David Gubbins

Thermal and electrical conductivity of iron at Earth/'s core conditions

The Earth acts as a gigantic heat engine driven by the decay of radiogenic isotopes and slow cooling, which gives rise to plate tectonics, volcanoes and mountain building. Another key product is the geomagnetic field, generated in the liquid iron core by a dynamo running on heat released by cooling and freezing (as the solid inner core grows), and on chemical convection (due to light elements expelled from the liquid on freezing). The power supplied to the geodynamo, measured by the heat flux across the core–mantle boundary (CMB), places constraints on Earth’s evolution. Estimates of CMB heat flux depend on properties of iron mixtures under the extreme pressure and temperature conditions in the core, most critically on the thermal and electrical conductivities. These quantities remain poorly known because of inherent experimental and theoretical difficulties. Here we use density functional theory to compute these conductivities in liquid iron mixtures at core conditions from first principles—unlike previous estimates, which relied on extrapolations. The mixtures of iron, oxygen, sulphur and silicon are taken from earlier work and fit the seismologically determined core density and inner-core boundary density jump. We find both conductivities to be two to three times higher than estimates in current use. The changes are so large that core thermal histories and power requirements need to be reassessed. New estimates indicate that the adiabatic heat flux is 15 to 16 terawatts at the CMB, higher than present estimates of CMB heat flux based on mantle convection; the top of the core must be thermally stratified and any convection in the upper core must be driven by chemical convection against the adverse thermal buoyancy or lateral variations in CMB heat flow. Power for the geodynamo is greatly restricted, and future models of mantle evolution will need to incorporate a high CMB heat flux and explain the recent formation of the inner core.

Melting of the Earth's inner core.

The Earth’s magnetic field is generated by a dynamo in the liquid iron core, which convects in response to cooling of the overlying rocky mantle. The core freezes from the innermost surface outward, growing the solid inner core and releasing light elements that drive compositional convection. Mantle convection extracts heat from the core at a rate that has enormous lateral variations. Here we use geodynamo simulations to show that these variations are transferred to the inner-core boundary and can be large enough to cause heat to flow into the inner core. If this were to occur in the Earth, it would cause localized melting. Melting releases heavy liquid that could form the variable-composition layer suggested by an anomaly in seismic velocity in the 150 kilometres immediately above the inner-core boundary. This provides a very simple explanation of the existence of this layer, which otherwise requires additional assumptions such as locking of the inner core to the mantle, translation from its geopotential centre or convection with temperature equal to the solidus but with composition varying from the outer to the inner core. The predominantly narrow downwellings associated with freezing and broad upwellings associated with melting mean that the area of melting could be quite large despite the average dominance of freezing necessary to keep the dynamo going. Localized melting and freezing also provides a strong mechanism for creating seismic anomalies in the inner core itself, much stronger than the effects of variations in heat flow so far considered.

Numerical solutions of the kinematic dynamo problem

The expansion method of Bullard & Gellman is used to find numerical solutions of the induction equation in a sphere of conducting fluid. Modifications are made to the numerical methods, and one change due to G. O. Roberts greatly increases the efficiency of the scheme. Calculations performed recently by Lilley are re-examined. His solutions, which appeared to be convergent, are shown to diverge when a higher level of truncation is used. Other similar dynamo models are investigated and it is found that these also do not provide satisfactory steady solutions for the magnetic field. Axially symmetric motions which depend on spherical harmonics of degree n are examined. Growing solutions, varying with longitude, 0, as e1^, are found for the magnetic field, and numerical convergence of the solutions is established. The field is predominantly an equatorial dipole with a toroidal field symmetric about the same axis. When n is large the problem lends itself to a two-scale analysis. Comparisons are made between the approximate results of the two-scale method and the numerical results. There is agreement when n is large. When n is small the efficiency of the dynamo is lowered. It is shown that the dominant effect of a large microscale magnetic Reynolds number is the expulsion of magnetic flux by eddies to give a rope-like structure for part of the field. Physical interpretations are given which explain the dynamo action of these motions, and of related flows which support rotating magnetic fields.

