Vector Spherical Harmonics Research Articles

Mie Scattering and Domain Structure of Ferroelectrics

A calculation procedure for obtaining the complete set of scattering coefficients and incident, scattered and internal electric fields in terms of the vector spherical harmonics for a multilayered nonlinear structure (as model of polydomain ferroelectric) is proposed by using the T-matrix method.

Mie Scattering and Domain Structure of Ferroelectrics

A calculation procedure for obtaining the complete set of scattering coefficients and incident, scattered and internal electric fields in terms of the vector spherical harmonics for a multilayered nonlinear structure (as model of polydomain ferroelectric) is proposed by using the T-matrix method.

The spherical horizontal and spherical vertical boundary value problem - vertical deflections and geoidal undulations - the completed Meissl diagram

In a comparison of the solution of the spherical horizontal and vertical boundary value problems of physical geodesy it is aimed to construct downward continuation operators for vertical deflections (surface gradient of the incremental gravitational potential) and for gravity disturbances (vertical derivative of the incremental gravitational potential) from points on the Earth's topographic surface or of the three-dimensional (3-D) Euclidean space nearby down to the international reference sphere (IRS). First the horizontal and vertical components of the gravity vector, namely spherical vertical deflections and spherical gravity disturbances, are set up. Second, the horizontal and vertical boundary value problem in spherical gravity and geometry space is considered. The incremental gravity vector is represented in terms of vector spherical harmonics. The solution of horizontal spherical boundary problem in terms of the horizontal vector-valued Green function converts vertical deflections given on the IRS to the incremental gravitational potential external in the 3-D Euclidean space. The horizontal Green functions specialized to evaluation and source points on the IRS coincide with the Stokes kernel for vertical deflections. Third, the vertical spherical boundary value problem is solved in terms of the vertical scalar-valued Green function. Fourth, the operators for upward continuation of vertical deflections given on the IRS to vertical deflections in its external 3-D Euclidean space are constructed. Fifth, the operators for upward continuation of incremental gravity given on the IRS to incremental gravity to the external 3-D Euclidean space are generated. Finally, Meissl-type diagrams for upward continuation and regularized downward continuation of horizontal and vertical gravity data, namely vertical deflection and incremental gravity, are produced.

Nonsolid rotation of structures in isothermal magnetized atmospheres

Possible equilibrium rotating magnetized plasma configurations in which angular asymmetry due to a nonuniform distribution of the molecular weight is absent are studied. The viscosity of the medium is assumed to be negligible. The variables in exact nonlinear equations are separated by representing the vector fields as series in orthogonal vector spherical harmonics. These series are normally divergent. However, the truncation of the series is also possible. Equilibrium models corresponding to the truncated series are generalizations of a simple rigidly rotating model. They are studied below. It is shown that nonaxisymmetric equilibrium structures with the magnetic field of a tilted dipole are possible. In the case of an isothermal atmosphere, conditions for the superrotation of the medium and meridional circulation of the matter may arise. The feasibility of such conditions in the Earth’s upper atmosphere is discussed.

A vector wavelet approach to iono- and magnetospheric geomagnetic satellite data

A multiscale method is introduced using spherical (vector) wavelets for the computation of the earth's magnetic field within source regions of ionospheric and magnetospheric currents. The considerations are essentially based on two geomathematical keystones, namely (i) the Mie representation of solenoidal vector fields in terms of toroidal and poloidal parts and (ii) the Helmholtz decomposition of spherical (tangential) vector fields. Vector wavelets are shown to provide adequate tools for multiscale geomagnetic modelling in form of a multiresolution analysis, thereby completely circumventing the numerical obstacles caused by vector spherical harmonics. The applicability and efficiency of the multiresolution technique is tested with real satellite data.

Sum rules for products of light scattering functions

Basis vectors of a spherical system are shown to be linear combinations of functions commonly used in light scattering. These expressions are used to obtain an expansion for the unit dyadic as a linear combination of products of these functions, and this expansion is used to motivate the use of matrix orthonormality to obtain a complete set of sum rules for products of scalar light scattering functions. As an example demonstrating the utility of these expressions, sum rules are obtained for dyadic, component, and inner products of vector spherical harmonics used in the theory of light scattering.

Formation of complex rotational structures in convective and isothermal magnetized zones

We study the development of a complex rotation law in the magnetized convective and isothermal zones of stars and planetary atmospheres through the decomposition of vector quantities in terms of orthogonal vector spherical harmonics. In the case of a solar-type extended convective zone, it is assumed that (a) the transformation of thermal into magnetic energy is favorable from the viewpoint of energy balance, (b) the state that is supported with minimum energy loss is realized, and (c) the condition of minimum entropy production consistent with the two previous requirements is satisfied. To find the rotation law of a zone, weak interaction between variations in the rotation and magnetic-field distributions is assumed. Two possible zones of generation of the solar magnetic field are considered. The first is located in the lower half of the solar convective zone and possesses a latitude dependence of the rotational velocity similar to that observed. The second zone is located just below the surface, and has a rotational velocity that decreases sharply with height and depends only weakly on latitude. We also study simple equilibrium structures, in particular, those describing the super-rotation of the medium in a convective or isothermal zone. Realization of such super-rotation in an isothermal zone is associated with the outflow of matter and fields toward upper layers.

