In this study, we use lidar observations collected during the OBLEX-F1 campaign at the FINO1 offshore platform in the southern North Sea (Fig. ), conducted between May 2015 and September 2016 . The wind profile data are collected by a pulsed scanning Doppler wind lidar (WindCube 100S from Leosphere, now Vaisala) operated in Doppler beam swing (DBS) mode. A single DBS scan was completed after about 15 s and then averaged over 1 h for this study. The range gate length of the lidar was set to 25 m, resulting in an effective vertical profile resolution of 23.5 m. For more details on the measurement setup, as well as data availability, the reader is referred to . For our analysis, we concentrate on the lower portion of the wind profile, specifically up to around 1 km, where data coverage is more reliable and pertinent to boundary-layer dynamics.

For comparison, we utilize data from ERA5, the fifth-generation European Reanalysis from the European Centre for Medium-Range Weather Forecasts (ECMWF) . ERA5 is a widely used global climate reanalysis dataset, known for its improved modeling and data assimilation techniques compared to its predecessor, ERA-Interim. It provides hourly estimates of a vast number of weather- and climate-related variables. The three-dimensional data are available on 137 vertical levels with a horizontal grid spacing of 0.28125° (approximately 31 km) . In this study, we extract wind profiles, including zonal and meridional wind components, alongside potential temperature, from the ERA5 grid cell closest to the FINO1 location. Given our focus on the lower atmospheric boundary layer and LLJ structures, we limit our data retrieval to the 18 lowest model levels, extending up to approximately 800 m. This selection offers a balance between computational efficiency and sufficient vertical resolution for our objectives.

Additionally, we use NORA3, which is a non-hydrostatic 3 km hindcast dataset for the North Sea, the Norwegian Sea, and the Barents Sea . The dataset was generated using the non-hydrostatic convection-permitting numerical weather prediction HARMONIE-AROME , forced at the lateral boundaries by ERA5 data. It is initialized four times daily, with each run producing 9 h forecasts. NORA3 data are available at 3 h intervals on all 65 model levels or at hourly intervals on a reduced set of seven height levels below 750 m. For this research, we opt for the hourly subset at the grid point nearest to FINO1. This choice prioritizes higher temporal resolution within the marine boundary layer, which is crucial for capturing the dynamics of transient events such as LLJs. All wind profiles from lidar, NORA3, and ERA5 are vertically interpolated to a common grid to ensure comparability across the datasets. This grid spans 80 to 740 m in 20 m increments, covering the primary region of interest for boundary-layer and LLJ analysis. Furthermore, the lidar data are temporally aggregated to an hourly resolution to match the temporal resolution of the NORA3 hourly subset and to facilitate direct comparisons.

Figure  presents an example of processed wind speed profiles in the form of time–height cross-sections derived from the three datasets, spanning 12–14 August 2015. This period was specifically chosen to illustrate a prominent LLJ event, which is distinctly observed in the lidar data, exhibiting a peak wind speed at approximately 300 m near midnight on 13 August. While both ERA5 and NORA3 capture the overall wind speed trend, including the LLJ structure, they exhibit a slight underestimation of the maximum LLJ wind speeds compared to the lidar observations. Furthermore, the lidar data reveal a low-wind-speed anomaly below 200 m, commencing in the evening of 13 August, which is attributed to the wake effect from the Alpha Ventus wind farm in the vicinity of the FINO1 platform, as documented by .

To investigate the climatological characteristics of LLJs, we analyze a continuous 50-year dataset of 1-hourly ERA5 wind profiles, spanning 1971 to 2020. For composite analyses focused on larger-scale atmospheric patterns, we additionally extract two-dimensional fields of geopotential height and horizontal wind components at the 975 and 500 hPa pressure levels from ERA5. These synoptic-scale variables are used to examine the broader atmospheric environment associated with each detected LLJ event, helping to elucidate the synoptic drivers and their effect on the characteristics and variability of offshore LLJs.

