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Some models of the geomagnetic field in western Europe from 1960 to 1980
Abstract
Achache et al.'s averages over 10 west European observatories of the annual mean magnetic components X, Y, Z are studied from 1953 to 1979. A 1970 jerk is fitted to the data by least squares, and the fit is quantitatively compared with that of a polynomial having the same number of free parameters as the jerk (a quintic). A crude correction for the 11-y sunspot cycle is also attempted. The jerk and quintic are equally good fits to X, the jerk is clearly better for Y, and the quintic is modestly better for Z. The jerk amplitudes of X and Z but not Y depend heavily on whether a sunspot correction is made, and such a correction may be required to estimate the energy of the internal part of the jerk in the manner of Malin and Hodder. The sunspot correction to Y probably cannot be brought above the noise because it is small and highly correlated with both the jerk and the quintic. The sunspot corrections to X and Z are statistically significant but depend on whether the core signal is taken as a jerk or a quintic. Courtillot and Le Mouël's physical model for the jerk predicts that the jerk amplitudes of X, Y, Z will be proportional to the longitudinal derivatives of X, Y and Z. This prediction is roughly verified for the magnitudes but not the signs; it is just possible that the signs in X and Z are not statistically significant.
There is much debate as to whether there was a worldwide geomagnetic jerk in 1969 or 1970. It is agreed that there was an unusual sharp change in the secular variation in the east component, Y, in Europe at that time. This note points out how a localized sudden change in the secular variation pattern of one component in Europe can occur without having any large worldwide effects in any of the components. The accompanying changes in the spherical harmonic coefficients for such a localized change are also discussed. -after Author
Magnetospheric modeling is the art of deriving the configuration of the magnetosphere and representing its behavior. Two main levels exist: descriptive modeling which tries to represent the observed magnetic field B and perhaps the electric field E , and physical modeling which also includes plasma effects. Descriptive modeling must only satisfy Maxwell's equations, in particular ∇ · B = 0; physical modeling adds density ρ, pressure p etc. and must also satisfy conservation laws of magneto‐hydrodynamics (MHD).
Discussed below are descriptive models and simple physical ones, up to the level of the magneto‐hydrostatic (MHS) approximation; full MHD simulations are covered in a companion report. In the MHS approximation the MHD momentum balance condition reduces to
∇p = j × B
The theory of surface operators is described and applied to four surface operators on spheres: the dimensionless surface gradient, ▽1=r▽-r∂r; the dimensionless surface curl, Λ1=rˆ×▽1; the dimensionless surface Laplacian, ▽1²=▽1 · ▽1; and the Funk-Hecke operators, integral operators with axisymmetric kernels. Three methods are given for solving ▽1²g=f as g=▽1−2f; one method works numerically when f has a rapidly convergent expansion in spherical harmonics, the second works when f is smooth in longitude but not latitude, and the third (a Funk-Hecke operation) works when f is rough in all directions. With this apparatus, a complete proof is given of the Helmholtz representation of an arbitrary vector field vS (r), the spherical surface of radius r centered on the origin: there are unique scalar fields f, g, h on S(r) such that v=rˆf+▽1g+Λ1h and 〈g〉r=〈h〉r=0. Here 〈g〉r is the average value of g on S (r). From the Helmholtz representation on spherical surfaces, the Mie or poloidal-toroidal representation in spherical shells is deduced. Suppose S(a,c) is the spherical shell whose inner and outer boundaries are S(a) and S(c). Suppose B is solenoidal in S (a,c), i.e., ▽·B=0 and 〈Br〉a=0. Then there are unique scalar fields P and Q in S(a,c) such that B=▽ × Λ1P + Λ1Q and 〈P〉r=〈Q〉r=0 for a ⪕ r ⪕ c. The fields P = ▽ × Λ1P and Q=Λ1Q are the poloidal and toroidal parts of B. Applications of this formalism to geomagnetic field modeling are discussed. Gauss's resolution of the geomagnetic field B on S(b) into internal and external parts is generalized; if the radial current Jr does not vanish on S (b), then to Gauss's expression must be added a toroidal field on S(b) due entirely to Jr on S(b). A simple proof is given of Runcorn's theorem that to first order in susceptibility no external magnetic field results from magnetization in a horizontally homogeneous spherical shell polarized by sources inside the shell. A Funk-Hecke-based method of modeling ionospheric currents is described, which may be more accurate than truncated spherical harmonic expansions and easier to use than Biot-Savart integrals. Finally, the formalism makes possible the modeling of satellite samples of the geomagnetic field in a spherical shell S (a,c) where electric currents cannot be neglected. Two approximation schemes are described. One is a truncated power series expansion in (c-a)/H, where H is the radial length scale of the currents. The other assumes that most of B in S(a,c) is not due to the currents between S (a) and S(c), and that the currents in S(a,c) are field-aligned. Then the collection of physically possible magnetic fields in S(a,c) is only 50% larger, in a well-defined sense, than the collection of vacuum fields there. Methods of calculation are given explicitly.
