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Steady state toroidal magnetic field at Earth’s core-mantle boundary
Abstract
Measurements of the dc electrical potential near the top of earth's mantle have been extrapolated into the deep mantle in order to estimate the strength of the toroidal magnetic field component at the core-mantle interface. Recent measurements have been interpreted as indicating that at the core-mantle interface, the magnetic toroidal and poloidal field components are approximately equal in magnitude. A motivation for such measurements is to obtain an estimate of the strength of the toroidal magnetic field in the core, a quantity important to our understanding of the geomagnetic field's dynamo generation. Through the use of several simple and idealized calculation, this paper discusses the theoretical relationship between the amplitude of the toroidal magnetic field at the core-mantle boundary and the actual amplitude within the core. Even with a very low inferred value of the toroidal field amplitude at the core-mantle boundary, (a few gauss), the toroidal field amplitude within the core could be consistent with a magnetohydrodynamic dynamo dominated by nonuniform rotation and having a strong toroidal magnetic field.
... The dashed curves correspond to the upper-lower mantle boundary located midway in the mantle. and the solid curves correspond to a boundarylocated midway in the mantle, and the solid curves correspond to a boundary location 20% of the mantle's thickness above 9 the core-mantle boundary (Levy and Pearce, 1991). . . . . . . . . . 72 ...
... This assumption is commonly made as little physical insight is gained from the additional computational expense incurred from a continuously variable conductivity model (i.e., Levy and Pearce, 1991). In the case of vanishing diffusivity the magnetic field evolves according to the first term on the right hand side which states that any differential fluid motion, non-parallel to B, will stretch B in that direction is the mobile differential operator, :Fe = 2n x v is the Coriolis force, :FL = B· 'VB / po is the Lorentz force, :FA = pg is the Archimedean force and : ...
... The analytic calculations were performed by Levy and Pearce (1991) and the results are displayed in Figure (2.1). Here is plotted the ratio of the toroidal magnetic field strength at the core-mantle boundary to the maximum value occurring at the shell of shear as a function of the radial position of the shear layer for the outer .50% of the core. ...
The effect of sudden changes in the Earth's moment of inertia on the hydromagnetic state of the core is studied. Rapid changes in georotation, due to ice age transgression and regression, are described as varying boundary conditions in an axisymmetric Earth model containing both viscous and electromagnetic coupling. The deterministic equations describing the limit of rapid rotation are employed in conjunction with restricted 2-D predictive magneto-fluid equations. A kinematic description is devised for both buoyancy driven mass motions and the regeneration of the poloidal magnetic field. A pseudo-spectral method is used to solve the incompressible magneto-fluid equations. The variables are collocated in radius using Chebyshev polynomials and the pseudospectral evaluations in colatitude are done using associated Legendre polynomials. Time dependence and magnetic diffusion are controlled by a modified second order semi-implicit Runge Kutta scheme. Deterministic steady state solutions were found in full agreement with Hollerbach and Jones (1993a,b; 1995). Steady state boundary layers, arising from differential motion of the outer core boundaries, were found to induce significant departures for both alpha^2 - and alphaomega-dynamo steady state configurations. The hydromagnetic communication time of the core, determined the predictive magneto-fluid equations, is found to be consistent with the deterministic calculations. Within the context of this model, it is concluded that a causal connection is plausible between geomagnetic transients and significant changes in the Earth's moment of inertia.
... The toroidal component of the geomagnetic field is presumed to be located somewhere deep inside the Earth, being inaccessible for contemporary observations, because the Earth's mantle prevents its propagation to the surface. In regards to initial ideas on the Earth's toroidal magnetic field observations, see [23]. ...
Observations of exoplanets open a new area of scientific activity and the structure of exoplanet magnetospheres is an important part of this area. Here we use symmetry arguments and experiences in spherical dynamo modeling to obtain the set of possible magnetic configurations for exoplanets and their corresponding host stars. The main part of our results is that the possible choice is much richer than the basic dipole magnetic field of both exoplanets and stars. Other options, for example, are quadrupole configurations or mixed parity solutions. Expected configurations of current sheets for the above mentioned exoplanet host star systems are presented as well.
