Wednesday, July 13, 2011

Black-Body Tropopause

Our CC7 program provides six different ways to heat the atmospheric cell array. In CC8, we eliminate most of these and replace them with only three: Day, Night, and Cycle. The others were useful in checking the performance of the simulation, but are no longer necessary. Because we now have a good understanding of the relationship between simulation time, impetus for circulation, and program iterations, we are now able to express the Sun's heat in W/m2 instead of the less realistic K/iteration of our earlier programs. We represent the incoming solar power with Q_sun instead of the previous Q_heating. Furthermore, we can use Stefan's Law directly upon the top cells, as if they were black and the gas above them were transparent. Our program now contains a value for Stefan's Constant, which we set to 5.7×10−8.

For our convenience, we display the time of day in hours in the main window. Time 12.0 hr is noon, when the Sun is certain to shine, and midnight is 0.0 hr. We display the current solar power in W/m2. Previously, we applied a sinusoidal variation in the Sun's power during the day, but now we simply turn the Sun's power on to Q_sun during daylight hours, and to zero during the night. As before, however, the Sun will shine for a fraction of the day given by day_fraction.

We run the program with Day heating and the Sun's power set to 350 W/m2. We have ke_fraction at 0.0 and we un-check left_only. Our cells have mass 333 kg/m2. Their specific heat capacity at constant pressure is 1 kJ/K. Thus the lower cells warm up at 0.001 K/s, which matches our previous Q_heating of 0.001 K/iteration. We allow the array to reach equilibrium, which takes a long time: five million iterations, or one hours on our lap-top. You will find the equilibrium state in Day_1.txt. You can load it into CC8 with the Load button. At equilibrium, the top row's average temperature is 280.0 K. Applying Stefan's Law, the top cells should radiate 350 W/m2, which is what we expect, since that's what we are putting in. The surface cell average temperature is 335.4 K, giving us the 55-K drop from the surface to the tropopause. This drop is consistent with our previous results.

We run the program with Q_sun set to 700 W/m2 and Cycle heating to simulate day and night. We have day_fraction set to 0.5. After a million iterations we see the temperature of the bottom and top rows varying by a few degrees during night and day, as we did in Rotating Greenhouse.

The equilibrium surface temperature of 59°C (335 K) is much hotter than the surface of the Earth (around 14°C), even though 350 W/m2 is the average power of the Sun. The Earth is cooler because its surface and lower atmospheric layers are able to radiate almost half their heat directly into space, assuming there is no cloud cover (see Total Escaping Power and subsequent posts).

Our next step is to allow the surface cells to radiate directly into space, and we will see how the surface cools down as a result. We must implement the surface and tropopause radiation before we can model the effect of clouds.