How Do I Calculate Channel Slope/Gradient?

In geometry class, we learn that the slope of a line is calculated by dividing the vertical change (rise) by the horizontal change (run). On a piece of graph paper, that's pretty straightforward. But how does that work when we're looking at a river or stream in the real world?

Why Does Channel Slope Matter?

Channel slope is a key metric for understanding whether a river is eroding or depositing sediment. A steep slope means that water is moving faster, which can mobilize larger particles of sediment. In contrast, a shallower slope means the current doesn't have enough energy to keep those larger grains suspended and they deposit on the river bed. In the absence of disturbance, most rivers move towards a slope that allows for a dynamic equilibrium between erosion and deposition.

How Do I Calculate Channel Slope in the Geosciences?

Maximum Channel Slope

The maximum channel slope or gradient is the vertical drop between an upstream point and a downstream point divided by the straight line distance between the two. You'll only see this channel slope in a perfectly straight channel.  Such an idealized case is shown in Fig. 1.

1. First, we need to know "rise" for the feature. Rise is the difference in elevation from the top to bottom of the feature we're interested in.
   Show me rise for the sample problem in Fig. 1
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   In the image to the right, the rise is the difference in elevation between the upper node and the lower. The top end of the river is at 140 m elevation. The elevation of the lower end is 100 m. The difference in elevation is the top minus the bottom (140 m – 100 m) so "rise" = 40 m
2. For the "run", simply measure the straight line distance between the two points.  In Fig. 1, the "run" distance is labeled as 1000 m.
3. Take the difference in elevation and divide it by the horizontal difference (always making sure you keep track of units).
   Show me rise for the sample problem in Fig. 1
   Hide
   In Fig. 1, the river's rise = 40 m and run is 1000 m. So we set up the problem like this:
   
   `text{slope}=\frac{40\ m}{1000\ m}=0.04\ \frac{m}{m}=40\ \frac{m}{km}`
4. You can also express the gradient as a percent, as long as the rise and run are both in the same units.
   Show me rise for the sample problem in Fig. 2
   Hide
   Above, we found that the slope was 
   
   `text{slope}=0.04\ \frac{m}{m}`
   
   Since the top and bottom of the fraction have the same units, they can cancel each other out leaving us with a unitless number: slope = 0.04
   
   To get to % we need to multiply the calculated slope by 100.
   
   `text{slope}=0.04\ \times100=4.0%text{slope}`

Meandering Channel Slope

A more realistic stream will curve or "meander" on the landscape, rather than run in a straight line.  That means that the distance traveled will be longer, even if the change in elevation is the same, which will change the channel slope.  This kind of situation is shown in Fig. 2.

1. In Fig. 2, the rise is the same as it was in Fig. 1. The elevation is 140 m at the top and 100 m at the bottom. So the "rise" = 40 m.
2. In this case, we need to measure the length of the stream channel, rather than the straight line distance from the top to the bottom.  Measure along the path of the stream to get this distance.  In Fig. 2, the channel length is labeled as 1600 m.
3. Take the difference in elevation and divide it by the horizontal difference (always making sure you keep track of units).
   Show me rise for the sample problem in Fig. 2
   Hide
   In Fig. 2, the river's rise = 40 m and run is 1600 m. So we set up the problem like this:
   
   `text{slope}=\frac{40\ m}{1600\ m}=0.025\ \frac{m}{m}=25\ \frac{m}{km}`
4. You can also express the gradient as a percent, as long as the rise and run are both in the same units.
   Show me rise for the sample problem in Fig. 1
   Hide
   Above, we found that the slope was 
   
   `text{slope}=0.025\ \frac{m}{m}`
   
   Since the top and bottom of the fraction have the same units, they can cancel each other out leaving us with a unitless number: slope = 0.025
   
   To get to % we need to multiply the calculated slope by 100.
   
   `text{slope}=0.025\ \times100=2.5%text{slope}`

Channel Slope from a Topographic Map

Now let's practice getting the rise and run information off of a topographic map.  Using Fig. 3, let's find the channel slope between the two points A and B along the creek.  (Additional help for using Topographic Maps.)

1. First get comfortable with the features of the topographic map in Fig. 3. Make sure you know a few things:
   • What is the contour interval (sometimes abbreviated CI) of this map?
     The contour interval tells you "rise," specifically the change in elevation between each of the contours (the brown lines). In this case, the contour interval is shown in the key in the upper right and is 20 ft.
   • What is the scale of the map?
     The scale tells you the "run," or the distance on the ground. On this map, it is also shown graphically in the key. If you print out the map (Acrobat (PDF) 336kB Apr2 26), you will find that the scale is 0.75 inch = 1 mile.
   • What is the feature for which you want to know the slope?
     We want to find the channel slope between points A and B on the map.
2. Next we'll use the contour lines on the map to figure out the "rise" between the two points.  A is at the point where the 680 ft contour crosses the creek.  B is where the 580 ft contour crosses the creek.  That means that the difference in elevation is 
   `text{rise}=680\ ft\-\580\ ft=100\ ft`
3. In this example, the "run" is the distance along the course of the creek between A and B. Use a flexible measuring tape or piece of string to find out how long the channel is between the points.  Then use the scale bar to convert that length into a distance.  The length on the map should come out to be about 10.5 inches.  Using the scale bar, that means that the channel's run = 14 miles.
4. Now comes the "rise over run" part. There are two ways that you may be asked to make calculations relating to slope. Make sure you know what the question is asking you and follow the steps associated with the appropriate process:
   • If you are asked to calculate slope (as in a line or a hillside), a simple division is all that is needed. Just make sure that you keep track of units!
     Show me how to calculate slope
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     1. Take the difference in elevation and divide it by the horizontal difference (always making sure you keep track of units). 
        On the map of Math State Park, the creek drops by 100 ft over 14 mi of channel. So we set up the problem like this:
        
        `text{slope}=\frac{text{rise}}{text{run}}=\frac{100\ ft}{14\ mi}=7.14\ \frac{ft}{mi} `
        
        The units you end up with might be feet/mile or m/km or feet/foot (slope can be expressed in all these ways). It just depends on what you started with.
   • You may also be asked to calculate percent (or %) slope. This calculation takes a couple of steps. And it mostly has to do with paying attention to units. The units on both rise and run have to be the same.
     Show me how to calculate % slope
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     1. To calculate percent slope, both rise and run must be in the same units (for example, feet or meters). If your rise and run are in different units (like feet and miles), it usually makes the most sense to convert the run into the units used for the rise. To convert from miles to feet, multiply by 5280 ft/mi; km to m, multiply by 1000m/km. (If you need more help with this or need to convert other units, check out the Unit Conversions module from The Math You Need When You Need It.)
        Show me how to do this for the sample problem
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        Right now you have rise in feet and run in miles. Let's convert the miles to feet by multiplying by the appropriate conversion factor: 1 mile = 5280 feet. So, we should multiply "run" by `\frac{5280\ ft}{1\ mi}`: 
        
        `14\ mi\times\frac{5280\ ft}{1\ mi}=73,920\ ft`
     2. Once you have converted so that both elevation and distance have the same units, we can write an equation for slope: rise over run (implying rise divided by run).
        Show me how to do the calculation
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        We know that the rise is 100 ft and the run is 73,920 ft: 
        
        `\frac{rise}{run}text { for this question is } \frac{100\ ft}{73,920\ ft}`
        
        and we get to % slope by multiplying that result by 100. 
        
        `%text{slope}=\frac{100\ ft}{73,920\ ft}\times100=0.14%`
        
        Note that % slope does not have any units because ft cancel in the calculation. Make sure that you indicate that it's % however!