... Some attempts to observationally constrain the toroidal Correspondence: takahashi.futoshi.386@m.kyushu-u.ac.jp Department of Earth and Planetary Sciences, Faculty of Sciences, 33 Kyushu University, 6-10-1 Hakozaki, Higashi-ku, 812-8581 Fukuoka, Japan magnetic field at the CMB have been pursued by electric potential measurements over distances larger than 1,000 km (Lanzerotti et al. 1993;Shimizu et al. 1998), whereas there are also some discussions on the consistency of such observations with dynamo theory (Levy and Pearce 1991;Shimizu and Utada 2004). ...
I quantitatively test a method of toroidal field imaging at the core-mantle boundary (CMB) using a synthetic magnetic field and core surface flow data from a 3-D self-consistent numerical dynamo model with a thin electrically conducting layer overlying the CMB, like the D ″ layer. With complete knowledge of the core flow, the imaged toroidal field well reproduces the magnitude and pattern of the dynamo model toroidal field. However, quality of the imaging depends strongly on latitude. In particular, the amplitude and correlation between the dynamo model and the imaged toroidal fields decline substantially at low latitude. Such degradation in imaging quality is due to inability to account for the radial derivative of the toroidal field, that is, an effect of magnetic diffusion, which is not incorporated in the method.
... For B , = 20 mT and using the relatively short length-scale (35), then for mantle conductivity model 3, the toroidal field in the interior of the core is some 70 times greater than our magnetostrophic estimate (32), namely B , = 2900 mT. This is greater than the upper bound of-600 mT for the average field strength set by Backus (1975) using thermodynamic arguments, and exceeds conventional estimates-10 mT based on numerical models of strong-field dynamos (see, for example, Levy & Pearce 1991). For mantle-conductivity models with less conductance than model 3, or for a weak dynamo, the conclusion that excessive field strengths are necessary for a misfit of-1 is only reinforced. ...
Owing to exchanges of angular momentum between the Earth's fluid outer core and the overlying solid mantle, the Earth's rotational rate fluctuates on periods of a few years to a few decades. However, the mechanism which allows the exchange of angular momentum is not understood. Here we examine the possibility that core-mantle coupling is predominantly electromagnetic and thus responsible for the decadal length of day variations. the electromagnetic couple on the mantle can be divided into poloidal and toroidal parts, and, by requiring continuity of the horizontal component of the electric field at the core-mantle boundary, the toroidal couple can be divided into separate advective and leakage parts. the poloidal couple results entirely from the interaction of the poloidal field with currents induced by its time variation; the advective couple results from the dragging of poloidal field lines through a conducting mantle; and the leakage couple results from the diffusion of toroidal magnetic field from the core's interior into the mantle. the poloidal and advective couples can be estimated by using models of the downward continued poloidal field and models of the core velocity. We find that neither the poloidal couple nor the advective couple exhibit sufficient variability to account for the decadal length of day variations. If this is true, and if core-mantle coupling is indeed predominantly electromagnetic, then most of the variability in the length of day must result from the leakage couple, which, unfortunately, cannot be calculated directly from surface observations.
We assume that the horizontal component of the magnetic field is continuous across the core-mantle boundary, that the frozen-flux approximation adequately describes the time dependence of the horizontal component of the magnetic field at the core surface, and that most of this time dependence results from steady core motion. Then by treating the determination of the toroidal field at the core-mantle boundary as an inverse problem, we find that only very strong and spatially complex toroidal field models are consistent with both advection in the core and the decadal length of day variations. We argue that strong toroidal fields are necessary to account for the length-of-day variations since there is significant cancellation when the magnetic stress is integrated over the core-mantle boundary (CMB), the necessary time-dependent torque resulting from the slight and temporary non-cancellation of magnetic stress established by a slowly varying and spatially complex toroidal field. But, since our toroidal field models are too strong according to dynamo theory, produce electric fields at the Earth's surface which are stronger than measured values, and produce ohmic heating which either exceeds or contributes an unacceptably large fraction of the Earth's surface heat flow, we deem the toroidal-field models consistent with our analysis of electromagnetic coupling to be physically unreasonable. Thus, we argue that core-mantle coupling is not predominantly electromagnetic. However, this conclusion may not hold if, for example, core motion is highly time dependent, or if a strong diffusive boundary layer is present beneath the core-mantle boundary, the presence of which may allow for a significant discontinuity in the horizontal component of the magnetic field and the breakdown of the frozen-flux approximation.
