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GG 612 Excercise 1

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 = Areasphere Te4

where is the Stefan-Boltzmann constant, and Te 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 Te4

or, rearranging,

L (1-) / 4 = Te4

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

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 (IRe), using the Stefan-Boltzmann law as before. This energy (IRe) is completely absorbed by the atmospheric layer, which itself radiates energy, both upward (IRa,up) and downward (IRa,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 = IRa,up

which implies

L (1-) / 4 = Ta4

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

IRe = 2 IRa,up

which means that

Te4 = 2 Ta4

or

Te = 21/4 Ta

Now, notice that the temperature of the top layer in this calculation (Ta) is the same as the temperature of the bare earth in the Case I calculation (Te). 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.

Problems

  1. The basic calculation

    The basic greenhouse effect calculation for a 1-layer blackbody atmosphere model is so fundamental that you should practically be able to do it in your sleep. So, to start out this lab, make sure that you can begin with a blank sheet of paper, and show (without referring to the notes) that a simple 1-layer blackbody atmosphere that is transparent to solar radiation increases the surface temperature by a factor of 21/4 above what the temperature would be on a no-atmosphere planet. Try a minor modification of the derivation by assuming that the surface has nonzero albedo , reflecting a proportion of solar radiation back upwards. Would your answer change if it were the atmosphere rather than the surface that were doing the reflecting?

  2. Variant 1: 1 layer model with solar absorption in the atmosphere

    Modify the 1-layer model by assuming that the atmosphere isn't transparent, but instead absorbs a fraction of the incoming solar radiation. You may still assume that the surface albedo is zero.

    The energy budget of the atmospheric layer is now:

    L / 4 + Te4 = 2Ta4

    What is the corresponding energy budget of the surface? Solve for the temperatures and discuss how they depend on . How do your results change if the surface has a nonzero albedo instead of being completely absorbing?

  3. Variant 2: A 2-layer model

    In this exercise, you can go back to assuming that the atmosphere is transparent in the visible spectrum. Now, instead of representing the atmosphere as a single blackbody layer, we represent it as two layers which have different temperatures. They are coupled to each other and to the ground only by radiation (i.e. no heat transfer by convection).

    Write down an energy budget for each layer and for the ground, and determine all the temperatures as a function of L. How does the ground temperature compare with that of the 1-layer case?

    Example: The energy budget for the middle layer is:

    Te4 + T24 = 2 T14

    Explain this equation, and write a similar budget for the ground and for the top layer.

    If you know that the absorbed solar radiation is (1-) L, can you determine the temperature of the top layer without knowing what the other layers are doing? Explain why or why not. You may assume that the system is in equilibrium.

    How do your results change if you assume that convection couples the two atmospheric layers so strongly that their temperatures are always identical to each other?

  4. Variant 3: Nuclear Winter

    Suppose that some kind of catastrophe injected a layer of black soot into the upper atmosphere. The soot layer is perfectly absorbing both in the IR and the solar parts of the spectrum. The soot layer is so high up that it is essentially thermally isolated from the ground (i.e. there is no energy exchange between it and the ground except by radiation). Approximate the atmosphere as an IR-absorbing, solar-transparent black-body. What is the effect of the soot layer on the surface temperature? On the atmospheric temperature?

12 Sep 1999
rmyneni@bu.edu