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FowlerHeat2 - 226 7 Heat_—_— Table 7.2 Relative...

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Unformatted text preview: 226 7 Heat ___-______—___—_________ Table 7.2. Relative abundance of isotopes and crustal heat generation in the past relative to the present Relative abundance Heat generation Age (Ma) “3U “5U U“ Th K Model A” Model B‘ Present [.00 1.00 1.00 1.00 1.00 1.00 1.00 500 1.08 1.62 1.10 1.03 1.31 1.13 1.17 1000 1.17 2.64 1.23 1.05 1.70 1.28 1.37 1500 1.26 4.30 1.39 1.08 2.34 1.48 1.64 2000 1.36 6.99 1.59 1.10 2.91 1.74 1.98 2500 1.47 11.4 1.88 1.13 3.79 2.08 2.43 3000 1.59 18.5 2.29 1.16 4.90 2.52 3601 3500 1.71 29.9 2.88 1.19 6.42 3.13 3.81 "This assumes a present isotopic composition of99.2886% 238U and 0.7114“,J 235U. I'Model A based on Th/U = 4, K/U = 20,000. ‘Model B based on Th/U = 4, K/U = 40,000. Source: Jessop and Lewis (1978). are in the ranges 150—260 ppm, 80—100ppb and 15725 ppb, respectively. These abundances result in a total radioactive heat production for the crust and mantle of 1.4727 x 10‘3W, with a best guess value of 2.1 x10‘3W. 7.3 Calculation of Simple Geotherms 7.3.1 Equilibrium Geotherms As can be seen from Eq. 7.18, the temperature in a column of rock is controlled by several parameters, some internal and some external to the rock column. Internal parameters are the conductivity, specific heat, density and radioactive heat generation. External factors include heat flow into the column, the surface temperature and the rate at which material is removed from or added to the top of the column (erosion or deposition). Temperature—depth profiles within the earth are called geatherms. If we consider a one-dimensional column with no erosion or deposition and a constant heat flux, the column may eventually reach a state of thermal equilibrium in which the temperature at any point is steady. In that case, the temperaturefidepth profile is called an equilibrium geotherm. In this equilibrium situation, BT/dt = 0, and Eq. 7.16 applies: 2 a—T = 7 5 (7.20) v 622 k Since this is a second-order differential equation, it can be solved given two boundary conditions. Assume that the surface is at z = O and that 2 increases downwards. Let us consider two pairs of boundary conditions. One possible pair is 7.3 Calculation of Simple Geotherms 227 ——-_——_*_— (i) T=Oon z=0and (ii) a surface heat flow Q = — kaT/dz = — QO on 2 =0. The surface heat flow Q = —Q0 is negative because heat is assumed to be flowmg 'upwards out of the medium and z is positive downwards. Integrating Eq. 7.20 once gives ~= ——+cl (7.21) where cl is the constant of integration. Because 37702 = Qu/k on 2 = 0 is boundary condition (ii), the constant c1 is given by =% k Substituting Eq. 7.22 into Eq. 7.21 and then integrating the second time gives Cl (7.22) A Q — __ 2 _0 l 2k2 + k z + £2 (7.23) where cz is the constant of integration. However, since 7 = 0 on 2 = 0 was specified as boundary condition (i), c2 must equal zero. The temperature Within the column is therefore given by __ A Qo 7".— —i22 + 72 (7-24) An alternative pair of boundary conditions could be (i) T=0 on 2:0 and (ii) Q=—Qd on z=d. This could, for example, be used to estimate equilibrium crustal geotherms if d was the depth of the crust/mantle boundary and Qd was the mantle heat flow into the base of the crust. For these boundary conditions integrating Eq. 7.20 gives, as before, , (3T A E = —~];z + Cl (7.25) where {:1 is the constant of integration Because 6T 6 ' i _ . z = k on = boundary condition (ii), c1 is given by / Qd/ Z d IS _ Qd Ad _ 7 + 7 (up) Ci Substituting Eq. 7.26 into Eq. 7.25 and then integrating again gives _ A Q +Ad T— —Ezz+( dk )2“; (7.27) where c2 is theconstant of integration. Because T=D on z=0 was boundary condition (1), c2 must equal zero. The temperature in the column Temperature (C) D 500 1000 1500 N 0 Depth (km) b D ea db: Figure 7.3. Equilibrium geotherms calculated from Eq. 7.28 for 8 50km thick column of rock. Curve a: standard model with conductivity 2.5 W m"1 °C' 1, radioactive heat generation 1.25pW m' and basal heat flow 21 x10"’Wm“1. Curve b: standard model with conductivity reduced to 1.7 W m’ ‘ “C‘ ‘. Curve c: standard model with radioactive heat generation increased to 2.5 [1w rn ’ 3. Curve d: standard model with basal heat flow increased to 42 x 10‘ 3 W m'z. Curve e: standard model with basal heat flow reduced to 10.5 x10" Wm". (From Nisbet and Fowler 1982.) 