Terms Of Spherical Harmonics Research Articles

A Double Fourier Series (DFS) Dynamical Core in a Global Atmospheric Model with Full Physics

Abstract This study describes an application of the double Fourier series (DFS) spectral method developed by Cheong as an alternative dynamical option in a model system that was ported into the Global/Regional Integrated Model System (GRIMs). A message passing interface (MPI) for a massive parallel-processor cluster computer devised for the DFS dynamical core is also presented. The new dynamical core with full physics was evaluated against a conventional spherical harmonics (SPH) dynamical core in terms of short-range forecast capability for a heavy rainfall event and seasonal simulation framework. Comparison of the two dynamical cores demonstrates that the new DFS dynamical core exhibits performance comparable to the SPH in terms of simulated climatology accuracy and the forecast of a heavy rainfall event. Most importantly, the DFS algorithm guarantees improved computational efficiency in the cluster computer as the model resolution increases, which is consistent with theoretical values computed from a dry primitive equation model framework. The current study shows that, at higher resolutions, the DFS approach can be a competitive dynamical core because the DFS algorithm provides the advantages of both the spectral method for high numerical accuracy and the gridpoint method for high performance computing in the aspect of computational cost.

Modern global Earth’s gravity field models and their errors

The methods for obtaining a priori and a posteriori estimates of the accuracy of the global Earth’s gravity field (EGF) models in terms of geopotential spherical harmonics are considered and classified. The ways of how to accomplish this task, taking into consideration the latest advances in the creation of EGF models, are discussed. The main requirements to the database and software used to estimate the gravity field error model are formulated.

Seeking inflation fossils in the cosmic microwave background

If during inflation the inflaton couples to a ``fossil'' field---some new scalar, vector, or tensor field---it typically induces a scalar-scalar-fossil bispectrum. Even if the fossil field leaves no direct physical trace after inflation, it gives rise to correlations between different Fourier modes of the curvature or, equivalently, a nonzero curvature trispectrum, but without a curvature bispectrum. Here we quantify the effects of a fossil field on the cosmic microwave background temperature fluctuations in terms of bipolar spherical harmonics (BiPoSHs). The effects of vector and tensor fossils can be distinguished geometrically from those of scalars through the parity of the BiPoSHs they induce. However, the two-dimensional nature of the cosmic microwave background sky does not allow vectors to be distinguished geometrically from tensors. We estimate the detectability of a signal in terms of the scalar-scalar-fossil coupling for scalar, vector, and tensor fossils, assuming a local-type coupling. We comment on a divergence that arises in the quadrupolar BiPoSH from the scalar-scalar-tensor correlation in single-field slow-roll inflation.

The quadrupole model for rigid-body gravity simulations

We introduce two new models for gravitational simulations of systems of non-spherical bodies, such as comets and asteroids. In both models, one body (the “primary”) may be represented by any convenient means, to arbitrary accuracy. In our first model, all of the other bodies are represented by small gravitational “molecules” consisting of a few point masses, rigidly linked together. In our second model, all of the other bodies are treated as point quadrupoles, with gravitational potentials including spherical harmonic terms up to the third degree (rather than only the first degree, as for ideal spheres or point masses). This quadrupole formulation may be regarded as a generalization of MacCullagh’s approximation.Both models permit the efficient calculation of the interaction energy, the force, and the torque acting on a small body in an arbitrary external gravitational potential. We test both models for the cases of a triaxial ellipsoid, a rectangular parallelepiped, and “duplex” combinations of two spheres, all in a point-mass potential. These examples were chosen in order to compare the accuracy of our technique with known analytical results, but the ellipsoid and duplex are also useful models for comets and asteroids. We find that both approaches show significant promise for more efficient gravitational simulations of binary asteroids, for example. An appendix also describes the duplex model in detail.

Fox-Wolfram moments in Higgs physics

Geometric correlations between jets as part of hard processes or in addition to hard processes are key ingredients to many LHC analyses. Fox--Wolfram moments systematically describe these correlations in terms of spherical harmonics. These moments, either computed from the tagging jets or from all jets in each event, can significantly improve Higgs searches in weak boson fusion. Applications of Fox--Wolfram moments in LHC analyses obviously surpass jets as analysis objects as well as Higgs searches in terms of analyses.

