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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 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

Vertical transport of water in the Martian boundary layer

We are continuing our examination of the transport of H2O through the martian boundary layer, and we have written a one-dimensional numerical model of the exchange of H2O between the atmosphere and subsurface of Mars through the planetary boundary layer (PBL). Our goal is to explore the mechanisms of H2O exchange, and to elucidate the role played by the regolith in the local H2O budget. The atmospheric model includes effects of Coriolis, pressure gradient, and frictional forces for momentum, as well as radiation, sensible heat flux, and advection for heat. The model differs from Flasar and Goody by use of appropriate Viking-based physical constants and inclusion of the radiative effects of atmospheric dust. We specify the pressure gradient force or compute it from a simple slope model. The subsurface model accounts for conduction of heat and diffusion of H2O through a porous adsorbing medium in response to diurnal forcing. The model is initialized with depth-independent H2O concentrations (2 kg M(exp -3)) in the regolith, and a dry atmosphere. The model terminates when the atmospheric H2O column abundance stabilizes at 0.1 percent per sol.

Zent, Aaron P.↗
NASA NTRS1997

The 4-Day Wave as Observed from the Upper Atmosphere Research Satellite Microwave Limb Sounder

The "4-day wave" is an eastward moving quasi-nondispersive feature with period near 4 days occurring near the winter polar stratopause. This paper presents evidence of the 4-day feature in Microwave Limb Sounder (MLS) temperature, geopotential height, and ozone data from the late southern winters of 1992 and 1993. Space-time spectral analyses reveal a double-peaked temperature structure consisting of one peak near the stratopause and another in the lower mesosphere, with an out-of-phase relationship between the two peaks. This double- peaked structure is reminiscent of recent three-dimensional barotropic/baroclinic instability model predictions and is observed here for the first time. The height variation of the 4-day ozone signal is shown to compare well with a linear advective-photochemical tracer model. Negative regions of quasigeostrophic potential vorticity (PV) gradient and positive Eliassen-Palm flux divergence are shown to occur, consistent with instability dynamics playing a role in wave forcing. Spectral analyses of PV derived from MLS geopotential height fields reveal a 4-day signal peaking near the polar stratopause. The three-dimensional structure of the 4-day wave resembles the potential vorticity "charge" concept, wherein a PV anomaly in the atmosphere (analogous to an electrical charge in a dielectric material) induces a geopotential field, a vertically oriented temperature dipole, and circulation about the vertical axis.

Allen, D. R.↗
DOE OSTI2022

Turbulence Structures in the Very Stable Boundary Layer Under the Influence of Wind Profile Distortion

Abstract In very stable boundary layers (VSBL), a “cocktail” of submeso motions routinely result in elevated mean wind speed maxima above the ground, acting as a new source of turbulence generation. This new source of turbulent kinetic energy enhances turbulent mixing and causes mean wind profile distortion (WPD). As a results, this transient distortion in the wind profile adjusts the classical log‐law. Addressing how WPD‐induced turbulence regulates flow structures, turbulent fluxes, and transitions in stability regimes across layers remains a challenge. Eddy covariance data measured at four levels on a 62‐m tower are employed to address these questions. It is shown that the WPD initiates large turbulent eddies that penetrate downward, leading to enhanced vertical mixing and comparable turbulent transport efficiencies across layers. As a consequence, turbulence intensity and fluxes are increased. As the WPD is intensified, turbulent fluxes and turbulent flux transport caused by large eddies are also enhanced, leading to a transition from very stable to weakly stable regimes. Due to the influence of WPD‐induced large eddies, the large‐eddy turbulent Prandtl number does not deviate appreciably from unity and the partitioning between turbulent kinetic and potential energies is linearly related to the gradient Richardson number.

