For the investigation of the impact of mesoscale wind direction changes, we first define three different scales determined by the distance of prominent offshore locations (wind farms, an offshore substation and two meteorological masts) in the German Bight: the intra-cluster scale which corresponds to the distance between the offshore substation DolWin1 and the met mast FINO1, the inter-cluster scale defined by the distance between the wind farms Deutsche Bucht (DBU) and Global Tech I (GT I), and the German Bight scale by met masts FINO1 and FINO3. Figure  shows the locations and distances between these locations.

The comparison of wind directions at different spatial scales was performed using mesoscale model time series of wind direction and speed at the grid points named above over a period of 30 years. The difference in wind direction between the two time series was calculated for each time step. To examine the dependency on the wind speed, they were then binned by the wind speeds at a reference location (WSREF): 1ΔWDbini=ΔWDWSREF∈bini, with i∈n and n as the number of bins. A bin width of 2.5 m s-1 was chosen, as this was found to depict the dependency of wind direction change ΔWD on the wind speed while still reducing the number of bins to a reasonable amount. For each wind speed bin, the median, quartiles and the percentiles of wind direction differences were calculated.

In a second step, the whole German Bight area was investigated to examine the horizontal patterns of wind direction changes for a climatology of 30 years. The wind direction data were again binned by wind speed at the reference location, and the mean values of the yearly median, upper and lower quartiles of ΔWD at each grid point were calculated. The wind direction change was calculated as a difference in wind direction between each grid point Pj and the reference point PREF, and the statistical parameters described above were calculated: 2ΔWDPj,bini=ΔWDPj,WSREF∈bini-WDPREF,WSREF∈bini.

In order to account for the distances to the reference location, the wind direction changes per 100 km were evaluated using the great circle distance between the Pj and PREF.

The area in Fig.  marked in gray was chosen for analyzing mean wind direction changes over the whole German Bight, as it excludes onshore sites. This area has a minimum distance to the nearest coastline of 5.3 km southwards and 7.7 km eastwards. After the mean of the yearly percentiles was taken for each grid point, the mean of the grid points in the offshore area was calculated. In all cases where wind direction differences were calculated, we accounted for the circular nature of the wind direction variable.

Wind direction changes are expected to be more pronounced on shorter timescales during the passage of synoptic pressure systems. Cyclones are usually transient and characterized by a strong curvature of the isobars. The passage of their frontal system is related to a drop in surface pressure and high wind speeds . Therefore, the wind direction changes were investigated for single cyclone events by choosing time frames of 2 d around time periods when wind speeds above 15 m s-1 were persisting at FINO1 for each event. Anticyclones in contrast are often prevailing over longer time periods and are often characterized by relatively constant high atmospheric pressure at the surface. Thus, the examined high-pressure situations were filtered by sea level pressure constantly exceeding 1020 hPa at FINO1. These conditions were found to last about 4 d in the selected examples. It has to be noted that in this part of the study, particularly pronounced synoptic pressure systems were investigated to examine extreme situations.

As before, the analysis was based on mesoscale model data. The wind direction at the mesoscale grid point closest to FINO1 was subtracted from the one at each grid point in the German Bight, and the resulting direction changes were visualized in maps. Furthermore, the mean wind direction change ΔWD‾ over these above-mentioned time frames was calculated for different spatial scales as shown in Fig. .

For the different spatial scales, the distances between the locations presented in Fig.  are investigated. For the following analysis, we consider different wind speed ranges: below cut-in wind speed (0–2.5 m s-1), the partial load range (2.5–14 m s-1) and above the rated wind speed (>14 m s-1) (Table ). Those ranges originate from the technical specifications of the different turbine types currently (winter 2022) operational in the German Bight.

Especially for low wind speeds which are associated with high thrust coefficients, the variability in ΔWD is large while reducing towards higher wind speeds. This is exemplarily shown for the inter-cluster scale as a boxplot in Fig. . It can be observed that the median of ΔWD reduces towards the upper partial load range while increasing again towards the cut-out wind speed. Overall, the median varies between about ±1∘. However, in 50 % of the cases as denoted by the quartiles in Fig. , considerable wind direction changes of more than 9∘ in the lower operational range (see, e.g., 2.5–5.0 m s-1) can be found. Averaged wind direction changes for the different operational ranges are summarized in Table .

