Throughout the present work, the three-dimensional filtered velocity field is described by the incompressible Navier–Stokes equations. The Boussinesq approximation is used, and we employ a transport equation for the filtered potential temperature. The set of equations is explicitly given in Appendix , and an in-depth description of the equations can be found in .

Within this paper, we focus on barotropic flows, in which a constant background pressure gradient across the domain balances the Coriolis force above the capping inversion, resulting in a geostrophic wind in the free atmosphere that is constant with height. Moreover, the forcing exerted by the turbines on the flow (see Sect. ) is accounted for through an actuator disk model (ADM). With regard to the subgrid-scale model, we use the stability-dependent Smagorinsky model developed by . The corresponding Smagorinsky coefficient is set to Cs=0.14, similar to previous works carried out with SP-Wind (; ; ; ). Moreover, we use the damping approach of near the wall, which is a well-established technique for neutral atmospheric boundary layers (ABLs) (see also ).

In order to solve the set of equations, we use the in-house SP-Wind solver (see, e.g., ; ; ; ; ; ). This software relies on a classical fourth-order Runge–Kutta scheme with a Courant–Friedrichs–Lewy number of 0.4 to integrate the system in time. At every stage of the numerical scheme, the Poisson equation is solved to ensure continuity. Further, SP-Wind provides pseudo-spectral Fourier schemes that are used to discretize the equations along the streamwise and spanwise directions. We note that aliasing errors are prevented thanks to the 3/2 dealiasing rule of . Finally, the vertical direction is discretized following an energy-preserving fourth-order finite-difference scheme, as discussed in . The reader can refer to, e.g., for further details about the discretization employed.

In this paper, we model the performances of the IEA 15 MW offshore turbine detailed by . It is equipped with a rotor diameter D = 240 m located at hub height zH = 150 m that delivers a rated power of Pr = 15 MW.

We model the turbine rotor as a non-rotating actuator disk, similar to the LES studies of , , and , among others. In the actuator disk model, the turbine acts as an infinitely thin disk that extracts momentum from the flow. However, in order to prevent numerical instabilities associated with abrupt gradients of forces, we smooth out the force distribution by means of a Gaussian filtering operation (; ). We define the three-dimensional Gaussian filter as 1G(x)=6πΔf23/2exp⁡-6x2+y2+z2Δf2, where x denotes the coordinate vector, and Δf is the filter width. In SP-Wind, the filter width relates to the grid spacing through Δf=max{fxΔx,fyΔy,fzΔz}, with fx=fy=fz=1.5 being the filter parameters and Δx, Δy and Δz being the cell dimensions discussed in Sect. . Consequently, the footprint for a turbine centered at xt corresponds to () 2R(x)=∫∫∫ΩG(x-x′)δ(x′-xt)⋅e⟂×H(D/2-||x′-xt||2)dx′, with e⟂ being the unit vector orthogonal to the turbine and Ω being the three-dimensional space. In Eq. (), the symbols δ and H represent the Dirac delta distribution and the Heaviside function, respectively.

Then, the velocity at the location of the kth rotor and perpendicular to it is computed as the spatial average over the footprint: 3ud,k=MA∫∫∫ΩR(x)u⋅e⟂dx, where A=πD2/4, and M is the velocity correction factor (). For coarse grids, the filtering operation may lead to a power overestimation as the rotor diameter appears to be artificially increased. Therefore, we use the velocity correction factor M proposed by as a function of the filter width Δf and the disk-based thrust coefficient CT′: 4M=1+CT′213πΔfD-1.

Furthermore, the magnitude of the thrust force exerted by the kth rotor, 5Fk=12ρ0CT,k′ud,k2π4D2, is distributed over the turbine footprint as done by , 6fk(x)=FkR(x), where ρ0 is the reference air density and ud,k the local disk-averaged velocity (Eq. ). Further, the force intensity is set through the disk-based thrust coefficient denoted CT′, the value of which is given as input to SP-Wind (see Sect. ). Similarly to and , we employ a simple yaw controller that maintains the actuator disk perpendicular to the flow direction measured 1 D upstream. Consequently, we emphasize that the total force Fk (Eq. ) is the magnitude of a vector that generally has components along both the spanwise and the streamwise directions. Finally, the total power the kth turbine extracts from the flow, denoted Pk, is computed as follows: 7Pk=Fkud,k=12ρ0CT,k′ud,k3π4D2.

