By Niklas Manz, Hannah Phillips, Conner Herndon, Abigail E. Ambrose, and Flavio H. Fenton
Nonlinear dynamical behavior giving rise to complex oscillations is found in many biological, chemical, and physical systems with purely temporal dynamics—such as the “chemical clock” of the Briggs-Rauscher reaction—or spatial dynamics, as in the Turing patterns on animals’ skin and fur. However, some of the most fascinating oscillatory patterns occur in spatiotemporal nonlinear systems. For example, one can mathematically model moving fronts from steady-state changes in chemical precipitations via a dynamical system with one unstable and two stable fixed points between the phase space’s two basins of attraction. Researchers can describe propagating pulses—such as the human waves in a stadium, electrical depolarization waves on the heart, or fire fronts—as excitable systems coupled in space, with one stable fixed point in phase space. These systems experience a threshold of excitation and can only be excited after returning to the fix point, thus exhibiting an unexcitable refractory period. One can model such behavior with a reaction-diffusion (RD) system, described by at least two coupled nonlinear differential equations in space that balance a reaction-type mechanism with a diffusion-like transport process. Ilya Prigogine coined the term “dissipative structures” and received the Nobel Prize in Chemistry for his pioneering work on thermodynamic systems far from equilibrium.
Fire is a good example—with great visual value—to exemplify an excitable system. For example, one can consider an oil lamp with no fire to be in rest state (a stable fixed point of the system); this will not change unless perturbed with a flame. The lamp then burns (activation) until all of the oil in the containers is gone. Or—if the oil is very viscus and takes time to diffuse up to the lamp’s tip—the lamp will burn until it finishes the available oil from the wick (see Figure 1). Then a refractory period ensures, a time when the wick cannot burn again until enough oil is re-absorbed up the wick.
Figure 1. Oil candle as an excitable system where no fire is considered at rest. Once ignited, it stays in an excited state until the oil in the wick is consumed, at which point it remains in a refractory period and cannot be reignited unless the oil is replenished.
In space, the dynamics of fire as an excitable system can yield propagating waves with interesting dynamics, as evidenced in the subsequent examples.
Example 1
In 1972, Richard Rothermel derived an equation for the speed of fire propagation along a slope using large wooden tripods on an incline, and determined a \(\tan^2\)-dependence of the fire speed up the slope angle \(q\) [1]. In 2015, Christian Punckt, Pablo S. Bodega, Prabha Kaira, and Harm H. Rotermund developed a model to examine forest fire in a laboratory setting, and published experimental and computational results for planar match stick arrays [2]. We combined the match stick array with the sloped condition and created diamond-like match stick patterns on three-dimensional-printed models with \(3.3\) millimeter (mm) holes, which kept a constant match distance of \(5.0\) mm along the horizontal (\(\Delta x=\textrm{const.}\)). We are using the larger \((l = (58±1)\) mm\()\) Diamond Strike on Box matches (see Figure 2a).
Figure 2. 2a. Example of two-dimensional match grids with a constant distance along the horizontal, arranged on a base that has an incline from 0 to 45 degrees. 2b. Setup of the array and camera. 2c. Example of fire propagating on the array.
After initiating a fire front and determining its velocity using the
Tracker Video Analysis and Modeling Tool (see Figure 2b), we confirmed the expected speed-slope dependence of fire fronts propagating up or down the hill, with a cutoff slope value above which no fire front can exist. Rothermel’s \(v = \tan^2(\theta)\)-relationship was confirmed if fitting for negative and positive slopes separately. Combining all propagation speeds from \(-40°\) (downhill) to \(+40°\) (uphill) was best fit by an exponential function of \(v=(13.0±0.8)+(-1.4±0.9)^{(0.05±0.01)\theta}\) in mm/s. Keeping the distance between the matchstick heads constant in vertical direction (\(\Delta z = \textrm{const.}\)) or along the slope (\(\Delta r=\textrm{const.}\)) significantly changed the propagation dynamics. We found a quadratic curve fit \(v=(9±1)+(0.01±0.04)\theta\)\(+(0.003±0.002)\theta^2\) for the \(\Delta z\)-models, and two separate fit functions for \(\theta<0\) \((v=(2.9±0.8)\tan^2θ+(6.3±0.7))\) and \(\theta>0\) \((v=(2.8±0.6)\tan^2\theta+(8.1±0.4))\). We will need to obtain more experimental data in order to draw final conclusions.
