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Energy Sources Possible sources of energy in the earth may, for heuristic purposes, be classified as follows: 1. "Original" heat 2. Gravitational energy 3. Tidal dissipation of kinetic energy 4. Radioactivity This classification is somewhat arbitrary, in the sense that, as we shall presently see, much of the original heat is probably gravitational energy released at the time of formation of the earth. ORIGINAL H EAT By original heat we mean the heat content of the earth very soon after formation. In the nineteenth century, cooling of an originally hot earth was thought to account for the whole of the surface heat flow as well as for the formation of mountains, believed to result from contraction of the cooling body. This hypothesis, which led, as is well known, to serious difficulties regarding the age of the earth, was gradually aban- doned after the discovery of radioactivity, but some of it still survives, as there is no way of showing that the earth is not indeed cooling. As discussed in Chapter 4, secular cooling of the core is thought by some to provide, through the mechanism of crystallization of the inner core, the energy needed to generate the magnetic field. 15

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16 ENERGETICS OF THE EARTH The earth is now generally thought to have formed from the "solar nebula," a slowly rotating, disk-shaped mass of gas left over from the explosion of an earlier star in which heavy elements had recently been synthesized ("recently" here means not more than 107-108 yr before the formation of the earth). We picture the solar nebula, at the center of which the sun itself is in the process of forming, as cooling gradually by radiation, mostly in the direction not offal to the plane of the disk; the radial temperature profile in the central plane of the disk is assumed to be adiabatic. The pressure also decreases radially outward and may be of the order of 10-3-10-4 bar (1 bar = 105 Pa) at the radius of the earth's orbit. The composition of the nebula is assumed to be roughly the same as that of the sun, with hydrogen predominating. The first stage in the formation of the earth and other planetary bodies consists in condensation of cooling nebular gas into solid "dust" particles, which then somehow come together in the second stage of accretion to form the planetary bodies. Condensation is reasonably easy to follow from thermodynamic data. The first solids to appear in the range 1750-1600K would be oxides, silicates, and titanates of calcium and aluminum (e.g., corundum, Al2O3; perovskite, CaTiO3; melilite, Ca2Al2Si2074. At a total pressure of 10-3 bar, metallic iron begins to condense at 1471K. The bulk of the magnesian silicates (forsterite, enstatite) condenses around 1400K. Oxidation of iron to the ferrous state (as in olivine, pyroxenes, etc.) and formation of FeS begin around 700K (at P = 10-4 bar), while hydrated silicates appear only below 500K. From a variety of thermodynamic equilibria it has been inferred that the volatile-rich carbonaceous chondrites (a kind of meteorite) have formation temperatures in the range 300~00K. It would thus seem that since the earth obviously contains volatiles such as water and CO2, the initial temperature of the material from which it formed could not have exceeded a few hundred degrees Kelvin. This, however, is not necessarily the temperature of the earth itself just after forming, which depends on what happens during accretion. Accretion is much less easy to analyze than condensation. It is domi- nated by the consideration that a very large amount of gravitational energy must be released as particles fall onto the growing earth. This gravitational energy is of the order of 1032 J for the whole earth, quite enough to raise the temperature by tens of thousands of degrees and cause the earth to evaporate back into space as fast as it forms. The manner in which the earth manages to get rid of most of this energy, and in particular the fraction of it that is retained, determines the tempera- ture and heat content of the growing planetary body. Hanks and Anderson (1969) have made some calculations of initial

