Wildfires in the Lab: Simple Experiment and Models for the

Christian Punckt†, Pablo S. Bodega‡, Prabha Kaira‡, and Harm H. Rotermund§. † Department of Chemical & Biological Engineering, Princeton Univ...
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Wildfires in the Lab: Simple Experiment and Models for the Exploration of Excitable Dynamics Christian Punckt,†,∥ Pablo S. Bodega,‡ Prabha Kaira,‡ and Harm H. Rotermund*,§ †

Department of Chemical & Biological Engineering, Princeton University, Princeton, New Jersey 08544, United States Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany § Department of Physics and Atmospheric Science, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada ‡

S Supporting Information *

ABSTRACT: Wildfires lead to the loss of life and property in many parts of the world. Understanding their dangers and, more particularly, the underlying dynamics which lead to fires of catastrophic scale contributes to better awareness as well as prevention and firefighting capabilities within the affected areas. In order to enable a basic understanding of the mechanisms of fire propagation for high school and college students, we present here simple model experiments using match sticks that demonstrate the dangers of spatial coupling between individual reactive (combustible) elements leading to rapid spreading of the combustion reaction. Supplemented with easy-to-grasp numerical simulations, our work illustrates the dynamics of wildfires in impressive but controllable ways and puts wildfire propagation into the fascinating framework of excitable active media which have many equivalents in chemical, ecological, biological or societal systems. KEYWORDS: Upper-division Undergraduate, Graduate Education/Research, Demonstration, Interdisciplinary, Physical Chemistry, Public Understanding/Outreach, Safety/Hazards, Analogies

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INTRODUCTION Wildfires, sometimes called brushfires, bushfires, or forest fires, depending on the type of vegetation being burned, have occurred on earth on a periodic base for as long as vegetation has existed.1 Though destructive on the short-term, these fires are essential for soil fertilization and “clean-up” of the underbrush2 and play a role in the evolution of the animal world3,4 and carbon sequestration.5,6 Their occurrence and spreading is highly irregular on a local scale, depending on vegetation, wind, temperature, moisture, and so forth.7 On a global scale, however, wildfires follow an annual activity pattern and occur in the tropical areas of each continent.8 To illustrate the vast areas of those fires and their periodic nature, we compiled data from fires in Africa from an animation by The Visible Earth Project of NASA showing the distribution of fires as seen by satellites from December 1992 to November 1993 (Figure 1). In December of 1992, a broad continuous stripe in central Africa was on fire. An activity minimum was observed in April of 1993, followed by a large burst in activity in the equatorial region that propagated from west to east, eventually reaching Madagascar. In November of 1993, the cycle begins again starting with fires south of the Sahara. The vast majority of these fires occur naturally and are caused by lightning. Under worst-case conditions, such as dry period and high winds, wildfires can reach average propagation rates on the order of meters per second;9,10 In the rare case of the emergence of a firestorm, local propagation speeds of up to 250 km/h (“fire tornadoes”) have been reported, e.g. in the disastrous Peshtigo Fire of 1871, whichwith estimated 1200 to 2400 casualtiesis among the 10 worst natural disasters in North America in terms of lives lost.11 Since, depending on the circumstances, the loss of property and even lives can be very © XXXX American Chemical Society and Division of Chemical Education, Inc.

Figure 1. Satellite images of wildfire distribution on the African continent from December 1992 to November 1993. Images taken from link for Africa fires from NASA, The Visible Earth is part of the EOS Project Science Office located at NASA Goddard Space Flight Center. http://eoimages.gsfc.nasa.gov/ve//8876/a000387.mpg.

high, wildfires are in the focus of intense experimental and theoretical research efforts aiming at understanding their dynamics, impact on wildlife and climate, and their control.2−7,12−16 Such studies include the initiation of controlled wildfires, and the characterization of fire properties on such large scale has been conducted in the past by various research organizations and government agencies. One of the most thoroughly studied fires was set ablaze by the International Crown Fire Modeling

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encourage the reader to first study the videos of our experiments to assess the very real dangers of performing experiments described in this work.

