Series Of Spherical Harmonics Research Articles

Global evaluation of new GRACE mascon products for hydrologic applications

AbstractRecent developments in mascon (mass concentration) solutions for GRACE (Gravity Recovery and Climate Experiment) satellite data have significantly increased the spatial localization and amplitude of recovered terrestrial Total Water Storage anomalies (TWSA); however, land hydrology applications have been limited. Here we compare TWSA from April 2002 through March 2015 from (1) newly released GRACE mascons from the Center for Space Research (CSR‐M) with (2) NASA JPL mascons (JPL‐M), and with (3) CSR Tellus gridded spherical harmonics rescaled (sf) (CSRT‐GSH.sf) in 176 river basins, ∼60% of the global land area. Time series in TWSA mascons (CSR‐M and JPL‐M) and spherical harmonics are highly correlated (rank correlation coefficients mostly >0.9). The signal from long‐term trends (up to ±20 mm/yr) is much less than that from seasonal amplitudes (up to 250 mm). Net long‐term trends, summed over all 176 basins, are similar for CSR and JPL mascons (66–69 km3/yr) but are lower for spherical harmonics (∼14 km3/yr). Long‐term TWSA declines are found mostly in irrigated basins (−41 to −69 km3/yr). Seasonal amplitudes agree among GRACE solutions, increasing confidence in GRACE‐based seasonal fluctuations. Rescaling spherical harmonics significantly increases agreement with mascons for seasonal fluctuations, but less for long‐term trends. Mascons provide advantages relative to spherical harmonics, including (1) reduced leakage from land to ocean increasing signal amplitude, and (2) application of geophysical data constraints during processing with little empirical postprocessing requirements, making it easier for nongeodetic users. Results of this product intercomparison should allow hydrologists to better select suitable GRACE solutions for hydrologic applications.

Hybrid Monte Carlo and continuum modeling of electrolytes with concentration-induced dielectric variations.

The distribution of ions near a charged surface is an important quantity in many biological and material processes, and has been therefore investigated intensively. However, few theoretical and simulation approaches have included the influence of concentration-induced variations in the local dielectric permittivity of an underlying electrolyte solution. Such local variations have long been observed and known to affect the properties of ionic solution in the bulk and around the charged surface. We propose a hybrid computational model that combines Monte Carlo simulations with continuum electrostatic modeling to investigate such properties. A key component in our hybrid model is a semianalytical formula for the ion-ion interaction energy in a dielectrically inhomogeneous environment. This formula is obtained by solving for the Green's function Poisson's equation with ionic-concentration-dependent dielectric permittivity using a harmonic interpolation method and spherical harmonic series. We also construct a self-consistent continuum model of electrostatics to describe the effect of ionic-concentration-dependent dielectric permittivity and the resulting self-energy contribution. With extensive numerical simulations, we verify the convergence of our hybrid simulation scheme, show the qualitatively different structures of ionic distribution due to the concentration-induced dielectric variations, and compare our simulation results with the self-consistent continuum model. In particular, we study the differences between weakly and strongly charged surfaces and multivalencies of counterions. Our hybrid simulations conform particularly the depletion of ionic concentrations near a charged surface and also capture the charge inversion. We discuss several issues and possible further improvement of our approach for simulations of large charged systems.

On the exactness of estimates for irregularly structured bodies of the general term of Laplace series

The main form of the representation of a gravitational potential V for a celestial body T in outer space is the Laplace series in solid spherical harmonics \((R/r)^{n+1}Y_n(\theta ,\lambda )\) with R being the radius of the enveloping T sphere. The surface harmonic \(Y_n\) satisfies the inequality $$\begin{aligned} \langle Y_n\rangle < Cn^{-\sigma }. \end{aligned}$$ The angular brackets mark the maximum of a function’s modulus over a unit sphere. For bodies with an irregular structure \(\sigma = 5/2\), and this value cannot be increased generally. However, a class of irregular bodies (smooth bodies with peaked mountains) has been found recently in which \(\sigma = 3\). In this paper, we will prove the exactness of this estimate, showing that a body belonging to the above class does exist and $$\begin{aligned} 0<\varlimsup n^3\langle Y_n\rangle <\infty \end{aligned}$$ for it.

