Cross-Correlation between Global Temperature and Atmospheric CO 2 with a Temperature-Leading Time Lag

The temperature change and rate of CO ₂ change are correlated with a time lag, as reported in a previous paper. The correlation was investigated by calculating a correlation coefficient r of these changes for selected ENSO events in this study. Annual periodical increases and decreases in the CO ₂ concentration were considered, with a regular pattern of minimum values in August and maximum values in May each year. An increased deviation in CO ₂ and temperature was found in response to the occurrence of El Niño, but the increase in CO ₂ lagged behind the change in temperature by 5 months. This pattern was not observed for La Niña events. An increase in global CO ₂ emissions and a subsequent increase in global temperature proposed by IPCC were not observed, but an increase in global temperature, an increase in soil respiration, and a subsequent increase in global CO ₂ emissions were noticed. This natural process can be clearly detected during periods of increasing temperature specifically during El Niño events. The results cast strong doubts that anthropogenic CO ₂ is the cause of global warming.

• Global Warming
• Thermally-Induced CO ₂
• Soil Respiration
• Cross-Correlation Coefficient
• Time Lag
• El Niño

1. Introduction

Murry Salby analyzed the change in temperature (ΔT) and the change rate of the CO₂ concentration (rco₂) and suggested that the change rate of the CO₂ concentration (drco₂/dt) can be expressed as Equation (1) [1] [2] :

$drc{o}_2/dt=\gamma \Delta T$ (1)

where γ is a constant, or Equation (1) is also expressed by Equation (2):

$r_{C{O}_2}=\gamma \underset{0}{\overset{t}{{\displaystyle \int }}}\Delta Tdt$ (2)

The major natural functions affecting drco₂/dt are (1) plant photosynthesis, (2) CO₂ solubilization into the ocean, and (3) plant decomposition or respiration [3] . For a small temperature range, drco₂/dt may be considered to be approximately proportional to the temperature change [4] :

$\begin{array}{l}drc{o}_2/dt\\ \fallingdotseq \text{rate of photosynthesis}+\text{solubilization rate into sea}\\ +\text{plant decomposition rate}\\ \fallingdotseq \gamma \Delta T\end{array}$ (3)

Recent direct observations of CO₂ concentrations have shown that the CO₂ concentration steadily increases every year [5] . The change rate of the CO₂ concentration, drco₂/dt, is the change in the CO₂ concentration during a predetermined time and can be expressed as an annual CO₂ growth rate, “ppm/year”.

The annual mean CO₂ growth rates are reported by the National Oceanic and Atmospheric Administration (NOAA) [6] . Since 1979, NOAA satellites have been carrying instruments that measure natural microwave thermal emissions from oxygen in the atmosphere [7] . Every month, the University of Alabama in Huntsville (UAH) updates global temperature datasets from different satellites. Satellite-based temperatures, the 13-month average of lower troposphere anomaly values, and CO₂ growth rates were compared to evaluate Equation (1) during 1979-2022, which revealed that temperature changes and the CO₂ growth rate are correlated over 40 years. Therefore, Equation (1) is supported by the observed results [8] .

For a short period of time, Equation (1) may be expressed as Equation (4),

$\Delta rc{o}_2\propto \Delta T$ (4)

(Δrco₂: a change in CO₂ concentration).

In other words, the CO₂ concentration is determined by temperature changes, and generally, the higher the temperature is, the higher the CO₂ concentration, which may be called “thermally induced CO₂” [1] [2] . As temperatures change due to seasons and weather events, CO₂ can also change with temperature [8] . This was confirmed by ENSO events, which drive temperature changes that lag behind a change in an ENSO index by approximately one year. Furthermore, the rates of CO₂ absorption and emission vary with temperature changes during ENSO events, and CO₂ changes lag behind temperature changes by approximately 0.5 - 1 year [8] :

$\Delta T\to \left(0.5\text{-}1\text{year}\right)\to \Delta rc{o}_2$ (5)

CO₂ emissions during El Niño events were interpreted as an increase in plant respiration (decomposition) due to increased temperatures, as reported in previous papers [9] - [11] . The proposed process for strong El Niño events is shown in Figure 1(a) [8] .

