#
On Hens, Eggs, Temperatures and CO_{2}: Causal Links in Earth’s Atmosphere

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## Abstract

**:**

_{2}]) has been enormous. According to the commonly assumed causality link, increased [CO

_{2}] causes a rise in T. However, recent developments cast doubts on this assumption by showing that this relationship is of the hen-or-egg type, or even unidirectional but opposite in direction to the commonly assumed one. These developments include an advanced theoretical framework for testing causality based on the stochastic evaluation of a potentially causal link between two processes via the notion of the impulse response function. Using, on the one hand, this framework and further expanding it and, on the other hand, the longest available modern time series of globally averaged T and [CO

_{2}], we shed light on the potential causality between these two processes. All evidence resulting from the analyses suggests a unidirectional, potentially causal link with T as the cause and [CO

_{2}] as the effect. That link is not represented in climate models, whose outputs are also examined using the same framework, resulting in a link opposite the one found when the real measurements are used.

Science is generated by and devoted to free inquiry: the idea that any hypothesis, no matter how strange, deserves to be considered on its merits. The suppression of uncomfortable ideas may be common in religion and politics, but it is not the path to knowledge; it has no place in the endeavor of science. We do not know in advance who will discover fundamental new insights.Carl Sagan [1]

## 1. Introduction

_{2}]) challenged the conventional, and well established, wisdom that increased [CO

_{2}] causes an increase in temperature. The study examined whether the commonly assumed causal chain is supported by data or, alternatively, whether a hen-or-egg (HOE) causal relationship is more plausible. The phrase “hen or egg” (originally in Greek, ὄρνις ἢ ᾠὸν) was first used in a philosophical context by Plutarch [3] to describe situations in which it is not clear which of two interrelated events or processes is the cause and which the effect.

_{2}] data at several sites (Mauna Loa, HI, USA; Barrow, AK, USA; South Pole; global average) with monthly time steps for the period 1980–2019. An innovative element of this study was that it explained the reasons why using the original T and [CO

_{2}] data series yielded spurious results, and it proposed using the changes (differences in time) thereof instead. We note that differencing is of very common use in economics literature (e.g., [4,5]). In particular, for the [CO

_{2}] it proposed taking the logarithm before differencing (something resembling techniques used in economics [5]) and thus the time series that were correlated were $\mathsf{\Delta}T$ and $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$, where the differences are taken over 12 months. By studying lagged correlations of the two, the study asserted that, while both causality directions exist, the results support the hypothesis that the dominant direction is T → CO

_{2}. Changes in [CO

_{2}] follow changes in T by about six months on a monthly scale or about one year on an annual scale. In turn, the study attempted to interpret this mechanism by referring to biochemical reactions, since at higher temperatures soil respiration, and hence CO

_{2}emission, increases.

_{2}] concentration (again in terms of differences $\mathsf{\Delta}T$ and $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ over 12 months) was included in the thirty case studies presented. In brief, the related analyses pointed to the following (quoting from [7]):

Clearly, the results […] suggest a (mono-directional) potentially causal system with T as the cause and [CO_{2}] as the effect. Hence the common perception that increasing [CO_{2}] causes increased T can be excluded as it violates the necessary condition for this causality direction.

[…] in other words, it is the increase of temperature that caused increased CO_{2}concentration. Though this conclusion may sound counterintuitive at first glance, because it contradicts common perception […], in fact it is reasonable. The temperature increase began at the end of the Little Ice Period, in the early nineteenth century, when human CO_{2}emissions were negligible […].

_{2}] relationship, here we will go deeper into this.

- To expand the time frame of the investigation backward and forward by exploiting the longest available data series (Section 4).
- To check whether seasonality, as reflected in different phases of [CO
_{2}] time series at different latitudes, plays any role that could modify or possibly reverse the detected causality relationship (Section 5). - To propose and apply a method for investigating the effect of the timescale in causality detection (Section 6).
- To extend the methodology for disambiguating cases in which the type of causality, HOE or unidirectional, is not quite clear (Section 7).
- To exploit the methodology in defining a type of data analysis that, regardless of the detection of causality per se, could shed light on modeling performance by comparing observational data with model results (Section 8).
- To discuss possible extensions of the scope of the methodology, i.e., from detecting possible causality to building a more detailed model of stochastic type (Section 9).
- To provide logical support for the findings by revisiting the carbon balance in the atmosphere (Appendix A.1) and investigating additional processes that may have caused increases in temperature (Appendix A.2, Appendix A.3 and Appendix A.4).

## 2. Summary of the Stochastic Approach to Causality

- Potentially HOE causal if we have ${g}_{j}>0$ for both some positive and some negative lags j,
- Potentially causal if ${g}_{j}=0$ for all $j<0$, and
- Potentially anticausal if ${g}_{j}=0$ for all $j>0$

## 3. Data and Case Studies

_{2}] relationship, case studies #23 and #24 in [7] used satellite-based temperature series (UAH) for the lower troposphere and [CO

_{2}] data from Mauna Loa. Temperature data of the other two satellite levels for the troposphere were also examined, where the results were very similar to those reported for case studies #23 and #24.

_{2}concentration, in addition to the Mauna Loa data set, which we updated to include the latest measurements of more than one year, we also added the South Pole data set compiled by the US National Oceanic and Atmospheric Administration (NOAA). The measurements began in 1975 and are taken for two cases, flask and in situ, of which we used the former on a mean monthly basis, except in a few cases of missing data where we filled in with in situ data.

_{2}] time series used in climate models for scenario SSP2-4.5 have also been retrieved and analyzed. We note, though, that these time series are given on an annual timescale, unlike all other data that are provided on a monthly scale.

## 4. Investigating the Maximum Time Span That Modern Data Allow

_{2}] measurements is that of Mauna Loa, which began in 1958. The global temperature at 2 m of the NCEP/NCAR reanalysis series goes back to 1948 and thus allows studying the T-[CO

_{2}] relationship for the period 1958–2022 (two additional decades of data in comparison to those studied in [7]).

## 5. Investigating the Possible Effect of Seasonality

_{2}] time series at different latitudes, could modify or possibly reverse the detected causality relationship, we have conducted an additional analysis with the South Pole [CO

_{2}] measurements, which began in 1975.

- The system T-[CO
_{2}] appears to be potentially causal (unidirectional) in the direction $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$, rather than hen-or-egg causal. - All characteristic time lags (${h}_{c},{\mu}_{h},{h}_{1/2}$) are positive in the direction $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ (ranging from about 7 to about 10 months), and negative in the direction $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$.
- The explained variance ratio is greater in the direction $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ (about 1/3) than in the direction $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ (around 1/5).

## 6. On the Timescale of Validity of Results

## 7. Possible Ambiguities and Disambiguation

## 8. Comparing Observational Data with Model Results

_{2}] causality analysis, which is based on measurements. To this aim, we use climate model outputs as specified in Section 3, in the case studies #16 to #23 detailed in Table 1. Numerical results of our analysis are shown in Table 1, and graphical depictions of IRFs are shown in Figure 9 for the cases in which no roughness constraint is used and in Figure 10 for the cases in which the roughness constraint is used.

