Editor's Note: This article originally appeared in SIAM News on November 21, 2023 (https://sinews.siam.org/Details-Page/unprecedented-rates-of-atmospheric-warming-trigger-zombie-fires).

So-called “zombie fires” (also known as overwinter fires) in the peatlands of Alaska, Canada, and Siberia disappear from the Earth’s surface and smolder underground during the winter before “coming back to life” the following spring, ultimately releasing gigatons of carbon into the atmosphere. While most scientists believe that surface fires (such as campfires or wildfires) lead to zombie fires, we have identified an alternative primary cause of these fires: rate-induced tipping to a subsurface hot metastable state in bioactive peat soils, which occurs in response to too-fast rates of atmospheric warming. We employed a soil-carbon ordinary differential equation model and subjected it to realistic global warming scenarios and summer heat waves in order to form our hypothesis. We then combined a special compactification technique with geometric singular perturbation theory to uncover the existence of a hot metastable state in peat soils and ultimately explain rate-induced instabilities from the cold base state to this hot metastable state.

The Concept of R-tipping

Tipping points—critical transitions from a base state to some alternative state—occur in natural and human systems subjected to changing external inputs. Put simply, a tipping point is a sudden, large change in the state of a system in response to a small or slow change in the external input. This concept has garnered significant interest in recent years, due in large part to theorized and observed tipping points in the Earth system that stem from contemporary climate change. A particularly interesting—albeit not fully understood—case is that of rate-induced tipping (R-tipping) [2]. R-tipping from the base state to an alternative state is solely due to the rate of change of the external input (e.g., when the external input varies more quickly than some critical rate) and usually occurs without any loss of stability in the base state. Although scientists and policymakers seldomly address R-tipping, the subject will likely become increasingly prevalent in literature on the contemporary climate and human-dominated Earth, which is characterized by systems that are subject to unprecedented rates of atmospheric warming, weather extremes, and fast-changing environmental and human inputs [8].

Temperature-sensitive Arctic Peatlands

The primary motivation for our study is the combination of three global environmental features: the distribution of carbon-rich bioactive peat soils, the rate of atmospheric warming, and increasing trends in summer heat waves (see Figure 1). An estimated teraton of carbon—more than the current estimate of the total carbon content in the atmosphere—is trapped in temperature-sensitive Arctic peat soils. The Arctic is also the fastest-warming region on the planet, with Arctic amplification causing warming rates that are approximately double the global mean. Could this dangerous combination be responsible for the recent zombie fires in the Arctic [1]? 

Zombie Fires and the Hot Metastable State

We use a modified process-based soil-carbon model [6] to study bioactive peat soils that we subject to realistic changes in weather and climate patterns, like global warming scenarios and summer heat waves. This model describes the time evolution of the soil temperature \(T\) and soil carbon concentration \(C\) in response to changing atmospheric temperature input \(T_a\):

\[\begin{eqnarray} \epsilon \frac{dT}{dt}&=&-\frac{\lambda}{A}(T-T_a(rt))+C \; R_s(T), \tag1 \\ \frac{dC}{dt}&=&\Pi - C \; R_s(T), \tag2 \end{eqnarray}\]

where we introduce the small parameter \(\epsilon = \mu/A\). The model incorporates three soil processes. The first describes the balancing of soil temperature \(T\) and atmospheric temperature \(T_a\) towards a thermal equilibrium—according to Newton’s law of cooling—at a rate that depends on the soil-to-atmosphere heat transfer coefficient \(\lambda\) and the specific heat capacity of the soil \(\mu\). The second—referred to as the gross primary production—describes a linear increase in the soil carbon concentration \(C\) over time at a rate \(\Pi\) due to the generation of carbon from decaying plant litter and other processes. The third and only nonlinear process in the model describes temperature-sensitive microbial activity in the soil in terms of the soil respiration function \(R_s(T)\), which combines the universal \(Q_{10}\) exponential growth due to chemical reactions at low soil temperatures and the exponential decay due to the death of soil microbes at high soil temperatures.

We use realistic values for the soil parameters, including the small parameter \(0<\epsilon \ll 1\) which quantifies the difference between the fast timescale of the soil temperature \(T\) and the slow timescale of the carbon concentration \(C\). The rate parameter \(r\), which quantifies the (third) timescale of the atmospheric temperature input \(T_a(rt)\), is key parameter in our study.

