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NASA NTRS1994

On the determination of the cosmic infrared background radiation from the high-energy spectrum of extragalactic gamma-ray sources

In a recent paper Stecker, De Jager, & Salamon have suggested using the observed approximately MeV to TeV spectra of extragalactic gamma-ray sources as probes of the local density of the cosmic infrared background radiation (CIBR) and have subsequently claimed a first possible measurement of the CIBR from the analysis of the gamma-ray spectrum of Mrk 421 (De Jager, Stecker, & Salamon). The CIBR from normal galaxies consists of two components: a stellar emission component (CIBRs), and a thermal dust emission component (CIBRd). Photons with energies in the approximately 0.1-2 TeV range interact primarily with the CIBRs, whereas interactions with CIBRd dominate the absorption of photons in the approximately 2-100 TeV energy range. SDS 92 and DSS94 considered only the interaction of the gamma-rays with the dust emission component of the CIBR. We present here an improved analysis of the absorption of extragalactic TeV gamma rays by the CIBR, taking the dual nature of its origin into account. Applying the analysis to the observed gamma-ray spectrum of Mrk 421, a BL Lac object at z = 0.031, we find agreement with DSS94 tentative evidence for absorption by the CINRs. Our analysis therefore limits the detection of the CIBR to the approximately 15-40 micron wavelength regime which, considering the uncertainties in the highest energy (greater than 4 TeV) data and ion the possibility of absorption inside the source, many turn out to be an upper limit on its energy density. At shorter wavelengths (lambda approximately = 1-15 microns), where the gamma-ray interactions are dominated by the CIBRs, our analysis definitely yields only an upper limit on the energy density of the CIBR. In contrast, DSS94 have claimed a possible first measurement of the CIBR over the entire 1-120 micron wavelength region. The upper limit on the CIBRs and tentative detection of the CIBRd are consistent with normal galaxies contributing most of the energy to the CIBR, and constrain the contribution of some exotic sources. With careful modeling of infrared foreground emissions, these constraints on the CIBR are above the values measurable by the DIRBE experiment on board the Cosmic Background Explorer (COBE) satellite.

Dwek, Eli↗
NASA NTRS1994

Satellite measurements through the center of a substorm surge

Measurements have been made of electric and magnetic fields, plasma drifts, and electron precipatation within a surge at the westward, leading edge of the auroral 'bulge' at the peak of the substorm expansion phase. The trajectroy of the Dynamics Explorer 2 (DE 2) satellite over the auroral emissions is determined from nearly simultaneous observations with the imager on the DE 1 satellite at a higher altitude. The electric field and plasma drift measurements have enabled us to deduce the basic configuration of the ionospheric electric potential, or plasma convection, around the surge. The electric potential shows that the bulge is associated with a protrusion of the dawn convection cell into the dusk cell, poleward of the 'Harang discontinity.' This protrusion conains a westward electric field that strongly enhances the westard electrojet current by the creation of a "Cowling channel.' This westward electric field, and the associated Cowling current, appear to terminate within the surge, which contains an intense, upward field-aligned current. The magneitc field measurements show that the region containing this field-aligned current is shaped more like a cylinger rather than a long sheet. The total is found to exceed one-half million amperes.

Weimer, D. R.↗
NASA NTRS1993

Performance assessment of the TDRSS Onboard Navigation System (TONS) experiment on EP/EUVE

The National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC) is currently developing an operational Tracking and Data Relay Satellite (TDRS) System (TDRSS) Onboard Navigation System (TONS) to provide onboard knowledge of high-accuracy navigation products autonomously to users of TDRSS and its successor, TDRS-2. A TONS experiment has been implemented on the Explorer Platform/Extreme Ultraviolet Explorer (EP/EUVE) spacecraft, launched June 7, 1992, to flight qualify the TONS operational system using TDRSS forward-link communications services. This paper assesses the performance of the TONS flight hardware, an ultrastable oscillator (USO) and Doppler extractor (DE) card in one of the TDRSS user transponders, and the protoype flight software, based on the TONS experiment results. An overview of onboard navigation via TDRSS is also presented for both the EP/EUVE experiment and for future users of TONS. USO and DE short-term and long-term stability performance has been excellent. TONS Flight Software analysis indicates that position accuracies of better than 25 meters root-mean-square are achievable with tracking every one to two orbits, for the EP/EUVE 525-kilometer altitudes, 28.5-degree inclination orbit. The success of the TONS experiment demonstrates the flight readiness of TONS, which is scheduled to provide autonomous navigation for the Earth Observing System (EOS)-AM mission.

