SIAM News Blog

Predicting the Unexpected: Early Warning of Impending Climate Tipping Points

By Chris A. Boulton and Joshua E. Buxton

Over the last few decades, tipping points—the thresholds at which small changes in the stress on a system can cause large, nonlinear responses—have garnered attention in climate and ecological sciences. Because devastating global effects can occur when a climate system crosses its tipping point, understanding which systems are approaching the threshold is of paramount importance. Recent research suggests that some climate systems—such as the Greenland and West Antarctic ice sheets—could “tip” if global warming exceeds 1.5 degrees Celsius above pre-industrial levels. Systems such as the Amazon rainforest and the Atlantic Meridional Overturning Circulation (AMOC) could tip within 2 to 4 degrees Celsius of global warming, which might be likely under current conditions [1]. Figure 1 illustrates a number of potential climate tipping points.

In a mathematical setting, tipping points are called bifurcations. They occur when the stability of the system’s occupied state degrades, causing the system to rapidly transition to a new state. Figure 2 envisions the potential landscape of the system as a “ball in a well” analogy; wells represent the stable states and peaks represent the unstable states. When the system faces an external force, the potential well shallows — representing a loss of stability and degradation of the restoring feedbacks that would return the system to equilibrium. In essence, the ball would take longer to return to the bottom of the well. With enough forcing, the occupied state’s stability is completely lost. 

Figure 1. A map of climate systems that could cross their respective tipping points under projected future climate change. Figure courtesy of Globaïa and the Earth Commission.

Given the stochastic nature of real-world systems, such systems will likely reach this tipping point before the actual bifurcation takes place. In fact, the probability of tipping increases as the system approaches the bifurcation. Researchers often assume that the external forcing transpires on a slow time scale, thus allowing them to observe the movement of the system as it approaches the tipping point. However, it is worth noting that this might not always be true in climate systems, which can experience relatively fast anthropogenic forcing; instead, a scenario known as rate-induced tipping may occur. During rate-induced tipping, the system cannot keep up with the movement of the well and thus tips to a new state.

Early warning signals (EWS) that measure resilience—i.e., a system’s ability to recover from perturbations—serve as a way to monitor these systems. In the context of climate, the perturbations consist of short-term weather effects. As a system approaches a tipping point, it begins to respond more sluggishly to perturbations and moves further away from equilibrium (the bottom of the well of the potential landscape in Figure 2a) as the restoring feedbacks continue to weaken. This process is called critical slowing down (CSD). For example, if the Amazon rainforest was approaching a tipping point, we would expect it to be affected more (i.e., have a longer recovery time) in the instance of a drought than if it was more resilient and further away from the tipping point. EWS provide quantitative evidence of this behavior by looking for changes in the statistical properties of a system’s time series.

We can describe the potential landscape of the system in Figure 2 as a function \(U(x)\), such that

\[\frac{dx}{dt}=-U\prime (x(t))+ \eta(t).\]

Figure 2. Visualization of a system that is externally forced towards tipping, along with an early warning signal of the impending tipping point. 2a. The evolution of the system’s potential well (from blue to purple to red) indicates a shallowing of the left-hand state. 2b. This shallowing eventually causes a tipping point in the system. 2c. The scatter plots depict the increasing lag 1 autocorrelation in the colored regions of the time series. 2d. The time series of the autoregressive model \(AR(1)\) is calculated on a moving window. Colored points depict the individual \(AR(1)\) values from each scatter plot. Figure courtesy of Chris Boulton.
Here, \(\eta(t)\) is additive white noise (the aforementioned short-term weather effects). Close to equilibrium (at the bottom of the well), we can approximate the system as a quadratic function: \(U(x)\approx\frac{-\lambda x^2}{2}.\) The system’s leading eigenvalue \(\lambda\) is directly linked to its negative restoring feedbacks, and it approaches \(0\) from below as the system moves towards the tipping point. When we discretize the system into time steps of \(\Delta t\), it becomes an autoregressive process with lag-\(n\) autocorrelation function

\[AR(n)=e^{n\lambda \Delta t}\]

and variance 

\[\textrm{var}(\Delta x)= \frac{\sigma^2}{1-e^{2\lambda \Delta t}} \approx -\frac{\sigma^2}{e^{2 \lambda \Delta t}}.\]

As resilience is lost and \(\lambda\) increases towards \(0\) from below, both \(AR(1)\) and variance are expected to increase. To put it simply, an increase in \(AR(1)\) suggests that as CSD occurs over time, the previous time point becomes more similar to the present one. Variance also increases as time passes because the restoring feedbacks weaken and the system can sample more of the surrounding area. We use these indicators as EWS and measure them on a sliding window across a time series of the system in question. Doing so provides a time series of the EWS indicator, which helps us determine the significance of its increase. A significant rise in this indicator suggests that the system is losing resilience and approaching a tipping point.