The boundary conditions of the numerical domain are specified as follows. On the bottom face, we model the development of shear stresses by means of the classic Monin–Obukhov similarity theory for a neutral boundary layer (; ), for which a surface roughness z0 = 1 × 10⁻⁴ m, representative of offshore conditions, is selected.

Further, both the streamwise and the spanwise lateral sides of the domain are assigned periodic boundary conditions. This allows us to model an infinitely wide domain, provided that no farm-induced effects reach the edge of the domain. The choice of an appropriate domain size is discussed in Sect. . Along the streamwise direction, we employ the wave-free fringe region technique developed by , in which a body force is applied to ensure that the desired inflow conditions are imposed at the front of the domain. The generation of spurious gravity waves arising from this non-physical body force is prevented thanks to a damping of the vertical momentum above the ABL. The wave-free fringe region technique is used together with a concurrent precursor approach, from which the fully developed turbulent flow field can be imposed (see Sect. ).

At the top of the domain, a rigid-lid condition ensures zero shear stress, zero vertical velocity and a fixed potential temperature. Without particular treatment, however, this boundary condition significantly reflects gravity waves. Therefore, we use a Rayleigh damping layer (RDL) to curtail this wave reflection effect, similar to and , among others. The method consists of applying a body force in the upper part of the free atmosphere, with an intensity proportional to the difference between the local velocity field and the geostrophic wind.

Prior to simulating the flow in the wind farm, we run a precursor simulation in which the turbulent flow fully develops and reaches a statistically steady behavior. When running the wind farm simulation, the precursor is concurrently advanced in time so as to provide the inflow, as discussed in Sect. . In the scope of this work, we select a precursor domain with dimensions Lxp=Lyp = 10 km and Lzp = 3 km, as done by and . Next, we set the dimensions of the main domain on the basis of the observations of . We note that in the latter study, the authors consider a wind farm about 4 km longer but 2 km narrower than the one investigated in the present work (see Sect. ). Therefore, we use the same main domain length and height as in , i.e., Lx×Lz = 50 km × 25 km, but we increase the domain width by 10 km so that the main domain has dimensions Lx×Ly×Lz = 50 km × 40 km × 25 km. While the domain height may initially appear overly large, it is required to allow for the non-reflecting radiation of gravity waves and to accommodate the Rayleigh damping layer described in Sect. . The farm is symmetrically positioned along the spanwise direction, resulting in a distance of Lside = 14.3 km between the edges of the farm and the lateral sides of the domain. Eventually, the distance upstream of the farm is taken equal to Lind = 18 km to allow for a full representation of the induction zone .

Because periodicity is imposed over the four lateral sides of the precursor domain, the tiling technique of is employed to extend the 10 km long and 10 km wide precursor field to the horizontal dimensions of the main domain. The resulting field is used as the initial state in the wind farm simulations. The same tiling operation, limited to the spanwise direction however, is carried out to generate the concurrent precursor with horizontal dimensions Lxcp×Lycp = 10 km × 40 km. Additionally, we artificially extend the height of the precursor field by imposing the geostrophic flow field from 3 to 25 km for all the considered atmospheric conditions. The characteristics of the precursor field are further discussed in Sect. .

