Page 229 - Physical Principles of Sedimentary Basin Analysis

7.7 The surface heat flow of the McKenzie model 211

The Fourier coefficients are therefore given by
1 1/β 1
b n = (β − 1)ˆz sin(nπ ˆz) dˆz + (1 −ˆz) sin(nπ ˆz) dˆz, (7.56)
2 0 1/β
where the integrations on the right-hand side finally lead to

sin(nπ/β)
b n = 2β . (7.57)
(nπ) 2
The three integrations needed to find the Fourier coefficients are given as Exercise 7.9.

Exercise 7.8 Derive the orthogonality relationship (7.55).

Exercise 7.9
(a) Derive the following integrals:

1 n
(−1)
ˆ z sin(nπ ˆz) dˆz = (7.58)
0 nπ
1/β
cos(nπ/β) sin(nπ/β)
ˆ z sin(nπ ˆz) dˆz =− + 2 (7.59)
0 nπβ (nπ)
(−1) cos(nπ/β)
1 n
sin(nπ ˆz) dˆz = + . (7.60)
1/β nπ nπ
(b) Use the integrals to verify that equation (7.56) gives the Fourier coefficients b n (7.57).

7.7 The surface heat flow of the McKenzie model

Estimates for the surface heat flow are interesting because the heat flow can be measured
at the surface, as opposed to the heat flow at large depths. The temperature solution (7.46)
gives the surface heat flow

ˆ
∂T T a ∂T
q s = λ = λ (7.61)
∂z z=0 a ∂ ˆz ˆ z=0
when it is expressed by the dimensionless temperature solution, and where λ is the heat
conductivity. The heat flow through the lithosphere before stretching is q 0 = λT a /a, where
T a /a is the thermal gradient of the steady state temperature. The surface heat flow relative
to the steady state heat flow is therefore

∞
ˆ
q s sin(nπ/β) −(nπ) ˆ t
2
ˆ q = = ∂T = 1 + 2β e . (7.62)
q 0 ∂ ˆz nπ
ˆ z=0 n=1
The first term 1 corresponds to the steady-state heat flow and the sum over n is the transient
part of the surface heat flow, which decays to zero with time. Figure 7.12 shows the surface
heat flow for β = 2. It is at its maximum immediately after instantaneous stretching, when