The Physics of Energy, page 131
As depth beneath the surface increases, both density and pressure increase. Earth’s average density is given by
(32.1)
Estimates for density and pressure as functions of depth based on a simple Earth model (PREM, or Preliminary Reference Earth Model [216]) are shown in Figure 32.3. The density distribution of material inside the planet has a significant effect on the variation of gravitational force with depth. For a solid sphere of uniform mass density and radius R, the gravitational acceleration at a point inside the sphere at radius r from the center is given by where is the gravitational acceleration on the surface. This follows simply from the variation of Newton’s gravitational force law, combined with the fraction of total mass inside a sphere of radius r. Thus, in a sphere of uniform mass, the gravitational force decreases linearly with decreasing r. Within Earth’s interior, in the upper mantle the density is so much smaller than the core density that the term dominates, however, and the gravitational force actually increases with increasing depth, decreasing only within the radius of the core (Problem 32.1). The radial dependence of gravitational acceleration is depicted in Figure 32.3(b).
Figure 32.3 (a) Density and (b) pressure and the acceleration due to gravity as functions of distance from Earth’s center in a simplified model of Earth’s interior. After [216].
32.1.2 Sources of Thermal Energy
Along with density and pressure, temperature also increases with depth beneath Earth’s surface, rising rapidly through the lithosphere and more slowly through the mantle and towards the center of the core. This thermal gradient is associated with a constant flow of heat out to Earth’s surface. The relatively steep temperature gradient in the lithosphere arises from the low thermal conductivity of solid rock. Within the mantle and fluid outer core, convection plays an important role in moving thermal energy outward, so that a similar rate of energy flow outward can be sustained by a weaker thermal gradient than that of the lithosphere.
Averaged over Earth’s surface, geothermal energy flows outward at a rate of
(32.2)
While this is many orders of magnitude smaller than the average solar energy input, integrating over Earth’s surface area of roughly km2 gives a total power of roughly 44 TW.
Where does this thermal energy come from? One substantial source for Earth’s internal energy is radioactivity. Measurements of radioactive isotope densities in near-surface rocks, combined with other considerations, suggest that in continental crust, roughly 60% of the thermal energy flowing outward has its source in radioactive decays in the crust. The density of radioactive elements is much lower, however, deeper in the continental crust and in oceanic crust.
The source of the fraction of geothermal energy flux that does not originate in crustal radioactive decay is not entirely understood. A complete picture would require a better understanding of the conditions present at the initial formation of the planet and more information about the present distribution and properties of material in Earth’s interior. Laboratory experiments on iron and minerals at high temperatures and pressures, coupled with seismic and other empirical studies have, however, progressed to the point that a reasonably coherent picture of the approximate interior configuration and temperature profile of the planet can be drawn.
It is generally agreed that most of the geothermal energy flux through the surface that does not come from radioactive elements in the crust and mantle is associated with the gradual cooling of the planet’s interior. At the time of Earth’s formation, as a mass of dust was pulled together through gravitational attraction, the gravitational potential energy released as the material contracted heated the growing planet, presumably raising its temperature until the material became a molten mass. Heavier elements such as iron gravitated to the core, releasing more energy. After some hundreds of millions of years, the surface of the newly formed planet cooled sufficiently to form a crust, which substantially slowed the rate of release of internal energy. This storehouse of initial energy is still largely present within the planet, maintaining the high temperatures that fuel the internal dynamics of the core and mantle.
It may seem surprising that thermal energy could be stored within the planet for many billions of years, but the relatively small thermal gradients and associated heat flow in the crust give a very long time scale for cooling of the interior. This is similar to the slow rate of downward heat flux between the ocean surface layer and the deep ocean estimated in eq. (27.1). The total thermal energy content of Earth is currently estimated at 12.6 J (Problem 32.2). At a rate of outward energy flow of 44 TW, it would take on the order of ten billion years for this thermal energy to dissipate. Given that a substantial fraction of the outward energy flow is replaced by radioactive decay, the time scale for cooling of the planet is clearly longer than the time since Earth’s formation.
In addition to the energy coming from radioactive decay and residual thermal energy from the initial formation of the planet, it is likely that further thermal energy is continuously added in the interior as further stratification, thermal contractions, and phase changes occur in the core and mantle. The precise details of these processes are the subject of current research.
Direct borehole measurements of temperatures at depths of up to several kilometers beneath the surface, as well as empirical data on thermal conductivity of rock, show that typical temperature gradients in continental crust range between roughly 10 ℃/km and 50 ℃/km. Thermal conductivity in the crust is typically in the range 2–4.5 W/mK. For example, granite at surface temperature and pressure has a thermal conductivity of
(32.3)
From this we see that a temperature difference of 30 ℃ over a 1 km thickness of granite would give a heat flow of mW/m2. Since granite also contains radioactive elements, the added heat from nuclear decays within the granite must be added in a more realistic calculation (see Example 32.1).