_{2}] time series of measurements, which are available on a monthly scale, the SSP2 [CO

_{2}] data series is provided on an annual scale. Therefore, case studies #16 to #23 had to be made on an annual scale. If we did the analysis for the period 1958–2021, as in case studies #3 and #4 (NCEP/NCAR Reanalysis temperature at 2 m; and Mauna Loa [CO

_{2}] time series), the annual data would be too few to support estimation of the IRFs (63 data values to estimate 41 coefficients). Therefore, in case studies #16 to #23 we extended the period back to 1850, which is covered by the climate model outputs. We performed separate analyses for the periods 1850–2100 (entire period covered by climate models) and 1850–2021 (only the past).

- There is no essential difference between the results for the periods 1850–2100 and 1850–2021.
- While, as expected, the IRFs differ if they are calculated with or without constraining roughness, the characteristic lags are similar in the two cases (with the exception of ${h}_{1/2}$ in cases #17 and #21).
- In all cases, the lags are negative in the direction $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ and positive in the direction $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$, suggesting a HOE causality with principal direction $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$.
- Clearly, the finding in point 3, resulting from climate model outputs, is opposite to the results found when real measurements are used (NCEP/NCAR Reanalysis temperature and Mauna Loa [CO
_{2}] time series). - Oddly, while the principal direction suggested by the models is $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$, the explained variance is impressively low (10–15%) in this direction and impressively high (reaching 90%) in the opposite direction, $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$.

_{2}] with the CMIP6 time series for the same period, 1958–2021, while keeping the NCEP/NCAR time series for T, there is still agreement of the annual with the monthly cross-correlations. However, when we also replace the NCEP/NCAR time series for T with the CMIP6 series (lower left plot), the two plots decouple. The decoupling is even more prominent if we go to the longer period 1850–2021 (lower right panel).

## 9. Discussion and Further Results

_{2}] → T makes a compelling narrative, as everything is blamed on a single cause, the human CO

_{2}emissions. Indeed, this has been the popular narrative for decades. However, popularity does not necessarily mean correctness, and here we have provided strong arguments against this assumption. Since we have identified atmospheric temperature as the cause and atmospheric CO

_{2}concentration as the effect, one may be tempted to ask the question: What is the cause of the modern increase in temperature? Apparently, this question is much more difficult to reply to, as we can no longer attribute everything to any single agent.

_{2}emissions as the main cause and regards everything else as feedback of the single main cause, can explain what happened on Earth for 4.5 billion years of changing climate.

- The dependence of the carbon cycle on temperature is quite strong and indeed major increases of [CO
_{2}] can emerge as a result of temperature rise. In other words, we show that the natural [CO_{2}] changes due to temperature rise are far larger (by a factor > 3) than human emissions (Appendix A.1). - There are processes, such as the Earth’s albedo (which is changing in time as any other characteristic of the climate system), the El Niño–Southern Oscillation (ENSO) and the ocean heat content in the upper layer (represented by the vertically averaged temperature in the layer 0–100 m), which are potential causes of the temperature increase, unlike what is observed with [CO
_{2}], their changes precede those of temperature (Appendix A.2, Appendix A.3 and Appendix A.4). - On a large timescale, the analysis of paleoclimatic data supports the primacy of the causal direction T → [CO
_{2}], even though some controversy remains about this issue (Appendix A.5).

_{2}emissions due to the burning of fossil fuels have largely increased since the beginning of the industrial age. However, the global temperature increase began succeeding the Little Ice Period, at a time when human CO

_{2}emissions were very low. To cast light on the problem, we examine the issue of CO

_{2}emissions vs. atmospheric temperature further in the Supplementary Information, where we provide evidence that they are not correlated with each other. The outgassing from the sea is also highlighted sometimes in the literature among the climate-related mechanisms. On the other hand, the role of the biosphere and biochemical reactions is often downplayed, along with the existence of complex interactions and feedback. This role can be summarized in the following points, examined in detail and quantified in Appendix A.1.

- Terrestrial and maritime respiration and decay are responsible for the vast majority of CO
_{2}emissions [32], Figure 5.12. - Overall, natural processes of the biosphere contribute 96% to the global carbon cycle, the rest, 4%, being human emissions (which were even lower in the past [33]).
- The biosphere is more productive at higher temperatures, as the rates of biochemical reactions increase with temperature, which leads to increasing natural CO
_{2}emission [2]. - Additionally, a higher atmospheric CO
_{2}concentration makes the biosphere more productive via the so-called carbon fertilization effect, thus resulting in greening of the Earth [34,35], i.e., amplification of the carbon cycle, to which humans also contribute through crops and land-use management [36].

_{2}]. Generally, the time lags shown in Figure 13 complete a consistent picture of potential causality links among climate processes and always confirm the $T\to \left[{\mathrm{C}\mathrm{O}}_{2}\right]$ direction.

_{2}] system is potentially causal with direction $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$, we estimate $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ as

## 10. Conclusions

- All evidence resulting from the analyses of the longest available modern time series of atmospheric concentration of [CO
_{2}] at Mauna Loa, Hawaii, along with that of globally averaged T, suggests a unidirectional, potentially causal link with T as the cause and [CO_{2}] as the effect. This direction of causality holds for the entire period covered by the observations (more than 60 years). - Seasonality, as reflected in different phases of [CO
_{2}] time series at different latitudes, does not play any role in potential causality, as confirmed by replacing the Mauna Loa [CO_{2}] time series with that in South Pole. - The unidirectional $T\to \mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ potential causal link applies to all timescales resolved by the available data, from monthly to about two decades.
- The proposed methodology is simple, flexible and effective in disambiguating cases where the type of causality, HOE or unidirectional, is not quite clear.
- Furthermore, the methodology defines a type of data analysis that, regardless of the detection of causality per se, assesses modeling performance by comparing observational data with model results. In particular, the analysis of climate model outputs reveals a misrepresentation of the causal link by these models, which suggest a causality direction opposite to the one found when the real measurements are used.
- Extensions of the scope of the methodology, i.e., from detecting possible causality to building a more detailed model of stochastic type, are possible, as illustrated by a toy model for the T-[CO
_{2}] system, with explained variance of [CO_{2}] reaching an impressive 99.9%. - While some of the findings of this study seem counterintuitive or contrary to mainstream opinions, they are logically and computationally supported by arguments and calculations given in the Appendices.

## Supplementary Materials

_{2}emissions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

_{2}time series are readily available on a monthly scale from the climexp (http://climexp.knmi.nl/ (accessed on 1 January 2023) platform, namely from http://climexp.knmi.nl/data/inair_0-360E_-90-90N_n.dat and http://climexp.knmi.nl/getindices.cgi?WMO=CDIACData/maunaloa_f&STATION=Mauna_Loa_CO2 (accessed on 1 January 2023). The South Pole CO

_{2}concentration data are provided by the Global Monitoring Laboratory of the USA’s National Oceanic and Atmospheric Administration (NOAA) at https://gml.noaa.gov/dv/data/index.php?parameter_name=Carbon%2BDioxide&site=SPO (accessed on 1 January 2023). The data retrieved are “Monthly Averages” for Type “Flask” and “Insitu”. The CO

_{2}time series used in climate models have been downloaded from https://gmd.copernicus.org/articles/13/3571/2020/gmd-13-3571-2020-supplement.zip (accessed on 1 January 2023); from the Excel file provided, the data from the column “CO

_{2}ppm World” of the tabs “T2—History Year 1750 to 2014” and “T5—SSP2-4.5” have been retrieved. The climate model outputs were downloaded from the climexp platform, http://climexp.knmi.nl/selectfield_cmip6.cgi (accessed on 1 January 2023); specifically, from the “Monthly CMIP6 scenario runs”, the globally averaged time series on “CMIP6 mean over all members” and “UKESM1-0-LL f2” have been derived through the platform. The CERES data were downloaded from https://ceres-tool.larc.nasa.gov/ord-tool/jsp/SSF1degEd41Selection.jsp (accessed on 17 March 2023). The SOI data were downloaded from https://www.ncdc.noaa.gov/teleconnections/enso/indicators/soi/ (accessed on 17 March 2023). The data on monthly global upper ocean mean temperature were downloaded from http://climexp.knmi.nl/getindices.cgi?WMO=NODCData/temp100_global&STATION=global_upper_ocean_mean_temperature (accessed on 17 March 2023).

