Fun with ice melting math

[Mr Bean]

No idea how to go about this to produce accurate numbers but here's the hypothetical event in question.


Tomorrows for reason unknown, the earth's rotation changes, no one dies, but the South pole becomes much more angled towards the sun, and as a result the South poll now gets Florida style temp. As in right now the temp jumps to 70*F at night, 90*F during the day during summer. It's winter ATM so it's only going to be 60*F in the day and 45*F during the night.

How quickly will the ice melt, and what are the effects 1 day, 1 week and 1 month later. Assume the temp holds steady at an average of 55*F.

Go!

[Hawkwings]

We all die?

Actually, isn't the north pole no longer covered by ice?

[Mr Bean]

We all die?

Actually, isn't the north pole no longer covered by ice?

Fine, changed it to the South pole then, that's still covered in ice to be sure, hell it's even more ice.


I'm looking at how fast the ice would melt, how much run-off we could expect, how long it would take, what local and global effects it would have.

The pole change is not the issue, I'm just trying to find out if we raised the temperature of the South pole to an average 55*F during the Winter, how quick how much energy would be released and how high sealevel would climb because of it.

[Ariphaos]

The pole change is not the issue, I'm just trying to find out if we raised the temperature of the South pole to an average 55*F during the Winter, how quick how much energy would be released and how high sealevel would climb because of it.

It's been far to long since I did ODE's to run the math for convection loss, but it would also help to know the average angle of sunlight involved, as well as cloud cover levels - or should we assume they stay the same?

[Sikon]

Tomorrows for reason unknown, the earth's rotation changes, no one dies, but the South pole becomes much more angled towards the sun, and as a result the South poll now gets Florida style temp. As in right now the temp jumps to 70*F at night, 90*F during the day during summer. It's winter ATM so it's only going to be 60*F in the day and 45*F during the night.

How quickly will the ice melt, and what are the effects 1 day, 1 week and 1 month later. Assume the temp holds steady at an average of 55*F.

Actually, you don't know that the air temperature will reach anything close to those arbitrary figures, even under magical change to the orientation of earth. There's the cooling effect of the ice, like a 30 quadrillion-ton giant heat sink, plus its reflectivity, as the ice will remain around for a long time.

This would have vastly less effect than you appear to expect in short timeframes like those mentioned in the opening post.

The southern pole already receives a lot of sunlight during part of the year. An illustration:

During the winter at the South Pole, the Sun never rises. (The 6 months of night is another reason this site is good for astronomical observations.) During the summer, the Sun never sets! It goes all the way around the sky every day. (For more information, see the tour page on Life at the South Pole and the Sun.) All this light can actually be dangerous. Because the South Pole is a high altitude site, the sunlight is very intense. And, in addition, you have lots of reflected light from all the snow. You can't go outside at all without sunglasses (with uv-blocking coating). In fact, there is a substantial risk of snow blindness, where you literally sunburn your eyes -- it can be serious and painful.

From here

The following shows how in January, for example, the 24-hour sunlight gives the south pole more incoming solar radiation than even the equator, though the perpetual night during the opposite portion of the year is the opposite:



However, the sunlight is largely reflected, especially with the high albedo of ice and snow, reducing the amount of solar energy actually absorbed, e.g.:



Looking at available versus absorbed solar insolation data, the most average available solar insolation at any location on earth's surface is around 400 W/m^2, but the maximum absorbed is around 300 W/m^2. And the amount absorbed at equatorial regions is higher than it would be if those regions were covered with up to thousands of meters thickness of high-reflectivity ice.

(Although the solar constant in space is 1370 W/m^2 at earth's distance from the sun, one major factor is that a sphere has an area of 4 <pi>r^2, versus the starting amount of sunlight intercepted being over an area of <pi>r^2, already no more than a quarter as much on average; after the atmosphere, global mean available solar radiation is 180 W/m^2, though some of that is reflected).

At this point, let's stop answering the opening post scenario of magical change in earth's orientation.

