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Energetics of the Earth (1980)

Chapter: TEMPERATURES WITHIN THE EARTH

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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Suggested Citation:"TEMPERATURES WITHIN THE EARTH." National Research Council. 1980. Energetics of the Earth. Washington, DC: The National Academies Press. doi: 10.17226/9579.
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Temperatures Within the Earth The present temperature distribution in the earth depends on (1) the original temperature distribution shortly after formation; (2) the dis- tribution and intensity of heat sources, all of which are time dependent; and (3) the mechanism of internal heat transfer, whether by conduction or convection or both. Conversely, if the present temperature were known, it should be possible, at least in theory, to extract from it some information on, and possibly set upper or lower limits to, the distribu- tion of internal heat sources. Much effort has been devoted in the past to calculating the tempera- ture distribution in the-mantle, particularly the upper mantle, from the heat conduction equation. The main datum is the surface heat flow; a steady state is commonly assumed. Examples of such calculations are the often quoted "shield" and "oceanic" geotherms of Clark and Ring- wood (1964) and the calculations by MacDonald (1965), which are now only of historical interest, if only because it is now certain that heat is also carried in the upper mantle by convection; not all heat transfer takes place by conduction in the radial direction. It is clear that, for instance, much of the heat flowing across the oceanic floor has been brought there by horizontal motion of hot material rising under, and moving away from, oceanic ridges. Curiously, it escaped everyone's (or almost everyone's) notice that these calculations were self- contradictory, insofar as the large horizontal temperature variations between subcontinental and suboceanic mantle that emerged from 29

30 ENERGETICS OF THE EARTH them (e.g., MacDonald, 1965) were sufficient to cause the very convec- tion that had been assumed not to take place. But when convection is included, there is no simple way of calculat- ing the temperature, short of solving the full set of time-dependent equations expressing conservation of mass, momentum, and energy, which contain all the parameters of the conduction equation plus some poorly known parameters describing theological properties and their dependence on pressure and temperature. Only partial solutions for idealized models, generally two-dimensional ones, are available at this time; there is still some debate as to whether convection embraces the whole mantle or is restricted to its upper few hundred kilometers. Furthermore, to get a solution one must first postulate what and where the heat sources are; thus we obviously cannot use these calculations to determine the heat sources, which is our main objective at this stage. We must thus turn to other, more direct methods for estimating what the temperature may be at a given place. Broadly speaking, there are two methods for doing this. In the first method, one attempts to determine the temperature at a given depth from a known property of material at that depth. The property may be physical (e.g., velocity of elastic waves, electrical conductivity) or chemical (e.g., phase equilibrium between solid and liquid, or between polymorphs). In the second method, one tries to determine the tem- perature gradient in a homogeneous layer from an observed variation with depth of a physical property of the layer. In both methods, one must first allow for the effect of pressure, which, of course, also in- creases with depth. Results, as we shall see, are on the whole rather disappointing. The main difficulty generally lies in estimating the effects of pressure. Pres- sure in the earth is tolerably well constrained, to better than 1 percent, but pressure coefficients of physical properties are hard to measure at the very high pressures of the earth's interior. Physical properties in the earth are generally not known very precisely. Density at a given depth, for instance, is not known to better than a few percent, which is also the magnitude of any likely temperature effect. The exact composition and nature of the mineral phases present in the lower mantle are still debated, and so is the composition of the core. Some physical proper- ties (e.g., electrical conductivity) may be sensitive to variables (oxidation state, grain size, lattice defects, impurities) that, in the ab- sence of representative samples, cannot be estimated at all. We now review, in descending order from the top of the mantle to the inner core, some of the results.

Temperatures Within the Earth 31 THE UPPER MANTLE THE LOW-VELOCITY ZONE The low-velocity layer, as the name implies, is a layer in which the seismic velocities vp and vs are slightly lower than they are in the layers immediately above or below. The decrease in velocity with increasing depth is more pronounced for vat (shear wave) than for vp (compres- sional wave), and more pronounced under oceanic areas than under continents, where the low-velocity layer may either be absent or occur at greater depth. In oceanic areas the top of the low-velocity layer almost reaches the surface under ridges. Its depth increases progres- sively with distance from the ridge to a maximum of about 100 km; nowhere does it coincide with a sharp discontinuity. There is some evidence that the low-velocity layer coincides with the asthenosphere or "weak" zone, so named for its enhanced ability to flow as compared with the more rigid lithosphere above it. Lithospheric thickness is generally determined from seismic surface-wave dispersion; it appears to be well correlated with surface heat flow (Chapman and Pollack, 1977). There are several possible interpretations for the low-velocity layer. Thomsen (1971) has shown that it could be due to a steep temperature gradient. The effects on seismic velocities of increasing temperature and increasing pressure being generally of opposite signs, the normal increase in velocity with increasing depth (pressure) could be reversed in a zone where the temperature gradient is sufficiently steep to over- come the pressure effect. The required temperature gradient is of the order of 5°-10°/km for vs and 10°-20°/km for vp. At the moment, the preferred interpretation is that the reduced seis- mic velocities in the low-velocity layer are caused by the presence of a very small amount, of the order of 1 percent or so, of melt that forms a thin fluid film separating the grains of an otherwise solid rock. The evidence in favor of this interpretation is summarized by Solomon (1976~. The temperature at the top of the low-velocity layer must then be the temperature at which melting begins at the corresponding pressure. The temperature of incipient melting depends of course on the composition of the rock but does not depart much from 1100°C (at P = 1 bar) for the several types of peridotite whose density and elastic properties make them likely constituents of the upper mantle. The melting point for dry peridotites rises with pressure at a rate of about 10°/kbar, bringing it to about 1400°-1500°C at a depth of 100 km, where

32 ENERGETICS OF THE EARTH the pressure P -30 kbar. In the presence of water, however, the incipient melting point first decreases with increasing pressure, reach- ing a minimum of about 1000°C at 30 kbar for a water content of only 0.1 percent (Wyllie, 1971, Figure ~181. The amount of water present in the mantle (presumably mainly in amphiboles and micas) is not exactly known. Carbon dioxide may also be present. Mysen and Boettcher (1975) have reported experiments on the melting of four different perid- otites (two spinet and two garnet lherzolites) in the presence of water and CO2 in variable proportions, and in the pressure range from 7.5 kbar to 30 kbar. When present alone, CO2 does not lower much the solidus temperature from its value for the dry system, in contrast to water, which may lower the solidus to less than 900°C at 15-20 kbar. Mysen and Boettcher's experimental conditions are such that there is enough water present to saturate the melt at all pressures, a condition that may not be met in the mantle. Furthermore, the lherzolite speci- mens used in these experiments (inclusions in Hawaiian lavas and/or kimberlite pipes) may represent the fraction of the original mantle rock that is left after partial melting and removal of the most easily fusible components, and are therefore not necessarily typical of undepleted mantle or pyrolite. What may be concluded from this is that if we accept the hypothesis that the top of the low-velocity zone or bottom of the lithosphere is indeed the depth of incipient melting, the temperature there is probably in the range 950°-1300°C where the lithosphere is 50 km thick, and in the range 900°-1400°C where its thickness is lOO km. In continental areas where the lithospheric thickness is 200 km, the temperature of incipient melting is probably not greater than 1200°C. The lower limit of the ranges quoted here applies when there is enough water present to saturate the melt, while the upper limit is for the "dry" case of no water at all. If the bottom of the low-velocity zone is defined as the depth at which the shear velocity resumes the value it has at the top of the low-velocity zone, the thickness of the low-velocity zone comes out at about 10~150 km. The important point is that the thickness is finite, implying that at depths greater than 200 250 km the mantle is again entirely solid, or that the temperature gradient has dropped sufficiently for the effect of pressure on velocity to once more become preponderant, as it is above the low-velocity zone. Return to subsolidus temperatures could be caused either by a sharp drop in the gradient temperature, as in Figure 3-la, by a sharp rise in the solidus temperatures, as in Figure 3-lb, or by a combination of the two. Melting in the presence of water entails a minimum solidus temperature.

Temperatures Within the Earth 33 (a) SOLI DUS ACTUAL TEMP. (b) 1 l B DEPTH- t ACTUAL TEN SOLIDUS// I CL ,'_ , ! A B DEPTH- FIGURE ~1 Possible temperature profiles in the upper mantle. The low-velocity layer extends from A to B. In (a), the temperature of incipient melting (solidus) rises linearly with increasing pressure or depth ("dry mantle"); the finite thickness of the low-velocity layer is due to a drastic reduction in slope of the temperature curve. In (b), temperature increases linearly while the solidus exhibits a min~rnum ("wet mantle"). The situation in the mantle is probably somewhere between cases (a) and (b). GEOTHERMS FROM NODULES IN KIMBERLITE Geotherms (i.e., temperature-depth curves) for subcontinental areas in the depth range 10~250 km have been estimated from the inferred temperature and pressure of equilibration of the several minerals pres- ent in lherzolite inclusions in kimberlites. The equilibrium distribution of the major elements calcium and magnesium between coexisting clinopyroxenes and orthopyroxenes in lherzolites is a function of tem- perature but is relatively insensitive to pressure. On the other hand, the distribution of aluminum between coexisting garnet and orthopyroxene is sensitive to pressure. The distribution coefficients are then compared with distribution coefficients measured on natural samples to determine the temperature and pressure at which the samples last equilibrated. Experiments are difficult because of the great length of time needed to reach equilibrium; the influence of other ubiquitous elements (e.g., iron) on distribution coefficients must also be ascertained. Investigating in this manner a number of nodules from kimberlite pipes in Lesotho, Boyd and Nixon (1973) reported temperatures in the range 900°-1400°C at depths ranging from 140 km to a little over 200 km. The individual points fall nicely on two curves ("inflected" geotherms) indicating a sharp steepening of the temperature gradient at about 180 km. Specimens from below that depth are generally sheared, whereas

