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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 649 records · Page 36
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NASA NTRS1997

The Aggregate Description of Semi-Arid Vegetation with Precipitation-Generated Soil Moisture Heterogeneity

Meteorological measurements in the Walnut Gulch catchment in Arizona were used to synthesize a distributed, hourly-average time series of data across a 26.9 by 12.5 km area with a grid resolution of 480 m for a continuous 18-month period which included two seasons of monsoonal rainfall. Coupled surface-atmosphere model runs established the acceptability (for modelling purposes) of assuming uniformity in all meteorological variables other than rainfall. Rainfall was interpolated onto the grid from an array of 82 recording rain gauges. These meteorological data were used as forcing variables for an equivalent array of stand-alone Biosphere-Atmosphere Transfer Scheme (BATS) models to describe the evolution of soil moisture and surface energy fluxes in response to the prevalent, heterogeneous pattern of convective precipitation. The calculated area-average behaviour was compared with that given by a single aggregate BATS simulation forced with area-average meteorological data. Heterogeneous rainfall gives rise to significant but partly compensating differences in the transpiration and the intercepted rainfall components of total evaporation during rain storms. However, the calculated area-average surface energy fluxes given by the two simulations in rain-free conditions with strong heterogeneity in soil moisture were always close to identical, a result which is independent of whether default or site-specific vegetation and soil parameters were used. Because the spatial variability in soil moisture throughout the catchment has the same order of magnitude as the amount of rain failing in a typical convective storm (commonly 10% of the vegetation's root zone saturation) in a semi-arid environment, non-linearitv in the relationship between transpiration and the soil moisture available to the vegetation has limited influence on area-average surface fluxes.

White, Cary B.↗
NASA NTRS1996

Kinetics of Static Strain Aging in Polycrystalline NiAl-based Alloys

The kinetics of yield point return have been studied in two NiAl-based alloys as a function of aging time at temperatures between 300 and 700 K. The results indicate that the upper yield stress increment, Delta sigma(sub u) (i.e., stress difference between the upper yield point and the final flow stress achieved during prestraining), in conventional purity (CP-NiAl) and in high purity carbon-doped (NiAl-C) material first increased with a t(exp 2/3) relationship before reaching a plateau. This behavior suggests that a Cottrell locking mechanism is the cause for yield points in NiAl. In addition, positive y-axis intercepts were observed in plots of Delta sigma(sub u) versus t(exp 2/3) suggesting the operation of a Snoek mechanism. Analysis according to the Cottrell Bilby model of atmosphere formation around dislocations yields an activation energy for yield point return in the range 70 to 76 kJ/mol which is comparable to the activation energy for diffusion of interstitial impurities in bcc metals. It is, thus, concluded that the kinetics of static strain aging in NiAl are controlled by the locking of dislocations by Cottrell atmospheres of carbon atoms around dislocations.

Weaver, M. L.↗
NASA NTRS1997

Invariance of Hypersonic Normal Force Coefficients with Reynolds Number and Determination of Inviscid Wave Drag from Laminar Experimental Results

Observations have been made and reported that the experimental normal force coefficients at a constant angle of attack were constant with a variation of more than 2 orders of magnitude of Reynolds number at a free-stream Mach number M(sub infinity) of 8.00 and more than 1 order of magnitude variation at M(sub infinity) = 6.00 on the same body-wing hypersonic cruise configuration. These data were recorded under laminar, transitional, and turbulent boundary layer conditions with both hot-wall and cold-wall models. This report presents experimental data on 25 configurations of 17 models of both simple and complex geometry taken at M(sub infinity) = 6.00, 6.86, and 8.00 in 4 different hypersonic facilities. Aerodynamic calculations were made by computational fluid dynamics (CID) and engineering methods to analyze these data. The conclusions were that the normal force coefficients at a given altitude are constant with Reynolds numbers at hypersonic speeds and that the axial force coefficients recorded under laminar boundary-layer conditions at several Reynolds numbers may be plotted against the laminar parameter (the reciprocal of the Reynolds number to the one-half power) and extrapolated to the ordinate axis to determine the inviscid-wave-drag coefficient at the intercept.

