Problem 11
The exponential depth dependence of heat production is preferred because it is self-preserving upon erosion. However, many alternative models can be prescribed. Consider a two-layer model with \(H=\) \(H_{1}\) and \(k=k_{1}\) for \(0 \leq y \leq h_{1},\) and \(H=H_{2}\) and \(k=k_{2}\) for \(h_{1} \leq y \leq h_{2} .\) For \(y>h_{2}, H=0\) and the upward heat flux is \(q_{m} .\) Determine the surface heat flow and temperature at \(y=h_{2}\) for \(\rho_{1}=2600 \mathrm{~kg} \mathrm{~m}^{-3}\), \(\rho_{2}=3000 \mathrm{~kg} \mathrm{~m}^{-3}, k_{1}=k_{2}=2.4 \mathrm{~W} \mathrm{~m}^{-1} \mathrm{~K}^{-1}\) \(h_{1}=8 \mathrm{~km}, h_{2}=40 \mathrm{~km}, \rho_{1} H_{1}=2 \mu W \mathrm{~m}^{-3}, \rho_{2} H_{2}=\) \(0.36 \mu W \mathrm{~m}^{-3}, T_{0}=0^{\circ} \mathrm{C},\) and \(q_{m}=28 \mathrm{~mW} \mathrm{~m}^{-2}\).
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 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 48
Suppose that upon entering the Earth's atmosphere, the surface of a meteorite has been heated to the melting point and the molten material is carried away by the flow. It is of interest to calculate the rate at which melting removes material from the meteorite. For this purpose, consider the following problem. The surface of a semi-infinite half-space moves downward into the half-space with constant velocity \(V\), as indicated in Figure \(4.39 .\) The surface is always at the melting temperature \(T_{m},\) and melted material above the instantaneous surface is removed from the problem. Assume that the surface of the half-space is melted by a constant heat flux \(q_{m}\) into the half-space from above the surface. Assume also that far from the melting surface the temperature is \(T_{0}\); that is, \(T \rightarrow T_{0}\) as \(\zeta \rightarrow \infty\). Find the temperature distribution in the half-space as a function of time \(T(\zeta, t),\) and determine \(V\) in terms of \(q_{m}\) and the thermodynamic properties of the rock. Account for the latent heat \(L\) required to melt the material.
Problem 56
Assume that the continental lithosphere satisfies the half-space cooling model. If a continental region has an age of \(1.5 \times 10^{9}\) years, how much subsidence would have been expected to occur in the last 300 Ma? Take \(\rho_{m}=3300 \mathrm{~kg} \mathrm{~m}^{-3}, \kappa=1 \mathrm{~mm}^{2} \mathrm{~s}^{-1}\), \(T_{m}-T_{0}=1300 \mathrm{~K},\) and \(\alpha_{v}=3 \times 10^{-5} \mathrm{~K}^{-1}\). Assume that the subsiding lithosphere is being covered to sea level with sediments of density \(\rho_{s}=2500 \mathrm{~kg} \mathrm{~m}^{-3}\).
Problem 58
Assume that the continental crust and lithosphere have been stretched by a factor \(\alpha=2\). Taking \(h_{c c}=35 \mathrm{~km}, y_{L 0}=125 \mathrm{~km}, \rho_{m}=3300\) \(\begin{array}{lllll}\mathrm{kg} \mathrm{m}^{-3}, & \rho_{c c}=2750 & \mathrm{~kg} \mathrm{~m}^{-3}, & \rho_{s}=2550 & \mathrm{~kg} \mathrm{~m}^{-3}\end{array}\) \(\alpha_{v}=3 \times 10^{-5} \mathrm{~K}^{-1}\), and \(T_{1}-T_{0}=1300 \mathrm{~K},\) deter- mine the depth of the sedimentary basin. What is the depth of the sedimentary basin when the thermal lithosphere has thickened to its original thickness?
