The basis of our co-evolutionary model is the conceptual carbon cycle model by . The model covers the most basic interactions between terrestrial, atmospheric and marine carbon pools, and was developed specifically to enable a bifurcation analysis of carbon-related planetary boundaries and their interactions. We modified atmosphere–land interactions for a better representation of empirically observed Earth system carbon dynamics and extended the model by a stylised societal management feedback loop mimicking the current focus of international policy processes on climate change. We calibrated the model in order to represent global carbon cycle dynamics consistent with observational data and simulations from detailed high-resolution Earth system models (Sect. ). In the following, we provide an overview of the fundamental model equations. A detailed motivation of the model design and underlying assumptions are given in .

The adapted model consists of five interacting carbon pools: land Ct (t), atmosphere Ca(t), upper-ocean Cm (t), geological fossil reservoirs Cf(t) and a potential CE carbon sink CCE(t) (Fig. ). All model equations are summarised in Table . Note that only the upper-ocean carbon pool is included because the movement of carbon into the deep ocean occurs on longer timescales relative to those of interest, as discussed by . The land carbon pool combines soil and vegetation carbon pools, implying a simple proportional partitioning of aboveground and belowground carbon pools . These simplifications have been adopted because they reduce the number of state variables and we were able to qualitatively reproduce the dynamics of observed carbon pool evolution with the adapted model.

The co-evolutionary dynamics of the system is determined by Eqs. (1)–(5). Conservation of mass (Eq. 1) dictates that the active carbon in the system, i.e. the sum of terrestrial, atmospheric and maritime carbon is equal to the active carbon at pre-industrial times (C0) plus carbon released from fossil reservoirs (Cr(t)) minus carbon extracted via tCDR (CCE(t)) to permanent stores. Fossil carbon release (Eq. 2) is approximated by a logistic function parameterised by the maximum emitted carbon cmax⁡ and rate of carbon release ri.

The social management feedback loop is motivated by proposals of CE as a management intervention in response to intolerable levels of global warming. It comprises atmospheric carbon monitoring and tCDR action conditional on the proximity to a critical threshold of atmospheric carbon content (Eq. 3). CE action is implemented via a tCDR carbon offtake from terrestrial carbon (HCE(t)) and storage in a permanent (geological) sink CCE. Carbon offtake for tCDR (Eq. 11) is defined analogous to human offtake for agriculture or land-use change (Eq. 13), however, with a dynamic offtake rate αCE(Ca(t)) (Eq. 12).

The tCDR characteristics are governed by three parameters: (i) implementation threshold (Cã) in terms of atmospheric carbon content, representing societal foresightedness, (ii) maximally achievable rate of tCDR (αmax⁡), a measure of societies' efforts, as well as biogeochemical constraints and (iii) the slope of tCDR implementation (sCE), parameterising social and economic implementation capacities. Figure  depicts an exemplary tCDR trajectory for constant terrestrial carbon in Eq. (11) for two values of sCE. The implementation time can be computed from the slope of tCDR implementation by using current increase rates of atmospheric carbon as a conversion factor. With current increase rates of approximately 2 ppmv a-1 , the two depicted values of sCE correspond to tCDR ramp-up times of approximately 20 and 40 years (from 10 to 90 % capacity) for sCE=0.1 ppmv-1 (solid) and sCE=0.05 ppmv-1 (dashed), respectively.

The atmosphere–ocean carbon feedback (Eq. 4) is governed by diffusion, which in the model is assumed to depend on the difference between atmospheric and maritime carbon pools.

Land–atmosphere interaction is determined by both ecological and social processes: the net ecosystem productivity (Eq. 6), tCDR offtake (Eq. 11) and other human offtake for agriculture and other land use (Eq. 13), respectively.

