Page 210 - Physical Principles of Sedimentary Basin Analysis

192 Heat flow

compared with expression (6.321) for the change in entropy. The adiabat is therefore
given as
dT T α ∂T T α g
= or = (6.338)
dp c p ∂z c p
◦
when using p = gz. The adiabatic temperature gradient becomes dT/dz ≈ 0.5 C/km
◦
using the parameters T = 1300 C (or 1573 K), α = 3·10 −5 K −1 and c p = 1kJ kg −1 K −1 .
The gradient (6.338) can be rewritten as
dT α g
= dz (6.339)
T c p
which is straightforward to integrate:

α g z − z 0
T = T 0 exp (z − z 0 ) = T 0 exp (6.340)
c p l 0
where T 0 is the temperature at the reference position z 0 .(Both T and T 0 are defined as
absolute temperatures in kelvin.) The second equality expresses the adiabat in terms of
the characteristic length scale l 0 = c p /(α g). We can approximate the adiabat as a linear
geotherm

z − z 0
T = T 0 1 + (6.341)
l 0
as long as |z − z 0 |
l 0 . The numbers above give that the length scale is typically l 0 ≈
5
3·10 km, which suggests that the geotherm is linear over several hundred km. Figure 6.41
shows a mantle adiabat in the upper asthenosphere and a geotherm through the lithosphere.
The geotherm in the lithosphere is at 1300 C at the depth 120 km, through radioactive heat
◦

0
crust
lithospheric mantle
100
depth [km] 200 asthenospheric mantle

300

400
0 500 1000 1500
temperature [°C]
Figure 6.41. The figure shows the geotherm through the lithosphere and into the upper part of the
asthenosphere.