3 228 7 Heat ____________—_————— 0 g z < d is therefore given by T: A 2 +(Qd+ Ad); _ _ 7.28 Zkz k ( ) Comparison of the second term in Eqs. 7.24 and 7.28 shows that a column of material of thickness d and radioactive heat generation A makes a contribution to the surface heat flow of Ad. Similarly, the mantle heat flow Qd contributes de/k to the temperature at depth 2. 7.3.2 One-Layer Models Figure 7.3 illustrates how the equilibrium geotherm for a model rock column changes when the conductivity, radioactive heat generation and basal heat flow are varied. This model column is 50 km thick, has conductivity 2.5Wm'1 °C'1, radioactive heat generation 1.25;1Wm'3 and a heat flow into the base of the column of 21 x10‘3Wm'2. The equilibrium geotherm for this model column is given by Eq. 7.28 and is shown in Figure 7.33; at shallow levels the gradient is approximately 30°Ckm'1, whereas at deep levels the gradient is 15°C km"1 or less. Conductivity Reducing the conductivity of the whole column to 1.7Wm’l "C.1 has the effect of increasing the shallow-level gradient to about 45°C krri'l (see Fig. 7.3b). Increasing the conductivity to 3.4Wm'1 "C’1 would have the opposite eflect of reducing the gradient to about 23°Ckm'1 at shallow levels. Heat Generation Increasing the heat generation from 1.25 iiW m’3 to 2.5 [.iW m’ 3 raises the shallow-level gradient to over 50°C km"1 (Fig. 7.30); in contrast, reducing the heat generation to 0.4 ,uW m' 3 reduces this shallow-level gradient to about 16°C km’ 1. Basal Heat Flow If the basal heat flow is doubled from 21 to 42 ><10‘3Wm'2 the gradient at shallow level is increased to about 40°Ckm”1 (Fig.7.3d). If the basal heat flow is halved to 10.5X IO'JWm“2, the shallow-level gradient is reduced to about 27°C km‘l (Fig. 7.3e). 7.3.3 Two-Layer Models The models described so far have been very simple with a 50km thick surface layer of uniform composition. This is not appropriate for the real earth but is a mathematically simple illustration. More realistic models have a layered crust with the heat generation concentrated towards the top (see, e.g., Sect. 7.6.1). The equilibrium geotherm for such models is calculated exactly as described in Eqs. 7.20—7.28 except that each layer must be considered separately and temperature and temperature gradients must be matched across the boundaries. Consider a two—layer model: 7.3 Calculation of Simple Geotherms 229 A = A1 for 0 s z < 21 A=A2 for 21<z<22, T = O on 2 = 0 with a basal heat flow Q = — Q2 on 2 = 22. In the first layer, 0 s z < 21, the equilibrium heat conduction equation is 62 T _ A 1 In the second layer, 21 s z < 22, the equilibrium heat conduction equation is 62 _ A: 622 k The solution to these two difl'erential equations, subject to the boundary conditions and matching both temperature T and temperature gradient 6T/dz on the boundary 2 = 21, is (7.30) _~A,2 Q2 A2 A2 T— flz +[T+T(22—z,)+ L1} for O<z<z1 (7.31) A; 2 Q2 A222 A —A T 2k2 +[k+ k z+[ 12k 2}} for 21<z<22 (7.32) Figure 7.4 shows an equilibrium geotherm calculated for a model Archaean crust. The implication is that Archaean crustal temperatures may have been relatively high (compare with Fig. 7.3.). 7.3.4 The Time-Scale of Conductive Heat Flow Geological structures such as young mountain belts are not usually in thermal equilibrium because the thermal conductivity of rock is so low that equilibrium takes many millions of years to attain. For example consrder the model rock column with the geotherm shown in Figure 7 3a, If the _basal heat flow were suddenly increased from 21 I to 42 x 10 3 Wm”, the temperature of the column would increase until the new'equilibrium temperatures (Fig. 7.3d) were attained. That this process is very slow can be illustrated by considering a rock at depth 20 km. The initial temperature at 20 km would be 567°C, and 20 Ma after the basal heat flow increased, conduction would have raised the temperature at 20km to about 580°C. Only after 100 Ma would the temperature at 20 km be over 700°C and close to the new equilibrium value of 734°C. This can be estimated quantitatively from Eq. 7.17: 87‘ BIT _ = K fi 61‘ 022 The characteristic time 1 = lz/K gives an indication of the amount of time necessary for a temperature change to propagate a distance of I in a medium haVing thermal diffusivity ic. Likewise, the characteristic thermal Tennerature ('0) o son 1000 150i: CRUST l MANTLE O .63 Figure 7.4. Two-layer model Archaen crust and equilibrium geotherm. Heat generation A in nWm”; heat flow from mantle Q in 10‘3 Wm”. (After Nisbet and Fowler 1982.) ,L W‘s-(96b 7t ‘0 - g . v\ I. , K [9 WWI 9" iv“) ‘5“ fir LP W \o" 34%,“ “Vi-w It » .r—P“ ...
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