Experiment at a site in the Northwest Territories, Canada, near Fort Providence.17 A snapshot of it has been taken by the United States Department of Agriculture’s (USDA) and is depicted in Figure 2. A whole array of measuring devices in

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EXPERIMENTAL SETUP WARNING: Do not underestimate the fire intensity when working with large numbers of match sticks! The videos available as Supporting Information online can serve as a substitute for actually performing the experiments described here and give a first impression regarding the necessary safety measures. A photo of our experimental setup is shown in Figure 3. The setup consists of a 5 mm thick aluminum plate of dimensions

Figure 2. Photograph taken during the Northwest Crown Fire Experiment. Arrays of measurement devices within and nearby the forest allowed for an unprecedented level of fire characterization. Furthermore, during the experiment the effectiveness of fire protection equipment was tested under realistic conditions. (Source: Bunk, S. World on Fire. PLoS Biol. 2004, 2, e54, DOI: 10.1371/journal.pbio.0020054.).

close vicinity to the fire can be seen. The aluminum-covered tents to the right are firefighter shelters whose performance is measured as well. A quite informative video about the Northwest Crown Fire Experiment can be found online.18 Despite the abundant information in the scientific literature as well as in news media about the dangers of wildfires, we contend that it is difficult to grasp the violence of a propagating fire on a large scale without personally experiencing it. On the other hand, we certainly do not wish or recommend that any reader of this article should put themselves into a potentially life-threatening situation involving a real wildfire. We have therefore developed a match stick-based laboratory-scale model system that is capable of illustrating the mechanisms that lead to the rapid and catastrophic spreading of large wildland fires. Similar systems have been used in the past to study the impact of wind on fire propagation,9 however, those were not designed to reflect the rapid propagation of fire or the emergence of propagation patterns. When used with appropriate care, our setup is suitable for effective and impressive demonstrations of wildfire dynamics in a high school or college laboratory environment. In the first part of this article, we will use our system to show fire propagation modes and simple principles of the underlying so-called excitable system dynamics. In the second part, we will present approaches for the numerical modeling of wildfire propagation and present our topic in the general framework of excitable dynamical systems. These modeling approaches are kept simple enough to be implemented in easy-to-use programming environments by senior undergraduate, Master’s or first-year Ph.D. students. However, with proper guidance also high school students may appreciate some of the aspects presented here. We give examples of our simulation codes in the form of MATLAB scripts in the Supporting Information of this article. Further, two video recordings of our experiments are available for download as Supporting Information. We

Figure 3. Experimental setup (“match forest”) used for demonstration of wildfire propagation.

250 by 190 mm with a 47 × 34 array of 2 mm diameter holes at a distance of 5 mm (center to center) fabricated by our institute’s machine shop. Aluminum was chosen for both its machinability and its heat reflection properties leading to improved fire propagation and protection of the surface on which the plate was situated. At the four corners of the plate, screws serve as about 15 mm tall posts allowing air to access the plate from below and preventing possible heat damage of the surface below. The holes were outfitted with regular household safety matches as seen in Figure 3. Depending on the desired fire properties, we placed matches in 50% (high density) or 25% (low density) of the holes in a regular pattern. A match density of 100% produced undesirably rapid fire propagation and a short but very intense fire. The experiments were recorded with a hand-held digital video camera and subsequently transferred to a computer for further processing and single frame extraction. Experiments were conducted on a table with nonflammable surface in an open laboratory space with a ceiling height of approximately 4.5 m. Fire extinguishing equipment was at hand and all participants maintained a safe distance of at least 1 m to the experimental setup after a fire was initiated. Great caution should be taken regarding a safe laboratory environment (absence of flammable materials etc.) since at high match density for a few seconds a flame height in excess of 1 m can be observed. Experiments are initiated by bringing a burning match into contact with one or several matches of the array. B

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comes to a stop after about 15 s, and eventually a small number of matches remain “unharmed”. The ignition of only a single match at the edge of the array, i.e., without the addition of extra matches, results in the burning of a handful of matches in direct vicinity of the ignition site but no propagation is observed (Figure 4c). Figure 5 shows three additional experimental configurations and the resulting fire propagation. If a single match in the

Simulations were performed using MATLAB. Details of the simulation algorithms are presented in the Results Section, and the MATLAB scripts used for numerical simulation are included in the Supporting Information of this article.

