The simulations in this paper are carried out using WRF v4.3.0 with two PBL schemes: MYNN and the NCAR 3DPBL . To avoid confusion regarding nomenclature of new turbulence models, we note that the NCAR 3DPBL is different from the 3DTKE PBL scheme . The WRF v4.3.0 code in this study was modified to include the NCAR 3DPBL code, which is being prepared for public release. For simplicity, we refer to the NCAR 3DPBL as simply the “3DPBL.” Both MYNN and the 3DPBL share a common origin – they are fundamentally rooted in the turbulence modeling of . Here, we use the level 2.5 MYNN and 3DPBL schemes, which both treat TKE as a prognostic variable, thus improving their utility for wind turbine modeling, because generated TKE is advected by the PBL schemes. This behavior stands in contrast to other PBL schemes, such as YSU, which does not treat TKE as a prognostic variable.

MYNN and the 3DPBL treat turbulent mixing differently. MYNN computes the vertical turbulent mixing by calculating the vertical turbulent stress divergence, and it allows the horizontal turbulent mixing to be handled externally with a Smagorinsky-type approach Sect. 4.2 therein. In contrast, the 3DPBL directly accounts for horizontal turbulent mixing by explicitly computing the turbulent flux divergences for momentum, heat, and moisture. The 3DPBL has been implemented into WRF to allow for three different configurations following the original Mellor–Yamada developments: (i) a full 3D model, (ii) a quasi-3D model using the so-called PBL-approximation, and (iii) a 1D model using the PBL-approximation. In this analysis, we employ the second option, as the full 3D parameterization is currently too computationally expensive for yearlong wind resource assessments. When using the second option, the 3DPBL scheme handles both the vertical and horizontal turbulent mixing by computing the 3D turbulent stress divergence, in addition to the 3D turbulent flux divergence of heat and moisture. The vertical turbulent fluxes in the 3DPBL are calculated similarly to MYNN, and the horizontal turbulent fluxes are calculated analytically following after applying the PBL approximation (i.e., neglecting the horizontal derivatives of mean quantities in addition to the vertical derivative of vertical velocity).

Aside from different approaches for horizontal mixing, the two PBL schemes also employ different master length scales and closure constants. Both schemes employ one “master” length scale, although they calculate them differently. In the simulations in this study, the 3DPBL master length scale follows , whereas the MYNN master length scale uses a different approach that simultaneously accounts for length scales that characterize buoyancy, the surface layer, and the PBL depth. The closure constants for the 3DPBL length scale come from , whereas the MYNN closure constants were updated in .

While the values of empirical constants are different, MYNN and the quasi-3DPBL use the same formulation to parameterize turbulent momentum, heat, and moisture fluxes. For example, they parameterize the vertical flux of the u component of wind speed as 1〈uw〉=-LqSm∂U∂z, where L is the master length scale, q is 2TKE, Sm is a stability function, and U is zonal velocity .

To simulate wakes with the 3DPBL, we first integrated the Fitch WFP with the 3DPBL inside the WRF code. The Fitch WFP modifies flow in two key manners : by slowing winds 2∂uk∂t=-12AkCTUkukzk+1-zk3∂vk∂t=-12AkCTUkvkzk+1-zk and by adding TKE 4∂TKEk∂t=12AkαCTKEUk3zk+1-zk. In the above equations, k is the vertical level that intersects the rotor, Ak is the area of the rotor on this vertical level, CT is the thrust coefficient, Uk is the wind speed, uk is the zonal wind, vk is the meridional wind, and zk is the height. The turbulence coefficient CTKE is calculated as the difference between the thrust coefficient CT and the power coefficient CP. The thrust and power coefficients are functions of wind speed that are unique to a particular wind turbine, and their values are specified in the input file wind-turbine.tbl. The coefficient α was introduced by to empirically modify the amount of explicit TKE addition, and, in this study, we set it to either 0 or 1.

The major challenge in integrating the Fitch WFP and the 3DPBL is that the 3DPBL code is housed in the dynamics (dyn_em/) part of the code, as opposed to the physics (phys/) part of the code, where most other PBL schemes reside. As such, the code base was substantially modified to account for the user-selected PBL scheme. A call to the Fitch WFP's dragforce subroutine was added to the end of dyn_em/module_first_rk_step_part2.F. When called for the 3DPBL, the velocity tendencies and TKE tendencies are additionally scaled by the column mass in order to match the identical scaling that happens to the phys/-calculated tendencies earlier within dyn_em/module_first_rk_step_part2.F. Additionally, whereas the Fitch WFP code modifies the MYNN TKE field directly (including a time step factor of ∂t), the new code modifies the 3DPBL TKE tendency field (omitting a factor of ∂t and letting the rest of the code carry out the time integration).

