2.1. Eocene, 50 − 35 Ma BP
First let us think of designing an experiment to measure the impact of the atmospheric CO2 concentration onto the surface-air temperature. The CO2 concentration needed to be changed and for each change, its value and the corresponding temperature recorded. Other temperature influences needed to be neglectable or well controlled. It turns out that Earth has performed such an experiment in the past. During the Eocene, in the period 35–50 Ma BP, atmospheric CO2 has steadily been removed by sequestration while recording its concentration and the corresponding temperature via proxies. Other temperature influences are judged negligible. This assumption is considered a first-order approximation subjected to potential amendment as the time horizon and the data base widen in the course of the further analysis. The span of the CO2 concentration has been from 1600 to 500 ppmv in the considered period, the temperature span from about 28 to 20°C.
An interpretation of the ‘measurement’ data (i.e. the proxy reconstructions) has previously been presented [2, 3]. In the present studies, these reconstruction data are found to follow a simple relationship between the CO2 concentration (hereafter 𝑝CO2 in the unit ppmv) and the entailed temperature (TCO2), in the further course referred to as the Eocene (CO2-temperature) relationship:
TCO2 = ln(𝑝CO2/22) * 6.68°C.
From the historical CO2 concentrations of  (here used in course representation), the related temperatures are determined according to the preceding Eocene relationship. A slight correction is applied to account for the steady solar luminosity increase with time (ΔTsol) by approximating  via
ΔTsol = -0.01514 * t °C,
with t the time from present into the past in million years, and by applying 0.75 °C/(W/m2) for the radiative forcing-to-temperature sensitivity (see e.g. ).
In Figure 1, the resulting T = TCO2 + ΔTsol (smooth blue line) is compared with the ‘measured’ data given in  (orange wiggly line). The simple logarithmic function (equation 1) for the temperature impact from the atmospheric CO2 concentration is well able to reproduce the temperatures of the considered period 50-35 Ma BP and beyond, extending to 60 Ma BP. As a sensitivity test, the two coefficients in TCO2 (equation 1) are changed by ±1 % and the resulting temperature boundaries depicted in Figure 1 by the dotted bright-blue lines.
Conclusion from the Eocene: As the primary change process, the atmospheric CO2 concentration was steadily reduced in the period of 50 to 35 Ma BP. Roughly, a difference of 1100 ppmv in the CO2 concentration is followed by a temperature difference of 8°C. This causal relationship is well explained by simulation programs [2, 3]. At the same time, the simple 2-parameter logarithmic function of Eq. (1), the Eocene relationship, is able to reflect the compound effect of all underlying processes.
2.2. Late Quaternary, 420 ka BP until present
To explore the general applicability of the simple Eocene relationship, it is examined for a period with heavy disturbances to the pure CO2 influence: the Late Quaternary with its dominant waxing and waning ice sheets, in cause alternating the surface albedo and thus, the absorbed surface insolation. The present study is based on the Vostok ice core data . The herein reported CO2 concentrations are used to derive the CO2-effected temperature contributions according to the Eocene relationship (TCO2). The albedo effect (ΔTice-Quaternary) is approximated with help of the also reported proxy-determined temperature variabilities (ΔTVostok) of  by adapting the linear δ18O-sea level-albedo relationship of  via:
ΔTice−Quaternary = (0.2 * ΔTVostok – 2.5) °C.
The factor 0.2 has the meaning of αT/αp where αp the polar amplification (in this work taken as 2) and αT the proportionality factor for the global mean surface temperature, hence 0.4.
In Fig. 2, the resulting temperatures T = TCO2 + ΔTice−Quaternary are compared with the proxy-measured temperatures. The computed temperatures T (orange solid curve) are in good accordance with the measured temperatures (long-dashed dark blue from  and short-dashed bright blue from ).
The two contributions to the computed temperature T, originating from CO2 and predominantly ice albedo, are depicted in Fig. 3. Each, CO2 and ice albedo, influence the surface temperature at similar size. In a more general (and correct) view, ΔTice−Quaternary represents all terms not covered by TCO2. From Fig. 2, it is inferred that the aggregate non-CO2 temperature contribution largely follows a linear relationship to the global mean surface temperature.
Conclusion from the Late Quaternary, part 1: By switching on ice albedo as a massive second temperature determinant in addition to CO2, the observed temperatures are also well reproduced with help of the Eocene CO2-temperature relationship. The Eocene relationship is indicated as independent of other temperature-driving forces.