"The mechanism for generating the geomagnetic field remains one of the
central unsolved problems in geoscience." So states the report on the
National Geomagnetic Initiative (NGI) prepared by the U.S. Geodynamics
Committee, et al [1993], with advice from the NGI Workshop held in
Washington D.C. in March 1992. All analyses of the geomagnetic data
point to the core as containing the source of the field and "The basic
premise that virtually everyone accepts is that the Earth's magnetism is
created by a self-sustaining dynamo driven by fluid motions in Earth's
core" (NGI, p.135). Dynamical questions at once arise, such as "What is
the energy source driving those motions?" Jacobs [1953] proposed that
the solid inner core (SIC) is the result of the freezing of the fluid
outer core (FOC). Verhoogen [1961] noticed that the release of latent
heat at the inner core boundary (ICB) during freezing would help drive
thermal convection in the FOC, and Braginsky [1963] pointed out that the
release of the light alloying elements during fractionation at the ICB
would provide compositional buoyancy. These two sources suffice to
supply the geodynamo with energy throughout geological time, even in the
absence of dissolved radioactivity in the core [Braginsky and Roberts,
1994a; Kuang et al, 1994]. Stevenson [1991] argues that potential
differences on the core-mantle boundary (CMB) of electrochemical origin
may be partially responsible for the geomagnetic field.
For the computation of the electromagnetic (EM) core-mantle coupling torque, the geomagnetic field must be known at the core-mantle boundary (CMB). It can be divided into linearly independent poloidal and toroidal parts. As shown by previous investigations, the toroidal field produces more than 90% of the EM torque. It can be obtained by solving the associated (toroidal) induction equation for the electrically conducting part of the mantle, i.e. an initial boundary value problem (IBVP). The IBVP differs basically from that for the poloidal field by the boundary values at the interface between lower conducting and upper insulating parts of the mantle: the toroidal field vanishes, the poloidal field continues harmonically as potential field towards the Earth surface. The two major subjects are to find a suitable algorithm to solve the IBVP and to compute the toroidal geomagnetic field at the CMB. Compared to the poloidal field, the toroidal field at the CMB cannot be inferred from geomagnetic observations at the Earth’s surface. In this study, it is inferred from the velocity field of the fluid core flow and the poloidal field at the CMB using an approximation which is consistent with the frozen-field approximation of the geomagnetic secular variation. This investigation differs from earlier ones by: (i) inferring the poloidal field at the CMB from the observed geomagnetic field using a rigorous inversion of the associated (poloidal) induction equation on which the fluid-flow inversion is based to determine consistently the surface flow velocities at the CMB, (ii) applying orthonormal spherical harmonic functions for the representation of the fields and torques, (iii) solving the IBVP numerically by a modified Crank-Nicolson algorithm, which (iv) allows us to highlight the influence of this approach on the resulting EM coupling torques. In addition to an outline of the derivations of the theoretical formalism and numerical methods, the time-variable toroidal field at the CMB is presented for different conductivity models.
Measurements of the value of the large-scale (4050 km) mean earth potential between California and Hawaii are reported as determined over a one year interval in 1990-91. The mean electric field measured during quiet geomagnetic intervals, 0.183 +/- 0.056 mV/km, is about six times larger than that reported for a shorter interval over a slightly longer distance in the Atlantic (Lanzerotti et al., 1985), but of the same sign. The present results are also about three standard deviations from a null value, in contrast to the previous results, which were consistent with a null mean. If the mean potential is entirely attributable to leakage of a poloidal electric field from the core-mantle boundary, then under some models of the mantle conductivity one can conclude that the deduced toroidal magnetic field intensity in the core would be of the same order as the poloidal magnetic field at the core-mantle boundary. The measured potential gradient is smaller than that which might be expected from thermoelectric emfs at the core-mantle boundary. The potential could also correspond to a south to north quasi-steady water flow of about 0.6 cm/sec.
