Past exam of the physics of the Earth as a Planet course of the University of Cambridge / 2014 / Paper 1 / 2 / c / Solution

Assume the original ice sheet had time to reach local isostatic compensation, melted rapidly relative to subsequent rebound, and left one exponentially relaxing depression mode. Neglect elastic layering and assign the whole present gravity deficit to that unrelaxed depression. Load balance givesThe supplied sheet-gravity approximation gives the residual amplitudeFrom $h(t)=h_0{e}^{-t/\tau }$,For an order-of-magnitude single-mode estimate, choose the ice-sheet diameter as wavelength, $\lambda \simeq 2R=6000\mathrm{km}$. Taking $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, the more accurate specified relation then givesA circular load has a spectrum, not one uniquely specified wavelength. Choosing $\lambda \sim R$ instead gives about $1.3\times 10^{21}\mathrm{Pa}\mathrm{s}$; another wavelength convention rescales the answer linearly. The robust inference is viscosity of order $10^{21}\mathrm{Pa}\mathrm{s}$ within this simple model. A realistic estimate would also model gradual melting, other gravity anomalies, density interfaces, lithospheric flexure and depth-dependent mantle viscosity.