This raises the question about the CO2-temperature relationship in the other direction: It is well known that temperature is viably directing the atmospheric CO2 concentration. On the sceptics’ side, there is remarkable supposition that the CO2 concentration is predominantly driven by temperature, rather than by human emissions during the industrial age. For an examination, let us think of an experiment to measure the CO2 concentration entailed by different temperatures. Again, nature has done such an experiment: in the Late Quaternary. By increasing and reducing ice coverage, albedo is being varied, by this the absorbed surface insolation and in turn, the surface temperature. Temperature and CO2 concentration have been recorded via proxies educed from ice cores (see before), and the associated time via the ice core depth. During the Late Quaternary, temperature is considered the predominant CO2 change agent, other CO2-determining processes judged disregardable.
Looking at the Vostok ice core data , the local temperature has varied by about 10°C between glacial and inter-glacial maxima, and the CO2 concentration by 100 ppmv. 10°C temperature difference in the Vostok ice core data roughly relate to 5°C in the global average temperatures (see factor of 0.5 in Fig. 2). Thus, a change of 1°C of the global annual mean temperature is followed by a change of 20 ppmv in CO2 concentration. This is a factor of 2 higher then resulting from theoretical research , where the CO2 concentration (pCO2) varies per 1°C of temperature change according to pCO2/27 (ppmv). For pre-industrial pCO2, this roughly results in 10 ppmv CO2 concentration change caused by a 1°C temperature change.
Application of this theorical relationship to the temperature variabilities in the Vostok ice core data results in the CO2 concentrations as depicted by the dashed orange and dotted gray lines of Fig. 4, for Vostok temperatures times 0.5 and raw Vostok temperatures, respectively; the solid blue line shows the CO2 concentrations as reported from the ice cores.
Conclusion from the Late Quaternary, part 2: Nature reveals different CO2-temperature relationships for either direction: (a) temperature driving CO2, (b) CO2 driving temperature. In direction (a), the atmospheric CO2 concentration follows temperature changes by 10–20 ppmv per 1°C temperature change. In direction (b), a change of 10 ppmv in CO2 concentration causes a temperature change of about 0.07°C. Regarding for instance a CO2 concentration increase of 100 ppmv, the Eocene relationship indicates an induced temperature increase of 0.7°C. Since this temperature increase, in turn, causes a concentration change of 7–14 ppmv, about 7–14 % of the 100 ppmv-increase is to be attributed to the entailed temperature increase.
2.3. PETM, 56 Ma BP, and Devonian to Triassic, 400 − 200 Ma BP
So far, the Eocene CO2-temperature relationship has proven applicable for two geological ages, the Eocene and the Late Quaternary. The next sections shall turn to other eons with yet different conditions. The first is the time of the Paleocene-Eocene Thermal Maximum (PETM), circa 56 Ma BP. In a previous computer simulation study , temperature and CO2 conditions have been analyzed by varying the CO2 concentration up to 9 times pre-industrial levels. In Fig. 5, the results of the simulation study (blue dots connected by the solid line) are compared with the Eocene relationship results, corrected by ΔTsol (Eq. 2) for 56 Ma (orange dots connected by the dashed line); the black circle depicts the PETM condition according to .
Conclusion from the PETM-study: The simple Eocene CO2-temperature relationship is well able to reflect the comprehensive understanding of nature as implemented in simulation programs.
In a further earlier study , the period of 400 to 200 Ma BP has been analyzed. Based on observed CO2 concentrations , the related radiative forcings have been determined. In Fig. 6, these forcings (solid blue line) are compared to those given by the Eocene relationship (dashed orange line) by applying a sensitivity of 1.2°C/(W/m2).
Conclusion from the 400 − 200 Ma-period: The pattern of the radiative forcing from earlier computer studies is well reproduced by the simple Eocene relationship. It is noted that a sensitivity of 1.2°C/(W/m2) is required for the agreement, whereas 0.75°C/(W/m2) are perceived as a generally applicable standard. At this point, no interpretation can be given on the sensitivity specifics of this case; as hypothesis, the difference may predominantly be attributed to water vapor.
2.4. Late Paleozoic, 420 Ma BP until present
So far, the considerations have each focused on rather specific periods. In the various periods, the Eocene CO2-temperature relationship has proven as a viable tool to quantify the CO2-induced temperature variabilities. In this paragraph, the entire Late Paleozoic from 400 Ma BP to present will be analyzed utilizing the Eocene relationship. The CO2 data are now taken from  (as in the previous 400 − 200 Ma study, context of Fig. 6), and the temperature data from . Either data are judged coherent state-of-the-art reconstructions for the considered period. Both data are shown together in Fig. 7, the blue (mostly upper) line for the temperature and the orange line for the CO2 concentration.
