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Wind fluctuations on time scales of several minutes to several hours are interesting because they fall in the part of the wind speed spectrum which separates turbulent flow from the mean flow (Stull, 1988). This part of the spectrum is often referred to as the ‘spectral gap’, since many authors have identified a local minimum in the wind speed spectrum at a period close to half an hour (for example, Van der Hoven, 1957; Petersen, 1975; Courtney and Troen, 1990; Yahaya et al., 2003). Based on the assumption of the spectral gap, we might expect there to be little variability in the wind speed on these time scales, but it has been shown that the spectral gap is not always well defined, and that in certain atmospheric conditions it may not exist at all (Gjerstad et al., 1995; Heggem et al., 1998). Wind speed fluctuations on these time scales are not a constant feature of the wind, and therefore exhibit climatological patterns with regard to time of year and other explanatory variables. Atmospheric processes that contribute to the generation of variance in the spectral gap region of the spectrum include large cumulus clouds (Stull, 1988), convective cells (Gjerstad et al., 1995; Heggem et al., 1998) and horizontal roll vortices (Heggem et al., 1998).
Large fluctuations in wind speed on time scales of minutes to hours also present important practical challenges, particularly for offshore engineering and construction projects in locations such as the North Sea. For example, episodes of severe wind variability can have serious implications for large offshore wind farms, because the high concentration of turbines within a small geographical area means that fluctuations in wind speed can be closely coupled to severe power fluctuations (Akhmatov et al., 2007). In particular, large amplitude wind speed variability with periods of several minutes up to several hours has been identified as a recurring and problematic feature in the wind speeds at the Horns Rev offshore wind farm off the west coast of Denmark (Akhmatov, 2007). The time evolving nature of the spectral properties of wind speed and other meteorological variables are also of relevance to statistical forecasting models. For example, the time evolving fluctuations in wind speeds on time scales of minutes to hours must be considered in the application of models which rely on a basic assumption of stationarity of the time series. Therefore, the existence and implications of severe wind fluctuations on time scales of minutes to hours cannot be ignored.
Turbulent fluctuations and wind speed profiles in the North Sea region have been the subject of a number of investigations, due to the importance of turbulent loading on wind turbines and other structures, and to the importance of knowing the hub height wind speed based on measurements at a different height (e.g. Tambke et al., 2005; Peña et al., 2008). Wind resource mapping over the sea (or the climatological average of the wind speed) has also been studied in detail due to its importance for the siting and planning of wind farms (e.g. Barthelmie and Pryor, 2006; Hasager et al., 2006). However, wind variability on time scales of several minutes to several hours falls between these two areas and has received less attention. The problem needs to be studied in a different framework from turbulence (which could be well described by classical spectral analysis given a nearly stationary set of atmospheric conditions), and from climatological averages (in which temporal variation does not play a role).
Several authors have recently addressed the problem of wind and power fluctuations on time scales of minutes to hours, and each has chosen a different technique to study the problem. Akhmatov (2007) investigated power fluctuations (which are assumed to be mainly driven by wind speed fluctuations) at the Horns Rev wind farm by calculating the maximum 10-min change in power and discovered that the largest power fluctuations were experienced when the wind direction was from the westerly sector. Sørensen et al. (2008) used the spectral properties of wind speed time series to create models of power production to be used in planning and siting of wind farms, focusing on scales of several minutes to several hours. Vigueras-Rodríguez et al. (2010) studied the importance of including both low-frequency wind speed fluctuations and the spatial correlations between them in an aggregated model of wind farm power fluctuations. Studying the problem of fluctuating wind power production on the east coast of Australia, Davey et al. (2010) created a metric of wind variability using a moving average of the standard deviation of the band-pass filtered wind speed, then related their metric to large-scale meteorological fields using a random forest model. They found that important predictors of wind variability included fields such as planetary boundary layer height, vertical velocity, wind speed and geopotential height.
