This model is inspired in the model posted by David Evans in his blog page. The model is based in the concept of emission layers of the atmosphere. The different active gases that are part of the composition of the the atmosphere, each emits infra red radiation at characteristic wavelengths and from different atmospheric layers.

The active gases of the atmosphere, sometimes called greenhouse gases, are H2O, CO2, O3 and CH4 in order of decreasing thermal emissions. Apart from the active gases some radiation is emitted directly from Earth’s surface and the top of the clouds through what is called “The atmospheric IR window“, the spectrum to which the atmosphere is transparent in the IR. In David’s nomenclature these 6 possible sinks for the incoming heat are called “pipes”, of these two are of minor importance O3 and CH4, leaving 4 main pipes. Energy can redistribute though the other pipes if one of them gets blocked, as for example by adding CO2.

David does a very good job at summarizing the available data on the highs of emission of the different gases and the top of the clouds, here. The gases are supposed ti be almost black body emitters in the window through which each is active, meaning the emitted energy is only a function of the temperature of the layer of the atmosphere from which the emission takes place. Since the temperature of the atmosphere decreases with altitude (in the troposphere), a higher layer emits less power than one closer to the Earth’s surface.

David’s OLR (outgoing long-wave radiation) model is only concerned on how the variation of various parameters modify the distribution of heat through the pipes, how these parameters may be dependent of the temperature or other independent variables is outside his scope.

Here I am going to layout a thermal model, based in well known physics to try to explain some of these missing relations. The first step is to build a model that fits the data, so to that purpose I am going to use the numbers from David Evans’ post:

• Lapse rate 6.5ºC/km, surface temperature= 288K
• Cloud cover = 62%, albedo = 30%, solar constant = 1367.7 W/m²
• Water emission layer: height=8km, output power = 33%
• Carbon Dioxide layer: height=7km, output power = 20%
• Cloud top emission layer: height=3.3km, output power = 20%
• Methane emission layer: height=3km, output power = 2%
• Ozone emission layer: height=16km, output power= 5.8%
• Surface emission layer: height=0km, output power=18.2%

Note: for now I have treated the CO2 as emitting from a constant average hight, I liked David’s treatment of the wights of the spectral emission on this spectrum, and I am planning on taking a similar approach on my next refinement. (End note)

The model uses a 2 surfaces representation of The Earth: surface 0 the ground surface (the origin) and the top of the atmosphere surface which is characterized by the maximum height of the convective Hadley. Temperatures are assumed to be linear throughout the atmosphere, so once the convective overturn is specified and the temperature at the top of the Hadley cell is known, the temperature of any other layer is linearly interpoled. The amount of energy that flows through each pipe is controlled by six additional parameters that represent the spectral width of the different spectral windows for each pipe. In the analogy of flow coming out of a damp through a set of pipes in parallel, these parameters represent the widths of the pipes.  For now these values have been adjusted to fit the percentages specified above, but I pretend to deduce their dependence with the height of the emission layers and the wave-lengths of the windows in the next post of the series.

The complete equations of the model and the values of the different parameters are on the link. The core of the model is equations 41, 50 and 51; representing the energy balance in both regions, the surface and the atmosphere.

Fig 2. Model schematic. One surface and one band model. Two balance equations one on the surface and one on the upper atmosphere as a whole. The atmosphere emits from different layers which are at different temperatures

The incoming solar power, modified by albedo, is the heat source of planet Earth and this heat is assumed to be absorbed on the surface. The surface balances the heat by radiation and convection mechanisms. The surface radiates either directly to space (about 18%) or to the clouds, this makes a total of three heat sinks for the surface: the two radiation and the convective mechanism.

$Q_{Solar}=Q_{Conv}+Q_{Direct}+Q_{ToClouds}$

The atmosphere on the other hand is heated by the surface, through the convention and the radiation to clouds mechanisms, which being heat sinks for the surface, become sources for the atmosphere. The atmosphere is balanced by its own sinks which is the radiation to space from the different active layers: clouds, H2O, CO2, CH4 and O3.

$Q_{Conv}+Q_{ToClouds}=Q_{FromClouds}+ Q_{H2O}-Q_{CO2}+Q_{CH4}+Q_{O3}$

Each of the radiative emission layers is modeled like so:

$Q_i=A_i \epsilon f_i \sigma T_i^4$

$T_i=T_0-\alpha h_i$

where $A_i$ is the surface area, $\epsilon$ is the emittance of the atmosphere (0.996), $\sigma$ is Stephan-Boltzmann constant, $T_i$ is the temperature of the emission layer in K, $f_i$ is the window factor, $T_0$ is the temperature of Earth’s surface, $\alpha$ is the lapse rate and $h_i$ the height of the emission layer.

The convective heat is modeled as so:

$Q_{Conv}=A_0 h_{conv} (T_0-T_1)$

The lapse rate is then:

$\alpha=(T_0-T_1)/H$

where $A_0$ is the area of Earth’s surface, $h_{conv}$ is the convection film coefficient, $T_1$ is the temperature at the top of the Hadley convective cell, and H is the height of the convective cell.

The direct radiation to space is then:

$Q_{Direct}=A_0 \epsilon f_{direct}(1-c)\sigma T_o^4$

where c is the cloud cover and f_{direct} the direct atmospheric window.

$Q_{ToClouds}=A_0 \epsilon f_{direct}c\sigma T_o^4-A_1 \epsilon f_{clouds}c\sigma T_1^4$

with $f_{clouds}$ being the atmospheric window from the top of the clouds and $A_1$ the surface of a sphere which encompass the convective layer of Earth.

$Q_{Solar}=A_0 G_s/4 (1-a)$

Whit $G_s$ as the solar constant and $a$ as the albedo.

The model has then 8 parameters that can be adjusted to fit the experimental data: all 6 window factors, the convective coefficient and the height of the convective cell. These parameters are set by imposing the experimental outgoing power distribution, the experimental mean lapse rate and the surface mean temperature which are a total of 7 restrictions. This leaves an extra degree of freedom which I chose as setting the height of the convective cell as 8.2 km arbitrarily.

There are several problems with the current model, that will be addressed in the next post of the serie:

1. The temperature of the stratosphere increases with height from the tropopausa at about 10-12 km so the ozone temperature layer is not correct. The actual ozone layer is above 20-30 km high but I chose to leave it at 16km so that it’s temperature not fall drastically when using the linear lapse rate. The stratosphere increases temperature  because the O3 captures part of the UV light from the sun and is heated. In future models I may include this effect.
2. Although the physical meaning of the window factors is clear, these factors can be deduced mathematically from the temperature of the emission layer and the wavelength interval as the fraction of the Planck distribution at the temperature that is emitted through the window. This will be tried on next model, once done the factor will be linked to the height of the layer, the lapse rate and the surface temperature through the temperature of the layer. The fact that the model has an extra degree of freedom (the height of the convection cell) increases my confidence that once the theoretical window fractions are calculated, which inevitably will be different from those obtained from the adjustment, the model will still fit the experimental data within reason.
3.   CO2 emits radiation from a whole range of heights in the atmosphere through the weights of the spectral window (see figure 1), the treatment of this feature will be studied. I think it is the result of a lower opacity (larger optical length) of the CO2 at those wavelengths so the solution is only partly lowering the emission height but also the emittance at those wavelengths, since a lower absorption (opacity) will always be accompanied by a lower emittance at a same wavelength (Kirchhoff Law of radiation)

This has been a very interesting post for me. I look forwards to the continuation. Any comment, or doubt or correction is welcomed.