# Dynamic Response of Slender Vertical Structures Subjected to Thunderstorm Outflows
## 2 Analytical Formulation
### Wind Velocity  
Let us consider the horizontal non-directional wind speed of a thunderstorm outflow along a vertical linear domain defined by the spatial coordinate \(z\) (with \(z=0\) m at the ground level). The wind speed can be decomposed as follows [18; 26; 27; 28]:  
\[v(z,t)=\overline{v}(z,t)+v^{\prime}(z,t) \tag{1}\]  
where \(\overline{v}(z,t)\) and \(v^{\prime}(z,t)\) are the slowly varying mean wind speed and the residual turbulent fluctuations, respectively. They can be expressed as:  
\[\overline{v}(z,t)=\overline{v}_{\max}(h)\alpha(z)\gamma(t) \tag{2}\]  
\[v^{\prime}(z,t)=\overline{v}_{\max}(h)\overline{I}_{v}(h)\alpha(z)\theta(z) \gamma(t)\widetilde{v}^{\prime}(z,t) \tag{3}\]  
where \(\alpha(z)\) and \(\theta(z)\) are the vertical profiles of the mean wind velocity and turbulence intensity (Appendix A), respectively, \(\gamma(t)\) is the modulating function of the slowly varying mean speed, \(\overline{I}_{v}(h)\) is the mean value of the turbulence intensity [13; 18; 20; 29] at the reference height \(h\) at which the wind velocity is assigned, and \(\widetilde{v}^{\prime}(z,t)\) are the reduced turbulent fluctuations, dealt with as a stationary and Gaussian process with zero mean and unitary standard deviation. It is worth pointing out that in Equations (2) and (3), the vertical profiles of the mean speed and turbulence intensity are assumed to be independent of time [27; 30; 31]. This is somehow in contrast with the experimental observations, which indicate that both these quantities evolve over time during the development of the downburst [5; 32; 33], and that in particular, the characteristic nose-shaped profile of the mean speed is clearly noticeable solely during the ramp-up phase of the wind speed. However, since there are no well-established analytical models in the literature that suitably describe their evolution over time, such vertical profiles are assumed as independent of time. The reliability of this assumption for the mean wind speed, widely used in the literature [15; 18; 24; 26; 34], is partially justified by the fact that the nose-shaped profile is noticeable within a time interval in which the mean wind velocity reaches its maximum, and is thus in the neighborhood of the instant when the maximum response is expected [13; 32; 35]. The residual fluctuations in Equation (3) are a uniformly modulated process and can be characterized through their cross-evolutionary power spectral density (CEPSD):  
\[S_{\
u^{\prime}\
u^{\prime}}(z,z^{\prime},t,n)=\overline{v}_{\max}^{2} \overline{t}_{\
u}^{2}(h)\gamma^{2}(t)\alpha(z)\alpha(z^{\prime})\theta(z) \theta(z^{\prime})S_{\bar{\
u}^{\prime}\bar{\
u}}(z,z^{\prime},n) \tag{4}\]  
where \(S_{\bar{\
u}^{\prime}\bar{\
u}^{\prime}}(z,z^{\prime},n)\) is the cross-power spectral density (CPSD) of the reduced turbulent fluctuations, which can be written as follows:  
\[S_{\bar{\
u}^{\prime}\bar{\
u}^{\prime}}(z,z^{\prime},n)=\sqrt{S_{\bar{\
u}^{ \prime}}(z,n)S_{\bar{\
u}^{\prime}}(z^{\prime},n)}Coh_{\bar{\
u}^{\prime}\bar {\
u}^{\prime}}(z,z^{\prime},n) \tag{5}\]  
In Equation (5), \(Coh_{\bar{\
u}^{\prime}\bar{\
u}^{\prime}}(z,z^{\prime},n)\) is the coherence function of the reduced turbulent fluctuations \(\bar{\
u}^{\prime}(z,t)\) between two points, \(z\) and \(z^{\prime}\); it is assumed to be time-invariant. In accordance with [18], the coherence function is defined as:  
\[Coh_{\bar{\
u}^{\prime}\bar{\
u}^{\prime}}(z,z^{\prime},n)=\exp\biggl{\{} \frac{2 nc_{z}|z-z^{\prime}|}{\overline{v}_{\max}(z)-\overline{v}_{\max}(z^{ \prime})}\biggr{\}} \tag{6}\]  
where \(c_{z}\) is the exponential decay coefficient of \(\bar{\
u}^{\prime}(z,t)\) along \(z\). It is important to notice that the hypothesis of the time-invariant vertical profile allows us to treat the coherence function between the processes \(v^{\prime}(z,t)\) and \(v^{\prime}(z^{\prime},t)\) as independent of time, being the modulating function \(\gamma(t)\), which is independent of height.