The study of wind fluctuations lends itself to treatment in the spectral domain because we are interested in the variability over a certain range of frequencies. However, to isolate certain times when there is enhanced variability on these frequencies, an adaptive spectral analysis method which can uncover the time-evolving spectral behaviour of the time series is needed. A classical Fourier spectrum can either be applied to a long time series to give the best fit, on average, of a set of harmonics to the data, or can be applied to short segments of the data where it is limited by the difficulty of finding particular atmospheric regimes that last long enough to create a meaningful spectrum. Neither of these options is satisfactory if one wishes to describe the evolution in time of certain classes of statistical behaviour. Therefore, for this analysis, an adaptive spectral analysis method called the Hilbert-Huang transform (HHT) was adopted (Huang et al., 1998). The HHT is based on an empirical decomposition of the time series in such a way that instantaneous frequencies and amplitudes in the data can be calculated and combined to form a time-evolving spectrum. It is ideal for this analysis, since it describes the changing statistical properties of the time series. The potential of using the HHT for analysing wind speed time series has been demonstrated and extensively discussed in Vincent et al. (2010).
Here, the HHT is used to study wind fluctuations at the Horns Rev wind farm on temporal scales of 1 min to 10 h. To analyse fluctuations with periods of 1–10 h, a 4-year time series of 10 min wind speed observations is used, while the analysis of fluctuations with periods of 1–60 min is based on a 10-month time series of sonic anemometer measurements of frequency of 12 Hz. The flexibility of the method in creating time-evolving and conditionally averaged spectra is exploited to demonstrate climatological patterns between wind fluctuations and several potential explanatory variables including wind direction, time of year, pressure tendency, precipitation and stability. These explanatory variables are footprints of the physical conditions which may be the real underlying cause of particular classes of wind speed behaviour. For example, the wind direction relates to upstream surface conditions as well as the synoptic situation, heavy precipitation is suggestive of generally convective conditions, pressure tendency is a simple proxy for the synoptic cycle and stability is a measure of the buoyancy and shear in the boundary layer. Wind variability as a function of wind speed was also explored, but it was found that there was no strong relationship between wind variability on scales of 1–10 h and wind speed.
The structure of the paper is as follows. In Section 2, the measurement site and the data are described. In Section 3, the background of the HHT and its specific use in this study are discussed. In Section 4, the results are presented. In particular, it is shown that there is a strong directional dependence in wind variability, that the most severe wind variability occurs in autumn and winter, that variability is enhanced when precipitation is observed and that the spectral gap is better defined for stable than for unstable conditions.
2. Site description and data
The Horns Rev wind farm is situated in the Danish North Sea, with the nearest wind turbines approximately 14 km from the Danish coast. The wind farm consists of 80 2-MW turbines. Meteorological observations were performed at a 62-m-high mast located to the north west of the wind farm. The mast is instrumented with cup anemometers at heights of 15, 30, 45 and 62 m, wind vanes at heights of 60, 43 and 28 m, temperature sensors at heights of 55 m, 13 m and at a water depth of 4 m, pressure and relative humidity sensors, and for limited periods, a sonic anemometer at a height of 50 m. The locations of the wind farm and the mast are shown in Figure 1.
Data from the mast has already been described and used in several studies, including Peña et al. (2008), Peña and Gryning (2008) and Vincent et al. (2010). This study focused on two time series. For consideration of wind fluctuations on time scales longer than 1 h, a 4-year time series (2000–2003) of 10 min average wind speeds from the 62 m cup anemometer was used. For time scales shorter than 1 h, a 10-month time series from the sonic anemometer at a height of 50 m covering the months January–October 2004 was used. This time series had a measurement frequency of 12 Hz, and was averaged to a resolution of 10 s for the analysis presented here. The choice of time periods was motivated by the availability of the sonic anemometer data, and by the nearly gap-free time series of cup anemometer measurements in the years 2000–2003.
For the study of the relationship between wind variability and wind direction, measurements from the 43-m wind vane were used. Every effort was made to check that the measurements of wind direction did not contain an offset, but due to the difficulty in finding a suitable reference data set, the uncertainty in the direction measurements is around ± 10°. For the study of wind variability in different stability classes, wind measurements at 15 m, temperature measurements at 13 m and sea water temperature measurements at a depth of 4 m were used. Precipitation and pressure measurements from sensors mounted on the mast were used to study the relation between wind variability and rain and pressure, respectively.
