Page 247 - Physical Principles of Sedimentary Basin Analysis

7.11 Lithospheric extension, phase changes and subsidence/uplift 229

and show that equations (7.106) and (7.107) give the subsidence/uplift caused by a change
in the average crust or mantle density, respectively.

Exercise 7.20 How large a reduction in the average mantle density is needed to create
2 km of uplift when the initial average mantle density is m = 3300 kg m −3 , the thickness
of the lithospheric mantle is a = 150 km and the uplift takes place in the air (above sea
level)?

Exercise 7.21
(a) Verify expressions (7.112)to(7.114)for β 1 , z bot , z top and β 2 .
(b) Verify the density difference (7.116)to(7.118).
Exercise 7.22
(a) Show that a change A r in the gradient of the phase boundary makes a change
( x,0 − x,0 ) (T 0 − T r )
s =− A r (7.121)
( m,ref − m ) (A r − A 0 β) 2
in the subsidence. Hint: use the chain rule of differentiation
ds d m dz 2
s = A r . (7.122)
d m dz 2 dA r
(b) What is s when A r = 0.01? Use the numbers in the text above for the other
parameters.

Exercise 7.23 Radioactive heat production makes the lithosphere hotter than it otherwise
would have been. Lithospheric extension thins the heat producing crust, and the litho-
spheric mantle becomes less hot when it returns to a steady state after a sufficiently long
time. The reduction in the steady-state mantle temperature implies thermal contraction and
subsidence. This exercise looks at the amount of subsidence one could expect.
In case the radioactive heat production should disappear the corresponding increase in
the average mantle density from thermal contraction becomes
1 z a
ρ m = m α T (z) dz (7.123)
z a 0
where T (z) is the difference in the geotherms for non-zero and zero radioactive heat pro-
duction. The subsidence from the increase in density is s = ρ m z a /( m − w ).
(a) Use the geotherm (6.59)–(6.60) and show that the integral over the temperature
difference is
a
S 0 z 2
1 1 1
m
T (z) dz = − z m + z a + z 1 (7.124)
0 λ c 6 4 4
where
λ m
z 1 = z m + T a − T m (7.125)
λ c
is the depth to the asthenosphere for the geotherm with heat production. The geotherm
without heat production has the larger depth z a to the asthenosphere. The asthenosphere is
assumed to be isothermal.