If all the world's ice were to melt, would this shift the earth's centre of gravity with consequential effects on sea levels?

This question on Maths SE asked how we might calculate the effect on sea level if all the world's ice (ie including the whole Antarctic ice sheet) were to melt. I posted an answer containing calculations suggesting - and this came as a surprise to me - that the rise in sea level would be up to about 40 metres more at high northern than at high southern latitudes. The key qualitative assumptions underlying this conclusion are:

1. Most of the world's ice is in the Antarctic ice sheet.
2. The melting of all the world's ice and spread of the resulting liquid water across the oceans would result in a shift in the earth's centre of gravity. Given 1 above, the shift would be towards the north.
3. The shift in the earth's centre of gravity would not significantly affect the position of the earth's land masses and ocean bed which in this context can be regarded as a solid mass around which the ocean water would position itself.
4. The distribution of liquid water across the world's oceans, though affected by other factors, is largely determined by the gravitational force pulling it towards the earth's centre of gravity. Hence a northward shift in the centre of gravity would pull a slightly larger proportion of the ocean water to the northern hemisphere.

• – Keith McClary May 21 '19 at 20:40

As a prescript, your question is firmly within the field of Glacio-Isostatic Adjustment (GIA) modeling of the Earth and the sea level. The main scientific names doing research in this direction are Jerry Mitrovica, W.R. Peltier, Roblyn Kendall, Glenn Milne, Kurt Lambeck, Giorgio Spada, and probably some more that I'm forgetting. It all comes down to, basically, the 'Sea Level Equation' (SLE), https://en.wikipedia.org/wiki/Sea_level_equation .

Copying straight from Wikipedia, the simplest form of the sea level change $$S$$ is given by the difference between the shift in the geoid (i.e., the effects you describe) $$N$$ and the changes in vertical land positions ($$U$$): $$S = N-U .$$

It is this second factor, the response of the Earth and ocean bottom to the changing loads, that you didn't take into account, that can lead to some serious local deviations from the gravitational aspect alone.

See Figure 2-2 from http://skb.se/upload/publications/pdf/TR-09-11.pdf, copied here below: Figure 2-2. (a) Ocean height is perturbed upwards due to the gravitational attraction of the ice sheet. Solid Earth is depressed downwards due to the mass of the ice sheet. (b) Local relative sea-level fall due to a decrease in gravitational attraction by the ice (red dashed line shows previous position), combined with solid Earth rebound (green line shows previous positions) due to the ice sheet having a smaller mass. Far-field sea-level rise is due to the addition of meltwater into the ocean. Adapted from Tamisiea et al. 2003.

After staring at this figure for a while, you may appreciate that, indeed, from state (a) to (b) as the ice cap melts, the far-field sea level rises through addition of fluid water and a drop in the gravitational attraction nearby. However, also, now that the ice cap is smaller and less heavy, the solid Earth responds by moving up! Hence, 'close' to the ice cap, the sea level doesn't drop as much as you would expect from amount of fluid water + gravity alone. NB: the same happens with the oceans: adding more water to the oceans depresses them, see page 26, figure 11 here https://arxiv.org/pdf/1212.5061.pdf with the 'relative sea level change' around the Mediterranean Sea, compared to the present. Copied here below is one for 10 thousand years ago: Apparently, the sea level has risen by 30 meters in the center of the sea, but only about 22 meters on the edges. The reason is that the Mediterranean sea, from 10 thousand years ago to today, gained more mass, which caused it to depress the sea bottom / Earth's crust in the center! Hence, it had to rise even more in the center (compared to the coast lines) to create a flat surface today.

The difficult thing is that these are non-local processes. You kind of need to know about such local Mediterranean changes to make an accurate prediction for the sea level in, e.g., the Gulf of Mexico or so, because it changes the center of mass and the fluid sea water budget that can be distributed!

So with respect to your question, I absolutely think you are right that gravity is an important aspect, and 40 meters relative sea level change differences due in the hemispheres are absolutely possible for the reason you mention. If the melting is instantaneous, this is the main aspect. However, if you wait a century or so, the Earth's deformations start to kick in, can play a serious role, and can dramatically affect the sea level distribution again, in not easily foreseeable ways!