The long eccentricity cycle with a periodicity of 405 kyr is missing in most climate signals for the last 5 Myr, e.g. in the benthic δ18O , but is imprinted on the global carbon cycle, since significant power in it is contained in benthic δ13C e.g.. For earlier parts of the Cenozoic, the 405 kyr cycle is not only found in δ13C but also in δ18O . This difference between climate and carbon cycle makes the interpretation of the Plio-Pleistocene ¹³C cycle, and related interpretations with respect to processes responsible for changes in atmospheric CO₂, challenging.

Some interpretations of the impact of long eccentricity on δ13C have been put forward. used a carbon cycle box model to analyse how oceanic processes, e.g. small changes in nutrient availability or in the ratio of organic matter to CaCO₃ in the export production, might have caused these δ13C changes. They already point out that changes in only the ¹³C cycle but not in atmospheric CO₂, and therefore global climate, are difficult to explain. used a slightly different box model and proposed that eccentricity impacts on the weathering intensity and nutrient supply ultimately changed δ13C in ∼ 400 kyr periodicity during the Miocene. used a conceptual model and suggested that riverine organic carbon inputs into the ocean caused by sea-level-related erosion at river mouths might be responsible for ∼ 400 kyr variability in δ13C of the last 4 Ma. In a data review, also proposed that the origin of this low δ13C frequency is related to riverine input but suggested that the underlying process is the monsoon intensity.

While the long-term changes in δ13C in the ocean have long been known e.g., the record of δ13CO₂ from ice cores is still relatively new and, at 155 kyr, rather short . New ice core data across Termination IV (around 340 kyr BP) offer another ∼ 10 kyr long snapshot of changes in δ13CO₂ . When δ13CO₂ data across Termination II were first measured , an apparent offset between the Penultimate Glacial Maximum and the Last Glacial Maximum (PGM and LGM, respectively) of +0.45‰ prevented a straightforward interpretation of the data. Among other possibilities, the authors speculated that geological processes, namely changes in the isotopic ratio of carbonate weathering or in the contributions from volcanic input fluxes, might explain this offset in δ13CO₂. However, it had not yet been discussed in the ice core community if the PGM-to-LGM offset in δ13CO₂ and the long eccentricity cycle in benthic δ13C were related to each other.

Here, I use the well-established global carbon cycle box model BICYCLE-SE to test in detail if these geological processes proposed by might indeed be responsible for the PGM-to-LGM offset in δ13CO₂ and eventually also for the changes in the ¹³C cycle related to the long eccentricity. For that effort, I first reanalyse previously published simulations which were performed with an updated ¹³C cycle over the time window covered by δ13CO₂ . In that study, atmospheric δ13CO₂ was already prescribed by the ice core data in some simulations and not internally calculated in the model. This approach indirectly introduced some long-term variability in δ13C to the whole ¹³C cycle, while the boundary conditions changing the main carbon cycle and CO₂ were not modified. I first analyse if and how these changes in ¹³C necessary for δ13CO₂ to agree with ice core reconstructions might have been caused by the two geological processes of interest. I then add further scenarios to my assessment, in which one variable of the ¹³C cycle in the model is not prescribed by data but only weakly nudged to a δ13C time series. This nudging approach reduces the amount of necessary adjustments considerably and is first applied to atmospheric δ13CO₂ and then expanded to deep-ocean δ13C in order to be able to cover simulations longer than 155 kyr. Building on earlier work, I then extend my investigation to simulations covering the ice core time window of the last 800 kyr and finally the last 5 Myr covering most of the Plio-Pleistocene . Such an approach of nudging the model to data is an inverse investigation, which offers new insights not directly (or only with a time-consuming effort) available from classical forward modelling. While in the latter well-defined hypotheses are tested by changing the boundary conditions or processes beforehand in simulated scenarios, the inverse approach offers solutions one might not have been looking for (e.g. unexpected frequencies in the variables of investigation). As a side effect, my nudging approach might also increase the simulated glacial–interglacial (G-IG) amplitudes in δ13C in the earlier parts of the Plio-Pleistocene and bring them in better agreement with reconstructions .

Atmospheric CO₂ as simulated in my scenarios is shown as a control (Fig. , first row) against data . This is chosen here as an indicator of the dynamics of the carbon cycle, in depth discussions of which are covered in previous papers .

