heat flow in the Earth

heat flow in the Earth From its molten outer core to its temperate surface, the temperature of the Earth varies by several thousand degrees. The heat of the Earth's interior accumulated by means of a poorly understood combination of mechanisms: (1) potential energy acquired as the planet accreted from infalling meteorites and asteroidal fragments, (2) the conversion of gravitational to thermal energy as metallic iron segregated from silicate rock and sank to form the planetary core, and (3) the energy released by the continuing decay of radioactive elements within rocks and the core. Internal heat has been gradually escaping to outer space during the Earth's history, impeded by the thermal insulation of the planet's outer layers of silicate rock. This thermal energy is transported primarily by two mechanisms: conduction and advection.

In conductive heat flow, thermal energy flows from warmer to cooler materials. The energy flux is proportional to the temperature difference ΔT = T₁−T₂ over a distance interval ΔZ, where T₁ and T₂ are the temperatures of the warmer and cooler materials respectively. For any particular material, there is a proportionality constant k that scales (ΔT/ΔZ). This constant, k, is known as the thermal conductivity. The thermal conductivity of silicate rock is very low, typically 10 times smaller than metal sulphide minerals like pyrite (FeS₂) and 100 times smaller than pure metals like copper (Cu). A steady conductive heat flow near the Earth's surface is maintained by a gradual increase of temperature with depth, roughly one degree Kelvin for every 50 metres near the outer surface. The conductive vertical heat flow Q = k(ΔT/ΔZ) can be estimated by measuring temperatures at two depths Z₁ and Z₂ within a borehole, and estimating the thermal conductivity k from the rock-types within the interval ΔZ =  Z₁–Z₂.

Because conduction in silicate rocks is inefficient, thermal energy in the Earth is commonly advected, that is, transported by the motion of material. Heat can be advected by water that percolates along fissures or through porous rock. Rainwater in volcanic areas can descend as far as several kilometres into the crust, later rising to the surface in hot springs or geysers. A spectacular form of heat advection occurs when molten rock, or magma, erupts from a volcano. Although the silicate rock of the Earth's mantle and crust is predominantly solid, on geological timescales it can flow in response to buoyancy forces. Positive and negative variations in temperature cause rocks to expand and contract. Just like a hot-air balloon, but much more slowly, heated rock expands, experiences a proportional decrease in its density, and tends to rise within the planetary interior. Similarly, cooled rock contracts, experiences an increase in density, and tends to sink. The Earth's mantle advects heat upward by a continuous exchange of rising warmer rock for sinking cooler rock, in the process called convection.

How do geophysicists determine whether the mantle convects heat from the molten-iron core to the surface crust, rather than conducting heat through motionless rock? Consider a sizable volume of heated rock. The tendency of heated rock to become buoyant is scaled by its coefficient of thermal expansion α. Typical values of a for silicate rocks imply expansion by two parts in 1000 for every 100 °K increase. Buoyancy is also favoured if the temperature difference ΔT between the top and bottom of the mantle is large. Average surface temperatures are close to 290 °K. The temperature at the base of the mantle must be higher than the melting point of iron in the underlying core. This melting point is imperfectly known, but is likely to exceed 3500 °K. The buoyancy forces must overcome the resistance of the rock to flow, as determined by its viscosity η. Buoyancy forces must also overcome the tendency of heat to escape by conduction from the buoyant rock, as determined by its thermal conductivity k. The total size of the (potentially) convecting region, measured by a length scale, is also important, because opposing flows of warmer and cooler rock are difficult to maintain if the system is of limited extent. The balance of parameters can be combined into a single dimensionless ratio, called the Rayleigh number Ra, that governs whether overall heat flow is convective or conductive. Above a critical value of Ra, which can be estimated using model experiments in the laboratory, a system will experience a convective ‘instability’ and be subject to advective heat transport. By the mid-twentieth century, geophysicists had determined that the Earth's mantle was likely to advect heat from the core, because its Rayleigh number far exceeds the convective threshold. In fact, convective instabilities are probable on regional scales within the mantle, not just for the system as a whole.

