## 2 Previous Models
### Second Model: Rosseland Approximation for Local Radiation  
The second model is again a one-phase 2D model, but it addresses radiation as the dominant thermal transfer mechanism, without forgetting convection, which represents the effect of wind and slope. In [23], radiation was incorporated in the fire spread model by using a local radiation term using the Stefan-Boltzmann law and approximating the fourth power of the temperature by a Taylor expansion [24]. This is known as the Rosseland radiation model; it is valid when the medium is optically thick. It is a diffusion approximation for the radiation, and it introduces a nonlinearity in the diffusive term of the energy equation. Another modification compared to the first model is the simplification of the vertical heat loss term or natural convection. In this second model, the one-phase simplification leads to introducing a logic expression depending on the phase change temperature, which we can assimilate with the pyrolysis temperature. The PDEs in this model are written as follows:  
\[\rho C\bigg{(}\frac{\partial}{\partial t}T+\mathbf{v}\cdot\
abla T\bigg{)}- \
abla\Big{(}(4\sigma\delta T^{3}+K)\
abla T\Big{)}+H(T-T_{\infty})=(T>T_{p}) \,Q\,M\,A\,e^{-\frac{E_{q}}{\delta t}}, \tag{7}\]  
\[\frac{\partial}{\partial t}M=-(T>T_{p})\,M\,A\,e^{-\frac{E_{q}}{\delta t}}, \tag{8}\]  
where \(\sigma\) is the Stefan-Boltzmann constant, \(\delta\) is the optical path length for radiation, and \(T_{p}\) is the pyrolysis temperature. A more rational change of variables is also incorporated for the non-dimensionalization of the equations, the Frank-Kamenetskii change of variables [25]. This change of variables involves setting reference values for temperature and fuel in which an equilibrium state can be assumed, as well as spatial and temporal variables, and permits elucidating the significant parameters. The reference values are the ambient temperature \(T_{\infty}\) and the initial solid fuel load \(M_{0}\). Then, the dimensionless temperature is now \(u=(T-T_{\infty})/\varepsilon\,T_{\infty}\), and the dimensionless solid fuel load is \(c=M/M_{0}\). We kept again the notation for the spatial and temporal variables. The non-dimensional PDEs of this second model are  
\[\frac{\partial}{\partial t}u+\mathbf{w}\cdot\
abla u-\
abla(K(u)\
abla u)+ \alpha u=f(u,c), \tag{9}\]  
\[\frac{\partial}{\partial t}c=-\frac{\varepsilon}{q}f(u,c), \tag{10}\]  
where  
\[K(u) = \varkappa(1+\varepsilon u)^{3}+1,\] \[f(u,c) = (u>u_{p})c\varepsilon^{\frac{u}{1+\varepsilon u}},\]  
\(\alpha\) is the dimensionless natural convection coefficient, and it depends on the natural convection coefficient, the density, and the specific heat. The nonlinear function \(K(u)\) in the diffusive term represents the local radiation. The parameter \(\varkappa\) is the dimensionless inverse of the conductivity coefficient and depends on the Stefan-Boltzmann constant, the optical path length for radiation, the thermal conductivity, and the ambient temperature. \(\varepsilon\) is the dimensionless inverse of the activation energy and depends on the activation energy, the universal constant of gases, and the ambient temperature. The function \(f(u,c)\) is given by an Arrhenius type law, where \(u_{p}\) is the non-dimensional pyrolysis temperature, and the logic expression is equal to 1 if it is true and 0 if it is false. \(q\) is the non-dimensional reaction heat and depends on the heat of combustion, the initial fuel, and the specific heat. For a deep understanding of the dimensionless process, see [23]. Equations (9) and (10), together with appropriate initial and boundary conditions, provide challenging mathematical and numerical problems. Facing these types of mathematical questions is one of the challenges that should not be forgotten in the development of simulating models of processes as complex as wildfire spread. Some results about the existence and uniqueness of weak solutions of the non-convective version, that is the nonlinear reaction-diffusion problem, can be found in [23].