34 ENERGETICS OF THE EARTH those from above are granular. Boyd and Nixon point out that in the absence of marked differences in thermal properties above and below the inflection, continuity of steady-state heat flow across the inflection requires a continuous temperature gradient. They see in the inflected geotherm evidence for an event that caused major heating in the depth range 15~200 km. (The eruption of kimberlite that carried the nodules to the surface is in itself evidence for a perturbation of some sort, since kimberlite is not erupted continuously everywhere.) Figure 3-2 shows results of many investigators, as plotted by Harte (19781. Although there is much regional variation, the plot suggests possible tempera- tures of about 1000°C at 150 km, and of 1200°-1400°C at 200 km (65 kbar) and that the Mg/Fe ratio is 9; from Akimoto's data we would estimates for a given temperature, from 65 kbar to 40 kbar in one instance; corresponding depth estimates are reduced by 6(}65 km. Harte (1978) suggests lowering the estimates shown in Figure 3-2 by perhaps 30~5 km. The matter can apparently not be resolved until we get more reliable experimental data on the distribution coefficients of major elements in coexisting phases. Perhaps all that can be said at the moment is that nodules in kimberlite suggest temperatures of about 1000° + 200°C at 100-km depth, and 1200° + 200°C at 150 km. 1400 1200 C' (a) ~i// / '1 - (b) i: L 8001 100 150 200 100 depth/km 1000 ~ / of o o ° o )~` oo 2/ o Ye · · · . . . . · . . . · . 150 200 FIGURE ~2 Estimated temperatures and depths of equilibration of garnet-lherzolite xenoliths in kimberlite pipes. Plot (a) is for pipes in northern Lesotho; (b) represents pipes in southern and southwest Africa and in Udachnaya (USSR). Symbols are as follows: triangles, Premier; squares, Kimberly; dots, Louwrencia; diamonds, Udachnaya. The depth scale may need reduction, in some instances by perhaps as much as 60 km (see text). Reproduced with permission from Harte (1978).

Temperatures Within the Earth 35 THE MANTLE'S TRANSITION ZONES Ever since Bullen's early work (Bullen, 1936, 1940), seismologists have noted that the rate of increase with depth of the body-wave velocities vp and as is particularly rapid in what Bullen called zone C, roughly between 350 and 900 km. Details of the seismic velocity distribution have long been, and to some degree still are, obscure. After muc h debate, it is now generally recognized that there is no discontinuity of the first order (discontinuous jump in velocity), or even of the second order (discontinuous change in velocity gradient), in this depth range. There do seem to exist, however, two narrow zones, at around 400 and 650 km, respectively, in which the velocity of both vp and vs increases with depth more rapidly than it does either immediately below or above these transition zones; the thickness and depth of these zones are still rather uncertain. Bullen had already recognized that velocities in zone C are incompatible with the assumption of a homogeneous mantle. Birch (1952) confirmed that heterogeneity might be due to phase changes rather than to changes in gross chemical composition. Most of the minerals likely to be present in an upper mantle peridotite (olivine, pyroxenes, garnet) have now been found experimentally to undergo phase transformations at pressures such as exist in the mantle's C zone. For instance, at 10(10°C olivine undergoes a major transformation at a pressure in the neighborhood of 11~120 kbar for olivines with a molar fraction of Mg2SiO4 (forsterite) greater than 0.8. The olivine transformation is now generally believed to account for the velocity transition zone near 400 km. Common olivine being a two- component system (Mg2SiO4-Fe2SiO4), any phase transition (e.g., melting) will take place, at fixed temperature, over a finite range of pressure. This spreading out of the transformation over a finite depth range accounts for the absence, near 400 km, of any first-order discon- tinuity in either density, elastic properties, or seismic velocities. The transformation has been studied experimentally by Ringwood and Major (1970) and by Akimoto (19721. Olivines with less than about 70 mol percent Mg2SiO4 invert to a so-called ~ phase with the spinet structure; but magnesium-rich olivines invert to a phase ~) with a much distorted spinet structure that, according to Ringwood and Major, remains stable up to very high pressures. Akimoto, on the contrary, thinks that the ,B phase transforms to a By phase at a pressure not greatly in excess of that of the olivine-,6 transition. At 1000°C, and for an olivine with 90 percent Mg2SiO4, the olivine-,`3 transition begins at about 119 kbar, and the beginning of the ,l3-~y transition is at about 135 kbar. With 80 percent Mg2SiO4, the olivine-^y transition begins at about

36 ENERGETICS OF THE EARTH 100 kbar; at 110 kbar the iron-rich By phase inverts to ad phase that disappears again at 135 kbar. Pure Mg2SiO4 inverts to the ,6 phase at 125 kbar and 1000°C; it is not known at what pressure it transforms to the phase, if indeed it ever does. From Akimoto's phase diagrams at 800° and 1000°C, rough values of the temperature coefficient of the transfor- mation pressure can be determined. It appears to be about 45 bar/de" (dT/dP = 20.8°/kbar) for pure Mg2SiO4, somewhat less (30 bar/de", dT/dP = 33.3°/kbar) for 80 percent Mg2SiO4. The depths in the mantle at which the transformation begins and ends are not precisely known, and are likely to vary from place to place. Suppose that the transformation starts somewhere at 400 km (P = 132 kbar) and that the Mg/Fe ratio is 9; from Akimoto's data we would infer a temperature of about 1350°C. Ringwood and Major find a some- what higher temperature (1600°C) at 400 km if that is the mean depth of the transformation zone and the olivine has 11 percent Fe2SiO4. Considering the high uncertainty (50 percent) affecting the tempera- ture coefficient of the transformation pressure, and our ignorance of the exact iron content of mantle olivines, of the precise depths at which the transformation begins and ends, and of whether the olivine-spinel phase transformation is indeed responsible for the seismic anomaly near 400-km depth, all we can say is that the temperature at that depth is probably in the range 1300°-1600°C. This is not a very precise estimate, but it is so far the best we have. It agrees with an earlier estimate by Graham (1970) of 1450° + 120°C at 370 km. Near 650-km depth (P-250 kbar) there is another thin zone in the mantle in which the seismic velocities increase downward abnormally quickly. This has been variously attributed to a breakdown of olivine (spinet form) to MgSiO3 + MgO (Liu, 1975b), to a transformation of pyroxenes to phases with ilmenite or perovskite structure, or to a breakdown of all silicates to their constitutent oxides, with SiO2 in the form of stishovite. The situation is summarized in Figure 3-3 taken from Liu (19771. None of the relevant phase equilibria is sufficiently well known to allow even an approximate determination of the tem- perature at which the phase change occurs. THE LOWER MANTLE LAYER D From differences in the gradient of seismic velocities vp and vs. Bullen (1950) divided the lower mantle into two zones, namely D' and D". D' extends from 984 km to 2700 km and is characterized by a monotonic

Temperatures Within the Earth 37 300 2SO . 200 Periclase + Perovski e ~ ~- ll l 1 ! c ! Spinel + Stishovite I ~ + Stishovite c 150 1 100~ TO ~1 1 o A I ~I , 1 ~I 05 Mole fraction SiO2 FIGURE ~3 Schematic phase diagram for the system MgO-SiO2 at about 1000° C. The dashed line represents a mixture 60 mol per- cent olivine plus 40 mol percent pyroxene. Reproduced with permis- sion from Liu (1977). downward increase in velocity. D" extends from 2700 km to the core boundary (2885 km). It is now generally recognized that the velocity gradients are much smaller in D" than in D' and may in fact be negative (Bolt, 1972; Cleary, 19741. The lower mantle, from 700 km to the top of D", is commonly con- sidered to be mineralogically homogeneous even though some seismic observations have suggested to Johnson (1969) the possible presence of anomalous zones corresponding perhaps to phase transitions with small changes in density and elastic constants. For the time being, we shall

38 ENERGETICS OF THE EARTH ignore them and take D' to be homogeneous. We will return to the matter later. In a homogeneous layer, in hydrostatic equilibrium, the variation in density p with depth is due to the effects of pressure P and temperature T. The pressure varies as dP = -g p dr, where r is distance from the center of the earth and g = gory is the acceleration of gravity. The variation of density is then given by the relation dp gp2 dT = - -ap dr KT dr (3.1) where KT = p~dP/8p)T is the isothermal incompressibility and cY = - (l/p)~6p/8T)p is the coefficient of thermal expansion. If ~ and KT were known from the equation of state p = p(P, T) for the material forming the layer, a knowledge of the density variation with depth dp/dr would be sufficient to determine the temperature gradient dT/dr. The difficulty is that the equation of state for the lower mantle is not known a priori, since we aren't quite sure what the lower mantle is made of. A major step was taken by Birch in his classical paper of 1952. An ingenious and rigorous transformation of Equation (3.1) leads to Birch's celebrated generalization of the Adams-Williamson equation: 1 _ ~ d!+ = (8KT) + Ta{-YA + (TCY)12B + g , (3.2) where ~ is the seismic parameter vp2 -~413)vs2. The left-hand side of this equation is thus derivable from seismic observations. ~ is Grueneisen's ratio °EKT y= = = pcv pep cp (3.3) KS is the adiabatic (isentropic) incompressibility, and cv and cp are, respectively, the specific heat at constant volume and constant pres- sure; ~ is the superadiabatic gradient defined by dT _ Tag _ = - -7, dr cp (3.4) where the first term on the right-hand side of Equation (3.4) is the