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Energy Sources 17 1 temperature on the assumption that at every stage of accretion the surface temperature of the earth is such that it radiates energy back into the dust cloud at precisely the rate at which gravitational energy is released by dust particles free-falling onto its surface. There are several uncertain parameters in the relevant equation. The first is the tempera- ture of the dust cloud itself, and the second is the rate of accumulation, or rate of growth of the earth's radius. This rate is unlikely to remain constant during the accretion process, since it would be essentially zero at the beginning, when the earth was so small that its gravitational attraction was very weak, and would be zero again at the end of the process, when accretion stopped because of depletion of the dust cloud. The mechanics of accretion are not sufficiently well known to allow a reliable estimate of the time required for accretion; estimates are commonly in the range 105-106 yr. There is also much debate as to whether accretion was heterogeneous or homogeneous. In the first case, it is supposed that accretion starts at essentially the same time as condensation. As the temperature drops in the nebula, the chemical composition of condensing particles changes, and so does the chemical composition of the growing planet. Thus iron is the first to aggregate, forming the earth's core before common silicates have begun to con- dense. The homogeneous model supposes, on the contrary, that con- densation is complete before aggregation starts; the earth then has initially a uniform composition, and processes that occur later are re- quired to separate the mantle from the core and the crust from the mantle. Hanks and Anderson (1969), assuming homogeneous accretion in 106 yr, calculate an initial temperature that barely exceeds 1000K halfway down the mantle and drops off to about 500K at the earth's center. Since present-day temperatures are far in excess of these values, it would seem that original heat cannot contribute much to the heat budget. Heterogeneous accretion, on the other hand, allows higher initial temperatures. Considerable heating could also be generated, and higher internal temperatures reached, if the earth accumulated partly by the continuing capture of a planetesimal swarm of meteoritic bodies (Safronov, 1969; Wetherill, 1976, 19774. These bodies, hitting the earth at orbital veloc- ities considerably greater than the velocity of free-fall, would cause much melting (as seems to have occurred on the moon) and would also, by means of shock waves, generate heat throughout the impacted body. In brief, very little can be ascertained regarding the original heat content of, and temperature distribution in, the earth at the time of formation. Chemists assure us that the temperature of the original dust cloud must have been rather low, less than 1750K, which is the temper

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18 ENERGETICS OF THE EARTH ature at which solid particles would begin to condense from the gaseous nebula. The subsequent events are quite uncertain and conclusions tentative. It seems nevertheless rather likely that initial temperatures were lower than they are today (see Chapter 31. Yet original heat is probably not entirely negligible. Sharpe and Pettier (1978) have recently worked out the thermal history of a convecting earth with a "crust" 64 km thick, in which radioactive sources presently generate one half of the surface heat flow, and with no deep-seated heat sources other than original heat. Although their model cannot be accepted literally, if only because there is surely radioactivity below 64 km, it nevertheless shows that original heat might indeed still contribute significantly to the surface heat flow. GRAVITATIONAL ENERGY A source of heat comparable in magnitude to radioactive heating (see below) is the release of gravitational energy inherent in changes in density distribution inside the earth. The gravitational energy Q of a body is defined as the sum of the gravitational interactions of all its past. Consider a shell of inner radius r, thickness dr, in a spherically symmetrical body in which the density p is a function of r only. This shell exerts no gravitational attraction on masses inside it, which in turn attract the shell as if their whole mass mr were concentrated at the center, where r = 0; here r mr = J 4=pr2 dr. o Thus the gravitational energy do for the shell is d52 G mrdmr 7 where dmr = Herr 2 pdr is the mass of the shell. Summing for all shells, we get 52 = G jR mrdmr= G iM o r 2 o where R is the outer radius of the body and M is its mass. , (2.1)

