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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.↗

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.↗

NASA NTRS1988

Statistics of the geomagnetic secular variation for the past 5 m.y

Constable, C. G.↗

NASA NTRS2004

Magnetic Probing of Core Geodynamics

To better understand geomagnetic theory and observation, we can use spatial magnetic spectra for the main field and secular variation to test core dynamical hypotheses against seismology. The hypotheses lead to theoretical spectra which are fitted to observational spectra. Each fit yields an estimate of the radius of Earth's core and uncertainty. If this agrees with the seismologic value, then the hypothesis passes the test. A new way to obtain theoretical spectra extends the hydromagnetic scale analysis of Benton to scale-variant field and flow. For narrow scale flow and a dynamically weak field by the top of Earth's core, this yields a generalized Stevenson-McLeod spectrum for the core-source field, and a secular variation spectrum modulated by a cubic polynomial in spherical harmonic degree n. The former passes the tests. The latter passes many tests, but does not describe rapid dipole decline and quadrupole rebound; some tests suggest it is a bit hard, or rich in narrow scale range. In a core geodynamo, motion of the fluid conductor does work against the Lorentz force. This converts kinetic into magnetic energy which, in turn, is lost to heat via Ohmic dissipation. In the analysis at length-scale 1/k, if one presumes kinetic energy is converted in either eddy-overturning or magnetic free-decay time-scales, then Kolmogorov or other spectra in conflict with observational spectra can result. Instead, the rate work is done roughly balances the dissipation rate, which is consistent with small-scale flow. The conversion time-scale depends on dynamical constraints. These are summarized by the magnetogeostrophic vertical vorticity balance by the top of the core, which includes anisotropic effects of rotation, the magnetic field, and the core-mantle boundary. The resulting theoretical spectra for the core-source field and its SV are far more compatible with observation. The conversion time-scale of order 120 years is pseudo-scale-invariant. Magnetic spectra of other planets may differ; however, if a transition to non-conducting fluid hydrogen in Jupiter acts as a barrier to vertical flow, as well as current, then the shape of the jovi-magnetic spectrum could be remarkably Earth-like.

Voorhies, Coerte V.↗

NASA NTRS2004

Magnetic Probing of Core Geodynamics

To better understand geomagnetic theory and observation, we can use spatial magnetic spectra for the main field and secular variation to test core dynmcal hypotheses against seismology. The hypotheses lead to theoretical spectra which are fitted to observational spectra. Each fit yields an estimate of the radius of Earth's core and uncertainty. If this agrees with the seismologic value, then the hypothes pass the test. A new way to obtain theoretical spectra extends the hydromagnetic scale analysis of Benton to scale-variant field and flow. For narrow scale flow and a dynamically weak field by the top of Earth's core, this yields a generalized Stevenson-McLeod spectrum for the core-source field, and a secular variation spectrum modulated by a cubic polynomial in spherical harmonic degree n. The former passes the tests. The latter passes many tests, but does not describe rapid dipole decline and quadrupole rebound; some tests suggest it is a bit hard, or rich in narrow scale change. In a core geodynamo, motion of the fluid conductor does work against the Lorentz force. This converts kinetic into magnetic energy which, in turn, is lost to heat via Ohmic dissipation. In the analysis at lentgh-scale l/k, if one presumes kinetic energy is converted in either eddy- overturning or magnetic free-decay time-scales, then Kolmogorov or other spectra in conflict with observational spectra can result. Instead, the rate work is done roughly balances the dissipation rate, which is consistent with small scale flow. The conversion time-scale depends on dynamical constraints. These are summarized by the magneto-geostrophic vertical vorticity balance by the top of the core, which includes anisotropic effects of rotation, the magnetic field, and the core- mantle boundary. The resulting theoretical spectra for the core-source field and its SV are far more compatible with observation. The conversion time-scale of order l20 years is pseudo-scale-invarient. Magnetic spectra of other planets may differ; however, if a transition to non-conductmg fluid hydrogen in Jupiter acts as barrier to vertical flow, as well as current, then the shape of the jovi-magnetic spectrum could be remarkably Earth-like.

Voorhies, Coerte V.↗

NASA NTRS1976

On the question of the energy of the precessional dynamo

The various estimates for the coupling mechanism by which precession transfers rotational, kinetic energy of earth into the energy of its magnetic field are generally considering hydromagnetic stresses that originate between mantle and core. Estimates of the energy of the geomagnetic field calculated from the data of spherical harmonic analysis derive precession energy values in reasonable agreement with the observed external energy of the geomagnetic field and with the rate of ohmic dissipation of energy in the core.

