Climate 2017, 5, 76; doi:10.3390/cli5040076 www.mdpi.com/journal/climate
The Relationship between Atmospheric Carbon
Dioxide Concentration and Global Temperature for
the Last 425 Million Years
W. Jackson Davis 1,2
1 Environmental Studies Institute, Boulder, CO 80301, USA; acksonDavis@EnvironmentalStudiesInstitute.org
2 Division of Physical and Biological Sciences, University of California, Santa Cruz, CA 95064, USA
Received: 8 August 2017; Accepted: 22 September 2017; Published: 29 September 2017
Abstract: Assessing human impacts on climate and biodiversity requires an understanding of the
relationship between the concentration of carbon dioxide (CO2) in the Earth’s atmosphere and
global temperature (T). Here I explore this relationship empirically using comprehensive,
recently-compiled databases of stable-isotope proxies from the Phanerozoic Eon (~540 to 0 years
before the present) and through complementary modeling using the atmospheric
absorption/transmittance code MODTRAN. Atmospheric CO2 concentration is correlated weakly
but negatively with linearly-detrended T proxies over the last 425 million years. Of 68 correlation
coefficients (half non-parametric) between CO2 and T proxies encompassing all known major
Phanerozoic climate transitions, 77.9% are non-discernible (p > 0.05) and 60.0% of discernible
correlations are negative. Marginal radiative forcing (ΔRFCO2), the change in forcing at the top of
the troposphere associated with a unit increase in atmospheric CO2 concentration, was computed
using MODTRAN. The correlation between ΔRFCO2 and linearly-detrended T across the
Phanerozoic Eon is positive and discernible, but only 2.6% of variance in T is attributable to
variance in ΔRFCO2. Of 68 correlation coefficients (half non-parametric) between ΔRFCO 2 and T
proxies encompassing all known major Phanerozoic climate transitions, 75.0% are non-discernible
and 41.2% of discernible correlations are negative. Spectral analysis, auto- and cross-correlation
show that proxies for T, atmospheric CO2 concentration and ΔRFCO2 oscillate across the
Phanerozoic, and cycles of CO2 and ΔRFCO2 are antiphasic. A prominent 15 million-year CO2 cycle
coincides closely with identified mass extinctions of the past, suggesting a pressing need for
research on the relationship between CO2, biodiversity extinction, and related carbon policies. This
study demonstrates that changes in atmospheric CO2 concentration did not cause temperature
change in the ancient climate.
Keywords: Phanerozoic climate; radiative forcing; marginal forcing; MODTRAN; glacial cycling;
anthropogenic global warming; climate policy; biodiversity; mass extinctions; carbon regulation;
mitigation of CO2
Understanding the role of atmospheric carbon dioxide (CO2) in forcing global temperature is
essential to the contemporary debate about anthropogenic global warming. Atmospheric CO2 and
other trace gases emitted to the atmosphere by the combustion of fossil fuels and changes in land use
patterns capture infrared energy radiated from the Earth’s surface, warming the atmosphere and
surface by the “greenhouse” effect. Anthropogenic emissions of CO2 accelerated at the start of the
Industrial Age in the mid-18th century and are now increasing the atmospheric concentration of CO2
by 1–2 parts per million by volume (ppmv) annually , a rate of increase that may be
unprecedented in recent climate history. Over this same period the Earth has warmed by ~0.8 °C .
Climate 2017, 5, 76 2 of 34
A central question for contemporary climate policy is how much of the observed global warming is
attributable to the accumulation of atmospheric CO2 and other trace greenhouse gases emitted by
The purpose of this study is to explore the relationship between atmospheric CO2 and global
temperature in the ancient climate, with the aim of informing the current debate about climate
change. The effect of atmospheric CO2 on climate has been investigated for nearly two centuries (for
historical reviews see [3–5]), beginning with the insight by Fourier [6,7] that the atmosphere absorbs
longwave radiation from the Earth’s surface. This insight was confirmed empirically by laboratory
experiments demonstrating selective absorption of longwave radiation by both CO2 and especially
water vapor [8,9]. The possible influences of atmospheric CO2 on the Earth’s energy balance and
temperature were refined throughout the 19th century [8,9], presaging the first successful model of
the Earth’s radiation budget, the demonstration of the logarithmic dependence of radiative forcing
on the atmospheric concentration of CO2, and the first estimate of the sensitivity of temperature to a
doubling of atmospheric CO2 concentration . These empirical studies led to the hypothesis that
past glacial cycles were regulated by changes in atmospheric CO2. The earliest suggestion that CO2
emitted by human activities might warm the Earth [10,11] prompted numerous calculations and
projections of anthropogenic global warming, and this hypothesis prevailed throughout most of the
20th century .
The role of CO2 in climate was complimented by theoretical advances in the early twentieth
century. Building on previous work [13,14], Milankovitch calculated that fluctuations in solar
insolation caused by variations in the Earth’s orbit around the Sun are a central cause of past global
glacial cycles . This theoretical foundation, combined with the empirical demonstration by
paleoclimate scientists during the mid-20th century of past and impending “ice ages”, focused
renewed attention on global cooling, including any that might be induced by anthropogenic aerosols
(, reviewed in ). The consensus among climate scientists in the mid-20th century remained,
however, that greenhouse warming was likely to dominate climate on time scales most relevant to
human societies (, reviewed in ).
The role of atmospheric CO2 in regulating global temperature came under renewed scrutiny at
the turn of the 21st century in respect to the climate of the Phanerozoic Eon beginning about 540
million years before present (Mybp). One group of investigators supported the prevailing consensus
that atmospheric CO2 played the central role in forcing the Phanerozoic climate [18–21] based on
what they interpreted as a “pervasive tight correlation” between temperature and CO2 proxies that
implied “strong control ” of global temperature by atmospheric CO2  (p. 5665). The posited
association between atmospheric CO2 and T was inferred from visual examination of CO2 time series
(modeled and proxy) and their relationship with stratigraphically-sourced glacial and cold periods,
however, rather than computed correlation coefficients between CO2 and T.
Other investigators of the Phanerozoic climate hypothesized a decoupling of global
temperature from the atmospheric concentration of CO2, suggesting that atmospheric CO2 played
little [22–25] or no [26,27] role in forcing the ancient climate. One group concluded that an updated
and exhaustive stable-isotope database of a range of Phanerozoic climate proxies completed in 2008
“does not provide unambiguous support for a long-term relationship between the carbon cycle [CO2] and
paleoclimate [T].”  (p. 132). The posited absence of coupling between atmospheric CO2 and T
during the Phanerozoic Eon was again, however, not supported by computed correlation
coefficients, and the renewed debate over the role of atmospheric CO2 in the Phanerozoic climate
remained, therefore, unresolved. Contemporary investigators, particularly in the climate modeling
community, have largely embraced the hypothesis that atmospheric CO2 plays a significant if not the
predominant role in forcing past and present global temperature [5,29–32].
The role of atmospheric CO2 in climate includes short- and long-term aspects. In the short term,
atmospheric trace gases including CO2 are widely considered to affect weather by influencing
surface sea temperature anomalies and sea-ice variation, which are key leading indicators of annual
and decadal atmospheric circulation and consequent rainfall, drought, floods and other weather
extremes [33–37]. Understanding the role of atmospheric CO2 in forcing global temperature
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therefore has the potential to improve weather forecasting. In the long term, the Intergovernmental
Panel on Climate Change (IPCC) promulgates a significant role for CO2 in forcing global climate,
estimating a “most likely” sensitivity of global temperature to a doubling of CO2 concentration as 2–4
°C [29–31]. Policies intended to adapt to the projected consequences of global warming and to
mitigate the projected effects by reducing anthropogenic CO2 emissions are on the agenda of local,
regional and national governments and international bodies.
The compilation in the last decade of comprehensive empirical databases containing proxies of
Phanerozoic temperature and atmospheric CO2 concentration enables a fresh analytic approach to
the CO2/T relationship. The temperature-proxy databases include thousands of measurements by
hundreds of investigators for the time period from 522 to 0 Mybp [28,38,39], while proxies for
atmospheric CO2 from the Phanerozoic Eon encompass 831 measurements reported independently
by hundreds of investigators for the time period from 425 to 0 Mybp . Such an unprecedented
volume of data on the Phanerozoic climate enables the most accurate quantitative empirical
evaluation to date of the relationship between atmospheric CO2 concentration and temperature in
the ancient climate, which is the purpose of this study.
I report here that proxies for temperature and atmospheric CO2 concentration are generally
uncorrelated across the Phanerozoic climate, showing that atmospheric CO2 did not drive the
ancient climate. The concentration of CO2 in the atmosphere is a less-direct measure of its effect on
global temperature than marginal radiative forcing, however, which is nonetheless also generally
uncorrelated with temperature across the Phanerozoic. The present findings from the Phanerozoic
climate provide possible insights into the role of atmospheric CO2 in more recent glacial cycling and
for contemporary climate science and carbon policies. Finally, I report that the concentration of
atmospheric CO2 oscillated regularly during the Phanerozoic and peaks in CO2 concentration closely
match the peaks of mass extinctions identified by previous investigators. This finding suggests an
urgent need for research aimed at quantifying the relationship between atmospheric CO2
concentration and past mass extinctions. I conclude that that limiting anthropogenic emissions of
CO2 may not be helpful in preventing harmful global warming, but may be essential to conserving
2.1. Data Sources
Temperature proxies employed here include raw, non-detrended δ18O measurements (sample
size or n = 6680; data from ), which were multiplied by negative unity to render isotope ratios
directly proportional to temperature. This operation is expressed throughout this paper as δ18O*(−1)
and does not change the absolute values of correlation coefficients reported here. Oxygen isotope
ratios are widely used as a paleoclimate temperature proxy (e.g., [41–45]), although this proxy also
reflects salinity, ice volume and other environmental variables and encompasses variance associated
with location (e.g., proximity to the ocean) and time . It is estimated that approximately half of
the δ18O signal reflects changes in past temperatures and numerous equations for converting δ18O to
temperature (“paleothermometers”) have been developed and are in wide use (e.g., [44,45,47]). It is
widely accepted that oxygen isotope ratios are proportional to past temperature, the central
prerequisite for valid regression analysis.
2.2. Temperature Proxies
The present analysis is limited to proxy isotopic values (‰)  generally without converting
to temperature. Samples used here included measurements from tropical, temperate and Arctic
latitudes  in the respective proportions ~20:6:1. Temperature-proxy data were therefore
dominated by paleotropical data. Radiative forcing varies with latitude , and the present
MODTRAN modeling exercises (described below) therefore included tropical latitudes to enable the
most accurate possible comparison of modeled with empirical climate data. Forcing of T by CO2
Climate 2017, 5, 76 4 of 34
computed using MODTRAN was used for all correlation analyses with paleoclimate proxies
contained in the empirical databases evaluated here.