Net ecosystem productivity is given by the net carbon flux of photosynthesis (Eq. 8) and respiration (Eq. 9), multiplied by the terrestrial carbon pool and a logistic dampening function which represents competition for space, sunlight, water or nutrients. Both photosynthesis and respiration are continuous functions of global land temperature (T(t), Eq. 10), which in turn depends linearly on atmospheric carbon content. It is important to note that in our model, respiration exceeds photosynthesis for higher temperatures (Fig. ). The state of equilibrium of the terrestrial carbon pool is thus determined by the land surface temperature, as well as the terrestrial carbon carrying capacity (Eq. 7) in the density function. In contrast to , we implement a dynamic terrestrial carbon carrying capacity as a function of atmospheric carbon content. This is motivated by a number of factors such as CO2 fertilisation and a higher water-use efficiency under higher atmospheric carbon concentrations, as well as higher average vegetation density in a warmer world e.g.. For low atmospheric carbon we assume a rapid increase in terrestrial carbon storage capacity as a function of atmospheric carbon concentration and a saturation of storage capacity for high atmospheric carbon, in line with assessments of coupled carbon cycle climate models . The functional relationship in Eq. (7) follows these constraints for chosen parameter values (Sect. ).

A sufficiently suitable application of a conceptual model in the context of the planetary boundaries as in  requires the model's ability to simulate credible transients of global carbon dynamics. In order to achieve this, we calibrated model parameters to observed carbon fluxes and pools, as well as simulation results of detailed high-resolution Earth system models.

Because we simulate relative dynamics between the different carbon compartments and do not aim at prognostics of actual time evolution of carbon pools, all carbon fluxes and pools are normalised to the active carbon at pre-industrial times, i.e. the total sum of pre-industrial carbon in the year 1750 AD (3989 Gt C, Fig. ). All normalised parameter values are summarised in Table .

We use the carbon-related planetary boundaries (climate change, ocean acidification and land system change) to define the desirability of given trajectories of carbon pool evolution. The proposed locations of these boundaries are normalised to match the normalisation of our model.

The planetary boundary for climate change is proposed at 350 ppmv CO2 equivalents in the atmosphere with an uncertainty range to 450 ppmv . For our study we take the middle of the uncertainty range (400 ppmv) because critical atmospheric thresholds are likely to be located somewhere within the uncertainty range and obtain a normalised climate change boundary is at 0.21 atmospheric carbon. Ocean acidification is measured via the saturation state of aragonite and its boundary is set at 80 % of the pre-industrial average annual global saturation state of aragonite . Since chemical processes are not explicitly represented in our model, this measure is not directly transferable to maritime carbon content. This measure is not directly transferable to maritime carbon content because it largely depends on chemical variables such as pH-value, ocean alkalinity and dissolved inorganic carbon that are not included in the model. At the current carbon content (1150 Gt C), the saturation state of aragonite is at 84 % of the pre-industrial value . We therefore estimate the normalised ocean acidification boundary at 0.31, about 5 % higher than the current value of the marine carbon stock (0.29). The land system change boundary is defined in terms of the amount of remaining forest cover, motivated by critical biogeophysical feedbacks of forest biomes to the physical climate system . The global boundary has been specified as 75 % of global forest cover remaining . Due to the lack of biogeophysical feedbacks in the model, we translate deforestation into carbon content by measuring the loss of vegetation carbon with deforestation. We thereby neglect vegetation carbon of all non-forest biomes, while at the same time neglecting soil carbon changes by deforestation , thus approximating that soil carbon changes by deforestation are of the same order of magnitude as vegetation carbon pools of non-forest biomes. With vegetation carbon of 550 Gt C , we obtain a normalised land system change boundary at 0.59.

Note that the exact location and normalisation of the boundaries is not decisive for our results because we qualitatively analyse the influence of tCDR management on the existence of desirable trajectories. Slightly different sets of planetary boundaries would not qualitatively change the systemic effects reported in this study.

