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 9
Determine the rate at which magma flows up a twodimensional channel of width \(d\) under the buoyant pressure gradient \(-\left(\rho_{s}-\rho_{l}\right) g\). Assume laminar flow.
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?
Problem 36
Apply the two-dimensional boundary-layer model for heated-from-below convection to the entire man- tle. Calculate the mean surface heat flux, the mean horizontal velocity, and the mean surface thermal boundary-layer thickness. Assume \(T_{1}-T_{0}=\) \(3000 \mathrm{~K}, b=2880 \mathrm{~km}, k=4 \mathrm{~W} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \kappa=\) \(1 \mathrm{~mm}^{2} \mathrm{~s}^{-1}, \alpha_{v}=3 \times 10^{-5} \mathrm{~K}^{-1}, g=10 \mathrm{~m} \mathrm{~s}^{-2},\) and \(\rho_{0}=4000 \mathrm{~kg} \mathrm{~m}^{-3}\). A MATLAB solution to this problem is provided in Appendix \(D\).
