If
the Paris Agreement targets are to be met, there may be very few years left
for policy makers to start cutting emissions. Here we calculate by what year,
at the latest, one has to take action to keep global warming below the

The Earth system is currently in a state of rapid warming that is
unprecedented even in geological records

The climate system is characterized by positive feedbacks causing
instabilities, chaos and stochastic dynamics

We define pre-industrial temperature as the 1861–1880 mean temperature, in accordance with IPCC AR5.

) have turned out to be salient. TheA range of studies has appeared to provide insight into the safe level of
cumulative emissions to stay below either the

In this paper we pose the following question: assume one wants to limit
warming to a specific threshold in the year 2100, while accepting a certain
risk tolerance of exceeding it, then when, at the latest, does one have to
start to ambitiously reduce fossil fuel emissions? The point in time when it
is “too late” to act in order to stay below the prescribed threshold is
called the point of no return (PNR;

The models used in

We let

As all CMIP5 models are designed to represent similar (physical) processes but use different
formulations, parameterizations, resolutions and implementations, the results from different models
offer a glimpse into the (statistical)
properties of future climate change, including various forms of uncertainty. We perceive each model
simulation as one possible, equally likely, realization of climate change. Applying
ideas and methods from statistical physics

We only use those ensemble members from CMIP5 for which the control run and
at least one perturbation run are available, leading to 34 members for the
abrupt (

The

Ensemble mean (

Beyond finding the temperature change as a result of

The full (temperature and carbon) LRT model is summarized as

The constant

Stochastic state-space model. Carbon model on the left, temperature model on the right.

Internally, emissions need to be converted from GtC year

Stochastic state-space model parameters. All timescales are in years, the carbon model
amplitudes

In Fig.

Reconstruction of RCP results using the response
function model. In all panels, solid lines refer to RCP4.5,
dotted to RCP6.0 and dashed lines to RCP8.5. Black lines show
RCP data while colors (blue: RCP4.5, orange: RCP6.0, green: RCP8.5)
give our reconstruction.

The model outlined above still contains a data-based temperature response
function and it informs only about the

The response function

These are the fossil
fuel and cement production emissions from

The major benefit of this formulation is that we can include stochasticity.
We introduce additive noise to the carbon model such that the standard
deviation of the model response to an emission pulse as reported by

The SSSM described in the previous section is forced with fossil

In addition, negative emission technologies may be employed. They cause a
direct reduction in atmospheric

For long timescales, these (after a transient) constant negative emissions may not be realistic. However, we are interested in the period until the year 2100.

With the emission scenarios and the SSSM – returning

Concretely, let the temperature target

Then, in the context of Eq. (14), the PNR is the earliest

Both SCB and PNR depend on temperature
target, climate uncertainties and risk tolerance, but the PNR also depends on the aggressiveness of the
climate action considered feasible (here given by the value of

Clearly there is a close connection between the PNR and the SCB. Indeed, one could define a PNR also in terms of the ability to reach the SCB. The one-to-one relation between cumulative emissions and warming gives the PNR in “carbon space”. Its location in time, however, depends crucially on how fast a transition to a carbon-neutral economy is feasible.

For details on the scenarios, we refer to

To demonstrate the quality of the SSSM we initialize it at pre-industrial
conditions, run it forward and compare the results with those of CMIP5
models. The SSSM is well able to reproduce the CMIP5 model behavior under the
different RCP scenarios (Fig.

Stochastic state-space model applied to RCP scenarios.

The safe carbon budget.

To determine the SCB, 6000 emission reduction strategies (with

The temperature anomaly in 2100 (

Safe Carbon Budget (in GtC since 2015) as a function of threshold and safety probability

Allowable emissions are drastically reduced when enforcing the target with a
higher probability (following the horizontal lines from right to left in
Fig.

