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Analytical and numerical insights into wildfire dynamics: Exploring the advection-diffusion-reaction model

Cordula Reisch, Adrián Navas-Montilla, Ilhan Özgen-Xian

Abstract

Understanding the dynamics of wildfire is crucial for developing management and intervention strategies. Mathematical and computational models can be used to improve our understanding of wildfire processes and dynamics. This paper presents a systematic study of a widely used advection-diffusion-reaction wildfire model with non-linear coupling. The importance of single mechanisms is discovered by analysing hierarchical sub-models. Numerical simulations provide further insight into the dynamics. As a result, the influence of wind and model parameters such as the bulk density or the heating value on the wildfire propagation speed and the remaining biomass after the burn are assessed. Linearisation techniques for a reduced model provide surprisingly good estimates for the propagation speed in the full model.

Analytical and numerical insights into wildfire dynamics: Exploring the advection-diffusion-reaction model

Abstract

Understanding the dynamics of wildfire is crucial for developing management and intervention strategies. Mathematical and computational models can be used to improve our understanding of wildfire processes and dynamics. This paper presents a systematic study of a widely used advection-diffusion-reaction wildfire model with non-linear coupling. The importance of single mechanisms is discovered by analysing hierarchical sub-models. Numerical simulations provide further insight into the dynamics. As a result, the influence of wind and model parameters such as the bulk density or the heating value on the wildfire propagation speed and the remaining biomass after the burn are assessed. Linearisation techniques for a reduced model provide surprisingly good estimates for the propagation speed in the full model.
Paper Structure (22 sections, 4 theorems, 49 equations, 24 figures, 7 tables)

This paper contains 22 sections, 4 theorems, 49 equations, 24 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The wildfire propagation model
  3. Advection--diffusion--reaction equation-based wildfire model
  4. Numerical solver
  5. Analysis of hierarchical sub-models
  6. Dynamical system analysis for the spatially lumped sub-model
  7. Combustion-free sub-models
  8. Energy dissipation in a combustion-free reaction--diffusion sub-model
  9. Combustion-free advection--diffusion--reaction sub-model
  10. Advection-free sub-models
  11. Travelling wave speed
  12. Terminal biomass
  13. Summary of the analysis of the hierarchical sub-models
  14. Insights from the full model
  15. One-dimensional wildfire simulation
  16. ...and 7 more sections

Key Result

Theorem 1

The initial value problem in Eq. eq:modelODE has a unique solution for $(T(0), Y(0)) \in [T_\mathrm{pc}, T_\mathrm{max}] \times [0,1]$ and $t\in [0, \tau]$ where $\tau$ is the maximal time with $T(\tau)\geq T_\mathrm{pc}$. $T_\mathrm{max}$ is a finite maximum value for the temperature.

Figures (24)

  • Figure 1: Representation of the two-dimensional spatial domain $\Omega$ and heat fluxes relevant to wildfire spreading: the conductive flux $\dot{{Q}}_{\mathrm{cond}}$, the convection heat flux $\dot{{Q}}_{\mathrm{conv}}$, radiation heat flux $\dot{{Q}}_{\mathrm{rad}}$, and the reaction heat flux $\dot{{Q}}_{\mathrm{reac}}$. The convective flux is driven by the wind velocity $\mathbf{v}$. All heat fluxes can be defined for an arbitrary control volume $V \subset \Omega$. Colours inside the domain are meant to represent the temperature distribution.
  • Figure 2: Phase space portrait of the system, showing the heating (red) and cooling (blue) regions of the trajectories, separated by the dashed gray line. Trajectories of the solutions for the initial conditions $T_0=470$ K and $Y_0=\left\{ 0.2, 0.4, 0.6, 0.8, 1.0 \right\}$ are plotted with black continuous lines.
  • Figure 3: Evolution in time of the temperature (left) and biomass concentration (right) for the initial conditions $T_0=470$ K and $Y_0=\left\{ 0.2, 0.4, 0.6, 0.8, 1.0 \right\}$.
  • Figure 4: Terminal biomass $Y^{\star}$ as a function of the initial temperature and biomass.
  • Figure 5: Sensitivity analysis of the tipping line in Eq. \ref{['eq:tipping']}. Left: Change of the factor $m$ in $m\rho_0 H A$. Right: Change of the activation temperature $T_\mathrm{ac}$. Note that the range of variation is larger than the realistic parameter ranges in Table \ref{['table:general-parameters']} for highlighting the changes.
  • ...and 19 more figures

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Lemma 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Remark 5
  • Remark 6