The general definition of an LLJ remains loosely defined, typically referring to a wind speed maximum within the lowest few kilometers of the atmosphere , and there is no universal established method or set of criteria for LLJ detection . One widely used metric in previous studies is the wind speed fall-off, defined as the difference between the wind speed maximum and the minimum above it where different studies apply either an absolute or a relative fall-off threshold. The absolute fall-off threshold typically ranges from 2 to 5 m s⁻¹ , while the relative fall-off threshold varies between 10 % and 25 % . Some studies combine both absolute and relative thresholds . The rationale for using a relative threshold is that it helps identify the characteristic nose shape of LLJs, while a weak absolute threshold (e.g., 2 m s⁻¹) helps filter out spurious LLJ detections caused by small errors when wind speeds are low. The detection height, which can range from a few hundred meters to 2 km, also influences LLJ identification . If the wind speed minimum above the jet maximum is higher than the predefined detection height, the wind speed at the top level is used instead, generally resulting in a lower fall-off ratio for higher-altitude LLJs.

In this study, we adopt a relative fall-off threshold of 20 %, following , , and . However, before applying this threshold, we fit the wind profile U(z) using the sum of a logarithmic profile and a jet component: 1Ufitted(z)=Ulog⁡(z)+Ujet(z), where 2Ulog⁡(z)=u*κlog⁡zz0 is a logarithmic profile. Here, κ=0.41 is the von Kármán constant. The friction velocity u* (m s⁻¹) and the roughness length z0 are the fitting parameters to be estimated.

The second term represents a jet profile, defined by three parameters characterizing the LLJs: 3Ujet(z)=Umzzmexp⁡1S1-zzmS. Um represents the maximum wind speed; zm the height at which this maximum occurs; and S a shape parameter, defining the vertical structure and extension of the jet component (see Fig. a). This formulation is adapted from the tangential wind profile model for tropical cyclones proposed by .

While a standard logarithmic profile (Eq. 2) is theoretically restricted to neutral stratification, in our decomposition method, it primarily serves as a background component that increases with height monotonically. The empirical jet profile (Ujet) provides the mathematical flexibility to account for stability-induced deviations from this background. As a result, combining these two components reduces the need for explicit atmospheric stability corrections while still capturing the overall shape of the observed wind profile.

The optimal set of the five log–jet parameters (Um,zm,S,u*,z0) is determined by minimizing the mean squared error (MSE) between the approximated profile and the original data. Because the objective function is non-linear and may contain local minima, we utilize the differential evolution algorithm , a robust global optimization method. To ensure the physical realism of the fitted profiles and to accelerate convergence, the search space is constrained by the following predefined parameter bounds: Um∈[0,30] m s⁻¹, zm∈[80,1000] m, S∈[0.1,8], u*∈[0.01,1] m s⁻¹, and z0∈[10-5,0.02] m.

Furthermore, to ensure the physical validity of the extracted parameters and to avoid characterizing noisy or highly irregular profiles, we evaluate the goodness of fit using the coefficient of determination (R2), defined as 4R2=1-∑iU(zi)-Ufitted(zi)2∑iU(zi)-U‾2, where U(zi) is the original wind speed at height zi, Ufitted(zi) is the corresponding fitted value, and U‾ is the mean wind speed of the profile. Profiles with an R2 below 0.90 are considered insufficiently good fits and are discarded prior to applying the 20 % fall-off criterion.

This decomposition method offers several advantages for LLJ analysis. First, it effectively filters out minor wind speed variations, isolating the primary jet structure (see Fig. b). Second, it facilitates gap filling by addressing common lidar data gaps (e.g., Fig. ), thereby enabling wind profile extrapolation across the full range. Third, it permits direct comparison of LLJ characteristics across datasets via the three key jet component parameters. Finally, a bias correction can be applied directly to the five parameters to align model data with observations. This bias correction can then be extended to longer reanalysis datasets to quantify LLJ characteristics over extended periods.

To improve the representation of LLJs over a climatological period, we developed a quantile-mapping procedure to bias correct a 50-year ERA5 dataset (1971–2020). As shown in the subsequent evaluation section, ERA5 performs comparably to or better than NORA3 for this specific location, making it a highly suitable choice given its long historical record and ease of access.

