<p>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.</p>
<p>Based on the image below temperatures <span class="math-container">$T_s$</span> and <span class="math-container">$T_a$</span> are derived by a fairly simple calculation:</p>
<p><a href="https://i.sstatic.net/gkAhH.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/gkAhH.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$$T_s = T_e \left(\frac{2}{2-\epsilon}\right)^{1/4} \tag{1}$$</span></p>
<p><span class="math-container">$$T_a = T_e \left(\frac{1}{2}\right)^{1/4} \tag{2}$$</span></p>
<p>where <span class="math-container">$T_s$</span> denotes the ideal earth temperature without atmosphere.</p>
<p>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.</p>
<p>My conclusion would be, that under equilibrium conditions the part of the atmosphere which contributes most to outgoing radiation (&quot;the single layer&quot;) 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 <span class="math-container">$\tau \approx 1$</span> <em>measured from TOA down</em>, 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 &quot;last emitting&quot; layer must be.</p>
<p>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 <span class="math-container">$\tau \approx 1$</span> <em>measured from surface level up</em>. 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 &quot;single layer&quot; and why does it give correct numbers?</p>
<p>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.</p>

CC BY-SA 4.0
2022-11-22T00:29:15.943Z
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.
AtmosphericPrisonEscape
CC BY-SA 4.0
2022-11-22T00:30:55.773Z
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..
AtmosphericPrisonEscape
CC BY-SA 4.0
2022-11-22T07:09:20.480Z
I understand the symbols exactly in the way you described. If there would be only a spatially very thin atmospheric layer which is optically thick, then a single temperature makes sense. However, I have no freedom where to place this layer, because it must have the temperature Ta as calculated above and also the temperature  according adiabatic law, given temperature at the surface. However, when the layer is assumed to be thin, there is nothing at the surface which can have a temperature. So this model is somehow illogical.
MichaelW
CC BY-SA 4.0
2022-11-22T07:13:11.180Z
In the lecture we learned, that this is the reason, why higher atmospheres are colder than lower layers. In another lecture we learned, that adiabatic processes within an air parcel are responsible for the temperature gradient. Those two are completely different processes and I see not a common reason, how the temperature gradient given by 1) and 2) should be in accordance to the adiabatic law. What I wanted to express is, that we have no freedom of choice, where to place the layer.
MichaelW
CC BY-SA 4.0
2022-11-22T07:21:19.890Z
With $\epsilon=0,97$, $\alpha=0,3$, $T_e=18,2°C$ I get a value of $T_a=-20°C$ and $T_s=27,7°C$.  So this layer must be at a height of about $z=8km$ to be consistent with known temperature gradient. We learned that atmosphere will have Ta, when Ts is given. But no z was assumed. If I would place this layer at z=1km, it would give complete nonsense. So why can we deduce atmospheric temperature alone by radiation considerations? Isn't here an argumentative flaw? Maybe you understand my question better now.
MichaelW
CC BY-SA 4.0
2022-11-22T10:32:49.630Z
The only reasonable interpretation for me would be, that the height of the effective radiation "single layer" is fixed (by the optical thickness). For this height we have a well defined temperature relative to surface just by adiabatic law - so surface temperature must finally adapt so that equations above are satisfied. But that doesn't mean that temperature of the atmospheric layer  is determined by radiation equilibrium. I think, that this was a mistake of the lecturer.
MichaelW
CC BY-SA 4.0
2022-11-22T17:41:11.727Z
"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: https://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.
AtmosphericPrisonEscape
CC BY-SA 4.0
2022-11-22T17:43:07.410Z
In short: This model is inconsistent with an adiabatic atmosphere, but that doesn't matter, that's not the point of the model.
AtmosphericPrisonEscape
CC BY-SA 4.0
2022-11-22T21:18:42.287Z
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.
MichaelW
CC BY-SA 4.0
2022-11-23T21:05:50.130Z
I guess we'd call this a toy model then - those are the ones made particularly to understand concepts. A 'real' model should however give good results.
AtmosphericPrisonEscape
<meteorology><climate-change><greenhouse-gases><atmospheric-radiation><radiation-balance>
How can a single layer atmospheric model for the greenhouse effect be consistent with adiabatic temperature gradient and optical depth considerations?
MichaelW
# How can a single layer atmospheric model for the greenhouse effect be consistent with adiabatic temperature gradient and optical depth considerations?

(Asked by: username_0 on 2022-11-21)

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](https://i.sstatic.net/gkAhH.png)](https://i.sstatic.net/gkAhH.png)

$$T\_s = T\_e \left(\frac{2}{2-\epsilon}\right)^{1/4} \tag{1}$$

$$T\_a = T\_e \left(\frac{1}{2}\right)^{1/4} \tag{2}$$

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.

**Tags:** meteorology, climate-change, greenhouse-gases, atmospheric-radiation, radiation-balance

> **Comments:**
> **username_1**: 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.  
> **username_1**: 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..  
> **username_0**: I understand the symbols exactly in the way you described. If there would be only a spatially very thin atmospheric layer which is optically thick, then a single temperature makes sense. However, I have no freedom where to place this layer, because it must have the temperature Ta as calculated above and also the temperature  according adiabatic law, given temperature at the surface. However, when the layer is assumed to be thin, there is nothing at the surface which can have a temperature. So this model is somehow illogical.  
> **username_0**: In the lecture we learned, that this is the reason, why higher atmospheres are colder than lower layers. In another lecture we learned, that adiabatic processes within an air parcel are responsible for the temperature gradient. Those two are completely different processes and I see not a common reason, how the temperature gradient given by 1) and 2) should be in accordance to the adiabatic law. What I wanted to express is, that we have no freedom of choice, where to place the layer.  
> **username_0**: With $\epsilon=0,97$, $\alpha=0,3$, $T_e=18,2°C$ I get a value of $T_a=-20°C$ and $T_s=27,7°C$.  So this layer must be at a height of about $z=8km$ to be consistent with known temperature gradient. We learned that atmosphere will have Ta, when Ts is given. But no z was assumed. If I would place this layer at z=1km, it would give complete nonsense. So why can we deduce atmospheric temperature alone by radiation considerations? Isn't here an argumentative flaw? Maybe you understand my question better now.  
> **username_0**: The only reasonable interpretation for me would be, that the height of the effective radiation "single layer" is fixed (by the optical thickness). For this height we have a well defined temperature relative to surface just by adiabatic law - so surface temperature must finally adapt so that equations above are satisfied. But that doesn't mean that temperature of the atmospheric layer  is determined by radiation equilibrium. I think, that this was a mistake of the lecturer.  
> **username_1**: "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: https://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.  
> **username_1**: In short: This model is inconsistent with an adiabatic atmosphere, but that doesn't matter, that's not the point of the model.  
> **username_0**: 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.  
> **username_1**: I guess we'd call this a toy model then - those are the ones made particularly to understand concepts. A 'real' model should however give good results.  

---

## 0 answers

