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According to NASA, one of the main reasons for the rising of sea levels is the increase in ocean temperature. The increase was of $0.4^\circ \text{F}\sim 0.2^\circ C$ for waters with depth $\sim700\text{ m}$. The observed sea level rise in that period was of around $\sim 10\text{ mm}$.

If the radius of earth is $R$, sea level heigth is $h$, and $\beta$ is the volumetric temperature coefficient at $17^\circ C$, a very simple model gives the volume change by $$ \Delta V = 4\pi R^2h\beta\Delta T. $$ The volume of the thin spherical shell due to volume change is $$ \Delta V = 4\pi R^2\Delta h. $$ Hence $\Delta h = h\beta\Delta T$. Considering that $\beta = 1.7\times 10^{-4}/^\circ C$, we find $$ \Delta h \sim (700\times 10^3\text{ mm})\times 1.7\times 10^{-4}\times 0.2 =23.8\text{ mm}. $$

This is huge, much larger than the observed sea level rise. What is the greatest source of error of this calcultation? I want to make this calculation in class and then present the reasons why it is not precise.

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    $\begingroup$ My guess is that the 0.2°C change is for the ocean surface, the change will be less than that at 700 m. $\endgroup$ – PM 2Ring Jun 24 '19 at 15:09
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    $\begingroup$ The calculation is within the same order of magnitude. The proper statement is not that this is a huge and much larger value. Also, when you drop the mean temperature to 10 $^o$C, you will cut your relative expansion in half. In summary, the very fact that you can do this type of simple calculation and end up with a comparable order of magnitude (e.g. mm rather than km) is in itself informative both in terms of the geometry approximation (a thin shell) let alone the application of properties of sea water. $\endgroup$ – Jeffrey J Weimer Jun 24 '19 at 15:31
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The first source of error I noticed was the temperature difference. it's listed as approximately .2 - .4 degrees. This is not known very accurately and your equation is linear in the temp difference. Also, β is given for 17º C while the water was much colder. β could also depend on pressure which is much greater at 700' depth. The last possible source of error I noticed was based on my assumption. i assumed from your description that you were looking at how much the volume of the first 700 meters of water expanded. To this, I would comment that the Volume increase would be less since parts of oceans are not 700' deep. Or maybe I don't understand how you're doing the calculation.

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