Figure 1

Figure 1. (a) Proposed process for strong El Niño events: an increase in global temperature, an increase in soil respiration, and subsequent global CO 2 emissions [8] ; (b) the anthropogenic CO 2 emission ratio in the global CO 2 cycles shown in the IPCC report [3] [11] ; and (c) changes in the carbon cycle (ΔCO 2 ) due to soil respiration (ΔRs) and global temperature (ΔT) [11] .

Equations (1)-(5) cast strong doubts that anthropogenic CO₂ is the cause of global warming, although the concept of global warming due to anthropogenic CO₂ has been proposed by the Intergovernmental Panel on Climate Change (IPCC) [12] . Anthropogenic CO₂ emissions constitute a small portion of the global CO₂ cycles shown in the IPCC report [3] [11] (see Figure 1(b) ):

$\begin{array}{l}\text{anthropogenic}{\text{CO}}_{\text{2}}\text{ratio}\\ \fallingdotseq \left(\text{fossil}\text{fuel}\text{combustion}\right)/(\text{fossil}\text{fuel}\text{combustion}\\ +\text{respiration}\text{and}\text{decomposition}+\text{ocean}\text{atmosphere}\text{exchange})\\ \fallingdotseq 7.8/\left(7.8+107.2+79.2\right)\left(\text{unit}:\text{GtC}\right)\\ \fallingdotseq 0.04\end{array}$ (6)

As summarized in Figure 1(c) , our previous paper [11] suggested that temperature changes affect plant decomposition and soil respiration, followed by a change in CO₂ generation. The higher the temperature is, the more CO₂ is generated. On the basis of our results, we concluded that changes in plant decomposition and soil respiration due to global temperatures control global CO₂ cycles. The impact of CO₂ emissions from fossil fuel combustion on global warming is extremely low. In this work, a cross correlation between drco₂/dt or Δrco₂ and ΔT with a time lag is further clarified.

2. Global Data

Since 1979, the UAH has updated global temperature datasets that represent the piecing together of temperature data from a total of fifteen instruments flying on different satellites over the years. Further details are available [7] . Temperatures here were obtained from the datasets, and monthly data and the 13-month average of lower troposphere anomaly values were used.

The annual mean growth rate of CO₂ in a given year is the difference in concentration between the end of December and the start of January of that year reported by NOAA. Further details are available on their website [6] . Because of the seasonal changes in CO₂ concentrations, the annual means and the monthly data are compared.

These datasets can be downloaded on the websites [6] [7] , and only two datasets, temperatures and CO₂ concentrations, are analyzed in this paper.

3. Results and Discussion

On the basis of Equation (1), a cross correlation between drco₂/dt and ΔT with a time lag may be considered, where a correlation coefficient r can be defined as follows (x = drco₂/dt and y = ΔT):

$r=\frac{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n\left(xi-\bar{x}\right)\left(yi-\bar{y}\right)}{\sqrt{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(xi-\bar{x}\right)}^2}\sqrt{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(yi-\bar{y}\right)}^2}}$ (7)

It has been reported that temperature change and the rate of CO₂ increase are correlated over 40 years, as shown in Figure 2 from a previous paper [8] . Its correlation coefficient r can be calculated via Equation (7). The value of r is 0.73, and the correlation is relatively good, where optionally, the calculation can be easily performed via Excel’s built-in function. Since the 13-month average of temperature change and the annual average of the rate of CO₂ increase are used in Figure 2 , a time lag within 12 months between drco₂/dt and ΔT cannot be effectively analyzed.

