Electromagnetic core–mantle coupling—I. Explaining decadal changes in the length of day

Measured changes in the Earth's length of day on a decadal timescale are usually attributed to the exchange of angular momentum between the solid mantle and fluid core. One of several possible mechanisms for this exchange is electromagnetic coupling between the core and a weakly conducting mantle. This mechanism is included in recent numerical models of the geodynamo. The 'advective torque', associated with the mantle toroidal field produced by flux rearrangement at the core–mantle boundary (CMB), is likely to be an important part of the torque for matching variations in length of day. This can be calculated from a model of the fluid flow at the top of the outer core; however, results have generally shown little correspondence between the observed and calculated torques. There is a formal non-uniqueness in the determination of the flow from measurements of magnetic secular variation, and unfortunately the part of the flow contributing to the torque is precisely that which is not constrained by the data. Thus, the forward modelling approach is unlikely to be useful. Instead, we solve an inverse problem: assuming that mantle conductivity is concentrated in a thin layer at the CMB (perhaps D"), we seek flows that both explain the observed secular variation and generate the observed changes in length of day. We obtain flows that satisfy both constraints and are also almost steady and almost geostrophic, and therefore assert that electromagnetic coupling is capable of explaining the observed changes in length of day.     

Magnetic induction fields (2–30 cpd) on Hawaii Island and their implications regarding electrical conductivity in the oceanic mantle

Summary. Geomagnetic time-variations observed at several sites on the island of Hawaii are analysed for the effects of island bathymetry as well as for the inductive response of the deeper mantle. The data are generally consistent with the deep conductivity profile derived using lower frequency, electromagnetic data from the Island of Oahu. Hawaii data fit better if that model is modified to give the upper 200 km of the mantle a lower conductivity of 0.02 S/m compared to 0.1 S/m for Oahu. The data are represented by a complex, frequency-dependent function of location, T _u, relating the vertical variation Z to a component U of the horizontal variation ( T_u = Z/U ). The direction of U is nearly frequency independent at each site but is different for each site. Below a frequency of about 30 cycles per day, the functions, T _u, at any two sites are found to be related by a real constant. This suggests that the deeper conductivity structure is the same beneath each site. This result is consistent with quasi-static induction in a non-uniformly conducting thin sheet above a stratified conductivity structure. The response of such a model can be written as T _u= Aq , where q is a quasi-uniform, complex, frequency-response function characterizing the effect of the deep conductivity and A is a spatially dependent parameter parameterizing the effect of variable conductivity in the thin sheet. The parameter A may be estimated by fitting observational estimates of T _u to models of deep conductivity structure.     

Geomagnetic field analysis - I. Stochastic inversion

Summary. Most of the Earth's magnetic field and its secular change originate in the core. Provided the mantle can be treated as an electrical insulator, stochastic inversion enables surface observations to be analysed for the core field. A priori information about the variation of the field at the core boundary leads to very stringent conditions at the Earth's surface. The field models are identical with those derived from the method of harmonic splines (Shure, Parker & Backus) provided the a priori information is specified appropriately.
The method is applied to secular variation data from 106 magnetic observatories. Model predictions for fields at the Earth's surface have error estimates associated with them that appear realistic. For plausible choices of a priori information the error of the field at the core is unbounded, but integrals over patches of the core surface can have finite errors. The hypothesis that magnetic fields are frozen to the core fluid implies that certain integrals of the secular variation vanish. This idea is tested by computing the integrals and their standard and maximum errors. Most of the integrals are within one standard deviation of zero, but those over the large patches to the north and south of the magnetic equator are many times their standard error, because of the dominating influence of the decaying dipole. All integrals are well within their maximum error, indicating that it will be possible to construct core fields, consistent with frozen flux, that satisfy the observations.     

