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SUMMARYA novel approach to computing the Earth's normal modes is presented. With the Langer approximation incorporated into the framework of the theory of the generalized reflection/transmission coefficients, our method, when applied to calculate the toroidal modes, produces results that agree extremely well with the exact results from Mineos and successfully overcomes the limitations of the WKBJ analysis. Given any 1-D earth model, regardless of the number of discontinuities, our method can perform the calculations reliably with satisfactory accuracy at high frequencies. The success achieved with the toroidal modes encourages us to tackle in a future study the asymptotic computation of the spheroidal modes, especially those high-frequency trapped modes for which the accuracy of Mineos is demonstrably inadequate.

Recent cosmochemical studies have shown that most of Earth’s building blocks were close to enstatite meteorites in isotopic compositions. This implies the formation of an impact-induced proto-atmosphere enriched in H2 and CH4 on accreting Earth. Such a reduced proto-atmosphere would have been largely lost by hydrodynamic escape, but its flux and time-scale for hydrogen depletion remain highly uncertain. Here we carry out 1D hydrodynamic escape simulations for such an H2–CH4 proto-atmosphere by incorporating expanded chemical networks and radiative cooling processes for estimation of the duration of the H2-rich surface environment on early Earth. In the escape outflow, CH4 is dissociated effectively by direct photolysis and chemical reactions with photochemically produced ion species. On the other hand, radiative cooling by photochemical products such as H$_{3}^{+}$, CH, and CH3 significantly suppresses atmospheric escape. Even though CH4 and their concentrations are small, the heating efficiency decreases to $\sim 5\, {{\ \rm per\ cent}}$ when CH4/H2 = 0.007 in the lower atmosphere and CH4 would suffer negligible escape when CH4/H2≳ 0.01. The time-scale for H2 escape consistent with the constraints of the isotopic compositions and the amount of C and N on the present Earth is possibly more than several hundred million years. Our results suggest that a long-lived hydrogen-rich reduced environment played important roles in climate warming and the generation of organic matters linked to the emergence of living organisms during the first several hundred million years of Earth.

SUMMARY Tsunamis are often modelled as surface gravity waves of incompressible homogenous water propagating over a rigid seafloor. Previous studies have noted that when computing long-period tsunamis travelling at trans-oceanic distances with dominant periods of thousands of seconds, we need to consider four factors that are not included in the surface gravity wave theory: compressibility of seawater, density stratification of oceans, elasticity of the Earth and gravitational potential change associated with the tsunami motion. However, their effects on short-period tsunamis with dominant periods below 1000 s have not been examined. Here, we investigate how the four factors influence short-period tsunamis. Theoretical analyses and 1-D simulations using phase speeds of different tsunami models indicate that the resultant phase delay of short-period tsunamis becomes apparent after ∼1000 km propagation, mainly because of the first three factors. We then introduce a new phase correction method for dispersive short-period tsunamis with consideration of period-dependent ray paths and apply it to a 2-D simulation of a short-period tsunami from a submarine volcanic earthquake near Japan in 2015. The correction of the traveltime of a synthetic waveform by including the four factors amounts to ∼40 s at a distant station 1430 km away from the source, whereas the effects of the four factors on the waveforms are negligibly small at stations < ∼500km from the source. The observed traveltime at the ocean bottom pressure (OBP) gauge with a sampling interval at 15 s of the distant station can be explained only when these factors are incorporated into synthetic waveforms, indicating the effects due to the four factors are detectable by high-sampling OBP gauges that are deployed over broad oceanic regions.

Previous studies have reported that coronal mass ejections (CMEs) and solar flares lead to the development of huge storms and high-speed streams. Our aim in this paper is to investigate the response of the geomagnetic index SYM/H to the solar wind parameters, such as solar wind speed V, dynamic pressure P, input energy IE and the interplanetary magnetic field (IMF) Bz component, associated with solar flares and CME events. The response of the ground geomagnetic field (H-component) to the solar wind parameters and the IMF Bz component at three low-latitude stations has also been analysed. Our findings show that the delay of the solar wind changes in the Earth’s magnetosphere in response to the weak geomagnetic storm (SYM/H = −30 nT) at the beginning of 2014 December 21. A weak storm of SYM/H = −30 nT in the middle of 2014 November 5 is suggested by a low magnetic reconnection process in the magnetosphere. In addition, the strong southward IMF Bz component and high solar wind changes in the magnetosphere system, which were a result of the X2.0 solar flare event and the CMEs on 2014 October 27, responded to the moderate storm (SYM/H = −60 nT) at the beginning of 2014 October 28. This dynamic physical process in the magnetosphere caused by solar wind variation is seen to excite the Earth’s H-component through ultra low frequency at the ground-based magnetometers at the BCL (Vietnam), TIR (India) and SCN (Indonesia) stations during the geomagnetic storm. This study relates to seismic investigations and geomagnetic-induced current on the ground.