Geoid modelling based on EGM08 and recent Earth gravity models of GOCE

In this paper an estimator for geoid is presented and applied for geoid computation which considers the topographic and atmospheric effects on the geoid. The total atmospheric effect is mathematically developed in terms of spherical harmonics to degree and order 2,160 based on a recent static atmospheric density model. Also the contribution of its higher degrees is formulated. Another idea of this paper is to combine one of the recent Earth gravity models (EGMs) of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) mission with EGM08 and the terrestrial gravimetric data of Fennoscandia in an optimum way. To do so, the GOCE EGMs are compared with the Global Positioning System (GPS)/levelling data over the area for finding the most suited one. This comparison is done in two different ways: with and without considering the errors of the EGMs. Comparison of the computed geoids with the GPS/levelling data shows that a) considering the total atmospheric effect will improve the geoid by about 5 mm, b) GOCO03S is the most suited GOCE EGM for Fennoscandia, c) the errors of some of the GOCE EGMs are optimistic and far from reality. Combination of GOCO03S from degree 120 to 210 and EGM08 for the rest of degrees shows its good quality in these frequencies.

Study of the partial wave structure of π0η photoproduction on protons

Analysis of the partial wave structure of gamma p --> pi0 eta p reaction is presented in the energy region from threshold up to the total center of mass energy W = 1.9 GeV. Angular distributions measured with the Crystal Ball/TAPS hermetic detector system at the Mainz Microtron MAMI are expanded in terms of spherical harmonics. The relation of the extracted moments to the partial wave structure of the reaction amplitude is discussed and compared with predictions from model calculations.

Reflection and transmission coefficients for finite-sized aperiodic aggregates of spheres

Generalized Mie theory is employed to study the reflection and transmission properties of finite-sized spherical arrays of nanoparticles based on aperiodic geometries. To simulate realistic experimental conditions, a circular aperture is used to create an incident field with a finite beamwidth. The diffracted fields from the circular aperture are expanded in terms of vector spherical wave harmonics, which are then employed to derive the scattered fields using the generalized Mie theory. Expansion of diffracted fields in terms of spherical harmonics also led to new analytical expressions for two important integrals involving Bessel, associated Legendre, and trigonometric functions, which arise in electromagnetic diffraction problems. Subsequently, generalized scattering parameters were defined in terms of far-field specular energy fluxes. To verify the results, the method was applied to a truncated periodic array of spherical gold nanoparticles for which the generalized scattering parameters were compared and found to agree with the scattering parameters obtained for an infinite planar structure subject to periodic boundary conditions. The method was then applied to an aperiodic array of gold nanoparticles based on a Penrose geometry, and the lowest-order photonic resonances were observed in the predicted regions. Furthermore, it was shown that by proper scaling, the photonic resonances can be strategically placed in the plasmonic region of the gold, where they are enhanced due to strong coupling between the plasmonic and photonic modes.

Weighted Sobolev orthogonal polynomials on the unit ball

For the weight function Wμ(x)=(1−|x|2)μ, μ>−1, λ>0 and bμ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product 〈f,g〉=bμ[∫Bdf(x)g(x)Wμ(x)dx+λ∫Bd∇f(x)⋅∇g(x)Wμ(x)dx] are constructed in terms of spherical harmonics and a sequence of Sobolev orthogonal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to 〈⋅,⋅〉, can be generated through a recursive formula.

Elastic fields and effective moduli of particulate nanocomposites with the Gurtin–Murdoch model of interfaces

A complete solution has been obtained for periodic particulate nanocomposite with the unit cell containing a finite number of spherical particles with the Gurtin–Murdoch interfaces. For this purpose, the multipole expansion approach by Kushch et al. [Kushch, V.I., Mogilevskaya, S.G., Stolarski, H.K., Crouch, S.L., 2011. Elastic interaction of spherical nanoinhomogeneities with Gurtin–Murdoch type interfaces. J. Mech. Phys. Solids 59, 1702–1716] has been further developed and implemented in an efficient numerical algorithm. The method provides accurate evaluation of local fields and effective stiffness tensor with the interaction effects fully taken into account. The displacement vector within the matrix domain is found as a superposition of the vector periodic solutions of Lamé equation. By using local expansion of the total displacement and stress fields in terms of vector spherical harmonics associated with each particle, the interface conditions are fulfilled precisely. Analytical averaging of the local strain and stress fields in matrix domain yields an exact, closed form formula (in terms of expansion coefficients) for the effective elastic stiffness tensor of nanocomposite. Numerical results demonstrate that elastic stiffness and, especially, brittle strength of nanoheterogeneous materials can be substantially improved by an appropriate surface modification.

