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ID: past-exam-of-the-physics-of-the-earth-as-a-planet-course-of-the-university-of-cambridge/2014/paper-1/2/c/solution

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

Codex  0 2026-10-20

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 gives

h_{0} ≃ \frac{ρ _{i}}{ρ _{m}} H_{i} = 900 m .

(1)

The supplied sheet-gravity approximation gives the residual amplitude

h (t) = \frac{∣Δ g ∣}{2 π G ρ _{m}} = \frac{2.0 \times 1 0 ^{- 4}}{2 π ( 6.67 \times 1 0 ^{- 11} ) ( 3000 )} ≃ 159 m .

(2)

From

h (t) = h_{0} e^{- t / τ}

τ = \frac{1 0 ^{4} yr}{ln ( 900/159 )} ≃ 5.77 \times 1 0^{3} yr .

(3)

For an order-of-magnitude single-mode estimate, choose the ice-sheet diameter as wavelength,

λ ≃ 2 R = 6000 km

. Taking

g = 9.81 m s^{- 2}

, the more accurate specified relation then gives

η = \frac{ρ _{m} g λ τ}{4 π} ≃ 2.6 \times 1 0^{21} Pa s .

(4)

A circular load has a spectrum, not one uniquely specified wavelength. Choosing

λ \sim R

instead gives about

1.3 \times 1 0^{21} Pa s

; another wavelength convention rescales the answer linearly. The robust inference is viscosity of order $1 0^{21} Pa s$ within this simple model. A realistic estimate would also model gradual melting, other gravity anomalies, density interfaces, lithospheric flexure and depth-dependent mantle viscosity.

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