91.3% of ice in the world is found in Antarctica, 8.1% is in Greenland and the remaining 0.6% is in mountain glaciers, snowfields etc.
Both Antarctica and Greenland are land-masses covered with ice. Their respective highest points are 4,892 metres (Mount Vinson) and 3,694 metres (Ginnbjorn).
The adiabatic lapse rate (ALR) provides that temperature decreases with altitude, in dry air this is 0.97?C per 100 metres (DALR). Because Antarctica and Greenland are bitterly cold the air is very dry, Antarctica being the largest desert on Earth.
The DALR means that at an altitude of 4,000 metres the temperature will be 38.8?C less than at sea-level. The average global temperature (AGT) is 14.5?C.
As ice extent recedes it affects the local and regional climatologies through reduced reflectance, increase solar radiation absorption and disruption to the associated feedback mechanisms. This increases sensitivity to temperature fluctuation such that the ALR isn't absolute. In short, despite the extent and altitude of the ice-caps, it would only require an increase in the AGT of about 12?C sustained over 100,000 years to melt all the ice on Earth.
Even in the worst case scenario, global warming isn't going to raise the AGT enough for this to happen. There are however, situations in nature which can; to wit the orbit of the solar system around the galactic centre. Not that we need to concern ourselves with this as it's not going to happen for another 70 million years or so.
To experience an iceless world means going beyond global warming, but let's do it anyway.
The amount of water vapour in the atmosphere is dictated by the saturation fraction - essentially a measure of how much water vapour a parcel of air can contain before it condenses out and falls as precipitation. The only variable we need concern ourselves with is temperature (others such as the presence of CCN, supercooling, pressure etc have minimal effect).
For air at 14.5?C this is about 1.3% by mass. Global warming is predicted to increase the AGT by 2 to 3?C by 2100, this would push the saturation fraction up to 1.55% (17.5?C). In our runaway scenario we're increasing the AGT by 12?C, the SF would then be just over 2.0% (26.5?C).
Our atmosphere isn't at saturation vapour point (SVP), instead of 1.3% by mass of water vapour it's 1.0% or 77% SVP capacity (or 13 trillion tonnes).
At 17.5?C SF is 1.55%, at 77% SVP capacity this means 1.19% by mass (15Tt) of water vapour. At 26.5?C SF is 2.00%, at 77% SVP capacity this means 1.54% by mass (20Tt) of water vapour.
Clouds are three-dimensional therefore the cross-sectional area (XSA) will not increase in the same ratios. Assigning a value of 1.000 to present cloud XSA means that at 17.5?C the XSA will be 1.123 and at 26.5?C it will be 1.334.
Now we need to convert these figures into energies. Earth receives 174 petawatts of energy from the Sun each year (Total Solar Irradiance or TSI) of which 35PW are reflected by the clouds. If XSA increases by 12.3% then 39.3PW will be reflected, an increase of 33.4% would see 46.7PW being reflected.
There's a further 17PW that's reflected by the atmosphere and Earth's surface, we need to discount this and concentrate on the fraction of TSI that's relevant to the climate. The figures would be 122PW at 14.5?C, 117.7PW at 17.5?C and 110.3PW at 26.5?C.
The 122PW we currently receive provides the 34?C of warming we experience on Earth, as the figure decreases we're going to get less warming.
But, we also need to proportionally account for equilibrium climate sensitivity (ECS) - the amount by which greenhouse gas concentrations would have to rise in order to result in an increase in the temperature by X. ECS is generally held to be about 3.0?C for an increase of CO2 from 280 to 560 parts per million by volume. In other words, that 3?C rise will only happen if concentrations of greenhouse gases double. To get the 12?C increase they'd need to increase by 1600%.
At 17.5?C / 26.5?C there will be 12.3% / 33.4% more clouds, 39.3PW / 46.7PW of TSI will be reflected, the AGT would fall by 1.052?C / 3.862?C, ECS would necessitate a 3.0?C / 12.0?C rise in temps. The net effect would be an increase in the AGT of 1.948?C / 8.138?C.
Note: The above is overly simplified, only major factors have been considered. Most of the stated figures can be accurately calculated except for ECS which could be anywhere between 1.5?C and 4.5?C. Allow a further 50% margin of error due to all the variables not taken into account.
SUMMARY: Cloud cover will increase as the temperature rises, the mechanism by which the temperature increases is the presence of greenhouse gases in the atmosphere. The warming consequent to an enhanced greenhouse effect exceeds the cooling from additional cloud cover.