The control simulations (without prescription or nudging to δ13C) contain typical multi-millennial variations of the order of 0.5 ‰ in atmospheric δ13CO₂ (Fig. , second row) and 0.3 ‰ in δ13CDIC of the wider tropical surface ocean (Fig. , third row) and G-IG variations of 0.5 ‰ in δ13CDIC of the deep Indo-Pacific (Figs. , fourth row) during the Late Pleistocene. As shown before , atmospheric δ13CO₂ and δ13CDIC of the wider tropical surface ocean are highly correlated on orbital timescales of 41 kyr periodicities. In scenarios C1 (prescribed δ13CO₂) and C2 (nudged to δ13CO₂), there is also a loose correlation between atmospheric δ13CO₂ and deep Indo-Pacific δ13C on these timescales (Fig. S5), with a coherence that rose from below 0.2 in scenario SEi to 0.6 (C1) or 0.4 (C2). A tighter correlation between δ13C in the deep ocean and the atmosphere is not expected, since the marine carbon pumps introduce vertical gradients in δ13C e.g., with opposite effects on atmospheric δ13CO₂ (or surface-ocean δ13C) and deep-ocean δ13C on G-IG timescales . Therefore, there has to be a certain anti-correlation between atmospheric δ13CO₂/surface-ocean δ13C on the one hand and deep Pacific δ13C on the other hand. In other words, prescribing or nudging to atmospheric δ13CO₂ or deep Pacific δ13C always increases the agreement of the simulations to the reconstructions in the other variable, but a perfect fit in the other endmember of the ¹³C cycle is not expected. In scenarios C2 and D2, the long-term evolution from PGM to LGM in both atmospheric δ13CO₂ and wider surface-ocean δ13C is covered in the simulations, but the local maxima in the data around 80 kyr BP are not contained, suggesting that internal processes in the atmosphere–ocean–biosphere subsystem rather than these solid Earth fluxes are responsible for these anomalies, which occur on shorter timescales. Millennial-scale changes in atmospheric δ13CO₂ between 70 and 60 kyr BP have already been successfully explained by such internal processes, namely rapid changes in terrestrial carbon pools and/or Southern Ocean air–sea gas exchange , which were probably also active around 80 kyr BP.

The introduction of the long-term periodicities into my simulated ¹³C cycle can be seen in the spectral analysis. Wavelets show, apart from the time window 4–3 Myr BP, no power in periodicities beyond 100 kyr in the deep Indo-Pacific δ13C in the control simulations SEi and SEi++V6 (Fig. a, b), but simulations contain power comparable to the data around 400 kyr and also in 200 kyr periodicities when nudged to the reconstructions (Figs. c, d and c, d). I therefore conclude that, on these slow timescales, my nudged simulations are in reasonable agreement with the data.

The time series of δ13Crockhypo or δ13Cvhypo necessary to close the ¹³C cycle in the long eccentricity timescale are found in the two lowermost rows of Fig.  for the nudged scenarios (C2, D2, D2-L, D2-P), while those of the prescribed scenarios (C1, D1, D1-L) are due to the larger data scatter plotted with different y axes in Fig. S6, which otherwise repeats Fig. . The resulting density distributions of both δ13Crockhypo and δ13Cvhypo, together with the data constraints, are compiled in Fig. . When prescribing one δ13C record from data (scenarios C1, D1, D1-L), it is clearly seen that the necessary distributions of the parameter values are with 1σ>14 ‰ (δ13Crockhypo) and 1σ>9 ‰ (δ13Cvhypo) much wider than what reconstructions are suggesting (1σ<4 ‰ for δ13Crock and 1σ<3 ‰ for δ13Cv from present-day arc volcanoes). This finding indicates that a prescription of δ13C, during which the model would follow the reconstruction not only on orbital timescales but also during shorter multi-millennia anomalies, cannot be explained by the isotopic signature of either carbonate rock or volcanic outgassing. In other words, the scenarios C1, D1, and D1-L are unrealistic with respect to an explanation of the 405 kyr periodicity in the ¹³C cycle and can be discarded here.