To estimate the integrity of temperature-proxy data, δ18O values were averaged into bins of 2.5
million years (My) and the coefficient of variation (CV; the standard deviation divided by the mean)
was plotted against age for the updated δ18O database . The Pearson correlation coefficient
between age and CV is non-discernible (R = 0.08, p = 0.11, n = 427; Figure 1a). Enhanced variance
characterizes two sampling bandwidths, 0–50 Mybp and 100–200 Mybp (Figure 1), while older
sample ages are associated with low and stable variance. The resolution of temperature-proxy
measures is relatively high—6680 samples across 522 My, or a mean of 12.8 sample datapoints per
My corresponding to a mean sampling interval of 78,125 thousand years (Ky)—and does not decline
with sample age (Figure 2a). Assuming that repeat-measure variance is inversely related to sample
quality, these findings do not support the hypothesis that older δ18O data are compromised in
comparison with more recent δ18O data by age-related degeneration of information quality from, for
example, diagenetic settling [49,50]. Analysis of the more comprehensive databases used here,
therefore, supports the same conclusions about variation of data quality with age as reached earlier
on the basis of more limited datasets [28,38,51].
Figure 1. Variance in repeat proxy measurements over the Phanerozoic Eon. The coefficient of
variation (the standard deviation divided by the mean) is plotted against sample age using data
grouped in bins of 2.5 My for the temperature (a) and atmospheric CO2 concentration (b) proxies
used in this study. These graphs suggest that older samples are affected minimally by degeneration
of data quality (e.g., diagenesis).
Stratigraphic and age uncertainty for temperature proxies are from Prokoph et al. 2008 .
According to these investigators, maximum stratigraphic uncertainty was <5% of the mean sample
age, while a minimum 1σ uncertainty of 0.5% of the mean age was applied to all samples  (p.
115). Error analysis of a smaller but overlapping dataset yielded isotopic error bars for T that were
Climate 2017, 5, 76 5 of 34
“smaller than datapoint symbols”  (Figure 1, p. 380). In this case uncertainties in the isotopic proxies
of δ18O are unlikely to have altered the conclusions of the present study.
2.3. CO2 Proxies
Proxies of atmospheric CO2 (n = 831) were obtained from the expanded and updated CO2
database compiled by Royer  and composed of measurements from six sources, predominantly
δ13C (61%) consisting of paleosols (39%), plankton (foraminiferans, 20%) and liverworts (2%). The
balance of CO2 proxy data are from measurements of stomatal indices/ratios (29%), marine boron
(10%) and sodium carbonates (<1%). Variance of mean atmospheric CO2 proxy values averaged in
2.5 My bins showed a small but discernible positive correlation with age (R = 0.33, p = 0.002, n = 77;
Figure 1b). The number of measurements per mean (sample resolution) declines with age (Figure
2b), however, i.e., sample resolution is lower for older portions of the record. These statistical
characteristics of the dataset, rather than deterioration of information quality with age, may explain
the weak increase in sample CV with age. Error analysis suggests that the maximum uncertainty
associated with CO2 concentration proxies is ~17–50% for foraminifera at an atmospheric CO2
concentration of 1500 ppmv, less at lower CO2 concentrations, and less for other proxies such as
boron  (Figure 3, p. 255). The uncertainties in proxies of atmospheric CO2 concentration (2σ, 96%
confidence limits) are generally <200 ppmv  (Figure 1, p. 380). In this case uncertainties in the
isotopic proxies of atmospheric CO2 concentration in the ancient atmosphere are unlikely to have
compromised the conclusions drawn here.
Figure 2. Sample resolution of climate proxies over the Phanerozoic Eon. Shown are sample
datapoints per My for δ18O (a) and atmospheric CO2 concentration (b). Zero values correspond to
time periods in which no measurements are available.
Averaging of T proxies, δ18O*(−1), was accomplished by computing means across different bin
widths from 0.5 to four My in duration. Densely-sampled portions of the temperature record
enabled matched-pair analyses at resolutions down to a mean sampling interval of 59 Ky. Averaging
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of δ18O data was also performed with the moving average method used by Veizer et al. , i.e.,
mean δ18O*(−1) proxies were computed over windows of different widths from six to 50 My and
advanced in time increments of 3–10 My. Calculation of the moving average excludes initial values
less than half the width of the averaging window, or 25 My for the 10–50 averages, requiring
substitution of one-My averaged means in this period of the 10–50 mean temperature time series for
the recent Phanerozoic. Temperature-proxy data processing was done on both non-detrended and
linearly-detrended data. Detrended T-proxy data are considered more reliable for correlation
analysis because they are not encumbered by the large (8–9 °C) continuous cooling of the globe that
took place across the Phanerozoic Eon, which could in principle dominate or at least influence
computed correlation coefficients.
2.4. Analytic Approaches and Limitations
Averaging of proxies for atmospheric CO2 concentration during the Phanerozoic was
necessarily performed differently for different time periods owing to differences in sampling
resolution (Figure 2). Averages were computed across bin widths of one My or less for the time
period from 85 to 0 Mybp, over which every one-My bin contains at least one sample datapoint.
From 173 to 85 Mybp, sample resolution of CO2 concentration proxies is lower and a small number
of one-My bins lack a representative proxy datapoint. Across this time period averages were either
computed in larger bin widths, or empty bins were filled with linearly-interpolated CO2
concentration values following Prokoph et al.  (p. 123). In the oldest part of the Phanerozoic
climate record, from 173 to 425 Mybp, approximately 100 one-My bins contain at least one datapoint,
while 252 one-My bins are empty. Across this sparsely-sampled time period, bins were likewise
either filled with linearly-interpolated datapoints or data were averaged in larger bin widths, but the
strength of possible inferences is compromised using interpolated data in sparsely-sampled time
Given these sampling regimens for atmospheric CO2 concentration proxies, inferences drawn
from correlation analysis between CO2 concentration and T are strongest for the most recent portion
of the Phanerozoic (85 to 0 Mybp), moderate for the middle portion of the record (173 to 86 Mybp),
and weak or nil for the older Phanerozoic (425 to 174 Mybp). Evaluation of the tables of correlation
coefficients presented in the Results illustrates that these three levels of confidence in correlation
data comprise respectively 41.2%, 11.8% and 47% of the correlation coefficients evaluated.
Conclusions drawn from the strong, moderate and weak segments of the record are nonetheless
similar. Higher-resolution analyses—down to a sampling interval of 59 Ky and including
matched-pair correlations analysis—were enabled by relatively high sample resolution during the
recent- to mid-Phanerozoic, and these all yielded conclusions identical to those drawn from the
Spectral power analysis was done on both linearly-detrended and non-detrended data. All
results shown here are based on linearly-detrended data, following the analytic approach of
Prokoph et al.  (p. 126). Spectral analysis was done using SAS JMP software. This software
employs Fisher’s Kappa Test Statistic (κ) as a white-noise test, and returns the probability that the
distribution analyzed is generated by Gaussian (random) white noise against the alternative
hypothesis that the spectral distribution is non-random. Kappa is the ratio of the maximum value of
the periodogram, I (f(Ʃ)), to its average value. The probability (Pr) of observing a larger κ if the null
hypothesis is true is given by:
Pr 1 ( 1) max(1 ,0)
where q = n/2 if n is even and q = (n − 1)/2 if n is odd. This probability is reported for all spectral
analyses done here in the corresponding figure legends. Auto- and cross-correlation were performed
to evaluate possible non-random periodicity in all records and to estimate phase relations between
cyclic variables, respectively. The sampling frequencies of temperature and CO2 concentration
Climate 2017, 5, 76 7 of 34
proxies were sufficient to detect cycles down to 120 Ky in period, depending on the time segment of
the record analyzed. All cycles identified here in proxies for both atmospheric CO2 concentration
and T are more than three orders of magnitude larger in period than the Nyquist-Shannon sampling
frequency minimum of two samples per cycle [53–56], eliminating frequency alias as a possible
source of error.
All hypotheses were tested using conventional two-sided parametric statistics and a minimum
alpha level of 0.05. Regressions were computed and corresponding best-fit trendlines fitted using the
method of least squares. Pearson product-moment correlation coefficients were computed and
evaluated using the distribution of the t-statistic under the conservative assumption of unequal
population variance (two-sided heteroscedastic t-tests). Tests for equal variance and normal
distribution of data were not, however, conducted, and datapoints from both CO2 and T proxies
clearly violate the parametric assumption of independence of measurements. To compensate for
these limitations of parametric statistics, the distribution-free non-parametric Spearman Rho
correlation coefficient was also computed across all major climate transitions (Tables), with
conclusions identical to those drawn using the Pearson correlation coefficient.
2.5. MODTRAN Computations
Several atmospheric absorption/emission codes have been developed to compute radiative
forcing (RF) from atmospheric trace gases [29–31,48,57,58]. In this study, RF from CO2 was computed
using MODTRAN, selected from available atmospheric forcing codes in part because MODTRAN is
“the most widely used and accepted model for atmospheric transmission.”  (p. 179). Additionally,
MODTRAN is implemented online at the University of Chicago website in simple, user-friendly
format , facilitating replication of the present modeling results. MODTRAN calculations were
done using several parameters, including default settings, as reported with the results. In all cases
forcing parameters were computed for altitudes known to correspond to the tropopause at the
Radiative forcing can take several forms, depending on the surface considered (e.g., the top of
atmosphere, the tropopause, or the surface of the Earth), the time scale of the forcing (instantaneous,
pulsed, delayed, integrated) and the molecular species of the greenhouse gas considered (CO2, water
vapor, all greenhouse gases). This study is explicitly limited to instantaneous forcing measured at
the tropopause, as is conventional [29–31]. This study is constrained further to forcing in the absence
of all climate feedbacks, since T is held constant in computational iterations of the MODTRAN code
and therefore feedbacks from changing temperature are eliminated. This approach has the
advantage of constraining forcing to changes induced at the tropopause by the single molecular
species of interest, in this case CO2. Details of model settings used are presented with the results.
2.6. Pilot Studies
To assess the robustness of the findings and conclusions reported here, more than a dozen
exploratory data analyses of the correlation between the concentration of atmospheric CO2 and T
over the Phanerozoic were conducted. These pilot studies employed the original T  and
atmospheric CO2 concentration  proxy databases. Pilot studies also included temperatures
adjusted for the pH of seawater versus atmospheric CO2 concentration across the Phanerozoic ,
temperatures projected using the carbon-cycle model GEOCARB-III versus CO2 concentration ,
and both detrended and non-detrended temperature records computed from δ18O data drawn from
the older proxy database [24,38] versus both the old  and updated  atmospheric CO2
Pilot studies also utilized three forms of the updated δ18O database in different combinations
with the old and new atmospheric CO2 concentration databases, including updated low-resolution
(10–50 moving averages) and high-resolution averaged δ18O data (3–6 moving averages, four-bin
averages). In pilot studies, atmospheric CO2 concentration ranges were incremented in steps of 5, 10,
15, 20, 30 and 40 datapoints for purposes of correlation analysis. Warming and cooling episodes
Climate 2017, 5, 76 8 of 34
were defined in pilot studies from three different averaged temperature proxy records, 10–50, 3–6,
and four-bin means.
