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NASA NTRS1996

A Note on Substructuring Preconditioning for Nonconforming Finite Element Approximations of Second Order Elliptic Problems

In this paper an algebraic substructuring preconditioner is considered for nonconforming finite element approximations of second order elliptic problems in 3D domains with a piecewise constant diffusion coefficient. Using a substructuring idea and a block Gauss elimination, part of the unknowns is eliminated and the Schur complement obtained is preconditioned by a spectrally equivalent very sparse matrix. In the case of quasiuniform tetrahedral mesh an appropriate algebraic multigrid solver can be used to solve the problem with this matrix. Explicit estimates of condition numbers and implementation algorithms are established for the constructed preconditioner. It is shown that the condition number of the preconditioned matrix does not depend on either the mesh step size or the jump of the coefficient. Finally, numerical experiments are presented to illustrate the theory being developed.

Maliassov, Serguei↗

NASA NTRS1994

Assimilation by Lunar Mare Basalts: Melting of Crustal Material and Dissolution of Anorthite

We discuss techniques for calculating the amount of crustal assimilation possible in lunar magma chambers and dikes based on thermal energy balances, kinetic rates, and simple fluid mechanical constraints. Assuming parent magmas of picritic compositions, we demonstrate the limits on the capacity of such magmas to melt and dissolve wall rock of anorthitic, troctolitic, noritic, and KREEP (quartz monzodiorite) compositions. Significant melting of the plagioclase-rich crustal lithologies requires turbulent convection in the assimilating magma and an efficient method of mixing in the relatively buoyant and viscous new melt. Even when this occurs, the major element chemistry of the picritic magmas will change by less than 1-2 wt %. Diffusion coefficients measured for Al2O3 from an iron-free basalt and an orange glass composition are 10(exp -12) m(exp 2) s(exp -1) at 1340 C and 10(exp -11) m(exp 2) s(exp -1) at 1390 C. These rates are too slow to allow dissolution of plagioclase to significantly affect magma compositions. Picritic magmas can melt significant quantities of KREEP, which suggests that their trace element chemistry may still be affected by assimilation processes; however, mixing viscous melts of KREEP composition with the fluid picritic magmas could be prohibitively difficult. We conclude that only a small part of the total major element chemical variation in the mare basalt and volcanic glass collection is due to assimilation/fractional crystallization processes near the lunar surface. Instead, most of the chemical variation in the lunar basalts and volcanic glasses must result from assimilation at deeper levels or from having distinct source regions in a heterogeneous lunar mantle.

Finnila, A. B.↗

NASA NTRS1994

Mathematical Model for the Mineralization of Bone

A mathematical model is presented for the transport and precipitation of mineral in refilling osteons. One goal of this model was to explain calcification 'halos,' in which the bone near the haversian canal is more highly mineralized than the more peripheral lamellae, which have been mineralizing longer. It was assumed that the precipitation rate of mineral is proportional to the difference between the local concentration of calcium ions and an equilibrium concentration and that the transport of ions is by either diffusion or some other concentration gradient-dependent process. Transport of ions was assumed to be slowed by the accumulation of mineral in the matrix along the transport path. The model also mimics bone apposition, slowing of apposition during refilling, and mineralization lag time. It was found that simple diffusion cannot account for the transport of calcium ions into mineralizing bone, because the diffusion coefficient is two orders of magnitude too low. If a more rapid concentration gradient-driven means of transport exists, the model demonstrates that osteonal geometry and variable rate of refilling work together to produce calcification halos, as well as the primary and secondary calcification effect reported in the literature.

Martin, Bruce↗

NASA NTRS1994

Vertical Transport Rates in the Stratosphere in 1993 from Observations of CO2, N2O and CH4

Measurements of CO2, N2O and CH4 are analyzed to define hemispheric average vertical exchange rates in the lower stratosphere from November 1992 to October 1993. Effective vertical diffusion coefficients were small in summer, less than or equal to 1 m(exp 2)/sec at altitudes below 25 km; values were similar near the tropopause in winter, but increased markedly with altitude. The analysis suggests possibly longer residence times for exhaust from stratospheric aircraft, and more efficient transport from 20 km to the middle stratosphere, than predicted by many current models. Seasonally-resolved measurements of stratospheric CO2 and N2O provide significant new constraints on rates for global-scale vertical transport.