Spectral analysis is an obvious way to study fluctuations in the wind. For example, the Fourier spectrum shows the set of frequencies which dominate the variance of the wind speed in a global sense. This is appropriate for a stationary time series—that is, one where the mean and variance remain constant with time. There are several arguments, however, to suggest that wind speed time series must be non-stationary on multiple time scales. For example, Post and Kärner (2008) illustrated that synoptic scale daily air flow characteristics in the Baltic Sea region were non-stationary using the Hurst Exponent, Andreas et al. (2008) used the definitions of first- and second-order stationarity to detect non-stationarity in turbulence time series, and Pinson and Madsen (2009) used adaptive Markov-switching autoregressive models to show that on short time scales, wind power time series undergo sudden changes in their statistical properties denoted as ‘regime changes’. As argued by Vincent et al. (2010), wind speed time series should be treated as non-stationary, even though the classification of such complex time series is difficult because it depends on the time scale under consideration.
For analysis of the spectral properties of a non-stationary time series, a local spectral technique is required. Available techniques include wavelets, a moving Fourier transform, or the relatively new HHT. The HHT, first presented in Huang et al. (1998), has been shown to be an excellent tool for analysing non-stationary time series, and was applied to the problem of describing the fluctuations in long time series of wind speed in Vincent et al. (2010). In the HHT, the time series is decomposed in such a way that the Hilbert transform can be used to meaningfully calculate the instantaneous frequency of each component. The result is a time-evolving spectrum that responds quickly to sudden changes in statistical properties of the time series.
The HHT has been described in detail elsewhere [for example, in Huang and Wu (2008)], so it will be discussed only briefly here. The HHT consists of three steps: empirical mode decomposition (EMD) of the time series into a set of components called ‘intrinsic mode functions’ (IMFs), normalization of the IMFs and application of the Hilbert transform to each component to calculate instantaneous frequencies.
The EMD begins by extracting the fastest oscillations from the time series, U(t). Two cubic splines are defined, one which passes through all the local maxima of the time series and the other which passes through all the local minima. The average of the two splines is considered the local mean of the data and is subtracted from the original time series. The result now has zero mean, but does not necessarily form an IMF since it can still contain introduced local extrema. Therefore the process is repeated until the remaining signal satisfies the conditions of being an IMF—in practice, that each maximum–minimum pair is separated by an axis crossing. When the first IMF, x1(t) has been found, it is subtracted from U(t) such that
U1(t) is the same as U(t), but with the fastest oscillations removed. The second IMF, x2(t), is then found from U1(t), such that
and so on. The result of the decomposition is the set of IMFs, xi(t), and the remaining low-frequency trend, ε(t), which may be added to recover the original time series, U(t).
where N is the number of IMFs into which the time series is decomposed.
Although Huang et al. (1998) originally recommended applying the Hilbert transform directly to the IMFs, later work showed that better results were obtained by normalizing the IMFs so that they had constant amplitude of unity (Huang, 2005; Huang and Wu, 2008). This avoided the problem of erroneous results if the spectrum of an IMF overlapped the spectrum of its envelope. Normalization is achieved by dividing each IMF by the cubic spline that passes through the absolute value of its extrema (that is, the envelope of the IMF) repeatedly until the amplitude converges to unity. Normalization separates each IMF into an amplitude modulation part, Ai(t), and a frequency modulation part, Fi(t) such that
The normalized IMFs satisfy the conditions for calculation of instantaneous frequencies using the Hilbert transform. That is, each IMF contains one unique frequency at each time. The Hilbert transform of a time series is its convolution with 1/t. For each IMF, we can write
where hi(t) is the Hilbert transform of xi(t), and PV denotes the principal value of the integral due to the singularity at t = τ. Rewriting Hi(t) in polar coordinates shows that it is a function of the instantaneous phase, θi(t), and the instantaneous amplitude, ai(t).
Defining the instantaneous frequency, ωi(t), as the first derivative of the instantaneous phase, we can write
The instantaneous frequencies and amplitudes of each IMF can be combined to construct a time-evolving spectrum which shows amplitude on a frequency–time axis. An example of the IMFs and time-evolving spectrum for a 14-day period of wind speed observations is shown in Figure 2.