Symmetry properties of the dynamo equations for palaeomagnetism and geomagnetism

Abstract Recently both palaeomagnetists and geomagnetists have searched for symmetries in their data which would give some guide to the nature of the Earths dynamo, most frequently quoting analyses of mean-field (αω) kinematic dynamos. The separable solutions of the fully nonlinear, convective dynamo with spherically symmetric buoyancy forces and boundary conditions arise from the group of symmetry operations that leave a rotating sphere unchanged; they are more general than the rather specialised solutions usually quoted in the geomagnetic literature. The full set of symmetry operations is an Abelian Lie group but two simple, finite subgroups contain all the symmetries we have found in the recent literature. The smaller subgroup contains both reflection in the equatorial plane, which gives rise to the so-called ‘dipole/quadrupole’ separation, and inversion, or field reversal. The full group also includes rotations about the polar axis; these rotations would not normally be significant but current interest in core-mantle interactions, which can make the core longitude sensitive, demands that we include them. This larger subgroup includes, in addition to field reversal and equatorial reflection, rotation by an angle π about the polar axis and reflection through the origin. We give the spherical harmonic expansions for each separable solution and indicate the type of data required to discriminate between different symmetries.

Convection in a rotating spherical fluid shell with an inhomogeneous temperature boundary condition at infinite Prandtl number

We examine thermal convection in a rotating spherical shell with a spatially non-uniformly heated outer surface, concentrating on three distinct heating modes: first, with wavelength and symmetry corresponding to the most unstable mode of the uniformly heated problem; secondly, with the critical wavelength but opposite equatorial symmetry; and thirdly, with wavelength much larger than that of the most unstable mode. Analysis is focused on boundary-locked convection, the associated spatial resonance phenomena, the stability properties of the resonance solution, and time-dependent secondary convection. A number of new forms of instability and convection are found: the most interesting is perhaps the saddle-node bifurcation, which is the first to be found for realistic fluid systems governed by partial differential equations. An analogous Landau amplitude equation is also analysed, providing an important mathematical framework for understanding the complicated numerical solutions.

Transport properties for liquid silicon-oxygen-iron mixtures at Earth's core conditions

We report on the thermal and electrical conductivities of two liquid silicon-oxygen-iron mixtures (Fe0.82Si0.10O0.08 and Fe0.79Si0.08O0.13), representative of the composition of the Earth’s outer core at the relevant pressure-temperature conditions, obtained from density functional theory calculations with the Kubo-Greenwood formulation. We find thermal conductivities k = 100(160) W m −1 K −1 , and electrical conductivities σ = 1.1(1.3) × 10 6 � −1 m −1 at the top (bottom) of the outer core. These values are between two and three times higher than previous estimates, and have important implications for our understanding of the Earth’s thermal history and the functioning of the Earth’s magnetic field, including rapid cooling rate for the whole core or high level of radiogenic elements in the core. We also show results for a number of structural and dynamic properties of the mixtures, including the partial radial distribution functions, mean square displacements, viscosities, and speeds of sound.

Geomagnetic polarity reversals: A connection with secular variation and core‐mantle interaction?

Geomagnetic polarity reversal remains one of Natures most enigmatic phenomena. Dynamo theory admits solutions in pairs with reversed magnetic fields B and −B, but detailed calculations are required to understand how the field can change sign. Theory also admits separate solutions with different symmetry across the equatorial plane, the symmetric (ES) “quadrupole” and the antisymmetric (EA) “dipole” solutions, which may be important in the reversal process and which offer a simple framework for interpreting small paleomagnetic data sets. Ordinary secular variation leads to very large changes in the magnetic field over several centuries and could easily develop into full reversal; the theory of secular variation, which is relatively well developed, may therefore help in understanding reversals. Other clues to geomagnetic reversals come from the Sun, whose magnetic field reverses every 11 years. Paleomagnetic data show the Earth’s magnetic field reverses every million years or so, with each transition taking about a thousand years, during which the intensity may fall by as much as 1 order of magnitude. Reversal frequency undergoes a modulation on the long timescale (107 years) of mantle convection, and there have been two long intervals in the past with no reversals. Such behavior is typical of a highly nonlinear dynamical system, but the very long timescale of changes in reversal frequency, and its close proximity to the overturn time of mantle convection, suggests some control of the dynamo by the mantle. Short-term phenomena, such as change in the length of day and secular variation, have been studied extensively for evidence of core-mantle interactions, and we may draw on this body of evidence in order to understand long-term effects. Three physical mechanisms have been proposed: topographic, electromagnetic, and thermal, with the last two being most significant for long-term effects. Symmetries allow the dynamo to generate an EA field, with the major component a dipole, but lateral variations on the core-mantle boundary may lead to magnetic fields with no symmetry, reflecting the structure of the boundary anomalies. Changes in reversal frequency on the mantle convection timescale could arise either from changes in total heat flux from the core to the mantle or from instabilities associated with lateral variations at the core-mantle boundary. Neither mechanism is well understood, but the former involves significant changes to the Earths overall heat budget, whereas the latter must always arise as a natural consequence of deep-mantle convection. Recent measurements of transition fields show pole paths that lie close to two preferred longitudes near 90°W and 90°E; if substantiated, the result would provide the first definitive evidence of long-term mantle control of the geomagnetic field. Further evidence suggests that the geomagnetic pole during stable polarity also lies along these two longitudes and that magnetic flux at the core surface tends to concentrate along the same longitudes, as does the present field. A simple theory is proposed relating changes in the core field to apparent transition paths measured at the Earths surface. The model shows that longitude confinement of the transition paths can occur for quite complicated core fields and that surface intensities can drop by 1 order of magnitude, on average, simply because of the reduction in length scale of the transitional field. Simple transition paths may be an indication of some organization of flux at the core surface but not of large-scale or small-amplitude core fields.

