The MYNN and 3DPBL schemes share a common origin with the governing equations: 1∂Ui∂t+Uj∂Ui∂xj=-ϵijkfjUk-1ρ∂P∂xi-∂<ujui>∂xj-gi,2∂Θ∂t+Uj∂Θ∂xj=-∂<ujΘ>∂xj, where the ensemble mean momentum per unit mass is Ui, potential temperature is θ, P describes the mean kinematic pressure, gi captures the influence of gravity, and fk is the Coriolis vector.

With a level-2.5 closure, these governing equations become 3∂q2∂t+Uk∂q2∂xk=∂∂xk[lqSq∂q2∂xk]+2(Ps+Pb-ϵ),4<uiuj>=δij3q2-3l1q[(<ukui>-C1q2δki)∂Uj∂xk+(<ukuj>-C1q2δkj)∂Ui∂xk-23δij<ukun>∂Un∂xk]-3l1qβ[gj<uiθv>+gi<ujθv>-23δijgn<unθv>],5<θv2>=-Λ2q<ukθv>∂Θv∂xk,6<ujθv>=-3l2q[<ujuk>∂Θv∂xk+<ukθv>∂Uj∂xk+βgj<θv2>], such that q2=2TKE=<u2>+<v2>+<w2>, where the TKE budget includes shear production Ps=-<uiuj>∂Ui∂xk, buoyant production Pb=-βgi<uiθv>, and dissipation ϵ=q3Λ1. β, Sq, and C1 are all constants. Finally, the four length scales – l1, l2, Λ1, and Λ2 – are proportional in such a way that 7[l1Λ1l2Λ2]=l[A1B1A2B2], where l represents a master length scale, and A1, A2, B1, and B2 represent another set of constants.

In this level-2.5 closure, TKE is a prognostic variable, advecting TKE generated within the scheme. This TKE treatment for both MYNN and 3DPBL differs from that for several other PBL schemes (i.e., YSU) and allows for both the MYNN and 3DPBL schemes to integrate with the Fitch WFP (described below) to analyze wind farm wake effects on TKE.

From this common origin, the MYNN and 3DPBL schemes differ. First, the two schemes differ in terms of how they represent horizontal turbulent mixing. The MYNN scheme uses Smagorinsky mixing , while the 3DPBL scheme explicitly calculates the horizontal turbulent flux divergences.

A “boundary layer” approximation of the 3DPBL scheme retains the same 3DPBL length scale formulation (and corresponding closure constants) but instead sets the horizontal gradients of the mean quantities and the vertical gradient of the vertical velocity to zero. This 3DPBL approximation is the most commonly used version of the 3DPBL scheme , and we employ it in this analysis for consistency.

The two PBL schemes also rely on differing master length scales and corresponding closure constants. For this analysis, the 3DPBL scheme uses a master length formulation as follows: 8l=kz1+kz/l0, such that k is the von Kármán constant, z is the height above ground level, and l0 is defined by 9l0=α(∫0hqzdz)(∫0hqdz), with α as an empirical constant and with the length scale constants determined experimentally under neutrally stratified conditions. These constants are those used in .

The version of MYNN used in this analysis is best characterized by the default (i.e., bl_pbl_opt = 5, bl_mynn_mixlength = 2) configuration of the MYNN-EDMF, as described in . As a result, the closure constants used by the MYNN scheme in this analysis reflect the update in . The MYNN scheme for this analysis also then calculates its master length scale based on a blending of PBL depth, buoyancy, and surface layer length scales .

The two PBL schemes are also located within different sections of the WRF code base. While the MYNN scheme is located within the physics (/phys/) directory, the 3DPBL scheme resides in the dynamics (/dyn_em/) directory. As a result, the 3DPBL scheme introduces an additional subroutine that ensures both that velocity and those tendencies in the 3DPBL are scaled in the same manner as that for the MYNN scheme and that the Fitch scheme (described in Sect. 2.3) appropriately modifies the 3DPBL TKE tendency. Further details on the implementation of the 3DPBL Fitch integration can be found in .

We now describe other calculated quantities used to understand site performance and physical mechanisms.

The wind speed deficit, λ, characterizes wake strength: 10λijk=UNWFijk-UFitchijk, where UNWFijk is the horizontal wind speed at a specific grid cell ij and specific height index k with no wind farms, and UFitchijk is the horizontal wind speed with the corresponding wind farm simulation.

The “drag proxy”, D, for a constant density and rotor area is given by 11D=CTV2, where CT and V are the corresponding manufacturer-specified proper thrust coefficients and wind speeds, respectively (Figs. a, b).