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Notes on Carbon Cycle and its Dependence on Temperature

_{2}emissions and 1.2 ± 0.7 Gt C/year from land-use change) is imposed and then distributed into several components. This representation is misleading, missing the large changes (by orders of magnitude) in the historical evolution of the abundance of CO

_{2}in Earth’s atmosphere. A different approach and account has recently been provided by Stallinga [53], according to whom humans add 38 Gt of per year to the atmosphere-ocean system, a quantity equivalent to 10.4 Gt C/year.

_{2}inflows to the atmosphere.

**Figure A1.**Annual carbon balance in the Earth’s atmosphere in Gt C/year, based on the IPCC [32] estimates. The balance of 5.1 Gt C/year is the annual accumulation of carbon (in the form of CO

_{2}) in the atmosphere.

_{2}, releasing energy and consuming molecular oxygen [32]. As seen in Figure A1 (and in several publications, e.g., [55]), respiration has increased in recent years, the main reason for this being the increased temperature. Photosynthesis, the biochemical process that removes CO

_{2}from the atmosphere, producing carbohydrates in plants, algae and bacteria using the energy of light [32], has also increased, resulting in the greening of Earth [34,35,36] due to the increased atmospheric concentration of CO

_{2}, which is plants’ food.

**Figure A2.**Evolution of global land (terrestrial) and sea (maritime) temperature at 2 m from the NCEP/NCAR Reanalysis data set, retrieved from the ClimExp platform, and resulting slopes of linear trends.

_{2}emission by fossil fuel combustion (9.4 Gt C /year including cement production).

#### Appendix A.2. Investigation of Causality between Albedo and Atmospheric Temperature

^{2}, this implies a difference (imbalance) of received energy by Earth of $0.004\times 340=1.4$ W/m

^{2}. This result does not disagree with that of Hansen et al. [54], who found that in the period January 2015 through March 2022, the global absorbed solar energy is +1.01 W/m

^{2}higher than the mean for the first 10 years of data (2000–2009). These figures are greater than the average imbalance (net absorbed energy) of the Earth, which, if calculated from the ocean heat content data, is about 0.4 W/m

^{2}[33]. According to mainstream science, this imbalance is attributed to the increase of [CO

_{2}], but the analyses in this study do not support this hypothesis. Moreover, it is hard to see how the albedo fall could be caused by increased [CO

_{2}] (and for this reason, it is usually blamed on aerosols).

**Figure A3.**TOA albedo time series (continuous line), as provided by NASA’s Clouds and the Earth’s Radiant Energy System (CERES), along with linear trend (dashed line).

**Table A1.**Summary indices of the case studies related to albedo. Data are on a monthly timescale and the time step of differencing is 1 year; for explanation of symbols see Table 1.

Case System | # | Direction | ${\mathit{h}}_{\mathbf{c}}$ | ${\mathit{\mu}}_{\mathit{h}}$ | ${\mathit{h}}_{1/2}$ | ${\mathit{r}}_{\mathit{y}\mathit{x}}\left({\mathit{h}}_{\mathit{c}}\right)$ | $\mathit{e}$ | $\mathit{\epsilon}$ |
---|---|---|---|---|---|---|---|---|

Albedo, α: CERES, TERRA; T: NCEP/NCAR; period: 2000–2022 | 24 | $-\mathsf{\Delta}\alpha \to \mathsf{\Delta}T$ | 3 | 1.08 | 2.90 | 0.24 | 0.13 | 9.1 × 10^{–4} |

25 | $\mathsf{\Delta}T\to -\mathsf{\Delta}\alpha $ | –3 | –0.31 | –2.46 | 0.24 | 0.06 | 3.6 × 10^{–4} |

**Figure A4.**IRFs for albedo–temperature based on the CERES albedo time series and the NCEP/NCAR Reanalysis temperature at 2 m, respectively—case studies #24 (

**left**; $-\mathsf{\Delta}\alpha \to \mathsf{\Delta}T]$;) and #25 (

**right**; $\mathsf{\Delta}T\to -\mathsf{\Delta}\alpha $).

#### Appendix A.3. Investigation of Causality between El Niño, Atmospheric Temperature and CO_{2}

**Figure A5.**SOI time series (continuous line) along with rolling (right-aligned) 10-year average (dashed line). Negative and positive values indicate the El Niño and La Niña phases, respectively.

_{2}]. In both cases, we tested differences with a time step of differencing of 1 year (thus reducing the effect of seasonality) and to make the correlation positive, we used $-\mathsf{\Delta}\mathrm{S}\mathrm{O}\mathrm{I}$ (as in the albedo case).

**Table A2.**Summary indices of the case studies related to ENSO. Data are on a monthly timescale and the time step of differencing is 1 year; for explanation of symbols see Table 1.

Case System | # | Direction | ${\mathit{h}}_{\mathbf{c}}$ | ${\mathit{\mu}}_{\mathit{h}}$ | ${\mathit{h}}_{1/2}$ | ${\mathit{r}}_{\mathit{y}\mathit{x}}\left({\mathit{h}}_{\mathit{c}}\right)$ | $\mathit{e}$ | $\mathit{\epsilon}$ |
---|---|---|---|---|---|---|---|---|

$\mathrm{S}\mathrm{O}\mathrm{I}$: NOAA; T: NCEP/NCAR; period: 1951–2022 | 26 | $-\mathsf{\Delta}\mathrm{S}\mathrm{O}\mathrm{I}\to \mathsf{\Delta}T$ | 3 | 4.14 | 3.85 | 0.46 | 0.33 | 8.1 × 10^{–4} |

27 | $\mathsf{\Delta}T\to -\mathsf{\Delta}\mathrm{S}\mathrm{O}\mathrm{I}$ | 3 | –2.15 | –0.93 | 0.46 | 0.30 | 2.3 × 10^{–3} | |

$\mathrm{S}\mathrm{O}\mathrm{I}$: NOAA; [CO_{2}]: Mauna Loa, period: 1958–2022 | 28 | $-\mathsf{\Delta}\mathrm{S}\mathrm{O}\mathrm{I}\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 11 | 11.62 | 11.15 | 0.32 | 0.24 | 6.6 × 10^{–4} |

29 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to -\mathsf{\Delta}\mathrm{S}\mathrm{O}\mathrm{I}$ | –11 | −3.73 | –3.84 | 0.32 | 0.08 | 0 |

**Figure A6.**IRFs for ENSO–temperature based on the SOI time series and the NCEP/NCAR Reanalysis temperature at 2 m, respectively—case studies #26 (

