Spherical Harmonic Expansion Research Articles

Nonlocal thermal transport in magnetized plasma along different directions

Nonlocal thermal transport in magnetized plasmas is studied theoretically and numerically with the Vlasov–Fokker–Planck (VFP) model, in which the magnetic field has nonzero components both perpendicular to and along the temperature gradient. Nonlocal heat transport is found in both the longitudinal and transverse directions, provided the temperature gradients are sufficiently large. The magnetic field tends to reduce the nonlocality of the thermal transport in the direction perpendicular to the magnetic field, i.e., the difference between the heat fluxes predicted by the Braginskii theory and the VFP simulation decreases with increasing magnetic field strength. When the initial temperature gradient is steep, the nonlocal heat flux depends not only on the present temperature profile, but also on its time history. Moreover, the contribution of high-order terms in the spherical harmonic expansion of the electron distribution function becomes important for a magnetized plasma, in particular for thermal transport in the direction perpendicular to the temperature gradient.

The Nonsmooth Dynamics of Combined Slip and Spin Motion Under Dry Friction

We investigate the motion of rigid bodies subject to combined slipping and spinning over a rough flat surface in the presence of dry friction. Integration of Coulomb friction forces over the contact area gives rise to a dynamical system with an isolated discontinuity of codimension 3. Recent results about such vector fields are applied to the motion of flat bodies under the assumption of known, time-independent distributions of normal contact forces and to general bodies where kinematic constraints enforce a state-dependent normal contact force distribution with a discontinuity at the sticking state. In both cases, the equations of motion are transformed into a smooth slow–fast dynamical system. The fixed points of the fast flow indicate the possible directions of combined slip–spin motion immediately before a body stops. We also introduce an approximation of the frictional forces and moments by the leading-order term of a spherical harmonic expansion, which allows for an explicit formulation of the equations of motion. The approximate model captures important empirically observed features of the motion. It is proven analytically and illustrated by examples that the number of fixed points in the approximate model is 2, 4, or 6.

A novel adaptive mesh free approach for even- parity neutron transport equation

In this paper, a novel spatially adaptive Mesh free approach using local nodes and radial basis functions, based on weak-form formulation is presented to solve the even-parity neutron transport equation. A residual based error indicator is proposed and used in an adaptive scheme for error estimation. The local balance of neutrons is used as an indicator to determine the appropriate regions of problem for node refinement. In this regard our previous computer program, MFreent, is upgraded for solving neutron transport equation using K+ variational principle in an adaptive scheme for two-dimensional X-Y geometry. The multi-quadrics radial basis function is used to construct the radial point interpolation shape functions for spatial approximation of even-parity angular flux. Also, the angular part of flux is implemented via spherical harmonics expansion. Some numerical test cases are investigated to evaluate the performance of the locally adaptive node-refinement approach for both fixed source and eigenvalue problems solution. The obtained results confirm the solution accuracy and efficiency of the proposed method.

Modelling the Earth's magnetic field

The Earth's magnetic field can be decomposed into spherical harmonics, and the exact coefficients of the decomposition can be determined through a few measurements of its value at different locations. Using measurements from a magnetometer on board the International Space Station, we computed the multipolar moments of an expansion in spherical harmonics to order 3, calculated the location of the magnetic dip poles, and produced an intensity map of the magnetic field across the globe. The accuracy of the results was evaluated by comparison with data from the International Geomagnetic Reference Field.

Model2SAS: software for small-angle scattering data calculation from custom shapes

To meet the challenges in resolving the complex morphologies of emergent nanoparticles, a program with a user-friendly graphical user interface has been developed for calculating small-angle scattering curves from custom shapes. The software allows STL-format 3D models, models defined by mathematical functions or combinations of the two as initial input. As a transitional stage, lattice models are generated and the orientation-averaged small-angle scattering data can be calculated using typical spherical harmonics expansion. The validity of the protocol is verified by demonstration models with Protein Data Bank structures and known scattering functions. The software is applied to successfully calculate the scattering curves of a porous spherical shell model where traditional mathematical derivation fails.