Conduction still plays a large role in the Earth's heat flow, mainly in the top and bottom 100–200 km of the mantle, roughly 10 per cent of its 2900 km thickness. The conductive zones form boundary layers where heat is absorbed from the underlying core and, through the crust, released to the overlying atmosphere. The upper boundary layer is known as the lithosphere. The lower boundary layer has been named D by seismologists. It is characterized by unusual wave propagation behaviour (scattering, for example) that suggests a structural complexity comparable to that of the lithosphere. Substantial increases in temperature occur with depth in the boundary layers, enhancing the heat flow through them. Between the boundary layers, however, the near-absence of conductive heat transport is reflected in a weaker ‘adiabatic’ temperature profile, along which the increase in temperature with depth is associated with compression of the rock by increasing overburden pressure, not an increase in its intrinsic heat content.

The famous 1944 textbook Principles of physical geology by Arthur Holmes, written prior to the advent of plate tectonic theory, illustrated the advection of mantle heat as a steady migration of silicate rock within a convection cell, in which regions of upwelling warmer material were matched by symmetrical regions of downwelling cooler material. The real mantle is more complex. Its high Rayleigh number encourages a sporadic progression of upwellings and downwellings in both space and time. If Earth history could be accelerated to fit within a typical television advertisement, the planet's underlying mantle flow would look quite turbulent. The viscosity of silicate rock depends strongly on temperature; cooler regions therefore tend to be stiffer, and warmer regions more prone to flow. As a result, the surface of the Earth has, as it cooled, formed stiff tectonic plates (the lithosphere) that are subject to brittle fracture (which is expressed as earthquakes) along their mutual boundaries. When these cooling plates founder and sink, they descend as sheets, or ‘slabs’, through the upper mantle at ocean trenches, forming earthquake-prone Benioff–Wadati zones. Upwelling mantle rock is less viscous, typically rising as localized ‘plumes’ or ‘hot spots’. The collision of a plume with the lithosphere gives rise to volcanic islands in the world's oceans, as well as progressions of volcanoes on the continents. Although plumes probably represent most of the advective heat that is transported from the core–mantle boundary to the surface boundary layer, most surface volcanism occurs elsewhere, along the weak boundaries of the thick surface plates. One of the most prominent hot spots has formed the Hawaiian Island chain, which can be traced across the north-west Pacific Ocean as a crooked line of extinct underwater volcanoes. The motion of the hot spot is only apparent. The tectonic plate that underlies most of the Pacific Ocean has instead been sliding across the mantle, at a speeds of 5 to 10 cm per year, for more than 100 million years. By contrast, the mid-Atlantic Ridge has hovered above the hot spot that formed Iceland for a comparable length of time without significant drift.

The lower part of the mantle is estimated to be 10–30 times more viscous than its upper layers, presumably because of mineral phase changes from low- to high-pressure crystal structures at a depth of about 670 km. The increase in viscosity with depth probably forms a barrier to downward convective flow, causing some slabs to stagnate at a depth of 600–750 km. The high-viscosity lower mantle probably discourages the lateral drift of upwelling plumes from the core–mantle boundary. The hot spots therefore form a fixed reference frame on which the motions of the surface plates can be tracked. Computer models of convection that incorporate large Rayleigh numbers and temperature-dependent viscosity can reproduce the gross features of observed mantle convection, such as stiff surface boundary layers, downgoing slabs, and rising plumes. Computer simulations also indicate that cool slabs that stagnate at 600–700 km depth are occasionally flushed catastrophically into the deeper mantle. Such overturns of mantle rock may help to explain episodes of widespread intense volcanism in Earth history.