The maximum strength of toroidal magnetic field in the Earth's core is estimated on the basis of the stability of the magnetic field. In our model we take our basic magnetic field BO to be composed of both toroidal and poloidal axisymmetric decay modes of lowest order. While the strength of the poloidal component, BP is taken consistent with observation, the maximum strength of the toroidal field, |BT|max, is regarded as a parameter of the model. By demonstrating that viscous dissipation is of secondary importance and therefore that the results are independent of the parameter associated with viscosity, our model is eventually dependent on only one parameter: the ratio A of the maximum strength of the invisible toroidal field to the strength of the poloidal field at the pole of the core-mantle boundary. It is shown that |BT|max < 10|BP(θ = 0, r = ro)| in order that the basic magnetic field BO = BP + BT is stable, giving an estimated upper bound on strength of the invisible toroidal field of order 50 gauss.
In rapidly rotating systems, a (and, in certain circumstances, the) most important nonlinear effect is the geostrophic flow Vg(s)1 associated with Taylor's (1963) constraint. Its role has been extensively studied in the context of 2 - and -dynamos, and, to a lesser extent in magnetoconvection problems. Here, we investigate its role in the magnetic stability problem, using a cylindrical geometry. First, we investigate the influence of a representative variety of arbitrarily prescribed flows Vg(s)1phi, with V(s) = somega(s), and find that there can be a significant reduction in the critical field strength for flows having a negative outward gradient (domega/ds < 0). We then choose a typical such flow (V= -Rms2) and focus attention on the interaction between the magnetic instability present (or not) when the flow is absent (Rm = 0) and the instability driven by differential rotation when the flow is stronger. It is found that instability (even when driven only by the differential rotation) exists only above a minimum field strength. Finally, having gained an understanding of the roles that differential rotation can play, we investigate the nonlinear magnetic stability problem, where the nonlinear effect is the geostrophic flow. We find cases where the geostrophic flow has the property of destabilising the system. This can happen for the most unstable mode, so the nonlinear effect of the geostrophic flow can be subcritical. Corresponding nonlinear calculations at finite Ekman number E (Hutcheson and Feam, 1995a, b) did not find subcriticality so there must be some value of E < 10 -4 below which the geostrophic flow dominates the other nonlinear effects and subcriticality becomes possible. What that value is may influence how low E must be taken in full geodynamo simulations to correctly qualitatively describe the dynamics of the core.
The core-mantle boundary (CMB) is the most significant internal boundary within our planet, buried at remote depths and probably forever hidden from direct observation; yet this region is very important to our understanding of the dynamic Earth system. The thermal and chemical processes operating near the CMB have intimate relationships to fundamental events in Earth history, such as core formation, and continue to play a major role in the planet's evolution, influencing the magnetic field behavior, chemical cycling in the mantle, irregularities in the rotation and gravitation of the planet, and the mode of thermal convection of the Earth. This overview highlights the progress and future directions in geophysical investigations of the CMB region. -from Authors
Considerable publicity has accompanied Song and Richards' recent measurement of the rotation of the Earth's inner core and the observation of a reversal of the magnetic field in Glatzmaier and Roberts' computer model of the geodynamo. Additionally, the Galileo spacecraft has returned data suggesting the existence of magnetic fields in Io, Ganymede and perhaps Europa. These have given further impetus to the growing interest and activity in the problem of planetary magnetic-field generation and the core flows responsible for it. Here, the current understanding of this problem is reviewed in the light of many recent developments from theory, experiment and observation. The essential aspects of geodynamo theory are included, with further details being available in many recent reviews. The fundamental results of hydromagnetic flow in rapidly rotating systems are followed by a discussion of the latest thinking on the power source for the geodynamo. This replenishes the energy of some basic state but it may be instabilities of this basic state that result in the complex motions responsible for field generation. We discuss instabilities deriving their energy from buoyancy, the magnetic field and shear in the core flow. Our computational models can usually be run using an arbitrary choice of parameters. From this we learn of potential pitfalls when planetary values are used. In particular, the problems associated with the low values of the Ekman number E and Roberts number q are highlighted. Approaches to overcoming these are discussed and the resulting progress in producing full numerical dynamo models is reviewed. Finally, we discuss how the ideas and models developed primarily with the Earth in mind can be adapted and applied to the other planets and satellites known to have magnetic fields.