We present a 3D geometrical model to describe the propagation and expansion of coronal mass ejections (CMEs) in the interplanetary space based on radiative proxies to be implemented in previous procedures that use SXR and microwave emissions to estimate the Earth-directed CME propagation speed. We carefully selected a sample of 45 well-defined CME-ICME events to evaluate our model. We computed this 3D geometrical model for each event as a tool to improve the arrival time predictions based on radiative proxies. We conducted a different analysis for each radiative proxy: SXR emission and microwave emission at 9 GHz, and we compared the results separately with the observations by the Wind spacecraft. In general, the results showed that the implementation of our 3D geometrical model improves the predictions and provides an important complement to the arrival time prediction method based on radiative proxies, especially for CME events whose source origin were located at helio longitudes far from the central meridian at least 10°. Improvements for this tool based on observations by Parker Solar Probe and Solar Orbiter must be developed in the future work.

SUMMARYWe present a theory of modern, thermally induced deformation in a realistic Earth. The heat conduction equation is coupled with standard elastic deformation theory to construct a boundary-value problem comprised of eighth-order differential equations. The accurate and stable dual variable and position propagating matrix technique is introduced to solve the boundary-value problem. The thermal load Love numbers are defined to describe the displacements and potential changes driven by thermally induced deformation. The proposed analytical method is validated by comparing the present results with exact solutions for a homogeneous sphere, which are also derived in this paper. The analytical method is then applied to a realistic Earth model to evaluate the effects of layering and self-gravitation of the Earth on displacement and changes of potential. Furthermore, the frequency dependence in the thermal load is illustrated by invoking different thermal periodicities in the computation. Thermal anisotropy is also considered by comparing the results obtained using isotropic and transversely isotropic Earth models. Results show that, when simulating thermally induced deformation, invoking a homogeneous spherical Earth leads to results that substantially differ from those obtained using a more realistic Earth model.

SUMMARYWe derive exact asymptotic solutions for the static deformation due to a concentrated or point-like dislocation in a spherical, layered, elastic, isotropic and self-gravitating Earth. The exact asymptotic solutions are quite general and can provide the dislocation Love numbers on the Earth's surface, near the dislocation or ‘source’, and close to any layer interface or boundary. We also discuss the special case where both the source and field points are located on the Earth's surface. We compare our exact asymptotic solutions with previous results obtained from the analytical dual variable and position (DVP) method via curve fitting. Our comparison confirms that the analytical DVP results converge to the exact asymptotic solutions. These new exact asymptotic solutions are particularly helpful when evaluating slowly convergent series of Green's functions using a Kummer transformation, anywhere within the layered Earth, especially for field points located very close to the point dislocation or source.

SUMMARY The energy changes corresponding to the axis and speed of the Earth’s rotation are the polar motion energy, ΔEpm and the spinning energy, ΔEss. Here we develop an explicit approach to compute the coseismic ΔEpm and ΔEss based on a spherical-Earth elastic dislocation theory. We estimate the cumulative changes induced by global major earthquakes from 1976 to 2019 using the derived expression. The results show that the rate of ΔEpm and ΔEss is 0.45 × 1012 and 0.20 × 1019 J −1, corresponding to 0.01 MW (1 MW = 106 Watt) and 0.06 TW (1 TW = 1012 Watt), respectively. The χ2 test shows that global earthquakes tend to strongly increase the spinning energy rather than the polar motion energy. Comparing the coseismic rotational energy to the coseismic gravitational energy and the radiated seismic-wave energy induced by every single event, we found that coseismic rotational energy is distinctly at least 7.6 times larger than the radiated seismic-wave energy, demonstrating the importance of rotational energy in the energy budget of the faulting.

Previous work has demonstrated orbital stability for 100 Myr of initially near-circular and coplanar small bodies in a region termed the 'Earth-Mars belt' from 1.08 au $< a, Comment: 8 pages, 4 figures. Accepted for publication in MNRAS

SUMMARYPlanetary scale interannual deformations of the Earth’s surface, of millimetric amplitude, have recently been related to both geomagnetic field changes and motion within the fluid outer core. We calculate the temporal variations of the dynamical pressure at the surface of the core associated with core flow models inverted from geomagnetic observations. From these we compute predictions of the changes in Earth’s topography in response to elastic deformations in the mantle. We show that at decadal periods, the predicted changes in Earth’s topography are at most of the order of 0.3 mm. Focused at interannual periods between 4 and 9.5 yr, the predicted topography variations are smaller than 0.05 mm, at least an order of magnitude smaller than the reported observations. These amplitudes are only weakly sensitive to the choice of hypothesis used to reconstruct fluid motions at the core surface. We conclude that surface deformations induced by dynamical pressure changes in the core are below the detection level at present-day. Alternative geophysical sources must be sought to explain the observed millimetric interannual variations of the planetary scale topography, and its associated gravity variations. We currently see no justification for a physical relationship between interannual fluctuations of the geomagnetic field and of Earth’s observed deformations. We conjecture that the largest gravity signal of core origin is potentially associated with decadal longitudinal oscillations of the inner core. It might be detectable as longer series will become available.