Hawkins, Richard↗
NASA NTRS1997

Future Aerospace Technology in the Service of the Alliance: Sustained Hypersonic Flight - Volume 3

Sustained Hypersonic Flight capability is envisioned for several future military and civilian applications, such as long range immediate reaction reconnaissance, high speed interception of air targets, long range precision strike against hardened or time critical targets, and access to space. The inherent reduction in time-to-target and low vulnerability will permit new operational tactics. The symposium outlined mission needs and operational scenarios for hypersonic vehicles. Addressed were: Technological issues and challenges in external hypersonic aerodynamics and design, propulsion and engine/airframe integration, military ramjet applications, overall system design including structures and materials development needs, and test facilities. Ground based test facilities cannot provide full simulation at Mach numbers above 5. Therefore, the use of Computational Fluid Dynamics design tools is essential. For routine use of the computational tools in analysis, design and optimization, it is necessary to reduce the time for the entire computational process by two orders of magnitude. Hypersonic boundary layer transition remains a critical design issue because of the important impact on engine drag and on heating, which can affect the choice of materials and thermal protection systems. In addition, reliable prediction of scramjet net thrust is an absolute must in resolving hypersonic air breathing vehicle design issues. Due to current facility and computational shortfalls, the development of future hypersonic flight systems requires research flight tests in the technology areas of boundary layer transition and air-breathing propulsion engine performance. For sustained hypersonic flight beyond Mach 6, the supersonic combustion ramjet (scramjet) engine is the only choice for the near future. Only this air-breathing concept offers a significant promise of large reductions in required propellant fractions, increased payload fractions, and reduced size vehicles, together with a foreseeable technological feasibility. Airframe/engine integration, combustor design and thermal management are the predominant engineering tasks. Fuels, hydrogen or hydrocarbon, must be matched to the operational needs of military or civil use. Experience in existing ramjet propelled missiles capable of speeds up to Mach 4 can support the development effort. The potential mission and cost benefits of sustained hypersonic flight to both military and civil applications are tremendous. From the budget point of view, the possibility of sharing development costs between military and civil programs offers a specific advantage.

Source record↗
NASA NTRS1998

Polar Hydra Data Analysis

The science activities are: 1) Hydra is still operating successfully on orbit. 2) A large amount of analysis and discovery has occurred with the Hydra ground data processing this past year. 3) Full interdetector calibration has been implemented and documented. This intercalibration was necessitated by the incorrect installation of bias resistors in the pre-acceleration stage to the electron channeltrons. This had the effect of making the counting efficiency for electrons energy dependent as well as channeltron specific. The nature of the error had no impact on the ion detection efficiency since they have a different bias arrangement. This intercalibration is so effective, that the electron and ion moment densities are routinely produced with a level of agreement better than 20%. 4) The data processing routinely removes glint in the sensors and produces public energy time spectrograms on the web overnight. 6) Routine, but more intensive computer processing codes are operational that determine for electrons and ions, the density, the flow vector, the pressure tensor and the heat flux by numerical integration. These codes use the magnetic field to sustain the quality of their output. To gain access to this high quality magnetic field within our data stream we have monitored Russell's web page for zero levels and timing files (since his data acquisition is not telemetry synchronous) and have a local reconstruction of B for our use. We have also detected a routine anomaly in the magnetometer data stream that we have documented to Chris Russell and developed an editing algorithm to intercept these "hits" and remove them from the geophysical analysis.

Scudder, J. D.↗
NASA NTRS1997

Energy from Redox Disproportionation of Sugar Carbon Drives Biotic and Abiotic Synthesis

To identify the energy source that drives the biosynthesis of amino acids, lipids, and nucleotides from glucose, we calculated the free energy change due to redox disproportionation of the substrate carbon of: (1) 26-carbon fermentation reactions and (2) the biosynthesis of amino acids and lipids of E. coli from glucose. The free energy (cal/mmol of carbon) of these reactions was plotted as a function of the degree of redox disproportionation of carbon (disproportionative electron transfers (mmol)/mmol of carbon). The zero intercept and proportionality between energy yield and degree of redox disproportionation exhibited by this plot demonstrate that redox disproportionation is the principal energy source of these redox reactions (slope of linear fit = -10.4 cal/mmol of disproportionative electron transfers). The energy and disproportionation values of E. coli amino acid and lipid biosynthesis from glucose lie near this linear curve fit with redox disproportionation accounting for 84% and 96% (and ATP only 6% and 1 %) of the total energy of amino acid and lipid biosynthesis, respectively. These observations establish that redox disproportionation of carbon, and not ATP, is the primary energy source driving amino acid and lipid biosynthesis from glucose. In contrast, we found that nucteotide biosynthesis involves very little redox disproportionation, and consequently depends almost entirely on ATP for energy. The function of sugar redox disproportionation as the major source of free energy for the biosynthesis of amino acids and lipids suggests that sugar disproportionation played a central role in the origin of metabolism, and probably the origin of life.