There are further extensions and improvements to the HHT which are not used here, but which should be mentioned. First, problems relating to the uniqueness of the decomposition and the lack of physical meaning in an IMF which can contain frequencies from different physical processes at different times have been addressed in the work by Wu and Huang (2009). They suggest decomposing a set of pre-whitened time series into IMFs and treating the ensemble average as the true decomposition. Although this would be a good method for studying individual physical features, it was too computationally expensive for the long data sets in question in this analysis. Second, problems in the use of the Hilbert transform for certain classes of IMFs have been addressed in Huang et al. (2009), where it is suggested that the instantaneous frequency could be calculated directly from the IMF using the arctangent of the frequency modulation part of the signal divided by its quadrature. This development could also be important for detailed analysis of individual features, but is unlikely to affect the climatologically averaged results.
For climatological studies it is necessary to summarize the spectrum to find patterns and trends in long time series. As shown in Vincent et al. (2010), climatological patterns can be explored by integrating the spectrum over a range of interesting frequencies to create a time series of ‘variability’, or by creating conditionally averaged spectra based on some additional data such as wind direction, or time of day. These techniques are used here to demonstrate the patterns in wind variability with respect to time of year, wind direction, pressure tendency, precipitation and atmospheric stability.
4.1. Seasonal dependence of wind variability
The Hilbert spectrum was conditionally averaged according to time of year by dividing the year into 28 segments, each of 13 days, or 1872 observations. As 4 years of data were available, each spectrum was an average of about 7488 spectra. The 28 average spectra are presented in Figure 3 as a contour plot of average amplitude as a function of frequency and time of year for temporal scales between 1 and 10 h. The amplitudes have been scaled by frequency to emphasize the high-frequency contributions to the variability. As with any climatological result based on a limited number of years of data, it is possible that a single extreme event could have an unreasonable impact on the spectrum. However, the large amount of data used in the analysis probably means that the trends seen in Figure 3 have a high certainty.
The plot shows enhanced variability in autumn and winter compared with spring and summer. Autumn and the first part of winter are the time of year when the water in the North Sea is on average warmer than the air, as shown for 2001 in Figure 4, which means that there is a prevalence of thermally unstable conditions. The increase in variability in October and November is present to some extent in each of the 4 years of data that were analysed, and is not simply an artefact on one strong event.
4.2. Directional dependence of wind variability
Wind variability was expected to be dependent on wind direction due to both the preferred directions in certain synoptic conditions and to the different local surface conditions experienced when the flow comes from particular directions. At the Horns Rev meteorological mast, wind from directions of 174° to 13° approaches the wind farm from the North Sea, while wind from 13° to 174° approaches the wind farm from the land. The measurement mast used in this study is located to the north-west of the wind farm, so the wind directions between about 105° and 174° could be affected by the wake of the wind farm. It has been shown that there is both an increase in turbulence and a decrease in the mean wind in the wake of the wind farm (Christiansen and Hasager, 2005). However, the fluctuations studied here are slower than what is normally referred to as turbulence, and by definition of the HHT, the fluctuations are not influenced by the mean wind speed. Therefore, the wake is not expected to have a large impact on the results.
The average Hilbert spectrum as a function of wind direction was created by categorizing the individual spectra into 18 direction bins of width 20°. Synoptic patterns and surface characteristics both change on an annual cycle, so the direction-dependent spectra were further categorized according to time of year. The total bin sizes for each season for 4 years of data are shown in Figure 5, where it is seen that despite classification by both time of year and wind direction, each bin still contains at least 1000 individual spectra.
The average Hilbert spectra as a function of frequency and wind direction for the four seasons of the year are shown in Figure 6. The three direction sectors are labelled on the first plot, and are the same for the other seasons. There is no discernible projection of the number of observations in each bin on the spectrum, suggesting that enough data have been used in the analysis to ensure reasonable certainty in the results. There are some strongly preferred directions for the development of intense wind fluctuations, and these patterns are not stationary through the year. In autumn, when the maximum variability is observed, wind variability in flow from the sea is greatly exaggerated compared with that from the land. In winter, the same pattern is seen, except that the maximum variability is found in a narrower range of directions in the north westerly sector. In summer and spring, variability is generally less than for autumn and winter, and the most noticeable feature is the enhanced variability for periods longer than 3 h for summer time flow from the land. This is either an artefact of the sea–land breeze circulation, or a reflection of summer time surface-driven convection over the land.