In this paper, time lags for selected ENSO events, ①-④, shown in Figure 3 [8] , were analyzed in detail. Figure 4 shows the changes in CO₂ and temperature anomalies between 2014 and 2018, corresponding to the El Niño ④ events in Figure 3 . The CO₂ concentration is increasing, but it increases and decreases periodically each year, showing a regular pattern of minimum values in August and maximum values in May each year. Therefore, for each year, we consider the change in CO₂ over the course of a year starting from September as increased CO₂ (ΔCO₂), as shown in Figure 5 . Temperatures rise in response to the occurrence of El Niño. CO₂ concentrations also increase in response to temperature increases. This is shown as a larger ΔCO₂ from 2015-2016 in Figure 5 . Here, we consider the deviation of ΔCO₂ between 2015-2016 and 2014-2015 in Figure 5 . The deviations in ΔCO₂ and ΔT between 2015-2016 and 2014-2015 are plotted in Figure 6(a) . The same data are plotted in Figure 6(b) , but ΔT is shifted to the right by five months. The correlation coefficient r is improved from “0.47” to “0.94” by shifting. This means that the CO₂ concentration is determined by temperature changes but with a 5-month time lag. The results prove Equations (4)-(5).

Figure 2

Figure 2. Correlations between temperature and CO 2 changes during 1979-2022. Temperature (˚C, red line): 13-month average of lower troposphere anomaly values by UAH with scales on the left. CO 2 (ppm/year, blue vertical lines): difference from the previous year in annual averages by NOAA with scales on the right (a correlation coefficient r = 0.73).

Figure 3

Figure 3. Correlation between temperature and the ENSO index during 1979-2022. Temperature (˚C, red line): 13-month average of lower troposphere anomaly values by UAH with scales on the right. ENSO index (blue vertical lines): two-month average by NOAA with scales on the left [8] .

Figure 4

Figure 4. Seasonal change in global CO 2 (blue dots; scale: left axis; unit: ppm) between 2014 and 2018. The red curve shows satellite-based temperatures (scale: right axis, unit: ˚C) during the same period: 13-month average lower troposphere anomaly values by the UAH.

Figure 5

Figure 5. Changes in the annual CO 2 concentration (ppm) from September 2014, 2015, or 2016 to August 2015, 2016, or 2017.

Figure 6

Figure 6. (a) The deviations of ΔCO 2 and ΔT between 2015-2016 and 2014-2015 in Figure 5, and (b) the same data are plotted, but ΔT is shifted to the right by five months. The correlation coefficient r is improved from “0.47” to “0.94” by shifting.

Next, other ENSO events, ①-③, were similarly analyzed. Larger deviations in ΔCO₂ were not found for La Niña events ① and ③ but were again found for El Niño events ②, as shown in Figure 7 . The deviations in ΔCO₂ and ΔT are plotted in Figure 8 , similar to Figure 6(a) . The correlation coefficient r for El Niño ② in Figure 8(b) also improved from “0.21” to “0.92” when ΔT was shifted to the right by five months. This means that the CO₂ concentration is determined by temperature changes but with a 5-month time lag. The results also prove equations (4)-(5).

Figure 7

Figure 7. Changes in the annual CO 2 concentration (ppm) from September each year during other ENSO periods: (a) La Niña event ① in Figure 3 , (b) El Niño event ② in Figure 3 , and (c) La Niña event ③ in Figure 3 .

Figure 8

Figure 8. Deviations in ΔCO2 and ΔT for ①-③ in Figure 7.--Figure 8. Deviations in ΔCO2 and ΔT for ①-③ in Figure 7.

Figure 8

Figure 8. Deviations in ΔCO2 and ΔT for ①-③ in Figure 7.--Figure 8. Deviations in ΔCO2 and ΔT for ①-③ in Figure 7.