The theory of CSD and its indications is sound, and EWS are evident in many small, closely monitored systems — such as those in ecology, with data mainly stemming from field or lab experiments. However, the identification of EWS in climate systems was originally limited to climate models that range from simple box models to complex general circulation models. While these approaches demonstrate the potential of EWS with the hindsight of knowing whether or not a system eventually tips, our main priority is the real world. Data for large-scale, real-world climate systems often come from remote sensing, typically from satellites. The temporal extent of some of these records allows us to use a sliding window in our EWS calculation that is large enough to capture the long-term dynamics of the system to monitor substantial changes in resilience. Recent studies have utilized remotely sensed data to show the potential movement towards tipping in the AMOC [2] and reveal increases in \(AR(1)\) across more than 75 percent of the deep Amazon rainforest [3]. These indicators signify a loss of resilience when the mean state of the system itself remains relatively stable, ultimately highlighting the importance of these higher-order statistics in determining overall system health.

The last several years have also seen an increase in the use of machine learning techniques to learn more about tipping points. For example, researchers have employed deep learning to identify the type of bifurcation that a system is approaching [4] and implemented random forests to understand the drivers that force systems towards tipping points [5]. Advances in this field can complement more traditional EWS and help us determine the severity of threats.

In addition, recent efforts have begun to consider the possibility of positive tipping points: the idea that these nonlinear responses in various social systems could actually aid in our transition to a greener society [6]. However, the necessary data to monitor these systems and test this approach is often much more difficult to acquire at a suitable temporal extent and resolution. The perturbations of these social systems may also differ from natural systems. For instance, we are likely to place a stronger emphasis on specific events that drive social systems rather than short-term “weather-type” effects that impact natural systems. Our focus then shifts to the return time from these perturbations as a measure of the system’s resiliency. Sufficiently early detection of a system’s loss of resiliency could provide an early opportunity to tip the system into a new, more desirable state.

Crossing tipping points in climate systems will likely have dangerous, severe consequences both locally and globally. Nevertheless, the techniques that we outline here may provide advanced warning for systems that are losing resilience and moving towards a threshold — allowing us time to act before it is too late.

[1] Armstrong McKay, D.I., Staal, A., Abrams, J.F., Winkelmann, R., Sakschewski, B., Loriani, S., … Lenton, T.M. (2022). Exceeding 1.5°C global warming could trigger multiple climate tipping points. Science, 377(6611), abn7950. 
[2] Boers, N. (2021). Observation-based early-warning signals for a collapse of the Atlantic Meridional Overturning Circulation. Nat. Clim. Change, 11, 680-688. 
[3] Boulton, C.A., Lenton, T.M., & Boers, N. (2022). Pronounced loss of Amazon rainforest resilience since the early 2000s. Nat. Clim. Change, 12, 271-278. 
[4] Bury, T.M., Sujith, R.I., Pavithran, I., Scheffer, M., Lenton, T.M., Anand, M., & Bauch, C.T. (2021). Deep learning for early warning signals of tipping points. Proc. Natl. Acad. Sci., 118(39), e2106140118. 
[5] Forzieri, G., Dakos, V., McDowell, N.G., Ramdane, A., & Cescatti, A. (2022). Emerging signals of declining forest resilience under climate change. Nature, 608, 534-539.
[6] Otto, I.M., Donges, J.F., Cremades, R., Bhowmik, A., Hewitt, R.J., Lucht, W., … Schellnhuber, H.J. (2020). Social tipping dynamics for stabilizing Earth’s climate by 2050. Proc. Natl. Acad Sci., 117(5), 2354-2365.

Chris A. Boulton is a research fellow at the Global Systems Institute at the University of Exeter, where his research focuses on the detection of movement towards climate tipping points. He is also a keen data scientist who enjoys exploring any interesting dataset that he can get his hands on. 

Joshua E. Buxton is a research associate at the Global Systems Institute at the University of Exeter. His research focuses on resilience change and tipping points across a wide range of systems, from vegetation monitoring with satellites to technological diffusion.
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