# Sea level change as ocean surface warms

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1. Aug 18, 2015

### bwilson4web

1. The problem statement, all variables and given/known data

I am attempting to replicate the IPCC estimates of the amount of sea level rise due to thermal effects on the seawater.

The earth, black-body temperature is 16C but there are projections of a 1C increase in less than 100 years. Ignoring ice melt and land flooding, how much higher would the sea level rise from a 1C increase in the top meter of the oceans?

2. Relevant equations

71% - sea coverage
A = 4πr*r - surface area of sphere
.000223 m/mK - sea water coefficient of expansion @16C

3. The attempt at a solution

a) calculated earth surface area, 5.10064E+14 m**2
b) remove the 71% land, 3.62146E+14 m**2
c) calculate the sea only, earth radius, 5,368,300 m
d) calculate the sea only circumference, 33,730,023.8 m @16C
e) raise temperature 1C and calculate new circumference, 33,737,545.6 m @17C
f) calculate the sea only, earth radius @17C, 5,369,497 m
g) calculate the change, 5,369,497 - 5,368,300 = 1,197 m

This number is two orders of magnitude greater than projected sea level increases, ~1-2 m, that already includes land-ice melt. It would still take substantial flooding to reduce the thermal sea level rise to the projected levels.

The only out might be to add the land area previously taken out to shrink the globe to a water-only earth. But back of the envelope suggests this won't reduce the temperature induced, sea level change enough to get back to the IPCC projected levels. Only flooding the land gives a chance of adjusting the ratios.

Thanks,
Bob Wilson

2. Aug 18, 2015

### Bystander

You might want to check your work one more time.

3. Aug 18, 2015

### bwilson4web

I double-checked my excel spreadsheet and don't see an obvious error. The most complex calculations are surface area:

H1 - earth radius in meters
=4*PI()*H1*H1 # earth sea surface area, H1 = earth radius in meters,

H3 = 71% of earth sea surface area

I also double-checked the coefficient of thermal expansion from two sources and they are within normal error limits.

My approach assumes a 1 m**2 ring of sea water heated from 16C to 17C. The radius increase should be the sea level change from the temperature change.

Thanks for the suggestion. Would having the spreadsheet help?
Bob Wilson

4. Aug 18, 2015

### Bystander

If this means what I think it means, you've calculated the change in radius for a planet that is 100% water changing temperature from center to surface by one degree.

5. Aug 18, 2015

### haruspex

Your calculation effectively takes the expansion from the 1C warming all the way to the centre of the Earth. If a layer 1m deep were to expand in the way you have calculated, it would still be only 1m deep but hovering kms above the rest of the ocean.
Instead, just consider a 1m x 1m x 1m cube at the surface. If this expands in volume, but not in area, what is its increase in height?
Btw, to get the sort of expansion that would happen in practice, you would need to consider a warmed layer more like 100-200m deep. (The rate at which heat reaches the depths has turned out to be greater than expected.)

6. Aug 18, 2015

### bwilson4web

This approach dates back to a 1970, material course where we calculated the dimension changes of a hollow sphere and solid ball of the same diameter. So reducing the earth's sea surface to a ring seemed an acceptable approach. Just the results suggest I've either done something trivially wrong or the shore-line flooding has a greater effect than originally expected.

Bob Wilson

#### Attached Files:

• ###### sea_level_temp_010.xlsx
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7. Aug 18, 2015

### haruspex

Follow that model through. You have treated the top 1m of water as a 1m thick hollow sphere. Yes, if that were to expand as though a solid it wouldn't get much thicker but its radius would increase considerably. It is now separated from the water below by a huge air gap. That is not going to happen, right?

8. Aug 18, 2015

### bwilson4web

We agree I'm calculating what happens to the heated surface layer. Descriptive Physical Oceanography, An Introduction, Talley, L.D. Pickard, George L., Emery, William J., Switft, James H. indicates the solar induced heating primarily occurs in the first meter. Heating at lower levels comes from mixing, not direct effects. So it seemed a reasonable approach. Still, the data being so far off means a more sophisticated model than a simple sea-water ring is needed.

My next version will add the land mass with a flood gradient. A meter of water and the low coefficient of expansion, it won't take much slope to absorb the expanded volume. But getting credible shore metrics might take a little more work.

You've given me some good things to mull over and helped a lot.

The reason I'm interested in sea level effects of surface temperature change is GRACE and other satellite systems have already given some initial metrics of land, ice-melt. But I'd like something a little more current. Fortunately, NOAA and others provide sea surface metrics giving world-wide, sea surface temperatures (i.e., google 'unisys sst'.) So if we can model the sea surface temperature effects and compare them to the sea-level data, the difference should be the lost melt-water.

My concern was whether or not I made some silly mistake in calculating the thermal expansion of the surface.

Thanks, again!
Bob Wilson

9. Aug 18, 2015

### bwilson4web

Correct but I had taken out the land mass, the slope of the shore. Adding a sloped shore is the type of buffer needed to moderate the sea level rise.

In the real world, heating a solid, especially one with poor tensile strength and thermal conductivity, can lead to fractured surfaces. In the real seas, any tensile stress will cause the water to become turbulent, mixing the upper and lower layers but this doesn't happen instantly. Still, density will lead to warmer surface over a colder core. You don't have to dive down far to find the cold water even in a lake.

Thanks,
Bob Wilson

10. Aug 18, 2015

### haruspex

I don't think you are getting what I'm telling you. Your model is utterly wrong. That t 1m of water will not behave anything like a 1m solid crust. Try the method I mentioned.

11. Aug 18, 2015

### bwilson4web

I see it and that makes sense. The volume will increase because of the temperature but with only one degree of freedom, altitude, that becomes the sea level rise. In effect, analysis of one element of the ring.

Thanks,
Bob Wilson

Last edited: Aug 18, 2015