The backward integration of the mantle's energy balance in section 3.7,

PHYSICS · Multi-part · Fermi University Physics · Earth, Climate & Energy Physics · The solid Earth: structure, fields and thermal history · The solid Earth: structure, fields and thermal history

The Solid Earth, Ch Dating, heat flow and the thermal history of the Earth

Problem

The backward integration of the mantle's energy balance in section 3.7, Cm dTm/dt=H(t)−Q(Tm)C_m\,\mathrm{d}T_m/\mathrm{d}t = H(t) - Q(T_m), gave a runaway rather than a gentle cooling history, because Q(Tm)∝(Tm−Ts)4/3e−Ea/3RTmQ(T_m) \propto (T_m-T_s)^{4/3}e^{-E_a/3RT_m} depends steeply on TmT_m. ((a)) Show that the logarithmic sensitivity of QQ to TmT_m is dln⁡QdTm=43(Tm−Ts)+Ea3RTm2.\frac{\mathrm{d}\ln Q}{\mathrm{d}T_m} = \frac{4}{3(T_m-T_s)} + \frac{E_a}{3RT_m^2}. ((b)) Evaluate this sensitivity at today's mantle temperature Tm,0=1623 KT_{m,0}=1623\,\mathrm{K}, for Ts=300 KT_s=300\,\mathrm{K} and Ea=300 kJ mol−1E_a = 300\,\mathrm{kJ\,mol^{-1}}. ((c)) For a present Urey ratio of 0.50.5, the text gives today's net cooling rate as 114 K Gyr−1114\,\mathrm{K\,Gyr^{-1}}. Using the sensitivity from part (b), estimate the fractional change in QQ implied by a single 100 Myr100\,\mathrm{Myr} backward step taken at that rate, and say whether a single such step looks safe. ((d)) The chapter states that the runaway is not an artefact of the 0.1 Myr0.1\,\mathrm{Myr} step used there. Using your answer to part (c), explain in one or two sentences why making the step size smaller could not remove it.

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