The growing need for clean and sustainable energy sources has led to a demand for large-scale and potentially concentrated deployment of wind farms . Mesoscale models represent the atmospheric state and its dynamics and can now include the effect of wind farms and wind farm clusters on the atmospheric flow. These models are being used in the wind farm planning phase, as they consider the potential impacts of a proposed farm or a cluster of wind farms on the surrounding wind energy resources, together with the regional climate of the site . Given the economical, political, and environmental significance of these applications, it is essential that these models accurately represent turbine–atmosphere interactions.

The Weather Research and Forecasting (WRF) model is a mesoscale model that has become an essential tool for wind resource assessments , as reflected in its use for the creation of modern wind atlases . The WRF model can incorporate the effect of wind farms through a wind farm parameterization (WFP). Several WFPs have been implemented, with the Fitch WFP and the explicit wake parameterization EWP; being the two most commonly used . Both WFPs have been extensively discussed in the literature, and research findings are usually framed in comparisons of wind-speed-based wake extension, turbulent kinetic energy (TKE), and power production . Across these studies, a common finding is that the Fitch WFP typically simulates deeper wakes and higher TKE than the EWP, with the latter often underestimating TKE compared to limited measurements .

The differences in outcomes from these WFPs rely on their methods to parameterize the impact of wind turbines on the flow. The two parameterizations differ in how they vertically distribute the wind speed deficit and whether they include an explicit source of TKE. Each WFP relies on a single controlling parameter that impacts one of the behaviours of the WFP outlined above. In the Fitch WFP, the correction factor cf, initially implemented as a bug fix by , modulates the TKE production from the turbines. The EWP, without an explicit TKE source, defines an initial length scale (σ0) that partially controls how the velocity deficit is vertically distributed . Understanding the WFPs, their controlling parameters, and their interactions with the modelled atmospheric conditions is crucial for their effective use and future improvements.

The importance of validating the WFPs under different atmospheric conditions cannot be overstated, as the state of the atmosphere impacts the wind farm's performance. Atmospheric stability is known to influence both wake behaviour and turbine-induced turbulence . However, validating these two aspects for WFPs poses considerable challenges under realistic conditions . A primary issue is the scarcity of extensive, accessible, and long-term datasets for comparison. This often leads to WFPs that are validated within brief atmospheric episodes representing a limited range of conditions. Furthermore, validating the WFPs against measurements greatly relies on the mesoscale model's ability to accurately represent the background conditions . Any inaccuracies in this representation can significantly impact the performance of the WFPs, introducing an additional layer of complexity to the validation process.

In response to these validation challenges, some studies evaluate the WFP results against high-fidelity wake simulations, e.g. from large-eddy simulations LES;. LES, using actuator discs (ADs), or actuator lines methods can provide a high-resolution and detailed representation of wind turbine wakes with a high computational cost. In particular, the WRF model can also operate in LES mode with the option of using ADs and has been used to study non-neutral boundary layers . These LES methods, including WRF-LES with ADs, can improve the accuracy of WFPs by providing a detailed reference for their refinement .

The Fitch and EWP WFPs and their recommended parameter values have been calibrated under neutral atmospheric conditions using LES representations or averaged stability-filtered measurements . However, the accuracy of these parameterizations under non-neutral atmospheric stability conditions has not been clearly determined and requires substantial exploration. Therefore, our study's primary objective is to deepen our understanding of WFPs in mesoscale models by evaluating them against LES using ADs under different atmospheric stability conditions. Specifically, our study seeks to answer two key research questions. (1) How comparable are mesoscale inflow conditions to those simulated by LES under different atmospheric stabilities before adding wind turbines? And (2) how accurately can the EWP and Fitch WFPs represent the impact of wind turbines on the flow across different atmospheric stability conditions compared to an analogous LES-AD-based simulation?

To address these questions, our study conducts controlled simulations that characterize the structure of neutral, unstable, and stable atmospheric boundary layers (ABLs). We focus on the wake effects under these atmospheric conditions while maintaining, as close as possible, the inflow conditions in the LES and mesoscale simulations. More importantly, the LES and mesoscale runs are performed with the same version of the WRF model, thus ensuring consistency in both physics and numerical approximations between the two modelling approaches.

