# Unfrozen Skewed Turbulence for Wind Loading on Structures
## 3 Computation of Turbulent Time-Histories  
The generation of turbulent wind histories is based on the spectral representation method (e.g., [64; 65; 23; 66]). The wind histories are generated in \(N_{p}\) scattered points. Since the present algorithm does not rely on a computational grid to achieve convergence, wind histories can be conveniently computed directly at the integration points of a 5 km long floating bridge. Popular turbulence generators for wind loading modelling, e.g., TurbSim [67] or those relying on stochastic wave-based models (e.g., [68]) typically use gridded data points, which can lead to a prohibitive computational cost if the structure studied has a size of several kilometres as in the case previously mentioned. On the other hand, stochastic wave-based models may be better suited than Cholesky-based models for structures with dimensions smaller than a few hundred meters or when the focus is on dynamic characteristics associated with higher modes of vibrations. Indeed, following [PERSON] and [PERSON] [69], Cholesky-based models may produce large numerical errors when the separation distances become close to zero, due to the high correlation between neighbour locations. However, in the present case, the roundoff error associated with the Cholesky decomposition was found to be negligible, even for densely gridded data points. It can also be noted that in [PERSON] and [PERSON] [69], an analytical solution of the frequency-wavenumber spectrum was used, which can significantly reduce the computational costs, although at the expense of some realistic turbulence characteristics, which are accounted for by e.g., the uniform shear model [68]. The uniform shear model can also be modified to relax [PERSON]'s hypothesis with possible applications for wind turbine wake modelling or wind-turbine control [70]. In summary, the use of the spectral representation method was adopted in the present study as it suits well the spatial and temporal scales investigated, which are of the order of 1 km and 1 h (or more), respectively. At each frequency step, the cross-spectral density matrix \(\mathbf{S}\) is  
\[\mathbf{S}= \begin{bmatrix}\mathbf{S}_{\mathbf{uu}}&\mathbf{S}_{\mathbf{uv}}& \mathbf{S}_{\mathbf{uw}}\\ \mathbf{S}_{\mathbf{vu}}&\mathbf{S}_{\mathbf{vv}}&\mathbf{S}_{\mathbf{vw}}\\ \mathbf{S}_{\mathbf{wu}}&\mathbf{S}_{\mathbf{wv}}&\mathbf{S}_{\mathbf{ww}} \end{bmatrix}  \tag{19}\]  
where \(\mathbf{S}_{\mathbf{uu}}\), \(\mathbf{S}_{\mathbf{vv}}\), \(\mathbf{S}_{\mathbf{ww}}\) and \(\mathbf{S}_{\mathbf{uw}}\) are \(N_{p}\) by \(N_{p}\) matrices defined as  
\[\mathbf{S}_{\mathbf{uu}} =\sqrt{S_{t}(f_{k})S_{u}(f_{k})\mathsf{\tau}}\mathrm{coh}_{uu}(f_ {k},d_{x},d_{y},d_{z}) \tag{20}\] \[\mathbf{S}_{\mathbf{vv}} =\sqrt{S_{v}(f_{k})S_{v}(f_{k})\mathsf{\tau}}\mathrm{coh}_{vv}(f_ {k},d_{x},d_{y},d_{z})\] (21) \[\mathbf{S}_{\mathbf{ww}} =\sqrt{S_{w}(f_{k})S_{w}(f_{k})\mathsf{\tau}}\mathrm{coh}_{ww}(f_ {k},d_{x},d_{y},d_{z})\] (22) \[\mathbf{S}_{\mathbf{uw}} =\sqrt{Co_{uw}(f_{k})Co_{uw}(f_{k})\mathsf{\tau}}\mathrm{coh}_{uw }(f_{k},d_{x},d_{y},d_{z}) \tag{23}\]  
where \(f_{k}\) is the discretized frequency with \(k=\{1,2,...,N\}\) and \(N\) a natural number power of two. As mentioned before, the matrices \(\mathbf{S}_{\mathbf{uv}},\mathbf{S}_{\mathbf{wu}}\), \(\mathbf{S}_{\mathbf{vw}}\) and \(\mathbf{S}_{\mathbf{vw}}\) are assumed to contain only zeros. It implies that the covariances between the \(u\) and \(v\) components and \(v\) and \(w\) components are negligible. While this assumption may not be true in mountainous terrain, it is adopted here for the sake of simplicity and will be further addressed in Section 4.1.