On Some Aspects of the Inversion Problem of Magnetostatics

Debye potentials and vector spherical harmonics were applied to a magnetostatic problem. By these mathematical methods it is shown that the inverse problem of the magnetostatics cannot be solved though the divergence of vectorMis given.

Vector spherical harmonics: Concepts and applications to the single centre expansion method

In this paper the basic formulation of the Vector Spherical Harmonics (VSH) is recalled from the literature and the details on the various notations clarified. By means of compact definitions of differential operators in terms of VSH, a new formulation for the gradient and Laplacian of a generic Single Centre Expanded (SCE) function is developed. The new formulae are so derived to be directly computable and implementable in numerical codes to face problems in atomic and molecular physics. A set of numerical computations have been carried out to assess the proposed formulae where the behaviour of the gradient and Laplacian of the electron density and static potential of the water molecule is exploited in detail within the SCE framework of reference.

Series in vector spherical harmonics: An efficient tool for solution of nonlinear problems in spherical plasmas

The series expansion of the plasma fields and currents in vector spherical harmonics has been demonstrated to be an efficient technique for solution of nonlinear problems in spherically bounded plasmas. Using this technique, it is possible to describe the nonlinear plasma response to the rotating high-frequency magnetic field applied to the magnetically confined plasma sphere. The effect of the external magnetic field on the current drive and field configuration is studied. The results obtained are important for continuous current drive experiments in compact toruses.

A new procedure for specifying nonradiating current distributions and the fields they produce

This paper reports a new procedure for specifying monochromatic nonradiating (NR) current distributions (NR sources) and the electric and magnetic fields they produce (NR fields). Vector spherical harmonics and a Fourier–Bessel series are used to derive a new vector spherical-wave expansion for continuous NR fields confined within a spherical volume. The analysis yields complete orthogonal sets in terms of which all such NR fields can be expanded. By making use of a Maxwell operator representation for NR current distributions, we obtain a new series expansion for NR current distributions confined within a spherical volume. The analysis also yields complete sets for such NR current distributions. The developed theory is illustrated with special cases.

Study of the equilibrium and motions of the medium in a rotating star with a magnetic field using tools from angular momentum theory

The complicated processes occurring in a rotating magnetized medium are examined by representing all the vector quantities in the form of expansions in a complete system of orthogonal vector spherical harmonics. A separation of variables is ultimately achieved without a loss of accuracy despite the presence of nonlinear forces (the magnetic and Coriolis forces). The distribution of the rates of rotation and circulation motions in an adiabatically stratified, slowly rotating star or an atmospheric convection zone is studied as an example. The postulate of minimum entropy production from nonequilibrium thermodynamics is employed to find the most probable steady-state configuration. One solution satisfactorily describes the differential rotation observed on the Sun. The preliminary data support the notion that the superfast rotation of the type observed in Venus’ atmosphere can also be explained within the theory discussed.

A Continuous Model for Molecular Vibrations in C60

A continuous model for molecular vibration in C60 is developed which presents a physical picture easy to understand. The molecular vibration modes are expanded by vector spherical harmonics. With the choice of suitable parameters, the calculated frequencies are fit the experimental values very well. In combination with the pseudopotential model for the valence electrons in C60, the interaction between the electron and intramolecular phonon in solid C60 can be analysed and calculated based on our model.

Determination of Transition Matrices for Inomogeneous Dielectric Bodies by a Wave Propagator Method

A method for determining transition matrices for inhomogeneous dielectric shells is presented. The source can be either inside or outside the shell. The method is based upon expansions of the electric and magnetic fields in vector spherical harmonics. The expansion coefficients in the shell satisfy a system of linear ordinary differential equations in the radial direction. The expansion coefficients for the incident and scattered fields are related via transition matrices, defined in the same way as in the T-matrix or null-field method. Numerical examples showexcellen t agreement with results obtained by the T-matrix method. Numerically the method is particularly strong for the case of a source at the center of large, but thin, inhomogeneous spherical shell. (Less)

Determination of Transition Matrices for Inomogeneous Dielectric Bodies By a Wave Propagator Method - Abstract

A method for determining transition matrices for inhomogeneous dielectric shells is presented. The source can be either inside or outside the shell. The method is based upon expansions of the electric and magnetic fields in vector spherical harmonics. The expansion coefficients in the shell satisfy a system of linear ordinary differential equations in the radial direction. The expansion coefficients for the incident and scattered fields are related via transition matrices, defined in the same way as in the T-matrix or null-field method. Numerical examples show excellent agreement with results obtained by the T-matrix method. Numerically the method is particularly strong for the case of a source at the center of large, but thin, inhomogeneous spherical shell.