In the nudged simulations (scenarios C2, D2, D2-L, D2-P), the resulting δ13Crockhypo and δ13Cvhypo vary on orbital (and sometimes faster) timescales over the whole parameter range provided by the reconstructions (Fig. ). While these dynamics challenge our understanding of which processes might have been responsible for them (see Discussion), the resulting statistics (density distributions) are not unreasonable (Fig. ). In detail, the nudged 800 kyr long simulations overlap well with the Phanerozoic reconstructions of δ13C in carbonate rock (D2: 0.8 ± 3.0 ‰), while, in longer simulations, I find even tighter distributions but with lower mean values (D2-L: 0.2 ± 2.6 ‰; D2-P: -0.4± 3.0 ‰) more in agreement with the Precambrian δ13C rock data (Fig. a, b). Similarly, the mean in δ13Cvhypo is lower in nudged multi-million-year simulations than in those covering only 800 kyr (D2: -5.1± 2.0 ‰; D2-L: -6.2± 1.7 ‰; D2-P: -6.6± 2.0 ‰), and the distributions are, at 1σ≤2 ‰, sufficiently narrow and overlap with data of δ13C in volcanic CO₂ (Fig. c, d). Remember that the nudging strength ηDP was determined from the width of the distribution of δ13Crockhypo in scenario D2 to be comparable to the data but that its mean value was never used as a tuning target: it freely evolved from the simulations and the post-processing analysis. The shift towards lower mean values in δ13Crockhypo or δ13Cvhypo for longer simulations is dominantly caused by δ13C in ODP846 (used for nudging in scenario D2-P) being on average lower than δ13C in the deep Pacific stack (used for nudging in scenarios D2 and D2-L).

When nudging to δ13CO₂ (scenario C2), I obtain a rather skewed distribution of δ13Crockhypo (2.7±2.6 ‰) which is wider than the reconstruction from carbonate rock (Fig. a). The width of the solution does not improve if I revise the nudging strength ηA within reasonable bounds. Thus, it seems that the shortness of the δ13CO₂ time series, which covers not even half of one long eccentricity cycle, is responsible for this unsatisfactory result. The distribution for δ13Cvhypo (-4.5±1.7 ‰) in scenario C2 has a slightly larger mean value, but its width is similar to what I obtain in scenario D2 from longer runs and nudging to deep Pacific δ13C (-5.7±1.9 ‰). However, the shape of the distribution for C2 is, without a clear maximum, very different from a bell-shaped Gaussian curve of a normal distribution (Fig. c). As this analysis initially starts from understanding δ13CO₂, the available data do not yet cover a sufficiently long period for a complete understanding of the 405 kyr cycle in them, although the recently published new data around 340 kyr BP fully support my findings so far, where δ13CO₂ ranges between -6.8‰ and -7.3‰, much lower than in scenario SEi but in agreement with my scenario D2 (Fig. 4b).

As a side effect, my nudging approach also increases the G-IG amplitudes in the simulated ¹³C cycle before the MPT. In the data and in the prescribed scenario D1-L, the G-IG amplitudes in deep Indo-Pacific δ13C were only reduced to 75 % of their post-MPT size prior to the MPT, while, in the control run SEi++V6, they were reduced to 24 % (Figs. a, c). My nudging approach increases these pre-MPT G-IG amplitudes to 41 % (D2-L) and 32 % (D2-P) of their amplitudes during the last 1 Myr, thus partially explaining this feature of the 41 kyr world (Fig. c).

Altogether, I conclude that the carbon isotopic signature of either carbonate rocks or volcanic CO₂ outgassing might be the process which closes the ¹³C cycle on periodicities slower than 100 kyr in the Plio-Pleistocene. Furthermore, the PGM-to-LGM offset in atmospheric δ13CO₂ and the long eccentricity in benthic δ13C are potentially caused by the same processes and are therefore related to each other.

A frequency analysis of the derived solutions for δ13Crockhypo and δ13Cvhypo (Fig. e–h) finds that only little power is contained in periodicities beyond 100 kyr. Results for both variables are dominated by power in the obliquity band and some contributions to the 100 kyr band. Between 2–3 Myr BP, some enhanced power in the 200 kyr periods is contained in both solutions obtained from scenario D2-P. Modifications in either δ13Crockhypo or δ13Cvhypo, both with little power around 405 kyr, lead nevertheless to a persisting occurrence of the long eccentricity in the resulting ¹³C cycle. A transition of power to other frequencies was suggested to be caused by the long residence time of carbon in the ocean during earlier quasi-ice-free periods of the Cenozoic . However, the power transition was probably happening for other reasons, since the main carbon cycle and CO₂ contain no power in 405 kyr periodicity (Fig. g–i), which was not the case in these other studies. Looking closer at the spectra of the resulting simulated deep Indo-Pacific δ13C and the inversely obtained δ13Crockhypo (or δ13Cvhypo) for the same scenario, I find that the relative power in obliquity (41 kyr) and long eccentricity (405 kyr) is actually similar in both variables (Fig. S7). This similarity, not necessarily obtained from the related wavelets (Fig. d, f), supports the connection between both variables and highlights that obliquity is indeed the most important in both time series, while 405 kyr is only weakly contained. Thus, it is not surprising that little 405 kyr power in deep Indo-Pacific δ13C is derived from δ13Crockhypo (or δ13Cvhypo), with similarly little relative power in the same frequency band.

