For any turbulent wind field, the wind velocity time series Ui can be expressed as a function of mean velocity 〈Ui〉 and velocity fluctuation ui, with i=1,2,3 in three-dimensional space: 1Ui=〈Ui〉+ui, where the angular brackets 〈.〉 denote an ensemble average. Throughout this paper, the ensemble average is partly replaced by spatial or temporal averages for practical purposes which will be indicated in the respective cases below. In our case, we assume only a mean velocity in the x1 direction, i.e., 〈U〉=(〈U1〉,0,0), which is the inflow direction in the following. From this decomposition, the turbulence intensity (TI) can be calculated as follows: 2TIi=σi〈U1〉=〈ui2〉〈U1〉, where i indicates the direction in which the turbulence intensity is calculated with respect to the mean velocity in the inflow component 〈U1〉 and σi is the variance of ui. The turbulence intensity is a one-point statistics in space and time of wind fields. However, for information on the spatial structures, one-point statistics is not sufficient and two-point statistics should be used for more information about the wind fields. As an example, the co-variance tensor contains two-point statistics: 3Rij(r,x,t)=〈ui(x,t)uj(x+r,t)〉, where Rij is the two-point correlation which is independent of the position x in the case of homogeneous turbulence and r is the displacement vector between the two points. Since Eq. () is a theoretical equation, there are no restrictions on selecting the two points for the two-point statistics. However, in practice, it is limited to the size of the data set of the wind fields. The spectral velocity tensor for homogeneous turbulence resulting from a Fourier transform of Eq. () gives () 4Φij(κ,t)=1(2π)3∫-∞∞∫-∞∞∫-∞∞Rij(r,t)exp⁡(-iκ⋅r)dr. In this equation, κ=(κ1,κ2,κ3) represents a three-dimensional wavenumber vector for the three directions. The one-dimensional spectrum follows from Eq. () by integration: 5Fi(κ1,t)=∫-∞∞∫-∞∞Φii(κ,t)dκ2dκ3=12π∫-∞∞Rii(r1,0,0,t)exp⁡(-iκ1r1)dr1, where the ii index refers to the respective diagonal element of the tensor. The one-dimensional spectra with respect to κ2 and κ3 are computed accordingly. The spectral coherence for the wavenumber κ1 with respect to two separate points in the x2-x3 plane reads () 6cohij(κ1,Δx2,Δx3,t)≡|χij(κ1,Δx2,Δx3,t)|2Fi(κ1,t)Fj(κ1,t), with Fi(κ1,t) from Eq. () and the cross spectra χij as defined below: 7χij(κ1,Δx2,Δx3,t)=12π∫-∞∞Rij(r1,Δx2,Δx3)exp⁡(-iκ1r1)dr18=∫-∞∞∫-∞∞Φij(κ,t)exp⁡(i(κ2Δx2+κ3Δx3))dκ2dκ3, where Δx indicates the spatial step in three-dimensional space with its components Δx1, Δx2, and Δx3 in the x1, x2, and x3 directions, respectively. Another example of two-point statistics is the increment statistics in space described by 9ṽi(x,Δx,t)=ui(x+Δx,t)-ui(x,t). Instead of spatial increments, one can also consider temporal increments as 10vi(x,t,τ)=ui(x,t+τ)-ui(x,t), where τ indicates the time lag. The probability density functions (PDFs) of the introduced temporal increments vi have been investigated () as part of a more detailed characterization of wind turbulence beyond the standard guidelines. The Gaussian distribution is completely described by the mean and the variance. Further statistical information is required for characterizing a non-Gaussian distribution. The so-called kurtosis of a distribution allows quantification of its deviation from a Gaussian distribution. For the temporal increments vi, the kurtosis is defined as 11Kurt(vi)(x,t,τ)=〈vi(x,t,τ)4〉〈vi(x,t,τ)2〉2. For a Gaussian distribution, the kurtosis equals 3. Higher values indicate a heavy-tailed distribution in which extreme velocity increment values have a higher probability than predicted by a Gaussian distribution. We assume statistically stationary turbulence and, hence, omit the time t in the equations in the following.

