Finite strain theories and comparisons with seismological data

All the finite strain equations that we are aware of that are worth considering in connection with the interior of the Earth are given, with the assumptions on which they are based and corresponding relationships for incompressibility and its pressure derivatives in terms of density. In several cases, equations which have been presented as new or independent are shown to be particular examples of more general equations that are already familiar. Relationships for deriving finite strain equations from atomic potential functions or vice versa are given and, in particular it is pointed out that the Birch-Murnaghan formulation implies a sum of power law potentials with even powers. All the equations that survive simple plausibility tests are fitted to the lower mantle and outer core data for the PEM earth model. For this purpose the model data are extrapolated to zero temperature, using the Mie-Grüneisen equation to subtract the thermal pressure (at fixed density) and the pressure derivative of this equation to substract the thermal component of incompressibility. Fitting of finite strain equations to such zero temperature data is less ambiguous than fitting raw earth model data and leads immediately to estimates of the low temperature zero pressure parameters of earth materials. On this basis, using the best fitting equations and constraining core temperature to give an extrapolated incompressibilityK ₀=1.6×10¹¹Pa, compatible with a plausible iron alloy, the following numerical data are obtained: Core-mantle boundary temperature 3770 K Zero pressure, zero temperature densities: lower mantle 4190 kg m^–3 outer core (solidified) 7500 kg m^–3 Zero pressure, zero temperature incompressibility of the lower mantle 2.36×10¹¹PaHowever, an inconsistency is apparent betweenP() andK() data, indicating that, even in the PEM model, in which the lower mantle is represented by a single set of parameters, it is not perfectly homogeneous with respect to composition and phase.     

Rare-earth and LIL element fractionation in high-grade charnockitic gneisses,south Norway

High-Fe, intermediate-acid, charnockitic gneisses in the Arendal-Tromøy area of the Svecofennian terrain of southeast Norway comprise two chemically contrasting zones - one with normal large-ion-lithophile (LIL) element characteristics, and the other IIL-deficint. The noramal -LiLL-deficient varitties also have low ΣEEE, commonly with positive Eu anomalies. The normal-LIL rocks are enriched in ?REE, exhibit fractionated patterns and have negative Eu anomalies. Modelling shows that both the LIL and REE patterns are consistent with an essentially primary fractionation process involving the separation of cumulus (LIL-deficient) phases from andesitic-dacitic magma emplaced directly under the high-grade conditions, with the normal-LIL rocks crystallising from the residual melt. This process is interpreted as a deep-seated component of the magma system which culminated in the emplacement of some higher level rapakivi granite late in the Svecofennian event. The model presented does not require anorthosite to be part of the same magma system.     

The chemistry and origins of Proterozoic low-potash,high-iron,charnockitic gneisses from Tromøy,south Norway

New major and trace element data for 79 acid-intermediate charnockitic gneisses (the Tromøy gneisses) and 16 associated metabasites from the island of Tromøy show that this part of the 1200–900-m.y. Sveconorwegian zone is occupied by rocks of unusual composition. Overall values for K and Rb are the lowest yet reported for any granulites, and K/Rb ratios are very high. Cs and Th are also low and, abnormally for granulites, so are Ba, Sr and Zr. Ba/Sr ratios are similar to those in other suites, but K/Ba and K/Sr are higher. These features may partially be reflecting unusual pre-metamorphic lithologies, but it is considered more likely that they are largely the product of metamorphically induced depletion processes involving metasomatism. There is some indication that the Na₂O/CaO and normative Ab/An ratios may also have been modified during metamorphism.Data for the presumed relatively immobile elements Cr, Co, Ni and V support an igneous origin for the Tromøy gneisses, but the presence of a paragneiss component cannot be ruled out. A characteristic of the gneisses is their high iron content, and spatial and temporal considerations point towards a genetic link with the iron-rich, intrusive rapakivi suites of Finland, Sweden and south Greenland. If the Tromøy gneisses do represent material of this type, it would seem to follow that potash fractionation has been extreme.     

A thermal model of the earth

A thermal model, consistent as far as possible with the parameterised earth model of Dziewonski et al. and with thermodynamic principles and relevant equations of state, is tabulated. This is made more secure by two recent developments, an experimental study of the FeS eutectic to 100 kbar by Usselman and a calculation by Bukowinski which reveals an electronic phase collapse of potassium in the 200–300 kbar range and explains the core heat source. Use is made of the Vashchenko-Zubarev formulation of the Grüneisen ratio, and Lindemann's melting law, both of which have been shown recently to have particular relevance at very high pressures. Values of electronic specific heat and the Grüneisen ratio, which contribute significantly to core properties are calculated from the electron equation of state of Zharkov and Kalinin.     

