In December 2019, the International Association of Geomagnetism and Aeronomy (IAGA) Division V Working Group (V-MOD) adopted the thirteenth generation of the International Geomagnetic Reference Field (IGRF). This IGRF updates the previous generation with a definitive main field model for epoch 2015.0, a main field model for epoch 2020.0, and a predictive linear secular variation for 2020.0 to 2025.0. This letter provides the equations defining the IGRF, the spherical harmonic coefficients for this thirteenth generation model, maps of magnetic declination, inclination and total field intensity for the epoch 2020.0, and maps of their predicted rate of change for the 2020.0 to 2025.0 time period.

In December 2019, the 13th revision of the International Geomagnetic Reference Field (IGRF) was released by the International Association of Geomagnetism and Aeronomy (IAGA) Division V Working Group V-MOD. This revision comprises two new spherical harmonic main field models for epochs 2015.0 (DGRF-2015) and 2020.0 (IGRF-2020) and a model of the predicted secular variation for the interval 2020.0 to 2025.0 (SV-2020-2025). The models were produced from candidates submitted by fifteen international teams. These teams were led by the British Geological Survey (UK), China Earthquake Administration (China), Universidad Complutense de Madrid (Spain), University of Colorado Boulder (USA), Technical University of Denmark (Denmark), GFZ German Research Centre for Geosciences (Germany), Institut de physique du globe de Paris (France), Institut des Sciences de la Terre (France), Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation (Russia), Kyoto University (Japan), University of Leeds (UK), Max Planck Institute for Solar System Research (Germany), NASA Goddard Space Flight Center (USA), University of Potsdam (Germany), and Université de Strasbourg (France). The candidate models were evaluated individually and compared to all other candidates as well to the mean, median and a robust Huber-weighted model of all candidates. These analyses were used to identify, for example, the variation between the Gauss coefficients or the geographical regions where the candidate models strongly differed. The majority of candidates were sufficiently close that the differences can be explained primarily by individual modeling methodologies and data selection strategies. None of the candidates were so different as to warrant their exclusion from the final IGRF-13. The IAGA V-MOD task force thus voted for two approaches: the median of the Gauss coefficients of the candidates for the DGRF-2015 and IGRF-2020 models and the robust Huber-weighted model for the predictive SV-2020-2025. In this paper, we document the evaluation of the candidate models and provide details of the approach used to derive the final IGRF-13 products. We also perform a retrospective analysis of the IGRF-12 SV candidates over their performance period (2015–2020). Our findings suggest that forecasting secular variation can benefit from combining physics-based core modeling with satellite observations.

This paper tries to give a clear summary of the various fields that can contribute, intentionally or not, to the IGRF Gauss coefficients at present, leading to some ambiguities. It points out how modellers have developed various practices without any explicit external notification, and argues that it would be logical and useful if we could agree how best to tackle some of the problems that have arisen, and to convey our solution as simply as possible to the user community. It suggests a simple wording for a possible future definition of the IGRF.

Background: The 12th revision of the International Geomagnetic Reference Field (IGRF) was issued in December 2014 by the International Association of Geomagnetism and Aeronomy (IAGA) Division V Working Group V-MOD (http://www.ngdc.noaa.gov/IAGA/vmod/igrf.html). This revision comprises new spherical harmonic main field models for epochs 2010.0 (DGRF-2010) and 2015.0 (IGRF-2015) and predictive linear secular variation for the interval 2015.0-2020.0 (SV-2010-2015).Findings: The models were derived from weighted averages of candidate models submitted by ten international teams. Teams were led by the British Geological Survey (UK), DTU Space (Denmark), ISTerre (France), IZMIRAN (Russia), NOAA/NGDC (USA), GFZ Potsdam (Germany), NASA/GSFC (USA), IPGP (France), LPG Nantes (France), and ETH Zurich (Switzerland). Each candidate model was carefully evaluated and compared to all other models and a mean model using well-defined statistical criteria in the spectral domain and maps in the physical space. These analyses were made to pinpoint both troublesome coefficients and the geographical regions where the candidate models most significantly differ. Some models showed clear deviation from other candidate models. However, a majority of the task force members appointed by IAGA thought that the differences were not sufficient to exclude models that were well documented and based on different techniques.Conclusions: The task force thus voted for and applied an iterative robust estimation scheme in space. In this paper, we report on the evaluations of the candidate models and provide details of the algorithm that was used to derive the IGRF-12 product.

