A study on the Fennoscandian post-glacial rebound as observed by present-day uplift rates and gravity field model GOCO02S

Abstract

Repeated absolute gravity measurements in Fennoscandia have revealed that the on-going post-glacial rebound can be regarded as a pure viscous flow of mantle mass of density 3390 kg/m³ towards the central part of the region caused by a gravity/uplift rate of −0.167 μGal/mm. Our model estimates the rebound induced rates of changes of surface gravity and geoid height to have peaks of −1.9 μGal/yr and 1.6 mm/yr, respectively, the former being consistent with absolute gravity observations. The correlation coefficient of the spherical harmonic representations of the geoid height and uplift rate for the spectral windows between degrees 10 and 70 is estimated to −0.99±0.006, and the maximum remaining land uplift is estimated to the order of 80 m. Both the (almost) linear increase of relaxation time with degree and the linear relation between geoid height and uplift rate support a model with mass flow in the major part of the mantle and disqualify the model with a flow in a thin channel below the crust. The mean viscosity of the flow in the central uplift region is estimated to 4×10²¹ Pa s.

Appendix

Let us assume that the elastic deformation of the crust of thickness D and density ρ _c is negligible in the vertical direction. Then the disturbing potential T _total at sea-level caused by the glacial depression of the crust is composed of the sum of the potential contributions from the deformations of the upper and lower surfaces (T _up and T _low ) by their depression d:

$$ T_{total} = - G\rho_{c}\iint_{\sigma} \int _{R - d}^{R} \frac {r^{2}dr}{l} d\sigma- G\Delta\rho \iint_{\sigma} \int_{R - D - d}^{R - D} \frac{r^{2}dr}{l} d\sigma= T_{up} + T_{low}. $$

(A.1)

Here G is the gravitational constant Δρ is the difference between the mantle minus crust density contrast, σ is the unit sphere, R is sea-level radius, D is crustal thickness, l is the distance between computational point and integration point. The first integral is the effect on the Earth’s potential due to the mass deficit between sea-level and the depressed crust. The second term is the effect due to the depressed bottom of the crust into the mantle. Expanding the inverse distance in an exterior type of Legendre series, one obtains for the two potential contributions:

$$\begin{aligned} T_{up} =& - G\rho_{c}\sum_{n = 0}^{\infty} \iint _{\sigma} \int_{R - d}^{R} \frac{r^{n + 2}}{R^{n + 1}}dr P_{n} ( \cos\psi )d\sigma \\ =& - G\rho _{c}R^{2}\sum_{n = 0}^{\infty} \frac{1}{n + 3}\iint_{\sigma} \biggl[ 1 - \biggl( 1 - \frac{d}{R} \biggr)^{n + 3} \biggr]P_{n} ( \cos\psi )d\sigma \\ \approx&- G\rho_{c}4\pi R\sum_{n = 0}^{\infty} \frac{d_{n}}{2n + 1} = - 4\pi G\rho_{c}R\sum_{n = 0}^{\infty} \frac{1}{2n + 1} \sum_{m = - n}^{n} d_{nm}Y_{nm}, \end{aligned}$$

(A.2)

and

$$\begin{aligned} T_{low} =& - G\Delta\rho\sum_{n = 0}^{\infty} \iint _{\sigma} \int_{R - D - d}^{R - D} \frac{r^{n + 2}}{R^{n + 1}}dr P_{n} ( \cos\psi )d\sigma= \\ =&- G\Delta\rho R^{2}\sum_{n = 0}^{\infty} \frac{1}{n + 3}\iint_{\sigma} \biggl[ \biggl( 1 - \frac{D}{R} \biggr)^{n + 3} - \biggl( 1 - \frac{D - d}{R} \biggr)^{n + 3} \biggr]P_{n} ( \cos\psi )d\sigma \\ \approx&- 4\pi G\Delta\rho R\sum_{n = 0}^{\infty} \frac{d_{n}}{2n + 1} = - 4\pi G\Delta\rho R\sum_{n = 0}^{\infty} \frac{1}{2n + 1} \sum_{m = - n}^{n} d_{nm}Y_{nm}. \end{aligned}$$

(A.3)

Summing up, one obtains the total disturbing potential at sea-level:

$$ T_{total} \approx- G\rho_{m}R\sum_{n = 0}^{\infty} \iint_{\sigma} dP_{n} ( \cos\psi )d\sigma= - 4\pi G \rho_{m}R\sum_{n = 0}^{\infty} \frac{1}{2n + 1} \sum_{m = - n}^{n} d_{nm}Y_{nm}. $$

(A.4)