To better understand how variations in δ13Crock or δ13Cv with orbital frequencies impact on deep-ocean δ13C, I performed some tests. Using time series of the orbital parameters eccentricity, climate precession, and obliquity , artificial changes in δ13Crock or δ13Cv were generated with patterns (mean, σ) similar to the reconstructions and then applied to some forward simulations modifying the control run SEi. The results (Fig. S8) show that δ13Cv has a larger effect than δ13Crock on deep Indo-Pacific δ13C, probably because the related carbon influx (CO₂ outgassing) is about twice as large as that of carbonate rock during weathering. Furthermore, the resulting deep-ocean δ13C anomalies with respect to SEi are lagging the forcing by about a quarter of the forcing periodicities. This is happening for all orbital forcings and frequencies indicating the time delay by which the deep-ocean δ13C reacts to changes in the geological isotopic signatures. In other words, the change in δ13Crock or δ13Cv obtained in the nudging approach, which is suggested to be responsible for the slow variations (405 kyr periodicity) in deep-ocean δ13C, is delayed accordingly. Although this lag might not be important for the spectra of δ13Crock or δ13Cv it might need consideration if the hypothesis here is tested in future studies in forward simulations (see below). Furthermore, this lag suggests that the proposed solutions in δ13Crock or δ13Cv might lead to some extent to overshooting/undershooting in the target variable, which would influence the amplitudes but not the frequencies. However, since the obtained distributions in both variables are already in agreement with the reconstructions, smaller ranges would only increase their match to the data.

The chosen setup restricted my analysis to the inversely obtained changes in the isotopic signatures of the geological sources. Therefore, the changes in δ13Crockhypo and δ13Cvhypo are not correlated to the underlying carbon fluxes (r2<0.01), although volcanism was partly a function of the obliquity-dominated changes in sea level/land ice volume and the strength of the carbonate weathering in multi-million-year-long runs was related to CO₂, which is also dominated by obliquity (Fig. e). Such relationships between fluxes and isotopic signatures, if found, might give further support to this hypothesis, since such connections might be expected from a process-based perspective. However, previous model results have shown that changing the strength of fluxes without compensating effects very likely changes not only the ¹³C cycle but also the underlying main carbon cycle . Thus, such results are not easily obtained but might be searched for in future studies.

Obliquity has an influence on incoming solar radiation in the high latitudes and is therefore thought to be the main control for land ice dynamics in the Northern Hemisphere e.g.. Thus, obliquity, together with some influence of precession and internal feedbacks e.g., determines the G-IG cyclicity of the Plio-Pleistocene. Therefore, the dominance of obliquity in my solutions for either δ13Crockhypo or δ13Cvhypo suggests that land ice might also play an important role in my closure of the ¹³C cycle for slow timescales of the Plio-Pleistocene.

Previously, the Pleistocene 400–500 kyr periodicity in marine δ13C was suggested to have been caused by continental weathering and monsoon activity , and climate simulations also highlight the role of the 405 kyr periodicity in tropical precipitation . The Asian monsoon during the last 650 kyr as deduced from detrended δ18O in speleothems has significant power in obliquity but in antiphase to summer insolation at 65° N, which suggests that the ultimate cause for the 41 kyr related changes are situated in the Northern Hemisphere ice sheets . Furthermore, present-day reconstructions of the continental weathering rates indicate that highly active regions dominate the global signal, with 10 % of the area contributing about 50 % of the global weathering fluxes . This study and others also identified runoff, which is highly related to precipitation, as one of the dominant controls of local weathering fluxes. Obliquity-controlled monsoon systems therefore have a leverage on weathering, potentially activating contributions from different regions on Earth, with variable δ13Crock, accordingly. I therefore suggest that obliquity-driven changes might also easily control the weathering strength of different areas, finally ending in changes in δ13Crockhypo without the need for large variations in the global weathering flux itself. This influence of obliquity might operate via monsoon, but a variable contribution of available weatherable rock around dynamic ice sheets in North America and Eurasia or from continental shelves during sea level low stands also seems possible e.g..

