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All sources NASA NTRS DOE OSTI

DOE OSTI2022

Writable spin wave nanochannels in an artificial-spin-ice-mediated ferromagnetic thin film

We report magnonics, which employs spin-waves to transmit and process information, is a promising venue for low-power data processing. One of the major challenges is the local control of the spin-wave propagation path. Here, we introduce the concept of writable magnonics by taking advantage of the highly flexible reconfigurability and rewritability of artificial spin ice systems. Using micromagnetic simulations, we show that globally switchable spin-wave propagation and locally writable spin-wave nanochannels can be realized in a ferromagnetic thin film underlying an artificial pinwheel spin ice. The rewritable magnonics enabled by reconfigurable spin wave nanochannels provides a unique setting to design programmable magnonic circuits and logic devices for ultra-low power applications.

42 ENGINEERING↗

NASA NTRS1992

Submillimeter heterodyne spectroscopy and remote sensing of the upper atmosphere

Submillimeter-wavelength heterodyne spectroscopy by microwave limb sounding (MLS) from orbiting satellite is a measurement technique for studying Earth's stratosphere, mesosphere and lowerr thermosphere on a global scale. Development and deployment of such tools are timely, as stratospheric ozone shileds life from solar ultraviolet radiation but is depleted by pollution from industrial activities. MLS experiments on NASA's Upper Atmosphere Research Satellite (UARS) and Earth Observing System (EOS) are now being developed and implementend for global monitoring. Atmospheric thermal-emission spectra at millimeter and submillimeter wave-lengths are measured as the instrument field of view (FOV) is scanned through the limb from above. Atmospheric profiles of molecular abundances, temperature, pressure, wind and magnetic field can be determined from the measured emission spectra. Intensity of the emission can provide abundance and temperature. Measured linewidths, and emission from temperature-insensitive O2 lines, can provide pressure. Differentitaion of measured pressure with repsect ot measured height differential (obtaiend from the instrument FOV scan encoder) can also provide temperature through atmospheric hydrostatic equilibrium (which relates temperature to pressure and height differential). Doppler shifts of spectral lines can provide wind, and Zeeman splitting of the magnetic-dipole lines of O2 can provide magnetic field.

Waters, J. W.↗

DOE OSTI2020

Theory for transport in magnetized plasmas

We present a two fluid model for anomalous thermal, particle and momentum transport which has been derived from kinetic theory using a small parameter ε~10-2. This small parameter can come from either the ratio of wave frequency to cyclotron frequency or the ratio of potential to thermal energies. This fluid model is strongly nonlinear and has an exact closure within the accuracy ε. It can be used to simulate the full radius of a tokamak including internal and edge transport barriers at the same time without introducing any local assumptions for physics model or grid size. It involves a full transport matrix, thus giving possibility for both density and heat pinches. The model can be derived from the Vlasov equation, adding also close collisions and includes particle pinches of the type found in the magnetosphere and in the Massachusetts Institute of Technology elevated dipole machine. Pinches are also in agreement for the heat pinch experiments on Doublet III D including simultaneous particle and heat pinches. For tokamaks it gives the right scaling of confinement time versus heating power. Finally, we have also recovered the Turbulent Equipartition result that the density scales as inverse safety factor q in typical simulations. Thus this model is both fundamental and of high practical use.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

Nanoscale Imaging of Magnetotransport around a Circular 𝑝−𝑛 Junction in Graphene

Magnetoresistance studies of 2D systems are often shaped by the motion of electrons that occupy spatially confined wave functions, such as topological edge modes and disorder-induced bound states. Directly probing how such states form and behave in situ , under applied currents, provides a clear way of connecting microscopic physics to the macroscopic transport response. Here, in this Letter, scanning tunneling potentiometry is used to probe the local, current-induced electrochemical potential of carriers in graphene near circular 𝑝−𝑛 junctions in an out-of-plane magnetic field ranging from 0 to 1.4 T. These measurements provide detailed information about the motion of carriers at the nanometer scale, revealing how it evolves with increasing field. The electrochemical potential displays distinct patterns, such as dipoles, spirals, and concentric disks in weak, moderate, and high fields, respectively. The size and orientation of these patterns can be used to understand how local carrier dynamics change in different transport regimes as well as to directly extract physical parameters such as the electron mean free path and cyclotron diameter.