The overall workflow of this procedure is illustrated in Fig. , divided into a calibration phase and a long-term reconstruction phase. The calibration phase involves applying the log–jet decomposition to the 1-year datasets (lidar and ERA5) and estimating their statistical distributions, resulting in the quantile pairs of the five log–jet parameters (u*,z0,Um,zm,S). The reconstruction phase uses these pairs to bias correct the log–jet parameters from the 50-year ERA5 dataset; then, the bias-corrected ERA5 wind speed profiles are analytically reconstructed using Eq. (1). Regarding the practical implementation, it is important to clarify that we do not intervene in the full raw model outputs or rerun the model. Instead, the bias correction is entirely parameter centric.

During the calibration phase, 100 percentiles were calculated for each parameter and stored for use in the quantile-mapping process (Fig. a). However, due to the limited number of ERA5–lidar data pairs, using the raw data directly could lead to overfitting, resulting in an adjusted distribution with unrealistically small variability (Fig. b). To address this issue, we applied a kernel density estimation (KDE) smoothing step to mitigate sampling limitations in the lidar dataset. Specifically, we used an ad hoc KDE bandwidth of 0.2 for parameters Um, zm, and S and 0.5 for parameters u* and z0. These bandwidths are smaller than the default KDE values, ensuring that small variations in the final probability density function (PDF) are smoothed while preserving the overall shape of the distribution. Although the quantile–quantile plots in Fig. a appear visually similar for both the original and the KDE-smoothed data, applying a relatively small bandwidth in the KDE step leads to noticeable differences in the final PDF (Fig. b). This highlights the effectiveness of the smoothing procedure in refining the histogram shape, even when the corresponding quantile values differ only slightly.

Finally, it should be noted that this quantile mapping is applied globally to the dataset, regardless of season, wind direction, or atmospheric stability. We acknowledge that correcting for an overall bias may mask the cancellation of compensating errors that occur under different atmospheric regimes. However, because the observational lidar dataset spans just over 1 year, conditionally dividing the data into multiple seasonal or directional bins would result in sample sizes that are too small to construct robust PDFs, thereby introducing severe overfitting. Therefore, the global correction serves as a robust baseline for this climatological study, though extending this method to conditionally dependent corrections remains a valuable objective for future studies with longer observational records.

After bias correcting the 50-year ERA5 log–jet parameters using the quantile-mapping method, wind profiles are reconstructed to facilitate LLJ detection. LLJs are identified using a 20 % threshold for the fall-off rate. The detected LLJs are then classified using the K-means method , a clustering algorithm that minimizes within-cluster variance. Clustering is performed on the horizontal wind components (U and V) across all vertical levels, thereby capturing the shape, magnitude, and direction of the entire wind profile for each cluster. To determine the optimal number of clusters, we apply the elbow method e.g.,, which utilizes the total within-cluster sum of squares (WCSSs, also known as inertia) as its evaluation score across varying cluster counts. A clear inflection point (the “elbow”) at three clusters suggests this number optimally balances complexity and explanatory power.

To further account for the role of atmospheric stability, each of the three clusters is subdivided into two sub-clusters using the K-means method based on the vertical gradient of potential temperature (dθ/dz). This gradient serves as a proxy for stratification, which influences LLJ formation and persistence. To investigate the large-scale weather patterns associated with each LLJ cluster, composite maps of wind speed and geopotential height at 925 and 500 hPa are generated. Wind vectors are overlaid to visualize prevailing synoptic flow regimes.

Before applying the bias correction to the historical 50-year dataset, we first evaluate the baseline performance of the raw ERA5 and NORA3 models against the 1-year concurrent lidar observations. As established in the methodology, the log–jet decomposition enables a direct comparison of the underlying LLJ properties independent of the specific fall-off threshold used for LLJ detection. Figure  presents this evaluation through quantile–quantile (Q–Q) plots and corresponding histograms for the three key jet parameters.