Mapping Magnetic Signals of Individual Magnetite Grains to Their Internal Magnetic Configurations Using Micromagnetic Models.

Micromagnetic tomography (MMT) is a technique that combines X‐ray micro computed tomography and scanning magnetometry data to obtain information about the magnetic potential of individual grains embedded in a sample. Recovering magnetic signals of individual grains in natural and synthetic samples provides a new pathway to study the remanent magnetization that carries information about the ancient geomagnetic field and is the basis of all paleomagnetic studies. MMT infers the magnetic potential of individual grains by numerical inversion of surface magnetic measurements using spherical harmonic expansions. The magnetic potential of individual particles in principle is uniquely determined by MMT, not only by the dipole approximation, but also more complex, higher order, multipole moments. Here, we show that such complex magnetic information together with both particle shape and mineral properties severely constrains the internal magnetization structure of an individual grain. To this end, we apply a three dimensional micromagnetic model to predict the multipole signal from magnetization states of different local energy minima. We show that for certain grains it is even possible to uniquely infer the magnetic configuration from the inverted magnetic multipole moments. This result is crucial to discriminate single‐domain particles from grains in more complex configurations such as multi‐domain or vortex states. As a consequence, our investigation proves that by MMT it is feasible to select statistical ensembles of magnetic grains based on their magnetization states, which opens new possibilities to identify and characterize stable paleomagnetic recorders in natural samples.

Spectral consistency of satellite and airborne data: Application of an equivalent dipole layer for combining satellite and aeromagnetic data sets

We present a novel approach for the combination of aeromagnetic and satellite data applying an equivalent dipole layer and spherical harmonic (SH) expansion of the dipoles. The method involves two steps: (1) inversion for the magnetic parameters of the equivalent dipole layer and (2) conversion of the magnetic parameters into SH coefficients. With this approach, SH analysis can be used for a regional study area, and, for example, the long-wavelength range of aeromagnetic data can be replaced with satellite data. We test our approach on the third, fourth, and fifth editions of the Magnetic Anomaly Map of Australia, which were releveled using independent aeromagnetic data sets for long-wavelength correction. The results indicate that the magnetic compilations leveled to long-haul control lines have a good agreement with the LCS-1 satellite model in the range of SH degree 40 to 110 (corresponding to half-wavelengths between 180 to 500 km), whereas the third edition of the Magnetic Anomaly Map of Australia demonstrates poor control over long- wavelengths in this spectral range. Our analysis suggests that even the carefully processed fifth edition would benefit from the replacement of long-wavelength data with satellite data.

Assessing the potential for simplification in global climate model cloud microphysics

Abstract. Cloud properties and their evolution influence Earth's radiative balance. The cloud microphysical (CMP) processes that shape these properties are therefore important to represent in global climate models. Historically, parameterizations in these models have grown more detailed and complex. However, a simpler formulation of CMP processes may leave the model results mostly unchanged while enabling an easier interpretation of model results and helping to increase process understanding. This study employs sensitivity analysis of an emulated perturbed parameter ensemble of the global aerosol–climate model ECHAM-HAM to illuminate the impact of selected CMP cloud ice processes on model output. The response to the perturbation of a process serves as a proxy for the effect of a simplification. Autoconversion of ice crystals is found to be the dominant CMP process in influencing key variables such as the ice water path and cloud radiative effects, while riming of cloud droplets on snow has the most influence on the liquid phase. Accretion of ice and snow and self-collection of ice crystals have a negligible influence on model output and are therefore identified as suitable candidates for future simplifications. In turn, the dominating role of autoconversion suggests that this process has the greatest need to be represented correctly. A seasonal and spatially resolved analysis employing a spherical harmonics expansion of the data corroborates the results. This study introduces a new application for the combination of statistical emulation and sensitivity analysis to evaluate the sensitivity of a complex numerical model to a specific parameterized process. It paves the way for simplifications of CMP processes leading to more interpretable climate model results.

