Time harmonic, spherical harmonic, and power series expansion of the Boltzmann equation.
Time harmonic, spherical harmonic and power series expansion of Boltzmann equation
Morrone, T.↗
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Quantification of Global Cloud Properties with Use of Spherical Harmonic Functions
Spherical harmonic (SH) expansion is a useful tool to study any variable that has valid values at all latitudes and longitudes. The variable can be quantified as a sum of different spherical harmonic components, which are the spherical harmonic functions multiplied by their expansion coefficients. We find that the SH components of cloud radiative effect (CRE) have correlations with El Niño-Southern Oscillation (ENSO) and the Hadley Circulation (HC). In particular, the expansion degree 2 (l = 2) SH power spectrum component anomaly of CRE is strongly correlated with ENSO. The two dipole patterns appearing in the l = 2 SH component anomaly map can be reasonably explained by a known mechanism of ENSO’s impact on cloud properties. The l = 3 and l = 5 SH power spectrum components are correlated with HC intensity, whereas the l = 6 and l = 8 components are correlated with HC latitudinal widths. In ENSO warm and cold phases, the HC-correlated SH components have opposite anomalies, which suggests the impact of ENSO on HC. This study illustrates that the SH expansion technique provides a different perspective to study the impacts of large-scale atmospheric circulation on global cloud properties and radiative effects.
clouds↗
Gravitational potential energy of the earth: A spherical harmonic approach
A spherical harmonic equation for the gravitational potential energy of the earth is derived for an arbitrary density distribution by conceptually bringing in mass-elements from infinity and building up the earth shell upon spherical shell. The zeroth degree term in the spherical harmonic equation agrees with the usual expression for the energy of a radial density distribution. The second degree terms give a maximum nonhydrostatic energy in the mantle and crust of -2.77 x 10 to the twenty-ninth power ergs, an order of magnitude. If the earth is assumed to be a homogeneous viscous oblate spheroid relaxing to an equilibrium shape, then a lower limit to the mantle viscosity of 1.3 x 10 to the twentieth power poises is found by assuming the total geothermal flux is due to viscous dissipation. If the nonequilibrium figure is dynamically maintained by the earth acting as a heat engine at one per cent efficiency, then the viscosity is ten to the twenty second power poises, a number preferred by some as the viscosity of the mantle.
Rubincam, D. P.↗
Gravitational potential energy of the earth - A spherical harmonic approach
A spherical harmonic equation for the gravitational potential energy of the earth is derived for an arbitrary density distribution by conceptually bringing in mass-elements from infinity and building up the earth shell upon spherical shell. The zeroth degree term in the spherical harmonic expansion agrees with the usual expression for the energy of a radial density distribution. The second degree terms give a maximum nonhydrostatic energy in the crust and mantle of -2.77 x 10 to the 29th ergs, an order of magnitude below McKenzie's (1966) estimate. McKenzie's result stems from mathematical error. Our figure is almost identical with Kaula's (1963) estimate of the minimum shear strain energy in the mantle, a not unexpected result on the basis of the virial theorem. If the earth is assumed to be a homogeneous viscous oblate spheroid relaxing to an equilibrium shape, then a lower limit to the mantle viscosity of 1.3 x 10 to the 20th P is found by assuming that the total geothermal flux is due to viscous dissipation of energy. This number is almost six orders of magnitude below MacDonald's (1966) estimate of the viscosity and removes his objection to convection. If the nonequilibrium figure is dynamically maintained by the earth acting as a heat engine at 1% efficiency, then the viscosity is 10 to the 22nd P, a number preferred by Cathles (1975) and Peltier and Andrew (1976) as the viscosity of the mantle.
Rubincam, D. P.↗
Gravity Degree-Depth Relationship Using Point Mass Spherical Harmonics
Relationships between the degree of a spherical harmonic model of the gravitational field of a body and the depth of a source expressed as a density contrast can be used to study the structure of features. Here, we show that the gravitational acceleration per spherical harmonic degree of a constant density source has an extremum that depends on the depth of the source. Using the spherical harmonics expansion for a point mass source, we use this to derive a degree-depth relationship. Our relationship resembles an earlier one derived by Bowin (1983), with substantial differences at the lower degrees. We also find that a recent relationship derived by Deng et al. (2022) over-estimates the source depth. The relationship that we derive relates spherical harmonic degree n to depth d for a planet of radius R according to d = (1 - e -1/n+1 )R, which simplifies to d = R/(n + 1) for high degrees. We support our new relationship with synthetic models of a density contrast in a planet. We also show how the differences between our relationship and that of Bowin (1983) affect band-filtered gravity, for example when inspecting the upper 100 km of the Moon. Using point masses in our modeling results in an approximate relationship where in reality sources can be deeper than estimated, since any source contributes to all spherical harmonic degrees. The use of the contribution per individual degree however provides an intuitive relationship between spherical harmonic degree and depth that can be used to place relative bounds on source depths or to determine the bounds on spherical harmonic expansions when band-filtering gravity field models.
