## 2 Analytical Framework
### Turbulence Intensity  
It is conventional to treat the turbulence ratio (the ratio of the standard deviation of longitudinal wind velocity component \(\sigma_{u}\) to the friction velocity \(u_{*}\)) as terrain-independent in the equilibrium boundary layer [36; 37]. On the other hand, [PERSON] and [PERSON] [38] proposed an empirical model to consider the variation of turbulence ratio with height as:  
\[\frac{\sigma_{u}}{u_{*}}=2.63\eta\left[0.538+0.090\ln\left(\frac{z}{z_{0}} \right)\right]^{\eta^{16}}, \tag{8}\]  
where \(\eta=1-z/h;h=u_{*}/(6f)\); \(f=1.458\times 10^{-4}\sin\phi\) is the Coriolis parameter; and \(\phi\) denotes the latitude of the observation site. Due to the fact that the derivation of Equation (8) was partly based on non-equilibrium-condition data, the estimation of maximum turbulence ratio, \(\left[\sigma_{u}/u_{*}\right]_{max}\) obtained from Equation (8) is dependent on the terrain roughness length, which is in contradiction to the equilibrium assumption. To correct this issue, the empirical variation of \(\sigma_{u}/u_{*}\) with respect to terrain roughness is introduced in ESDU [39] to obtain approximately a constant \(\left[\sigma_{u}/u_{*}\right]_{max}\) for various terrain roughness lengths:  
\[\left[\frac{\sigma_{u}}{u_{*}}\right]\left(z_{0}\right)=1+0.156\ln\left(\frac {u_{*}}{fz_{0}}\right), \tag{9}\]  
Since the field measurements give \(\left[\sigma_{u}/u_{*}\right]_{max}=2.85\) for the terrain with a roughness length of 0.03 m, Equation (8) can be corrected by factoring \(2.85/\{1+0.156\ln\left[u_{*}/(fz_{0})\right]\}\), resulting in an improved model to calculate turbulence ratio as in ESDU [39]:  
\[\frac{\sigma_{u}}{u_{*}}=\frac{7.496\eta\left[0.538+0.090\ln\left(\frac{z}{z_{ 0}}\right)\right]^{\eta^{16}}}{1+0.156\ln\left(\frac{u_{*}}{fz_{0}}\right)}. \tag{10}\]  
As expected, Equation (9) gives a maximum turbulence ratio \(\left[\sigma_{u}/u_{*}\right]_{max}\) of approximately 2.85 for various roughness lengths. However, field measurements show that turbulence ratios in tropical cyclones are usually greater than the values in extratropical storms [12; 24; 40], making the selection of \(\left[\sigma_{u}/u_{*}\right]_{max}=2.85\) inapplicable to tropical cyclones. On the other hand, a height-independent relation between turbulence ratio \(\sigma_{u}/u_{*}\) and underlying surface roughness length \(z_{0}\) was proposed by [PERSON] et al. [13] based on the analysis of field measurements in typhoons. In this study, this height-independent relation proposed by [PERSON] et al. [13] is adopted as:  
\[\left[\frac{\sigma_{u}}{u_{*}}\right]_{max}=2.72-0.25\log z_{0}. \tag{11}\]  
As a result, the turbulence ratio in tropical cyclone will be corrected by multiplying Equation (8) by the following factor:  
\[\frac{2.72-0.25\log z_{0}}{1+0.156\ln\left[u_{*}/(fz_{0})\right]}. \tag{12}\]  
Then the turbulence ratio could be expressed as:  
\[\frac{\sigma_{u}}{u_{*}}=\frac{2.63\eta\left[0.538+0.009\ln\left(\frac{z}{z_{ 0}}\right)\right]^{\eta^{16}}\left[2.72-0.25\log(z_{0})\right]}{1+0.156\ln \left(\frac{u_{*}}{fz_{0}}\right)}. \tag{13}\]  
For a standard terrain condition (\(z=10\) m; \(z_{0}=0.03\) m; assuming \(u_{*}=1\) m/s and \(\phi=25^{\circ}\)), the turbulence ratios estimated by Equations (9) and (11) are 2.55 and 2.78, respectively. In thisstudy, Equation (13) is utilized to convert the measured turbulence ratio to the standard condition. Accordingly, the longitudinal turbulence intensity in a standard exposure can be calculated as:  
\[\left[I_{u}\right]_{s}=\frac{2.63\eta_{s}\Big{[}0.538+0.090\ln\Big{(}\frac{z}{z_ {0}}\Big{)}\Big{]}^{\eta_{s}}\big{[}2.72-0.25\ln(z_{08})\big{]}\big{[}u_{s} \big{]}_{s}}{\Big{[}1+0.156\ln\Big{(}\frac{u_{s}}{fz_{0}}\Big{)}\Big{]}\big{[}U \big{]}_{s}}. \tag{14}\]