SUMMARYIn this paper, we derive analytical solutions for the dislocation Love numbers (DLNs) and the corresponding Green's functions (GFs) within a layered, spherical, transversely isotropic and self-gravitating Earth. These solutions are based on the spherical system of vector functions (or the vector spherical harmonics) and the dual variable and position matrix method. The GFs for displacements, strains, potential and its derivatives are formulated in terms of the DLNs and the vector spherical harmonics. The vertical displacement due to a vertical strike-slip dislocation and the potential change (nΦ) due to a vertical dip-slip dislocation are found to be special, with an order O(1/n) on the source level and O(n) elsewhere. Numerical results are presented to illustrate how the internal fields depend on the particular type of dislocation. It is further shown that the effect of Earth anisotropy on the strain field can be significant, about 10 per cent in a layered PREM model and 30 per cent in a homogeneous earth model.

SUMMARYThe current set of reference models for the radial variation of Earth structure have been in use for several decades, and provide a good representation of many aspects of the seismic wavefield. Nevertheless, strong constraints from the differential times between pairs of SmKS phases indicate the need to modify the P wave speed profile in the upper part of the outer core. In order to incorporate such a change and maintain the representation of the full suite of seismic phases compensatory adjustments have to be made, dominantly in the mantle. Using multi-objective optimization, a new preferred radial model ek137 has been generated that provides a good representation of the traveltimes of all core phases. An adiabatic profile can be maintained through most of the outer core, but departures are needed at the base, as in the ak135 model. The latest estimates for inner core shear wave speed are included in ek137.

SUMMARY Planetary-scale mass redistributions occur on Earth for certain spatiotemporal periods, and these surface mass changes excite the global periodic loading deformations of a viscoelastic Earth. However, the characteristics of periodic viscoelastic deformations have not been well investigated even in a simple earth model. In this study, we derive the semi-analytical Green's functions (fully analytical Love numbers) for long-standing point sources with given periods using a modified asymptotic scheme in a homogeneous Maxwell spherical earth model. Here, the asymptotic scheme is needed in order to obtain accurate semi-analytical time-dependent Green's functions. The amplitudes and phases of the Green's functions may be biased if only the series summations of the Love numbers are used because the influence of viscoelasticity is degree-dependent. We compare the viscoelastic and elastic periodic Green's functions with different material viscosities and loading periods and investigate the amplitude increase percentage and phase delay of the periodic displacement and geoid change. For example, our analysis revealed that the viscosity increases the amplitude by 40–120 per cent and delays the phase approximately −100° to 60° for the displacement and geoid change when bearing a 10-yr loading period, assuming a viscosity of 1018 Pa s and a shear modulus 4 × 1010 Pa.

SUMMARY This study establishes the proper framework in which to compare seismic observations with mineral physics constraints for studies of the inner core by determining how the elements of the elasticity tensor are sampled by the normal modes of the Earth. The obtained mapping between the elements of the elasticity tensor and the seismic wave speeds shows that the choice of averaging scheme used to calculate isotropic properties is crucial to understand the composition of the inner core, especially for comparison with the shear wave speed such as that provided in PREM. For example, the appropriate shear wave speed calculated for an Fe-Ni-Si hcp alloy at inner-core conditions differs from the shear wave speed obtained by taking a Reuss average by as much as $27\, {\rm per\, cent}$. It is also shown for the first time that by combining the isotropic observations based upon normal-mode characteristic frequencies and anisotropic parameters from their splitting, the five independent elastic parameters (A, C, F, L and N) that fully describe a transversely isotropic inner core can be uniquely constrained. The elastic values based upon a variety of mode-splitting studies are reported, and the differences between models from various research groups are shown to be relatively small ($\lt 10\, {\rm per\, cent}$). Additionally, an analogous body-wave methodology is developed to approximately estimate the five independent elastic constants from observations of compressional wave traveltime anomalies. The body-wave observations are utilized to consider the depth dependence of inner-core anisotropy, in particular, the structure of the innermost inner core. Finally, we demonstrate that substantial errors may result when attempting to relate seismically observed P and S wave speeds from Debye velocities obtained through nuclear resonant inelastic X-ray scattering. The results of these experiments should be compared directly with the Debye velocity calculated from seismically constrained elastic constants. This manuscript provides a new set of formulae and values of seismic observations of the inner core that can be easily compared against mineral physics constraints for better understanding of the inner-core composition.

