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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Directional finite difference method for directly solving 3D gyrokinetic field equations with enhanced accuracy

The gyrokinetic (GK) field equation is a three-dimensional (3D) elliptic equation, but it is often simplified to a set of two-dimensional (2D) equations by assuming that the field does not vary along a specific direction. However, this simplification can introduce inevitable 0th-order numerical errors, as nonlinear mode coupling in toroidal geometry can produce undesirable harmonic modes that violate the assumption. In this work, we propose a novel directional finite difference method (FDM) with a local coordinate transformation to better resolve the target field of interest. The directional FDM can accurately solve 3D GK field equations without simplifications, which can overcome the limitations of conventional methods. The accuracy and efficiency of different FDMs are analyzed in great detail for a variety of geometries, from simple 2D Cartesian coordinates to realistic 3D curvilinear coordinates. The 0th-order numerical errors of simplified 2D GK equations were found to be more problematic for low-harmonic modes and low aspect ratio geometries such as spherical tokamaks. On the other hand, the directional 3D FDM can accurately resolve a much wider range of harmonic modes aligned to the direction of interest, including the low-harmonic modes. In conclusion, we demonstrate that the directional 3D FDM is a highly effective algorithm for solving the 3D GK field equations, achieving accuracy improvements of 10 to 100 times or more, particularly for low-harmonic modes in spherical tokamaks.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1997

Modeling of the Coupled Magnetospheric and Neutral Wind Dynamos

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

Thayer, Jeffrey P.↗
NASA NTRS2013

Region Spherical Harmonic Magnetic Modeling from Near-Surface and Satellite-Altitude Anomlaies

The compiled near-surface data and satellite crustal magnetic measured data are modeled with a regionally concentrated spherical harmonic presentation technique over Australia and Antarctica. Global crustal magnetic anomaly studies have used a spherical harmonic analysis to represent the Earth's magnetic crustal field. This global approach, however is best applied where the data are uniformly distributed over the entire Earth. Satellite observations generally meet this requirement, but unequally distributed data cannot be easily adapted in global modeling. Even for the satellite observations, due to the errors spread over the globe, data smoothing is inevitable in the global spherical harmonic presentations. In addition, global high-resolution modeling requires a great number of global spherical harmonic coefficients for the regional presentation of crustal magnetic anomalies, whereas a lesser number of localized spherical coefficients will satisfy. We compared methods in both global and regional approaches and for a case where the errors were propagated outside the region of interest. For observations from the upcoming Swarm constellation, the regional modeling will allow the production a lesser number of spherical coefficients that are relevant to the region of interest

Modeling↗
NASA NTRS1974

Error enhancement in geomagnetic models derived from scalar data

Models of the main geomagnetic field are generally represented by a scalar potential gamma expanded in a finite number of spherical harmonics. Very accurate observations of F were used, but indications exist that the accuracy of models derived from them is considerably lower. One problem is that F does not always characterize gamma uniquely. It is not clear whether such ambiguity can be encountered in deriving gamma from F in geomagnetic surveys, but there exists a connection, due to the fact that the counterexamples of Backus are related to the dipole field, while the geomagnetic field is dominated by its dipole component. If the models are recovered with a finite error (i.e. they cannot completely fit the data and consequently have a small spurious component), this connection allows the error in certain sequences of harmonic terms in gamma to be enhanced without unduly large effects on the fit of F to the model.

Stern, D. P.↗
NASA NTRS2016

Normalization and Implementation of Three Gravitational Acceleration Models

Unlike the uniform density spherical shell approximations of Newton, the consequence of spaceflight in the real universe is that gravitational fields are sensitive to the asphericity of their generating central bodies. The gravitational potential of an aspherical central body is typically resolved using spherical harmonic approximations. However, attempting to directly calculate the spherical harmonic approximations results in at least two singularities that must be removed to generalize the method and solve for any possible orbit, including polar orbits. Samuel Pines, Bill Lear, and Robert Gottlieb developed three unique algorithms to eliminate these singularities. This paper documents the methodical normalization of two of the three known formulations for singularity-free gravitational acceleration (namely, the Lear and Gottlieb algorithms) and formulates a general method for defining normalization parameters used to generate normalized Legendre polynomials and Associated Legendre Functions (ALFs) for any algorithm. A treatment of the conventional formulation of the gravitational potential and acceleration is also provided, in addition to a brief overview of the philosophical differences between the three known singularity-free algorithms.

