Page 326 - Physical Principles of Sedimentary Basin Analysis

308 Flexure of the lithosphere

10

8

log 10 (t 0 /t e ) [−] 6 4

2

0
−3 −2 −1 0 1 2 3
log (λ/λ ) [−]
10
c
Figure 9.13. The scaled time constant t 0 /t e as a function of the scaled wavelength λ/λ c .
increases with decreasing wavelength, see Figure 9.13. For example, a wavelength that is
4
a factor 1/10 less than λ c will last 10 longer than t e .
The deflection (9.122) can be decomposed into an elastic part (w e ) and a viscous part
(w v ) as shown in Section 9.8, and we get

q 0 t e −t/t 0 (k)
w e (x, t) = e cos(kx) (9.125)
g t 0 (k)
q 0
−t/t 0 (k)
w v (x, t) = 1 − e cos(kx). (9.126)
g
The elastic part of the deflection is initially equal to the flexure of a purely elastic plate,
and it then decays to zero with the half-life t 0 ln2. The viscous part is initially zero, but it
increases with time and approaches isostatic subsidence as time goes far beyond t 0 .The
following table summarizes some of the wave-dependent properties of the solution:

Wavelength Characteristic time Initial deflection
4
λ
λ c t 0 ≈ (λ c /λ) t e w e
w iso
λ
λ c t 0 ≈ t e w e ≈ w iso

where isostatic subsidence is w iso = (q 0 / g) cos(kx).
A simple estimate for the time constant t e is 0.3 Ma, using that E = 100 GPa and
μ = 10 24 Pa s. This implies that loads with a typical wavelength similar to the critical
wavelength (like volcanic islands) should not remain partly supported by the elastic forces
of the plate for several tens of Ma. But there are loads, like for example the Emperor
Seamounts, that seems to have been supported by the plate for more than 60 Ma. Such