Simplified atmospheric model

I want to examine three extreme models of the atmosphere: entirely transparent, entirely opaque and entirely transparent to visible and opaque to IR. From this I will show that the presumed adiabatic lapse through the atmosphere of “33C” must be too low and a better figure is closer to 36C
For information on models etc. see previous articles:

(1) Entirely transparent Model

If the atmosphere is entirely transparent, then it has no interaction with either visible or IR. It therefore neither gains nor loose heat from the atmosphere. Therefore as the only place that gains or looses heat is the surface, then the temperature will be determined by this surface. Assuming a uniform temperature (not true – but good enough for this), we come down to a simple blackbody:

Model of earth

Model of earth

Incoming radiation
As the diagram to right shows. Sunlight falls on only one side of the earth (daylight) so if the fraction of reflectance or albedo is A, the solar energy per unit area S, Es is the total solar (shortwave) energy collected by the planet per unit time (in units of W), the amount of sunlight absorbed is:

Es= (1-A) SπR2

Outgoing radiation is emitted from the entire surface. According to the Stefan-Boltzman law the power emitted is proportional to the fourth power of its temperature. So emitted heat energy is given by:

Ep= 4πR2 σT4

Where Ep is the planetary (longwave) radiation (in W) and σ is the Stefan-Boltzman constant.
The temperature increases until a radiative balance is reached when EP=Es so:

4πR2 σT4 = (1-A) S πR2

rearranging and eliminating terms we obtain:

T= ((1-A) S / 4σ)-1/4

For the earth S is about 1368 Wm-2
σ is 5.6704 10-8
A of  about 0.3

And this leads to a value of -18C as the theoretical temperature of a sphere the size of the earth, with the same distance from the sun and at a uniform temperature.
However, because when a gas expands it cools, and because the pressure reduces as we rise up the atmosphere, we’d expect the temperature to drop** and so the atmosphere would be colder** and we would see a temperature profile** but this would be determined solely by the surface temperature of -18C
**if there is no heat loss, in practice there can be no adiabatic gradient as the whole atmosphere is only in contact with one heat source: the ground.

(2) Entirely Opaque Model

The entirely opaque atmosphere is a model that assumes all radiation is stopped. As such the atmosphere now behaves like the surface of the earth, and except for being slightly bigger, the calculation is the same and so ~ -18C as the surface temperature.

(3) Transparent-Opaque Model (Entirely Transparent to visible & Opaque to IR)

In this model I will assume all incoming radiation is visible (not true as there is IR) and that all outgoing radiation is IR.

Simplified atmospheric model: transparent to visible, opaque to IR

Simplified atmospheric model: transparent to visible, opaque to IR

Here we have the atmosphere which is now opaque to IR. For simplicity, I’ve shown a infinitesimal small gap between the earth and lowest part of the atmosphere so that I can show that the IR from the earth matches the BACK RADIATION from this atmosphere.
This also means I must add conduction from the ground to the atmosphere (which is the same size as incoming solar). The temperature of the bottom of the atmosphere is the same as the surface. We now need heat flow through the atmosphere which comes from convection and transevaporation (heat from plants sweating  going up to produce white fluffy things called clouds)
Now the earth & atmosphere have a simple radiation balance where all the incoming solar must equal the IR emitted from the top of the atmosphere. So, the top of the atmosphere (being within a smidgen of the size of the earth) will be at -18C.
Pressure pushing down causes heat to rise.

Pressure pushing down causes temperature to increase.

And because increasing air pressure increases the temperature of the air we now see an adiabatic increase which is usually quoted as 33C making the earth’s surface

-18C + 33C = 15C

Which is within a midges dongle of the right answer.

But what about the real atmosphere

The real atmosphere stops around 90% of all IR, but 10% escapes from the ground without ever passing GO. (Gaseous … Oh I can’t think of anything)
So, the real atmosphere is somewhere between the 100% transparent (1) and transparent-opaque (3). You really can’t average temperatures – but I will so that means the actual system is something like:

10% (1) + 90%(3)

Under the opaque model (1) the surface is -18C. Under the transparent-opaque, the top of the atmosphere is -18C. We know the surface is really at around 15C. So:

15C = 10% x -18C + 90% (-18C + Adiabatic lapse from top to bottom).

Which gives:

Adiabatic lapse from (opaque) top of atmosphere to bottom of atmosphere (i.e. surface)

= (15 + 0.1 x 18) / .9 + 18

= 36.6C

Simplified Atmospheric Model

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