Our analysis of the co-evolutionary system aims at assessing transient dynamics of carbon pools with respect to planetary boundaries. First, we run the model and exemplarily show the influence of socially controlled parameters of tCDR implementation on the transient carbon pool evolution (Sect. ). It is of particular relevance under what circumstances the simulated carbon pool trajectories (atmosphere, ocean and land) do not cross their respective planetary boundaries. We refer to the regions on the safe side of the planetary boundaries as “safe regions”. All carbon pool trajectories remaining in the respective safe region at all times are considered “safe trajectories”. For example, all atmospheric carbon trajectories that do not cross the planetary boundary for climate change (i.e. trajectories that are in the safe region of atmospheric carbon) are safe atmospheric carbon trajectories. System states with each carbon pool remaining in its respective safe region are referred to as carbon system states within the SOS, i.e. “safe states”.

In a nonlinear dynamical system, trajectories can be sensitive to initial conditions. The pre-industrial distribution of carbon pools, as well as carbon dynamics in the Earth system are relatively well-assessed, while still subject to high uncertainty . Furthermore, considerable uncertainty remains with respect to our conceptual model structure and the exact values of planetary boundaries. Bearing in mind these inherent uncertainties, we explore how robust the existence of safe trajectories is under a variation of the initial conditions, i.e. the initial carbon pool distribution and different tCDR characteristics (Sect. ).

Such a variation of initial conditions is also a common approach to conceptualising and measuring resilience of social–ecological systems as the ability to return to an attracting state after a perturbation . A suitable approach to quantifying the likelihood of a complex system to return to an attracting state under finite perturbations is basin stability analysis .

In the context of planetary boundaries, not necessarily all trajectories that approach a “safe attractor” (i.e. an attractor within the SOS associated to all three planetary boundaries) would be considered safe because they could temporarily leave the safe region. The concept of constrained basin stability and related methods provide generalisations of basin stability that allow taking transient phenomena into account. Similarly to the constrained basin stability approach, we classify different domains in the initial-condition state space based on transient dynamics of carbon pools. The set of initial conditions resulting in safe carbon trajectories form the “safe domain”. We refer to this domain as the manageable core of the safe operating space (MCSOS), as it depends on the tCDR management characteristics and the emission pathway. The “undesirable domain” is formed by all initial conditions resulting in a transgression of all three carbon boundaries at some point in time. Remaining state space domains are formed by initial conditions leading to a transgression of a subset of planetary boundaries. They are referred to as the respective partially manageable domains (MDs) (e.g. the land manageable domain is the state space domain of initial conditions with trajectories without a transgression of the land boundary).

The computational efficiency of our model allows for a systematic analysis of the MCSOS and other domains under variation of societal parameters (tCDR management and fossil fuel emissions). We analyse how the size of all domains (MCSOS, partially MDs and the undesirable domain) varies with different tCDR characteristics (Sect. ) and emission pathways (Sect. ). In the spirit of , the size of (partially) manageable domains can be interpreted as a resilience-like measure of the opportunities to stay within the carbon-related SOS, taking into account inherent structural uncertainties of our model, the location of planetary boundaries, and the pre-industrial carbon pool distribution. Note that the maximum extent of the MCSOS is constrained by the planetary boundaries, but it may differ from the SOS (i.e. the “safe” region) as the safety of the domain is determined by transient system dynamics, whereas the SOS is defined within static planetary boundaries.

To illustrate how the co-evolutionary social–environmental system evolves with respect to carbon-related planetary boundaries, Fig.  depicts trajectories of the major carbon pools with tCDR adhering to different management characteristics. All trajectories start at their respective normalised pre-industrial state. The normalised planetary boundaries (Sect. ) are indicated as dotted lines and the safe region of each boundary (refer to Sect. ) is shaded in the respective colours. Variation of tCDR characteristics reflects uncertainty about possible tCDR rates related to overall biomass harvesting potentials and societies' implementation capacities (Sect. ).

