A simple model of the CO2 greenhouse effect

For some time now I have wanted to better understand exactly  how the CO2 greenhouse effect works. The only real way to do this  is to calculate how  heat is radiated out from the Earth’s surface through an IR absorbing atmosphere to space.  I set myself the task of writing from scratch  a simple programme to do this, focussing only on CO2 and  ignoring other greenhouse gases. – You can download the code here..

Fig 1. IR spectra of Earth showing dominance of a single CO2 band.

I soon realised that some basic assumptions need to be made to make any progress. At the heart of the problem is yet again the lapse rate. We have to assume a pre-existing lapse rate, since without one there can be no greenhouse effect. At the same time it is claimed that a lapse rate cannot be sustained without a greenhouse effect, since the top of the atmosphere must radiate energy to space in order to to enable heat flow through it. The lapse rate itself is caused  by convection and depends only on Cp of mainly N2 and O2. To make progress on a model of the real atmosphere some assumptions must be made:

  1. A flat stationary Earth surface with constant incident solar radiation and  Ts=288K
  2. An environmental lapse rate DT/DZ = 6.5 C/km up to a height of 11km with constant temperature from 11 to 15km.
  3. Hydrostatic equilibrium: P_0 e^{-\frac{z}{Scale}}  where Scale is the scale height of the atmosphere
  4. The Beer-Lambert law for absorption of radiation:  \tau = e^{-kPL} ,   where \tau is the transmission coefficient (0-1)  and A=(1-\tau ) is the absorption by CO2, k is the absorption coefficient, P=partial pressure of CO2, and L is the distance travelled by radiation through air.
  5. Kirchoff’s Law:  This simply states that a black body in thermal equilibrium will emit the same amount of energy as it absorbs. We will assume this to be the case for a layer of air containing CO2 at a given height z.

To simplify things further we only  take the main absorption band for CO2 between 13-17 microns. Then we simply guestimate that 20% of the total black body radiation emitted by the surface lies within this CO2 band – see figure 1.

The absorption coefficient k for the 13-17 micron  band has been measured to be 1.48 m-1 atm-1 (Essenhigh 2001). For the partial pressure of CO2 we take the concentration in ppm scaled to the atmospheric pressure at a given height. Atmospheric pressure then varies according the hydrostatic balance, and the partial CO2 pressure is in atmospheres i.e.

P = CO_2P_0e^{\frac{-z}{scale}}  where scale is the atmospheric scale height.

The Model

The atmosphere is divided into 100m thick layers above the ground with a surface temperature of 288K. As the surface radiates IR upwards, it is partly absorbed in each layer by CO2 molecules. It is assumed that Kirchoff’s law applies and that each layer re-emits an equal amount of IR in CO2 bands as that absorbed for a given temperature T. The atmosphere is assumed to be dry with no other greenhouse gas present. How does the radiation balance vary with CO2 concentration ?  Figure 1 shows a schematic.

Figure 2: Model schematic of radiation transfer from surface at 288K through 100m wide slices of atmosphere. Each slice is a grey body absorbing and emitting IR. A lapse rate of 6.5K/km and a hydrostatic scale height of 8.6km is assumed.

The surface radiates heat as a black body of temperature 288K upwards. The atmosphere is divided into 150×100 m layers. Each level absorbs photons from any direction  according to its local temperature, and numbers of CO2 molecules( partial CO2 pressure). The absorption(transmission) rates  are governed by the Beer-Lambert law depending on T(lapse rate) and P(hydrostatic balance). The lapse rate is  taken as 6.5 deg.C /km and the Pressure variation as exp(-z/scale), where scale is the “scale height” in the atmosphere where the pressure drops by the factor e (2.71828..). The scale height varies with temperature but on Earth the average scale height is about 8.6km.  The calculation starts at level 0 and works upwards. At each level the transmitted and absorbed IR from below are calculated. The actual net IR flux upwards is the difference between the up-going radiation and the integral of all the higher levels of down-going radiation.  Thermal equilibrium is assumed for each level, so at level j we must sum up all the contributions from all higher levels applying Beer-Lambert’s law.  The IR flux up and down are then stored for each level.

