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At least 613 records · Page 34

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NASA NTRS2015

Understanding and Mitigating Tip Leakage and Endwall Losses in High Pressure Ratio Cores

Reducing endwall and tip secondary flow losses will be a key enabler for the next generation of commercial and military air transport and will be an improvement on the state-of-the-art in turbine loss reduction strategies. The objective of this research is three-fold: 1) To improve understanding of endwall secondary flow and tip clearance losses 2) To develop novel technologies to mitigate these losses and test them in low-speed cascade and rig environments 3) To validate predictive tools To accomplish these objectives, Pratt & Whitney (P&W) has teamed with Pennsylvania State University (PSU) to experimentally test new features designed by P&W. P&W will create new rim-cavity features to reduce secondary flow loss and improve purge flow cooling effectiveness and new blade tip features to manage leakage flows and reduce tip leakage secondary flow loss. P&W is currently developing technologies in these two areas that expect to be assimilated in the N+2/N+3 generation of commercial engines.

Turbomachinery↗

NASA NTRS2016

Potential Impact of a US Climate Policy and Air Quality Regulations on Future Air Quality and Climate Change

We have investigated how future air quality and climate change are influenced by the US air quality regulations that existed or were proposed in 2013 and a hypothetical climate mitigation policy that aims to reduce 2050 CO2 emissions to be 50% below 2005 emissions. Using the NASA GISS ModelE2 general circulation model, we look at the impacts for year 2030 and 2055. The US energy-sector emissions are from the GLIMPSE project (GEOS-Chem LIDORT Integrated with MARKAL (MARKet ALlocation) for the Purpose of Scenario Exploration), and other US emissions data sets and the rest of the world emissions data sets are based on the RCP4.5 scenario. The US air quality regulations are projected to have a strong beneficial impact on US air quality and public health in year 2030 and 2055 but result in positive radiative forcing. Under this scenario, no more emission constraints are added after 2020, and the impacts on air quality and climate change are similar between year 2030 and 2055. Surface particulate matter with a diameter smaller than 2.5 micron PM(sub 2:5) is reduced by 2 approximately μg/m(sup -3) on average over the USA, and surface ozone by approximately 8 ppbv. The improved air quality prevents about 91 400 premature deaths in the USA, mainly due to the PM(sub 2:5) reduction approximately (74 200 lives saved). The air quality regulations reduce the light-reflecting aerosols (i.e., sulfate and organic matter) more than the light-absorbing species (i.e., black carbon and ozone), leading to a strong positive radiative forcing (RF) over the USA by both aerosols' direct and indirect forcing: the total RF is approximately 0.04 W m(sup -2) over the globe, and approximately 0.8 W m(sup -2) over the USA. Under the hypothetical climate policy, a future CO2 emissions cut is achieved in part by relying less on coal, and thus SO2 emissions are noticeably reduced. This provides air quality co-benefits, but it could lead to potential climate disbenefits over the USA. In 2055, the US mean total RF is +C02 W m(sup -2) due to positive aerosol direct and indirect forcing, while the global mean total RF is -0.06 W m(sup -2) due to the dominant negative CO2 RF (instantaneous RF). To achieve a regional-scale climate benefit via a climate policy, it is critical (1) to have multinational efforts to reduce greenhouse gas (GHG) emissions and (2) to simultaneously target emission reduction of light-absorbing species (e.g., BC and O3) on top of long-lived species. The latter is very desirable as the resulting climate benefit occurs faster and provides cobenefits to air quality and public health.

Air quality↗

NASA NTRS2017

The Role of DYNAMO in Situ Observations in Improving NASA Ceres-like Daily Surface and Atmospheric Radiative Flux Estimates