The use of equivalent current systems in the interpretation of Geomagnetic Deep Sounding data

Summary. Many geomagnetic variation anomalies are probably caused by the channelling, through small-scale bodies, of electric currents induced in much larger conductors elsewhere. Consequently, the direct interpretation of anomalous magnetic fields by modelling the electromagnetic response of conductive structures may give misleading results. It is suggested that, rather than attempting to proceed directly from the electromagnetic fields to conductivity models, we should instead take the intermediate step of determining the distribution of anomalous current flow.
Maps of the anomalous fields over a conductive structure can be generated from inter-station transfer functions. If it is assumed that the internal currents are concentrated in a thin sheet at a specified depth, the equivalent current system in the sheet can be computed directly from the vertical magnetic field. The most straightforward method of performing this calculation is to compute the Fast Fourier Transform of the magnetic field data, and then to apply a wavenumber filter.
The presence of any vertical currents invalidates the thin sheet model. However, if the spatial distribution of a horizontal component of the anomalous magnetic field is also known, the presence of any vertical currents can be detected directly, and their position determined. The value of the methods is illustrated by applying them to the interpretation of a Geomagnetic Deep Sounding survey of the Kenya rift valley.     

Geomagnetic field analysis—IV. Testing the frozen-flux hypothesis

Summary. We present a model of the magnetic field at the core–mantle boundary, for epoch 1959.5, based on a large set of observatory and survey measurements. Formal error estimates for the radial field at the core are 50 μT, compared with 30 and 40 μT for our previous MAGSAT (1980) and POGO (1970) models.
Current work on the determination of the velocity of the core fluid relies on the assumption that the core behaves as a perfect conductor, so that the field lines remain frozen to the fluid at the core surface. This frozen-flux condition requires that the integrated flux over patches of the core surface bounded by contours of zero radial field remain constant in time. A new method is presented for constructing core fields that satisfy these frozen-flux constraints. The constraints are non-linear when applied to main field data, unlike the case of secular variation which was considered in an earlier paper. The method is applied to datasets from epochs 1969.5 and 1959.5 to produce fields with the same flux integrals as the 1980 model.
The frozen-flux hypothesis is tested by comparing the changes in the flux integrals between 1980/1969.5, 1969.5/1959.5 and 1980/1959.5 with their errors. We find that the hypothesis can be rejected with 95 per cent confidence. The main evidence for flux diffusion is in the South Atlantic region, where a new null flux curve appears between 1960 and 1970, and continues to grow at a rapid rate from 1970 to 1980. However, the statistical result depends critically on our error estimates for the field at the core surface, which are difficult to assess with any certainty; indeed, doubling the error estimates negates the statistical argument. The conclusion is therefore, at this stage, tentative, and requires further evidence, either from older data, if good enough, or from future satellite measurements.     

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.     

A comparison of analytic and numerical results for a two-dimensional control model in electromagnetic induction – I. B-polarization calculations

Summary. A conducting slab of finite thickness divided into three segments of different conductivities and overlying a perfect conductor is proposed as a suitable two-dimensional 'control' model for testing the accuracy of the various numerical modelling programs that are available for calculating the fields induced in the Earth by an external, time-varying magnetic source. An analytic solution is obtained for this control model for the case of the magnetic field everywhere parallel to the conductivity boundaries ( B -polarization). Values of the field given by this solution for a particular set of model parameters are calculated at selected points on the surface and on a horizontal plane inside the conductor, and are tabulated to three figure accuracy for reference. They are used to check the accuracy of the results given by the finite difference program of Brewitt-Taylor & Weaver and the finite element program of Kisak & Silvester for the same model. Improved formulae for calculating the derived electric field components in B -polarization are first developed for incorporation in the finite difference program, and these give surface electric fields within 1 per cent of the analytic values, while all three field components inside the conductor are calculated to better than 96 per cent accuracy by the finite difference program. The results given by the finite element program are not quite so satisfactory. Errors somewhat greater than 10 per cent are present and although the program requires much less disk space it takes rather more CPU time to complete the calculations.     

H-polarization induction over an ocean edge coupled to the mantle by a conducting crust

Summary. The Wiener—Hopf technique is used to obtain an exact analytical solution for the problem of H -polarization induction over the edge of a perfectly conducting thin sheet, representing an ocean, electrically connected to a perfectly conducting mantle through a slab of finite conductivity and thickness, which represents the Earths crust. It is shown that the induced currents resulting from this type of induction process are drawn up into the sea from the cust and mantle with the greatest concentration of current near the ocean edge. The surface impedance over the land surface, is calculated for various mantle depths and is shown to increase sharply as the coastline is approached. The magnetic field along the ocean floor is also plotted as a function of distance from the coastline, and the results are found to agree very well with those calculated previously by approximate and numerical methods.     

