Page 137 - Physical Principles of Sedimentary Basin Analysis

6.4 Stationary 1D temperature solutions with heat generation 119

0 0
−6
S 0 = 2 × 10 W/m 3
20 −6 3 20
S 0 = 1 × 10 W/m
λ = 3W/Km
m
−6
S = 0 × 10 W/m 3 = 4W/Km
40 0 40 λ m λ m = 5W/Km
depth [km] 60 depth [km] 60
80 80
100 100
(a) (b)
120 120
0 500 1000 1500 0 500 1000 1500
temperature [°C] temperature [°C]
Figure 6.5. The geotherm (6.59)–(6.60) is plotted for T m and q m given by equations (6.65) and (6.66),
respectively, which is the case of a constant temperature T a at the depth z a . (a) The geotherm for dif-
ferent values of heat generation (S 0 ); (b) the geotherm for different mantle heat conductivities (λ m ).
Other parameters are S 0 = 1 · 10 −6 Wm −3 , λ c = 3 Wm −1 K −1 , λ m = 3.5 Wm −1 K −1 , crustal
◦
thickness z m = 35 km, lithospheric thickness z a = 120 km and the temperature T a = 1300 Catthe
base of the lithosphere.
and

1 2
T a − T m λ c (T a − T 0 ) − S 0 z m
2
q m = λ m = . (6.66)
z a − z m (λ c /λ m )(z a − z m ) + z m
Figure 6.5 shows examples of the temperature solution (6.59)–(6.60) when the boundary
conditions are fixed temperatures at the surface and at the base of the lithosphere. Not only
the temperature at the base of the crust varies, but also the mantle heat flux varies, when a
fixed temperature at the base of the lithosphere is used as a boundary condition. Note 6.1
shows an alternative and more direct solution of the stationary temperature equation (6.49)
in the case of constant heat conductivities and heat production.

Note 6.1 The stationary temperature equation (6.49) can also be solved directly for the
case of constant heat conductivities for the crust and mantle, and a constant crustal heat
production. The temperature is first written as two parts, one for the crust and one for the
mantle:
1 2
T 1 (z) =− (S 0 /λ c ) z + a 1 z + b 1 , z in crust
T (z) = 2 (6.67)
T 2 (z) = a 2 z + b 2 , z in mantle
where there are four coefficients in the solution (a 1 , b 1 , a 2 and b 2 ). We have only two
boundary conditions, so we need two more before we have four equations for the
four unknown coefficients. The two remaining boundary conditions are for the interface
between the crust and the mantle. The temperature at the interface is continuous and the
heat flow is also continuous, which is written

dT 1 dT 2
T 1 (z m ) = T 2 (z m ) and λ c (z m ) = λ m (z m ). (6.68)
dz dz