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Energy Sources 19 Integrating by parts gives 52 = GM +G J mr dr (2.2) Let ~ be the gravitational potential such that gr = d'4/dr =-Gm~/r 2, where gr is the gravitational acceleration at r. Then (2.2) can also be written as 5~ = GM _ 1 Him d ' or JM 52=- ~dmr. 2 o The function ~ is, of course, a solution of Poisson's equation V2 = - 47rGp. If the body is in hydrostatic equilibrium, the pressure P at any point satisfies dP = pd9, and l2 can be written, from (2.1), as OR lo= - 4~ ~rap dig= - 47T Or 3dP (2.4) Jo If the pressure is zero on the outer boundary, integrating by parts gives rR rv 12 = 12q, ~ Pr2dr = 3| P dV, Jo au (2.5) where V is the volume of the body. The apparent simplicity of (2.5) disguises the fact that P is a function of p and Hi. For a body with uniform density, Q = 2GM2/5R. Equation (2.4) shows that 51 is easily calculated if the pressure distribution P = P (r) is known. For the earth with the density distribution given by his model one, Birch (196Sb) calculates t1 = 2.49 x 1032 J. CORE FORMATION The change in density distribution most frequently considered corre- sponds to the separation of a dense core and lighter mantle in an earth formed by homogeneous accretion of material with uniform uncom

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20' ENERGETICS OF THE EARTH pressed density (uncompressed density means density at P = 01. The calculation is difficult, because any local change in p entails changes in g, , and P everywhere; a change in P entails a further change in p because of compressibility. Furthermore, since the gravitational energy released is ultimately dissipated as heat, the temperature of the differ- entiated earth will be different from, and presumably higher than, that of the undifferentiated body; the effect of this change in temperature on the density must also be taken into account. Thus the equation of state p = p (P. 1) must be known. In a preliminary calculation, Birch (1 96Sb) used for the undifferentiated state the pressure-density relation for the present (hot) earth and obtained for the gravitational energy AT re- leased by formation of the core /~52 = 1.7 x 103~ J. In a later paper (Flasar and Birch, 1973), a correction was made for the fact that the temperature would be lower in the undifferentiated earth than in the differentiated earth, precisely because of the release of gravitational energy; the corrected value of /~52 now comes out at 1 x 103i J. which is enough to raise the temperature of the whole earth by some 1500C. Not all of this energy is available for heating, however; part of it (about 15 percent) goes into strain energy involved in compression under changing pressure. A negligible amount would go into kinetic energy of rotation, as the moment of inertia of the undifferentiated earth is larger than that of the differentiated one. Curiously, the radius of Birch and Flasar's undifferentiated earth is slightly smaller (6220 + 20 km) than the present earth's radius, as if condensation of matter toward the center caused the earth to expand. For comparison, note that 1 x 103~ J is more than the present total heat flux at the surface (4 x 10~3 W) multiplied by the age of the earth. Formation of the core would thus be an energetic event of the first magnitude. There is of course no unanimous agreement as to when this event took place. It might have occurred at the time of formation of the earth (heterogeneous accretion), which may have had a dense core from the start; in that case the AT would be part of the 2.5 x 1032 J of the energy released at the time of formation, most of which was presumably lost by radiation from the accreting surface. More frequently, separation is assumed to have occurred at the time, of the order of half a billion years after formation, when radiogenic heating would have raised the tem- perature inside the undifferentiated earth to the melting point of iron, or of an Fe-FeS mixture, allowing liquid core material to trickle down much faster than solid material would. It has also been suggested that the process may be a slow, continuous one still in progress. Monin (1978), for instance, assumes that the rate of accretion at the core's