Weber, Arthur L.↗
NASA NTRS1998

Stereo-Video Data Reduction of Wake Vortices and Trailing Aircrafts

This report presents stereo image theory and the corresponding image processing software developed to analyze stereo imaging data acquired for the wake-vortex hazard flight experiment conducted at NASA Langley Research Center. In this experiment, a leading Lockheed C-130 was equipped with wing-tip smokers to visualize its wing vortices, while a trailing Boeing 737 flew into the wake vortices of the leading airplane. A Rockwell OV-10A airplane, fitted with video cameras under its wings, flew at 400 to 1000 feet above and parallel to the wakes, and photographed the wake interception process for the purpose of determining the three-dimensional location of the trailing aircraft relative to the wake. The report establishes the image-processing tools developed to analyze the video flight-test data, identifies sources of potential inaccuracies, and assesses the quality of the resultant set of stereo data reduction.

Alter-Gartenberg, Rachel↗
NASA NTRS1999

Seasonal and Interannual Variations of Evaporation and their Relations with Precipitation, Net Radiation, and Net Carbon Accumulation for the Gediz Basin Area

A model combining the rate of carbon assimilation with water and energy balance equations has been run using satellite and ancillary data for a period of 60 months (January 1986 to December 1990). Calculations for the Gediz basin area give mean annual evaporation as 395 mm, which is composed of 45% transpiration, 42% soil evaporation and 13% interception. The coefficient of interannual variation of evaporation is found to be 6%, while that for precipitation and net radiation are, respectively, 16% and 2%, illustrating that net radiation has an important effect in modulating interannual variation of evaporation. The mean annual water use efficiency (i.e., the ratio of net carbon accumulation and total evaporation) is ca. 1 g/sq m/mm, and has a coefficient of interannual variation of 5%. A comparison of the mean water use efficiency with field observations suggests that evaporation over the area is utilized well for biomass production. The reference crop evaporation for irrigated areas has annual mean and coefficient of variation as, respectively, 1176 mm and 3%. The total evaporation during three summer months of peak evaporation (June-August) is estimated to be about 575 mm for irrigated crops like maize and cotton. Seasonal variations of the fluxes are presented.

Choudhury, Bhaskar J.↗
NASA NTRS1997

Energy from Redox Disproportionation of Sugar Carbon Drives Biotic and Abiotic Synthesis

To identify the energy source that drives the biosynthesis of amino acids, lipids, and nucleotides from glucose, we calculated the free energy change due to redox disproportionation of the substrate carbon of: (1) 26-carbon fermentation reactions and (2) the biosynthesis of amino acids and lipids of E. coli from glucose. The free energy (cal/mmol of carbon) of these reactions was plotted as a function of the degree of redox disproportionation of carbon (disproportionative electron transfers (mmol)/mmol of carbon). The zero intercept and proportionality between energy yield and degree of redox disproportionation exhibited by this plot demonstrate that redox disproportionation is the principal energy source of these redox reactions (slope of linear fit = -10.4 cal/mmol of disproportionative electron transfers). The energy and disproportionation values of E. coli amino acid and lipid biosynthesis from glucose lie near this linear curve fit with redox disproportionation accounting for 84% and 96% (and ATP only 6% and 1%) of the total energy of amino acid and lipid biosynthesis, respectively. These observations establish that redox disproportionation of carbon, and not ATP, is the primary energy source driving amino acid and lipid biosynthesis from glucose. In contrast, we found that nucleotide biosynthesis involves very little redox disproportionation, and consequently depends almost entirely on ATP for energy. The function of sugar redox disproportionation as the major source of free energy for the biosynthesis of amino acids and lipids suggests that sugar disproportionation played a central role in the origin of metabolism, and probably the origin of life.