Generalized thin sheet analysis in magnetotellurics: an extension of Price's analysis

Summary. Price's thin sheet analysis for electromagnetic fields has been extended in order to model the effects of crustal resistivity and conductivity variations on magnetotelluric fields. These extensions allow for a general layered medium below the crust and also account for the vertical resistance of the crust as well as its horizontal conductance. An important parameter emerges from the analysis which determines the distance it takes for the crustal current levels to readjust to changes in the crustal conductance. This adjustment distance is given by the square root of the conductivity thickness product multiplied by the resistivity thickness product. Approximate analytical solutions were developed for two-dimensional geometries in order to demonstrate these effects as well as the modifications produced by finite source wavelengths.     

Time series analysis of the 10 000 yr geomagnetic secular variation record from SE Australia

Summary. A 10 000 yr continuous secular variation record from intensively dated lake sediments in SE Australia has been subjected to periodogram and maximum entropy method analysis. Tests on synthetic data reveal some of the limitations of the latter method, particularly when applied to complex number series. Anticlockwise precession of the magnetic vector at a period of 5000 ± 1000 yr is tentatively ascribed to dipole precession, and clockwise precession at a period of about 2800 yr is probably due to westward drift of features of the non-dipole field.
The effect of calibrating the radiocarbon time-scale is important and results in periodicity shifts of up to 25 per cent. Even for well-dated lacustrine sequences power spectra are poorly constrained: it is thus possible that the geomagnetic secular variation on a time-scale of thousands of years is more uniform than often supposed. Mismatches between declination and inclination spectra can arise as a natural consequence of certain types of source mechanism and should not be simply attributed to figments of the analysis employed.     

B-polarization induction in two generalized thin sheets at the surface of a conducting half-space

Summary. A new closed-form solution is obtained analytically for a B- polarization induction problem of geophysical interest, in which a local region of the Earth is represented by a generalized thin sheet at the surface of and in electrical contact with a uniformly conducting half-space. The generalized sheet, first introduced by Ranganayaki & Madden, is a mathematical idealization of a double layer which consists, in this problem, of two adjacent half-planes with distinct conductances representing a surface conductivity discontinuity such as an ocean—coast boundary, underlain by a uniform sheet of finite integrated resistivity representing the lower crust. The resistive sheet exerts a considerable mathematical influence on the solution causing, under certain conditions, an additional pole to appear in one of the forms of contour integral by which the solution can be expressed; it also weakens or eliminates field singularities that would otherwise occur at the conductance discontinuity. A numerical calculation is made for model parameters typifying an ocean—coast boundary underlain by a highly resistive crust. It is found that the residue of the pole associated with the resistive sheet dominates the solution for this example, the main consequence of which is a huge increase in the horizontal range over which the induced currents adjust themselves between the different 'skin-effect' distributions at infinity on either side of the model. Moreover the solution shows that this 'adjustment distance' has a more complicated dependence on the conductance and integrated resistivity of the sheet than that given simply by the square root of their product which was the length parameter proposed by Ranganayaki & Madden.     

Using geomagnetic secular variation to separate remanent and induced sources of the crustal magnetic field

Magnetic fields originating from magnetized crustal rocks dominate the geomagnetic spectrum at wavelengths of 0.1–100 km. It is not known whether the magnetization is predominantly induced or remanent, and static surveys cannot discriminate between the two. Long‐running magnetic observatories offer a chance, in principle, of separating the two sources because secular variation leads to a change in the main inducing field, which in turn causes a change in the induced part of the short‐wavelength crustal field. We first argue that the induced crustal field, b ^I( t ), is linearly related to the local core field, B ( t ), through a symmetric, trace‐free matrix A : b ^I( t )= A B ( t ). We then subtract a core field model from the observatory annual means and invert the residuals for three components of the remanent field, b ^R( t ), and the five independent elements of A . Applying the method to 20 European observatories, all of which have recorded for more than 50 years, shows that the most difficult task is to distinguish b ^R from the steady part of b ^I. However, for nine observatories a time‐dependent induced field fits the data better than a steady remanent field at the 99 per cent confidence level, suggesting the presence of a significant induced component to the magnetization.     