Wofsy, Steven C.↗

NASA NTRS1994

Mathematical Model for the Mineralization of Bone

A mathematical model is presented for the transport and precipitation of mineral in refilling osteons. One goal of this model was to explain calcification 'halos,' in which the bone near the haversian canal is more highly mineralized than the more peripheral lamellae, which have been mineralizing longer. It was assumed that the precipitation rate of mineral is proportional to the difference between the local concentration of calcium ions and an equilibrium concentration and that the transport of ions is by either diffusion or some other concentration gradient-dependent process. Transport of ions was assumed to be slowed by the accumulation of mineral in the matrix along the transport path. ne model also mimics bone apposition, slowing of apposition during refilling, and mineralization lag time. It was found that simple diffusion cannot account for the transport of calcium ions into mineralizing bone, because the diffusion coefficient is two orders of magnitude too low. If a more rapid concentration gradient-driven means of transport exists, the model demonstrates that osteonal geometry and variable rate of refilling work together to produce calcification halos, as well as the primary and secondary calcification effect reported in the literature.

Martin, Bruce↗

NASA NTRS1995

Photochemistry of Triton's Atmosphere and Ionosphere

The photochemistry of 32 neutral and 21 ion species in Triton's atmosphere is considered. Parent species N2, CH4, and CO (with a mixing ratio of 3 x 10(exp -4) in our basic model) sublime from the ice with rates of 40, 208, and 0.3 g/sq cm/b.y., respectively. Chemistry below 50 km is driven mostly by photolysis of methane by the solar and interstellar medium Lyman-alpha photons, producing hydrocarbons C2H4, C2H6, and C2H2 which form haze particles with precipitation rates of 135, 28, and 1.3 g/sq cm/b.y., respectively. Some processes are discussed which increase the production of HCN (by an order of magnitude to a value of 29 g/sq cm/b.y.) and involve indirect photolysis of N2 by neutrals. Reanalysis of the measured methane profiles gives an eddy diffusion coefficient K = 4 x 10(exp 3)sq cm/s above the tropopause and a more accurate methane number density near the surface, (3.1 +/- 0.8)x IO(exp 11)/cu cm. Chemistry above 200 km is driven by the solar EUV radiation (lambda less than 1000 A) and by precipitation of magnetospheric electrons with a total energy input of 10(exp 8) W (based on thermal balance calculations). The most abundant photochemical species are N, H2, H, 0, and C. They escape with the total rates of 7.7 x 10(exp 24)/ s, 4.5 x 10(exp 25)/s, 2.4 x 10(exp 25)/s, 4.4 x 10(exp 22)/s, and 1.1 x 10(exp 24), respectively. Atomic species are transported to a region of 50-200 km and drive the chemistry there. Ionospheric chemistry explains the formation of an E region at 150-240 km with HCO(+) as a major ion, and of an F region above 240 km with a peak at 320 km and C(+) as a major ion. The ionosphere above 500 km consists of almost equal densities of C(+) and N(+) ions. The model profiles agree with the measured atomic nitrogen and electron density profiles. A number of other models with varying rate coefficients of some reactions, differing properties of the haze particles (chemically passive or active), etc., were developed. These models show that there are four basic unknown values which have strong impacts on the composition and structure of the atmosphere and ionosphere. These values and their plausible ranges are the CO mixing ratio f(sub co) = 10(exp -4) - 10(exp -3), the magnetospheric electron energy input (1 +/- 0.5) x 10(exp 8) W, the rate coefficient of charge-exchange reaction N2(+) + C(kappa) = 10(exp -11) - 10(exp -10)cu cm/s, and the ion escape velocity upsilon(sub i) approx. equals 150 cm/s.

