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I'm completely lost...In an elementary course on meteorology, we recently learned how to explain the greenhouse effect in terms of a simple single-layer atmosphere model.

Based on the image below temperatures $T_s$ and $T_a$ are derived by a fairly simple calculation:

enter image description here

$$T_s = T_e \left(\frac{2}{2-\epsilon}\right)^{1/4} \tag{1}$$

$$T_a = T_e \left(\frac{1}{2}\right)^{1/4} \tag{2}$$

where $T_s$ denotes the ideal earth temperature without atmosphere.

Although quite logical I feel a slight kind of inconsistency in it: We calculate the temperature of the atmosphere to be lower than that from earth by a fixed factor 0,84. But nothing is said about the height of this atmospheric layer. How can this be, because the temperature in an atmosphere's layer is (at least to some degree) given by an adiabatic temperature gradient and, therefore, there is no additional freedom in temperature in a given height when temperature on ground is given.

My conclusion would be, that under equilibrium conditions the part of the atmosphere which contributes most to outgoing radiation ("the single layer") corresponds to a height, where temperature matches equation (2). OK. But on the other hand, the portion of atmosphere, which radiates directly into open space must be within a layer of optical thickness of about $\tau \approx 1$ measured from TOA down, because layers below should be opaque from the outside view. So there is also no freedom in the height of the emitting layer, because it is solely given by optical thickness and the more greenhouse gases there are, the higher this "last emitting" layer must be.

Additionally, to make my confusion complete, when viewed from earth's surface, radiation received by surface must be from a layer within about optical thickness $\tau \approx 1$ measured from surface level up. But this height must be significantly lower as compared to the height of the layer which radiates into space - otherwise atmosphere would be transparent for IR. So how can we speak of a "single layer" and why does it give correct numbers?

So I don't get along with this description at all, although I would like it for its simplicity, not least because it gives a result consistent with data. Where is my misconception? I've been pondering this for a good month now and nobody can tell me what I'm doing wrong. Up to now, the field of meteorology appears a bit alchemistic for me.

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    $\begingroup$ To me, this looks like a misunderstand of the symbols used: $T_e$ seems to be the the equilibrium temperature for a zero albedo planet. $T_a$ is the temperature of the optically thin atmosphere, usually not a function of any optical properties at all (in simple models). And the surface temperature then, $T_s$ is the atmospheric temperature at the surface, that is the one which experiences the greenhouse effect. The strength of the greenhouse effect is encapsulated into the parameter $\epsilon$, but in truth this is some function of the optical depth $\tau$, i.e. physical quantities. $\endgroup$
    – AtmosphericPrisonEscape
    Nov 22, 2022 at 0:29
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    $\begingroup$ The height does not play a role here, as the actual values of $\tau$ are omitted, and energy transport is assumed to be purely radiative and not adiabatic convection. On Earth the greenhouse effect is complicated by the convection, but the basic principle stands. Furthermore it is clear from the model that the surface layer is optically thick, otherwise the right-hand arrows would not exist, $\epsilon$ would be zero, and the middle arrows would just go through from bottom to top. I hope this helps? If you want another shot at this from the astrophysics point of view, I can provide a paper.. $\endgroup$
    – AtmosphericPrisonEscape
    Nov 22, 2022 at 0:30
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    $\begingroup$ "But that doesn't mean that temperature of the atmospheric layer is determined by radiation equilibrium." - But it is! This is, by construction, a purely radiative atmosphere. T is only a function of $\tau$, nothing else, no z exists, no adiabatic temperature gradients exist. This isn't a model for a real, adiabatic-radiative atmosphere, but a purely radiative toy model to give you some intuition about greenhous. Maybe this paper clarifies it: ui.adsabs.harvard.edu/abs/2010A%26A...520A..27G/abstract their solution for T($\tau$) is a multi-layer model, but you can view it equivalently. $\endgroup$
    – AtmosphericPrisonEscape
    Nov 22, 2022 at 17:41
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    $\begingroup$ In short: This model is inconsistent with an adiabatic atmosphere, but that doesn't matter, that's not the point of the model. $\endgroup$
    – AtmosphericPrisonEscape
    Nov 22, 2022 at 17:43
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    $\begingroup$ What you say gives sense. I was too involved in real world scenarios and have overseen, that a model doesn't need to give good results - it is more for understanding concepts. $\endgroup$
    – MichaelW
    Nov 22, 2022 at 21:18

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