Wind Spectra 1. Definitions \(z\) height above calm water level \(z_r\) wind reference height, normally 10 m \(\bar{u}_r\) average velocity at the reference height \(z_r\) above the calm water level \(\theta\) average wind propagation direction When using parametric spectra to describe the wind field the wind is assumed to propagate in the horizontal plane. Further, any spatial variation is neglected except in the vertical direction. The time varying part of the wind is assumed to be a Gaussian stochastic process. For a given wind propagation direction, \(\theta\), the wind speed is divided into a mean and a fluctuating component which varies with time: \[u_\theta(z, t) = \bar{u}(z) + u_{f,\theta}(z, t)\] Optionally, a fluctuating component transverse to the the mean wind direction may be included: \[u_t(z, t) = u_{f,t}(z, t)\] 2. Mean wind profile The mean wind profile, \(\bar{u}(z)\) is given by \[\bar{u}(z)=\bar{u}_r(\frac{z}{z_r})^\alpha\] where \(\alpha\) height coefficient (0.10 - 0.14) 3. Admittance function Due to the fact that high frequency wind fluctuation has low spatial correlation, an admittance function is introduced. The admittance function is structure-dependent, and serves mainly as a low-pass filter for the gust spectra. The admittance function, proposed by (Davenport, 1977), is given by \[x(\omega ,z)=(1+(\frac{\omega \sqrt{A}}{\pi \bar{u}_r})^{4/3})^{-1}\] where \(\mathrm {A}\) is the characteristic area of the structure. For bodies with Quadratic Wind Coefficients, the admittance function is applied to the wind spectrum both in the mean direction and, if included, for the transverse wind spectrum: \[\begin{array}{l}S_u^{+'}(\omega )=x^2(\omega )S_u^+(\omega )\\\\S_v^{+'}(\omega )=x^2(\omega )S_v^+(\omega )\end{array}\] 4. Wind spectra for the main direction The fluctuating component in the mean direction is realized from a spectrum function. Multiple spectrum functions are available: Davenport wind spectrum Harris wind spectrum Wills spectrum Sletringen wind spectrum ISO 19901-1 wind spectrum API wind spectrum ESDU wind spectrum 5. Transverse wind spectrum The wind spectrum normal to the mean wind direction may be combined with any of the above spectra and is described according to (Simiu, 1978) by \[S_v^+(z,\omega )=\frac{\kappa\bar{u}_r^2Kx'}{\omega (1+9.5x')^{5/3}}\] where \[x' = \frac{\omega z}{2 \pi \bar{u}_r} \label{eq-simiu-2}\] \[K = 17 \label{eq-simiu-3}\] Wind Davenport