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Energy Sources 21 surface is at all times proportional to the remaining average concentra- tion of heavy material in the mantle; this leads to a core that is still growing and will continue to grow for another 1.9 billion years. Energy released to date is 1.61 x 103i J. according to Monin, who uses a very approximate and unreliable equation of state. On the whole it appears unlikely that the core should still be growing, as its growth would be reflected in a secularly decreasing moment of inertia and a correspond- ing increase in the angular velocity of the earth, of which there is not much sign. Rates of rotation in the distant past, as determined from growth patterns of invertebrate shells, seem to be in good accord with deceleration rates from tidal friction, leaving but little room for any steady continuous acceleration. We also know, from paleomagnetic observations, that the earth already had a magnetic field of approxi- mately the present intensity 2.6 billion years ago, a time when, accord- ing to Monin, the mass of the core would have been less than 15 percent of its final mass. Finally, seismic wave velocities and densities in the lower mantle do not seem to allow the presence in it of much metallic iron, except perhaps in the lower 100 km, just above the core boundary. The density of the solid inner core being greater than that of the liquid outer core, progressive crystallization of a cooling core must release gravitational energy. It has been suggested (e.g., Braginsky, 1964; Gubbins, 1977; Loper, 1978) that this gravitational energy contributes significantly to the generation of the magnetic field. We shall return to this subject in Chapter 4. TIDAL FRICTION The gravitational attractions of the sun and moon produce a tidal defor- mation of the oceans and of the solid earth. Frictional dissipation of rotational kinetic energy accelerates the earth by an amount - , a magnitude that has been a matter for debate for two centuries. The subject is by no means closed, as much uncertainty remains as to the magnitude of the tidal deceleration and the mechanism of dissipation, whether in ocean tides or in solid-earth tides. The acceleration ~ of lunar origin can, in principle, be determined from the acceleration n of the moon's orbital motion, if conservation of angular momentum for the earth-moon system is assumed. The lunar acceleration n can, again in principle, be determined from observations of the moon's position in ephemeris or atomic time. Until a few years ago, the accepted value of n was -22 arc sec/century2, with a corresponding (AT = -5 X 1O-22/S2. Modern observations have generally tended to increase (almost double) the value of n (Rochester, 1973), but a recent determination from lunar

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22 ENERGETICS OF THE EARTH range measurements puts it back to -24.6 arc sec/century2 (Calame and Mulholland, 19781. These figures are difficult to check against the historical record of past eclipses because the earth's mantle also sus- tains accelerations from a variety of sources, including electromagnetic torques associated with fluctuations of the magnetic field. Yukutake (1972) has shown that the long-period (~8000 years) oscillation of the magnetic field may have caused in the last 2000 years a mantle accelera- tion of the order of 1 x 10-22/s2. The matter is further complicated by the possibility of a secular change in G. the gravitational constant. Counting daily growth rings in fossil corals and other shell-building invertebrates has provided estimates of the number of days in a lunar month and in a year at known times in the geological past. When Newton reviewed the data in 1968 (Newton, 1968), he was led to con- clude that G may be decreasing at a rate of the order of 1 part in 108 per century, while the moment of inertia C of the earth must have increased by a few percent in the last 350 million years. The uncertainties in- volved in counting growth rings are, however, so large that no great weight can be attached to these results. It would seem now that all that can be said is that the tidal decelera- tion of the earth is probably between 5 x 10-22 and 10 x 10-22/s2, with a corresponding decrease in rotational kinetic energy in the range 3 x 10~2 - x 10~2 W. roughly 10 percent of the present surface heat flow. There has been a long-lasting debate as to whether dissipation, and hence heat production, occurs mainly in the oceanic tides (particularly shallow-sea tides) or in an imperfectly elastic mantle (solid-earth tides). Lambeck (1975) has reviewed the matter while calculating the accelera- tion of the moon's orbital motion from available models of the oceanic tides. The value of n he thus finds (-35 + 4 arc sec/century2) is suf- ficiently close to recent astronomical determinations to allow him to conclude that a very major part, if not all, of the moon's acceleration is caused by dissipation in the oceans. The total loss of rotational energy is, according to him, about 5.7 x 10~2 W. the oceans accounting for about 5 x 10~2 W. The balance of 0.7 x 10~2 W could be interpreted as a source of heat in the mantle but is hardly significant, as it repre- sents the small difference between two large and uncertain numbers. A well-known consequence of the tidal slowing of the earth's rotation is that the moon's mean distance c from the earth must increase at a rate that is presently of the order of a few centimeters per year. As the tidal dissipation varies as c-6, it must have been much larger in earlier times. It is thus possible that in the early days of the earth-moon system, when the moon was much closer to the earth, internal heat generation by

