Exercise 10 - Exploring power law and exponential functions:

1) The objective of this first exercise is to see how well a stream profile gradients fit a non-linear relationship. Follow these steps.

a) Choose a section of drainage profile to look at in Google Earth. You can use topographic maps or other sources of information, but Google Earth measurement tools allow you do this fairly easily. Using the profile tool makes this exercise especially easy. Make sure you chose your points carefully, down in the lowest part of the valley. Since most of the change occurs in the headwater region (often mountainous) that is where you should focus. You will need scale information, and should hand in a copy of the image you use. If you can't find coverage or an area, come find me.

b) Derive x, y data where x = map distance from the highest point on the profile section as measured along the drainage path, and y = elevation. Get at least 20 or so x, y points.

c) Insert these numbers into a Excel Spread sheet and then create a simple profile scatter plot of x versus y. Hand in a copy of that plot.

d) Manipulate your date if necessary and find the best fit mathematical relationship. One form you can try is:

ln y = ln (y(at x=o)) - lambda * x .

Take the natural log of both x and y in order to make this transformation. In this case lambda is not a half life decay rate, but an elevation decay rate. Remember that Excel has a number of best-fit options. Hand in at least two plots of different best-fit curves or lines for the same data.

d) Compare the various attempts at a best-fit description and discuss the results from a geologic and process perspective in a couple of sentences.

2) This exercise is designed to allow you to see a very interesting implication of the equation that models the thermal subsidence of oceanic crust with time. Complete the following using an Excel sheet. See your lecture notes and the reading for support material.

a) Model the profile elevation relative to the ridge crest in increments 1 km away from the spreading center for a spreading ridge that spreads at 2 cms/year for 20 million years using the formulas and values given in Turcotte. x should be your distance away from the ridge and y is the elevation relative to the ridge crest. Remember the admonition in the lecture notes - make sure your units are all the same. I would suggest Kelvin, kilograms, meters, and years. Do the same for a spreading ridge that spreads at 3 cms per year. If you set it up right, all you will need to do is change the cell with the spreading rate in it from 2 to 3 and the change will propagate through the sheet. Plot the two profiles together and compare. Hand in that comparative plot. Please also hand in the excel sheet so that I will have a better chance of seeing where you went wrong if you did go wrong. See the diagram below as an example of how the sheet might be arranged.

b) Compute the average rate of vertical subsidence relative to the ridge crest in cm/yr for the same 1 km increments for your 2 cms/year case. Basically, it would be the difference in elevation between adjacent1 km increment divided by the amount of time that it would take the crustal material to traverse that 1 km distance (you may need to read that sentence 2 or 3 times). For example, 1 km away at 2 cms per year is equal to 1000 m divided by .02 m per year or 50,000 years. What is the difference in elevation for that time?

c) Consider a transform fault fracture zone boundary for a ridge spreading at 2 cms/yr where crust different by 2 Ma is juxtaposed. The distance between the two would be 2,000,000 years multiplied by .02 m per year or 40,000 m or 40 km. On both sides of the fracture zone crust will be subsiding, but at different rates because they are at different distances (and different ages) form their respective spreading ridge segments. Compute the difference in subsidence rates across the transform fault for the oceanic crust of different age for a distance of 20 km along the fracture zone and in 1 km increments. In other words compare the subsidence rate for the one side at 1 km away from the ridge, and the other at 41 km away. Plot the difference as function of x, distance from the ridge. It helps to visualize the problem. Two schematic diagrams below may help to visualize what is going on.

Map view image of offset spreading ridge. The offset between the ridges can be expressed in distance (kilometers), but if the spreading rate is known this is also a maximum age difference cross the transform.

Cross section and profile view looking along the axis of spreading of two offset ridges that are linked by a transform. There are 2 drivers of motion of the crust. First is the much larger plate movement rate which is in the range of cms/yr, which is represented by the red arrows. Second there is the smaller rate of subsidence as the oceanic crust formed at the ridge moves away and gets older and colder, which is represented by the blue arrows. This rate is mms/yr or less and will decrease as a square root function with distance from the ridge that crust formed at. For the transform component the strike-slip component will dominate, but for the fracture zone part farther from the crests there will be only the dip component due to the different rates at which the two sides subside.

There are many ways to skin this modeling beast in terms of Excel sheet arrangement. Below is just one example. This has in its favor that you can change input parameters (e.g. such as the spreading velocity) and see what type of difference it makes. Note that k has already been computed for you using the units suggested above. Of course the equations are not included here, but you can insert those. For the cells in columns with the following headings you should have an appropriate equation: spreading velocity (msec-1), depth beneath crest, age of crust, subsidence in the last increment, subsidence difference with crust 2 Ma older. As always, ask if you have questions.

Copy of a portion of an Excel sheet that models the thermal subsidence of ridges.

For ease the equation is repeated from the preceding page below:

Y depth = k * {(km * x )/(pi * uo)}^.5 where k = [( 2 * pm * am * (Tm-To))/ (pm-pw)]

where km = thermal diffusivity of mantle rocks. Typically 1 mm^2 s^(-1)

x = distance away from ridge center