Acoustic radiation force and torque on a spheroid near an interface

Acoustic radiation force and torque on a spheroid of arbitrary size near a rigid or pressure release interface are calculated using expansions of the pressure field in terms of both spherical and spheroidal wave functions. The spheroidal expansion coefficients are obtained by satisfying the boundary conditions at the interface and at the surface of the spheroid. Conditions at the interface are satisfied using a virtual (image) spheroid, while satisfying conditions at the spheroid surface require the use of addition theorems for spheroidal wave functions. The radiation force and torque are expressed as a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields [Ilinskii et al., J. Acoust. Soc. Am. 144, 568–576 (2018)]. Far-field asymptotes are used to relate the spheroidal and spherical harmonic expansion coefficients analytically. Results from the present analytical model are compared with those from a finite-element model. The effect of the interface on the radiation force and torque relative to that on a spheroid in a free field is discussed. [B.E.S. is supported by the Applied Research Laboratories Chester M. McKinney Graduate Fellowship in Acoustics.]

Incorporation of anisotropic scattering into the method of characteristics

In this study, we incorporate an anisotropic scattering scheme involving spherical harmonics into the method of characteristics (MOC). The neutron transport solution in a light water reactor can be significantly improved because of the impact of an anisotropic scattering source with the MOC flat source approximation. Several problems are selected to verify the proposed scheme and investigate its effects and accuracy. The MOC anisotropic scattering source is based on the expansion of spherical harmonics with Legendre polynomial functions. The angular flux, scattering source, and cross section are expanded in terms of the surface spherical harmonics. Later, the polynomial is expanded to achieve the odd and even parity of the source components. Ultimately, the MOC angular and scalar fluxes are calculated from a combination of two sources. This paper presents various numerical examples that represent the hot and cold conditions of a reactor core with boron concentration, burnable absorbers, and control rod materials, with and without a reflector or baffle. Moreover, a small critical core problem is considered which involves significant neutron leakage at room temperature. We demonstrate that an anisotropic scattering source significantly improves solution accuracy for the small core high-leakage problem, as well as for practical large core analyses.

Applicability of multipole decomposition to plasmonic- and dielectric-lattice resonances.

Periodic nanoparticle arrays have attracted considerable interest recently since the lattice effect can lead to spectrally narrow resonances and tune the resonance position in a broad range. Multipole decomposition is widely used to analyze the role of the multipoles in the resonance excitations, radiation, and scattering of electromagnetic waves. However, previous studies have not addressed the validity and accuracy of the multipole decomposition around the lattice resonance. The applicability of the exact multipole decomposition based on spherical harmonics expansion has not been demonstrated around the lattice resonance with the strong multipole coupling. This work studies the two-dimensional periodic arrays of both plasmonic and dielectric nanospheres and compares the multipole decomposition results with the analytic ones around their lattice resonances. We study both the effective polarizabilities of multipoles and the scattering spectra of the structures. The analytical results are calculated from the coupled dipole-quadrupole model. This study demonstrates that the exact multipole decomposition agrees well with the numerical simulation around lattice resonances. Only a small number of multipoles are required to represent the results accurately.