ΔTKE represents the time-averaged difference in TKE between the two PBL schemes at a given i, j, and k location. 12ΔTKE=TKE3DPBL-TKEMYNN This difference field, consistent with all subsequent difference fields, is defined as 3DPBL - MYNN.

Additional spatial and temporal averaging was applied, with care being taken to reflect site differences. For the FINO1 region, 10 min WRF output data for the hours of 12:00:00–00:00:00 UTC were averaged to compare with the FINO1 data. For the FINO1 region, the 10 min model output data were subset to the closest ij grid cell and k model level in relation to the 10 min observations. Each observational height was evaluated separately.

Additional processing was necessary to capture the temporal and spatial variability of the aircraft transect paths. Because each transect represented approximately 10 min of observations, each transect could be reasonably compared to a single 10 min model output. At the same time, observations within a single transect spanned multiple model grid cells. Given these additional considerations, we processed the aircraft region data with the following process, based on and . First, we calculated the horizontal wind speed as follows: 13U=u2+v2, where u and v are the zonal and meridional components, respectively, of the wind in m s⁻¹ for a given transect with the 100 Hz observations. Then, we resampled the horizontal wind speeds with a moving 2 km window. This 2 km window was first determined by and was later implemented in for this case study. This window was selected based on the aircraft speed to yield an average turbulent timescale on the order of a couple of minutes. This integral timescale appropriately separates the small-scale fluctuations from the large-scale turbulent motions . Additional averaging was then performed, this time across grid cells. The 2 km resolution wind speed calculations were mapped to a corresponding model grid cell based on their latitude and longitude, and all of the 2 km resolution wind speed calculations for a given model grid cell were averaged together. The number of 2 km resolution wind speed calculations for a given model grid cell depended on the amount of time that the aircraft spent in that grid cell. Well-sampled grid cells may contain close to 3000 points, whereas less sampled grid cells may contain only 10 points. These grid-cell-averaged values could then be compared to the relevant model cell value at the closest time step (Table ).

We employed a similar process to calculate TKE, also based on and . We again isolated the 100 Hz observations for a given transect and resampled the TKE based on a 2 km moving (standard deviation) window : 14TKE=12(σu2+σv2+σw2), where σu, σv, and σw correspond to the standard deviations of the u, v, and w components, respectively. We then averaged these values across each grid cell to compare to the WRF model (Table ).

Model data and observations from the aircraft vertical profiles were compared to assess the model's ability to capture important local characteristics like atmospheric stability. Because each profile traversed several model grid cells, observations for a given profile were compared to a designated “middle” model grid cell (Table ). Potential temperature, θ [K], was calculated – both from the 100 Hz observations and from 10 min model data – according to the following equation: 15θ=(p0p)0.2854, where p0 is the reference pressure 10 000 hPa, and p (hPa) is the pressure surrounding the aircraft at a given moment. Horizontal wind speed and TKE for the aircraft profile observations were calculated using Eqs. () and (), as used with the aircraft transect flights. However, the rolling-averaging window for the aircraft profile flights was 10 m (as opposed to 2 km), again, as in .

These wind speed (and corresponding TKE) retrievals were subject to errors in aircraft yaw alignment. analyzed experimental influences on retrievals from a drone platform and concluded yaw misalignment to be the dominant source of error for wind speed (and TKE) retrievals. also noted that errors in yaw alignment have opposite signs depending on whether the sensor interacts with the wind from the starboard or backboard side. Because odd transects expose the sensor to one side of the plane and even transects expose the sensor to the other side, this error in yaw alignment could be reflected in systematic differences in wind speed (and TKE) between even transects and odd transects.

The standard error metrics of bias, centered root mean square error (cRMSE), correlation squared (R2), and earth movers' distance (EMD) were calculated for each available variable at each site. The bias, cRMSE, and EMD all have an optimal value of 0, whereas R2 has an optimal value of 1. For the FINO1 region, the averages represented time averages, and, for the aircraft region, the averages were across grid cells. FINO1 time averages included all 10 min data points for the 12:00:00–00:00:00 UTC period. The bias represents the difference between the modeled and observed means: 16bias=(p‾-o‾), where p‾ represents the modeled mean, and o‾ represents the observed mean. The cRMSE represents the unbiased component of the model error. The cRMSE, in this case, is as follows: 17cRMSE=[1N∑n=1N[(pn-p‾)-(on-o‾)]2]1/2, where N is the number of data points. The Pearson correlation coefficient (R) represents the correspondence between two variables: 18R=1N∑n=1N[(pn-p‾)-(on-o‾)]σpσo, where σp and σo represent the standard deviations of the predictions and observations, respectively. Here, we reported the coefficient of determination, R2, as recommended in . Finally, EMD, also known as the Wasserstein distance, represents the area between two cumulative distribution functions. The EMD can be described by the following: 19EMD(u,v)=inf⁡π∈Γ(u,v)∫R×R|x-y|dπ(x,y), where Γ(u,v) is the set of probability distributions on R×R, whose marginals are u and v on the first and second factors, such that u(x) is the probability of u at position x, and v(x) is the probability of v at position x . Here, the EMD was calculated with the Python function wasserstein_distance() from the SciPy library .