Gramling, C. J.↗
NASA NTRS1994

Elevated CO2: Impact on diurnal patterns of photosynthesis in natural microbial ecosystems

Algae, including blue-green algae (cyanobacteria), are the major source of fixed carbon in many aquatic ecosystems. Previous work has shown that photosynthetic carbon fixation is often enhanced in the presence of additional carbon dioxide (CO2). This study was undertaken to determine if this CO2 fertilization effect extended to microbial mats, and, if so, at what times during the day might the addition of CO2 affect carbon fixation. Four microbial mats from diverse environments were selected, including mats from a hypersaline pond (area 5, Exportadora de Sal, Mexico), the marine intertidal (Lyngbya, Laguna Ojo de Liebre, Mexico), an acidic hotspring (Cyanidium, Nymph Creek, Yellowstone National Park), and an acidic stream at ambient temperature (Zygogonium, Yellowstone National Park). Carbon fixation in the absence of additional CO2 essentially followed the rising and falling sunlight levels, except that during the middle of the day there was a short dip in carbon fixation rates. The addition of CO2 profoundly enhanced carbon fixation rates during the daylight hours, including during the midday dip. Therefore, it is unlikely that the midday dip was due to photoinhibition. Surprisingly, enhancement of carbon fixation was often greatest in the early morning or late afternoon, times when carbon fixation would be most likely to be light limited.

Rothschild, L. J.↗
NASA NTRS1995

Investigation of plasma instabilities in the polar cusp

This technical paper describes the occurrence of magnetosonic waves in the dayside and nightside polar magnetosphere and the roles that these waves can play in the heating of ions in the ionosphere. Plasma waves near the auroral kilometric radiation (AKR) source regions is studied, using wave data obtained from both the Galileo and DE-1. During the Galileo encounter with the Earth in 1992, very intense auroral kilometric radiation was observed. The high spectral resolution obtained by the wide-band receiver of the plasma wave instrument on board the spacecraft often show distinctive, dispersive, arc-like features each extending over a period of several seconds. Similar features were also observed by De-1. These features may be due to an intrinsic velocity and/or to a source simulation by different wave modes. A reply to the comment by Orlowski and Russell on the earlier paper 'Electron Beam Excitation of Upstream Waves in the Whistler Frequency Mode Frequency Range' is also included.

Wong, H. K.↗
NASA NTRS1996

A Study of Large Droplet Ice Accretions in the NASA-Lewis IRT at Near-Freezing Conditions

This report documents the results of an experimental study on large droplet ice accretions which was conducted in the NASA-Lewis Icing Research Tunnel (IRT) with a full-scale 77.25 inch chord Twin-Otter wing section. This study was intended to: (1) document the existing capability of the IRT to produce a large droplet icing cloud, and (2) study the effect of various parameters on large droplet ice accretions. Results are presented from a study of the IRT's capability to produce large droplets with MVD of 99 and 160 microns. The effect of the initial water droplet temperature on the resultant ice accretion was studied for different initial spray bar air and water temperatures. The initial spray bar water temperature was found to have no discernible effect upon the large droplet ice accretions. Also, analytical and experimental results suggest that the water droplet temperature is very nearly the same as the tunnel ambient temperature, thus providing a realistic simulation of the large droplet natural icing condition. The effect of temperature, droplet size, airspeed, angle-of attack, flap setting and de-icer boot cycling time on ice accretion was studied, and will be discussed in this report. It was found that, in almost all of the cases studied, an ice ridge formed immediately aft of the active portion of the de-icer boot. This ridge was irregular in shape, varied in location, and was in some cases discontinuous due to aerodynamic shedding.

Miller, Dean R.↗
NASA NTRS1994

Recombination of H(3+) and D(3+) ions with electrons

Flowing-afterglow measurements in decaying H3(+) or D3(+) plasmas suggest that de-ionization does not occur by simple binary recombination of a single ion species. We find that vibrational excitation of the ions fails to provide an explanation for the effect, contrary to an earlier suggestion. Instead, we suggest that collisional stabilization of H3** Rydberg molecules by ambient electrons introduces an additional dependence on electron density. The proposed mechanism would permit plasma de-ionization to occur without the need for dissociative recombination by the mechanism of potential-surface crossings.