Temperatures Within the Earth 39 "adiabatic" gradient derived from the thermodynamic relation for the change in temperature with pressure at constant entropy S. Tc' pep (3.5) In equation (3.2), A, B. and C are dimensionless functions of param- eters such as (dKT/8P)T, (8KT/8T)P, (~°~/6T)P' (dcp/dT)p; all these quantities are, in principle, derivable from the equation of state, pro- vided that the equation of state be known in the pressure-temperature range of the lower mantle; then Equation (3.2) can, again in principle, be used to find T and r. The difficulty is that the temperature terms, for instance ATya, are small compared to the first term, (6KT/dP)T, which must therefore be known very accurately. As things stood in 1952, Birch concluded that all one could say was that a temperature of 5000°C produces no "serious discrepancy." Since Birch's early work, some progress has been made in determin- ~ng an equation of state applicable to the lower mantle. An interesting empirical discovery was made of a linear relationship between density and compressional wave velocity Up, or bulk sound velocity c = (Ks /p) I/2, for materials of the same mean molecular weight, regardless of crystal structure. Shock wave experimental data on a variety of minerals and rocks have also provided much information on density at very high pressures. Graham and Dobrzykowski (1976), and more recently Watt and O'Connell (1978), have calculated the temperature distribution in the mantle between 700 and 1200 km that best fits density and seismic velocities, using third-order finite strain theory and assuming adiaba- ticity. Results depend on assumed composition (e.g., the ratio of (Mg,Fe)O to SiO2), on mineralogy (e.g., mixed oxides versus phases with the perovskite structure), and, of course, on which of the many proposed density distributions is used. Graham and Dobrzykowski (1976) exclude pyroxene composition ((Mg,Fe)O/SiO2 = 1), as it leads to "a temperature profile which is clearly outside the range of reason- able solutions" (~2500°C at 600 km). Peridotite and dunite composi- tions lead to a temperature of 1600° + 400°C at the 671-km seismic discontinuity. Watt and O'Connell suggest temperatures about 1300°-1700°C at 700 km and an adiabatic gradient of about 0.3°/km. In what is probably the best study of this kind to date, Wang (1972) used shock wave data to calculate temperatures in the lower mantle between 1300 and 2800 km. He starts from an acceptable density distri

o at 11 Ado It ~-_: i,,, _ USE 11 1 ret - _\ m. 11 :~0 I' ~1 ~-,r] U. ~ 8 CU a) o In o 0 o o 8 ~D 11 :,~o ~1 - o o o o o ~n ~ o o o o o o o o ~ o ,0 o (>10) aJnID'adwa1 40 8 0 \~- 8 o o~ o o U. o eC ,~ _ . o c . ~ _ ~ en 3 ° c: ~ =: .m ~ ~ ~ ox O'o ~ ~o ~ 3 ~ ~ o _ ~ ~ ~ ~, o _ U: ·_ U. ~ =,u, o =, Ct ~ o C~ _ ' - ~V - ~ <, 3 . - e~ ~ ~ .^ o ° C X _ ~ D o Cd .= c Ce ~ o ~ D 00 '- U~ O Ct' ~ - U' '- ~ ~ C~ ~ ._ .O - ~ O ~C s~ 01) ~ ~ 'Ct - U, _ U' - _ 11 O ~, _ ~ ~ ~ C~ O

Temperatures Within the Earth 41 button that, when compared to the sound velocity c in the lower mantle, provides an estimate of gross chemical composition (mean molecular weight m = 21.31. By interpolation between shock-wave experiments, on dunite and fayalite, he determines a likely Hugoniot for material with m = 21.3; this gives him a density-pressure relation for mantle material at the temperature TH of the Hugoniot. Hugoniots can be reduced to isentropes if the Grueneisen ratio ~ and its volume dependence (taken to be of the form ~ = ~y0 (V/Vo)77) are known. Wang chooses a set of likely values of ye and A, and calculates for each set a number of isentropes, which he plots on a P-T diagram. Lines connecting points of equal density on different isentropes are drawn. Along any line of constant density Pi there is a point with pressure Pi corresponding to the depth in the mantle where Pi occurs; the temperature Ti at Pi and Pi is then read from the diagram (Figure 3~. There is, of course, a different Ti for each set of values of ~y0 and 7~. Wang noticed, however, that only certain combinations of ye and 71 (e.g., ~y0 = 1.3 ford = 0; ye = 1.5 ford = 1) yield temperatures that, when plotted against P. fall on an isentrope. Thus the additional hy- pothesis that the lower mantle is isentropic permits selection of a pre- ferred A, and hence of temperature. The temperatures found in this manner do not strongly depend on the value of a, the difference be- tween them being less than 100° at any depth. The mean of those curves shows a temperature of 2800°K at 1300 km, increasing almost linearly to 3300°K at 2800 km with an average gradient of about 0.33°/km. Wang estimates the uncertainty on the temperature to be less than +800°. Wang's results extrapolated upward give a temperature of about 2000°C at 700 km, which is also the upper limit of Graham and Dobrzy- kowski's (1976) estimate of 1600° + 400°C. All the previously mentioned studies assume an isentropic (=adiabatic) lower mantle. The reason for doing so is as follows. The viscosity of the lower mantle has now been shown to be of the order of 1 x 1022 poise (Cathles, 1975; Pettier, 19761. As discussed in Chapter 5, this relatively low viscosity corresponds to a Rayleigh number much in excess of that needed for convective instability. Transport of heat by convection is so much more effective than transport by conduction that convection, when it starts, will tend to reduce the vertical temperature gradient to the minimum value needed to maintain convection, which is precisely the adiabatic gradient. Numerical solutions to the equations governing steady-state convection (e.g., Turcotte and Oxburgh, 1967) indeed commonly show a convective pattern consisting of a thick adia- batic core between thin top and bottom boundary layers, so that the average temperature gradient over a large fraction of the volume of the

42 ENERGETICS OF THE EARTH fluid does not depart significantly from its adiabatic value. But should we discover that the gradient in the lower mantle is significantly greater or smaller than the adiabatic, we would have to conclude that the lower mantle is not convecting; this would lead to serious difficulties regard- ing transfer through the mantle of the large amount of heat that, as we shall show in Chapter 4, must be leaving the core. Somerville (1977) has recently re-examined in much detail the prop- erties of the lower mantle and has extracted values for the superadia- batic gradient ~ from Birch's (1952) generalization of the Adams- Williamson equation, -= g ~ 1 - -I dz ~ ~g J (3.6) where z is depth and the other symbols are as before. He selects a number of density models satisfying the usual constraints for total mass, moment of inertia, and normal oscillation modes, each model leading to a different a. (The different models selected from the litera- ture generally differ from each other, as regards density at a given depth, by no more than 1 or 2 percent.) The parameter ~ is known from seismic data, and p and dp/dz are taken from the model; g is also calculated from the density model. Thus ~ could be calculated directly from (3 .6) if ~ were known. There is no direct way of determining c' for the lower mantle; usually one starts from (3.3), Ace ~ = , noting that at high temperature cp is insensitive to eitherP or T. and that quantities related to, but not exactly equal to, Grueneisen's ratio ~ can be derived from seismic data (Verhoogen, 1951) by use of Debye's theory of specific heat or the Mie-Grueneisen equation of state. Noting that Cp = CV (1 + Trays, and hence ~ = ~cvl(¢-TT2CV), Somerville calculates c' by taking ~ = EYE, AD being the "Debye fre- quency" or "acoustic" Grueneisen parameter, V do AD = ~ dV ' (3.7) where V, the specific volume, is 11p; ~ is the Debye temperature. The Debye temperature in the deep mantle is calculated from Debye's formula

Temperatures Within the Earth 43 h ~ = -ED k (3.8) (h = Plank's constant, k = Boltzmann's constant), where AD, the cutoff Debye frequency of the vibrational spectrum, is ~ 9N ~ \4=vJ (3.9) with 1/v3 = (1/vp3) + (2/Vs3~. Here N/V is the number of atoms for unit volume; while vp and v, are, respectively, the velocities of the compres- sional and shear elastic waves. The heat capacity at constant volume cv has its classical Debye value slightly corrected, by empirical means, for anharmonicity of lattice vibrations. As a starting point, a temperature of 1900°K at a depth of 670 km is assumed. This temperature was first suggested by Ahrens (19731. Starting from an upper mantle consisting of garnet, orthopyroxene, clinopyroxene, and olivine, Ahrens first computed the composition of the phases that would be in equilibrium at various depths, using Akimoto's (1972) phase diagram for the olivine- spinel transformation. Likely values of the seismic velocity vp were then computed from laboratory ultrasonic experiments. The iron con- tent, proportion of garnet, and temperatures were then adjusted to give the best fit to the actual vp distribution down to 670 km. Somerville finds that (1) the adiabatic gradient in the lower mantle is typically 0.4°/km, (2) the superadiabatic gradient varies from +0.5 to -0.8°/km, depending on the density model used at the start. It is very close to zero for Jordan and Anderson's (1974) model B. If ~ = 0, the temperature at 2770 km then comes out at 1900° + 0.4 x 2100° = 2740°K, or roughtly 2500°C. Somerville has also examined multiIayer models of the lower mantle in which part of the density increase with depth is attributed to phase transformations rather than to elastic compression. The superadiabatic gradients for multilayer models range from 0.6°/km to 1.4°/km. Note, however, that since the nature of the phase transformations is not known, it is impossible to estimate the corresponding entropy change; a "superadiabatic" gradient in Somerville's sense does not necessarily imply that the lower mantle is not isentropic. All that can be said is that the temperature gradient in the lowermost mantle is likely to be higher by about 1°Ikm if the mantle is multilayered; this would add about 2000° to the temperature at 2800 km. We note, however, that the layered structure of the lower mantle is not universally accepted by seismolo- gists.