Imaging of Broadband Noise from Rotating Sources in Uniform Axial Flow

A method is presented for imaging rotating broadband sources with a microphone array. The microphone array rotates virtually with the same angular frequency as the source such that the pressure data are mathematically transformed into a rotating reference frame and the influence of the moving source (Doppler shift) is compensated. In contrast with other works, this method works completely in the frequency domain by using the spherical harmonic series expansion method to calculate a modified free-space Green function in the rotating system. An analytical solution for the sound radiation from the rotating sound source, which considers motion compensation and a uniform subsonic axial flow, is deduced. Simulations on a rotating point source considering axial flow are performed. Measurements with a fan are examined for different frequencies with standard delay and sum beam-forming and high-resolution algorithms.

Tectonics of the terrestrial litosphere in spherical harmonics

According to the classical tectonics theory, lithosphere is divided into more than twenty plates that are considered as absolutely rigid. However, modern astronomical and geodetic observations reveal disagreements with the standard theory. First of all, there are already more than two hundreds plates and newer ones are still being discovered. Then, it was found out that lithospheric plates cannot be regarded as absolutely rigid, since they are evidently subject to deformations. This work is aimed at expansion of velocities of the global geodetic network stations into vector spherical harmonic series on the assumption of division of the velocity field into toroidal and poloidal components. Spherical harmonics up to fourth degree are obtained for corresponding components and a preliminary attempt of their geophysical interpretation is submitted.

Evaluation of nonlinear effects at the focus of a hemispherical acoustic source

Several theoretical models exist for simulating nonlinear generation of harmonics for weakly or moderately focused waves. Under these conditions, the acoustic field can be considered as a directed beam. In the case of hemispherical sources, this assumption is no longer valid, so development of different models is needed. Here, a theoretical approach is presented for evaluating the amplitude level of the second harmonic generated in hemispherically focused fields. The approach is based on the successive approximation method and includes several steps. First, the linear acoustic problem is solved for a hemispherical source using a spherical harmonics series expansion. Next, the volume sources for the second harmonic generation are calculated using the solution to the linear problem. Finally, these sources are used for obtaining an analytic solution for the second harmonic amplitude at the focus. The developed method was applied to study the acoustic field of the ExAblate 3000 clinical transducer that has a hemispherical shape of 30 cm diameter, operating frequency of 670 kHz, and acoustic power of several hundred watts. Calculations predict that nonlinear effects are weak even for the free-field focusing in water at acoustic power levels up to 1 kW. [Work supported by RSF 14-15-00665.]

Fundamental solutions for general anisotropic multi-field materials based on spherical harmonics expansions

A unified method to evaluate the fundamental solutions for generally anisotropic multi-field materials is presented. Based on the relation between the Rayleigh expansion and the three-dimensional Fourier representation of a homogenous partial differential operator, the proposed technique allows to obtain the fundamental solutions and their derivatives up to the desired order as convergent series of spherical harmonics. For a given material, the coefficients of the series are computed only once, and the derivatives of the fundamental solutions are obtained without any term-by-term differentiation, making the proposed approach attractive for boundary integral formulations and efficient for numerical implementation. Useful general relationships for the computation of derivatives of various order of the fundamental solutions are presented. Furthermore, no particular treatment is needed for mathematically degenerate cases. The fundamental solutions of the Laplace equation and isotropic elastic solids are exactly retrieved as special cases. Numerical results are presented to demonstrate the accuracy of the approach for isotropic elastic, generally anisotropic elastic, transversely isotropic and generally anisotropic piezo-electric and magneto-electro-elastic materials.

Incorporating directivity in the Fourier pseudospectral time-domain method using spherical harmonics.

