Wednesday, May 5, 2010

Dissipation by Convection

Now we return to our Extreme Greenhouse thought experiment. In our previous post on the subject, Work by Convection, we showed that atmospheric convection does work. We wondered where the energy for that work comes from, and where it goes to.

In our Atmospheric Convection post, the air taking part in our convection cycle warmed by 15 K while in contact with the warm surface of the Body, and cooled by 10 K while in the tropopause radiating its heat into space. The warming and cooling took place at constant pressure. The amount of heat required to warm 1 kg of a gas by 1 K at constant pressure is the heat capacity at constant pressure of the gas. Air is a close to an ideal gas. Its heat capacity at constant pressure is close to 1 kJ/kg/K at all the temperatures and pressures we are likely to encounter in the atmosphere of our Extreme Greenhouse. Each 1 kg of air absorbs 15 kJ of heat from the Body and radiates 10 kJ of heat into space. There are 5 kJ unaccounted for. These 5 kJ are converted into work.

Each kilogram of air taking part in the cycle rushes upwards with such force that it will do 5 kJ of work for us and still reach the tropopause. As it rushes upwards, it mixes with other air. Its momentum is diverted into whirlwinds. The viscosity of air is low, but it is not zero. The air moves upwards and sideways and in circles, rubbing against itself until the viscous friction it encounters consumes all the work available from convection. Viscosity turns the 5 kJ into heat.

As a result, the air in the tropopause will be warmer by 5 K than we expected with perfect, adiabatic expansion. Given that the air in the tropopause is over 100 K cooler than the air at the surface of the Body, this 5 K makes little difference to the size of our greenhouse effect. But it does mean that the air must cool in the tropopause by 15 K instead of 10 K, or else our convection cycle will not be closed. All 15 kJ of heat absorbed by the air from the Body will be radiated into space at the tropopause.

We know from our previous calculations that the atmosphere of our Extreme Greenhouse must transport 350 W/m2 from the Body surface to the tropopause (the boundary between the Filter Gas and the Upper Gas). Of these 350 W/m2, roughly 230 W/m2 will be carried directly by the heat capacity of the atmosphere, but the remaining 120 W/m2 will first be turned into work, later converted into heat by viscous action, and finally radiated into space.

The atmosphere of our Extreme Greenhouse has 120 W/m2 available to cause weather. This work is available only because the upper atmosphere is radiating heat into space. If there were no greenhouse gases in our Extreme Greenhouse, there would be no heat flow to the upper atmosphere, and no convection cycle to generate weather.

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