Weber, Arthur L.↗
NASA NTRS1996

Investigation of the Physical Processes Governing Large-scale Tracer Transport in the Stratosphere and Troposphere

This report reviews the second year of a three-year research program to investigate the physical mechanisms which underlie the transport of trace constituents in the stratosphere- troposphere system. The primary scientific goal of the research is to identify the processes which transport air masses within the lower stratosphere, particularly between the tropics and middle latitudes. The SASS program seeks to understand the impact of the present and future fleets of conventional jet traffic on the upper troposphere and lower stratosphere, while complementary airborne observations under UARP seek to understand the complex interactions of dynamical and chemical processes that affect the ozone layer. The present investigation contributes to the goals of each of these by diagnosing the history of air parcels intercepted by NASA research aircraft in UARP and AEAP campaigns.

Selkirk, Henry B.↗
NASA NTRS1999

Simulation of 1986 South China Sea Monsoon with a Regional Climate Model

A Regional Land-Atmosphere Climate Simulation System (RELACS) project is being developed at NASA Goddard Space Flight Center. One of the major goals of RELACS is to use a regional scale model with improved physical processes and in particular land-related processes, to understand the role of the land surface and its interaction with convection and radiation as well as the water/energy cycles in the IndoChina/South China Sea (SCS) region. The Penn State/NCAR MM5 atmospheric modeling system, a state of the art atmospheric numerical model designed to simulate regional weather and climate, has been successfully coupled to the Parameterization for Land-Atmosphere-Cloud Exchange (PLACE) land surface model. The original MM5 model (without PLACE) includes the option for either a simple slab soil model or a five-layer soil model (MRF) in which the soil moisture availability evolves over time. However, the MM5 soil models do not include the effects of vegetation, and thus important physical processes such as evapotranspiration and interception are precluded. The PLACE model incorporates vegetation type and has been shown in international comparisons to accurately predict evapotranspiration and runoff over a wide variety of land surfaces. The coupling of MM5 and PLACE creates a numerical modeling system with the potential to more realistically simulate atmosphere and land surface processes including land-sea interaction, regional circulations such as monsoons, and flash flood events. In addition, the Penn State/NCAR MM5 atmospheric modeling system has been: (1) coupled to the Goddard Ice Microphysical scheme; (2) coupled to a turbulent kinetic energy (TKE) scheme; (3) modified to ensure cloud budget balance; and (4) incorporated initialization with the Goddard EOS data sets at NASA/Goddard Laboratory for Atmospheres. The improved MM5 with two nested domains (60 and 20 km horizontal resolution) was used to simulate convective activity over IndoChina and the South China Sea, during the monsoon season, from May 6 to May 20, 1986. The model results captured several dominant observed features, such as twin cyclones, a depression system over the Bay of Bengal, strong south-westerly winds over IndoChina before and during the on-set of convection over the SCS, and a vortex over the SCS. Two additional MM5 runs with different land process models, Blackadar and MRF, were performed, and their results are compared to the run with PLACE. The preliminary results indicate that the MM5 results using PLACE and Blackadar are in very good agreement, but the results using MRF do not contain the south-westerly wind over IndoChina prior to the on-set of convection over the SCS.

Tao, W. -K.↗
NASA NTRS1998

Carbon-Water-Energy Relations for Selected River Basins

A biophysical process-based model was run using satellite, assimilated and ancillary data for four years (1987-1990) to calculate components of total evaporation (transpiration, interception, soil and snow evaporation), net radiation, absorbed photosynthetically active radiation and net primary productivity over the global land surface. Satellite observations provided fractional vegetation cover, solar and photosynthetically active radiation incident of the surface, surface albedo, fractional cloud cover, air temperature and vapor pressure. The friction velocity and surface air pressure are obtained from a four dimensional data assimilation results, while precipitation is either only surface observations or a blended product of surface and satellite observations. All surface and satellite data are monthly mean values; precipitation has been disaggregated into daily values. All biophysical parameters of the model are prescribed according to published records. From these global land surface calculations results for river basins are derived using digital templates of basin boundaries. Comparisons with field observations (micrometeorologic, catchment water balance, biomass production) and atmospheric water budget analysis for monthly evaporation from six river basins have been done to assess errors in the calculations. Comparisons are also made with previous estimates of zonal variations of evaporation and net primary productivity. Efficiencies of transpiration, total evaporation and radiation use, and evaporative fraction for selected river basins will be presented.