Meteorology & Atmospheric Sciences↗
DOE OSTI2020

Azimuthally averaged wind and thermodynamic structures of tropical cyclones in global climate models and their sensitivity to horizontal resolution

Characteristics of tropical cyclones (TCs) in global climate models (GCMs) are known to be influenced by details of the model configurations, including horizontal resolution and parameterization schemes. Understanding model-to-model differences in TC characteristics is a prerequisite for reducing uncertainty in future TC activity projections by GCMs. This study performs a process-level examination of TC structures in eight GCM simulations that span a range of horizontal resolutions from 1° to 0.25°. A recently developed set of process-oriented diagnostics is used to examine the azimuthally averaged wind and thermodynamic structures of the GCM-simulated TCs. Results indicate that the inner-core wind structures of simulated TCs are more strongly constrained by the horizontal resolutions of the models than are the thermodynamic structures of those TCs. As expected, the structures of TC circulations become more realistic with smaller horizontal grid spacing, such that the radii of maximum wind (RMW) become smaller, and the maximum vertical velocities occur off the center. However, the RMWs are still too large, especially at higher intensities, and there are rising motions occurring at the storm centers, inconsistently with observations. The distributions of precipitation, moisture, radiative and surface turbulent heat fluxes around TCs are diverse, even across models with similar horizontal resolutions. At the same horizontal resolution, models that produce greater rainfall in the inner-core regions tend to simulate stronger TCs. Further, when TCs are weak, the radial gradient of net column radiative flux convergence is comparable to that of surface turbulent heat fluxes, emphasizing the importance of cloud-radiative feedbacks during the early developmental phases of TCs.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Using Ensemble Data Assimilation to Estimate Transient Hydrologic Exchange Flow Under Highly Dynamic Flow Conditions

Abstract Quantifying dynamic hydrologic exchange flows (HEFs) within river corridors that experience high‐frequency flow variations caused by dam regulations is important for understanding the biogeochemical processes at the river water and groundwater interfaces. Heat has been widely used as a tracer to infer steady‐state flow velocities through analytical solutions of heat transport defined by the diurnal temperature signals. Under sub‐daily dynamic flow conditions, however, such analytical solutions are not applicable due to the violation of their fundamental assumptions. In this study, we developed a data assimilation‐based approach to estimate the sub‐daily flux under highly dynamic flow conditions using multi‐depth temperature observations at a 5‐min resolution. If the hydraulic gradient is measured, Darcy's law was used to calculate the flux with permeability estimated from temperature responses below the riverbed. Otherwise, flux was estimated directly by assimilating multi‐depth temperature data at 1‐ or 2‐hr time intervals assuming one‐dimensional flow and heat transport governing equation. By comparing estimated fluxes with model‐generated synthetic truth, we demonstrated that both schemes have robust performance in estimating fluxes under highly dynamic flow conditions. This data assimilation‐based flux estimation method was able to capture the vertical sub‐daily fluxes using multi‐depth high‐resolution temperature data alone, even in the presence of multi‐dimensional flow. This approach has been successfully applied to real field temperature data collected at the Hanford site, which experiences highly dynamic HEFs. Our study shows the promise of adopting distributed 1‐D temperature monitoring to capture spatial and temporal exchange dynamics in river corridors at a watershed scale or beyond.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1989

Ozone budgets from the Dynamics and Chemistry of Marine Stratocumulus experiment

Measurements from the Dynamics and Chemistry of marine Stratocumulus experiment have been used to study components of the regional ozone budget. The surface destruction rate is determined by eddy correlation of ozone and vertical velocity measured by a low-flying aircraft. Significant variability is found in the measured surface resistance; it is partially correlated with friction velocity but appears to have other controlling influences as well. The mean resistance is 4190 s/m which is higher (slower destruction) than most previous estimates for seawater. Flux and mean measurements throughout the marine boundary layer are used to estimate the net rate of in situ photochemical production/destruction of ozone. Averaged over the flights, ozone concentration is found to be near steady state, and a net of photochemical destruction of 0.02-0.07 ng/cu m per sec is diagnosed. This is an important confirmation of photochemical model results for the remote marine boundary layer. Ozone vertical distributions above the boundary layer show a strongly layered structure with very sharp gradients. These distributions are possibly related to the stratospheric ozone source.