The pseudospectral time-domain method (PSTD) provides an efficient way to solve the linear acoustics equations. With regards to acoustic modeling and auralization, source directivity as well as head-related directivity have a clear influence on the perceived sound field and have to be included in computations. In this paper directive sources are implemented in the time-domain method PSTD. First, a given frequency dependent source directivity is decomposed onto spherical harmonic functions. The directive source is then implemented through spatial distributions in PSTD that relate to the spherical harmonic functions, and time-dependent functions are assigned to the spatial distributions in order to obtain the frequency content of the directivity. Since any directivity function can be expressed as a summation of series of spherical harmonics, the approach can be used to model any type of directive source. For the evaluation of the method, a directivity function was designed analytically and then modeled in PSTD. Octave band analysis was performed and results show a good agreement between the analytical and simulated directivity. A distance related error was observed. However, for distances above 17.5 grid cells from the source center the average error was small (<0.9 dB) at all octave-bands.

Computing forces on interface elements exerted by dislocations in an elastically anisotropic crystalline material

Driven by the growing interest in numerical simulations of dislocation–interface interactions in general crystalline materials with elastic anisotropy, we develop algorithms for the integration of interface tractions needed to couple dislocation dynamics with a finite element or boundary element solver. The dislocation stress fields in elastically anisotropic media are made analytically accessible through the spherical harmonics expansion of the derivative of Green’s function, and analytical expressions for the forces on interface elements are derived by analytically integrating the spherical harmonics series recursively. Compared with numerical integration by Gaussian quadrature, the newly developed analytical algorithm for interface traction integration is highly beneficial in terms of both computation precision and speed.

Elastic interaction of point defects in cubic and hexagonal crystals

The elastic interaction of two point defects in cubic and hexagonal structures has been considered. On the basis of the exact expression for the tensor Green’s function of the elastic field obtained by the Lifschitz–Rozentsveig for a hexagonal medium, an exact formula for the interaction energy of two point defects has been obtained. The solution is represented as a function of the angle of their relative position on the example of semiconductors such as III-nitrides and α-SiC. For the cubic medium, the solution is found on the basis of the Lifschitz–Rozentsveig Green’s tensors corrected by Ostapchuk, in the weak-anisotropy approximation. It is proven that the calculation of the interaction energy by the original Lifschitz–Rozentsveig Green’s tensor leads to the opposite sign of the energy. On the example of the silicon crystal, the approximate solution is compared with the numerical solution, which is represented as an approximation by a series of spherical harmonics. The range of applicability of the continual approach is estimated by the quantum mechanical calculation of the lattice Green’s function.

Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method.

A rigorous theoretical treatment to analyze the electromagnetic scattering of an inhomogeneous elliptically polarized plane wave by a sphere buried in a lossy half-space is presented. To consider the losses in the media an inhomogeneous plane wave is considered. The incident and the scattered electric field components are expanded in series of vectorial spherical harmonics using the Legendre functions generalized via hypergeometrical and gamma functions, with unknown expansion coefficients. The spectral-domain method to represent the scattered electric field is used in order to compute the scattered-reflected and scattered-transmitted fields, considering the reflection and transmission of each elementary plane wave by the interface. Finally, the unknown coefficients of the scattered field are computed by imposing the boundary condition on the spherical surface. In order to validate the model, a homemade code has been implemented. Comparisons with the simulations performed with a commercial software and the results in the literature are presented.

