## 3 Current Model and Numerical Algorithm
### Model Description
Section 4 details some aspects related to the cartographic and meteorological information necessary to apply the PhyFire model in real cases. The zero-order term \(\alpha u\) in Equation (31) represents the energy lost by natural convection in the vertical direction, as in the initial models. Here, parameter \(\alpha\) depends on the natural convection coefficient \(H\) through the following expression:  
\[\alpha=\frac{H[t]}{MC} \tag{36}\]  
where \([t]\) is a time scale. The convective term, \(\beta\mathbf{v}\cdot\
abla e\), represents the energy convected by the gas pyrolyzed through the elementary control volume. \(\mathbf{v}\) is the wind velocity re-scaled by a correction factor \(\beta\). The idea behind this parameter is how the one-phase model, PhyFire (mainly a solid phase model), can be understood as a simplification of a gas-solid two-phase model where we retained the enthalpy transported in the gas phase. The enthalpy transported in the gas phase is retained, and \(\beta\) represents the fraction of transported heat that must be taken into account in Equation (31). For typical values of the heat capacity for the air and for the fuel (wood), \(\beta\) can be estimated to have an order of magnitude of \(10^{-2}\) inside the flame and \(10^{-4}\) away from the flame. See [41] for more details. Furthermore, we have to mention that the surface wind velocity \(\mathbf{v}\) can be considered as the given data or can be computed, for example, by means of the high-definition wind field model HDWind specifically developed to be used coupled with the PhyFire model (see [14,15]). As we explained in Section 2.4, the multivalued Equation (32) models the influence of moisture content and depends on the fuel moisture content \(M_{v}\) and solid fuel pyrolysis temperature \(T_{p}\). The dimensionless enthalpy \(e\) is, thus, an element of the multivalued maximal monotone operator \(G\) defined by Equation (19), where, remember, \(u_{v}\) is the non-dimensional water evaporation temperature and \(u_{p}\) is the non-dimensional solid fuel pyrolysis temperature. It remains to specify the expression defining the non-dimensional evaporation heat \(\lambda_{v}\):  
\[\lambda_{v}=\frac{M_{v}\Lambda_{v}}{CT_{\infty}} \tag{37}\]  
where the latent heat of water evaporation \(\Lambda_{v}=2.25\times 10^{6}\ Jkg^{-1}\) and the fuel moisture content \(M_{v}\) (kg of water/kg of dry fuel) is one of the fuel-type-dependent input variables in Table 2. We should clarify that the fuel moisture content depends not only on the fuel-type, \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Symbol** & **Units** \\ \hline Natural convection coefficient & \(H\) & J s\({}^{-1}\)m\({}^{-2}\)K\({}^{-1}\) \\ Convective term factor & \(\beta\) & \(-\) \\ Mean absorption coefficient & \(a\) & m\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model parameters. but also on other factors that vary both spatially and temporally. Fuel moisture content is driven by several weather factors, such as long-term droughts, recent rains, or immediate temperature and relative humidity [42]. Topographic aspect, i.e., the direction that the land faces, is another factor that affects the fuel moisture content [43]. Depending on the landscape position, there are areas that receive more radiation, will dry more quickly, and will, therefore, have lower fuel moisture. One of the latest enhancements to the PhyFire model is the ability to use space-varying fuel moisture content data, replacing the fuel-type-dependent input variable \(M_{\text{\tiny{p}}}\) with a raster containing the spatial distribution of the \(M_{\text{\tiny{0}}}\) values, using remote sensing data. The non-local radiation term \(r\) in the right-hand side of Equation (31) represents the radiation from the flames above the surface where the fire takes place, as we explained in Section 2.5. This term depends on the incident energy due to radiation \(R\) received in each point of the surface \(S\) given by Equation (20).