Choudhury, B. J.↗
NASA NTRS1999

Sensitivity of Climate to Changes in NDVI

The sensitivity of global and regional climate to changes in vegetation density is investigated using a coupled biosphere-atmosphere model. The magnitude of the vegetation changes and their spatial distribution are based on natural decadal variability of the normalized difference vegetation index (ndvi). Different scenarios using maximum and minimum vegetation cover were derived from satellite records spanning the period 1982-1990. Albedo decreased in the northern latitudes and increased in the tropics with increased ndvi. The increase in vegetation density revealed that the vegetation's physiological response was constrained by the limits of the available water resources. The difference between the maximum and minimum vegetation scenarios resulted in a 46% increase in absorbed visible solar radiation and a similar increase in gross photosynthetic C02 uptake on a global annual basis. This caused the canopy transpiration and interception fluxes to increase, and reduced those from the soil. The redistribution of the surface energy fluxes substantially reduced the Bowen ratio during the growing season, resulting in cooler and moister near-surface climate, except when soil moisture was limiting. Important effects of increased vegetation on climate are : (1) A cooling of about 1.8 K in the northern latitudes during the growing season and a slight warming during the winter, which is primarily due to the masking of high albedo of snow by a denser canopy. and (2) A year round cooling of 0.8 K in the tropics. These results suggest that increases in vegetation density could partially compensate for parallel increases in greenhouse warming . Increasing vegetation density globally caused both evapotranspiration and precipitation to increase. Evapotranspiration, however increased more than precipitation resulting in a global soil-water deficit of about 15 %. A spectral analysis on the simulated results showed that changes in the state of vegetation could affect the low-frequency modes of the precipitation distribution and might reduce its low frequency variability in the tropics while increasing it in northern latitudes.

Bounoua, L.↗
NASA NTRS1999

"Driverless" Shocks in the Interplanetary Medium

Many interplanetary shocks have been detected without an obvious driver behind them. These shocks have been thought to be either blast waves from solar flares or shocks due to sudden increase in solar wind speed caused by interactions between large scale open and closed field lines of the Sun. We investigated this problem using a set of interplanetary shock detected {\it in situ} by the Wind space craft and tracing their solar origins using low frequency radio data obtained by the Wind/WAVES experiment. For each of these "driverless shocks" we could find a unique coronal mass ejections (CME) event observed by the SOHO (Solar and Heliospheric Observatory) coronagraphs. We also found that these CMEs were ejected at large angles from the Sun-Earth line. It appears that the "driverless shocks" are actually driver shocks, but the drivers were not intercepted by the spacecraft. We conclude that the interplanetary shocks are much more extended than the driving CMEs.

Gopalswamy, N.↗
NASA NTRS1999

High Power Intermodulation Measurements up to 30 W of High Temperature Superconducting Filters

We have demonstrated a high power intermodulation measurement set-up capable of delivering 30 W in each of two fundamental tones. For closely spaced frequencies (less than 35 MHz), the dynamic range of the measurement is limited by the nonlinear performance of the mixer in the front end of the HP71210C spectrum analyzer. A tunable TE(sub 011) mode copper cavity was fabricated in which one of the endwalls could be adjusted shifting its resonant frequency between 5.7 and 6.6 GHz. Since the Q-value of this cavity is high, greater than 10(exp 4), and its bandwidth is small, less than 1 MHz, it can be used to attenuate the two fundamental tones relative to one of the harmonic tones, which greatly enhances the dynamic range of the measurement. This set-up can be used to measure the two-tone intermodulation distortion of any passive microwave device, e.g. a HTS filter, a connector, a cable, etc., over a frequency range of 5.9 to 6.4 GHz and a power range of 0.1 to 30 W. The third order intercept (TOI) of a prototype HTS filter measured at powers up to 30 W was +81.3 dBm.