SUMMARY Parker–Oldenburg's method is perhaps the most commonly used technique to estimate the depth of density interface from gravity data. To account for large density variations reported, for instance, at the Moho interface, between the ocean seawater density and marine sediments, or between sediments and the underlying bedrock, some authors extended this method for variable density models. Parker–Oldenburg's method is suitable for local studies, given that a functional relationship between gravity data and interface geometry is derived for Earth's planar approximation. The application of this method in (large-scale) regional, continental or global studies is, however, practically restricted by errors due to disregarding Earth's sphericity. Parker–Oldenburg's method was, therefore, reformulated also for Earth's spherical approximation, but assuming only a uniform density. The importance of taking into consideration density heterogeneities at the interface becomes even more relevant in the context of (large-scale) regional or global studies. To address this issue, we generalize Parker–Oldenburg's method (defined for a spherical coordinate system) for the depth of heterogeneous density interface. Furthermore, we extend our definitions for gravity gradient data of which use in geoscience applications increased considerably, especially after launching the Gravity field and steady-state Ocean Circulation Explorer (GOCE) gravity-gradiometry satellite mission. For completeness, we also provide expressions for potential. The study provides the most complete review of Parker–Oldenburg's method in planar and spherical cases defined for potential, gravity and gravity gradient, while incorporating either uniform or heterogeneous density model at the interface. To improve a numerical efficiency of gravimetric forward modelling and inversion, described in terms of spherical harmonics of Earth's gravity field and interface geometry, we use the fast Fourier transform technique for spherical harmonic analysis and synthesis. The (newly derived) functional models are tested numerically. Our results over a (large-scale) regional study area confirm that the consideration of a global integration and Earth's sphericty improves results of a gravimetric forward modelling and inversion.

SUMMARY Antipodal diffracted, compressional wave (Pdiff) data analysed diametrically opposite three large earthquakes have uniformly sampled 99% of the laterally heterogeneous zone above Earth's core–mantle boundary (D″ in seismic nomenclature). These antipodal data offer a fundamental gross Earth datum—a robust, global constraint on the average compressional velocity at the base of the mantle. We use for the first time the seismic phase PKPAB as a reference, which travels an identical mantle path as Pdiff, thereby cancelling common mantle heterogeneity. Differential traveltimes between Pdiff, PKPAB and PKIKP are measured, appropriately making allowance for the phase shifts acquired in propagation. We have independently confirmed the $\pi /4$ polar phase shift of Pdiff at the antipode. The global mean PKPAB − Pdiff time is 136.5 ± 0.6 s. The global mean apparent velocity (13.05 km s−1) and ray parameter (4.65 ± 0.01 s deg−1) are within the margin of error of prior Pdiff studies—which were dominated by Northern Hemisphere paths—indicating that complementary, southern hemisphere paths have a comparable, mean Pdiff apparent velocity. The seismic velocity constraints afforded by antipodal Pdiff and PKPAB suggest that the heterogeneous processes already observed in D″ may be broadly ascribed where D″ coverage has been lacking or poorly resolved.

Most known porphyry Cu ± Au deposits are associated with moderately oxidized and sulfur-rich, calc-alkaline to mildly alkalic arc-related magmas in the Phanerozoic. In contrast, sodium-enriched tonalite–trondhjemite–granodiorite–diorite (TTG) magmas predominant in the Archean are hypothesized to be unoxidized and sulfur-poor, which together preclude porphyry Cu deposit formation. Here, we test this hypothesis by interrogating the causative magmas for the ~2·7 Ga TTG-related Côté Gold, St-Jude, and Clifford porphyry-type Cu ± Au deposit settings in the Neoarchean southern Abitibi subprovince. New and previously published geochronological results constrain the age of emplacement of the causative magmas at ~2·74 Ga, ~2·70 Ga, and ~ 2·69 Ga, respectively. The dioritic and trondhjemitic magmas associated with Côté Gold and St-Jude evolved along a plagioclase-dominated fractionation trend, in contrast to amphibole-dominated fractionation for tonalitic magma at Clifford. Analyses of zircon grains from the Côté Gold, St-Jude, and Clifford igneous rocks yielded εHf(t) ± SD values of 4·5 ± 0·3, 4·2 ± 0·6, and 4·3 ± 0·4, and δ18O ± SD values of 5·40 ± 0·11 ‰, 3·91 ± 0·13 ‰, and 4·83 ± 0·12 ‰, respectively. These isotopic signatures indicate that, although these magmas are mantle-sourced with minimal crustal contamination, for the St-Jude and Clifford settings the magmas or their sources may have undergone variable alteration by heated seawater or meteoric fluids. Primary barometric minerals (i.e. zircon, amphibole, apatite, and magnetite–ilmenite) that survived variable alteration and metamorphism (up to greenschist facies) were used for estimating fO2 of the causative magmas. Estimation of magmatic fO2 values, reported relative to the fayalite–magnetite–quartz buffer as ΔFMQ, using zircon geochemistry indicates that the fO2 values of the St-Jude, Côté Gold, and Clifford magmas increase from ΔFMQ –0·3 ± 0·6 to ΔFMQ +0·8 ± 0·4 and to ΔFMQ +1·2 ± 0·4, respectively. In contrast, amphibole chemistry yielded systematically higher fO2 values of ΔFMQ +1·6 ± 0·3 and ΔFMQ +2·6 ± 0·1 for Côté Gold and Clifford, respectively, which are consistent with previous studies that indicate that amphibole may overestimate the fO2 of intrusive rocks by up to 1 log unit. Micro X-ray absorption near edge structure (μ-XANES) spectrometric determination of sulfur (i.e. S6+/ΣS) in primary apatite yielded ≥ΔFMQ −0·3 and ΔFMQ +1·4–1·8 for St-Jude and Clifford, respectively. The magnetite–ilmenite mineral pairs from the Clifford tonalite yielded ΔFMQ +3·3 ± 1·3 at equilibrium temperatures of 634 ± 21 °C, recording the redox state of the late stage of magma crystallization. Electron probe microanalyses revealed that apatite grains from Clifford are enriched in S (up to 0·1 wt%) relative to those of Côté Gold and St-Jude (below the detection limit), which is attributed to either relatively oxidized or sulfur-rich features of the Clifford tonalite. We interpret these results to indicate that the deposits at Côté Gold and Clifford formed from mildly (~ΔFMQ +0·8 ± 0·4) to moderately (~ΔFMQ +1·5) oxidized magmas where voluminous early sulfide saturation was probably limited, whereas the St-Jude deposit represents a rare case whereby the ingress of externally derived hydrothermal fluids facilitated metal fertility in a relatively reduced magma chamber (~ΔFMQ +0). Furthermore, we conclude that variable modes of formation for these deposits and, in addition, the apparent rarity of porphyry-type Cu–Au deposits in the Archean may be attributed to either local restriction of favorable metallogenic conditions, and/or preservation, or an exploration bias.

