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NASA NTRS1982

A problem in representing the core magnetic field of the earth using spherical harmonics

Although there are computational advantages to the representation of the earth's magnetic field by spherical harmonic coefficients of the magnetic potential, up to the thirteenth degree and order, the following disadvantages emerge: (1) the use of spherical harmonics of up to a certain degree does not remove wavelengths greater than a certain value from the surface fields, and (2) the total field magnitudes represented by spherical harmonics up to a certain degree have minimum wavelengths equal to the circumference of the earth divided by twice the maximum degree of the harmonic used. The implications of the ways in which surface fields are separated into core and crustal components are discussed, and it is concluded that since field signals are generated in the core, the representation of the core field by spherical harmonics of potential does not adequately represent all core field components.

Carle, H. M.↗

NASA NTRS1984

Fundamental and harmonic emission in interplanetary type 2 radio bursts

Three interplanetary type II radio bursts which show two prominent and long duration bands in their dynamic spectra were analyzed in detail and compared to similar bands in meter wavelength type II events. These bands, which differ by a factor of about two in frequency, were interpreted in terms of fundamental and harmonic emission. The fundamental component has a greater average intensity than the harmonic, due largely to short intense brightenings. The fundamental spectral profile is more narrow than that of the harmonic, with harmonic band typically exhibiting a larger bandwidth to frequency ratio than the fundamental by a factor of two. The fundamental has a larger source size than the harmonic, 160 degrees versus 110 degrees, on average, as viewed from the Sun. Two of the events have source positions which correlate well with the associated flare positions.

Lengyel-Frey, D.↗

NASA NTRS1985

Fundamental and harmonic emission in interplanetary type II radio bursts

Three interplanetary type II radio bursts which show two prominent and long duration bands in their dynamic spectra were analyzed in detail and compared to similar bands in meter wavelength type II events. These bands, which differ by a factor of about two in frequency, were interpreted in terms of fundamental and harmonic emission. The fundamental component has a greater average intensity than the harmonic, due largely to short intense brightenings. The fundamental spectral profile is more narrow than that of the harmonic, with harmonic band typically exhibiting a larger bandwidth to frequency ratio than the fundamental by a factor of two. The fundamental has a larger source size than the harmonic, 160 degrees versus 110 degrees, on average, as viewed from the sun. Two of the events have source positions which correlate well with the associated flare positions.

Lengyel-Frey, D.↗

NASA NTRS2007

The Local Stellar Velocity Field via Vector Spherical Harmonics

We analyze the local field of stellar tangential velocities for a sample of 42,339 nonbinary Hipparcos stars with accurate parallaxes, using a vector spherical harmonic formalism.We derive simple relations between the parameters of the classical linear model (Ogorodnikov-Milne) of the local systemic field and low-degree terms of the general vector harmonic decomposition. Taking advantage of these relationships, we determine the solar velocity with respect to the local stars of (V(sub X), V(sub Y), V(sub Z)) = (10.5, 18.5, 7.3) +/- 0.1 km s(exp -1) not for the asymmetric drift with respect to the local standard of rest. If only stars more distant than 100 pc are considered, the peculiar solar motion is (V(sub X), V(sub Y), V(sub Z)) = (9.9, 15.6, 6.9) +/- 0.2 km s(exp -1). The adverse effects of harmonic leakage, which occurs between the reflex solar motion represented by the three electric vector harmonics in the velocity space and higher degree harmonics in the proper-motion space, are eliminated in our analysis by direct subtraction of the reflex solar velocity in its tangential components for each star...

kinematics↗

NASA NTRS2014

Driving an Active Vibration Balancer to Minimize Vibrations at the Fundamental and Harmonic Frequencies

Vibrations of a principal machine are reduced at the fundamental and harmonic frequencies by driving the drive motor of an active balancer with balancing signals at the fundamental and selected harmonics. Vibrations are sensed to provide a signal representing the mechanical vibrations. A balancing signal generator for the fundamental and for each selected harmonic processes the sensed vibration signal with adaptive filter algorithms of adaptive filters for each frequency to generate a balancing signal for each frequency. Reference inputs for each frequency are applied to the adaptive filter algorithms of each balancing signal generator at the frequency assigned to the generator. The harmonic balancing signals for all of the frequencies are summed and applied to drive the drive motor. The harmonic balancing signals drive the drive motor with a drive voltage component in opposition to the vibration at each frequency.