Focusing first on the jet magnitude, the Q–Q plot and histogram of Um (Fig. a, d) show that the distribution of Um is remarkably similar for NORA3 and ERA5. This is a surprising result because NORA3 is a non-hydrostatic hindcast with a much higher spatial resolution (3 km) specifically designed to dynamically downscale ERA5 and better capture mesoscale and boundary-layer processes. However, both datasets exhibit a systematic bias, as evidenced by their deviations from the 1 : 1 line. Specifically, these deviations increase at higher quantile levels. From the 0.6 quantile level (corresponding to a quantile value of roughly 2 m s⁻¹) onward, both models consistently underestimate the lidar observations across all larger quantiles by approximately 30 %.

A similar systematic underestimation is observed for the shape parameter S (Fig. c, f). A larger S corresponds to a broader jet profile with a less aggressive wind speed fall-off above the maximum. Both reanalysis datasets struggle to capture the full frequency of these broader jets, but their performance diverges notably. As seen in the Q–Q plot (Fig. c), ERA5 aligns much more closely with the lidar quantiles than NORA3. The histograms (Fig. f) reveal that the lidar data exhibit a bimodal distribution for S, with a notable peak at the algorithm's upper boundary (S=8). While ERA5 successfully captures this bimodal structure, NORA3 heavily overestimates the frequency of sharp, narrow jets (low S values) and entirely fails to reproduce the broader jet structures.

For the jet height zm (Fig. b, e), both models demonstrate a tendency to place the LLJs too close to the surface compared to the observations. NORA3 exhibits a severe overestimation of jet frequencies in the lowest height bins (near 100 m), while underestimating occurrences in the middle boundary layer. In contrast, ERA5 captures the overall vertical distribution of the jet heights much more accurately, with its Q–Q curve (Fig. b) tracking significantly closer to the 1 : 1 reference line. It is also worth noting that all three datasets show a minor accumulation of data points at the 1000 m mark, which is an expected artifact of the predefined upper-boundary constraint during the optimization process.

Overall, the model evaluation reveals a critical finding: despite NORA3 being a high-resolution downscaling designed to improve boundary-layer representation, it actually underperforms compared to the global ERA5 reanalysis in capturing the structural shape (S) and vertical placement (zm) of offshore LLJs at this location. Both models similarly underestimate the magnitude of strong jets (Um). Given that ERA5 provides a closer match to the observed spatial and structural jet distributions, alongside its vast historical record and computational accessibility, it is clearly the superior dataset for long-term LLJ analysis. Consequently, for the remainder of this study, we apply the quantile-mapping procedure exclusively to bias correct the 50-year ERA5 dataset, utilizing the observed lidar distributions to ensure a highly reliable climatological analysis of LLJ behavior.

After bias correcting the log–jet parameters in the 50-year ERA5 dataset, the wind profiles were reconstructed and LLJs were identified using a 20 % fall-off rate threshold. To illustrate the differences between the overall climatological wind conditions and the conditions under LLJs, we present wind roses at 200 m above sea level – near the peak height of the LLJ distribution (Fig. e) – for the full dataset (Fig. a) and for the LLJ subset (Fig. b). In Fig. a, the prevailing winds are predominantly west-southwesterly, with mainly moderate speeds in the range of 5–15 m s⁻¹. The occurrence of higher wind speeds (15–20 m s⁻¹) is notable, though not dominant, and only a small fraction of winds exceed 20 m s⁻¹. By contrast, the LLJ subset (Fig. b) also features speeds primarily between 5–15 m s⁻¹ but shows fewer occurrences of 15–20 m s⁻¹ winds, and no LLJs exceed 20 m s⁻¹. Additionally, while the full dataset is characterized by a prevailing west-southwesterly direction, the LLJs are nearly reversed, predominantly blowing from the east-southeast. This directional shift highlights the distinct synoptic or local conditions under which LLJs form at a given site and underscores the importance of separately analyzing LLJ behavior for applications such as wind resource assessment or boundary-layer studies.

Figure  presents three distinct wind profile clusters identified using the K-means clustering method for the identified LLJs. Each cluster corresponds to a different prevalent wind regime, potentially associated with specific synoptic or mesoscale conditions. Cluster 1 (Fig. a) represents the most frequent regime, accounting for 38.8 % of the cases, and is characterized by a southeasterly wind direction. Cluster 2 (Fig. b) accounts for 33.0 % of cases and exhibits a southwesterly wind direction. Finally, cluster 3 (Fig. c) covers the remaining 28.1 % of cases and is associated with a northeasterly flow.