Krasnopolsky, Vladimir A.↗

NASA NTRS1995

Photochemistry of Triton's Atmosphere and Ionosphere

The photochemistry of 32 neutral and 21 ion species in Triton's atmosphere is considered. Parent species N2, CH4, and CO (with a mixing ratio of 3 x 10(exp -4) in our basic model) sublime from the ice with rates of 40, 208, and 0.3 g/sq cm/b.y., respectively. Chemistry below 50 km is driven mostly by photolysis of methane by the solar and interstellar medium Lyman-alpha photons, producing hydrocarbons C2H4, C2H6, and C2H2 which form haze particles with precipitation rates of 135, 28, and 1.3 g/sq cm/b.y., respectively. Some processes are discussed which increase the production of HCN (by an order of magnitude to a value of 29 g/sq cm/b.y.) and involve indirect photolysis of N2 by neutrals. Reanalysis of the measured methane profiles gives an eddy diffusion coefficient K = 4 x 10(exp 3) sq cm/s above the tropopause and a more accurate methane number density near the surface, (3.1 +/- 0.8) x 10(exp 11)/cc cm. Chemistry above 200 km is driven by the solar EUV radiation (lambda less than 1000 A)) and by precipitation of magnetospheric electrons with a total energy input of 10(exp 8) W (based on thermal balance calculations). The most abundant photochemical species are N, H2, H, O, and C. They escape with the total rates of 7.7 x 10(exp 24)/ s, 4.5 x 10(exp 25)/ s, 2.4 x 10(exp 25)/ s, 4.4 x 10(exp 22)/ s, and 1.1 x 10(exp 24)/ s, respectively. Atomic species are transported to a region of 50-200 km and drive the chemistry there. Iono- spheric chemistry explains the formation of an E region at 150-240 km with HCO(+) as a major ion, and of an F region above 240 km with a peak at 320 km and C(+) as a major ion. The ionosphere above 500 km consists of almost equal densities of C(+) and N(+) ions. The model profiles agree with the measured atomic nitrogen and electron density profiles. A number of other models with varying rate coefficients of some reactions, differing properties of the haze particles (chemically passive or active), etc., were developed. These models show that there are four basic unknown values which have strong impacts on the composition and structure of the atmosphere and ionosphere. These values and their plausible ranges are the CO mixing ratio f(sub co) = 10(exp -4)- 10(exp -3), the magnetospheric electron energy input (1 +/- 0.5) x 10(exp 8) W, the rate coefficient of charge-exchange reaction N2(+) + Ck = 10(exp -11)-10(exp -10)cc cm/s, and the ion escape velocity upselon(sub i) approx. 150 cm/s.

Krasnopolsky, Vladimir A.↗

NASA NTRS1998

Comments on the Diffusive Behavior of Two Upwind Schemes

The diffusive characteristics of two upwind schemes, multi-dimensional fluctuation splitting and locally one-dimensional finite volume, are compared for scalar advection-diffusion problems. Algorithms for the two schemes are developed for node-based data representation on median-dual meshes associated with unstructured triangulations in two spatial dimensions. Four model equations are considered: linear advection, non-linear advection, diffusion, and advection-diffusion. Modular coding is employed to isolate the effects of the two approaches for upwind flux evaluation, allowing for head-to-head accuracy and efficiency comparisons. Both the stability of compressive limiters and the amount of artificial diffusion generated by the schemes is found to be grid-orientation dependent, with the fluctuation splitting scheme producing less artificial diffusion than the finite volume scheme. Convergence rates are compared for the combined advection-diffusion problem, with a speedup of 2.5 seen for fluctuation splitting versus finite volume when solved on the same mesh. However, accurate solutions to problems with small diffusion coefficients can be achieved on coarser meshes using fluctuation splitting rather than finite volume, so that when comparing convergence rates to reach a given accuracy, fluctuation splitting shows a speedup of 29 over finite volume.