Figure  shows characteristics of temporal velocity increment statistics of atmospheric wind speed measurements. The data were collected at the site of a Nordex wind turbine, located in northern Germany. The wind sensor was mounted at a height of 125 m. Time series of 10 min in length were considered for the analysis. These time series correspond to the horizontal magnitude of the wind speed measured at the hub height with a sampling rate of 50 Hz; the vertical component (U3) was neglected here. In total, 494 time series were provided. Similar to Eq. (1), the measured wind speed can be decomposed as Umeas=〈Umeas〉+umeas, where 〈Umeas〉 is calculated over the 10 min period. The turbulence intensity for the measured data is calculated as TImeas=〈umeas2〉/〈Umeas〉. All the mean values of the wind direction, calculated over each of the 10 min periods, are distributed within a range of 165∘. The selection and preparation of the data set were performed by Nordex according to internal objectives from the analysis. Here, we used the wind data set for illustration purposes rather than for a rigorous investigation of its characterization. Nevertheless, for our study, we investigate the time series of the wind velocity with similar first- and second-order moments. Accordingly, we selected a subset of time series whose values of mean wind speed and turbulence intensity are contained within a specific range. The range is defined as 9±1 m s-1 for the mean wind speed and 0.15±0.02 for the turbulence intensity. Further, we also constrain the data in terms of their wind direction. For that, we consider the mean direction calculated over the 10 min period. Then, we define a range of ±20∘ from the main wind direction at the specific site. After the selection process, 20 out of the 497 available time series are analyzed.

The increments for the atmospheric measured data vmeas=umeas(x,t+τ)-umeas(x,t) were calculated similar to Eq. (). Then, the statistics of vmeas are computed for the whole subset of 20 conditioned 10 min time series. Figure a presents the PDFs of vmeas for different time lags τ from 1 to 60 s. For clarity of presentation, individual distributions are depicted with different markers and shifted vertically. For comparison, all the PDFs are normalized to a standard deviation equal to 1, and the corresponding Gaussian distributions are shown by a solid line.

As can be observed, the PDFs of vmeas for timescales τ equal to 1, 5, and 10 s deviate from the respective Gaussian distributions. Specifically, the distributions of vmeas show heavy tails which represent the high probability of extreme events, which is higher than for the Gaussian PDFs. This effect, which is called intermittency, is a well-known feature of atmospheric wind as , , , and and many others have found. Similar to the results from the investigations mentioned earlier, Fig. a shows that the temporal increments vmeas are intermittent over a broad range of time lags τ. The evolution of the intermittency with τ can be quantified from the kurtosis. Figure b shows the calculated kurtosis of the PDFs according to Eq. () of the atmospheric increments vmeas as a function of τ. The values of Kurt(vmeas) higher than 3 reveal the intermittent behavior of the atmospheric data over a wide range of τ. For the analyzed data, intermittent features are distinguishable up to timescales of τ≈11s when Kurt(vmeas) converges to the Gaussian value of 3.

The basis of the Mann model () is a properly modeled spectral tensor according to Eq. (). In this model, the turbulent velocity field is assumed to be incompressible, and the velocity fluctuations are assumed to be homogeneous in space. The resulting velocity field depends on only three parameters, namely a length scale L; a non-dimensional parameter Γ, which is related to the eddy lifetime and, hence, to the shear gradient; and a parameter cKϵ2/3 with cK as a constant and ϵ as the turbulent dissipation rate per unit mass. Within these assumptions, the second-order statistics such as variances and cross spectra of real wind can be met. In the case of isotropic turbulence (where assuming no shear), the spectral tensor takes the form () 12Φijiso(κ)=E(κ)4πκ4(δijκ2-κiκj), with κ as the magnitude of κ and E(κ) as the energy spectrum. E(κ) can be expressed with cKϵ2/3 and L that leads to () 13Φijiso(κ)=cKϵ2/3L17/34πδijκ2-κiκj[1+(Lκ)2]17/6. The corresponding one-dimensional spectra result in : 14Fi(κ1)={955cKε2/31(L-2+κ12)5/6fori=13110cKε2/33L-2+8κ12(L-2+κ12)11/6fori∈{2,3}, and, accordingly, the variance becomes 15σiso2=σ12=σ22=σ32=955πΓ13Γ56cKϵ2/3L2/3≈0.688cKϵ2/3L2/3, where in this equation the Γ denotes the mathematical Gamma function. The value of the cKϵ2/3 parameter can be found by fitting the model to specific site data assuming isotropic turbulence and an infinitely large domain. Another possible way to avoid such assumptions was suggested by , who suggest choosing the value of cKϵ2/3 arbitrarily and then re-scaling the velocity field with a scaling factor. The value of cKϵ2/3 was taken according to the Engineering Sciences Data Unit () spectral model to achieve TI1≈10% as will be explained later in detail.

