Continental Geotherms
Dependence of surface heat flow q0 on the radiogenic heat production per unit volume in surface rock ρ H0 in selected geological provinces: Sierra Nevada (solid squares and very long dashed line), east- ern U.S. (solid circles and intermediate dashed line), Norway and Sweden (open circles and solid line), eastern Canadian shield (open squares and short dashed line). In each case the data are fit with the linear relationship Equation dioactivity can explain the linear surface heat flow–surface radioactivity relation. The exercises to follow show that the exponential radioactivity distribution is not unique in its ability to model the linear q0 versus ρH0 relation; other dependences of H on depth that confine radioactivity near the surface are consistent with observations. However, the exponential distribution is the only one that preserves the linear q0 versus ρH0 relation under differential erosion, a strong argument to support its relevance to the continental crust. The exponential depth dependence is also consistent with magmatic and hydrothermal differentiation processes, although a detailed understanding of these processes in the continental crust is not available. Problem 4.7 Table 4–4 gives a series of surface heat flow and heat production measurements in the Sierra Nevada Mountains in California. Determine the reduced heat flow qm and the scale depth hr. Consider one-dimensional steady-state heat conduction in a half-space with heat production that decreases exponentially with depth. Surface Heat Flow and Heat Production Data for the Sierra Nevada Mountains q0 ρ H0 q0 ρ H0 (mW m–2) (µWm–3) (mW m–2) (µW m–3) 18 0.3 31 1.5 25 0.8 34 2.0 25 0.9 42 2.6 29 1.3 54 3.7 The surface heat flow–heat production relation is q0 = qm + ρH0hr. What is the heat flow–heat production relation at depth y = h∗? Let q∗ and H∗be the upward heat flux and heat production at y = h∗.Problem 4.9 Assume that the radioactive elements in the Earth are uniformly distributed through a near-surface layer. The surface heat flow is 70 mW m−2, and there is no heat flow into the base of the layer. If k = 4 W m−1 K−1, T0 = 0◦C, and the temperature at the base of the layer is 1200◦C, determine the thickness of the layer and the volumetric heat production.
Problem 4.10 Consider one-dimensional steady state heat conduction in a half-space. The heat sources are restricted to a surface layer of thickness b; their concentration decreases linearly with depth so that H = H0 at the surface y = 0 and H = 0 at the depth y = b.
For y > b, H = 0 and there is a constant upward heat flux qm. What is the q0 (upward surface heat flow)–H0 relation? Determine the temperature profile as a function of y.