Geopotential theory↗
Spherical Harmonic Analysis of Particle Velocity Distribution Function: Comparison of Moments and Anisotropies using Cluster Data
This paper presents a spherical harmonic analysis of the plasma velocity distribution function using high-angular, energy, and time resolution Cluster data obtained from the PEACE spectrometer instrument to demonstrate how this analysis models the particle distribution function and its moments and anisotropies. The results show that spherical harmonic analysis produced a robust physical representation model of the velocity distribution function, resolving the main features of the measured distributions. From the spherical harmonic analysis, a minimum set of nine spectral coefficients was obtained from which the moment (up to the heat flux), anisotropy, and asymmetry calculations of the velocity distribution function were obtained. The spherical harmonic method provides a potentially effective "compression" technique that can be easily carried out onboard a spacecraft to determine the moments and anisotropies of the particle velocity distribution function for any species. These calculations were implemented using three different approaches, namely, the standard traditional integration, the spherical harmonic (SPH) spectral coefficients integration, and the singular value decomposition (SVD) on the spherical harmonic methods. A comparison among the various methods shows that both SPH and SVD approaches provide remarkable agreement with the standard moment integration method.
Gurgiolo, Chris↗
Contributions of Spherical Harmonics to Gravitational Moment
A scalar gravitational potential function expressed as a series of spherical harmonics frequently serves as the basis for a model of an astronomical body's gravitational field. The contribution of a generic spherical harmonic to gravitational gradient is expressed as a dyadic, which is then used to obtain an analytical expression in vector-dyadic form for the contribution to the moment of gravitational forces about the mass center of a small body such as a spacecraft. The expression developed for a harmonic's contribution to gravitational gradient can be applied in areas beyond the scope of the paper; for example, gravitational gradient plays an important role in the state propagation matrix and the state transition matrix that are used in spacecraft trajectory targeting and Kalman filtering. Additionally, it can be employed in numerical simulations of orbit determination based on measurements obtained with a gradiometer in low-Earth orbit. Contributions of spherical harmonics to gravitational moment may be of interest in connection with attitude control of a spacecraft in the vicinity of a body with an irregular shape, such as an asteroid. Normalized spherical harmonic coefficients up to degree and order 10 are obtained for the asteroid 216 Kleopatra and used in numerical evaluations of contributions to gravitational moment.
Roithmayr, Carlos M.↗
GRACE Fast Mascons from Spherical Harmonics and a Regularization Design Trade Study
Mass concentration (mascon) solutions have become a prominent medium for investigating time-variable gravity recovered by GRACE and GRACE Follow-On. While GRACE Level-2 spherical harmonic products require various post-processing techniques to eliminate correlated noise, mascon formulations employ spatial regularization strategies during the estimation step to improve signal recovery. However, mascon estimation has traditionally required large computing resources and GRACE Level-1B processing capabilities. In this study, we show that a typical mascon estimation system can be reformulated to allow for the estimation of regularized mascons from Level-2 spherical harmonics. Provided that spherical harmonic solution covariances are available, the computed mascons will be mathematically equivalent to similar mascons estimated from Level-1B observations. This method is computationally efficient, better leverages GRACE spherical harmonics than past methods, and matches the performance of typical mascon solutions without locking scientists into pre-determined regularization designs. We develop a proof-of-concept solution using ITSG-Grace2018 and compare results with traditional mascons from JPL and NASA GSFC. We then assess the effects of spherical harmonic truncation and use of regularization correlations on basin signal recovery. We find that spherical expansions to degree and order 60 provide the minimum expansion necessary to study most basins, while larger expansions help further localize signals. We also find that diagonal regularizations (i.e., regularizations that do not contain inter-mascon correlations) are adversely affected by leakage, especially across boundaries such as coastlines where signals are not highly correlated, whereas including inter-mascon correlations and regional boundaries in the regularization greatly improves signal recovery.
M. J. Croteau↗
Perturbation of the Geomagnetic Field - a Spherical Harmonic Expansion
Perturbation of geomagnetic field - spherical harmonic expansion
SPHERICAL HARMONICS↗
A problem in representing the core magnetic field of the earth using spherical harmonics
Although there are computational advantages to the representation of the earth's magnetic field by spherical harmonic coefficients of the magnetic potential, up to the thirteenth degree and order, the following disadvantages emerge: (1) the use of spherical harmonics of up to a certain degree does not remove wavelengths greater than a certain value from the surface fields, and (2) the total field magnitudes represented by spherical harmonics up to a certain degree have minimum wavelengths equal to the circumference of the earth divided by twice the maximum degree of the harmonic used. The implications of the ways in which surface fields are separated into core and crustal components are discussed, and it is concluded that since field signals are generated in the core, the representation of the core field by spherical harmonics of potential does not adequately represent all core field components.