Eckman, Randy A.↗
NASA NTRS1985

The magnetic field at the core-mantle boundary

Models of the geomagnetic field are, in general, produced from a least-squares fit of the coefficients in a truncated spherical harmonic expansion to the available data. Downward continuation of such models to the core-mantle boundary (CMB) is an unstable process: the results are found to be critically dependent on the choice of truncation level. Modern techniques allow this fundamental difficulty to be circumvented. The method of stochastic inversion is applied to modeling the geomagnetic field. Prior information is introduced by requiring that the spectrum of spherical harmonic coefficients to fall-off in a particular manner which is consistent with the Ohmic heating in the core having a finite lower bound. This results in models with finite errors in the radial field at the CMB. Curves of zero radial field can then be determined and integrals of the radial field over patches on the CMB bounded by these null-flux curves calculated. With the assumption of negligible magnetic diffusion in the core; frozen-flux hypothesis, these integrals are time-invariant.

Bloxham, J.↗
NASA NTRS2011

Normalization of Gravitational Acceleration Models

Unlike the uniform density spherical shell approximations of Newton, the con- sequence of spaceflight in the real universe is that gravitational fields are sensitive to the nonsphericity of their generating central bodies. The gravitational potential of a nonspherical central body is typically resolved using spherical harmonic approximations. However, attempting to directly calculate the spherical harmonic approximations results in at least two singularities which must be removed in order to generalize the method and solve for any possible orbit, including polar orbits. Three unique algorithms have been developed to eliminate these singularities by Samuel Pines [1], Bill Lear [2], and Robert Gottlieb [3]. This paper documents the methodical normalization of two1 of the three known formulations for singularity-free gravitational acceleration (namely, the Lear [2] and Gottlieb [3] algorithms) and formulates a general method for defining normalization parameters used to generate normalized Legendre Polynomials and ALFs for any algorithm. A treatment of the conventional formulation of the gravitational potential and acceleration is also provided, in addition to a brief overview of the philosophical differences between the three known singularity-free algorithms.

Eckman, Randy A.↗
NASA NTRS1981

Voyager 1 assessment of Jupiter's planetary magnetic field

An estimate of Jupiter's planetary magnetic field is obtained from the Voyager 1 observations of the Jovian magnetosphere. An explicit model for the magnetodisc current system is combined with a spherical harmonic model of the planetary field with both sets of parameters determined simultaneously using a nonlinear generalized inverse methodology. The resulting model fits the observations extremely well throughout the analysis interval (r 20 Jovian radii). The Jovian internal field model obtained from the Voyager 1 data is very similar to the octopole Pioneer 11 models. The best fitting magnetodisc lies in the centrifugal equator, 2/3 of the way between the rotational and magnetic equators, as appropriate for centrifugal loading of the magnetosphere by a cold plasma.

Connerney, J. E. P.↗
NASA NTRS1992

Analysis of crustal structure of Venus utilizing residual Line-of-Sight (LOS) gravity acceleration and surface topography data. A trial of global modeling of Venus gravity field using harmonic spline method

To construct Venus' gravity disturbance field (or gravity anomaly) with the spacecraft-observer line of site (LOS) acceleration perturbation data, both a global and a local approach can be used. The global approach, e.g., spherical harmonic coefficients, and the local approach, e.g., the integral operator method, based on geodetic techniques are generally not the same, so that they must be used separately for mapping long wavelength features and short wavelength features. Harmonic spline, as an interpolation and extrapolation technique, is intrinsically flexible to both global and local mapping of a potential field. Theoretically, it preserves the information of the potential field up to the bound by sampling theorem regardless of whether it is global or local mapping, and is never bothered with truncation errors. The improvement of harmonic spline methodology for global mapping is reported. New basis functions, a singular value decomposition (SVD) based modification to Parker & Shure's numerical procedure, and preliminary results are presented.

Fang, Ming↗
NASA NTRS1996

Observations of Thermospheric Horizontal Winds at Watson Lake, Yukon Territory (lambda=65 Deg N)

Fabry-Perot interferometer observations of the thermospheric O I (6300 A) emission have been conducted from an airglow observatory at a dark field site in the southeastern Yukon Territory, Canada, for the period November 1991 to April 1993. The experiment operated in unattended, remote fashion, has resulted in a substantial data set from which mean neutral winds have been determined. Dependent upon geomagnetic activity, the nocturnal location of the site is either equatorward of the auroral oval or within oval boundaries. The data set is rich enough to permit hourly binning of neutral winds based upon the K(sub p) geomagnetic disturbance index as well as the season. For cases of low geomagnetic activity the averaged vector horizontal neutral wind exhibits the characteristics of a midlatitude site displaying antisunward pressure-gradient-driven winds. As the geomagnetic activity rises in the late afternoon and evening winds slowly rotate sunward in an anticlockwise direction, initially remaining near 100 m/s in speed but eventually increasing to 300 m/s for K(sub p) greater than 5. For the higher levels of activity the observed neutral wind flow pattern resembles a higher-latitude polar cap pattern characterized by ion drag forcing of thermospheric neutral gases. In addition, rotational Coriolis forcing on the dusk side enhances the ion drag forcing, resulting in dusk winds which trace out the clockwise dusk cell plasma flow. On the dawn side the neutral winds also rotate in an anticlockwise direction as the strength of geomagnetic disturbances increase. Since the site is located at a transition latitude between the midlatitude and the polar cap the data set provides a sensitive test for general circulation models which attempt to parameterize the contribution of magnetospheric processes. A comparison with the Vector Spherical Harmonic (VSH) model indicates several regions of poor correspondence for December solstice conditions but reasonable agreement for the vernal equinox.