The exact transformation from spherical harmonic to ellipsoidal harmonic coefficients for gravitational field modeling

The spherical and ellipsoidal harmonic series of the external gravitational potential for a given mass distribution are equivalent in their mutual region of uniform convergence. In an instructive case, the equality of the two series on the common coordinate surface of an infinitely large sphere reveals the exact correspondence between the spherical and ellipsoidal harmonic coefficients. The transformation between the two sets of coefficients can be accomplished via the numerical methods by Walter (Celest Mech 2:389–397, 1970) and Dechambre and Scheeres (Astron Astrophys 387:1114–1122, 2002), respectively. On the other hand, the harmonic coefficients are defined by the integrals of mass density moments in terms of the respective solid harmonics. This paper presents general algebraic formulas for expressing the solid ellipsoidal harmonics as a linear combination of the corresponding solid spherical harmonics. An exact transformation from spherical to ellipsoidal harmonic coefficients is found by incorporating these connecting expressions into the density integral. A computational procedure is proposed for the transformation. Numerical results based on the nearly ellipsoidal Martian moon, Phobos, are presented for validation of the method.

On computation and use of Fourier coefficients for associated Legendre functions

The computation of spherical harmonic series in very high resolution is known to be delicate in terms of performance and numerical stability. A major problem is to keep results inside a numerical range of the used data type during calculations as under-/overflow arises. Extended data types are currently not desirable since the arithmetic complexity will grow exponentially with higher resolution levels. If the associated Legendre functions are computed in the spectral domain, then regular grid transformations can be applied to be highly efficient and convenient for derived quantities as well. In this article, we compare three recursive computations of the associated Legendre functions as trigonometric series, thereby ensuring a defined numerical range for each constituent wave number, separately. The results to a high degree and order show the numerical strength of the proposed method. First, the evaluation of Fourier coefficients of the associated Legendre functions has been done with respect to the floating-point precision requirements. Secondly, the numerical accuracy in the cases of standard double and long double precision arithmetic is demonstrated. Following Bessel’s inequality the obtained accuracy estimates of the Fourier coefficients are directly transferable to the associated Legendre functions themselves and to derived functionals as well. Therefore, they can provide an essential insight to modern geodetic applications that depend on efficient spherical harmonic analysis and synthesis beyond [ $$5~\times ~5$$ ] arcmin resolution.

Jerks and conductivity anisotropy of lower mantle

Conductivity anisotropy of the lower mantle presumably caused by phase change of dielectric magnesiowustite at depths of 1500–2000 km is detectable from jerks. Jerks are induced by currents in the fluid outer core, propagate upward from the CMB through anisotropic conducting mantle, and appear on the Earth’s surface. The surface jerk patterns are studied theoretically from the potential of the geomagnetic field presented as a sum of magnetic and electric modes. Equations for the fields of both modes and their relationship in a weakly anisotropic earth are obtained by the perturbation method. The field potential is expanded into a series of spherical harmonics, and the equations are solved in the frequency and time domains. The surface jerk responses can be inverted to retrieve anisotropy parameters; the goal function in the inversion may correspond to misfit between the model and experimental values either along the horizontal or vertical components.

Analytical Acoustic Power Spectrum Formulations for Rotating Monopole and Dipole Point Sources

Analytical acoustic power spectrum formulations for the rotating monopole and dipole point sources are proposed by employing the spherical harmonic series expansion method. Both the analytical acoustic power spectra and the overall acoustic power show a good agreement with the results obtained from other methods. A nondimensional acoustic power ratio (APR) is employed to investigate the effects of the rotational Mach number, the direction of the dipole source, and the number of sources on the acoustic power output.

Representation of Signals as Series of Orthogonal Functions

This paper gives an introduction to the theory of orthogonal projection of functions or signals. Several kinds of decomposition are explored: Fourier, Fourier-Legendre, Fourier-Bessel series for 1D signals, and Spherical Harmonic series for 2D signals. We show how physical conditions and/or geometry can guide the choice of the base of functions for the decomposition. The paper is illustrated with several numerical examples.