Kawa, S. R.↗
DOE OSTI2026

NGEE Arctic Integrated Modeling (IM2): Improved subgrid hillslope hydrologic connectivity

This data product represents the integration of new code capability for arctic tundra hillslope hydrologic processes into the Energy Exascale Earth System Model (E3SM), through the E3SM Land Model (ELM) component. This code integration is the result of collaborative effort between the NGEE Arctic project and the E3SM project. The current ELM represents water movement primarily through vertical processes, such as precipitation, canopy interception, evaporation, infiltration, and soil water movement. Lateral water movement—such as surface runoff, subsurface flow, and river transport—plays a significant role in the hydrological cycle, especially in regions with varied topography. While E3SM includes a runoff routing component representing water transport in the river network, the lateral transport of water at the subgrid scale within the land model has previously not been taken into account. With the recent development of topographic units within the ELM subgrid data structure, there is an opportunity to simulate hillslope hydrologic connectivity by introducing water transport along topographic gradients. We expect that more realistic representation of hillslope hydrologic processes will lead to improved predictions of both soil water content and river network flows. Lateral transport of water at and near the surface is represented as a sub-grid process in this new code development. Water is tracked as it moves from higher to lower elevations within a gridcell. This capability uses the nested hierarchical sub-grid scheme within ELM to connect water fluxes from sub-grid elements with higher elevation to those with lower elevation. This data record consists of a single document (pdf format) that describes the theoretical basis for the hillslope hydrology processes added to ELM, and describes the modifications made to the ELM code. The Methods section of this metadata record includes a link to the public E3SM code repository where the exact code modifications as integrated in E3SM can be accessed. The Next-Generation Ecosystem Experiments: Arctic (NGEE Arctic), was a research effort to reduce uncertainty in Earth System Models by developing a predictive understanding of carbon-rich Arctic ecosystems and feedbacks to climate. NGEE Arctic was supported by the Department of Energy's Office of Biological and Environmental Research. The NGEE Arctic project had two field research sites: 1) located within the Arctic polygonal tundra coastal region on the Barrow Environmental Observatory (BEO) and the North Slope near Utqiagvik (Barrow), Alaska and 2) multiple areas on the discontinuous permafrost region of the Seward Peninsula north of Nome, Alaska. Through observations, experiments, and synthesis with existing datasets, NGEE Arctic provided an enhanced knowledge base for multi-scale modeling and contributed to improved process representation at global pan-Arctic scales within the Department of Energy's Earth system Model (the Energy Exascale Earth System Model, or E3SM), and specifically within the E3SM Land Model component (ELM).

Thornton, Peter E [ORNL] (ORCID:0000000247595158)↗
NASA NTRS2001

Constraining the Mean Crustal Thickness on Mercury

The topography of Mercury is poorly known, with only limited radar and stereo coverage available. However, radar profiles reveal topographic contrasts of several kilometers over wavelengths of approximately 1000 km. The bulk of Mercury's geologic activity took place within the first 1 Ga of the planet's history), and it is therefore likely that these topographic features derive from this period. On Earth, long wavelength topographic features are supported either convectively, or through some combination of isostasy and flexure. Photographic images show no evidence for plume-like features, nor for plate tectonics; I therefore assume that neither convective support nor Pratt isostasy are operating. The composition and structure of the crust of Mercury are almost unknown. The reflectance spectrum of the surface of Mercury is similar to that of the lunar highlands, which are predominantly plagioclase. Anderson et al. used the observed center-of-mass center-of-figure offset together with an assumption of Airy isostasy to infer a crustal thickness of 100-300 km. Based on tidal despinning arguments, the early elastic thickness (T(sub e)) of the (unfractured) lithosphere was approximately equal to or less than 100 km. Thrust faults with lengths of up to 500 km and ages of about 4 Ga B.P. are known to exist on Mercury. Assuming a semicircular slip distribution and a typical thrust fault angle of 10 degrees, the likely vertical depth to the base of these faults is about 45 km. More sophisticated modelling gives similar or slightly smaller answers. The depth to the base of faulting and the elastic layer are usually similar on Earth, and both are thought to be thermally controlled. Assuming that the characteristic temperature is about 750 K, the observed fault depth implies that the heat flux at 4 Ga B.P. is unlikely to be less than 20 mW m(exp -2) for a linear temperature gradient. For an elastic thickness of 45 km, topography at 1000 km wavelength is likely to be about 60% compensated. There are thus likely to be considerable lateral variations in crustal thickness. Additional information is contained in the original extended abstract.