This analysis is highly site specific, because it relies on not only the synoptic patterns, but also the surface conditions of various direction sectors. Compared to other offshore wind farm sites, such as the Nysted wind farm in the Baltic Sea, the land surrounding Horns Rev is simple. Flow from the west basically comes from the sea, while flow from the east comes from the relatively flat land. Moreover, most of the intense weather systems approach the wind farm site from the sea, so they are not complicated by topographic effects.
Since there is a good representation of all wind directions at the Horns Rev site, there was sufficient data in each direction bin during the 10 months of available sonic anemometer data to extend the spectra shown in Figure 6 to periods of 1 min, using the 12-Hz data averaged to a time resolution of 10 s. The average Hilbert spectra for periods of 1 min to 2 h for the four seasons of the year are shown in Figure 7. Note that the variability has generally smaller amplitudes on these time scales, so the contour shading scheme is not the same as for the 1–10 h time scale plots (Figure 6). Periods of 1–2 h overlap with the fastest hour of the plots as shown in Figure 6, although on this axis time scales are compressed to the bottom section of the plot. Because the data only cover the dates of 1 January–27 October, the autumn and winter seasons are both incomplete, containing only about 8 weeks of data. The high-frequency plots are therefore less representative than the 1–10 h time scale plots. They are also based on measurements from a different instrument, and in a different year to any of the years included in the 1–10 h plots. Despite these differences between the plots, the trend of enhanced variability for flow from the sea in autumn continues throughout all time scales. Variability in spring and summer is generally suppressed, apart from a narrow band of enhanced variability for flow directions of around 300°. A further interesting feature of these plots is the narrow bands of enhanced variability around the edges of the wake region for time scales shorter than about 4 min, seen most clearly in autumn and spring. This result is consistent with the recent work of Larsen et al. (2009), who showed that the wind speed in the wake of an 80-m diameter turbine rotor contains enhanced variance for frequencies greater than about 0.004 Hz. Given the uncertainty in direction measurements, the precise location of the region of the spectrum that is influenced by the wake may be shifted slightly from that shown in Figures 6 and 7.
4.3. Wind variability and the synoptic cycle
The well-defined patterns between wind variability and wind direction and season suggest that there may be some synoptic influences. To test this hypothesis in a simple way, the variability spectrum conditional on the three hourly pressure tendency was calculated. Positive values mean that the pressure is rising, and it is likely that the synoptic situation is post-frontal. Negative values mean that the pressure is falling, and that the synoptic situation may be pre-frontal. This is obviously an over simplification of the large-scale weather patterns, which are not just an easily identifiable progression of low-pressure troughs and high-pressure ridges. The analysis could be extended to include a more complex system of classifying the synoptic type, using methods such as those suggested by Soriano et al. (2006) or Bower et al. (2007), where it is expected that similar results would be obtained.
The centred pressure tendency was divided into 12 bins, each of which had a width of 0.5 hPa per 3 h. There were at least 2900 observations in each bin, although the number of observations in each bin was very uneven. The averaged spectrum (Figure 8) shows some tendency for increased variability when the pressure is rising fast (that is, in post-frontal situations), and to a lesser extent when it is falling fast (that is, in pre-frontal conditions). When the pressure tendency is close to zero, variability appears to be suppressed. This result shows that the greatest wind variability is experienced around the time of the passage of a front.
4.4. Wind variability and precipitation
There are several reasons to believe that wind variability is linked with precipitation. First, the regular patterns of rain showers seen on rain radars show that a location may be subjected to a nearly regular sequence of passing showers for some hours. As each rain shower passes, the gustiness associated with the outflow of the convective cell should create a periodic fluctuation in the wind speed. Further, according to Weusthoff and Hauf (2008), a distinct precipitation ‘track’, consisting of a collection of forming, splitting and dissipating precipitation cells under post-frontal conditions over land, can have life times of 29–40 min so that the changes in wind speed with the passage of showers should also follow these or slower time scales.
The results already shown in this paper also suggest a link with rainfall. Rain showers tend to approach the west coast of Denmark from the North Sea, and the rainiest months in western Denmark are August to December (http://www.dmi.dk/dmi/index/danmark/klimanormaler.htm). Although there were not enough rainfall events during the 4-year study period to create conditionally averaged spectra based on rain or rain-rate, it was possible to create a simple scatter plot of variability against rain rate.