The results and conclusions of these preliminary studies were in all cases similar to those
reported here from the updated and expanded proxy databases [28,40]. Conclusions were similar
also using temperature averages computed at high-resolution (four-bin) and smoothed using the
lowest-resolution (10–50), and the high-intermediate resolution for increments in CO2 concentration
for correlation analysis. The similarity of results and conclusions across all exploratory data analyses
conducted demonstrates that the findings and interpretations reported here are robust across
analytic datasets, protocols and platforms.
Following completion of most of this research, a new and more comprehensive database of
58,532 Phanerozoic temperature proxies from low-Mg calcite marine shells from the last 512 My was
published . This database incorporates the preceding compilation  but is 2–3 times larger.
The expanded sample did not alter visibly the time series of T over the Phanerozoic, however (cf.
 Figure 1, with Figure 3 of this paper). Moreover, this expanded database does not include
additional proxy data for atmospheric CO2 concentration. The correlation coefficients reported here
were, therefore, not re-computed.
3.1. Overview of the Phanerozoic Climate
Temperature proxies show a steady decline across the Phanerozoic Eon modulated by slow
temperature cycles, both in raw data (Figure 3a) and in curves fitted to these data using diverse
filtering algorithms (Figure 3a,b). Non-detrended temperature-proxy data provide the clearest and
most accurate view of the temperature profile of the ancient climate, declining by an estimated 8–9
°C over 522 My  (Figure 3). However, linearly-detrended temperature-proxy data provide a
clearer indication of the periodicity of the climate during the Phanerozoic (lower curves in Figure
3b). The original analysis of Veizer et al.  shows a temperature-proxy periodicity of 135–150 My
and a cycle amplitude (trough-to-peak) convertible to ~4 °C (Figure 3b, purple curve). The expanded
database of Prokoph et al.  shows approximately the same periodicity and a slightly reduced
cycle temperature-proxy amplitude (Figure 3b, red curve). These findings collectively demonstrate
that temperature oscillated during the Phanerozoic on a long time scale, with an average periodicity
estimated visually as 135–150 My. Spectral analysis  placed the energy density peak at 120 My,
but as noted by the authors, that estimate was based on a single cycle  (p. 127).
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Figure 3. Temperature proxies over the Phanerozoic Eon. (a) Raw temperature data (red datapoints)
and fitted black curve (3–6 My moving average). Original data are from Prokoph et al. . (b)
averaged non-detrended (upper curves) and linearly-detrended (lower curves) temperature proxy
data. The curves of moving averages are color coded following the key in My. In (b), detrended older
data from Veizer et al.  and inclusive newer and more comprehensive data from Prokoph et al.
 are compared (lower curves).
Proxies for the atmospheric concentration of CO2 across the Phanerozoic Eon show a weak
declining trend over the Phanerozoic Eon (Figure 4). The baseline remains relatively constant near
1000 parts per million by volume (ppmv) with numerous smaller troughs and larger peaks as high
as nearly 6000 ppmv at 200 Mybp (Figure 4). Therefore, the steady and steep decline in mean global
temperature during the Phanerozoic is not attributable to a corresponding much weaker decline in
atmospheric CO2 concentration. As discussed in the Methods, the smaller number of proxy
datapoints for atmospheric CO2 concentration yields a less complete record than for T, with the
consequence that the averaged curve (red curve in Figure 4) exhibits intermittent gaps over the
represented span of the Phanerozoic. The most complete continuous proxy record of atmospheric
CO2 concentration is from ~174 to 0 Mybp. The most densely- and continuously-sampled record is
the last 80 My, which encompasses the Paleocene-Eocene Thermal Maximum starting at
approximately 56 Mybp and includes the steady global cooling that has taken place since that time.
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Figure 4. Atmospheric carbon dioxide (CO2) concentration proxies over the Phanerozoic Eon.
Original raw data (green datapoints, n = 831) are from the database compiled by Royer . The
fitted red curve was computed as a 3–6 My moving average.
3.2. Temperature versus Atmospheric Carbon Dioxide
Temperature and atmospheric CO2 concentration proxies plotted in the same time series panel
(Figure 5) show an apparent dissociation and even an antiphasic relationship. For example, a CO2
concentration peak near 415 My occurs near a temperature trough at 445 My. Similarly, CO2
concentration peaks around 285 Mybp coincide with a temperature trough at about 280 My and also
with the Permo-Carboniferous glacial period (labeled 2 in Figure 5). In more recent time periods,
where data sampling resolution is greater, the same trend is visually evident. The atmospheric CO2
concentration peak near 200 My occurs during a cooling climate, as does another, smaller CO2
concentration peak at approximately 37 My. The shorter cooling periods of the Phanerozoic, labeled
1–10 in Figure 5, do not appear qualitatively, at least, to bear any definitive relationship with
fluctuations in the atmospheric concentration of CO2.
Regression of linearly-detrended temperature proxies (Figure 3b, lower red curve) against
atmospheric CO2 concentration proxy data reveals a weak but discernible negative correlation
between CO2 concentration and T (Figure 6). Contrary to the conventional expectation, therefore, as
the concentration of atmospheric CO2 increased during the Phanerozoic climate, T decreased. This
finding is consistent with the apparent weak antiphasic relation between atmospheric CO2
concentration proxies and T suggested by visual examination of empirical data (Figure 5). The
percent of variance in T that can be explained by variance in atmospheric CO2 concentration, or
conversely, R2 × 100, is 3.6% (Figure 6). Therefore, more than 95% of the variance in T is explained by
unidentified variables other than the atmospheric concentration of CO2. Regression of
non-detrended temperature (Figure 3b, upper red curve) against atmospheric CO2 concentration
shows a weak but discernible positive correlation between CO2 concentration and T. This weak
positive association may result from the general decline in temperature accompanied by a weak
overall decline in CO2 concentration (trendline in Figure 4).
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Figure 5. Temperature (T) and atmospheric carbon dioxide (CO2) concentration proxies during the
Phanerozoic Eon. Time series of the global temperature proxy (δ18O*(−1), red curve, n = 6680) are
from Prokoph et al.  while concurrent atmospheric CO2 concentration proxies (green curve, n =
831) are from Royer . The red curve plots moving averages of the non-detrended T proxy
averaged in 50 My windows incremented in ten-My steps (the 10–50 My moving average in Figure
3b). The green curve shows mean CO2 concentration values in one-My bins averaged over
high-resolution portions of the CO2 record (the most recent Phanerozoic) and linearly interpolated
over low-resolution portions (the older Phanerozoic). Glaciations based on independent sedimentary
evidence are designated by vertical blue cross-hatched areas, while putative cool periods are
designated by vertical solid blue bars. Major cooling and warming cycles are shown by the colored
bars across the top while geological periods and evolutionary milestones are shown across the
bottom. Abbreviations: CO2, atmospheric concentration of carbon dioxide based on various proxies
(Methods); ppmv, parts per million by volume; Silu, Silurian; Neo, Neogene; Quatern, Quaternary.
The three major glacial periods and ten cooling periods identified by blue cross-hatches and solid
lines, respectively, are (after ): Glacial periods. 1. late Devonian/early Carboniferous; 2.
Permo-Carboniferous; 3. late Cenozoic. Cooling periods. 1. late Pliensbachian; 2. Bathonian; 3. late
Callovian to mid-Oxfordian; 4. Tithonian to early Berriasian; 5. Aptian; 6. mid-Cenomanian; 7.
mid-Turonian; 8. Campanian-Maastrichtian boundary; 9. mid-Maastrichtian; 10. late-Maastrichtian.
Climate 2017, 5, 76 12 of 34
Figure 6. Scatter plot of linearly-detrended temperature-proxy data averaged in one-My bins from
425 to 0 Mybp versus atmospheric carbon dioxide concentration. The negative correlation coefficient
(R = −0.19, n = 206) is discernibly different from zero (p = 0.006) but weak (R2 = 0.036). The trendline
was fitted by the method of least squares.
The correlation coefficients between the concentration of CO2 in the atmosphere and T were
computed also across 15 shorter time segments of the Phanerozoic. These time periods were selected
to include or bracket the three major glacial periods of the Phanerozoic, ten global cooling events
identified by stratigraphic indicators, and major transitions between warming and cooling of the
Earth designated by the bar across the top of Figure 5. The analysis was done separately for the most
recent time periods of the Phanerozoic, where the sampling resolution was highest (Table 1), and for
the older time periods of the Phanerozoic, where the sampling resolution was lower (Table 2). In
both cases all correlation coefficients between the atmospheric concentration of CO2 and T were
computed both for non-detrended and linearly-detrended temperature data (Tables 1 and 2, column
D1). The typical averaging resolution in Table 1 is one My, although resolutions down to 59 Ky were
obtained over some time periods of the recent Phanerozoic (see below). Table 2 is also based on
one-My interval averaging, although CO2 values were interpolated in about half the cases and
therefore inferences are correspondingly weaker as noted in the Methods.
For the most highly-resolved Phanerozoic data (Table 1), 12/15 (80.0%) Pearson correlation
coefficients computed between atmospheric CO2 concentration proxies and T proxies are
non-discernible (p > 0.05). Of the three discernible correlation coefficients, all are negative, i.e., T and
atmospheric CO2 concentration are inversely related across the corresponding time periods. Use of
the distribution-free Spearman Rho correlation coefficient yields similar conclusions: 10/13
Spearman correlation coefficients computed (76.9%) are non-discernible and all discernible
correlation coefficients are negative (Table 1). The similarity of results obtained using parametric
and non-parametric statistics suggests that conclusions from the former were not affected by the
underlying assumptions (normality, independence, equal variance).
The most recent Phanerozoic was sampled most frequently in both the temperature and
atmospheric CO2 concentration proxy databases used here (Figure 2). Correlation coefficients over
these “high-resolution” regions are designated by the superscript “c” in Table 1, and include Entries
# 1 and # 3. The average sample resolutions over these time periods are 105 Ky (Entry # 1) and 59 Ky
(Entry # 3), a three order-of-magnitude improvement over the one-My resolution that characterizes
most paleoclimate data evaluated here. The respective correlation coefficients between temperature
and atmospheric CO2 concentration proxies are nonetheless non-discernible, consistent with the
majority of correlation analyses. Correlation analysis for the highest-resolution data, therefore, yield
the same conclusions as from the broader dataset.
