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Given a magnetic field B = -del(vector differential operator)(sub gamma) with gamma expanded in spherical harmonics, it is shown that analytic Euler potentials may be derived for B if gamma is asymmetrical but contains only the contribution of a single index n. This work generalizes a result for sectorial harmonics with n = m, derived by Willis and Gardiner (1988).
Full disk Doppler observations of solar oscillations were used to compare the amplitudes of sectoral modes propagating along the equator with those of similar modes propagating along a great circle aligned with the poles. It is found that the amplitudes are generally not equal for the two classes of modes, but the results are sensitive to analysis procedures attempting to isolate the different modes of oscillation. Spatial filtering of the data using spherical harmonics suggests that greater amplitudes are associated with 'polar' sectoral modes than with 'equatorial' sectoral modes.
The paper reports on a general circulation model being developed with the ultimate objective of assessing the effect of SST operations on the atmospheric ozone distribution. The model variables are represented in the spectral domain using 79 spherical harmonics and 26 vertical levels between the ground and 70 km. Initial calculations have attempted to simulate the seasonal variations of the unperturbed global ozone distribution. The model shows the presence of a westerly stratospheric jet in the winter hemisphere with amplitudes similar to those observed in the polar night jet. Easterlies dominate in the summer hemisphere. The model also shows ozone to be transported polewards and downwards. Furthermore the annual cycle in ozone concentrations at mid-latitudes has been simulated. Preliminary comparisons are also made with other features of the dynamics and the ozone distribution in the stratosphere.
The TOPEX/POSEIDON (T/P) prelaunch Joint Gravity Model-1 (JGM-1) and the postlaunch JGM-2 Earth gravitational models have been developed to support precision orbit determination for T/P. Each of these models is complete to degree 70 in spherical harmonics and was computed from a combination of satellite tracking data, satellite altimetry, and surface gravimetry. While improved orbit determination accuracies for T/P have driven the improvements in the models, the models are general in application and also provide an improved geoid for oceanographic computations. The postlaunch model, JGM-2, which includes T/P satellite laser ranging (SLR) and Doppler orbitography and radiopositioning integrated by satellite (DORIS) tracking data, introduces radial orbit errors for T/P that are only 2 cm RMS with the commission errors of the marine geoid for terms to degree 70 being +/- 25 cm. Errors in modeling the nonconservative forces acting on T/P increase the total radial errors to only 3-4 cm root mean square (RMS), a result much better than premission goals. While the orbit accuracy goal for T/P has been far surpassed geoid errors still prevent the absolute determination of the ocean dynamic topography for wavelengths shorter than about 2500 km. Only a dedicated gravitational field satellite mission will likely provide the necessary improvement in the geoid.
This report describes a new method for determination of the geopotential, or the equivalent geoid. It is based on Satellite-to-Satellite Tracking (SST) of two co-orbiting low earth satellites separated by a few hundred kilometers. The analysis is aimed at the GRACE Mission, though it is generally applicable to any SST data. It is proposed that the SST be viewed as a mapping mission. That is, the result will be maps of the geoid or gravity, as contrasted with determination of spherical harmonics or Fourier coefficients. A method has been developed, based on Geophysical Inverse Theory (GIT), that can provide maps at a prescribed (desired) resolution and the corresponding error map from the SST data. This computation can be done area by area avoiding simultaneous recovery of all the geopotential information. The necessary elements of potential theory, celestial mechanics, and Geophysical Inverse Theory are described, a computation architecture is described, and the results of several simulations presented. Centimeter accuracy geoids with 50 to 100 km resolution can be recovered with a 30 to 60 day mission.
Conventional gravity and magnetic anomaly continuation invokes the standard Poisson boundary condition of a zero anomaly at an infinite vertical distance from the observation surface. This simple continuation is limited, however, where multiple altitude slices of the anomaly field have been observed. Increasingly, areas are becoming available constrained by multiple boundary conditions from surface, airborne, and satellite surveys. This paper describes the implementation of continuation with multi-altitude boundary conditions in Cartesian and spherical coordinates and investigates the advantages and limitations of these applications. Continuations by EPS (Equivalent Point Source) inversion and the FT (Fourier Transform), as well as by SCHA (Spherical Cap Harmonic Analysis) are considered. These methods were selected because they are especially well suited for analyzing multi-altitude data over finite patches of the earth such as covered by the ADMAP database. In general, continuations constrained by multi-altitude data surfaces are invariably superior to those constrained by a single altitude data surface due to anomaly measurement errors and the non-uniqueness of continuation.
Conventional gravity and magnetic anomaly continuation invokes the standard Poisson boundary condition of a zero anomaly at an infinite vertical distance from the observation surface. This simple continuation is limited, however, where multiple altitude slices of the anomaly field have been observed. Increasingly, areas are becoming available constrained by multiple boundary conditions from surface, airborne, and satellite surveys. This paper describes the implementation of continuation with multi-altitude boundary conditions in Cartesian and spherical coordinates and investigates the advantages and limitations of these applications. Continuations by EPS (Equivalent Point Source) inversion and the FT (Fourier Transform), as well as by SCHA (Spherical Cap Harmonic Analysis) are considered. These methods were selected because they are especially well suited for analyzing multi-altitude data over finite patches of the earth such as covered by the ADMAP database. In general, continuations constrained by multi-altitude data surfaces are invariably superior to those constrained by a single altitude data surface due to anomaly measurement errors and the non-uniqueness of continuation.
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.