From visual impression, the extremes exhibit rather consistent patterns: nearly the same CO2 concentrations correspond to the respective temperatures at the minima and maxima (except at the maxima of 90 and 55 Ma BP). In between, CO2 may lead temperature by circa 20 Ma (400 − 320 Ma BP) or lag by 20 Ma (280 − 220 Ma BP). From this, it is expected improbable to extract a statistically significant correlation between the two variables – if not artificially adapted for the 20 Ma-time shifts. Since there is no explanation in sight for a potential time lead / lag of this order, such statistical analysis is disregarded.
Instead, the Eocene relationship is applied to the CO2 concentrations. The resulting temperatures are depicted in Fig. 8 (dashed orange line) with a constant subtraction of 3°C, and compared to the reconstructed (measured) temperatures (solid blue line). Besides the artificial 3°C-offset, the agreement between the two curves is perceived remarkably good. One may infer that the Eocene relationship represents the major temperature driving force.
However, it is known that the absorbed insolation is subject to modulations with time. Significant variability is to be expected from the constantly increasing solar luminosity (see ΔTsol of Eq. 2), from surface albedo via snow and ice coverage (e.g. regarding the Late Paleozoic icehouse at around 300 Ma), and proposedly from the cyclic cosmic ray intensities . Further significant temperature influence is expected from tectonic changes (the entire considered period covered by supercontinent Pangea assembly to break-up).
The cosmic ray intensity φ(t)/φ(0) is taken from  and its temperature influence approximated via fit by
ΔTcrf = -4 * φ(t)/φ(0) °C.
The resulting variability of ~ 3 °C is found in consistency with .
The tectonic changes are apparent in the paleogeographic evolvement; Fig. 9 shows a course reconstruction of . The temperature impact is approximated via multiplying the coverages (in percent) of landmass, mountains, and ice sheets by -0.2°C/%, and the coverages of water (shallow waters and deep ocean) by + 0.2°C/%, and applying a constant offset of -7°C:
ΔTtec = (Σi fi * Ci -7) °C,
with i indicating the tectonic types, fi the coverage-temperature impact described before, and Ci the respective coverages (Figure 9).
This approach means for instance: In case, land gives 1 % to water, then 0.2°C is contributed by the reduction of land coverage and another 0.2°C by the simultaneous increase of the water area, in total 0.4°C. Originally introduced to explore the tectonic influences, ΔTtec in its given form is interpreted as predominantly reflecting albedo variabilities and in addition, overall land/water-driven climate variabilities (shift in the coverage ratio of continental vs. warm-humid climates).
To put this into perspective, a 1 % land increase from today’s tectonics – with ocean and land coverages 0.71 and 0.29, respectively, the ocean and land solar surface absorptions of , and a sensitivity of 0.75°C/(W/m2) – results in a temperature reduction of 0.26°C. More qualitatively, the albedo of water clouds is about 10 % higher over land than over oceans, 0.46 versus 0.42 , contributing to higher surface insolation at oceans than at land. In conclusion, the albedo interpretation of ΔTtec and the chosen parameter set are viewed as principally supported by separate studies. For further instance, in the Late Paleozoic icehouse at around 300 Ma BP, the ice sheet contribution to ΔTtec is -2.9°C if the ice area is recruited from water areas.
In summary, the total temperature is determined by
T = TCO2 + ΔTsol + ΔTcrf + ΔTtec.
The result is depicted in Fig. 10 by the dashed orange line and compared to the reconstructed (measured) temperatures (solid blue line). The agreement is perceived fair, particularly regarding the extensive period of about 400 Ma covering a large variety of disparate conditions. The pattern of the agreement remains principally unchanged (not shown) if considering the 68 % confidence boundaries for the CO2 concentrations of , the temperature discussion of , and a potential sensitivity dependency on the climate state by varying the non-CO2-terms in Eq. (6) by ±⅓. The agreement of the present high-level consideration with observations is seen as confirmation that the major temperature-determining components have been identified and that their respective contributions can be quantified by simple approximations.