The emission baseline used for all results displayed in Fig.  is a business-as-usual scenario with cumulative emissions as in RCP8.5 . Without tCDR (Fig. a), all that fossil carbon societies emit into the atmosphere is distributed to ocean, land and atmosphere. This results in more active carbon (red), leading to carbon accumulation in all pools and a transgression of the atmosphere and ocean boundaries. In this emission scenario, the land system accumulates carbon and, thus, moves away from its planetary boundary in our model setting (note that the actual control variable of the planetary boundary of land system change as defined by is the remaining forest cover, which would not be directly modified by changing atmospheric carbon concentrations). Moreover, higher emission baselines (results not shown here) can lead to decreasing terrestrial carbon stocks when respiration dominates over photosynthesis due to strong global warming.

In Fig. b) and c), the societal tCDR response via harvesting from the terrestrial carbon stock and subsequent storage starts just before the atmospheric boundary is reached (Cã=0.18∼340 ppmv). With a low tCDR rate (maximal storage flux of about 7 GtCa-1, αmax⁡=0.0025), the CE sink is filled relatively slowly (Fig. b). Thus, a transient transgression of the atmosphere and ocean boundaries cannot be prevented. However, all trajectories re-enter their respective “safe” region after about 150 years. A higher tCDR rate (αmax⁡=0.025, corresponding to very high-potential storage fluxes of 26 GtCa-1 or 5 % of global biomass per year) can prevent a large increase in active carbon and thus prevents the transgression of both atmosphere and ocean boundaries (Fig. c). However, extensive harvest from the land carbon pool then leads to a temporary transgression of the land boundary. The implementation of tCDR was thus effective in its purpose of preventing entry into a dangerous region of climate change, but at the cost of exploiting the land system to an extent that crossed the land system change boundary.

These results show that small tCDR rates (Fig. b) (or implementation that is too late, results not shown here) do not necessarily keep the system in the SOS. High tCDR rates (Fig. c) could seem successful when focusing on the climate change boundary, but might in fact not be feasible if other components of the carbon system are taken into account. In light of ongoing deforestation for the purpose of bioenergy production , this simulated collateral transgression of the land system change boundary with large-scale tCDR is an important and plausible feature of the model.

In the actual Earth system, a transgression of the land system change boundary might evoke additional trade-offs to the biogeophysical climate system , which are not represented in the model. For example, large tCDR rates can only be achieved by large-scale land-use change that could alter atmospheric circulations and rainfall patterns even though the carbon-related climate change boundary might not be transgressed with high tCDR rates.

The carbon values stated here are primarily given as an orientation for the reader, and should not be directly interpreted with respect to tCDR feasibility assessments. However, tCDR rates of 7 GtCa-1 are in line with more conservative biomass harvest potentials considering biodiversity conservation and agricultural limits . More idealistic assessments of tCDR rates of more than 35 GtCa-1 – assuming high biomass yields of more than one-quarter of global land area – have been reported as well . In this context, the range of tCDR rates studied in this paper reflects both conservative and highly optimistic tCDR potentials reported in the literature.

We compute the state space domain structure (refer to Sect. ) from a sample of initial conditions around the pre-industrial carbon state. We sample approximately 66 000 initial conditions from a regular grid by variation of each carbon pool by ±0.2 around the pre-industrial conditions. This range is a pragmatic choice which does not influence the following qualitative analysis. To compute the existing domains, we evolve each initial condition for 600 years in time and colour it according to the domains following from the transient properties of the trajectories of land, atmosphere and ocean carbon, as described above. The mapping of initial conditions sheds light on possible domains in the carbon system and potential transitions into other state space domains in our model of the carbon cycle. In this context, the vicinity of the pre-industrial and current Earth system states to such domain boundaries in the model's initial-carbon-condition state space is of particular relevance.

Figure  shows the existing domains without tCDR (a), with intermediate tCDR rates (b) and with very high tCDR rates (c). The emission baseline is the same for all variations of tCDR characteristics, with cumulative emissions of approximately 880 Gt C, which is comparable to RCP2.6 cumulative emissions . The current state of the carbon cycle is located in proximity to domain borders, highlighting that it is close to a transgression of the land system and climate change boundaries. Historical emissions and land system changes have moved the state of the carbon cycle closer towards the undesirable domain, and remaining on an emission trajectory similar to RCP2.6 without tCDR results in the non-existence of the MCSOS (Fig. a). Thus, the manageable core does not exist if the implementation of tCDR management is not considered by society, even in a relatively low-emission scenario.

