Excitability in ramped systems: the compost-bomb instability

S. Wieczorek () 0 P. Ashwin 0 C. M. Luke 0 P. M. Cox 0 0 Mathematics Research Institute, University of Exeter , Exeter EX4 4QF , UK Articles on similar topics can be found in the following collections applied mathematics (384 articles) biogeochemistry (3 articles) climatology (15 articles) Receive free email alerts when new articles cite this article - sign up in the box at the top right-hand corner of the article or click here - Subject collections Email alerting service Excitability in ramped systems: the compost-bomb instability The paper studies a novel excitability type where a large excitable response appears when a systems parameter is varied gradually, or ramped, above some critical rate. This occurs even though there is a (unique) stable quiescent state for any fixed setting of the ramped parameter. We give a necessary and a sufficient condition for the existence of a critical ramping rate in a general class of slowfast systems with folded slow (critical) manifold. Additionally, we derive an analytical condition for the critical rate by relating the excitability threshold to a canard trajectory through a folded saddle singularity. The general framework is used to explain a potential climate tipping point termed the compost-bomb instabilityan explosive release of soil carbon from peatlands into the atmosphere occurs above some critical rate of global warming even though there is a unique asymptotically stable soil carbon equilibrium for any fixed atmospheric temperature. 1. Introduction An excitable system remains in a quiescent state, if undisturbed, produces a small response to a small stimulus but fires a large transient response when the small stimulus exceeds a certain threshold. The notion of excitability was first introduced in biology and physiology in an attempt to understand the spiking behaviour of neurons (Hodgkin & Huxley 1952; FitzHugh 1955, 1961) and their electronic simulators (Nagumo et al. 1962). Soon after this work, excitability was found in chemical reactions (Zaikin & Zhabotinsky 1970; Ruoff 1982). More recently, there have been a number of theoretical and experimental demonstrations of excitability in liquid crystals (Coullet et al. 1994), optical systems including lasers (Dubbeldam et al. 1999; Yacomotti et al. 1999; Tredicce 2000; Wieczorek et al. 2002; Wnsche et al. 2002; Krauskopf et al. 2003; Goulding et al. 2007) and photonic crystals (Yacomotti et al. 2006). A hallmark of excitability is a genuine or apparent discontinuity in the systems response versus the stimulus strength (FitzHugh 1955; Hoppensteadt & Izhikevich 1997; Doi et al. 1999; Izhikevich 2006). It has become clear that this strongly nonlinear S. Wieczorek et al. phenomenon is mathematically intriguing and relevant to different fields of science. In this paper, we demonstrate a new form of excitability that is relevant to the stability of peatlands under global warming. From the original notion of excitability, which is purely phenomenological, one can conclude that an excitable system has to have the following properties: (P1) A quiescent state. (P2) An excitability threshold above which the system initially evolves away from the quiescent state giving rise to an excitable response. (P3) A return mechanism that specifies the type of excitable response and, when the stimulus is off, brings the system back to the quiescent state so it can be excited again. A typical candidate for a stimulus that perturbs the system from the quiescent state to above the excitability threshold is a fast and large enough change in one of the system parameters, a short impulse, for example (Wnsche et al. 2002). Other possibilities include stochastic fluctuations in the system variable(s) (Lindner et al. 2004). This work studies excitability owing to parameter rampinga steady, slow and monotonic change in one of the system parameters referred to as the ramped parameter. In the problem considered here, a (unique) quiescent state exists for any fixed setting of the ramped parameter. However, a very large excitable response appears when the parameter is ramped sufficiently fast from one setting to another. (a) Summary of main results In 2, we define the excitability phenomena in rigorous terms of dynamical systems, give a brief survey on classification of excitability, and describe two different types of excitability, namely type A and type B. This work focuses on an in-depth analysis of type B excitable models, and 3 describes excitability properties in these models with state jumps. In 4, we study a general class of type B excitable models with parameter ramping, of which the compost-bomb problem is a representative. We show that if a suitable parameter is ramped in a slowfast system with an asymptotically stable equilibrium and locally folded critical (slow) manifold, then there may be a critical value of the ramping rate above which an excitable response appears. This result is obtained in two steps using concepts from singular perturbation theory. In the first step, we show that a necessary and sufficient condition for the existence of a critical ramping rate is a folded saddle singularity in the corresponding desingularized system. In the second step, we derive an analytical condition for calculating the critical ramping rate. The general analysis in 4 is motivated by a need to understand the response of peatlands to global warming (or atmospheric temperature ramping), which represents a potential tipping point in the response of the climate system to anthropogenic forcing (Lenton et al. 2008). It is estimated that peatland soils contain 4001000 billion tonnes of carbon, which is of the same order of magnitude as the carbon content of the atmosphere. Peat carbon is increased by plant litter and reduced by microbial decomposition in the soil. Peat decomposition is expected to accelerate under global warming, leading to concerns that carbon Excitability in ramped systems could be released to the atmosphere, accelerating the rate of carbon dioxide increase and providing a positive feedback to global warming (Cox et al. 2000; Khvorostyanov et al. 2008b). There is also strong empirical evidence of peat instability in response to warm and dry climate anomalies, such as the peatland fires in Russia in summer 2010. Although many global climate-carbon cycle models predict loss of soil carbon under global warming (Friedlingstein et al. 2006), few properly deal with organic soils such as peats, and all ignore the effects of biochemical heat release associated with microbial decomposition (Khvorostyanov et al. 2008a). In their recent work, Luke & Cox (in press) define the peatland soil system in terms of its vertically integrated soil carbon content, C (kg m2) and soil temperature, T (C). Soil carbon is increased by litter fall from plants, P = 1.055 (kg m2 yr1), and reduced by microbial decomposition, which depends on the store of carbon, C , and also the temperature sensitivity of (...truncated)