**left**; $-\mathsf{\Delta}\mathrm{S}\mathrm{O}\mathrm{I}\to \mathsf{\Delta}T]$;) and #27 (

**right**; $\mathsf{\Delta}T\to -\mathsf{\Delta}\mathrm{S}\mathrm{O}\mathrm{I}$).

**Figure A7.**IRFs for ENSO–[CO

_{2}] based on the SOI and the Mauna Loa time series, respectively—case studies #28 (

**left**; $-\mathsf{\Delta}\mathrm{S}\mathrm{O}\mathrm{I}\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]]$;) and #29 (

**right**; $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to -\mathsf{\Delta}\mathrm{S}\mathrm{O}\mathrm{I}$).

#### Appendix A.4. Investigation of Causality between Ocean Heat Content, Atmospheric Temperature and CO_{2}

_{2}] is the heat content of the upper-layer ocean. This is indirectly represented by data of the upper ocean mean temperature of the layer 0–100 m [63] (OMT0–100), also known as Vertically Averaged Temperature Anomaly (0–100 m layer) [64]. These are based on historical ocean temperature data, bathythermograph data and Argo data [65] and are made available at the timescale of three months by the National Oceanographic Data Center of the US NOAA [64], as well as by the ClimExp platform, which we used to download the data. The time series, starting in 1955, is plotted in Figure A8.

**Figure A8.**OMT0–100 time series (continuous line) along with rolling (right-aligned) 10-year average (dashed line).

**Table A3.**Summary indices of the case studies related to OMT0–100. Data are on a three-month timescale and the time step of differencing is 1 year; for explanation of symbols see Table 1.

Case System | # | Direction | ${\mathit{h}}_{\mathbf{c}}$ | ${\mathit{\mu}}_{\mathit{h}}$ | ${\mathit{h}}_{1/2}$ | ${\mathit{r}}_{\mathit{y}\mathit{x}}\left({\mathit{h}}_{\mathit{c}}\right)$ | $\mathit{e}$ | $\mathit{\epsilon}$ |
---|---|---|---|---|---|---|---|---|

OMT0–100: NOAA; T: NCEP/NCAR; period: 1955–2022 | 30 | ΔOMT0–100 $\to \mathsf{\Delta}T$ | 0 | 2.42 | 0.98 | 0.68 | 0.53 | 7.1 × 10^{–3} |

31 | $\mathsf{\Delta}T\to $ ΔOMT0–100 | 0 | –2.15 | –0.93 | 0.68 | 0.52 | 3.8 × 10^{–3} | |

OMT0–100: NOAA; [CO_{2}]: Mauna Loa; period: 1958–2022 | 32 | ΔOMT0–100 $\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 2 | 2.22 | 2.93 | 0.46 | 0.35 | 5.8 × 10^{–4} |

33 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to $ ΔOMT0–100 | –2 | −2.73 | –2.82 | 0.46 | 0.21 | 5.6 × 10^{–3} |

**Figure A9.**IRFs for upper ocean temperature—atmospheric temperature based on the OMT0–100 and the NCEP/NCAR Reanalysis data, respectively—case studies #30 (

**left**; ΔOMT0–100 $\to \mathsf{\Delta}T]$;) and #31 (

**right**; $\mathsf{\Delta}T\to $ ΔOMT0–100).

**Figure A10.**IRFs for upper ocean temperature—[CO

_{2}] based on the OMT0–100 and the NCEP/NCAR Reanalysis data, respectively—case studies #32 (

**left**; ΔOMT0–100 $\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]]$;) and #33 (

**right**; $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to $ ΔOMT0–100).

#### Appendix A.5. Notes on the T-[CO_{2}] Relationship on Large Timescales

_{2}concentration measurements (about six decades), it may be useful to refer to studies that used paleoclimatic proxies to assess the T-[CO

_{2}] relationship on much larger timescales. Berner and Kothavala [66] studied the entire phanerozoic (the last 530 million years) and asserted that “over the long term there is indeed a correlation between CO

_{2}and paleotemperature”, while their Figure 13 showed that the atmospheric [CO

_{2}] was much higher (up to 27 times) than the current one for most of the time during the phanerozoic. They also emphasized the “importance of considering ALL factors affecting CO

_{2}when modelling the long term carbon cycle and not concentrating [on] only one cause”. On the other hand, Veizer et al. [67] presented evidence for decoupling atmospheric CO

_{2}and global climate during the phanerozoic, questioning the role of the (partial pressure of) CO

_{2}as the main driving force of past global (long-term) climate changes, at least during two of the four main cool climate modes of the phanerozoic.

_{2}] as the effect, with estimates of the time lag varying from 50 to 1000 years, depending on the time period and the particular study [23,70,71,72]. Claims that CO

_{2}concentration leads (i.e., a negative lag) have not generally been made in these studies. At most, a synchrony claim has been sought on the basis that the estimated positive lags are often within the 95% uncertainty range [73], while in one of them [72], it has been asserted that a “short lead of CO

_{2}over temperature cannot be excluded”.

_{2}]. For the same period, another study by Shakun et al. [75] gave different lead-lag relationships for the north and south hemispheres. Generally, it appears that the issue remains controversial, as illustrated, for example, in a recent report (2021) by NOAA (in the framework of Paleo Perspectives [76]), which states: “While it might seem simple to determine cause and effect between carbon dioxide and climate from which change occurs first, […] the determination of cause and effect remains exceedingly difficult.”

_{2}concentration follows has been given by Roe [40], who demonstrated that in the Quaternary it is the effect of Milanković cycles, rather than of atmospheric CO

_{2}concentration, that explains the glaciation process. Specifically, he found that “variations in atmospheric CO

_{2}appear to lag the rate of change of global ice volume. This implies only a secondary role for CO

_{2}—variations in which produce a weaker radiative forcing than the orbitally-induced changes in summertime insolation—in driving changes in global ice volume.”

_{2}concentration, as obtained by proxy data, appears to be of HOE type with principal direction T → [CO

_{2}]”, same as in the records of the modern period, but with a much higher time lag, of the order of 1000 years.

