All Lakes Are Temporary

… for a very good reason.

I read an interesting article in the DeseretNews about Lake Powell — the impoundment behind the Glen Canyon dam on the Colorado River. The dam is in Arizona, but most of the reservoir is in Utah. (As an aside: this area, and the dam in particular, is the overarching target in the novel The Monkey Wrench Gang; written by Edward Abbey and published in 1975. If you are interested in the roots of eco-terrorism, this fictional account may be worth a bit of your time.)

Anyway, the basis of the DeseretNews article is that increased sedimentation —exposed by dropping water levels during the recent drought cycle — is filling the impoundment with sand, silt, and mud, and reducing the amount of storage capacity behind the dam. This in turn imperils the water supply to the southwestern states, as well as Mexico.

From where I sit — and despite all the OMG and WTF comments — this infilling is absolutely to be expected. Indeed, and as the title of this post suggests, it is essentially guaranteed. So, why is this happening?

In order to understand the hydrodynamics of the system, we need to look at some math for a bit. Sorry, but I promise to keep it simple enough for all of us — especially me — to understand. (Important note: most of this post’s graphics and images are from my G102 course on streams and fluid dynamics, where I attempt to explain this to a room full of mathophobes.)

We all love fractions, which is a real good thing since the damn things crop up all over the place. Look, here’s one now…

Fractions are easy, and they are all the same in one important way: there is a number above the line (called the numerator) and a second number below (the denominator). The value of the fraction is calculated by dividing the numerator by the denominator, which also ends up giving us what is called the “decimal equivalent.” In the example above, the fraction ½ has a decimal equivalent of 0.5 after doing the division.

Fiddling with the relationship between the numerator and denominator changes the value of the fraction, and in a predictable way. Fortunately, it’s much simpler than it should be: make the numerator larger relative to the denominator, and the value of the fraction increases. Make the denominator larger and the value of the fraction is reduced. It works this way with all fractions.

I also love three-variable equations, mostly because they can also be fractions, and as such follow the same rules regarding the relationship between the value above the line relative to the variable below. These marvels of math and science are used to define many relationships — we already discussed the three-variable formula that describes density in an earlier post (cleverly titled The Magic of Density).

(As an aside: all three-variable equations can be expressed to solve for any of the values. Sure, D=m/V, but solve for volume and V=m/D. Solve for mass, and m=DV.)

So yeah, three-variable equations are just fractions and are therefore really easy, and the good news is that there are a trio of them which, when taken together, define the reality of stream dynamics and sediment transport (and get us most of the way to understanding why Lake Powell is filling with sediment).

The first of them is Sir Isaac Newton’s 2^nd Law of Motion, which puts in mathematical form his 1^st Law of Motion. The long-winded version of his 1^st Law may sound complex (look it up if you don’t believe me), but it’s really simple: if you want to move something (like a pebble in the Colorado River), you need to apply a force to it (like is available from the water flowing over the stream bed).

This is something we already understand, and — even better — Newton’s a=f/m formula works just like all fractions: with the same quantity of flow (and therefore the same amount of force), a smaller rock (the mass value below the line) will move, but a larger boulder with a greater mass will not. We live this concept everywhere in our lives, and it is so obvious we don’t usually appreciate how it follows the rules of fractions.

The second equation relates to how the velocity of the water is affected by two things: how much water there is (the Quantity of water), and how large is the opening it’s moving through (called the cross-sectional Area).

This is why you stick your thumb over the end of the hose when you want to wash dirt off your driveway. Most of us think this increases the pressure (I know I used to, but you’re probably smarter than me), but it doesn’t — the pressure is fixed by the water department (or the pressure tank in your well). All your thumb does is reduce the cross-sectional area (the value below the line in the equation), thereby increasing the velocity. Don’t you just love how math always works like it’s supposed to?

One more equation, and we’re done.

Any discussion of sediment transfer by moving water is very much tied up with the reality of kinetic energy. The short version: kinetic energy is the force an object has simply because it is in motion, and the amount of energy is based upon two things: how much mass is involved, and how fast it’s going. This formula only has three variables, but the equation also has a constant (the ½ multiplier), and one of the values has an exponent (the V²), so it looks scarier than the others. Click here for a more detailed introduction to K_e from an earlier post.

Anyway, put all three equations together, and we can explain what controls the flow of sediment in any river or stream, including the Colorado. (One more aside: note that in the equations, the variable A is used in two of them: one is capitalized but the other is not. Why? It’s simple: they mean different things. In Newton’s equation the lower-case a is for acceleration, but in the formula for the shape of the channel, the upper-case A represents the cross-sectional area.)

Got all that?

So why is Lake Powell filling with sediment? Again it’s easy and is due to a rapid change in the shape of the channel. When the rampaging, sediment-choked Colorado River reaches the lake, the cross-section of the river’s channel (the value below the line) suddenly gets much greater, the velocity drops, the water essentially stops moving, and the force required to keep the dirt and rock being carried by the river is gone. The sediment drops to the bottom of the lake.

This abrupt increase in cross-sectional area and loss of velocity (and therefore energy) is also why a delta will build up at the mouth of a river when it reaches the ocean, and why alluvial fans form when a stream flows out of a narrow canyon into a wide desert basin.

In the case of a river flowing into an artificial lake, this loss of sediment also affects the erosional power of the flow beyond the dam. Carry a bag of potato chips home from the store, and then go back for two cases of your favorite beverage. It becomes obvious very quickly that the more weight you have to carry, the slower you go.

It works the same with moving water. Having been relieved of the added material it was carrying as it entered the lake, the river below the dam is now packing less mass and moves faster, resulting in additional energy to erode the valley and pick up more sediments. This may re-balance the flow vs. load relationship, but definitely has an impact on the topography downstream of the impoundment.

The sedimentation and filling of a lake can significantly impact the storage capacity of the reservoir behind the dam, and, given enough time, will completely fill the basin. The cross-sectional graph above is from Lake Meade, also along the Colorado but further downstream from Glen Canyon dam. Note the loss of capacity due to the sedimentation (as of 2001). The good news is that the rate of infilling has likely lessened of late — much of the sediment coming from the upper reaches of the river are now being caught by Lake Powell.

Unfortunately, there really isn’t much that can be done to fix this — it is, after all, controlled by the math (and you can’t argue with the way math works). The only viable option is to somehow dredge the sediments from the bottom of the lake, move them to below the dam, and put them back into the river.

Good luck with that (although it would somewhat restore the natural balance of the river).

So, a final question may be in order: Are dams good or bad? This is one of those posers that is impossible to answer with any degree of confidence. Sorry, but once again it is up to all of us to decide what works for our personal value system — do you prefer a natural flow, or the myriad benefits that proponents claim a dam can provide: flood control, water storage, hydroelectric power, and water skiing… to mention only a few — and then vote accordingly.