particles observed in the present simulations as well as those of others.

Squires, Kyle D.↗
NASA NTRS1980

Simulator study of conventional general aviation instrument displays in path-following tasks with emphasis on pilot-induced oscillations

A study of the use of conventional general aviation instruments by general aviation pilots in a six degree of freedom, fixed base simulator was conducted. The tasks performed were tracking a VOR radial and making an ILS approach to landing. A special feature of the tests was that the sensitivity of the displacement indicating instruments (the RMI, CDI, and HSI) was kept constant at values corresponding to 5 n. mi. and 1.25 n. mi. from the station. Both statistical and pilot model analyses of the data were made. The results show that performance in path following improved with increases in display sensitivity up to the highest sensitivity tested. At this maximum test sensitivity, which corresponds to the sensitivity existing at 1.25 n. mi. for the ILS glide slope transmitter, tracking accuracy was no better than it was at 5 n. mi. from the station and the pilot aircraft system exhibited a marked reduction in damping. In some cases, a pilot induced, long period unstable oscillation occurred.

Adams, J. J.↗
NASA NTRS1983

Spatial variation of corn canopy temperature as dependent upon soil texture and crop rooting characteristics

A soil plant atmosphere model for corn (Zea mays L.) together with the scaling theory for soil hydraulic heterogeneity are used to study the sensitivity of spatial variation of canopy temperature to field averaged soil texture and crop rooting characteristics. The soil plant atmosphere model explicitly solves a continuity equation for water flux resulting from root water uptake, changes in plant water storage and transpirational flux. Dynamical equations for root zone soil water potential and the plant water storage models the progressive drying of soil, and day time dehydration and night time hydration of the crop. The statistic of scaling parameter which describes the spatial variation of soil hydraulic conductivity and matric potential is assumed to be independent of soil texture class. The field averaged soil hydraulic characteristics are chosen to be representative of loamy sand and clay loam soils. Two rooting characteristics are chosen, one shallow and the other deep rooted. The simulation shows that the range of canopy temperatures in the clayey soil is less than 1K, but for the sandy soil the range is about 2.5 and 5.0 K, respectively, for the shallow and deep rooted crops.

Choudhury, B. J.↗
NASA NTRS1984

Higher order derivative correlations of velocity and temperature in isotropic and sheared numerical turbulence

Through a combination of small scale statistics and flow visualization it is shown that the small scales of a numerically generated turbulent flow are strongly aligned around vortex tubes. Movies of the time developing flow will be used to illustrate the nature of these structures. An inertial subrange is found with a Kolmogorov constant, alpha, equal to 2.45. This is consistent with simulated and measured values for the skewness, but disagrees with experimental values for the Kolmogorov constant, which usually range between 1.4 to 1.7.

Kerr, R. M.↗