## References

- Sagan, C. Cosmos; Ballantine Books: New York, NY, USA, 1985. [Google Scholar]
- Koutsoyiannis, D.; Kundzewicz, Z.W. Atmospheric temperature and CO
_{2}: Hen-or-egg causality? Sci**2020**, 2, 83. [Google Scholar] [CrossRef] - Πλούταρχος, Συμποσιακά Β’ (Plutarch, Quaestiones Convivales B’)—Βικιθήκη. Available online: https://el.wikisource.org/wiki/Συμποσιακά_Β΄ (accessed on 5 February 2023).
- Chan, K.H.; Hayya, J.C.; Ord, J.K. A note on trend removal methods: The case of polynomial regression versus variate differencing. Econometrica
**1977**, 45, 737–744. [Google Scholar] [CrossRef] - Estrella, A. Why does the yield curve predict output and inflation? Econ. J.
**2005**, 115, 722–744. [Google Scholar] [CrossRef] - Koutsoyiannis, D.; Onof, C.; Christofides, A.; Kundzewicz, Z.W. Revisiting causality using stochastics: 1. Theory. Proc. R. Soc. A
**2022**, 478, 20210836. [Google Scholar] [CrossRef] - Koutsoyiannis, D.; Onof, C.; Christofides, A.; Kundzewicz, Z.W. Revisiting causality using stochastics: 2. Applications. Proc. R. Soc. A
**2022**, 478, 20210835. [Google Scholar] [CrossRef] - Young, P.C. Recursive Estimation and Time Series Analysis; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Young, P.C. Refined instrumental variable estimation: Maximum likelihood optimization of a unified Box-Jenkins model. Automatica
**2015**, 52, 35–46. [Google Scholar] [CrossRef] - Papoulis, A. Probability, Random Variables and Stochastic Processes, 3rd ed.; McGraw-Hill: New York, NY, USA, 1991. [Google Scholar]
- Kestin, T.S.; Karoly, D.J.; Yang, J.I.; Rayner, N.A. Time-frequency variability of ENSO and stochastic simulations. J. Clim.
**1998**, 11, 2258–2272. [Google Scholar] [CrossRef] - Wills, R.C.; Schneider, T.; Wallace, J.M.; Battisti, D.S.; Hartmann, D.L. Disentangling global warming, multidecadal variability, and El Niño in Pacific temperatures. Geophys. Res. Lett.
**2018**, 45, 2487–2496. [Google Scholar] [CrossRef] - Granger, C.W. Investigating causal relations by econometric models and cross-spectral methods. Econometrica
**1969**, 37, 424–438. [Google Scholar] [CrossRef] - Granger, C.W. Testing for causality: A personal viewpoint. J. Econ. Dyn. Control.
**1980**, 2, 329–352. [Google Scholar] [CrossRef] - Moraffah, R.; Sheth, P.; Karami, M.; Bhattacharya, A.; Wang, Q.; Tahir, A.; Raglin, A.; Liu, H. Causal inference for time series analysis: Problems, methods and evaluation. Knowl. Inf. Syst.
**2021**, 63, 3041–3085. [Google Scholar] [CrossRef] - Pearl, J. Causal inference in statistics: An overview. Stat. Surv.
**2009**, 3, 96–146. [Google Scholar] [CrossRef] - Pearl, J.; Glymour, M.; Jewell, N.P. Causal Inference in Statistics: A Primer; Wiley: Chichester, UK, 2016. [Google Scholar]
- Pearl, J. and Mackenzie, D., The Book of Why, The New Science of Cause and Effect, Basic Books: New York, NY, USA, 2018.
- Kalnay, E.; Kanamitsu, M.; Kistler, R.; Collins, W.; Deaven, D.; Gandin, L.; Iredell, M.; Saha, S.; White, G.; Woollen, J.; et al. The NCEP/NCAR 40-year reanalysis project. Bull. Am. Meteorol. Soc.
**1996**, 77, 437–472. [Google Scholar] [CrossRef] - Meinshausen, M.; Nicholls, Z.R.J.; Lewis, J.; Gidden, M.J.; Vogel, E.; Freund, M.; Beyerle, U.; Gessner, C.; Nauels, A.; Bauer, N.; et al. The shared socio-economic pathway (SSP) greenhouse gas concentrations and their extensions to 2500. Geosci. Model Dev.
**2020**, 13, 3571–3605. [Google Scholar] [CrossRef] - Sellar, A.A.; Jones, C.G.; Mulcahy, J.P.; Tang, Y.; Yool, A.; Wiltshire, A.; O’Connor, F.M.; Stringer, M.; Hill, R.; Palmieri, J.; et al. UKESM1: Description and evaluation of the UK Earth System Model. J. Adv. Model. Earth Syst.
**2019**, 11, 4513–4558. [Google Scholar] [CrossRef] - Koutsoyiannis, D. Stochastics of Hydroclimatic Extremes—A Cool Look at Risk, 2nd ed.; Kallipos Open Academic Editions: Athens, Greece, 2022; 346p, ISBN 978-618-85370-0-2. [Google Scholar] [CrossRef]
- Koutsoyiannis, D. Time’s arrow in stochastic characterization and simulation of atmospheric and hydrological processes. Hydrol. Sci. J.
**2019**, 64, 1013–1037. [Google Scholar] [CrossRef] - Strotz, R.H.; Wold, H.O.A. Recursive vs. nonrecursive systems: An attempt at synthesis (Part I of a triptych on causal chain systems). Econometrica
**1960**, 28, 417–427. Available online: https://www.jstor.org/stable/1907731 (accessed on 15 March 2023). [CrossRef] - Hannart, A.; Pearl, J.; Otto, F.E.L.; Naveau, P.; Ghil, M. Causal counterfactual theory for the attribution of weather and climate-related events. Bull. Am. Met. Soc.
**2016**, 97, 99–110. [Google Scholar] [CrossRef] - Hannart, A.; Naveau, P. Probabilities of causation of climate changes. J. Clim.
**2018**, 31, 5507–5524. [Google Scholar] [CrossRef] - Koutsoyiannis, D.; Efstratiadis, A.; Mamassis, N.; Christofides, A. On the credibility of climate predictions. Hydrol. Sci. J.
**2008**, 53, 671–684. [Google Scholar] [CrossRef] - Anagnostopoulos, G.G.; Koutsoyiannis, D.; Christofides, A.; Efstratiadis, A.; Mamassis, N. A comparison of local and aggregated climate model outputs with observed data. Hydrol. Sci. J.
**2010**, 55, 1094–1110. [Google Scholar] [CrossRef] - Koutsoyiannis, D.; Christofides, A.; Efstratiadis, A.; Anagnostopoulos, G.G.; Mamassis, N. Scientific dialogue on climate: Is it giving black eyes or opening closed eyes? Reply to “A black eye for the Hydrological Sciences Journal” by D. Huard. Hydrol. Sci. J.
**2011**, 56, 1334–1339. [Google Scholar] [CrossRef] - Tyralis, H.; Koutsoyiannis, D. On the prediction of persistent processes using the output of deterministic models. Hydrol. Sci. J.
**2017**, 62, 2083–2102. [Google Scholar] [CrossRef] - Scafetta, N. CMIP6 GCM validation based on ECS and TCR ranking for 21st century temperature projections and risk assessment. Atmosphere
**2023**, 14, 345. [Google Scholar] [CrossRef] - Masson-Delmotte, V.; Zhai, P.; Pirani, A.; Connors, S.L.; Péan, C.; Berger, S.; Caud, N.; Chen, Y.; Goldfarb, L.; Gomis, M.I.; et al. (Eds.) IPCC, Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change; Cambridge University Press: Cambridge, UK; New York, NY, USA, 2021; 2391p. [Google Scholar] [CrossRef]
- Koutsoyiannis, D. Rethinking climate, climate change, and their relationship with water. Water
**2021**, 13, 849. [Google Scholar] [CrossRef] - Zhu, Z.; Piao, S.; Myneni, R.B.; Huang, M.; Zeng, Z.; Canadell, J.G.; Ciais, P.; Sitch, S.; Friedlingstein, P.; Arneth, A.; et al. Greening of the Earth and its drivers. Nature Climate Change
**2016**, 6, 791–795. [Google Scholar] [CrossRef] - Li, Y.; Li, Z.L.; Wu, H.; Zhou, C.; Liu, X.; Leng, P.; Yang, P.; Wu, W.; Tang, R.; Shang, G.F.; et al. Biophysical impacts of earth greening can substantially mitigate regional land surface temperature warming. Nat. Commun.
**2023**, 14, 121. [Google Scholar] [CrossRef] - Chen, C.; Park, T.; Wang, X.; Piao, S.; Xu, B.; Chaturvedi, R.K.; Fuchs, R.; Brovkin, V.; Ciais, P.; Fensholt, R.; et al. China and India lead in greening of the world through land-use management. Nat. Sustain.
**2019**, 2, 122–129. [Google Scholar] [CrossRef] - Milanković, M. Nebeska Mehanika; Udruženje “Milutin Milanković”: Beograd, Serbia, 1935. [Google Scholar]
- Milanković, M. Kanon der Erdbestrahlung und Seine Anwendung auf das Eiszeitenproblem; Koniglich Serbische Akademice: Beograd, Serbia, 1941. [Google Scholar]
- Milanković, M. Canon of Insolation and the Ice-Age Problem; Agency for Textbooks: Belgrade, Serbia, 1998. [Google Scholar]
- Roe, G. In defense of Milankovitch. Geophys. Res. Lett.
