The model we use is the COnsortium for Small-scale MOdelling-CLimate Mode (COSMO-CLM) non-hydrostatic limited-area atmospheric model . COSMO-CLM is a community model that is continuously maintained and developed by its users under the coordination of the German Weather Service (DWD). The dynamical core of this model solves the primitive thermo-hydrodynamical equations describing a compressible flow in a moist atmosphere with a time step of 10 s. The COSMO model uses an Arakawa-C grid and a staggered Lorenz vertical grid with terrain following Gal-Chen coordinates. The horizontal grid is mapped out in rotated coordinates with a spacing of 0.0135°. This corresponds to a horizontal distance of approximately 1.5km, which allows for explicit representation of deep convection in the model. For shallow convection, the dynamical core of the model is expanded with a shallow-convection parametrisation of . Sub-grid-scale turbulence is parametrised by a one-dimensional diagnostic level 2.5 closure scheme based on a prognostic turbulent kinetic energy (TKE) equation . Further parametrisations that take sub-grid-scale processes regarding micro-physical cloud processes and radiative transfer into account are present in the model. COSMO-CLM has proven to be an adequate tool for long-term convection-permitting simulations, allowing for a statistical analysis of mesoscale weather systems . It has also shown its value for studying wind speed metrics . The model is directly driven by the 31 km resolution ERA5 reanalysis data . The sea surface temperature (SST) is also provided by ERA5 and updated on an hourly basis. The diurnal cycle of the SSTs is relatively limited at 1 to 2 K, and this is comparable to the diurnal cycle found in potential sea surface temperature of the Baltic Sea Physics Analysis and Forecast provided by the Copernicus Marine Environment Monitoring Service CMEMS;. More information about the nesting strategy can be found in Appendix . Some deficiencies in ERA5, like meridional variability in surface winds and moist convection , are better represented at the kilometre-scale resolution. For practical reasons the calculation of the 10-year simulation has been divided into smaller periods, but using the restart files generated by COSMO, these periods were initialised with a warm start. To account for the relaxation and spin-up of the forcing data, 20 grid points from every side of the domain are excluded from the analysis domain. The setup for this simulation has been used before to investigate the effect of wind farms on the regional climate in the German Bight .

The model was evaluated with L3 scatterometer data from the ASCAT instrument on the Metop-A satellite available from the Copernicus Marine Environment Monitoring Service CMEMS;. The Metop-A satellite has a Sun-synchronous orbit and scans the Kattegat at approximately 09:00 and 21:00 UTC. The ASCAT instrument from Metop-A infers the 10 m wind vector over sea via the backscatter of the microwave radiation it emits in three different directions. Validation with ASCAT data has already been used for a variety of offshore wind datasets . Points flagged by CMEMS for poor quality are removed from the dataset. The scatterometer does not cover the Kattegat on every overpass it makes, yet a total of 299 503 wind vector cells (WVC), which is equivalent to ≈20800000 aggregated (i.e. averaged) model grid cells, is available for comparison. As the scatterometer only works over water surfaces, it is not available in the vicinity of coastlines . In order to allow for a comparison, the COSMO model output is aggregated to the 12 km grid of the scatterometer data. A direct point-to-point comparison between ASCAT and model output, like the root-mean square deviation (RMSD), might underestimate the quality of the model in representing the mesoscale variability. In the absence of strong forcing over the sea, a model can easily reproduce a mesoscale system albeit slightly shifted in time and/or space. The reproduced mesoscale system then produces two errors. First, it causes an error over the place where it should have been but is not reproduced now, and secondly, it causes an error over the place where it is now but should not have been, a situation which is referred to as a double penalty . That is why, apart from the root-mean-square deviation (RMSD), statistical methods to assess the distribution of wind speeds like the 25th, 50th, and 75th percentile wind speeds are also used. The comparison of these distribution parameters has an added value compared to an evaluation of the mean, as an erroneous distribution can still have a good representation of the mean due to error compensation. Evaluating distribution parameters is therefore more rigorous than only evaluating the mean.