We carry out a series of idealized simulations to study the effect of the PBL scheme on simulated wake dynamics in a simple offshore environment. All simulation inputs can be found on Zenodo (10.5281/zenodo.5565399). We use the neutral idealized simulations of as inspiration for our simulations, but we make a number of modifications. All simulations use two domains, each 202×202 grid points in the horizontal. MYNN is always used in the outer domain, whereas the inner domain is either MYNN or the 3DPBL. The outer domain uses a horizontal grid spacing of 3 km and a time step of 9 s, whereas the inner domain uses a horizontal grid spacing of 1 km and a time step of 3 s. The vertical grid uses 81 cells, up to a height of 20 km. Vertical grid stretching is employed to provide finer resolution near the surface, thereby allowing 28 vertical levels below a height of 300 m, following the recommendation of for nominally 10 m of resolution near the surface. All simulations have a roughness length of 0.0001 m, which is characteristic of offshore environments .

In order to eventually simplify wake comparisons, we force all turbine-free simulations in such a manner that average hub-height wind speeds are roughly equal (∼9.35 m s-1) after they are spun up (Table ). In principle, we could have matched the geostrophic winds instead of the hub-height winds in the idealized simulations, but the resulting different hub-height wind speeds would have made it more difficult to isolate the different turbulent recovery effect that comes with using the 3DPBL instead of MYNN.

Simulations for each stability case are initialized with a neutral temperature profile of 285 K within the boundary layer up to 500 m. The boundary layer is capped with a two-layer inversion: a strong inversion (5 K warming between 500 and 600 m) and a weaker inversion (3 K km-1 lapse rate above 600 m). Depending on the case, each simulation was forced with either 9 or 10 m s-1 geostrophic winds. Stable simulations are additionally forced with -15 W m-2 surface cooling, and unstable simulations are forced with 20 W m-2 surface heating. These sensible heat flux values were chosen based on typical simulated conditions at a planned offshore plant in the US mid-Atlantic and are smaller than typical values over land. After spinup, the boundary layer height as determined through the turbine-free temperature profile (Fig. ) is approximately 250 m in the stable simulations, 550 m in the neutral simulations, and 600 m in the unstable simulations.

After spinning up turbine-free simulations, we run three cases of simulations for each of the stabilities and each of the PBL schemes for 24 h. The first case is simply a continuation of the turbine-free simulations and is referred to as the no-wind-farm (NWF) case. The second case (100TKE) starts after the respective NWF simulation has spun up and shares its boundary conditions, but it includes a 10×10 grid of turbines based on the 12 MW International Energy Agency IEA; reference offshore wind turbines placed in the center of the inner domain. The turbine hub height is 138 m, and the rotor diameter is 215 m. Cut-in speed is 3 m s-1, rated speed is 10.9 m s-1, and cut-out speed is 30 m s-1. Turbines are placed 2 km apart, which is close to 1-nautical-mile spacing. In this case, 100 % of explicit TKE is generated by the Fitch WFP (α=1). In the third case (0TKE), we explore the sensitivity to explicitly added TKE by duplicating the setup of the second case, but turning off explicit TKE generation (α=0).

We spin up the idealized turbine-free simulations so that hub-height wind speeds achieve a pseudo-steady state as well as a value of approximately 9.35 m s-1 (Fig. ). As was observed in , inertial oscillations occur in neutral conditions, but they sufficiently dampen out in our simulations after 4 d. Unstable simulations initially show hub-height wind speed behavior that is similar to the neutral simulations. However, surface warming initiates thermal turbulence during the first day of spinup, and after 24 h of spinup, the hub-height wind speed behavior becomes stationary. Stable simulations show an initial hub-height wind speed spike due to the development of a low-level jet (LLJ; Fig. ), but the wind speeds linearly decay over time as the nose of the LLJ moves upward. The stable MYNN and 3DPBL simulations achieve the target wind speed after 6 and 5.25 d, respectively.

Having discussed the initial transient phase of the idealized simulations, it is also necessary to characterize the baseline wind speeds and TKE values in the turbine-free simulations (NWF) before analyzing turbine impacts (Fig. ). The differences and similarities in the wind and TKE profiles of the NWF simulations will dictate the comparison of the wake effects between the PBL schemes in turbine-including simulations. In general, MYNN and the 3DPBL will predict differing wake effects because of two primary factors: different predictions of turbine-free wind speed profiles and differing wake recovery behavior, which is linked to parameterizations of turbulent fluxes . Due to the experimental configuration of our idealized simulations, the plant inflow wind speeds are similar, and thus we expect the largest wake differences to arise from differing turbulent recovery behavior.