It becomes clear that with increasing distance, the mesoscale direction changes become more relevant. However, also in the intra-cluster scale, 50 % of the cases exceed about 1.5∘ on average in the partial load range. Considering that far-reaching cluster wakes have been measured at distances between the inter-cluster and the German Bight scales, the focus needs to be on these scales concerning cluster wake deflection. Here, wind direction changes are commonly exceeding about 5∘ and about 10∘ in operational conditions, where direction changes reach twice the magnitude in the lower operational wind speed range of 2.5–5.0 m s-1 (not shown). Even though the wake-associated wind speed deficit does not impact the power production above the partial load range, the mechanical loads due to the increased turbulence in the wake can reduce the turbines' lifetime and thus are of relevance.

In the next step, we evaluated the large-scale horizontal patterns of wind direction changes over the 30-year NEWA period (1989–2018) as shown in Fig. . The wind direction changes are defined with respect to the reference location FINO1. The wind speed used to bin the results for the median and the quartiles is also derived at the same reference location. The wind direction changes for all wind speeds (see Fig.  – upper row) show a dipole pattern for the median, with positive ΔWD on the seaward side of the reference location and negative ones towards the coastal regions. Negative direction changes can be interpreted as clockwise-turning winds (veering), while positive values correspond to counterclockwise-turning winds (backing).

Considering the area relevant for offshore wind farms (see Fig. ), the median of ΔWD ranges between ±5∘. Nevertheless, the upper and lower quartiles show values reaching up to -13 and +10∘ in this offshore region. For the wind speed bin representing the lower partial load range (Fig.  – central row), the medians show a similar range, while the quartiles show much larger variation in the data, reaching up to -30 and +25∘. Towards the upper end of the partial load range (Fig.  – lower row), this variation reduces to a range of -7 and +5∘. With increasing wind speeds, the dipole pattern of ΔWD becomes more distinct (Fig.  – lower row), which aligns well as the main wind direction at FINO1 shifts to the west-southwest for higher wind speeds see, e.g.,. This is related to common routes of cyclones over the North Sea .

The asymmetry in the wind direction changes is mainly caused by the coastal transition and can thus in particular be observed at the eastern and southern coastlines of the German Bight and for moderate to high wind speeds (see, e.g., Fig. ). There, comparably large direction changes can reach up to some tens of kilometers offshore. One explanation is the sudden change in surface roughness which leads to a new balance of Coriolis, pressure gradient and frictional forces and thus a turning of the wind direction at the coastline. In the case of sea surface temperatures colder than the advected air mass, an induced stable stratification develops, which can in the end lead to the formation of low-level jets .

Table  shows the values of the binned wind direction changes averaged for the selected area (gray area in Fig. ). It can be summarized that 50 % of all cases show on average a wind direction change of more than 11∘ (first row of Table ). Focusing on the partial load range (2.5–14 m s-1), the distance between the values of the lower and upper quartiles decrease with increasing wind speed until the upper end of the partial load range. In the lower partial load range, the wind direction changes easily exceed ±20∘ for almost 50 % of the data. Table  also reflects that, while the mean wind direction change can be quite low, a large share of situations in particular for wind speeds in the partial load range (<10.0 m s-1) exist with significant wind direction changes. Very high wind speeds, despite being less frequent, come along with larger positive direction changes. One explanation for this could be the passage of cold fronts of low-pressure systems that typically are associated with a strong change in wind direction from southerly to more westerly or even northwesterly winds.

As mentioned, far-reaching cluster wakes were found under both neutral and stable atmospheric stratification. These conditions occur with a probability of 29.2 % at the location of the wind farm Deutsche Bucht, indicated by a positive inverse Obukhov length. Only conditions with wind speeds exceeding 2.5 m s-1 were considered to exclude wind speeds that are below the cut-in wind speed and thus irrelevant for wake formation. Figure  shows the magnitude and probability of wind direction changes exceeding the quartiles of -3.2 and +4.2∘ at the inter-cluster scale (DBU and GT I). Conditions under which wind direction changes exceeding the lower and upper quartiles align with atmospheric stratification that benefits far-reaching wakes have a probability of 14.6 %.

To investigate the situation during the passage of synoptic pressure systems over the German Bight, shorter time periods were selected for further investigation.

Figure  is a particular example of the wind direction changes happening as the frontal system of Cyclone Sebastian (in September 2017) passed over the German Bight. Typically, low-pressure systems move quite fast from west to east in the northern mid-latitudes, and their frontal systems are associated with strong and sudden wind direction changes. The left plot shows the median of the wind direction changes between each NEWA grid point in the German Bight and the one closest to the reference location FINO1 for a time frame of 48 h around the wind speed maximum.