This study is structured as follows. Section  presents the configurations of the WRF model needed for the LES and mesoscale frameworks to represent the three atmospheric stability conditions. This section also details the WFPs, the AD model, and the methods used for comparison. Section  evaluates the similarities between the LES and mesoscale inflow conditions. Section  details our wind turbine simulation results. Finally, Sects.  and  discuss our results and summarize the key conclusions of the study.

We perform a series of WRF model simulations at different time and spatial scales to verify the behaviour of the WFPs (Fitch and EWP) under different atmospheric stability conditions and WFP-specific parameter values. These simulations are divided into coarse-resolution runs using PBL schemes (i.e. mesoscale, from now on referred to as PBL simulations) and high-resolution runs (i.e. LES) as reference. We first simulate the atmospheric flow for three distinct atmospheric stability regimes with LES without turbines. Then, we try to match the LES inflow conditions as closely as possible in the PBL simulations. Afterwards, we apply the WFPs, with different WFP-specific parameter values, in the PBL simulations and the AD in the LES.

The two modelling approaches use the same version of the WRF model (version 4.2.2), which includes an implementation of an AD model and two WFPs: Fitch and EWP . We investigate two simple turbine layouts: one with an isolated turbine and the second with two flow-aligned turbines with a separation of 5.6 D (rotor diameters). The two layouts use the DTU 10 MW reference turbine with a hub height of 119 m and D=178.3 m.

In the following subsections, we present the configurations for the WRF model simulations, the details used for developing the different atmospheric boundary layers in the LES and PBL simulations, the parameter values used in the WFPs, and finally the skill metric used in the assessments.

We analyse the differences between the LES and PBL vertical profiles of wind speed and direction, potential temperature, and TKE obtained for the three ABL types before simulating the turbines. We also compute the boundary-layer height by finding the height of the maximum vertical temperature gradient, and the Obukhov length (WRF model output of the surface-layer scheme) is computed from the surface fluxes. Lastly, the degree of similarity between the PBL and LES inflow vertical profiles of wind speed and direction is quantified with the RMSD using Eq. () over the 32 vertical levels intersecting the area where the wind turbine rotor will be placed later.

Figure  compares the inflow vertical profiles from the time-averaged PBL and the temporal and spatial averaged LES over Area 1 in Fig. b. In the neutral case, the vertical profiles of potential temperature in the PBL and LES frameworks (Fig. c) closely resemble each other, with boundary-layer heights of 747.2 m for LES and 754.2 m for PBL (Table ). The TKE is similar between the LES and PBL runs within the upper part of the boundary layer. However, the closer to the surface the larger the differences, with lower TKE in the PBL than in the LESs (Fig. d). In contrast, the neutral ABL wind speed shows similar values and shear below hub height and tends to show more differences in the upper part of the domain. This behaviour also applies to the wind direction. Although the PBL height is nearly the same for both LES and PBL, the location of the jet's nose is about 100 m apart.

In the unstable case, vertical profiles generally agree between the PBL and LES runs, and the largest differences appear in the upper part of the domain for the wind speed and direction (Fig. a and b). Within the rotor area, the PBL and LES results show a RMSD of 0.01 m s-1 and 0.6° for the wind speed and direction, respectively (Table ). Note that the values of the inverse Obukhov length (Table ) are similar between the PBL and LES runs, depicting an unstable atmospheric surface layer. The PBL run shows the largest differences in TKE compared to the LES run in this unstable case.

The stable case, as expected, exhibits a shallower boundary layer than the neutral and unstable cases. The PBL simulation agrees with the LES throughout the boundary layer for wind speed and direction with a RMSD of 0.02 m s-1 and 0.07°, respectively. However, the PBL simulation consistently shows less TKE than the LES. Furthermore, the jet's nose in the PBL simulation (at about 250 m) is higher than that in the LES (at about 230 m), which is consistent in the three ABL types.

