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Studying wildfire fronts using advection-diffusion-reaction models

Koondanibha Mitra, Qiyao Peng, Cordula Reisch

TL;DR

The paper addresses wildfire front propagation using advection--diffusion--reaction models that couple temperature $T$ and biomass $Y$, and introduces a persistent-fire variant to capture on/off combustion behavior. By non-dimensionalizing and applying a traveling-wave ansatz in one dimension, the authors reduce the problem to a semi-autonomous ODE system in $(u,v,z)$ with an unknown wave speed $c$ and develop a shooting method to locate traveling waves. Numerical experiments compare the full EF and PF PDEs with a linearized approximation, showing two TWs in the no-wind or mild-wind regime (one in each direction) and a single forward TW under strong wind, with solver agreement at the few-percent level. The work yields a practical framework for rapid prediction of wildfire front speeds under varying wind and parameter conditions and sets the stage for rigorous mathematical proofs and extensions to more complex geometries.

Abstract

In this work, we study the propagation of wildfires using an advection--diffusion--reaction model which also includes convective and radiative heat loss. An existing model is discussed \cite{asensio_2002} and a physically consistent modification of the model is proposed. Using this, the existence of travelling waves (TWs) in the one-dimensional case is investigated. Prior numerical studies reveal the existence of TWs \cite{reisch_2023}. Under the travelling wave ansatz and certain approximation, the model is reduced to a semi-autonomous dynamical system with three unknowns which can be analyzed by a shooting algorithm. It is hypothesized that under mild wind speeds, TWs in both directions exist, and under strong tailwinds only TWs in the direction of wind are possible. The theoretical implications are investigated using both solvers for the PDE models and the shooting algorithm. The results match, and unveil the dependence of the fronts on the parameters consistent with the predictions.

Studying wildfire fronts using advection-diffusion-reaction models

TL;DR

The paper addresses wildfire front propagation using advection--diffusion--reaction models that couple temperature and biomass , and introduces a persistent-fire variant to capture on/off combustion behavior. By non-dimensionalizing and applying a traveling-wave ansatz in one dimension, the authors reduce the problem to a semi-autonomous ODE system in with an unknown wave speed and develop a shooting method to locate traveling waves. Numerical experiments compare the full EF and PF PDEs with a linearized approximation, showing two TWs in the no-wind or mild-wind regime (one in each direction) and a single forward TW under strong wind, with solver agreement at the few-percent level. The work yields a practical framework for rapid prediction of wildfire front speeds under varying wind and parameter conditions and sets the stage for rigorous mathematical proofs and extensions to more complex geometries.

Abstract

In this work, we study the propagation of wildfires using an advection--diffusion--reaction model which also includes convective and radiative heat loss. An existing model is discussed \cite{asensio_2002} and a physically consistent modification of the model is proposed. Using this, the existence of travelling waves (TWs) in the one-dimensional case is investigated. Prior numerical studies reveal the existence of TWs \cite{reisch_2023}. Under the travelling wave ansatz and certain approximation, the model is reduced to a semi-autonomous dynamical system with three unknowns which can be analyzed by a shooting algorithm. It is hypothesized that under mild wind speeds, TWs in both directions exist, and under strong tailwinds only TWs in the direction of wind are possible. The theoretical implications are investigated using both solvers for the PDE models and the shooting algorithm. The results match, and unveil the dependence of the fronts on the parameters consistent with the predictions.
Paper Structure (8 sections, 17 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 8 sections, 17 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Advection--diffusion--reaction models for wildfire
  3. The existing ADR model
  4. Model variations
  5. Non-dimensionalization and effective model
  6. One-dimensional fronts and travelling waves
  7. Numerical results
  8. Discussion and future directions

Figures (4)

  • Figure 1: The exponential function $\exp(-T_{\rm ac}/T)$ and its least-square fitted linear approximation in the interval $300\leq T\leq 1500$. Here $T_{\rm ac}=400$.
  • Figure 2: The evolution of temperature $T$ (left column) and fuel $Y$ (right column) profiles are shown over (part of) the computational domain. The transparency of the profiles decreases as the time increases ($t \in \{0, 40, 120, 280, 400\}$). Here, (a)-(b) represents the no wind ($w=0 \,m/s$) case; (c)-(d) mild wind ($w=0.03m/s$); (e)-(f) strong wind ($w=0.3m/s$); and (g)-(h) no wind case where TWs do not exist (dimensional $h=19.85 kW/(m^3\cdot K)$, dimensionless $h=1.5$).
  • Figure 3: The comparison between profiles of ADR-EF (black solid), ADR-PF (blue dashed), and the approximated (red dashed) models shown at various times ($t\in\{0,40,400\}$).
  • Figure 4: Comparison of temperature profiles between the PDE and the ODE simulations of the approximated ADR-PF model.

Theorems & Definitions (1)

  • Claim 1: Existence of travelling waves for the ADR-PF model