The Mann model provides a spatially correlated synthetic turbulent wind field where spatial homogeneity is one of the basic assumptions in deriving this model. However, since this model is based on the spectra but not on increment statistics of the atmospheric wind fields, it fails to display intermittency which should have effects on several turbine loads as shown in different previous research studies; see Sect. .

introduced the idea of generating a synthetic velocity time series u(s), on an intrinsic timescale s, based on a coupled Ornstein–Uhlenbeck process () for u(s) and a reference wind speed ur(s). Instead of a mean velocity, this reference wind speed ur(s) is used in the modeling process which influences the whole velocity field ().

After generating the velocity time series u(s) for all grid points in a plane perpendicular to the flow velocity, a stochastic process is applied to map the field from the intrinsic time s to the physical time domain t. This mapping process is the crucial feature of the model for generating intermittent characteristics in the wind field. The mapping process is defined as () 16dt(s)ds=τα(s), where τα follows an α-stable Lévy distribution with characteristic exponent α (); for general parameterization, cf. : 17p(τα)=1πRe∫0∞dzexp⁡[-izτα-zαexp⁡(-iπα2)]. generated the random variable τα according to the implementation introduced by : 18τα=sin⁡(α(V+π2))cos⁡(V)1α(cos⁡(V-α(V+π2))W)1-αα, where V is a uniformly distributed random variable that takes a value in the range between ]-π2,π2[ and W is an exponentially distributed variable with a mean value = 1. The Lévy distribution is truncated for 0<τα<c if 0<α<1, where c is a cutoff. The α-stable Lévy process with the parametrization as in Eq. () ensures that the τα does not take negative values. showed that when continuous stable Lévy processes are used for the transformation from the intrinsic time s to the physical time t, the process is dominated by “waiting” regions. In this context, these regions are periods during which the wind speed is constant, a feature that is not observed in the atmospheric wind. In order to limit those periods to a realistic length, the Lévy distribution is truncated at the cutoff c. When α=1, p(τα) is not a Lévy distribution anymore. It becomes a δ-correlated distribution with τ1=1. In that case, the mapping process from s to t is linear, which makes u(s)=u(t), and no intermittent behavior is introduced to the field.

As mentioned before, the main advantage of the CTRW model is that it manages to generate an intermittent wind field. However, the original CTRW model proposed by assumes that the spatial correlations of the velocity fluctuations between different points of the grid decay exponentially with the distance between them. This assumption must be understood as a simplification of atmospheric turbulence. A modified version of the original CTRW model was implemented by as introduced in Sect. . They generated individual CTRW time series and arranged them to generate three different spatial correlations in the wind field. Firstly, they set up fully correlated wind fields by repeating the same velocity time series in the whole x2–x3 plane (perpendicular to the inflow direction). For the second arrangement, they generated delta-correlated wind fields by setting up an independent Ornstein–Uhlenbeck process at each point of the grid in the x2–x3 plane. Finally, they considered an intermediate version of a 3×3 sub-divided fully correlated wind field. In this third scenario, the rotor plane was divided into a 3×3 grid in which each cell corresponds to a fully correlated area, but the nine areas are uncorrelated between them. They found different effects of the wind fields on the turbine loads depending on the resolution (i.e., fully correlated, delta-correlated, or partially correlated). However, no conclusive recommendation was provided on which resolution is most realistic.

This section provides a thorough comparison of statistical quantities such as the spectral tensors, the coherence, and the velocity increment distribution of both fields, the regular Mann and the new Time-mapped Mann wind field. The dimensions of the grid for the generation of the wind fields are selected according to Table , such that they can be used for the BEM simulations in Sect. . In this case, the defined grid has 32×32 nodes with Δx2,m=Δx3,m=2.6 m covering an 80.6m×80.6m area. The original Mann field is generated using the Mann model at mean wind speed 〈U1〉=20 m s-1; turbulence intensity in the flow direction, which is the x1 direction in this case; TI1=10%; Γ=0; and cKϵ2/3=0.62 m4/3 s-2 according to the parameters of the Mann model as introduced in the ESDU spectral model. The value of TI1=10% was used in this work to keep the same standard deviation value as used for the class B turbine (average class) in the IEC 61400-1 standard () at 20 m s-1 mean wind speed. Corresponding to Γ=0, we assume homogeneous turbulence and wind shear are not taken into consideration in this work. Also, based on this assumption, the velocity components can be averaged over each slice to calculate the wind field statistics. According to , the L parameter can be calculated by 20L=zTI1a, where z is the height of the calculation point (which is the turbine hub height in this work) and a is a shear exponent. Equation () is used to calculate the length scale for multi-megawatt wind turbines (). However, this equation is also used in the present work for a 1.5 MW wind turbine in order to enable a generalization for larger wind turbines. Even though we assume no shear in this work, the shear exponent value a is only used to calculate the value of L with no effect on the wind velocities. According to the hub height of the studied 1.5 MW turbine with z=84 m and for neutral conditions with a shear exponent of a=1/7 (), Eq. () results in L=58.8 m.