By nature of the approximations, the regarded contributions subsume all relevant underlying processes. This particularly applies to the Eocene CO2-temperature relationship comprising e.g. atmospheric water vapor variations with temperature, changing ocean-atmosphere interaction with varying atmospheric CO2 concentration and temperature, and the temperature influence on the CO2 concentration (see above, Late Quaternary). TCO2 in Eq. (6) gives the near-surface temperature if CO2 was the only forcing. The further components of Eq. (6) act as correction terms, each again subsuming all underlying processes. These are explicitly incorporated in ΔTsol (Eq. 2) by applying the sensitivity of 0.75°C/(W/m2) and implicitly incorporated via the factors − 4 and fi in ΔTcrf (Eq. 4) and ΔTtec, (Eq. 5), respectively. Dependency of the sensitivity on the climate state is approximated as zero, cross-terms and higher-order terms in the forcing-to-temperature relationship are interpreted to be partly contained as averages in the insolation components of Eq. (6) (i.e. ΔTsol, ΔTcrf, ΔTtec) and to be partly attributed to the residuals.
To examine model alternatives, variations have been applied to Eq. (6). (A) First, the contribution from the cosmic ray flux is set to zero. With the parameters of ΔTtec changing from − 0.2 to -0.3°C/%, from + 0.2 to + 0.3°C/%, and the constant to -15°C, the temperatures are given as depicted by the dotted gray line in Fig. 10. (B) From here, ΔTtec is replaced by two components. (i) Snow/ice albedo is approximated by a linear relationship to temperature: for TCO2 + ΔTsol > 17°C, the relative albedo contribution is + 3°C; for lower temperatures, the contribution is (TCO2 + ΔTsol − 11.5) ∙ 0.545°C. (ii) A temperature contribution is introduced proportional to the ocean continental coverage , which is a measure for the eustatic sea level; this temperature contribution is taken proportional as 0.2°C per 1 % continental coverage difference with a constant offset of -6°C. This temperature contribution is interpreted to originate from albedo variabilities. The resulting temperatures are shown in Fig. 10 by the dot-dashed green line. (C) Introduction of effects from atmospheric oxygen variabilities leads to temperatures within the ranges exhibited in Fig. 10 (therefore not shown).
In general, the pursued selective and simple driving-force consideration cannot cater for the entirety of all related processes. Major contributions to the temperature variabilities are expected from strong volcanic activities (beyond the CO2 effects) as well as from wind and ocean currents. The latter may be the cause for the deviations between about 50 and 30 Ma BP in Fig. 10 which decrease by circa − 4°C during this period (differences between solid blue and dashed orange lines in Fig. 10). Such progressive cooling may well be ascribed to changes in the ocean currents . Also the model-to-reconstruction deviations before and after the center of the late Paleozoic icehouse (at about 300 Ma BP) are proposed to be predominantly attributed to warming contributions from – tectonically determined – ocean current specifics, these being largely reduced in the presence of wide-spread glaciation (i.e. at the center of the icehouse).
The proxy reconstructions used for the Late Paleozoic in this paragraph exhibit deviations from those used for the derivation of the Eocene relationship in § 2.1. Nevertheless, the original relationship of Eq. (1) reveals as best fit through the Late Paleozoic-analysis.
From comparison of Fig. 10 (dashed orange line) with Fig. 8, the summed effect of insolation variabilities (particularly from solar luminosity (ΔTsol) and albedo) roughly acts as a constant temperature reduction of 3°C. As example for detailed insight, the single temperature contributions to T (Eq. (6), dashed orange line in Fig. 10) are depicted in Fig. 11.
For an illustration of reconstruction uncertainty effects, the 68 %-pCO2 confidence envelope is used for TCO2 of the dotted gray line in Fig. 10 and the results depicted by the dotted gray lines of Fig. 12. The relative temperature uncertainties are emulated as 0.3 times the relative pCO2 uncertainties (68% confidence). By this, the uncertainty increase with depth into the past is accounted for; the absolute height (factor 0.3) has intuitive character. It is interpreted that detailed error treatment cannot substantially alter the preceding considerations.
Conclusion: The attempt is perceived successful to describe the fundamental climate determinants by simple means. The Eocene CO2-temperature relationship is revealed to be applicable throughout (at least) the past 400 Ma, as resulting from comparisons with paleo-reconstructions (Eocene, Late Quaternary, Late Paleozoic) together with plausibility considerations on the further major climate determinants. CO2 delivers the major contribution to the climate variabilities. The second major influence stems from the modulation of the absorbed insolation by the sun’s luminosity, the planetary albedo (via paleogeography/tectonics, or snow/ice and sea level), and potentially cosmic rays. The Milankovitch-cycles turn out to play a subordinate role for understanding the climate variabilities on the high level pursued in this study. However, there is room for other important contributions, particularly from ocean currents. At the very least, the benefit of the present analysis is to have a handy tool for estimates, particularly to quickly size risk from the CO2-temperature relationship.