Page 234 - Physical Principles of Sedimentary Basin Analysis

216 Subsidence

t

z(t) = z 0 exp − G(t ) dt . (7.77)
0
The distance z(t) is the depth of a point in the lithosphere during stretching that had the
initial depth z 0 before stretching started at t = 0. The vertical velocity at the depth z during
stretching becomes
dz
v z = =−Gz (7.78)
dt
when the vertical velocity at the surface (z = 0) is set to zero. Mass conservation, in case
of a constant density, requires that ∇· v = 0, where v = (v x ,v z ). We therefore have

∂v x ∂v z
=− = G (7.79)
∂x ∂z
and the lateral velocity becomes after integration
dx
v x = = G (x − x ref ) (7.80)
dt
where x ref is an arbitrary reference position where the lateral velocity is zero. Integration
of the velocity (7.80) in the same way as shown in Exercise (6.5) gives the lateral position
t

x(t) = x ref + (x 0 − x ref ) exp G(t ) dt (7.81)
0
where x 0 is the initial position at t = 0. The path

r(t) = x(t), z(t) (7.82)

tells where a particle (point in the lithosphere) moves during stretching. The path is also a
streamline, when the strain rate G is constant and the velocity field is independent of time.
It is then a solution of the equations for streamlines
dr
= v(r) (7.83)
dt
and the position vector r(t) becomes

Gt
(x(t), z(t)) = (x 0 e , z 0 e −Gt ) (7.84)
where the initial position at t = 0is (x 0 , z 0 ). The corresponding velocity is

(v x ,v z ) = (Gx, −Gz) (7.85)

when the velocity is measured relative to the origin. Figures 7.14a and 7.14b show the
velocity field and the streamlines, respectively, as dimensionless quantities, where the
x
dimensionless coordinates are ˆ = x/l x and ˆz = z/l z , where 2l x is the lateral extent of
the rectangle and l z is the depth. Time is scaled with t 0 = 1/G, which is the characteristic
time for the stretching process.