Climate 2017, 5, 76 13 of 34
Table 1. Correlation coefficients, temperature vs. atmospheric CO2 concentration (column D1) and
temperature vs. marginal forcing (column D2) for the most recent (highest resolution) time periods of
the Phanerozoic Eon.
C. Geological Periods
D: R (n, p)
D(1): CO2 v. T
D(2): ΔRFCO2 v. T
c 0.31 (19, 0.19)
c −0.31 (19, 0.20)
d −0.20 (11, 0.56)
d 0.02 (11, 0.96)
e −0.39 (11, 0.23)
e 0.39 (11, 0.23)
a 0.16 (31, 0.40)
a −0.06 (31, 0.75)
b 0.16 (31, 0.38)
b −0.05 (31, 0.80)
a 0.17 (31, 0.35)
a −0.12 (31, 0.52)
b 0.19 (31, 0.31)
b −0.12 (31, 0.52)
d 0.03 (110, 0.78)
d −0.13 (110, 0.19)
e −0.006 (110, 0.95)
e 0.03 (110, 0.72)
c 0.09 (17, 0.92)
c −0.02 (17, 0.93)
a −0.11 (41, 0.48)
a 0.14 (41, 0.37)
b 0.01 (41, 0.95)
b 0.06 (41, 0.70)
a −0.16 (41, 0.32)
a 0.10 (41, 0.55)
b 0.002 (41, 0.99)
b −0.03 (41, 0.84)
d −0.004 (116, 0.96)
d −0.12 (116, 0.20)
e −0.04 (116, 0.68)
e 0.05 (116. 0.59)
most recent global
glaciation (late Cenozoic)
a −0.63 (34, 0.00005)
a 0.63 (34, 0.001)
b −0.73 (34, 0.00001)
b 0.53 (34, 0.00006)
a −0.48 (34, 0.0041)
a 0.52 (34, 0.002)
b −0.58 (34, 0.0003)
b 0.62 (34, 0.0001)
d −0.25 (179, 0.0006)
d 0.20 (179, 0.0075)
e −0.26 (179, 0.0005)
e 0.26 (179, 0.0005)
CO2 range similar
a 0.043 (26, 0.83)
a −0.02 ( 26, 0.94)
b −0.16 (26, 0.44)
b 0.16 (26, 0.42)
a −0. 052 (26, 0.80)
a 0.12 (26, 0.55)
b −0.16 (26, 0.44)
b 0.23 (26, 0.25)
d −0.03 (154, 0.70)
d 0.03 (154, 0.71)
e −0.05 (154, 0.53)
e 0.05 (154, 0.53)
Red and black font designate discernible and non-discernible (p < 0.05, two-sided tests) Pearson
product moment correlation coefficients (R), respectively. Green and blue fonts designate discernible
and non-discernible Spearman Rho non-parametric (distribution free) correlation coefficients,
respectively. The sample size and p value for discernibility are in parentheses immediately following
each R value. The superscripts denote R calculated as follows: a. non-detrended temperature data
averaged into one-My bins; b. linearly detrended temperature data averaged into one-My bins; c.
non-detrended temperature data at the highest temporal resolution supported by the sampling
frequency for the temperature-proxy database (see text); d. detrended temperature data with CO2
datapoints organized as matched pairs (see text); and e. Spearman Rho correlation coefficient for
detrended temperature data with atmospheric CO2 concentration and T datapoints organized into
matched pairs. The first column (Entry #) is included to facilitate cross referencing with the text. N/A
designates non-applicable values owing to a sample size smaller than ten.
Sample datapoints are sufficiently frequent in the recent Phanerozoic that individual datapoints
of temperature proxies can be closely matched (±<1%) with individual datapoints of atmospheric
CO2 concentration proxies, eliminating the need for interpolation or averaging in bins. These
matched-pair data are the strongest available for correlation analysis and are designated by the
Climate 2017, 5, 76 14 of 34
superscript “d” (parametric) and “e” (non-parametric) in Table 1, comprising Entries # 1, # 2, and #
4–6. The sampling resolution over these regions is computed by dividing the duration of the
corresponding period by the sample size. To illustrate the period from 26 to 0 Mybp (Entry # 6 in
Table 1), the Pearson correlation coefficient is −0.03, the mean relative age difference is −0.22%, the
number of sample datapoints is 154, and, therefore, the mean sampling interval is 169 Ky. The mean
sampling interval for all of these high-resolution calculations is 199 Ky. Of the ten matched-pair
correlation coefficients computed over the early Phanerozoic (five non-parametric), eight are
non-discernible, two are discernible, and both discernible correlation coefficients are negative.
Therefore, the most powerful and highly-resolved matched-pair regression analysis possible using
these proxy databases yields the same conclusions as drawn from the entire dataset.
For the less highly-resolved older Phanerozoic data (Table 2), 14/20 (70.0%) Pearson correlation
coefficients computed between atmospheric CO2 concentration and T are non-discernible. Of the six
discernible correlation coefficients, two are negative. For the less-sampled older Phanerozoic (Table
2), 17/20 (85.0%) Spearman correlation coefficients are non-discernible. Of the three discernible
Spearman correlation coefficients, one is negative. Combining atmospheric CO2 concentration vs. T
correlation coefficients from both tables, 53/68 (77.9%) are non-discernible, and of the 15 discernible
correlation coefficients, nine (60.0%) are negative. These data collectively support the conclusion that
the atmospheric concentration of CO2 was largely decoupled from T over the majority of the
Table 2. As in Table 1, but for the most ancient (lowest resolution) time periods of the Phanerozoic Eon.
A. Entry #
C. Geological Periods
Covered and/or Description
D: R (n, p)
D(1): CO2 v. T
D(2): ΔRFCO2 v. T
most of Phanerozoic Eon
a 0.15 (191, 0.035)
a 0.10 (185, 0.16)
b −0.19 (206, 0.006)
b 0.16 (199, 0.02)
a −0.048 (191, 0.248)
a −0.037 (185, 0.61)
b −0.202 (191, 0.005)
b 0.193 (185, 0.0082)
(trough to trough)
a 0.22 (33, 0.23)
a −0.14 (35, 0.42)
b −0.16 (33, 0.37)
b 0.17 (35, 0.32)
a −0.04 (33, 0.86)
a 0.28 (33, 0.21)
b −0.095 (33, 0.61)
b 0.17 (33, 0.36)
(trough to trough)
a −0.17 (43, 0.26)
a 0.33 (39, 0.04)
b −0.16 (43, 0.30)
b 0.25 (39, 0.12)
a −0.14 (43, 0.36)
a 0.18 (39, 0.28)
b −0.22 (38, 0.10)
b 0.19 (38, 0.23)
a 0.28 (22, 0.28)
a 0.03 (21, 0.90)
b 0.28 (22, 0.20)
b 0.05 (21, 0.82)
a 0.09 (22, 0.69)
a 0.26 (21, 0.25)
b −0.03 (20, 0.90)
b 0.26 (20, 0.27)
a −0.30 (44, 0.05)
a 0.20 (42, 0.20)
b −0.17 (44, 0.26)
b 0.16 (42, 0.31)
a −0.22 (44, 0.16)
a 0.31 (42, 0.04)
b −0.12 (44, 0.44)
b 0.24 (42, 0.12)
Period of flat baseline with
~My ΔRF oscillations
a 0.15 (80, 0.17)
a −0.10 (70, 0.28)
b −0.11 ( 80, 0.25)
b −0.10 (77, 0.38)
a 0.23 (76, 0.042)
a −0.27 (72, 0.02)
b −0.03 (76, 0.79)
b −0.01 (70, 0.93)
early Jurassic to Cretaceous
(brackets all cool periods)
a 0.15 (71, 0.11)
a −0.24 (68, 0.03)
b −0.01 (71, 0.09)
b −0.05 (68, 0.68)
a 0.20 (71, 0.09)
a −0.25 (68, 0.04)
b 0.02 (71, 0.88)
b −0.07 (68, 0.56)
Jurassic to Cretaceous
a 0.21 (93, 0.02)
a −0.21 (89, 0.02)
b −0.11 (93, 0.30)
b −0.05 (89, 0.61)
a 0.25 (93, 0.02)
a −0.27, (89, 0.01)
b −0.10 (93, 0.33)
b 0.05 (89, 0.64)
Climate 2017, 5, 76 15 of 34
a 0.63 (12, 0.03)
a −0.71 (11, 0.01)
b 0.62 (12, 0.03)
b −0.70 (11, 0.02)
a 0.36 (12, 0.25)
a −0.26 (11, 0.43)
b 0.36 (12, 0.25)
b −0.27 (11, 0.41)
a 0.12 (43, 0.43)
a −0.27 (41, 0.08)
b 0.11 (43, 0.43)
b −0.29 (41, 0.07)
a 0.17 (43, 0.25)
a −0.23 (41, 0.14)
b 0.15 (43, 0.35)
b −0.22 (41, 0.17)
Conventions are the same as for Table 1 except that high-resolution data (subscripts c and d) and
matched pair analyses (subscript e) are absent.
Spectral analyses of time series of atmospheric CO2 concentration and T over the Phanerozoic
Eon (Figure 7) reveal non-random distribution of spectral density peaks at My time scales. Close
association of atmospheric CO2 concentration and T cycles would be indicated by spectral density
peaks at the same period in the respective periodograms. Instead, the periodogram for atmospheric
CO2 concentration shows spectral peaks at 2.6, 3.7, 5.3, 6.5 and 9.4 My, while the periodogram for T
shows similar but smaller peaks at lower frequencies, including 2.6, 3.9 and 5.2 My, but lower
frequency (higher period) peaks that are not matched in the atmospheric CO2 concentration
periodogram occur at 6.0, 6.8, 7.8, 8.9, 11.3 and 14.6 My. The finding that periodograms of
atmospheric CO2 concentration proxies and T proxies exhibit different frequency profiles implies
that atmospheric CO2 concentration and T oscillated at different frequencies during the Phanerozoic,
consistent with disassociation between the respective cycles. This conclusion is corroborated by the
auto- and cross-correlation analysis presented below.
Figure 7. Spectral power periodograms of atmospheric CO2 and temperature over the Phanerozoic
Eon. The green and red curves show CO2 and temperature spectral density profiles, respectively. The
temperature periodogram was constructed using detrended temperature proxy data. Time periods
covered and (in parentheses) Fisher’s Kappa and the corresponding probability that the atmospheric
CO2 concentration and T periodograms result from white (Gaussian) noise are 85 to 0 My (12.57, 1.96
× 10−5) and 522 to 0 My (28.98, 2.0 × 10−13), respectively.