Holliday, Ezekiel S.↗

NASA NTRS2018

Contributions of Spherical Harmonics to Gravitational Moment

A scalar gravitational potential function expressed as a series of spherical harmonics frequently serves as the basis for a model of an astronomical body's gravitational field. The contribution of a generic spherical harmonic to gravitational gradient is expressed as a dyadic, which is then used to obtain an analytical expression in vector-dyadic form for the contribution to the moment of gravitational forces about the mass center of a small body such as a spacecraft. The expression developed for a harmonic's contribution to gravitational gradient can be applied in areas beyond the scope of the paper; for example, gravitational gradient plays an important role in the state propagation matrix and the state transition matrix that are used in spacecraft trajectory targeting and Kalman filtering. Additionally, it can be employed in numerical simulations of orbit determination based on measurements obtained with a gradiometer in low-Earth orbit. Contributions of spherical harmonics to gravitational moment may be of interest in connection with attitude control of a spacecraft in the vicinity of a body with an irregular shape, such as an asteroid. Normalized spherical harmonic coefficients up to degree and order 10 are obtained for the asteroid 216 Kleopatra and used in numerical evaluations of contributions to gravitational moment.

Roithmayr, Carlos M.↗

NASA NTRS2021

GRACE Fast Mascons from Spherical Harmonics and a Regularization Design Trade Study

Mass concentration (mascon) solutions have become a prominent medium for investigating time-variable gravity recovered by GRACE and GRACE Follow-On. While GRACE Level-2 spherical harmonic products require various post-processing techniques to eliminate correlated noise, mascon formulations employ spatial regularization strategies during the estimation step to improve signal recovery. However, mascon estimation has traditionally required large computing resources and GRACE Level-1B processing capabilities. In this study, we show that a typical mascon estimation system can be reformulated to allow for the estimation of regularized mascons from Level-2 spherical harmonics. Provided that spherical harmonic solution covariances are available, the computed mascons will be mathematically equivalent to similar mascons estimated from Level-1B observations. This method is computationally efficient, better leverages GRACE spherical harmonics than past methods, and matches the performance of typical mascon solutions without locking scientists into pre-determined regularization designs. We develop a proof-of-concept solution using ITSG-Grace2018 and compare results with traditional mascons from JPL and NASA GSFC. We then assess the effects of spherical harmonic truncation and use of regularization correlations on basin signal recovery. We find that spherical expansions to degree and order 60 provide the minimum expansion necessary to study most basins, while larger expansions help further localize signals. We also find that diagonal regularizations (i.e., regularizations that do not contain inter-mascon correlations) are adversely affected by leakage, especially across boundaries such as coastlines where signals are not highly correlated, whereas including inter-mascon correlations and regional boundaries in the regularization greatly improves signal recovery.

M. J. Croteau↗

NASA NTRS1995

Simulation of 100-300 GHz solid-state harmonic sources

Accurate and efficient simulations of the large-signal time-dependent characteristics of second-harmonic Transferred Electron Oscillators (TEO's) and Heterostructure Barrier Varactor (HBV) frequency triplers have been obtained. This is accomplished by using a novel and efficient harmonic-balance circuit analysis technique which facilitates the integration of physics-based hydrodynamic device simulators. The integrated hydrodynamic device/harmonic-balance circuit simulators allow TEO and HBV circuits to be co-designed from both a device and a circuit point of view. Comparisons have been made with published experimental data for both TEO's and HBV's. For TEO's, excellent correlation has been obtained at 140 GHz and 188 GHz in second-harmonic operation. Excellent correlation has also been obtained for HBV frequency triplers operating near 200 GHz. For HBV's, both a lumped quasi-static equivalent circuit model and the hydrodynamic device simulator have been linked to the harmonic-balance circuit simulator. This comparison illustrates the importance of representing active devices with physics-based numerical device models rather than analytical device models.

