Homework #7 2012, Earth Surface Processes, Humphrey

1 The most common criteria for initiation of sediment motion (or loosely.. sediment transport competence) is the Shields plot. This uses the Shields criterion or parameter: Q= t_b / (( r_sed -r_fluid ) g D_sed). The Shields parameter predicts when the basal shear stress is just enough that sediment of a particular size will be transported. There is a set of plots in the readings that describe the behavior of the parameter. However if we stand back a bit we see that the parameter is approx. 0.05 over a wide range of flows (Reynolds numbers) typical in rivers and for a range of sediments also typical of rivers.

a) Use the Shields criteria to estimate the maximum size of sediment in motion in the Laramie river during a bankfull flood. [Just to introduce some consistency, assume the depth of the Laramie river in minor flood is 1.0m and the slope is 2x10^-4]

b) Use Manning’s equation to discuss how basal shear stress would change in a river that enters a reach that is twice as steep? At first glance it might appear that the slope goes up, but the depth goes down, so the shear stress might not change. Not true. This is a hard question… a couple of hints, use depth instead of hydraulic radius in Manning’s equation, and assume the width does not change, so that the discharge (Q=v*d*w) stays constant. (In a real world the width usually decreases when the slope steepens, which increases the shear stress).

c) Show that the Shields parameter (with the 0.05 approximation) implies that otherwise similar alluvial rivers (alluvial rivers are rivers that are reworking sediment that the river itself is moving) must be steeper for coarser sediment. (Hint, write out the terms for t_b and appeal to the previous question)

d) If discharge in a river increases downriver, as is usually the case, but the sediment size in transport stays the same: say something about what will likely happen to the river slope downstream?

e) Look back in your homework to where you calculated the velocity of the Laramie River.  Now use that velocity and Manning's equation to back-calculate the value of n for the Laramie River. Compare your result to a tabulated value, such as found on the web (try http://wwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/index.htm ) for comparison estimates of 'n'. 

2 In class we discussed that the ratio of settling velocity (w) to shear velocity (v*) as a discriminator of types of sediment transport. Sediment with a ratio of less than about 0.2 is predicted to move in suspension.

a) Calculate the shear velocity (v*) for the Laramie river.

b) Calculate the maximum size of particles that you would expect to find in suspension in the Laramie River. Fortunately you can get away with using the Stoke’s fall velocity for the settling velocity since the particles will be small enough to have a fairly low particle Reynolds number.

3 . What is the order of magnitude of the lift force on a 1cm diameter piece of gravel on a river bed, with a 0.5m/s water flow over it and stagnant water under it. Use the Bernoulli equation we developed in class, (it is also in the equation list on the web page). Is the force enough to lift the particle? (to get a scale for the lift force from the Bernoulli effect, you can assume the water velocity difference is the full 0.5m/s) [hint: the Bernoulli equation gives you a way to calculate the water pressure; that water pressure can be turned into a force by multiplying by the cross-sectional area of the particle]

4.      a. If a bedload particle gets up into the flow, in what direction (up or down) will the Bernoulli lift try to move the particle?  You may assume the particle has not been accelerated up to the full flow speed, and that the flow is a shear flow, i.e. faster at the river surface than the bed.  

     b. (a mini geomorph puzzle) Why do golf balls have dimples?  (just a sentence or two on why making something rougher, makes it go further)

5. (this weeks geomorphic puzzle) Many lakes in cold regions of the world turn-over.  The phrase describes a phenomena that occurs once (or twice) a year; the water in the lake moves so that in a short time the water at the top of the lake sinks and flows under the water at the lake bottom, while the water that was at the bottom circulates onto the top of the lake.  This process is hugely important for nutrient cycling.  Why does it occur, where does the head gradient come from?  If you can, explain why the event is often sudden, not gradual, and why it doesn’t occur in warmer regions.  And as a bonus, why doesn’t the ocean do the same thing.  (hint: think of temperature and density gradients)