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What will the effects be of a major water reservoir collape?

Physics Asked by Gobabis on April 9, 2021

Some time ago I heard a comment on some National Geographic or Discovery Channel program that the earth rotation is slowed down due to water trapped in dams I did not pay attention to this at the time as it was said to be a fraction of a second.

Planning my next vacation I came across this article and this made me wonder about the effects of a major dam failing at full capacity this dam has $180,mathrm{km}^3$ of water and will spill over and also take out this dam having another $55,mathrm{km}^3$ combined this is a substantial mass of water that will run into the ocean.

Will the release of these amounts of water affect the rotation of the earth?

2 Answers

The impact of such a sudden drop in water levels are tiny, several hundred times smaller than the variations in rotation and axial tilts caused by natural events, such as the combined alignment of the Sun and Moon gravitational pull on the Earth.

This reference below is to the Three Gorges Dam, which contains 43 trillion tons of water.

Raising 39 trillion kilograms of water 175 meters above sea level will increase the Earth’s moment of inertia and thus slow its rotation. However, the effect would extremely small. NASA scientists calculated that shift of such as mass would increase the length of day by only 0.06 microseconds and make the Earth only very slightly more round in the middle and flat on the top. It would shift the pole position by about two centimeters (0.8 inch). Note that a shift in any object’s mass on the Earth relative to its axis of rotation will change its moment of inertia, although most shifts are too small to be measured (but they can be calculated).

Correct answer by user140606 on April 9, 2021

The source cited by user140606 is incorrect. This is my Answer to a similar Question on earthscience.SE :

You need to calculate the change in the moment of inertia of the Earth and use conservation of angular momentum (the rotation period is proportional to the moment of inertia). Most of the water will ultimately come from the oceans, effectively removing a thin layer of water.

Jerry Mitrovica discusses this effect (in reverse) in a Nautilus interview:

Is water moving off glaciers, slowing the Earth’s rotation, this time analogous to a figure skater putting arms out?
Right. Glaciers are mostly near the axis. They’re near the North and South Poles and the bulk of the ocean is not. In other words, you’re taking glaciers from high latitudes like Alaska and Patagonia, you’re melting them, they distribute around the globe, but in general, that’s like a mass flux toward the equator because you’re taking material from the poles and you’re moving it into the oceans. That tends to move material closer to the equator than it once was.
So the melting mountain glaciers and polar caps are moving bulk toward the equator?
Yes. Of course, there is ocean everywhere, but if you’re moving the ice from a high latitude and you’re sticking it over oceans, in effect, you’re adding to mass in the equator and you’re taking mass away from the polar areas and that’s going to slow the earth down.

The contribution of the removed water to the moment of inertia depends on the distance from the axis and hence on the latitude. This is a simple calculation if we assume the world is all ocean.

The moment of inertia of the lake is $m(Rcos L)^2$ and the moment of inertia of a spherical shell is $frac{2}{3}mR^2$, where m is the mass of water, $R$ is the Earth's radius and $L$ is the latitude of the lake ($30.82305$ degrees for Three Gorges). The relative change in the moment of inertia $I$ of the Earth is then $$frac{mR^2}{I}(cos^2 L - frac{2}{3}) = frac{39 times 10^{12} times(6.37 times 10^{6})^2}{8.04×10^{37}}(cos^2 L - frac{2}{3})$$ $$=1.97×10^{-11}(cos^2 L - frac{2}{3})$$

Multiplying by the number of microseconds in a day ($8.64 times 10^{10}$) gives $$1.7(cos^2 L - frac{2}{3}) = 1.7 × 0.071 = 0.12$$ microseconds.

Why the difference from NASA's $0.06$ ? Note that the expression changes sign at $cos^2 L = frac{2}{3}$ or $L ≈ 35$ degrees (pretty close to the latitude of Three Gorges). The Earth will actually speed up if the lake is at high latitudes and slow down if it is at the equator. The $frac{2}{3}$ term comes from the "all ocean" assumption. As I understand this paper, the $frac{2}{3}$ term should be multiplied by $frac{1.414}{1.38}$ to account for the shape of the oceans (search in the PDF for those numbers), resulting in $0.09$ microseconds (.

Answered by Keith McClary on April 9, 2021

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