Wavelet analysis is applied to detect and characterize singular events, or singularities, or jerks, in the time series made of the last century monthly mean values of the east component of the geomagnetic field from European observatories. After choosing a well-adapted wavelet function, the analysis is first performed on synthetic series including an ``internal,'' or ``main,'' signal made of smooth variation intervals separated by singular events with different ``regularities,'' a white noise and an ``external'' signal made of the sum of a few harmonics of a long-period variation (11 years). The signatures of the main, noise, and harmonic signals are studied and compared, and the conditions in which the singular events can be clearly isolated in the composite signal are elucidated. Then we apply the method systematically to the real geomagnetic series (monthly means of Y from European observatories) and show that five and only five remarkable events are found in 1901, 1913, 1925, 1969, and 1978. The characteristics of these singularities (in particular, homogeneity of some derived functions of the wavelet transform over a large range of timescales) demonstrate that these events have a single source (of course, internal). Also the events are more singular than was previously supposed (their ``regularity'' is closer to 1.5 than to 2., indicating that noninteger powers of time should be used in representing the time series between the jerks).
This paper attempts to summarize briefly the recent and ongoing efforts of the geomagnetism and paleomagnetism community to understand both the earth's magnetic‐field secular variation, and its implications for the core dynamo process. Secular variation is defined in this paper as the general spatio‐temporal variation of the field; secular change will refer to the first time derivative of the spatial field (strict geomagnetic definition of secular variation). We will emphasize here the long‐term (10¹ year or longer) behavior of the historic secular variation (HSV), for this is the variation that is comparable with the paleomagnetic secular variation (PSV). Two important related topics, the 1969/1970 geomagnetic ‘jerk’ in secular acceleration and the spatial geomagnetic field components due to crustal or mantle sources, are treated more fully in other reviews.
A detailed study of the earth's magnetic field is presented. The
historical background of the topic and the instruments used in it are
briefly reviewed. Present knowledge of the external and internal
contributions to geomagnetic variations in the period range from less
than one day to several hundred years are then examined in detail.
Emphasis is placed on recent progress in the period range where these
two types of variations are now known to overlap, particularly on the
solar-cycle effects and on secular variation impulses. The use of the
variations to probe the electrical conductivity of the earth's crust and
mantle and to infer motions in the convecting fluid core is examined.
About 3300 satellite values of geomagnetic intensity and about 700 observatory values of annual mean magnetic vector components from 1960 to 1978 were fitted by three global models of the geomagnetic field B. Each model includes a spatially constant external field whose time dependence is a constant plus another constant times the Dst index, and each model accepts a time-independent station correction at each observatory. The time dependence of the internal Gauss coefficients is either cubic, quintic, or biquadratic (two independent quadratics, one before and one after January 1, 1970); and g1(0) also has an induced term proportional to the Dst index. The rms residual of the data fit is the same for the cubic and biquadratic models and insignificantly smaller for the quintic model. The quintic and biquadratic models have 1164 adjustable parameters, and the cubic has 1038. At a high level of significance the parameters of the best fitting biquadratic rule out a physical model for the magnetic impulse of 1969 in which the level surfaces of electrical conductivity in the lower mantle are approximately spherical, and the radial magnetic field at the core-mantle boundary goes from one quadratic time dependence to another in a year or less.
Information (obtained from Magsat and other sources) published by the U.S. authors between 1983 and 1986 on the large-scale geomagnetic field of deep internal origin and its secular variation (SV) is reviewed. Results on the main field modeling, including reference fields, and on the separaton of core and crustal fields are discussed together with the advances made in geomagnetic field theory and applications of novel methods. Consideration is also given to global SV, short-term global SV, and regional CV analyses; nonsecular impulses, featuring the sudden geomagnetic jerk of 1969; and the frozen-flux core approximation, including downward continuation to the core-mantle boundary constraints, westward drift, and estimates of the fluid velocity at the top of the core.
