Problem 5
For an asthenosphere with a viscosity \(\mu=4 \times 10^{19} \mathrm{~Pa} \mathrm{~s}\) and a thickness \(h=200 \mathrm{~km},\) what is the shear stress on the base of the lithosphere if there is no counterflow \((\partial p / \partial x=0) ?\) Assume \(u_{0}=\) \(50 \mathrm{~mm} \mathrm{yr}^{-1}\) and that the base of the asthenosphere has zero velocity.
Problem 11
Show that the constant of integration \(A\) in the above postglacial rebound solution is given by $$A=-\left(\frac{\lambda}{2 \pi}\right)^{2} \frac{\rho g W_{m 0}}{2 \mu} e^{-t / \tau_{r}} . \quad(6-106)$$ Quantitative information on the rate of postglacial rebound can be obtained from elevated beach terraces. Wave action over a period of time erodes a beach to sea level. If sea level drops or if the land surface is elevated, a fossil beach terrace is created, as shown in Figure \(6-15 .\) The age of a fossil beach can be obtained by radioactive dating using carbon 14 in shells and driftwood. The elevations of a series of dated beach ter- races at the mouth of the Angerman River in Sweden are given in Figure \(6-16 .\) The elevations of these beach terraces are attributed to the postglacial rebound of Scandinavia since the melting of the ice sheet. The elevations have been corrected for changes in sea level. The uplift of the beach terraces is compared with the exponential time dependence given in Equation \((6-104)\). We assume that uplift began 10,000 years ago so that \(t\) is measured forward from that time to the present.
Problem 14
Derive expressions for the lifting torques on the top and bottom of a slab descending into the mantle with speed \(U\) at a dip angle of \(60^{\circ}\).
Problem 16
Suppose that the \(660-\mathrm{km}\) density discontinuity in the mantle corresponds to a compositional change with lighter rocks lying above more dense ones. Estimate the minimum decay time for a disturbance to this boundary. Assume \(\rho=\) \(4000 \mathrm{~kg} \mathrm{~m}^{-3}, \Delta \rho=100 \mathrm{~kg} \mathrm{~m}^{-3},\) and \(\mu=10^{21} \mathrm{~Pa} \mathrm{~s}\).
Problem 18
In the examples of folding just considered we assumed that the competent rock adhered to the incompetent rock. If the layers are free to slip, show that the wavelength of the most rapidly growing disturbance in an elastic layer of rock contained between two semi-infinite viscous fluids is given by $$\lambda=\pi h\left[E / \sigma\left(1-v^{2}\right)\right]^{1 / 2}$$ The free slip condition is equivalent to a zero shear stress condition at the boundaries of the elastic layer.
Problem 35
Suppose that convection extends through the entire mantle and that \(10 \%\) of the mean surface heat flow originates in the core. If the surface thermal boundary layer and the boundary layer at the core-mantle interface have equal thicknesses, how does the temperature rise across the lower mantle boundary layer compare with the temperature increase across the surface thermal boundary layer?
