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Most subduction zones have associated long‐wavelength anomalies in the lithospheric magnetic field observed at satellite altitude. We model the 13 subduction zones defined by seismicity and seismic tomography using vertically integrated magnetizations that are increasing, level, or decreasing away from the trench. These mimic end members of a magnetized mantle wedge, a uniform layer, and a magnetized dipping lithospheric slab. They are added to a global model of vertically integrated magnetization based on continental and oceanic geology. We find the dipping slab places the anomaly too close to the trench, while the other two fit the data equally well and use the level model in the main part of the study. Anomalies at the Sunda, Aleutians, Cascadia, Central American, and Kamchatka‐Japan zones are well modeled by uniform magnetization of differing susceptibilities and spatial extents. We show the South American anomaly is weak because the magnetization lies mainly in the null space that produces no external potential magnetic field. There is no anomaly associated with the Ryukyu system, possibly because the present subduction started too recently for magnetization to have formed. The magnetic anomaly stretching down the Baja California peninsula is not present in the prediction because there is no seismicity on which to base a slab geometry, but recent tomography suggests a fossil slab there and we propose historic subduction as the origin of the Baja magnetic anomaly. Finally, we discuss the mineralogical origins of the magnetization and favor serpentinization of the region above the subducted plate. Many subduction zones are associated with long‐wavelength magnetic anomaliesThe anomalies are unlikely to be explained by magnetization solely in the subducting slabModeling supports the hypothesis of magnetized serpentinite in the mantle wedge

The Earth’s magnetic field is powered by energy supplied by the slow cooling and freezing of the liquid iron core. Efforts to determine the thermal and chemical history of the core have been hindered by poor knowledge of the properties of liquid iron alloys at the extreme pressures and temperatures that exist in the core. This obstacle is now being overcome by high-pressure experiments and advanced mineral physics computations. Using these approaches, updated transport properties for Fe–Si–O mixtures have been determined at core conditions, including electrical and thermal conductivities that are higher than previous estimates by a factor of two to three. Models of core evolution with these high conductivities suggest that the core is cooling much faster than previously thought. This implies that the solid inner core formed relatively recently (around half a billion years ago), and that early core temperatures were high enough to cause partial melting of the lowermost mantle. Estimates of core–mantle boundary heat flow suggest that the uppermost core is thermally stratified at the present day.

Mars has a magnetic field originating in its strongly magnetized crust that holds clues to the planet's interior. We apply vector spherical harmonic decomposition to simple candidate magnetic structures to separate the parts responsible for the anomalies from those that remain invisible. A uniform magnetic layer produces no anomalies: spatial variations are essential although secondary magnetization does produce a weak field that might reflect the primordial dynamo field. A hemispheric layer produces anomalies confined to the equator rather than the observed hemispheric difference. A uniformly magnetized crust with variable thickness determined from gravity and topography produces a crustal field with large anomalies at the major impact crater sites that are not observed. These anomalies are not present if the magnetic layer lies deeper than the crater floor. We conclude that decomposing magnetizations in this way is a useful tool in the interpretation of Martian magnetic anomalies. Four billion years ago Mars had a magnetic field generated by a dynamo operating in its liquid core, as Earth has today. It cooled faster than Earth and dynamo action ceased but not before it had magnetized the planet's crust. This study is made topical by the arrival of the Chinese rover Zhurong, which is capable of carrying out a ground magnetic survey. The lander InSight recorded a magnetic field some 10 times stronger than expected from measurements made by satellite in orbit. Here we use a relatively new technique to separate proposed magnetic structures into their “invisible” and “visible” parts. We show this while the magnetic field is stronger in the Southern Hemisphere than the North, this does not imply one hemisphere is more strongly magnetized than the other. Strong ground measurements can be explained by a strongly magnetized, invisible, shell that has been broken up into smaller, visible, fragments. Larger impact craters have no magnetic anomaly, an observation often attributed to removal of the original magnetized material; we show the anomaly remains if the surrounding crust is strongly magnetized and propose the source of the anomalies lies deeper than the bottom of these craters. Vector spherical harmonic analysis can help elucidate magnetic structures in Mars' crustSecondary magnetization in a uniform shell produces a magnetic anomaly that reflects the original dynamo fieldAbsence of anomalies at large impact craters cannot be explained by simply excavating magnetized crust Vector spherical harmonic analysis can help elucidate magnetic structures in Mars' crust Secondary magnetization in a uniform shell produces a magnetic anomaly that reflects the original dynamo field Absence of anomalies at large impact craters cannot be explained by simply excavating magnetized crust