Niciejewski, R. J.↗
NASA NTRS1980

Global gravity field to degree and order 30 from Geos 3 satellite altimetry and other data

A model of the geopotential field in spherical harmonics to degree and order 30 is obtained from Geos 3 satellite to sea surface altimetry data, terrestrial gravity measurements and satellite perturbation analysis. A general perturbation solution is employed for the calculation of the orbits of 10 satellites based on satellite laser ranging data, and 1 deg x 1 deg surface gravity data are used to compute 550 km x 550 km block anomalies by means of autocovariance analysis. Altimeter-determined sea-surface heights, which are taken as the geoid, are averaged for each 1 deg x 1 deg ocean surface area and treated by autocovariance analysis to obtain 550 x 550 km block undulations. Observation and normal equations are formed from the altimeter and surface gravity data, which together cover 1635 out of 1654 possible surface elements, and are combined with the available satellite-derived normal equations to obtain a solution for the spherical harmonics coefficients. In addition, a value of 6,378,138.23 + or - 1.3 m is obtained for the earth's semimajor axis.

Gaposchkin, E. M.↗
DOE OSTI2022

Implementation of hybrid finite element method based transport solver in GRIFFIN

A new transport solver option based on the hybrid FEM (HFEM) was implemented in GRIFFIN, the MOOSE-based reactor analysis code, as an effort to support routine core design calculations for advanced reactor applications. The HFEM formulation with P{sub N} (spherical harmonics expansion), akin to the variational nodal method, is effective for solving a spatially homogenized problem with strong transport effect. The residual and Jacobian evaluations of the HFEM weak form were derived and successfully implemented in GRIFFIN, having the diffusion and the PN options available in the new HFEM based transport solver. The performance was tested with the simplified ABTR benchmark problems. The results indicate that the HFEM-based transport solver is a feasible option for solving problems with spatially homogenized and strong streaming by providing superior accuracy with a proper p-refinement. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2022

3D spherical functional expansion tallies in Serpent 2 Monte Carlo code

This work extends the application of functional expansion tallies to 3D spherical geometries. The 3D Zernike polynomials are set as an orthonormal polynomials basis for the functional reconstruction. The study describes the construction of the complete set of polynomials, a natural expansion of the spherical harmonics polynomials where 3D Zernike moments can be evaluated as a linear combination of the geometrical moments. The 3D Zernike polynomials formulation and the computational approach implemented in Serpent 2 are presented and tested through the Godiva model from the ICSBEP criticality benchmark test cases. The implementation results are in agreement with a reference solution described in a fine-resolution mesh, enhancing also the performance and memory demand. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS2006

Anisotropic Developments for Homogeneous Shear Flows

The general decomposition of the spectral correlation tensor R(sub ij)(k) by Cambon et al. (J. Fluid Mech., 202, 295; J. Fluid Mech., 337, 303) into directional and polarization components is applied to the representation of R(sub ij)(k) by spherically averaged quantities. The decomposition splits the deviatoric part H(sub ij)(k) of the spherical average of R(sub ij)(k) into directional and polarization components H(sub ij)(sup e)(k) and H(sub ij)(sup z)(k). A self-consistent representation of the spectral tensor in the limit of weak anisotropy is constructed in terms of these spherically averaged quantities. The directional polarization components must be treated independently: models that attempt the same representation of the spectral tensor using the spherical average H(sub ij)(k) alone prove to be inconsistent with Navier-Stokes dynamics. In particular, a spectral tensor consistent with a prescribed Reynolds stress is not unique. The degree of anisotropy permitted by this theory is restricted by realizability requirements. Since these requirements will be less severe in a more accurate theory, a preliminary account is given of how to generalize the formalism of spherical averages to higher expansion of the spectral tensor. Directionality is described by a conventional expansion in spherical harmonics, but polarization requires an expansion in tensorial spherical harmonics generated by irreducible representations of the spatial rotation group SO(exp 3). These expansions are considered in more detail in the special case of axial symmetry.