The pattern of heat flow at the Earth's surface is governed partly by mantle convection beneath the lithosphere, partly by the location of plate boundaries, and partly by the concentration of radioactive elements in the crust. Geologists have identified several dozen active hot spots, each a centre of present-day volcanism. The heat flow at plate boundaries depends on their character. At transcurrent boundaries where plates slide past each other, such as the San Andreas Fault in California, surface volcanism is absent or minimal. Where plates spread apart, mantle rock at an estimated 1550 °K rises to fill the gap and undergoes pressure-release melting. The magma rises by porous flow to form new oceanic crust along linear ridges on the sea floor, as well as linear volcanic zones on land like the East African Rift. Tectonic plates develop at mid-ocean ridges, because the viscosity of mantle rock increases substantially once temperatures fall below 1350 °K. New lithosphere thickens and stiffens as it cools. Much of the near-surface cooling is accomplished by sea water, which circulates within the oceanic crust. Theoretical calculations predict, and measurements confirm, that the heat flow of a new tectonic plate decreases inversely with the square root of its age. The thermal buoyancy of the cooling lithosphere is reflected in the elevated sea-floor topography of mid-ocean ridges, whose crests are about 2 km above the abyssal sea floor. The near-constant depth of the abyssal sea floor indicates that the lithosphere thickens to roughly 100 km in 70 million years, but then ceases to grow. Small-scale convective instabilities cause any additional thickness of cooled rock to peel off and sink into the mantle.

At boundaries where plates converge, one plate typically underthrusts the other and sinks into the mantle, forming a deep oceanic trench in its wake. As it descends, volatile compounds, such as water (H₂O), carbon dioxide (CO₂), and sulphur dioxide (SO₂) are released from the descending slab and catalyse melting in the surrounding mantle. Rising magma forms lines of volcanoes in the overriding plate at a predictable distance from the trench. If the overriding plate is oceanic, an arc of islands is formed; for example, the Tonga–Fiji island arc north of New Zealand. If the overriding plate is continental, the volcanic arc expresses itself as an extended mountain range such as the Andes of South America.

The total conductive heat flux at the Earth's surface is estimated to be 32 × 10¹² W (32 billion billion watts). Heat flow from circulating water is estimated to add 10 × 10¹² W to the conductive heat flux, giving a total of 42 × 10¹² W. This massive outpouring of thermal energy can be put in perspective by noting that the average conductive heat flux is only 0.06 W m⁻². The conductive heat flux from an area the size of a football field would, according to this average, power three 100-W light bulbs.

On the continents, heat flow tends to be greater where the surface is underlain by younger rocks. In contrast to the oceans, the main factor in this trend is the age-dependent concentration of radioactive elements in the granitic rocks of the continental crust. Oceanic crust is thinner than continental crust and basaltic in composition, with lower concentrations of radioactive elements. In continental crustal rocks formed over a billion years ago, much of the uranium (U), thorium (Th), and the radioactive isotope ⁴⁰K of potassium has decayed, either to lead (Pb) or, in the case of ⁴⁰K, argon (Ar). Radioactive decay in old rocks therefore contributes less thermal energy to the crust. The large radiogenic heat production of continental crustal rocks compensates for a low conductive heat flow from the underlying continental mantle, which is estimated to be 0.02–0.025 W m⁻² on average. The greater thickness and age of continental lithosphere (much of it more than 200 km thick and older than a billion years) as compared with oceanic lithosphere (which is no thicker than 100 km and no older than 200 million years) may be the root cause of this heat-flow deficit. Continental lithosphere may simply have cooled for a longer time. However, why small-scale convective instabilities should not cause the base of the continental lithosphere to founder and sink, thereby limiting its growth, is thus far not understood well.

Jeffrey Park

Bibliography

Fowler, C. M. R. (1990) The solid Earth: an introduction to global geophysics. Cambridge University Press.
Sigurdsson, H. (1999) Melting the Earth: the history of ideas on volcanic eruptions. Oxford University Press.