Numerical models of mantle convection with secular cooling

A 2-D time-dependent finite-difference numerical model is used to investigate the thermal character and evolution of a convecting layer which is cooling as it convects. Two basic cooling modes are considered: in the first, both upper and lower boundaries are cooled at the same rate, while maintaining the same temperature difference across the layer; in the second, the lower boundary temperature decreases with time while the upper boundary temperature is fixed at 0°C. The first cooling mode simulates the effects of internal heating while the second simulates planetary cooling as mantle convection extracts heat from, and thereby cools, the Earth's core. The mathematical analogue between the effects of cooling and internal heating is verified for finite-amplitude convection. It is found that after an initial transient period the central core of a steady but vigorous convection cell cools at a constant rate which is governed by the rate of cooling of the boundaries and the viscosity structure of the layer. For upper-mantle models the transient stage lasts for about 30 per cent of the age of the Earth, while for the whole mantle it lasts for longer than the age of the Earth. Consequently, in our models the bulk cooling of the mantle lags behind the cooling of the core-mantle boundary. Models with temperature-dependent viscosity are found to cool in the same manner as models with depth-dependent viscosity; the rate of cooling is controlled primarily by the horizontally averaged variation of viscosity with depth. If the Earth's mantle cools in a similar fashion, secular cooling of the planet may be insensitive to lateral variations of viscosity.     

Geomagnetic field analysis—III. Magnetic fields on the core—mantle boundary

Summary. The method of stochastic inversion, previously applied to secular variation data, is applied to main field data. Adaptations to the method are required: non-linear, as well as linear, data are used; allowance is made for crustal components in the observatory data; and the prior information is specified differently. The requirement that the models should satisfy a finite lower bound on the Ohmic heating in the core provides strong prior information and gives finite error estimates at the core—mantle boundary.
The new method is applied to data from the epochs 1969.5 and 1980.0. The resulting field models are very much more complex than other models, such as the IGRF models extrapolated to the core, and show considerable small-scale detail which, on the basis of the error analysis, can be believed.
The flux integral over the northern hemisphere is computed at each epoch; the difference between the two epochs is approximately one standard deviation, suggesting that the question as to whether the decay of the dipole is consistent with the frozen-flux hypothesis has been resolved in favour of the hypothesis.     

Induction in a thin sheet of variable conductance at the surface of a stratified earth – II. Three-dimensional theory

Summary. The algorithm of Dawson & Weaver for modelling electromagnetic induction effects in a thin sheet at the surface of a uniform earth is modified to permit the use of a layered earth model. The theory is developed in Fourier space in terms of the toroidal and poloidal transfer functions instead of with the Green's function approach which was used by Dawson & Weaver. The integral equation for the surface electric field and most of the integral formulae for the derived field components are the same as before, except for the inclusion of additional integral the kernel of which has to be calculated numerically with the aid of fast Hankel transforms. The accuracy of the results is tested by comparing solutions with those obtained from a related 2-D algorithm and finally an example of 3-D modelling is presented.     

Monopoly

Summary. A model for the geomagnetic secular variation field is given, consisting of a series of magnetic monopoles at the surface of the Earth's core. These are distributed according to the density of the data to allow more detailed representation in areas where the density of observations is high, without introducing spurious detail where data are sparse. A monopole model is calculated from observatory secular change data for the epoch 1957.5–1962.5 and its usefulness assessed.     

Taking into account truncation problems and geomagnetic model accuracy in assessing computed flows at the core-mantle boundary

SUMMARY
Since the time Roberts & Scott (1965) first expressed the key 'frozen flux' hypothesis relating the secular variation of the geomagnetic field (SV) to the flow at the core surface, a large number of studies have been devoted to building maps of the flow and inferring its fundamental properties from magnetic observations at the Earth's surface. There are some well-known difficulties in carrying out these studies, such as the one linked to the non-uniqueness of the flow solution [if no additional constraint is imposed on the flow (Backus 1968)] which has been thoroughly investigated. In contrast little investigation has been made up to now to estimate the exact importance of other difficulties, although the different authors are usually well aware of their existence. In this paper we intend to make as systematic as possible a study of the limitations linked to the use of truncated spherical harmonic expansions in the computation of the flow. Our approach does not rely on other assumptions than the frozen flux, the insulating mantle and the large-scale flow assumptions along with some simple statistical assumptions concerning the flow and the Main Field. Our conclusions therefore apply to any (toroidal, steady or tangentially geostrophic) of the flow models that have already been produced; they can be summarized in the following way: first, because of the unavoidable truncation of the spherical harmonic expansion of the Main Field to degree 13, no information will ever be derived for the components of the flow with degree larger than 12; second, one may truncate the spherical harmonic expansion of the flow to degree 12 with only a small impact on the first degrees of the flow. Third, with the data available at the present day, the components of the flow with degree less than 5 are fairly well known whereas those with degree greater than 8 are absolutely unconstrained.     

