We use the Spectral Mapping for Atmospheric Radiative Transfer (SMART) code, written by David Crisp , for our radiative transfer calculations. This code works at a line-by-line resolution but uses a spectral mapping algorithm to treat different wave number regions with similar optical properties together, giving significant savings in computational cost. We evaluate the radiative transfer in the range 50–100 000 cm-1 (0.1–200 µm) as a combined solar and thermal calculation. Line data for all radiatively active gases are taken from the HITRAN 2012 and 2000 databases.

The RCM used in this work derives from and . A “hard convective adjustment” is used, whereby the tropospheric structure is set as the moist adiabatic lapse rate, so surface and tropospheric temperature are represented by a single degree of freedom. The stratosphere is radiatively adjusted and the tropopause position is adjusted.

The SMART radiative transfer code is very computationally expensive. Hence the Newton–Raphson method used previously is too expensive, requiring a separate radiative transfer run for each degree of freedom (model level). Therefore, we derive a new numerical method, which diagnoses a grey emissivity for each layer and solves a linearized set of equations to adjust the model temperature. The algorithm is described in full in the appendix. For test cases with a grey atmosphere radiative transfer code, this gave convergence in 3–4 iterations. Unfortunately, the diagnosis of pseudo-grey emissivities for each layer from the real-gas radiation field was not as effective as we hoped and introduced some numerical instabilities. We introduced numerical smoothing and damping at each iteration to control the instability. Convergence was typically obtained in 20–30 iterations.

Water vapour was parametrized as in . Relative humidity (h) is given by h=h⋆Q-0.021-0.02, where h⋆=0.77, Q=p/p⋆, and p⋆ is surface pressure. When Q is smaller than 0.02, the relative humidity becomes negative; thus, it is necessary to specify a minimum humidity distribution for small Q values. determine a minimum mixing ratio of water vapour to be 3×10-6gg-1ofair. We take this as the minimum mixing ratio for a “mid-H2O” set of calculations. Since the saturation vapour pressure is proportional to temperature and we expect significant warming in the upper troposphere and stratosphere from shortwave absorption by CH4, the relative humidity parametrization may significantly affect the amount of atmospheric H2O. The H2O concentrations would then affect the strength of the H2O greenhouse. Furthermore, elevated high troposphere and stratospheric water vapour concentrations would increase the emission level to space from H2O and thus would promote cooling of this level. A further complication is that methane oxidation is a significant stratospheric moisture source, and this would be enhanced with higher methane abundances.

To examine the sensitivity of our results to the parametrization of H2O, we perform a “low-H2O” set of calculations, for which we reduce the minimum mixing ratio of water vapour by a factor of 1000 to 3×10-9gg-1ofair, and a “high-H2O” set of calculations, for which we increase the minimum mixing ratio of water vapour by a factor of 10 to 3×10-5gg-1ofair. We attempted an additional set of runs in which the H2O mixing ratio above the tropopause was set at the tropopause value but found it to be unstable in our model.

Gas amounts are given in abundances, a, relative to the modern atmosphere (1 bar, molecular weight of 28.97 gmoles-1, total moles (n0) of ≈1.8×1020). Thus, a=ngas/n0. For our experiments we add gas abundances to background N2 partial pressure (0.8 bar), increasing the atmospheric pressure.

We calculate the equilibrium temperature profile over many CH4 abundances to be in the range of 10-6 to 10-2 with a solar constant of 0.8 S0. We perform sets of runs with background CO2 abundances of 10-3, 10-2, and 10-1 for the high-H2O, mid-H2O, and low-H2O water vapour parametrizations. These sets of runs are performed with both the HITRAN 2000 and HITRAN 2012 line data, giving a total of 18 sets of runs. The reason for running sets with varying CO2 abundances is that CO2 cools the upper atmosphere and, thus, we would like to examine whether cooling by CO2 or warming by CH4 dominates.

To test our RCM and diagnose the climate sensitivity, we calculate the equilibrium temperature profile with pre-industrial (280 ppmv) and doubled (560 ppmv) CO2. We find that our model performs well in recreating the pre-industrial atmospheric profiles (Fig. ). For low-H2O, mid-H2O, and high-H2O parametrizations, we find the pre-industrial surface temperature to be 288.4, 288.5, and 291.4 K. The temperature change for a doubling of CO2 is 1.76 K for low H2O, 1.75 K for mid-H2O, and 1.73 K for high H2O. These are within the range of climate sensitivities given by the IPCC (2013, 1.5–4.5 K) but are less than the best guess of 3 K. The climate sensitivity is largest for the low-H2O parametrization because the water vapour change is larger.