In summary, the wind speed and direction profiles within the rotor area in the PBL and LES frameworks for the three stability classes match well (Table ). This demonstrates the comparability between the simulation frameworks in capturing key features of the inflow conditions.

We present the analysis and comparison of the WFPs and the LES-AD results for the single-turbine and two-turbine layouts across the three ABL types. We show the results in two groups: a reference analysis and a joint analysis of wake interactions under the effect of atmospheric stability. First, we present the reference analysis, which consists of the results of the single turbine under neutral atmospheric conditions. The purpose of the reference analysis is to provide a common point of reference with previous studies that tested the Fitch WFP against LES-AD but now with the inclusion of the EWP. Later, we present the analysis of the wake interactions from a simple two-turbine layout under the three ABL types from both WFPs and the LES-AD results.

Our results show that the effectiveness of the WFPs varies depending on the variable analysed (wind speed or TKE) and the area of study (at the turbines' site or downstream). Particularly for TKE results, the EWP without an explicit TKE source, and regardless of the σ0 values, cannot match that of the LES-AD values. The Fitch WFP using cf=0.25 provides the closest results to the LES-AD under the range of ABL types studied, but the differences varied depending on the specific area, layout, and ABL type.

Regarding wind speed, the two WFPs offer a distinct and different representation of the wake; they either accurately depict the wake in turbine areas with deviations in the downstream areas or vice versa. The Fitch scheme with all tested cf values aligns well with the LES-AD results in the turbine areas but produces deeper wakes in the downstream areas. In contrast, the EWP's accuracy depends on the σ0 value: σ0=0.6r0 mirrors the Fitch results, while larger initial length scales (specifically, σ0=1.0r0, 1.7r0) show better performance downstream but a weaker agreement at the turbine area. Regardless of these differences, the performance of the WFPs remained relatively consistent across the three ABL types.

When translating our findings on the velocity differences to relative power differences (relative to those obtained from the LES-AD), we find small differences within the range of cf values used, ≈1 %–2 %. However, the variation in relative power difference due to the use of different σ0 values is much larger, ranging from 8.5 %, 13.1 %, and 15.4% under unstable, neutral, and stable atmospheric conditions, respectively. These variations underscore the importance of reference high-resolution wake simulations that provide the means to understand the performance of WFPs.

Among the three ABL types, the WFPs resemble the reference under stable conditions the least. This is likely related to the WFPs overlooking the effects of the shallow layer in the stable ABL. The WFPs either misrepresent the wind speed above the boundary layer (EWP with a σ0≥1.0r0) or do not adequately depict the vertical TKE distribution (Fitch). Flow acceleration near the surface of the stable ABL is partially captured when TKE is injected in the Fitch WFP with cf≥0.25. However, larger cf values lead to this acceleration under neutral and unstable conditions (with cf≥0.5 in the unstable ABL; not shown), which are absent in our LES-AD reference. This acceleration is also observed in Fitch simulations and is absent in horizontally averaged LES results for neutral conditions in previous works .

The results involving the EWP extend previous studies that examined the impacts of σ0 for neutral and stable cases in wind farm simulations . Although previous research calibrated and validated σ0 values (within the range 1.5r0:1.9r0) and found minor impacts on the wind speed field, our study further investigates the effect of σ0 within a broader range (σ0=0.6r0, 1.0r0, and 1.7r0). We find that low values of σ0 produce deeper wakes (with deficits concentrated at hub height), and higher values produce more diffuse wakes (the deficit is distributed vertically). When investigating EWP's TKE production, we find that all types of simulation produce negligible levels of TKE, a behaviour seen in previous studies when compared with airborne measurements . This comparison points to a potential area for future improvement in the EWP, which includes an explicit source of TKE.

Our Fitch results with one turbine under neutral conditions are consistent with those of and using cf=0.25. Their demonstrated good performance on wind speed in the turbine area and the growth of wind speed differences in downstream cells align with our findings. However, for the TKE analysis with two aligned turbines under non-neutral conditions, the performance varies depending on the analysed location and the ABL type. Under neutral conditions, cf=0.25 agrees well with the reference within the downstream turbine areas for single- and two-turbine layouts. But for the two-turbine layout under unstable and stable conditions, cf=0.5 and 0.75 seem better choices. These changes in performance depending on the location and atmospheric conditions highlight the limitations of current WFPs and underscore the need for further research to improve their parameters.

