GG 312: Global Climate Change and Environmental Impacts (Fall 2001)

 

The Layer Model Approximation to the Greenhouse Effect

Designed by Prof. David Archer, University of Chicago

Overview

This lab exercises your understanding of "layer models" of the greenhouse effect. These are also called "isothermal slab models," or "glass atmosphere models." They have serious shortcomings in their neglect of the way thermodynamics and convection alters the vertical temperature structure of real atmospheres, but they are still useful in understanding the basic way the greenhouse effect operates.

Case 1: A Bare Blackbody Earth

Consider a spherical planet with no atmosphere. Energy pours in as visible light from the sun, and energy is lost by emission of infrared light from the surface of the earth.

The amount of energy coming in to the planet is given by

Flux In = (1-) r² L

where is the albedo, defined as the proportion of the incident light which gets reflected back to space. Snow has a high albedo, and black velvet paintings of Elvis have generally a low albedo (unless they have sparkles on them). L is the intensity of sunshine, which at the mean radius of the earth's orbit equals 1380 W/m². The factor r² reflects the fact that the sphere intercepts a "disk" of radiation of radius r.

For outgoing energy, the process is blackbody radiation. We know from the Stephan-Boltzmann Radiation law that for a blackbody the rate at which energy is lost by outgoing infrared light can be calculated as a simple function of temperature, as

Flux Out = Area_sphere T_e⁴

where is the Stefan-Boltzmann constant, and T_e is temperature of the surface of the earth. The area of a sphere is 4 r². Of course we're assuming here that the surface of the planet is all of uniform temperature, but this kind of cavalier cowboy-style assumption making is what thought experiments are for.

If we set the incoming and outgoing energy fluxes equal to each other, we end up with

(1-) r² L = 4 r² T_e⁴

or, rearranging,

L (1-) / 4 = T_e⁴

which gives us a relation between the solar constant L, the albedo of the planet , and the temperature at the surface of the earth T_e.

Case II: A Single Layer of IR Absorber in the Atmosphere

Now let's clothe our planet with an atmosphere. We'll do it in a simplified "idealized" way to start with, using a thought experiment known as the "layer model". The idea is to wrap the earth in a layer which has the properties of optical transparency in the visible and complete absorbtion in the infrared.

This complicates things a bit, as we have to calculate the energy balance of both the earth and its atmospheric wrapper. The picture looks like this:

The energy coming in from the sun is the same as before, and it makes it through the atmospheric "layer" unchanged. Infrared light is emitted from the ground (IR_e), using the Stefan-Boltzmann law as before. This energy (IR_e) is completely absorbed by the atmospheric layer, which itself radiates energy, both upward (IR_a,up) and downward (IR_a,down). (In case you are wondering: There are just two directions, rather than a whole sphere's worth of directions, because the world extends infinitely far in both directions, and is completely uniform. So some light shines off at angles other than plain up and down, but what goes out of the picture by flowing sideways is balanced by what comes into the picture from next-door. If you weren't wondering about the two directions, don't worry about it.)

The crux of the whole thing, the whole idea of the greenhouse effect, is based on the assumption that some of the IR from the atmospheric layer shines downward. OK here's how it works. We know that the energy going into the layer has to balance the energy leaving the layer.

We know that half of the energy leaving the layer makes it to space. Therefore, the temperature of the layer is determined by the energy balance as we used it before

Solar in = IR_a,up

which implies

L (1-) / 4 = T_a⁴

But the energy flowing upward from the atmosphere (IR_a,up) is only half of the energy going in to the atmospheric layer, which comes from the surface of the earth (IR_e). That is to say,

IR_e = 2 IR_a,up

which means that

T_e⁴ = 2 T_a⁴

or

T_e = 2^1/4 T_a

Now, notice that the temperature of the top layer in this calculation (T_a) is the same as the temperature of the bare earth in the Case I calculation (T_e). We can therefore see that the planetary surface is warmed up by the greenhouse atmosphere than the no-atmosphere case by a factor of the fourth root of 2, Or about 20%.

This is the greenhouse effect.

 

 

Sep-03-2001
rmyneni@bu.edu