Nimmo, F.↗
NASA NTRS1992

A finite-volume Euler solver for computing rotary-wing aerodynamics on unstructured meshes

An unstructured-grid solver for the unsteady Euler equations has been developed for predicting the aerodynamics of helicopter rotor blades. This flow solver is a finite-volume scheme that computes flow quantities at the vertices of the mesh. Special treatments are used for the flux differencing and boundary conditions in order to compute rotary-wing flowfields, and these are detailed in the paper. The unstructured-grid solver permits adaptive grid refinement in order to improve the resolution of flow features such as shocks, rotor wakes and acoustic waves. These capabilities are demonstrated in the paper. Example calculations are presented for two hovering rotors. In both cases, adaptive-grid refinement is used to resolve high gradients near the rotor surface and also to capture the vortical regions in the rotor wake. The computed results show good agreement with experimental results for surface airloads and wake geometry.

Strawn, Roger C.↗
DOE OSTI2021

Can Simple Machine Learning Tools Extend and Improve Temperature-Based Methods to Infer Streambed Flux?

Temperature-based methods have been developed to infer 1D vertical exchange flux between a stream and the subsurface. Current analyses rely on fitting physically based analytical and numerical models to temperature time series measured at multiple depths to infer daily average flux. These methods have seen wide use in hydrologic science despite strong simplifying assumptions including a lack of consideration of model structural error or the impacts of multidimensional flow or the impacts of transient streambed hydraulic properties. We performed a “perfect-model experiment” investigation to examine whether regression trees, with and without gradient boosting, can extract sufficient information from model-generated subsurface temperature time series, with and without added measurement error, to infer the corresponding exchange flux time series at the streambed surface. Using model-generated, synthetic data allowed us to assess the basic limitations to the use of machine learning; further examination of real data is only warranted if the method can be shown to perform well under these ideal conditions. We also examined whether the inherent feature importance analyses of tree-based machine learning methods can be used to optimize monitoring networks for exchange flux inference.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS2021

An Operational Overview of the EXport Processes in the Ocean from RemoTe Sensing (EXPORTS) Northeast Pacific Field Deployment

The goal of the EXport Processes in the Ocean from RemoTe Sensing (EXPORTS) field campaign is to develop a predictive understanding of the export, fate, and carbon cycle impacts of global ocean net primary production. To accomplish this goal, observations of export flux pathways, plankton community composition, food web processes, and optical, physical, and biogeochemical (BGC) properties are needed over a range of ecosystem states. Here we introduce the first EXPORTS field deployment to Ocean Station Papa in the Northeast Pacific Ocean during summer of 2018, providing context for other papers in this special collection. The experiment was conducted with two ships: a Process Ship, focused on ecological rates, BGC fluxes, temporal changes in food web, and BGC and optical properties, that followed an instrumented Lagrangian float; and a Survey Ship that sampled BGC and optical properties in spatial patterns around the Process Ship. An array of autonomous underwater assets provided measurements over a range of spatial and temporal scales, and partnering programs and remote sensing observations provided additional observational context. The oceanographic setting was typical of late-summer conditions at Ocean Station Papa: a shallow mixed layer, strong vertical and weak horizontal gradients in hydrographic properties, sluggish sub-inertial currents, elevated macronutrient concentrations and low phytoplankton abundances. Although nutrient concentrations were consistent with previous observations, mixed layer chlorophyll was lower than typically observed, resulting in a deeper euphotic zone. Analyses of surface layer temperature and salinity found three distinct surface water types, allowing for diagnosis of whether observed changes were spatial or temporal.The 2018 EXPORTS field deployment is among the most comprehensive biological pump studies ever conducted. A second deployment to the North Atlantic Ocean occurred in spring 2021, which will be followed by focused work on data synthesis and modeling using the entire EXPORTS data set.

Biological pump↗
NASA NTRS1976

Determination of exospheric neutral gas temperatures

The Atmosphere Explorer satellites (AE-C, -D, and -E) were initially placed into highly elliptical orbits with perigees around 140 km and apogees of 4000 km. As a result of such an orbital geometry, measurements of neutral constituents at high altitudes represent mainly vertical changes in densities. The influence of horizontal density gradients on measurements above 400 km is small. Under geomagnetically quiet conditions, the density profiles can be used to derive scale-height temperatures of the exosphere. The open-source neutral mass spectrometer (OSS) flown on all three AE-satellites measured neutral constituents such as N2, O, and N well above 400 km. The temperatures derived from scale heights show a good agreement among the constituents and the expected close correlation with the F10.7-cm solar flux. Satellites with highly elliptical orbits provide the opportunity to measure simultaneously both densities and temperatures.

