Fall Velocity

It is assumed the particle fall speed is equal to  its terminal settling velocity, which for a spherical particle may be written as
\[V_{t} = \sqrt{\frac{\rho_{ap}d_{p}g}{3\rho_{a}C_{D}}} \]
where g is gravitational acceleration, and CD is the drag coefficient. As in previous studies (e.g., \citealt{tse1998flight,bhutia2010comparison}), we assume CD = 0.45. Our assumption that firebrand fall speed is immediately equal to Vt is reasonable considering that typically reach their terminal velocity within seconds.

Landing and ignition

Firebrands land when their vertical position reaches a defined height threshold (fs_firebrand_land_hgt) over an unignited grid point. A spotfire is ignited when the number of firebrands landing on a given grid point, where fuel load is greater than 0 Kg m-3, meets two specified criteria. The ignition criteria are defined to allow ignitions when (a) firebrands land on a grid point and adjacent neighbors, comprising up to 8 adjacent grid points, reflecting a spatial likelihood (Eq. 14), and (b) a specified number of firebrands land on a grid point and adjacent neighbors, reflecting an intensity likelihood (Eq. 15). The criteria are made of two independent thresholds: 1) the number of neighboring grid points (fs_ignneighb) and 2) the total number of firebrands landing on the grid point and adjacent neighbors (fs_ignthresh). 
\[N=\sum_{i-1}^{i+1}\sum_{j-1}^{j+1}1\text{, if }(f_{ij}>0)\text{ AND }(fuelload_{ij}>0)\]
\[T=\sum_{i-1}^{i+1}\sum_{j-1}^{j+1}f_{ij}\text{, if }(fuelload_{ij}>0)\]
where i and j are the grid point indices, N represents the number of neighbors, T is the threshold for the total number of landing firebrands, fij represents the number of landed firebrands on the corresponding grid point during the current timestep, and fuelload is the fuel load on the corresponding grid point during the current timestep.
The set value for fs_ignneighb can be as low as 1, referring to the central grid point ij itself, and as high as 9, referring to the central grid point and the 8 adjacent neighbors. When both fs_ignneighb and fs_ignthresh are set to zero,  ignitions do not occur.
At this stage of development, the firebrand parameterization in WRF-Fire is a framework that enables firebrand simulation and consequently more adequate fire propagation in WRF-Fire. The framework allows for future contributions, such as those described by \citealt{Manzello2020} and \citealt{wadhwani2022review}, including for example implementation of a more sophisticated generation process, advection and burnout of non-spherical particles, and empirical ignition criteria, which would certainly improve the model capability. This article is intended to validate the framework and serve as reference for continuing developments in future version releases.  

Idealized Simulations

In this section we illustrate the parameterization components through a set of idealized simulations using WRF-ARW v4.5.1 for two idealized scenarios:

Uncoupled

The first scenario is an “uncoupled” scenario in which the feedback from the fire to the atmosphere (i.e., the release of fire heat fluxes to the atmosphere) is turned off, and the surface boundary condition is set to be free-slip. In this configuration, the atmospheric conditions, including the wind speed and direction, remain constant during the simulation. This scenario is configured to eliminate environmental factors that can affect the firebrands generation and transport in order to validate the implemented parametrization. For this scenario, the atmosphere’s horizontal grid interval is set to 40 m, with refined fire grid interval of 5 m. The model top is set at 2 km with 51 equally spaced grids. The timestep of the simulation is set to 0.5 s with open lateral boundary conditions. The simulations are initialized with a surface temperature of 305K and an air temperature vertical profile constant in the lower 1 km (300 K) below a statically stable layer in the upper 1 km, with temperature increasing linearly to 310 K at the top. The fuel in the fire grid is homogeneous and set to Anderson’s 13 fuel category 10, which corresponds to timber litter with understory, and the fire is ignited using a 1 km long and 100 m wide ignition line 10 s after the simulation start time. 

Coupled 

The second idealized cases scenario is configured in large-eddy simulation (LES) with two-way feedback, in that the fire fluxes are transferred to the atmosphere allowing for turbulent eddies and a fire-induced atmospheric circulation. In this scenario, the atmosphere’s grid interval is set to 10 m and the refined fire grid interval to 5 m. The model top is set at 2 km with 51 vertically stretched grids. The timestep of the simulation is reduced to 0.125 s to account for the finer grid mesh and fire updrafts. The lateral boundary condition is periodic and the Deardorff’s turbulent kinetic energy subgrid-scale model is utilized with coefficient of 0.1. The simulations in this scenario are initialized using the same temperature profile employed in the uncoupled scenario. We impose an initial zonal wind speed of 10 m s-1 (unless otherwise specified), interacting with an idealized surface with heat flux of 100 W m2 and drag coefficient of 0.005. To facilitate the turbulence development in the model, a temperature perturbation bubble with magnitude of 0.5 K and depth of 40 m was utilized. The simulations were allowed to “spin-up” for 0.5 h prior to fire ignition in order to develop the turbulent boundary layer. After the spin-up, the fire is ignited using a 1 km long and 40 m wide ignition line on a homogenous fuel bed set to Anderson’s fuel type 10.
In both of these scenarios, fire spots remained off to reduce degrees of freedom and allow a proper assessment of the parameterization components. Unless otherwise specified, the parameterization settings used in these simulations were set to the values indicated in Table 1, column Ideal Scenarios. 
Table 1: Firebrand spotting parameterization settings and default values