Our study uses LES-derived data to assess the accuracy of the TKE and wind speed representations in both WFPs, providing the means of validation with limitations. Given the distinctive temporal and spatial length scales used when employing PBL approximations and LES, achieving identical inflow profiles remains a significant challenge, and, consequently, a perfect resemblance should not be expected. Developing these conditions in the WRF model requires running the model under different combinations of boundary conditions, initial profile soundings, and heat fluxes. Testing each combination is an expensive procedure. In addition, long simulations are needed to reach a point when the inertial flow oscillations have decayed while considering that the boundary layer might continuously grow. Here, we focused on matching similar conditions at hub height and found good agreement within the rotor area.

Despite these efforts, there were still differences in inflow conditions between the PBL and LES simulations. One of these differences is the higher production of TKE in the LESs compared to that of the PBL scheme. These differences are more prominent near the surface and extend vertically to hub height, which could potentially modify the wind speed deficit due to wake. Furthermore, high TKE values translate into high wake recovery rates (more mixing). Besides the different temporal and spatial scales between the LES and PBL frameworks, the higher background TKE and that added from the turbines could contribute to faster wind speed wake decay (or mixing) in the LES simulations, explaining some of the differences found.

The dependence of the WFPs performance in the various wake regions might be related to the choice of the MYNN scheme. Among the various PBLs available in the WRF model, MYNN is currently the only scheme coupled with the Fitch WFP, primarily due to the TKE advection that is required by this WFP. Consequently, most studies simulating wind farms in the WRF model use this PBL scheme. We used the EWP with this PBL for consistency, although it is not dependent on the TKE advection. Using different PBL schemes together with WFPs could potentially provide a more representative comparison against the LES-AD simulations. Only a few studies have employed the Fitch WFP with alternative PBLs (e.g. the NCAR 3DPBL; ) that theoretically could offer a more accurate representation than traditional 1D PBL schemes. Future studies should focus on investigating the wake behaviour using different PBL schemes.

The use of WFPs in mesoscale models plays an important role in understanding the response of wind farms to atmospheric flow. This work evaluated two WFPs in the WRF model, Fitch and EWP, under three distinct atmospheric stability conditions using the WRF model in LES mode as the reference. The evaluation reveals in detail the manner and consequences of using parameterizations that miss the intricacies of the flow provided by the LES reference.

Compared to the LES reference, the performance of the two WFPs evaluated depends heavily on the area of interest (range of downwind areas) and the variable analysed.

The WFPs can depict the vertical profiles of the wind speed differences in two distinct scenarios: (1) good agreement in the grid cells where the wind turbines are located but deeper wakes downstream or (2) good agreement in the grid cells in the downstream areas but weaker wakes at the turbine locations. Specifically, the Fitch WFP aligns with the first scenario. Meanwhile, the behaviour of the EWP depends on its parameter value, allowing it to emulate either of the two described behaviours.

The variations in the performance of the WFPs, as outlined above, are consistently observed under the three atmospheric stability conditions, with the WFPs being the least comparable in stable conditions. Various factors may influence the performance patterns listed above, including the specific PBL parameterization used. Recognizing these variations, our study identifies potential areas for improving the WFPs. Future studies may expand on this work by exploring alternative PBL schemes and considering factors such as varied turbine layout, spacing, and farm sizes.

In their current state, WFPs only offer a partial view of wake dynamics compared to the LES reference. Because of this, it is essential that the WRF model users understand how the performance of their chosen WFPs and the setup may influence the reliability of the results for a given use case. Assessing performance across downstream cells can help understand wind farms' effects on wind resources and nearby farms, and achieving accurate results at the turbine location can lead to more refined and effective power modelling. Ultimately, these insights emphasize the importance of using WFPs appropriately based on their intended application.