36 MATERIALS SCIENCE↗

NASA NTRS1961

A Study of the Free Oscillations of the Earth

Published observations on the toroidal oscillations of the earth are critically reviewed. A supplementary analysis of the record obtained by the Lamont strain seismometer is presented. Eleven toroidal modes are identified, and it is concluded that the periods are known to within 1 per cent. A perturbation scheme involving the ratio of the angular velocity of the earth to the resonant frequency is used in calculating the effects due to the rotation of the earth on the resonant frequency. The free oscillations are viewed as a superposition of traveling waves. In a nonrotating system two traveling waves combine to produce a stationary standing wave. In a rotating system, the rotation distinguishes between waves that travel in the direction of rotation and those that travel in the opposite direction. Rotation removes a degeneracy and results in a splitting of a spectral peak of order l into 2 times l plus 1 peaks. The fractional displacement in frequency for the lowest-order toroidal oscillations is 1/206 and of the same order as the Q of the peak, so that splitting will probably not be observed in the toroidal oscillations. Viewed locally, rotation causes a particle to precess about a direction parallel to the axis of rotation. This precession will cause a variation of amplitude with time if the motion is recorded by an instrument with an anisotropic response function. Care is therefore needed in studying the time decay of a given spectral peak. Rotation also couples the normal coordinates so that a motion that is initially purely horizontal will develop a vertical component. It is expected that vertical seismometers should record particle motion with the toroidal frequencies. The perturbations of the toroidal oscillations due to core-mantle interaction are treated in detail. An exact expression is obtained for the rate of energy dissipated by a finitely conducting plate oscillating across a magnetic field. The energy dissipated at the core-mantle boundary due to viscous and hydromagnetic coupling is shown to be insignificant as compared with the energy dissipated within the mantle. The toroidal magnetic field leaking into the lower mantle combines with the dipole field, resulting in a stress on the mantle, tending to stiffen the lower boundary. The stress is of sufficient magnitude to produce a displacement toward higher frequency in the lower-order toroidal oscillations. Observations on the (sub 0) T (sub 2) oscillations lead to an estimate of the toroidal magnetic field in the lower mantle. A calculation of elastic energy in the low-order oscillations suggests a value of 10 (sup 18) ergs per cycles per hour for the energy density at low frequencies in the Chilean earthquake. Each mode of oscillation has a characteristic radial distribution of elastic energy associated with it. This distribution determines which parts of the earth contribute most heavily in determining a particular resonant frequency. The distribution of energy for the lower 17 modes for a homogeneous and a Gutenberg model earth is calculated. The resonant frequencies for models of the earth based on the Gutenberg and Lehmann distribution of elastic properties are presented. It is shown that the Gutenberg model earth fits the observations more closely than the Lehmann model and that a slight alteration of the Gutenberg model gives a significantly better fit to the observations. The alteration involves a lower shear-wave velocity in the lower mantle while the Gutenberg velocity distribution is maintained in the upper mantle. Various studies of the earth's oscillations coupled with surface-wave investigations substantiate Gutenberg's hypothesis of a layer of low velocity in the upper mantle. The physical conditions required for the formation of a region of low velocity are examined in detail. The results confirm Birch's earlier statement that a temperature gradient in excess of 6 degrees to 7 degrees per kilometer is needed to produce a decrease in velocity. The low-velocity layer does not require that the temperature approach or exceed the melting temperature. If tile upper mantle is homogeneous, the region of lower velocity should commence at the base of the crust and extend to 150 kilometers under the oceans and about 100 kilometers under continental regions. The distribution of thermal conductivity and radioactivity consistent with the low-velocity layer is also considered.