Dispersion of P waves in subducted lithosphere: Evidence for an eclogite layer

Cold, subducted lithosphere has relatively fast seismic velocity which leads to early arrivals for some event-station paths. The effect is very large for events in the Tonga-Kermadec deep seismic zone recorded at certain New Zealand stations. These particular arrivals are very high-frequency (3 Hz or greater) and sometimes resemble two distinct phases, the later arrival appearing at about the time predicted by Jeffreys-Bullen tables. Data from the digital station SNZO in Wellington confirm the travel time results of the analog stations and furthermore show frequencies above 5 Hz, much higher than can be seen on analog records, and up to 4% dispersion in the range 1–8 Hz. Energy in the second phase (which is often absent at SNZO) is mainly 1–2 Hz. The digital data support the idea, proposed earlier, that the effect is caused by propagation through a thin slab which passes only short-wavelength waves. The essential features of the wave propagation are modeled by acoustic waves in a one-dimensional high-velocity slab; the waveforms produced by the model are discussed in terms of the leaky modes of the system and calculated by a reflectivity method. A very thin (< 15 km) uniform slab provides the required dispersion, but the waves are heavily attenuated and would never be observed at teleseismic distances; a thicker slab allows the energy through but does not give enough dispersion. Altering the variation of velocity across the slab provides the required dispersion if a thick high-velocity layer, with wave speed increasing gradually with height, is overlain by a thin lid of even higher velocity. For the models considered the lid thickness must lie in the range 6–15 km and be continuous from a depth of about 50 km to the bottom of the earthquake zone. The thick layer could arise from the thermal anomaly in the subducted lithosphere; the thin lid may be the gabbroic part of the subducted crust that has transformed to eclogite.

SKS splitting and the seismic anisotropy of the mantle beneath the Hikurangi subduction zone, New Zealand

Abstract The analysis of 12 SKS phases recorded on broadband stations above the Hikurangi subduction zone in New Zealand shows clear evidence of mantle anisotropy, with the fast direction (28° ± 5°) almost parallel to the strike of subduction and the dominant geology of the region. The slow shear-wave delay times show a systematic change with the azimuth of the arrivals which, if hexagonal symmetry is assumed, indicates that either the axis of symmetry of the anisotropic volume beneath the subduction zone is not horizontal, or that more than one anisotropic layer is present. The magnitude of the delays (1.5 ± 0.4s) suggests that the anisotropy is most probably confined to the top 300 km of the mantle.

Longitudinally confined geomagnetic reversal paths from non-dipolar transition fields

IT has long been thought that conditions at the boundary between the core and mantle influence the Earths magnetic field, but the supporting evidence is rather indirect1–3. Recent palaeomagnetic results, suggesting that there are persistent preferred longitudinal paths for the virtual geomagnetic pole (VGP) during reversals4, would provide the first direct evidence of the solid mantles influence on the core, although their statistical significance has been disputed5,6. The results are potentially exciting because the preferred paths lie close to the Pacific rim, where the present geomagnetic secular variation changes character2,7. Here we present a simple model, based on an extension of a previous theory8, that produces reversals with VGP paths confined within relatively narrow longitude bands despite the transition field having a substantially non-dipolar structure. Thus, although longitude bias of the VGP paths is definitive evidence for core-mantle interaction, simple VGP paths are not evidence of near-dipolar transition fields.