Figure b and c serve as an example of how human intervention and management by tCDR can influence the size and even the existence of the MCSOS and other domains. With an implementation of tCDR, the MCSOS can be re-established, potentially to its full extent, which is directly determined by the three planetary boundaries (Fig. b). Even for a relatively low-emission scenario, the tCDR threshold needs to be at sufficiently low atmospheric carbon content (Cã=0.16) to prevent potential boundary transgressions. Nevertheless, because of past land-use change, the current Earth system state is approaching domains with unsafe land system and climate change. If tCDR is applied under the same conditions but with a 10 times higher potential tCDR rate (αmax⁡=0.04), the MCSOS shrinks due to over-exploitation of the land system for tCDR (Fig. c). The land system is overexploited when the total human biomass offtake flux (HCE+H) exceeds net ecosystem productivity (NEP). This decreases terrestrial carbon pools (Eq. 5) which in turn limits the potential for tCDR (Eq. 11). In Fig. c this occurs under high initial atmospheric carbon concentrations, because these result in a higher tCDR flux for the same potential tCDR rate (αmax⁡, ref. to Fig. ). The current state of the carbon cycle of the Earth system is out of the MCSOS. In this case, large societal commitment to avoid a transgression of the climate change boundary leads to a collateral transgression of the land system change boundary in our model.

The size and existence of the MCSOS and other state space domains depends on tCDR characteristics (refer to Sect. ). We compute the size of the different initial-condition state space domains depending on the most decisive management parameters, i.e. on the implementation threshold Cã and on the potential maximum tCDR rate αmax⁡. The size of all domains is measured in relation to the size of the considered state space section as depicted in Fig. , which is given by a variation of pre-industrial conditions by ±0.2.

Figure  depicts the relative size of the MCSOS and the partially manageable domains under baseline emissions of cmax⁡=0.4, corresponding to cumulative emissions in the order of RCP6.0. The size of the MCSOS or partially MDs can be interpreted as a form of resilience of the system (i.e. the likelihood that the system stays within the carbon-related SOS). Thus, we measure the resilience of the carbon cycle by the size of MCSOS (i.e. the opportunity of success of tCDR to maintain safe trajectories). This strongly depends on the atmospheric carbon threshold at which tCDR is implemented. Obviously, only the anticipation of an approaching planetary boundary can prevent a transgression thereof. Thresholds higher than the atmospheric carbon boundary (bl=0.21) are not sufficient in sustaining a MCSOS, because the atmosphere MD disappears by definition at Cã=0.21 (grey line in Fig. a).

However, strong anticipation coupled with too early tCDR implementation does not necessarily maintain the system within the SOS. If tCDR is initialised at relatively low atmospheric carbon content (Cã=0.13 (approximately 330 ppmv) in Fig. a), the MCSOS is diminished due to a transgression of the land system change boundary at some point in time. Hence, the window of opportunity for using tCDR as a means of staying in the SOS under this exemplary fossil fuel emission scenario is limited to a relatively narrow range of tCDR implementation thresholds. The size of the land MD shows nonlinear dependence on the tCDR threshold. For thresholds between 0.2 and 0.25, the land MD is almost diminished (Fig. a), because the relatively high tCDR rate (αmax⁡=0.02) leads to an over-exploitation of the land system (ref. to Sect. ). However, higher tCDR thresholds avoid this over-exploitation and increase the land MD, because of a later onset of tCDR and overall higher NEP due to higher atmospheric carbon content and temperature (Eq. 6).