To mathematically model the oscillations of a spherical bubble, either the time-dependent volume V(t) or the radius R(t) can be used as the dependent variable. A recent exchange between the advocates of the volume approach and the radius approach concerns in apparent discrepancy in the predicted nonlinear acoustic emissions from a bubble subject to a time-periodic external pressure field. This letter attempts to resolve the discrepancy via an examination of the arguments offered by both sides. In addition, some general comments on nonlinear harmonic and subharmonic emissions from acoustically forced bubble oscillations are presented.
Analytic expressions are derived to model the surface topography and the normal to the surface of Phobos. The analytic expressions are comprised of a spherical harmonic expansion for the global figure of Phobos, augmented by addition terms for the large crater Stickney and other craters. Over 300 craters were measured in more than 100 Viking Orbiter images to produce the model. In general, the largest craters were measured since they have a significant effect on topography. The topographic model derived has a global spatial and topographic accuracy ranging from about 100 m in areas having the highest resolution and convergent, stereo coverage, up to 500 m in the poorest areas.
The HWM90 thermospheric wind model was revised in the lower thermosphere and extended into the mesosphere and lower atmosphere to provide a single analytic model for calculating zonal and meridional wind profiles representative of the climatological average for various geophysical conditions. Gradient winds from CIRA-86 plus rocket soundings, incoherent scatter radar, MF radar, and meteor radar provide the data base and are supplemented by previous data driven model summaries. Low-order spherical harmonics and Fourier series are used to describe the major variations throughout the atmosphere including latitude, annual, semiannual, and longitude (stationary wave 1). The model represents a smoothed compromise between the data sources. Although agreement between various data sources is generally good, some systematic differences are noted, particularly near the mesopause. Root mean square differences between data and model are on the order of 15 m/s in the mesosphere and 10 m/s in the stratosphere for zonal wind, and 10 m/s and 4 m/s, respectively, for meridional wind.
The HWM90 thermospheric wind model was revised in the lower thermosphere and extended into the mesosphere and lower atmosphere to provide a single analytic model for calculating zonal and meridional wind profiles representative of the climatological average for various geophysical conditions. Local time variations in the mesosphere are derived from rocket soundings, incoherent scatter radar, MF radar, and meteor radar. Low-order spherical harmonics and Fourier series are used to describe these variations as a function of latitude and day of year with cubic spline interpolation in altitude. The model represents a smoothed compromise between the original data sources. Although agreement between various data sources is generally good, some systematic differences are noted. Overall root mean square differences between measured and model tidal components are on the order of 5 to 10 m/s.
The purpose of this paper is to present a general model for the acceleration exerted on a spacecraft by the radiation coming from a planet. Both the solar radiation reflected by the planet and the thermal emission associated with its temperature are considered. The planet albedo and the planet emissive power are expanded in spherical harmonics with respect to an equatorial reference frame attached to the planet. The satellite external surface is assumed to consist of a juxtaposition of planar surfaces. A particular choice of variables allows to reduce the surface integrals over the lit portion of the planet visible to the satellite to one-dimension integrals.
We study a novel characterization of errors for numerical weather predictions. In its simplest form we decompose the error into a part attributable to phase errors and a remainder. The phase error is represented in the same fashion as a velocity field and will be required to vary slowly and smoothly with position. A general distortion representation allows for the displacement and a bias correction of forecast anomalies. In brief, the distortion is determined by minimizing the objective function by varying the displacement and bias correction fields. In the present project we use a global or hemispheric domain, and spherical harmonics to represent these fields. In this project we are initially focusing on the assessment application, restricted to a realistic but univariate 2-dimensional situation. Specifically we study the forecast errors of the 500 hPa geopotential height field for forecasts of the short and medium range. The forecasts are those of the Goddard Earth Observing System data assimilation system. Results presented show that the methodology works, that a large part of the total error may be explained by a distortion limited to triangular truncation at wavenumber 10, and that the remaining residual error contains mostly small spatial scales.
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).
The orbit accuracies needed to support the global crustal dynamics project and recent satellite altimeter missions have placed unique demands on the data analysis and orbit analysis systems. These demands include accurate and well distributed observations, improved computational techniques and substantial enhancements in the force models which represent the satellite's motion. For example, the satellite altimeter mission (TOPEX), whose objectives will be: (1) to measure the time variable ocean surface topography, and (2) to demonstrate the ability to map the general ocean circulation, requires that the radial component of the satellite's orbit be known with an rms accuracy of 13 cm for the three year mission lifetime. The primary force model uncertainty which limits the contemporary orbit computation accuracy is the inaccuracy in the values assigned to the spherical harmonic coefficients used to model the Earth's gravity field.
The spherical harmonic equation for the gravitational potential inside the earth is presented. The equation satisfies Poisson's equation and converges uniformly. It obviates the need for downward continuation of the exterior potential with its attendant convergence difficulties but of course requires some knowledge of the earth's density distribution. The equation is used to derive the general expression for the geophysical measurement of the gravitational constant G made inside the earth, such as in boreholes and mine shafts. Numerical evidence is also presented to show that the long- to intermediate-wavelength gravity anomalies can masquerade as the 'fifth force' if not properly corrected for.
A generalized inverse technique is applied to retrieve two types of balanced winds that characterize the large-scale dynamics of the atmosphere: rotational winds based upon the linear balance equation, and divergent winds based upon the vorticity budget equation. Both balance equations are singular at or near the equator. The balance equations are transformed in spherical harmonic function space to an under-determined system, for which the scale-weighed least-squares solution consists of a sum of principal and singular components. The principal components represent the response to the source function for the regular eigenmodes, while the singular components are determined by the projection of an independent measurement on the singular eigenmodes. The method was tested with the NCEP/NCAR reanalysis data in which a quasi-balance condition exists. A realistic balanced wind field is retrievable when the singular components are computed based upon the reanalyzed wind data.