Global Warming - a Simplified Greenhouse Effect Model

I have seen this extremely over simplified model used to try and explain the Greenhouse Effect. It is repeated here so that its shortcomings can be discussed.

The Model

Assumptions

The atmosphere has only one layer
The atmosphere is transparent to SW energy
The atmosphere absorbs 100% of the outgoing LW radiation
The single layer is at a uniform temperature
The atmosphere is not in contact with the surface (no conduction or convection)
100% of the SW energy (from the Sun) is absorbed by the surface

Obviously, these assumptions are ridiculous .. but they allow the development of a very simple model.

Sometimes, the model also uses a non-rotating Earth .. but that is going WAY too far.

The formulas below are based on Stefan's law which determines the blackbody temperature of an object at equilibrium.

P * (1-Albedo) = sigma * T^4

You can perform the associated calculations using my Javascript calculator

In this model,

S represents the shortwave (SW) energy from the Sun
L represents the longwave (LW) energy from the surface of the Earth
A represents the longwave (LW) energy absorbed by the surface from the atmosphere
sigma refers to Stefan's constant

Because the Earth rotates, the amount of energy heating the surface is S/4. In the absence of an atmosphere, that would produce a temperature proportional to the fourth root of S/4.

Ts = [1/sigma (S/4)]^0.25

When the atmosphere is considered, the total energy absorbed by the surface is equal to the Energy from the Sun plus the energy from the atmosphere.

Ts = [1/sigma (A + S/4)]^0.25

By conservation of energy, the total LW energy emitted from the top of the atmosphere MUST equal the amount of SW energy received from the Sun. In addition, the energy emitted by the atmosphere toward space must be equal to the energy it emits toward the surface. As a result, the temperature of the atmosphere must be

Ta = [1/sigma (2 * A)]^0.25

Since the total energy into the atmosphere equals the total emitted

2 * A = A + S/4
    A = S/4

The immediate implication of this result is that the temperature of the surface and the temperature of the atmosphere are identical and significantly above the temperature expected without an atmosphere.

With "real" numbers, the Sun provides 1361 W/m2 at the top of the atmosphere. Without an atmosphere, that should produce a surface temperature of 5.3°C and with the atmosphere, the temperature would be 58°C. (The current accepted surface temperature is 15°C.)

With an albedo of 30%, the surface would be -18.4°C without an atmosphere and 29.7°C with one.

Why the model gives the wrong results

Obviously, the model assumptions are ridiculous.

There are many layers in the atmosphere, not just one
The atmosphere actually absorbs a significant amount of SW energy
The atmosphere does not absorb 100% of the outgoing LW radiation
Because the atmosphere is actually in contact with the surface, a significant amount of energy enters the atmosphere via convection
Some of the SW radiation is reflected by the surface

Better Model

Still using the single layer model, and assuming a 30% albedo, and a partially transparent atmosphere, the following gives a ballpark result.

Assuming a 30% albedo and an atmospheric absorption of 80%, the surface temperature would be 16°C (397 W/m2).

S            = 1361          Sun TOA
S/4          = 340.25        Sun TOA/4

S/4 (1-0.30) = 238.175       Albedo = 30%


        2 * A = 80% L         The atmosphere absorbs 80% of the surface energy
                              But 'A' is emitted up AND 'A' is emitted down

  (20% L) + A = S/4 (1-0.30)  The amount of heat lost equals the amount from the Sun

20% L + 40% L = S/4 (1-0.30)  Just combine to remove 'A'

      0.6 * L = S (0.7/4)     Simplify

 L = S (0.7/2.4) = 397 W/m2 => 16.27°C   Close to 15°C


A = 0.4 * 397 = 158.8 W/m2

158.8 + 238.2 = 397.0 W/m2   The values check
 A   +   S    =  L

Using this model, increasing CO2 will increase the 80% value. For instance, increasing it to 81% yields 399 W/m2 and 16.6°C.

At current values, doubling CO2 from 350 ppm will increase the energy absorbed by the atmosphere by 10 W/m2 (based on a line-by-line model that I hope to complete soon). That increases L by 5 W/m2 and produces a new surface temperature of about 17.2°C ... an increase of about 1°C (assuming no cloud or albedo feedbacks).

By the way, if the atmosphere emits 158.8 W/m2, that is associated with an expected temperature of around -30°C (158.8/0.8 = 198.5 W/m2).

