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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 NTRS2007

The Local Stellar Velocity Field via Vector Spherical Harmonics

kinematics↗

NASA NTRS2010

VLBI2020: From Reality to Vision

The individual apparent motions of distant radio sources are believed to be caused by the effect of intrinsic structure variations of the active galactic nuclei (AGN). However, some cosmological models of the expanded Universe predict that systematic astrometric proper motions of distant quasars do not vanish as the radial distance from the observer to the quasar grows. These systematic effects can even increase with the distance, making it possible to measure them with high-precision astrometric techniques like VLBI. The Galactocentric acceleration of the Solar System barycenter may cause a secular aberration drift with a magnitude of 4 uas/yr. The Solar System motion relative to the cosmic microwave background produces an additional dipole effect, proportional to red shift. We analyzed geodetic VLBI data spanning from 1979 until 2009 to estimate the vector spherical harmonics in the expansion of the vector field of the proper motion of 687 radio sources. The dipole and quadrupole vector spherical harmonics were estimated with an accuracy of 1-5 as/yr. We have shown that over the next decade the geodetic VLBI may approach the level of accuracy on which the cosmological models of the Universe could be tested. Hence, it is important to organize a dedicated observational program to increase the number of measured proper motions to 3000.

Titov, Oleg↗

NASA NTRS1967

Expansions in spherical harmonics. IV.

Vector functions expansions in spherical harmonics of directions of individual vectors

Sack, R. A.↗

DOE OSTI2023

A minimum assumption approach to MEG sensor array design

Objective. Our objective is to formulate the problem of the magnetoencephalographic (MEG) sensor array design as a well-posed engineering problem of accurately measuring the neuronal magnetic fields. This is in contrast to the traditional approach that formulates the sensor array design problem in terms of neurobiological interpretability the sensor array measurements. Approach. We use the vector spherical harmonics (VSH) formalism to define a figure-of-merit for an MEG sensor array. We start with an observation that, under certain reasonable assumptions, any array of m perfectly noiseless sensors will attain exactly the same performance, regardless of the sensors' locations and orientations (with the exception of a negligible set of singularly bad sensor configurations). We proceed to the conclusion that under the aforementioned assumptions, the only difference between different array configurations is the effect of (sensor) noise on their performance. We then propose a figure-of-merit that quantifies, with a single number, how much the sensor array in question amplifies the sensor noise. Main results. We derive a formula for intuitively meaningful, yet mathematically rigorous figure-of-merit that summarizes how desirable a particular sensor array design is. We demonstrate that this figure-of-merit is well-behaved enough to be used as a cost function for a general-purpose nonlinear optimization methods such as simulated annealing. We also show that sensor array configurations obtained by such optimizations exhibit properties that are typically expected of 'high-quality' MEG sensor arrays, e.g. high channel information capacity. Significance. Our work paves the way toward designing better MEG sensor arrays by isolating the engineering problem of measuring the neuromagnetic fields out of the bigger problem of studying brain function through neuromagnetic measurements.

60 APPLIED LIFE SCIENCES↗

NASA NTRS1987

A computer model of global thermospheric winds and temperatures

Output data from the NCAR Thermospheric GCM and a vector-spherical-harmonic (VSH) representation of the wind field are used in constructing a computer model of time-dependent global horizontal vector neutral wind and temperature fields at altitude 130-300 km. The formulation of the VSH model is explained in detail, and some typical results obtained with a preliminary version (applicable to December solstice at solar maximum) are presented graphically. Good agreement with DE-2 satellite measurements is demonstrated.

Killeen, T. L.↗

NASA NTRS1991

A unified approach to the helioseismic forward and inverse problems of differential rotation

A general, degenerate perturbation theoretic treatment of the helioseismic forward and inverse problem for solar differential rotation is presented. For the forward problem, differential rotation is represented as the axisymmetric component of a general toroidal flow field using velocity spherical harmonics. This approach allows each degree of differential rotation to be estimated independently from all other degrees. In the inverse problem, the splitting caused by differential rotation is expressed as an expansion in a set of orthonormal polynomials that are intimately related to the solution of the forward problem. The combined use of vector spherical harmonics as basis functions for differential ratio and the Clebsch-Gordon coefficients to represent splitting provides a unified approach to the forward and inverse problems of differential rotation which greatly simplify inversion.