The geodynamo requires that at least part of the Earth’s liquid core be convecting so vigorously as to mix it thoroughly to uniform composition and adiabatic temperature. It is possible, however, that part of liquid core is stably stratified, either thermally because of a low temperature gradient or compositionally because light material has separated out. The top of the outer core is the most likely site for stability because the adiabat is steepest there and because light material will rise to the surface. Here I show that part of the observed secular variation, that associated with flux expulsion in the southern hemisphere, can only be caused by fluid upwelling in the electromagnetic boundary layer at the top of the core or by very strong poloidal field gradients at the top of the core. Any stratified layer is limited to roughly the uppermost 100 km if flux expulsion is the mechanism; if the layer is any thicker the region in the southern hemisphere where reverse flux patches are growing must also be the site of very strong field gradients caused by very strong near-surface shear flows.

The term dynamo wave was introduced by Parker to describe oscillatory solutions of the mean field induction equation containing differential rotation and an f -effect from helicity. In this article, we examine dynamo waves generated by a class of steady, three-dimensional flows in a sphere without using the mean-field approximation. The flows are defined by two parameters: D the differential rotation, and M the meridional circulation. Contrary to expectations based on the mean-field equations, most 3-dimensional solutions are stationary. Dynamo waves occupy a very narrow band in the parameter space of flows. They behave like solutions to the mean-field equations in three important ways: the magnetic Reynolds number approaches the asymptotic value, meridional circulation produces steady solutions, and radial and azimuthal components of flux tend to occupy the same region of the sphere. The last observation explains why steady solutions, in which radial and azimuthal flux occupy different parts of the sphere, dominate in 3-dimensions: the third dimension facilitates the separation of the two components. This kinematic result may apply to dynamical solutions in which any change in flow that tends to concentrate radial and azimuthal field in the same place leads to oscillations or reversal. This is a possible mechanism by which the Earth's magnetic field reverses.

The magnetic fields of the planets are generated by dynamo action in their electrically conducting interiors. The Earth possesses an axial dipole magnetic field but other planets have other configurations: Uranus has an equatorial dipole for example. In a previous paper we explored a two–parameter class of flows, comprising convection rolls, differential rotation (D) and meridional circulation (M), for dynamo generation of steady fields with axial dipole symmetry by solving the kinematic dynamo equations. In this paper we explore generation of the remaining three allowed symmetries: axial quadrupole, equatorial dipole and equatorial quadrupole. The results have implications for the fully nonlinear dynamical dynamo because the flows qualitatively resemble those driven by thermal convection in a rotating sphere, and the symmetries define separable solutions of the nonlinear equations.Axial dipole solutions are generally preferred (they have lower critical magnetic Reynolds number) for D> 0, corresponding to westward surface drift. Axial quadrupoles are preferred for D< 0, and equatorial dipoles for convection with little Dor M. No equatorial quadrupole solutions have been found. Symmetry selection can be understood if one assumes that the flow concentrates flux in certain places without reference to sign. Fields with dipole symmetry must change sign across the Equator; if flux is concentrated at the Equator, as tends to be the case for D< 0, they have a small length–scale and consequent high dissipation, making them harder to generate than axial quadrupoles. If flux is concentrated nearer the poles (D> 0), axial dipoles are preferred. The equatorial dipole must change sign between east and west hemispheres, and is not favoured by any elongation of the flux in longitude (caused by D) or polar concentrations (caused by M): they are preferred for small Dand M. Polar and equatorial concentrations can be related to dynamo waves and the sign of Parker's dynamo number. For the three–dimensional flow considered here, the sign of the dynamo number is related to the sense of spiralling of the convection rolls, which must be the same as the surface drift.