**2006**, 33, L24703. [Google Scholar] [CrossRef] - Markonis, Y.; Koutsoyiannis, D. Climatic variability over time scales spanning nine orders of magnitude: Connecting Milankovitch cycles with Hurst–Kolmogorov dynamics. Surv. Geophys.
**2013**, 34, 181–207. [Google Scholar] [CrossRef] - Stephens, G.L.; Hakuba, M.Z.; Kato, S.; Gettelman, A.; Dufresne, J.-L.; Andrews, T.; Cole, J.N.S.; Willen, U.; Mauritsen, T. The changing nature of Earth’s reflected sunlight. Proc. R. Soc. A
**2022**, 478, 1–37. [Google Scholar] [CrossRef] - Connolly, R.; Soon, W.; Connolly, M.; Baliunas, S.; Berglund, J.; Butler, C.J.; Cionco, R.G.; Elias, A.G.; Fedorov, V.M.; Harde, H.; et al. How much has the Sun influenced Northern Hemisphere temperature trends? An ongoing debate. Res. Astron. Astrophys.
**2021**, 21, 131.1–131.68. [Google Scholar] [CrossRef] - Scafetta, N.; Bianchini, A. The planetary theory of solar activity variability: A review. Front. Astron. Space Sci.
**2022**, 9, 937930. [Google Scholar] [CrossRef] - Kamis, J.E. The Plate Climatology Theory: How Geological Forces Influence, Alter, or Control Earth’s Climate and Climate Related Events. Available online: https://books.google.gr/books/?id=7lRqzgEACAAJ (accessed on 10 March 2023).
- Chakrabarty, D. The Climate of History in a Planetary Age; University of Chicago Press: Chicago, IL, USA, 2021; Available online: https://books.google.gr/books?id=ETQXEAAAQBAJ (accessed on 10 March 2023).
- Davis, E.; Becker, K.; Dziak, R.; Cassidy, J.; Wang, K.; Lilley, M. Hydrological response to a seafloor spreading episode on the Juan de Fuca ridge. Nature
**2004**, 430, 335–338. [Google Scholar] [CrossRef] [PubMed] - Urakawa, L.S.; Hasumi, H. A remote effect of geothermal heat on the global thermohaline circulation. J. Geophys. Res. Ocean.
**2009**, 114, C07016. [Google Scholar] [CrossRef] - Patara, L.; Böning, C.W. Abyssal ocean warming around Antarctica strengthens the Atlantic overturning circulation. Geophys. Res. Lett.
**2014**, 41, 3972–3978. [Google Scholar] [CrossRef] - Koutsoyiannis, D. A random walk on water. Hydrol. Earth Syst. Sci.
**2010**, 14, 585–601. [Google Scholar] [CrossRef] - Koutsoyiannis, D. Hydrology and change. Hydrol. Sci. J.
**2013**, 58, 1177–1197. [Google Scholar] [CrossRef] - Friedlingstein, P.; O’Sullivan, M.; Jones, M.W.; Andrew, R.M.; Gregor, L.; Hauck, J.; Le Quéré, C.; Luijkx, I.T.; Olsen, A.; Peters, G.P.; et al. Global Carbon Budget 2022. Earth Syst. Sci. Data
**2022**, 14, 4811–4900. [Google Scholar] [CrossRef] - Stallinga, P. Residence time vs. adjustment time of carbon dioxide in the atmosphere. Entropy
**2023**, 25, 384. [Google Scholar] [CrossRef] - Hansen, J.E.; Sato, M.; Simons, L.; Nazarenko, L.S.; von Schuckmann, K.; Loeb, N.G.; Osman, M.B.; Kharecha, P.; Jin, Q.; Tselioudis, G.; et al. Global warming in the pipeline. arXiv
**2022**, arXiv:2212.04474. [Google Scholar] - Bond-Lamberty, B.; Thomson, A. Temperature-associated increases in the global soil respiration record. Nature
**2010**, 464, 579. [Google Scholar] [CrossRef] [PubMed] - Arrhenius, S.A. Über die Dissociationswärme und den Einfluß der Temperatur auf den Dissociationsgrad der Elektrolyte. Z. Phys. Chem.
**1889**, 4, 96–116. [Google Scholar] [CrossRef] - Patel, K.F.; Bond-Lamberty, B.; Jian, J.L.; Morris, K.A.; McKever, S.A.; Norris, C.G.; Zheng, J.; Bailey, V.L. Carbon flux estimates are sensitive to data source: A comparison of field and lab temperature sensitivity data. Environ. Res. Lett.
**2022**, 17, 113003. [Google Scholar] [CrossRef] - Pomeroy, R.; Bowlus, F.D. Progress report on sulfide control research. Sew. Work. J.
**1946**, 18, 597–640. [Google Scholar] - Robinson, C. Microbial respiration, the engine of ocean deoxygenation. Front. Mar. Sci.
**2019**, 5, 533. [Google Scholar] [CrossRef] - CERES Data Products. SSF1deg—Level 3, Gridded Daily and Monthly Averages of the SSF Product by Instrument. Available online: https://ceres-tool.larc.nasa.gov/ord-tool/jsp/SSF1degEd41Selection.jsp (accessed on 12 March 2023).
- McPhaden, M.J.; Zebiak, S.E.; Glantz, M.H. ENSO as an integrating concept in earth science. Science
**2006**, 314, 1740–1745. [Google Scholar] [CrossRef] - Kundzewicz, Z.W.; Pinskwar, I.; Koutsoyiannis, D. Variability of global mean annual temperature is significantly influenced by the rhythm of ocean-atmosphere oscillations. Sci. Total Environ.
**2020**, 747, 141256. [Google Scholar] [CrossRef] - Levitus, S.; Antonov, J.I.; Boyer, T.P.; Baranova, O.K.; Garcia, H.E.; Locarnini, R.A.; Mishonov, A.V.; Reagan, J.R.; Seidov, D.; Yarosh, E.S.; et al. World Ocean heat content and thermosteric sea level change (0–2000 m). Geophys. Res. Lett.
**2012**, 39, L10603. [Google Scholar] [CrossRef] - National Oceanographic Data Center, NOAA, Global Ocean Heat and Salt Content. Available online: https://www.ncei.noaa.gov/access/global-ocean-heat-content/index3.html (accessed on 12 March 2023).
- Roemmich, D.; Johnson, G.C.; Riser, S.; Davis, R.; Gilson, J.; Owens, W.B.; Garzoli, S.L.; Schmid, C.; Ignaszewski, M. The Argo Program: Observing the global ocean with profiling floats. Oceanography
**2009**, 22, 34–43. [Google Scholar] [CrossRef] - Berner, R.A.; Kothavala, Z. GEOCARB III: A revised model of atmospheric CO
_{2}over Phanerozoic time. Am. J. Sci.**2001**, 301, 182–204. [Google Scholar] [CrossRef] - Veizer, J.; Godderis, Y.; François, L.M. Evidence for decoupling of atmospheric CO
_{2}and global climate during the Phanerozoic eon. Nature**2000**, 408, 698–701. [Google Scholar] [CrossRef] [PubMed] - Jouzel, J.; Lorius, C.; Petit, J.R.; Genthon, C.; Barkov, N.I.; Kotlyakov, V.M.; Petrov, V.M. Vostok ice core: A continuous isotope temperature record over the last climatic cycle (160,000 years). Nature
**1987**, 329, 403–408. [Google Scholar] [CrossRef] - Petit, J.R.; Jouzel, J.; Raynaud, D.; Barkov, N.I.; Barnola, J.-M.; Basile, I.; Bender, M.; Chappellaz, J.; Davis, M.; Delayque, G.; et al. Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica. Nature
**1999**, 399, 429–436. [Google Scholar] [CrossRef] - Caillon, N.; Severinghaus, J.P.; Jouzel, J.; Barnola, J.M.; Kang, J.; Lipenkov, V.Y. Timing of atmospheric CO
_{2}and Antarctic temperature changes across Termination III. Science**2003**, 299, 1728–1731. [Google Scholar] [CrossRef] - Soon, W. Implications of the secondary role of carbon dioxide and methane forcing in climate change: Past, present, and future. Phys. Geogr.
**2007**, 28, 97–125. [Google Scholar] [CrossRef] - Pedro, J.B.; Rasmussen, S.O.; van Ommen, T.D. Tightened constraints on the time-lag between Antarctic temperature and CO
_{2}during the last deglaciation. Clim. Past**2012**, 8, 1213–1221. [Google Scholar] [CrossRef] - Chowdhry Beeman, J.; Gest, L.; Parrenin, F.; Raynaud, D.; Fudge, T.J.; Buizert, C.; Brook, E.J. Antarctic temperature and CO
_{2}: Near-synchrony yet variable phasing during the last deglaciation. Clim. Past**2019**, 15, 913–926. [Google Scholar] [CrossRef] - Parrenin, F.; Masson-Delmotte, V.; Köhler, P.; Raynaud, D.; Paillard, D.; Schwander, J.; Barbante, C.; Landais, A.; Wegner, A.; Jouzel, J. Synchronous change of atmospheric CO
_{2}and Antarctic temperature during the last deglacial warming. Science**2013**, 339, 1060–1063. [Google Scholar] [CrossRef] - Shakun, J.D.; Clark, P.U.; He, F.; Marcott, S.A.; Mix, A.C.; Liu, Z.; Otto-Bliesner, B.; Schmittner, A.; Bard, E. Global warming preceded by increasing carbon dioxide concentrations during the last deglaciation. Nature
**2012**, 484, 49–54. [Google Scholar] [CrossRef] - NOAA National Centers for Environmental Information. Temperature Change and Carbon Dioxide Change; 2021. Available online: https://www.ncei.noaa.gov/sites/default/files/2021-11/8%20-%20Temperature%20Change%20and%20Carbon%20Dioxide%20Change%20-%20FINAL%20OCT%202021.pdf (accessed on 12 January 2023).