Johnsen, R.↗
NASA NTRS1995

U-groove aluminum weld strength improvement

Though butt-welds are among the most preferred joining methods in aerostructures, their strength dependence on inelastic mechanics is generally the least understood. This study investigated experimental strain distributions across a thick aluminum U-grooved weld and identified two weld process considerations for improving the multipass weld strength. The extreme thermal expansion and contraction gradient of the fusion heat input across the groove tab thickness produces severe peaking which induces bending under uniaxial loading. The filler strain-hardening deceased with increasing filler pass sequence, producing the weakest welds on the last pass side. Current welding schedules unknowingly compound these effects which reduce the weld strength. A de-peaking index model was developed to select filler pass thicknesses, pass numbers, and sequences to improve de-peaking in the welding process. Intent is to combine the strongest weld pass side with the peaking induced bending tension to provide a more uniform stress and stronger weld under axial tensile loading.

Verderaime, V.↗
NASA NTRS1996

Effects of plasmaspheric ion heating due to ionospheric and magnetospheric sources

In an initial study, the He(+) observations from the Retarding Ion Mass Spectrometer on Dynamics Explorer 1 (RIMS/DE 1) was examined for more than 120 transits of the plasmasphere in the fall of 1981. The He(+) to H(+) ratio was determined as it varied spatially over portions of the DE 1 orbit, and its variation with solar and magnetic activities and with local time, focusing specifically on the inner plasmasphere. These variations were compared along the L = 2 field line with calculations made by the Field Line Interhemispheric Plasma (FLIP) code. In a recently submitted paper, the He(+) to H(+) density ratio was examined for all the available data from 1981 to 1984 from the RIMS on DE 1. There are two basic characteristics of the ratio: one is that the ratio decreases with radial distance in the plasmasphere, and the other is the strong dependence of the density ratio on solar activity. In addition to the He(+)/H(+) ratio research, a phenomenon has been studied in the topside ionosphere which relates to the thermal coupling of the ionosphere to the plasmasphere. There is little or no correlation with magnetic and solar activity here. Another study has been directed toward the relation of plasma properties to the density gradients forming the plasmapause. The study has followed a two-pronged approach. First, the observations have been analyzed to determine what happens to the plasma properties across these boundary layers (density gradients). Second, comparisons were made with FLIP model calculations to determine how well the model is able to treat these conditions. Among the significant lessons learned in these studies are two that bear directly on the direction of future investigations in this area. First, composition cannot be viewed independently of thermal structure. Second, solar and magnetic activity effects are real; but the causal relationship between activity and effects is frequently quite complicated because several different processes appear to be operating in different ways and on different time scales. Under these circumstances, large correlation coefficients should not be expected and are not generally found.

Comfort, Richard H.↗
NASA NTRS1996

Validation of NASA Thermal Ice Protection Computer Codes: Program Overview - Part 1

The Icing Technology Branch at NASA Lewis has been involved in an effort to validate two thermal ice protection codes developed at the NASA Lewis Research Center. LEWICE/Thermal (electrothermal deicing & anti-icing), and ANTICE (hot-gas & electrothermal anti-icing). The Thermal Code Validation effort was designated as a priority during a 1994 'peer review' of the NASA Lewis Icing program, and was implemented as a cooperative effort with industry. During April 1996, the first of a series of experimental validation tests was conducted in the NASA Lewis Icing Research Tunnel(IRT). The purpose of the April 96 test was to validate the electrothermal predictive capabilities of both LEWICE/Thermal, and ANTICE. A heavily instrumented test article was designed and fabricated for this test, with the capability of simulating electrothermal de-icing and anti-icing modes of operation. Thermal measurements were then obtained over a range of test conditions, for comparison with analytical predictions. This paper will present an overview of the test, including a detailed description of: (1) the validation process; (2) test article design; (3) test matrix development; and (4) test procedures. Selected experimental results will be presented for de-icing and anti-icing modes of operation. Finally, the status of the validation effort at this point will be summarized. Detailed comparisons between analytical predictions and experimental results are contained in the following two papers: 'Validation of NASA Thermal Ice Protection Computer Codes: Part 2- The Validation of LEWICE/Thermal' and 'Validation of NASA Thermal Ice Protection Computer Codes: Part 3-The Validation of ANTICE'