44 ENERGETICS OF THE EARTH It is difficult to assess the confidence to be placed on these results. Much depends, in Somerville's work, on the identification of Gruenei- sen's ratio ~ with ~D, the acoustic Grueneisen parameter derived from Debye's theory. Debye's theory applies only to a "harmonic" solid, that is, a solid in which a displacement of an atom from its equilibrium position is resisted by a force strictly proportional to the displacement; such a Debye solid has ~ = ~ = (dKT/6P)T = 0. To account for finite thermal expansion, for the dependence of ~ on volume, and for the observed temperature dependence of KT, it is necessary to use "fourth order" theory, that is, to retain all terms up to order 4 in the Taylor expansion of the interatomic potential function (Debye's theory is a second-order theory) (Thomsen, 19711. To obtain a valid fourth-order equation of state from which ~ could be calculated at any T and P. it is necessary to measure precisely at least six quantities (e.g., p, a, K, (8KI8P)T, (82K/8P2)T, (8KI6T)P), some of which (e.g., the second de- rivative of the incompressibility) are not readily determined; further- more, the measurements must be made on the same high-pressure phases that are present in the mantle, the nature of which is still largely unknown. Yet it would seem, in spite of all the uncertainties, that the adiabatic gradient in the lower mantle cannot be far from 0.3°-0.4°/km on the average. Wang's estimate of 3300° + 800°K at a depth of 270~2800 km seems realistic. LAYER D" We now consider layer D", comprising the lowest 100-200 km of the mantle. The lower boundary of D" is the core-mantle boundary at a depth of approximately 2900 km; the upper boundary of D" is not sharply defined. Recall that layer D" is characterized by a small and possibly negative seismic velocity gradient for both vp and vs. Bolt (1972) interpreted the seismic data as requiring an anomalous down- ward density increase that he attributed to admixture in increasing proportion of core material. Jones (1977) has shown that the seismic data could also be accounted for by a temperature gradient in D" of about 12°/km. This high gradient, which contrasts with the much smaller adiabatic gradient of 0.3°-0.4°/km prevailing in the lower mantle just above D", may be an effect of the relatively large amount of heat that seeps from the core into the mantle. As will be shown later, generation of the earth's magnetic field almost certainly requires that an amount of heat variously estimated as between 2 x 10~2 and 10 x lOi2 W (see Chapter 4) be carried, mostly by convection, to the core's surface and into the

Temperatures Within the Earth 45 mantle. If a commonly accepted (1.4 x 10-2 car/cm s deg. or about 6 W/m deg.) value of the thermal conductivity of the lower mantle is used, it is easy to calculate that to carry, say, 8 x 10~2 W through the mantle by conduction alone, a temperature of about 14,000°C is re- quired at the core-mantle interface. Since this is much more than the melting point of mantle material (corrected for pressure) and is incon- sistent with the existence of a solid inner core, which requires the temperature at the boundary of the inner core to be less than some 8000°C (see below), it follows that transfer of heat through the lower mantle must be by convection. But the core and mantle are so different with respect to properties that govern convection (e.g., Prandtl number) that their flow patterns must necessarily also be very different, requiring the presence between them of a temperature boundary layer in which steep and horizontally variable temperature gradients allow for continuity of temperature and heat flow. The D" layer is thought to represent this thermal boundary layer (Verhoogen, 1973; Jones, 1977; Elsasser et al., 1979~. If the interpretation is correct, the temperature Tc at the core-mantle boundary would exceed the temperature at depth 2800 km by some 1200° if D" is assumed to be 100 km thick and if Wang's estimate is accepted. Much of the interpretation of D" as a thermal boundary layer with a steep temperature gradient rests, however, on the assumed value of the thermal conductivity of the lower mantle, k, which is, unfortunately, one of the less well known geophysical parameters. The value of k quoted above (1.4 x 10-2 car/cm s deg. or 6 W/m deg.) is taken from Horai and Simmons (l970), who calculated it in the following manner. They first note an empirical relation between k and seismic velocities for silicate minerals: vp = (0.17 + 0.02)k + (5.93 + 0.17), vs = (0.09 + 0.02)k + (3.31 + 0.16), (3. 10) where vp and vat are expressed in km/s and k is in meal/cm s deg. (Applied to the lower mantle at 2700 km, where vp 13.6 and vs-7.26, these relations give, respectively, k = 45.1 and k = 43 .9; but see below.) Noting the relations (3.8) and (3.9) between seismic velocities and Debye temperature 8, Horai and Simmons proceed to transform equa- tions (3.10) into an empirical relation between k and B: = (25.6 + 3.0)k + (385 + 28) (3.11)

46 ENERGETICS OF THE EARTH The Debye temperature for the mantle is calculated from seismic ve- locities, using (3.8) and (3.91; this gives, at 2898 km, ~ = 1242°K, and k = 34. They then proceed to correct for the temperature (unspecified by Horai and Simmons, but presumably room temperature) at which the empirical relations (3.10) and (3.11) are found to be valid. The temperature correction is based on theoretical work that predicts that k will be proportional to y2pI/3 T (forT>y', where ~ is Grueneisen's ratio and p is the density. The corrected value of k at 2898 km comes out, according to Horai and Simmons, at 14 meal/cm s deg; this is about one third of the value calculated by substi- tuting directly in (3.10) the observed seismic velocities of the lower mantle. Horai and Simmons' result may be open to criticism. For one thing, it is difficult to know what significance should be attached to the em- pirical relations (3.101. Thermal conductivity is indeed proportional to the velocity of phonons, which is an averaged sound velocity, but k depends also on factors unrelated to elastic-wave velocities. At T > 8, the thermal conductivity of a perfect crystal of large dimensions de- pends mostly on phonon-phonon scattering, which is a function of the anharmonicity of lattice vibrations; a crystal in which lattice vibrations are purely harmonic has infinite thermal conductivity as well as zero thermal expansion, and ~ = 0. Most theories of lattice conduction (see, for instance, Drabble and Goldsmid, 1961) take anharmonicity into account by introducing Grueneisen's ratio into the formulation; this leads, for instance, to Lawson's formula (Lawson, 1957) 372pI,2T ~ (3.12) where rO is the nearest-neighbor distance and KT is the isothermal bulk-modulus. Mao (1973a) has used (3.12) to predict that a lower mantle consisting of dense oxide phases (e.g., stishovite, mag- nesiowustite) with high incompressibility would have a thermal con- ductivity approaching 1 car/cm s deg (418 W/m de"), or 70 times Horai and Simmons' corrected value! Kieffer, using yet a different method though assuming the same composition (periclase ~ stishovite), es- timates the lattice conductivity at the core-mantle boundary to be about

Temperatures Within the Earth 47 10 meal/cm s deg (Kieffer, 1976), in broad agreement with Horai and Simmons' value. Add to this uncertainty the possible (but unknown) effects of phonon scattering by crystal defects and grain boundaries, and add a (possibly small) contribution to the conductivity from radiative transport of heat (Mao and Bell, 1972; Mao, 1973b; Schatz and Simmons, 1972) or from electronic excited states (Mao, 1973a), and it becomes clear that k is not known to better than one order of magnitude, with Horai and Simmons' value erring possibly on the low side. Note that if we accept Jones' (1977) interpretation of the seismic anomaly in layer D" as being caused by a steep temperature gradient of 12°/km, and assume the heat flow Q from the core to be 8 x 10~2 W. the thermal conductivity required to maintain that gradient (k =- (Q/4~rO2) After, where rO is the core radius) turns out to be 4.4 W/m deg (10.5 meal/cm s de"), close to Horai and Simmons' estimate of 14 meal/cm s deg. Mao's (1973a) value k = 1 car/cm s deg would require, to maintain a gradient of 12°/km, a heat flux into the mantle Q of 1.82 x 10~4 calls = 7.6 x 10~4 W. many times the earth's total heat output and therefore very unlikely; if Mao is right, the seismic anomaly in layer D" is not an effect of temperature, and the temperature is essentially the same at 2700 km and at the core boundary. If k in the lower mantle is about 10 meal/cm s deg. the conducted heat flow corresponding to the gradient above D" (0.4°/km) is 4 x 10-8 caVcm s. Thus either the heat inflow from the core is 30 times less than 2 x 10~2 calls (8 x 10~2 W), which corresponds to a heat flow at the core boundary of 1.25 x 10-6 cal/cm2 s, or the mantle convects with a Nusselt number N of the order of 30. (The Nusselt number is the ratio of total heat carried to heat carried by conduction alone.) The latter hypothesis is acceptable. Turcotte and Oxburgh (1967) have shown that the convective pattern in a viscous layer of thickness d heated from below will consist of a thick isothermal core between two horizontal boundary layers of thickness 6, with rising and falling plumes also of thickness &. If R is the Rayleigh number, they predict 6,_dR-~/3 W_KR NO-R~3 d where w is the vertical velocity and K iS the thermal diffusivity, K = kipcp. If k is about 10-2 caVcm s deg. K iS ~6 x 1O-3 cm2/s. We want a boundary layer thickness ~ ~ 100 km, which ford = 2900 km requires R ]/3-29, and consequently N 29; the corresponding vertical veloc