Clusters 1 and 2 display remarkably similar maximum wind strengths, with average profile peaks of 10.4 and 10.1 m s⁻¹, respectively. However, they differ significantly in their vertical placement; cluster 2 has a much higher average LLJ core height of 332 m compared to 218 m for cluster 1. In contrast, cluster 3 represents the weakest and lowest jet regime, with an average peak wind speed of only 6.7 m s⁻¹ occurring at a height of 144 m.

Furthermore, the three-dimensional hodographs reveal distinct veer characteristics. The horizontal projections (gray shadows) of clusters 1 and 2 exhibit notable wind veer, where the wind vector rotates clockwise with increasing height – a classic signature of Ekman turning and inertial oscillations in the Northern Hemisphere. Conversely, cluster 3 shows minimal directional change with height, maintaining a nearly uniform northeasterly orientation throughout the depth of the boundary layer.

The inertial oscillation is a key process in forming LLJs over flat surfaces. It is initiated when a stable inversion layer decouples the lower atmosphere from the surface friction. Once decoupling occurs, for example due to radiative cooling, the wind vectors rotate clockwise in the Northern Hemisphere, transitioning from sub-geostrophic to super-geostrophic states. This transition can produce a wind speed maximum (i.e., the LLJ) above the inversion layer. To further understand LLJ formation and its relation to boundary-layer stratification, each primary cluster is subdivided using the K-means method applied to the vertical gradient of potential temperature, dθ/dz, which serves as an indicator of atmospheric stability.

Figure  displays the vertical profiles of the average wind speed and dθ/dz for the three main clusters and their respective sub-clusters. Each primary cluster (Fig. a–c) is divided into two distinct stability regimes: sub-cluster a (middle row, Fig. d–f) and sub-cluster b (bottom row, Fig. g–i). Sub-cluster a exhibits weak stability near the surface, but dθ/dz gradually increases with altitude, indicating the presence of a stable layer aloft situated near the height of the jet core. In stark contrast, sub-cluster b is characterized by a pronounced and sharply defined inversion layer much closer to the surface, typically located just beneath the jet core. Notably, in sub-cluster 3b (Fig. i), the peak of this strong low-level inversion is exceptionally sharp and perfectly co-located with the low-altitude jet core, indicative of a very shallow, highly stratified boundary layer.

Overall, the a-type sub-clusters comprise the majority of the events, accounting for approximately 74 % of all cases, whereas the strongly stratified, surface-based b-type sub-clusters account for the remaining 26 %. Because a strong low-level inversion is absent below the LLJ core in the majority of instances (sub-cluster a), the trigger for the inertial oscillation in these cases is likely not local. Given the offshore location of FINO1, it is highly probable that these LLJs originate onshore and are subsequently advected over the sea. Conversely, the pronounced surface-based inversion layers observed in sub-cluster b suggest local forcing mechanisms, which may be linked to radiative cooling of the sea surface under clear skies or the advection of warm continental air over the colder ocean surface.

Figure  reveals distinct temporal signatures for each cluster and its sub-clusters. Cluster 1 (Fig. a), associated with southeasterly flow, exhibits a strong annual cycle with peak occurrences during the spring and early summer (March to July). Within this cluster, sub-cluster 1a (Fig. d) shows a broad diurnal distribution with a distinct peak in May during the early morning hours (00:00–08:00 UTC). Conversely, the strongly stratified sub-cluster 1b (Fig. g) is much more tightly constrained, displaying a sharp maximum in May during the evening and early night (18:00–22:00 UTC).

Cluster 2 (Fig. b), representing the southwesterly flow, demonstrates a dual seasonality. It occurs frequently throughout the day during the winter months (December to February) but shifts to a highly nocturnal pattern during the spring (March and April). This transition is clearly driven by sub-cluster 2b (Fig. h), which exhibits a remarkably intense, concentrated peak in March and April between 00:00 and 06:00 UTC. Sub-cluster 2a (Fig. e) is more distributed throughout the year but features a notable absence of LLJ events during the daytime hours (10:00–18:00 UTC) from May through August.

