Page 237 - Physical Principles of Sedimentary Basin Analysis

7.9 Lithospheric stretching of finite duration 219

way to do modeling with a laterally varying strain rate is to approximate the strain rate as a
piecewise constant. The lithosphere is then made discrete using columns, where each one
has its own (time-dependent) strain rate.
An x-dependent strain rate implies that the stretching factor β also becomes x-
dependent. In a Lagrangian coordinate frame (that follows the lateral movements) both
the strain rate and the β-factor are only dependent on t. We have that G = (1/β)(Dβ/dt)
in a Lagrangian coordinate frame. When the time derivative is brought from the Lagrangian
frame to the Euler coordinate x we get
1 Dβ 1 ∂β ∂β ∂x

G = = + (7.90)
β dt β ∂t ∂x ∂t
where ∂x/∂t = v x . From mass conservation (7.79)wealsohavethat G = ∂v x /∂x, which
gives the following equation for the β-factor:
∂β ∂β ∂v x
+ v x − β = 0. (7.91)
∂t ∂x ∂x
More complicated flow patterns result if the strain rate is allowed to vary with both the
x- and the z-coordinates.

Exercise 7.13 The upper and brittle part of the lithosphere may deform by normal faulting
and rotation of fault blocks during extension. Figure 7.16 shows how the “domino-type” of
fault block rotation accommodates the stretching. We notice that large displacement along
a few faults is the same as little displacement along many faults.
(a) Show that the β-factor is
1
β = (7.92)
sin φ
when the blocks make an angle φ with the horizontal.
◦
(b) What is the angle φ when β is 2? (Answer: 30 )
(c) Assume that the fault blocks rotate to an angle φ 1 during a first rift phase and then
further to an angle φ 2 <φ 1 during a second rift phase. (The angle φ decreases from π/2

φ

Δx Δl
(a) (b) (c)
Figure 7.16. (a) Normal faulting. (b) Domino style rotation of the fault blocks. (c) A large number of
smaller faults makes the displacement at each fault less.