NONLINEAR CIRCUITS↗

DOE OSTI2024

Harmonic and Subharmonic RF Injection Locking of THz Metasurface Quantum-Cascade VECSEL

Harmonic and subharmonic RF injection locking is demonstrated in a terahertz (THz) quantum-cascade vertical-external-cavity surface-emitting laser (QC-VECSEL). By tuning the RF injection frequency around integer multiples and submultiples of the cavity round-trip frequency, different harmonic and subharmonic orders can be excited in the same device. Modulation-dependent behavior of the device has been studied with recorded lasing spectral broadening and locking bandwidths in each case. In particular, harmonic injection locking results in the observation of harmonic spectra with bandwidths over 200 GHz. A semiclassical Maxwell-density matrix formalism has been applied to interpret QC-VECSEL dynamics, which aligns well with experimental observations.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2025

First measurement of symmetric cumulants of hexagonal flow harmonics in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV

Correlations between event-by-event fluctuations of anisotropic flow harmonics are measured in Pb-Pb collisions at a center-of-mass energy per nucleon pair of 5.02 TeV, as recorded by the ALICE detector at the LHC. This study presents correlations up to the hexagonal flow harmonic 𝑣 6 , which was measured for the first time. The magnitudes of these higher-order correlations are found to vary as a function of collision centrality and harmonic order. These measurements are compared to viscous hydrodynamic model calculations with EKRT initial conditions and to the iEBE-VISHNU model with T R ⁢ENTo initial conditions. The observed discrepancies between the data and the model calculations vary depending on the harmonic combinations. Due to the sensitivity of model parameters estimated with Bayesian analyses to these higher-order observables, the results presented in this work provide new and independent constraints on the initial conditions and transport properties in theoretical models used to describe the system created in heavy-ion collisions.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2026

Evidence of nonlinear coupling in the edge harmonic oscillation sustaining quiescent high confinement in a tokamak plasma

In a tokamak plasma, we measure nonlinear coupling between harmonics comprising a saturated edge oscillation, as predicted by nonlinear magnetohydrodynamic simulations. The coherent structure formed by this coupling suppresses bursty edge-localized modes that cause energy and particle loss. We also measure nonlinear coupling of the edge harmonic oscillation to core-resonant magnetic tearing modes. When tearing modes are present, the edge oscillation harmonics decouple, and edge-localized modes return. In this article, we conclude that nonlinear interaction with tearing modes interrupts the formation of the edge harmonic oscillation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

Harmonic suppression gratings for soft X-ray monochromators

We describe an approach to harmonic suppression in soft X-ray monochromators by engineering the reflection grating’s diffraction pattern to approximate a sinusoidal amplitude. At synchrotron and free-electron laser sources, X-ray beamlines powered by insertion devices produce a spectrum containing harmonic photon energies that can couple unwanted light into experiments. Beamlines in the soft X-ray energy range (100 eV to 2 keV) commonly employ energy-filtering elements to suppress these harmonics. Available approaches tend to be inefficient, significantly reducing the transmitted power. We show that with pseudo-grayscale binary halftone patterns, gratings can approximate a sinusoidal amplitude and suppress higher diffraction orders. Prototype demonstrations of lithographically fabricated gratings were conducted on a soft X-ray beamline with photon energies of 110 eV and 330 eV. Relative to a square-wave amplitude grating, the third-harmonic intensity was reduced by a factor of 9.0 with a first-order efficiency reduction of 38%.

Goldberg, Kenneth A. [Lawrence Berkeley National L↗

NASA NTRS1971

Determination of nonlinear resistance voltage-current relationships by measuring harmonics

Test configuration measures harmonic signal amplitudes generated in nonlinear resistance. Vacuum-type voltmeter measures low frequency sinusoidal input signal amplitude and wave-analyzer measures amplitude of harmonic signals generated in junction. Input signal harmonics amplitude must not exceed that of harmonics generated in nonlinear resistance.