3.3. Marginal RF of Temperature by Atmospheric CO2
The absence of a discernible correlation between atmospheric CO2 concentration and T over
most of the Phanerozoic, as demonstrated above, appears to contravene the widely-accepted view
about the relationship between atmospheric CO2 and temperature, by which increases in
atmospheric CO2 concentration cause corresponding increases in T owing to increased radiative
forcing. Moreover, this finding from the ancient climate appears to be inconsistent with the
well-established positive correlation between atmospheric CO2 concentration and T across glacial
cycles of the last 400–800 Ky [60,61]. I sought to resolve these apparent paradoxes by evaluating a
more direct functional measure of the warming effect of atmospheric CO2 concentration on T,
radiative forcing (RF), quantified using the well-known logarithmic relationship between RF by
Climate 2017, 5, 76 16 of 34
atmospheric CO2 (RFCO2) and its atmospheric concentration. The logarithmic RFCO2 curve,
established more than a century ago , implies a saturation effect, or diminishing returns, in
which the marginal forcing power of atmospheric CO2 declines as CO2 concentration in the
atmosphere increases. I hypothesized that the consequent decline in absolute and marginal forcing
at high atmospheric CO2 concentrations over the Phanerozoic climate might explain the absence of
discernible correlation between the atmospheric concentration of atmospheric CO2 concentration
and T simply because large swings in atmospheric CO2 concentration are then expected to have little
effect on marginal forcing.
To evaluate this possibility, the RFCO 2 forcing curve was first constructed using the MODTRAN
atmospheric absorption/transmittance code (Figure 8a). Six conditions were modeled, represented
by successively lower curves in each part of Figure 8a,b: tropical latitudes, mid-latitudes, and the
sub-Arctic, each modeled assuming one of two cloud conditions, clear-sky or cumulus clouds. The
respective altitudes of the tropopause are 17.0, 10.9 and 9.0 km. The global mean tropopause of the
International Standard Atmosphere is 10.95 km in altitude, corresponding to the mid-latitude
(green) curves in Figure 8, which is therefore considered most representative of mean global forcing.
Each latitude was modeled with no clouds or rain (upper curve of each color pair in Figure 8) and
with a cloud cover consisting of a cumulus cloud base 0.66 km above the surface and a top at 2.7 km
above the surface (lower curve of each color pair). Additional MODTRAN settings used to construct
Figure 8 are the model default values, namely: CH4 (ppm) = 1.7, Tropical Ozone (ppb) = 28,
Stratospheric Ozone scale = 1, Water Vapor Scale = 1, Freon Scale = 1, and Temperature Offset, oC = 0.
The RF curves in Figure 8a demonstrate that the absolute value of forcing at the tropopause
increases with atmospheric opacity (thickness) and is therefore greatest in tropical latitudes and least
in the sub-Arctic, as already well-established . The shape of the logarithmic curve is similar at
different latitudes, although the absolute value of forcing decreases at progressively higher latitudes,
also as expected.
The ΔRFCO2 curves (Figure 8b) were constructed by difference analysis of each radiative forcing
curve (Figure 8a). Each datapoint in every RF curve was subtracted from the next higher datapoint
in the same curve to compute the marginal change in forcing over the corresponding range of
atmospheric CO2 concentrations for the identified latitudes and cloud conditions. The resulting
ΔRFCO2 curves (Figure 8b) are shown here only for the natural range of atmospheric CO2
concentration, i.e., ~200–6000 ppmv CO2, because only these values are normally relevant to forcing
of T. Tropical latitudes were used to compute ΔRFCO2 for correlation analysis because the majority of
empirical datapoints in the databases used are from paleotropical environments (Methods). Forcing
is higher than the global mean in the tropics, however, owing to greater atmospheric opacity.
Mid-latitudes are more representative of the global mean, and were therefore used to compute the
decay rates of incremental forcing versus the atmospheric concentration of CO2.
The mid-latitude forcing curve in Figure 8a corresponding to a clear sky (arrow in Figure 8b) is
best fit (R2 = 0.9918) by the logarithmic function:
y = 3.4221ln(x) + 1.4926
This function explains 99.18% of the variance in forcing associated with CO2 concentration and
conversely. The marginal forcing curves (Figure 8b) can be fit by both exponential and power
functions. The corresponding mid-latitude curve in Figure 8b corresponding to a clear sky, for
example, is best fit (R2 = 0.9917) by the power function:
y = 562.43x−0.982
This power function explains 99.17% of the variance in ΔRFCO2 associated with CO2. An
exponential function, however, also provides a reasonable fit (R2 = 0.8265) to the same ΔRFCO2 data:
y = 0.9745e−4E−04x
Given that both power and exponential functions provide acceptable fits to the marginal forcing
curves, I used two corresponding measures to quantify the rate of decay of marginal forcing by CO2:
the half-life and the exponential decay constant, respectively. Half-life is the time required to decline
Climate 2017, 5, 76 17 of 34
to half the original value and is most appropriate for a power function. The exponential decay
constant is the time required for marginal forcing to decline to 1/e or 36.79% of its original
(maximum) value and is applicable only to an exponential function. Both the half-lives and the
exponential decay constants were calculated here using the best-fit mid-latitude clear-sky marginal
forcing curves and are expressed as the corresponding concentrations of atmospheric CO2 in units of
For mid latitudes, MODTRAN calculations show that ΔRFCO2 peaks at 3.7 W/m2 at an
atmospheric concentration of CO2 of 200 ppmv, near the minimal CO2 concentration encountered in
nature during glacial cycling (~180 ppmv) [60,61]. From that peak ΔRFCO2 declines continuously with
increasing atmospheric CO2 concentration (Figure 8b) as RF increases continuously and
logarithmically (Figure 8a). Using the above Equations (3) and (4) respectively, ΔRFCO2 declines to
half its initial value at an atmospheric concentration CO2 of 337.15 ppmv, and to 1/e or 36.79% of its
initial value at 366.66 ppmv (Figure 8b). Both the half life and the exponential decay constant are
similar across forcing curves computed here as expected (Figure 8b). The half-life and exponential
decay constant, therefore, are comparable across latitudes while absolute forcing varies by more
than 300%. At the current atmospheric concentration of CO2 (~407 ppmv) (Figure 8b), CO2 forcing
computed using Equation (3) above has declined by nearly two-thirds, to 41.56% of its maximum
natural forcing power.
Figure 8. Relationship among atmospheric CO2 concentration, radiative forcing (RF) and marginal
radiative forcing (ΔRFCO2) computed using MODTRAN. Each graph shows a family of curves for
three latitudes (key in (b)) and two cloud conditions (labeled only for the top pair of red curves in
(a)). For each latitude the upper curve pertains to a cloudless sky and the lower curve to a cloudy sky
containing cumulus clouds starting a base at 0.66 km and a top of 2.7 km. Remaining model
parameters used to generate curves are described in the text. (a) Atmospheric CO2 concentration
versus RF at the tropopause. (b) Difference analysis of the curves in (a) to show marginal radiative
forcing (ΔRFCO2), i.e., the change in forcing associated with a unit increment in atmospheric CO2.
Colored curves are in all cases computed from MODTRAN while the dashed black curves are best-fit
logarithmic (a) or power function trendlines (b). Abbreviations: ppmv, parts per million by volume;
W/m2, Watts per square meter. The arrow identifies the reference curve used for computing ΔRFCO2
decay constants because it most closely approximates values for the International Standard
Atmosphere (see text).
Climate 2017, 5, 76 18 of 34
If ΔRFCO2 is a more direct indicator of the impact of CO2 on temperature than atmospheric
concentration as hypothesized, then the correlation between ΔRFCO2 and T over the Phanerozoic Eon
might be expected to be positive and statistically discernible. This hypothesis is confirmed (Figure 9).
This analysis entailed averaging atmospheric CO2 concentration in one-My bins over the recent
Phanerozoic and either averaging or interpolating CO2 values over the older Phanerozoic (Methods).
Owing to the relatively large sample size, the Pearson correlation coefficient is statistically
discernible despite its small value (R = 0.16, n = 199), with the consequence that only a small fraction
(2.56%) of the variance in T can be explained by variance in ΔRFCO2 (Figure 9). Even though the
correlation coefficient between ΔRFCO2 and T is positive and discernible as hypothesized, therefore,
the correlation coefficient can be considered negligible and the maximum effect of ΔRFCO2 on T is for
practical purposes insignificant (<95%).
Figure 9. Scatter plot of detrended temperature-proxy data averaged in one-My bins from 425 to 0
My ago versus marginal RF by atmospheric CO2. The positive correlation coefficient (R = 0.16, n =
199) is discernibly different from zero (p = 0.02) at this large sample size, but weak, i.e., as ΔRFCO2
increased, the temperature proxy δ18O*(−1) increased only slightly. The trendline was fitted by the
method of least squares.
The correlation coefficients between ΔRFCO2 and T were computed also across the same 15
smaller time periods of the Phanerozoic Eon bracketing all major Phanerozoic climate transitions as
done above for atmospheric CO2 concentration for both non-detrended and linearly-detrended
temperature data. For the most highly-resolved Phanerozoic data (Table 1), 12/15 (80.0%) Pearson
correlation coefficients between δ18O*(−1) and ΔRFCO2 are non-discernible (p > 0.05). Of the three
discernible correlation coefficients, all are positive. Use of the distribution-free Spearman Rho
correlation coefficient yields similar conclusions: 10/13 Spearman correlation coefficients computed
(76.9%) are non-discernible and all discernible correlation coefficients are positive (Table 1).
High-resolution and matched-pair correlation analyses gave similar or identical results. The most
recent period of the Phanerozoic (Table 1, Entry # 5), where the sampling resolution is highest,
shows moderate-to-strong positive ΔRFCO2/T correlations for both detrended and non-detrended
temperature data. Therefore, for this most recent cooling period, where sample resolution is greatest,
about 24–28% of the variance in T is explained by variance in ΔRFCO2 and conversely.
For the less-resolved data from older time periods of the Phanerozoic Eon (Table 2), 14/20
(70.0%) Pearson correlation coefficients between δ18O*(−1) and ΔRFCO2 are non-discernible. Of the six
discernible correlation coefficients, four are negative. For the less-resolved older Phanerozoic time
periods (Table 2), 15/20 (75.0%) Spearman correlation coefficients are non-discernible. Of the five
discernible Spearman correlation coefficients, three are negative. Therefore, although the data from
this period of the Phanerozoic (Table 2) are less resolved, conclusions drawn from their analysis are
generally similar to those drawn using more highly-resolved data from the more recent Phanerozoic
(Table 1). The main exception is that discernible correlations computed using more highly-resolved
Climate 2017, 5, 76 19 of 34
data (Table 1) are more positive, while those computed from less-resolved data (Table 2) include
more negative values. Combining RFCO2 vs. T correlation coefficients from both tables, 51/68 (75.0%)
are non-discernible, and of the 17 discernible correlation coefficients, seven (41.2%) are negative.
