1 Introduction

Thanks in part to its linkage to the issue of global climate change, we are beginning to gain quite a reliable, albeit incomplete, understanding of the functioning of the Earth's highly complex climate system. Several large‐scale climate‐related changes have already been identified and listed, and their probability of occurrence quantified (Alley et al., 2005; Kriegler et al., 2009). ‘Tipping points’ (TPs) are defined as ‘critical thresholds at which a tiny perturbation can qualitatively alter the state or development of a system’ (National Research Council, 2002; Lenton et al., 2008), i.e. any event able to trigger a gradual (as a trend) or abrupt non‐stationnarity of the global climate system. At the same time, TPs are part of a larger system, and understanding the Earth's climate system as a whole would therefore involve assessing the effects of climate‐process interactions, including the possible cumulative effects of multiple crossings of TPs. Here, we propose simple qualitative Boolean models to handle TP interactions and discuss the possibility of a runaway change in the Earth system. Our goal is to develop a conceptual model complementary to those based on differential equations and system dynamics (Zaliapin and Ghil, 2010).

Several possibly major changes related to global warming forced by anthropogenic increase of greenhouse gases have been identified. They include the reorganization of the Atlantic meridional overturning circulation (CMOC; Rahmstorf and Ganopolski, 1999), the melt of the Greenland ice sheet (MGIS; North, 1984), the disintegration of the west Antarctic ice sheet (DAIS; Huybrechts and De Wolde, 1999), the dieback of the Amazon rainforest (AMAZ; Cox et al., 2000) and the shift to a more persistent El Niño regime (NINO; Collins and CMIP Modelling Groups, 2005) (Figure 1(a)). Additional TPs have been proposed, but their probabilities of occurrence were not reliably estimated (Lenton et al., 2008), mainly due to limited knowledge about them and about their possible interactions with external forcings. New TPs could be identified through physical models of the global climate sensitivity in the future (Zaliapin and Ghil, 2010). In the older works (e.g. Lenton et al., 2008), each of the five TPs quoted above when ‘triggered’ (i.e. crossed) have been found by experts to exhibit a non‐negligible probability (0.16 for medium − +2° to +4 °C – to 0.56 – above +4 °C – for strong global warming) of initiating a runaway degradation of the global Earth system, or, at least, of making such an initiation in the future inevitable, under global warming. Note that the current target of COP21 (i.e. +2 °C from the preindustrial era, that is the RCP 2.6 scenario from Intergovernmental Panel for Climate Change (IPCC)) is in the medium range and thus quotes the possible runaway as a non‐negligible event for the near future. The previous works also asserted that the probability of triggering a TP may increase (or decrease) depending on whether or not a TP in another sub‐system has already been crossed (Kriegler et al., 2009).

Figure 1 Open in figure viewerPowerPoint et al. ( 2009 et al., 2009 The global climate relational graph is shown in its simplified (a) and extended (b) versions. The simplified graph corresponds to the TP interactions (nodes) already identified in Kriegler), with five plus three colours symbolizing their different natures and influences (a, right). Nodes may be absent (empty) or present (full, in grey) in the following representations. Rules may model increasing (+), decreasing (–) or uncertain (±) interactions. The extended graph (b) schematizing the Earth's climate is gathering the nine physical components (nodes) involved into the dominant TPs (arrows, with the same colour typology than for the simplified graph). In addition to modified nodes and arrows, dashed elements of this graph (b) were explicitly added by authors to extend the previous study (Kriegler).

However, such conclusions might be viewed with scepticism as most of the processes and events identified are nonlinear, have so far revealed high‐measured uncertainties and are probably all scale‐dependent (Ghil et al., 1991). Additionally, the selected five TPs are clearly not exhaustive, and their exact roles in the fate of global climate are still unclear. These points will be addressed in the discussion after the analysis of simple Boolean models of TPs interaction based on the available expert elicitation (Kriegler et al., 2009). Hence, our main question is while TPs are often associated with positive feedbacks, leading to possible cumulative effect, could there also exist negative feedbacks (kinds of ‘stabilizing points’) that may mitigate a climate runaway?

Experts identified 12 among 20 possible combinations of preceding and succeeding TPs (Kriegler et al., 2009), highlighting the tangled network of climate process interactions in the Earth's climate system. However, no statistical or mechanistic models have so far confirmed these findings. We still lack integrated models that will allow us to handle such a complex system with our partial understanding of sub‐system processes (Nordhaus and Boyer, 2000), although steps in this direction are being taken (Ogutu et al., 2015). One way to proceed to develop integrated models of climate is to represent climate components by nodes of a graph and their interactions by edges connecting them. There is already substantial work in this track, applying network theory (Tsonis and Roebber, 2004), interaction graphs (Saltzman, 1983; Mysak et al., 1990) and Boolean methods, in particular Boolean delay equations (Ghil and Mullhaupt, 1985; Ghil et al., 2008), to global climate.

In this paper, we propose a simple, integrated model based on qualitative TP interactions represented in a Boolean graph and handling its qualitative dynamics by means of grammatical rules. The innovative part of this model concerns the grammar engine rather than the graph itself. An important property of this model lies in the possibility for TPs (i.e. the graph nodes) to appear (‘On’) or disappear (‘Off’) depending on whether they have been crossed or not, thus mimicking the ‘development’ (the interaction graph growth) of the system under study. Sometimes called the DS2 approach to model dynamical systems carried by dynamical structures (Giavitto and Michel, 2003; Sayama and Laramee, 2009), this formal method, based on graph grammar, has been used particularly in biology, through Boolean networks to model biochemical and genetic interactions in cells and organisms (Kauffman, 1993; Barbier et al., 2006), and in ecology to capture ecosystem dynamics (Campbell et al., 2011; Gaucherel et al., 2012).

The aim of the present study is to model TP interactions to answer specific questions, namely could the global climate run away due to cumulative TP crossings? Or, reversely, do TP interactions dampen the perturbations of the system? And further, what are the most sensitive TPs that should therefore be mostly monitored? As a first guess, more TPs crossed (i.e. the greater the number of TPs ‘On’) may lead to more catastrophic shifts of the global climate. The number of TPs indeed appeared as the simplest available indicator of a generalized climatic impact. As a second hypothesis, following the numerous studies on its impacts (Philander and Federov, 2003; Gaucherel, 2010), we predict that NINO is likely the most sensitive TP in the Earth's system. This point continues to be debated (Kosaka and Xie, 2013), and, if confirmed, such sensitivity would bring independent and interesting insights to the debate. Nevertheless, it is obviously extremely difficult and risky to guess the respective weights of TPs without an integrated model. A simple (i.e. interpretable) model of this complex system would not only provide some insights on this issue but would also help to highlight missing knowledge (i.e. a heuristic model) and to assess the extent of our understanding of the system.