MacDonald, Gordon J. F.↗

DOE OSTI2025

Improved Simulation of East Asian Summer Monsoon in the High-resolution CESM1 and Its Causes

Based on the high- and low-resolution Community Earth System Model, version 1 (CESM1), and corresponding simulations from phase 6 of the Coupled Model Intercomparison Project (CMIP6), we compare the interannual variability of the East Asian summer monsoon (EASM). The EASM interannual variability is characterized by the anomalous western North Pacific anticyclone (WNPAC) circulation and the dipole rainfall pattern with a negative southern lobe over the western North Pacific and a positive northern lobe along the Meiyu–Baiu region, which is better reproduced by the high-resolution models. Here, the reason for the improvement in the high-resolution models has been attributed to the better simulation of the warm temperature advection from the wind anomalies on the climatological temperature gradient. Positive sea surface temperature (SST) anomalies over the tropical Indian Ocean are the key to the improved wind anomalies featuring a WNPAC in the high-resolution models. The warm SST anomalies over the tropical Indian Ocean strengthen the WNPAC by triggering a Kelvin-wave response to the enhanced heat release induced by the increased precipitation. Based on the mixed-layer heat budget analysis, the warm SST anomalies over the western Indian Ocean in the high-resolution CESM1 are tied to the anomalous easterly wind along the equator, which reduces surface evaporation and upwelling. Therefore, the better simulations of air-sea feedback and the oceanic mesoscale eddy over the western Indian Ocean are the key for the improved simulation of the EASM interannual variations in the high-resolution CESM1.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

An intriguing intermediate state as a bridge between antiferroelectric and ferroelectric perovskites

The configuration of electric or magnetic dipoles embedded into a crystal lattice gives rise to the defining physical properties of a functional material. The prevailing antiferroelectric materials have so far been allusively assumed to be in a strict antiparallel electric polarization configuration, although they have shown huge potential applications for use in electronic devices. Herein, we have revealed for the first time the complex crystal structure of an intermediate state between the two conventional antiferroelectric (AFE) and ferroelectric (FE) states in PbZrO 3 -based perovskites by employing a combination of advanced synchrotron X-ray, neutron and three-dimensional (3D) electron diffraction techniques. We found that the intermediate state displays an intriguing flexible configuration of a sinusoidal wave functional with imbalanced polarization, different to the commonly believed antiparallel and balanced polarization. Such an intermediate state was concluded to act as a bridge between the conventional AFE and FE states. Our observations offer new insight into the AFE state and its transformation into the FE state. Such an intermediate state with a unique polarization configuration is believed to be a common feature that occurs in high-performance AFE perovskite materials.

36 MATERIALS SCIENCE↗

NASA NTRS1997

Physics of Rotation Powered Pulsars and Their Nebulae

This letter is my progress report for the Astrophysics Theory grant. The first goal of the research supported by this grant is to produce a time dependent theory of the unsteady relativistic collisionless shock wave terminating the relativistic wind from a pulsar, and compare the predicted surface brightness fluctuations to Hubble Space Telescope observations of the wisps in the Crab Nebula. The second goal is to model the production of electron-positron pairs over the polar caps of rotation powered pulsars, and use the results to predict the heating of the surface due to particle trapping and bombardment of the atmosphere at the base of the polar field lines. We have succeeded in creating a one dimensional hybrid code, in which the electron-positron pairs incident on the shock structure are modeled as a relativistic, adiabatic, ideal MHD fluid, while the heavy ions are treated as particles using a particle-in-cell algorithm. The electromagnetic fields are evaluated from the currents and charge densities in the pairs and the ions, while the particles and the fluid accelerate in response to the computed self-consistent electromagnetic fields. The results are promising, in that the underlying ion cyclotron instability generates finite amplitude, propagating magnetosonic waves in the pairs, whose wavelengths and frequencies, when translated into physical units, are comparable to the observed running waves of brightness observed by HST near the Crab pulsar. The code is undergoing a number of tests, to assure us that this preliminary correspondence is not an artifact. In the coming year, the observational appearance of the models will be computed and compared to the HST observations of the Crab now in hand, and used to predict the HST results which will be obtained the year after next. WE also developed a one dimensional cascade theory for pair creation over pulsars' polar caps. A linear integral equation describing the synchrotron cascade has been derived and solved by iterative techniques, in the case when a high energy electron moving parallel to a star centered dipole magnetic field initiates the cascade through curvature gamma ray emission.