Miller, Dean↗
NASA NTRS1997

Modeling of the Coupled Magnetospheric and Neutral Wind Dynamos

Over the past four years of funding, SRI, in collaboration with the University of Texas at Dallas, has been involved in assessing the influence of thermospheric neutral winds on the electric field and current systems at high latitudes. The initial direction of the project was to perform a set of numerical experiments concerning the contribution of the magnetospheric and neutral wind dynamo processes, under specific boundary conditions, to the polarization electric field and/or the field-aligned current distribution at high latitudes. To facilitate these numerical experiments we developed a numerical scheme that relied on using output from the NCAR Thermosphere-Ionosphere General Circulation Model (NCAR-TIGCM), expanding them in the form of spherical harmonics and solving the dynamo equations spectrally. Once initial calculations were completed, it was recognized that the neutral wind contribution could be significant but its actual contribution to the electric field or currents depended strongly on the generator properties of the magnetosphere. Solutions to this problem are not unique because of the unknown characteristics of the magnetospheric generator, therefore the focus was on two limiting cases. One limiting case was to consider the magnetosphere as a voltage generator delivering a fixed voltage to the high-latitude ionosphere and allowing for the neutral wind dynamo to contribute only to the current system. The second limiting case was to consider the magnetosphere as a current generator and allowing for the neutral wind dynamo to contribute only to the generation of polarization electric fields. This work was completed and presented at the l994 Fall AGU meeting. The direction of the project then shifted to applying the Poynting flux concept to the high-latitude ionosphere. This concept was more attractive as it evaluated the influence of neutral winds on the high-latitude electrodynamics without actually having to determine the generator characteristics of the magnetosphere. The influence of the neutral wind was then determined not by estimating how much electric potential or current density it provides, but by determining the contribution of the neutral wind to the net electromagnetic energy transferred between the ionosphere and magnetosphere. The estimate of the net electromagnetic energy transfer and the role of the neutral winds proves to be a more fundamental quantity in studies of magnetosphere- ionosphere coupling also showed that by using electric and magnetic field measurements from the HILAT satellite, the Poynting flux could be a measurable quantity from polar-orbiting, low- altitude spacecraft. Through collaboration with Dr. Heelis and others at UTD and their expertise of the electric field measurements on the DE-B satellite, an extensive analysis was planned to determine the Poynting flux from the DE-B measurements in combination with a modeling effort to help interpret the observations taking into account the coupled magnetosphere-ionosphere.

Thayer, Jeffrey P.↗
NASA NTRS1995

Adaptive Identification and Characterization of Polar Ionization Patches

Dynamics Explorer 2 (DE 2) spacecraft data are used to detect and characterize polar cap 'ionization patches' loosely defined as large-scale (greater than 100 km) regions where the F region plasma density is significantly enhanced (approx greater than 100%) above the background level. These patches are generally believed to develop in or equatorward of the dayside cusp region and then drift in an antisunward direction over the polar cap. We have developed a flexible algorithm for the identification and characterization of these structures, as a function of scale-size and density enhancement, using data from the retarding potential analyzer, the ion drift meter, and the langmuir probe on board the DE 2 satellite. This algorithm was used to study the structure and evolution of ionization patches as they cross the polar cap. The results indicate that in the altitude region from 240 to 950 km ion density enhancements greater than a factor of 3 above the background level are relatively rare. Further, the ionization patches show a preferred horizontal scale size of 300-400 km. There exists a clear seasonal and universal time dependence to the occurrence frequency of patches with a northern hemisphere maximum centered on the winter solstice and the 1200-2000 UT interval.