During the early morning, the cold front passed over the German Bight, causing large wind direction changes (see center and right panels). The comparably large wind direction changes of the instantaneous velocity fields in these panels demonstrate the extent of wind direction changes on shorter timescales. The difference between the mean directions at DBU and GT I for the whole time period from 12 September at 13:30:00 UTC to 14 September 2017 at 13:00:00 UTC is only about 2.3∘, while even smaller values occur for other cyclone passages. Contrasting the small mean difference in wind direction, peaks in ΔWD on the inter-cluster scale were found to reach up to 50 and -80∘ during the passage of the front shown in Fig. . This specific situation is further investigated with respect to large-scale wind direction changes in the following sections.

For anticyclones (not shown), comparably large wind direction changes were found for longer time periods. Over the course of 4 d, the difference in mean wind direction at DBU and GT I was, e.g., 6.1∘ (6 May at 00:00:00 UTC to 9 May 2008 at 23:30:00 UTC) and 4.3∘ (25 June at 00:00:00 UTC to 28 June 2018 at 23:30:00 UTC). This highlights that anticyclones can cause wind direction changes at larger scales, persisting over longer time frames.

Specifically, the strong wind direction changes during the passage of cyclones are often less relevant for the power production as the wind speed is mostly above the rated wind speed. Nevertheless, the wake can reduce the wind speed at downstream wind farms to U<Ucut-out. Furthermore, due to the increased turbulence in the wake, these conditions are relevant for load calculation and therefore for the prediction of the turbines' lifetime. Moreover, the warm air advection associated with the frontal system of cyclones influences the atmospheric stability, as it often leads to stable stratification and thus to increased wake lengths.

The wake effects of clusters N6 and N8 (see marked clusters in Fig. ) have been simulated during the passage of the cold front of Cyclone Sebastian (13 September 2017, see Fig. ).

Figure  shows the wind field at about hub height perturbed by the interaction with the wind farms. The three columns display three relevant stages, i.e., instances in time, of the phenomena. The BLM (upper row) shows unexpected wake behavior as soon as the cold front approaches the N6 cluster, with the wakes of the leftmost turbines crossing the ones of the other turbines. When the cold front passes over N6, the wake of this cluster reaches and impacts N8, despite the different wind direction to the right of the front preventing this from happening. In contrast, the lower panels of Fig.  show the wakes simulated with the SWM. Already at the first time step, the new model describes the wake propagation in a more sound manner, resolving the wake crossing problem. Also, the following snapshots reveal a more realistic adaptation of the wakes to the heterogeneous background flow.

To demonstrate that the SWM calculates the trajectory along which cluster wakes evolve with greater fidelity than the BLM, few comparisons to SAR imagery were performed. Here, only a single situation considered in the comparison is depicted in Fig. . The lower panel shows the normalized radar cross-section (NRCS) from Sentinel-1A, acquired at 05:48 UTC. In the eastward direction, the shadow areas and streaks downstream of the wind farms indicate low backscatter and correspond to the region of reduced wind speed that characterized the wakes. Quite long wakes and an evident wake turning induced by westerly winds that get a more northern component in the southern part of the German Bight are visible. The upper panels of Fig.  show the wind speed at hub height above the mean sea level in the German Bight simulated for 21 March 2018 at 06:30:00 UTC. All wind farms installed in the area on that date have been considered. However, precise information on the turbine operation states was missing. In this situation, the underlying WRF data suggest a westerly wind that is not particularly strong. Thus, the turbines operate at maximum thrust, offering long persisting wakes. Furthermore, as the focus in the comparison is the wake trajectories, the wake model coefficients are chosen so as to minimize wake recovery and show the wake evolution throughout the domain. The results of the simulations with the BLM (upper left panel) differ significantly from the one of the SWM (upper right panel). In comparison to the SAR image (bottom panel), the SWM captures with higher fidelity the trajectory of the cluster wakes of the southern clusters, allowing us to correctly represent the distinct anticyclonic turning of the N3 cluster wake. Also, when looking at the Gemini cluster, we observe a much better agreement of the model with SAR when the SWM is used instead of the BLM.

Note that the displayed SAR was taken about 45 min earlier than the mesoscale model data. The different time in the models was chosen as it provided the best agreement with the SAR data, acknowledging that phase errors up to several hours are a well-known phenomenon in mesoscale model data.