Example 32.1 Flow of Heat Through Granite
Consider as an example a 1 km thick slab of granite, assuming for simplicity that the thermal conductivity W/mKand mass density kg/m3 are uniform throughout the slab. Assume that the temperature on the bottom of the slab is330 K and the temperature at the top is 300 K. If the concentrations of uranium, thorium, and potassium are given bytypical values for granite (ppmm = parts per million by mass)
then what is the rate of heat flow at the top and bottom of the slab?
The rate of heat production from radioactivity is, from eq. (32.4),
The heat flow equation (32.5) then becomes
so
The solution to this equation is (taking at the surface, z negative below the surface, and measuring z in units of kilometers)
where we have solved for using the boundary conditions at and . The rate of heat flow on the top and bottom surfaces is
The difference between these rates of heat flow is 2.5 mW/m2, equal to the heat added from radioactive decay
32.1.3 Radioactive Decay in the Crust
The four isotopes that contribute most substantially to radioactive heat production in the crust are long-lived isotopes of uranium (, ), thorium (), and potassium (). The half-lives, primary decay channels, and average (near-surface) crustal abundances of these isotopes are given in Table 20.6. Note that potassium is quite abundant, composing some 2% of the mass of typical crustal rock, but the radioactive isotope is rare. The abundance of the radioactive elements is in general highly variable, depending not only on the type of rock but also upon specific location. Granite, for example, has particularly high concentrations of all these radioactive elements. The concentration of radioactive isotopes diminishes sharply below the upper crust. It is often convenient to express the net radioactive heat production in rock (in units of power/mass) as a function of the mass concentration ratios through (Problem 32.3)
(32.4)
The flow of thermal energy through rock containing radioactive isotopes is described by an extension of the usual heat equation in which a source is explicitly included. In an idealized situation where the conducting material is homogeneous in the horizontal directions and has a thermal conductivity depending only on the coordinate z in the direction of heat flow, a slight modification of the argument in §6.2.4 gives the heat flow equation
(32.5)
where is the rate of thermal energy added through radioactive decay, expressed in units of power/ volume.
As mentioned above, in continental crust, roughly 60% of the thermal flux reaching the surface comes from radioactive decay in the lithosphere, with most of the radioactive material residing in the upper crust. The more recently formed oceanic crust contains much less radioactive material; most of the thermal flux from beneath the ocean comes from cooling of the lithosphere, with only some 10% of the heat flow coming from beneath the lithosphere and 5% from radioactivity.
Of the total of 44 TW of geothermal energy passing out to the surface, current estimations indicate that roughly 12 TW comes from the decay of and another 13 TW from the decay of . (The contribution from decay of is negligible.) The contribution from in the crust is substantially smaller, on the order of 3 TW. While the contribution from potassium in the lithosphere and mantle is relatively small, it has been hypothesized that the core may contain a substantial quantity of potassium in the form of a potassium-iron alloy, which would contribute to geothermal flux and could impact core dynamo dynamics, though recent work [217] suggests that this quantity is actually fairly small. The estimates for radioactive fractions in standard Earth models are corroborated through neutrino measurements by particle physics experiments on Earth’s surface, and by comparison to fractions of radioactive material in chondrite meteorites that are believed to come from asteroids in the early solar system and to have element fractions representative of the material forming the solar system as a whole.
Geothermal Heat Flux
The average global rate of geothermal heat flux is
Some of this is energy from radioactive decay of elements in Earth’s crust, while some is residual thermal energy in the planet’s interior from gravitational energy released during Earth’s formation. The rate of heat flow is higher through oceanic crust than through continental crust and highest through freshly formed crust, and near plate margins and hotspots.
32.1.4 Dynamical Processes in Earth’s Interior
Extracting energy in usable quantities from the average rate of thermal energy flow at the surface, or from the distributed release of energy through radioactive decay is clearly impractical. Exploiting geothermal energy for human use requires identifying locations where high-temperature reservoirs of thermal energy are accessible to existing technologies. The distribution of thermal energy beneath Earth’s surface is far from uniform, so to understand where and how geothermal energy can be extracted we need to know something more about how processes within Earth affect the local distribution of geothermal energy.
Both conduction and convection play important roles in the transport of energy within the planet. In the solid lithosphere, conduction is the dominant means of outward heat transport. In the mantle, though motion is slow on human time scales, convective processes are nevertheless most effective in transferring heat outward. In the fluid outer core, convection dominates, though conduction is also important since the metallic outer core is highly electrically and thermally conducting. We give here a brief summary of the role of dynamics within the core and the mantle, and how these dynamics affect the lithosphere, temperature distribution, and magnetic field of the planet.