We also discovered a general decrease in fire propagation speed after the company switched from the red match heads to the “greenlight” matches, which is currently under further investigation in a planar system. We plan to extend the experimental investigation of match-type propagation speed dependence to Diamond Strike Anywhere and Ohio Blue Tip Strike on Box matches, and study the differences using a cellular automaton model and a continuous RD model with non-isotropic conductivity.
Example 2
Scientists have devoted extensive research to the discovery of simple media that display continuous propagating waves and chaotic properties. A candle wick that draws flammable oil from a reservoir is one type of simple system. By varying oil viscosity and wick material, the system exhibits refractoriness and nonlinear restitution dynamics under constant periodic reignition. When arranged in a row, these candles can ignite their neighbors and create propagating fire waves that beautifully display complex spatial dynamics.
Figure 3. A one-dimensional square with fire everywhere at once (too excitable).
In 1913, George Ralph Mines published two articles describing electrical excitation waves as a source for heartbeat [3-4], followed by an article in 1914 that linked abnormal excitation waves to tachycardia and fibrillation [5]. In 2016, a special issue in the
Journal of Physiology commemorated Mines’ seminal work on cardiac nonlinear dynamics.
We used fire-retardant canvas, aluminum tracks, and a viscous oil mixture to create a one-dimensional oil-candle ring to visualize some of Mines’ described electrical excitation wave behaviors. The square aluminum tracks (side length of \(15\) cm) pressed the flat canvas (height of \(30\) mm) to form a square-ring that exposed five mm of the canvas. We placed the ring in a baking sheet soaking in an oil mixture (comprised of Fluka mineral oil (light) and Paraffin oil of different viscosities) at the bottom of an aluminum pan. If the wick in the oil bath is too short or the oil is too thick, the system is below the excitation threshold to start a fire and cannot produce a wave. In the opposite case of too much oil or oil with a density that is too low, the oil diffusion into the one-dimensional wick occurs too quickly and the whole system ignites and burns along the circumference — as desired in normal candle systems (see Figure 3). With the right conditions, the system is above the threshold and a short segment of the canvas ignites, creating two fire fronts that moving in opposite directions. If one direction is extinguished, a single flame moves around the one-dimensional candle ring (see the upper-left corner of the path in Figure 4).
Figure 4. A one-dimensional square with one propagating flame.
Due to the slow diffusion of oil into the candle wick after the flame has passed, the same location can be excited again when the flame reaches the point after one “rotation.” We have therefore reproduced experiments of “reentrant” waves in cardiac tissue using a fire model. Experimentally, we obtained a maximum of three to four fire front revolutions before the wick burned down too much to create a flame.
As an extension, we will employ the experimental one-dimensional fire ring to create a fire spiral, as shown in Jan Totz’s (Technical University Berlin) simulation in StarCraft II on YouTube. Using a “wick grid” will create a rotating fire spiral: another table-top analogue to wave phenomena found on the heart muscle and many other biological, chemical, and physical systems.
Niklas Manz presented this work during a minisymposium at the 2019 SIAM Conference on Applications of Dynamical Systems, which took place last month in Snowbird, Utah.
Acknowledgments: This work was supported by the National Science Foundation through grant DMR-1560093 and CMMI-1762553, and by the College of Wooster’s Sophomore Research Program.
References
[1] Rothermel, R.C. (1972). A mathematical model for predicting fire spread in wildland fuels (USDA Forest Service Research Paper INT-115). Ogden, Utah: Intermountain Forest and Range Experiment Station.
[2] Punckt, C., Bodega, P.S., Kaira, P., & Rotermund, H.H. (2015). Wildfires in the Lab: Simple Experiment and Models for the Exploration of Excitable Dynamics. J. Chem. Educ., 92(8), 1330-1337.
[3] Mines, G.R. (1913). On functional analysis by the action of electrolytes. J. Physiol., 46(3), 188-235.
[4] Mines, G.R. (1913). On dynamic equilibrium in the heart. J. Physiol., 46(4-5), 349-383.
[5] Mines, G.R. (1914). On circulating excitations in heart muscles and their possible relation to tachycardia and fibrillation. Trans R. Soc. Can. (Series III, Section IV), 8, 43-52.
Niklas Manz is an assistant professor in the Department of Physics at the College of Wooster. His research is focused on experimental reaction-diffusion systems. Hannah Phillips is an undergraduate student and Conner Herndon is a graduate student at Georgia Institute of Technology. Abigail E. Ambrose is an undergraduate student majoring in physics at the College of Wooster. Flavio H. Fenton is a professor in the School of Physics and Astronomy at Georgia Institute of Technology. His research interests are complex systems and excitable media, especially experimental physiology.