Numerical simulations of the 3D MHD-equations that describe rotating
magnetoconvection in a Cartesian box have been performed using the code
NIRVANA. The characteristics of averaged quantities like the turbulence
intensity and the turbulent heat flux that are caused by the combined action of
the small-scale fluctuations are computed. The correlation length of the
turbulence significantly depends on the strength and orientation of the
magnetic field and the anisotropic behavior of the turbulence intensity induced
by Coriolis and Lorentz force is considerably more pronounced for faster
rotation. The development of isotropic behavior on the small scales -- as it is
observed in pure rotating convection -- vanishes even for a weak magnetic field
which results in a turbulent flow that is dominated by the vertical component.
In the presence of a horizontal magnetic field the vertical turbulent heat flux
slightly increases with increasing field strength, so that cooling of the
rotating system is facilitated. Horizontal transport of heat is always directed
westwards and towards the poles. The latter might be a source of a large-scale
meridional flow whereas the first would be important in global simulations in
case of non-axisymmetric boundary conditions for the heat flux.
Electric field observations by using 1,000 km scale submarine cables have been performed since early 1990's. One of the main purposes of the observations is to obtain observational constraints on the dynamics of Earth’s core such as the strength and the distribution of the toroidal magnetic field and its variation at the core mantle boundary. Several constraints have been obtained until present, but the electromagnetic plausibility of them have not been examined. In this paper, electromagnetic field variations generated by a simple spherical mean- field kinematic dynamo within an electrically conducting mantle are discussed. The field variations are assumed to be generated by perturbing a steady α^2-dynamo with torsional oscillation type zonal flows having period of 30 years. It is confirmed that the kinematic dynamo can generate the observed amplitude of electric voltage variation (～100mV) naturally. The amplitude of voltage variation is controlled mainly by the energy state of the dynamo, i.e., the magnetic Reynolds numbers, and the strength of the toroidal field variation at the CMB is determined by the magnetic Reynolds numbers and the conductance of the D" layer. Potential obstacles for the detection of the 100mV signal in 1,000km scale submarine cable voltages are the electric voltages induced by external magnetic field variations (magnetotelluric induction) and that induced by the ocean flow (motional induction). Although the magnetotelluric current with decadal time scales seems negligibly small, the motionally induced electric field variation can be as much as 100mV for 1,000km scale. It is necessary to know the time variation of ocean flux in order to discuss the electric voltages generated in the deep interior of Earth correctly.
The current system flowing in the earth's core that is responsible for the dipole geomagnetic field is toroidal and does not appreciably extend into the mantle. However, the mechanism supporting these currents almost certainly generates in addition a poloidal current system that does extend into the mantle and that could, in principle, be measured at the earth's surface. In this paper the current distribution in the mantle is examined for several assumed distributions of conductivity in the mantle. The effect of the oceans is also briefly considered. It is found that the potential gradient at the surface of the earth associated with these deep earth currents may be of the order of a millivolt/kilometer if thermoelectric emf's at the core boundary contribute appreciably to the magnetic field, but will be smaller if the primary emf's are inside the core.
The electrical conductivity of liquid (Fe90Ni10)3S2 saturated with 2.6 weight percent carbon averages 2.7·105 mho/m at 1000°C and zero pressure. This may imply a slightly lower electrical conductivity for the earth's core than that obtained by extrapolating the properties of pure liquid iron and solid iron alloys to core pressures and temperatures. Although a sulphur-rich core would have a smaller proportion of sulphur, the effect of lowering the sulphur content of the FeNiSC liquid to about 15 weight percent would be unlikely to increase the conductivity above 5·105 mho/m.