During the performance phase, all simulations have similar average hub-height wind speeds: between 9.3 and 9.4 m s-1. The wind speed profiles for both PBL schemes match expected canonical behavior for each stability . Across the rotor disk, the neutral and unstable wind speed profiles have similar values for both MYNN and the 3DPBL. However in the stable simulations, wind speed profiles slightly differ between the two PBL schemes. The nose of the MYNN low-level jet achieves a wind speed of 11.6 m s-1 and sits at the top of the rotor disk. In contrast, the nose of the 3DPBL LLJ achieves a wind speed of 12.3 m s-1 and sits about 40 m below the top of the rotor disk. Thus, we later see that the height of maximum wind speed deficits differs between the two simulations (Fig. ).

While MYNN and the 3DPBL produce near-identical TKE profiles in neutral conditions, their TKE profiles differ in stable and unstable conditions. In stable conditions, the MYNN TKE profile linearly decays when moving from the surface to the capping inversion, whereas the 3DPBL profile shows an irregular shape that somewhat resembles the wind speed profile of an LLJ. In unstable conditions, the TKE profiles are relatively constant over the height of the rotor disk, but the 3DPBL TKE values are 2–3 times larger than the MYNN values. Contrary to what might be expected, we note that hub-height values of TKE are weaker in the unstable MYNN simulations than in the neutral MYNN simulations. We hypothesize that these low TKE values occur due to the very weak heat fluxes.

Wakes within the extent of the plant are sensitive to the choice of PBL scheme, presence of explicit TKE generation, and stability (Fig. ). We quantify wind speed deficits inside the plant by finding the daylong time-averaged hub-height wind speeds within the plant in the turbine-including simulations (“WFP”, which is a generic stand-in for “100TKE” or “0TKE”) relative to hub-height winds in the turbine-free simulations (“NWF”). We also calculate the percentage of wind speed loss with reference to the NWF winds inside the plant. Before discussing the impact of PBL scheme, we reiterate that previous work at offshore wind farms demonstrates that stability impacts waking , and our idealized wakes follow expected trends: stable wakes are strongest (1.17–1.54 m s-1, 12.4 %–16.3 %), followed by neutral wakes (0.93–1.27 m s-1, 10.0 %–13.5 %), followed by unstable wakes (0.89–1.19 m s-1, 9.4 %–12.7 %).

Average wind speed deficits inside the plant can vary quite substantially between MYNN and the 3DPBL. Across all simulations, MYNN predicts internal waking that differs from the 3DPBL by between -0.20 m s-1, or -2.2 percentage points (pp; in the stable 100TKE simulations), to +0.22 m s-1, or +2.4 pp (in the unstable 100TKE simulations). This large spread induces significantly different predictions of power production (Sect. ).

While these simulations show that wakes within the plant can substantially differ, they do not reveal any obvious patterns of how they will differ across conditions. At times, the MYNN simulations produce stronger wakes internally than the 3DPBL, whereas MYNN wakes are weaker at other times. Sometimes, turning explicit TKE addition off decreases the internal wake magnitude (e.g., stable conditions), whereas other times it increases internal wake strength (e.g., neutral and unstable conditions). Sometimes MYNN internal wake strength changes more substantially when explicit TKE addition is turned off (e.g., neutral and unstable conditions), whereas 3DPBL internal wake strength changes more substantially at other times (e.g., stable conditions). Thus, this variability within the idealized runs suggests that real-world case studies should be tailored to a specific region and turbine configuration.

In addition to characterizing wakes within the extent of the plant, we analyze wakes outside the plant. There is no singular standard approach that is used to characterize wakes external to a plant , so we adopt three approaches: by identifying the contours of the 1 m s-1 deficit, by identifying the contours of the 0.5 m s-1 deficit, and by identifying the e-folding contour. We calculate the e-folding contour as 1/e times the average internal wake strength, or about 36 % , and as such, this uses a relative metric, whereas the other contours use an absolute metric. We employ the 1 m s-1 contour to highlight regions of strong external waking and the 0.5 m s-1 contour to emphasize moderate external waking. We only include the 1 m s-1 deficit contour in the stable simulations, as this contour obscures internal wakes in the neutral and unstable simulations. We note that choosing one definition versus the other can lead to definitions of wake lengths that differ by tens of kilometers.