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RESULTS The ignition of a single match stick is probably an event that every reader has experienced and which we therefore do not document with photos. After striking the match on a suitable surface or bringing it close to a flame, the head bursts into a bright white flame for about a second, after which activity drops and the splint burns down slowly. The rapidity of the initial combustion event is due to the special composition of the match heads: Potassium chlorate oxidizer (which releases oxygen while decomposing) and extra fuel (sulfur, starch and glue) cause a strong initial fire as the match ignites, which then spreads to the paraffin-impregnated splint.19 Although burning a single match is not a particularly earth-shaking event, assembling a large number of matches in an array and lighting one of those matches leads to a completely different picture: Figure 4 depicts snapshots from three different experiments conducted at two different match densities. At high match

Figure 5. Fire propagation in laboratory system: (a) Subcritical and (b) supercritical excitation in the middle of the “match forest” (see Video 2 in the Supporting Information). (c) Transition of fire from a high excitability (high match density) to a low excitability (low match density) region.

center of the low match density array is ignited (Figure 5a) no fire propagation occurs, as already seen in Figure 4c. However, addition of only one extra match is sufficient to cause fire propagation as seen in Figure 4b. In Figure 5c, we show a scenario where the upper half of the plate is outfitted with a high density of matches, while the lower half exhibits low density. The fire is initiated at the top right corner and propagates through the upper half as seen in Figure 4a. Interestingly, subsequent propagation through the low density area is now significantly more rapid and regular compared to the cases shown in Figures 4b and 5b. From these observations, we can deduce a few key properties of our system. (a) At high match density, a small excitation of the system (a single burning match) can cause a large event (whole array on fire). (b) At low match density, a single burning match does not give rise to any large event. Only strong excitation through 5 burning matches in close vicinity to one another causes propagation. (c) At low match density, propagation proceeds more slowly and more irregularly than at high match density. Such properties are typical for so-called autowaves in active media,20−22 examples of which are falling dominoes, nerve excitations,23,24 certain chemical systems,25−29 spread of infectious diseases,30 and others (In the Supporting Information, we briefly explain the analogies between these examples and a wildfire). Active media can generally show three classes of dynamics: (i) oscillatory, (ii) bistable, or (iii) excitable behavior22 of which excitable dynamics describe the effects observed here.20 If we consider an excitable system to be composed of individual elements (e.g., the trees of a forest) it is

Figure 4. Fire propagation in laboratory system: (a) Densely spaced matches sustain the propagation of a fast-moving fire front (see Video 1 in the Supporting Information). (b) More sparsely placed matches result in a more irregular spreading of the fire. Note the extra matches placed at the boundary which are used for ignition. (c) With no extra match at the boundary, the excitation is subcritical, and the fire does not propagate.

density (Figure 4a) the ignition of a single match at the corner of the array results in the rapid propagation of fire across the complete plate within less than 8 s. A quite regular circular fire front is seen burning bright white, followed by an orange-red fire caused by the slow burning of the paraffin-impregnated splints. All matches catch fire and burn down. A surprisingly strong fire arises for a short time (few seconds) with flames more than 1 m tall, illustrating the necessity of proper precautions. At low match density (Figure 4b), the addition of 4 extra matches is necessary to cause fire propagation from the edge of the array. An irregular propagation of the fire is seen, in strong contrast to the observations at high match density: Flames spread seemingly randomly across the match array, propagation C