48 ENERGETICS OF THE EARTH ity w is 0.55 cm/yr. Similarly, Moore and Weiss (1973) find, for convec- tion dominated by viscosity (high Prandtl number), N _ 2(R/RC)~/3, where Rc is the critical Rayleigh number at which convection starts. Since Rc for the mantle is ~ 103-104, a Nusselt number of 30 requires a Rayleigh number of order 3 x 106-3 x 107, which does not appear impossible. (See also McKenzie et al., 19t4.) To recapitulate, the interpretation of the seismic anomaly ins " as an effect of temperature requires a temperature gradient in D" of about 12°/km (Jones, 19771. This gradient equals q0/k = Q/47rrO2k, where Q is the total heat escaping from the core. If k = 10-2 car/cm s deg (4.18 W/m de"), Q = 2 x 10~2 calls = 8 x 10~2 W. There seem to be no major inconsistencies in these figures, or between them and the existence of an adiabatic gradient of 0.3-0.4°/km in most of the lower mantle; in particular, the Nusselt number of about 30 predicted from these figures for the lower mantle seems in line with various theoretical estimates. The temperature at the core-mantle boundary is then 4500°K if we start from Wang's estimate (Wang, 1972), or close to 6000°K if we take Somerville's result for a multilayered lower mantle. Much depends, however, on the value of k. It is not impossible, for theoretical reasons (Mao, 1973a), that k could be much larger than the value adopted here. In that case, interpretation of D" as a thermal boundary layer would have to be abandoned, since it would require Q to be equal to or greater than the total heat flux at the earth's surface. Supposing that the temperature gradient ins" is the same as in the rest of the lower mantle, the temperature at the core-mantle boundary is then approximately 3350°K (Wang), or 4800°K (Somerville, for layered mantle). Perhaps some support of the interpretation of D" as a thermal bound- ary layer with a steep temperature gradient may be provided by the observation that the temperature at the core-mantle boundary is not likely to be as low as 3350°K, judging from the fact that the outer core is liquid. As will be discussed in greater detail in the next chapter, the melting point of iron at the pressure (1.4 Mbar) of the core-mantle boundary cannot be much less than 5000°K (Leppaluoto, 1972a,b). Addition of sulfur lowers the melting point. Very little can be said on the matter, since the exact sulfur content of the core (10 percent, 15 percent?) is unknown, as is the phase diagram for the Fe-FeS system at the relevant pressure. A sulfur content of 15 percent lowers the liquidus by about 200° at ordinary pressure and probably by not more

Temperatures Within the Earth 49 than 1000° at P = 1.4 Mbar; thus the liquidity of the core seems to require a temperature not lower than 4000°K at the core-mantle boundary. THE OUTER CORE When it comes to guessing at the temperatures in the core, roughly three lines of approach are used: 1. Determine the temperature from seismic data, an acceptable den- sity distribution, and an equation of state. 2. Assume the temperature to be known at the core-mantle boundary (CMB), and assume further the outer core to be isentropic. Temperatures then fall on the adiabat through the known point. This calculation requires only Grueneisen's ration. 3. Assume the temperature at the inner corrupter core boundary to be the melting temperature of inner-core material at the relevant pres- sure. Since the core does not consist of a pure substance (e.g., iron), melting temperatures depend on composition, a topic to which we now turn. COMPOSITION OF THE CORE It has been known for some time that the density of the outer core is noticeably less than the density predicted from shock wave experi- ments on iron. Neither do experimental sound velocities at a given density agree with observed velocities in the core (Birch, 19631. Elements besides iron that could be present in the core in sufficient proportion to reduce its density by about 10 percent are few: oxygen, magnesium, silicon, sulfur (carbon and hydrogen are readily soluble in iron but do not much affect its density). Silicon, once considered a likely candidate, now seems to have fallen out of favor, mainly because of its high affinity for oxygen. It is indeed difficult to see how the highly reducing environment required by the presence in the core of metallic silicon could be compatible with the oxidizing conditions implied by the presence of FeO in the mantle. When in contact, silicon in the core would reduce FeO in the mantle to form SiO2 (or a silicate) and metallic iron. Alder (1966) made some rough thermochemical calculations suggest- ing that MgO could be soluble in liquid iron under the pressures and temperatures of the core. This suggestion has not been followed up, perhaps for lack of experimental data. Oxygen (as FeO) may be present in the core in sufficient amount to account for its density (Dubrovskii

-l: Am ·0130~ uo!laldaa (P;34 = 50

Temperatures Within the Earth 51 and Pankov, 19721. Ringwood, once a staunch supporter of silicon has recently endorsed this view (Ringwood, 1977), which again lacks exper- imental support. At ordinary pressure, a mixture of metallic iron and iron oxides containing, say, 10 percent oxygen would melt at a temper- ature only a few degrees below the melting point of pure iron to produce two immiscible liquids, one of which is a metallic solution containing very little oxygen (approximately 0.2 percent), while the other has a composition close to FeO. Although the solubility of oxygen in the metallic liquid increases rapidly with increasing temperature, it is not known at what temperature the immiscibility gap vanishes, or even whether it closes at all. Pressure would presumably increase the solu- bility of FeO. Ringwood estimates that above 500 kbar, complete mis- cibility would occur. The problem is further complicated by the tend- ency of Fez+, the usual form of iron in FeO, to disproportionate at high pressure (Mao and Bell, 19771: 3 Fez+ = 2 Fe3+ + Fe (metal). The arguments for or against the presence of sulfur in the core are mainly based on consideration of natural abundances, apart from the well-known affinity of iron for sulfur. In the crust and mantle, there is very little sulfur (about 600 atoms per 106 atoms of silicon), in sharp contrast to chondritic meteorites (about 105 atoms for 106 of silicon) or to the sun, where the atomic ratio of sulfur to silicon is somewhere between 0.2 and 0.6, with a probable value of about 0.35 (Ross and Aller, 19761. For the earth to have the same S/Si ratio as the sun, the earth's core would have to contain about 18 percent sulfur by weight. While there is no reason to expect precisely the same ratio in the sun, earth, and meteorites, it is difficult to explain, if the core contains no sulfur, why the earth should be so much more strongly depleted in sulfur than in the other "volatile" elements (carbon, nitrogen, fluorine, chlorine, bromine, and iodine and the rare gases neon, argon, krypton, and xenon), relative to chondritic meteorites (Murthy and Hall, 1970, 1972), which are themselves not significantly depleted in sulfur with respect to the sun (Figure 3-5~. Many decades ago, Goldschmidt had stated on purely geochemical grounds that the earth must contain much unseen sulfur in the form of FeS, a suggestion that was later revived by Mason (1966) and Murthy and Hall (1970~. Arguments based on abun- dances are, however, not entirely convincing because, as Brett (1976) noted, the sulfur content of the mantle is essentially unknown except for its upper 100 or 200 km, which might not be representative of the whole. More convincing arguments are those of Murthy and Hall (1970,

52 ENERGETICS OF THE EARTH 1972) and Murthy (1976), who argue that. in any model of homogeneous accretion of the earth, sulfur is bound to follow iron into the core, if only because the Fe-FeS eutectic liquid is the first liquid to form, at the lowest temperature, for all reasonable terrestrial compositions. If there is any sulfur at all in the earth, some of it is bound to be in the core. It is, of course, possible, and even likely, that the earth's core might contain, in addition to sulfur and to nickel, which is ubiquitous in iron meteorites, some proportion of carbon, magnesium, oxygen, and still other elements. As we shall see later, there is a fair chance that it might contain small amounts, of the order of 0.1 percent or less, of potassium. But as something is known of the phase relationships in the Fe-FeS system, while nothing (or almost nothing) is known of the Fe-FeO or Fe-MgO systems at very high pressure and temperature, we will as- sume in what follows that the earth's core consists essentially of Fe + FeS, with an FeS/Fe ratio somewhere near 10-15 percent. We now return to the matter of determination of temperature by the three methods outlined above. TEMPERATURE FROM AN EQUATION OF STATE The problem has been examined with much care by Stewart (1973), who proceeds as follows: Assume that the density, the pressure, and the seismic parameter ~ (which is the square of the sound velocity, ~ = Ks/p) are known at all depths in the outer core (but note that the density is usually calculated, as by the Adams-Williamson equation, on the assumption of adiabaticity). Assume that the Grueneisen ratio varies as {V\ )' 3° \\voJ where the subscript 0 refers to zero pressure and V is specific volume. Assume further that core material exhibits under shock compression the same linear relation between shock wave velocity us and particle velocity up that is observed in many substances: us = Sup + CO. Here C0 and S are characteristic constants of the material, C0 is the sound velocity at P = 0, and S is a dimensionless parameter related to the pressure derivative of the incompressibility. Stewart shows how, given values of the "Hugoniot parameters" p0, C0, and S and of y0 and A, it is possible to calculate the sound velocity and a corresponding seismic parameter ~ at any density. Stewart then attempts to determine the values of the Hugoniot parameters that will give the best fit, at all

