Rising sea levels due to thermal expansion

Question

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.

Answer 1

Your formula $\Delta h = h\beta\Delta T$ seems to be Ok as does the estimate for the thermal expansion coefficient , but it seems to be very sensitive to the value of h chosen. If we choose a value of h = 100m instead of 700m, the result is 7 times lower.

The NASA link you provided suggests the temperature rise per year for the top 100m is $ 0.33 ^{\circ}C/year$, so using those figures for the top 100m gives:

$$ \Delta h \approx (100\times 10^3\text{ mm})\times 1.7\times 10^{-4}\times 0.33 \approx 5.6\text{ mm/year}. $$

This is within the ball park of an average of about 4mm/year observed over the last 10 years and about 12 mm/year over the last year. (note: This does not mean sea level is rising exponentially. It is normal for sea level to fluctuate and vary significantly from year to year and longer term averages have to be taken.) Some of the measured sea level rise is due to melting of ice at the poles$^*$, so I would suspect this estimate for rise due to thermal expansion to still be a bit on the high side.

*This NASA webpage suggests 1/3 of the observed sea level rise is due to thermal expansion and the other 2/3 is due to other factors, so further research required....

As an aside, can you imagine the difficulty of measuring the global average sea water temperature and level? Locally, sea level changes by the hour due to tides, currents and wind driven waves and temperatures vary according to time of day or night and cold inflows from rivers etc and varying mixing and stratification of water thermal layers which is level dependent and also with global undersea currents. This suggests it is very difficult to measure these parameters accurately on short time scales. The calculations introduced in this post suggest that the average rise in sea water temperature is a bit less than indicated or the average rise in sea water level is a bit more than indicated or both.

Answer 2

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.