For the Time-mapped Mann wind field, the three components of the velocity fluctuations of each transverse plane (x2–x3 in this case) are shifted according to the resulting t(s). When applying the time-mapping process to the wind field, the same stochastic process (Cα,c,Δstτα(sn)) is applied to each point of the x2–x3 plane, and it is not a function of the location of different points in the x2–x3 plane. Figure  shows the resulting relationship between intrinsic timescale (s) and real-time scale (t) used in the time-mapping process of the original Mann field to the new Time-mapped Mann field according to Eq. () for different values of α (in the Lévy distribution). As noticed from this figure, the relationship between s and t is not an exact straight line, which means that the resulting field will be different from the original field as explained in Sect. . The values of τα in Eq. () follow a Lévy's distribution with the corresponding PDFs presented in Fig.  at different values of α. In the following analysis and simulations, we will only consider α=0.6, which is highlighted with red lines in Figs.  and . Additionally, values of c=20 and Δst=8 s were considered for the calculation of the scaling factor Cα,c,Δst in Eq. (). These parameters were selected through an iterative process to achieve values of Kurt(v1), defined in Eq. (), in a comparable range to the data shown in Fig. and to other literature values like . The iteration aimed not only to find values of the parameters that match the desired values of Kurt(v1) but also to keep other statistical values such as the standard deviation and the spectrum of the time series of the wind velocity.

The comparison between the Mann and the new Time-mapped Mann wind fields regarding statistical properties is performed in terms of one-point and two-point statistics from Sect. . Starting with one-point statistics, the values of the mean wind speed in the longitudinal direction U1 and the corresponding standard deviation 〈u12〉 are presented in Table . These values are calculated from the time series at the hub point (location of the hub of the turbine in the BEM simulations; see Sect. ). For better comparison, the relative differences (Rel.Diff) calculated as the difference between both results over the result from Mann field (i.e., (Time-map.-Mann)/Mann) are also presented in the table.

The values of the relative differences for 〈U1〉 and 〈u12〉 are lower than 2%. This means, in terms of one-point statistics at the hub point, that the two generated wind fields are comparable. The slight differences between the two fields are caused by the interpolation procedure explained in Fig. . However, one-point statistics are not enough to show the effects of intermittency.

To show the effect of the time mapping on the spectral properties of the new field, Fig.  shows a comparison between the spectra κ1〈Fi(κ1)〉I of three components of the original Mann field and the new Time-mapped Mann field where κ1=2πf/U1. In these figures, 〈.〉I denotes spatial averaging over the plane I={(x2,x3)|0≤x2≤l2,0≤x3≤l3}. Figure a shows the wavenumber spectrum of the u1 velocity component in the longitudinal (x1) direction, while Fig. b and c show the same comparison but for the u2 component in the x2 direction and the u3 component in the x3 direction, respectively. The theoretical spectra of the Mann field from Eq. () are plotted in the same figures to show how the spectra should be distributed over the different wavenumbers. Further to that, an uncorrelated, intermittent wind field was also added for the comparison. This uncorrelated wind field is generated by arbitrarily shuffling the grid points of the intermittent Time-mapped Mann field in the transverse directions (x2 and x3 directions). Of course, that still means that all points in the x2–x3 plane are shifted by the same time step in the temporal direction, but now the spatial correlations in the x2–x3 directions are disturbed by mixing. This uncorrelated field is generated to show the difference between a correlated field in the transverse direction (which is the Time-mapped Mann field) and a completely uncorrelated turbulent field in the transverse direction.

Figure  shows the change in spectra due to the application of time mapping on a turbulent Mann field. Discrepancies between the two fields arise from the time-mapping process and the interpolation procedure. The fields analyzed in Fig. 6 are the fields generated without re-scaling as explained in Sect.  to make sure that the analyzed fields keep their original characteristics before applying any mathematical scaling. It can be noticed from the figures that the peak values of κ1F1(κ1) occur at lower κ1 values compared to the theoretical and the Mann spectra, and the time-mapped spectra curves are shifted to the lift, i.e., towards lower κ1 values. Such a shift results from the time-mapping process that slightly changes the characteristics of the resulting field. Also, the time-mapped field has a lower peak value of κ1F1(κ1). It is expected that such a decrease happens due to the interpolation introduced to the Mann model during the time-mapping process.