Wake behavior outside the plant varies just as much as it did inside the plant (Fig. ). The most severe waking, demarcated by the 1 m s-1 deficit contour, varies with stability as expected from previous work, with the strongest wakes in stable conditions. The 1 m s-1 contours extend the farthest in stable conditions, whereas they travel at most about 10 km downwind in neutral and unstable conditions. We note that MYNN predicts wakes that are tens of kilometers longer than the 3DPBL does in stable conditions. The addition of explicit TKE consistently increases the wake length, regardless of what metric is used to define the boundary of the wake. This increase is seen most clearly in the neutral MYNN simulations (Fig. b and e), where wake length grows by dozens of kilometers. All stable and all unstable simulations show a growth in wake lengths, roughly on the scale of about 10 km. We also note that neither MYNN nor the 3DPBL show consistently longer wake lengths across all stabilities. Stability impacts on moderate-intensity wakes (either the 0.5 m s-1 contour or the e-folding contour) are more varied. For example, the e-folding contour is smaller in the stable 3DPBL simulations than in the neutral 3DPBL or unstable 3DPBL simulations.

We briefly digress from the discussion on wakes to discuss two effects that are secondary to the primary analysis of this study: upwind blockage and flow acceleration. Upwind blockage occurs in some of the idealized simulations. Blockage is strongest in the stable conditions, where 0.5 m s-1 deficits extend 5–10 km upwind of the plant. Under neutral conditions, blockage of up to 0.25 m s-1 extends 3–5 km upwind of the plant. Blockage does not appear in the unstable simulations. In general, blockage here is a function of stability but not PBL scheme or TKE addition. Tangential flow accelerations, similar to the speed-ups seen by , can be observed adjacent to the wakes. The hub-height wind acceleration neighboring the wakes is also a function of stability (strongest in stable conditions, weakest in unstable conditions), but it also varies with TKE addition (stronger acceleration when TKE addition is turned on).

While hub-height winds are particularly important to quantify, it is also helpful to characterize wakes over the vertical extent of the rotor disk. We calculate the wind speed deficit averaged across the y extent (predominantly crosswind) of the plant for each simulation (Fig. ). Just as the top-down view (Fig. ) of wind speed deficits suggested, the stable simulations produce the strongest wind speed deficit profiles. Blockage is also visible upwind of the plant in stable conditions. In contrast, the neutral and unstable simulations produce wind speed deficits that are relatively similar to one another. The stronger stable wind speed deficits occur, in part, because of the shallow capping inversion that sits just above the top of the rotor disk. The wakes in the neutral and unstable simulations are able to mix with stronger ambient winds above the plant, thereby eroding the wake, whereas this behavior is not possible in the stable simulations.

The side view of wind speed deficits shows that vertical mixing of wind speed deficits increases when explicit TKE generation is turned on. This behavior consistently occurs across all simulations. The wind speed deficits above the wind plants are stronger in the neutral 100TKE simulations and in the unstable 100TKE simulations than in their counterparts with 0TKE. As a result, the neutral and unstable 0TKE simulations have stronger maximum wind speed deficits within the rotor disk than their 100TKE counterparts. For example, the neutral 100TKE MYNN simulation shows a maximum wind speed deficit of 1.125 m s-1 within the rotor disk, whereas the neutral 0TKE MYNN has a maximum deficit of 1.625 m s-1. While the shallow capping inversion in the stable simulations obscures the effects of explicit TKE addition above the plant, the TKE effects can be seen below the plant. When explicit TKE addition is turned on in the stable simulations, flow acceleration occurs below the rotor disk, but this acceleration does not occur when TKE addition is turned off. We note that acceleration under the rotor disk was observed in but not in . Correlating with the presence of flow acceleration below the rotor disk, the stable 100TKE simulations show stronger wind speed deficits within the rotor disk than the stable 0TKE simulations.

Finally, the side view of wind speed deficits also shows that the choice of PBL scheme can be important. The most pronounced differences between PBL schemes occur in stable conditions. For example, the wind speed deficit in the 0TKE 3DPBL simulation stays stronger than 2 m s-1 for 50 km downwind of the plant, whereas the wake recovers more quickly in the 0TKE MYNN simulation.

