# Formula to calculate the Maximum Ocean Level Rise due to melting Ice

The data I could find by searching the web: (Please correct any errors)

The volume of ice in the polar caps are about $26,500,000 Km^3$, most of which is in the Antarctic. After melting this will become $23,850,000 Km^3$

Earth's surface is about $510,000,000 Km^2$

The current Ocean surface is about $362,100,000 Km^2$ (71% of the surface of the planet)

I do not think the fact that earth is not a perfect sphere makes much of a difference to the result, so we can (probably) work on a sphere for the sake of the calculation.

The ocean surface will increase as the water covers more of the land (and to a lesser degree due to the radius of the sphere increasing with the rising ocean level). Are there any numbers that tell us how much? Can we use some guesses as to the average incline of land in coastal regions? Otherwise if we can get a formula based on an assumption that an average slope (angle) can be found. Failing any real numbers maybe we can work out the worst-case (90-degree incline) and best case (no incline at all) as I do not want to over complicate the formula for no real benefit in accuracy.

Secondly of course the shape of the continents is irregular. Can we ignore the capes and inlets and twists and turns in the coastline? Can we build in a factor for this? How much does it affect the end result? Can we approximate the continents as a number of simple geometric shapes, and would that actually give a more accurate answer than say assuming all land mass is in a single round island with a constant slope? Once again over complicating the formula for no real benefit in accuracy makes no sense.

We can discard the effect of any floating ice since it is already displacing the same amount of water that it would when melted. I don't know what percentage of the total ice this constitutes though, but I assume it is negligible.

This leaves the question of what amount of the WAIS ice and the EAIS ice are in fact above sea level. Since the shelves are resting on the bedrock we know that there is more above sea level than the weight of the water displaced.

Some data about the thickness of the ice shelves are here: http://www.antarcticglaciers.org/antarctica/west-antarctic-ice-sheet/

I have not found answers about the actual average thickness of the ice or the amount of ice above sea level. We may be able to work this out using the volume and surface area of the ice as well as the average height above sea level, but I have not yet found these numbers.

Any ice which is completely below sea level will cause a negative rise in sea level when melting. In fact only the part above the sea which is more than the "10%" by which water expands will contribute towards sea level rise, at 90% of the volume of that part.

The way I see it we have 2 questions to answer:

• What volume of ice is available to cause a rise in the sea levels (the part of the ice which is more than the 10% above sea level)

• What is the volume of the space that needs to be filled (the result of the shape of the earth causing an increase in volume for every meter we rise above current sea level.

Are there other factors that I forgot about?

How can we construct a formula to to use all the variables in order to get an answer to the question "how much will the ocean level rise if all the ice melted"

We could model the shapes of the continents using geometric shapes to get a more accurate estimate of the rise in level. We could even go so far as to use actual topographic data but I believe a good average will give us an accurate enough result.

Note: This question is purely about satisfying my own curiosity. I have no agenda to prove/disprove global warming. I've seen claims of numbers between 45 and 65m, etc., and I wondered "How did they work that out!" and "Is that number realistic" Are we going to arrive at the same number? Probably not, there are just too many assumptions and guesses. So please save any trolls/flames/discussions on global climate change that doesn't add to finding the answer for an appropriate forum.

Note 2: I realise that an element of this relates to finding the numbers to plug into the formula, rather than to making up the mathematical formula, but I imagine this is the kind of thing that mathematical minds do enjoy.

• It is not obvious what your mathematical question is, you seem to have a quite thorough understanding of what you want to do and just have to do it now... – Joce Jan 5 '17 at 9:57
• I'm looking for the formula, but also for input on what should go into the forumla, eg are there other factors that I forgot about, factors that I can ignore because they are negligible, and so on. – Johan Jan 5 '17 at 9:59

The ocean is a complex system, and climate scientists use very sophisticated models that take into account hundreds of factors that affect other factors in myriad ways (how does salinity change as ice sheets melt? How does this affect the surface temperature/density? How does it affect evaporation? Precipitation? What about river runoff/glacial melting? How does snow-covered ice melt differently to open ice? Will this have an effect? and on and on and on.)

I think the most widely used model for this is the Max Planck Institute's global ocean/sea ice model, which needs to be run on a computer. Take a look at chapter four "Numerical Background" of this draft document for a look at the various formulae used by this model: http://www.mpimet.mpg.de/fileadmin/models/MPIOM/DRAFT_MPIOM_TECHNICAL_REPORT.pdf

The model is continually tweaked and updated as we learn more about how these systems function. This is how climate scientists arrive at their estimates.