A sphere containing electrically conducting fluid can generate a magnetic field by dynamo action, provided the flow is sufficiently complicated and vigorous. The dynamo mechanism is thought to sustain magnetic fields in planets and stars. The kinematic dynamo problem tests steady flows for magnetic instability, but rather few dynamos have been found so far because of severe numerical difficulties. Dynamo action might, therefore, be quite unusual, at least for large–scale steady flows. We address this question by testing a two–parameter class of flows for dynamo generation of magnetic fields containing an axial dipole. The class of flows includes two completely different types of known dynamos, one dominated by differential rotation (D) and one with none. We find that 36% of the flows in seven distinct zones in parameter space act as dynamos, while the remaining 64% either fail to generate this type of magnetic field or generate fields that are too small in scale to be resolved by our numerical method. The two previously known dynamo types lie in the same zone, and it is therefore possible to change the flow continuously from one to the other without losing dynamo action. Differential rotation is found to promote large–scale axisymmetric toroidal magnetic fields, while meridional circulation (M) promotes large–scale axisymmetric poloidal fields concentrated at high latitudes near the axis. Magnetic fields resembling that of the Earth are generated by D > 0, corresponding to westward flow at the surface, and M of either sign but not zero. Very few oscillatory solutions are found.

Fluid motions driven by convection in the Earth's fluid core sustain geomagnetic fields by magnetohydrodynamic dynamo processes. The dynamics of the core is critically influenced by the combined effects of rotation and magnetic fields. This paper attempts to illustrate the scale–related difficulties in modelling a convection–driven geodynamo by studying both linear and nonlinear convection in the presence of imposed toroidal and poloidal fields. We show that there exist three extremely large disparities, as a direct consequence of small viscosity and rapid rotation of the Earth's fluid core, in the spatial, temporal and amplitude scales of a convection–driven geodynamo. We also show that the structure and strength of convective motions, and, hence, the relevant dynamo action, are extremely sensitive to the intricate dynamical balance between the viscous, Coriolis and Lorentz forces; similarly, the structure and strength of the magnetic field generated by the dynamo process can depend very sensitively on the fluid flow. We suggest, therefore, that the zero Ekman number limit is strongly singular and that a stable convection–driven strong–field geodynamo satisfying Taylor's constraint may not exist. Instead, the geodynamo may vacillate between a strong field state, as at present, and a weak field state, which is also unstable because it fails to convect sufficient heat.

Models of the magnetic field at the core–mantle boundary at selected epochs from 1715.0 to 1980.0 reveal novel features in the field at the core. These suggest that the flow of core fluid is coupled to the mantle, and that magnetic diffusion is significant.

By W. Menke. Academic Press, 1984, 280 pages. Price: U.S. $42.50. ISBN 0 124 90920 5.

There are many examples of fluid motions that give kinematic dynamos in an infinite medium, but very few cases of dynamos in an electrical conductor of bounded extent. This curious observation is investigated by numerical calculation. Two significant effects are revealed. An insulating boundary is found to cause a current sheet to form near it and this inhibits magnetic field generation. Also the ratio of poloidal to toroidal component of fluid flow must lie in a certain range for dynamo action to occur, because otherwise the diffusive effects of the separate components dominate the regeneration mechanism which requires a combination of both. These two effects help account for some of the difficulties encountered in constructing kinematic dynamo models, and are relevant to the problem of generation of magnetic field by thermal convection in a rotating system.

Seismic waves may be attenuated in the Earth's outer core if it contains a slurry. The damping will be appreciable only if the time for the transition between solid and liquid is comparable with the period of the wave. This reaction time has been estimated for certain regions of the Earth's mantle as about 1 s. Here the same calculation is done for iron in the core giving a reaction time lower than 10--9s. This means that waves will pass unattenuated at the phase speed appropriate to equilibrium between liquid and solid.