Coley, W. R.↗
NASA NTRS1996

Results of Detailed Modeling of the Narrow-Line Region of Seyfert Galaxies

We present model line profiles of [O II] lambda3727, [Ne III] lambda3869, [O I] lambda5007, [Fe VII] lambda6087, [Fe X] lambda6374, [O I] lambda6300, H(alpha) lambda6563, and [S 2] lambda6731. The profiles presented here illustrate explicitly the pronounced effects that collisional de-excitation, and that spatial variations in both the ionization parameter and cloud column density, have on Narrow-Line Region (NLR) model profiles. The above effects were included only qualitatively in a previous analytical treatment by Moore and Cohen. By making a direct correspondence between these model profiles and the analytical model profiles of Moore and Cohen, and by comparing with the observed profiles presented in a companion paper and also with those presented elsewhere in the literature, we strengthen some of the conclusions of Moore and Cohen. Most notably, we argue for constant ionization parameter, uniformly accelerated outflow of clouds that are individually stratified in ionization, and the interpretation of emission-line width correlations with ionization potential as a column density effect. For comparison with previous observational studies, such as our own in a companion paper, we also calculate profile parameters for some of the models, and we present and discuss the resulting line width correlations with critical density (n(sub cr)) and Ionization Potential (IP). Because the models we favor are those that produce extended profile wings as observed in high spectral resolution studies, the line width correlations of our favoured models are of particular interest. Line width correlations with n(sub cr) and/or IP result only if the width parameter is more sensitive to extended profile wings than is the Full Width at Half-Maximum (FWHM). Correlations between FWHM and n(sub cr) and/or IP result only after convolving the model profiles with a broad instrumental profile that simulates the lower spectral resolution used in early observational studies. The model in agreement with the greatest number of observational considerations has electron density decreasing outward from n(sub e) approx. equals 10(exp 6)/cu cm to n(sub e) approx. equals 10(exp 2)/cu cm and, due to collisional de-excitation effects in the lowest velocity clouds, it generates broad flat-topped profile peaks in the lines of lowest critical density (e.g., [O II] lambda3727 and [S II] lambda(lambda)6716, 6731). Because the observed profile peaks of both low and high critical density lines are often very similar, our favored model requires a contribution to NLR emission-line spectra from low-velocity, low-density, and low-ionization gas not included in the model NLR.

Moore, David↗
NASA NTRS1995

Large Enhancements in the O/N2 Ratio in the Evening Sector of the Winter Hemisphere During Geomagnetic Storms

In this paper, we have looked for enhancements of the O/N2 ratio in data measured by the Dynamics Explorer 2 (DE 2) satellite in the middle latitudes of the winter hemisphere, based on a prediction that was made by the National Center for Atmospheric Research thermosphere/tonosphere general circulation model (NCAR-TIGCM) that such increases occur. The NCAR-TIGCM predicts that these enhancements should be seen throughout the low latitude region and in many middle latitude locations, but that the enhancements in O/N2 are particularly strong in the middle-latitude, evening-to-midnight sector of the winter hemisphere. When this prediction was used to look for these effects in DE 2 NACS (neutral atmosphere composition spectrometer) data, large enhancements in the O/N2 ratio (approx. 50 to 90%) were seen. These enhancements were observed during the main phase of a storm that occurred on November 24, 1982, and were seen in the same region of the winter hemisphere predicted by the NCAR-TIGCM. They are partially the result of the depletion of N2 and, as electron loss is dependent on dissociative recombination at F(sub 2) altitudes, they have implications for electron densities in this area. Parcel trajectories, which have been followed through the NCAR-TIGCM history file for this event, show that large O/N2 enhancements occur in this limited region in the winter hemisphere for two reasons. First, these parcels of air are decelerated by the antisunward edge of the ion convection pattern; individual parcels converge and subsidence occurs. Thus molecular-nitrogen-poor air is brought from higher to lower heights. Because neutral parcels that are found a little poleward of the equatorial edge of the eveningside convection pattern are swept inward toward the center of the auroral oval, the enhancements occur only in a very limited range of latitudes. Second, nitrogen-poor air is transported from regions close to the magnetic pole in the winter hemisphere. During geomagnetic storms, enhanced meridional winds are driven by the increased pressure-gradient force that is associated with intensified Joule heating in the auroral oval. These pressure-driven winds decrease rapidly on the dayside beyond the auroral oval where the parcels originate, limiting the region into which the parcels can be transported. Thus these two processes drive values of O/N2 in a limited region of the winter hemisphere, and reinforce only in the evening sector, causing large changes in this region.