On this occasion the selection of the IGRF for 2000 was left to a small Task Force. Before it was accepted by the Task Force as IGRF 2000, the final candidate model (a truncated version of Ørsted(10c/99)) was compared with a comprehensive set of independent surface and satellite data. The method, data selection, and results of this comparison are described.

We investigate the orthogonality of the potential distributions that are the basis solutions of Laplace's equation appropriate to 3-D ellipsoidal (including spheroidal) coordinate systems, and also the orthogonality of the corresponding vector gradient fields, both over the surface of the ellipsoid, and for integration over the volume of the annular shell between two confocal ellipsoids. The only situation for which there is orthogonality is for the vector gradients when integrated over the annular shell. In the other three cases (potential over surface or annulus, and field over surface) orthogonality can be restored by using an appropriate geometrical weighting factor applied to the integrand; it is therefore still possible to perform the equivalent of a classical spherical harmonic analysis. In the special case of the sphere, there is real orthogonality in all four cases; in effect the weighting factors are all unity. In geodesy, spheroidal harmonic analysis is done using a method that relies on a particular result valid only for potential; it cannot be extended to the corresponding vector field, or to ellipsoidal geometry. The lack of orthogonality over the surface means that care must be taken when interpreting conventional geomagnetic power spectra, and geodetic degree variance, as these no longer correspond exactly to the mean-square values over the actual ellipsoidal surface. We illustrate some of the problems by comparing different versions of the power spectrum for a spheroidal analysis of the global lithospheric magnetic field. We use only simple vector algebra, and do not need to know the details of the actual basis solutions, only that they are the product of three functions, one for each coordinate and involving only that coordinate, and that they satisfy Laplace's equation. Similarly, our results do not depend on the normalization used in the basis functions.

This article describes how a succession of events starting in 1946 led to the world's first laboratory self-exciting simply-connected dynamo in 1963. In Manchester, in l947, an interest in the magnetic deflection of cosmic rays and Babcock's measurement of the magnetic field of a remote star, led Blackett to resurrect the idea of a “fundamental rotation theory” for the production of the Earth's magnetic field. Bullard suggested a geophysical test of the theory – measure the variation of magnetic field with depth inside the Earth. Blackett gave the job of doing this to a newly arrived Junior Lecturer, Runcorn. As the mine work was coming to an end, in 1949 Runcorn took on a Ph.D. student, Lowes, and started him off on a project based on a 1948 suggestion by Bullard that an individual non-dipole field focus could be produced by electromagnetic induction in a rotating eddy in the Earth's conducting fluid core. When Bullard produced his paper, to allow him to get an algebraic solution, he had to assume that his eddy was electrically insulated from the rest of the core. Blackett pointed out that more field might be obtained if the eddies were in electrical contact with the core, and got Lowes to build a laboratory model to simulate and investigate this. The model used a solid-body rotation of a (half) cylindrical “eddy” in a solid “core” having an asymmetrical boundary. A fellow (Theoretical Physics) Ph.D. student Herzenberg became interested, and eventually produced the corresponding theory, for a single rigid-body spherical eddy off-centre in a spherical core. In 1954, Bullard and Gellman produced the first crude numerical “solution” for a kinematic dynamo model in the core; however, there was still great reluctance to accept that there could be self-excitation in a simply-connected homogeneous conductor. Herzenberg then extended his theory to a two-eddy situation in a spherical core, and in 1957 showed rigorously that it could produce a self-exciting dynamo. Now at Newcastle, in 1959, Lowes took on Wilkinson as a Ph.D. student. To test Herzenberg's theory, and look for reversals, Wilkinson built a laboratory dynamo model that eventually achieved self-excitation in 1963; in 1968 another model showed spontaneous reversal.