From turbulence theory, there exist many other models which address more statistical characteristics and are based on a proper superposition of Gaussian statistics (). A similar approach has also recently been applied to the Mann model where the small-scale Gaussian statistics of Mann wind fields with varying covariances were superposed to yield small-scale intermittency (). However, our intention is to combine well-established models in wind energy which we did by combining the Mann model with the time-mapping procedure from the CTRW model. As a consequence, our model shows spatial correlations in the x2–x3 plane similar to the Mann model and allows us to investigate the influence of temporal intermittency. For the uncorrelated intermittent field, a comparable agreement as in the correlated case between this field and the Time-mapped Mann field in the longitudinal direction x1 can be seen since the arbitrary shuffling was only applied in the transverse directions.

The new model allows a detailed comparison between Gaussian Mann and non-Gaussian Time-mapped Mann fields. This comparison is shown in detail in this work. For further investigation of these deviations between Mann and Time-mapped Mann fields, Fig.  shows the same spectra as shown in Fig.  but with an equal number of grid points in all directions (in this case, nx1=nx2=nx3=512). This case is referred to as the “cubic case”. The spectra of these fields in Fig.  show that for a lower number of grid points in the x1 direction, the differences in spectra between Mann and the Time-mapped Mann field are lower.

As indicated in Sect. , the velocity spectrum is not sufficient to show intermittency. Therefore, increment statistics should be used in this case to show the effects of intermittency. The coherence of velocity components (Eq. ) of the two wind fields in the three spatial directions is plotted versus κ1Δx2 in Fig.  with Δx2=2.6 m. In this figure, the difference in the coherence of the two fields, as we move in the x2 direction, is obvious due to the time mapping in the longitudinal direction. The Mann model is based on the von Kármán energy spectrum, which is reflected by the coherence of the Mann-modeled wind field. On the other hand, a comparison between the coherence of the two wind fields versus κ2Δx3 for the cubic case is shown in Fig.  in the Appendix. In the latter case, the differences in coherence between the two fields are small. This is due to the unchanged x2–x3 planes during the time-mapping process. This shows that the coherence with respect to the longitudinal direction is highly influenced by the time mapping, whereas it is not with respect to the transverse direction. That shows that the time mapping goes at the cost of the spatial correlations in the x1 direction, which is expected since the x1 axis is stretched and compressed, which is also reflected by the spectrum in the x1 direction in Fig. a. By comparing Figs.  and , it can be noticed that the plots in Fig.  look smoother. This is also due to the difference in the number of grid points in the x1 direction between the two cases.

Following on from the two-point statistics analysis, the velocity increments v1 defined in Eq. () and the degree of intermittency of the wind fields are analyzed. Figure  shows the comparison between the PDFs of the velocity increment of both the Mann and the new Time-mapped Mann wind fields. It is obvious in this figure how extreme events in the case of the time-mapped wind field (the red markers) have a higher probability than the corresponding events in the case of the Gaussian Mann field (the black markers). Also, it can be noticed that the intermittency increases with the decrease in τ. Note that this τ refers to the temporal increment in Eq. () and not to the τα introduced in the context of the Lévy distribution. Another important measure of the increment statistics is the kurtosis calculated from Eq. (). In the case of the kurtosis of the wind fields, the angular brackets denote averaging over time, and kurtosis, in this case, is only a function of τ. Figure  shows a comparison between the kurtosis of the velocity increment from the Mann field and the Time-mapped Mann fields at different τ values. As explained earlier, higher values of τ show a Gaussian distribution with kurtosis of 3, and lower values of τ show more deviation from the Gaussian PDF of the Mann field.

After analyzing the effect of the time mapping on the different statistics of the wind fields in the previous section, here the effect on exemplary turbine loads is studied. The main point of this section is to see whether the intermittency from the wind field is carried to the turbine structure. If the turbine loads are proven to reflect intermittency, this might have a significant effect on the dynamic loads acting on the wind turbine. For example, the damage equivalent loads (DEL) calculated from fatigue analysis might be affected due to intermittency because the probability for extreme values in the load increments is larger than for a Gaussian distribution.

