Energetics of the Earth (1980) / Chapter Skim
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3 TEMPERATURES WITHIN THE EARTH
3 TEMPERATURES WITHIN THE EARTH
Pages 29-66
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From page 29... ...
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 distribution of internal heat sources. Much effort has been devoted in the past to calculating the temperature distribution in the-mantle, particularly the upper mantle, from the heat conduction equation.
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From page 30... ...
. In the second method, one tries to determine the temperature gradient in a homogeneous layer from an observed variation with depth of a physical property of the layer.
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From page 31... ...
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 overcome the pressure effect.
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From page 32... ...
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.
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From page 33... ...
The individual points fall nicely on two curves ("inflected" geotherms) indicating a sharp steepening of the temperature gradient at about 180 km.
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From page 34... ...
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.
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From page 35... ...
This spreading out of the transformation over a finite depth range accounts for the absence, near 400 km, of any first-order discontinuity in either density, elastic properties, or seismic velocities. The transformation has been studied experimentally by Ringwood and Major (1970)
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From page 36... ...
THE LOWER MANTLE LAYER D From differences in the gradient of seismic velocities vp and vs. Bullen (1950)
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From page 37... ...
The lower mantle, from 700 km to the top of D", is commonly considered 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.
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From page 38... ...
incompressibility, and cv and cp are, respectively, the specific heat at constant volume and constant pressure; ~ 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)
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From page 39... ...
used shock wave data to calculate temperatures in the lower mantle between 1300 and 2800 km. He starts from an acceptable density distri
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From page 40... ...
o at 11 Ado It ~-_: i,,, _ USE 11 1 ret - _\ m.
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From page 41... ...
indeed commonly show a convective pattern consisting of a thick adiabatic core between thin top and bottom boundary layers, so that the average temperature gradient over a large fraction of the volume of the
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From page 42... ...
Somerville (1977) has recently re-examined in much detail the properties of the lower mantle and has extracted values for the superadiabatic gradient ~ from Birch's (1952)
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From page 43... ...
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.
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From page 44... ...
are not readily determined; furthermore, 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.
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From page 45... ...
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.
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From page 46... ...
the observed seismic velocities of the lower mantle. Horai and Simmons' result may be open to criticism.
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From page 47... ...
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.
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From page 48... ...
. Perhaps some support of the interpretation of D" as a thermal boundary 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.
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From page 49... ...
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 experiments on iron. Neither do experimental sound velocities at a given density agree with observed velocities in the core (Birch, 19631.
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From page 50... ...
-l: Am ·0130~ uo! laldaa (P;34 = 50
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From page 51... ...
, which again lacks experimental support. At ordinary pressure, a mixture of metallic iron and iron oxides containing, say, 10 percent oxygen would melt at a temperature 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)
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From page 52... ...
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.
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From page 53... ...
Five different proposed 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.
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From page 54... ...
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 impossible 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.
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From page 55... ...
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.
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From page 56... ...
to 1261°K (for x = 31.4~. Since addition to a pure substance of any component 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.
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From page 57... ...
, and recall that from Debye's theory ~ _ Vl/3 v, where v is the mean elastic velocity defined in Equation (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.
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From page 58... ...
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.
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From page 59... ...
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 dependence of viscosity.
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From page 60... ...
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.
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From page 61... ...
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.
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From page 62... ...
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 temperature curve is essentially zero up to 55 kbar.
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From page 63... ...
That temperature 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) composition.
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From page 64... ...
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 represents solely the effect of melting at constant composition.
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From page 65... ...
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.
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From page 66... ...
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°.
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