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Energy Sources 23 solid-earth tides may have been much larger than it is now. The early history of the earth-moon system is, however, not well understood. Although it is agreed that capture of the moon may have constituted a thermal event of the first magnitude for the earth, it is not agreed as to when capture occurred, if indeed it did. Just one more uncertainty in the thermal history of our planet! RADIOGENIC HEAT The important heat-producing radioactive isotopes in the earth are U935, U238, Th232, and K40. Other radioactive isotopes (e.g., Rb87) decay so slowly, with such a small release of energy, and are present in such minute amounts that their contribution to the heat budget is negligible. The rate of heat generation is 0.97 x 10-7 W/g of natural uranium with the usual U235/U238 ratio; 36 x 1O-~3 W/g of potassium, and 0.27 10-7 W/g of thorium. But how much uranium, thorium, and potassium is there in the earth? The problem can be approached, but not solved, from different sides. One may assume, for instance, that the gross composition of the earth is the same as that of analyzed meteorites. But all meteorites do not have the same composition; eucrites, for instance, contain about 130 ppb uranium, while ordinary chondrites contain only 12 ppb. Alterna- tively, one could assess the uranium content of the upper mantle by measuring uranium in lavas that rise from the upper mantle (e.g., oceanic basalt), determining what fraction (5 percent?, 10 percent?) of the solid mantle melts to form these lavas, and more or less guessing by what amount uranium is enriched in the melt with respect to the solid from which it forms. Such calculations usually put the uranium content of the upper mantle at about 1~20 ppb. Measurements on lherzolite inclusions in some Australian lavas (Green et al., 1968) give figures ranging, depending on the specimen, from 3 to 114 ppb, with an average for six specimens of 34 ppb. Larimer (1971) sets a lower limit for uranium in the earth as equal to the amount of uranium known or believed to be present in the crust, and an upper limit based on geo- chemical considerations regarding the degree of enrichment that is likely to occur in formation of the crust. He thus sets the uranium content of the earth at 12 + 6 ppb. Ganapathy and Anders (1974) take the uranium content to be 18 ppb. Since the mass of the earth is close to 6 x 1027 g, its total uranium content would thus be about 1.1 x 102 g, producing heat at a global rate of 1 x 10~3 W. One would judge that the figure is uncertain by a factor of at least 2. Once the uranium content has been selected, the thorium content

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24 ENERGETICS OF THE EARTH follows from the observation that in most rocks and meteorites the Th/U atomic ratio is rather uniform, in the range from 3 to 4. Thus Ganapathy and Anders (1974) set the thorium content of the earth at 3.9 x 102 g. The problem of potassium is a difficult one. The abundance of a number of common elements (e.g., iron, calcium, aluminum, magne- sium, and sodium) relative to silicon seems to be much the same on earth and in the sun and, by inference, in the original nebula from which the sun, earth, and other planets were formed. If the same rule applied to potassium, its abundance on earth should be about 800 Ann, which is close to the median for ordinary chondrites. Since, however, some elements, called "volatiles," are present in less abundance on the earth than in the sun, some fractionation must have occurred at the time of formation, and the question arises as to the degree to which potassium has also been fractionated. It has long been known that the ratio K/U is much lower on crustal rocks (_1 x 104) than in meteorites (8 x 1041. A minimum can be set for terrestrial potassium by noting that radio- active decay of K40 produces A40, as well as Ca40. On the assumption that all the A40 produced in the earth since its formation has now escaped into the atmosphere, knowledge of the amount of A40 presently in the atmosphere leads to an estimate of the minimum amount of potassium in the earth. This is a minimum because there is of course no guarantee that all the internally produced A40 has indeed reached the surface; it would in fact be surprising if it had, since volcanic gases today still contain nonradiogenic He3 coming from the mantle (Craig et al., 1978), where it must have resided ever since the formation of the earth. Various geochemical arguments regarding the probable degree of outgassing of the mantle thus lead to estimates of terrestrial potassium as 130 ppm (Larimer, 1978) or 170 ppm (Ganapathy and Anders, 19741. But this is not the end of the story, because there is a possible connection between potassium and sulfur, first noticed by Lewis (19711. The S/Si ratio is considerably lower in crustal and upper-mantle rocks than in the sun, a fact that has been attributed to the chemical affinity of sulfur for iron and the consequent concentration of sulfur in the earth's core. It is indeed fairly certain that the density and sound speed in the core do not agree with the values expected for liquid iron under core conditions of pressure and temperature, and it has long been surmised that the core must contain substantial amounts (1~20 per- cent?) of an element much lighter than iron; sulfur, silicon, oxygen, and a few others have been suggested, with sulfur a favored candidate (see Chapter 41. Lewis, noticing that potassium tends to form strong bonds with sulfur (the compound K2S, for instance, is quite stable), has sug

