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session1A

Session 1, Extras, part A: Cooling of the Earth

The (potential) temperature T of the Earth’s upper mantle today is about 1350oC1350oC , but heat released from accretion of the Earth and differentiation into a core-mantle layered system probably heated the early internal Earth to several 100K100K hotter than that. Since then, radio-active decay added (and still adds) another significant amount of heat, while the only way for the Earth to lose its heat is through its surface. A simple heat balance for the Earth states that the difference between the total radioactive heat source for the whole Earth, HH, and the total heat sink for the Earth, QQ, (surface heat flow) leads to cooling (or heating) of the Earth. Mathematically, this can be expressed as: CdTdt=H(t)−Q(t)CdTdt=H(t)−Q(t), with C=7×1027C=7×1027 JK−1JK−1 the Earth’s total heat capacity. TT, HH, and QQ generally are all functions of time tt.

A.1) The following table lists the four most important decay systems for the Earth’’s thermal history, using H=H0exp(−λt)H=H0exp⁡(−λt), with λλ related to the half-life of a decay system as λ=ln(2)/τλ=ln⁡(2)/τ:

Construct a finite difference code to calculate the thermal history of the Earth. Hint: t=0t=0 can be chosen to be either today (in which case time runs backward) or at the time of Earth’s accretion (so that time runs forward). Note that this forward or reverse sense of time has nothing to do with Forward or Backward Euler timestepping.

A.2) Plot the resulting temperature TT as a function of time. According to this (simple) model, did the Earth always cool down over time?

A.3) Test the code with a convergence test, and compare results with those of your colleagues.

A.4) The Urey ratio UrUr is defined as the present-day ratio of H/QH/Q, and is thought to be anywhere between Ur=0.2Ur=0.2 and 0.50.5. Vary UrUr by multiplying the total heat production with a constant prefactor, and explore how that affects the Earth’s thermal evolution. Plot multiple UrUr results in a single plot.

A.5) QQ is probably not constant, but mantle temperature dependent (e.g. a hotter, weaker mantle might lead to faster mantle convection and therefore faster cooling). Adjust QQ to be linearly proportional to TT. Note that this will make the governing equation a differential equation (since now dTdtdTdt is dependent on TT). Examine the different solution for Forward Euler, Backward Euler, and Crank-Nicholson timestepping.