The Day After Tomorrow - Crazy Science

The Day After Tomorrow is a big Hollywood movie portraying devastating weather phenomena as the world's climate is suddenly flipped over to a new ice age by global warming. Tornadoes in Los Angeles, huge hailstones in Tokyo and a giant storm surge flooding Manhattan. Like most big-budget movies, it's better to leave your brains at the door and just enjoy the ride (and a pretty good ride it is too for The Day After Tomorrow). However, there was one effect in particular that is just begging for some critical analysis.

In the movie huge cyclonic systems form in the polar regions, wreaking frozen devastation across Europe and North America. The eyes of the cyclones funnel very cold (approximately -100 degrees Celcius) upper stratospheric down to the surface, freezing everything within seconds. There has to be something wrong with this!

Early on in the movie some of the main protagonists visit the New York Museum of Natural History, viewing a stuffed mammoth which, according to the movie, froze during the last ice age in a matter of seconds. We are supposed to believe that an object that size can equilise its temperature difference with the air in a very, very short period of time. Surely, this would require huge energy flows.

We can get some sort of idea of how much energy is lost by constructing a theoretical model of the mammoth. Let the mammoth be a water filled sphere (maybe 70% of flesh is water, so this probably isn't too bad an assumption) of radius 2 metres and weighs 1 metric tonne (= 1 x 108g ). Suppose the mammoth has a body temperature of 36 degrees celcius, similar to humans and that the water drops to 0 degrees celcius and forms ice in 60 seconds.

The specific heat for water, H, is 4.18 J/gC. That is the amount of energy required to heat a gram of liquid water by one degree celcius.

The formula for calculating the energy released by decreasing the temperature of the mass of water is:

q = mH(T2 - T1) = 1 x 108g x 4.18 J/gC x (36C - 0C) = 1.5 x 108 joules

Now, undergoes a change of state when if freezes. The heat of formation of water is -2.9 x 105 J/mol and water has a mass of 18 g/mol. So, one tonne of water freezing will release:

q = 2.9 x 105 J/mol x 1mol/18g x 1 x 108g = 1.6 x 1012 joules, making our first calculation irrelevant as far as orders of magnitude are concerned.

This energy is released over a period of 60 seconds, outputing power:

P = 1.6 x 1012 joules / 60s = 2.7 x 1010 watts, which is almost half a billion 60 watt lightbulbs or about 1/20th the total electricity generating capability of all the powerplants in the US.

The above calculations show that a phenomenal amount of energy would have to be liberated the objects snap frozen by the super-cyclones. We are talking about whole cities and landscapes in the movie. Where would this energy go? Into heating the air. I suspect that the air (maybe the cooling matter itself) could not transport the heat away fast enough to cool down as quickly as portrayed in the movie as heat tranfer in gases is relatively slow. Yet within the eye of the cyclone the air is still and great gusts don't rage as the surrounds freeze.

For all its faults, The Day After Tomorrow is great entertainment and has some useful messages about science. While no serious climatologist is predicting a sudden descent into a radically different world climate over a period of weeks (disregarding events such as comet impacts or a huge series of volcanic eruptions), it is certainly possible that we may one day reach a critical point which sees dramatic weather phenomena before the climate settles down into a new stable state, be that hotter greenhouse world or an ice age planet.

Additional Reference

Brown, T. L. & LeMay, H. E., (1991), Chemistry: The Central Science, (Englewood Cliffs: Prentice-Hall)