Arons, Jonathan↗

NASA NTRS1997

Effective Collision Strengths for Electron Impact Excitation of Inelastic Transitions in S III

We have calculated electron collisional excitation strengths for all electric dipole forbidden, semi-forbidden, and allowed transitions among the lowest 17 LS states 3s(exp 2)3p(exp 2) P-3, D-1, S-1, 3s3p(exp 3)S-5(exp 0), D-3(exp 0), P-3(exp 0), P-1(exp 0), S-3(exp 0), D-1(exp 0), 3S(exp 2)3p3d D-1(exp 0), F-3(exp 0), P-3(exp 0), D-3(exp 0), F-3(exp 0), P-1(exp 0), and 3S(exp 2)3p4S P-3(exp 0), P-l(exp 0) of S III using the R-matrix method. These S m states are represented by fairly extensive configuration-interaction wave functions that yield excited state energies in close agreement with recent laboratory measurements. Rydberg series of resonances converging to the excited state thresholds are explicitly included in the scattering calculation. The effective collision strengths are determined assuming Maxwellian distribution of electron energies. These are listed over a wide temperature range ([0.5-10] x 10(exp 4) K) and compared, where possible, with other available calculations. Subject headings: atomic data - atomic processes

Tayal, S. S.↗

NASA NTRS2017

Electric Potential and Electric Field Imaging with Dynamic Applications: 2017 Research Award Innovation

The technology and methods for remote quantitative imaging of electrostatic potentials and electrostatic fields in and around objects and in free space is presented. Electric field imaging (EFI) technology may be applied to characterize intrinsic or existing electric potentials and electric fields, or an externally generated electrostatic field may be used for illuminating volumes to be inspected with EFI. The baseline sensor technology (e-Sensor) and its construction, optional electric field generation (quasi-static generator), and current e- Sensor enhancements (ephemeral e-Sensor) are discussed. Critical design elements of current linear and real-time two-dimensional (2D) measurement systems are highlighted, and the development of a three dimensional (3D) EFI system is presented. Demonstrations for structural, electronic, human, and memory applications are shown. Recent work demonstrates that phonons may be used to create and annihilate electric dipoles within structures. Phonon induced dipoles are ephemeral and their polarization, strength, and location may be quantitatively characterized by EFI providing a new subsurface Phonon-EFI imaging technology. Initial results from real-time imaging of combustion and ion flow, and their measurement complications, will be discussed. These new EFI capabilities are demonstrated to characterize electric charge distribution creating a new field of study embracing areas of interest including electrostatic discharge (ESD) mitigation, crime scene forensics, design and materials selection for advanced sensors, combustion science, on-orbit space potential, container inspection, remote characterization of electronic circuits and level of activation, dielectric morphology of structures, tether integrity, organic molecular memory, atmospheric science, and medical diagnostic and treatment efficacy applications such as cardiac polarization wave propagation and electromyography imaging.

Generazio, Ed↗

NASA NTRS1993

Source characteristics of Jovian hectometric radio emissions

Direct confirmation that low-frequency Jovian hectometric (HOM) radio emissions centered near 0 deg central meridian longitude consist of distinct, oppositely polarized northern and southern beams has been achieved using data from the Unified Radio and Plasma Wave (URAP) experiment on the Ulysses spacecraft during the Ulysses-Jupiter encounter in early February 1992. Distinct northern and southern beams were observed in the frequency range from approximately 300 kHz to 1 MHz for at least eight Jovian rotations during the Ulysses inbound pass at distances from 100 to 40 R(sub j). The radiation from the two magnetic hemispheres was measured from different Jovigraphic longitudes and magnetic (or centrifugal) latitudes. Observed temporal variations in the radio intensities, with time scales on the order of 30 min, may result either from longitudinal variations of the HOM sources or from longitudinal density variations in the Io plasma torus. Using the URAP direction-finding capabilities and assuming a tilted dipole planetary magnetic field model, the three-dimensional HOM source locations, the L shell through these source locations, and the beam opening angles were independently deduced. The HOM sources were found to originate at approximately 3 R(sub j) and on low L shells (L approximately 4 to 6), with beam opening angles ranging from 10 to 50 deg.

