The Experts below are selected from a list of 165 Experts worldwide ranked by ideXlab platform
Zdeněk Martinec - One of the best experts on this subject based on the ideXlab platform.
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shavel a program for the Spherical Harmonic analysis of a horizontal Vector field sampled in an equiangular grid on a sphere
Computer Physics Communications, 2018Co-Authors: Zdeněk Martinec, David EinspigelAbstract:Abstract A method for performing a Spherical Harmonic analysis, using observed horizontal components of a tangent Vector on a sphere, is presented. The Vector data samples are assumed to be provided in an equiangular grid, which essentially simplifies the least-squares analysis by making use of (1) the block diagonal structure of the normal equations of least squares, (2) the even–odd symmetry of the associated Legendre functions, and (3) the fast Fourier transform of mix-radix. The correct function of the program and its numerical precision is verified by applying it to a data set, derived by evaluating a given set of Vector Spherical Harmonic coefficients. That the program works correctly is demonstrated by the excellent agreement between the input and output Spherical Harmonic coefficients. Program summary Program Title: SHAVEL Program Files doi: http://dx.doi.org/10.17632/nppz4y7wg7.1 Licensing provisions: GPLv3 Programming language: Fortran 2003, Linux External routines: FFTPACK5.1, CPC program library SPHAN Classification: 4.9, 4.10, 4.11 Nature of problem: The least-squares analysis of horizontal Vector field sampled in an equiangular grid on a sphere in terms of horizontal Vector Spherical Harmonics. Solution method: The Vector Spherical Harmonic coefficients of a horizontal Vector field are estimated by the method of least-squares adjustment of data samples distributed in an equiangular grid on a sphere. For such a regular grid the normal matrix is sparse and allows the system of the normal equations to be decomposed into a series of subsystems according to azimuthal order m . The solution of each subsystem is sought by the Gauss elimination. The fast Fourier transform of mix-radix is implemented in (i) setting up the right-hand sides of the normal equations, and (ii) performing the Spherical Harmonic synthesis where the series of Spherical Harmonics are summed.
C. G. Phillips - One of the best experts on this subject based on the ideXlab platform.
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Scalar and Vector Spherical Harmonic spectral equations of rotating magnetohydrodynamics
Geophysical Journal International, 2008Co-Authors: D. J. Ivers, C. G. PhillipsAbstract:SUMMARY Vector Spherical Harmonic analyses have been used effectively to solve laminar and mean-field magnetohydrodynamic dynamo problems with product interactions, such as magnetic induction, anisotropic alpha-effect and anisotropic magnetic diffusion, that are difficult to analyse spectrally in Spherical geometries. Spectral forms of the non-linear rotating, Boussinesq and anelastic, momentum, magnetic induction and heat equations are derived for Spherical geometries from Vector Spherical Harmonic expansions of the velocity, magnetic induction, vorticity, electrical current and gravitational acceleration and from scalar Spherical Harmonic expansions of the pressure and temperature. By combining the Vector Spherical Harmonic spectral forms of the momentum equation and the magnetic induction equation with poloidal–toroidal representations of the velocity and the magnetic field, non-linear Spherical Harmonic spectral equations are also derived for the poloidal–toroidal potentials of the velocity or the momentum density in the anelastic approximation and the magnetic field. Both compact and spectral interaction expansion forms are given. Vector Spherical Harmonic spectral forms of the linearized rotating magnetic induction, momentum and heat equations for a general basic state can be obtained by linearizing the corresponding non-linear spectral equations. Similarly, the Spherical Harmonic spectral equations for the poloidal–toroidal potentials of the velocity and the magnetic field may be linearized. However, for computational applications, new alternative hybrid linearized spectral equations are derived. The algorithmically simpler hybrid equations depend on Vector Spherical Harmonic expansions of the velocity, magnetic field, vorticity, electrical current and gravitational acceleration of the basic state and scalar Spherical Harmonic expansions of the poloidal–toroidal potentials of the perturbation velocity, magnetic field and temperature. The spectral equations derived herein may be combined with the corresponding spectral forms of anisotropic diffusion terms.
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A Vector Spherical Harmonic spectral code for linearised magnetohydrodynamics
ANZIAM Journal, 2003Co-Authors: D. J. Ivers, C. G. PhillipsAbstract:Linearised rotating magnetohydrodynamic stability code for the steady axisymmetric basic states of an electrically conducting fluid sphere is described. The code generates compact hybrid angular spectral forms of the magnetic induction, heat and Boussinesq Navier-Stokes equations, using toroidal and poloidal representations of the perturbation Vector fields, and Vector or scalar Spherical Harmonic expansions of all fields. The momentum equation may include inertial, Coriolis, buoyancy, viscous and magnetic Lorentz forces. Three subroutines evaluate the spectral interactions of products. There are only six radial functions, which are discretised using uniform second-order finite differences. The resulting large scale complex non-hermitian generalised eigen- and critical-value problems are solved using inverse and Newton-Raphson iteration methods, respectively. Test results are presented for several models.