Cambon, Claude↗
NASA NTRS2019

Electromagnetic Scattering by Discrete Random Media. I: The Dispersion Equation and the Configuration-Averaged Exciting Field

We consider the scattering of a plane electromagnetic wave obliquely incident on a plane-parallel layer of discrete random medium with non-scattering boundaries. We solve the Lax integral equation for the conditional configuration-averaged exciting field coefficients by assuming a special-form solution, that is, by representing the conditional configuration-averaged exciting field coefficients as a linear combination of the coefficients corresponding to an up-going and a down-going wave. This solution representation is supposed to be valid within the whole domain occupied by the particles, even in the close proximity of the boundaries. By balancing the waves with different propagation directions and wavenumbers we derive two homogeneous systems of equations corresponding to the generalized Lorenz–Lorentz law and two inhomogeneous systems of equations corresponding to the generalized Ewald–Oseen extinction theorem. It is shown that (i) the two homogeneous systems of equations of the generalized Lorenz–Lorentz law reduce to a single homogeneous system of equations corresponding to a semi-infinite discrete random medium at normal incidence; (ii) the dispersion equation is direction and polarization independent; and (iii) the two inhomogeneous systems of equations of the generalized Ewald–Oseen extinction theorem can be reduced to two scalar equations by means of the addition theorem for vector spherical harmonics. It is also shown that the same dispersion equation can be obtained without assuming a special-form solution representation in the proximity of the boundaries.

Adrian Doicu↗
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 corrected 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. The Oort parameters determined by a straightforward least-squares adjustment in vector spherical harmonics are A=14.0 +/- 1.4, B=13.1 +/- 1.2, K=1.1 +/- 1.8, and C=2.9 +/- 1.4 km s(exp -1) kpc(exp -1). The physical meaning and the implications of these parameters are discussed in the framework of a general linear model of the velocity field. We find a few statistically significant higher degree harmonic terms that do not correspond to any parameters in the classical linear model. One of them, a third-degree electric harmonic, is tentatively explained as the response to a negative linear gradient of rotation velocity with distance from the Galactic plane, which we estimate at approximately -20 km s(exp -1) kpc(exp -1). A similar vertical gradient of rotation velocity has been detected for more distant stars representing the thick disk (z greater than 1 kpc), but here we surmise its existence in the thin disk at z less than 200 pc. The most unexpected and unexplained term within the Ogorodnikov-Milne model is the first-degree magnetic harmonic, representing a rigid rotation of the stellar field about the axis -Y pointing opposite to the direction of rotation. This harmonic comes out with a statistically robust coefficient of 6.2 +/- 0.9 km s(exp -1) kpc(exp -1) and is also present in the velocity field of more distant stars. The ensuing upward vertical motion of stars in the general direction of the Galactic center and the downward motion in the anticenter direction are opposite to the vector field expected from the stationary Galactic warp model.

Galaxy↗
NASA NTRS1971

Preliminary Goddard geopotential using optical tracking data and a comparison with SAO models

A preliminary Goddard Space Flight Center (GSFC) geopotential and center of mass station coordinate solution was obtained from satellite orbital data using numerical integration theory. This geodetic solution is a prelude to a more general solution which will combine the 1971 International Satellite Geodesy Experiment (ISAGEX) laser data with the present data being employed. The present GSFC geopotential solution consists of the spherical harmonic coefficients through degree and order eight with higher order satellite resonant coefficients. The solution represents a first iteration result from 17 satellites with approximately 150 weekly orbital arcs containing some 40,000 optical observations. The GSFC preliminary result is compared with final results from the Smithsonian Astrophysical Observatory (SAO) solutions including the 1969 SAO Standard Earth II solution. One aspect of interest for the comparison is that SAO uses an analytic theory for the orbital solution whereas GSFC uses a numerical integration theory. The comparison of geopotential results shows that good agreement exists in general but that there are some areas of minor differences.

Lerch, F. J.↗
NASA NTRS1986

Statistics of the geomagnetic secular variation for the past 5Ma

A new statistical model is proposed for the geomagnetic secular variation over the past 5Ma. Unlike previous models, the model makes use of statistical characteristics of the present day geomagnetic field. The spatial power spectrum of the non-dipole field is consistent with a white source near the core-mantle boundary with Gaussian distribution. After a suitable scaling, the spherical harmonic coefficients may be regarded as statistical samples from a single giant Gaussian process; this is the model of the non-dipole field. The model can be combined with an arbitrary statistical description of the dipole and probability density functions and cumulative distribution functions can be computed for declination and inclination that would be observed at any site on Earth's surface. Global paleomagnetic data spanning the past 5Ma are used to constrain the statistics of the dipole part of the field. A simple model is found to be consistent with the available data. An advantage of specifying the model in terms of the spherical harmonic coefficients is that it is a complete statistical description of the geomagnetic field, enabling us to test specific properties for a general description. Both intensity and directional data distributions may be tested to see if they satisfy the expected model distributions.

Constable, C. G.↗