When the atmospheric temperature \(T_a\) is fixed in time—i.e., when \(r=0\)—the ensuing frozen system has just one globally attractive equilibrium that is called the cold base state. Somewhat surprisingly, the dynamics become very interesting when \(T_a\) varies in time, i.e., when \(r>0\). Analysis of the nonautonomous system reveals three remarkable results:

• Bioactive peat soils have a subsurface hot metastable state that is reminiscent of the zombie fires in Arctic peatlands. In this context, metastable means that the state lasts for a long but finite time.
• Realistic climate patterns—including the summer heat wave and global warming scenarios in Figure 2—can trigger R-tipping from the cold base state to the hot metastable state.
• R-tipping to the hot metastable state in \((1)\)-\((2)\) shows qualitative and quantitative agreement with the results of intermediate complexity partial differential equation models [3].

Based on these results, we hypothesize that R-tipping to the hot metastable state, due to too-fast rates of atmospheric warming, is a main cause of zombie fires.

The Mathematics of Quantitative R-tipping

At first glance, equations \((1)\)-\((2)\) appear deceptively simple. But upon closer inspection, analysis of R-tipping in system \((1)\)-\((2)\) turns out to be quite challenging.

Due to the explicit time dependence of \(T_a(rt)\), the system does not contain any equilibria or other compact invariant sets in the phase space (which would typically serve as starting points for traditional stability analysis). A natural approach would involve identifying (nonautonomous) pullback attractors and pullback repellers, then studying their bifurcations. However, the R-tipping in Figure 2 is a quantitative rather than qualitative instability because the system always settles back to the cold base state in the long term, meaning that it cannot be captured by the classical bifurcation theory techniques or bifurcations of pullback attractors [4, 10].

We overcome this obstacle in three steps. First, we restrict inputs \(T_a(rt)\) to those that limit to a constant as \(t \rightarrow \pm \infty\) (see Figures 3a and 3b); our problem thus becomes asymptotically autonomous. As such, we employ a special compactification technique for asymptotically autonomous dynamical systems [11]. We introduce an additional dependent variable \(s=g(rt)\) (which is bounded between \(−1\) and \(1\)), incorporate the autonomous dynamics from \(t=\pm \infty\) \((s=\pm1)\), and work with the ensuing compactified system:

\[\begin{eqnarray} \epsilon \frac{dT}{dt} &=& -\frac{\lambda}{A}(T-T_a^\nu(s))+C \; R_s(T), \tag3 \\ \frac{dC}{dt} &=& \Pi-C \; R_s(T), \tag4 \\ \frac{1}{r} \frac{ds}{dt} &=& \frac{\nu}{2}(1-s^2). \tag5 \end{eqnarray}\]

In this way, we bring equilibrium points from infinity into the phase space of our problem; the compactified system contains equilibrium base states of the autonomous limit systems within the flow-invariant subspaces \(\{s=\pm 1\}\). 

In the second step, we take advantage of multiple timescales and utilize ideas from geometric singular perturbation theory [5, 9]. We identify the hot metastable state with an attractive component of a slow manifold, as well as an elusive quasi-threshold with a repelling/saddle component of the slow manifold. We can thus quantitatively define the R-tipping instability in terms of interactions between slow manifolds and the real, unstable manifold of the base state \(\tilde{e}^-\) from the past limit \(\{s= -1\}\). Most importantly, we reveal special points for \(\epsilon=0\) that do not exist in the original system \((1)\)-\((2)\), where \(0< \epsilon \ll 1\), and refer to these points as singular edge states. In the case of global warming, these special points are different varieties of folded singularities for the reduced problem (also known as canard points) that are found on the fold \(F_1\) of the (gray) critical manifold in Figure 3a. In the case of a summer heat wave, the special point is a new saddle equilibrium \(\tilde{m}_2^-\) for the layer system that is found in the future limit \(\{s=1\}\) within the (gray) layer in Figure 3b.