Until recently there has been nothing in the geomagnetic literature giving the Gauss coefficients (equivalent to magnetic multipole moments) for the magnetic scalar potential produced outside a finite-sized region of electric current. Nor has there been an expression for the corresponding magnetic vector potential. This paper presents a simple expression for the Gauss coefficients in terms of a volume integral over the current, and also a series expansion of the vector potential in terms of these coefficients. We show how our result is related to the classical expressions for the scalar potential given by a spherical current sheet, and to the results of the recent papers by Engels and Olsen (1998), Stump and Pollack (1998) and Kazantsev (1999).

The eleventh generation of the International Geomagnetic Reference Field (IGRF) was agreed in December 2009 by a task force appointed by the International Association of Geomagnetism and Aeronomy (IAGA) Division V Working Group V-MOD. New spherical harmonic main field models for epochs 2005.0 (DGRF-2005) and 2010.0 (IGRF-2010), and predictive linear secular variation for the interval 2010.0–2015.0 (SV-2010-2015) were derived from weighted averages of candidate models submitted by teams led by DTU Space, Denmark (team A); NOAA/NGDC, U.S.A. (team B); BGS, U.K. (team C); IZMIRAN, Russia (team D); EOST, France (team E); IPGP, France (team F); GFZ, Germany (team G) and NASA-GSFC, U.S.A. (team H). Here, we report the evaluations of candidate models carried out by the IGRF-11 task force during October/November 2009 and describe the weightings used to derive the new IGRF-11 model. The evaluations include calculations of root mean square vector field differences between the candidates, comparisons of the power spectra, and degree correlations between the candidates and a mean model. Coefficient by coefficient analysis including determination of weighting factors used in a robust estimation of mean coefficients is also reported. Maps of differences in the vertical field intensity at Earth’s surface between the candidates and weighted mean models are presented. Candidates with anomalous aspects are identified and efforts made to pinpoint both troublesome coefficients and geographical regions where large variations between candidates originate. A retrospective analysis of IGRF-10 main field candidates for epoch 2005.0 and predictive secular variation candidates for 2005.0–2010.0 using the new IGRF-11 models as a reference is also reported. The high quality and consistency of main field models derived using vector satellite data is demonstrated; based on internal consistency DGRF-2005 has a formal root mean square vector field error over Earth’s surface of 1.0 nT. Difficulties nevertheless remain in accurately forecasting field evolution only five years into the future.

DC electric railways produce magnetic fields, not only from the intended traction currents, but also from unintended earth-leakage currents; these fields, particularly those from the leakage currents, are becoming an increasing problem for geomagneticians. This paper introduces the relevant properties of DC-railway traction-power circuits, and the various ways in which earth-leakage currents are produced, and discusses models of how these leakage currents vary along the track and with train position. It describes the geometry of the resultant magnetic fields, and gives the formal algebra for calculating the magnetic field when these leakage currents are known, but also suggests some simple approximations that could be used when the current distribution is not known in detail. This paper also summarises previous relevant papers.

Fellow of the RAS and pioneer of palaeomagnetism.