Using scatterometer data, only the near-surface wind speeds at 09:00 and 12:00 UTC of our model can be evaluated, which is not necessarily representative of the hub-height wind speed. Complementary to the scatterometer, a validation by a light detection and ranging (lidar) device located 2 km west of the Anholt wind farm is used. This is not an ideal location, since it might be affected by the wind farm, which is not implemented in our model, but it is the only measurement at hub height in the Kattegat area. Therefore, a sub-period of the measurement campaign was used, one in which the Anholt wind farm was still under construction, and the normalised availability of operational wind farms increased from 12 % to 36 % (see Fig.  for the availability of the Anholt wind farm). Note that eventually 111 wind turbines became operational in the Anholt wind farm. Moreover, the wind mainly blows from the west in this area , resulting in a lidar signal that is largely unwaked. The lidar provides 10 min averages of the wind speed and direction between 2013 and 2014 at different height levels and is offered by Ørsted. The 10 min averages are compared to the instantaneous wind speeds of the COSMO grid cell corresponding to the location of the lidar. Note that the COSMO wind speed values are a grid cell average. This spatial averaging, similar to the temporal averaging of the lidar data, should diminish the impact of wind gusts on the analysis. The model wind speeds are interpolated to the measurement height level of the lidar using the wind profile power law given by 1Vhlidar=Vhm⋅hlidarhmα, where the shear coefficient α is given by 2α=ln⁡Vhm+1/Vhmln⁡hm+1/hm. Here hlidar, hm, and hm+1 are measurement heights of the lidar and the two COSMO output levels closest to the measurement height of the lidar. The agreement between the lidar data and the model output is quantified using the Perkins skill score PSS;. The PSS quantifies the overlap between two equally binned probability distributions and is calculated by taking the sum of the minimum of the two probability distributions over all the bins. Formally, this is expressed as 3PSS=∑i=1Nbinsminlidari,modeli, with lidari and modeli being the bin values of the lidar and the model probability distributions, respectively.

The spectral density of a signal is estimated using a periodogram, also referenced to as a spectrum, calculated using Welch's method . Due to the uncertainty in a signal, such as a time series of the wind speed, we can only make an estimation of the underlying spectrum. In Welch's method, the 10-year time series of 10 min interval wind speeds is cut into overlapping sections of approximately 7 days using a Hann window (1024 output intervals) in our case, with an overlap of 50 %. This window length allows for part of the synoptic peak in the periodogram to be seen. Subsequently, the spectral density of every section is calculated using a fast Fourier transform (FFT) algorithm. The Hann window reduces the reflections arising from performing an FFT on a finite time series . The final mean spectrum is then calculated as the average of the spectrum of every section. This last step averages out the fast natural variability in the spectrum and results in an estimate for the true 10-year mean spectral density of the signal.

The periodogram is calculated for the four meteorological seasons, allowing for a comparison between different seasons. The typical periodogram for a certain season (for instance spring, comprising the months March, April, and May) is then obtained by taking the mean of the periodogram of that season over every year of the simulation (in this example taking the mean over all the spring periods in the years 2010–2019). For the winters of 2010 and 2019, the months of December and January–February, respectively, are not included in the spectrum. Calculating the combined periodogram of these disjunctive time periods is useful to depict the seasonal effect in wind variability. Student's t test at the 95 % confidence level is used to quantify whether the differences between the winter and the summer spectra are significant over a given period.

Using the method described above, an average periodogram of the 100 m wind speed is calculated for every grid point for each season. These averaged periodograms can be integrated over a time interval of interest, resulting in one single value per grid point that quantifies the temporal variability over that time interval. This makes it possible to visually compare the different grid points for each season and allows us to focus on specific scales, which would be challenging in the time domain.

Periodograms can also be used to examine the fluctuations in potential wind power. The power curve of a wind turbine converts a wind speed time series to a wind power time series, and from the latter time series a periodogram can be calculated. The power curve of the Siemens SWT-3.6-120 wind turbine is available in table form , and a cubic spline interpolation is used to obtain the power for every possible wind speed. As the hub height for this type of turbine is 90 m, the 80 and 100 m model wind speeds are interpolated to 90 m via the power law given in Eq. ().