Model of the geoelectrical structure of the mid- and lower mantle in the Europe–Asia region

The conductivity structure of the Earth's mantle was estimated using the induction method down to 2100  km depth for the Europe–Asia region. For this purpose, the responses obtained at seven geomagnetic observatories (IRT, KIV, MOS, NVS, HLP, WIT and NGK) were analysed, together with reliable published results for 11  yr variations. 1-D spherical modelling has shown that, beneath the mid-mantle conductive layer (600–800  km), the conductivity increases slowly from about 1  S  m⁻¹ at 1000  km depth to 10  S  m⁻¹ at 1900  km, while further down (1900–2100  km) this increase is faster. Published models of the lower mantle conductivity obtained using the secular, 30–60  yr variations were also considered, in order to estimate the conductivity at depths down to the core. The new regional model of the lower mantle conductivity does not contradict most modern geoelectrical sounding results. This model supports the idea that the mantle base, situated below 2100  km depth, has a very high conductivity.     

Long-period (30 days1 year) electromagnetic sounding and the electrical conductivity of the lower mantle beneath Europe

The C -response connects the magnetic vertical component and the horizontal gradient of the horizontal components of electromagnetic variations and forms the basis for deriving the conductivitydepth profile of the Earth. Time-series of daily mean values at 42 observatories typically with 50 years of data are used to estimate C -responses for periods between 1 month and 1  yr. The Z : Y method is applied, which means that the vertical component is taken locally whereas the horizontal components are used globally by expansion in a series of spherical harmonics.
In combination with results from previous analyses, the method yields consistent results for European observatories in the entire period range from a few hours to 1  yr, corresponding to penetration depths between 300 and 1800  km.
1-D conductivity models derived from these results show an increase in conductivity with depth z to about 2  S  m^-1 at z =800  km, and almost constant conductivity between z =800 and z =2000  km with values of 310  S  m^-1, in good agreement with laboratory measurements of mantle material. Below 2000  km the conductivity is poorly resolved. However, the best-fitting models indicate a further increase in conductivity to values between 50 and 150  S  m^-1.     

南极布兰斯菲尔德海峡沉积物的地磁场长期变与定年

对位于布兰斯菲尔德海峡东北部的沉积物岩心D1-7进行了初步环境磁学和古地磁研究,获得了连续的相对地磁场强度和方向。沉积物岩性总体为灰色软塑性粘土,中下部出现一层黑色火山灰层。结合该孔沉积物和有孔虫AMS¹⁴C测年结果（王汝建等未发表资料）,相对地磁场强度及其方向提供了12 ka以来连续的定年标尺,其中相对地磁场强度与具有相似沉积速率的南美Laguna Potrok Aike湖泊记录对比提供了六个对比点;同时特征剩磁倾角和磁偏角提供了另外六个对比点。磁化率各向异性分析揭示了全新世早、中和晚期岩心所在位置底流（南极深层水~1000 m）发生了阶段性显著变化,同时磁性矿物含量、粒度、沉积物湿密度等也发生了相应的变化。这些环境磁学和沉积学的变化主要受控于南极相应纬度处的太阳辐射量以及与辐射量相关的夏季季风降水量的变化,降水量增加导致磁性矿物粒度变细。D1-7也记录了一些千年尺度旋回变化,但是南极地区目前缺乏与之相似分辨率的气候参数记录,无法进一步探讨这些气候事件的缘起和分布范围。对比发现,地磁场长期变产生的年龄与沉积物全样有机碳AMS¹⁴C测年结果之间存在系统的差异,在6 ka以来比有机碳年龄年轻,在6 ka之前则比有机碳年龄老。南极布兰斯菲尔德海峡内各种不同水体的混合以及在全新世全球升温背景下冰川的动态变化和由此引起的中、深层水变化,都是造成水体性质复杂的原因。而地磁场强度和方向变化则不受水体性质的影响,因此可以提供更合理的年龄信息。自12 ka至今,25 cm/ka的平均沉积速率表明南极布兰斯菲尔德海峡地区整个全新世内稳定和丰富的沉积物供应。