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Energy Sources 25 gested that the low potassium concentration of surface rocks could easily be accounted for without calling on complicated fractionation processes during condensation and accretion if some or most of the original potassium were now attached to sulfur and hidden from view in the earth's core, from which much of the radiogenic A40 would not yet have reached the earth's surface. Goettel (1972) has made some experiments to determine just how potassium would be partitioned between silicates and an iron sulfide melt. Oversby and Ringwood (1972) deny that such partitioning would occur to any appreciable degree, but Goettel (1976) insists that much of the terrestrial potassium could indeed be in the core, in which the potassium concentration would be of the order of 0.1 percent. His argument has received support from Bukowinski's (1976) calculations on the effect of pressure on potassium. Bukowinski shows that at core pressure, potassium would undergo an electronic transition that would effectively make it one of the transition elements, whose affinity for sulfur is well known. The matter of the concentration of potassium in the core is, as we shall see, of some interest in connection with the generation of the geomagnetic field. Let us return now to radiogenic heat. The figures for potassium, uranium, and thorium proposed by Ganapathy and Anders (1974) are as shown in Table 2-1. If we retain the same figures for uranium and thorium, but substitute 800 ppm for the abundance of potassium, the rate of heat generation comes to 3.8 x 10~3 W. much closer to the 4 x 10~3 W presently required. The difference between the two models for potassium is very impor- tant. If the lower figure (170 ppm) is correct, there seems to be a good probability that the earth may be cooling, whereas Goettel's higher TABLE 2-1 Abundance and Heat Generation of Potassium, Uranium, and Thorium Rate of heat Abundance generation for whole earth Element (ppm) (W) Potassium (K) 170 3.7 x 10~2 Uranium (U) 18 x 1O-3 1 X 10~3 Thorium (Th) 65 x 10-3 1.05 x 10~3 Total 2.42 x 10~3 Source: Ganapathy and Anders (1974)

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26 ENERGETICS OF THE EARTH figure (800 ppm) allows for a steady state or, if gravitational heat sources are included (see below), for the possibility that the earth may be heating up. No definite statement can be made in that respect, the more so if we remember that the abundance of uranium and thorium in the earth may be uncertain by a factor of 2 or more. CRUSTAL HEAT VERSUS MANTLE HEAT It seems that observed local variations in heat flow at the surface of continents provide a datum by which crustal and mantle contributions to the heat flow can be separated. The observation is (Roy et al., 1972) that a plot of surface heat flow q against the rate of heat generation (radioactive) ~ in the rocks in which the heat flow is measured shows a remarkable linear relation (Figure 2-11. q = qO + be ~me. . . . . .. (2.6) mere ~ IS measured on the same pluton~c bodies in which q iS measured; qO and b are two constants that characterize a region or geologic prov- ince. The heat flow qO iS that which would be observed, in the same region, in near-surface rocks with ~ = 0; it represents therefore the heat coming from the lower crust and mantle. The constant b, which has the dimension of length, measures the vertical distance over which the heat sources are distributed. If qO were zero, and ~ were uniformly distrib- uted over a depth b, the steady-state heat flow would of course be precisely be, as given by (2.6~. Lachenbruch (1968) was the first to point out that equation (2.6) seems to be obeyed regardless of the amount of erosion that has taken place since emplacement of the pluton in which q and ~ are measured. This requires that ~ be an exponentially decreasing function of depth z below the original surface. This follows, as noted by Albarede (1975), from the fact that if ~ is a function of z, so is q; by differentiating (2.6), dq de = b dz dz (2.7) But the continuity of heat flow requires, in the steady state, that div q = e. With the one-dimensional geometry in which q depends on z only (and not on the other coordinates x and y) and with z counted positive downward, div q = -dq/dz, so that, from (2.7), dq de = b = _e dz do