SUMMARY Linear and non-linear viscoelastic (VE) models such as the standard linear solid (SLS) and the generalized SLS (GSLS) are broadly used to represent the anelasticity of materials and Earth's media. However, although the VE approach is often satisfactory for any given observation, the inferred physical causes of anelasticity may be significantly misrepresented by this paradigm, and its predictions may be wrong or inaccurate in other cases. This problem is particularly important in heterogeneous media, including most cases of interest for seismology. For example, in homogenous media, VE and mechanics-based models predict identical quality-factor Q(f) and phase velocity c(f) spectra, but in heterogenous media, these models yield different time-stepping equations and interactions with material–property boundaries. The commonly used VE algorithms for modelling seismic waves rely on postulated convolutional integrals in time, whereas physically, models of rock rheologies should still be based on spatial interactions. To understand how VE models relate to mechanics, it is instructive to consider which physical properties of the medium are constrained reliably and which of them remain unconstrained by a pair of Q(f) and c(f) spectra, that is by VE properties. Despite its popular association with ‘attenuation,’ the peak value of Q−1(f) is actually a purely elastic property representing the existence of two (for SLS) or multiple (for GSLS) elastic moduli. These moduli are analogous to the drained and undrained moduli in poroelasticity or isothermal and adiabatic moduli in thermodynamics. By virtue of the Kramers–Krönig relations, the peak Q−1 is related to the total velocity dispersion, which is also caused by the difference between elastic moduli. By contrast, true anelasticity-related physical properties like viscosity are represented not by Q−1 values but by the frequencies of Q−1(f) peaks in the data. However, these frequencies also depend on multiple material properties that are not recognized or arbitrarily selected in the SLS and GSLS models. Inertial, body-force friction and the corresponding boundary effects are also ignored in VE models, which may again be improper for layered media. Thus, for physically accurate interpretation of laboratory experiments and numerical modelling of seismic waves, first-principle equations of mechanics should be used instead of VE models.

SUMMARYOver the past few decades, seismic studies have revealed complex structural anomalies in the Earth’s deep interior at various scales, such as large low-shear-velocity provinces (LLSVPs) and ultra-low velocity zones (ULVZs) in the lowermost mantle, and small-scale scatterers in the mid-mantle. These structures which are critical for better understanding of the geodynamics and evolution of the deep Earth, need to be further resolved by high-resolution imaging techniques. The spectral-element method (SEM) can be used to accurately simulate seismic wave propagation in heterogeneous Earth models, and its application in full-waveform inversion (FWI) provides a promising high-resolution and high-fidelity imaging technique. But it can be computationally prohibitive when used to model small scale structures in the deep Earth based upon high-frequency seismic waves. The heavy computational cost can be circumvented by using hybrid methods, which restrict the main computation by SEM solver to only a small target region (e.g. above the CMB) encompassing possible 2-D/3-D anomalies, and apply efficient analytical or numerical methods to calculate the wavefield for 1-D background models. These forward modelling tools based on hybrid methods can be then used in the so-called ‘box tomography’ approach to resolve fine-structures in the deep Earth.In this study, we outline the theory of a hybrid method used to model small scale structures in the deep Earth and present its implementation based on SEM solvers in a three-step workflow. First, the wavefield generated by the source is computed for the 1-D background model with traction and velocity saved for the virtual boundary of the target region, which are then used as boundary inputs to simulate the wavefield in the target region based on absorbing boundary condition in SEM. In the final step, the total wavefield at receivers is reconstructed based upon the total wavefield on the virtual boundary computed in the previous step. As a proof-of-concept study, we demonstrate the workflow of the hybrid method based on a 2-D SEM solver. Examples of the hybrid method applied to a coupled fluid–solid model show that our workflow can accurately recover the scattered waves back to the surface. Furthermore, we benchmark the hybrid method on a realistic heterogeneous Earth model built from AK135-F and show how teleseismic scattered waves can be used to model deep Earth structures. By documenting the theory and SEM implementation of the hybrid method, our study lays the foundation for future two-way coupling of 3-D SEM solver with other efficient analytic or numerical 1-D solvers.