A current sheet model for the Earth’s magnetic field

As an example in magnetostatics we consider the main magnetic field of the Earth and its current sources. The measured field on the surface is accurately given, in tables of the International Geological Reference Field, in terms of Gaussian coefficients. By applying Maxwell’s equations to these data we calculate the extended field, inside the Earth, and give graphical representations of it. We also construct a simple theoretical model of the source of the field, in which the field is the result of currents flowing on the surface of a sphere inside the Earth. The current sources which give the observed field are calculated in terms of vector spherical harmonics. The stream function and currents are displayed on a Mercator projection for a sphere whose radius is half the Earth’s radius. Interesting properties of vector operations on the Mercator plane are analytically and graphically described.

A theoretical discussion of vector pick-up systems for SQUID magnetometers

The magnetic induction field produced by non-homogeneously magnetized samples of arbitrary shape is expanded in terms of a complete set of functions, the vector spherical harmonics. The weighing coefficients of this expansion - the multipole moments - are computed and are given up to fifth order for the most common sample shapes and homogeneous magnetization. The magnetic flux threading a general detection coil is computed analytically and pick-up systems of rotational symmetry as well as transverse systems are discussed. The flux profiles of dipole and higher order multipole moments along the direction of the applied field are plotted for common coil geometries as functions of the sample position. Both the influence of position errors on the flux profiles and the crosstalk between mutually perpendicular components of the dipole vector are analysed.

Experimental and theoretical results of light scattering by aggregates of spheres

We present laboratory microwave scattering measurements for complex amplitude scattering matrices of three aggregates of 2, 8, and 27 identical spheres and compare them with theoretical predictions. Electromagnetic multiparticle-scattering calculations involve the determination of a large number of vector translation coefficients introduced by the addition theorems for vector spherical harmonics. For one of the two classes of vector translation coefficients there is an overall-sign discrepancy between two groups of formulations that exist in the literature. We compare our experimental data with the theoretical results from scattering calculations using the two different sets of formulas for computation of the translation coefficients. This comparison of experiment with theory reveals that Cruzan's original research on the vector addition theorems [Q. Appl. Math. 20, 33-40 (1962)] is correct, although many authors believe that Cruzan's formulation contains an overall-sign error.

The Magnetic Field of the Earth: Paleomagnetism, the Core, and the Deep Mantle

History of Geomagnetism and Paleomagnetism: Discovery of the Main Magnetic Elements. Fossil Magnetism and the Magnetic Field in the Past. Investigations of the External Magnetic Field. Origin of the Earth's Magnetic Field. The Present Geomagnetic Field: Analysis and Description from Historical Observations: Magnetic Elements and Charts. Spherical Harmonic Description of the Earth's Magnetic Field. Uniqueness and Other Mathematical Problems. Geomagnetic Secular Variation. The External Magnetic Field. Foundations of Paleomagnetism: Rock Magnetism. Magnetic Mineralogy. Paleomagnetic Directions and Poles. Paleointensity Methods. Age Determinations. The Recent Geomagnetic Field: Paleomagnetic Observations: Archeomagnetic Results. Analysis of Recent Lake Sediments. Geomagnetic Excursions. The Geomagnetic Power Spectrum. Reversals of the Earth's Magnetic Field: Evidence for Field Reversal. Marine Magnetic Anomalies. Analysis of Reversal Sequences. Polarity Transitions. The Time-Averaged Paleomagnetic Field: Geocentric Axial Dipole Hypothesis. Second-Order Terms. Variation in the Earth's Dipole Moment. Paleosecular Variation from Lavas (PSVL). Processes and Properties of the Earth's Deep Interior: Basic Principles: Seismic Properties of the Earth's Interior. Chemical and Physical Properties. Thermodynamic Properties of the Earth's Deep Interior. Thermal History Models. Non-dynamo Models for the Earth's Magnetic Field. Fluid Mechanics Fundamentals. Energy Sources. Introduction to Dynamo Theory: The Dynamo Problem. The Magnetic Induction Equation. The a and w Effects of Dynamo Theory. Waves in Dynamo Theory. Symmetries in Dynamo Theory. Theories for Geomagnetic Secular Variations and magnetic Field Reversals. Dynamo Theory: Vector Spherical Harmonics. Kinematic Dynamos. Cowling's Theorem and Other Constraints. Turbulence in the Core. Dynamo Waves. Dynamics of the Geodynamo. The Magnetic Fields of the Sun, Moon, and Planets: Origin of the Solar System. The Sun. The Moon. Meteorites. Magnetic Fields of the Planets. Geomagnetic Relevance. Examples of Synthesis: Fluid Velocities in the Core. Core-Mantle Coupling: Length of Day. Paleomagnetism and Dynamo Theory. Variations at the Core-Mantle Boundary and the Earth's Surface. Appendices. References. Subject Index.

Quasar Proper Motions and Low‐Frequency Gravitational Waves

We report observational upper limits on the mass-energy of the cosmological gravitational wave background, from limits on proper motions of quasars. Gravitational waves with periods longer than the time span of observations produce a simple pattern of apparent proper motions over the sky, composed primarily of second-order transverse vector spherical harmonics. A fit of such harmonics to measured motions yields a 95% confidence limit on the mass-energy of gravitational waves with frequencies ν < 2 × 10-9 Hz, of less than 0.11 h-2 times the closure density of the universe.