Fluid motions are driven by a random isotropic body force in a rotating system. The fluid is incompressible and infinite in extent. The dynamo action of the resulting fluid motions is investigated by Fourier decomposition. The global effects of the turbulent motions are expressed in terms of gradients of the local mean magnetic field, Bo. Two aspects of the problem are novel. The spectral approach is used to solve for the turbulent quantities while gradients of Boare retained, and secondly a non‐zero mean emf. is obtained from an entirely isotropic forcing.

Recent studies of the magnetic field at the core-mantle boundary have revealed fixed sites of either static magnetic features or persistent secular variation. This suggests part of the magnetic field behaviour is controlled by the mantle. The most plausible mechanism for core-mantle interaction is thermal coupling, although topography may also be significant. The magnetic sites coincide with anomalies in lower mantle seismic velocity, as determined from tomography, and density, as determined by flow models of mantle convection constrained by tomography and the geoid. Some magnetic features coincide with subduction zones, particularly those near Indonesia; they may be caused by “bumps” on the core-mantle boundary beneath trenches. Paleomagnetic pole positions suggest the magnetic behaviour has persisted for at least 5 Myr, as would be expected if it were controlled from the mantle.

Recent studies of old magnetic observations1and the seismic velocity of the lower mantle2suggest that at least part of the geomagnetic secular variation is driven by temperature anomalies in the solid mantle3. In particular, a patch of flux of opposite sign to that expected for a dipole field occurs beneath southern Africa. It has intensified throughout the twentieth century, by a process that is believed to be flux expulsion associated with a particularly hot part of the lowermost mantle, and is drifting westwards. A similar patch lies beneath South America. Here I show that the present fall in the dipole moment is directly related to the intensification and southward movement of these patches and suggest the fall occasionally leads to polarity reversal. The almost linear increase in the frequency of polarity reversals that has occurred since the Cretaceous quiet interval is attributed to the growth of the hot patch of mantle presently under southern Africa.

Maps of the magnetic field at the core–mantle boundary for 1715–1980 reveal static features in the field and an absence of westward drift from much of the core–mantle boundary. The static features suggest that flow in the core is coupled to the mantle. We examine different coupling mechanisms and predict lateral inhomogeneity at the core–mantle boundary. Comparison with models of the lower mantle and core–mantle boundary derived seismically and from the geoid is encouraging, though their resolution is currently inadequate for a full comparison to be made with the much better-resolved magnetic field.

Maps of the radial component of the magnetic field at the surface of Earth's core for 1715–1980 show, on average, a pattern of four lobes, placed symmetrically about the Equator, and zero flux near the poles. Time variations are largely confined to the Atlantic hemisphere (90° E to 90° W). We propose this average field to be the true dynamo-generated field; that equatorial symmetry is a fundamental consequence of the dynamo equations; and that the polar patches are because of the dynamical influence of the inner core. The main lobes are 120° apart and we propose that the flux from a missing third pair of lobes, which should appear symmetrically near the Greenwich meridian, has been re-distributed by near-surface fluid flow to generate secular variation in the Atlantic hemisphere.

The earth's magnetic field is generated by induction in fluid motions in the liquid iron core, a dynamo process that is not yet properly understood. The field changes on time-scales of years to centuries, and this change probably forms part of the fundamental regeneration process. A kinematic approach is used whereby the induction of fluid motions is modelled by choosing a flow and investigating the magnetic field that it generates. Using a novel matching condition, which is appropriate when the upper part of the liquid iron core is static, a set of kinematic dynamos is generated that belongs to the geophysically interesting class of flows with one cell in radius. The morphology of the radial component of the induced magnetic field is then mapped at the boundary of the core and compared with equivalent maps derived from observation to retrieve some essential, symmetric static features in the geomagnetic field.