For this analysis, ‘total variability’ was defined as the sum of all amplitude contributions between periods of 1 and 3 h, which is a scalar time series that gives the upper bound on the amplitude of variability if all oscillatory motions were in phase (Vincent et al., 2010). ‘Rain rate’ was a scalar time series, defined at time t as the total rainfall observed within a 90-min window before and after time t. This definition is justified by the idea that rain showers can be used as a proxy for generally convective conditions which are likely to last longer than the briefly observed passing rain shower.
The scatter plot of variability against rain rate is given in Figure 9. The points in the scatter plot are aligned on particular rainfall values because precipitation is measured in discrete 0.25 mm quantiles. Further, the graph is hard to read because there are fewer and fewer values for increasing rain amounts. To aid interpretation, contours that show the probability density function of variability for each given rain rate have been added. From the contours, it is seen that with increasing rainfall, there tends to be an increase in the observed wind variability. There are also many high variability events when rain is not observed. Unfortunately an observation of no rain at a single point does not mean that there are not nearby rain showers, or that conditions of cellular convection and precipitation are not occurring. To draw a stronger relationship between rain showers and wind variability, spatial observations of precipitation (for example, from a rain radar) would be ideal.
4.5. Wind variability as a function of atmospheric stability
The effect of stability on offshore wind regimes has been considered in detail with regard to the characterization of wind profiles. For example, Lange et al. (2004) studied the performance of the wind resource assessment program WAsP at offshore sites in the North Sea, and suggested that both internal boundary layers and the dependence of sea surface roughness on wind speed were complicating factors in the calculation of offshore wind profiles. Tambke et al. (2005) found large errors in wind profiles predicted by numerical weather prediction models at Horns Rev, and suggested improved modelling of the profile by considering the Ekman layers of the air and sea. More recently, Peña and Gryning (2008) found that non-dimensional wind profiles at Horns Rev could be modelled by including a scaling based on Charnock's model for sea surface roughness, and Peña et al. (2008) studied models of wind profiles at Horns Rev using lidar measurements that extended the wind profile to 160 m. Nissen (2009) used a mesoscale model to study seasonal variation in wind profiles at a Danish coastal site near Horns Rev, and found that the change in surface momentum flux divergence introduced on the England–North Sea coastline could cause an inertial oscillation to form during the spring and winter seasons. He suggested that this effect could be partly responsible for large variations in the surface layer wind profiles observed during spring and winter.
Here, we are not studying wind profiles, but the difficulties that have been encountered in describing the instantaneous marine wind profile are indicative of the diversity of largely unobserved boundary layer structures that are encountered at a site such as Horns Rev. These structures will have a strong influence on the mesoscale variance. For example, the boundary layer height, the structure of mesoscale convection and the amount of buoyancy will affect the time and length scales of eddies that exist in the boundary layer.
Wind speed and temperature measurements from the meteorological mast at Horns Rev were used to classify the stability according to the Obukhov length, L. L was calculated from the bulk Richardson number, Rib, as defined by Grachev and Fairall (1997) and used by Peña et al. (2008) to classify the stability at Horns Rev.
where Δθv is the difference between the sea surface temperature and the virtual potential temperature at height z. Uz and Tz are the wind speed and temperature at height z, respectively. Here, wind speed measurements at 15 m and temperature measurements at 13 m were used. Sea water temperature measurements at a depth of 4 m were available, which were shown by Peña et al. (2008) to be close to satellite-derived sea surface temperature observations. Additionally, measurements of pressure and relative humidity were used to calculate the virtual potential temperature. Gaps in the measurements meant that Rib could be calculated for most of 2001, and for parts of 2002 and 2003.
Following Grachev and Fairall (1997), the Obukhov length is related to Rib according to
for stable conditions, and
for unstable conditions. C and α are constants, for which Grachev and Fairall (1997) suggest the values C = 10 and α = 5. Following a similar classification to that in Peña et al. (2008), L was classified as stable for 50 < L < 500, unstable for − 500 < L < − 50 and near-neutral for |L|> 500.
The nonlinear nature of the Obukhov length means that it does not make sense to make a contoured conditional spectrum of amplitude as a function of frequency and stability. Instead, a ‘marginal spectrum’ was created for each stability class. The marginal spectrum, as defined in Huang et al. (1998), is the time averaged Hilbert spectrum, and yields a relationship between amplitude and frequency of the same form as a Fourier spectrum. However, because of its instantaneous rather than global formulation, Huang et al. (1998) argues that it has a ‘totally different meaning’ to the Fourier spectrum. Nonetheless, key features such as the synoptic and diurnal cycle should be present in both analyses.
