Wind Reduction Factor
The rate of spread models often require a wind input at a certain height. Generally, they need the wind of mid-flame height, representing the average influence of the wind on the flame. However, it is only sometimes possible to get this information. The wind information can come from stations at a specific height (20 ft, 10 m) or an atmospheric model. The latter vertical resolution is often larger than the midflame height itself. Therefore, a method is needed to estimate the midflame wind speed from wind speed information at a different height.
The wind reduction factor, noted \(\alpha\) [-], is introduced to get the wind at the midflame height \(U_f\) from the wind at a different height, noted $U_r$:
\[ U_f = \alpha \, U_r.\]
To compute the wind reduction factor, the method is generally based on a specific vertical profile that describe the wind speed as a function of height, considering the vegetation cover. The following sections describe methods to compute the wind reduction factors from a vertical profile and vegetaiton properties.
To apply a wind reduction factor in a workflow, you can use the function firebench.wind_interpolation.apply_wind_reduction_factor. The block for this function can be found in firebench/docs/assets/diagram_blocks/process/apply_wind_reduction_factor.svg.
Baughman and Albini (1980)
This section is based on [1, 2]. The wind profile above the vegetation cover is given by: \[ U (z) = \frac{U_*}{\kappa} \ln \left ( \frac{z - D_0}{z_0} \right ), \] where \(U(z)\) is the wind speed at the height above ground level \(z\), \(U_*\) is the friction velocity, \(\kappa\) is the Von Karman constant, \(D_0\) is the zero-plane displacement, and \(z_0\) is the reoughness length. [1] sets \(D_0 = 0.64\,h\), and \(z_0 = 0.13\,h\), where \(h\) is the vegetation height.
Unsheltered wind reduction factor for 20-ft reference wind
When considering that the reference wind speed is 20 ft above the fuel top, the wind reduction factor is given by: \[ \alpha (h, h_f) = \frac{1 + 0.36 h / h_f}{\ln \left ( \frac{20 + 0.36h}{0.13h} \right )} \left [ \ln \left ( \frac{h_f/h + 0.36}{0.13} \right ) -1 \right ],\] where \(h\) is the fuel height [ft], and \(h_f\) is the flame height [ft]. This wind reduction factor does not interpolate the value to midflame height but allows to calculate the average wind speed over the flame length.
Note This formula can be applied to Anderson fuel model, considering \(h_f/h=1\), to retrieve the wind reduction factor contained in the dataset. However, the values for fuel category 7 and 8 are different from the values in [1], where the value 0.36 is given in [1] whereas the formula gives 0.28.
Generalized unsheltered wind reduction factor
The methodology described in [1, 2] can be applied to any reference height for the input wind, noted \(h_r\). The advantage of this formulation is that is does not force the input wind to a certain height and it works with any units (m or ft) as long as they are the same for all input variables. The definition of \(h_r\) can be different depending on the context, explicited in the following paragraphes. In any case, the following integral needs to be computed:
\[ \int_h^{h+h_f} U(z) dz = \mathcal F (h+h_f) - \mathcal F (h), \]
where
\[ \mathcal F (z) = (z - D_0) \ln \left( \frac{z - D_0}{z_0} \right ) - z.\]
Reference height is defined above the ground level
When the reference wind height \(h_r\) is given from the ground level, the wind reduction factor is given by:
\[ \alpha (h, h_f, h_r) = \frac{1}{h_f U (h_r)} \int_h^{h+h_f} U(z) dz. \]
Fig. 1 : Interpolation of midflame wind from a wind at height defined above ground level.
Reference height is defined above the vegetation level
When the reference wind height \(h_r\) is given from the top of the vegetation layer, the wind reduction factor is given by:
\[ \alpha (h, h_f, h_r) = \frac{1}{h_f U (h+h_r)} \int_h^{h+h_f} U(z) dz. \]
Fig. 2 : Interpolation of midflame wind from a wind at height defined above vegetation top.