Ritzwoller, Michael H.↗

NASA NTRS1976

Representation of magnetic fields in space

Several mathematical methods which are available for the description of magnetic fields in space are reviewed. Examples of the application of such methods are given, with particular emphasis on work related to the geomagnetic field, and their individual properties and associated problems are described. The methods are grouped in five main classes: (1) methods based on the current density, (2) methods using the scalar magnetic potential, (3) toroidal and poloidal components of the field and spherical vector harmonics, (4) Euler potentials, and (5) local expansions of the field near a given reference point. Special attention is devoted to models of the magnetosphere, to the uniqueness of the scalar potential as derived from observed data, and to the L parameter.

Stern, D. P.↗

NASA NTRS1986

Transfer function analysis of thermospheric perturbations

Applying perturbation theory, a spectral model in terms of vectors spherical harmonics (Legendre polynomials) is used to describe the short term thermospheric perturbations originating in the auroral regions. The source may be Joule heating, particle precipitation or ExB ion drift-momentum coupling. A multiconstituent atmosphere is considered, allowing for the collisional momentum exchange between species including Ar, O2, N2, O, He and H. The coupled equations of energy, mass and momentum conservation are solved simultaneously for the major species N2 and O. Applying homogeneous boundary conditions, the integration is carred out from the Earth's surface up to 700 km. In the analysis, the spherical harmonics are treated as eigenfunctions, assuming that the Earth's rotation (and prevailing circulation) do not significantly affect perturbations with periods which are typically much less than one day. Under these simplifying assumptions, and given a particular source distribution in the vertical, a two dimensional transfer function is constructed to describe the three dimensional response of the atmosphere. In the order of increasing horizontal wave numbers (order of polynomials), this transfer function reveals five components. To compile the transfer function, the numerical computations are very time consuming (about 100 hours on a VAX for one particular vertical source distribution). However, given the transfer function, the atmospheric response in space and time (using Fourier integral representation) can be constructed with a few seconds of a central processing unit. This model is applied in a case study of wind and temperature measurements on the Dynamics Explorer B, which show features characteristic of a ringlike excitation source in the auroral oval. The data can be interpreted as gravity waves which are focused (and amplified) in the polar region and then are reflected to propagate toward lower latitudes.

Mayr, H. G.↗

NASA NTRS1987

A full simulation of a vortex ring

A three-dimensional spectral method is developed for the solution to the incompressible Navier-Stokes equations in an unbounded domain. The spectral method relies on divergence-free basis functions as proposed by Leonard (1981). The basis functions are formed using vector spherical harmonics and Jacobi polynomials together with a mapping in the radial direction. An axisymmetric code was written and is verified using an exact solution of the Stokes equations. Preliminary results for the evolution of a vortex ring according to the Navier-Stokes equations are presented.

Stanaway, S. K.↗

NASA NTRS1988

Empirical global model of upper thermosphere winds based on atmosphere and dynamics explorer satellite data

Thermospheric wind data obtained from the Atmosphere Explorer E and Dynamics Explorer 2 satellites have been used to generate an empirical wind model for the upper thermosphere, analogous to the MSIS model for temperature and density, using a limited set of vector spherical harmonics. The model is limited to above approximately 220 km where the data coverage is best and wind variations with height are reduced by viscosity. The data base is not adequate to detect solar cycle (F10.7) effects at this time but does include magnetic activity effects. Mid- and low-latitude data are reproduced quite well by the model and compare favorably with published ground-based results. The polar vortices are present, but not to full detail.

