Atmospheric carbon dioxide levels for the last 500 million years
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Communicated by Paul F. Hoffman, Harvard University, Cambridge, MA (received for review October 9, 2001)
Abstract
The last 500 million years of the strontiumisotope record are shown to correlate significantly with the concurrent record of isotopic fractionation between inorganic and organic carbon after the effects of recycled sediment are removed from the strontium signal. The correlation is shown to result from the common dependence of both signals on weathering and magmatic processes. Because the longterm evolution of carbon dioxide levels depends similarly on weathering and magmatism, the relative fluctuations of CO_{2} levels are inferred from the shared fluctuations of the isotopic records. The resulting CO_{2} signal exhibits no systematic correspondence with the geologic record of climatic variations at tectonic time scales.
The longterm carbon cycle is controlled by chemical weathering, volcanic and metamorphic degassing, and the burial of organic carbon (1, 2). Ancient atmospheric carbon dioxide levels are reflected in the isotopic content of organic carbon (3) and, less directly, strontium (4) in marine sedimentary rocks; the former because photosynthetic carbon isotope fractionation is sensitive to CO_{2} levels, and the latter because weathering and degassing are associated with extreme values of the abundance ratio ^{87}Sr/^{86}Sr. However, attempts to use these geochemical signals to estimate past CO_{2} levels (5–8) are hindered by the signals' additional relationships to various tectonic (9, 10) and biological (11) effects. Moreover, the strontium signal has proven especially difficult to parse (12–15).
Here, I attempt to resolve these ambiguities in the isotopic signals of carbon and strontium. First, it is shown that the last 500 million years of the strontium signal, after transformation to remove the effects of recycled sediment (16, 17), correlate significantly with the concurrent record of isotopic fractionation between inorganic and organic carbon (3). This empirical result is supplemented by the theoretical deduction that the two records are linked by their common dependence on rates of continental weathering and magmatic activity. The assumption that CO_{2} levels fall with the former and rise with the latter then indicates that an appropriate average of the two records should reflect the longterm fluctuations of the partial pressure of atmospheric CO_{2}. The CO_{2} signal derived from this analysis represents fluctuations at time scales greater than about 10 million years (My). Comparison with the geologic record of climatic variations (18) reveals no obvious correspondence.
Strontium and Carbon Isotopic Signals
Fig. 1 shows the strontium and carbon isotopic signals for the last 500 My. The data for the strontium isotope ratios ^{87}Sr/^{86}Sr were compiled by Veizer et al. (4) and Walter et al. (19); the former source accounts for all ^{87}Sr/^{86}Sr data younger than 520 My, whereas the latter was used for data extending back to 608 My (not shown). Both sets of ^{87}Sr/^{86}Sr data were first averaged in nonoverlapping time windows of 10 My. The resulting unevenly spaced record was then transformed (20) to an evenly spaced record with time increments of approximately 10 My and no contributions to the power spectral density at periods less than 21 My.
The second record in Fig. 1, compiled by Hayes et al. (3) derives from the isotopic composition of marine organic carbon and carbonate carbon. From isotopic abundance ratios R_{x} = (^{13}C/^{12}C)_{x} for carbon in sample x, the isotopic fractionation between sample x and a standard (STD) sample, δ_{x} = 1,000[(R_{x} − R_{STD})/R_{STD}], is obtained for carbonate (δ_{a}) and organic (δ_{o}) carbon. The isotopic fractionation ɛ_{toc} between total organic carbon and sedimentary carbonates is then given approximately by ɛ_{toc} = δ_{a} − δ_{o}, which is the second signal plotted in Fig. 1.
Fig. 1 shows a surprising similarity between the two records for fluctuations with periods less than about 100 My. However, the correlation between the two time series is not statistically significant‡ [Spearman rank correlation coefficient (21) R_{s} = −0.40, P = 0.17, N = 46], because the longerperiod fluctuations are not in phase. It is therefore interesting to ask why the two records appear similar at shorter time scales and dissimilar at longer time scales.
Recent work has shown that the measurements of ɛ_{toc} approximately fit the empirical relation (3, 22) 1 The parameter ɛ_{0} represents the isotopic effects of photosynthesis and secondary biological processes along with the isotopic depletion of dissolved CO_{2} in surface waters relative to sedimentary carbonate. Because ɛ_{0} is approximately constant throughout the period plotted in Fig. 1, the main source of ɛ_{toc}'s fluctuations is contained in isotopic effects due to changing algal physiology—the permeability, surfacetovolume ratio, and growth rate of algal cells—represented by Γ, and the concentration Φ of dissolved carbon dioxide in surface waters (3, 8).