Are there any other factors? Here is one ...

The obvious effect of melting of all the ice would be to increase the volume of water in the oceans and therefore raise its surface level. A less obvious effect, given that most of the ice is in the Antarctic while water spreads around the globe, would be to shift the centre of gravity of the Earth (which is determined by the spatial distribution of its entire mass, including that of ice and water) away from the Antarctic, that is, northwards. This shift in the centre of gravity would in turn affect the distribution of water in the oceans, which is determined primarily by the gravitational pull of the Earth (and to a lesser extent by the centrifugal force due to the Earth's rotation and by the gravitational pull of the Moon and Sun). Hence the rise in surface level would be more at high northern than at high southern latitudes.

I've never seen the above stated, but it seems to follow from basic physical principles.

A very approximate estimate of the size of this effect can be obtained as follows:

Total mass of Earth = $6 \times 10^{24}$ kg

Volume of Antarctic ice sheet = $3 \times 10^7$ km$^3$

Volume of Arctic (ie Greenland) ice sheet = $3 \times 10^6$ km$^3$

Excess volume of Antarctic over Arctic ice sheet = $(3 \times 10^7) - (3 \times 10^6) = 2.7 \times 10^7$ km$^3$

Density of ice = $900$ kg/m$^3$ = $9 \times 10^{11}$ kg/km$^3$

Mass of excess of Antarctic over Arctic ice = $(2.7 \times 10^7) \times (9 \times 10^{11}) = 2.4 \times 10^{19}$ kg (Call this Mass A)

Mass of whole of Earth other than excess of Antarctic over Arctic ice = $(6 \times 10^{24}) - (2.4 \times 10^{19}$ kg, which is not materially different from $6 \times 10^{24}$ kg. (Call this Mass B)

Radius of Earth = $6 \times 10^3$ km

We now find the combined centre of gravity of Masses A and B, focussing on the dimension through the Earth's poles and assuming that the centre of gravity of Mass A is at the South pole while that of Mass B is at the centre of the Earth. Using the formula for the centre of gravity of two masses and measuring from the centre of the Earth, the distance of the combined centre from the centre of the Earth is given (in km) by:

$$\frac{[(2.4 \times 10^{19}) \times (6 \times 10^3)] + [(6 \times 10^{24}) \times 0]}{6 \times 10^{24}}$$

$$= \frac{1.4 \times 10^{23}}{6 \times 10^{24}}$$

$$= \frac{1.4}{60} = 0.02$$

So the shift in centre of gravity would be very approximately 20 metres. This implies that the rise in the ocean surface level would be up to 40 metres more at high northern than high southern latitudes. If, as has been estimated, the average rise in ocean level if all the ice were to melt were about 60 metres, that would imply a rise of up to 80 metres at high northern latitudes, but "only" just over 40 metres at high southern latitudes.

• This is interesting. But what about the shift in weight of the ocean water as the center of gravity moves? – Johan Aug 23 '18 at 10:49
• @Johan I think you are implying that the weight of a given mass of water may change if the Earth's centre of gravity shifts. This is correct, since weight depends not only on mass but also on distance from where the gravitational pull is effectively coming from. But I can't see that that affects my reasoning. The key point is that (leaving aside centrifugal force and pull of Moon and Sun), the distribution of the total mass of liquid water will be determined by the location of the Earth's centre of gravity. – Adam Bailey Aug 23 '18 at 18:00
• Sorry I'm not being clear. With the caps melting, the bigger mass of water from the southern cap is spread over the entire ocean surface. I gather this is what is in your calculation. This then, according to your calculation, shifts the center of mass north by 20 meters. Which would then result in more water being puled up north, including some of the existing ocean water. Which would again shift the center of gravity. My question is whether this last part is dealt with in your formula. – Johan Aug 27 '18 at 9:49
• @Johan Thank you for the clarification. No, my calculation does not allow for a further shift in the centre of gravity due to the initial shift pulling water north. Without having attempted to calculate it I think that secondary effect would be much smaller than the initial effect and probably immaterial. – Adam Bailey Aug 27 '18 at 18:07
• I would assume the opposite - there is much more water in the oceans available to shift than that which would come from the ice caps. – Johan Aug 27 '18 at 22:42