Real World Model

When the model is further modified to allow multiple layers and the other real world properties, the change in surface temperature due to an increase in CO2 approaches zero ... even if the CO2 concentration is increased about 10 times over today's value. For a complete description of this, please read my paper - How Greenhouse Gases Work.

Venus

I have provided a fairly complete analysis of Venus on another page. However, this model provides a simple way to determine the maximum possible Greenhouse temperature of Venus. Using NASA's Venus Fact Sheet to get the insolation and albedo,

2613.9 W/m2  Sun TOA

yields a surface temperature of 190°C without an atmosphere and 278°C with one that absorbs 100% of the longwave radiation. The fact that the measured surface temperature is 464°C proves, without a doubt, that the temperature of Venus has nothing to do with the Greenhouse Effect.

Just to be clear, the computed values above assumed that 100% of the available energy reached the surface and that only one side of the planet faced the Sun. (Each side actually faces the Sun for 58 days before being dark for 58 days.) Since it is known that about 75% of the solar radiation is reflected by the clouds (and, therefore, never reaches the surface) the actual maximum Greenhouse temperature is much less.

The bottom line is the same - Venus is heated internally and not via some Greenhouse Effect.

Venus - Try 2

The model above has a single atmosphere layer with a constant temperature. In a real atmosphere, the temperature decreases with height until some minimum value is reached, and then the temperature begins to increase. Assuming that this minimum value is determined by physics, it can be used to compute a more realistic surface temperature.

On Venus, the atmosphere temperature decreases for about 60 km with a lapse rate of 7.9°K/km. At that point, there are clouds. Above the clouds, the atmosphere cools at about 3.0°K/km for the next 30 km. Unfortunately, I do not know of any way to determine heat loss based on the slope of the lapse rate. My calculator says that CO2 will emit 136 W/m2 at 60 km and 263°K (-10°C). If the clouds are decent blackbodies, they will emit 271 W/m2.

From this rough computation, it is possible that only 136 W/m2 reaching the surface could conceivably produce a 464°C surface temperature because the energy must travel the slow way through the atmosphere before being released to space.

136 W/m2 / 2613.9 W/m2 = 0.052

60 km * 7.9°K/km = 474°K

474°K + 263°K = 737°K (464°C)  -> Surface temperature

Implying that only 5% of the available solar radiation could produce a high surface temperature if the atmosphere is considered opaque, thermally conductive, and emits energy at a low temperature.

This also shows how ridiculous the single layer model is. (Climate models only use 4 to 7 constant temperature layers ... and you wonder why I don't trust them.)

Reference: Atmospheric Flight on Venus provides the temperature verses altitude data. The computations are made with simple programs I wrote. These computations should be considered speculative, but they do show how a small amount of energy could create a very high surface temperature.

By the way, because the day to night temperature difference is so small, it is not clear that any heat reaches the surface from the Sun. It is probably more likely that the energy comes from an internal heat source.

Wikipedia version

The Wikipedia page - Idealized greenhouse model - basically says exactly what is above ... except ...

The article claims (without a reference) that the atmospheric emissivity is 0.78. (I used 0.80 above.)

Assuming a temperature of 15°C and an emissivity of one, the surface should emit 390.08 W/m2 (using Stefan's equation). Using my line-by-line program, the atmosphere should emit 378.68 W/m2 (assuming 15°C and 1.0% water vapor) producing an effective emissivity of 0.97.

This is a major difference.

If I use their atmospheric temperature of 242.5 °K (-30.5°C), then my program states that 184.25 W/m2 is emitted in each direction (up and down).

    184.25 / 390.08 = 0.47
2 * 184.25 / 390.08 = 0.94

To be sure, you can play with the amount of water in the atmosphere to get any values you want. Without extensive data (which I have not been able to find), there is no right or wrong answer. The point is that the models are good teaching tools, but are simply worthless if you want to predict anything.

The test of these models is to see if they work on Venus. (Hint - theirs doesn't.)

The other main flaw in this model is that only radiative heat is considered. As a result, they claim that both the atmosphere and surface temperatures will increase as more Greenhouse gases are added. (And the model does predict that.) However, when convective heat is added to the model (ie, if the model is more real world), then adding more Greenhouse gases will always cause the temperature of the atmosphere to decrease.

Author: Robert Clemenzi