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Energy Sources 27 3.5 3.0 2.5 to 1 lo 2.0 1.5 _ 1 1, / / / _ / //- / / / / , / / ' // / / - / / / He ~/ / / / - / / ./ //~ // //~/ i/// // 10 1 1 1 1 0 5 10 15 20 A, 10 cal/cm3 sec FIGURE 2-1 Heat flow Q plotted against radioactive heat produc- tion rate A for the southern Rocky Mountain region (large dots) and the Basin and Range province (small dots). Reproduced with permis- sion from Roy et al. (1972). which by integration yields c0 exp ~-rib). (2.8) There is no generally recognized mechanism by which radioactive elements become heavily concentrated in the uppermost continental crust (see also Heier, 19781. What is of interest here is that the differ- ence q-qO can be interpreted to represent the contribution of crustal radioactivity to the surface heat flow. This contribution turns out to be appreciable. Thus Roy et al. (1972) find that in the United States east of the Rockies the "reduced" heat flow qO from the deep crust and mantle amounts to no more than 0.8 HFU; corresponding figures for the Basin and Range province and for the Sierra Nevada are 1.4 and 0.4

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28 ENERGETICS OF THE EARTH HFU, respectively. By a slightly different method, Rybach et al. (1977) find that in the central Alps in Switzerland, only 0.6 HFU, slightly less than one half of the surface heat flow of about 1.5-1.7 HFU, enters the crust from the mantle where Moho is deepest (~55 km); mantle heat flow is slightly higher (0.8 HFU) under the foreland, where Moho is also somewhat shallower (40 km). In Precambrian areas of Norway the average surface heat flow is 1.0 HFU and the reduced heat flow about 0.4 HFU (Heier and Grdnlie, 1977~. Thus one can probably estimate that radioactive heat sources in the crust generate about 50 percent of the continental surface heat flow. Since continents cover about one third of the earth's surface, roughly 15 percent of the global heat flow is thus accounted for. SUMMARY What emerges from this morass of fragmentary and uncertain data is that radioactivity by itself could plausibly account for at least 60 per- cent, if not 100 percent, of the earth's heat output. If one adds the greater rate of radiogenic heat production in the past by present ra- dioactive elements to that of elements now extinct (Al26, Pu244), pos- sible release of gravitational energy (original heat, separation of core, separation of inner core), tidal friction in the early days of the earth- moon system, and possible meteoritic impact in the early days of the earth itself, the total supply of energy may seem embarrassingly large. But, as we have repeatedly noted, most if not all of the figures men- tioned above are uncertain by a factor of at least 2, so that disentangling contributions from the several sources is not an easy problem. It is, in fact, two problems, one in space (e.g., is there potassium in the core?), and one in time (when did the core separate from the mantled. Not only must we know where the heat sources are, we must also know when they came into effect and what the rate of heat transfer may be. We must, in short, reconstruct the whole thermal history of the earth. Some clues may be provided by the present internal temperature distribution, a subject we turn to in the next chapter.