Hedin, A. E.↗

NASA NTRS1988

A numerical study of viscous vortex rings using a spectral method

Viscous, axisymmetric vortex rings are investigated numerically by solving the incompressible Navier-Stokes equations using a spectral method designed for this type of flow. The results presented are axisymmetric, but the method is developed to be naturally extended to three dimensions. The spectral method relies on divergence-free basis functions. The basis functions are formed in spherical coordinates using Vector Spherical Harmonics in the angular directions, and Jacobi polynomials together with a mapping in the radial direction. Simulations are performed of a single ring over a wide range of Reynolds numbers (Re approximately equal gamma/nu), 0.001 less than or equal to 1000, and of two interacting rings. At large times, regardless of the early history of the vortex ring, it is observed that the flow approaches a Stokes solution that depends only on the total hydrodynamic impulse, which is conserved for all time. At small times, from an infinitely thin ring, the propagation speeds of vortex rings of varying Re are computed and comparisons are made with the asymptotic theory by Saffman. The results are in agreement with the theory; furthermore, the error is found to be smaller than Saffman's own estimate by a factor square root ((nu x t)/R squared) (at least for Re=0). The error also decreases with increasing Re at fixed core-to-ring radius ratio, and appears to be independent of Re as Re approaches infinity). Following a single ring, with Re=500, the vorticity contours indicate shedding of vorticity into the wake and a settling of an initially circular core to a more elliptical shape, similar to Norbury's steady inviscid vortices. Finally, we consider the case of leapfrogging vortex rings with Re=1000. The results show severe straining of the inner vortex core in the first pass and merging of the two cores during the second pass.

Stanaway, S. K.↗

NASA NTRS1991

On the dynamics and composition of the high-latitude thermosphere

An overview is presented of some of the more recent progress made in the study of the dynamics and composition of the high-latitude thermosphere at the University of Michigan, with emphasis on the interpretation of experimental measurements made from the DE-2 spacecraft and from ground-based observatories in Greenland. Attention is focused on perturbations to the solar-driven upper thermospheric winds, temperatures, and compositional structures due to forcing associated with high-latitude ionosphere-thermosphere coupling processes. The nature of the experimental measurements made on DE-2 is discussed. The vector spherical harmonic model, some sample results, and experiment-model comparisons are described using both satellite and ground-based measurements of high-latitude thermosphere dynamics. Considerations pertaining to compositional perturbations at high latitudes are presented.

Killeen, T. L.↗

NASA NTRS1995

Modeling of the coupled magnetospheric and neutral wind dynamos

Work at SRI involved modeling the exchange of electromagnetic energy between the ionosphere and magnetosphere to help interpret the DE-B Poynting flux observations. To describe the electrical properties of the high-latitude ionosphere, we constructed a numerical model, from the framework provided by the Vector Spherical Harmonic (VSH) model, that determines the ionospheric currents, conductivities, and electric fields including both magnetospheric inputs and neutral wind dynamo effects. This model development grew from the earlier question of whether an electrical energy source in the ionosphere was capable of providing an upward Poynting flux. The model solves the steady-state neutral wind dynamo equations and the Poynting flux equation to provide insight into the electrodynamic role of the neutral winds. The modeling effort to determine the high-latitude energy flux has been able to reproduce many of the large-scale features observed in the Poynting flux measurements made by DE-2. Because the Poynting flux measurement is an integrated result of energy flux into or out of the ionosphere, we investigated the ionospheric properties that may contribute to the observed flux of energy measured by the spacecraft. During steady state the electromagnetic energy flux, or DC Poynting flux, is equal to the Joule heating rate and the mechanical energy transfer rate in the high-latitude ionosphere. Although the Joule heating rate acts as an energy sink, transforming electromagnetic energy into thermal or internal energy of the gas, the mechanical energy transfer rate may be either a sink or source of electromagnetic energy. In the steady state, it is only the mechanical energy transfer rate that can generate electromagnetic energy and result in a DC Poynating flux that is directed out of the ionosphere.

Thayer, J. P.↗

NASA NTRS1995

Upper Thermosphere Winds and Temperatures in the Geomagnetic Polar Cap: Solar Cycle, Geomagnetic Activity, and Interplanetary Magnetic Field Dependencies