The daily surface and atmospheric radiative fluxes from NASA Clouds and the Earths RadiantEnergy System (CERES) Synoptic 1 degree (SYN1deg) Ed3A are among the most widely used data to studycloud-radiative feedback. The CERES SYN1deg data are based on Fu-Liou radiative transfer computations thatuse specific humidity (Q) and air temperature (T) from NASA Global Modeling and Assimilation Office (GMAO)reanalyses as inputs and are therefore subject to the quality of those fields. This study uses in situ Q and Tobservations collected during the Dynamics of the Madden-Julian Oscillation (DYNAMO) field campaign toaugment the input stream used in the NASA GMAO reanalysis and assess the impact on the CERES dailysurface and atmospheric longwave estimates. The results show that the assimilation of DYNAMOobservations considerably improves the vertical profiles of analyzed Q and T over and near DYNAMO stationsby moistening and warming the lower troposphere and upper troposphere and drying and cooling themid-upper troposphere. As a result of these changes in Q and T, the computed CERES daily surface downwardlongwave flux increases by about 5 W m(exp -2), due mainly to the warming and moistening in the lowertroposphere; the computed daily top-of-atmosphere (TOA) outgoing longwave radiation increases by2-3 W m(exp -2) during dry periods only. Correspondingly, the estimated local atmospheric longwave radiativecooling enhances by about 5 W m(exp -2) (7-8 W m(exp -2)) during wet (dry) periods. These changes reduce the bias inthe CERES SYN1deg-like daily longwave estimates at both the TOA and surface and represent animprovement over the DYNAMO region.

DYNAMO↗

NASA NTRS2019

Hyper-cooling Limit, Heat of Fusion, and Temperature-Dependent Specific Heat of Fe-Cr-Ni Melts

Thermophysical properties of Fe-Cr-Ni melts are studied using electrostatic levitation and rapid solidification techniques. Six hypoeutectic Fe(sub 0.72)Cr(sub w)Ni(sub(0.28−w)) alloys with a Cr/Ni ratio of around 0.8 were melted and solidified at different degrees of undercooling. From the observed relationship between the undercooling and thermal plateau time, the hyper-cooling limit and heat of fusion of Fe(sub 0.72)Cr(sub w)Ni(sub(0.28−w)) melts are determined as a function of Cr mass fraction. A ratio of specific heat and total hemispherical emissivity of the Fe-Cr-Ni melts is calculated using the time-temperature profiles. A new method is presented to evaluate the temperature dependence of specific heat for undercooled melts and applied to this alloy family.

Sangho Jeon↗

NASA NTRS2019

Investigation of Stacking Fault and Antiphase Boundary Segregation in Ni-Based Superalloys Using Density Functional Theory

Ni-base superalloys have a long history of use in jet turbine engines, and efforts to improve their performance in that application are ongoing. It is known that the precipitation of the Ni3Al γ' phase within the disordered FCCγ phase strengthens the overall material. However, in the high-temperature environment found inside a turbine engine during operation, creep can cause the γ' phase to transform to different, weaker phases along stacking faults, leading to a deterioration of performance. In the γ' phase one mode of creep deformation is the formation of stacking fault ribbons, which consist of intrinsic stacking faults further shearing into antiphase boundaries (APBs). It is also known that certain alloying additions exhibit segregation to stacking faults. If segregating elements could be identified which segregate to the intrinsic stacking fault, but not to the APB, the inclusion of such elements could lead to improved creep strength in these alloys. To investigate this possibility, a density functional investigation of the segregation of W, Mo and Cr to both a superlattice intrinsic stacking fault (SISF) and an APB was performed. It was found that W, Mo and Cr all exhibit segregation to the SISF. In contrast, for the APB, Cr was either energy-neutral, or segregates, depending on the presence of additional nearby Cr, or on the specific lattice site upon which it was placed, while Mo and W did not segregate. Because W and Mo segregate to the SISF but not to the APB, the inclusion of these elements could provide a degree of protection against creep-related deterioration.

Good, Brian S.↗

NASA NTRS2018

Simulation of Sound Absorption by Scattering Bodies Treated with Acoustic Liners Using a Time-Domain Boundary Element Method

Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for aircraft noise reduction, and are characterized by a complex valued frequency-dependent impedance, Z(w). Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering of geometric bodies treated with acoustic liners. This work uses an admittance boundary condition where the admittance, Y(w), is defined to be the inverse of impedance, i.e., Y(w) = 1/Z(w). An admittance boundary condition will be derived and coupled with a time domain boundary integral equation. The solution will be obtained iteratively using spatial and temporal basis functions and will allow for acoustic scattering problems to be modeled with geometries consisting of both unlined and soft surfaces. Stability will be demonstrated through eigenvalue analysis.