Temperatures Within the Earth 53 depths in the outer core, between the calculated and observed seismic parameters ~ and +. Once a choice of best-f~tting Hugoniot parameters has been made, the temperature and the adiabatic gradient at any depth can be calculated, provided a value is chosen for cv, the specific heat at constant volume. It turns out, however, that partly because of the uncertainty affecting seismic velocities and ¢>, there is a wide range of acceptable values of the Hugoniot parameters. Estimated temperatures vary widely with assumed Hugoniot parameters and assumed density distribution in the core. Stewart, using experimental parameters for Fe-Si alloys, shows that the temperature at the core-mantle boundary changes from about 1000°K to more than 5000°K when the assumed silicon content of the core is reduced from 20 percent to 10 percent. Although these calculations are not directly applicable to the core, which probably does not consist of Fe-Si, they do serve to point out that whatever the core happens to consist of, its temperature cannot presently be calculated with any degree of accuracy, partly because of the uncertainty that affects the density and, to a lesser extent, the seismic velocity. The fact is that a change in temperature of 1000° probably changes density in the core by less than 1 percent. The same remark applies to the adiabatic gradient, which cannot be determined from seismic data to better than within a factor of 2 or more. An entirely different approach is taken by Bukowinski (1977) to determine the temperature in the solid inner core, assumed to consist of pure iron. A quantum-mechanical electronic band structure calcula- tion is made for ~ (fcc) iron. The equation of state is taken to be of the Mie-Grueneisen type, with an appropriate term to represent the contri- bution to the thermal pressure from conduction electrons. Bukowinski then seeks the temperature at which his theoretical equation of state fits the geophysical data (P. p, ¢) for the inner core. Five different pro- posed density distributions are examined, for which the temperature T at the inner core-outer core boundary ranges from a low of 4400° + 40°K to a high of 8900° + 100°K. Values of By range from 0.41 + 0.1 to 3.93 + 0.1. Bukowinski's preferred solution, corresponding to the PEM model of Dziewonski et al. (1975), has Ti = 5450° + 65°K and ~ = 1.87 + 0.1. Surprisingly, the inner core turns out not to be isothermal, the temperature at the center being 234° higher than Ti; this implies heat sources in the inner core, which we shall consider later (Chapter 41. Bukowinski ignores the possibility that the stable phase of iron under inner core conditions might be the ~ (hcp) phase rather than the ~ (fcc) phase he exclusively considers (Liu, 1975a). The density of the ~ phase exceeds that of the ~ phase by a few percent, enough perhaps to make a noticeable difference in the core temperature.

54 ENERGETICS OF THE EARTH TABLE 3-1 Predicted Density with Depth Density (g/cm3) Depth (km) Model CAL SIlAa Model PEM. 2885 (top of outer core) 9.95 9.909 5155 (bottom of outer core) 12.34 12.139 5155 (top of inner core) 13.11 12.794 6371 (center of earthy 13.35 13.01 a From Bolt and Uhrhammer (1975) From Dziewonski et al. (l 975) Clearly an accurate determination of temperature in the inner core requires a very precise knowledge of its density. Permissible models of the earth (i.e., models that satisfy constraints on total mass, moment of inertia, surface waves and body waves, and eigenvibrations) differ by notable amounts, as can be seen from Table 3-1. Although densities in these two models are almost identical at the top of the outer core, they differ by 3 percent at the top of the inner core and by 2.6 percent at the center. We conclude, then, that partly because of uncertainty concerning the density and seismic velocity distributions in the core, it remains im- possible to assess temperatures in the core from equations of state and geophysical data to better than within a few hundred or perhaps even a few thousand degrees. THE ADIABATIC GRADIENT A second method for determining temperatures in the core starts from the supposedly known temperature at some point (e.g., the core-mantle boundary) and assumes an adiabatic (isentropic) distribution. The adiabatic gradient is dT dr ~ ' (3.13) where ~ = aKslpcp is the Grueneisen thermal ratio, ~ = vp2 is the seismic parameter determined from the sound velocity vp, and g is the acceleration of gravity. All three quantities are functions of r. The temperature at radius r is then Tr = To exp gy dr [at ~ ] (3.14)

Temperatures Within the Earth 55 where To is the reference temperature at r = rO. ~ (r) is known from seismic observations, and g (r) may be calculated for any density model of the earth. It remains to calculate y. A recent example of this method of determining the temperature may be found in Stacey (1977), whose calculations illustrate the many pit- falls into which one is likely to fall and to which Knopoff and Shapiro (1969) drew attention. After making a number of dubious approxima- tions, Stacey finds that ~ in the outer core varies from 1.26 at the bottom to 1.42 at the top. These numbers fall in a plausible range, but so would numbers 50 percent bigger or smaller. A surprising feature of many of the papers on the outer core's Grueneisen ratio is their insistence of referring it to the experimental values ye of pure solid iron at normal pressure and temperature. The outer core is, of course, liquid, a fact that should throw at least some doubt on the validity of applying to it results straight out of crystal lattice dynamics. There seems to be a secret hope that properties of liquid iron might not, after all, be very different from those of solid iron. But this does not seem to be the case. At the melting point (1809°K) at normal pressure, the thermal expansion of solid iron is 69.5 x 10-6/deg, while that of liquid iron is 119.2 x 10-6/deg (Kirshenbaum and Cahill, 19621. The Grueneisen ratio of liquid iron at the melting point can be calculated from the sound velocity c since ~ = c'K,,/pcp = ac2/cp. The value of c is 4.4 km/s according to Filipov et al. (1966), and 3.93 km/s according to Kurz and Lux (19691. The former value gives ~ = 2.8,* the latter gives by = 2.23; both values are much larger than the value for solid iron at room temperature (l.591. Whether the difference between the values for liquid and solid will increase or decrease with increasing pressure or density is not known. It is noteworthy that whereas By for solids usually decreases with increasing density, the opposite is true for water and mercury (Knopoff and Shapiro, 19693. However that may be, an average value By = 1.3 is not impossible. In the outer core, g varies from about 10 m/s2 at the top to about 4.5 m/s2 at the bottom, and ~ varies from 66 km2Is2 to about 100 km2/s2. Taking averages, g _ 7 m/s2 and ~ = 85 km2/s2, gives 1.275 for the ratio of the temperature at the inner core boundary (ICB) to the temperature at the outer core boundary (CMB). If the latter is taken to be 4500°K, the former is ~ 5740°K. The adiabatic gradient is ~1°/km at the top, and 0.3°/km at the bottom. The temperature of 5740°K at the ICB may be compared to Bukowinski's (1977) estimate of 5,450° + 65°K referred to above. The general agreement between the two figures should not, *Using cp = 11.0 cal/mol deg (Anderson and lIultgren, 1962).

56 ENERGETICS OF THE EARTH however, be considered to be much more than a coincidence, consider- ing the amount of guessing that goes into the choice of y. THE TEMPERATURE AT THE INNER CORE-OUTER CORE B OUNDARY Since the inner core is solid and presumably consists of iron (plus, possibly, some nickel and small amounts of a few other less abundant elements), while the outer core, consisting mostly but not entirely of iron, is liquid, it is rather natural to assume that the temperature at the boundary must be such that solid iron is in equilibrium with its melt, at the prevailing pressure. The temperature at which a pure substance is in equilibrium with its melt is called the melting point Tm. It depends only on pressure, as dTm AV dP (3.15) where AV = Vl - Vet is the difference in volume between liquid and solid, and AS is the corresponding difference in entropy. In a multi- component system, the temperature at which a solid phase is in equilibrium with liquid depends not only on pressure but also on the composition of the liquid. For instance, in a system consisting of iron end x percent of sulfur atP = 1 bar, solid iron can be in equilibrium with liquid at any temperature between 1809°K (for x = 0, pure iron) to 1261°K (for x = 31.4~. Since addition to a pure substance of any com- ponent that is soluble in the melt necessarily lowers the melting point of the pure substance, the melting point of pure iron at the pressure of the inner core boundary sets an upper limit to the temperature there. THE MELTING POINT OF IRON The pressure dependence of the melting point of iron has been the subject of much debate in recent years (for good summaries see Jacobs, 1975, and Boschi, 1975~. Some measurements of doubtful accuracy have been carried to 200 kbar. The most frequently cited data are those of Sterrett e' al. (1965) up to 40 kbar. The problem is to extrapolate those results to the pressure (3.3 Mbar) of the inner core. It is curious that we are still unable to explain why solids melt, or, for that matter, why the liquid state exists at all. All theories of melting are essentially empirical. A theory that has had much success is that of Lindemann. Lindemann supposes that melting occurs when the ampli

Temperatures Within the Earth 57 tude of thermal vibrations reaches a fraction ha of the lattice spacing a . In that state, at temperature Tm, the kinetic energy of vibration 3/2 kTm = i/2 m (6a)2~2, assuming that atoms of mass m vibrate harmonically with frequency co; k is Boltzmann's constant. The eject of pressure on Tm will arise from its effect on a, which is proportional to Vi/3 (V iS volume), and on w. It is convenient here to introduce the Debye tem- perature ~ = h cu/k (h is Planck's constant), and recall that from Debye's theory ~ _ Vl/3 v, where v is the mean elastic velocity defined in Equa- tion (3.91. Thus Tm is directly proportional to v2, and the ratio of the square of the elastic velocity in the inner core to that of iron at P = 1 bar gives the ratio of the melting temperatures. Alder (1966) obtained in this manner a melting temperature of 7720°K. Attempts have been made to refine the Lindemann law by reformu- lating it in less empirical terms. Boschi (1974), using a melting criterion by Ross, derives the relation Tm = lo ~ l 3 Vo 6 (3 ) ( VO ) . . . } , (3 16) where Tm is the melting point at a pressure at which the volume of the solid is V, To is the melting point in some reference state (e.g., P = 0) at which the volume is VO, and 1` V = VO - V. Here n is the exponent in the expression for the potential ~ of the repulsive interatomic force, assumed to be of the form ~ r-n, where r is the interatomic distance. For iron, n _ 8.4, as determined from isothermal compression tests. The validity of (3 .16) depends on how well the power law r-n represents the interatomic potential; different expressions for that potential lead to widely different values of Tm when /`V/V is large. Kraut and Kennedy (1966) suggested that melting points can very generally be represented as Tm = To ~ 1 + C-~ (3.17) where TV and VO have the same significance as above, and C is a constant that can be determined from the initial slope of the melting point curve. Higgins and Kennedy (1971) extrapolated the data of Sterrett et al. (1965) to 3.3 Mbar to obtain a melting point of 4250°C (~4500°K).* How the volume V of the solid at its melting point at 3.3 *But the initial slope of the melting curve of Sterrett et al. (2. 85°C/kbar) does not agree with the slope (3.8°/kbar) calculated from measurements of ~ V and AS and the Clapeyron equation (3.15).

