Mathematics of Planet Earth

This past week, the International Centre for Mathematical Sciences (ICMS) hosted a workshop in Edinburgh, United Kingdom. The workshop brought together an international group of mathematicians, statisticians, climate scientists, and ecologists to address the topic of tipping points [1]. In particular, the workshop provided a forum for researchers to discuss current topics related to tipping point phenomena, such as signatures of tipping in conceptual models versus in large-scale or data-driven models, how results concerning the time scale or plausibility of tipping may affect policy decisions, and the merits and pitfalls of different early warning signs of tipping.

Tipping points, a term popularized through the writing of Malcolm Gladwell, have been defined in a broad spectrum of applications, from sociology, to economics, to climatology. The group which met at ICMS consisted of representatives from several of these backgrounds and included members of the Mathematics and Climate Research Network (MCRN), CliMathNet, the Nonlinear Dynamics in Natural Systems (NDNS+) cluster, and the Pacific Institute for Mathematical Sciences (PIMS). These groups are based in the United States, the United Kingdom, the Netherlands, and Canada, respectively.

Throughout the workshop, the researchers discussed a qualitative definition of a tipping point. We began with the idea that a climate tipping point must:

1. indicate a rapid change in a climate system or subsystem;
2. be considered irreversible, or only reversible on a very long timescale; and
3. have a significant potential impact on the earth system.

Mathematically, we distinguish between three types of tipping, namely bifurcation-induced tipping, noise-induced tipping, and rate-induced tipping. That is not to say, however, that tipping can only occur through these three phenomena, and possible other phenomena which induce tipping are currently being investigated. Bifurcation-induced tipping is indicated by the presence of a bifurcation of an attractor as a parameter passes through a critical value. In particular, saddle-node bifurcations are usually thought of as describing tipping points. Noise-induced tipping, which is possible when there are at least two stable states, occurs when a perturbation away from a stable attractor pushes a solution out of that attractor’s basin of attraction and into that of another attractor. Rate-induced tipping can occur when varying a parameter in time too swiftly causes a sudden transition.

The workshop consisted of several days of research presentations, two discussion/breakout sessions, and a poster session. The presentations and posters discussed approaches toward identifying and using early warning signals for sudden regime shifts, examples of how systems exhibiting the different types of tipping have been analyzed, and the relative time scales involved in critical transitions. Non-climate systems that were posed as potential insights into tipping point phenomena included financial crises, ecological regime shifts, and theoretical dynamical systems. The presentations were recorded and will be posted on the ICMS Tipping Points Workshop Website.

During the discussion sessions, small groups deliberated current problems that were of high interest to researchers, discussed the issues related to these topics, and then reported on their discussion or conclusions. Discussion topics included:

• Robust/generic indicators to tipping in complex systems;
• Mechanisms for tipping beyond saddle-nodes (higher than 1-dimension bifurcations);
• Connections between the predictions in conceptual models and real-world systems;
• Estimating the radius of the basin of attraction;
• Nonsmooth phenomena in the climate system;
• Testing for bi-/multistability in complex systems;
• Spatial vs. temporal indicators of tipping;
• Incorporating multiple time series into tipping analysis; and
• The interaction between noise and attraction.

Most of these discussions are still ongoing. However, one point of consensus among those at the workshop was the need to test and develop multiple indicators of early warning signs in climate systems, as well as further explore their implications and potential applications. Additionally, we discussed several paths and tools one may use to navigate between the conceptual, intermediate, and large-scale climate models, but the process must be explored further.

Many workshop participants also emphasized the importance of improving the interface between science and policy. Although it is important to communicate between the different disciplines studying climate, it is equally vital to acknowledge that policy-makers need information that they can implement on a reasonable scale. Informing policymakers about climate assessments in a realistic fashion, which policymakers can then transform into policy decisions, is crucial in working toward decreasing the impact humans are making on the earth system.

References:

[1] Lenton, T.M. et al. Tipping elements in the Earthʼs climate system. Proceedings of the U.S. National Academy of Sciences 105, 1786-1793 (2008).

Kaitlin Hill
Mathematics and Climate Research Network (MCRN)