Page 231 - Physical Principles of Sedimentary Basin Analysis

7.8 The thermal subsidence of the McKenzie model 213

It is also possible for the transient surface heat flow to be reduced right after the stretching
event too, if the crust is sufficiently heat producing, as shown in Exercise 7.10.

Exercise 7.10
(a) Show that the surface heat flow (7.64) after instantaneous stretching increases compared
to the initial heat flow (7.63) only if
Sc
β> . (7.66)
q m
We see that we will always have an increased heat flow after stretching when q m > Sc
because β is never less than 1.
(b) How large must β be for the heat flow to become larger right after stretching when
S = 10 −6 Wm −3 , c = 30 km and q m = 15 mW m −2 ?
Note 7.3 (Mathematical curiosity) The surface heat flow immediately after stretching
relative to the steady state heat flow is ˆq = β. (It is simply the thermal gradient of the tem-
perature immediately after stretching T I (z) over the gradient of steady state temperature
T 0 (z).) From equation (7.62)wealsohave
∞
sin(nπ/β)
ˆ q = 1 + 2β (7.67)
nπ
n=1
for time t = 0. It then follows that the sum of the series has to be
∞
sin(nπ/β) β − 1

= . (7.68)
nπ 2β
n=1
Expression (7.68) can be used to find out how many terms are needed in the Fourier
series (7.62) to obtain a certain accuracy.

7.8 The thermal subsidence of the McKenzie model
A lithosphere that is hot after instantaneous stretching returns slowly (over more than
100 Ma) to its steady state temperature by conductive cooling. The lithosphere contracts
(gets more dense) because of the cooling, which leads to thermal subsidence. The amount
of thermal subsidence is found assuming isostasy, and using solution (7.46) for the transient
temperature after rifting. The principle of isostasy states that the pressure at the same depth
in the asthenosphere remains the same through time. The pressure at the depth a + s T is

a a
m (T (z, t)) dz + s s T (t) = m (T I (z)) dz + m (T a )s T (t) (7.69)
0 0
where s T (t) is thermal subsidence as a function of time. The mantle density m is a func-
tion of temperature as given by equation (7.24), and the temperature is a function of depth.
The sediment density s is constant, and the sedimentary basin is taken to be sufficiently
thin compared to the thickness of the lithosphere so that the basin can be ignored in the