Stafford, J. M.↗

NASA NTRS1976

Nonlinear interaction between two electrostatic harmonics in a plasma

The problem of harmonic and subharmonic generation of electrostatic waves in a general collisionless plasma is treated using coupled-mode theory based on two time scales. A novel feature is that one of the two interacting waves may be a negative-energy wave. Since the model describing the medium need not be specified, only a general linear and nonlinear conductivity or an equivalent description is required. Just by invoking wave-energy conservation, the coupled-mode equations are obtained in such a way that unequivocal conclusions can be drawn. When both waves have positive energy, they exchange part of it in a periodic fashion, provided that both have some energy initially. If initially all the energy is in the fundamental, all of it will eventually end up (irreversibly) in the second harmonic. If all the energy is in the harmonic initially, no generation of the fundamental (or subharmonic) will take place. If one of the two waves is a negative-energy wave, an explosive instability develops, regardless of initial values. For comparable conditions, the instability time depends on whether the negative-energy wave is in the fundamental or in the upper harmonic.

Verheest, F.↗

NASA NTRS1978

Application of higher harmonic blade feathering for helicopter vibration reduction

Higher harmonic blade feathering for helicopter vibration reduction is considered. Recent wind tunnel tests confirmed the effectiveness of higher harmonic control in reducing articulated rotor vibratory hub loads. Several predictive analyses developed in support of the NASA program were shown to be capable of calculating single harmonic control inputs required to minimize a single 4P hub response. In addition, a multiple-input, multiple-output harmonic control predictive analysis was developed. All techniques developed thus far obtain a solution by extracting empirical transfer functions from sampled data. Algorithm data sampling and processing requirements are minimal to encourage adaptive control system application of such techniques in a flight environment.

Powers, R. W.↗

NASA NTRS1981

Spherical harmonic analysis of a model-generated climatology

Monthly mean fields of 850 mb temperature (T850), 500 mb geopotential height (G500) and sea level pressure (SLP) were generated in the course of a five-year climate simulation run with a global general circulation model. Both the model-generated climatology and an observed climatology were subjected to spherical harmonic analysis, with separate analyses of the globe and the Northern Hemisphere. Comparison of the dominant harmonics of the two climatologies indicates that more than 95% of the area-weighted spatial variance of G500 and more than 90% of that of T850 are explained by fewer than three components, and that the model adequately simulates these large-scale characteristics. On the other hand, as many as 25 harmonics are needed to explain 95% of the observed variance of SLP, and the model simulation of this field is much less satisfactory. The model climatology is also evaluated in terms of the annual cycles of the dominant harmonics.

Christidis, Z. D.↗

NASA NTRS1982

A harmonic analysis method for unsteady transonic flow and its application to the flutter of airfoils

A finite difference method for solving the unsteady transonic flow about harmonically oscillating wings is investigated. The procedure is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady differential equation for small disturbances. The differential equation for the unsteady velocity potential is linear with spatially varying coefficients and with the time variable eliminated by assuming harmonic motion. A study is presented of the shock motion associated with an oscillating airfoil and its representation by the harmonic procedure. The effects of the shock motion and the resulting pressure pulse are shown to be included in the harmonic pressure distributions and the corresponding generalized forces. Analytical and experimental pressure distributions for the NACA 64A010 airfoil are compared for Mach numbers of 0.75, 0.80 and 0.842. A typical section, two-degree-of-freedom flutter analysis of a NACA 64A010 airfoil is performed. The results show a sharp transonic bucket in one case and abrupt changes in instability modes.

Ehlers, F. E.↗

NASA NTRS1983

Ordinary mode auroral kilometric radiation, with harmonics, observed by ISIS 1

Topside-sounder receiver observations by ISIS 1 that reveal examples of o-mode auroral kilometric radiation (AKR) are presented. They correspond to locations outside of the low density source region of intense AKR x-mode emission. The propagation modes are identified by comparing the natural radiation wave cutoffs with the local resonant and wave cutoff phenomena stimulated by the sounder transmitter. The o-mode AKR is the dominant emission in these regions of relatively high electron density, but it is considerably weaker than the intense x-mode AKR observed to emanate from low density cavities above the auroral regions. In addition to the fundamental o-mode, 2nd and 3rd harmonic bands of radiation have also been detected. Harmonics associated with these o-mode AKR are less intense than the harmonics associated with x-mode AKR. It is difficult to explain the variety of harmonic AKR observations (x as well as o-mode) based on present AKR theories.

Benson, R. F.↗