The Width of Water – Cairns #76

Both my editors thought I should drop this section but this is me practicing how to communicate part of an idea that I am still wrestling with that I believe might become important. Feel free to skip over it at any time.

A stream’s width is always changing. In some places, the stream narrows and in other places, it broadens out. This change of width, easy to see, connects our mind to many other aspects of the flowing stream.

The easiest place to enter these connections is with the simplistic but still functional equation for stream discharge (how much water is flowing along a stream):

Discharge = width x depth x velocity

Width x depth is basically the familiar formula for finding the area of a rectangle (length x height). Width x depth give us the cross-sectional area of the flow. Velocity tells us how fast the water is flowing through this area. The distance the current can flow in a second gives us the length which when multiplied by the width and depth gives us the volume of a rectangular prism representing the volume of water flowing by in a second. In the United States, the units we measure by are feet x feet x feet/second = cubic feet per second or cfs.

This formula is simplistic because a stream’s depth is not uniform so the cross-sectional area is not rectangular. Furthermore, a stream’s velocity is not uniform. The outside of a river bend flows swiftly while the inside of the bend might actually be eddying upstream. However, this simple formula is good enough to give us a powerful introduction into changes happening within flow.

The key thought experiment is to imagine a stretch of stream where no additional water is joining the stream or leaving it. In that case, the discharge remains the same throughout that stretch. If the discharge remains the same, then that means that width x depth x velocity at one place must be equal to width x depth x velocity at another place along that stretch.

If the stream narrows to half its width at one point, then the product, depth x velocity must double at that point for discharge to remain constant. Depth could double with the velocity remaining constant or the velocity could double with the depth remaining the same. More likely, both the depth and velocity will increase somewhat so that the product depth x velocity doubles.

Width, depth, and velocity are interconnected in a fundamental way. This malleable interplay between the three is what you see when you hang out around streams. A change in one absolutely requires the others to change. Drop a big rock in the middle of a current, thereby diminishing the cross-sectional area of the current and the water will deepen and speed up as it races around the rock. All three stream variables adjust to the change. More peacefully, if one floats down a river, one notices that when the river speeds up without narrowing, the river bottom is only a few feet away whereas when the river slows down with no change in width, the bottom is way below.

So depth and velocity are the first flow attributes connected with stream width. The connection between width and velocity, however, also connects width with the stream’s gradient. If the slope steepens, the water will speed up. This increase in velocity must create a decrease in the width x depth. The channel will become narrower. On the other hand, if the stream gradient flattens out, the current will slow down which will require that the width x depth to increase. So when we see a stream channel narrow or widen, it often signals that the stream’s gradient is steepening or flattening in those places.

But now, a stream’s velocity plays an exponential role in a stream’s kinetic energy. KE=1/2 mv². Kinetic energy is one half the mass times the square of its speed. If a flow doubles in speed, its kinetic energy quadruples. Kinetic energy can pick up material and carry it downstream. Kinetic energy gives the flowing water the wherewithal to shape the land which will give rise to several feedback spirals.

For example, if a streambed steepens, the current speeds up, increasing in kinetic energy, allowing the stream to start eroding that area, carrying fragments from the streambed downstream. When the gradient gentles out, the stream slows down, losing some of its kinetic energy. It no longer contains sufficient energy to carry all of its load so some of the fragments drop out, creating deposition. Therefore, steep areas become places of erosion that are worn down and flatter areas become places of deposition that are built up. The steep stream sections grow less steep while the flatter sections grow more steep. The stream develops toward a smooth continuous gradient called stream equilibrium.

This process can be watched in miniature. One place is where a sandy bottom stream opens into a deeper, quiet pool. Sand grains come rolling along the streambed. When they come to the pool, the increased depth slows the water and the sand grain stops moving, raising the streambed so that the sand grain coming down behind it can now roll past that dropped stream grain where it stops rolling. Over time, this process builds an underwater delta at the head of the pool.

I love watching tiny deltas an inch high that are actively growing one sand grain at a time. A sand grain comes rolling along to the abrupt downstream edge of the delta where it rolls over the edge and tumbles down the inch high steep slope and comes to rest at the bottom of the slope. The next sand grain tumbles afterwards and comes to rest next to that sand grain. Those two grains together now allow a following sand grain to come to rest on top of them. A steady stream of sand grains fall over the edge and come to rest on the steep slope below. Gradually, a layer of sand one grain thick builds across and rises up the slope until it reaches the top, to the level of the incoming stream bed. Now the incoming sand grains can roll one sand grain length further before dropping over the edge and starting to build up the next layer (called a foreset bed). Here’s a diagram on the web: http://www.denniskalma.com/river/deltaxsect.gif

This interplay between the current and individual sand grains is a very peaceful process to watch. The change in stream depth changes the sand grain from tumbling to coming to rest but the grain’s coming to rest then causes the stream depth to change which affects every sand grain coming thereafter. Each sand grain follows a different path because of the sand grain that came before it.

But this movement towards “stream equilibrium” affects the other variables. When the stream slows down and drops some of its load, for example, the dropped load raises the streambed, decreasing the depth. Therefore, with both the speed and the depth decreasing, the width must increase. On the other hand, when a stream steepens, speeds up and erodes, it lowers its streambed, increasing its depth. The stream width must decrease. So the oscillations in stream width that we see tell us where the channel is cutting down and where it is building up. It reveals all the small alternating sections of erosion and deposition by which sands and gravels are being moved from section to section over time. When we start seeing a stream in this way, then we start seeing a stream’s widest place being the dynamic beginning in the stream’s narrowing and a stream’s narrowest place as the beginning of the stream’s widening.

The ability of water to shape the land creates feedback spirals. The head of an eroding gully eroding its way upslope is like a black hole, bending runoff towards it, increasing rates of convergence so that even more runoff flows through that section, increasing its mass and kinetic energy. Is the shape of the deepening gully shaping the stream flow or is the shape of the narrowing stream flow shaping the gully?

On the other hand, a widening channel leads to deposition which then fills in channels, forcing the water to flow over an even broader area which slows the flow still more and an alluvial fan begins to build. Water flows broadly and slowly through an alluvial fan so more soaks in here. Plants grow thick, sponging up even more of the runoff, slowing the runoff even more, as even more of the sediment comes to rest among the plant stems. Roots spread upwards into the accumulating sediment; the alluvial fan grows lush green, spreading and slowing the water even more.

Life’s ability to do a Galton machine spreading and thereby slowing runoff alters the entire stream equilibrium of a drainage. By spreading and slowing the runoff, the plants allow the slopes to rise more steeply against the forces of downward erosion. The land rises steeper and higher – or rather the land is worn down more slowly.

The land, its soil, the stream, its width, its depth, its speed, its energy, its load, its steepness, and life –  all fit together to form a writhing brown and green serpent of flowing mass and energy slithering over the land, with cause and effect undulating back and forth.