Wilker, Charles↗
NASA NTRS1999

Lidar Altimeter Measurements of Canopy Structure: Methods and Validation for Closed Canopy, Broadleaf Forests

Lidar altimeter observations of vegetated landscapes provide a time-resolved measure of laser pulse backscatter energy from canopy surfaces and the underlying ground. Airborne lidar altimeter data was acquired using the Scanning Lidar Imager of Canopies by Echo Recovery (SLICER) for a successional sequence of four, closed-canopy, deciduous forest stands in eastern Maryland. The four stands were selected so as to include a range of canopy structures of importance to forest ecosystem function, including variation in the height and roughness of the outer-most canopy surface and the vertical organization of canopy stories and gaps. The character of the SLICER backscatter signal is described and a method is developed that accounts for occlusion of the laser energy by canopy surfaces, transforming the backscatter signal to a canopy height profile (CHP) that quantitatively represents the relative vertical distribution of canopy surface area. The transformation applies an increased weighting to the backscatter amplitude as a function of closure through the canopy and assumes a horizontally random distribution of the canopy components. SLICER CHPs, averaged over areas of overlap where lidar ground tracks intersect, are shown to be highly reproducible. CHP transects across the four stands reveal spatial variations in vegetation, at the scale of the individual 10 m diameter laser footprints, within and between stands. Averaged SLICER CHPs are compared to analogous height profile results derived from ground-based sightings to plant intercepts measured on plots within the four stands. Tbe plots were located on the segments of the lidar ground tracks from which averaged SLICER CHPs were derived, and the ground observations were acquired within two weeks of the SLICER data acquisition to minimize temporal change. The differences in canopy structure between the four stands is similarly described by the SLICER and ground-based CHP results, however a Chi-square test of similarity documents differences that are statistically significant. The differences are discussed in terms of measurement properties that define the smoothness of the resulting CHPs and Lidar Altimeter Measurements of Canopy Structure - Harding et al. canopy properties that may vertically bias the CHP representations of canopy structure. The statistical differences are most likely due to the more noisy character of the ground-based CHPs, especially high in the canopy where ground-based sightings are rare resulting in an underestimate of canopy surface area and height, and to departures from the assumption of horizontal randomness which bias the CHPs toward the observer (upward for SLICER and downward for ground-based CHPs). The results demonstrate that the SLICER observations reliably provide a measure of canopy structure that reveals ecologically interesting structural variations such as those characterizing a successional sequence of closed-canopy, broadleaf forest stands.

Harding, D. J.↗
NASA NTRS2000

Cylindrical Optic Polishing Dwell Time Optimization

The Constellation-X x-ray telescope consists of replicated grazing incidence optics. The polishing of the "master' mandrel is accomplished by moving a polishing tool along a helical path on the surface of this almost cylindrical surface. The measurement of the surface is, however, performed along "axial" scan lines which intercept this helical path. This approach to polishing and measuring permits a relatively simple scheme to be implemented for the determination of the optimal dwell times of the polishing tool. These optimal dwell times are determined by a deconvolution which approaches the problem as a linear programming problem and uses the simplex method. This approach maximizes the amount of material removed at any point subject to inequality constraints. The effects of using these "optimum" dwell times is to significantly improve the "tools" effectiveness at removing the higher spatial frequencies while staying (strictly) within the bounds and constraints imposed by the hardware. In addition the "ringing" at the edges of the optic, frequently present in deconvolution problems, is completely eliminated.

Waluschka, Eugene↗
NASA NTRS1997

The Interplay Between Transpiration and Runoff Formulations in Land Surface Schemes Used with Atmospheric Models

The Project for Intercomparison of Land-surface Parameterization Schemes (PILPS) has shown that different land surface models (LSMS) driven by the same meteorological forcing can produce markedly different surface energy and water budgets, even when certain critical aspects of the LSMs (vegetation cover, albedo, turbulent drag coefficient, and snow cover) are carefully controlled. To help explain these differences, the authors devised a monthly water balance model that successfully reproduces the annual and seasonal water balances of the different PILPS schemes. Analysis of this model leads to the identification of two quantities that characterize an LSM's formulation of soil water balance dynamics: (1) the efficiency of the soil's evaporation sink integrated over the active soil moisture range, and (2) the fraction of this range over which runoff is generated. Regardless of the LSM's complexity, the combination of these two derived parameters with rates of interception loss, potential evaporation, and precipitation provides a reasonable estimate for the LSM's simulated annual water balance. The two derived parameters shed light on how evaporation and runoff formulations interact in an LSM, and the analysis as a whole underscores the need for compatibility in these formulations.

Koster, Rindal D.↗