Page 226 - Physical Principles of Sedimentary Basin Analysis

208 Subsidence
2
∂U ∂ U
− κ = 0. (7.38)
∂t ∂z 2
The initial condition for U(z, t) is the difference between the temperature immediately
after stretching and the steady state temperature,

⎧ z
⎨ T a β − 1 , 0 ≤ z < a/β
a
U(z, t = 0) = T I (z) − T 0 (z) =
(7.39)
z
⎩
T a 1 − a , a/β ≤ z < a.
Boundary conditions for U(z, t) are
U(z = 0, t) = 0 and U(z = a, t) = 0 (7.40)
because T 0 (z) = T I (z) for z = 0 and z = a. The solution of the temperature
equation (7.38) is simplified when it is turned into dimensionless form. The dimension-
ˆ
ˆ
less temperatures are T = T/T a and U = U/T a , because T a is a natural choice for
a characteristic temperature. The thickness of the lithosphere is the characteristic length
and the dimensionless z-coordinate becomes ˆz = z/a. The scaled version of the steady
ˆ
state temperature T 0 (z) is simply T (ˆz) =ˆz. The characteristic length a and the thermal
diffusivity gives the characteristic time
a 2
t 0 = . (7.41)
κ
ˆ
The dimensionless temperature equation for U becomes the parameterless equation
2 ˆ
ˆ
∂U ∂ U
− = 0 (7.42)
∂ ˆ t ∂ ˆz 2
when time is scaled as ˆ t = t/t 0 . The initial condition becomes simplified to

(β − 1)ˆz, 0 ≤ˆz < 1/β
ˆ
U(ˆz, t = 0) = (7.43)
(1 −ˆz), 1/β ≤ˆz < 1
while the boundary conditions are almost the same
ˆ
U(ˆz = 0, ˆ t) = 0 and U(ˆz = 1, ˆ t) = 0 (7.44)
ˆ
in dimensionless variables. The solution of the temperature equation (7.42) with the initial
condition (7.43) and the boundary conditions (7.44) is (as shown in Note 7.2)
∞
sin(nπ/β)
2
−(nπ) ˆ t
ˆ
U(ˆz, ˆ t) = 2β sin(nπ ˆz) e . (7.45)
(nπ) 2
n=1
2
ˆ
The transient solution U(ˆz, ˆ t) approaches zero because the factors e −(nπ) ˆ t approach zero
as time ˆ t goes to infinity. The full dimensionless solution of the temperature equation is
found by adding the transient solution to the steady state solution,
ˆ
T (ˆz, ˆ t) =ˆz + U(ˆz, ˆ t). (7.46)
ˆ