# Geometry for Problem

Figure 4.8 Temperature as a function of depth within the Earth assuming heat transport is by conduction (conduction geotherm). Also included are the solidus and liquidus of basalt and the solidus of peridotite (olivine).

Write a steady-state heat balance for the triangle ABC in Figure 4–7a. Use Fourier’s law of heat conduction to evaluate qζ and qη. Note q0 = −qy. Assume no heat sources. 4.7 A Conduction Temperature Profile for the Mantle 253 4.7 A Conduction Temperature Profile for the Mantle

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We can use Equation (4–17) to determine the temperature as a function of depth in the Earth, that is, the geotherm, assuming heat is transported by conduction. The depth profile of the temperature is given in Figure 4–8, assuming T0 = 0◦C, q0 = 70 mW m−2, ρ = 3300 kg m−3, H = 7.38× 10−12 W kg−1, and k = 4 W m−1 K−1. Also included in Figure 4–8 are the liquidus and solidus of basalt and the solidus of eridotite. Basalt is the low-melting temperature fraction of the mantle. When the temperature of the mantle exceeds the basalt solidus, this fraction starts to melt, resulting in volcanism. This is the cause of the extensive basaltic volcanism that forms theoceanic crust. When the temperature reaches the basalt liquidus, this fraction is entirely melted, leaving a high-melting-temperature residuum that is primarily composed of the mineral olivine. When the mantle temperature reaches the olivine solidus, the remainder of the mantle rock melts. The ability of seismic shear waves to propagate through the mantle indicates that substantial melting does not occur. The conclusion is that this conduction analysis does not predict the temperature in the Earth’s mantle. In an attempt to assess the failure of the conductive mantle geotherm to model the Earth, one may ask whether the near-surface concentration of radioactive elements in crustal rocks can modify the analysis. (The partialmelting processes that lead to the formation of the crust concentrate theradioactive elements.) The only way in which this could have an effect is through a reduction in the amount of the surface heat flow q0 attributed to mantle heat sources. Thus we must assess the contribution of crustal radioactivity to surface heat flow. It is appropriate to do this for the oceanic crust because the suboceanic mantle geotherm dominates the temperature distribution of the mantle. To determine the contribution qc to the surface heat flow of a layer of crust of thickness hc and heat production per unit mass Hc, we proceed as indicated in Figure 4–9. Equation (4–13) applies to this case also, with ρ = ρc and H = Hc (subscript c refers to the crust), ρcHcy = −kdT dy

+ c1 = q + c1. (4.21) To evaluate c1, we note that q = −qc on y = 0 and c1 = qc.

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