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Let's rather address the possible intent of the opening post by just implausibly pretending that the south pole magically receives up to 200 W/m^2 more average solar insolation, neglecting albedo and pretending up to that much is absorbed.

The volume of ice in Antarctica is about 29 million cubic kilometers, with an area of 13 million square kilometers, so its average thickness is around 2200 meters.

Unrealistically pretend 200 watts per square meter goes into melting ice. (The actual amount of heat absorbed, and ice melted would be less; indeed, in the short-term, the temperature would not initially even rise above freezing in inner Antarctica).

Neglecting energy requirements beyond the heat of fusion alone, consider that the density of ice is around 920 kg/m^3, while its heat of fusion is 334 kJ/kg. So up to 200 W/m^2 melts under 6.5E-7 meters of ice per second, under 0.06 meters per day, and under 20 meters per year.

The area of earth's oceans is 340 million square kilometers, compared to the 13 million square kilometer area of the ice sheet. So average sea level rise is estimated as under 0.7 meters per year.

Nominally it takes upwards of 100 years to melt all of the ice.

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Since one is calculating for impossible scenarios anyway, what about the opening post's air temperature of 55 degrees Fahrenheit?

The ice sheet could be treated as a semi-infinite solid for a simplified model. Actually, let's just note that the heat transfer may be approximated as q = h * A * (T2-T1), with heat transfer per unit area being h * (T2-T1) where h is the heat transfer coefficient and T2-T1 the temperature difference. A heat transfer coefficient could be estimated from a correlation, e.g. if one treated wind as creating forced convection with a corresponding Reynold's number, etc. But this is just an order of magnitude estimate anyway.

Let's neglect wind and give an approximate estimate. To the nearest order of magnitude, the heat transfer coefficient in air between freezing and this temperature is around 10 W/m^2*K for natural convection. Ice that is warmed up to the point that its heat of fusion must be delivered to melt it is 32 degrees Fahrenheit. So the temperature difference is ~ 13 degrees Celsius then. Convective heat transfer without wind is on the order of 130 W/m^2.

That's a very inexact figure, to say the least, but, still, one can see the general idea; it's not thousands of watts per square meter.

Like the imaginary scenario of up to 200 W/m^2 calculated before, melting ice that averages kilometers thick would take a long time.

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All of the preceding like the arbitrary 200 W/m^2 is impossible and different by orders of magnitude than real-world global warming. Overall world average radiative forcing is around 1.6 W/m^2 as mentioned in past posts. There is absolutely no possible way for radiative forcing orders of magnitude greater, not now, not a century from now, not centuries from now. The most radiative forcing that can be reached in the future is a few W/m^2. There's only so much remaining fossil fuel in the ground and emissions possible, and greenhouse gases affect only some of the spectrum.



In reality, the decrease to Antartica's ice sheet has been "152 (plus or minus 80) cubic kilometers of ice annually between April 2002 and August 2005 [... which] represents a change of about 0.4 millimeters (.016 inches) per year to global sea level." (From here). The annual amount melted is about 1/200000th of the total of 30 million cubic kilometers.

That's undesirable, but it is very much unrelated to the opening post's imagined scenario.

[Sikon]

Actually, isn't the north pole no longer covered by ice?

Part of the year, there's thin sea ice around much of the north pole, with a thickness up to several meters and usually thinner, like this illustration.

Most fluctuation is seasonal, but, since the north pole ice is literally three orders of magnitude thinner on average than the kilometers-thick Antarctic ice sheet, that sea ice has been vastly more affected by global warming.

[Mr Bean]

The pole thing is un-important. What I'm trying to find out is if we gave the South Pole Florida style temperatures(5 Million Heat lambs, a heat ray, act of Q whatever) how quickly the ice would melt relatively speaking.

Let me rephrase my time definitions then, 1 month, 1 year, 10 years.

[Sikon]

The pole thing is un-important. What I'm trying to find out is if we gave the South Pole Florida style temperatures(5 Million Heat lambs, a heat ray, act of Q whatever) how quickly the ice would melt relatively speaking.

Let me rephrase my time definitions then, 1 month, 1 year, 10 years.

