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GEOS 419/519 Physics of the Earth Spring 2018
Problem Set: Extra credit
Due: May 2, 2018 (Wednesday), 11:59pm
Instructions: These two problems are extra credit for the homework.
Please be neat and organized in what you hand in. For example, it would be a good idea to put a box around your final answer, so that I can find it. I do not need to see all your scratch work. Once you have found a way to the answer, rewrite it in an orderly fashion so that I can follow the steps. When plots are requested, do not use hand-drawn plots. Use Matlab!
1. Convection speeds in the mantle. Given our formula for the plate speeds derived in class, and the following dimensions and material constants, deduce the expected plate speeds for whole-mantle driven convection. These will be equivalent to the half-spreading rates. The formula is:
Take D = 3000 km, g = 9.81 m/s2, ρo = 4000 kg/m3, α = 2 × 10−5/◦C, T = 1400 ◦C, κ = 1 mm2/s, and η = 1022 Pa − s. Give the rates in cm/year.
a) What is the plate speed today using this formula?
b) If in the early days of Earth history the mantle viscosity was 2 orders of magnitude smaller (sinceit was hotter then), what would the plates speeds have been then?
c) Given the reduction in viscosity, by what factor would the lithospheric plate thickness at the timeof subduction have been reduced from that at the present? (a factor of 2?, a factor of 100?, and factor of 3.6?...)
2. Time scale for isostatic response to deglaciation. Given the table of values of the elevations of ancient beach terraces vs. age shown below, deduce the time constant for the isostatic adjustment for ice unloading. This classic data set (the left two columns) comes from the beach terraces of the Angerman River, in Sweden, near the center of the late Pleistocene Fennoscandian ice sheet.
uplift since 8.5 ka (m)
time since 8.5 (ka)
a) Plot uplift since the load was removed (say 8.5 ka) vs time, and make it aestically pleasing.
b) Fit a best fitting curve through these points to deduce the time scale, also called the response time,for unloading. The curve fit should look like U = Umax × (1 − e−t/τ).
Hint: you can do the fitting and plotting in Matlab using the command cftool, and specifying a custom equation. Give an equation like y = a*(1-exp(-x/b)). File –>“Print to figure” will transfer the plot to a regular plot window which you can then add labels to.
c) Now estimate the upper mantle dynamic viscosity, η, given the following formula (which is different from the one given in class): η = ∆ρgRτ, where τ is the response time deduced from the data above, ∆ρ is the density difference between the ice and the mantle (call this 3400 − 900 = 2300 kg/m3) and R is the radius of the ice sheet (take this to be 1000 km).
d) Finally, from your best fitting curve for the total uplift vs time, determine the uplift rate vs. time, and calculate the expected uplift rate at the present. I want to see your equation first, and then the evaluation of the equation at 8.5 ka after the load was removed.