energy budget of the earth To understand the thermal and tectonic history of the Earth it is necessary to recognize the enormous release of gravitational energy that took place in the process of its formation. This accretion energy dwarfs all other energy sources in the entire life of the Earth, including radiogenic heat, to which the high temperature of the interior is sometimes erroneously attributed. A comparison of the important contributions to global energy is given in Table 1, in which the first entry is the total gravitational energy released by the accretion of the Earth and segregation to its present density structure. The following four items are components of this total accretion energy. The reason for identifying them separately is that they are derived from processes that were not simultaneous. They therefore have a bearing on our understanding of the evolution of the Earth.
Of the total accretion energy, 90 per cent would be accounted for by the accumulation of originally widely dispersed material into a homogeneous sphere with the mass and radius of the Earth. The remaining 10 per cent is due to the concentration of mass towards the centre and, of this, 80 per cent was due to the separation of the dense core. The progressive growth of the solid inner core by freezing the material from the liquid outer core and the release of gravitational energy by separation of the light crust at the surface are geophysically important phenomena, but appear as minor items in Table 1.
The estimate of the total radiogenic heat given in Table 1 is based on the abundances of the four thermally important isotopes of the elements uranium, thorium, and potassium. The present rate of release of this heat is about 30 × 10
12 Watts (W). At the time of formation of the Earth, 4.6 × 10
9 years ago, these elements would have been producing four times as much heat, and the average over the subsequent life of the Earth is twice the present value. It is possible that the early Earth incorporated also some short-lived isotopes that contributed to its heating. They cannot, however, affect the conclusion that radioactivity provides only a topping-up of the Earth's internal heat. The fact that the Earth is hot inside is due to the early gravitational energy release. This is emphasized by a comparison of the total radiogenic heat produced in the Earth during its lifetime with present stored heat, which is twice as great.
Rotational energy is dissipated by the tides. Angular momentum is conserved by expansions of the Earth–Moon and Earth–Sun orbits, but rather little of the energy lost by the Earth's slowing rotation goes into the orbits. Most is dissipated in the Earth. The sea is, however, the principal sink of tidal energy and only about 5 per cent of it heats the deep interior.
Table 1. Contributions to global energy
Gravitational energy of accretion | |
Present density structure | 2.49×1032 J |
Uniform Earth | 2.33×1032 J |
Core separation | 1,61×1031 J |
Inner core formation | 8.3×1028 J |
Separation of crust | 7.6×1028 J |
Radiogenic heat | 8.0×1030 J |
Residual (stored) heat | 1.8×1031 J |
Tidal dissipation | 2 to 3×1030 J |
Present rotational energy | 2.1×1029 J |
Much of the gravitational energy would have been radiated away during the accretion process itself. The fraction retained as heat would have depended somewhat on the speed of this process. Core separation would, however, have required the Earth to be hot and substantially complete. It has even been suggested that core formation was delayed by up to 100 million years. Almost all the gravitational energy of core formation would thus have appeared as heat, and it would have sufficed to raise the temperature of an already hot Earth by a further 3000 °C. This is the source of the residual heat that the Earth is still losing at a rate of about 11 × 10
12 W.
Cooling of the Earth
Our current best estimate of the total heat lost through the Earth's surface is 44 × 10
12 W. With an estimated 29 × 10
12 W of radiogenic heat with 2.5 × 10
12 W of gravitational energy due to thermal contraction and 1.5 × 10
12 W of gravitational separation, the deficiency is 11 × 10
12 W. This represents a net loss by cooling of the mantle at about 70 °C per billion years, with slower cooling of the core. The inevitability of this continuing net loss of heat can be seen by considering the mechanism by which the heat is transported to the surface from the interior.
The mantle is undergoing thermal convection. Hot material rises, being replaced by cooler, sinking material, resulting in a net upward transfer of heat. The rate of convection is controlled by the viscosity or stiffness of the mantle material, and this depends very strongly on temperature. The temperature-dependence of mantle viscosity has a stabilizing effect on the convection. If convection proceeds too fast, then the mantle cools too fast, becoming stiffer and so slowing the convection, allowing the heat sources to catch up again. Conversely, if the convection were to stop or even slow down, the mantle would heat up and soften, accelerating the convection. But the principal continuing source of heat, being due to radioactive decay, is decreasing with time. The self-stabilizing effect of the temperature-dependent viscosity therefore requires that the convected heat flux must also be slowing down; and since this is controlled by temperature, the temperature must be falling.
Mechanical energy and earthquakes
The mantle of the Earth behaves as a heat engine. The process of transferring heat from the hot interior to the cool surface generates mechanical power with an efficiency that is readily calculated from the thermal properties of the mantle materials. It is about 15 per cent, which means that the mechanical power generated is this fraction of the con-vected heat flux. With a mantle heat flux of 32 × 10
12 W, the total mechanical power driving global tectonics is about 4.8 × 10
12W. This power is used in deforming the solid mantle material; that is, the convection itself consumes the power that it generates. In doing so it heats up the deformed material, but this is not an independent heat source; it is part of the mantle heat that is converted to mechanical energy and back again.