Pizzo, Michelle E.↗

NASA NTRS2020

Toward a Consistent Definition between Satellite and Model Clear-Sky Radiative Fluxes

A new method of determining clear-sky radiative fluxes from satellite observations for climate model evaluation is presented. The method consists of applying adjustment factors to existing satellite clear-sky broadband radiative fluxes that make the observed and simulated clear-sky flux definitions more consistent. The adjustment factors are determined from the difference between observation-based radiative transfer model calculations of monthly mean clear-sky fluxes obtained by ignoring clouds in the atmospheric column and by weighting hourly mean clear-sky fluxes with imager-based clear-area fractions. The global mean longwave (LW) adjustment factor is −2.2 W m−2 at the top of the atmosphere and 2.7 W m−2 at the surface. The LW adjustment factors are pronounced at high latitudes during winter and in regions with high upper-tropospheric humidity and cirrus cloud cover, such as over the west tropical Pacific, and the South Pacific and intertropical convergence zones. In the shortwave (SW), global mean adjustment is 0.5 W m−2 at TOA and −1.9 W m−2 at the surface. It is most pronounced over sea ice off of Antarctica and over heavy aerosol regions, such as eastern China. However, interannual variations in the regional SW and LW adjustment factors are small compared to those in cloud radiative effect. After applying the LW adjustment factors, differences in zonal mean cloud radiative effect between observations and climate models decrease markedly between 60°S and 60°N and poleward of 65°N. The largest regional improvements occur over the west tropical Pacific and Indian Oceans. In contrast, the impact of the SW adjustment factors is much smaller.

Norman G Loeb↗

NASA NTRS2018

Characterization Testing of Lockheed Martin Micro1-2 Cryocoolers Optimized for 220 K Environment

The Mapping Imaging Spectrometer for Europa (MISE) Instrument on the Europa Clipper mission has baselined a Lockheed Martin high-power Micro1-2 pulse tube cryocooler operating at 135 Hz with a 220 K heat rejection temperature. This paper describes the testing and results of two Lockheed Martin Micro1-2 coolers optimized for these conditions. The thermal performance of the microcoolers was measured in vacuum for heat reject temperatures between 220 and 230 K. The coolers were operated with input powers ranging from 5 W to 40 W and drive frequency between 125 Hz and 145 Hz. The optimal drive frequency was strongly dependent on heat reject temperature. In addition, the exported forces and torques of the coolers were measured at 300 K heat rejection for input powers ranging from 5 W to 60 W and drive frequency between 120 Hz and 160 Hz. The exported forces were dependent on both piston amplitude and drive frequency. Moreover, the following were measured on one of the coolers: DC and AC magnetic fields at various locations, the off-state conductance of the pulse tube, and the effect of inclination angle of the pulse tube relative to gravity on the performance of the cooler. Finally, one of the coolers was subjected to random vibration testing.

Rodriguez, J.I.↗

NASA NTRS2019

A mechanically pumped two-phase fluid loop for thermal control based on the capillary pumped loop

A novel two-phase Mechanically Pumped Fluid Loop (MPFL) for thermal control is presented. The operating principles are outlined and results from a prototype test facility are shown. The system is a variant of the Capillary Pumped Loop (CPL) where a mechanical pump and bypass line have been added to make a pump-assisted CPL. The benefits of this design are discussed, and test results are shown that demonstrate the feasibility of the system architecture. Stable performance for heat loads between 30 W and 850 W are reported along with the capability of handling a heat flux of up to 13 W/cm2. In addition, the testbed incorporates a novel planar Additively Manufactured (AM) evaporator. Preliminary test results show that the evaporator can maintain isothermality within 0.5 C for heat loads up to 100 W and within 3 C for heat loads up to 325 W.

Sunada, Eric↗

NASA NTRS2019

Spatial Distribution of Aerosol Microphysical and Optical Properties and Direct Radiative Effect from the China Aerosol Remote Sensing Network