Mauersberger, K.↗
NASA NTRS1985

Vapor Growth of Alloy-Type Semiconductor Crystals

The present effort is part of a continuing research program directed towards the investigation of basic vapor transport phenomena and of crystal growth properties of electronic materials. The primary purpose of ground-based studies is the development and definition of optimum experimental parameters for flight experiments. The ground-based effort includes the investigation of gravity-driven convection effects on mass transport rates and on crystal morphology for different orientations of the density gradient with respect to the gravity vector, and as a function of pressure and of temperature. In addition to the experimental tasks, theoretical efforts involve the quantitative thermodynamic analysis of the systems under investigation, the computation of fluid dynamic parameters, and the consideration of other possible effects on fluid flow under vertical, stabilizing and microgravity conditions. The specific experiments to be performed in a microgravity environment include the investigation of vapor transport and crystal growth phenomena of the GeSe-Xenon system and of the mass flux and growth of bulk and layer-type crystals of Hg sub (1-x) Cd sub x Te using HgI as a transport agent.

Wiedemeier, H.↗
DOE OSTI2023

Mixing across stable density interfaces in forced stratified turbulence

Understanding how turbulence enhances irreversible scalar mixing in density-stratified fluids is a central problem in geophysical fluid dynamics. While isotropic overturning regions are commonly the focus of mixing analyses, we here investigate whether significant mixing may arise in anisotropic statically stable regions of the flow. Focusing on a single forced direct numerical simulation of stratified turbulence, we analyse spatial correlations between the vertical density gradient $\partial \rho /\partial z$ and the dissipation rates of kinetic energy $\epsilon$ and scalar variance $\chi$ , the latter quantifying scalar mixing. The domain is characterized by relatively well-mixed density layers separated by sharp stable interfaces that are correlated with high vertical shear. While static instability is most prevalent within the mixed layers, much of the scalar mixing is localized to the intervening interfaces, a phenomenon not apparent if considering local static instability or $\epsilon$ alone. While the majority of the domain is characterized by the canonical flux coefficient $\varGamma \equiv \chi /\epsilon =0.2$ , often assumed in ocean mixing parametrizations, extreme values of $\chi$ within the statically stable interfaces, associated with elevated $\varGamma$ , strongly skew the bulk statistics. Our findings suggest that current parametrizations of turbulent mixing may be biased by undersampling, such that the most common, but not necessarily the most significant, mixing events are overweighted. Having focused here on a single simulation of stratified turbulence, it is hoped that our results motivate a broader investigation into the role played by stable density interfaces in mixing, across a wider range of parameters and forcing schemes representative of ocean turbulence.

Couchman, Miles M. P. (ORCID:0000000246676829)↗
DOE OSTI2025

Assessment of integral models for non-Boussinesq lazy plumes using numerical simulations

Integral modelling of turbulent buoyant plumes is crucial for rapid predictions of plume characteristics. While the governing equations are typically derived using self-similarity and a Boussinesq approximation, these assumptions may not hold for plumes originating from finite-area sources with large density ratios. Here, this work evaluates the accuracy of integral-scale models for non-Boussinesq lazy plumes using high-fidelity numerical simulations of turbulent helium plumes. We analyse the plume kinematics by computing vertical fluxes, plume radius and radial profiles, establishing some disparities between common practice and physical accuracy. We identify how the definition of the plume radius changes the perception of the plume structure when the flow is not self-similar and derive a relationship between the flux-based and threshold-based definitions without requiring self-similarity. We then examine the plume dynamics by evaluating the source terms from the governing plume equations. Our results support neglecting diffusive and viscous effects but emphasise the importance of the mean pressure gradient, even in the self-similar regime. Two coefficients need to be modelled: the well-known entrainment coefficient and the lesser-known momentum correction coefficient, which is a correction required for the momentum equation to account for self-similar and slender approximations. The momentum correction coefficient is found to be approximately constant and slightly greater than the assumed value of 1. The standard entrainment coefficient models perform well up to a local Richardson number three times the asymptotic value but overpredict entrainment for larger Richardson numbers. We propose a correction using the known finite limit of entrainment at infinite Richardson number.