Please see the preceding discussion, as a middle part of it already covers even the impossible scenario of high air temperature and the resulting convective heat transfer. The time for more than a portion to melt is longer than this century.

Although calculations are the proper way of looking at this, the general idea can be put intuitively too. Anyone who has lived in an area where there are large piles of snow and ice up to a half-dozen feet high left by snowplows may have seen how those can take a number of days or weeks to melt, even while thin snow on the grass melts more quickly. Now, imagine they were literally 1000 times thicker compressed ice (kilometers thick), influencing the whole local climate with their reflectance. If quadrillions of tons covering the continent instead of a several ton pile, they would take a long time to melt.

[Sikon]

Hypothetical scenarios aside, one can emphasize how much more sophisticated analysis exists for the actual situation. For example, a large report here estimates in detail the effects on the U.S. of a 1 meter sea level rise by 2100 (more than the recent IPCC estimate), determining that cumulatively ~ 7000 square miles (0.2% of U.S. land area) would be flooded, with a cost on the order of $1 billion per year due to that plus rebuilding roads, elevating structures, constructing levees, etc., from $73-$111 billion spread over 100 years.* There are other effects of climate change, such as a mixture of harm and benefit to agriculture, with decreased precipitation in some areas while increase in others, a beneficial CO2 fertilization effect but a temperature change mostly harmful in lower latitudes, etc.

Overall, net harm from climate change has been estimated as around 1% to 4% of U.S. GDP lost in economic damage in event of an eventual doubling in atmospheric CO2-equivalent radiative forcing change.

* The figure could be moderately adjusted for dollar conversion to 2007-dollars, but just the order of magnitude is of relevance here anyway.

The impossible, imaginary opening post scenario is an interesting exercise in illustrating the enormous thermal capacitance of that much ice, but it is irrelevant to the real world, unlike the much more advanced analysis that already exists in available publications.

[Ariphaos]

The pole thing is un-important. What I'm trying to find out is if we gave the South Pole Florida style temperatures(5 Million Heat lambs, a heat ray, act of Q whatever) how quickly the ice would melt relatively speaking.

Let me rephrase my time definitions then, 1 month, 1 year, 10 years.

The method is important. If it's an act of Q, and he holds the atmospheric temperature to a constant 300 Kelvin, there is still the question of average wind speed. Does it rain, now that it's warm enough to do so?

If Q magically makes the entire antarctic 300 kelvin, then I think the total flooding would be around 180 meters. Pretty colossal. But if all he's doing is warming the atmosphere, you need to crack open Newton's Law of Cooling and run the differential equation, but individual atmospheric effects are still important. Especially the rain question.

The amusing thing is, if precipitation is allowed, it could mean that the size of the ice cap initially increases. There are so many factors involved here that in order to get a straight answer, you are going to have to be REALLY SPECIFIC, and probably ask for decade to decade plots.

[Mr Bean]

OK lets be specific
Next Friday at 0001 AM, the temperature of the south pole(Defined as physical south out to 2000 miles) will climb as high as 90*F during the day, and 55*F during the night. Average air temperature will be 70*F
How quickly will the ice melt, and what local(IE south America) affects can we expect?

How quickly will the sea-level rise?
And what effect would this have on the South Pole's physical landmass.

[Sikon]

If Q magically makes the entire antarctic 300 kelvin, then I think the total flooding would be around 180 meters.

The figure's not right as that would be flooding of coastal areas up to 80 meters rather than 180 meters altitude above sea level, from the volume of total ice in Antarctica (which is 3.0E7 km^3) divided by the ocean surface area (which is 3.4E8 km^2), aside from adjustment for the melted water having around 90% the volume of the original ice due to the slight density difference ... though it would take magic to do that anytime soon.

Although the above is seen from calculating with the preceding figures, let's give another reference to illustrate:

Sea Level Rise Potential:

Greenland: ~7 meters
West Antarctica (WAIS): ~5 meters
East + West Antarctica: ~70 meters [...]

Sea level rise projection for 21st Century, all sources: 9-88cm [...]