Zeeman-Doppler imaging of five young solar-type stars

Context. The magnetic activity of the Sun changes with the solar cycle. Similar cycles are found in other stars as well, but their details are not known to a similar degree. Characterising stellar magnetic cycles is important for the understanding of the stellar and solar dynamos that are driving the magnetic activity. Aims. We present spectropolarimetric observations of five young, solar-type stars and compare them to previous observations, with the aim to identify and characterise stellar equivalents of the solar cycle. Methods. We use Zeeman-Doppler imaging (ZDI) to map the surface magnetic field and brightness of our targets. The magnetic field is decomposed into spherical harmonic expansions, from which we report the strengths of the axisymmetric versus non-axisymmetric and poloidal versus toroidal components, and we compare them to the Rossby numbers of the stars. Results. We present five new ZDI maps of young, solar-type stars from December 2017. Of special interest is the case of V1358 Ori, which had gone through a polarity reversal between our observations and earlier ones. A less evident polarity reversal might also have occurred in HD 35296. There is a preference for a more axisymmetric field, and possibly a more toroidal field, for the more active stars with lower Rossby number, but a larger sample should be studied to draw any strong conclusions from this. For most of the individual stars, the amounts of toroidal and poloidal field have stayed on levels similar to those in earlier observations. Conclusions. We find evidence for a magnetic polarity reversal having occurred in V1358 Ori. An interesting target for future observations is χ1 Ori, which may have a short magnetic cycle of a few years. The correlation between the brightness maps and the magnetic field is mostly poor, which could indicate the presence of small-scale magnetic features of different polarities that cancel one another out and are thus not resolved in our maps.

Scattering by a rigid sphere of audio sound generated by a parametric array loudspeaker.

This work investigates the scattering by a rigid sphere of audio sound generated by a parametric array loudspeaker (pal). A computationally efficient method utilizing a spherical harmonic expansion is developed to calculate the quasilinear solution of audio sound fields based on both Kuznetsov and Westervelt equations. The accuracy of using the Westervelt equation is examined, and the rigid sphere scattering effects are simulated with the proposed method. It is found the results obtained using the Westervelt equation are inaccurate near the sphere at low frequencies. Contrary to conventional loudspeakers, the directivity of the audio sound generated by a pal severely deteriorates behind a sphere, as the ultrasounds maintaining the directivity of the audio sound are almost completely blocked by the sphere. Instead, the ultrasounds are reflected and generate audio sound on the front side of the sphere. It means that a listener in front of the pal will hear the audio sound scattered back after introducing the sphere as if it is reflected by the sphere. The experiment results are also presented to validate the numerical results.

Ionospheric Nighttime Enhancements at Low Latitudes Challenge Performance of the Global Ionospheric Maps

In this study, two ionospheric nighttime enhancement (INE) events at low latitudes are selected to investigate their spatial features through the observations from Global Navigation Satellite System (GNSS) receivers and ionosondes. For the first time, we present the detailed spatial pictures of premidnight and postmidnight INEs under geomagnetically quiet conditions. The two INE events have the maximum extents of about 11° × 34° and 17° × 25° (longitude × latitude), respectively. Dramatic latitudinal and longitudinal features are revealed in the two INEs. We perform a comparison between the products of Global Ionospheric Maps (GIMs) and total electron content (TEC) measurement from GNSS receivers. However, GIMs fail to capture the TEC distribution during INEs owing to their limited spatial and temporal resolution. Considering the extent of INEs from the observations, the spherical harmonic (SH) expansion adopted by the GIM models needs to upgrade the degree and order to 36. The pixel-based methods developed from two GIM models are required to reduce their grid size for higher spatial resolution. The recommended time interval is shorter than 30 min. Among seven GIMs, CODG and JPLG maps generally have the best performance in reproducing the latitudinal structure of the ionosphere.

Computer-Assisted Image Processing System for Early Assessment of Lung Nodule Malignancy.