These results collectively suggest a somewhat stronger effect of ΔRFCO2 on T than observed above for
the effects of atmospheric concentration of CO2 on T, but the cumulative results nonetheless require
the conclusion that ΔRFCO2 was largely decoupled from T, or at most weakly coupled with T, over
the majority of the Phanerozoic climate.
The spectral periodograms of ΔRFCO2 and T show substantial similarities both in peak periods
and relative energy (Figure 10). Both profiles show clusters of peaks at 1–4, 5–9 and 14 My periods.
These similarities signify comparable oscillations of ΔRFCO2 and T across the Phanerozoic, unlike the
relationship between atmospheric CO2 concentration and T (cf. Figure 10 with Figure 7). Oscillation
at similar periods in turn implies the possibility of an association between the corresponding cycles,
although a mutual, reciprocal or common influence of a third variable cannot be excluded without
further analysis. These conclusions are generally corroborated using auto- and cross-correlation
analysis as described next.
Figure 10. Spectral power periodograms of ΔRFCO2 and T in the Phanerozoic climate. The orange and
red curves show ΔRFCO2 and T spectral density profiles, respectively. The temperature periodogram
was constructed using linearly-detrended proxy data. Time periods covered and (in parentheses)
Fisher’s Kappa and the corresponding probability that the atmospheric CO2 concentration and T
periodograms result from white (Gaussian) noise are 85 to 0 My (8.26, 0.005) and 522 to 0 My (28.98, 2
× 10−13), respectively.
3.4. Auto- and Cross-Correlation
The typical approach to the detection of oscillations in time series is spectral analysis as
illustrated above. Alternatively, qualitative inspection of time series followed by auto- and
cross-correlation to confirm qualitative impressions quantitatively can be used to test non-random
periodicity and phase relationships between oscillating variables that are not evident from spectral
analysis. Qualitative inspection of time series of δ18O*(−1), atmospheric CO2 concentration and
ΔRFCO2, particularly during the high-resolution and relatively flat period from 175 to 80 Mybp
(Figure 11) shows apparent non-random oscillation of all three climate variables over time. The
temperature-proxy oscillation (δ18O*(−1), red curve in Figure 11) is coupled tightly with peaks in the
oscillation of the strontium ratio (87Sr/86Sr, purple arrows in Figure 11) as reported by Prokoph et al.
 and estimated here visually. Strontium ratios are typically interpreted as a proxy for riverine
influx to the ocean attributable to climate change  and are therefore expected to be coupled with
temperature as confirmed in Figure 11. The CO2 oscillation does not appear tightly coupled with
temperature-proxy oscillations, however, which is consistent with the correlation analysis presented
above, but is clearly antiphasic with the oscillation of ΔRFCO2 (yellow arrows in Figure 11). This
antiphasic relationship is expected given the derivation of ΔRFCO2 from atmospheric CO2
Climate 2017, 5, 76 20 of 34
concentration, although detailed waveforms and phase relationships are difficult to predict a priori
owing to the power and/or exponential relationship between atmospheric CO2 concentration and
Figure 11. Time series of temperature proxies, atmospheric CO2 concentration and ΔRFCO2 over the
Phanerozoic Eon. Purple arrows correspond to peaks in the strontium ratio determined independently
from visual inspection of time series in . Temperature proxies (red curve) are from Prokoph et al.
. CO2 proxies (green curve) are from Royer , represented here using a 3–6 My moving
average. Yellow double-headed arrows highlight the expected antiphasic relationship between
atmospheric CO2 concentration and marginal forcing computed for tropical latitudes under clear-sky
conditions. The scale for the temperature proxy is modified here by a constant offset for graphical
presentation on the same scales as the other proxies. The correct temperature-proxy scale is shown in
Figure 5. In this and following graphs, climate variables are color-coded: red represents the
temperature proxy, δ18O*(−1); green represents atmospheric CO2 concentration proxies converted to
ppmv; and orange represents the marginal forcing calculated for each atmospheric CO2
concentration datapoint using the best-fit power function equation corresponding to the empirical
curve for tropical latitudes under clear-sky conditions (Figure 8b).
To evaluate these qualitative impressions quantitatively, auto- and cross-correlation coefficients
between different time series were used. Autocorrelation can be used to detect non-random
periodicity in time series by comparing a progressively lagged time series to itself. In this
comparison the time series are shifted relative to each other by one datapoint at a time (successive
lag orders) and the correlation coefficients between the variables are computed for each shifted
dataset. If the autocorrelation coefficient oscillates as a function of order, and if individual
autocorrelation coefficients are discernibly different from zero at peaks and/or troughs, then
non-random periodicity is indicated. Cross-correlation between different time series is done
similarly, except the time series of two different variables are compared to each other. Discernible
correlation coefficients that oscillate with increasing shift order confirm non-random periodicity
(autocorrelation) and at the same time establish phase relationships between oscillating variables
(cross-correlation), which are beyond the capacities of spectral analysis.
Autocorrelation across the time period from 174 to 0 Mybp demonstrates non-random
periodicity in all three major variables evaluated here (Figure 12) as anticipated from the
corresponding time series (Figure 11). The correlation profile across increasing lag order for
δ18O*(−1) (Figure 12a) is qualitatively dissimilar from the profile of atmospheric CO2 concentration
(Figure 12b) and ΔRFCO2 (Figure 12c). The profiles for atmospheric CO2 concentration and ΔRFCO2 are
Climate 2017, 5, 76 21 of 34
similar, as expected from derivation of the latter from the former. In both the atmospheric CO2
concentration and ΔRFCO2 profiles, both short and long cycles can be detected qualitatively by
modulation of the corresponding correlation coefficients, demarcated by double-headed open
arrows in Figures 12–16. For autocorrelation across the longer time period, the short and long cycles
of atmospheric CO2 concentration average 16.9 and 66.8 My, respectively, similar to the comparable
values for ΔRFCO2 (17.2 and 66.2 My). Cyclic variation of the autocorrelation coefficients from
discernibly negative to positive as lag order increases demonstrates non-random periodicity in all
three variables. Dominant cycles of forcing and temperature appear in both spectral periodograms
in (Figure 10) and autocorrelelograms (Figure 12) at ~15 My.
Figure 12. Lagged (serial) autocorrelation coefficient, R, of time series of Phanerozoic climate
variables over the time period from 174 to 0 Mybp. The periodicity in the autocorrelation profiles
demonstrates non-random periodicity in the corresponding climate variables. (a) δ18O*(−1). Average
cycle duration, 22.4 My. (b) Atmospheric CO2 concentration. Average short cycle, 16.9 My. Average
long cycle, 66.8 My. (c) ΔRFCO2. Average short cycle, 17.2 My. Average long cycle, 66.2 My.
Double-headed arrows demarcate the approximate times of cycles and sub-cycles identified visually.
Similar autocorrelation analysis restricted to the period of high sampling resolution and
relatively flat baseline during the Phanerozoic (174 to 87 Mybp) gives similar conclusions (Figure
13). All three climate variables show non-random periodicity, and the autocorrelation profile for
δ18O*(−1) is qualitatively dissimilar from corresponding profiles of both atmospheric CO2
concentration and ΔRFCO2, while the autocorrelation profiles of atmospheric CO2 concentration and
ΔRFCO2 are qualitatively similar. Estimates of cycle duration from autocorrelelograms of all three
variables over this shorter analytic time period are similar, namely, 16.8, 18.0 and 19.0 My for
δ18O*(−1), atmospheric CO2 concentration and ΔRFCO2, respectively.
Climate 2017, 5, 76 22 of 34
Figure 13. As in Figure 12, but over the shorter time period from 174 to 87 Mybp. Average cycle
durations: (a) 16.8 My (b) 18.0 My (c) 19.0 My. Otherwise as in the legend of Figure 12.
The cross-correlation analyses of the same three time series for the same time periods of the
Phanerozoic (Figures 14–16) enable assessment of the relationship between the three climate
variables studied here. Cross-correlation between δ18O*(−1) and atmospheric CO2 concentration
(Figure 14) shows periodicity on long (Figure 14a) and short (Figure 14b) time scales at respective
cycle periods of 70.0 and 16.0 My. The cross-correlation between δ18O*(−1) and ΔRFCO2 (Figure 15)
show similar periodicity with long and short cycles at about 55.0 and 17.0 My (Figure 15a,b
respectively). The cross-correlation between atmospheric CO2 concentration and ΔRFCO2, in contrast,
shows strongly negative correlation coefficients at low-order lags that maximize at zero order lag,
signifying the precise antiphasic relationship between these variables that is qualitatively evident in
Figure 11. Both cross-correlation profiles show short-period cycles of about 16.0 My, and the broader
time scale (Figure 16a) again shows the long cycle of about 70.0 My. The short cycle is evident also in
the spectral periodograms described above (Figures 7 and 10).
Climate 2017, 5, 76 23 of 34
Figure 14. Lagged (serial) cross-correlation coefficient, R, of temperature and atmospheric CO2
concentration proxies. (a) Time period from 174 to 0 Mybp. Average short cycle duration, 17.9 My.
Average long cycle duration, 70.2 My. (b) Time period from 174 to 87 Mybp. Average short cycle
duration, 15.4 My. Otherwise as in the legend of Figure 12.
Figure 15. As in Figure 14, but for δ18O*(−1) and ΔRFCO2. (a) Time period from 174 to 0 Mybp.
Average short and long cycle durations, 19.7 and 54.8 My respectively. (b) Time period from 174 to
87 Mybp. Average short cycle duration, 16.2 My. Otherwise as in the legend of Figure 12.
Climate 2017, 5, 76 24 of 34
Figure 16. As in Figure 14, but for atmospheric CO2 concentration and ΔRFCO2. (a) Time period from
174 to 0 Mybp. Average short and long cycle durations, 14.5 and 70.2 My respectively. (b) Time
period from 174 to 87 Mybp. Average short cycle duration, 17.8 My. The strong negative correlation
coefficients at low shift orders demonstrate the antiphasic (inverse) relation between atmospheric
CO2 concentration and ΔRFCO2. Otherwise as in the legend of Figure 12.
4. Discussion and Conclusions
The principal findings of this study are that neither the atmospheric concentration of CO2 nor
ΔRFCO2 is correlated with T over most of the ancient (Phanerozoic) climate. Over all major climate
transitions of the Phanerozoic Eon, about three-quarters of 136 correlation coefficients computed
here between T and atmospheric CO2 concentration, and between T and ΔRFCO2, are non-discernible,
and about half of the discernible correlations are negative. Correlation does not imply causality, but
the absence of correlation proves conclusively the absence of causality . The finding that
atmospheric CO2 concentration and ΔRFCO2 are generally uncorrelated with T, therefore, implies
either that neither variable exerted significant causal influence on T during the Phanerozoic Eon or
that the underlying proxy databases do not accurately reflect the variables evaluated.