Wood, William A.↗

NASA NTRS1998

Transport Between the Tropical and Midlatitude Lower Stratosphere: Implications for Ozone Response to High Speed Civil Transport Emissions

Several recent studies have quantified the air exchange rate between the tropics and midlatitudes in the lower stratosphere using airborne and satellite measurements of chemical species. It is found that the midlatitude air is mixed into the tropical lower stratosphere with a replacement timescale of 13.5 months (with 20% uncertainty) for the region from the tropopause to 21 km and at least 18 months for the region of 20-28 km. These estimates are used to adjust the horizontal eddy diffusion coefficients, K(sub yy), in a two-dimensional chemistry transport model. The value of K(sub yy) previously used to simulate the subtropical barrier, 0.03 x 10(exp 6)sq m/s, generates an exchange time of about 4 years, and the model without subtropical barrier (K(sub yy) = 0.3 x 10(exp 6)sq m/s) has an exchange time of 5 months. Adjusting the K(sub yy) to 0.13 x 10(exp 6) sq m/s from the tropopause to 21 km and 0.07 x 10(exp 6)sq m/s above 21 km produces the exchange timescales which match the estimates deduced from the measurements. The subtropical barrier prevents the engine emissions of the high-speed civil transport (HSCT) aircraft from being transported into the tropics and subsequently lifted into the upper atmosphere or mixed into the southern hemisphere. The model results show that the calculated ozone response to HSCT aircraft emissions using the K(sub yy), adjusted to observed mixing rates is substantially smaller than that simulated without the subtropical barrier.

Shia, R.-L.↗

NASA NTRS1999

Oxygen Diffusion and Reaction Kinetics in Continuous Fiber Ceramic Matrix Composites

Previous stressed oxidation tests of C/SiC composites at elevated temperatures (350 C to 1500 C) and sustained stresses (69 MPa and 172 MPa) have led to the development of a finite difference cracked matrix model. The times to failure in the samples suggest oxidation occurred in two kinetic regimes defined by the rate controlling mechanisms (i.e. diffusion controlled and reaction controlled kinetics). Microstructural analysis revealed preferential oxidation along as-fabricated, matrix microcracks and also suggested two regimes of oxidation kinetics dependent on the oxidation temperature. Based on experimental results, observation, and theory, a finite difference model was developed. The model simulates the diffusion of oxygen into a matrix crack bridged by carbon fibers. The model facilitates the study of the relative importance of temperature, the reaction rate constant, and the diffusion coefficient on the overall oxidation kinetics.

Halbig, Michael C.↗

NASA NTRS1999

The Measurement of Sulfur Oxidation Products and Their Role in Homogeneous Nucleation

An improved version of a transverse ion source was developed which uses selected ion chemical ionization mass spectrometry techniques inside of a particle nucleation flow tube. These new techniques are very unique, in that the chemical ionization is done inside of the flow tube rather than by having to remove the compounds and clusters of interest which are lost on first contact,with any surfaces. The transverse source is also unique because it allows the ion reaction time to be varied over more than an order of magnitude, which in turn makes possible the separation of ion induced cluster growth from the charging of preexisting molecular clusters. As a result of combining these unique capabilities, the first ever measurements of prenucleation molecular clusters were performed. These clusters are the intermediate stage of growth in the gas-to-particle conversion process. This new technique provides a means of observing clusters containing 2, 3, 4, ... and up to about 8 sulfuric acid molecules, where the critical cluster size under these measurement conditions was about 4 or 5. Thus, the nucleation process can now be directly observed and even growth beyond the critical cluster size can be investigated. The details of this investigation are discussed in a recently submitted paper, which is included as Appendix A. Measurements of the diffusion coefficient of sulfuric acid and sulfuric acid clustered with a water molecule have also been performed. The measurements are also discussed in more detail in another recently submitted paper which is included as Appendix B. The empirical results discussed in both of these papers provide a critical test of present nucleation theories. They also provide new hope for resolving many of the huge discrepancies between field observation and model prediction of particle nucleation. The second part of the research conducted under this project was directed towards the development of new chemical ionization techniques for measuring sulfur oxidation products.