ON THE EFFICIENCY AND GAIN OF ANTENNAS

The fundamental limits of the gain and e±ciency of an antenna are explored. These are very important quantities for, e.g., superdirective arrays. The antenna is in this paper con¯ned in a sphere and all of the currents are assumed to run in a material with a given conductivity. It is shown that one can ¯nd the current distribution in the sphere that optimizes the gain, given the frequency and the radius of the sphere. The results indicate the distribution of antenna elements in an antenna array in order to maximize gain, or e±ciency. The analysis is based on the expansion of the electromagnetic ¯elds in terms of vector spherical harmonics. Explicit expressions for the limits of gain and e±ciency, and the corresponding current densities, are derived for di®erent types of antennas. (Less)

Green's function of the time-dependent radiative transport equation in terms of rotated spherical harmonics

The time-dependent radiative transport equation is solved for the three-dimensional spatially uniform infinite medium which is illuminated by a point unidirectional source using a spherical harmonics transform under rotation. Apart from the numerical evaluation of a spherical Hankel transform which connects the spatial distance with the radial distance in Fourier space, the dependence on all variables is found analytically. For the special case of a harmonically modulated source, even the spherical Hankel transform can be carried out analytically. Additionally, a special solution for the isotropically scattering infinite medium is given. The Monte Carlo method is used for a successful verification of the derived solution.

The geomagnetic field gradient tensor

We develop the general mathematical basis for space magnetic gradiometry in spherical coordinates. The magnetic gradient tensor is a second rank tensor consisting of 3 × 3 = 9 spatial derivatives. Since the geomagnetic field vector B is always solenoidal $${(\nabla \cdot {\rm B}=0)}$$ there are only eight independent tensor elements. Furthermore, in current free regions the magnetic gradient tensor becomes symmetric, further reducing the number of independent elements to five. In that case B is a Laplacian potential field and the gradient tensor can be expressed in series of spherical harmonics. We present properties of the magnetic gradient tensor and provide explicit expressions of its elements in terms of spherical harmonics. Finally we discuss the benefit of using gradient measurements for exploring the Earth’s magnetic field from space, in particular the advantage of the various tensor elements for a better determination of the small-scale structure of the Earth’s lithospheric field.

Solution of the 3D-Helmholtz equation in exterior domains using spherical harmonic decomposition

This work is devoted to a finite element formulation for the Helmholtz equation in exterior domains. The proposed formulation uses a separation of variables, combining a 2D FE discretization on an intermediate spherical boundary and an ‘a priori’ analytical pattern for the radial direction. Using the analytical radial pattern and the series expansion of trial and test functions in terms of spherical harmonics, an efficient semi-analytical technique is obtained for the direct calculation of the global FE matrices. The accuracy and reliability of the formulation are illustrated through numerical examples of radiation and scattering in the exterior domain.

Contribution from cosmological scalar perturbations to the angular velocity spectrum of extragalactic sources

The possibility of the influence of adiabatic scalar perturbations on the angular velocity spectrum of extragalactic sources is considered. The multipole expansion coefficients of the angular velocity field in terms of vector spherical harmonics are calculated. We show that there is no contribution from adiabatic perturbations to the angular spectrum for a spatially flat Universe at the dusty stage, while there is a contribution only to the electric multiple coefficients at the stage of Λ-term domination. The cases of long-wavelength and short-wavelength perturbations are considered separately. The relationship between the multipole angular velocity spectrum and the primordial scalar perturbation spectrum is discussed.

Analytic Tangent Irradiance Environment Maps for Anisotropic Surfaces

AbstractEnvironment‐mapped rendering of Lambertian isotropic surfaces is common, and a popular technique is to use a quadratic spherical harmonic expansion. This compact irradiance map representation is widely adopted in interactive applications like video games. However, many materials are anisotropic, and shading is determined by the local tangent direction, rather than the surface normal. Even for visualization and illustration, it is increasingly common to define a tangent vector field, and use anisotropic shading. In this paper, we extend spherical harmonic irradiance maps to anisotropic surfaces, replacing Lambertian reflectance with the diffuse term of the popular Kajiya‐Kay model. We show that there is a direct analogy, with the surface normal replaced by the tangent. Our main contribution is an analytic formula for the diffuse Kajiya‐Kay BRDF in terms of spherical harmonics; this derivation is more complicated than for the standard diffuse lobe. We show that the terms decay even more rapidly than for Lambertian reflectance, going as l–3, where l is the spherical harmonic order, and with only 6 terms (l = 0 and l = 2) capturing 99.8% of the energy. Existing code for irradiance environment maps can be trivially adapted for real‐time rendering with tangent irradiance maps. We also demonstrate an application to offline rendering of the diffuse component of fibers, using our formula as a control variate for Monte Carlo sampling.

