Page 136 - Physical Principles of Sedimentary Basin Analysis

118 Heat flow

(a) (b)
0 0

5 5

depth [km] 10 depth [km] 10

15
(z) (n) 15
(3)
20 20
(2)
(1)
25 25
0 100 200 300 400 0 100 200 300 400 500
temperature [°C] temperature [°C]
Figure 6.4. (a) The geotherm (6.59)for q m = 0.025 Wm −2 , λ = 2.5 Wm −1 K −1 ,z m = 25 km,
S 0 = 1 · 10 −6 Wm −3 . The linear geotherm in the case of zero heat production is also shown. (b)
The geotherms in the case of the three different distributions (6.54)to(6.56) of the heat production
in the crust. The average crustal heat production is the same for the three distributions.

The temperature T m at the base of the crust depends on the distribution of the heat
production with depth. For example the linearly changing heat productions (6.54)to(6.56)
1 2
◦
give a temperature difference T m = S 0 z ≈ 83 C between S 1 and S 3 ,asshown in
6 m
Figure 6.4b. (See Exercise 6.10 for the details.)
◦
The temperature T a ≈ 1300 C specifies the transition zone between the lithosphere and
the asthenosphere. The depth to the base of the lithosphere is the solution of T (z a ) = T a ,
using the mantle geotherm (6.60), which gives

λ m λ m S 0 z m
z a = z m + (T a − T 0 ) − 1 + z m . (6.62)
q m λ c 2q m
It is also possible to rewrite the depth to the asthenosphere using the surface heat flux
instead of the heat production S 0 as shown in Exercise 6.6.
We will now look at the other situation where the temperature (T a ) at the base of the
lithosphere (z = z a ) is the boundary condition instead of the heat flow q m . In the case
of constant heat conductivities for the mantle and the crust and a constant crustal heat
production we have
q m S 0 2
T m = T 0 + z m + z m (6.63)
λ c 2λ c
q m
T a = T m + (z a − z m ) (6.64)
λ m
which are two equations for the two unknowns T m and q m . We find that

2
1
λ c T 0 (z a − z m ) + λ m T a z m + S 0 z (z a − z m )
m
T m = 2 (6.65)
λ c (z a − z m ) + λ m z m