Core Dynamics and Earth’s Magnetic Dipole Field Understanding the dynamics of Earth’s fluid outer core represents a significant challenge for geophysicists. In particular, while it has been believed for decades that the dynamics of the core is responsible for generating Earth’s magnetic field, the details of this process are still poorly understood. We outline some of the basic features of Earth’s magnetic field here; a much more detailed description, along with a summary of the current understanding of the source of the geomagnetic field, can be found in [218].
The precise distribution of the magnetic field is rather complicated and time-dependent, but the dominant contribution is from a magnetic dipole field (Box 3.2). The strength and orientation of this magnetic dipole change with time. The dipole moment at present is approximately A m2, with a tilt of about 11° relative to Earth’s axis of rotation. Geological evidence, which we discuss further below, shows that Earth’s magnetic field occasionally flips its direction of polarity. (A flip of polarity from the current configuration would mean, for example, that all compasses would point south instead of north.) Over the last 80 million years such polarity reversals have occurred on average several times every million years, with a frequency that has increased since a preceding period of almost 40 million years in which polarity flips apparently did not occur (the Cretaceous superchron). Polarity reversals occur relatively quickly, and with no obvious regularity.
The magnetic dipole of Earth arises primarily from electric currents circulating through its liquid outer core. The material in the outer core is mostly iron, which conducts well at the high temperatures and pressures achieved thousands of kilometers beneath the surface. From the estimated resistivity of this material, the currents that produce Earth’s magnetic field would dissipate over less than 20 000 years, so they must be continually generated by a self-exciting dynamo mechanism. The basic idea of the dynamo is similar to the electromagnetic generator (§3), though in the dynamo the current produced by motion of charges through a magnetic field follows a trajectory that feeds back into the magnetic field. A simple model of a rotating disk dynamo is shown and described in Figure 32.4. Earth’s dynamo is, however, significantly more complex than this simple model. In particular, the dynamics of the core (like the observed magnetic field) does not have an exact axial symmetry, and must operate in a regime where the observed polarity flips are possible. The basic idea is that internal torques within the core combined with the Coriolis effect drive the dynamo. Even the source of the energy driving the dynamo is not generally agreed upon, however. The power driving the dynamo is estimated at between 0.1 and 1 TW. Energy of this order of magnitude is in principle available from several sources: tidal friction in the core, thermally driven convection, and convection driven by compositional changes. While for some time it was believed that the dynamo is driven by thermal convection, current understanding suggests that the primary driver is energy released through compositional convection – such as energy released when iron from the outer core solidifies and increases the size of the inner core. Irrespective of the energy source, the detailed dynamics of the dynamo process in the core are rather poorly understood. The magnetohydrodynamic equations describing electromagnetic fields in a moving fluid become quite complicated when applied to a rotating system in a gravitational field with an inhomogeneous temperature distribution, and have solutions of a somewhat chaotic nature. Analytic models and numerical simulations are not yet able to reproduce the observed time-dependence of Earth’s magnetic field even in a probabilistic sense, although simulations can produce regimes in which polarity reversal occurs with varying frequencies.
Figure 32.4 A very simple dynamo based on a rotating, conducting disk. An external magnetic field initially runs parallel to the axis of the disk. A torque acting on the disk causes it to rotate. The rotation of the disk in the magnetic field produces a Lorentz force that generates an outward-directed current in the disk. The current is conducted to wires that carry it back to the central axis after following a circular path. This circulating current enhances the magnetic field. Even after the original magnetic field is removed, this results in a self-sustaining system in which energy input through the external torque is converted to resistive losses as the current flows through the dynamo.
Over the last 500 years Earth’s dipole moment has decreased by some 15%, with the rate of decrease accelerating over the last century. Our understanding of core dynamics at this time is insufficient to know whether this indicates an impending polarity flip over the next few thousand years, or just a random fluctuation in the magnitude of the dipole.
Earth’s Magnetic Field
Earth’s magnetic field is dominated by a dipole field, currently tilted at 11° to Earth’s rotation axis. The magnetic field is believed to be produced by dynamic currents in the outer core through a complex dynamo action. The magnitude and direction of the field change dramatically over geological time scales.
Heat Transport Within the Mantle Although the viscosity of the mantle is much greater than that of the outer core, its thermal conductivity is also much lower. Convection within the mantle occurs over long time periods, and is believed to dominate the transfer of energy from the core–mantle boundary to the lithosphere. From the material properties and approximate temperature gradient it is believed that convective flow in the mantle gives rise to motion of the mantle material at velocities on the order of 5–10 cm/y. While slow from the point of view of the human time scale, mantle convection plays an important role in driving the motion of the crustal plates. The details of mantle convection are only partially understood. Heat added from radioactive decay is distributed throughout the mantle. Thus, unlike a boiling pot where convection is driven by heating from below, mantle convection is largely driven by surface cooling. This is discussed further below in the context of plate subduction. In addition, there are phase boundaries associated with changes in crystal structure in the mantle at depths of around 410 km and 670 km, which also affect the structure of mantle convection.