In this work, BEM simulations of the NREL WindPACT 1.5 MW () virtual wind turbine are performed using the aeroelastic simulator NREL FAST (v8.16) (). The main characteristics of the turbine and input parameters for the simulations are summarized in Table . The wind fields used in these simulations are the same Mann and Time-mapped Mann fields analyzed and compared in Sect. .

Four different load sensors were selected for analyzing the effect of the intermittent wind: the bending moment at the root of the blade in the flapwise direction (RootFlap), the rotor torque (Torque), the rotor thrust (Thrust), and the bending moment at the base of the tower in the fore–aft direction (TwrForeAft). According to , these four sensors are expected to be highly sensitive and mainly dominated by aerodynamic forces in the direction of the flow and not by other sources of load such as gravitational forces.

Firstly, the mean values and standard deviations calculated over the length of the simulation are presented in Table . Here, the results for the four load sensors of both the Time-mapped Mann (Time-map.) and the Mann (Mann) wind field are shown. Moreover, the relative differences (Rel.Diff) were also calculated (i.e., (Time-map.-Mann)/Mann).

In Table , the relative differences of the mean values and standard deviations of the analyzed sensors are less than ±1% between the Mann and the Time-mapped Mann wind field. It can be seen that this difference is in the same order as the differences in the mean wind speed 〈U1〉 and standard deviation 〈u12〉 at the hub point presented in Table . To investigate whether the intermittency in the wind field carries over to the turbine loads, the incremental statistics of the loads are evaluated. The load increments Mτ are defined similarly to velocity increments in Eq. (): 21Mτ(t,τ)=M(t+τ)-M(t), where M is the time series of the load sensor and τ corresponds to the time lag. Figure  shows the PDFs of Mτ for the four selected load sensors, for τ=3 s (top) up to τ=30 s (bottom) for the blade root flapwise bending moment and τ= 1 s (top) up to τ=30 s for the rotor thrust, rotor torque, and tower fore–aft bending moment. For a direct comparison to the PDFs of the velocity increments v1, the values of τ are selected analogously to Fig. . Similar to Figs.  and , all the individual PDFs are normalized to standard deviation 〈Mτ2〉=1. Here, the angular brackets denote moving averages over the time series.

The resulting load increments Mτ follow the characteristics of the wind velocity increments shown in Fig. . The PDFs of the Mann load increments are very well described by the Gaussian distribution. On the contrary, the PDFs of the load increments resulting from the Time-mapped Mann field deviate significantly from the Gaussian distribution for a range of scales. The observed heavy tails shown by the time-mapped PDFs reflect, as mentioned earlier, a higher probability of extreme load fluctuations than the probability of a Gaussian distribution.

Figure a shows a special case for the RootFlap load. The shown τ values differ from the other loads to make the intermittency visible in this case. The RootFlap moment is dominated by the 1P frequency while the other loads are dominated by the 3P frequency. Consequently, showing the same τ values in Fig. a as for the other loads would not reveal the intermittency in our special case. The energy spectra of the RootFlap and Thrust are shown in Appendix  for a deeper understanding of the different frequencies of the loads.

Also, it turns out that the frequencies of the energy spectra carry over to the frequencies in the kurtosis, as shown in Fig.  for the thrust. A more detailed description of the evolution of the intermittency of Mτ with τ is presented in Fig. . Here, the results of Kurt(Mτ) from Eq. () are calculated exemplarily for the Thrust signal. It is visible that the values of the kurtosis for the load case with the Time-mapped Mann wind field are higher than 3 (the value of 3 indicates a Gaussian distribution) for all the values of τ up to around 20 s. This proves the intermittent characteristics of the loading. In contrast, the corresponding values for the Mann case fluctuate around 3 for all the considered timescales, agreeing with the Gaussian statistics. Additionally, for timescales below 1 s, a decreasing intermittent behavior of the load is visible in the time-mapped case when decreasing τ. The kurtosis of the thrust in Fig. 11 is overall lower than that of the corresponding wind field as shown in Fig. 8. This has been also shown by , who have used an atmospheric wind measurement as input to calculate the torque.

Further investigations should be done to explain the potential sources of the observable bumps in the kurtosis of the Time-mapped Mann case. A strong dependence of the peaks on the rotational frequency of the rotor has been recognized by the authors. However, this first turbine study aimed at investigating whether the intermittency which was introduced to the wind field by deriving the Time-mapped Mann field carries over to turbine loads in general. This has clearly been shown here.