The speed of convection and the mantle stresses involved, including the stresses apparent in earthquakes, are directly related to this mechanical power. If we consider a block of deformable material and apply stresses to its faces, the power that is applied to it is the product of stress, strain rate, and volume. This principle is a variant of the familiar equation ‘work = force × distance’. It can be applied directly to the convective deformation of the Earth's mantle. Observations of the motions of the surface plates indicate the rate, and we know the volume involved. With the thermodynamic value of total power, we can then estimate the average magnitude of tectonic stress. The answer is about 5 MPa (50 bar), with an uncertainty factor of 2 because of the variability of flow rates in the mantle. This value falls neatly into the range of stresses deduced from the elastic waves radiated by earthquakes, which are usually in the range 1 to 10 MPa. Since the stress release in earthquakes is the best indicator that we have of the general magnitude of tectonic stress, we have a satisfying coincidence of numbers to demonstrate that tectonics, including earthquakes, are driven by thermal convection.
The annual average energy release by earthquakes is about 4 × 10
17 joules (J), corresponding to a mean power of 1.3×10
10 W. This is about 0.25 per cent of the total tectonic power. Earthquakes are localized irregularities in the tectonic motion of the mantle and crust, and through most of the mantle the motion is aseismic. Occasional very large earthquakes release more energy than the annual average. The largest well-recorded shock occurred in Chile in May 1960, with an estimated energy of 1.6 × 10
19 J.
Core energy and the geomagnetic field
The magnetic field of the Earth is produced by electric currents in the fluid, metallic outer core, driven by turbulent convection motion. The energy source for this motion must be identified with the slow cooling of the Earth. Radiogenic heat in the core is discounted because none of the thermally important elements, uranium, thorium, or potassium, separates into the iron in melting experiments on mixtures of iron and mantle-type silicates. These elements are completely absent from iron meteorites.
Thermal convective cooling of the core is possible in principle but appears not to be very effective, primarily because the high thermal conductivity of the core causes a conductive loss of heat, amounting to about 3.7 × 10
12 W; only the core heat flux in excess of this would cause convection and that only with an efficiency of 12 per cent. It is not in any case, likely that the core heat flux is greater than this. S. I. Braginsky, then in Moscow, first pointed out that cooling of the core would cause the solid inner core to grow by the solidification on to it of outer core material but that the solid does not have precisely the same composition as the liquid. Light solutes in the outer core (which may be some mixture of oxygen, silicon, sulphur, carbon, and even hydrogen) are rejected by the solid and so remain in the fluid at the boundary as an excess of the light components. Convective mixing of this excess into the whole outer core releases the gravitational energy of inner-core formation listed in Table 1. The present rate is estimated to be 3 × 10
11 W, at least half of which, 1.5 × 10
11 W, is available to drive the geomagnetic dynamo.
The kinetic energy of convective motion is very small, even in the fluid core where the motion is a million times faster than in the mantle. The convective energy in the core is converted directly to magnetic energy by the dynamo action. The total magnetic energy is about 10
22 J. This energy is continuously lost by the resistive heating of the core by the electric currents. We can thus identify the power input, 1.5 × 10
11 W, with the rate of loss and so calculate the time that it would take an unmaintained field to disappear,10
22 J/1.5 × 10
11 W = 6.7 × 10
11 seconds = 2000 years.
This is a rough but reasonable estimate of the characteristic time-scale for changes in the field.
The surface energy balance
The average flux of heat through the Earth's surface from the interior is 0.086 W m
−2. The corresponding temperature gradient in the crust, about 25 °C/km, is very noticeable in deep mines and boreholes, but at the surface itself this heat is completely insignificant. The power in the solar radiation is 1367 W m
−2, or 342 W m
−2 when averaged over the whole surface of the Earth, including the dark side. The internal heat has no effect on climate, except indirectly when it causes volcanic eruptions.
A look into the future
To the extent that radioactivity maintains the internal heat that drives the tectonic engine, it has been likened to a spring-driven clock that is winding down. But the analogy is very imperfect. The energy of a spring has a definite end-point, but radioactive decay does not. Moreover, as the figures in Table 1 show, the Earth's stored heat is twice as great as the total radiogenic heat release since its formation; it is six times as great as the radiogenic heat release from now until the end of time. Radioactivity slows the cooling of the Earth, but the stored energy source will suffice to maintain convection and all the tectonic processes for at least another 10
10 years; and in less time than that the Earth will be engulfed by a profligate expansion of the dying Sun.
Frank D. Stacey
Bibliography
Stacey, F. D. (1992) Physics of the Earth (3rd edn). Brookfield Press, Brisbane.