Although the mechanisms responsible for the fluctuations of s = ^{87}Sr/^{86}Sr are subject to much debate (4, 9, 10, 12–15), some aspects of the signal's evolution are nevertheless clear. Because ^{87}Rb decays to ^{87}Sr with a halflife of ∼48 billion years, the supply of ^{87}Sr may be taken to be approximately constant over the last 500 My. However, it is not uniformly distributed: the fluvial input to the oceans is derived in part from rocks—both silicates and metamorphosed carbonates (12–15)—that are relatively enriched in radiogenic Sr (s ≃ 0.712 or greater) compared to the nonradiogenic Sr of mantle origin supplied at hydrothermal vents (s ≃ 0.7035) (23). The value of s(t) at some particular time t represents, to first approximation, the relative fluxes of these two extreme values.
However, this first approximation ignores the recycling of rocks in the sedimentary cycle (16). As pointed out by Brass (17), about 75% of the strontium input to the oceans should come from the weathering of exposed carbonates of marine origin. Because these carbonates retain a memory of s at the time of deposition, their contribution to sedimentary Sr significantly damps the signal coming from hydrothermal vents (low s) and rocks containing radiogenic Sr (high s) (9).
Denoting the fluvial flux of radiogenic Sr by r, the input from hydrothermal vents by v_{h}, and the memory effect by m, a simple expression for the strontium isotope ratio s is 2 where 0 ≤ μ ≤ 1 is the fraction of sedimentary Sr deriving from the memory flux, and g is a function that depends on both r and v_{h}. The memory term can be approximated by assuming that sedimentary strontium is weathered at a rate proportional to its mass (16). Then m(t) is simply a weighted (exponentially decaying) average of s(τ), τ < t (17), which we express by the convolution 3 The parameter λ is related to the halflife t_{1/2} of sedimentary Sr; i.e., λ = −(ln 2)/t_{1/2}. Brass estimates 57 My < t_{1/2} < 102 My (17). Therefore, the memory effect should strongly influence the fluctuations of s at long time scales (greater than about 100 My), whereas the short time scale fluctuations should be relatively unaffected.
The shorttime correlations may be explained in part by noting the common dependence of ɛ_{toc} and g on the global weathering rate w (i.e., the flux of all weathered products from the continents to the oceans) and the rate of magmatic activity v associated not only with the hydrothermal flux v_{h} but also with volcanic degassing. Assuming that r = r(w) and v_{h} = v_{h}(v), g's dependence on w and v may be written§ 4 With respect to ɛ_{toc}, the physiological term Γ may likewise depend on w because of changing nutrient concentrations in the oceans. Γ could also depend on CO_{2} levels, which in turn depend on degassing, weathering rates—because of the net uptake rate u = u(w) of atmospheric CO_{2} associated with silicate weathering (1, 2, 24)—and other processes, such as organic carbon burial, which we collectively designate by b. Aggregating these dependencies then yields 5 Comparison of Eqs. 4 and 5 directly reveals the joint dependence of g and ɛ_{toc} on v and w.
Removal of the Effects of Sedimentary Recycling
Because the strontium signal s contains a memory effect, whereas ɛ_{toc} does not, removal of the memory effect should reveal correlations between s and ɛ_{toc} at both short and long time scales, thereby confirming the joint dependence on v and w. To test this hypothesis, I first assume it is true and use it to estimate g. For a given λ and μ, m is calculated by discretizing Eq. 3 for τ > −608 My. Eq. 2 is then solved for g: 6 Here the subscripts make explicit the dependencies on λ and μ.
Let R_{λμ}(ɛ_{toc}, g_{λμ}) be the Pearson productmoment correlation coefficient (21) that quantifies the similarity of ɛ_{toc} and g_{λμ}. From the foregoing argument, the best estimate of λ and μ should be that which minimizes R_{λμ} (because the expected correlation is negative). Fig. 2 shows contours of R as a function of μ and t_{1/2} = −(ln 2)/λ. The minimum R_{λμ} = −0.81 occurs at t_{1/2} = 54 My and μ = 0.99. However, the corresponding estimate of g_{λμ} includes values below the hydrothermal minimum of 0.7035. Such nonphysical ratios probably derive from the amplification of any measurement noise resulting from division by the small quantity 1 − μ in Eq. 6. Indeed, the dark gray region in Fig. 2 indicates that all such results occur only for μ close to unity. I therefore define the best estimates λ*, μ* of λ, μ by minimizing R_{λμ} subject to the constraint that g_{λμ} > 0.7035. One then finds t*_{1/2} = −ln 2/λ* = 41 My and μ* = 0.83, corresponding to R_{λ*μ*} = −0.80 (within 1% of the unconstrained minimum). These results are in reasonable accord with Brass's estimate of the halflife of sedimentary Sr (57 − 102 My) (17) and his conclusion that “strontium leached from limestones is about 75% of the total input” (17).
