Homework #1 Humphrey Geology 4880 Fall 2015

These are due next Tuesday (Sept 15). Answers only need to be (or for that matter can only be) approximate. There are 5 questions, 1 and 2 are very straightforward. 4 is an exercise in neat and tidy graphing. You will need to have access to the internet for several of these questions. This homework set will take several hours at least(!), make sure you have time. Note, Questions #1 and #3 depend on the lectures given on Tuesday and Thursday, therefore you may not be able to complete these two until after Thursday.

Notes:

Try to couch your answers at the appropriate level of accuracy that is implied by the question. You are free to use data from any source or to make reasonable assumptions, but state what you assume, or show the data you use and give the source of the data. Be aware that data and ideas from the Web are of highly variable quality. To do the homework you will have to take 4 steps: figure out how to do the problem, decide on the data you need (if any), collect the necessary information, and finally produce the answer.

Your work must be neat, legible, and organized. I am not willing to wade through a mess and I will just give it back to you to re-do. You can work with others, but be aware that you will have to do similar questions on exams and quizzes, by yourself.

Often the homework will include one or more questions that are quite hard. Usually the last one(s) are virtually impossible to get correct, however I expect an attempt, since I want to see how you approach a difficult question.

1 On the slopes above I-80, (East up to the Lincoln Memorial) are limestone blocks sitting on the steep side hills, which are composed of the underlying Casper sandstone.  We will assume that one of these blocks is on a steeper slope than really exists (they are actually only about 25 degrees).  To be precise, assume a limestone block that is 2m thick, 20m long and 20m wide is lying on the sandstone which is at an angle of 35 degrees. The block lies at this angle 100m above (vertically) the road, and it is a simple straight 35 degree slope down to the road. The effective angle of friction between the limestone and the sandstone is 30 degrees. The density of the limestone is 2500kg m^-3. (since everything you need is included, your answer to this question should have little error)

a) What would the factor of safety be, if the cohesion is 0?

b) Calculate a minimum value of cohesion, for the block to be stable?

c) If the mass fails (and cohesion goes to 0), what velocity will the block reach by the time it hits the road?

d) Will the chicken (block) cross the road (5 lane highway, plus shoulder and median)?

2 Rivers currently carry about 10¹⁰ cubic meters of soil and eroded rock to the sea each year, throughout the world.

a) Roughly how long will it take, at the present rate, for all the continents to shrink by an average of 1 meter in elevation (if all other processes, such as tectonics, are ignored)

b) This implies a world averaged erosion rate of what (in mm per year)?

c) If the Platte River basin above the Seminoe reservoir (see Google Earth) is lowering at the average rate of 0.2 mm per year from erosion, what is the approximate total sediment and dissolved load (kg per year) entering the reservoir? You can assume that all the material leaves the upper basin via the Platte (a fairly good assumption), but you will need to find the drainage area of the basin above the reservoir. (You may find data at the USGS site water.usgs.gov to be useful, although it is not an easy site to navigate). Note you will have to convert from volume eroded to mass, which will require a density for the material removed by surface lowering.

d) (Tricky) Most of the eroded material will be weathered rock and soil from the surface Should you use the density of unweathered rock, or the lesser density of soil, or what density(?) in part c?

3 We argued in class that mountains are mostly held together by friction.

a) Check out this idea by finding the approximate average slope of the Grand Teton (south side gives longest steep slopes), Everest (east face is probably the longest and steepest), Denali (see if you can find the longest steepest face) and a typical volcano such as Rainer or Fuji. (Google Earth is a good tool for this). (Don’t look for the steepest local slope, but try to find the steepest slopes that are several 1000 ft high). How do these crude estimates of slopes compare with the typical friction angle of rocks of 30-35degrees.

b) See if you can find slopes for a steep crater on the moon.

c) (Hard) Although for solid to solid friction, the angle of repose is independent of gravity, it is not so obvious that this is true for the complex interactions of a granular pile of material. Question: Should the slope angle of a sand pile or a talus change with changing gravity?

4. Many processes in geomorphology are non-linear in the sense that the results are not linearly related to some of the controlling variables. We have already talked about exponential decay. I want you to get more familiar with some of the typical geomorphic non-linearities, which are often logarithmic, exponential or power-law relationships in time or space.

a) Plot the following 4 functions of X and Y, with X going from about 0 to 10 (use more than 10 steps, 100 would be good):

linear Y = X, logarithmic Y = 10* log₁₀ X, exponential Y = e^(X/4.4), power law Y = X² /10

(linear, logarithmic, exponential, power law (square))

Plot 3 different graphs (with the 4 functions on each graph) as follows:

1- a graph with linear x-y axes that go from 0 to 10,

2- a graph with horizontal linear scale (0-10) and a vertical logarithmic scale (0-1), and

3- a log-log plot with a scale from 0-1 on both axes

(in all cases x will go from about 0 to near 10, however the values of y will not all plot on some graphs).

Use different colors or symbols for each curve, and label them, either by hand or have the computer do it. The purpose of this question is to get you to think about non-linear relations, but also to get you to figure out how to plot with various axes. I recommend either MATLAB or EXCELL (or some other computer or spreadsheet program).

5 A major jargon phrase in Geomorphology is “Magnitude and Frequency.” It refers to the idea that huge, infrequent events may dominate the evolution of a landscape, or conversely that small very frequent events may dominate. To get you thinking concretely, contemplate your possible death by infrequent, but large magnitude events:

The Cretaceous probably ended with a major asteroid impact on earth. Large (10km. diameter or more) impacts appear to occur randomly, but with an average time spacing of order 75 million years. Most of the larger life forms all over the globe suffer near complete mortality rates as a result of such an impact. Meteor impacts are an extreme example of a process dominated by high magnitude, not high frequency events.

a) What are the odds, or more precisely the probability, of a catastrophic impact somewhere on earth in your life time? (that would probably end your lifetime)

b) Since we are now talking about Landslides, which is more likely: that you will die from a landslide, or that you will die (with the rest of us) by an asteroid impact? (Data on landslide deaths in the US can be found on the Web, or I found one number in ‘Environmental Geology” by Keller, in the library.)

c) (Hard) A good example of a process that is dominated by Frequency is the background chemical erosion of the near surface rocks of the world. If you refer back to question 2c, not all the 0.2mm of yearly erosion is physical, some is chemical. Some of the chemical erosion products leave the basin dissolved in the water, not as particulates. If 15% percent of the erosion of the Platte river basin is by dissolution and by solution of chemical weathering products, what is the dissolved load concentration in the water in the Seminoe reservoir (in milligrams per liter)? (Note this a ‘back of the envelope’ type of calculation, since chemical weathering often produces products that weigh more or less than the original rocks, [however, pure dissolution would lead to an accurate answer]. Here we ignore the chemical details.)