Dolginov, S. S.↗

NASA NTRS1994

Radiation from advanced solid rocket motor plumes

The overall objective of this study was to develop an understanding of solid rocket motor (SRM) plumes in sufficient detail to accurately explain the majority of plume radiation test data. Improved flowfield and radiation analysis codes were developed to accurately and efficiently account for all the factors which effect radiation heating from rocket plumes. These codes were verified by comparing predicted plume behavior with measured NASA/MSFC ASRM test data. Upon conducting a thorough review of the current state-of-the-art of SRM plume flowfield and radiation prediction methodology and the pertinent data base, the following analyses were developed for future design use. The NOZZRAD code was developed for preliminary base heating design and Al2O3 particle optical property data evaluation using a generalized two-flux solution to the radiative transfer equation. The IDARAD code was developed for rapid evaluation of plume radiation effects using the spherical harmonics method of differential approximation to the radiative transfer equation. The FDNS CFD code with fully coupled Euler-Lagrange particle tracking was validated by comparison to predictions made with the industry standard RAMP code for SRM nozzle flowfield analysis. The FDNS code provides the ability to analyze not only rocket nozzle flow, but also axisymmetric and three-dimensional plume flowfields with state-of-the-art CFD methodology. Procedures for conducting meaningful thermo-vision camera studies were developed.

Farmer, Richard C.↗

NASA NTRS2004

Estimating Antarctic Near-Surface Magnetic Anomalies from Oersted and CHAMP Satellite Magnetometer Observations

Significant improvement in predicting near-surface magnetic anomalies can result from the highly accurate magnetic observations of the CHAMP satellite that is orbiting at about 400 km altitude. In general, regional magnetic signals of the crust are strongly masked by the core field and its secular variations due to wavelength coupling in the spherical harmonic representation and thus are difficult to isolate in the satellite measurements. However, efforts to isolate the regional lithospheric from core field components can exploit the correlations between the CHAMP magnetic anomalies and the pseudo magnetic effects inferred from gravity-derived crustal thickness variations. In addition, we can use spectral correlation theory to filter the static lithospheric field components from the dynamic external field effects. Employing these procedures, we processed the CHAMP magnetic conservations for an improved magnetic anomaly map of the Antarctic crust. Relative to the much higher altitude Oersted and noisier Magsat observations, CHAMP magnetic anomalies at 400 km altitude reveal new details on the effects of intra-crustal magnetic features and crustal thickness variations of the Antarctic. Moreover, these results greatly facilitate predicting magnetic anomalies in the regional coverage gaps of the ADMAP compilation of Antarctic magnetic anomalies from shipborne, airborne and ground surveys. Our analysis suggests that considerable new insights on the magnetic properties of the lithosphere may be revealed by a further order-of-magnitude improvement in the accuracy of the magnetometer.

vonFrese, Ralph R. B.↗

NASA NTRS1985

Hopf bifurcation in the presence of symmetry

Group theory is applied to obtain generalized differential equations from the Hopf bifurcation theory on branching to periodic solutions. The conditions under which the symmetry group will admit imaginary eigenvalues are delimited. The action of the symmetry group on the circle group are explored and the Liapunov-Schmidt reduction is used to prove the Hopf theorem in the symmetric case. The emphasis is on simplifying calculations of the stability of bifurcating branches. The resulting general theory is demonstrated in terms of O(2) acting on a plane, O(n) in n-space, and O(3) and an irreducible model for spherical harmonics.

Golubitsky, M.↗

NASA NTRS1991

Use of non-spherical gravity harmonics for relative motion GN/C (category 2)

Detailed analysis of the Automatic Rendezvous and Capture problem indicate a need for three different regions of mathematical description for the GN&C algorithms: (1) multi-vehicle orbital mechanics to the rendezvous interface point, i.e., within 100 nm; (2) relative motion solutions (such as Clohessy-Wiltshire type) from the far-field to the near-field interface, i.e., within 1 nm and; (3) close proximity motion - the near-field motion where the relative differences in the gravitational and orbit inertial accelerations can be neglected from the equations of motion. Limit boundaries to these regions can be precisely defined by further analysis and will be functions of the tracking measurement accuracies and the computer resources available for the solution of the algorithms. This paper analyzes the relative motion in Regions 2 and 3 above and present the derivation and discussion of the general case of non-spherical gravitational perturbed relative motion. Mathematical deviations from the numerically integrated spherical gravity case and solutions from the Clohessy-Wiltshire equations are presented in the analysis. Based upon this preliminary analysis, it is recommended that further efforts be used to assess the relative position and velocity differences in Region 2 due to non-spherical gravity harmonics and that viable GN&C algorithms be developed to include these gravity perturbations (especially the effects of the first gravity harmonic, J2).

Henderson, David M.↗

NASA NTRS2007

The Local Stellar Velocity Field via Vector Spherical Harmonics

kinematics↗

NASA NTRS2013

Global Propagation of Gravity Waves Generated with the Whole Atmosphere Transfer Function Model

A brief review is presented of the Transfer Function Model (TFM) [e.g., Mayr et al., Space Science Reviews, 1990], which describes acoustic gravity waves (AGW) that propagate across the globe in a dissipative and static (no winds) background atmosphere with globally uniform temperature and density variations extending from the ground to 700 km. Unique among existing models, the TFM can be placed between the analytical approach on one end, and the rigorous numerical approach of general circulation models (GCM). The time consuming numerical integration of the conservation equations is restricted to compute the transfer function (TF) for a broad range of frequencies and spherical harmonics. Given TF, the atmospheric response for a chosen source configuration is then obtained in short order. Computationally efficient, the model is well suited to serve as experimental and educational tool for simulating propagating wave patterns across the globe. By design, the TFM is also semi-analytical and therefore well suited to explore the different wave modes that can be generated under different dynamical conditions.

Transfer Function Model↗