scholarly journals Active compensation of magnetic field distortions based on vector spherical harmonics field description

AIP Advances ◽  
2017 ◽  
Vol 7 (3) ◽  
pp. 035216
Author(s):  
G. Wyszyński ◽  
K. Bodek ◽  
S. Afach ◽  
G. Bison ◽  
Z. Chowdhuri ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Spherical Harmonics ◽  
Vector Spherical Harmonics ◽  
Active Compensation

Algorithm 1018: FaVeST—Fast Vector Spherical Harmonic Transforms

2021 ◽  
Vol 47 (4) ◽  
pp. 1-24
Author(s):  
Quoc T. Le Gia ◽  
Ming Li ◽  
Yu Guang Wang
Keyword(s):  
Spherical Harmonics ◽  
Spherical Harmonic ◽  
Fourier Coefficients ◽  
Computational Cost ◽  
Tangent Vector ◽  
Tangent Vector Field ◽  
Vector Spherical Harmonics ◽  
Numerical Examples ◽  
Vector Spherical Harmonic ◽  
Tangent Fields

Vector spherical harmonics on the unit sphere of ℝ 3 have broad applications in geophysics, quantum mechanics, and astrophysics. In the representation of a tangent vector field, one needs to evaluate the expansion and the Fourier coefficients of vector spherical harmonics. In this article, we develop fast algorithms (FaVeST) for vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional to N log √ N for N number of evaluation points. The adjoint FaVeST, which evaluates a linear combination of vector spherical harmonics with a degree up to ⊡ M for M evaluation points, has cost proportional to M log √ M . Numerical examples of simulated tangent fields illustrate the accuracy, efficiency, and stability of FaVeST.

Modelling the core magnetic field of the Earth

1982 ◽  
Vol 306 (1492) ◽  
pp. 179-191 ◽  
Keyword(s):  
Magnetic Field ◽  
Spherical Harmonics ◽  
Secular Variation ◽  
Long Wavelength ◽  
Core Field ◽  
The Core ◽  
The Earth ◽  
High Degree ◽  
Current Systems ◽  
The Magnetic Field

The magnetic field generated in the core of the Earth is often represented by spherical harmonics of the magnetic potential. It has been found from looking at the equations of spherical harmonics, and from studying the values of the spherical harmonic coefficients derived from data from Magsat, that this is an unsatisfactory way of representing the core field. Harmonics of high degree are characterized by generally shorter wavelength expressions on the surface of the Earth, but also contain very long wavelength features as well. Thus if it is thought that the higher degree harmonics are produced by magnetizations within the crust of the Earth, these magnetizations have to be capable of producing very long wavelength signals. Since it is impossible to produce very long wavelength signals of sufficient amplitude by using crustal magnetizations of reasonable intensity, the separation of core and crustal sources by using spherical harmonics is not ideal. We suggest that a better way is to use radial off-centre dipoles located within the core of the Earth. These have several advantages. Firstly, they can be thought of as modelling real physical current systems within the core of the Earth. Secondly, it can be shown that off-centred dipoles, if located deep within the core, are more effective at removing long wavelength signals of potential or field than can be achieved by using spherical harmonics. The disadvantage is that it is much more difficult to compute the positions and strengths of the off-centred dipole fields, and much less easy to manipulate their effects (such as upward and downward continuation). But we believe, along with Cox and Alldredge & Hurwitz, that the understanding that we might obtain of the Earth’s magnetic field by using physically reasonable models rather than mathematically convenient models is very important. We discuss some of the radial dipole models that have been proposed for the nondipole portion of the Earth’s field to arrive at a model that agrees with observations of secular variation and excursions.

scholarly journals Effects of Dipole Magnetic Fields on Axisymmetric p-Mode Pulsations

2002 ◽  
Vol 185 ◽  
pp. 294-295
Author(s):  
Hideyuki Saio ◽  
Alfred Gautschy
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Angular Dependence ◽  
Spherical Harmonics ◽  
Field Interaction ◽  
Dipole Magnetic Field ◽  
Derivatives Of

We investigated nonradial pulsations in the presence of a dipole magnetic field in a non-rotating 1.7 M⊙ ZAMS star. Formally, like in the case of pulsation-rotation coupling (Lee & Saio, 1986), the angular dependence of the pulsations is expanded into a series of spherical harmonics of different latitudinal degrees l. To start with, we considered only axisymmetric (m = 0) modes under the adiabatic and the Cowling approximations. In contrast to previous studies of pulsation-magnetic field interaction (Dziembowski & Goode, 1996; Bigot et al., 2000; Cunha & Gough, 2000), we retained the latitudinal derivatives of the perturbed quantities.

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