A numerical optimization approach is introduced to the subject of dynamo theory. Conventional kinematic dynamo studies treat the induction equation as an eigenvalue problem by choosing a candidate velocity field and solving for a marginally stable solution of magnetic field and critical magnetic Reynolds number. The conventional approach has told us something about dynamo action and magnetic field morphology for specific velocities, but the arbitrary choice of fluid flow is a hit-or-miss affair; not all velocities sustain dynamo action, and of those that do, few yield mathematically tractable solutions. As a result, progress has been slow. Here we adopt a new approach, a non-linear numerical variational approach, which allows us to solve the induction equation simultaneously for both the magnetic field and the velocity field. The induction equation is discretized following the Bullard-Gellman formalism and the resulting algebraic equations solved by an iterative, globally convergent, Newton-Raphson method. The particular choice of optimization constraints allows one to design a dynamo which satisfies certain conditions; in this paper we minimize a linear combination of the kinetic energy (magnetic Reynolds number) and a smoothness norm on the magnetic field to produce efficient (low magnetic Reynolds number) well-converged (smooth magnetic field) solutions. We illustrate the optimization method by designing two dynamos based on a Kumar-Roberts velocity parametrization; a specific choice of the velocity parameters, KR, sustains a 3-D kinematic model of the geodynamo. Compared with KR, one of our new models, LG1, is designed to have a higher magnetic Reynolds number but smoother magnetic field, and the other, LG2, a lower magnetic Reynolds number and somewhat rougher magnetic field. We suggest that dynamo efficiency, defined by the magnetic Reynolds number, is achieved through reduced differential rotation and a favourable spatial distribution of the helicity. These examples demonstrate the value of the optimization method as a tool for exploring dynamo action with geophysically realistic flows. It can be extended to the dynamic dynamo problem and, by changing the constraints, be used to design dynamos with good numerical convergence, which match the observed geomagnetic surface field morphology and which place useful quantitative constraints on the physical nature of the geodynamo.

The Leeds Tararua array of nine broad-band, three-component seismometers was deployed in southern North Island, New Zealand, from 1991 January to 1992 September and recorded many local events. We selected 111 events with local magnitude ≥ 2.5 and good station coverage, and attempted to find their source mechanisms by using conventional first motions and by matching of the vertical component waveforms to Rayleigh-mode synthetics. 90 source mechanisms were found: 32 from the method of first motions, 18 by waveform matching, and 40 using both methods. A comparison of mechanisms for events using both methods suggests that waveform matching is reliable even for small events where the coverage on the focal sphere does not allow a first-motion mechanism to be found. The mechanisms are of all types but with a predominance of normal faulting with one of the planes aligned approximately NE–SW. There are many dip-slip events. Deep (60–80 km) events are mostly thrusts, and there is some evidence of a trend from normal faulting in the shallow part of the subducted slab and upper seismic plane to thrusts in the deeper part, although the pattern is not clear cut. A complex stress pattern is to be expected in this region, which is near the southern end of the Tonga–Kermadec–Hikurangi subduction zone. The magnitude–moment relation ML=(0.84 ± 0.23) log M0− (7.53 ± 1.48) is found for the region, where ML is the local magnitude and M0 the seismic moment in N m.