**Figure 1.**Explanatory sketch for the definition of the different potential causality types. For each graph, the mean ${\mu}_{h}$ is also plotted with a dashed line.

**Figure 2.**IRFs for temperature–CO

_{2}concentration based on the NCEP/NCAR Reanalysis temperature at 2 m and Mauna Loa [CO

_{2}] time series, respectively—case studies #3 (

**left**; $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$; potentially causal system) and #4 (

**right**; $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$; potentially anticausal system).

**Figure 3.**IRFs for temperature–CO

_{2}concentration based on the NCEP/NCAR Reanalysis temperature at 2 m and the South Pole time series, respectively—case studies #14 (

**left**; $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$; potentially causal system) and #15 (

**right**; $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$; potentially anticausal system).

**Figure 4.**(

**Left column**) Empirical autocorrelation function for the period 1958–2021 and for monthly timescale of (

**upper**) the NCEP/NCAR $\mathsf{\Delta}T$ time series and (

**lower**) the $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ time series for Mauna Loa. (

**Right column**) Empirical cross-correlation function of the two time series (continuous lines in blue), compared with those reconstructed from the IRF and the autocorrelation function on the left panel using the discretized version of Equation (7) (dashed line), for case studies (

**upper**) #3 and (

**lower**) #4.

**Figure 5.**IRFs for temperature–CO

_{2}concentration based on the NCEP/NCAR Reanalysis temperature at 2 m and Mauna Loa time series, respectively, as in Figure 2, but for differencing time steps equal (from upper to lower) 2, 4, 8 and 16 years;

**left**: $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ (potentially causal system);

**right**: $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ (potentially anticausal system).

**Figure 6.**IRFs for temperature–CO

_{2}concentration based on the NCEP/NCAR Reanalysis temperature at 2 m and Mauna Loa time series, respectively, also enabling negative lags (HOE) for causality direction $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ and for differencing time step of 1 year (

**left**, corresponding to Figure 2, left) and 16 years (

**right**, corresponding to Figure 5, bottom left).

**Figure 7.**IRFs for temperature–CO

_{2}concentration based on the NCEP/NCAR Reanalysis temperature at 2 m and Mauna Loa time series, for 21 time lags, differencing time step of 16 years and direction $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$. The lowest nonzero lag of each IRF is marked at the upper-right end of its curve.

**Figure 9.**IRFs for temperature–CO

_{2}concentration based on the CMIP6 mean temperature and SSP2-4.5 CO

_{2}time series, respectively, calculated without using the roughness constraint;

**upper row**: period 1850–2100—case studies #16 (

**left**; $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$) and #17 (

**right**; $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$);

**lower row**: period 1850–2021—case studies #18 (

**left**; $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$) and #19 (

**right**; $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$).

**Figure 10.**IRFs for temperature–CO

_{2}concentration based on the CMIP6 mean temperature and SSP2-4.5 CO

_{2}time series, respectively, as in Figure 9 but calculated using the roughness constraint;

**upper row**: period 1850–2100—case studies #20 (

**left**; $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$) and #21 (

**right**; $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$);

**lower row**: period 1850–2021—case studies #22 (

**left**; $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$) and #23 (

**right**; $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$).

**Figure 11.**Empirical cross-correlation functions for monthly and annual timescales (continuous lines in blue without markers and red lines with circles, respectively) for the data sets indicated in each panel. In all panels, the plot for the monthly scale is that of the NCEP/NCAR data for T and Mauna Loa data for [CO

_{2}], for the period 1958–2021. The upper-left panel also shows the cross-correlation function reconstructed from the IRF and the autocorrelation function using the discretized version of Equation (7) (dashed line).

**Figure 12.**Empirical autocorrelation functions for monthly and annual timescales (continuous lines in blue without markers and red lines with circles, respectively) for the data sets indicated in each panel. In all panels, the plot for the monthly scale is that of the NCEP/NCAR data for T and Mauna Loa data for [CO

_{2}], for the period 1958–2021.