If the electrical conductivity kappa of the mantle MU depends only on radius r, then for each surface spherical harmonic component of the poloidal geomagnetic field, MU behaves like a causal, time-invariant, real, linear filter. The nature of the filter and its effects are discussed. Mixing permits the magnetic field to change on a time-scale shorter than any smoothing time, so rapid changes do not require a low conductivity. The 1969 magnetic jerk in W Europe appears to have characteristics which demand significant mixing. The starting time to for the 1969 jerk may be the 1956 elbow in Morrison's length-of-day data. Then the data contain either significant mode mixing or an external (sunspot) signal. -from Author
But before I get on to catastrophe theory, let me first describe briefly René Thom’s early work on the classification of manifolds up to cobordism. Two n-manifolds X and Y are said to cobound if together they bound an (n+1)-manifold.
The ratios of the internal to external source terms of the earth's surface potential for certain geomagnetic fluctuations are indicative of the electrical conductivity of the earth's interior. A nonlinear first-order differential equation is derived for these ratios as functions of depth and conductivity in a spherically symmetric earth. This equation may be solved numerically to match an observable surface ratio with a conductivity-depth profile, or it may be solved analytically for a few of the infinite number of possible profiles. This approach provides a good qualitative insight into the problem of geomagnetic induction in a concentrically stratified earth.
French and United Kingdom workers have published reports describing a sudden change in the secular acceleration, called an impulse or a jerk, which took place in 1969. They claim that this change took place in a period of a year or two and that the sources for the alleged jerk are internal. This paper questions their method of analysis, pointing out that their method of piecemeal fitting of parabolas to the data will always create a discontinuity in the secular acceleration where the parabolas join and that the place where the parabolas join is an a priori assumption and not a result of the analysis. It is also shown that the jerk defined as an approximation to the third time derivative is mostly of external origin. A short review of the uses of the terms impulses and jerks is given in the introduction.
Observations of the thickness and convective regime of the outer core layers within the earth are examined to find connections to the geomagnetic secular variation. Correlations are displayed between the earth's angular rotation and the first derivative of declination of the geomagnetic field, showing decade variations on the order of magnitude of 3 x 10 to the minus 12th rad/sec and 10 arcmin/yr, respectively. The earth is regarded as having three layers: an inner core, outer core, and mantle. Each layer rotates with different angular velocities which are also different than the mean body rotation of the whole earth. A dipolar magnetic field is assumed generated by the inner core, and torques are calculated for each layer, resulting from Lorentz forces arising from an interaction between the electric currents sustaining the toroidal fields and the poloidal main field. The mantle conductivity is estimated, and is found to determine the phase lag between rotation and geomagnetic variations.
The observations of magnetic declination and dip made in London over the past 400 years comprise one of the longest and most complete series of such data for anywhere in the world. They give an indication of secular change on a time scale intermediate between that of recent observatory measurements and of archaeomagnetic measurements. We have attempted to catalogue all direct measurements of declination and dip, paying particular attention to completeness in the earlier years, and tracing the data to their original source wherever possible. Sufficient details of the observations are given for an assessment of their reliability and accuracy. The data are then corrected to a single site, fitted with a continuous smooth curve, and discussed.
In July 1971 a report on 'An Experimental, Comprehensive Flare Index and its Derivation for 'Major' Flares, 1955-1969' was published by the present authors as Report UAG-14 of World Data Center A for Solar-Terrestrial Physics. The index apparently proved to be a useful tool for some investigators, and values of the indices for the later years, 1970-1974, were published in Report UAG-52, 'Experimental Comprehensive Solar Flare Indices for Certain Flares, 1970-1974'. Requests have been received for values of the Comprehensive Flare Index (CFI) for still later years. Accordingly, the present report has been prepared for 1975 through 1979, years that complete data for solar cycle 20 and provide information for the ascending branch of cycle 21. (For November-December 1979, it was necessary to use preliminary summaries of flare data. Final Ha flare evaluations were not available for these months at the time of preparation of this report.)
A sudden change of the secular geomagnetic variation was observed in
1970. Differences of annual means of declination, horizontal intensity
and vertical downward intensity were examined for 184 world wide
observatories. It is seen from the spherical harmonic coefficients that
the jerk is limited to the low order harmonics. The phenomenon may be
interpreted as an impulsive jerk to the eccentric dipole in the
direction 90 deg E, 6 deg S, and an increase in the strength of the
dipole moment.
The physics of large-scale magnetic fields in fluids of high electrical
conductivity is examined, with emphasis on the generation of magnetic
fields in astronomical bodies and interactions of the fields with those
bodies. The role and nature of cosmic magnetic fields are described and
solutions to the basic electromagnetic equations are developed.
