## Numbers that were larger than I had imagined.

### February 4, 2012

From a Slashdot discussion of how fast ice caps could or could not melt, this time with all the arithmetic corrected and embedded references.

Radius of earth, r = 6.4 × 10^{6} m

Size of illuminated disk = PI × r^{2} = 1.3 × 10^{14} m^{2}

Sunlight at top of atmosphere = 1366 W/m^{2}

Continuous solar watts, p = 1366 W/m^{2} × 1.3 × 10^{14} m^{2} = 1.8 × 10^{17} W

Solar energy per year = p × seconds/year = 1.8 × 10^{17} × 3.15 × 10^{7} = 5.6 × 10^{24} J

Volume of Greenland and Antarctic ice caps = 33 × 10^{6} km^{3} = 33 × 10^{15} m^{3} = 33 × 10^{18} l)

(10^{9} m^{3} per km^{3}; 1000 l per m^{3})

Weight of ice caps = 0.92 kg/l × 33 × 10^{18} l = 3 × 10^{19} kg

Heat to melt the ice caps = 333.55 × 10^{3} J/kg × 3 × 10^{19} kg = 1 × 10^{25}J

**Years of total solar power received by earth required to melt the ice caps =
1 × 10 ^{25}J / 5.6 × 10^{24} J = 1.8 year**

And you might think, that’s a lot, right? But consider the size of the ocean and its heat capacity:

Volume of oceans = 1.3 × 10^{9} km^{3} = 1.3 × 10^{18} m^{3} = 1.3 × 10^{21}l = 1.3 × 10^{24}g

To heat oceans by 1 degree C = 1.3 × 10^{24} calories = 5.4 × 10^{24}J — about the same as total annual solar energy illuminating earth.

That is, to melt the ice caps entirely (assuming only heat of fusion, but that’s a lot) would only cool the oceans by less than 2 degrees C.

And this is why global warming is slow, and why changes in ocean currents can produce much larger effects in the short-term. The oceans store a boggling amount of heat energy.

Also to note — the average ocean depth is 12,000 feet, and water’s volume coefficent of thermal expansion is 0.000207, so cooling the oceans by 2 degrees C would lower sea levels by about 5 feet — but melting the ice caps would raise them by a couple of hundred feet (the crude number is 300, but parts of Greenland and Antarctica are below sea level, and hundreds of feet of sea level rise expands the surface area of the ocean a bit, so I understand it works out to about 200 feet if we melt all the ice through global warming).