The deformation of subducted oceanic lithosphere is revealed by detailed studies of the distribution of seismicity in Benioff zones and by the increasingly well-resolved images of the upper mantle obtained by seismic P-wave tomography. By using numerical experiments based on a simplified dynamical model of a subducted lithospheric slab, we test the idea that the observed deformation of the subducted slab results from a balance between internal buoyancy forces and the viscous stresses associated with deformation. The simplified model assumes that the lithosphere has uniform physical properties, it is denser and much more viscous than the upper mantle, and its penetration below 670 km depth is resisted by a step-like density increase at that level. The primary determinant of the style of deformation is the buoyancy number, F, a ratio of buoyancy stress generated by the mass anomaly in the slab to the stress associated with viscous deformation at a strain rate U0/L, defined in terms of subduction rate U0 and slab thickness L. For a small buoyancy number, deformation of the slab is dominated by viscous flexure. For large F, down-dip extension of the slab appears to dominate, and a buckling instability of the slab is observed if there is resistance to penetration of the 670 km level. The transition value of F at which this change in behaviour is observed is about 0.05 for constant viscosity, and about 0.2 for stress-dependent viscosity with a stress versus strain-rate exponent of n= 3. With n= 3, a boudinage-type instability of the slab is observed for large F. Penetration of the slab below the 670 km level depends on the density contrast between slab and lower mantle. Resisted only by the density difference, the slab may initially penetrate several hundred kilometres below the 670 km level (for plausible density parameters) before a buckling instability causes this part of the slab to rotate and ascend. Comparison of the deforming slab geometry and stress field with seismicity distribution and tomographic images from the Tonga subduction zone suggests that the effective buoyancy number F for this slab is approximately equal to the transition value that we determined experimentally. We use this constraint to estimate average rheological parameters for the Tonga slab which, when compared with published creep deformation laws for olivine, are consistent with the average temperature of the slab being about 0.4 to 0.45 times the melting temperature, based on constitutive laws for olivine. At shallow depths (< 100 km) these temperatures correspond to around 590°C for wet olivine or around 700°C for dry olivine.

The modern geomagnetic field is usually expressed as a spherical harmonic expansion. Although the palaeomagnetic record is very incomplete in both space and time, sufficient data are available from a span of ages to generate time-averaged spherical harmonic field models with many degrees of freedom. Here three data sets are considered: directional measurements from lavas, inclination measurements from ocean sediments, and intensity measurements from lavas. Individual data are analysed, as well as site-averages, using the same methods that have been developed for the modern field, to give models for the past 5 Myr. The normal-polarity field model has an axial-dipole intensity similar to that of the modern-day field, whilst the equatorial-dipole component is very much smaller. The field is not axisymmetric, but shows flux concentrations at the core's surface under Canada and Siberia similar to those observed in the field over historical timescales. Tests on synthetic data show that it is unlikely that these similarities result from the overprinting of the palaeomagnetic field due to inadequate cleaning of the samples. The reverse-polarity field model does not show such obvious features, but this may be due to the sparsity of the data. The patterns observed in the normal-polarity field, with persistent features in the northern hemisphere and a smooth southern hemisphere, could be explained if the present pattern of secular variation is typical of the past several million years. This would reveal itself as large variations over time in the direction of the magnetic vector in regions of high secular variation, with relatively little change over quieter regions. However, we have been unable to find any evidence for a geographical pattern of secular variation in the data.

Secular variation of the Earth's magnetic field is attributed to motion of fluid in the core, which can be taken as a perfect electrical conductor for sufficiently short time periods. This assumption is tested by direct calculation of models of secular variation that are consistent with a perfectly conducting core and that satisfy annual mean data from magnetic observatories for 1959 onwards, and found to be acceptable. Models for 1959, 1964, 1969, 1974 and 1979 are used to show evidence of a complex change in core motions associated with the secular acceleration impulse or ‘jerk’ that occurred in 1969.

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.

Geomagnetic secular variation is caused by flow of fluid in the Earth's core. If electrical diffusion can be ignored, field lines are frozen to the core fluid and can be used as tracers for the flow at the top of the free stream, which lies beneath the diffusive boundary layer. In this paper we assume the flux is frozen-in and investigate the jump in the horizontal field across the boundary layer. Theoretical studies indicate that such a jump will be small (about 1000 nT); we test the hypothesis that the jump is zero within the errors of measurement of the magnetic field. Integral constraints, which are conserved if the horizontal components of field at the base of the mantle can be equated to the components at the top of the free stream, are derived and computed using seven main-field models from epochs 1935.5 to 1980.0. These main-field models have been forced to satisfy all the conditions of frozen-flux theory. The integral constraints are found to be constant within their expected errors, and we conclude that the observations are consistent with zero jump in horizontal field across the boundary layer. This result implies that use can be made of the horizontal components of field in determining core flow. We attempt to compute the flow along null-flux curves, which is not determined at all by the radial component of the field, for epoch 1970. The result is very noisy because of its reliance on gradients of the secular variation at the core-mantle boundary, which are poorly determined. We do not attempt to compute reasonable core flows consistent with these observations, although presumably many such flows exist, because they would be so poorly constrained by observation.

