The truth is that no one knows what this value is - but speculations go from negative values up to about 8°C with the presumed most likely value being 3°C per doubling.

There are lots of references and models - you can basically support any value you like. This page tries to find a value using MODTRAN.

The projected increase of 40 ppm in atmospheric carbon dioxide to 2030 is calculated to contribute a global atmospheric temperature increase of 0.04°C |

Spoiler - I think they are both wrong.

- the modtran model of atmospheric infrared radiation [is] an early example of a line-by-line code
- No it is not. It uses 1 cm
^{-1}bins to compute the spectra. The MODTRAN documentation is very clear about this.

In his 2007 paper, Archibald claims that the current CO2 level is 380 ppm and that it is projected to rise to 420 ppm by 2030, and then implies that those are the values he used with MODTRANS.

In his rebuttal, Archer says

Run the model once with 375 ppm CO2 and another time with 415 ppm, and compare the Iout values in Watts / m2. |

Archibald says he used **MODTRANS** - I am not sure how that is different
from MODTRAN 3.5. Archer did not comment on this and an internet search found nothing.
Therefore, I am simply assuming that they both refer to the same basic application (model).

The projected 40 ppm increase reduces emission from the stratosphere to space from 279.6 watts/m2 to 279.2 watts/m2. |

Using the default MODTRAN 3.5 values and setting CO2 to 380 ppm and 420 ppm, the upward IR heat flux, and related change, are provided in the table.

CO2 ppm | IR heat flux looking down from 70 km W/m2 | ||
---|---|---|---|

Archibald's | Tropical | US Standard | |

380 | 279.6 | 298.174 | 267.277 |

420 | 279.2 | 297.672 | 266.837 |

delta | 0.40 | 0.50 | 0.44 |

I used the model defaults CH4 (ppm) 1.7 Trop Ozone (ppb) 28 Strat Ozone scale 1 Water Vapor Scale 1 Freon Scale 1 Temperature offset 0 Locality Tropical Atmosphere & 1976 U.S. Standard Atmosphere No Clouds or Rain Altitude (km) 70 Looking down |

- All the
**Localities**looking down from 70 km and 11 km - All the
**Localities**looking up from 0 km - Setting water vapor to zero - significantly increased the outgoing heat
- Setting all gases to zero except water vapor and CO2 - this was the closest when using the U.S. Standard Atmosphere

At this point, I have no confidence in Archibald's paper.

CO2 ppm | W/m2 |
---|---|

400 | 297.923 |

800 | 294.595 |

This is obviously different than the 3°C/doubling that Archer and the IPCC claim is most likely.

In all fairness, these computations do not include any "feedbacks" .. which is fine because the Archer response does not mention them. However, if the water vapor scale increases from 1.0 to 1.1 (and there is no documentation explaining what this value represents), then the 800 ppm temperature offset becomes 1.38°C, a bit higher, but still nowhere near 3°C.

The following analysis assumes the MODTRAN default configuration
(detailed above).
To return to the original upward IR heat flux after increasing the CO2,
the ground temperature must be increased by some value
which is entered via **Temperature Offset, C**.
Using the tropical atmosphere and Archibald's CO2 values,
the adjustment is 0.11°C which
would yield an increase for doubling of 0.76°C.

dy/dx is what is needed x is either CO2 or log(CO2) - the base does not matter y is normally surface temperature since dy/dx = slope then dy = slope * dx 0.11 / [ln(420) - ln(380)] = 0.11 / ln(420/380) = 1.099 1.099 * ln(2) = 0.76°C/doubling 0.76 / [ln(800) - ln(400)] = 0.76 / ln(800/400) = 1.096 1.096 * ln(2) = 0.76°C/doubling |

(Reversing the calculation to get the same slope, the temperature increase would have to be 0.109691.. instead of 0.11.)

Just for fun, let's assume that the change is linear.

0.11 / (420 - 380) = 0.00275 0.00275 * (800 - 400) = 1.1°C for doubling 0.76 / (800 - 400) = 0.0019 0.0019 * (800 - 400) = 0.76°C for doubling |

Thus, it is clear that the change in temperature with respect to changes in CO2 is logarithmic, not linear.

Since the model clearly indicates that the expected change in temperature for a doubling of CO2 is probably a little less than 1°C, it appears that both Archer and Archibald are wrong.

CO2 ppm | Scenario Default surface temperature | Units | ||||
---|---|---|---|---|---|---|

Tropical 299.7 K | US Standard 288.2 K w/ clouds | SubArtic Winter 257.2 K | Midlatitude Summer 294.2 K | |||

400 | 297.923 | 267.057 | 228.027 | 202.216 | 289.100 | W/m2 |

800 | 294.595 | 264.105 | 226.206 | 200.489 | 286.117 | W/m2 |

0.76 | 0.74 | 0.51 | 0.53 | 0.71 | Δ°C to restore |

I included one with clouds - US Standard with Altostratus (just marked *clouds* in the table).

This (very small) sample indicates that the true value for doubling CO2 is closer to 0.7°C than it is to 3°C.

But you be the judge.

The data in this table was collected using the US Standard Atmosphere, no clouds or rain, 70 km looking down. delta-T is the change in surface temperature required to return the Upward IR Heat Flux to 267.057 W/m2, the 400 ppm value.

CO2 ppm | 300 | 400 | 500 | 600 | 700 | 800 |
---|---|---|---|---|---|---|

W/m2 | 268.282 | 267.057 | 266.115 | 265.33 | 264.671 | 264.105 |

delta T | -0.3 | 0 | 0.24 | 0.43 | 0.60 | 0.74 |

CO2 ppm | 300 | 400 | 500 | 600 | 700 | 800 |
---|---|---|---|---|---|---|

W/m2 @ 1km | 371.148 | 371.148 | 370.834 | 370.834 | 370.834 | 370.834 |

W/m2 @ 5km | 313.435 | 312.713 | 312.147 | 311.676 | 311.268 | 310.923 |

The other three data sets, when plotted in a spreadsheet, indicate that the relationship is almost exactly logarithmic.

This is an area where I need to do further research.

I do not know if the MODTRAN results are accurate or not, but they do not support the claims by either David Archer or David C. Archibald.

And obviously, whenever you post data from a model, be sure to include enough information so that others can replicate your work.

Author: Robert Clemenzi