From IPCC-AR5

To determine the PNR, we resort to three illustrative choices to model the
abatement and mitigation rates with

Following Eq. (14) we construct fast mitigation (FM) and moderate mitigation
(MM) scenarios with

When varying

As an example,

Figure

Point of no return as a function of threshold and safety probability

The point of no return. Probability of staying below the
1.5 K

Including strong negative emissions delays the PNR by 6–10 years, which may
be very valuable especially for ambitious targets. For example, one can then
reach

The PNR varies substantially for slightly different temperature targets. This
also illustrates the importance of the temperature baseline relative to which

It is clear that an energy transition more ambitious than RCP2.6 is required
to stay below

The parameter sensitivities of SCB and PNR were determined by varying each
parameter by

The biggest sensitivities are found for the radiative forcing parameter

The sensitivity of SCB and PNR to the noise amplitudes is small, with largest
values found for the multiplicative noise amplitude

It is useful to remember that the stochastic formulation of our model is designed with the explicit purpose to incorporate parameter uncertainty in a natural way via the noise term, without having to make specific assumptions on the uncertainties of individual parameters.

Sensitivity of the safe carbon budget (SCB) and point of no return (PNR) to selected
parameter variations. Values as difference in GtC (SCB) and number of years (PNR)
relative to the undisturbed value (top row). The PNR values all refer to the EM scenario.
First and second numbers give

We have developed a novel stochastic state-space model (SSSM) to accurately capture the basic statistical properties (mean and variance) of the CMIP5 RCP ensemble, allowing us to study warming probabilities as a function of emissions. It represents an alternative to the approach that contains stochasticity in the parameters rather than the state. Although the model is highly idealized, it captures simulations of both temperature and carbon responses to RCP emission scenarios quite well.

A weakness of the SSSM is the simulation of temperature trajectories beyond
2100 and for high-emission scenarios. The large multiplicative noise factor
leads – especially at high mean warmings – to immensely volatile
trajectories that in all likelihood are not physical (on the individual
level, the distribution is still well-behaved). It might be worthwhile to
investigate how this could be improved. Another weakness in the carbon
component of the SSSM is that the real carbon cycle is not pulse-size independent.
Hence, using a single constant response function has
inherent problems, in particular when running very high-emission scenarios.
This is because the efficiency of the natural carbon sinks to the ocean and
land reservoirs is a function of both temperature and the reservoir sizes.
The SSSM therefore has slight problems reproducing

Taking account of non-

In

The concept of a point of no return introduces a novel perspective into the discussion of carbon budgets that is often centered on the question of when the remaining budget will have “run out” at current emissions. In contrast, the PNR concept recognizes the fact that emissions will not stay constant and can decay faster or slower depending on political decisions.

With these caveats in mind, we conclude that, first, the PNR is still
relatively far away for the

Third, we can clearly show the effects of changing

Fourth, negative emissions can offer a brief respite but only delay the PNR
by a few years, not taking into account the possible decrease in
effectiveness of these measures in the long term

In this work a large ensemble of simulations was used in order to average over stochastic internal variability. This allows us to determine the point in time where a threshold is crossed at a chosen probability level. Such an ensemble is not possible for more realistic models, nor do GCMs agree on details of internal variability. Therefore, in practice, the crossing of a threshold will likely be determined with hindsight and using long temporal means. This fact should lead us to be more cautious in choosing mitigation pathways.

We have shown the constraints put on future emissions by restricting GMST increase below 1.5 or 2 K, and the crucial importance of the safety probability. Further (scientific and political) debate is essential on what are the right values for both temperature threshold and probability. Our findings are sobering in light of the bold ambition in the Paris Agreement, and add to the sense of urgency to act quickly before the PNR has been crossed.

The study is based on publicly available data sets as described in the Methods section. Model and analysis scripts and outputs are available on request from the corresponding author.

MA and HAD developed the research idea, MA developed the model and performed the analysis. All authors discussed the results and contributed to the writing of the paper.

The authors declare that they have no conflict of interest.

We thank the focus area “Foundations of Complex Systems” of Utrecht University for providing the finances for the visit of Frederick van der Ploeg to Utrecht in 2016. Matthias Aengenheyster is thankful for support by the German Academic Scholarship Foundation. Henk A. Dijkstra acknowledges support by the Netherlands Earth System Science Centre (NESSC), financially supported by the Ministry of Education, Culture and Science (OCW), Grant no. 024.002.001. Edited by: Christian Franzke Reviewed by: two anonymous referees