Whereas conductive temperature profiles fail to describe the mantle geotherm, they successfully model the geotherm in the continental crust and lithosphere, where the dominant thermal processes are radiogenic heat production and conductive heat transport to the surface. Because of the great age 4.8 Continental Geotherms 255 of the continental lithosphere, time-dependent effects can, in general, be neglected.
The surface rocks in continental areas have considerably larger concentrations of radioactive elements than the rocks that make up the oceanic crust. Although the surface rocks have a wide range of heat production, a typical value for a granite is Hc = 9.6 × 10−10 W kg−1 (H for granite was calculated in Problem 4–4). Taking hc = 35 km and ρc = 2700 kg m−3, one finds that the heat flow from Equation (4–23) is qc = 91 mW m−2. Since this value is considerably larger than the mean surface heat flow in continental areas (65 mW m−2), we conclude that the concentration of the radioactive elements decreases with depth in the continental crust. For reasons that we will shortly discuss in some detail it is appropriate to assume that the heat production due to the radioactive elements decreases exponentially with depth, H = H0e −y/hr . (4.24) Thus H0 is the surface (y = 0) radiogenic heat production rate per unit
mass, and hr is a length scale for the decrease in H with depth. At the depth y = hr, H is 1/e of its surface value. Substitution of Equation (4–24) into the equation of energy conservation (4–12) yields the differential equation governing the temperature distribution in the model of the continental crust:
0 = k d2T
dy2 + ρH0e
−y/hr . (4.25)
Beneath the near-surface layer of heat-producing elements we assume that the upward heat flow at great depth is qm; that is, q → −qm as y → ∞. This model for heat production in the continental crust is sketched.
An integration of Equation (4–25) yields c1 = k dT dy − ρH0hre −y/hr = −q − ρH0hre −y/hr . (4.26)
The constant of integration c1 can be determined from the boundary condition on the heat flux at great depth, that is, from the mantle heat flux to the base of the lithosphere c1 = qm. (4.27) Thus the heat flux at any depth is q = −qm − ρH0hre −y/hr . (4.28) 256 Heat Transfer Figure 4.10 Model of the continental crust with exponential radiogenic heat source distribution. The surface heat flow q0 = −q(y = 0) is obtained by setting y = 0 with the result q0 = qm + ρhrH0. (4.29)
With an exponential depth dependence of radioactivity, the surface heat flow is a linear function of the surface radioactive heat production rate. In order to test the validity of the linear heat flow–heat production relation (4–29), determinations of the radiogenic heat production in surfacerocks have been carried out for areas where surface heat flow measurements have been made. Several regional correlations are given in Figure 4–11. In each case a linear correlation appears to fit the data quite well. The corresponding length scale hr is the slope of the best-fit straight line and the mantle (reduced) heat flow qm is the vertical intercept of the line. For the Sierra Nevada data we have qm = 17 mW m−2 and hr = 10 km; for theeastern United States data we have qm = 33 mW m−2 and hr = 7.5 km; for the Norway and Sweden data, qm = 22 mW m−2 and hr = 7.2 km; and for the eastern Canadian shield data, qm = 30.5 mW m−2 and hr = 7.1 km. In all cases the length scale hr is near 10 km. The values of the mantle or
reduced heat flow qm are reasoably consistent with the mean basal heating of the continental lithosphere qm = 28 mW m−2 given in Section 4–5.