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Dynamo Problems in Spherical and Nearly Spherical Geometries
Dynamo and Dynamics a Mathematical Challenge, 2001Co-Authors: D. J. Ivers, C. G. PhillipsAbstract:Hybrid Vector Spherical Harmonic / poloidal-toroidal Spherical spectral forms of the linearised magnetohydroynamic equations are described. The equations are highly structured with relatively few terms and form the basis of computer codes, which implement a wide range of dynamo problems in Spherical and nearly Spherical geometries.
D. J. Ivers - One of the best experts on this subject based on the ideXlab platform.
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Scalar and Vector Spherical Harmonic spectral equations of rotating magnetohydrodynamics
Geophysical Journal International, 2008Co-Authors: D. J. Ivers, C. G. PhillipsAbstract:SUMMARY Vector Spherical Harmonic analyses have been used effectively to solve laminar and mean-field magnetohydrodynamic dynamo problems with product interactions, such as magnetic induction, anisotropic alpha-effect and anisotropic magnetic diffusion, that are difficult to analyse spectrally in Spherical geometries. Spectral forms of the non-linear rotating, Boussinesq and anelastic, momentum, magnetic induction and heat equations are derived for Spherical geometries from Vector Spherical Harmonic expansions of the velocity, magnetic induction, vorticity, electrical current and gravitational acceleration and from scalar Spherical Harmonic expansions of the pressure and temperature. By combining the Vector Spherical Harmonic spectral forms of the momentum equation and the magnetic induction equation with poloidal–toroidal representations of the velocity and the magnetic field, non-linear Spherical Harmonic spectral equations are also derived for the poloidal–toroidal potentials of the velocity or the momentum density in the anelastic approximation and the magnetic field. Both compact and spectral interaction expansion forms are given. Vector Spherical Harmonic spectral forms of the linearized rotating magnetic induction, momentum and heat equations for a general basic state can be obtained by linearizing the corresponding non-linear spectral equations. Similarly, the Spherical Harmonic spectral equations for the poloidal–toroidal potentials of the velocity and the magnetic field may be linearized. However, for computational applications, new alternative hybrid linearized spectral equations are derived. The algorithmically simpler hybrid equations depend on Vector Spherical Harmonic expansions of the velocity, magnetic field, vorticity, electrical current and gravitational acceleration of the basic state and scalar Spherical Harmonic expansions of the poloidal–toroidal potentials of the perturbation velocity, magnetic field and temperature. The spectral equations derived herein may be combined with the corresponding spectral forms of anisotropic diffusion terms.
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Geomagnetism and Schmidt quasi-normalization
Geophysical Journal International, 2005Co-Authors: D. E. Winch, D. J. Ivers, J. P. R. Turner, Robert J. SteningAbstract:SUMMARY Spherical Harmonic analysis of the main magnetic field of the Earth and its daily variations is the numerical determination of coefficients of solid Spherical Harmonics in the mathematical expressions used for the magnetic scalar potential of fields of internal and external origin. The coefficients are determined from Vector components of the field and their purpose is to represent the Vector field, not to reconstruct the magnetic scalar potential. An alternative interpretation of the Spherical Harmonic analysis is presented: namely the determination of the coefficients of a series representation of the magnetic Vector field on a Spherical surface in orthonormal real Vector Spherical Harmonics, which correspond to the internal and external fields, and an additional non-potential toroidal field. The numerical values of the coefficients of an orthonormal Vector Spherical Harmonic series have a direct physical significance, which is not obscured by some arbitrary normalization of the Vector Spherical Harmonics. Therefore, we propose a Schmidt Vector normalization to be used in conjunction with the Schmidt quasinormalization of associated Legendre functions. A property of orthonormalized functions is that the standard deviations of the coefficients determined by the method of least squares from ideal data, which are uniformly accurate and uniformly globally distributed, are constant for all coefficients. The real Vector Spherical Harmonic analysis of the geomagnetic field is extended to a Spherical shell and conditions that restrict the radial dependence of the Vector Spherical Harmonic coefficients are examined. In particular, two hypotheses for the current systems deriving from the non-potential toroidal component of the magnetic field over the surface of a sphere are presented, namely, Earth‐air currents and field-aligned currents.