58 ENERGETICS OF THE EARTH Mbar is determined is not said, though the authors do state that the density at 1.5 Mbar is assumed to be 11.4 g/cm3. Although Equation (3.17) represents fairly well the melting point behavior of some substances at small compression, it is unlikely that it will apply to all substances, as Kraut and Kennedy (1966) had originally claimed. Validity of the simple linear relation is now claimed only for metals (Higgins and Kennedy, 1971~. If (3.17) is valid it follows that dips = To Cp-, where ,B is compressibility. Since,l3 and V are necessarily positive quantities, dTm IdP cannot change sign and Tm can go through neither a maximum nor a minimum. Yet there are many metallic elements (cesium, barium, gallium, silicon, antimony, bismuth, selenium, tel- lurium, cerium, uranium) that do just that; for lithium and potassium the slope of the melting point almost goes to zero, even though ,8 does not. (For a compilation of phase diagrams at high pressures, see Cannon, 1974.) It is true that many of the observed changes in slope of the melting curve (Tm, P) occur in conjunction with a change of phase of the solid, yet the maximum melting point for barium occurs between 10 kbar and 20 kbar, whereas the nearest phase change occurs along the melting curve only above 60 kbar. The melting point curve of selenium has a maximum near 50 kbar, but no reported phase change (note that selenium is mentioned by Higgins and Kennedy as an example of an element satisfying the linear law). Attributing changes in slope to phase changes does not help much for iron, since between the pressure range of the observations by Sterrett et al. and of the inner core, iron under- goes at least two phase changes: ~ ~ ~ and ~ ~ ~ (Liu, 1975a). On the basis of the "y-e transition, Birch (1972) estimated that even if the linear extrapolation were correct, the melting point at core pressures might have to be raised by some 700°. A comparison of (3.17) and (3.16) makes it clear that the linear rela- tion (3 .17) is an approximation that can be valid only in the limit of small 1\ V/VO or if n is small, as it would be for soft metals with high compres- sibility. Quite apart from the effect of phase changes, the linear law for iron almost certainly breaks down, perhaps seriously, at pressures in the megabar range. Boschi (1974), using Equation (3; 16), finds a melting point at 3.2 Mbar of 6600°C, as against Higgins and Kennedy's 4250°C. Note, however, that the figure arrived at by Boschi strongly depends on his choice for the repulsive potential. The volume dependence of the

Temperatures Within the Earth 59 melting point, as with A, to which the melting point is related in Lindemann's theory, is a sensitive function of the interatomic potential. Most theories of melting suffer from the defect that they attempt to predict the melting point by considering only properties of the solid phase. Since the melting point is, by definition, the temperature at which the free energy of the solid equals that of the liquid, it should reflect properties of both the solid and liquid phases. Leppaluoto (1972a,b) has attempted to do this, using Eyring's significant-structure theory to predict properties of liquid iron. It will be recalled that this theory represents a liquid as consisting on the one hand of "solidlike" atoms with vibrational degrees of freedom as in the solid, and on the other hand, of "gaslike" atoms that have acquired translational degrees of freedom that allow them to jump into unoccupied sites or "holes," the number of which determines the difference in volume between liquid and solid. A partition function is then written to represent the "solidlike" and "gaslike" structures and any other structures (magnetic, diatomic, etc.) that may be significant. Leppaluoto's work has been dismissed by Boschi (1974) as "lack- ing credibility" because an early attempt by Tuerpe and Keeler (1967) to apply significant-structure theory had led to anomalous results. Leppaluoto's work was designed, however, to circumvent these dif- ficulties. He first writes down a partition function for liquid iron at P = 1 bar in which the parameters are chosen so as to give the correct TV and AS of melting and the correct Gibbs free energy change 2\G = 0 at the melting point (1809°K). The lattice or "solidlike" proper- ties of solid iron at high temperature are calculated in the Einstein approximation. The partition function so designed predicts within 5 percent the thermal expansion and compressibility of liquid iron, a slightly high (20 percent) specific heat, and a good temperature depen- dence of viscosity. The melting point at high pressure is then determined as the tempera- ture at which the free energies of solid and liquid are equal. Properties of the solid are determined from shock wave data. The calculations involve a quantity AV*, an activation volume, which measures the pressure dependence of the free energy of activation a gaslike particle must have to move into a hole; to occupy a hole a molecule of liquid must do work P /` V* against the external pressureP. lo V* is similar, but not exactly equal, to the activation volume for self-diffusion, and is not well determined. Leppaluoto finds, however, that agreement between his calculated melting point and the melting point calculated from the Clapeyron equation requires AV* to be constrained between 0.07 and

60 ENERGETICS OF THE EARTH 0.18 cm3/mol at pressures up to 3.3 Mbar. The lower limit for ~ V* gives Tm = 7000°K at 3.3 Mbar; the upper limit gives 9500°K. For AV* = 0, which leads to a melting temperature inconsistent with the Clapeyron equation, the melting point at 3.3 Mbar is close to the value predicted by Higgins and Kennedy (1971~. There is also some uncertainty in the melting point arising from the fact that shock wave data presumably refer to the ~ (hop) phase of iron, not to the ~ (bcc) phase present at the melting point at P = 1 bar. Taking this into account, Leppaluoto sug- gests Tm = 7500° + 2000°K for iron at 3.3 Mbar. Alder's (1966) and Boschi's (1974) results fall within these limits. MELTING IN THE FE- S SYSTEM As mentioned earlier, the melting point of pure iron at 3.3 Mbar (~7500°K) is an upper bound on the temperature at the inner core-outer core boundary, since the outer core presumably does not consist of pure iron. The temperature at which solid iron is in equilibrium with a multicomponent melt will always be less than the melting point of pure iron by an amount that depends on the nature and proportion of the other components. Oxygen, for instance, has very little effect on the melting point of iron until its proportion reaches about 23 percent by weight; but even then the lowering of the melting point is less than about 150°. On the other hand, addition of sulfur may lower the melting point of iron by more than 500°. The lowest temperature at which solid iron can be in equilibrium with an Fe-S melt is 988°C at P = 1 bar (eutectic temperature); the melt then contains 31.4 percent of sulfur. At higher sulfur content, the solid phase in equilibrium with the melt (above 988°C) is FeS. The phase diagram for the Fe-S system at 1 bar is shown in Figure 3~. Just as the melting point of pure iron is an upper bound to the temperature at the inner core boundary, so the eutectic temperature is a lower bound. The effect of pressure on the eutectic temperature Te has been measured at 30 kbar by Brett and Bell (1969) and at from 30 to 100 kbar by Usselman (1975a). From an initial value of 988°C at P = 1 bar, Te first rises very slowly, to 998° + 5°C at 55 kbar, then much more rapidly, to 1190°C at 100 kbar. The break in slope, which occurs near 52 kbar, is presumably connected with a phase change in solid FeS, or with the substitution of FeS2 for FeS as the stable form of iron sulf~de; this is possible because the reaction 2FeS = FeS2 + Fe

Temperatures Within the Earth 61 1600 1 539° 1400 1200 1 000 800 600 400 J I l L i q u i d / Two Liquids 083° 7430 a> 11 1 ~ 1 . 1 1_ He 20 40 60 80 S FIGURE 3 6 Phase diagram for the Fe-S system at ordinary pressure. has a large negative volume change is V = -5.31 cm3 at ordinary pres- sure. As the pressure rises, the sulfur content of the eutectic mixture first decreases from 31.4 percent at P = 1 bar to 24 percent at 55 kbar, and remains essentially constant from there on. Attempts to extrapolate the eutectic temperature to core pressures have been made by Usselman (197Sb) and Stacey (1977~. Usselman uses the Kraut-Kennedy linear extrapolation, but since the compres- sibility of solid mixtures of Fe and FeS is very poorly known, the volume of the solid phases at 3.3 Mbar has to be guessed. Stacey uses a form of Lindemann's theory that requires knowledge of A, the Grueneisen ratio for the solid phase; since this is not known, Stacey uses the value he derived for the liquid outer core. Both calculations ignore the ~y-e transition in iron, which is likely to change the slope