**Figure 13.**Schematic of the examined possible causal links in the climatic system, with noted types of potential causality, HOE or unidirectional, and its direction. Other processes, not examined here, could be internal of the climatic system or external.

**Figure 14.**Modified IRF for temperature–CO

_{2}concentration based on the NCEP/NCAR Reanalysis temperature at 2 m and Mauna Loa time series, respectively, similar to Figure 2 but with IRF coordinates simultaneously optimized with the parameters of Equation (9).

**Figure 15.**Comparison of the actual $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ (

**upper**) and $\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ (

**lower**) with those simulated by the model of Equations (8) and (9).

**Table 1.**Main case studies and resulting summary indices. $\mathsf{\Delta}t$ is the time step of differencing; ${h}_{\mathrm{c}}$ is the time lag maximizing the cross-covariance ${c}_{yx}\left(h\right)$, or equivalently the cross-correlation ${r}_{yx}\left(h\right):={c}_{yx}\left(h\right)/\sqrt{{c}_{xx}\left(0\right){c}_{yy}\left(0\right)}$; ${\mu}_{h}$ is the mean (time average) of the function $g(h)$; ${h}_{1/2}$ is the median of the function $g(h)$; $e$ is the explained variance ratio; and $\epsilon $ is the roughness ratio. The case studies #1 and #2 are contained in [7] as case studies #23 and #24 and are included in the table only for comparison.

Case System | # | Direction | ${\mathit{h}}_{\mathbf{c}}$ | ${\mathit{\mu}}_{\mathit{h}}$ | ${\mathit{h}}_{1/2}$ | ${\mathit{r}}_{\mathit{y}\mathit{x}}\left({\mathit{h}}_{\mathit{c}}\right)$ | $\mathit{e}$ | $\mathit{\epsilon}$ |
---|---|---|---|---|---|---|---|---|

Monthly timescale, varying Δt | ||||||||

T$:\mathrm{UAH};\left[{\mathrm{CO}}_{2}\right]:\mathrm{Mauna}\mathrm{Loa},1979\u20132020(\mathrm{from}[7]),\mathsf{\Delta}t=1$ year | 1 | $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 5 | 7.70 | 6.35 | 0.48 | 0.31 | 1.3 × 10^{–5 *} |

2 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ | –5 | –5.67 | –5.49 | 0.48 | 0.23 | 7.3 × 10^{–4 *} | |

T$:\mathrm{NCEP}/\mathrm{NCAR};\left[{\mathrm{CO}}_{2}\right]:\mathrm{Mauna}\mathrm{Loa},1958\u20132021,\mathsf{\Delta}t=1$ year | 3 | $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 8 | 7.75 | 6.86 | 0.56 | 0.34 | 3.1 × 10^{–4 *} |

4 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ | –8 | −6.31 | –6.30 | 0.56 | 0.23 | 4.4 × 10^{–3 *} | |

$\mathrm{As}\#3\mathrm{and}\#4,\mathsf{\Delta}t=2$ years | 5 | $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 8 | 8.19 | 7.08 | 0.57 | 0.31 | 3.4 × 10^{–4 *} |

6 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ | –8 | −6.31 | –6.31 | 0.57 | 0.21 | 4.5 × 10^{–3 *} | |

$\mathrm{As}\#3\mathrm{and}\#4,\mathsf{\Delta}t=4$ years | 7 | $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 9 | 10.65 | 10.32 | 0.53 | 0.29 | 1.0 × 10^{–4 *} |

8 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ | –9 | −6.28 | –6.28 | 0.53 | 0.14 | 3.8 × 10^{–3 *} | |

$\mathrm{As}\#3\mathrm{and}\#4,\mathsf{\Delta}t=8$ years | 9 | $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 8 | 11.00 | 11.00 | 0.47 | 0.27 | 5.6 × 10^{–5 *} |

10 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ | –8 | −6.55 | –6.54 | 0.47 | 0.11 | 3.6 × 10^{–3 *} | |

$\mathrm{As}\#3\mathrm{and}\#4,\mathsf{\Delta}t=16$ years | 11 | $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 6 | 11.74 | 12.15 | 0.45 | 0.31 | 3.4 × 10^{–5 *} |

12 | $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 6 | 9.98 | 11.13 | 0.45 | 0.33 | 7.6 × 10^{–6} | |

13 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ | –6 | −6.33 | –6.31 | 0.45 | 0.12 | 7.7 × 10^{–3 *} | |

T$:\mathrm{NCEP}/\mathrm{NCAR};\left[{\mathrm{CO}}_{2}\right]:\mathrm{South}\mathrm{Pole},1975\u20132021,\mathsf{\Delta}t=1$ year | 14 | $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 10 | 9.76 | 8.91 | 0.40 | 0.35 | 2.0 × 10^{–4 *} |

15 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ | –10 | –8.51 | –8.35 | 0.40 | 0.18 | 1.1 × 10^{–3 *} | |

Annual timescale,$\mathsf{\Delta}t=1$ year | ||||||||

T: CMIP6 mean, SSP2-4.5; [CO_{2}]: SSP2-4.5, 1850–2100, w/o roughness constraint | 16 | $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 0 | –3.75 | –6.20 | 0.36 | 0.90 | 0.095 |

17 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ | 0 | 9.95 | 15.30 | 0.36 | 0.15 | 0.46 | |

As #16 and #17 but for 1850–2021 | 18 | $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 0 | –6.23 | –8.58 | 0.31 | 0.72 | 0.10 |

19 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ | 0 | 16.18 | 16.16 | 0.31 | 0.10 | 0.295 | |

As #16 and #17 but with roughness constraint | 20 | $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 0 | –3.65 | –5.55 | 0.36 | 0.84 | 3.5 × 10^{–5} |

21 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ | 0 | 6.86 | 1.63 | 0.36 | 0.13 | 7.7 × 10^{–3} | |

As #18 and #19 but with roughness constraint | 22 | $\mathsf{\Delta}T\to \mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]$ | 0 | –7.34 | –8.99 | 0.31 | 0.64 | 8.3 × 10^{–5} |

23 | $\mathsf{\Delta}\mathrm{l}\mathrm{n}\left[{\mathrm{C}\mathrm{O}}_{2}\right]\to \mathsf{\Delta}T$ | 0 | 11.26 | 14.77 | 0.31 | 0.13 | 9.4 × 10^{–3} |

^{*}The roughness was calculated without considering the second derivative at zero.

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## Share and Cite

**MDPI and ACS Style**

Koutsoyiannis, D.; Onof, C.; Kundzewicz, Z.W.; Christofides, A.
On Hens, Eggs, Temperatures and CO_{2}: Causal Links in Earth’s Atmosphere. *Sci* **2023**, *5*, 35.
https://doi.org/10.3390/sci5030035

**AMA Style**

Koutsoyiannis D, Onof C, Kundzewicz ZW, Christofides A.
On Hens, Eggs, Temperatures and CO_{2}: Causal Links in Earth’s Atmosphere. *Sci*. 2023; 5(3):35.
https://doi.org/10.3390/sci5030035

**Chicago/Turabian Style**

Koutsoyiannis, Demetris, Christian Onof, Zbigniew W. Kundzewicz, and Antonis Christofides.
2023. "On Hens, Eggs, Temperatures and CO_{2}: Causal Links in Earth’s Atmosphere" *Sci* 5, no. 3: 35.
https://doi.org/10.3390/sci5030035