Page 233 - Physical Principles of Sedimentary Basin Analysis

7.9 Lithospheric stretching of finite duration 215

The last factor 1 − cos(nπ) is 2 for n = 1, 3, 5 and all other odd indices n, and it is 0 for
n = 2, 4 and all even indices n. Only the odd indices are accounted for by replacing n by
2m + 1for m = 0, 1, 2, .... We then get expression (7.70) for thermal subsidence.

Exercise 7.11 Maximum thermal subsidence. Derive expression (7.33) for maximum
thermal subsidence using the steady state temperature (7.25) T 0 (z) and the initial tem-
perature (7.26) T I (z) immediately after stretching.
Solution: From equation (7.69) we get

1
ˆ
ˆ
( m,0 − s )s T,max = m,0 αT a a T I (ˆz) − T 0 (ˆz) dˆz (7.73)
0
where the integral is
1
1
1/β 1
(β − 1)ˆzdˆz + (1 −ˆz) dˆz = 1 − (7.74)
0 1/β 2 β
which is equation (7.33).
Exercise 7.12 (Mathematical curiosity) Show that

∞
sin((2m + 1)π/β) 1 1
= 1 − . (7.75)
((2m + 1)π) 3 8β β
m=0
Hint: let the maximum thermal subsidence expressed by equation (7.70) be equal to the
maximum thermal subsidence given by equation (7.33).

7.9 Lithospheric stretching of finite duration

We have seen how the β-factor measures the amount of stretching of the lithosphere. The
length of a rectangular-shaped section increases by a factor β and the (vertical) depth thins
by a factor 1/β (see Figure 7.6). The McKenzie model in its simplest version approximates
the stretching by an instantaneous event, although the extension process may take several
tens of million years. We will now look at the McKenzie model for finite duration stretch-
ing, where a realistic strain rate gives the extension. Strain rate is a local property defined
at any point by
1 dl
G(t) = (7.76)
l dt
where l is a small (infinitesimal) distance at the point. The rate (7.76) becomes constant for
any vertical length l when the strain rate is independent of the vertical position (see Exer-
cise 7.15). We therefore have that G =−(1/z)(dz/dt), where a minus sign has been added
because dz/dt is negative during extension. The strain rate may be time dependent, and it
is normally different from zero only during rifting phases. This equation is straightforward
to integrate (see Exercise 6.5), and we get