Geometric distortion and shimming considerations in a rotating MR‐linac design due to the influence of low‐level external magnetic fields

This work investigates with simulation the effect of external stray magnetic fields on a recently reported MRI-linac hybrid, which by design will rotate about the patient axis during therapy. During rotation, interactions with magnetic fields from the earth or nearby ferromagnetic structures may cause unacceptable field distortions in the imaging field of view. Optimal approaches for passive shimming implementation, the degree and significance of residual distortion, and an analysis of the active shimming requirements for further correction are examined. Finite element simulations were implemented on two representative types of biplanar magnet designs. Each of these magnet designs, consisting of a 0.2 T four-post and a 0.5 T C-type unit, was simulated with and without an external field on the order of the earth's field (0.5 G) over a range of rotated positions. Through subtraction, the field distribution resulting from the external field alone could be determined. These measured distributions were decomposed into spherical harmonic components, which were then used to investigate the effect of their selective removal to simulate the effects of passive and active shimming. Residual fields after different levels of shim treatment were measured and assessed in terms of their imaging consequence. For both magnet types, the overall success of a passive shim implementation was highly dependent on the orientation for which it was based. If this orientation was chosen incorrectly, the passive shim would correct for the induced fields at that location, but the overall maximal distortion at other locations was exacerbated by up to a factor of two. The choice of passive shim orientation with the least negative consequence was found to be that where the magnet B(0) axis and transaxial component of the external field were aligned. Residual fields after passive shimming and frequency offset were found to be low in the simulated scenarios, contributing to <1 mm of distortion for most standard imaging sequences (based on a 0.5 G external field). However, extremely rapid single-shot sequences could be distorted by these residual fields to well over 5 mm. These residuals when analyzed were found to correspond primarily to second-order spherical harmonic terms. One term in particular was found to account for the vast majority of these residual fields, defined by the product of the two axes perpendicular to the axis of rotation. The implementation of this term would allow the resulting geometric distortion to fall to the order of 1 mm, even for single-shot sequences. After appropriate passive shimming, the imaging distortion due to an external field of 0.5 G was found to be important only in rapid single-shot sequences, which are especially susceptible to field inhomogeneity. Should it be desirable to use these sequences for real-time tracking, made conceivable due to the lower susceptibility concerns at low field, these residual fields should be addressed. The ability to use only one second-order term for this correction will reduce the cost impact of this decision.

Recurrence relations between kernels of the nonlinear Boltzmann collision integral

A new approach to the solution of the nonlinear Boltzmann equation based on distribution function expansion in terms of spherical harmonics is considered. Owing to this approach, the complex five-fold collision integral can be replaced by a set of rather simple integral operators. Recurrence relations between nonlinear kernels Gl1,l2l(c,c1,c2) of the integral operators are derived on the basis of the collision integral invariance with respect to choice of the reference frame velocity. Recurrence formulae allow one to find kernels with arbitrary indices provided the kernel G0,00(c,c1,c2) is known. Explicit analytical expressions for kernels G1,01, G0,11, and G1,10 for the cases of hard spheres and Maxwellian molecules are presented.

3D vector tomography using vector spherical harmonics decomposition

The article presents the reconstruction method of the flow fields from vector tomography passive ion Doppler spectroscopy in a plasma experiment. The method is based on series expansion in terms of vector spherical harmonics for volumetric, divergent-free vector fields and intended for three-dimensional diagnostic in the spherical tokamak devices. An inversion of spectral experimental data is known in tomography as an inversion of vectorial ray transform. The relation of the vectorial ray transform with Doppler spectroscopy measurements are given in Balandin and Ono (2001, 2003) [7,8]. The effectiveness of the proposed method is tested on a range of model 3D divergent-free vector fields.

Vector Slepian basis functions with optimal energy concentration in high numerical aperture focusing

A series expansion method is proposed for high numerical aperture focusing problems. The angular spectrum of the focused field is expressed using orthogonal vector basis functions that are obtained by solving Slepian's concentration problem for a spherical cap and expressed in terms of the vector spherical harmonics. This newly obtained Slepian-type basis automatically satisfies the transversality condition and a subset exhibits excellent energy concentration inside the solid angle of illumination. This property makes the Slepian-type basis very useful in inverse focusing problems. The corresponding vector basis for focused fields is constructed as well. Examples are presented for linearly and radially polarized illumination, revealing the fact that, compared to an expansion using the vector spherical harmonics themselves, a lower number of terms is sufficient to achieve the same accuracy, suggesting that the Slepian-type basis is preferable even in direct focusing problems.