**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

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.

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-) r^{2} 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/m^{2}. The factor r^{2} 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_{e}^{4}

where is the Stefan-Boltzmann
constant, and T_{e} is temperature of the surface of the
earth. The area of a sphere is 4
r^{2}. 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-)
*r*^{2} L = 4 r^{2}
T_{e}^{4}

or, rearranging,

L (1-) / 4 = T_{e}^{4}

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}.

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_{a}^{4}

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_{e}^{4} = 2 T_{a}^{4}

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.