The dynamical and thermal structure of deep mantle plumes

If the interpretation of the D″ layer at the base of the mantle as a thermal boundary layer, with a temperature increment in the order of 800 K, is correct, then the formation of deep-mantle plumes to vent material from it appears inevitable. We demonstrate quantitatively that the strong temperature dependence of viscosity guides the upward flow into long-lived chimneys that are ～ 20 km in diameter near the base of the mantle and decrease in width with progressive upward softening and partial melting of plume material. The speed of flow up the axis of the plume is correspondingly fast; 1.6 m y^?1 at the base and 4.8 m y^?1 at 670 km depth. Thermal diffusive spreading of a heated plume is compensated by a slow horizontal convergence of mantle material toward the chimney in response to the lower pressure there. This convergence, which contributes only a small increment to the flux of material up the plume, also serves to throttle the flow in the chimney. The global plume mass flux necessary to transport 1.6 × 10¹² W of core heat upward through the mantle is 1.8 × 10⁶ kg s^?1. At its base, plume material is probably still significantly below its solidus or eutectic temperature, but substantial partial melting is very likely as it rises. We speculate that a small fraction of this fluid component eventually emerges at the surface in “hot spots”, with the fate of the remainder being unknown. The behaviour and properties of D″ and of plumes are closely coupled. Not only are plumes a necessary consequence of a thermal boundary layer, but their existence is impossible without that layer.     

The cooling Earth: A reappraisal

By treating the lithosphere as a diffusive boundary layer to mantle convection, the convective speed or mantle creep rate, $\text{?}\text{?}$, can be related to the mantle-derived heat flux, $\text{Q}\text{?}$. If cell size is independent of $\text{Q}\text{?}{}^2$ then $\text{?}\text{?}\text{∝}\text{Q}\text{?}$. (If cell size varies with $\text{Q}\text{?}$, then a different power law prevails, but the essential conclusions are unaffected.) Then the factthat for constant thermodynamic efficiency of mantle convection, the mechanical power dissipation is proportionalto $\text{Q}\text{?}$, gives convective stress $\text{σ ∝}\text{Q}\text{?}{}^{\mathrm{?1}}$, i.e. the stress increases as the convection slows. This means an increasing viscosityor stiffness of the mantle which can be identified with a cooling rate in terms of a temperature-dependent creep law. If we suppose that the mantle was at or close to its melting point within 1 or 2 × 10⁸ years of accretionof the Earth, the whole scale of cooling is fixed. The present rate of cooling is estimated to be about 4.6 × 10^?8 deg y^?1 for the average mantle temperature, assumed to be 2500 K, but this very slow cooling rate represents a loss ofresidual mantle heat of 7 × 10¹² W, about 30% of the total mantle-derived heat flux. This conclusion requires theEarth to be deficient in radioactive heat, relative to carbonaceous chondrites. A consideration of mantle outgassing and atmospheric argon leads to the conclusion that the deficiency is due to depletion of potassium, and that the K/U ratio of the mantle is only about 2500, much less than either the crustal or carbonaceous chondritic values. Thetotal terrestrial potassium is estimated to be about 6 × 10²⁰ kg. Acceptance of the cooling of the Earth removes the necessity for potassium in the core.     

Anhydrous melting and crystallization of peralkaline obsidians

Experimental determinations of the dry liquidus temperatures of two pantellerite, and two pantelleritic trachyte glasses in the pressure range 0–2 kilobars, show minima in the liquidus curves between 0.1 and 0.2 kb. The pantellerite minima are 830°–850° C; the trachyte minima are 920°–940° C. At pressures below the minima a separate vapour phase co-exists with liquid, at higher pressures the intrinsic volatiles are completely soluble in the liquid and the liquidus curves have a positivedT/dP. Similar results have been obtained from a range of other pantelleritic glasses, and together with consistent alkali feldspar compositions (from a wide range of experimental conditions) are indicative of a close approach to equilibrium. The form of the liquidus curves above the minima, if rellecting natural conditions, offers a ready explanation of the near-or super-liquidus aspect of many peralkaline lavas. The temperatures in these anhydrous experiments are 100°–150° C higher than those for similar compositions in the presence of excess water. (Also, in the presence of excess water, the crystallization sequences in the natural glasses are profoundly modified, with pyroxene appearing on the liquidus). At lower pressures, feldspar is the liquidus phase in the dry pantellerites, but is joined by quartz around 1 kb, and superseded by quartz at higher pressures. As pantellerites with quartz phenocrysts are uncommon, low pressure equilibration is perhaps normal in these magmas. Feidspar is the usual liquidus phase in the trachytes, except at very low pressures where it is preceded by iron oxide. Preliminary studies at 5 kb indicate that the pantelleritic and trachytic liquidus curves are converging (in the range 950°–1000° C). Crystallization sequences, and the forms and positions of the solidus curves are therefore of vital importance. These, together with the vapour-present/vapour-absent conditions, are currently under investigation.