Reiner, M. J.↗

NASA NTRS2007

Antenna for Measuring Electric Fields Within the Inner Heliosphere

A document discusses concepts for the design of an antenna to be deployed from a spacecraft for measuring the ambient electric field associated with plasma waves at a location within 3 solar radii from the solar photosphere. The antenna must be long enough to extend beyond the photoelectron and plasma sheaths of the spacecraft (expected to be of the order of meters thick) and to enable measurements at frequencies from 20 Hz to 10 MHz without contamination by spacecraft electric-field noise. The antenna must, therefore, extend beyond the thermal protection system (TPS) of the main body of the spacecraft and must withstand solar heating to a temperature as high as 2,000 C while not conducting excessive heat to the interior of the spacecraft. The TPS would be conical and its axis would be pointed toward the Sun. The antenna would include monopole halves of dipoles that would be deployed from within the shadow of the TPS. The outer potion of each monopole would be composed of a carbon-carbon (C-C) composite surface exposed to direct sunlight (hot side) and a C-C side in shadow (cold side) with yttria-stabilized zirconia spacers in-between. The hot side cannot view the spacecraft bus, while the cold side can. The booms also can be tilted to minimize heat input to spacecraft bus. This design allows one to reduce heat input to the spacecraft bus to acceptable levels.

Sittler, Edward Charles↗

DOE OSTI2022

Electronic structure of strongly correlated systems: recent developments in multiconfiguration pair-density functional theory and multiconfiguration nonclassical-energy functional theory

Strong electron correlation plays an important role in transition-metal and heavy-metal chemistry, magnetic molecules, bond breaking, biradicals, excited states, and many functional materials, but it provides a significant challenge for modern electronic structure theory. The treatment of strongly correlated systems usually requires a multireference method to adequately describe spin densities and near-degeneracy correlation. However, quantitative computation of dynamic correlation with multireference wave functions is often difficult or impractical. Multiconfiguration pair-density functional theory (MC-PDFT) provides a way to blend multiconfiguration wave function theory and density functional theory to quantitatively treat both near-degeneracy correlation and dynamic correlation in strongly correlated systems; it is more affordable than multireference perturbation theory, multireference configuration interaction, or multireference coupled cluster theory and more accurate for many properties than Kohn–Sham density functional theory. This perspective article provides a brief introduction to strongly correlated systems and previously reviewed progress on MC-PDFT followed by a discussion of several recent developments and applications of MC-PDFT and related methods, including localized-active-space MC-PDFT, generalized active-space MC-PDFT, density-matrix-renormalization-group MC-PDFT, hybrid MC-PDFT, multistate MC-PDFT, spin–orbit coupling, analytic gradients, and dipole moments. We also review the more recently introduced multiconfiguration nonclassical-energy functional theory (MC-NEFT), which is like MC-PDFT but allows for other ingredients in the nonclassical-energy functional. We discuss two new kinds of MC-NEFT methods, namely multiconfiguration density coherence functional theory and machine-learned functionals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1972

Ionization of multielectron atoms by fast charged particles.

Using plane waves to describe the incident and scattered particles, and screened hydrogenic and Coulomb functions to describe the atomic electrons before and after ejections, we have calculated the differential and total ionization cross sections of 11 atoms and one ion by electron impact, and ionization of helium by proton impact. The effective charges of the screened hydrogenic functions are fixed by the Hartree-Fock calculations. Calculations have been carried out for the atomic s, p, and d electrons. For low atomic numbers, we find reasonable agreement with the experimental data. For intermediate atomic numbers, we expect our results to overestimate the actual cross sections, since our choice of a unit charge for the Coulomb function of the ejected electrons will overestimate the atomic dipole potential strength, and in turn the high-energy cross sections. The advantage of the method presented here is that the ionization amplitude is given in analytic form. This may allow further analysis on this amplitude, and facilitates extension of the numerical integration for the cross section to high impact energies.

Omidvar, K.↗

NASA NTRSDate not supplied

Phased Array Characterization of Slat Noise Radiation from a High-Lift Common Research Model

