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

You must remember this!

Just Play it again!

Cassa Blanca?

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14 Responses to Simplified atmospheric model

  1. Derek Alker says:

    Mike writes –
    “(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:”
    i) The surface would conduct sensible heat to the atmosphere, so the atmosphere would be warmed.
    ii) “Assuming a uniform temperature (not true – but good enough for this)”
    EXACTLY, not true…. P/4 is unphysical, AND, you are ignoring the heat capacity of the surface too. THIS is NOT a black body, nor is it the black body concept you refer to, which is an imaginary surface, of no mass, and in a vacuum.
    I have to ask, good enough for what? Imaginary scenarios that are admitted as being unphysical!!
    Not a good start…..

    • Scottish-Sceptic says:

      Derek, if the atmosphere is a dead end with no way to gain or loose heat, then it must tend toward the temperature of the one and only thing it has any interaction with. Therefore the atmosphere will be at the temperature of the surface and to say otherwise is just stupid.
      Heat capacity: not relevant for what is clearly labelled as a simple model which I’ve clearly said is not true because it doesn’t model changes in surface and so I clearly don’t need Heat Capacity in.
      Black body: shorthand for a body modelled using Stephan Bolzman equation.

      • Derek Alker says:

        Mike, mass has heat capacity, end of story… To suggest otherwise IS unphysical.
        The atmosphere in your scenario would keep warming until it reached the maximum possible temperature induced at the solar zenith, to say it would only reach the P/4 average figure / temperature is WRONG. Very very wrong based upon a false interpretation and misrepresentation of what the black body concept actually IS.
        I do not care if you trust my opinion, knowledge, and understanding or not, just ask any self respecting physicist if a black body surface has mass. AND, you might as well also ask if it is possible to have mass without heat capacity too at the same time. But, I would suggest you do not, unless you like being laughed at.

        • Derek Alker says:

          btw the simplified model you suggest, for earth’s solar input would mean the atmosphere would rise to a temperature of 123C, NOT the -18C you say it would. THAT is quite a misrepresentation on your part..

          • Scottish-Sceptic says:

            The blackbody model, using the equations for the blackbody will produce the output from those equations which is -18C.
            That is the model. The model is a blackbody – you have to use the equations that go with it.
            Think about it this way – if a bank gives you an account – you have to use the cards they give you with the account – because if you use cards that don’t go with the account you won’t get any money.

        • Scottish-Sceptic says:

          Derek, this black body approximation works.
          Even if there is a better way of modelling black bodies, it must still explain the same experimental data and so will simplify to something so close to what I am using that it matters not a jot.
          Show me your theory, show me how it fits the available data matching the current blackbody model and then show me the data that shows why I have to change my view about a perfectly sensible and reliable way of calculating heat flow and assume your assertions which have absolutely no backing except your dislike of black bodies (is that racist?)

          • Derek Alker says:

            Mike writes –
            “The model is a blackbody”
            NO it is not. You have given a black body surface mass, the black body surface you use is NOT in a vacuum. You state the black body surface conducts heat to the atmosphere. ALL are WRONG.
            With respect, you do not know what the black body concept IS. Please go and ask a physicist whom you trust, he or she will tell you I am correct in my criticisms of your “model”.
            I do not know what model you are using Mike, it is black body based, but IT IS NOT a black body.
            You are actually inventing physics, like it or not. Voodoo science, literally.
            As I stated the model you use would produce an atmosphere with a temperature of 123C, NOT -18C, because that is the maximum possible temperature according to the Stefan Boltzman equation for a black body surface, of no mass, in a vacuum, receiving the same power of sunlight that earth does at the solar zenith.
            Derek Alker.

  2. The discrete solar rays are a time honored graphic, but I picture the earth as sitting a big cosmic soup, uniformly heated. Consider how the earth appears from the sun, a tiny dot.

    • Scottish-Sceptic says:

      “Consider how the earth appears from the sun”
      Would we even be visible?

      • Derek Alker says:

        If the sun is represented by a 1 meter beach ball then the earth is a marrow fat pea, 107.4 meters away.
        ie, Beach ball in one goal mouth, marrow fat pea in the other goal mouth of a normal sized football pitch. Seasonally, the pea moves 1.4 meters back and forth of the goal line.
        Some (hopefully) useful climate science related similes

        • Scottish-Sceptic says:

          But is it not more like standing on the stage in the stage light looking through the theatre doors and trying to see the pea in the pitch black street outside?

  3. All these popular graphical representations of the earth’s, heat balance, W/m2 gozintaz & gozoutaz, are they for sun light only? the net for 24 hours? is there a completely separate balance for night time? How come they don’t specify? Does the earth lose heat at the same rate day or night? Inquiring minds want to know.

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