Spherical harmonic representation of the gravitational potential of discrete spherical mass elements

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We present expressions in a spherical harmonic framework for the gravitational potential of discrete point, surface, and volume mass elements located at any depth within a sphere. Through analysis of the spherical harmonic spectrum, insight is gained into the properties of the potentials arising from a variety of mass distributions. A point mass at the surface of a sphere displays the richest harmonic spectrum in all degrees; spectra become increasingly reddened as the source mass is distributed through larger elements of area or volume, or is located at greater depths below the surface of the reference sphere. The spectra of dipolar distributions, useful in representing compensated masses, are depressed, especially in the low harmonic degrees, relative to the spectra of monopole elements.     

Numerical aspects of a spline-based multiresolution recovery of the harmonic mass density out of gravity functionals

We show the numerical applicability of a multiresolution method based on harmonic splines on the 3-D ball which allows the regularized recovery of the harmonic part of the Earth's mass density distribution out of different types of gravity data, for example, different radial derivatives of the potential, at various positions which need not be located on a common sphere. This approximated harmonic density can be combined with its orthogonal anharmonic complement, for example, determined out of the splitting function of free oscillations, to an approximation of the whole mass density function. The applicability of the presented tool is demonstrated by several test calculations based on simulated gravity values derived from EGM96. The method yields a multiresolution in the sense that the localization of the constructed spline basis functions can be increased which yields in combination with more data a higher resolution of the resulting spline. Moreover, we show that a locally improved data situation allows a highly resolved recovery in this particular area in combination with a coarse approximation elsewhere which is an essential advantage of this method, for example, compared to polynomial approximation.     

Measuring magnetic field in the 'diamagnetic' ionosphere

Satellite magnetometers sometimes pass through regions of plasma, such as the terrestrial ionosphere, where the ionization is large enough that some of the original ambient field is excluded from the plasma. This reduction of field inside the plasma region comes from the 'diamagnetic' effect of the charged particles in their helical trajectory around the magnetic field lines. The (container of the) magnetometer will exclude the plasma, and a simple-minded approach, treating the ionosphere in the same way as for a conventional diamagnetic fluid, predicts that the field seen by the magnetometer will be somewhat larger than the (reduced) field in the plasma. However, the 'diamagnetic' properties of the ionosphere are quite different from those of a conventional diamagnetic. In particular, there is a 'reflection' of the ionospheric charged particles at the surface of the magnetometer, and the overall effect is that the magnetometer does actually measure the field present in the plasma before the magnetometer is inserted. Similarly, any leakage fields from the magnetometer have no effect in the magnetosphere.     

Electric field at the seafloor due to a two-dimensional ionospheric current

The relation between the seafloor electric field and the surface magnetic field is studied. It is assumed that the fields are created by a 2-D ionospheric current distribution resulting in the E-polarization. The layered earth below the sea water is characterized by a surface impedance. The electric field at the seafloor can be expressed either as an inverse Fourier transform integral over the wavenumber or as a spatial convolution integral. In both integrals the surface magnetic field is multiplied by a function that depends on the depth and conductivity of the sea water and on the properties of the basement. The fact that surface magnetic data are usually available on land, not at the sea surface, is also considered. Test computations demonstrate that the numerical inaccuracies involved in the convolution method are negligible. The theoretical equations are applied to calculate the seafloor electric fields due to an ionospheric line current or associated with real magnetic data collected by the IMAGE magnetometer array in northern Europe. Two different sea depths are considered: 100 m (the continental shelf) and 5 km (the deep ocean). It is seen that the dependence of the electric field on the oscillation period is weaker in the 5 km case than for 100 m.     

The frequency response of the horizontal magnetic field for a conductive channel

Summary. Babour & Mosnier's results showing no frequency dependence between the anomalous horizontal magnetic field above a conductor and the difference between the horizontal magnetic fields above and below the conductor over a wide range of frequencies led them to conclude that this effect is due to current channelling. A two-dimensional numerical model of a conductive channel, with a uniform horizontal source field, shows the same effect over a wide range of frequencies. Thus local induction can show the same effect.     

The magnetic field of an infinite plane current sheet uniform except for two circular insertions of different uniform conductivities

Summary. The electrical system of currents excited by a uniform electric field of arbitrary direction in an infinite plane sheet of uniform conductivity except for two non-overlapping circular areas is obtained analytically. Using the method introduced by Ashour, the magnetic field of the system is also obtained. The components of this additional field are expressed as line integrals which are suitable for computation. The results reduce, in the special case of one insertion, to those obtained earlier by Ashour & Chapman.
As an illustration, numerical results are obtained for the special case of two equal insertions of zero conductivity.
The analysis and results obtained are useful in estimating the modification of the currents flowing in an ocean and their magnetic field by two islands.     

Modelling of solar quiet magnetic field variations near a conductivity anomaly

Summary. A simplified model of the solar quiet-time ionospheric current system is used to calculate the induced currents in a model earth. The conductivity is assumed to be constant below a depth of about 400 km and zero above that depth. The current induced in the north—south conductivity anomaly under the Rocky Mountains is then estimated from the time-varying potential difference between points at 30 and 45° latitude at the surface of the conducting sphere. The purpose of these calculations is to investigate whether variations in the latitude of the northern hemisphere current system vortex will substantially alter the relationship between the observed magnetic field components at the Earth's surface and the local magnetic field gradient caused by the conductivity anomaly. We find that a 10° shift in the latitude of the ionospheric current focus causes a change of 6 per cent or less in the transfer function from the field components to the gradient in the total field. Thus such latitude shifts cannot explain much of the magnetic field gradient variation at periods near 24 hr that has been observed near Boulder, Colorado.