Similar to the tCDR threshold, the parameter governing the maximal achievable rate of tCDR plays a decisive role for the existence of the MCSOS. With a tCDR implementation threshold not far below the atmospheric carbon boundary (Cã=0.2), high tCDR rates are required in order to maintain a MCSOS. The tCDR starts being effective in maintaining a MCSOS at a rate of αmax⁡>0.007 (corresponding to approximately 16.5 Gt C a-1 with a fixed land carbon pool of 0.6). Rates smaller than that are not sufficient because of a lacking atmospheric MD (grey line in Fig. b).

As the tCDR threshold, the tCDR rate has a strong influence on the size of the land MD. For small tCDR rates, the land MD is sustained because of high atmospheric carbon concentrations and small biomass extraction. Rates higher than αmax⁡=0.0075 result in a smaller land MD due to the over-exploitation of the photosynthetic productivity of the system which is reduced by both biomass removal and decreasing atmospheric carbon concentrations driving NEP. Higher rates, however, lead to overall smaller reductions of the land MD. This nonlinearity is evoked by the co-evolutionary feedbacks between society and the carbon cycle, which lead to a deceasing tCDR flux if the system is in the atmosphere MD. Thus, sufficiently high tCDR rates lead to fast atmospheric carbon decrease and tCDR is switched off before the land system boundary is transgressed.

This analysis of the size of initial-condition state space domains suggests that the success of tCDR in sustaining the Earth system's persistence in the carbon SOS nonlinearly depends on the characteristics of tCDR implementation. On the one hand, foresightedness and anticipation of planetary boundaries are required to maintain the MCSOS, while on the other hand, too-early or too-intensive management could trigger co-transgressions of other planetary boundaries.

While anticipation and appropriate management are necessary, the underlying emission scenario plays a major role in the resulting carbon dynamics. Figure  exemplarily depicts the relative MCSOS size for variations of tCDR characteristics (threshold and potential maximum rate) for emission pathways in accordance with RCP cumulative-emission scenarios. The window of opportunity for successful tCDR (i.e. the size of the MCSOS) decreases with increasing emission baselines and depends on the tCDR rate and threshold. In the case of the low-emission RCP2.6 scenario (cmax⁡=0.2), the MCSOS can be sustained for a broad range of parameter values (Fig. a). The medium emission scenarios RCP4.5 (cmax⁡=0.31; ) and RCP6.0 (cmax⁡=0.36; ) show a narrower range of tCDR characteristics that have the potential to sustain a MCSOS (Fig. b and c). In a business-as-usual RCP8.5 scenario, the room for manoeuvring to maintain a MCSOS is very small (Fig. d).

Besides the dependence on the emission scenario, Fig.  highlights that for most emission scenarios the range of tCDR thresholds sustaining the MCSOS is narrow and depends on the tCDR rate. As discussed in Sect.  (for a fixed tCDR rate), tCDR thresholds higher than the atmospheric carbon boundary (0.21) are not sufficient in preventing a boundary transgression in the medium-to-high emission scenarios (Fig. b–d), whereas small tCDR thresholds lead to a transgression of the land system change boundary (unless tCDR rates are within a very narrow range smaller than 0.001). The variation of both the tCDR rate and threshold shows that smaller tCDR rates require a smaller minimal tCDR threshold as well as a smaller maximal threshold (Fig. b–d). This dependence of the success of tCDR on both the tCDR characteristics and the underlying emission scenarios highlights the relevance of societal intervention for global carbon dynamics. Essentially, tCDR intervention can trigger a nonlinear carbon system response through the land system when human carbon offtake exceeds NEP, which in turn causes a further reduction in NEP and tCDR potentials.

In our conceptual framework, tCDR can be effective in complementing climate change mitigation strategies as employed in low-emission scenarios. However, already an RCP4.5 emission scenario narrows the range of potentially successful management options significantly in comparison to RCP2.6 emissions. Under a business-as-usual pathway, tCDR cannot be applied to maintain a MCSOS in a resilient way. In contrast to prevailing reasoning of CE as an emergency action in case of dangerous climate change , tCDR would most likely not function as an emergency option under high-emission scenarios when additional sustainability dimensions reflected by other planetary boundaries are taken into account.