In the third step, we reduce a quantitative R-tipping instability in the original nonautonomous system \((1)\)-\((2)\) to an “exotic” but qualitative heteroclinic bifurcation in the compactified system \((3)\)-\((5)\) with \(\epsilon=0\). Specifically, we consider heteroclinic orbits that connect the base state \(\tilde{e}^-\) from the past limit \(t = - \infty\) \(\{s= -1\}\) to a singular edge state. For example, in the case of global warming, we unfold a codimension-two heteroclinic connection to a folded saddle-node type-I singularity in order to explain different cases of R-tipping instabilities observed in numerical simulations. More details of this analysis are available in our corresponding paper [7].

Sebastian Wieczorek delivered a minisymposium presentation on this research at the 2023 SIAM Conference on Applications of Dynamical Systems, which took place in Portland, Ore., this May.

References
[1] CBBC Newsround. (2021, May 20). What are ’zombie fires’ and why is the arctic circle on fire? Retrieved from https://www.bbc.co.uk/newsround/57173570.
[2] Francis, M.R. (2021, November 1). It’s not the heat, it’s the rate: Rate-induced tipping’s relation to climate change. SIAM News, 54(9), p. 6. Retrieved from https://sinews.siam.org/Details-Page/its-not-the-heat-its-the-rate.
[3] Khvorostyanov, D.V., Ciais, P., Krinner, G., Zimov, S.A., Corradi, C., & Guggenberger, G. (2008). Vulnerability of permafrost carbon to global warming. Part II: Sensitivity of permafrost carbon stock to global warming. Tellus B: Chem. Phys. Meteorol., 60(2), 265-275.
[4] Kloeden, P.E., & Rasmussen, M. (2011). Nonautonomous dynamical systems. In Mathematical surveys and monographs (Vol. 176). Providence, RI: American Mathematical Society.
[5] Kuehn, C. (2015). Multiple time scale dynamics. In Applied mathematical sciences (Vol. 191). Cham, Switzerland: Springer International Publishing.
[6] Luke, C.M., & Cox, P.M. (2011). Soil carbon and climate change: From the Jenkinson effect to the compost-bomb instability. Eur. J. Soil Sci., 62(1), 5-12.
[7] O’Sullivan, E., Mulchrone, K.F., & Wieczorek, S. (2023). Rate-induced tipping to metastable zombie fires. Proc. R. Soc. A, 479(2275).
[8] Ritchie, P., Alkhayuon, H., Cox, P.M., & Wieczorek, S. (2023). Rate-induced tipping in natural and human systems. Earth Syst. Dynam., 14(3), 669-683.
[9] Wechselberger, M., Mitry, J., & Rinzel, J. (2013). Canard theory and excitability. In P.E. Kloeden & C. Pötzsche (Eds.), Nonautonomous dynamical systems in the life sciences (pp. 89-132). Cham, Switzerland: Springer International Publishing.
[10] Wieczorek, S., Xie, C., & Ashwin, P. (2023). Rate-induced tipping: Thresholds, edge states and connecting orbits. Nonlin., 36(6), 3238.
[11] Wieczorek, S., Xie, C., & Jones, C.K.R.T. (2021). Compactification for asymptotically autonomous dynamical systems: Theory, applications and invariant manifolds. Nonlin., 34(5), 2970.
[12] Xu, J., Morris, P.J., Liu, J., & Holden, J. (2018). PEATMAP: Refining estimates of global peatland distribution based on a meta-analysis. CATENA, 160, 134-140.

	Eoin O'Sullivan holds a Ph.D. in applied mathematics from University College Cork in Ireland, where he researched rate-induced tipping in peat soils. He is currently a quantitative analyst at Nanook Advisors.
	Kieran Mulchrone is a senior lecturer in applied mathematics at University College Cork, where he works on problems in geoscience, mathematical modeling, machine learning, and numerical computing. Mulchrone also collaborates with industry in areas such as health insurance and the application of machine learning to emergency vehicle driver behavior and inventory control.
	Sebastian Wieczorek is a professor (chair) and Head of Applied Mathematics at University College Cork. He specializes in a mathematical theory of instability and develops concepts and techniques to study instabilities in physical problems, including rate-induced tipping points in natural and human systems. Prior to his current position, Wieczorek worked at the University of Exeter and Sandia National Laboratories. He holds a Ph.D. in mathematical physics from the Vrije Universiteit Amsterdam.