Fluid motions are driven in a sphere by an applied body force. The fluid is electrically conducting and incompressible, and is surrounded by an insulator. The vorticity equation, in which inertial terms have been omitted but which includes magnetic forces, is solved in conjunction with the pre-Maxwell equations which govern the magnetic field. The model allows the magnetic field to grow to a finite limit. Some solutions are obtained when the body force is axisymmetric and drives fluid motions which are similar to those used in a previous kinematic study. The time evolution of the magnetic field is calculated and in most cases the magnetic field is found to approach a steady state in an oscillatory fashion. There is a critical limit of forcing below which no magnetic field can be sustained. When the dynamo is driven well above this critical limit the magnetic field undergoes large oscillations which may ultimately lead to Geomagnetic-type reversal behaviour. Possible modifications and improvements to the model, including the addition of rotation, are discussed, and future calculations are outlined.

The most likely origin of the Earth's magnetic field is a dynamo process acting in the liquid core, with motions of the fluid driven by thermal convection. This hypothesis is examined in relation to the constraints imposed by a knowledge of the surface magnetic field, the surface heat flux, and also to some extent the properties of the core. Assuming the Weidemann—Franz relation between the electrical and thermal conductivities for the core, it is possible to obtain lower bounds on the heat flux through the base of the mantle. The minimum heat flux from the core is found to be 2.1010W, independent of either the electrical conductivity or the temperature gradient. By requiring that these bounds be less than the surface heat flux, the product of the electrical conductivity and the temperature gradient may be bounded both from above and from below. The heat flux estimates are improved by considering ohmic heating from the toroidal magnetic field, which cannot be observed directly, and the convected heat flux. An expression is obtained for a bound of the ohmic heating of the toroidal field in terms of the radial fluid velocity. A comparison with existing dynamo models reveals that the true value of the ohmic heating is much larger than the ideal bound. This is evidence against the existence of a toroidal field as large as, say, 10 mT. The heat fluxes are larger for heating by uniformly distributed heat sources than for latent heat from growth of the inner core. Unless the adiabatic gradient is below about 0.1 K km-1, it is not possible to generate the magnetic field by distributed heat sources. The latent heat case does not suffer the same difficulty, but then it is possible for the outer regions of the core to be density-stratified. Gravitational energy released by accretion of the inner core may also be an appreciable or even dominant factor.

Travel times are observed at a network of seismometers from a number of earthquakes. The seismic velocity of the medium through which the waves pass is allowed to vary in all three dimensions and is represented in terms of a finite number (P) of parameters. The travel-time data is inverted for the velocity parameters and the hypocentre coordinates of the earthquakes. A linearized least squares inversion technique is used and both the stochastic and generalized inverses are considered. The method is computationally efficient and requires inversion (or decomposition) of only P × P and 4 × 4 matrices. The method is applied to a dataset of travel times reported to the I. S. C. from 56 earthquakes recorded at stations on the North Island of New Zealand. The velocity structure is characterized by only five parameters, so that only 5 × 5 matrices need be inverted. The inversion gives a 30 per cent reduction in residuals. The resulting velocity model is substantially different from that obtained when the hypocentres are kept fixed and only the velocity parameters are inverted for. A parallel study of synthetic data gives insight into the interpretation of results. In particular, the depth and origin time of some of the earthquakes, together with two of the five velocity parameters, are very poorly determined.