Using the Obukhov lengths calculated as described above, the marginal (or time-averaged) spectrum was calculated for the classes of stability given in Table I. The number of 10 min observations in each class for the 4-year time series of cup anemometer observations and for the 10 months during which higher-frequency sonic anemometer observations were available is also shown. The number of observations is influenced by both the frequency of the stability class and by the data availability. For the latter period, the stability class was treated as constant for each 10-min time period, in which there were 60 wind speed observations. While the wind speed time series used for the variability analysis had very few missing data, there were more missing data in the measurements used for calculation of the Obukhov length. These values are omitted from the analysis.
Table I. Stability classes and the number of 10-min observations (N) during the two study periods
Obukhov length, L (m)
N (Jan–Oct 2004)
− 500 < L < − 50
50 < L < 500
The marginal spectra for the stability classes are shown in Figure 10. The part of the spectrum calculated from the 10 min cup anemometer data in 2001–2003 is separated from the part calculated from the sonic anemometer data in 2004 by the grey line with a frequency of 2.7 × 10−4 Hz, corresponding to a period of 1 h. There is a good match between the two segments, particularly considering that the data come from different instruments mounted at different heights, and were measured in different years. The Fourier spectrum for the same two data sets is shown for comparison, although the Fourier spectrum for the sonic anemometer data in 2004 covers a 100-day period starting on January 1, 2004, during which period the time series was nearly free of gaps. Both the marginal Hilbert and the Fourier spectra were smoothed using a constant log-window smoothing algorithm.
Uncertainty in the calculation of the Obukhov length arises from uncertainty in the meteorological measurements, and from uncertainty in the parametrized calculation of the Bulk Richardson number. Error analysis shows the uncertainty in the measurements to be largest for neutral conditions, where the small difference between the virtual potential temperature at 13 m and the sea water temperature is sensitive to small errors in either measurement. This will result in some incorrect classification between near-neutral and stable events, and between near-neutral and unstable events. However, events with a well-defined temperature difference will be classified correctly.
There are several different ways in which the stability could have been calculated. For example, a gradient Richardson number which used the wind speed difference between two levels could have been used, although this would have introduced extra sensitivity to errors in the wind speed measurement. Instead of the equations of Grachev and Fairall (1997), the Obukhov length could have been calculated directly using stability functions as defined in Stull (1988). It was found that while the choice of stability calculation results in some points being placed in different stability bins, the averaged results show little sensitivity to the specific method of stability calculation.
The marginal Hilbert spectra in Figure 10 show that there is a minimum in wind fluctuations for time scales of around 6 min for stable conditions, but that no such minimum exists for unstable or neutral conditions. The Fourier spectrum, which reflects the combined effects of all stability conditions, shows a slight minimum in variance for time scales of around 12 min. These values are within the range of frequencies for the spectral gap suggested by authors such as Petersen (1975), Courtney and Troen (1990), Gjerstad et al. (1995), Heggem et al. (1998), Yahaya et al. (2003) and Metzger et al. (2007). Previous analysis of the spectral gap has addressed a variety of flow conditions and surface types. For example, Yahaya et al. (2003) analysed wind speed time series for an inland site in North Spain, and found a very well-defined spectral gap with a period of around 12 min. Courtney and Troen (1990) calculated the wind speed spectrum in a flat, agricultural area of Denmark, and found a modest spectral gap, with only a factor of two separating the variance in the spectral gap from that in the turbulent peak. Gjerstad et al. (1995) calculated spectra based on carefully chosen 12-h segments of maritime wind observations in stable to neutral and unstable conditions. It was found that there was an obvious spectral gap at a period of around 20 min for stable to neutral conditions, but that there was no discernible spectral gap for unstable conditions.
The variety of results for the spectral gap suggests that it is not a universal feature of all atmospheric spectra. At Horns Rev, the spectral gap is only clearly seen for stable atmospheric conditions, a result consistent with the findings of Gjerstad et al. (1995). The wind variability on time scales shorter than about 3 h occurs predominantly in unstable and near neutral conditions.