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instructive to first describe the individual dynamics of one such single element which will exhibit the following characteristics (in parentheses, the corresponding situation will be described for a single tree): It possesses a stable rest state (intact tree), but a sufficiently large perturbation of the rest state (e.g., lightning or uncontrolled camp fire) exceeding a certain threshold value can cause the transition of the system into the “active” state (tree on fire). The active state decays, and a period is entered in which the system is not excitable, called the refractory period (burnt treeunable to sustain another fire). After a certain amount of time, the excitable system returns into its original rest state (A new tree may grow, or the burnt tree may grow new shoots). The perturbation necessary to initiate the active state, called the perturbation threshold, is a measure for the excitability of the system. In case of a small threshold, i.e. high excitability (dry leaves in the tree and/or on the ground), a small perturbation (burning cigarette) can trigger the active state (fire). Conversely, a high threshold leading to low excitability (rainy conditions, low density of combustible material) can effectively prevent the system from entering the active state. These excitable dynamics are also mimicked by our match stick setup, with the exception that the refractory period is ended by the conductor of the experiment replacing the burnt match sticks. If our system were to consist of only a single match stick or tree, the dynamics of the system would be only dependent on time and could be described by a set of coupled, ordinary nonlinear differential equations. Such a model only considers the temporal self-organization of a system and thus covers spatially homogeneous conditions.22 However, our experimental setup as well as real wildfires show us that besides the temporal component of the observed dynamics, there is also a spatial pattern: The activity propagates in the form of excitation waves. Propagation of the excited state in a spatially extended system consisting of a large number of matches (each exhibiting excitable dynamics) becomes possible if the individual elements are coupled. In our case, this coupling is provided predominantly by heat radiation (enhanced through reflection at the aluminum plate) as well as convection of hot gases that both can cause an unburnt match in the vicinity of a burning one to ignite. In other chemical systems, the coupling may be strictly by diffusion of one or more of the reactants, such as in the famous Belousov−Zhabotinskyi reaction29,31 or the catalytic oxidation of carbon monoxide on Pt single crystal surfaces,32,33 although at high reaction rates heat propagation might be involved in the spatiotemporal dynamics of such systems as well.34,35 In this context, we can describe the observations presented in Figures 4 and 5 in terms of the dynamics of a spatially extended excitable system consisting of individual elements which are coupled: The setting in Figure 4a represents a highly excitable system with strong coupling as the match sticks are close to each other. Consequently, a small perturbation (ignition of one match in the corner of the array) leads to a strong, quickly propagating excitation. In the settings presented in Figure 4b−c and 5a−b, the excitability is decreased (as evidenced by the higher excitation threshold necessitating 5 extra matches at the edge of the array and 1 extra match in the center) and also coupling between the matches is reduced due to their larger separation (slower, more irregular propagation). In these latter cases, excitation of the system with only one single match does

not cause the propagation of the excited state; excitation with a single burning match is subcritical. While the direct experimental exploration of fire propagation is certainly exciting and can be expected to leave a lasting impression on any spectator, it is informative to attempt a mathematical description of excitable dynamics in order to study the impact of certain system parameters in an isolated fashion. Excitable dynamics are straightforward to simulate numerically on the computer, and we will in the following present increasingly complex approaches starting with a simple binary model. A flowchart describing the operation of a rudimentary binary model of fire propagation is shown in Figure 6. The system is

Figure 6. Simple flowchart for the simulation of fire propagation in a binary model.

thought to consist of a 2-dimensional array of numeric elements representing individual matches (trees). Initially, each element is set to zero, which represents the stable initial state. Individual elements within the array are then set to 1 (excited state). Such events could be regarded as an experimenter lighting a match or lightning striking a tree. The simulation then proceeds through a loop, which runs continuously until a stop criterion is met. In the shown example, during each iteration of the loop all elements in the array that have a value of zero and that have N or more elements with a value of 1 in their 3 × 3 neighborhood are set to a value of 1 until none such elements exist anymore. The propagation of activity patterns in such a system is illustrated in Figure 7 for different values of N. It can be seen

Figure 7. Simulation results on a 5 × 5 system for different excitation thresholds. The numbers within the fields indicate the time step at which the excitation wave reaches the field. N is the number of neighboring fields in the excited state necessary to cause excitation of a nonexcited field.

that with increasing N, the number of initially active elements necessary for propagation of system activity as well as the time needed to propagate the excitation from the top left to the bottom right corner of the system increases. For N ≥ 4, propagation of the excitation is limited to the special case of an inward traveling concave wave. Though conceptually very simple, the binary model can thus already illustrate many basic D

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Unsurprisingly, the activity spreads faster in the case of lower activation threshold. The result seen in Figure 8c−d somewhat resembles our experimental observations at high match density (Figure 4a) where we obtained the propagation of a nearly spherical wave, and can also be used to illustrate sub- and supercritical excitation. But our model fails to show irregular behaviors such as the one seen in Figure 4b at lower excitability. The reason for this is the absence of system heterogeneity in the model. In the experiment, matches are not perfectly identical, and the separation distance between adjacent matches is not exactly constant. In order to mimic such conditions, we chose to introduce heterogeneity in our model through a random value of Tc for each individual element. The result of such a configuration, where Tc follows Gaussian distributions with different widths σ, is shown in Figure 9. For the narrowest

concepts seen in the experiments as well, such as the excitation threshold (N) and propagation speed as a function of system excitability. A more realistic model describing the dynamics of our experiment is presented in Figure 8 (see Supporting

Figure 8. Time-dependent dynamics of the individual matches and simulation of resulting fire propagation. (a) Structure of neighborhood of individual elements (see text), (b) temporal evolution of individual element upon ignition, (c−d) spatiotemporal pattern propagation at indicated temperature thresholds for system of size 128 × 128 elements. Activity was initiated by simultaneous ignition of a 20 × 20 array in the top left corner.

