Discharge and Sediment Transport of a Meltwater Stream:
An Example of a Mathematical Expedition in Polar Science
VCTM Conference
March 14, 2014
Lynn Foshee Reed (lynn.foshee.reed@gmail.com)
Einstein Educator Fellow (NSF – Polar Programs)
Adapted from a field problem and activities suggested by Dr. Asa Rennermalm, Andreas Beck Mikkelsen, Kasper Busk,
and Dan Frost.
Background: Glacier Mass Balance and Glacial Erosion
Measuring melt river discharge from the Greenland Ice Sheet is interesting because it can be used in melt
models and therefore in sea level rise projections. Similarly, calculating the sediment transport from the
Greenland Ice Sheet provides a better understanding of fluvial landscape formation and thus a parallel to
previous environments.
Water balance equation: P + E + R + ΔS = 0, where P is precipitation, E is evaporation, R is runoff, and ΔS is
Δstorage (the change in ice volume).
Part I: Discharge measurements at Watson River, Kangerlussuaq, Greenland
This is an excellent location because the U.S. Army Corps blasted a nearly perfectly rectangular channel
through the bedrock. A simple way to find the discharge in a river is to multiply the average velocity with
the cross sectional area: Q = V * A, where Q = discharge (m3/s), V = average velocity (m/s), and A = Crosssection area (m2).
The width and the depth to the bottom of the channel is known; all that needs to be measured on a given
day is the distance to the surface of the water. (See the partial profile below.) You can calculate the water
level by measuring the distance from the bridge to the water. For example, if the distance from the bridge
to the surface is 12 meters, then the depth of the water in the channel is approximately 2.3 meters.
Figure 1: Northern bridge cross section. Water level is measured from the top of the wood construction (~30 cm above the road).
Image provided by Kasper Busk.
Surface velocity can be roughly measured by the following methods, both of which rely on the relationship:
surface velocity = float length/time.
1.
The float method. Your team will need to determine the distance (float length) and the time using
GPS and stopwatches. Because the river can be quite violent, your float will not be retrievable, so
you will need to decide on what to use as a float (willow branch, piece of wood, an orange, etc.)
2. The Right Triangle variation. Tie a specific length of rope to a float. Carefully lower the float to the
river’s surface. Measure the time it takes for the float to travel until the rope is stretched taut
(forming the hypotenuse of a right triangle).
Once the surface velocity of the river is measured, the average velocity can be calculated by multiplying the
surface velocity by 0.95.
At the average velocity you have calculated, how much water moves through the cross section in 1 hour? 1
day? 1 year? (This is assuming constant flow.) When calculating the discharge per year, consider that there
is only discharge from the river from May to September and that the discharge is peaking around July and
August.
Over time, many daily or weekly calculations of the water discharge can be accumulated to better estimate
total seasonal discharge:
Images courtesy of Andreas Bech Mikkelsen.
Part II: Sediment transport by the Watson River
Stream velocity, which increases as the volume of the water in the stream increases, affects the amount of
silt and sediment carried by the stream. Sediment introduced to quiet, slow-flowing streams will settle
quickly to the stream bottom. Fast moving streams will keep sediment suspended longer in the water
column. The Watson River current is so strong that all the sediment is assumed to be carried in
suspension.
To find the sediment carried by a river simply multiply the river’s discharge by the sediment concentration:
Qsediment (kg/s)= Qw (m3/s) * C (kg/m3)
Use the Q/C graph (below) to read the sediment concentration. Note: kg/m3 = g/l. The data used to create
the model came from water samples.
Graph provided by Andreas Bech Mikkelsen.
For the discharge you calculated in Part I, determine the corresponding sediment transported in 1 hour, 1
day, and 1 year.
In 2010, the maximum transport rate was equivalent to one full standard dump truck every 3 seconds. The
2010 accumulated sediment transported by the river is approximately 11,900,000 tons. If extrapolated to
the entire Greenland Ice Sheet, the sediment transport is 1,350,000,000 tons/year. Compared to the
estimate of global sediment transport to the oceans, the Greenland Ice Sheet would deliver 7% of the total.
Image courtesy of Andreas Bech Mikkelsen.
Part III. Discharge of a small melt water stream, Russell Glacier.
In this exercise, you will use data collected by
students near Kangerlussuaq, Greenland, to
calculate the discharge of a melt water stream.
The video clip will place this problem into
context, and you will familiarize yourself with the
data collection methods.
Make a scatter plot of the depth data that gives an approximation of the actual river cross-section. (You
may do this by hand or you may use Excel, but there must be a hard copy.)
Calculate an approximation for the discharge (volume rate of water flow) using RSUM, LSUM, or TRAP SUM
using the data gathered by the students. You will be basing your calculations on the discharge formula:
Q = A*V, where A is the cross-sectional area and V is the velocity. Give your answer in cubic meters per
second.
Show all your work, including dimensional analysis. Summarize your findings in one or two paragraphs.
Resources:
http://www.classzone.com/books/earth_science/terc/content/visualizations/es1303/es1303page01.cfm
NSTA “Watershed Investigations: 12 Labs for High School Science” Jennifer Soukhome, Graham Peaslee, Carl Van
Faasen, and William Statema
Chapter 7: Calculating Stream Discharge
http://www.nsta.org/store/product_detail.aspx?id=10.2505/9781933531489.7
Melt water stream data collected by Joint Science Education Program participants on July 9, 2012
Water depth cm
015
100
150
175
200
225
250
275
300
325
350
375
400
425
450
475
500
525
550
575
625
675
725
775
Height cm
024
122
155
145
164
194
190
232
236
252
254
250
262
255
265
260
240
235
220
204
193
120
122
120
130
Length cm
275
300
325
350
375
400
425
450
475
500
525
Count rotations/40 sec
156
122
129
138
160
157
153
122
103
89
90
Velocity m/s
2.626
2.055
2.173
2.324
2.694
2.637
2.576
2.055
1.736
1.413
1.514
20
25
45
45
40
50
45
55
60
40
35
25
5