Multi-year observations of aerosol microphysical and optical properties, obtained through ground-based remote sensing at 50 China Aerosol Remote Sensing Network (CARSNET) sites, were used to characterize the aerosol climatology for representative remote, rural, and urban areas over China to assess effects on climate. The annual mean effective radii for total particles (ReffT) decreased from north to south and from rural to urban sites, and high total particle volumes were found at the urban sites. The aerosol optical depth at 440 nm (AOD440 nm) increased from remote and rural sites (0.12) to urban sites (0.79), and the extinction Ångström exponent (EAE440–870 nm) increased from 0.71 at the arid and semi-arid sites to 1.15 at the urban sites, presumably due to anthropogenic emissions. Single-scattering albedo (SSA440 nm) ranged from 0.88 to 0.92, indicating slightly to strongly absorbing aerosols. Absorption AOD440 nm values were 0.01 at the remote sites versus 0.07 at the urban sites. The average direct aerosol radiative effect (DARE) at the bottom of atmosphere increased from the sites in the remote areas (−24.40 W m−2) to the urban areas (−103.28 W m−2), indicating increased cooling at the latter. The DARE for the top of the atmosphere increased from −4.79 W m−2 at the remote sites to −30.05 W m−2 at the urban sites, indicating overall cooling effects for the Earth–atmosphere system. A classification method based on SSA440 nm, fine-mode fraction (FMF), and EAE440–870 nm showed that coarse-mode particles (mainly dust) were dominant at the rural sites near the northwestern deserts, while light-absorbing, fine-mode particles were important at most urban sites. This study will be important for understanding aerosol climate effects and regional environmental pollution, and the results will provide useful information for satellite validation and the improvement of climate modelling.

Huizheng Che↗

NASA NTRS2021

Capturing the Footprints of Ground Motion in the Spatial Distribution of Rainfall-Induced Landslides

The coupled effect of earthquakes and rainfall is rarely investigated in landslide susceptibility assessments although it could be crucial to predict landslide occurrences. This is even more critical in the context of early warning systems and especially in cases of extreme precipitation regimes in post-seismic conditions, where the rock masses are already damaged due to the ground shaking. Here, we investigate this concept by accounting for the legacy of seismic ground shaking in rainfall-induced landslide (RFIL) scenarios. We do this to identify whether ground shaking plays a role in the susceptibility to post-seismic rainfall-induced landslides and to identify whether this legacy effect persists through time. With this motivation, we use binary logistic regression and examine time series of landslides associated with four earthquakes occurred in Indonesia: 2012 Sulawesi (M(sub w) = 6.3), 2016 Reuleut (M(sub w) = 6.5), 2017 Kasiguncu (M(sub w) = 6.6) and 2018 Palu (M(sub w) = 7.5) earthquakes. The dataset includes one co-seismic and three post-seismic landslide inventories for each earthquake. We use the peak ground acceleration map of the last strongest earthquake in each case as a predisposing factor of landslides representing the effect of ground shaking. We observe that, at least for the study areas under consideration and in a probabilistic context, the earthquake legacy contributes to increase the post-seismic RFIL susceptibility. This positive contribution decays through time. Specifically, we observe that ground motion is a significant predisposing factor controlling the spatial distribution of RFIL in the post-seismic period 110 days after an earthquake. We also show that this effect dissipates within 3 years at most.

Landslides↗

NASA NTRS2023

Surface Shortwave Radiative Fluxes Derived from the U.S. Air Force Cloud Depiction Forecast System World-Wide Merged Cloud Analysis

We present a global-scale evaluation of surface shortwave (SW↓) radiative fluxes as derived with cloud amount information from the U.S. Air Force (USAF) Cloud Depiction Forecast System (CDFS) II World-Wide Merged Cloud Analysis (WWMCA) and implemented in the framework of the NASA Land Information System (LIS). Evaluation of this product is done against ground observations, a satellite-based product from the Moderate Resolution Imaging Spectroradiometer (MODIS), and several reanalysis outputs. While the LIS/USAF product tends to overestimate the SW↓ fluxes when compared to ground observations and satellite estimates, its performance is comparable or better than the following reanalysis products: ERA5, CFSR, and MERRA-2. Results are presented using all available observations over the globe and independently for several regional domains of interest. When evaluated against ground observations over the globe, the bias in the LIS/USAF product at daily time scale was about 9.34 W m −2 and the RMS was 29.20 W m −2 while over the United States the bias was about 10.65 W m −2 and the RMS was 35.31 W m −2 . The sample sizes used were not uniform over the different regions, and the quality of both ground truth and the outputs of the other products may vary regionally. It is important to note that the LIS/USAF is a near-real-time (NRT) product of interest for potential users and as such fills a need that is not met by most products. Due to latency issues, the level of observational inputs in the NRT product is less than in the reanalysis data.