Information for MATLAB script). We again define our system as a two-dimensional array. However, now each element mimics the behavior of a single match which is solely described by its temperature evolution as a function of time. In order to determine whether a given element (match) should have its burning cycle initiated, we examine in each time step the neighborhood of the respective element (Figure 8a). The impact factor F of the neighboring matches is calculated as follows: 4

Figure 9. Simulated fire propagation as a function of system heterogeneity (randomly varying threshold temperature) for an average critical. (a, c, e) Histogram of temperature thresholds of all elements, (b, d, f) resulting propagation pattern. System size and initiation are chosen as in Figure 8

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F = a ∑ T1, i + b ∑ T2, i i=1

i=1

where the Tj,i represent the temperatures of the jth nearest neighbors of the element and a and b are weighing factors. Throughout this work, we use a = 0.5 and b = 0.25. The burning cycle is initiated if F exceeds a predefined threshold value Tc. In case of ignition, the excitation of each element follows a given temperature profile (“burning cycle”) over time (Figure 8b): The temperature initially quickly rises from zero (rest state) to 1000 °C (burning of the potassium chlorate) followed by a rapid decrease to 600 °C (subsequent burning of the splint) lasting for several seconds before the activity ceases. Examples for the propagation of activity are shown in Figure 8c−d where the system behavior in response to an initiation in the top left corner is compared for two different values of Tc.

distribution (σ = 5%, Figure 9a−b), a regular propagation pattern is seen, while with increasing width, the propagating waves become more irregular until they eventually break apart and follow a more random propagation pattern (σ = 15%, Figure 9e−f). To show the impact of the activation threshold on wave propagation in the presence of heterogeneity, we ran simulations with σ = 15% for three different values of Tc (Figure 10). It can be seen that for the highest degree of excitability (Tc = 600) wave propagation remains rather regular, and all matches in the system burn. With larger values of Tc, that is, at lower excitability, the impact of the system E

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Figure 11. Simulated pattern formation with automatic “replacement” of burnt matches. System size 256 × 256 elements, matches are “replaced” 5 s after their activity ceases, parameters as in Figure 10 with (a) Tc = 895 (b) Tc = 600. While the pattern in (a) evolved naturally after initialization in a 15 × 15 array in the top left (exact same conditions as in MATLAB script, Supporting Information), in (b) the spiral pattern was forced by “artificially” erasing all activity in the right half of the system after the wave had passed the center. The remaining “open end” promptly “curled up” into a spiral.

same time beautifully simple approach to incorporate temporal changes in excitability has been found by the Danish physicist Per Bak and co-workers38 known in a broader context under the term “self-organized criticality”.39−41 In Bak’s original paper on the forest fire model,38 the following rules define the dynamics of his lattice model of size N × N: (i) On an empty site (State 0) a tree can grow with probability P1 in each time step, switching the site to State 1. (ii) Any tree (State 1) can be spontaneously ignited (e.g., through lightning) with a probability P2 in each time step, switching the site to State 2. (iii) Trees on fire (State 2) burn down within one time step (immediate transition from State 2 to State 0 after one time step). (iv) Fire (State 2) spreads to trees (State 1) at its nearest neighbor sites in the next time step. A flow-chart representation of this model is shown in Figure 12, and the MATLAB script is available in the Supporting Information. The spatiotemporal dynamics found in such a seemingly simple model are highly complex and have been discussed exhaustively in the scientific literature.42,43 In the framework of this educational paper, we only compare two distinctly different states of the model which can be obtained through careful choice of its parameters (Figure 13). The system shown in Figure 13a has a low probability of fire initiation P2. As the number of trees increases over time, initially lightning strikes do not cause major fires, as the trees have not formed a sufficiently dense (percolated) network yet. However, around t = 4360 a lightning event causes the majority of the trees to burn down within a short time. The density of trees had exceeded a critical density and the fire could propagate throughout the whole system. The system shown in Figure 13b, on the other hand, has a high probability of lightning strikes. Here, we see that fires occur frequently, resulting in the total number of trees to fluctuate around a value of 800. A dynamic equilibrium is obtained in which no catastrophic events occur since the density of trees can never reach the critical value necessary for propagation of a fire throughout the whole system. To some extent, this result can be applied to real-life wildfires: In short, it needs to be avoided that a forest attain a critical state such as seen in Figure 13a, where fires can quickly spread throughout vast areas. Instead, if possible, it would be desirable to maintain a more steady state such as in the equilibrium system (Figure 13b). This is, of course, hard to