I agree with Henry Kolm that we need to “stand on the shoulders of giants” (Physics Today, October 2006, page 14), but he needs to put his giants in the right place, and get his facts right.Much of my 1949–52 PhD work at the University of Manchester in the UK was a laboratory-model simulation and extension of an early idea of Edward Bullard’s that a single eddy in Earth’s core could perturb the dipole field sufficiently to give a local focus of the nondipole field. Toward the end of that work, my fellow student Arvid Herzenberg became interested and produced a formal algebraic solution; in 1957 the two aspects were reported together. 1 1. A. Herzenberg, F. J. Lowes, Philos. Trans. R. Soc. London Ser. A 240, 507 (1957). Herzenberg then extended his work on a single eddy to two eddies, and showed that such a two-eddy situation could give a self-exciting dynamo. 2 2. A. Herzenberg, Philos. Trans. R. Soc. London Ser. A 250, 543 (1958). https://doi.org/10.1098/rsta.1958.0007 But Herzenberg’s paper was highly mathematical; it also considered only a steady-state situation, while the geomagnetic dynamo was reversing intermittently. By then I was teaching at Newcastle, so I selected a promising graduating student, Ian Wilkinson, and we set about building a laboratory-scale, self-exciting dynamo based on Herzenberg’s geometry. In 1963 we reported steady-state self-excitation. 3 3. F. J. Lowes, I. Wilkinson, Nature 198, 1158 (1963). https://doi.org/10.1038/1981158a0 Our dynamo was made of Perminvar, a magnetically soft steel alloy, and used 3.5-cm-diameter half-cylinders (not the “two meters” Kolm states). The cylinders rotated with their axes at right angles in a larger stationary block, with a thin layer of mercury (not just “equatorial”) between the rotors and the block. By going to a larger system made of annealed mild steel, with 5-cm-diameter rotors and variable geometry, we eventually produced a reversing dynamo. Kolm says that the field “reversed its direction every 20 minutes,” giving a period of 40 minutes. However, we reported that the reversals “mostly had a period of the order of 5 minutes, but that periods up to 20 minutes have been observed.”In a separate approach in 1955, Bullard published the results of numerical integration of a very simple lumped-constant dynamo model. In that model, based on a Faraday-disk dynamo, the output current flowed through a coil to give positive feedback to the initial imposed axial magnetic field. 4 4. E. C. Bullard, Proc. Cambridge Philos. Soc. 51, 744 (1955). https://doi.org/10.1017/S0305004100030814 Essentially it was a single-stage amplifier with positive feedback; depending on the conditions, it could give periodic oscillation but no reversals. In 1958 Tsuneji Rikitake (in Tokyo, and never Bullard’s student) extended the lumped-constant model to two Faraday disks in series. 5 5. T. Rikitake, Proc. Cambridge Philos. Soc. 54, 89 (1958). https://doi.org/10.1017/S0305004100033223 His model was equivalent to a two-stage amplifier with positive feedback; under certain conditions, it could give oscillations of increasing amplitude, which led to reversals. But the reversals were not periodic; that model is now recognized as an example of a chaotic system. I have some other comments: The best-fit dipole is currently about 500 km from the geocenter—about 4%, not 10%, of Earth’s diameter. Also, the geomagnetic field does not reverse periodically; it is because the reversal record is so erratic that it can be used for dating rocks.REFERENCESSection:ChooseTop of pageREFERENCES <<1. A. Herzenberg, F. J. Lowes, Philos. Trans. R. Soc. London Ser. A 240, 507 (1957). Google ScholarCrossref2. A. Herzenberg, Philos. Trans. R. Soc. London Ser. A 250, 543 (1958). https://doi.org/10.1098/rsta.1958.0007 , Google ScholarCrossref3. F. J. Lowes, I. Wilkinson, Nature 198, 1158 (1963). https://doi.org/10.1038/1981158a0 , Google ScholarCrossref4. E. C. Bullard, Proc. Cambridge Philos. Soc. 51, 744 (1955). https://doi.org/10.1017/S0305004100030814 , Google ScholarCrossref5. T. Rikitake, Proc. Cambridge Philos. Soc. 54, 89 (1958). https://doi.org/10.1017/S0305004100033223 , Google ScholarCrossref© 2007 American Institute of Physics.

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.