Consideration is given to magnetic field stress and energy and the
problem of magnetic equilibrium. The propagation of disturbances in a
magnetic fluid is discussed, and the basic properties and interactions
of isolated flux tubes and twisted flux tubes are considered. The
topology of magnetic lines of force, the nonequilibrium of invariant and
non-invariant fields, the breakup and escape of submerged magnetic
fields, the rapid reconnection of magnetic lines of force, the exclusion
of magnetic fields from closed circulation patterns, and the generation
and nature of magnetic fields in turbulent fluids, including the dynamo
problem, are treated, and planetary, solar, stellar and galactic
magnetic fields are examined.
Variations in the distribution of mass within the atmosphere, and changes in the pattern of winds produce fluctuations in all three components of the angular momentum of the atmosphere on time-scales upwards of a few days. It, has been shown that variations in theaxial component of atmospheric angular momentum during the Special Observing Periods in the recent First GARP Global Experiment (FGGE, where GARP is the Global Atmospheric Research Programme) are well correlated with short-term changes in the length of the day. They are consistent with the total angular momentum of the atmosphere and solid Earth being conserved on short timescales (allowing for lunar and solar effects), without requiring significant angular momentum transfer between the Earth's liquid core and solid mantle on timescales of weeks or months. It has also been shown that fluctuations, in the equatorial components of atmospheric angular momentum make a major contribution to the observed wobble of the instantaneous pole of the Earth's rotation with respect to the Earth's crust. A necessary step in the investigation was a re-examination of the underlying theory of non-rigid body rotational dynamics and angular momentum exchange between the atmosphere and solid Earth. Since only viscous or topographic coupling between the atmosphere and solid Earth can transfer angular momentum, no atmospheric flow that everywhere satisfied inviscid equations (including, but not solely, geostrophic flow) could affect the rotation of a spherical solid Earth. New effective angular momentum functions were introduced in order to exploit the available data and allow for rotational and surface loading deformation of the Earth. A theoretical basis has now been established for future routine determinations of atmopheric, angular momentum fluctuations for the purpose of meteorological and geophysical research, including the assessment of the extent to which movements in the solid Earth associated with very large earthquakes contribute to the excitation of the Chandlerian wobble.
Annual mean data from worldwide magnetic observatories show that there
was a jerk (step-change in the second time derivative) in the
geomagnetic field, which took place over an interval of less than 2 yr
around 1970. If such a short-lived phoenomenon originated in the earth's
core, and was still detectable after passing through the mantle to the
surface, this would imply a much lower conductivity for the lower mantle
than had been believed previously. Here it is shown by an objective test
that most of the jerk has an internal origin.
Models of the main (core) magnetic field are limited in accuracy by the nature of the data. Anomaly fields from the crust are a noise source in these calculations which restrict the accuracy to which the main field coefficients can be determined. For meaningful coefficients the degree and order of the model is limited to about 8 for observatory and other surface data and to about 13 for satellite data. Utilizing satellite and surface data together has permitted the incorporation of a solution for the anomaly field at each observatory. These fields vary from a few tens of nT(nanotesla) to several thousand nT. The residuals of the observatory measurements to such models is no longer dominated by the anomaly fields and so is commensurate with the actual measurement accuracy (about 5-20 nT, depending on the observatory). Incorporation of the anomaly estimation has made possible the inclusion of stable time derivatives of the spherical harmonic coefficients up to the third derivative.-from Authors
Long-cally stratified earth 68: 6273—6278. period geomagnetic variations and mantle conductivity: an Hide, R., 1984. Excitation of short-term length of day changes inversion using Bailey's method. Geophys A discussion of impulses and jerks in the Union in Cincinnati, Ohio.). geomagnetic field
- D H Eckhardt
- J Achache
- Le Mouël
- J L Courtillot
Eckhardt, D.H., 1963. Geomagnetic induction in a concentri-Achache, J., Le Mouël, J.L. and Courtillot, V., 1981. Long-cally stratified earth. J. Geophys. Res., 68: 6273—6278. period geomagnetic variations and mantle conductivity: an Hide, R., 1984. Excitation of short-term length of day changes inversion using Bailey's method. Geophys. J.R. Astron. and polar motion. EOS, 65: 196. (Abstract of invited paper Soc., 65: 579—602. G41-02 at the 1984 meeting of the American Geophysical Alldredge, L.R., 1984. A discussion of impulses and jerks in the Union in Cincinnati, Ohio.). geomagnetic field. J. Geophys. Res., 89: 4403—4412.