We take the model to be converged when the net flux above the tropopause (ΔF) is less than 0.2 Wm-2 for every layer above the tropopause. In cases for which this could not be achieved within reasonable limits on computational cost, we estimate the precision of runs with a lower convergence threshold.

To do this we examined the difference in temperature for unconverged iterations of a converged run. The error in surface temperature for each unconverged iteration was found by taking the difference in surface temperature between the converged and unconverged iterations (ΔT). The maximum net flux above the tropopause was plotted as a function of surface temperature error (Fig. ). To estimate the largest surface temperature error for a given maximum net flux above the tropopause, we found a linear slope which contained all of the points. All of the points fit within a region bounded by a slope of ΔF=2ΔT. Thus, uncertainty in temperature is taken to be ΔT=12ΔF.

We examine surface temperature as a function of CH4 for each set of runs (Fig. ). For all cases, there are significant differences between runs with HITRAN 2000 and HITRAN 2012 line data. At low CH4 abundances, the surface temperature is slightly warmer (≈1 K) using the HITRAN 2012 database relative to the HITRAN 2000 database, due to additional longwave absorption lines added to the HITRAN 2012 database. Additional CH4 shortwave absorption in HITRAN 2012 starts to become evident at CH4 above 10-4 and becomes significant at concentrations above 10-3. For a CH4 increase from 10-3 to 10-2, the warming is 4.8–6.4 K using HITRAN 2000 line data and -0.6–2.5 K using 2012 line data. Thus, the ability of CH4 to warm the surface is significantly diminished above 10-3.

The difference in absorption by CO2 and H2O is quite small between the two databases. Although, many new lines have been added to both CO2 and H2O databases, they do not provide a large radiative effect in the regime we examined. The differences between HITRAN versions result in a small increase to the greenhouse strength between versions, increasing the surface temperature by roughly 1 K in the regimes we examined.

The surface temperature is sensitive to the H2O parametrization used. At low CH4 abundances, the atmosphere is cold and the H2O abundances decrease rapidly with altitude to the minimum allowed abundance (Fig. ), resulting in large differences in the amount of atmospheric H2O between the difference parametrizations. Thus, the water vapour greenhouse effect is much stronger for higher minimum allowed abundances, which results in warmer surface temperatures.

Large atmospheric methane abundances cause atmospheric warming above a pressure level of 0.3p⋆ (see Sect. ). For our low- and mid-H2O parametrizations, this causes the amount of water here to increase, strengthening the greenhouse effect and causing surface warming. However, for our high-H2O parametrization, the water vapour is already at a maximum here, and we see the consequence of methane absorption in isolation. Between methane abundances of 3×10-4 and 1×10-2, there is a surface cooling of 2 K with the HITRAN 2012 database as compared to a warming of 7 K using the HITRAN 2000 database.

The parametrizations of relative humidity used here are simplistic and may not properly represent the relative humidity structure for an Archean atmosphere with high CH4. Water vapour moves from the troposphere to the stratosphere though complicated dynamical processes. On the modern Earth, water vapour enters the stratosphere through the extremely cold tropical tropopause . Resolving atmospheric dynamics would be required to correctly estimate stratospheric water vapour. Furthermore, methane oxidation is a significant source of stratospheric water vapour, and this will be a much larger source when there is more methane. Hence, photochemistry should be treated too.

The surface temperatures calculated using HITRAN 2000 line data agree well with the results of . For a CO2 abundance of 10-2 and a CH4 increase from 10-5 to 10-2, found a temperature increase of 11.5 K and we find a temperature increase of 12.6 K (low H2O), 11.2 K (mid-H2O), and 9.8 K (high H2O). This is much diminished with the HITRAN 2012 line data; for the same scenario we find a temperature increase of 8.1 K (low H2O), 6.7 K (mid-H2O), and 3.4 K (high H2O).

Increased shortwave absorption by CH4 warms the stratosphere (altitudes above ≈0.3p⋆) relative to the HITRAN 2000 line data (Fig. ). The warming as a function of CH4 is roughly 2–5 K for 10-4, 10–20 K for 10-3, and 20–35 K for 10-2. The warming also causes the tropopause to lower with increasing CH4.

