 Solution

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

Subtract the background hydrostatic pressure. A relief amplitude

h

p^{'} \sim ρ g h

and horizontal gradient

ρ g h / λ

. Incompressibility makes the disturbance penetrate a depth of order its horizontal scale. For sinusoidal viscous rebound, with relaxation speed

v \sim h / τ

, the viscous force density is

η v / λ^{2}

. Balancing gives

η \frac{h}{τ λ ^{2}} \sim \frac{ρ g h}{λ}, τ \sim \frac{η}{ρ g λ} .

This is a small-amplitude viscous half-space estimate; numerical factors from the sinusoidal geometry are absent from a scaling argument.

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