SUMMARYWe present an accurate approach for calculating the point-dislocation Green's functions (GFs) for a layered, spherical, transversely-isotropic and self-gravitating Earth. The formalism is based on the approach recently used to find analytical solutions for the dislocation Love numbers (DLNs). However, in order to make use of the DLNs, we first analyse their asymptotic behaviour, and then the behaviour of the GFs computed from the DLNs. We note that the summations used for different GF components evolve at different rates towards asymptotic convergence, requiring us to use two new and different truncation values for the harmonic degree (i.e. the index of summation). We exploit this knowledge to design a Kummer transformation that allows us to reduce the computation required to evaluate the GFs at the desired level of accuracy. Numerical examples are presented to clarify these issues and demonstrate the advantages of our approach. Even with the Kummer transformation, DLNs of high degree are still needed when the earth model contains very fine layers, so computational efficiency is important. The effect of anisotropy is assessed by comparing GFs for isotropic and transversely isotropic media. It is shown that this effect, though normally modest, can be significant in certain contexts, even in the far field.

The c.3·3 Ga Commondale komatiites located south of the Barberton greenstone belt in the Kaapvaal Craton are different from other komatiites, possessing compositional and textural features unique to this occurrence. Unlike almost all other known komatiite occurrences, they are not associated with komatiitic basalts or basalts. The komatiite flows are 0·5–25 m thick and are made up of a marginal zone of spinifex-textured and fine-grained aphyric rocks (low-Mg group) and an inner zone of olivine cumulates (high-Mg group), arranged in such a way to give highly symmetrical compositional profiles for many flows. Olivine is the dominant phase in all rocks, but orthopyroxene occurs as spinifex and elongate laths in the marginal zone. Clinopyroxene and plagioclase are entirely absent. The olivine cumulates formed from Mg-rich magma (36·1% MgO, 6·8% FeO) which caused inflation of the thicker flows. The maximum observed olivine composition in cores (Fo 96·6) is the highest recorded for any komatiite worldwide. The high-Mg magma would have erupted at a temperature close to 1670°C, the highest inferred temperature for an anhydrous terrestrial lava. The marginal zone is enriched in incompatible elements compared with the inner zone and formed by fractionation of the parental melt. However, all rock-types in the marginal zone are depleted in FeO (some as low as 3·5%) which could not have been derived by any primary magmatic process. The marginal zone rocks were modified by assimilation and/or alteration by seawater (or brine) components causing migration of iron and strong enrichment of sodium (up to 1·6 wt % Na2O) and chlorine (up to 2400 ppm). Zirconium has an identical distribution to sodium, with both elements greatly enriched above what would result from fractional crystallization, and may result from speciation of these elements at high temperature followed by post-crystallization alteration. Rare earth elements, Y and Nb have contents commensurate with fractionation of the primitive parental magma. Dendritic-textured olivine-rich rocks with orthopyroxene spinifex spatially and compositionally transitional between the marginal zone and the olivine cumulates resulted from interaction of the high temperature parental magma in the centre of the flows with the fractionated melt at the flow margins. A further manifestation of this association is the development of highly regular fine-scale (5–15 cm) layering (up to 45 layers) of alternating olivine cumulate and spinifex near the base of thick flows. This is overlain by olivine cumulates in which the melt/crystal-mush became arranged into a 3-dimensional network controlled by re-distribution of the trapped melt manifest by a spectacular knobbly texture in outcrop. Over 200 flow units are recognized and detailed chemical and mineralogical studies were carried out on drill cores intersecting 375 m of stratigraphy. The parental magma was highly depleted (in ppm Nb 0·017, Zr 1·18, total REE 1·7 and Gd/YbN=0·3, La/YbN=0·038) and although generally regarded to fall into the rare category of Al-enriched komatiites (AEKs), it is considered that these lavas are a unique class of their own of ultra-depleted komatiites. Relative to other AEKs the Commondale komatiites are both enriched in Al as well as being markedly depleted in Ti (390 ppm), giving rise to the extremely high Al2O3/TiO2 (81). The high temperature and low viscosity of the magma resulted in emplacement processes previously unrecognized in komatiites. The primary melt was derived by melting of mantle peridotite in equilibrium with olivine and orthopyroxene. The initial source was depleted in incompatible elements by small degrees of melting (3–4%) followed by high degrees of partial melting (70%) of the subsequent refractory source at 5 GPa (∼150 km).

One of the fundamental features of space plasmas is the observation of non-Maxwellian particle velocity distributions. In the present study, we observe electron velocity distributions in the Earth's magnetosphere at times when the electron density is low, typical of cusp values, and when it is enhanced as a result of disturbances by the solar wind. We find that electron distributions are flat-topped and have two populations: one cold and one hot. We fit the observed electron distributions by a generalized $( {r,q} )$ distribution, and derive and plot expressions for the real frequency and growth rate using fitted and observed parameters. We show that enhancement in the density of hot electrons enhances the growth rate of whistler waves, which play an important role in energy transport in the Earth's magnetosphere.