Elevated CO2 acts to cool the stratosphere and thus counteracts warming by CH4. In general, warming by CH4 is the dominant effect, although differences in the temperature structure at different CO2 concentrations are apparent. The most significant differences are as follows: (1) the difference in stratospheric temperature between the HITRAN databases is largest at low CO2; (2) a temperature inversion appears for a CO2 abundance of 10-3 but does not form at higher abundances; and (3) the tropopause is lower with less CO2 (for example, with a CH4 abundance of 10-2, the tropopause is at ≈0.4p⋆ for 10-3 of CO2, ≈0.3p⋆ for 10-2 of CO2, and ≈0.2p⋆ for 10-1 of CO2).

Longwave emissions to space from H2O act to cool the stratosphere. At low CH4, higher parametrized H2O abundances result in a cooler stratosphere. In contrast to the surface temperature, the H2O parametrization has only a minor effect on the temperature structure in the stratosphere at high CH4 concentrations. This is because the H2O concentration is similar for all cases at high CH4 (Fig. ).

In the contemporary modelling of the Archean atmosphere, the removal of O2 and O3 and increased CO2 result in decreased static stability of the stratosphere . Thus, deep convective mass and water fluxes are enhanced for the Archean . However, as shown above, high CH4 results in a stable stratosphere. Thus, it is worth discussing the effect that this increased stability would have on studies of the Archean climate.

Modelling studies have found that the removal of O2 and O3 results in a decrease in static stability and higher penetration of convection, which produces increased cirrus ice clouds in the stratosphere . found that replacing O2 and O3 with N2 in the present-day atmosphere produces a 3.9 Wm-2 longwave radiative forcing from clouds. However, the contribution of ice clouds to the greenhouse effect in the Archean is reduced due to the saturation of longwave spectral bands from elevated CO2 concentrations despite increased cloud fractions. Thus, it is unlikely that the absence of these clouds would have a large effect on the Archean climate.

Photochemical models have found that an organic haze is produced by photolysis as CH4 concentrations approach the CO2 concentration in a low-O2 atmosphere . Organic haze has been predicted by photochemical modelling at CH4/CO2 ratios larger than 1 (Zahnle, 1986), and laboratory experiments have found that organic haze could form at CH4/CO2 ratios as low as 0.2–0.3 . The organic haze would likely produce a significant anti-greenhouse effect by reflecting solar radiation while being transparent to infrared radiation, although the organic haze may also have shielded greenhouse gases from photolysis such as NH3 and produced other greenhouse gases such as C2H6,.

The precise radiative effect that an organic haze would have had on the early Earth's climate is poorly quantified. Further, the relative humidity at which the haze formed may have effected the radiative impacts of the haze. performed laboratory experiments on the formation of haze particles via ultraviolet photolysis over a range of relative humidities and found that increasing relative humidity increases the cooling effect of the haze particles. In contemporary Archean climate simulations, exceedingly low temperatures above the tropopause mean that saturation vapour pressure is quite low. Thus, despite having little water vapour, relative humidities grow large above the tropopause . However, stratospheric warming would increase the saturation vapour pressure and lower the relative humidities, which would effect the formation of an organic haze; higher relative humidity may cause fractal particles to collapse into spheres, while lower relative humidity would allow the fractal shape to be better preserved .

Geological constraints, based on the mass balance of weathering palaeosols, have suggested that the atmospheric CO2 partial pressure was in the range of 0.003–0.02 bar in the late Archean 2.69 Gyr ago;. Given that an organic haze could form at CH4/CO2 ratios as low as 0.2–0.3, this would imply that an organic haze would form at CH4 abundances greater than 6×10-4–6×10-3. In the presence of an organic haze, shortwave absorption by CH4 would likely be of less importance. However, at the upper limit of this range, a CH4 abundance of 6×10-3 results in a significant (3–4 K) difference in surface warming between HITRAN versions. Thus, given the constraints on atmospheric CO2 and organic haze, the calculated reduction in surface warming due to improved line data may have been radiatively important throughout the Archean. Furthermore, atmospheric CO2 constraints only exist for the latest Archean 2.69 Gyr ago;. The solar luminosity used in this study (80 % of today's value) occurred 2.86 Gyr ago Eq. 1,, which is 170 Myr before the earliest constraint on CO2 2.69 Gyr ago,. Thus, CO2 may have been significantly higher than 0.02 bar at this time, meaning atmospheric CH4 concentrations larger than 6×10-3 bar could have existed without haze formation.