Page 135 - Physical Principles of Sedimentary Basin Analysis

6.4 Stationary 1D temperature solutions with heat generation 117

Equation (6.51) shows that the surface heat flow remains the same regardless of the
vertical distribution of heat production S(z) as long as average heat production is constant.
For example the following three distributions of heat production:

S 1 (z) = 2S 0 (1 − z/z m ) (6.54)
S 3 (z) = S 0 (6.55)
S 2 (z) = 2S 0 z/z m (6.56)
have the same average heat production S 0 and they therefore give the same surface heat
flow. But, we will soon see that the temperature at the base of the crust can be quite different
for different distributions of heat production although the surface heat flow is the same.
An integration of the heat flow from the surface (z = 0) to the depth z yields the
geotherm
z z
du 1
z m
T (z) = T 0 + q m + S(u) du dv (6.57)
0 λ(u) 0 λ(v) v
in the crust (z < z m ). There is negligible heat production in the mantle (z > z m ) and this
part of the geotherm can therefore be written as
z du
T (z) = T m + q m for z > z m (6.58)
λ(u)
z m
where T m = T (z m ) is the temperature (6.57) at the base of the crust (z = z m ). The solu-
tion (6.57) uses the temperature T 0 at the surface z = 0 as one boundary condition and the
mantle heat flux q m at the base of the crust z = z m as the second boundary condition. The
integrals in solution (6.57) are straightforward to calculate for constant heat conductivities
λ c and λ m for the crust and the mantle, respectively, and a constant heat production S 0 in
the crust. It gives the geotherm
q m S 0 2
T (z) = T 0 + z + (2z m z − z ) (crust) (6.59)
λ c 2λ c
q m
T (z) = T m + (z − z m ) (mantle) (6.60)
λ m
where T m is the temperature

q m S 0 2
T m = T 0 + z m + z (6.61)
m
λ c 2λ c
at the base of the crust (z = z m ). It often convenient to express the geotherm using
the surface heat flow q s instead of the heat production S 0 as shown by Exercise 6.6.
The geotherm (6.60) is linear through the lithospheric mantle because of constant heat
flow and the assumption of a constant heat conductivity. Figure 6.4 shows an exam-
ple of the geotherm through a 25 km thick crust that has a constant heat generation
S 0 = 1 μWm −3 . The figure also shows the geotherm in the case of zero crustal heat
generation, and the temperature difference at the base of the crust for the two geotherms is
2
◦
T m = S 0 z /2λ c = 125 C, in accordance with equation (6.61).
m