62 ENERGETICS OF THE EARTH dTe/dP just as the phase transition in FeS steepens it at 52 kbar. At 3.3 Mbar, Te is between 3750° and 4050°K according to Usselman, and 4168°K according to Stacey. The effect of pressure on the eutectic is formally described by the relations (Prigogine and Defay, 1954, p. 365) ~ xv +x2Av2 ~ v'-us dP xiAhi +X2Ah2 (3.18) dx2= _x ---1- ~ _-- 1 , (3 .19) dP RT (xi ~ h ~ + x2 Ah2 ~ (d On x2~21/dx2 ~ lkh~Av2-Ah2Av~ where x~ and x2 are, respectively, the mole fractions of components 1 and 2 at eutectic composition, GYM is the activity coefficient of com- ponent 2 in the melt, and Ahi = hi'-his, Avi = vi'-vis, (i = 1, 2), where hi and vi are, respectively, the partial molar enthalpy and volume of component i, and superscripts l and s refer to liquid phase and solid phase, respectively. It is important to note that although viS at any pressure could be calculated from an adequate equation of state, the partial molar volume ~ cannot because it depends on volume changes that occur, at constant P and T. when Fe and FeS mix. If the FeS melt were a perfect solution, with no volume change upon mixing, vi' = vie, the molar volume of pure liquid i. The same remark applies to enthalpy, which would be calculable only if there were not heat of mixing, as in a perfect solution. That FeS liquids are not perfect solutions is shown experimentally by the very fact that the slope of the eutectic tempera- ture curve is essentially zero up to 55 kbar. This requires the numerator of (3.18) to be zero, and since x~ and x2 are both positive quantities, either Ivy or Av2 must be smaller than 0. The volume of the eutectic liquid is less than the sum of the volumes of its two pure liquid com- ponents, so contraction occurs upon mixing. Because of the markedly nonideal nature of the FeS system, it is in fact impossible to predict what its phase diagram will look like at high pressure. The solid phase FeS may become unstable with respect to FeS2 or a high-pressure phase of it. Pyrite (FeS2) melts incongruently at 743°C at P = 1 bar. The very existence of a eutectic between Fe and FeS (or FeS2) may be in doubt. As noted by Kullerud (1970), most sulfur-metal systems exhibit liquid immiscibility, i.e., the existence of not just one but of two coexisting liquid phases. The systems Cu-S,

Temperatures Within the Earth 63 Pb-S, and Hg-S have in fact two immiscibility ranges, one of which occurs at low sulfur concentration. In the Cu-S system at P = 1 bar, for instance, there is a eutectic with only 0.77 percent sulfur at 1067°C, barely below the melting point of pure copper (1083°C). At 1105°C, a liquid with 1.5 percent sulfur is in equilibrium with a liquid containing 19.8 percent sulfur, the composition of which is close to that of Cu2S, which melts congruently at 1129°C. The second immiscibility gap occurs at much higher concentrations of sulfur. This led Verhoogen (1973) to suggest that immiscibility in the Fe-S system could perhaps account for the properties of the lower few hundred kilometers of the outer core, which were once thought to form a separate layer (Bullen's layer F) with properties (e.g., density) different from those of the rest of the outer core (Bolt, 19721. It now seems that the properties of layer F are not sufficiently different to warrant its recognition as a separate entity, even though the decrease in slope of the seismic velocity versus depth curve has not been satisfactorily explained. Quite apart from the highly questionable propriety of applying, as Usselman and Stacey do, to a eutectic questionable theories of melting in one-component systems, it would seem that, from the very nature of the Fe-S system (or of the Fe-O system, for that matter), the determi- nation of its eutectic temperatures at very high pressure is even more uncertain than determining the melting point of pure iron. For all it is worth, one could perhaps venture to guess that since the slope (4.2°/kbar) of the eutectic temperature curve above 55 kbar is slightly steeper than the initial slope of the melting curve of iron (3.8°/kbar), pressure would tend to reduce the difference between the melting point of iron and the eutectic temperature. But this extrapolation ignores possible differences in compressibility between liquid iron and Fe-S melts that could reduce the slope of the melting curve more quickly for the eutectic liquid. In spite of all this uncertainty, Stacey (1977) has recently proposed a temperature distribution for the whole earth that is anchored to a temperature of 4168°K at the boundary of the inner core. That tempera- ture of 4168°K is, it will be recalled, Stacey's estimate of the eutectic temperature at inner core pressure, the basic assumption being that the liquid outer core and solid inner core have the same (eutectic) compo- sition. This assumption is almost certainly false, for two reasons. The first reason is that the density of the inner core is compatible with it consisting of iron (or iron plus a small amount of nickel), but there is no evidence to support the contention that an Fe-FeS solid eutectic mix- ture (zero pressure density ~ 5.5 g/cm3) could attain a density of about 13 g/cm3 at 3.3 Mbar (see Table 3-1~. The second reason is that, if the

64 ENERGETICS OF THE EARTH 7000 6000 . 5000 A o `t 4000 ~ 3000 llJ 2000 1000 II 1 ~ MANTLE .,-OUTER CORE_,_lCo ER_ O1000 2000 3000 4000 5000 6000 6371 DEPTH, KM FIGURE ~7 Temperatures in the earth, estimated by methods explained in the text. Points shown refer to, from left to right: (1) xenoliths in kimberlite; (2) the olivine-spinel transition at 400 km; (3) the lower mantle at 1300 km, from Wang (1972); (4) the lower mantle at 2800 km, also from Wang; (5) the core-mantle boundary; (6) the inner core boundary; and (7) the center of the earth, from Bukowinski (1977). Estimates of uncer- tainties are somewhat arbitrary, as they include many diverse factors, such as regional variations at point (1). The error bar at point (7) (center of the earth) reflects only the uncertainty inherent in Bukowinski's calculations, which assume that the inner core density is exactly that given by the PEM model of Dziewonski et al. (1975). Bukowinski estimates that an error of 1 percent in the density adds about 500 ° to the uncertainty on the temperature. inner and outer cores have the same eutectic composition, the density jump of 0.6 0.8 g/cm3 at their interface (Table 3-1) necessarily repre- sents solely the effect of melting at constant composition. When intro- duced in Equation (3.18) for the pressure dependence of the eutectic

Temperatures Within the Earth 65 temperature, it leads to a very high value, of the order of several degrees per kilobar, for the slope of the eutectic line, which is incon- sistent with Stacey's value of the eutectic temperature itself. It seems safe to conclude that the sulfur content of the outer core is less than that of the eutectic and that, accordingly, the temperature Ti at the bound- ary of the inner core is higher than the eutectic temperature, whatever that temperature may be. In summary, then, all that can be said about Ti is that it is less than the melting point of pure iron at 3.3 Mbar (7500° + 2000°C), but how much less is not known. Bukowinski's estimate of 5450°K is not ex- cluded; but recall that it is calculated from the properties of ~ iron, which may be irrelevant to the inner core, and for a particular density model that assigns to the inner core a somewhat lower density than other models do (Table 3-1~. All in all, it seems likely that temperatures in the core are in the broad range between 4500° + 800°K at the core-mantle boundary and 6000° + 500°K at the center. Those temperatures are somewhat higher than the estimated initial temperatures calculated from accretion theory (Hanks and Anderson, 1969), which are particularly low for the core because the rate of release of gravitational energy is necessarily low at the beginning of accretion, when the gravitational pull of the accreting body is still quite small. Our temperatures suggest that the core has heated up in the course of time. Several sources immediately suggest them- selves (Chapter 2~. If the core now contains 0.1 percent of potassium, the total heat generated by it through 4.5 billion years amounts to 103° J. enough to heat up the core by some 700°; if one half of the gravita- tional energy of separation of the core (1 x 103} J. according to Flasar and Birch, 1973) were transformed into heat in the core, it would raise the core's temperature by some 3500°. SUMMARY In Figure 3-7 we have plotted as a function of depth the possible temperatures, or temperature ranges, deduced from the considerations outlined above. These temperatures can be summarized as follows: 1. From temperatures of incipient melting at base of lithosphere: oceanic lithosphere 50 km thick: 1075° + 225°C oceanic lithosphere 100 km thick: 1150° + 250°C continental lithosphere 200 km thick: <1200°C

66 ENERGETICS OF THE EARTH 2. From kimberlite nodules: at 100 km: 1000° + 200°C at 150 km: 1200° + 200°C Clearly it is not possible to draw a single solution through these points. Regional variations are considerable. A (possibly meaningless) average temperature gradient in the first 100 km might be 1~12°/km. 3. From olivine-spinel transition: at 400 km: 1450° + 150°C 4. Mean adiabatic (isentropic) gradient between 100 700 km: 1.5° + 0.5°/km (including entropy change due to phase transitions) err · ~ nls gives: at 700 km: 2000° + 350°C = 2300° + 350°K 5. Below 700 km, the isentropic gradient falls to 0.3°~.4°/km. The following temperatures are those calculated by Wang (1972), assuming adiabaticity: at 1300 km: 2800° + 800°K at 2800 km: 3300° + 800°K 6. In boundary layer D", the gradient is estimated at 10°-12°/km. The temperature at the core boundary is then 4500° + 800°~. The melting point of pure iron at the pressure of the core-mantle boundary is 5000° + 500°K (Leppaluoto, 1972a,b). 7. Guessing that an average value of ~ suitable to the outer core is about 1.3, the temperature at the ICB comes out as 5740°K, with an uncertainty possibly as large as 1000°. Bukowinski's (1977) estimate is 5450° + 60°K. Bukowinski's temperature at the center of the earth, plotted in Figure 3-7, is 5684° + 65°K, a value predicated on the as- sumptions (1) that the inner core consists of iron in its ~ (fcc) form, and (2) that the density of the inner core is that given by the so-called PEM model. Bukowinski estimates that a change in density of 1 percent from the PEM value entails a change in temperature of about 500°.

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