Magmas and other viscously deforming fluids in the Earth frequently contain embedded crystals or other solid inclusions. These inclusions generally rotate about their own axis and, under certain co ...

We present a novel numerical method to simulate global seismic wave propagation in realis- tic aspherical 3-D Earth models across the observable frequency band of global seismic data. Our method, named AxiSEM3D, is a hybrid of spectral element method and pseudo-spectral method. It describes the azimuthal dimension of global wavefields with a substantially reduced number of degrees of freedom via a global Fourier series parametrisation, of which the num- ber of terms can be locally adapted to the inherent azimuthal complexity of the wavefields. AxiSEM3D allows for material heterogeneities, such as velocity, density, anisotropy and at- tenuation, as well as for finite undulations on radial discontinuities, both solid-solid and solid- fluid, and thereby a variety of aspherical Earth features such as ellipticity, surface topography, variable crustal thickness, undulating transition zone and core-mantle boundary topography. Undulating discontinuities are honoured by means of the “particle relabelling transformation”, so that the spectral element mesh can be kept spherical. The implementation of the particle relabelling transformation is verified by benchmark solutions against a discretised 3-D spectral element method, considering ellipticity, topography and bathymetry (with the ocean approxi- mated as a hydrodynamic load) and a tomographic mantle model with an undulating transition zone. For the state-of-the-art global tomographic models with aspherical geometry but without a 3-D crust, efficiency comparisons suggest that AxiSEM3D can be 2 to 3 orders of magnitude faster than a discretised 3-D method for a seismic period at 5 s or below, with the speed-up in- creasing with frequency and decreasing with model complexity. We also verify AxiSEM3D for localised small-scale heterogeneities with strong perturbation strength. With reasonable com- puting resources, we have achieved a corner frequency up to 1 Hz for 3-D mantle models.

Determining the thermal conductivity of iron alloys at high pressures and temperatures are essential for understanding the thermal history and dynamics of the Earth's metallic cores. The authors summarize relevant high-pressure experiments using a diamond-anvil cell and discuss implications of high core conductivity for its thermal and compositional evolution.

The discovery of komatiites, first in South Africa and then in many other Archean greenstone belts, with MgO concentrations of 20–30% and eruption temperatures of more than ∼1600 °C, showed that some parts of the mantle were hotter in the Archean than they are now. Since their discovery there have been many speculative proposals as to how such magmas can form. At present melt is produced by mantle upwelling, because the solidus temperature gradient of the mantle is steeper than that of isentropic decompression gradient at depths of less than 300 km. In contrast, in the lower half of the upper mantle the solidus gradient is shallower than the isentropic gradient, and, therefore, isentropic upwelling cannot generate melt. At the base of the upper mantle limited melting can occur, either in the thermal boundary layer at the base of the upper mantle, or in the upper part of the lower mantle where the solidus gradient is steeper than the isentropic gradient. In both cases melting can occur at depths of more than 600 km, where Ca perovskite, CaPv, is a stable phase on the solidus. A surprising feature of the partitioning between melt and solid CaPv is that most trace elements are compatible in the solid. Partitioning into CaPv can, therefore, account for the low concentrations of such elements in komatiites. The temperatures required to generate such magmas in plumes need be no more than ∼50 °C above those of Phanerozoic plumes. The presence of komatiites in the Archean, therefore, requires plume temperatures in the first half of the Earth’s history to have been somewhat hotter than they are now, but does not constrain the average temperature of the Archean upper mantle.

The c. 1·85 Ga Sudbury Igneous Complex (SIC) is the igneous remnant of one of the oldest, largest and best-preserved impact structures on Earth and contains some of the world’s largest magmatic Ni–Cu–PGE sulfide deposits. Most of the mineralization occurs in Sublayer, Footwall Breccia and inclusion-bearing quartz diorite (IQD), all of which contain significant (Sublayer and IQD) to minor (Footwall Breccia) amounts of olivine-bearing mafic–ultramafic inclusions. These inclusions have only rare equivalents in the country rocks and are closely associated with the Ni–Cu–PGE sulfide mineralization. They can be divided into three groups on the basis of petrography and geochemical characteristics. Group I (n = 47) includes igneous-textured olivine melanorite and olivine melagabbronorite inclusions in the Whistle and Levack embayments on the North Range with Zr/Y, Zr/Nb, Nb/U and Zr/Hf similar to igneous-textured Sublayer matrix. Group I inclusions are interpreted to be anteliths that crystallized from a mixture of SIC impact melt and a more mafic melt, probably derived by melting of ultramafic footwall rocks. Group II includes Group IIA (n = 17) shock metamorphosed wehrlite and olivine clinopyroxenite inclusions in the Levack embayment and Group IIB (n = 2) shock metamorphosed olivine melanorite inclusions in the Foy Offset on the North Range. Group II inclusions have similar trace element patterns [e.g. negative Th–U, Nb–Ta–(Ti), Sr and Zr–Hf anomalies] and overlapping Nb/U to a layered mafic–ultramafic intrusion in the footwall of the Levack and Fraser deposits, which together with their limited distribution suggests that Group II inclusions are locally-derived xenoliths. Group III (n = 21) includes phlogopite lherzolite and feldspar lherzolite inclusions with igneous, recrystallized and shock-metamorphic textures in the Trill, Levack and Bowell embayments, and the Foy Offset dike on the North Range. They have no equivalents in the exposed country rocks. The calculated parental magma is similar to continental arc basalt formed by approximately 5% partial melting of garnet peridotite. Ol–Cpx–Pl thermobarometry of several Group III inclusions indicate equilibration at 900–1120 ºC and 210 ± 166 MPa to 300 ± 178 MPa, suggesting crystallization in the upper-middle crust (7·7 ± 6·6 to 10·9 ± 6·5 km), prior to being incorporated into the lower parts of the impact melt sheet during impact excavation. The exotic xenoliths provide information about the depth of impact and composition of upper-middle crust in the Sudbury region at 1850 Ma, the local xenoliths provide information about the thermomechanical erosion process that followed generation of the impact melt, the anteliths provide information about the early crystallization history of the SIC, and all of the inclusions provide constraints on the genesis of Sublayer, IQD, footwall breccia, and associated Ni–Cu–PGE mineralization.

