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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 × 106 m

Size of illuminated disk = PI × r2 = 1.3 × 1014 m2

Sunlight at top of atmosphere = 1366 W/m2

Continuous solar watts, p = 1366 W/m2 × 1.3 × 1014 m2 = 1.8 × 1017 W

Solar energy per year = p × seconds/year = 1.8 × 1017 × 3.15 × 107 = 5.6 × 1024 J

Volume of Greenland and Antarctic ice caps = 33 × 106 km3 = 33 × 1015 m3 = 33 × 1018 l

Weight of ice caps = 0.92 kg/l × 33 × 1018 l = 3 × 1019 kg

Heat to melt the ice caps = 333.55 × 103 J/kg × 3 × 1019 kg = 1 × 1025J

Years of total solar power received by earth required to melt the ice caps =
1 × 1025J / 5.6 × 1024 J = 1.8 year

And you might think, that’s a lot, right? But consider the size of the ocean: 1.3 × 109 km3, versus 3.3 × 107 km3. It takes 80 calories to melt a gram of ice, and it takes 1 calorie to heat a gram of liquid water by 1 degree C. That is, to melt all the ice (but not warm it to ocean temperature), the oceans would cool on average by (3.3 × 80)/130 = 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. There’s an enormous amount of heat stored in the oceans.

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