The frozen flux approximation of Roberts & Scott is a constraint on the core field that can be used to aid interpretation of the very sparse datasets that palaeomagnetism and archaeomagnetism provide. It gives bounds on the size of the components of the magnetic field at a point, of the Gauss coefficients, and, if valid over such long time periods, limits the shape of the field during transitions between normal and reversed polarities. The maximum intensity at a point, consistent with the present flux, is 281 μT or 4 times the maximum field observed today. The present dipole is about 50 per cent of its upper bound. Polarity reversal is impossible if the transition field is purely axisymmetric. None of the measurements we consider violate the frozen flux approximation.

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.

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.

Lateral variations in heat flux or temperature in the lowermost mantle, associated with mantle convection, will drive fluid flow in the liquid core. The effect is modelled by a rotating spherical shell with uniform lower boundary temperature and a laterally varying upper surface temperature and the solutions found by numerical calculation, some for a density-stratified fluid. The problem depends upon two dimensionless numbers, a horizontal Rayleigh number and a stratification parameter. The calculations are restricted to surface temperatures which are symmetric about the equator and steady solutions are found. Induced toroidal flows near the surface are closely linked with the applied temperature profile through the thermal wind equation and Coriolis forces make the convection penetrate into the shell. Stratification acts mainly to suppress radial flow but the surface toroidal flow is relatively unaffected. The results illustrate the powerful influence exerted by the surface temperature but are difficult to apply directly to the earth's core because molecular values of the thermal diffusivity entail a very long time-scale. Assuming a turbulent thermal diffusivity equal to the magnetic diffusivity gives a more realistic time-scale and it is possible to recover, for specific simple imposed temperature profiles, flows similar to those found in the earth's core from inversion of geomagnetic secular change.

Geomagnetic secular variation is a result of inductive effects of fluid motion at the top of the Earth's liquid core, which can be mapped using the frozen-flux approximation and inverting mathematical models of the secular variation. Non-uniqueness may be removed by assuming the flow to be stationary. The use of an intermediate mathematical model of secular variation makes comparison of the derived models with observation unsatisfactory, and here we devise a new method, inverting the geomagnetic measurements directly to obtain steady core motion. The method is applied to observatory annual mean data for the 30 yr period 1960–1990 to produce six separate models of steady flow: three that are stationary over 10 yr intervals (1960–1970, 1970–1980 and 1980–1990), two 20 yr models (1960–1980 and 1970–1990), and one 30 yr model (1960–1990). The method is not restricted to continuous time series or linear (X, Y, Z) data. The 10 yr models fit the weighted data with misfits of 1.0–1.3, the 20 yr models fit less well, and the 30 yr model has a misfit of 1.7 when constrained to have the same norm as the 10 yr models. All models appear to fit the central decade with a gross misfit close to unity, suggesting that changes in core motion may be associated with the two geomagnetic jerks that occurred around 1969 and 1978. Furthermore, close inspection of the fits at individual observatories shows that steady core motions with this norm cannot produce the secular accelerations required at a large number of high-quality observatories. Steady flows can predict magnetic fields that follow the trend at most observatories, but cannot follow the detailed time variations of the past three decades.

Braginskii&ibreve;'s dynamo theory of the Earth's magnetic field is reviewed, with particular attention to the relative magnitudes of the symmetric and asymmetric components of the fluid flow. A general approach to the problem of the ordering of the velocity makes it clear that Braginski&ibreve;'s choice is the only one consistent with the physical processes dominant in the Earth's core. A special class of flows, which resemble the Bullard-Gellman dynamo, is studied using Braginski&ibreve;'s approach. By making use of a symmetry property of these velocities, the lengthy algebra usually involved in these studies can be eliminated, and it becomes feasible to examine the higher order terms in the expansion for the mean emf which supports the axially symmetric field. Braginski&ibreve; proved that, for his ordering of the velocity, this class of flows cannot give dynamo action at large magnetic Reynolds number. It is shown here that by increasing the magnitude of the asymmetric flow relative to that of the differential rotation, this ‘anti-dynamo’ restriction may be removed. The implications for numerical dynamo models are discussed.