Simple SummaryLung cancer is the second most common cancer in men after prostate cancer and in women after breast cancer, but it is the leading cause of cancer death among both genders. This manuscript proposes a new computer-aided diagnosis system that uses only a single computed tomography scan to diagnose the pulmonary nodule as benign or malignant. This system helps in the early detection of the pulmonary nodules and shows its ability to identify the pulmonary nodules precisely.Lung cancer is one of the most dreadful cancers, and its detection in the early stage is very important and challenging. This manuscript proposes a new computer-aided diagnosis system for lung cancer diagnosis from chest computed tomography scans. The proposed system extracts two different kinds of features, namely, appearance features and shape features. For the appearance features, a Histogram of oriented gradients, a Multi-view analytical Local Binary Pattern, and a Markov Gibbs Random Field are developed to give a good description of the lung nodule texture, which is one of the main distinguishing characteristics between benign and malignant nodules. For the shape features, Multi-view Peripheral Sum Curvature Scale Space, Spherical Harmonics Expansion, and a group of some fundamental morphological features are implemented to describe the outer contour complexity of the nodules, which is main factor in lung nodule diagnosis. Each feature is fed into a stacked auto-encoder followed by a soft-max classifier to generate the initial malignancy probability. Finally, all these probabilities are combined together and fed to the last network to give the final diagnosis. The system is validated using 727 nodules which are subset from the Lung Image Database Consortium (LIDC) dataset. The system shows very high performance measures and achieves , , and for the accuracy, sensitivity, and specificity, respectively. This high performance shows the ability of the system to distinguish between the malignant and benign nodules precisely.

3-D Sound Image Reproduction Method Based on Spherical Harmonic Expansion for 22.2 Multichannel Audio

In this paper, we propose a three-dimensional (3-D) sound image reproduction method based on spherical harmonic (SH) expansion for 22.2 multichannel audio. 22.2 multichannel audio is a 3-D sound field reproduction system that has been developed for ultra-high definition television (UHDTV). This system can reproduce 3-D sound images by simultaneously driving 22 loudspeakers and two sub-woofers. To control the 3-D sound image, vector base amplitude panning (VBAP) is conventionally used. VBAP can control the direction of 3-D sound image by weighting the input signal and emitting it from three loudspeakers. However, VBAP cannot control the distance of the 3-D sound image because it calculates the weight by only considering the image’s direction. To solve this problem, we propose a novel 3-D sound image reconstruction method based on SH expansion. The proposed method can control both the direction and distance of the 3-D sound image by controlling the sound directivity on the basis of spherical harmonics (SHs) and mode matching. The directivity of the 3-D sound image is obtained in the SH domain. In addition, the distance of the 3-D sound image is represented by the mode strength. The signal obtained by the proposed method is then emitted from loudspeakers and the 3-D sound image can be reproduced accurately with consideration of not only the direction but also the distance. A number of experimental results show that the proposed method can control both the direction and distance of 3-D sound images.

Validation of regional and global ionosphere maps from GNSS measurements versus IRI2016 during different magnetic activity

Abstract This manuscript explores the divergence of the Vertical Total Electron Content (VTEC) estimated from Global Navigation Satellite System (GNSS) measurements using global, regional, and International Reference Ionosphere (IRI) models over low to high latitude regions during various magnetic activity. The VTEC is estimated using a territorial network consisting of 7 GNSS stations in Egypt and 10 GNSS stations from the International GNSS Service (IGS) Global network. The impact of magnetic activity on VTEC is investigated. Due to the deficiency of IGS receivers in north Africa and the shortage of GNSS measurements, an extra high interpolation is done to cover the deficit of data over North Africa. A MATLAB code was created to produce VTEC maps for Egypt utilizing a territorial network contrasted with global maps of VTEC, which are delivered by the Center for Orbit Determination in Europe (CODE). Thus we can have genuine VTEC maps estimated from actual GNSS measurements over any region of North Africa. A Spherical Harmonics Expansion (SHE) equation was modelled using MATLAB and called Local VTEC Model (LVTECM) to estimate VTEC values using observations of dual-frequency GNSS receivers. The VTEC calculated from GNSS measurement using LVTECM is compared with CODE VTEC results and IRI-2016 VTEC model results. The analysis of outcomes demonstrates a good convergence between VTEC from CODE and estimated from LVTECM. A strong correlation between LVTECM and CODE reaches about 96 % and 92 % in high and low magnetic activity, respectively. The most extreme contrasts are found to be 2.5 TECu and 1.3 TECu at high and low magnetic activity, respectively. The maximum discrepancies between LVTECM and IRI-2016 are 9.7 TECu and 2.3 TECu at a high and low magnetic activity. Variation in VTEC due to magnetic activity ranges from 1–5 TECu in moderate magnetic activity. The estimated VTEC from the regional network shows a 95 % correlation between the estimated VTEC from LVTECM and CODE with a maximum difference of 5.9 TECu.