Results

The model was run several times with varying concentrations of CO2. Figure 3 shows the net IR energy flux for varying for the different CO2 concentrations.

Fig3. Radiation fluxes versus height in the atmosphere for the 13-17 micron CO2 band and varying CO2 concentrations.

Notice how rapidly the reduction in ground radiation occurs for low densities of CO2. At around 100 ppm the atmosphere becomes opaque to IR, as >90% of outgoing radiation becomes absorbed. The decrease in radiation is now logarithmic and the shape of the distribution changes. The shape depends on the density profile and where the atmosphere absorbs up-going radiation. In this simple model the ground radiation actually increases slightly when concentrations reach 1000 ppm as more radiation gets absorbed in the atmosphere.

Figure 4 shows the back radiation component at each level in the atmosphere showing how it increases and changes shape with increasing CO2 concentrations.

Fig4: Back radiation at each level from the atmosphere above that level. For comparison the net upward IR is shown for 300ppm and 600ppm. Note how the shape changes at higher concentrations.

The total greenhouse effect can be seen easily by plotting how the radiation flux varies with CO2 concentration in the atmosphere. Figure 5 shows how both the ground radiation and  the TOA (Top of Atmosphere)  vary. The difference between the two is the energy absorbed by the atmosphere itself. There is a more or less logarithmic dependency of outgoing radiation on CO2 levels above 100ppm giving approximately TOA=13lnCO2   and  Ground = 17lnCO2.

Fig 5. Dependence of the radiation from for the ground and TOA on CO2 levels in the atmosphere. Also shown is the absorbed energy flux by the atmosphere. Above 100ppm the dependence becomes logarithmic with CO2 levels.

Now we can estimate the surface temperature change induced by the greenhouse effect to maintain energy balance.  Differentiating the Stefan-Boltzmann equation we get

\frac{DS}{DT} = 4\sigma T^3

DT = \frac{DS}{4\sigma T^3}    equation 1.

The  reduction in outgoing radiation from the surface to space caused by a doubling of CO2 levels from  300ppm to 600 ppm is ~5 6 watts/m2 – see fig 4. Alternatively the radiative forcing at the surface by the increased “back radiation” is also ~5 6 watts/m2. The surface temperature will increase to counteract this change by warming just enough to balance the outgoing radiation.

The predicted warming from a doubling of CO2 is  1.2 1.5 deg.C using equation 1.

  1. This “direct” warming caused by a doubling of CO2 to 600ppm is very similar to other estimates and is encouraging. Note also that this includes a “guestimate” of 20% of SB radiation falling within the main CO2 absorption band. This should be calculated more precisely.
  2. The flattening of the radiation profiles with height for higher CO2 levels implies a warming of the upper troposphere in line with more sophisticated models. This is of course yet to be confirmed experimentally.

Conclusion 

A simple model of the physics of the greenhouse effect due to CO2 gives results similar to more sophisticated radiative transfer codes. There is no need to calculate an effective radiation height as the shape of the radiation profile itself changes with CO2 levels.

The real atmosphere is far more complex than this simple model and H2O dominates weather and climate. The chief uncertainty in predicting future warming is the net feedback of water. Extra CO2 causes a declining radiative forcing with increasing concentration. It is essentially logarithmic dependence with  CO2 levels. Does extra evaporation of H2O enhance a small CO2 induced  warming or do increased  clouds counteract it ? What role if any do solar cycles play? Are these effects more dominant than the direct warming of ~1 degree due to a doubling of CO2? I hope we don’t have to wait 20 years for nature to answer these basic questions !

I welcome all criticism – including those who think  I have got this completely wrong !

Note: You can download the PERL code used for this model here

Reference: Check out also the previous inspired work of   !

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