Models of Antarctic Ice Sheet [...]

East Antarctic ice sheet (~70m sea level rise) may survive up to 20 °C local warming.

West Antarctic ice sheet predicted to melt irreversibly but slowly for 10 °C [eventual, post 21st-century] local warming, losing 3m in 1,000 years.

From here.

The amusing thing is, if precipitation is allowed, it could mean that the size of the ice cap initially increases.

Depending upon details and assumptions, perhaps indeed, e.g. in regard to the thicker inland parts; there's the analogy of the real world situation where ice in the interior of Antarctica is actually growing thicker, even +1.8 cm/yr in parts, even though greater ice loss at the coast results in ~ 1/200000th of the total melting annually for its net contribution of ~ 0.4 mm/yr to global sea level rise.

[Ariphaos]

OK lets be specific
Next Friday at 0001 AM, the temperature of the south pole(Defined as physical south out to 2000 miles) will climb as high as 90*F during the day, and 55*F during the night. Average air temperature will be 70*F
How quickly will the ice melt, and what local(IE south America) affects can we expect?

How quickly will the sea-level rise?
And what effect would this have on the South Pole's physical landmass.

-what- rises in temperature? Can we assume it's magically hot and remains that way?

Just the atmosphere? Does the ocean warm too? The landmass? I'm assuming you don't mean the ice cap itself, as that makes the answer easy.

The easy answer is 61 meters over the course of the next week as everything melts[/url] (I was mixing up feet and meters in my previous comment). Some pretty big tsunamis involved, there.

-

In general, though, the ice cap would probably grow initially and the sea level would fall - they're at 236 Kelvin on average and are going to start absorbing precipitation until the top portions at least reach melting points. Though if the landmass is warmed too then you'll have them sliding into the ocean at a much quicker pace, but that would still take tens of thousands of years.

[Ariphaos]

The figure's not right as that would be flooding of coastal areas up to 80 meters rather than 180 meters altitude above sea level, from the volume of total ice in Antarctica

Yeah, like I mentioned I mixed up feet and meters, apologies.

[Sikon]

And what effect would this have on the South Pole's physical landmass.

After the future time by which all of the ice melted (up to ~ 70 meters eventual sea level rise, in a timeframe not precisely determined but not this century at least), the physical landmass underneath would rebound upwards eventually up to a kilometer in parts. The weight of the current ice (up to 15000 feet thick in places) would no longer be pressing it downwards. Such is a greater version of what happened elsewhere after the last ice age. In time, wildlife would settle the new, temperate, inhabitable continent, like what happens with new volcanic islands: first the hardiest plants, then more vegetation, then animals ... although the scenario would obviously destroy the original polar ecosystem as well as indirectly devastate coastal areas worldwide.

[Sikon]

EDIT: The volcanic island analogy is imperfect, since Antarctica is not just rock underneath the ice. Fossils and coal indicate its past, as many millions of years ago it once had a sub-tropical climate (not always having been at the South Pole, moving with plate tectonics).

[Ariphaos]

After the future time by which all of the ice melted (up to ~ 70 meters eventual sea level rise, in a timeframe not precisely determined but not this century at least), the physical landmass underneath would rebound upwards eventually up to a kilometer in parts. The weight of the current ice (up to 15000 feet thick in places) would no longer be pressing it downwards. Such is a greater version of what happened elsewhere after the last ice age. In time, wildlife would settle the new, temperate, inhabitable continent, like what happens with new volcanic islands: first the hardiest plants, then more vegetation, then animals ... although the scenario would obviously destroy the original polar ecosystem as well as indirectly devastate coastal areas worldwide.

The Great Lakes region is still rising, isn't it?

Anyway, how lush it gets partly depends on whether or not the surrounding ocean remains so cold - Antarctica is a desert and unless the surrounding ocean is warmed, it's going to remain that way. When Chile was connected with the continent, warm water was forced around it and it kept the entire continent relatively warm - enough for forests, etc. When they separated the remaining warmth of what is now the antarctic ocean fueled the creation of the ice cap.