Fig. 3 shows g_{λ*μ*} compared to ɛ_{toc}. Compared with Fig. 1, one sees that both the long and short period fluctuations are now not only approximately equally correlated, but the correlation is also significant (R_{s} = −0.74, P < 10^{−3}, N = 46). The hypothesis that recycled sediment partially obscured an inherent correlation because of a shared dependence on weathering and volcanic processes is therefore confirmed.
Relationship to CO_{2} Levels
To understand further the correlation, I make the following assumptions concerning the functional dependencies contained in Eqs. 4 and 5: 7 The first two state that the flux of radiogenic Sr to the oceans and the uptake of atmospheric CO_{2} because of silicate weathering increase as the global weathering rate (of all minerals in all terranes) increases. The third formalizes the natural assumption that the hydrothermal flux of Sr varies with the same sign as all magmatic processes.
Because the Sr isotope ratios increase with increasing r (i.e., ∂g/∂r > 0) and decrease with increasing v_{h} (i.e., ∂g/∂v_{h} < 0), one has 8 Because g and ɛ_{toc} are negatively correlated, one expects that ɛ_{toc} depends on v and w in the opposite sense: 9 The inequalities 8 and 9 have been obtained without making any explicit assumption concerning any dependence of r on u nor Γ on v and w. Because they show that v and w each influence g and ɛ_{toc} with opposite signs, the relative fluctuations of any other quantity that also depends on v and w with opposite signs may be inferred from the shared fluctuations of g and ɛ_{toc}.¶ Specifically, the concentration Φ of oceanic CO_{2} responds positively to v while its response to weathering is negative: 10 Thus the shared fluctuations of g and ɛ_{toc} indicate the relative fluctuations of ancient CO_{2} levels.‖
Estimation of the CO_{2} Signal
I proceed to estimate the CO_{2} signal explicitly. First, the highest measurement of ɛ_{toc}, at −175 My, is dropped because of its statistical insignificance (3). I then transform g(t) → ɛ_{g}(t), a strontiumderived estimate of ɛ_{toc}, by mapping the left axis of Fig. 3 to the corresponding value on the right axis. The quantities ɛ_{toc} and ɛ_{g} are then averaged to obtain 11 The times t_{i} are given by the points where ɛ_{toc} is specified and ɛ_{g}(t_{i}) is obtained by linear interpolation. Here it is implicitly assumed that ɛ_{toc} and ɛ_{g} contain a common signal that is enhanced by averaging. Statistical arguments suggest that ζ's rms signaltonoise ratio lies between about 2 and 3.**
Assume now that Γ and ɛ_{0} are constant and define the dimensionless CO_{2} concentration φ = Φɛ_{0}/Γ (8). In terms of φ and ζ, Eq. 1 then yields 12 At long time scales such that the oceans are in equilibrium with the atmosphere, the partial pressure pCO_{2} is proportional to φ (1). The relative fluctuations of pCO_{2} are therefore given by φ(t)/φ(0), which is plotted in Fig. 4 for ɛ_{0} = 36 per mil (‰). The gray area surrounding the pCO_{2} curve in Fig. 4 brackets this result for ɛ_{0} = 35‰ and 38‰, a range consistent with previous estimates (3).
Fig. 4 reveals that CO_{2} levels have mostly decreased for the last 175 My. Prior to that point they appear to have fluctuated from about two to four times modern levels with a dominant period of about 100 My. The decline for the last 175 My is also present in several previous pCO_{2} reconstructions (7, 8, 26, 27), and the entire curve displays some similarity to a previous estimate derived from the geologic record of carbonate formation (26). Although the period before −175 My differs substantially from previous geochemical model calculations (7), an approximate error estimate lends considerable credence to the pCO_{2} curve of Fig. 4. Specifically, φ should inherit ζ's signaltonoise ratio of 2–3. This correspondence would be exact if φ(ζ) were linear. Because φ(ζ) is monotonic for the observed range of ζ, its nonlinearity does not affect the timing of the maxima and minima of the pCO_{2} curve. Thus the linear error estimate remains pertinent.
Comparison with the Climate Record
Using a variety of sedimentological criteria, Frakes et al. (18) have concluded that Earth's climate has cycled several times between warm and cool modes for roughly the last 600 My. Recent work by Veizer et al. (28), based on measurements of oxygen isotopes in calcite and aragonite shells, appears to confirm the existence of these longperiod (∼135 My) climatic fluctuations. Changes in CO_{2} levels are usually assumed to be among the dominant mechanisms driving such longterm climate change (29).
It is therefore interesting to ask what, if any, correspondence exists between ancient climate and the estimate of pCO_{2} in Fig. 4. The gray bars at the top of Fig. 4 correspond to the periods when the global climate was cool; the intervening white space corresponds to the warm modes (18). The most recent cool period corresponds to relatively low CO_{2} levels, as is widely expected (30). However, no correspondence between pCO_{2} and climate is evident in the remainder of the record, in part because the apparent 100 My cycle of the pCO_{2} record does not match the longer climatic cycle. The lack of correlation remains if one calculates the change in average global surface temperature resulting from changes in pCO_{2} and the solar constant using energybalance arguments (7, 26).