A rigorous singular perturbation theory is developed to estimate the electric field E produced in the mantle M by the core dynamo when the electrical conductivity σ in M depends only on radius r, and when |r ∂rln σ| ⪢ 1 in most of M. It is assumed that σ has only one local minimum in M, either (a) at the Earth's surface ∂V, or (b) at a radius b inside the mantle, or (c) at the core-mantle boundary ∂K. In all three cases, the region where σ is no more than e times its minimum value constitutes a thin critical layer; in case (a), the radial electric field Er ≈ 0 there, while in cases (b) and (c), Er is very large there. Outside the critical layer, Er ≈ 0 in all three cases. In no case is the tangential electric field ES small, nearly toroidal, or nearly calculable from the magnetic vector potential A as −∂tAS. The defects in Muth's (1979) argument which led him to contrary conclusions are identified. Benton (1979) cited Muth's work to argue that the core-fluid velocity u just below ∂K can be estimated from measurements on ∂V of the magnetic field B and its time derivative . A simple model for westward drift is discussed which shows that Benton's conclusion is also wrong.
An algorithm has been found for inverting the problem of geomagnetic
induction in a concentrically stratified Earth. It determines the
(radial) conductivity distribution from the frequency spectrum of the
ratio of internal to external magnetic potentials of any surface
harmonic mode. The derivation combines the magnetic induction equation
with the principle of causality in the form of an integral constraint on
the frequency spectrum. This algorithm generates a single solution for
the conductivity. This solution is here proved unique if the
conductivity is a bounded, real analytic function with no zeros.
Suggestions are made regarding the numerical application of the
algorithm to real data.
The magnitude of the large-scale direct-current earth potential was measured on a section of a recently laid transatlantic
telecommunications cable. Analysis of the data acquired on the 4476-kilometer cable yielded a mean direct-current potential
drop of less than about 0.072 ± 0.050 millivolts per kilometer. Interpreted in terms of a generation of the potential by the
earth's geodynamo, such a small value of the mean potential implies that the toroidal and poloidal magnetic fields of the
dynamo are approximately equal at the core-mantle boundary.
Analysis of geomagnetic secular variation data from world observatories demonstrates the occurrence of a secular variation impulse in the late 1960's. When a simple mantle conductivity model is used in conjunction with the impulse data, it can be shown that the average conductivity of the lower mantle does not exceed 150 omega-1 m-1. This is orders of magnitude smaller than recently published values.
The work reported on here is an investigation of the radial distribution of electrical conductivity σ in the earth's mantle. Previously this function had been inferred only to about the 800-km depth from the geomagnetic transient variations of external origin (see Lahiri and Price). Throughout the remaining lower portion of the mantle, we make use here of the longer wave periods which characterize the geomagnetic secular variation. Choosing a power law for σ, the wave attenuation and phase retardation after propagation through the mantle are investigated, using sinusoidal input functions, and an equivalent conductivity is established on the basis of amplitude attenuation. Aperiodic models at the core are solved by the method of Laplace transforms and a time discontinuity in Hr. (δ-function in Hr) is treated in detail. The elapsed time for a pulse to reach the earth's surface is expressed in terms of an equivalent conductivity. The latter quantity is then gotten indirectly from a study of the time-dependent magnetic observatory records. Judged somewhat better than an order of magnitude, in these calculations, σ is shown plotted throughout the mantle.
Theories of an electric current generated by the earth's magnetic field
are presented. Preliminary measurements d of the current were undertaken,
using telephone cables in the Pacific Ocean near Australia. Voltage at the ends
of the cables, one end being earthed'' under the ocean, was measured. The
voltage differences were taken to eliminat the dominant tidal currents. An
approximately sinusoidal earth current with a period of 12 hours and a potential
gradient of less than 0.1 mv/km was obtained. (M.J.T.)