The global CO₂ residence time can be estimated to be 4 years by the IPCC CO₂ cycle budget [3] [11] , as shown in Equation (8):

$\begin{array}{l}{\text{CO}}_{\text{2}}\text{residence}\text{time}\\ =\left({\text{CO}}_{\text{2}}\text{in}\text{the}\text{atmosphere}\right)/(\text{fossil}\text{fuel}\text{combustion}\\ +\text{respiration}\text{and}\text{decomposition}+\text{ocean-atmosphere}\text{exchange})\\ \fallingdotseq 829/\left(7.8+107.2+79.2\right)\left(\text{unit}:\text{GtC}\right)\\ \fallingdotseq 4\end{array}$ (8)

The major natural functions affecting ΔCO₂ are (1) plant photosynthesis, (2) CO₂ solubilization into the ocean, and (3) plant decomposition or respiration, as described in the introduction. The periodical annual change in ΔCO₂ is dependent on these functions. Therefore, a time lag for the cross correlation between drco₂/dt (or ΔCO₂) and ΔT may be determined by natural functions and the residence time. Although the detailed physical process determining the time lag is not known at this time, our analysis reveals a “5-month” time lag when the temperature leads to a change in CO₂. It has been considered that temperature and CO₂ simultaneously change or that a change in CO₂ leads to a change in temperature [12] . However, a cross-correlation between the global temperature and atmospheric CO₂ with a temperature leading to a “5-month” time lag is noticeable.

Significant deviations in ΔCO₂ were not found for La Niña events ① and ③, as shown in Figure 7 . The deviations in ΔCO₂ and ΔT are plotted in Figure 8(a) and Figure 8(c) , similar to Figure 8(b) . A correlation between ΔCO₂ and ΔT is not observed, unlike El Niño ②. A time lag within 12 months between drco₂/dt and ΔT cannot be effectively analyzed under regular climate periods as shown in Figure 2 . However, the 5-month time lag was clearly recognized by the analysis during El Niño events, while it was difficult to analyze a time lag during La Niña events.

Since temperature changes affect plant decomposition and soil respiration (Rs), followed by a change in CO₂ generation, the change rate of soil respiration (dRs/dt) may be expressed as Equation (9), similar to Equation (1) [11] :

$dRs/dt\fallingdotseq {\gamma }^{\prime }\Delta T\left(Rs:\text{soil respiration},{\gamma }^{\prime }:\text{constant}\right)$ (9)

Since the average satellite-based global temperature has changed by 0.16˚C/decade [7] , soil respiration (Rs) is also expected to increase annually. The temporal increase, dRs/dt, again confirms this proposition, as reported in a previous paper [11] . We need more data to calculate a correlation coefficient r between dRs/dt and temperature because the determination of Rs is not straightforward, and only approximate values are available [13] - [15] .

If the cross-correlation between global temperature and atmospheric CO₂ with a temperature leading time lag is applied for a current temperature and CO₂ trend, a near-future trend of temperature and CO₂ may be predicted. The El Niño event that started in 2022 is not over at this time (September 2024). The same analysis as that for El Niño ④ events shown in Figure 3 was conducted for 2022 and 2023 on the basis of the durations of 2021 and 2022. The results are shown in Figure 9 . The improved correlation coefficient r between ΔCO₂ and ΔT was “0.94” after shifting to 4 months. Although it is a demonstrative example at this time, the estimation of an approximate ΔCO₂ in the near future may be possible.

Figure 9

Figure 9. The deviation of ΔCO 2 and ΔT between 2022-2023 and 2021-2022 and ΔT shifted to the right by four months. The correlation coefficient r is improved to “0.97” by shifting.

