Page 240 - Physical Principles of Sedimentary Basin Analysis

222 Subsidence

where the mantle velocity is given by equation (7.85)as v = (Gx, −Gz), The streamlines
of mantle flow can be used to solve the convective temperature equation, because the tem-

perature follows the movement of the mantle. The temperature is initially T (z) = T a z/l m
at t = 0, where T a is the temperature at the depth l m at the base of the lithosphere. In
order to find the temperature at position (x, z) at the time t, we have to follow a streamline
backwards in time until t = 0, and see what the temperature was at the initial position.
From the streamline solution (7.84) it follows that a particle at position (x, z) at time t had
the initial vertical position z 0 = ze Gt (at time t = 0). The initial temperature at position z 0
(at time t = 0) is T = (T a /l m ) z 0 , and the temperature (x, z) at time t is therefore
Gt
T a ze Gt
, ze ≤ l m
T (x, z, t) = l m (7.100)
T a , ze Gt > l m .
It is straightforward to verify the solution by inserting it into the temperature
equation (7.99). (See Exercise 7.18.) The factor β = z exp(Gt) is also identified, and
the temperature (7.100) is therefore equal to the temperature (7.26) for instantaneous
stretching.
The assumption that convective heat flow dominates over heat conduction during stretch-
ing is important, because the flow of mantle and crust can then be approximated by
instantaneous stretching. How rapid stretching then has to be is estimated using the half-life
for decay of thermal transients by heat conduction. The half-life is approximated by
ln2l 0 2
t s = (7.101)
2
π κ
where l 0 is the characteristic length scale, as shown by equation (7.49). The characteristic
length scale is now taken to be l 0 = l m /β, which is the depth to the asthenosphere after
instantaneous stretching by a factor β. The duration of stretching has to be shorter than
this half-life for convective heat flow to dominate. We therefore have that the duration of
stretching must be shorter than
ln2l 2 m t 1/2
t s = = (7.102)
2
π κβ 2 β 2
where t 1/2 is the half-life for the length scale of the entire lithosphere. A lithospheric thick-
2 −1
ness l m = 120 km and a thermal diffusivity κ = 1 · 10 −6 m s gives that t 1/2 = 32 Ma.
The limitation of the duration of stretching can therefore be estimated as
32 Ma
t s < . (7.103)
β 2
2
Jarvis and McKenzie (1980), who first made such an estimate, suggested t s < 60 Ma/β .
Heat conduction cannot be ignored if stretching is over a time span of several tens of
million years or more. The convection–conduction equation for the temperature must then
be solved. This equation is