Page 225 - Physical Principles of Sedimentary Basin Analysis

7.6 The thermal transient of the McKenzie model 207

Exercise 7.7 The initial subsidence caused by crustal thinning is s I , the maximum subsi-
dence is s max and the maximum thermal subsidence is s T,max .
(a) Show that s I > s T,max implies that s max > 2 s T,max .
(b) Show that s max > 2 s T,max implies that
c ρ m,0
> αT a . (7.35)
a ρ m,0 − ρ c,0
(c) Compare condition (7.30) for when the initial subsidence is larger than the thermal
subsidence with condition (7.35) for when the initial subsidence is positive.

7.6 The thermal transient of the McKenzie model

Instantaneous and uniform stretching of the lithosphere leads to upwelling of the hot
asthenosphere of temperature T a and heating of the lithosphere by the mantle and crust that
move upwards. The entire lithosphere gets hotter as shown in Figure 7.10. The temperature
after stretching T I (z) then decays to the steady state temperature T 0 (z) by conductive cool-
ing. The temperature equation is now solved for the transient cooling back to the steady
state temperature. The temperature at any time after stretching is written

T (z, t) = T 0 (z) + U(z, t), (7.36)
where U(z, t) is the thermal transient that decays to zero, and where the steady state
temperature is T 0 (z) = T a z/a. The temperature equation for conductive cooling is
2
∂T ∂ T
− κ = 0 (7.37)
∂t ∂z 2
as shown in Section 6.1. When the temperature (7.36) is inserted into the temperature
equation (7.37) we get that U is also a solution of the temperature equation,

T m
T

T (z)
I
T (z)
0
a/ β

a
z
Figure 7.10. The steady state temperature T 0 (z) and the temperature T I (z) immediately after
instantaneous and uniform stretching.