Carle, H. M.↗
Expansion of the fokker-planck equation in spherical harmonics
Fokker-Planck equation expansion in spherical harmonics
SPHERICAL HARMONICS↗
Orthogonality of spherical harmonic coefficients
Orthogonality relations are obtained for the spherical harmonic coefficients of functions defined on the surface of a sphere. Following a brief discussion of the orthogonality of Fourier series coefficients, consideration is given to the values averaged over all orientations of the coordinate system of the spherical harmonic coefficients of a function defined on the surface of a sphere that can be expressed in terms of Legendre polynomials for the special case where the function is the sum of two delta functions located at two different points on the sphere, and for the case of an essentially arbitrary function. It is noted that the orthogonality relations derived have found applications in statistical studies of the geomagnetic field.
Mcleod, M. G.↗
Spherical harmonic analysis of a synoptic climatology generated with a global general circulation model
Spherical harmonic analysis was used to analyze the observed climatological (C) fields of temperature at 850 mb, geopotential height at 500 mb, and sea level pressure. The spherical harmonic method was also applied to the corresponding "model climatological" fields (M) generated by a general circulation model, the "GISS climate model." The climate model was initialized with observed data for the first of December 1976 at 00. GMT and allowed to generate five years of meteorological history. Monthly means of the above fields for the five years were computed and subjected to spherical harmonic analysis. It was found from the comparison of the spectral components of both sets, M and C, that the climate model generated reasonable 500 mb geopotential heights. The model temperature field at 850 mb exhibited a generally correct structure. However, the meridional temperature gradient was overestimated and overheating of the continents was observed in summer.
Christidis, Z. D.↗
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↗
Spherical harmonic analyses for the spheroidal earth, ii
Spherical harmonic analysis of earth magnetic field
EARTH SHAPE↗
Geomagnetic main field analysis at the core-mantle boundary - Spherical harmonics compared with harmonic splines
An optimum truncation level, N, in a spherical-harmonic analysis of the geomagnetic main field at the core-mantle boundary is determined by harmonic-spline analysis. Specifically, that value of N is found at which the two analyses are closest in a well defined sense and, for that value of N, the 'closeness' of two models is determined. Depending slightly on the definition of closeness, optimum N is found to be either 10 or 11. For those values the two analyses give remarkably similar results, showing that the conveniences of spherical harmonics can be retained with little penalty.
Benton, E. R.↗
Spectral Spherical Harmonics Discrete Ordinate Method
A new method for modeling the radiative transfer in inhomogeneous three-dimensional media illuminated by a Gaussian beam is described. This approach, called the Spectral Spherical Harmonics Discrete Ordinate Method (SSHDOM), uses the Fourier expansion method to transform the three-dimensional radiative transfer into an one-dimensional equation in the spectral domain, and the Spherical Harmonics Discrete Ordinate Method (SHDOM) for its solution. Specifically, (i) the source function is represented in the spectral domain through a spherical harmonic expansion, (ii) the spectral one-dimensional radiative transfer equation is integrated along discrete ordinates through a spatial grid, and (iii) the solution method is based on the Picard iteration. Both SSHDOM and SHDOM algorithms are implemented in a common computer code.
Gaussian beam↗
Ocean Domains and Maximum Degree of Spherical Harmonic and Orthonormal Expansions
Ocean domains used for the orthonormal (ON) systems are studied to determine the maximum degree of spherical harmonic and orthonormal expansions that can be constructed. Although it was shown that one domain was restricted to degree 24, others were shown could be constructed to determine expansions to at least degree 36. Since 1991 the maximum degree expansion used for several Ohio State studies has been 24. In this report it is shown that the maximum degree for the ocean domain used by Wang and Rapp [1994] was 32 and 29 for the domain used by Rapp, Zhang, Yi [1996]. A modification of the former domain was developed (D1e) that enabled a solution to degree 36 to be determined. A modification of the Rapp, Zhang, Yi domain (D7d) enabled a degree 30 solution to be made. Combination coefficients were developed for domain D1e, to degree 36, and to degree 30 for domain D7d. The degree 30 spherical harmonic expansion provided by Pavlis [1998] of the POCM_4B dynamic ocean topography (DOT), and the degree 30 part of the degree 360 expansion [Rapp. 1998] of the POCM_4B model was converted to an ON expansion valid for the D7d domain. The degree 36 part of the degree 360 expansion was converted to the ON expansion for the D1e domain. The square root of the degree variances of the various solutions were compared. The root mean square value of DOT from the Pai,lis expansion, after conversion to the ON system, was + or - 66.52 cm (D7d domain). The value from the degree 30 part of the 360 expansion was + or - 66.65 cm. The value based on the actual POCM-4B data, in the D7d domain, was + or - 66.74 cm showing excellent agreement with the ON results. If the spherical harmonic coefficients had been used the implied root mean square value was + or - 60.76 cm [Pavlis] and + or -59.70 cm [Rapp].
Rapp, R.↗