Negative displacement height (zd) in RSL

[vitorlavor]

The displacement height in the RSL $z_{d,RSL}$ is calculated as:

$$ Z_{d,RSL} = Z_{H,RSL} - \beta^2 L_c $$

where:

• $Z_{H,RSL}$ is the building height level,
• $\beta$ is a parameter from Harman and Finnigan (2007)
• $L_c (= \frac{(1-\lambda_P)}{\lambda_F} Z_H$) is the length scale for the absorption of momentum in the atmosphere
• $\lambda_P% is the plan area index of buildings
• $\lambda_F$ is the frontal area index of buildings

$Z_{d,RSL}$ then becomes negative when values $\lambda_F$ < $\beta^2 (1-\lambda_P)$. However, since the maximum value of $\beta$ is constrained to 0.5, this imposes a lower limit on $\lambda_F$, given by:

$$ \lambda_F = 0.25(1-\lambda_P) $$

However, the value of $\beta$ is subject to Atmospheric stability conditions at the moment.

[nakaoipeirc]

Original paper may not avoid negativity of $Z_{d,RSL}=Z_{H,RSL}-\beta^{2}L_{c}$.
Their drag centroid method is formed as follows (with the origin of $z$ at $Z_{H,RSL}$),

$$z_{d,RSL}=-\frac{\int_{-\infty}^{0} zU^2/L_{c}dz}{\int_{-\infty}^{0} U^2/L_{c}dz}$$

The integral limit $-\infty$ (infinite bottom) seems as if exponential wind speed exists below ground level. So in some cases such as sparse buildings and in-canopy wind speed is intense enough, the center of the mass (the concept of the eq.) goes below ground.

Another choice may be,

$$z_{d,RSL}=-\frac{\int_{-Z_{H}}^{0} zU^2/L_{c}dz}{\int_{-Z_{H}}^{0} U^2/L_{c}dz}$$

With setting an actual ground as limit, you may have (with the coordinate $Z$ of the actual ground set 0)

$$Z_{d,RSL}=\frac{Z_{H,RSL}}{1-\left\{exp\left(\frac{-Z_{H,RSL}}{2\beta^{2}L_{c}}\right)\right\}^{2} }-\beta^{2}L_{c}$$

(*Could anyone double check it)
It seems similar to the original but,

$$f = \frac{x}{1-\left\{exp\left(\frac{-x}{2}\right)\right\}^{2} }-1>0$$

seems to hold if $x(=Z_{H,RSL}/\beta^{2}L_{c})&gt;0$.