Page 241 - Physical Principles of Sedimentary Basin Analysis

7.10 Finite duration stretching and temperature 223

Rift duration 5 Ma Rift duration 50 Ma
0 0

20 20
40 Ma
4 Ma instantaneous
40 instantaneous 40
depth [km] 60 5 Ma depth [km] 60 50 Ma

80 80 30 Ma
3 Ma 20 Ma
2 Ma
100 1 Ma 100 10 Ma
0 Ma 0 Ma
120 120
0 500 1000 1500 0 500 1000 1500
temperature [°C] temperature [°C]
(a) (b)
Figure 7.18. The temperature of the lithosphere is shown during finite duration rifting, where the
final β-factor is 2. Left: rifting during 5 Ma. Right: rifting during 50 Ma.
2
∂T ∂T ∂ T
+ Gz − κ = 0 (7.104)
∂t ∂z ∂z 2
in the vertical direction, where the vertical velocity is v z = Gz. The temperature
equation (7.104) does not only apply at x = 0, where v x = 0. It applies for x = 0
too, as long as the initial isotherms are horizontal and the vertical velocity is independent
of the lateral position. It is not straightforward to give an exact solution to this equa-
tion, because of the z-dependent velocity term, see Jarvis and McKenzie (1980). But a
numerical (finite-difference) solution of equation (7.104) is a simple means to explore
the temperature during finite duration rifting. Figures 7.18a and 7.18b show numerical
◦
solutions where the boundary conditions are 0 C at the surface and 1300 C at the base
◦
of the lithosphere. Both figures show stretching with a final β-factor 2. Figure 7.18a
shows stretching for the time interval 5 Ma, which is sufficiently short for the upper
bound (7.103) to apply, and instantaneous stretching is a good approximation. Figure 7.18b
shows stretching over a time interval 50 Ma that is longer than the upper bound (7.103),
and heat conduction becomes important. Instantaneous stretching cannot be assumed in
this case.
Note 7.7 The Peclet number can be used to decide when flow is sufficiently fast for
heat convection to dominate over heat conduction. The Pe-number was introduced in
Section 6.11, where it was applied to processes that had reached a stationary state. The
stretching of the lithosphere will not last long enough to reach a steady state. Neverthe-
less, the Pe-number can be used to check whether convection would have dominated over
heat conduction if stretching had lasted for a sufficiently long time for the temperature
to be nearly stationary. The Pe-number for vertical flow is Pe = v z l m /κ, where κ is the