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Constable, CG.  2011.  Modelling the geomagnetic field from syntheses of paleomagnetic data. Physics of the Earth and Planetary Interiors. 187:109-117.   10.1016/j.pepi.2011.05.004   AbstractWebsite

This review examines results from time-varying geomagnetic field models that span several thousand years, and from variations in dipole moment strength up to million year time scales. For the past 400 years, twin magnetic flux lobes bordering the inner core tangent cylinder in both northern and southern hemispheres dominate the geomagnetic field and appear more or less fixed in location. In contrast, the millennial scale view shows that such features are quite mobile and subject to morphological changes on time scales of a few centuries to a thousand years, possibly reflecting large scale reorganization of core flow. The lobes rarely venture into the Pacific hemisphere, and average fields over various time scales generally reveal two or three sets of lobes, of diminished amplitude. Thus millennial scale models are suggestive of thermal core-mantle coupling generating a weak bias in the average field rather than a strong inhibition of large scale field changes. The recovery of variations in dipole moment on million year time scales allows frequency domain analyses to search for characteristic time scales for core dynamics that might be associated with excursion and reversal rate, time taken for reversals, or any signs of control by Earth's orbital parameters. The spectrum is characteristically red for the time interval 0-160 Ma, suggesting non-stationarity associated with average reversal rate changes, probably reflecting the impact of superchrons and a continually evolving core. Distinct regimes of power law decay with frequency may reflect different physical processes contributing to the secular variation. Evidence for non-stationarity at shorter time-scales is also present in dipole moment variations over 0-2 Ma with average growth rate faster than the decay process. Rates of change of dipole moment and rapid local field variations found in the paleomagnetic record are evaluated in the context of the 400 year historical record and the spectrum of geomagnetic variations for 0-160 Ma. (C) 2011 Elsevier B.V. All rights reserved.





Jackson, A, Constable CG, Walker MR, Parker RL.  2007.  Models of Earth's main magnetic field incorporating flux and radial vorticity constraints. Geophysical Journal International. 171:133-144.   10.1111/j.1365-246X.2007.03526.x   AbstractWebsite

We describe a new technique for implementing the constraints on magnetic fields arising from two hypotheses about the fluid core of the Earth, namely the frozen-flux hypothesis and the hypothesis that the core is in magnetostrophic force balance with negligible leakage of current into the mantle. These hypotheses lead to time-independence of the integrated flux through certain 'null-flux patches' on the core surface, and to time-independence of their radial vorticity. Although the frozen-flux hypothesis has received attention before, constraining the radial vorticity has not previously been attempted. We describe a parametrization and an algorithm for preserving topology of radial magnetic fields at the core surface while allowing morphological changes. The parametrization is a spherical triangle tesselation of the core surface. Topology with respect to a reference model (based on data from the Oersted satellite) is preserved as models at different epochs are perturbed to optimize the fit to the data; the topology preservation is achieved by the imposition of inequality constraints on the model, and the optimization at each iteration is cast as a bounded value least-squares problem. For epochs 2000, 1980, 1945, 1915 and 1882 we are able to produce models of the core field which are consistent with flux and radial vorticity conservation, thus providing no observational evidence for the failure of the underlying assumptions. These models are a step towards the production of models which are optimal for the retrieval of frozen-flux velocity fields at the core surface.

Jackson, A, Constable C, Gillet N.  2007.  Maximum entropy regularization of the geomagnetic core field inverse problem. Geophysical Journal International. 171:995-1004.   10.1111/j.1365-246X.2007.03530.x   AbstractWebsite

The maximum entropy technique is an accepted method of image reconstruction when the image is made up of pixels of unknown positive intensity (e.g. a grey-scale image). The problem of reconstructing the magnetic field at the core-mantle boundary from surface data is a problem where the target image, the value of the radial field B-r, can be of either sign. We adopt a known extension of the usual maximum entropy method that can be applied to images consisting of pixels of unconstrained sign. We find that we are able to construct images which have high dynamic ranges, but which still have very simple structure. In the spherical harmonic domain they have smoothly decreasing power spectra. It is also noteworthy that these models have far less complex null flux curve topology (lines on which the radial field vanishes) than do models which are quadratically regularized. Problems such as the one addressed are ubiquitous in geophysics, and it is suggested that the applications of the method could be much more widespread than is currently the case.

O'Brien, MS, Parker RL, Constable CG.  1999.  Magnetic power spectrum of the ocean crust on large scales. Journal of Geophysical Research-Solid Earth. 104:29189-29201.   10.1029/1999jb900302   AbstractWebsite

The geomagnetic power spectrum R-l is the squared magnetic field in each spherical harmonic degree averaged over a spherical surface. Satellite measurements have given reliable estimates of the spectrum for the part that originates in the core, but above I = 15, where the geomagnetic field arises primarily from crustal magnetization, there is considerable disagreement between various estimates derived from observation. Furthermore, several theoretical models for the spectrum disagree with each other and the data. We have examined observations from a different source, 5000-km-long Project Magnet aeromagnetic survey lines; we make new estimates of the spectrum which overlap with the wavelength interval accessible to the satellites. The usual way the spectrum is derived from observation is to construct a large spherical harmonic decomposition first, then square, weight, and add the Gauss coefficients in each degree, but this method cannot be applied to isolated flight lines. Instead, we apply a statistical technique based on an idea of McLeod and Coleman which relates the geomagnetic spectrum to the power and cross spectra of magnetic field components measured on the survey lines. Power spectra from the 17 aeromagnetic surveys, all of which were conducted over the oceans, are averaged together to improve geographic coverage and reduce variance, and the average spectra are then inverted for the geomagnetic spectrum R-l. Like most of the theoretical models, our spectrum exhibits a maximum, but at a wavelength of 100 km, about a factor of 2 smaller than the closest theoretical prediction. Our spectrum agrees quite well with the most recent estimates based on satellite observations in the range 20 less than or equal to l less than or equal to 50, but above l=50, our values increase slowly, while all the satellite data suggest a sharply rising curve. In this wavelength range we believe our measurements are more trustworthy. Further work is planned to confirm the accuracy of our spectrum when continental survey paths are included.