Cloud radiative effects↗

NASA NTRS2021

Design of 94/238 GHz Dual-Frequency Feed Horn for Atmospheric Radar Applications

A compact dual frequency feed horn is designed and optimized to operate at W-band (94.92 GHz) and G-band (238 GHz). The feed horn consists of a single spline profile horn, a diplexer having W- and G-band ports and a band pass filter at the W-band port. The filter provides isolation between the W- and G-band ports. Simulations show a directivity of 21.9 dB and -12 dB beam width of 28 degrees at 94.92 GHz and 26.27 dB and 17 degrees respectively at 238 GHz. The dual frequency horn will be used as part of a W/G-band breadboard atmospheric radar.

Focardi, Paolo↗

NASA NTRS2023

Thermal Infrared Dust Optical Depth and Coarse-Mode Effective Diameter Over Oceans Retrieved From Collocated MODIS and CALIOP Observations

In this study, we developed a novel algorithm based on the collocated Moderate Resolution Imaging Spectroradiometer (MODIS) thermal infrared (TIR) observations and dust vertical profiles from the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) to simultaneously retrieve dust aerosol optical depth at 10 µm (DAOD10μm) and the coarse-mode dust effective diameter (Deff) over global oceans. The accuracy of the Deff retrieval is assessed by comparing the dust lognormal volume particle size distribution (PSD) corresponding to retrieved Deff with the in-situ measured dust PSDs from the AER-D, SAMUM-2 and SALTRACE field campaigns through case studies. The new DAOD10μm retrievals were evaluated first through comparisons with the collocated DAOD10.6μm retrieved from the combined Imaging Infrared Radiometer (IIR) and CALIOP observations from our previous study (Zheng et al. 2022). The pixel-to-pixel comparison of the two DAOD retrievals indicates a good agreement (R~0.7) and a significant reduction of (~50%) retrieval uncertainties largely thanks to the better constraint on dust size. In a climatological comparison, the seasonal and regional (5°×2°) mean DAOD10um retrievals based on our combined MODIS and CALIOP method are in good agreement with the two independent Infrared Atmospheric Sounding Interferometer (IASI) products over three dust transport regions (i.e., North Atlantic (NA; R = 0.9), Indian Ocean (IO; R = 0.8) and North Pacific (NP; R = 0.7)). Using the new retrievals from 2013 to 2017, we performed a climatological analysis of coarse mode dust Deff over global oceans. We found that dust Deff over IO and NP are up to 20% smaller than that over NA. Over NA in summer, we found a ~50% reduction of the number of retrievals with Deff > 5 μm from 15°W to 35°W and a stable trend of Deff average at 4.4 μm from 35°W throughout the Caribbean Sea (90°W). Over NP in spring, only ~5% of retrieved pixels with Deff > 5 μm are found from 150°E to 180°, while the mean Deff remains stable at 4.0 μm throughout eastern NP. To our best knowledge, this study is the first to retrieve both DAOD and coarse-mode dust particle size over global oceans for multiple years. This retrieval dataset provides insightful information for evaluating dust long-wave radiative effects and coarse mode dust particle size in models.

dust optical depth↗

NASA NTRS2024

AC and DC Fault Management for Megawatt Electrified Aircraft Electrical Powertrains - Task 2: Power Quality Filtering Using Nanocrystalline Soft Magnetic Inductor

The NASA RTAPS program on AC and DC Fault Management for Megawatt Electrified Power Train is a multi-year joint project with Pratt & Whitney (P&W), Collins Aerospace (CA), and RTX Technology Research Center (RTRC). This research program focuses on the high-voltage distribution issues that present a significant technological obstacle in the adoption of Electrified Aviation Propulsion (EAP) systems. One challenge to the adoption of high-voltage distribution systems with power electronic converters is the need for filter elements to limit the generation and propagation of noise, protect the cable systems from premature aging and prevent against excessive heating within subcomponents due to high-frequency induced currents. While increased distribution voltages aide in reducing the cable mass for a fixed power system, the associated mass with the filtering elements for power electronic converter can grow with increasing distribution voltages – thereby mitigating any benefit associated with increasing the distribution system voltage. To enable high-voltage distribution systems with high system specific power densities, new magnetic materials must be developed. Therefore, the second task of the NASA RTAPS program is associated with the design and application of advanced soft magnetic components for Megawatt class electric propulsion systems, specifically the motor drive system. This report covers the collaborative work between NASA Glenn Research Center (GRC), RTRC, P&W and CA in the development of three types of magnetic components over the span of the three-year program. These critical magnetic components are the DC side EMI filter, which limits the propagation of harmful electromagnetic noise to the rise of the distribution system, and the AC side damping with the dv/dt filter, which limits the fast rise time of the power electronic converter output voltage to limit the degradation on the cable/motor insulation systems. Each of these components are investigated from component level design and are optimized at the system level with a combined modelling and testing effort. In the final experimental evaluation of the NASA developed soft magnetic material with a dv/dt filter, a commercial-off-the-shelf (COTS) magnetic core and the GRC magnetic core are optimized and loaded at 320Arms to evaluate their difference in performance. After a run time of 30 minutes in a MW-class motor driver at RTRC, the NASA GRC cores were found to not only offer a lower temperature rise of nearly 25°𝐶, but also a reduction in measured core loss of 25% (12.75W to 9.5W).