We introduce the Mesoscale Spatial Variability Index (MSVI), extracted from the horizontal wind field at 100 m, by sliding two square windows of different sizes grid point by grid point over the study area (Fig. ). Within both of these windows, the mean wind speed is calculated. The small window aims to estimate the mesoscale wind speed, while the large window follows more or less the synoptic background. We define the MSVI as the deviation of the ratio between these two wind speeds and 1, 4MSVI=〈v〉smallwindow〈v〉largewindow-1, which returns a dimensionless number quantifying how much larger mesoscale wind speeds are relative to the synoptic background.

One limitation is that this metric is derived using a model run with 1.5 km grid spacing, which is a slightly higher resolution than most wind atlases (e.g. DOWA; , and NEWA; ), so the smallest signals that can be captured are at the effective resolution of 10×10 grid points ∼15 km. This effective resolution is chosen as the size for the small window, thereby capturing most of the mesoscale variability, although the smallest mesoscale signals are averaged out by this or any other product based on state-of-the-art mesoscale model simulations. The synoptic scale starts at ∼100 km , but when taking this size for the large window, frontal systems were averaged out. Therefore, we took approximately half of this as dimensions for the large window to fully capture the synoptic background velocity field including the frontal systems. A size of 30 grid points (∼45 km) is chosen for the large window size, which is 9 times larger in area than the small window. Using the large window, thunderstorms and smaller mesoscale systems are averaged out, whereas cloud clusters and fronts are not. With these window sizes, the MSVI was able to identify mesoscale systems like convective systems and land–sea breezes (see Sect. ). This is also the scale of variability that can result in sudden power ramping events for wind farms. Even though this is not, strictly speaking, the mesoscale length scale (100 km), we will refer to it as such throughout the rest of the paper. As the small window is contained within the large window, an upper bound exists for the metric. Indeed, the small window has a mean wind speed of x m s-1 and the large window has a wind speed of 0 m s-1 everywhere else, but in the small window, the denominator of the MSVI will be 19⋅x m s-1, giving an upper bound of 8 to the MSVI.

Since the focus of this paper is on the offshore wind conditions, the MSVI is calculated only if at least 75 % of the large window is sea, while all the onshore wind speed values are set to not a number (NaN). The maximum value per time step is calculated, quantifying the intensity of a mesoscale weather system in the domain during that specific time step.

A comparison between ERA5 and our simulation output was performed using the MSVI, assessing the added value of convection-permitting simulations for mesoscale variability. This is done by regridding ERA5 to the grid of our COSMO simulation output and then applying the MSVI metric to the ERA5 data.

Given the offshore wind speed variability, wind power fluctuations are expected. Wind speed and wind power are related to each other via the power curve of a wind turbine, and given this relation, an MSVI analysis of the wind power could be made in principle. Yet the shape of this power curve imposes restrictions on our methodology: below the cut-in wind speed of a turbine (3 m s-1) no power is produced, prohibiting an MSVI calculation for power fluctuations as the denominator would become zero. Instead, we opt for an RMSD comparison between power time series for a stationary 10×10 window and a 30×30 window (in grid points). The stationary windows are positioned in the area of the Anholt wind farm. The small window is in area 9 times smaller than the large window. In order to cover the whole large window, nine small window power time series are calculated and compared to the large window power time series. The average over these 9 RMSD values is then taken to assess the differences in wind power.

Welch's method and the MSVI metric complement each other. Welch's method produces a spectrum with information about wind speed variations over different timescales for every pixel. Bundling this information in a spectrum does, however, remove the temporal resolution of that time series. The MSVI, on the other hand, aggregates spatial information into a metric and in doing so gives up spatial resolution. The temporal resolution in this method stays intact. As mesoscale systems are by definition bound in both space and time, these two methods together offer a more complete view on offshore mesoscale wind speed variability than one metric would yield on its own.