Cluster 3 (Fig. c), the northeasterly flow, is the most temporally isolated of the three. It occurs almost exclusively in the late spring and early summer (May and June), with a very strong preference for the late afternoon and evening hours (14:00–22:00 UTC). This is mirrored closely by its weakly stable component, sub-cluster 3a (Fig. f). Sub-cluster 3b (Fig. i) is even more sharply focused, appearing almost entirely in May between 16:00 and 20:00 UTC.

Overall, the b-type sub-clusters (bottom row) show much stronger, more sharply defined seasonal and diurnal signals than the broader a-type sub-clusters. The tight temporal constraints of the b-type events – whether peaking in the late evening (1b and 3b) or the early morning (2b) – strongly support the conclusion that these strongly stratified LLJs are driven by specific local forcing mechanisms, such as diurnal surface cooling cycles, rather than broader synoptic-scale advection.

To highlight the spatial patterns of each sub-cluster, Fig.  shows the composite wind speed at 975 hPa (overlaid with wind vectors) for the six sub-clusters. Sub-clusters 3a and 3b (Fig. c, f) feature notably the weakest wind speeds over most of the North Sea, characterized by a modest northeasterly flow near the FINO1 location. By contrast, sub-clusters 1a and 1b (Fig. a, d) exhibit stronger southeasterly winds originating from the European continent and blowing out over the North Sea. Finally, sub-clusters 2a and 2b (Fig. b, e) are dominated by a pronounced southwesterly flow – essentially orthogonal to cluster 1 – but with the highest overall wind speeds (often exceeding 10–12 m s⁻¹), covering a much broader spatial area than the other clusters.

Figure  illustrates the composite geopotential height at 975 hPa for each sub-cluster, providing insight into the synoptic drivers of the observed wind regimes.

For cluster 1 (Fig. a, d), the FINO1 station is situated on the southwestern flank of a high-pressure system centered over Scandinavia and the Baltic Sea, with lower pressure extending over the Atlantic. This configuration drives the prevailing southeasterly winds. Notably, in the strongly stratified sub-cluster 1b, the Scandinavian high is much more pronounced and expansive than in sub-cluster 1a.

In cluster 2 (Fig. b, e), FINO1 is positioned between a deep low-pressure system situated to the northwest (near Iceland and the UK) and a high-pressure ridge over southern Europe. This arrangement creates a very steep meridional pressure gradient across the North Sea, which perfectly aligns with the strong, broad southwesterly flow observed in the previous wind speed composites.

By contrast, cluster 3 (Fig. c, f) is characterized by very weak pressure gradients across the study area. While there is a diffuse area of higher pressure generally situated to the north and northwest of FINO1, the slope of the geopotential height surface (indicated by the wide spacing of the contours) is remarkably flat. This lack of synoptic forcing explains the very weak, localized northeasterly wind speeds associated with this cluster.

The composite analysis of ERA5 geopotential height at the 500 hPa isobaric surface and wind vectors (Fig. ) reveals distinct atmospheric regimes across the six sub-clusters, providing insight into the synoptic-scale flow patterns. In general, geopotential height increases toward the south, leading to a predominantly westerly flow. A ridge feature is evident over the FINO1 location in all cases, though its strength and structure differ between the sub-clusters.

For the a-type sub-clusters (Fig. a–c), the ridge is less pronounced, and the height contours are more zonally oriented, resulting in a predominantly westerly flow at this level. In contrast, for the b-type sub-clusters (Fig. d–f), the ridge is more sharply defined, with a stronger geopotential height gradient to the north, leading to enhanced westerly winds. Despite the differences in surface-level structures observed in Fig. d–f, the geopotential height fields at 500 hPa remain relatively similar, with FINO1 consistently positioned on the western flank of a well-defined ridge centered near 10° E. This suggests that while surface-level variations exist, the upper-level forcing remains coherent.