Eisele, F. L.↗

NASA NTRS1998

Transport Between the Tropical and Midlatitude Lower Statosphere: Implications for Ozone Response to High-Speed Civil Transport Emissions

Several recent studies have quantified the air exchange rate between the tropics and midlatitudes in the lower stratosphere using airborne and satellite measurements of chemical species. It is found that the midlatitude air is mixed into the tropical lower stratosphere with a replacement timescale of 13.5 months (with 20% uncertainty) for the region from the tropopause to 21 km and at least 18 months for the region of 20-28 km. These estimates are used to adjust the horizontal eddy diffusion coefficients, K(sub yy), in a two-dimensional chemistry transport model. The value of K(sub yy) previously used to simulate the subtropical barrier, 0.03 x 10(exp 6) sq m/s, generates an exchange time of about 4 years, and the model without subtropical barrier (K(sub yy) = 0.3 x 10(exp 6) sq m/s) has an exchange time of 5 months. Adjusting the K(sub yy) to 0.13 x 10(exp 6) sq m/s from the tropopause to 21 km and 0.07 x 10(exp 6) sq m/s above 21 km produces the exchange timescales which match the estimates deduced from the measurements. The subtropical barrier prevents the engine emissions of the high-speed civil transport (HSCT) aircraft from being transported into the tropics and subsequently lifted into the upper atmosphere or mixed into the southern hemisphere. The model results show that the calculated ozone response to HSCT aircraft emissions using the K(sob yy) adjusted to observed mixing rates is substantially smaller than that simulated without the subtropical barrier.

Shia, R.-L.↗

NASA NTRS1992

Variable Property Flow in Rectangular Ducts with Repeated Rectangular Rib Roughness

PHOENICS was used to simulate turbulent flow of supercritical hydrogen through asymmetrically heated rectangular ducts with one side rib-roughened, in order to provide design data for active cooling systems of hypersonic aircraft where the hydrogen fuel is used as coolant. Variable property and conjugate heat transfer effects were considered. The computations using PHOENICS required provision of property sub-routines for supercritical hydrogen, a low-Reynolds number form of the k-epsilon turbulence model, a rough-wall wall function and a procedure for calculating proper diffusion coefficients at the fluid-solid interface. The effect of each design variable such as width and height of duct and wall thickness, on pressure drop and maximum wall temperature was investigated. PHOENICS version 1.4 was used on an IBM 3090 computer.

Youn, B.↗

NASA NTRS2000

Vertical Distributions of PH3 in Saturn from Observations of Its 1-0 and 3-2 Rotational Lines

Far-infrared Fourier-transform spectrometer measurements of the 1-0 and 3-2 PH3 transitions in Saturn's disk near 267 and 800 GHz (8.9 and 26.7/cm), respectively, were analyzed simultaneously to derive a global mean profile for the PH3 vertical mixing ratio between 100 and 600 mbar total pressure. The far-infrared spectrum is relatively free from spectral interlopers, suffers minimal absorption or scattering by atmospheric particulates, and contains intrinsically weak PH3 lines that are sensitive to a range of atmospheric depths. The combined spectra are inconsistent with a uniform tropospheric mixing ratio, even with a stratospheric cut-off. They are consistent with a volume mixing ratio of PH3 that drops from 1.2 x 10(exp -5) at 645 mbar pressure to a value of 4.1 x 10(exp -7) at 150 mbar pressure, a decrease that is linear is log abundance vs log pressure. The mixing ratio could drop even more quickly at atmospheric pressures below 150 mbar and still be consistent with the data. The mixing ratio may well remain constant with depth for pressures above 630 mbar. The maximum PH3 mixing ratio in this model is consistent with a [P]/[H] ratio in the deep atmosphere that is about a factor of 10 higher than solar composition. Such a model is consistent with rapid mixing up to the radiative-convective boundary and transport by, for example, vertical waves just above this boundary. In the best fitting model, the eddy diffusion coefficient is approximately 10(exp 4) sq cm near 630 mbar, and it must increase with altitude. The predominant PH3 loss mechanisms are direct photolysis by UV radiation and scavenging by H atoms produced by the photolysis.