When deriving spherical harmonic models of the Earth’s magnetic field, low-degree external field contributions are traditionally considered by assuming that their expansion coefficient q 1 0 varies linearly with the D st -index, while induced contributions are considered assuming a constant ratio Q 1 of induced to external coefficients. A value of Q 1 = 0.27 was found from Magsat data and has been used by several authors when deriving recent field models from Èrsted and CHAMP data. We describe a new approach that considers external and induced field based on a separation of D st = E st + I st into external ( E st ) and induced ( I st ) parts using a 1D model of mantle conductivity. The temporal behavior of q 1 0 and of the corresponding induced coefficient are parameterized by E st and I st , respectively. In addition, we account for baseline-instabilities of D st by estimating a value of q 1 0 for each of the 67 months of Èrsted and CHAMP data that have been used. We discuss the advantage of this new parameterization of external and induced field for geomagnetic field modeling, and describe the derivation of candidate models for IGRF 2005.

As part of the 9th generation of the IGRF defined by IAGA, we proposed a candidate model for DGRF 1995 and two candidate models for DGRF 2000. These candidate models, the derivation of which is described in the present note, are based on the “ Comprehensive Model, Version 4 (CM4) ”, and on the “ Ørsted Main and Secular Variation Model (OSVM) ”; two parent models that have been published elsewhere (Olsen, 2002; Sabaka et al. , 2004; Lowes and Olsen, 2004). However, the main field part of OSVM is contaminated by “leakage” of the ionospheric field and its induced counterpart, which affects mainly the zonal coefficients g 1 0 , g 3 0 , …, by 1–2 nT. We describe the reason for this contamination, and present a method to correct for it. Since not only OSVM, but probably all main field models that are derived primarily from data around local midnight suffer from this effect, the presented scheme can also be applied to approximately correct these models.

The recent satellite magnetic missions, combined with high quality ground observatory measurements, have provided excellent data for main field modelling. Four different groups submitted seven main-field and eight secular-variation candidate models for IGRF-10. These candidate models were evaluated using several different strategies. Comparing models with independent data was found to be difficult. Valuable information was gained by mapping model differences, computing root mean square differences between all pairs of models and between models and the common mean, and by studying power spectra and azimuthal distributions of coefficient power. The resulting adopted IGRF main-field model for 2005.0, an average of three selected candidate models, is estimated to have a formal root mean square error over the Earth’s surface of only 5 nT, though it is likely that the actual error is somewhat larger than this. Due to the inherent uncertainty in secular variation forecasts, the corresponding error of the adopted secular-variation model for 2005.0–2010.0, an average of four selected candidate models, is estimated at 20 nT/a.

Most modern spherical harmonic geomagnetic models based on satellite data include estimates of the variances of the spherical harmonic coefficients of the model; these estimates are based on the geometry of the data and the fitting functions, and on the magnitude of the residuals. However, the accuracy of these estimates depends on the correctness of various assumptions, some of which are in practice not valid. Using the Orsted OSVM model as an example, we show that ignoring the serial correlation of the magnetospheric 'noise' in the data, and not solving separately for the ionospheric field, led to quite inaccurate variance estimates. We estimate correction factors which range from 1/4 to 20, with the largest increases being for the zonal, m = 0, and sectorial, m = n, terms.With no correction, the OSVM variances give a mean-square vector field error of prediction over the Earth's surface at sea level of 1.5 nT(2) (for the field of degrees n = 1-13); applying our correction increases this to 3.5 nT(2).The leakage of the ionospheric field into the solution also gave a large systematic error field, of magnitude about 6 nT.We discuss how our approach is applicable to other similar modelling approaches.

It is shown that comparatively large covariances between some of the coefficients produced in a spherical harmonic analysis of the geomagnetic field result from the use of scalar (intensity) data in a situation in which the field is approximately that of an inclined dipole. As with the Backus effect, such correlations are reduced as relatively more vector data is used.