Page 256 - Physical Principles of Sedimentary Basin Analysis

238 Subsidence

(a) Find an expression for the step in a Newton solver of equation (7.138).
(b) Use (a) to make a numerical implementation of function X = X(p, T ) in your favorite
programming language.
Solution: Equation (7.138) can be rewritten as F(U) = 0 where

F(U) = p − a 1 U − a 2 exp(a 3 U) and F (U) =−a 1 − a 2 a 3 exp(a 3 U) (7.154)
where U = T s − 1100, a 1 = 1/136, a 2 = 4.968 × 10 −4 and a 3 = 1.2 × 10 −2 .If U is
a reasonable guess for a solution then we search a step U such that F(U + U) = 0.
The latter expression gives to first order that F(U) + F (U) U = 0 and the Newton step

becomes U =−F(U)/F (U). A guess for U gives a step U and then an improved
U ← U + U, which is used to make a new step and so on.
Exercise 7.25 Show that adiabat (7.152) can be approximated by equation (7.153).
Hint: use that ln(1 + x) ≈ x for |x|
1.
Exercise 7.26 Assume that we have observations of underplating that suggest a total melt
of thickness M = 2 km has been generated under lithospheric extension. How large a β-
factor is needed to produce this melt thickness when we apply the parameters in Table 7.1
(Hint: Figure 7.24.)

Exercise 7.27 The mass fraction of melt is denoted by X.
(a) Show that the volume fraction of melt is related to the mass fraction by

X s
φ = . (7.155)
(1 − X) m + X s
(b) Show the inverse – that the mass fraction of melt is related to the volume fraction as

φ m
X = . (7.156)
(1 − φ) s + φ m
Solution: (a) The volumes of melt and solid are V m = XM/ m and V s = (1 − X)M/ s ,
respectively, where M is the total mass. The volume fraction φ = V m /(V m + V s ) then
becomes (7.155).
(b) The relation (7.155) is straightforward to invert to the relation (7.156). Alternatively,
the mass of melt and solid are M m = φV m and M s = (1 − φ)V s , respectively, when V
is the total volume. The mass fraction X = M m /(M m + M s ) then becomes (7.156).

7.13 Thermal subsidence of the oceanic lithosphere
Oceanic lithospheric plates are created at mid-ocean ridges, where the upwelling hot
asthenosphere cools and forms new plates. The asthenosphere moves the plates apart from
the ridge at the same rate as they are produced. The cooling of a plate becomes a 1D (verti-
cal) problem for a column of rock that follows the movement of the plate. See Figure 7.27
where the same column of rock is shown at two different times. The temperature in a col-
umn at the ridge, where the asthenosphere reaches the surface, can be assumed to have