Burns, A. G.↗
NASA NTRS1997

Electric Propulsion for Low Earth Orbit Communication Satellites

Electric propulsion was evaluated for orbit insertion, satellite positioning and de-orbit applications on big (hundreds of kilograms) and little (tens of kilograms) low earth orbit communication satellite constellations. A simple, constant circumferential thrusting method was used. This technique eliminates the complex guidance and control required when shading of the solar arrays must be considered. Power for propulsion was assumed to come from the existing payload power. Since the low masses of these satellites enable multiple spacecraft per launch, the ability to add spacecraft to a given launch was used as a figure of merit. When compared to chemical propulsion ammonia resistojets, ion, Hall, and pulsed plasma thrusters allowed an additional spacecraft per launch Typical orbit insertion and de-orbit times were found to range from a few days to a few months.

Oleson, Steven R.↗
NASA NTRS1997

Radiation Near 2f(sub p) and Intensified Emissions Near f(sub p) in the Dayside and Nightside Auroral Region and Polar Cap

This paper reports the discovery in the DE 1 data of propagating radiation near 2f(sub p) (the H component) and relatively intense electromagnetic waves near f(sub p) with fields typically less than or approximately equal to 1 mV/m (the PF or plasma frequency component) on both the dayside and the nightside of Earth. These emissions are observed at auroral and polar cap latitudes for radial distances ranging from 2.5 to 4.5 R(sub E). The H component is unique in that no other 2f(sub p) emissions are known to be generated where the electron gyrofrequency f(sub g) exceeds 2f(sub p). Since existing theories for 2f(sub p) radiation assume f(sub g)/f(sub p) less than 1, new theories will be required to explain the H component. The PF waves near f(sub p) are electromagnetic, but with large ratios E/cB approximately 20. On the basis of cold plasma theory, the wave frequencies and the ratios E/cB, the PF component plausibly consists of zeta-mode and/or whistler mode waves near f(sub p), presumably driven by an electron instability. The H emissions have modest bandwidths of approximately 50% at frequencies ranging from 5 to 20 kHz. Grounds for interpreting the H component as emissions generated near 2f(sub p) are provided by the very good frequency tracking of the PF and H components and typical frequency ratios near 2.0. Strong evidence exists that part of the H component is propagating, electromagnetic radiation, based on propagation effects and spin modulation patterns. However, no magnetic signals have yet been detected for the H component, so that it could be partly electrostatic. Cold plasma theory and the observed wave characteristics favor interpreting the H component as composed of omicron mode and/or zeta mode signals. Combining the DE 1 observations with rocket observations, it is predicted that the much more intense Langmuir-like fields observed in the low altitude auroral zones should also generate observable 2f(sub p) radiation. This prediction should be testable using Polar and future rocket flights. Directions for future research are also described.

Cairns, Iver H.↗
NASA NTRS1997

Modeling a simple coronal streamer during whole sun month

The solar minimum streamer structure observed during the whole sun month was modeled. The Van de Hulst inversion was used in order to determine the coronal electron density profiles and scale-height temperature profiles. The axisymmetric magnetostatic model of Gibson, Bagenal and Low was also used. The density, temperature, and magnetic field distribution were quantified using both coronal white light data and photospheric magnetic field data from the Wilcox Solar Observatory. The densities and temperatures obtained by the Van de Hulst and magnetostatic models are compared to the magnetic field predicted by the magnetostatic model to a potential field extrapolated from the photosphere.

Gibson, S. E.↗
NASA NTRS1999

Evolution of Plasmaspheric Refilling: A Comparison of Measurements with an Interhemispheric Plasmasphere Model

The evolution of plasmasphere is closely coupled to processes governing flux tube refilling and transport. In this study we follow this evolution through a series of sequential ion observations made by RIMS on the DE 1 satellite and with the intervening time intervals filled by simulations made with the FLIP model. The FLIP model solves the continuity and momentum equations for the major ion species as well as the energy equations for ions and electrons along entire flux tubes rom 100 k altitude in both hemispheres. Convection has recently been included, driven a dawn-dusk electric field model. Since the observations are necessarily restricted to two local time passes through the plasmasphere per orbit, the FLIP code will follow the observed flux tubes as they corotate and convect through the other local times. The period of DE 1 is such that nearly the same flux tubes are observed every second day or every sixth orbit. The composite of observations and simulations allows assessment of the degree to which the degree to which the FLIP simulations accurately describe the processes which govern flux tube refilling over the two-day period between observations of the same flux tubes.

Comfort, Richard H.↗