Heat in the Earth and heat flow (see Mussett & Khan, Chapter 17; various topics in Fowler, Chapter 7, especially p. 269-281) At the surface of the Earth we receive heat energy from the Sun and from inside. The heat from the Sun is measured by the solar constant i.e. power (energy/sec) delivered per sq.m. at all wavelengths to the top of the atmosphere. This is about 1370 W/m 2 over the crosssectional area πr 2 (R = radius of the Earth). Some of the solar radiation is reflected from clouds or the Earth s surface itself, or scattered by particles, but the remainder is distributed across the total area of the Earth, 4πR 2. The numbers on the left-hand diagram are in percentages of the total incoming radiation (far left). The right-hand diagram shows the spectral distribution of the incoming radiation from the Sun (approximately a black body at 5750K) and the outgoing long-wave radiation from the Earth (approximately a black body at 245K). Let us assume that the Earth absorbs all the radiation from the Sun (at the short wavelengths). This means that it behaves as a black body which is a fair approximation. Assume also that it reradiates the same energy at a low temperature (long wavelengths). It is then easy to calculate what its mean surface temperature would be.
We use the Stefan-Boltzmann law P = σt 4 Watts/m 2 where T is the mean temperature of the Earth and σ is Stefan s constant, 5,67.10-8 W.K -4 m -2. It would give a temperature around 245K = -28 C, cf. the diagram on the right. We can thank CO 2 and other gases in the atmosphere for stopping some of the outgoing long-wave radiation. However, this course is mostly about solid-earth geophysics so we shall concentrate on the heat coming from below. According to measurements of heat flow in many places, the total globally-averaged flux of this heat is 4.10 13 W or only 0.08 W/m 2. Before discussion of the origin of this heat we might recall how is heat energy is transported. There are three mechanisms: -electromagnetic radiation (important outside the Earth and at large depths in the mantle), -conduction through materials, and -heat being carried from one place to another by movement of materials. The last one is often split into convection (vertical circulation, important in the Earth s core) and advection (due often to horizontal movement; is important e.g. in meteorology and oceanography). Convection tends to occur in a liquid heated from below if there is a downward decrease in density (due to thermal expansion); for this the expansion first has to overcome the compaction of the lower caused by overlying material, and then the movement must overcome viscosity forces. Convection can be enormously more efficient at transporting heat than conduction.
In the continental crust, conduction is the main mechanism of heat transport. Convection of water plays a part in the oceanic crust, especially at the fractured mid-ocean ridges (without sediment cover) and in fault zones. Conducted heat up through the Earth s crust, in Watts per m 2, is the product of the temperature gradient dt/dz and the thermal conductivity k of the rocks. On shore, the gradient is measured in drill holes of order 100 m in depth. It is often disturbed by -climate fluctuations, especially the Ice age - erosion - flow of water in pores or fissures - the process of drilling a with large amounts of coolant fluid More reliable measurements are obtained in probes penetrating of the order of 5-10 m into sediments on the ocean bottom where conditions are stable. The temperatures are measured by electrical devices (thermistors). Thermal conductivities are measured on disks cut from cores, and are generally around 2-3 W m -1 K -1 (i.e. Watts per sq.m. per temp.- gradient in K/m) in the crust. It is found that at the surface of the continental crust, the heat flow increases with its content of radioactive elements. It is therefore possible to split the heat flow into two: one part coming from below the crust ( basal heat flow ), the other part being generated within the crust. See the left-hand diagram. On a global scale we may expect that old continental crust contains less U than young crust because the original U in the former has decayed, see the right-hand diagram.
A list of heat production due to various isotopes is in Table 7.1 in Fowler. For instance, the table gives an average content of 4 ppm of U in granite: when multiplied by the heat production 9,8. 10-5 W per kg of pure U on p. 275, we get 3,9.10-10 W per kg of granite coming from U. In these units, 4.1 comes from Th and 1.3 from K. The oceanic crust contains very little of the radioactive isotopes but it is much thinner than the continental one. If we assume that the base of the crust is maintained at the same temperature (say 1000 C) for both types, it follows that the heat flow per m 2 in the oceans may be greater than in the continents. In the ocean crust an average gradient of 1000 C/30 km, times a conductivity of 3 W K - 1 m -1, gives on average some 100 mw/m 2 (but decreasing rapidly with distance from the spreading axes) compared to 65 mw/m 2 for the continental crust including shelf areas, see Table 7.3 in Fowler. In the mantle, the gradients are much lower and the effective thermal conductivities (including convection) are higher. Where does the heat represented by the basal heat flow have its origin?
One candidate is the potential energy lost when the earth accreted from dust or fragments. If the Earth was created from small pieces of rocks of total mass M falling in from infinity, this would release energy of sufficient magnitude to heat the Earth to thousands of degrees. A simple calculation is thus: Assume that a sphere of radius r, density ρ and mass m = 4πρr 3 /3 has already formed in space. Then a small mass dm of the same density arrives from infinity and gets smeared all over the surface of that sphere in a layer of thickness dr. Evidently, dm= 4πρr 2 dr. The process will convert an original potential energy of magnitude de = Gmdm/r to heat; G is Newton s gravitational constant. Integrating de from r = 0 to R and inserting M = 4πρR 3 /3 gives the total converted energy as E =3GM 2 /5R. However, it is expected that much of this energy would soon be radiated into space again, especially if the accretion was slow. We should keep in mind that in the early days of the Earth the amount of 238 U was about twice what it is now and the amount of 235 U was a much larger proportion. There may also have been many short-lived isotopes around to heat the Earth, and it is generally assumed that it was at some early stage mostly fluid. The melting may have happened rapidly because large amounts of potential energy were converted to heat once the Earth began to segregate into layers of increasing density with depth. Some chemical energy was also released by crystallization and by chemical reactions within the Earth, a little also by tidal dissipation and other processes. It is thought (acc. to textbook by F.D. Stacey, 1992) that the total heat production in the Earth now is 28.10 12 W from radioactivity and 4.10 12 W from other mechanisms (chiefly loss of potential energy). By heat transport, the core is losing 3 in the same units to the mantle, the mantle is losing 31 units (beyond these 3) and the
crust is losing 8, indicating a total loss of (42-32) = 10.10 12 W. These numbers would imply that the mantle is getting cooler by some 0.05 C per million years. According to the book by Mussett & Khan, radioactivity in the continental crust is producing c. 10% of the total heat outflow from the Earth, and radioactivity in the mantle c. 30%. Heat flow in Iceland The curve shows the temperature gradient in a 100-m drill hole in N- Iceland. The map indicates low-temperature (dots) and high-temperature geothermal areas, the latter being mostly in the active volcanic zones. Iceland is a region of high heat flow, a hot spot. To appreciate this we may for simplicity convert continental and oceanic heat flow values given above into an old unit, 1 µcal/cm 2 /sec, called heat flow unit. HFU. One HFU = 42 mw/m 2. Thus continents have a heat flow of 1.5 HFU on average, and oceans have 2.5 HFU. One old estimate indicated that in Iceland, conductive heat accounts for 3 HFU on average outside the volcanic zones (75 C/km x 1.7 W.K -1 m -1 ), convection of ground water brings up 2 HFU on average over the area of Iceland, and erupted magma carries on average 1 HFU to the surface.