The present findings corroborate the earlier conclusion based on study of the Paleozoic climate
that “global climate may be independent of variations in atmospheric carbon dioxide concentration.”  (p.
198). The present study shows further, however, that past atmospheric CO2 concentration oscillates
on a cycle of 15–20 My and an amplitude of a few hundred to several hundreds of ppmv. A second
longer cycle oscillates at 60–70 My. As discussed below, the peaks of the ~15 My cycles align closely
with the times of identified mass extinctions during the Phanerozoic Eon, inviting further research
on the relationship between atmospheric CO2 concentration and mass extinctions during the
4.1. Qualifications and Limitations
The conclusion that atmospheric CO2 concentration did not regulate temperature in the ancient
climate is qualified by the finding here that over the most recent Phanerozoic, from 34 to 0 Mybp,
where sampling resolution is among the highest in the proxy databases used, all six correlation
coefficients computed between T and ΔRFCO2 proxies are positive and range from moderate to strong
(Table 1, Entry # 5). For the same time period, all six correlation coefficients between temperature
and the atmospheric concentration of CO2 are negative and also moderate to strong (Table 1, Entry #
Climate 2017, 5, 76 25 of 34
5). Both temperature and atmospheric CO2 concentration were lower in the recent Phanerozoic,
implying greater marginal forcing by CO2 and, therefore, a potentially greater influence on
temperature and climate. This cluster of discernible correlation coefficients could therefore be
interpreted as support for the hypothesis that atmospheric CO2 concentration plays a greater role in
temperature control when the concentration is lower. Correlations over the period from 26 to 0
Mybp (Table 1, Entry # 6), however, are uniformly non-discernible despite a comparable sample
resolution (Figure 2). The moderate-to-strong positive correlations between marginal forcing and
temperature realized by expanding the upper limit of the computation range to 34 Mybp may,
therefore, simply reflect the different trajectories of atmospheric CO2 concentration and T from 34 to
26 Mybp as governed by other, unidentified forces.
A limitation of the present study is the low sampling resolution across parts of the Phanerozoic
climate record. Sample atmospheric CO2 concentration datapoints in the older Phanerozoic are too
sparse to enable strong inferences or, for some time periods, any inferences at all. The mean
sampling interval for either atmospheric CO2 concentration or T is ~83 Ky over the whole
Phanerozoic, and ~50 Ky in the more recent Phanerozoic. At these sampling frequencies, the shorter
cycles known to characterize weather and climate on annual through millennial [60,61] time scales
cannot be detected. These undetectable cycles presumably exhibit the same covariance between T
and atmospheric CO2 concentration, and between T and ΔRFCO2, that is known to characterize recent
glacial cycles, including the 41 Ky Marine Isotope Stages that extend back in time several Mybp
[65,66]. These shorter, undetectable millennial-scale climate cycles are presumably superimposed on
the longer climate cycles of the Phanerozoic demonstrated here, but this hypothesis cannot be tested
presently given the limits of available sample resolution in Phanerozoic climate data.
The limitation of low sampling resolution is somewhat offset over some time periods of the
recent Phanerozoic where sampling was frequent enough to yield mean sampling intervals of
59–199 Ky. Ten of these correlation coefficients (five non-parametric) were based on matched-pair
analysis with minimum (0.22%) difference in age between T and atmospheric CO2 concentration.
Analysis of these high-resolution matched-pair subsets yielded the same conclusions as analysis of
the entire dataset, suggesting that differences in data resolution did not alter the conclusions of this
study. Prokoph et al. noted that “much of the carbon isotope variability occurs over intervals shorter than 1
Ma, not resolvable in this study”  (p. 132) and that higher resolutions would be preferable. In the
present analysis sampling resolutions exceeded one My by three orders of magnitude over limited
time periods of the Phanerozoic. Higher-resolution sampling is always preferable, but in the
meantime the sampling resolution in the databases used here is the best that is currently available
and is unlikely to be surpassed in the near future.
4.2. The Perspective of Previous Studies
Three recent studies examined the relationship between temperature proxies and atmospheric
CO2 concentration during the Phanerozoic Eon. The first  concluded that atmospheric CO2
concentration was the primary driver of early Cenozoic climate, from 50 to 34 Mybp, echoing the
earlier view of several investigators [18–21] (Introduction). This conclusion  was based, however,
on analysis of a short portion of the Phanerozoic (<3% of the cumulative time period), and the use of
a single proxy isotope (boron) from a single isotopic source (planktonic foraminifera) from a single
sample site (southern coastal Tanzania). It is possible that such a limited local/regional sample over
so short a time period missed trends and relationships that emerge only from analysis of the larger
and more inclusive and diverse databases evaluated here. Using these broader databases, the
Pearson correlation coefficient between atmospheric CO2 concentration and T over the same time
period as studied in , namely the early Cenozoic from 50 to 34 Mybp, is negative (R = −0.32) and
not discernibly different from zero (n = 17, p = 0.20). Therefore, the present study does not support
the conclusion  that atmospheric CO2 concentration drove the climate of the early Cenozoic.
A second study  concluded that the current high rate of CO2 emissions risks atmospheric
concentrations not seen for 50 My by the middle of the present century. In contrast to that
conclusion, the current rate of CO2 emissions is raising atmospheric concentration of CO2 by 1–2
Climate 2017, 5, 76 26 of 34
ppmv per annum , implying an atmospheric concentration of CO2 of 440 to 463 ppmv by the
middle of the present century—concentrations that appear repeatedly over the last ten My and as
recently as three Mybp in the CO2 databases used here (Figure 5). The same study  concluded
that if additional emissions of CO2 at current rates continued until the 23rd century, they would
cause large and potentially hazardous increases in RFCO2. This conclusion can be evaluated using the
MODTRAN forcing estimates calculated here, particularly the representative global forcing at
mid-latitudes under clear-sky conditions (Equation (2) above). Using this equation the projected
changes in forcing from 407 to 689 and 407 to 971 ppmv CO2 are 1.80 and 2.98 W/m2 respectively,
from 48.7% to 80.4% of the IPCC estimated doubling sensitivity to CO2 forcing of 3.7 W/m2. Inclusion
of a cloud cover reduces this estimate further. It is not clear that these projected increases in forcing
qualify as “hazardous”.
A clue to the discrepancy between MODTRAN computations and IPCC estimates may be
available from application of the IPPC equation for a change in forcing , namely:
ΔF = 5.35ln(C/CO)
where ΔF is the change in forcing in W/m2 at the tropopause, C is the atmospheric concentration of
CO2 in ppmv, and CO is the original atmospheric concentration of CO2. This equation projects
increases in forcing from the same increases in atmospheric CO2 concentration as 2.81–4.65 W/m2,
75.9–125.7% of the IPCC estimated doubling sensitivity of 3.7 W/m2. The difference between the
MODTRAN computations and those derived using the IPCC-endorsed Equation (5) above may be
attributable to different methods of computing forcing. The present estimates are based on clear-sky
mid-latitude forcings, where troposphere opacity is probably most representative of the global
mean. The estimate using Equation (5) exceeds the MODTRAN estimate of forcing at the equator,
where atmospheric opacity is the highest on Earth and not representative of the global mean. In
either case, however, the change in forcing represents less than 1% of total forcing from all sources.
The third study relevant to the present investigation  addresses uncertainties in
paleo-measurements of atmospheric CO2 concentration using “mathematical equations”, and
concludes that large variations in atmospheric CO2 concentration (1000–2000 ppmv) did not occur
during the Phanerozoic Eon since the Devonian Period from 416 to 358 Mybp. In contrast to that
conclusion , the empirical database used in the present study shows that variations in
atmospheric CO2 concentration of 1000–2000 ppmv were common throughout the Phanerozoic Eon
(Figure 5). Atmospheric CO2 concentration oscillated on a regular cycle of several hundreds of ppmv
in amplitude and periods of 10–20 My and 60–70 My. Much larger individual excursions are evident
in the empirical data. The same study  reported “new empirical support” for the hypothesis that
atmospheric CO2 concentration exerts greater influence on Phanerozoic climate than previously
thought. The conclusions of that study are based minimally on empirical findings, however, and
maximally on mathematical equations. The suggestion that CO2 exerts greater influence on the
Phanerozoic climate than previously thought  is not supported by the results of the present
4.3. Tectonic Activity and Climate in the Phanerozoic
The generally weak or absent correlations between the atmospheric concentration of CO2 and T,
and between ΔRFCO2 and T, imply that other, unidentified variables caused most (>95%) of the
variance in T across the Phanerozoic climate record. The dissimilar structures of periodograms for T
and atmospheric CO2 concentration found here also imply that different but unidentified forces
drove independent cyclic fluctuations in T and CO2. Since cycles in atmospheric CO2 concentration
occur independently of temperature cycles, the respective rhythms must have a different etiology. It
has been suggested that volcanic activity and seafloor spreading produce periodic CO2 emissions
from the Earth’s mantle ( and references therein) which could in principle increase radiative
forcing of temperature globally. On this basis it was proposed that “the parallel between the tectonic
activity and CO2 together with the extension of glaciations confirms the generally accepted idea of a primary
Climate 2017, 5, 76 27 of 34
control of CO2 on climate...”  (p. 167). A similar conclusion was reached for the late Cenozoic
The hypothesis that CO2 emitted into the atmosphere from tectonic emissions served as a
“primary control” of the ancient climate  requires that the concentration of CO2 in the atmosphere
is low enough to enable significant marginal forcing of T by CO2 at the time of increased emissions
from the Earth’s mantle, and that atmospheric CO2 concentration and temperature are discernibly
correlated over parts or all of the Phanerozoic Eon. The present study shows that neither of these
prerequisites is realized. Therefore, changes in the rate of seafloor spreading and the consequent
release of CO2 into the ancient oceans and atmosphere did not control the Phanerozoic climate. This
conclusion encourages a search for alternative mechanisms of Phanerozoic temperature cycling of ~4
°C on a period of 135–150 My (e.g., ). It remains possible that the cycles in atmospheric CO2
emissions and marginal forcing by CO2 demonstrated here are caused by cycles of volcanic activity
or seafloor spreading, although as shown here these cycles bear no causal relationship with global
4.4. The Physical Basis of CO2/T Decoupling
The gradual reduction in the level of atmospheric CO2 concentration across the Phanerozoic as
global temperature declined could in principle have resulted in part from the enhanced solubility of
CO2 in colder sea water [71,72]. Estimates of this enhanced solubility effect on atmospheric CO2
concentration during recent ice ages are, however, small (30 ppmv) in comparison to changes in
atmospheric CO2 concentration over the Phanerozoic (hundreds of ppmv) despite comparable
changes in temperature . Additional potential causes of reduced atmospheric CO2 concentration
during the late Phanerozoic include increased chemical weathering owing to enhanced tectonic
uplift [69,70] and reduced tectonic activity and, therefore, less transfer of subterranean carbon to the
atmosphere . Whatever the cause, the result is much lower atmospheric concentration of CO2
now than in the distant past.