A test campaign was conducted in the open-jet test section of the NASA Langley 14- by 22-Foot Subsonic Tunnel on a 10%-scale, semispan version of a High-Lift Common Research Model (CRM-HL) incorporating a leading-edge slat, trailing-edge flap and removable high-fidelity main landing gear. Far field acoustic measurements were obtained on baseline and acoustically treated model configurations using a traversing 97-microphone phased array that viewed the pressure side of the airframe. A modified version of the DAMAS deconvolution method incorporating corrections for shear layer acoustic wave decorrelation was employed to determine the locations and strengths of relevant noise sources along the span of the slat and in the vicinity of the flap edges and landing gear. The main goal of the test campaign was to use the array data to evaluate the effectiveness of various slat noise reduction concepts. It was found that slat-gap fillers produced significant broadband noise reduction for all Mach numbers and airfoil angles of attack that were investigated. Mach number scaling revealed an approximate velocity to the sixth power relationship for the slat noise at higher frequencies. Directivity measurements obtained from integration of DAMAS pressure-squared values over defined geometric zones around the model showed that baseline slat noise radiation approximated a dipole as a function of polar emission angle, becoming approximately omnidirectional with the introduction of the slat-gap filler. The test campaign clearly demonstrated the noise reduction benefits that can be obtained by applying appropriate treatments to leading edge slats on commercial transport-class aircraft.

William M Humphreys, Jr.↗

DOE OSTI2025

Estimating an airborne dipole source using 3-D wavefield simulations and seismic receivers on the ground

Acoustic signals can couple to the ground, providing an opportunity to use seismic stations to investigate airborne sources. The study of Bishop et al. used wavefield simulations in a fluid-solid medium to quantify the role of topography on the seismic (ground) recordings of a monopole source in the air. We build upon this study by linking wavefield forward modelling with the source estimation code MTUQ, which can accommodate point forces or moment tensors in a solid medium, as well as sources in the air (or water) if they are enabled by the forward-modelling solver. We perform a series of synthetic numerical experiments to demonstrate that a dipole airborne source can be estimated using ground-based receivers, even within the presence of realistic topography. We investigate the influence of receiver coverage, topography, and assumed source location on the estimated results. The established capabilities raise the prospects for future efforts to estimate dipole sources in 3-D models that include heterogeneity in the air and the earth in addition to topography.

58 GEOSCIENCES↗

DOE OSTI2021

Influence of the Indian Ocean Dipole on the large-scale circulation in South America

The influence of each phase of the Indian Ocean Dipole (IOD) on the largescale circulation in South America is investigated using rainfall observations, fully-coupled, large-ensemble, historical simulations (LENS), and forced experiments using the coupled model’s atmospheric component. IOD events often occur when El Niño Southern Oscillation (ENSO), the largest source of interannual variability of precipitation in South America, is active. To distinguish from effects of ENSO, only cases during neutral ENSO conditions are analyzed in LENS and observations. During the positive IOD polarity, a perturbation in the localWalker circulation leads to increased convection over equatorial South America, resulting in wet anomalies in the Amazon basin. This signal is the opposite of what is expected during El Niño events. Tropical convection anomalies in the Indian Ocean also force an extratropical Rossby wave train that reaches subtropical South America. During positive IOD, the moisture flux from the Amazon to central and southeastern Brazil weakens, resulting in a drying of the area associated with the South Atlantic Convergence Zone. Meanwhile, the South Atlantic Subtropical High strengthens, contributing to a drying in southeastern Brazil. During negative IOD, the induced wave train from the Indian Ocean leads to increased moisture transport to the La Plata basin, leading to wet anomalies in the region.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1999

Verification and Analysis of Formulation 4 of Langley for the Study of Noise From High Speed Surfaces

There are several approaches to the prediction of the noise from sources on high speed surfaces. Two of these are the Kirchhoff and the Ffowcs williams-Hawkings methods. It can be shown that both of these methods depend on the solution of the wave equation with mathematically similar inhomogeneous source terms. Two subsonic solutions known as Formulation 1 and 1A of Langley are simple and efficient for noise prediction. The supersonic solution known as Formulation 3 is very complicated and difficult to code. Because of the complexity of the result, the computation time is longer than the subsonic formulas. Furthermore, it is difficult to assess the accuracy of noise prediction. We have been searching for a new and simpler supersonic formulation without these shortcomings. In the last AIAA Aeroacoustics Conference in Toulouse, Farassat, Dunn and Brentner presented a paper in which such a result was presented and called Formulation 4 of Langley. In this paper we will present two analytic tests of the validity this Formulation: 1) the noise from dipole distribution on the unit circle whose strength varies radially with the square of the distance from the center and 2) the noise from dipole distribution on the unit sphere whose strength varies with the cosine of the angle from the polar axis. We will discuss the question of singularities of Formulation 4.

Farassat, F.↗