Superficially, this observation would seem to imply that pCO_{2} does not exert dominant control on Earth's climate at time scales greater than about 10 My. A wealth of evidence, however, suggests that pCO_{2} exerts at least some control [see Crowley and Berner (30) for a recent review]. Fig. 4 cannot by itself refute this assumption. Instead, it simply shows that the “null hypothesis” that pCO_{2} and climate are unrelated cannot be rejected on the basis of this evidence alone.
Discussion and Conclusion
One of the principal contributions of this study is methodological. From observations of a weak correlation between strontium and carbon isotopic signals (Fig. 1) and their shared dependence on global weathering rates and magmatic activity, weathering and magmatism are deduced to be the main processes driving the signals' fluctuations. Correction for the effects of sedimentary recycling enhances the correlation (Fig. 3), indicates a strong shared signal, and strengthens this conclusion.
A second, crucial step is to note that any quantity with a similar joint dependence on weathering and magmatic processes may be expected to display similar fluctuations. Here attention has been focused on CO_{2} levels; as for the strontium and carbon isotopic signals, CO_{2} levels depend on weathering and magmatism with opposite signs and should therefore fluctuate roughly in sync with the isotopic signals. Because the reasoning is general, it need not be limited to CO_{2}. Among the many possible applications, the case of oceanic phosphate concentrations is particularly interesting. Phosphate concentrations should increase with weathering and decrease with hydrothermal activity (31); thus the methodology in this paper may be applicable to their reconstruction. Moreover, because phosphorus is a limiting nutrient, oceanic productivity may be expected to covary positively with its concentration in seawater, suggesting that CO_{2} levels and productivity covary negatively at geologic time scales (8).
Such reasoning naturally raises the issue of cause and effect. This study indicates that degassing and silicate weathering were the primary controls on the carbon cycle for the last 500 My. But the results do not themselves indicate whether either of these mechanisms dominated, or whether weathering was driven by the diversification of land plants (8), continental collisions (9), or a complex combination of tectonic, biological, and geochemical processes (7). They do, however, offer a new view of the longterm fluctuations of pCO_{2} that will hopefully stimulate novel approaches to the study of biogeochemical cycles at evolutionary time scales.
Acknowledgments
I thank O. Aharonson, L. Derry, J. Hayes, P. Hoffman, A. Knoll, L. Kump, J. Sachs, R. Summons, and the late John Edmond for helpful remarks. This work was supported in part by National Science Foundation Grant DEB0083983.
Footnotes

↵† Email: dan{at}segovia.mit.edu.

↵‡ Correlations between ɛ_{toc}(t) and s(t) or g(t) are computed from the N equaltime pairs obtained after linearly interpolating the Sr signal so that it is sampled at the same times as ɛ_{toc}. The statistical significance P is onesided and was estimated by using the Monte Carlo technique described in ref. 8.

↵§ Although w and v could conceivably depend on one another, here such dependencies are assumed to be insignificant compared to any independent variations.

↵¶ One also requires that if changes in v or w dominate one process then they dominate all processes.

↵‖ Because Φ can also depend on other processes b such as organic carbon burial, the shared fluctuations of g and ɛ_{toc} do not necessarily reflect the full evolution of Φ but rather the “partial” contribution of v and w alone. However, the good correlation of Fig. 3 indicates that b's influence on Φ's fluctuations has been small, except possibly in the Carboniferous. Thus one may conclude that the fluctuations of g and ɛ_{toc} follow to good approximation the fluctuations of Φ with opposite signs.

↵** To estimate the signaltonoise ratio of the time series ζ(t) defined by Eq. 11, assume that ɛ_{toc} and ɛ_{g} are each composed of a shared but unknown signal z in the presence of zeromean noise η_{toc} and η_{g}, respectively: ɛ_{toc} = z + η_{toc} and ɛ_{g} = z + η_{g}. The quantity ζ is an estimate of z. Its signaltonoise ratio may be computed under the assumption that η_{toc} and η_{g} each have variance σ, z has variance σ, and η_{toc}, η_{g}, and z are each uncorrelated to the other. The expectation of the correlation coefficient R is then σ/(σ + σ) (25). The observed correlation R = 0.80 then yields σ/σ = 4, or that the rms signal fluctuation is twice that of the noise. The rms signaltonoise ratio of ζ can be as large as a factor of greater, i.e., it should range between about 2 and 3.
Abbreviation
 My,
 million years
 Received October 9, 2001.
 Accepted January 30, 2002.
 Copyright © 2002, The National Academy of Sciences
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