ESDU Tropical Storm

\[S(f, z) = \frac{4 I_u(z)^2 U_{w,1h}(z) L_{u,x}(z)}{\left[ 1 + 70.8 \cdot \left( f \frac{L_{u,x}(z)}{U_{w,1h}(z)} \right)^2 \right]^{5/6}}\]

\[L_{u,x}(z) = \frac{50 \cdot z^{0.35}}{z_0^{0.063}}\]

\[U_{w,1h}(z) = \frac{u_*}{0.4} \ln \left( \frac{z}{z_0} \right)\]

\[I_u(z) = \frac{u_* \cdot 7.5 \cdot \eta \left[ 0.538 + 0.09 \cdot \ln \left(\frac{z}{z_0} \right) \right]^{\eta^{16}}}{1 + 0.156 \cdot \ln\left(\frac{u_*}{\widehat{f_C} \cdot z_0}\right)} \cdot \frac{1}{U_{w,1h}(z)}\]

\[u_* = \sqrt{C_{d_{10}}} \bar{u}_r\]

\[z_0 = 10 \cdot e^{\frac{-0.4}{\sqrt{C_{d_{10}}}}}\]

\[C_{d_{10}} = \begin{cases} \left( 0.49 + 0.065 \bar{u}_r \right) \cdot 10^{-3} &, \bar{u}_r < 27.85 \mathrm{\ m/s} \\ 0.0023 &, \bar{u}_r \geq 27.85 \mathrm{\ m/s} \end{cases}\]

\[\eta = 1 - 6 \widehat{f_C} \frac{z}{u_*}\]

\[\widehat{f_C} = 2 \cdot 7.29 \cdot 10^{-5} \sin|\Psi|\]