The SWM agreed well with the SAR data in only a few other situations considered; however, we observe cases where the agreement between SAR and both models was poor (not shown). For these cases, the mismatch is mostly related to a fundamental disagreement between SAR and the WRF mesoscale data used as input of the engineering models. While found that on average the mesoscale atlas of the NEWA represents offshore wind conditions well, the generated wind fields may present local errors. Recently, the NEWA atlas has been subject to validation activities for different applications e.g.,. At the same time, SAR imagery has not been demonstrated yet as a reliable tool for quantitative wake model validation, as the data acquired have uncertainty of up to 2 m s-1 . In conclusion, the comparison of Fig.  cannot be considered a validation of the new method, but it highlighted that in no case was the wakes' directional propagation computed by the BLM superior to the SWM.

In this section, we discuss the differences across the SWM and BLM when applied to computing the energy yield at two wind farm clusters impacting each other. Assuming the proposed method as an improvement over the common BLM, these simulations provide an estimate of the new model's potential to partly reduce and especially understand the sources of uncertainty in engineering wake model applications.

As before, the selected clusters for the study are clusters N6 and N8 in the German Bight. The tool flappy is applied for calculating the power production at any time step in the last 10 years of the NEWA mesoscale data with the BLM and the SWM.

As a first analysis, the total energy yield of N6 and N8 when operating together is calculated. The simulation setup adopted is described in detail in Sect. . In addition to the two wake propagation methods, also two different superposition methods for the velocity deficit will be considered. Table  briefly summarizes the four simulations performed for this analysis.

The energy yield simulated over a period of 10 years shows that the SWM results are very similar to the BLM results. The relative difference in the yield of both clusters is 0.01 % and 0.004 % for the superposition models' wind product and wind linear, respectively. However, the differences in the calculation for single situations are significantly greater, as demonstrated by the standard deviation found to be 1.21 % and 1.05 % for wind product and wind linear, respectively. Thus, BLM and SWM would compute differently the energy production when considering shorter time frames.

Most interestingly, the largest differences happen systematically according to wind direction and wind speed, as demonstrated in the heatmap of Table . According to the color coding, the relative difference between the models is greater for very low wind speeds or for wind speeds in the partial load range and wind directions where the clusters align. This can be explained by effects on the intra-cluster and the inter-cluster scales. As already discussed in Sect. , very low wind speeds are often associated with very inhomogeneous wind fields. Therefore, significant wind direction changes are observed even at an intra-cluster scale. In this case, the models can provide different results even in situations where the two clusters should not interact through their wakes due to differing wake propagation within one cluster. For higher wind speeds, the differences between the models are grouped around the wind directions along which the clusters align. Once the wind speed exceeds the rated speed, the differences progressively reduce as pitching control reduces the intensity of the wakes. Looking at the mean relative difference between the models, the SWM predicts a larger energy production in all the occasions where the main wind direction is such that the clusters align: bins [40, 60]–[60, 80], [220–240]. This is coherent with the fact that accounting for wake turning, the waked cluster will produce more, experiencing more “free-stream” velocity. At the same time, the SWM computes a lower yield than the BLM when the wind direction has an offset. A trend of counterclockwise turning in the wake is highlighted by the fact that the SWM computes the lowest yield with respect to the BLM for the bins [80, 100], [240, 260].

The results presented so far suggest that the use of the SWM could be useful for two main practices that engineering models enable: site assessment for new wind farms and short-term forecasting for, e.g., a grid following control strategies. In both applications, it is important to guarantee that the engineering models compute with a sufficient level of accuracy on short timescales.

The quantitative differences discussed so far are comprised of differences in the wake development, both on the scale of a single farm and the cluster wake scale. However, the underlying mesoscale data represent the flow field at the scale of cluster wakes well while lacking the accuracy to correctly describe the wind direction changes at a scale of a few rotor diameters. Thus, the uncertainty introduced by the effects within a cluster is isolated by performing the same set of simulations (C1–4; Table ) for N8 without N6. In these simulations, the energy yield of N8 is only affected by the intra-cluster effects. Therefore, the difference in energy yield of only N8 between these and the previous simulations gains the yield loss due to the cluster wake of N6. This value is again compared across the SWM and BLM to estimate how the new model offers a different prediction of cluster wake effects between neighboring clusters. The resulting differences for the two wake superposition models' wind linear and wind product are 0.5 % and 0.7 %, respectively.

In conclusion, in our results, the proposed model offers a small but non-negligible difference when estimating the interaction between neighboring wind farm clusters, like N6 and N8. Despite the new model not yet being validated with production data, we speculate that such a difference could represent an actual improvement over the baseline in terms of model accuracy.

Finally, it may be worth remembering that the applied wake model, optimized on a single wind farm, is likely underestimating the extent of cluster wakes. This aspect, in addition to the fact that a new wind farm cluster will most likely be surrounded by more than just one, could easily increase the gap between the BLM and the SWM, making the second more attractive.