Statistical significance testing was also performed to determine the strength of the differences between simulations. This testing was performed for each error metric separately, with each transect and/or height representing a value of the appropriate sample. Noting the small number of data points, as well as the non-normality of each sample, we performed our statistical testing with a Mann–Whitney U test . The Mann–Whitney U test is a non-parametric, rank-sum test that relaxes traditional normality and variance requirements with many other statistical tests. Tests were performed with the corresponding Python function from the SciPy stats module , and a result is deemed to be statistically significant if |p|<0.05.

The modeled atmospheric stability both at the FINO1 location (Fig. a, b) and over the aircraft region (Fig. f, g) suggests a weakly stable profile near the surface with stronger stability aloft. MYNN simulations are slightly warmer (according to potential temperature) and more stable near the surface than 3DPBL simulations (Fig. b, g). For both sites, these modeled temperature profiles are consistent with available observations (Fig. a, f), and the observed potential temperature in the aircraft region is slightly warmer than the modeled potential temperature (Fig. g). Because the modeled air temperatures are almost identical between models (Fig. a, f), these slight differences in the surface stability could be a consequence of the greater TKE with the 3DPBL scheme that encourages slightly more mixing. The wind direction profiles at both FINO1 (Fig. c) and the aircraft transect (Fig. h) regions show veering wind (i.e., the wind direction rotates from southwesterly direction near the surface to westerly aloft), suggesting warm-air advection. Simulations with MYNN also tend to have slightly more southerly winds than simulations with the 3DPBL scheme (Fig. c, h). Wind speeds are similar near the surface and through the rotor layer, with slight variations between the aircraft and FINO1 regions (Fig. d, i). Peak wind speeds for both sites are slightly higher than the peak TKE, reflecting an LLJ, as in . The wind direction vertical profiles for both the FINO1 (Fig. c) and aircraft (Fig. h) regions also suggest inversions at 500 m. Although this separate air mass would not be considered to be an inversion according to the potential temperature vertical profiles (Fig. a, d), the variation in wind speed does support a distinct layer above 500 m (Fig. d, i). Further, the modeled PBL height, which, under stably stratified conditions, is determined as the height at which the TKE profile reaches 5 % of its surface value , corroborates this distinction. Thus, dynamically, the top of the stable boundary layer likely resides around 500 m.

The TKE peaks differentiate the two model vertical profiles at both sites. The 3DPBL TKE scheme is consistently larger beyond the surface, and this discrepancy is largest at and slightly above the rotor level. The fundamental differences in horizontal-mixing and length scales between the schemes lead the 3DPBL scheme to characterize more TKE. These differences are more accentuated with both an LLJ and the turbine-induced TKE contribution.

Simulated TKE also differs from observations. While MYNN TKE underpredicts TKE observed during both even and odd transects, the 3DPBL TKE overpredicts TKE during even transects and underpredicts TKE during odd transects. These deviations between modeled and observed TKE may, in part, reflect the model formulations. However, yaw (mis)alignment of the aircraft may also play a part in these discrepancies. The error in yaw alignment is expected to have alternating signs on whether the wind approached the sensor from the starboard or the backboard side . Because the even and odd transects involved opposite alignments of the aircraft, it would follow that the odd and even transects would then show distinct TKE (and wind speed, on a smaller scale) errors. Thus, the discrepancy between TKE observed with the odd and even transects may also play a part in the poor model agreement.

The aircraft profile flights suggest a similar vertical structure to that for the FINO1 and aircraft transect regions. Both MYNN and 3DPBL temperature and potential temperature profiles at the locations of the profiles (Fig. a, b) suggest a weakly stable profile near the surface, with stronger stability aloft. This stability is reinforced by the aircraft profile observations (Fig. f, g). An LLJ once again emerges in both wind speed (Fig. d, i) and TKE (Fig. e, j) maxima. Finally, once again, both modeled (Fig. c) and observed (Fig. h) wind direction from the aircraft profiles suggest warm-air advection with an inversion that is supported by the modeled PBL height.

