Problem 9
Assume that the radioactive elements in the Earth are uniformly distributed through a nearsurface layer. The surface heat flow is \(70 \mathrm{~mW} \mathrm{~m}^{-2}\), and there is no heat flow into the base of the layer. If \(k=4 \mathrm{~W} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, T_{0}=0^{\circ} \mathrm{C},\) and the temperature at the base of the layer is \(1200^{\circ} \mathrm{C}\), determine the thickness of the layer and the volumetric heat production.
Problem 15
Derive an expression for the temperature at the center of a planet of radius \(a\) with uniform density \(\rho\) and internal heat generation \(H\). Heat transfer in the planet is by conduction only in the lithosphere, which extends from \(r=b\) to \(r=a\). For \(0 \leq r \leq b\) heat transfer is by convection, which maintains the temperature gradient \(d T / d r\) constant at the adiabatic value \(-\Gamma\). The surface temperature is \(T_{0}\). To solve for \(T(r)\), you need to assume that \(T\) and the heat flux are continuous at \(r=b\).
Problem 16
It is assumed that a constant density planetary body of radius \(a\) has a core of radius \(b\). There is uniform heat production in the core but no heat production outside the core. Determine the temperature at the center of the body in terms of \(a, b, k, T_{0}\) (the surface temperature), and \(q_{0}\) (the surface heat flow).
Problem 27
Assume that the temperature effects of glaciations can be represented by a periodic surface temperature with a period of \(10^{4} \mathrm{yr}\). If it is desired to drill a hole to a depth that the temperature effect of the glaciations is \(5 \%\) of the surface value, how deep must the hole be drilled? Assume \(\kappa=1 \mathrm{~mm}^{2} \mathrm{~s}^{-1}\).
Problem 29
Estimate the effects of variations in bottom water temperature on measurements of oceanic heat flow by using the model of a semi-infinite half-space subjected to periodic surface temperature fluctuations. Such water temperature variations at a specific location on the ocean floor can be due to, for example, the transport of water with variable temperature past the site by deep ocean currents. Find the amplitude of water temperature variations that cause surface heat flux variations of \(40 \mathrm{~mW} \mathrm{~m}^{-2}\) above and below the mean on a time scale of 1 day. Assume that the thermal conductivity of sediments is \(0.8 \mathrm{~W} \mathrm{~m}^{-1} \mathrm{~K}^{-1}\) and the sediment thermal diffusivity is \(0.2 \mathrm{~mm}^{2} \mathrm{~s}^{-1}\).
Problem 37
A body of water at \(0^{\circ} \mathrm{C}\) is subjected to a constant surface temperature of \(-10^{\circ} \mathrm{C}\) for 10 days. How thick is the surface layer of ice? Use \(L=320 \mathrm{~kJ}\) \(\mathrm{kg}^{-1}, k=2 \mathrm{~J} \mathrm{~m}^{-1} \mathrm{~s}^{-1} \mathrm{~K}^{-1}, c=4 \mathrm{~kJ} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}, \rho=\) \(1000 \mathrm{~kg} \mathrm{~m}^{-3}\).
Problem 38
Scientists believe that early in its evolution, the Moon was covered by a magma ocean with a depth of \(50 \mathrm{~km}\). Assuming that the magma was at its melt temperature of \(1500 \mathrm{~K}\) and that the surface of the Moon was maintained at \(500 \mathrm{~K}\), how long did it take for the magma ocean to solidify if it was cooled from the surface? Take \(L=320 \mathrm{~kJ} \mathrm{~kg}^{-1}, \kappa=1 \mathrm{~mm}^{2}\) \(\mathrm{s}^{-1}\), and \(c=1 \mathrm{~kJ} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}\).
Problem 43
The mantle rocks of the asthenosphere from which the lithosphere forms are expected to contain a small amount of magma. If the mass fraction of magma is 0.05 , determine the depth of the lithosphere-asthenosphere boundary for oceanic lithosphere with an age of 60 Ma. Assume \(L=400 \mathrm{~kJ}\) \(\mathrm{kg}^{-1}, c=1 \mathrm{~kJ} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}, T_{m}=1600 \mathrm{~K}, T_{0}=275 \mathrm{~K},\) and \(\kappa=1 \mathrm{~mm}^{2} \mathrm{~s}^{-1}\).
