Page 320 - Physical Principles of Sedimentary Basin Analysis

302 Flexure of the lithosphere

deflection is lost after a time span t
t 0 (k). We see that the rebound is slowest for a
periodic load with very long wavelength (k ≈ 0) where we have
η
t 0 ≈ . (9.85)
g
A very long wavelength corresponds to the case of isostatic subsidence where there is
almost no flexure of the plate. A plate with very short wavelengths (large wave num-
bers) will rebound faster since there are stronger elastic forces to recover the shape of
the plate to what it was before it was loaded. If the initial deflection can be decomposed
into a superposition of deflections for a large range of wave numbers, then the time needed
for rebound will be dominated by the shortest wave number (longest wavelength). The
dashpot parameter η can be estimated using uplift data for a deglaciation. For example,
4
η = 5.7 · 10 15 Pa s m −1 if the time period for post glacial rebound is t 0 = 3 · 10 years
and = 600 kg m −3 . The mantle length scale l is a difficult parameter, because it is not
3
2
a precise number. However, assuming that it is in the range of 10 km to 10 km gives a
corresponding viscosity in the range of 5.7 · 10 20 Pa s to 5.7 · 10 21 Pa s.
Note 9.6 Equation (9.81) is solved by separation of variables, where it is taken as a starting
point that the solution has the form

w(x, t) = Y(t) cos(kx). (9.86)
This solution is inserted into equation (9.81) when q(x) = 0, and we get

4
(Dk Y + gY + ηY)w 0 cos(kx) = 0 (9.87)
˙
where the dot above Y denotes time differentiation. We then get the following equation
for Y(t):
1 1
˙ 4
Y =− (Dk + g) Y =− Y. (9.88)
η t 0 (k)
An integration gives that Y(t) = Y(0)e −t/t 0 , where the value of Y at t = 0 is found from
the initial condition
w(x, t=0) = Y(0) cos(kx) = w e (x). (9.89)

We then get solution (9.83).

9.7 The equation for viscoelastic flexure of a plate
A lithospheric plate does not only behave elastically, but it may also deform viscously. The
latter property implies that the lithosphere becomes deformed permanently as opposed
to the elastic deformations that are fully recoverable. The strain of a viscoelastic plate
therefore has to be decomposed into elastic (recoverable) strain e and viscous (permanent)
strain v . The total deflection w is decomposed in the same way into an elastic (recoverable)
deflection w e and a viscous (permanent) deflection w v , and we have