Small bodies non-uniform gravity field on-board learning through Hopfield Neural Networks

Small bodies environment is usually difficult to be modelled for a number of reasons. Among the others, the uncertainty associated to the non-uniform gravitational field requires in-situ observations for its refinement, or its identification. This operation becomes even more challenging in case the orbiting platform is a CubeSat or, in general, a platform with reduced computational power as well as a high autonomy requirement. In this paper, a new approach to reconstruct on-board the gravity field of either unknown or partially known bodies is presented. In particular, the use of a Hopfield Neural Network (HNN) to reconstruct the coefficients of a Spherical Harmonics Expansion (SHE), that is assumed to approximate the gravity field of the body, is described. A comparison with an Extended Kalman Filter (EKF) used for parameter estimation is presented and the differences of the two methods are critically discussed: due to the structure of the HNN, the former results to be computationally faster and lighter than a stand-alone EKF used for parameter estimation.

A machine learning-based particle-particle collision model for non-spherical particles with arbitrary shape

In this study, contacts between identical non-spherical particles are investigated by machine learning approaches. Artificial neural network models are developed to correlate relative positions and orientations of the pair with contact occurrence and properties (mean contact point, mean norm, and inter-penetration depth). To evaluate the idea, eight different particles are generated by super-ellipsoid and spherical harmonic expansion. With these methods, pairs of non-spherical particles (regular or irregular shaped) are generated, and the contact properties are computed through vertices-based calculation. Using random translation and rotation, 80000 datasets are collected for each case and are utilized for training, validating, and evaluating the models. After training the proposed models, they enable accurate and rapid predictions of the contact occurrence and properties of non-spherical particles with arbitrary shape. It requires a smaller amount of data than discrete element method (DEM) applications. This study provides collision models that can be applied to large-scale DEM simulations.

Inversion of the satellite observations of the tidally induced magnetic field in terms of 3-D upper-mantle electrical conductivity: method and synthetic tests

SUMMARY The interaction of the oceanic tidal flow with the Earth’s main magnetic field provides a powerful natural source of electromagnetic (EM) energy suitable for suboceanic upper-mantle electrical conductivity sounding. In this paper, we have developed and tested a new frequency-domain, spherical harmonic-finite element approach to the inverse problem of global EM induction. It is set up for an effective inversion of satellite-observed tidally induced magnetic field in terms of 3-D structure of the electrical conductivity in the suboceanic upper mantle. Before proceeding to the inversion of Swarm-derived models of tidal magnetic signatures, we have performed a series of parametric studies, using the 3-D conductivity model WINTERC-e as a testbed. The WINTERC-e model has been derived using state-of-the-art laboratory conductivity measurements of mantle minerals, and thermal and compositional model of the lithosphere and upper mantle WINTERC-G. The latter model is based on the inversion of global surface waveforms, satellite gravity and gradiometry measurements, surface elevation and heat flow data in a thermodynamically self-consistent framework. Therefore, the WINTERC-e model, independent of any EM data, represents an ideal target for synthetic tests of the 3-D EM inversion. We tested the impact of the truncation degree of the spherical-harmonic expansion of the M2 tidal signal, the effect of random noise in synthetic data and inclusion of the N2 and O1 tidal constituents on the ability to recover the suboceanic upper-mantle conductivity structure. We demonstrate that with suitable regularization we can successfully reconstruct the 3-D upper-mantle conductivity beneath world oceans. In the ideal noise-free case, the correlation coefficient between the target and recovered conductivity is greater than 0.8 in the 150–270 km depth range.