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A Vector Spherical Harmonic spectral code for linearised magnetohydrodynamics
ANZIAM Journal, 2003Co-Authors: D. J. Ivers, C. G. PhillipsAbstract:Linearised rotating magnetohydrodynamic stability code for the steady axisymmetric basic states of an electrically conducting fluid sphere is described. The code generates compact hybrid angular spectral forms of the magnetic induction, heat and Boussinesq Navier-Stokes equations, using toroidal and poloidal representations of the perturbation Vector fields, and Vector or scalar Spherical Harmonic expansions of all fields. The momentum equation may include inertial, Coriolis, buoyancy, viscous and magnetic Lorentz forces. Three subroutines evaluate the spectral interactions of products. There are only six radial functions, which are discretised using uniform second-order finite differences. The resulting large scale complex non-hermitian generalised eigen- and critical-value problems are solved using inverse and Newton-Raphson iteration methods, respectively. Test results are presented for several models.
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Dynamo Problems in Spherical and Nearly Spherical Geometries
Dynamo and Dynamics a Mathematical Challenge, 2001Co-Authors: D. J. Ivers, C. G. PhillipsAbstract:Hybrid Vector Spherical Harmonic / poloidal-toroidal Spherical spectral forms of the linearised magnetohydroynamic equations are described. The equations are highly structured with relatively few terms and form the basis of computer codes, which implement a wide range of dynamo problems in Spherical and nearly Spherical geometries.
P. Sorcik - One of the best experts on this subject based on the ideXlab platform.
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The gravitational field of topographic-isostatic masses and the hypothesis of mass condensation II-the topographic-isostatic geoid
Surveys in Geophysics, 1996Co-Authors: J. Engels, E. W. Grafarend, P. SorcikAbstract:In Part I we focussed on a convergent representation of the gravitational potential generated by topographic masses on top of the equipotential surface at Mean Sea Level , the geoid , and by those masses which compensate topography. Topographic masses have also been condensated, namely represented by a “ single layer ”. Part II extends the computation of the gravitational field of topographic-isostatic masses by a detailed analysis of its force field in terms of Vector-Spherical Harmonic functions . In addition, the discontinuous mass-condensated topographic gravitational force Vector (“head force”) is given. Once we identify the Moho discontinuity as one interface of isostatically compensated topographical masses, we have computed the topographic potential and the gravitational potential which is generated by isostatically compensated masses at Mean Sea Level , the geoid, and illustrated by various figures of geoidal undulations. In comparison to a data oriented global geoid computation of J. Engels (1991) the conclusion can be made that the assumption of a constant crustal mass density, the basic condition for isostatic modeling, does not apply. Instead density variations in the crust, e.g. between oceanic and continental crust densities, have to be introduced in order to match the global “real” geoid and its topographic-isostatic model. The performed analysis documents that the standard isostatic models based upon a constant crustal density are unreal .
David Einspigel - One of the best experts on this subject based on the ideXlab platform.
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shavel a program for the Spherical Harmonic analysis of a horizontal Vector field sampled in an equiangular grid on a sphere
Computer Physics Communications, 2018Co-Authors: Zdeněk Martinec, David EinspigelAbstract:Abstract A method for performing a Spherical Harmonic analysis, using observed horizontal components of a tangent Vector on a sphere, is presented. The Vector data samples are assumed to be provided in an equiangular grid, which essentially simplifies the least-squares analysis by making use of (1) the block diagonal structure of the normal equations of least squares, (2) the even–odd symmetry of the associated Legendre functions, and (3) the fast Fourier transform of mix-radix. The correct function of the program and its numerical precision is verified by applying it to a data set, derived by evaluating a given set of Vector Spherical Harmonic coefficients. That the program works correctly is demonstrated by the excellent agreement between the input and output Spherical Harmonic coefficients. Program summary Program Title: SHAVEL Program Files doi: http://dx.doi.org/10.17632/nppz4y7wg7.1 Licensing provisions: GPLv3 Programming language: Fortran 2003, Linux External routines: FFTPACK5.1, CPC program library SPHAN Classification: 4.9, 4.10, 4.11 Nature of problem: The least-squares analysis of horizontal Vector field sampled in an equiangular grid on a sphere in terms of horizontal Vector Spherical Harmonics. Solution method: The Vector Spherical Harmonic coefficients of a horizontal Vector field are estimated by the method of least-squares adjustment of data samples distributed in an equiangular grid on a sphere. For such a regular grid the normal matrix is sparse and allows the system of the normal equations to be decomposed into a series of subsystems according to azimuthal order m . The solution of each subsystem is sought by the Gauss elimination. The fast Fourier transform of mix-radix is implemented in (i) setting up the right-hand sides of the normal equations, and (ii) performing the Spherical Harmonic synthesis where the series of Spherical Harmonics are summed.