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We might call this the “Noah's Ark” calculation, but in the movie Waterworld (1995) they have the icecaps melting and take poetic license making the ~$220$ feet or so of sea water (one estimate I've seen – of how much the ocean would rise if all ice was melted) into approximately $27,000$ feet deep of water. (Everest is $29,035$ feet and the island in the movie looked to be maybe ~$2,000$, so let's make it a nice even $27,000$ estimate.)

Given the circumference of the earth at sea level as it is today, and increasing the radius out $27,000$ feet, and making as many assumptions as needed (like the volume of all land and structures above sea level today) what is the cubic foot measurement of the additional water needed?

ipso
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2 Answers2

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It is a reasonable assumption to model the earth as an oblate spheroid. An oblate spheroid has two radii, the major axis $a$ and the minor axis $b$, and has volume $\frac{4}{3}\pi a^2b$.

According to Wikipedia, for the Earth we have $a \approx 6378.13$ km and $b \approx 6356.75$ km, giving a volume of approximately $1.083204 * 10^{12}$ km$^3$.

$27000$ feet $\approx 8.23$km, and Everest is pretty near the equator, so the "flooded Earth" would have $a_f \approx 6386.46$ km.

Presumably the "flooded Earth" would have a similar flattening ($= \frac{a-b}{a}$) to the normal Earth, giving $b_f \approx 6365.05$ km.

This gives a volume of the flooded Earth of approximately $1.087454 * 10^{12}$ km$^3$.

The difference between the volumes, i.e. the volume of water, is $0.00425 * 10^{12}$ km$^3 = 4.25$ billion cubic kilometers $\approx 1.5 * 10^{20}$ cubic feet.

Christopher
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  • This makes some sense only all of the earth was just see. Obviously, some quantity such as the altitude distribution must come into the equation. – Peter Franek Dec 10 '14 at 12:32
  • Frankly, I picked this one based on “oblate spheroid”, and I guess someone could reduce via “altitude distribution” if one wanted to. Thanks to both answers (at time of bounty expiration), although I think you're not supposed to say thanks. – ipso Dec 16 '14 at 18:31
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Since $27000$ feet ($\approx 8200$ meters) is negligible against the radius of the Earth. So we can simply approximate the additional water volume as surface times height, i.e. $$\tag1 V\approx 8200\,\mathrm m\times 5.1\cdot 10^8\,\mathrm{km}^2\approx4.2\cdot 10^9\mathrm{km}^3.$$ That is: If all is at Sea level. If - as other extreme - we assume that all land masses ($<30\,\%$) are above even the new see level, the estimate gets lowered accordingly to $$\tag2 V=0.7\times 4.2\cdot 10^9\mathrm{km}^3\approx 2.9\cdot 10^9\mathrm{km}^3.$$ The first assumption is certainly closer to the reality than the second (relatively few parts of the land masses reach even half the Everest height), i.e. the result should be pretty much closer to $(1)$ than to $(2)$.

Then again, didn't Mr Costner dive down to a sunken city? Even if he can breathe with his gills, it would take four kilometers to reach even a high-levelled city like La Paz from the water surface.

Hagen von Eitzen
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  • Yes. What surely must have happened is the ice caps melting causes massive weight redistribution and tectonic reconfiguration such that all exposed land subsides, and all the sea trenches are filled and current sea levels +220' would: a:) cover the earth all except for at least one Island, b.) make it such that he could reasonably dive to Denver (the underwater city in the movie), and c.) this would explain why there is no real weather in the movie. A relatively shallow sea worldwide, and no mountains, you'd have no major currents\storms. It could happen.... ; ) – ipso Dec 12 '14 at 14:27