Stream channel curvature and erosion

Most geologists/geomorphologists are aware that water flowing around a meander bend leads to scouring and erosion of the outer bank, but unaware that the details of this process are understood pretty poorly. In the late 70s and early 80s, two researchers tried to isolate the effect of stream curvature on erosion rates. Nanson and Hickin documented long-term erosion rates on the Beatton River (W Canada) using dendrochronological methods--using tree ring dating, they determined how quickly  the banks of the river were migrating through time. These observations were somewhat counter-intuitive. Our intuition suggests that the tighter the bend that the water is forced to go around, the faster the migration rate, or erosion rate, should be. In reality, as showed by Nanson and Hickin, the migration rate reaches a maximum speed at around R/w ~ 2.5. The term R/w is the radius of curvature scaled by channel width (at bankfull). Subsequent work even showed that in really tight bends, where R/w < 2.0, a flow separation cell develops on the outer bank (not the inner bank as previously assumed), and the high velocity filament gets pushed toward the inner bank/point bar. This can actually lead to deposition on the outer bank and erosion on the inner bank, i.e., negative bank migration.

Following this work, in 1981, Begin tried to arrive at the same conclusion from first principles. He obtained an expression for the centripetal force exerted on the flow (and the equal and opposite force exerted on the bank by the flow) when the flow rounds a bend.  This expression involved several empirical constants, some of which were estimated. Although his results did seem to match up to Nanson and Hickin's earlier work, a discussion paper in the Journal of Geology doubted whether his method was broadly applicable, which Begin disputed. At the least, Begin showed that bank shear stress should be proportional to radius of curvature.

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I took away from all of this that bank erosion rates should not be a simple (i.e. linear) function of meander curvature (or radius of curvature, or any similar measurement). As Nanson and Hickin showed, the relationship is complex, and despite the work of Begin, its precise form is unknown. We can look at the data collected by Nanson and Hickin and make some observations.

First, the maximum migration rates do indeed coincide with R/w between 2 and 3. An upper envelope (the dashed line) represents the maximum effect of curvature on migration rates. In reality, the response of a streambank doesn't reach this potential. As Begin showed bank strength accounts for much of the scatter beneath the upper envelope. 

The form of the envelope doesn't seem to match Begin's analysis, which produced a smooth gaussian-like function. In contrast, the erosion rates decay exponentially on either side of the maximum value.

What follows is a somewhat stream-of-consciousness account of my thinking. I wanted to represent the distribution of R/w about this central value using a function that goes from 1 at the center, and approaches 0 at negative and positive infinity. First, I started thinking about it in terms of standard deviations away from the mean. If you assume a gaussian distribution (which is probably not correct, but maybe close enough?), you can calculate the z-statistic, the distance from the mean in terms of standard deviations. Then, the proportion of the data that lies within a certain z is gin by erf(z/sqrt(2)), where erf is the error function. This has the property that at the mean, the erf is 0 and as you move away from the mean, the result approaches 1.

I transformed R/w in the following way:

R*/w = 1 - erf( |z| / sqrt(2) ) = erfc( |z| / sqrt(2) )

where R*/w are the transformed values, erf is the error function, and erfc is the complimentary error function, equal to 1 - erf. 

The complimentary error function seems to be a pretty good representation of Nanson and Hickin's upper envelope. The absolute value reflects the fact that erosion rate decreases on either side of the curve. R*/w is equal to 1 when R/w = 2.5, and quickly approaches 0 as the radius of curvature deviates from that value. I interpret this as the meander curvature being 100% effective at the empirical value of R/w = 2.5, and increasingly less effective as curvature deviates from 2.5. Actually, I found that R/w = 2.2 was the best fit to our data, which is within the range given above. I use 2.2 as the central value from here on out.

 Left: Erosion rate plotted against untransformed R/w shows a trend very similar to Nanson and Hickin's above plots. Right: Erosion rates plotted by R*/w (erfc transformed R/w) show 2 trends.

Left: Erosion rate plotted against untransformed R/w shows a trend very similar to Nanson and Hickin's above plots. Right: Erosion rates plotted by R*/w (erfc transformed R/w) show 2 trends.

When I plotted R*/w against erosion rates, I got graph on the right. There seem to be 2 well-defined trends. One where the banks responded to this curvature effect and one where they didn't, perhaps because of bank strength. The scatter of our values compared to Nanson and Hickin's data result from our study design. Unlike Nanson and Hickin, whose measurements came from the same river, we generally only measured 1-2 banks per river, and measured several different rivers in the area. Our situation is more comparable to Nanson and Hickin's figure 3 (above), and our data show this pretty clearly (left graph).

Compositionally, banks in our study area are actually relatively homogeneous -- they are almost entirely composed of sandy alluvial sediment. Vegetation, however, gives the bank much of its strength, and  vegetation is likely to control the response of individual banks to shear stress. Some banks also have large amounts of woody debris within them, presumably from the burial of an older forested floodplain (an interesting topic in itself).

We've collected root percentages at each bank that range from 0-100% by inserting a metal rod systematically into each bank and counting the number of obstructions hit. You would expect a bank with 50% root volume to be harder to erode than one with 20% root volume. When I added these root percentages as color to the graph, the results looked interesting (right).

The 2 sites in the bottom right corner, the ones that should be the strongest, had almost 90% root density. The lower trend seems to have a higher root percentage in general (white circles have no root data). The southeast quadrant of the graph is populated by sites with root percentages greater than 50%, while the upper trend has root percentages that are all less than ~20%.

When I fit a linear model with interactions between the my "meander index" and root density, I got an R^2 of 0.652! If I multiply the "meander index" by bankfull width first, R^2 goes up to 0.78. Conceptually, this means that 3 variables can explain 78% of the variability in erosion rates that we observed. Comparing these numbers to what I was getting before I performed the R/w transformation, I was shocked. The same model with untransformed R/w gives an R^2 of only 0.16!

These results are encouraging. I was beginning to despair that erosion rates were inherently upredictable. Thanks to the work of Nanson and Hickin and Begin (and others), I now see that they actually follow a relatively predictable trend. You just have to look at the right data!