Differential speed rolling of twin-roll-cast 6xxx al alloy strips and its influence on the sheet formability

We demonstrate the feasibility of a technique that combines twin-roll strip casting, asymmetric rolling, and subsequent heat treatment, in obtaining Al alloy sheets with high-strength/high-formability. The precipitation- hardening Al alloy sheet thus obtained exhibited an excellent formability (\(\bar r\)= 1.20, Δr = 0.17) and mechanical properties (σTS = 265 MPa, e = 35%), which cannot be readily obtained via the conventional route based on direct-chill casting and heavy rolling operation. In this study, we examined the effects of the various process conditions used at different stages of the process that contribute to the development of specific textures. Simulation studies based on the generalized spherical harmonic series expansion method and the viscoplastic self-consistent (VPSC) model were conducted to arrive at a comprehensive understanding of the factors associated with the high formability realized in Al alloy sheet. It was found that specific textures evolved via twin-roll strip casting, asymmetric rolling, and heat treatment canceled out the anisotropic characteristics of the individual textures, resulting in the high sheet formability.

Accelerated Method for Predicting Acoustic Far Field and Acoustic Power of Rotating Source

Visualization of acoustic pressure and acoustic velocity field and computation of acoustic power are meaningful to investigate noise radiation and propagation mechanism of turbomachines. However, the conventional surface integral method is very time-consuming because the acoustic response for numerous source–observer pairs should be numerically computed one by one. This paper presents an accelerated method for predicting acoustic pressure and acoustic velocity in the acoustic far field radiated by rotating sources. The accelerated method is derived based on a spherical harmonic series expansion method, which explicitly describes the variation of the acoustic pressure and acoustic velocity with the radial coordinate and azimuthal angle of the observer in acoustic far field. In the proposed accelerated method, only the acoustic pressure of a few reference observer points with different polar angles is directly computed by numerical integration, whereas the acoustic pressure and acoustic velocity of the other observers are fast computed using the interpolation method. Moreover, an accelerated method for computing the acoustic power is also developed, which avoids both the numerical integral of the observer surface and computation of the acoustic velocity. Numerical illustrations indicate that the accelerated method has a good accuracy and is more efficient than the conventional numerical integral method.

Calculation of gravity due to a vertical cylinder using a spherical harmonic series and numerical integration

We developed a spherical harmonic series that represents the gravitational potential and its gravity field due to a buried right vertical cylinder. This series can be used at far- and intermediate-regions, and is fast and accurate, using only a few terms. We compared the values of the fields acquired by this new spherical harmonic series, with ones computed by direct numerical integrations, using a fine-mesh structure for a vertical cylinder. Results of the calculations are shown and performances of the two different methods are compared. Faithfulness of the spherical harmonic series is tested with an inversion example.

Non-linear shape oscillations of rising drops and bubbles: Experiments and simulations

This paper focuses on shape-oscillations of a gas bubble or a liquid drop rising in another liquid. The bubble/drop is initially attached to a capillary and is released by a sudden motion of that capillary, resulting in the rise of the bubble/drop along with the oscillations of its shape. Such experimental conditions make difficult the interpretation of the oscillation dynamics with regard to the standard linear theory of oscillation because (i) amplitude of deformation is large enough to induce nonlinearities, (ii) the rising motion may be coupled with the oscillation dynamics, and (iii) clean conditions without residual surfactants may not be achieved. These differences with the theory are addressed by comparing experimental observation with numerical simulation. Simulations are carried out using Level-Set and Ghost-Fluid methods with clean interfaces. The effect of the rising motion is investigated by performing simulations under different gravity conditions. Using a decomposition of the bubble/drop shape into a series of spherical harmonics, experimental and numerical time evolutions of their amplitudes are compared. Due to large oscillation amplitude, non-linear couplings between the modes are evidenced from both experimental and numerical signals; modes of lower frequency influence modes of higher frequency, whereas the reverse is not observed. Nevertheless, the dominant frequency and overall damping rate of the first five modes are in good agreement with the linear theory. Effect of the rising motion on the oscillations is globally negligible, provided the mean shape of the oscillation remains close to a sphere. In the drop case, despite the residual interface contamination evidenced by a reduction in the terminal velocity, the oscillation dynamics is shown to be unaltered compared to that of a clean drop.