4. Conclusion

The temperature change and rate of CO₂ change are correlated but with a time lag, as reported in a previous paper. The correlation was investigated by calculating a correlation coefficient r of these changes for selected ENSO events. Annual periodical increases and decreases in the CO₂ concentration were considered, with a regular pattern of minimum values in August and maximum values in May each year. An increased deviation in CO₂ and temperature was found in response to the occurrence of El Niño, but the increase in CO₂ lagged behind the change in temperature by 5 months. This pattern was not observed for La Niña events. An increase in global CO₂ emissions and a subsequent increase in global temperature proposed by IPCC were not observed, but an increase in global temperature, an increase in soil respiration, and a subsequent increase in global CO₂ emissions were noticed. This natural process can be clearly detected during periods of increasing temperature at El Niño events. The results support Equation (1) or Equation (4) and Equation (5) and cast strong doubts that anthropogenic CO₂ is the cause of global warming.

Abbreviations

ENSO Index	El Niño-Southern Oscillation Index
IPCC	Intergovernmental Panel on Climate Change (the United Nations body)
NOAA	National Oceanic and Atmospheric Administration
UAH	University of Alabama in Huntsville
drco2/dt	The change rate of the CO2 concentration or CO2 growth rate
Rs	Soil respiration
ΔT	Temperature change
r	correlation coefficient

References

1. Salby, M. (2018) What Is Really Behind the Increase of Atmospheric CO ₂? Video Presentation, Helmut-Schmidt-University Hamburg. >https://www.youtube.com/watch?v=rohF6K2avtY

2. Salby, M. (2016) Atmospheric Carbon. Video Presentation, University College London. >https://www.youtube.com/watch?v=3q-M_uYkpT0

3. IPCC Assessment Report.>https://www.ipcc.ch/site/assets/uploads/2018/02/Fig6-01-2.jpg

4. https://www.encyclopedie-environnement.org/en/life/effects-temperature-on-photosynthesis/#Effects%20of%20Temperature%20on%20Photosynthesis:%20Summary%20Diagram

5. NOAA Database. >https://gml.noaa.gov/ccgg/trends/gl_trend.html

6. Lan, X., et al. (2023) Trends in Globally Averaged CO ₂ Determined from NOAA Global Monitoring Laboratory Measurements. Version 2023-08. NOAA Global Monitoring Laboratory.

7. https://www.drroyspencer.com/latest-global-temperatures/

8. Nishioka, M. (2023) Changes in Temperature and CO ₂ Concentration Induced by El Niño. Scientific Reports Journal, 2, 192.

9. Nishioka, M. (2024) Changes in Temperature and CO ₂ in the Atmosphere at Various Latitudes. Current Research in Environmental Science and Ecology Letters, 1, 1-9. >https://dx.doi.org/10.33140/CRESEL.01.01.02

10. Nishioka, M. (2024) Effects of Plant Decomposition and Soil Respiration on CO ₂ in the Atmosphere via Global Temperature Changes. Current Research in Environmental Science and Ecology Letters, 1, 1-12. >https://dx.doi.org/10.33140/CRESEL.01.01.03

11. Nishioka, M. (2024) Effects of Anthropogenic Co ₂ and Thermally-Induced CO ₂ on Global Warming. Atmospheric and Climate Sciences, 14, 317-327. >https://doi.org/10.4236/acs.2024.143020

12. The Content of the IPCC Assessment Report AR6. >https://www.ipcc.ch/assessment-report/ar6/

13. Raich, J.W. and Schlesinger, W.H. (1992) The Global Carbon Dioxide Flux in Soil Respiration and Its Relationship to Vegetation and Climate. Tellus B: Chemical and Physical Meteorology, 44, 81-99. >https://doi.org/10.3402/tellusb.v44i2.15428

14. Hashimoto, S., Ito, A. and Nishina, K. (2023) Divergent Data-Driven Estimates of Global Soil Respiration. Communications Earth&Environment, 4, Article No. 460. >https://doi.org/10.1038/s43247-023-01136-2

15. Nissan, A., Alcolombri, U., Peleg, N., Galili, N., Jimenez-Martinez, J., Molnar, P., et al. (2023) Global Warming Accelerates Soil Heterotrophic Respiration. Nature Communications, 14, Article No. 3452. >https://doi.org/10.1038/s41467-023-38981-w