Orton, G. S.↗

NASA NTRS1999

Ocean Wave Studies with Applications to Ocean Modeling and Improvement of Satellite Altimeter Measurements

Combining analysis of satellite data (altimeter, scatterometer, high-resolution visible and infrared images, etc.) with mathematical modeling of non-linear wave processes, we investigate various ocean wave fields (on scales from capillary to planetary), their role in ocean dynamics and turbulent transport (of heat and biogeochemical quantities), and their effects on satellite altimeter measuring accuracy. In 1998 my attention was focused on long internal gravity waves (10 to 1000 km), known also as baroclinic inertia-gravity (BIG) waves. We found these waves to be a major factor of altimeter measurements "noise," resulting in a greater uncertainty [up to 10 cm in terms of sea surface height (SSH) amplitude] in the measured SSH signal than that caused by the sea state bias variations (up to 5 cm or so). This effect still remains largely overlooked by the satellite altimeter community. Our studies of BIG waves address not only their influence on altimeter measurements but also their role in global ocean dynamics and in transport and turbulent diffusion of biogeochemical quantities. In particular, in collaboration with Prof Peter Weichman, Caltech, we developed a theory of turbulent diffusion caused by wave motions of most general nature. Applied to the problem of horizontal turbulent diffusion in the ocean, the theory yielded the effective diffusion coefficient as a function of BIG wave parameters obtainable from satellite altimeter data. This effort, begun in 1997, has been successfully completed in 1998. We also developed a theory that relates spatial fluctuations of scalar fields (such as sea surface temperature, chlorophyll concentration, drifting ice concentration, etc.) to statistical characteristics of BIG waves obtainable from altimeter measurements. A manuscript is in the final stages of preparation. In order to verify the theoretical predictions and apply them to observations, we are now analyzing Sea-viewing Wide Field of view Sensor (SeaWiFS) and Field of view Sensor (SeaWiFS) and Advanced Very High-Resolution Radiometer (AVHRR) data on sea surface temperature (SST) and chlorophyll concentration jointly with TOPEX/POSEIDON data on SSH variations.

Glazman, Roman E.↗

NASA NTRS2000

COSIM: A Finite-Difference Computer Model to Predict Ternary Concentration Profiles Associated with Oxidation and Interdiffusion of Overlay-Coated Substrates

A finite-difference computer program (COSIM) has been written which models the one-dimensional, diffusional transport associated with high-temperature oxidation and interdiffusion of overlay-coated substrates. The program predicts concentration profiles for up to three elements in the coating and substrate after various oxidation exposures. Surface recession due to solute loss is also predicted. Ternary cross terms and concentration-dependent diffusion coefficients are taken into account. The program also incorporates a previously-developed oxide growth and spalling model to simulate either isothermal or cyclic oxidation exposures. In addition to predicting concentration profiles after various oxidation exposures, the program can also be used to predict coating fife based on a concentration dependent failure criterion (e.g., surface solute content drops to two percent). The computer code, written in an extension of FORTRAN 77, employs numerous subroutines to make the program flexible and easily modifiable to other coating oxidation problems.

Nesbitt, James A.↗

NASA NTRS2001

Inter-cusp Ion and Electron Transport in a Nstar-derivative Ion Thruster

Diffusion of electrons and ions to anode surfaces between the magnetic cusps of a NASA Solar Electric Propulsion Technology Application Readiness ion thruster has been characterized. Ion flux measurements were made at the anode and at the screen grid electrode. The measurements indicated that the average ion current density at the anode and at the screen grid were approximately equal. Additionally, it was found that the electron flux to the anode between cusps is best described by the classical cross-field diffusion coefficient.

Foster, John E.↗

NASA NTRS2001

Ion and Electron Transport in an Nstar-derivative Ion Thruster

Foster, John E.↗