Emission Height and the Greenhouse Effect

(c) 2023 by Barton Paul Levenson

The Greenhouse Effect.

Let's examine how the greenhouse effect works in terms of emission height and the lapse rate.

We start with an example of a planet with the same illumination and albedo as Earth, but no greenhouse gases in its atmosphere. Mean top of atmosphere illumination is 340 watts per square meter, but only 240 W m^-2 is absorbed because of the nonzero albedo. The surface gets all of that 240 W m^-2, so it must radiate an equal amount to be in energy balance. By the Stefan Boltzmann law, its temperature is 255 K. Radiation to space is directly from the surface, unimpeded by the atmosphere, so the emission height is 0.

The Effect of Greenhouse Gases

Now we introduce greenhouse gases into the atmosphere. Absorption varies radically with wavelength, but we'll simplify matters by using a semigray approximation (gray = same coefficient of absorption at all wavelengths, semigray = one coefficient in one range of wavelengths and another elsewhere). We'll approximate the absorption in Earth's atmosphere by having it take place only from wavelengths 4-8 and 12-200 microns, with absorption being total in those bands. Absorption is 0 from 0-4 and from 8-12 microns, approximating the shortwave window and the thermal infrared window.

The Planck curve indicates that 0.0727 of the flux put out by a 255 K object falls between 4 and 8 microns, while 0.7122 falls between 12 and 200 microns, for a total of 0.7849. 78.49% of the radiation from the surface is blocked (absorbed). That's 0.7849 x 240 = 188 watts per square meter. Only 52 watts per square meter is now reaching space from the surface.

But the atmosphere is absorbing energy, thanks to the greenhouse gases, so it must radiate as well. And unlike the ground, it can radiate both up and down. So it's radiating half of its 188 W m^-2 input up and half down, 94 W m^-2. This corresponds to a temperature of 202 K.

52 watts per square meter is reaching space from the ground and 94 from the atmosphere, for a total of 146 W m^-2. So the climate system is absorbing 240 W m^-2, but losing only 146 W m^-2. With more energy coming in than going out, it must heat up.

Equilibrium is achieved when the system is radiating as much out as it is getting in. At equilibrium, we have

240 = (1 - 0.7849) σ Tg⁴ + 0.5 x 0.7849 x σ Tg⁴

where the first term to the right of the equals sign is the ground contribution and the second term is the atmosphere contribution. Doing the math,

240 = 0.2151 σ Tg⁴ + 0.39245 σ Tg⁴

240 = 0.60755 σ Tg⁴

395 = σ Tg⁴

Tg = 289 K. Not spot on, but close. This is a very crude model, after all. The atmosphere is at a temperature of 229 K. For extra credit, figure out how I got that figure.

Introducing the Lapse Rate

Now, we assume our model planet has the same environmental lapse rate as Earth: 6.5 K km^-1. If the surface is at 289 K and the atmosphere is at 229 K, the atmosphere must be radiating to space from an altitude of 9.23 km. And since the radiation from the atmosphere is 0.39245 x 395 = 155 W m^-2, and the radiation reaching space from the ground is 0.2151 x 395 = 85 W m^-2, then 64.58% of the radiation is coming from the atmosphere and the rest from the ground. So the mean emission height is (1 - 0.6458) x 0 km + 0.6458 x 9.23 km or 5.96 km.

The presence of greenhouse gases in the atmosphere has raised the emission height.

As I've said before, this is a very crude model. In reality, absorption varies with wavelength, and with pressure and temperature (thus with altitude), different amounts of greenhouse gases are present at different altitudes, the principal absorber (water vapor) has a very spotty and rapidly changing distribution over the surface, radiation reaches space from all kinds of altitudes, and I haven't dealt with complicating factors like convection, conduction, evapotranspiration, absorption of shortwave radiation by the atmosphere, clouds, etc., etc., etc. But it illustrates the basic principle. To get everything more exactly you'd need to write a complete radiative-convective column model of the atmosphere, or better yet, a 3-D general circulation model.

But in general terms:

T_s = T_e + Γ H

(Hansen et al. 1981) where Ts is surface temperature, Te equilibrium temperature, Gamma lapse rate, and H mean emission height. For the actual situation on present day Earth, we have

288 = 255 + 6.5 (5.1)

For some other planets, we have

Venus: 735.3 = 227  + 8.8 (57.8)

Mars:   214   = 210   + 2.5  (1.6)

Titan:   93.7 =  82.7 + 1.38 (7.97)

Reference