The simulation is evaluated using scatterometer data. Over the 10-year integration, we find a spatiotemporal RMSD between the 10 m wind speeds of 1.35 m s-1 over the data points located away from coastlines and small islands. Note that the ASCAT wind speed error is about 0.5 m s-1 with no particular expected regional deviation in the COSMO domain . Moreover, the 1.35 m s-1 RMSD may be explained by the double penalty mentioned earlier, but there is no straightforward way of testing this. However, tests with spectral nudging did not substantially improve the performance, indicating that the lateral boundaries to a large extent control the timing and location of weather systems for this domain and model configuration. Distributions of wind speeds are compared via the 25th, 50th, and 75th percentiles (Fig. ). For the 25th and to a lesser extend the 50th percentiles, simulation and observation disagree, especially near the coastlines and over the islands. On the western shores, simulation output and the scatterometer agree better than on the eastern shore. The simulation also shows lower wind speeds over the Anholt and Læsø islands. Recently, ASCAT winds have been validated extensively close to the coast, and an operational product has been introduced providing good quality winds as close as 10 km from the coast , which could in the future be used to further evaluate the discrepancies between simulation output and scatterometer data. Relative differences between the simulation and the scatterometer further away from the islands and the coastline are smaller: -13 % for the 25th percentile, +1 % for the 50th percentile, and +5 % for the 75th percentile.

For validation of the model output above the 10 m level, a lidar device measuring the wind speeds at 102.6 m is used. The 10 min model wind speeds at the 80 and 100 m level are extrapolated to 102.6 m using the wind profile power law (Eq. ) and then compared to the lidar data. The nearby Anholt wind farm was being built during the lidar measurement campaign and was not fully operational (wind farm availability is shown in Fig. ). In order to minimise the impact of the Anholt wind farm on the validation of the model, we have also made a comparison taking into account only the first 100 d of the measurement campaign. Over the whole 2-year-long lidar dataset, the normalised PSS of the simulation output compared to the lidar data is 0.97 (Fig. ). During this period, the simulation output overestimates the lidar data by 0.08 m s-1 on average. Taking only the first 100 d into account results in a similar normalised PSS of 0.96, with COSMO overestimating the lidar wind speeds by slightly less, namely 0.02 m s-1. Even though there is some uncertainty about the effect of the wind farm on the lidar data, the small difference in performance between the two periods, together with the close correspondence between lidar data and COSMO, gives us some confidence that the model performance is adequate. It might, however, be possible that COSMO underpredicts winds in the simulation without turbines, which is masked by the wind farm wakes experienced by the lidar. However, the effect is likely small due to the wind mainly blowing from the west in this area .

Using Welch's method, the spectrum can be estimated for the 10-year 100 m wind speed time series of each pixel in our domain. Differences between winter and summer are investigated using the combined periodogram for all winter months, December, January, and February (DJF), and for all summer months, June, July, and August (JJA; Fig. ). The resulting periodograms resemble what is found in the literature by for 100 m offshore wind speed spectra. For the lower frequencies, the periodogram seems to level off, which is indicative of the synoptic weather peak . Comparing the simulation output with lidar data, it appears, however, that our simulation slightly underestimates the intensity at the higher frequencies (Figs.  and ). These periodograms are averaged over a subdomain of the Kattegat, excluding coastlines and islands. On the high-frequency end of the periodograms, a higher intensity is found in DJF compared to JJA. Relatively warm SSTs in winter result in turbulence and unstable conditions. Unstable conditions are a driver for convective cells. The systems are not bound to a specific place and are advected with the wind. Therefore, they can move relatively fast and result in short-timescale variability 20 min–1 h;. For wind speed variability on the long timescales (6–12 h), however, there is a larger intensity found for JJA. Using Student's t test, we find that the differences over these time slots are significant at the 0.05 level. The difference between DJF and JJA on the longer timescales may be due to the sun being higher in summer and heating the land more effectively. With the sun over land, the air above it expands and generates a breeze over the sea during the morning or afternoon. Nocturnal jets may be formed during the night due to the land cooling. Sea breezes are fed by the contrast in land surface temperature and sea surface temperature. This keeps them confined to the vicinity of the intersection between land and sea, and this results in longer timescale variability .