Fluctuations in wind speed with periods of several minutes to 10 h are not a constant feature of the near surface wind speed. This work demonstrated certain flow conditions observed at the Horns Rev wind farm in which wind speed fluctuations tend to be more intense. For example, the most variability in the wind speed is found in an unstable atmosphere, while for a stable atmosphere the variability is greatly suppressed. Further, the most intense fluctuations in the wind speed are found for flow coming from the sea, particularly in the autumn and winter seasons of the year, and the wind is more variable when the pressure is quickly rising or falling. Finally, there is a greater chance of observing intense wind speed fluctuations if there are rain showers in the vicinity of the location.
Presenting the wind speed spectrum as set of conditional spectra according to stability classes was a useful way of showing that the spectral gap is a much more prominent feature of a stable than an unstable atmosphere. Since the conditional spectra were averages of selected parts of the time-evolving spectrum, the stability classes did not need to fall in consecutive parts of the time series. This means that a lot more data were included in the analysis than would have been possible if we had been constrained to finding continuous periods of a certain stability class.
The finding that the variability of the wind speed is strongly dependent on the time of year and on the atmospheric conditions emphasizes the fact that wind speed time series are non-stationary. That is, their statistical properties change with time, and the time-dependent nature of the spectrum will not be captured in a global method such as the Fourier transform. The application of the HHT in this work was found to be an effective way to study the non-stationary wind speed time series. It allowed the separation of the spectral behaviour into certain classes of atmospheric conditions. These classes of atmospheric conditions do not necessarily occur in long, continuous segments of the time series, so calculation of Fourier spectra would have been impossible. The HHT is a relatively new methodology, and several recent extensions to the technique were not pursued here. However, applications such as the ensemble EMD of the data, which aides physical interpretation of the individual IMFs, and the direct quadrature frequency calculation, which improves on the Hilbert transform calculation in certain conditions, would be useful extensions to the work, particularly if direct physical interpretations of the IMFs were to be sought.
Some of the results found here are likely to be general, while some others may be site specific. For example, the relationship between variability and stability is likely to apply at other locations. Other results, such as the relationship between variability and direction, may signify a more general result about flow from the sea in contrast to flow from the land (for example, the result could reflect boundary layer structures such as open cellular convection or convective rolls which exist predominantly over the sea). The work could be repeated for other sites to test the generality of the results. Further, the general applicability of the results could be tested through modelling, both of idealized and real situations.
The results presented here have implications for both the statistical and physical modelling of wind speeds. For statistical modelling, the results show that episodes of intense wind variability can be related to certain criteria such as wind direction, which occur on a larger spatial scale and may be more predictable than the higher frequency part of the wind speed time series itself. The results suggest that it is important to consider the non-stationarity of wind speed time series when applying statistical models that have stationarity as a basic assumption. In fact, given the rapidly advancing theory of the HHT, it seems likely that nonparametric time series prediction methods could be developed based on theory similar to the HHT itself. Development of such models would not only be theoretically interesting, but would be of great technical value for the optimum management of wind power based on the predicted wind speed.
For physical modelling, the results raise interesting questions about why variability is more intense in some situations. The trends in wind variability with respect to time of year and wind direction show that while the wind variability itself is a meso- or microscale feature, it can be linked closely to larger-scale forcings. Therefore, through proper understanding of these forcings, good operational predictions of wind variability can probably be achieved without resorting to extremely high-resolution modelling. It also remains an interesting challenge to find whether results are directly reproducible in mesoscale or large eddy simulation models, in which case it should be possible to explicitly forecast the onset of severe wind variability, and to study directly the boundary layer structures responsible for the intense fluctuations in wind speed.
Measurement data were supplied by Vattenfall and DONG Energy as part of the Danish Public Service Obligation (PSO) fund project ‘HRensembleHR—High Resolution ensembles for Horns Rev’ (under contract PSO-6382), which is gratefully acknowledged. The work was partly supported by the PSO project ‘Mesoscale atmospheric variability and the variability of wind and production of offshore wind farms’ (under contract PSO-7141). The advice on the manuscript from Alfredo Peña, Rozenn Wagner and Erik Lundtang Petersen from Risø-DTU was greatly appreciated. The helpful and interesting comments of two anonymous reviewers are also acknowledged.