Dominant power in the obliquity band is also contained in the reconstructed activity of the circum-Pacific chain of arc volcanism, sometimes referred to as the “Ring of Fire”, during the last 1 Myr . In a revised analysis, this power in 41 kyr is reduced and is only second behind 100 kyr . This spectral analysis supports hypotheses that changes in land ice load or sea level might influence subaerial or submarine volcanism, respectively . Note that, in the model, the component of volcanic CO₂ outgassing that is influenced by either land ice or sea level changes is lagging them by 4 kyr . In light of known temporal changes in the isotopic signature of individual volcanoes e.g. and of the large heterogeneity of the mantle material e.g., the obliquity-related changes in δ13Cvhypo, as inversely deduced here from the analysis, are therefore also conceivable.

Possible ways of testing the proposed hypothesis of the obliquity-dominated influence on the isotopic signature of the geological sources might be the following. The most direct support might be derived from a forward modelling approach in which the isotopic signature of the geological sources and the strength of the related fluxes are correlated to each other. This would, however, need a data-mining effort, in which fluxes and isotopic signature are spatially connected for the present day. This approach might not generate long time series as done here, but it tries to establish if fast changes in global mean δ13Crockhypo or δ13Cvhypo, as suggested here, are feasible from a more process-based understanding. However, unknown mantle heterogeneity and unknown temporal evolution of both local fluxes and local isotopic signatures might make this a rather challenging task. Frequency analysis of various weathering proxies e.g. obtained from sediment cores in different ocean basins might give indications if obliquity played a dominant role on the regional scale, although this would only give indirect evidence on flux strength and not on δ13Crock. Monitoring programmes on present-day volcanoes might give a broader foundation if temporal changes in δ13Cv, as observed for the Etna, are an exception or the rule . Finally, the connection of the PGM and LGM offset in atmospheric δ13CO₂, and how this is related to the 405 kyr periodicity, might be verified with new ice core data of atmospheric δ13CO₂ further back in time.

My analysis, based on an inverse approach which nudged simulated δ13C time series to reconstructions, suggests that variations in the isotopic signatures of the geological sources (δ13C of weathered carbonate rock or δ13C of volcanic outgassed CO₂) might, without any further changes in the carbon cycle, be sufficient to explain the signals in δ13C related to the long eccentricity cycle during most of the Plio-Pleistocene. Furthermore, my suggested solution connects this periodicity around 405 kyr in deep-ocean δ13C with the PGM-to-LGM offset in atmospheric δ13CO₂, implying that both are caused by the same underlying processes. My analysis highlights the influence of obliquity on these changes in the ¹³C cycle, which I interpret as Northern Hemisphere ice sheets being the ultimate cause for the necessary variations in δ13Crock or δ13Cv. Studies show that both weathering and volcanic activity might, directly or indirectly, be influenced by land ice dynamics, giving independent support for my hypothesis. Taking the link to obliquity for granted, it is furthermore unlikely that the same mechanisms might have been responsible for the 405 kyr periodicity in marine δ13C found in earlier climates of the Cenozoic, since there has not been any substantial Northern Hemisphere ice sheet . During these times, not only δ13C, but also δ18O and potentially atmospheric CO₂, contained cyclicity related to the long eccentricity, which is missing here for the Plio-Pleistocene; this requires other solutions to the problem. However, both my possible answers, either via carbonate weathering or via volcanism, seem similarly likely. So far, I have not yet found a criterion which makes one more likely than the other, and a mixture of both is probably the most realistic solution to the closure of the ¹³C cycle on these slow timescales suggested here. However, as outlined in the Discussion, further investigations and proposed tests might favour one above the other. The dominance of obliquity in my solutions readily explains an improvement in simulated G-IG amplitudes in δ13C in the 41 kyr world of the Plio-Pleistocene, but their reconstructed sizes are not fully obtained. Thus, while the influence of the isotopic signature of the geological sources might be important for the decoupling of climate from the carbon cycle, there is room for other (not yet detected) processes for a complete understanding of the Plio-Pleistocene ¹³C cycle.