Next, periodograms for each grid cell are integrated over a chosen time slot, resulting in a map that quantifies the wind speed variability, allowing for a spatial analysis of the temporal wind speed variability. In the previous paragraph, a stronger variability was already found in JJA than in DJF for the long-timescale winds (Fig. ). In contrast, for the short-timescale winds, there is a stronger variability in DJF than in JJA (Fig. ), albeit an order of magnitude weaker than the long-timescale variability. During winter, the variability increases from southwest to northeast. This gradient is probably related to southwest prevailing winds and indicates that the convective systems need some spin-up time and distance to reach their maximal wind speed variability. Indeed, when isolating a time series of 11 consecutive winter days with only easterly winds (directions from 45 to 135°), a clear gradient is observed from east to west (see Fig. ), confirming that spatial spin-up of convective systems is an underlying cause for the large wind speed variability over the northeastern part of the Kattegat. Similarly, relate wind speed variability over the North Sea in winter on these short timescales to unstable conditions created by the relatively warm seawater and the relatively cold air above. The higher variability found towards the centre of the North Sea compared to the coastlines also indicates the spin-up time needed for the atmospheric conditions to reach maximal variability.

The MSVI metric quantifies spatial variations in wind speed, and high MVSI values should indicate the presence of a mesoscale weather system. Indeed, looking at the wind fields associated with these peaks, a variety of mesoscale systems are clear, such as convective and sea breeze systems developing over the Kattegat. Convective cells are initiated over land and over Læsø island in the afternoon, resulting in local showers and offshore wind speed variability (Fig. ). Other peaks relate to sea breeze systems also causing spatial wind speed variability (Fig. ). In the case study, no onshore front is formed due to strong synoptic winds. However, this system still has a large impact on offshore winds, with a substantial drop in wind speed where the sea breeze counteracts the synoptic flow. At sunset, the sea breeze systems detach, and the synoptic wind over the Kattegat recovers. Convective and sea breeze systems are quite different, yet they both create mesoscale wind speed variations quantified by the MSVI.

Both the mean MSVI and its diurnal amplitude are higher in summer (JJA) than in winter (DJF) (Fig. ). In winter, wind speed variability is found to be more or less constant throughout the day. In summer, the MSVI clearly peaks in the afternoon, when the surface temperature over land reaches its maximum , confirming an influence of the land on the mesoscale winds over the sea. Over at least one area over the sea, the mesoscale wind speed is on average 20 % larger than the synoptic wind speed at 17:00 UTC in summer. Moreover, when analysing individual cases in our dataset with a high mesoscale wind speed component, it can be seen that the systems detected in summer often originate over land and then travel over sea. Models that explicitly resolve convection feature more mesoscale spatial variability in wind speed compared to coarser model datasets, e.g. the ERA5 reanalysis. Calculating the mean MSVI for ERA5 over our simulation domain results in a substantially lower MSVI and a smaller diurnal amplitude. To confirm that this is indeed the inability of a coarser-scale gridded model to resolve mesoscale variability, we have conservatively remapped the output of our convection-permitting COSMO simulation to the ERA5 grid and then calculated the MSVI, which also resulted in a substantially smaller MSVI than the original simulation. The difference between the coarser-scale COSMO MSVI and the ERA5 MSVI on the same resolution might be due to the fact that the effective resolution of the model is substantially coarser than the native resolution. By aggregating to a coarser scale, part of the advantage of a high-resolution model still remains, as demonstrated earlier by for precipitation in convection-permitting simulations.

The power output over the 10-year simulation period is comparable for the small and the large window time series. The additional wind speed variations that the small windows capture cancel out over the simulation duration. The RMSD between the nine small windows on the one hand and the large window on the other hand is, however, quite substantial. At 286 kW, this is 14 % of the average power output. The wind speed variability captured by the mesoscale window translates to a non-trivial portion of the wind power variability. Therefore, accurate forecasting of power fluctuations requires accurate high-resolution wind forecasts.