For the aircraft profile flights, the vertical structure suggests discrepancies between the model and observations in terms of both values and shape (Fig. ). The wind direction (Fig. c, h) rotation is also modeled at a higher altitude than observed. The observed surface wind direction (Fig. h) also reflects an approximately 20° discrepancy from the modeled surface wind direction (Fig. c). Observed wind speeds (Fig. i) are also faster than modeled wind speeds (Fig. d), with an additional local peak lower than 200 m that is not captured by the model. Finally, observed TKE (Fig. j) suggests two peaks, with one peak within the turbine rotor layer and the other peak well above aircraft transect flights and the top of the boundary layer. In contrast, modeled TKE (Fig. c) suggests a single TKE peak with a more gradual decrease with height. These discrepancies in agreement could be explained by several factors, such as the larger number of available data points for the aircraft profile flights, a lack of a smoothing time average for the aircraft profile observations, and the spatial variability sampled with the aircraft profile flights.

The stable stratification may suppress some of the turbine-generated turbulence from extending aloft. Both FINO1 and the aircraft transect measurement regions show simulated TKE peaks at altitudes within the rotor region due to wind-farm-generated turbulence. Both the FINO1 and aircraft regions also show larger amounts of TKE with the 3DPBL scheme (Fig. a, b) than with the MYNN scheme (Fig. c, d). These differences in TKE between the PBL schemes, which are consistent with those shown under the idealized, stable conditions simulated in , reflect the fundamental differences between the models. The stronger TKE maxima in the 3DPBL (Fig. a, b) at both sites also lead to greater interfarm TKE overlap with 3DPBL than for MYNN (Fig. c, d), such that the TKE interactions between the wind farms are more pronounced for 3DPBL. The difference in both the intensity and degree of overlap between the 3DPBL and MYNN TKE maxima is stronger in the aircraft region (Fig. f) than in the FINO1 region (Fig. e), likely due to the larger number of wind turbines in the aircraft region (Fig. a, c). However, whereas the aircraft region's simulation suggests a higher maximum TKE than the FINO1 region, not all of the aircraft region TKE is captured by the measurements (Fig. b, d). The turbine-induced turbulence is sampled well by the FINO1 tower (Fig. a, c). However, some of this turbine-induced turbulence is suppressed from reaching the aircraft region measurement height (Fig. b, d). The two PBL schemes also do not respond to this stable stratification in the same manner. The 3DPBL scheme simulates more of this TKE reaching the aircraft measurement height, resulting in a comparatively larger 3DPBL - MYNN TKE difference at the aircraft measurement height (Fig. f).

These diverging patterns in TKE characterization between the two sites also have secondary influences on measured wind speeds (Fig. ). In the aircraft region, enhanced TKE implies greater momentum extraction and, consequently, reduced wind speeds at the measurement site (Fig. b, d, f). In the FINO1 region, however, TKE from aloft mixes more momentum from aloft into the measurement region, which increases wind speeds (Fig. a, c, e). Thus, the differing measurement altitudes between the two sites may impact both the TKE and wind speed assessments presented below.

Wind field behavior near the turbines differs from that for the rest of the simulation domain on average. Recall that the 3DPBL scheme incorporates more potential sources of TKE generation. MYNN average wind speeds are faster than 3DPBL average wind speeds outside of the turbine wakes (Fig. a). These MYNN average wind speeds likely exceed 3DPBL average wind speeds in this area because the 3DPBL scheme generates more TKE (Fig. b) due to the fact that the 3DPBL turbulence parameterization includes more terms that allow for horizontal mixing. Because the atmospheric boundary layer experiences more TKE in the 3DPBL simulation, the frictional forces are slightly larger, reducing the horizontal winds in this stably stratified flow case. This larger TKE with the 3DPBL scheme extracts more momentum from the mean wind, resulting in a greater reduction in wind speeds. This finding that MYNN wind speeds are faster than 3DPBL wind speeds for the same forcing or boundary conditions is consistent with other comparisons of these two PBL schemes, completed under both real and idealized conditions . This finding is also documented further in Fig.  and in Sect. . Note that this increased TKE in the 3DPBL scheme is particularly evident along the Danish coast, resulting in even slower winds in the 3DPBL scheme (Fig. a, b).