SEVERAL empirical values for the electrical resistivity of the Earth's fluid core have been proposed1. The latest estimates are those of Gardiner and Stacey2 which correct earlier estimates by Stacey3. They use data on the electrical resistivity and its temperature coefficient for liquid iron and its alloys at 1,800 K and combine this with the observed pressure coefficient of resistivity in solid iron-nickel alloys at 100 kbar. These results are then extrapolated to the core temperatures (∼ 4,000 K) and pressures (3,000 kbar) assuming the temperature and pressure effects to be additive. As the authors of ref. 2 admit, this latter assumption is of doubtful validity. Obviously such large extrapolations involve considerable uncertainties; since the core resistivity is a vital parameter in the theories of the origin of the geomagnetic field it is important to assess the validity of the existing estimates.
The following general question is addressed: what can be learned about a planetary interior from measurements of the global planetary magnetic field at (or near) its surface? The discussion is placed in the context of Earth, for clarity, but the considerations apply to terrestrial planets in general (so long as the observed magnetism is either predominantly of internal origin, or else external source effects can be successfully filtered out of the observations). Attention is given to the idealized but typical situation of a rotating but spherically symmetric planet containing a highly conducting uniform fluid core surrounded by a nearly insulating rigid mantle, whose conductivity, a function of at most radius only, falls monotonically from its largest value at the base of the planetary mantle to zero at the planetary surface; the largest value of mantle conductivity as well as the mean value for the whole mantle and the mantle conductance are assumed small compared to the corresponding values of the core. Exterior to the planet is vacuum in the sense of an electrically uncharged insulator. The core fluid is inviscid, Boussinesq and gravitationally driven.
A crucial step in the investigation of the energetics of motions in the Earth's core and the generation of the geomagnetic field by the hydromagnetic dynamo process is the estimation of the average strength of the magnetic field B = Bp + BT in the core. Owing to the probability that the toroidal field BT in the core, which has no radial component, is a good deal stronger than the poloidal field Bp, direct downward extrapolation of the surface field to the core-mantle interface gives no more than an extreme lower limit to . This paper outlines the indirect methods by which can be estimated, arguing that is probably about 10−2 T (100 Γ) but might be as low as 10−3 T (10 Γ) or as high as 5 × 10−2 T (500 Γ).
Values of electrical resistivity and its temperature coefficient for liquid iron and its alloys by B. A. Baumet al. (1967) are combined with P. W. Bridgman's (1957) data on the pressure coefficients of resistivity in solid iron alloys and are extrapolated to pressures, temperatures and probable compositions for the Earth's core. The results are grouped around the average value assumed by E. C. Bullard (1949), 3 × 10−6 ω · m, and therefore support its validity. However, remaining uncertainties leave the plausible range as 1 to 6 × 10−6 ω · m. The calculated depth dependence of resistivity is negligible.
Dynamo magnetic fields are self-excited and, once started, can perpetrate themselves with no outside source of magnetic flux, as long as the necessary fluid motions persist. Such dynamo fields behave completely independently of the field's overall polarity. In the presence of an external field of separate origin this polarity symmetry of the dynamo states is broken; the dynamo states become asymmetric with respect to polarity. In this paper a calculation is performed of the characteristics of a spherical shell dynamo in the presence of a fossil magnetic field penetrating into the dynamo from below. The asymmetric periodic states are found as a function of the strength of underlying fossil field. Applying these results to the sun, there appears to be no evidence of any intense large-scale primordial magnetic flux, having either dipole-like or quadrupole-like symmetry about the sun's equator, penetrating into the convection zone from the sun's radiative core. Indeed, the calculations indicate, even on the basis of the presently crude observations, that any such primordial field must have an intensity smaller than a few gauss.
Electrical conductivity measurements of Perovskite and a Perovskite-dominated assemblage synthesized from pyroxene and olivine demonstrate that these high-pressure phases are insulating to pressures of 82 GPa and temperatures of 4500 K. Assuming an anhydrous upper mantle composition, the result provides an upper bound of 0.01 S/m for the electrical conductivity of the lower mantle between depths of 700 and 1900 km. This is 2 to 4 orders of magnitude lower than previous estimates of lower-mantle conductivity derived from studies of geomagnetic secular variations.
The late 1960's secular variation impulse: Further constraints on the deep mantle conductivity, Phys. Earth _Planet. Inter The electric field produced in the man-tle by the dynamo in the core Inversion of the geomagnetic induction prob-lem
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