Ground-based Fabry-Perot interferometers located at Thule, Greenland (76.5 deg. N, 69.0 deg. W, lambda = 86 deg.) and at Sondre Stromfjord, Greenland (67.0 deg. N, 50.9 deg. W, lambda = 74 deg.) have monitored the upper thermospheric (approx. 240-km altitude) neutral wind and temperature over the northern hemisphere geomagnetic polar cap since 1983 and 1985, respectively. The thermospheric observations are obtained by determining the Doppler characteristics of the (OI) 15,867-K (630.0-nm) emission of atomic oxygen. The instruments operate on a routine, automatic, (mostly) untended basis during the winter observing seasons, with data coverage limited only by cloud cover and (occasional) instrument failures. This unique database of geomagnetic polar cap measurements now extends over the complete range of solar activity. We present an analysis of the measurements made between 1985 (near solar minimum) and 1991 (near solar maximum), as part of a long-term study of geomagnetic polar cap thermospheric climatology. The measurements from a total of 902 nights of observations are compared with the predictions of two semiempirical models: the Vector Spherical Harmonic (VSH) model of Killeen et al. (1987) and the Horizontal Wind Model (HWM) of Hedin et al. (1991). The results are also analyzed using calculations of thermospheric momentum forcing terms from the Thermosphere-ionosphere General Circulation Model TGCM) of the National Center for Atmospheric Research (NCAR). The experimental results show that upper thermospheric winds in the geomagnetic polar cap have a fundamental diurnal character, with typical wind speeds of about 200 m/s at solar minimum, rising to up to about 800 m/s at solar maximum, depending on geomagnetic activity level. These winds generally blow in the antisunward direction, but are interrupted by episodes of modified wind velocity and altered direction often associated with changes in the orientation of the Interplanetary Magnetic Field (IMF). The central polar cap (greater than approx. 80 magnetic latitude) antisunward wind speed is found to be a strong function of both solar and geomagnetic activity. The polar cap temperatures show variations in both solar and geomagnetic activity, with temperatures near 800 K for low K(sub p) and F(sub 10.7) and greater than about 2000 K for high K(sub p) and F(sub 10.7). The observed temperatures are significantly greater than those predicted by the mass spectrometer/incoherent scatter model for high activity conditions. Theoretical analysis based on the NCAR TIGCM indicates that the antisunward upper thermospheric winds, driven by upstream ion drag, basically 'coast' across the polar cap. The relatively small changes in wind velocity and direction within the polar cap are induced by a combination of forcing terms of commensurate magnitude, including the nonlinear advection term, the Coriolis term, and the pressure gradient force term. The polar cap thennospheric thermal balance is dominated by horizontal advection, and adiabatic and thermal conduction terms.

Killeen, T. L.↗

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 NTRS1996

Continuing Development of a Hybrid Model (VSH) of the Neutral Thermosphere

We propose to continue the development of a new operational model of neutral thermospheric density, composition, temperatures and winds to improve current engineering environment definitions of the neutral thermosphere. This model will be based on simulations made with the National Center for Atmospheric Research (NCAR) Thermosphere-Ionosphere- Electrodynamic General Circulation Model (TIEGCM) and on empirical data. It will be capable of using real-time geophysical indices or data from ground-based and satellite inputs and provides neutral variables at specified locations and times. This "hybrid" model will be based on a Vector Spherical Harmonic (VSH) analysis technique developed (over the last 8 years) at the University of Michigan that permits the incorporation of the TIGCM outputs and data into the model. The VSH model will be a more accurate version of existing models of the neutral thermospheric, and will thus improve density specification for satellites flying in low Earth orbit (LEO).

Burns, Alan↗

NASA NTRS2011

What Supergranule Flow Models Tell Us About the Sun's Surface Shear Layer and Magnetic Flux Transport

Models of the photospheric flows due to supergranulation are generated using an evolving spectrum of vector spherical harmonics up to spherical harmonic wavenumber l~1500. Doppler velocity data generated from these models are compared to direct Doppler observations from SOHO/MDI and SDO/HMI. The models are adjusted to match the observed spatial power spectrum as well as the wavenumber dependence of the cell lifetimes, differential rotation velocities, meridional flow velocities, and relative strength of radial vs. horizontal flows. The equatorial rotation rate as a function of wavelength matches the rotation rate as a function of depth as determined by global helioseismology. This leads to the conclusions that the cellular structures are anchored at depths equal to their widths, that the surface shear layer extends to at least 70 degrees latitude, and that the poleward meridional flow decreases in amplitude and reverses direction at the base of the surface shear layer (approx.35 Mm below the surface). Using the modeled flows to passively transport magnetic flux indicates that the observed differential rotation and meridional flow of the magnetic elements are directly related to the differential rotation and meridional flow of the convective pattern itself. The magnetic elements are transported by the evolving boundaries of the supergranule pattern (where the convective flows converge) and are unaffected by the weaker flows associated with the differential rotation or meridional flow of the photospheric plasma.

Hathaway, David↗