Meehan, Michael Alexander [Sandia National Laborat↗
DOE OSTI2024

Sensitivities of the West Greenland Current to Greenland Ice Sheet Meltwater in a Mesoscale Ocean/Sea Ice Model

Abstract Meltwater from the Greenland Ice Sheet can alter the continental shelf/slope circulation and cross-shelf freshwater fluxes and limit deep convection in adjacent basins through surface freshening. We explore the impacts on the West Greenland Current and eastern Labrador Sea with different vertical distributions of the meltwater forcing. In this study, we present the results from global coupled ocean/sea ice simulations, forced with atmospheric reanalysis, that are mesoscale eddy-active (∼2–3-km horizontal spacing) and eddy-permitting (∼6–7-km horizontal spacing) in the study region. We compare the West Greenland Current in mesoscale eddy-active and eddy-permitting without meltwater to highlight the role of small-scale features. The mesoscale eddy-active configuration is then used to assess the change in the eastern Labrador Sea when meltwater is added to the surface or vertically distributed to account for mixing within fjords. In both simulations with meltwater, the West Greenland and West Greenland Coastal Currents are faster than in the simulation with no meltwater; their mean surface speeds are the highest in the vertical distribution case. In the latter case, there is enhanced baroclinic conversion at the shelf break compared to the simulation with no meltwater. When meltwater is vertically distributed, there is an increase in baroclinic conversion at the shelf break associated with increased eddy kinetic energy. In addition, in the eastern Labrador Sea, the salinity is lower and the meltwater volume is greater when meltwater is vertically distributed. Therefore, the West Greenland Current is sensitive to how meltwater is added to the ocean with implications for the freshening of the Labrador Sea. Significance Statement Our goal is to understand how the flux of freshwater across the West Greenland continental slope into the Labrador Sea is modified by meltwater from the Greenland Ice Sheet. We compare the simulations of the ocean that capture key dynamics along the West Greenland continental slope that have no meltwater, meltwater added to the ocean surface, and meltwater distributed vertically to represent the mixing within fjords. When meltwater is added, the currents along the continental slope are faster, with the greatest increase when meltwater is vertically distributed. In that case, there is enhanced freshening of the Labrador Sea because modified density gradients generate more eddies. Proper representation of the vertical structure of meltwater is important for projecting the impact of freshwater on the subpolar North Atlantic.

58 GEOSCIENCES↗
NASA NTRS1986

Nonlinear interactions between convection, rotation and flows with vertical shear

The effects of a mean flow with vertical shear on the convective motions in a rotating layer are examined using a three-dimensional and time-dependent numerical model. In the absence of rotation, the convective motions are shown to be dominated by the shear flow when the Richardson number becomes greater than about -1.0. Both heat and momentum are carried down their respective gradients. For rotating cases with vertical rotation vectors, the Coriolis force turns the flow induced by the convection to produce a more complicated shear that changes direction with height. For rotating cases with tilted rotation vectors, the results depend on the direction of the shear. When the imposed flow is in the opposite direction, the convection motions are less energetic and are even suppressed entirely when the shear is strong. When the imposed flow is in the same direction, as that produced by the rotation, the convective motions are enhanced and a countergradient flux of momentum can be produced.

Hathaway, D. H.↗
NASA NTRS1986

The equilibrium structure of thin magnetic flux tubes. II

The thermal structure of the medium inside thin, vertical magnetic flux tubes embedded in a given external atmosphere is investigated, assuming cylindrical symmetry and a depth-independent plasma beta. The variation with tube radius of the temperature on the tube axis is computed and the temperature on the tube wall is estimated. The temperature variation across the flux tube is found to be due to the depth variation of the intensity and to the density stratification of the atmosphere. Since the temperature difference between the axis and the wall is small in thin flux tubes (of the order of 10 percent), the horizontal temperature gradient may often be neglected and the temperature in a tube of given radius may be described by a single function of depth. Thus, a more detailed numerical treatment of the radiative transfer within thin flux tubes can be substantially simplified by neglecting horizontal temperature differences within the flux tube proper.

Kalkofen, W.↗