Figure 10. Simulated fire propagation as a function of average excitation threshold (constant relative heterogeneity of 15% as in Figure 9e−f). The last column of frames represents a map of matches burnt at the end of the calculation.

heterogeneity becomes increasingly noticeable as waves break up and a significant number of matches remain intact. This scenario is in accordance with our experimental results, where we see the less regular propagation of fire in the cases of low excitability (small match density). It should be noted, that the consideration of local variations in the propagation of real wild fires is in fact essential for a prediction of their spreading. In wildfires parameters such as vegetation type and density, topographical features, wind patterns, and so forth need to be included.7 Unlike in the experimental system, where matches burn once and cannot be replaced while the experiment is running (for reasons of safety and speed), we can take our simulations one step further, and have elements return to their initial (unburnt) state after a certain time. This way, the system can sustain continuous activity over time and exhibit patterns typical for excitable dynamics such as propagating pulse trains and spiral waves (Figure 11).20 More advanced models of excitable systems, which we cannot discuss here, do not assign a predefined activity pattern to individual elements of the system. Instead, dynamics are described in the form of differential equations describing the underlying system physics or chemistry, one famous example from surface chemistry being the Krischer−Eiswirth−Ertl model for the catalytic oxidation of carbon monoxide.36 A similar model for forest fires would involve partial differential equations describing the spatially coupled combustion dynamics as a function of available “fuel” concentration and temperature, as well as the dynamics of vegetation growth. Due to the differences in time scale between active and refractory period, such wildfire models typically only describe the active part of excitable wildfire dynamics.37 So far, we have looked at systems with given excitability and analyzed the propagation of activity as a function of excitation thresholds and system heterogeneity. An interesting aspect of wildfire propagation that we have not mentioned thus far is the temporal evolution of excitability. A very powerful and at the F

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achieve in reality. However, the use of controlled small fires (“prescribed fires”, initiated under suitable weather conditions), targeted cutting of trees in high-risk areas (limiting propagation), or other methods can prevent catastrophic events of large magnitude to some extent.

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SUMMARY Using arrays of match sticks, we have developed a convenient experimental platform to study the dynamics of propagating wildfires on a laboratory scale. Experiments arewith proper precautionssuitable for high school or college laboratory environments and can serve to understand the violence of wildfire events that may erupt as a result of seemingly small causes such as careless handling of cigarettes. We put experimental observations into a mathematical context of excitable active media and present simple models that can be solved numerically with little effort and help to explore the impact of parameters such as excitability or system disorder on fire propagation dynamics. We hope this work will contribute to wildfire awareness and at the same time spark curiosity regarding the fascinating world of nonlinear dynamics and active media in general.

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ASSOCIATED CONTENT

S Supporting Information *

Examples of active medium, MATLAB script for simulation of match stick fire, MATLAB script for SOC model, and video recordings of the experiements. This material is available via the Internet at http://pubs.acs.org.

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Figure 12. Calculation flowchart for model based on self-organized criticality. The variable A is a two-dimensional array of size N × N representing the system state. P1 and P2 are the probabilities for growth of a new tree and lightning strike in each time step, respectively, and tmax is the maximum number of time steps the simulation is allowed to run.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: (902) 494 2342. Present Address ∥

Currently at the Institute of Nanotechnology, Karlsruhe Institute of Technology, PO Box 3640, 76021 Karlsruhe, Germany. Notes

The authors declare no competing financial interest.

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REFERENCES

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Figure 13. Self-organized criticality: System dynamics in the case of low (a) and high (b) lightning probability. The panels to the right show snapshots of the systems at indicated time. Empty sites are black, sites occupied by a tree gray, and burning sites white.

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DOI: 10.1021/ed500714f J. Chem. Educ. XXXX, XXX, XXX−XXX