Elecrified Aircraft Propulsion↗

NASA NTRS1988

In situ observations of ClO in the wintertime Northern Hemisphere: ER-2 aircraft results from 21 N to 61 N latitude

Measurements of lower stratospheric ClO taken during a NASA ER-2 flight between Moffett Field, CA (37 N, 122 W) and Great Slave Lake, Canada (61 N, 116 W) on 13 February 1988 are reported. Northbound, the aircraft was flown at about 20 km altitude from 39 N to 56 N, at 18 km from there to 58 N, in a descent to 15 km at 60 N, and in a rise and turn at the northernmost point. The southbound leg was flown in a gradual climb from 20 km to 21.5 km. On this day, the central position of the Arctic polar vortex, as determined by an NMC analysis of heights and temperatures at the 50 mb and 70 mb levels, was approximately 79 N, 100 W. Because the vortex was located on the North American side of the pole, the aircraft was able to reach a point slightly inside the maximum horizontal wind region where wind speeds were 80 to 90 knots. The general pattern for the observed ClO is that it increased with both latitude and altitude, and attained a maximum of about 55 pptv at 61 N latitude and 20.5 km altitude. This value is about 20 times smaller than the maxima observed over Antarctica, but is comparable to those seen just outside the chemical containment vessel located inside the Antarctic Polar vortex. On the other hand, in a comparison with northern midlatitude data taken on this and three other February flights, ClO mixing ratios observed north of 55 N latitude are 2 to 5 times larger at all flight altitudes (15 to 20 km). Possible reasons are discussed for this enhancement over midlatitude and the evidence is considered for whether or not the instruments sampled Arctic polar vortex air. A second feature of the data is the strong positive correlation between ClO and O3 during the entire flight.

Brune, W. H.↗

NASA NTRS1972

Reflectance of evaporated rhenium and tungsten films in the vacuum ultraviolet from 300 to 2000 A.

Discussion of the dependence of the reflectance of Re and W on the substrate temperature during deposition, film thickness, and aging during exposure to air. Re and W of 99.99% purity were evaporated with a 6-kW fine-focused electron gun and deposited on glass and fused quartz plates of various temperatures ranging from 40 to 500 C. With Re, films of highest reflectance were obtained by evaporation onto unheated substrates, whereas with W, heating of the substrate greatly increased the reflectance of the deposited films. For both metals, the reflectance losses during extended exposure to air remained rather small, indicating that the oxide films formed on both film materials at room temperature were very thin.

Cox, J. T.↗

NASA NTRS1973

Relativistic stellar stability: Preferred-frame effects

Possible preferred-frame effects on stellar stability were examined and no new instabilities were found. In particular, it is shown that: (1) Although terms linear in the preferred-frame velocity w (time-odd terms, analogous to viscosity and energy generation) change the shapes of the normal modes, their symmetry properties prevent them from changing the characteristic frequencies. Thus, no new vibrational or secular instabilities can occur. (2) Terms quadratic in w do not change either the shapes of the normal modes or the characteristic frequencies for radial pulsations. Thus, they have no influence on radial stability. (3) Terms quadratic in w do change both the normal modes and the characteristic frequencies of nonradial pulsations; but in the limit of a neutral mode these changes vanish. Hence, there is no modification of the criterion for convective stability, i.e., the standard Schwarzschild criterion remains valid.

Ni, W.↗