Once these faster winds, in the MYNN scheme, enter the region of a wind farm, the faster winds lead to a larger drag force exerted by the wind turbines (Fig. b) and therefore a larger wake effect, resulting in slower winds in the MYNN wakes (Fig. 8a). This distinct behavior in the wakes arises from differences in the drag forces for each PBL scheme. The drag forces are very sensitive to wind speed (Fig. b). Because the MYNN wind speeds are slightly faster when entering the wind farms, the resulting MYNN drag force (Eq. , Fig. b) is generally larger than the 3DPBL drag force. As a consequence, the MYNN scheme shows stronger and longer wakes than the 3DPBL scheme, on average (Fig. c). The MYNN average wind speed reduction is sufficiently strong such that 3DPBL average wind speeds exceed MYNN average wind speeds within the turbine wake (Fig. a). Further, because 3DPBL average wind speeds exceed MYNN average wind speeds in this region, the 3DPBL scheme also has more turbine-induced TKE than the MYNN scheme (Fig. b). This turbine-induced TKE can help erode the wake in the 3DPBL simulations.

This behavior emerges fundamentally because of the difference in ambient background turbulence between the 3DPBL scheme and the MYNN scheme under these stably stratified conditions. Other behavior would manifest in stably stratified conditions if winds exceeded the wind speed of the peak drag force (near 13 m s⁻¹ in Fig. b), such that faster winds would result in a smaller drag force.

Simulations capture the slowing winds observed at FINO1. Modeled FINO1 Fitch wind speeds appropriately capture the temporal shifts throughout the observational period (Fig. ).

Observational agreement for FINO1 modeled wind speeds differs by measurement altitude. The median FINO1 modeled wind speeds at the highest locations (81, 91, and 102 m) perform best (compared to those at the lower altitudes of 34, 41, 51, and 61 m) for R2 (Fig. b) and cRMSE (Fig. c) and the worst for the bias (Fig. a) and EMD (Fig. d). The low (34 and 41 m) and middle (51 and 61 m) heights show similar performance for the bias (Fig. a), R2 (Fig. b), and EMD (Fig. d), with some additional shaping to the cRMSE (Fig. c). Further, the wind speeds at lower altitudes are also slower, resulting in smaller bias. Whereas 3DPBL EMD shifts from outperforming MYNN EMD at the low and middle heights to underperforming compared to MYNN EMD for the higher heights, the physics-based trends are consistent across the FINO1 model heights, which are stronger than the variability between heights. The 3DPBL cases consistently show larger wind speed biases (Fig. a) and smaller wind speed cRMSE (Fig. c) than the MYNN cases.

Similar patterns emerge for the wake-affected region of the flight path. The general trends of increased TKE (Fig. a) and decreased wind speed (Fig. b) above the wind farm clearly emerge. TKE in the wake portion of the flight path is simulated to be stronger in the 3DPBL simulations than in the MYNN simulations (Fig. a), and the maximum value in the 3DPBL simulations is similar to the maximum value in the observations, although the location of the TKE maximum is shifted downwind in both simulations. As noted earlier, these differences in TKE between the PBL schemes reflect the fundamental differences between the models in length scales, empirical constants, and horizontal-mixing approaches. Because the 3DPBL scheme has higher TKE, the 3DPBL scheme consequently shows slower wind speeds (Fig. b, a).

Simulated aircraft region wind speeds and TKE are also subject to systematic influences between transects. First, a performance discrepancy between odd and even transects exists. Simulated wind speeds and TKE for southeast-to-northwest transects (1, 3, and 5) have a larger bias and EMD than those for northwest-to-southeast transects (2, 4, and 6) (Figs. a, d and a, d). This systematic transect variability corresponds to the reversed directions of the transect paths – whereas transects 1, 3, and 5 are performed in the northwesterly direction, transects 2, 4, and 6 are performed in the southeasterly direction.

This directional variability could potentially be explained by errors in yaw alignment. As noted earlier, the dominant source of error in one set of wind speed retrievals was previously determined to be the yaw measurements . The error in yaw alignment was also determined to have alternating signs on whether the wind approached the sensor from the starboard or the backboard side . Because the even and odd transects involved opposite alignments of the aircraft, it would follow that the odd and even transects would then show distinct wind speed (and TKE) errors.

Wind speed and TKE cRMSE and R2 in the aircraft region also reveal additional systematic influences. While the transect direction does suggest an influence on these error metrics (Figs. b, c and b, c), these error metrics (cRMSE and R2) (Figs. b, c and b, c) additionally suggest an improvement over time that is not evident with the bias or the EMD (Figs. a, d and a, d). This performance improvement occurs after transect 2 and continues until transect 4 or 5 and is most evident from the R2, although it does extend to the cRMSE, especially for the 3DPBL (Figs. c, c).