Why did higher past levels of atmospheric CO2 not drive increases in global temperature in the
ancient climate? The differences between ΔRFCO2 and atmospheric CO2 concentration as an index of
respective impact on temperature are muted in a CO2-rich environment such as the older
Phanerozoic, where atmospheric CO2 concentration reached many times the contemporary
atmospheric concentration of CO2. At such high atmospheric CO2 concentrations the marginal
forcing of CO2 becomes negligible (Figure 8b). At the estimated baseline concentration of
atmospheric CO2 across the recent Phanerozoic, 1000 ppmv, CO2 has already lost 85.7% of its
maximum radiative forcing power as computed using MODTRAN (Figure 8b). At this
concentration, even a relatively large increase in atmospheric CO2 concentration, therefore, yields a
small and even negligible increase in the marginal forcing of temperature.
As shown here, and as expected from the derivation of marginal forcing from concentration,
atmospheric CO2 concentration and marginal radiative forcing by CO2 are inversely related.
Diminishing returns in the forcing power of atmospheric CO2 as its concentration increases ensure
that in a CO2-rich environment like the Phanerozoic climate, large variations in CO2 exert little or
negligible effects on temperature. Therefore, decoupling between atmospheric CO2 concentration
and temperature is not only demonstrated empirically in Phanerozoic data, but is also expected from
first principles. This straightforward consequence of atmospheric physics provides a simple physical
explanation for the lack of correlation between atmospheric CO2 concentration and temperature
across most of the Phanerozoic.
Nonlinearities in the climate system imposed by the saturation of the strongest absorption
bands in the CO2 molecule have been recognized previously in connection with assessment of Global
Warming Potential (GWP) of greenhouse gases, including CO2 . It has been suggested that as
CO2 concentration in the atmosphere increases, the reduced marginal forcing attributable to the
saturation of CO2 absorption bands would be compensated exactly by increased global forcing
attributable to lower solubility of CO2 in a warmer ocean and therefore higher concentration of CO2
in the atmosphere  (p. 251). This conclusion is based, however, on the premise that increased
Climate 2017, 5, 76 28 of 34
atmospheric CO2 concentration causes increased global warming. As shown empirically in the
present study, this premise does not apply to the Phanerozoic climate, where atmospheric CO2
concentration and global temperature were decoupled. Whether the conclusion applies to the
contemporary climate depends on the relative magnitudes of the opposing effects. As noted above,
large changes in global temperature associated with glacial cycles are associated with small changes
in the solubility effect on atmospheric concentration of CO2, estimated as 30 ppmv . Moreover, in
view of the diminishing returns in marginal forcing by atmospheric CO2 identified in the present
study, changes in the concentration of contemporary atmospheric CO2 are expected to cause small or
negligible changes in global temperature. It may be fruitful to revisit the concept of GWP, which is
central to contemporary climate policy, in light of the present results.
4.5. Implications for Glacial Cycling
Application of the present findings to the more recent climate is conditioned by the relatively
low contemporary concentration of atmospheric CO2 in comparison with most of the Phanerozoic
Eon and the consequent higher marginal radiative forcing power of CO2. Particularly in the depths
of glacial maxima, where the concentration of atmospheric CO2 approached the natural minimum of
180 ppmv [60,61], the marginal forcing of temperature by CO2 was at its highest, 94% of its
maximum (computed using MODTRAN). As warming proceeded during the glacial termination,
however, the concentration of CO2 in the atmosphere increased sharply [60,61] and marginal
radiative forcing by CO2 declined accordingly, reaching 54.5% of its maximum (computed using
MODTRAN) by the time that atmospheric CO2 concentration reached the interstadial value of ~300
The physics of atmospheric CO2, therefore, creates variable positive-feedback induction and
amplification of temperature during natural glacial cycling (variable-gain feedback). This positive
feedback is strongest at the beginning of the glacial termination, which would trigger and/or
reinforce warming maximally, and weaker by half at the end of the glacial termination, which would
release temperature from positive-feedback reinforcement by atmospheric CO2 and, therefore,
enable or facilitate the onset of the next glaciation.
Variable-gain feedback during the glacial cycling provides a plausible physical basis for
relaxation oscillation [75,76] of global temperature as a causal contributory mechanism underlying
glacial cycling. Variable-gain positive feedback supports the hypothesis advanced by several
investigators that the glacial cycling may constitute, in whole or in part, a natural relaxation
oscillator [77–81]. In this case, atmospheric CO2 could play a significant role in glacial cycling. The
temperature during past interstadials was, however, higher than present despite higher
contemporary CO2 concentration, from which it has been concluded that “Carbon dioxide appears to
play a very limited role in setting interglacial temperature.”  (p. 6).
4.6. Risk Assessment and Contemporary Carbon Policies
At the current atmospheric concentration of CO2 (~407 ppmv ), the marginal forcing power is
smaller than during natural glacial cycles but still greater than during most of the Phanerozoic Eon.
The half-decay of CO2 marginal forcing (~337 ppmv) was surpassed in 1980, while the exponential
marginal forcing decay constant (~367 ppmv) was exceeded in 1999. At the current atmospheric CO2
concentration, which is approaching 410 ppmv, atmospheric CO2 has lost nearly two-thirds of its
cumulative marginal forcing power.
Such diminishing returns in marginal forcing of temperature by atmospheric CO2 have two
implications for contemporary climate policies. First, if anthropogenic CO2 emissions continue at
today’s levels or increase in the coming decades, the consequent increasing concentration of CO2 in
the atmosphere from anthropogenic sources will have exponentially smaller forcing impact on
global temperature. As implied by the decline in marginal forcing, when atmospheric CO2
concentration reaches 1000 ppmv, near the baseline value for most of the Phanerozoic Eon (Figure 5),
marginal forcing will decline to 14.3% of its maximum (computed using MODTRAN; Figure 8b).
Such diminishing returns ensure that additional increments in anthropogenic emissions from
Climate 2017, 5, 76 29 of 34
today’s level will have exponentially smaller marginal impact on global temperature. This
consequence of diminishing returns in marginal forcing by CO2 has been incorporated into policy
assessments in reference to national responsibilities for CO2 emissions under the UN Framework
Convention on Climate Change and Kyoto Protocol [82–85].
Second, and conversely, as atmospheric CO2 concentration increases, any unit reductions in
atmospheric CO2 concentration that may be achieved by deliberate mitigation of CO2 emissions will
yield exponentially smaller reductions of temperature forcing. Diminishing returns in marginal
forcing by atmospheric CO2 ensure, therefore, that efforts to mitigate global warming by reducing
emissions of CO2, exemplified by carbon sequestration, will become relatively more expensive per
unit of climate benefit returned. This consequence of atmospheric physics will increase the
cost-benefit ratio of CO2 mitigation policies exponentially, at least insofar as the cost-benefit ratio is
limited to climate.
Risk assessment for atmospheric CO2 is not, however, limited to its effects on climate.
Increasing emissions of CO2 can damage sea life by acidifying ocean water [86,87]. Such effects could
be amplified within the sea surface microlayer, where most sea plants and animals spend at least
part of their lifecycle in their most vulnerable stages: eggs, embryos and larvae . Evidence
developed here from the Phanerozoic climate record shows that from 225 to 0 Mybp, the
atmospheric concentration of CO2 oscillated on multi-million year cycles with a prominent spectral
peak at ~15 My. Phanerozoic mass extinctions have been shown independently to oscillate  at
dominant periods of ten and 60 My [89,90], similar within likely error limits to the major period
modes observed here for oscillations in atmospheric CO2 concentration. The peaks of the 15-My CO2
cycle identified here coincide closely with peaks of identified mass extinctions : of 14 such CO2
peaks from 225 to 0 Mybp (Figure 11), 12 match closely with peaks in identified mass extinctions and
six of these 12 peaks coincide precisely with the peaks of identified mass extinctions (cf. Figure 11
with  (Figure 1, p. 135).
A defining feature of the cycles in atmospheric CO2 concentration identified here during the
period from 225 to 0 Mybp is that their amplitude is ~500 ppmv from a base as low as 400 ppmv, and
yet this relatively small excursion of CO2 concentration above levels similar to today’s CO2
concentrations repeatedly left strong extinction imprints on biodiversity. These extinction events
were not caused by temperature changes, since as shown here CO2 did not drive temperature in the
Phanerozoic climate. As shown above, a comparable increase in CO2 from the current baseline of 407
ppmv CO2 to 971 ppmv could occur by end of the 23rd century at current CO2 emission rates.
Moreover, the residence time of CO2 in the atmosphere may be much longer than previously
believed—not centuries as often reported [29–31], but millennia  or tens of millennia . Once
emitted to the atmosphere, therefore, plausible increases in the concentration of CO2 resulting from
human activities could impact biodiversity adversely for tens of thousands of generations of sea life,
potentially enough CO2 over a long enough exposure time to induce mass extinctions similar to
those of the past.
While “no consensus has been reached on the kill mechanism for any marine mass extinction”  (p.
127), acidification of the sea surface microlayer by atmospheric CO2 is a plausible candidate. Ocean
acidification may affect the ecology of the sea surface microlayer adversely by proliferating
acidophilic bacteria that in turn degrade the nutritional environment for other organisms that
inhabit the microlayer . Substances associated with tectonic emissions of CO2 during the
Phanerozoic (sulfur, methane, chlorine, mercury) are also toxic and, therefore, are plausible
candidates for the elusive “kill mechanism” of past mass extinctions. The toxicity of CO2 in aqueous
environments, however, is well-established [86,87]. Possible links between past mass extinctions and
the carbon cycle, therefore, comprise an urgent future research direction. The results of the present
study suggest that regulation of anthropogenic CO2 emissions may be less important than
previously thought for controlling global temperature, but more important than previously
recognized for conserving biodiversity.
Climate 2017, 5, 76 30 of 34
Acknowledgments: This research was supported generally by the University of California, Santa Cruz, and the
Environmental Studies Institute, a non-profit (501)(3)(c) corporation headquartered in Boulder, CO, USA. This
research did not receive any specific grants from funding agencies in the public or commercial sectors. William
B. Davis produced the spectral periodograms and provided invaluable statistical counsel. I thank R. G. Plantz
and W. B. Davis for comments on earlier drafts, which improved the presentation; Professor Jan Veizer and
colleagues for helpful suggestions about data processing; and Professor Dana Royer for generously making
available the CO2 proxy database he compiled. I thank three anonymous reviewers for constructive comments
that significantly improved the manuscript.
Conflicts of Interest: The author declares no conflicts of interest.
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