In large part, the two PBL schemes produce different wind speed deficits in their wakes because the schemes parameterize turbulent fluxes differently, as we visualize here. The u and v components of wind speed are modified by mechanisms such as advection of the mean wind, the Coriolis force, and the divergence of the turbulent momentum fluxes Eq. 3.4.3c therein. We expect all these terms, aside from the divergence of turbulent momentum fluxes, to be similar for both MYNN and the 3DPBL, as the NWF wind speeds are similar, but two PBL schemes parameterize momentum fluxes uniquely. We calculate the u tendency due to the turbulent flux divergence as the vertical derivative of u′w′‾, neglecting the horizontal components of flux divergence because they are significantly smaller in the 3DPBL than u′w′‾, and they are not computed in the MYNN parameterization. We also omit visualizations of v tendency because they are substantially smaller than the u tendency in these idealized simulations forced with a u-geostrophic wind. We investigate the relationship of wind speeds and turbulent fluxes between the two PBL schemes in the stable 100TKE simulations by comparing two fields in the wakes – the wind speed deficits and the turbulent flux divergence u-tendency “deficits” (Fig. ). The u-tendency deficits are defined as tendencies in the turbine-free simulations subtracted from tendencies in the turbine-including simulations.

The differences in tendency deficits between the two PBL schemes drive the differences in the wind speed wakes. As winds advect primarily along the x direction, wind speed magnitudes are modified by the tendency. For example, the u tendency is more negative for MYNN above the rotor disk. Correspondingly, the MYNN wind speed deficits in this region as well as downwind of this region are more negative. The same pattern of behavior occurs in the upper half of the rotor disk, where the u tendency for the 3DPBL is more negative, and therefore the 3DPBL wind speed deficits are more negative. Thus, the modeled wind speed deficits in the wake of a plant depend on how the PBL scheme parameterizes turbulent momentum fluxes.

Just as wind speed deficits are sensitive to the choice of PBL scheme, TKE associated with the wind plant also varies as a function of PBL scheme, stability, and explicit TKE generation (Fig. ). The WFP induces changes in TKE, and the changes are primarily constrained within the horizontal extent of the plant and tend to not advect far downwind. In contrast, the real onshore WRF WFP simulations of saw substantial TKE changes 20–30 km downwind. The 100TKE simulations produce substantially more TKE than the 0TKE simulations, as would be expected. The 100TKE 3DPBL simulations also consistently predict stronger levels of additional TKE than their MYNN counterparts. For example, the maximum added TKE in the 100TKE stable simulations was 1.375 m2 s-2 for the 3DPBL and 0.750 m2 s-2 for MYNN.

The behavior of the 0TKE simulations was more varied. In neutral conditions, both the 0TKE MYNN and 0TKE 3DPBL simulations create a moderate amount of shear-generated TKE at the top of the rotor disk. However in unstable conditions, the 0TKE 3DPBL simulation shows shear-generated TKE, whereas the 0TKE MYNN simulation does not. In stable conditions, the 0TKE simulations lack shear-generated TKE at the top of the rotor disk due to the low capping inversion. However, the stable 0TKE 3DPBL turbine-including simulation actually has less TKE than the turbine-free simulation. The LLJ in the turbine-free simulation exhibits strong wind speed shear, and the presence of the wind farm reduces that shear, leading to this behavior.

Power production and power losses due to internal waking change with PBL scheme (Fig. ). We calculate the capacity factor for each turbine, the average capacity factor of the plant, and the average power deficit due to internal wakes with reference to the NWF hub-height wind speed. Capacity factor is defined as the ratio of actual power output relative to the maximum possible power output. Across all simulations, the average capacity factor for the plant ranged between 39.5 % and 53.8 %. Capacity factor losses due to internal wakes ranged between 16.7 and 31.6 pp.

Power production in the idealized simulation varies with the simulation parameters. As discussed earlier (Sect. ), when explicit TKE addition is turned off, hub-height wind speed deficits can either increase or decrease. Accordingly, turning off explicit TKE generation can either grow or shrink the capacity factor. Turning off explicit TKE generation changes internal wake losses to the capacity factor by between -6.9 pp (in the stable 3DPBL) and 5.2 pp (in neutral MYNN). Changing from one PBL scheme to another results in wake loss shifts of a similar magnitude – switching from MYNN to the 3DPBL changes internal wake losses by between -3.4 pp (in stable 100TKE simulations) and -9.6 pp (in unstable 100TKE simulations). Thus, these simulations emphasize the critical role of PBL scheme on power production.

In the end, these power calculations emphasize that the behavior of modeled power losses is complicated, even in a simple idealized environment. We stress that these idealized simulations have been carried out for one set of hub-height winds in one part of the power curve under pseudo-steady conditions. To better predict the cumulative non-linear interactions of the effects of these parameters on losses at a real-world location, it is critical to run real simulations.