Understanding the mineralogy of the Earth's interior is a prerequisite for unravelling the evolution and dynamics of our planet. Here, we conducted high pressure-temperature experiments mimicking the conditions of the deep lower mantle (DLM, 1800–2890 km in depth) and observed surprising mineralogical transformations in the presence of water. Ferropericlase, (Mg, Fe)O, which is the most abundant oxide mineral in Earth, reacts with H2O to form a previously unknown (Mg, Fe)O2Hx (x ≤ 1) phase. The (Mg, Fe)O2Hx has a pyrite structure and it coexists with the dominant silicate phases, bridgmanite and post-perovskite. Depending on Mg content and geotherm temperatures, the transformation may occur at 1800 km for (Mg0.6Fe0.4)O or beyond 2300 km for (Mg0.7Fe0.3)O. The (Mg, Fe)O2Hx is an oxygen excess phase that stores an excessive amount of oxygen beyond the charge balance of maximum cation valences (Mg2+, Fe3+ and H+). This important phase has a number of far-reaching implications including extreme redox inhomogeneity, deep-oxygen reservoirs in the DLM and an internal source for modulating oxygen in the atmosphere.

[abridged] In the typical giant-impact scenario for the Moon formation most of the Moon's material originates from the impactor. Any Earth-impactor composition difference should, therefore, correspond to a comparable Earth-Moon composition difference. Analysis of Moon rocks shows a close Earth-Moon composition similarity, posing a challenge for the giant-impact scenario, given that impactors were thought to significantly differ in composition from the planets they impact. Here we use a large set of 140 simulations to show that the composition of impactors could be very similar to that of the planets they impact; in $4.9\%$-$18.2\%$ ($1.9\%$-$6.7\%$) of the cases the resulting composition of the Moon is consistent with the observations of $��^{17}O, 15 pages, 12 figures, resubmitted to MNRAS after referee comments. Abstract abridged for arXiv submission

Filamentous microstructures in 3.46 billion-year-old rocks (Apex chert) from Western Australia have been claimed to represent the oldest morphological evidence of life on Earth [1]. However, the biological nature of these filaments has been questioned on numerous occasions [e.g., 2] and this has led to one of the longest running and most controversial debates in palaeontology. Here we use a combination of focused ion beam (FIB) milling, TEM and SEM to decode the detailed morphology and chemistry of the Apex filaments and determine their formation mechanism.

Anthelmintic resistance urges alternatives to control gastrointestinal nematodes (GIN). Copper oxide wire particles (COWP) are more efficacious when used with dewormers and little is known on efficacy of diatomaceous earth (DE) to control GIN. The objective was to examine the effects of DE and COWP on GIN control. Katahdin lambs (n = 32; ~150 d of age; 25.0 ± 1.8 kg) were randomly assigned to receive: 1) 2% dietary DE, 2) 1g COWP, 3) both 2% DE and 1g COWP, and 4) neither DE or COWP. Feces and blood were collected on d 0, 7, 14, 21 and 28 to determine fecal egg counts (FEC) and blood packed cell volume (PCV). Feces were cultured to determine GIN genera. FEC were log transformed; data were analyzed as a 2 x 2 factorial using proc mixed with day as repeated (SAS). There was a mixed GIN population including 58% Haemonchus contortus and 30% Trichostrongylus spp. FEC tended to be lower by d 28 in DE (175 eggs/g) than no DE (753 eggs/g) fed lambs (P = 0.09), but in general, were relatively low and no other differences among treatments detected. PCV tended to be higher in COWP than no COWP lambs by d 28 (P = 0.10), and no other differences detected. In conclusion, during a low GIN challenge, there may have been a small effect of DE on FEC but no advantage was observed in combination treatment.