Furthermore, the grid resolution is identical to that selected by , that is, Δx = 31.25 m and Δy = 21.74 m along the streamwise and spanwise directions, respectively. This corresponds to Nxp=320 and Nyp=460 grid points along the x and y axes of the precursor domain. In the main domain, the selected resolution leads to Nx=1600 and Ny=1840 points. Contrary to the regular grid spacing adopted in the horizontal plane, we use a height-dependent vertical discretization to reduce the computational cost. Firstly, a relatively fine constant spacing Δz = 5 m is retained below 1.5 km, with which the velocity gradients can be accurately captured. Consequently, the turbine rotor encompasses 11 and 48 grid points along the spanwise and vertical directions, respectively. We note that these values align with those of other recent similar studies (; ; ; ). Secondly, the grid is smoothly stretched over 180 points in the region between 1.5 and 15 km. Lastly, an additional stretch is applied over 10 grid points from 15 to 25 km. Overall, a total of Nz=490 grid points are used along the vertical direction. Note that the same vertical discretization, trimmed to Lzp = 3 km, however, is adopted for the initial precursor simulation. In the region where the vertical grid spacing is the finest, the spanwise/vertical aspect ratio is equal to Δy/Δz≃4.3. Although no detailed study on the aspect ratio impact has been performed with SP-Wind, values of the order of 3 to 4 have historically been retained (; ; ) in order to account for the differences between the discretization schemes used in the spanwise and vertical directions (see Sect. ).

In Sect. , the power production of an isolated turbine is compared to that of the wind farm for reference. Therefore, it is necessary to perform simulations of an identical turbine operating in standalone conditions. The horizontal dimensions of the corresponding main domain are the same as those of the precursor simulation so that only the vertical extension from 3 to 25 km is required.

Finally, as the wind farm set-up and the domain size considered here are very similar to those of , the same settings are selected for the Rayleigh damping layer and for the fringe region. The corresponding values are summarized in Tables  and , respectively. In Table , the first two parameters denote the magnitude and the growing rate of the RDL, whereas Lzra is the thickness of the layer. The first four parameters in Table  refer to the starting and ending points of the fringe region and the corresponding smoothness coefficients. The following four parameters denote the same quantities but related to the vertical-momentum-damped region. Eventually, hmax characterizes the strength of the fringe function. The mathematical expressions of the Rayleigh damping function, the fringe region forcing and the momentum damping function are given in Appendix .

The wind farm examined in the current work consists of 100 IEA 15 MW turbines arranged in a 10-by-10 configuration. The spacing between each turbine is set to Sx=Sy=5D in both the spanwise and the streamwise directions. We introduce an offset of Sy/2 between every downstream row to obtain a staggered layout. Given the turbine diameter specified in Sect. , the resulting power density is Pr/(SxSy)≃ 10.42 MWkm-2. Overall, the wind farm, starting at Lind = 18 km (Sect. ), is Lxf = 10.8 km long and Lyf = 11.4 km wide, leading to the following ratios: Lind/Lxf=1.67, Lx/Lxf=4.63 and Ly/Lyf=3.51. A sketch of the numerical domain is depicted in Fig. .

In order to explore the potential for power optimization and load reduction using axial-induction control, different disk-based thrust coefficients (Eq. ) are tested. From axial momentum theory, the value CT=8/9, or equivalently CT′=2 (), maximizes power extraction. In practice, the designed thrust set point is slightly lower to reduce the associated loads at rated wind speed (). In the scope of this work, we therefore select the theoretical optimal disk-based thrust coefficient, along with three other values of CT′ evenly spaced at intervals of 0.75: CT′={0.50,1.25,2.0,2.75}. Following classical momentum theory (), the corresponding values of the thrust coefficient are CT={0.40,0.73,0.89,0.97}. We emphasize that in all the simulations, the considered CT′ value is constant throughout the wind plant, representing the choice of a collective thrust set point. This allows us to restrict the number of parameters in the study.

To initialize the precursor simulation, a potential temperature profile is defined following the model of for conventionally neutral boundary layers (CNBLs). We denote the height of the capping inversion by H and set the constant potential temperature below it to θ0 = 288.15 K. Further, we refer to the strength and thickness of the capping inversion as Δθ and ΔH, respectively. Above the capping inversion, the potential temperature profile is controlled by the rate Γ in the free atmosphere.

Based on the observations of , we select a first set of parameters, {H = 150 m, Δθ = 8 K, Γ = 1 Kkm-1} (referred to as H150–Δθ8–Γ1), for which strong blockage effects are expected to have a substantial influence on the wind farm efficiency. reported a non-local efficiency lower than 0.3, however partially counterbalanced by a strong favorable pressure gradient within the farm, with which the wake efficiency becomes larger than 1. In spite of this, the H150–Δθ8–Γ1 case was observed to result in a low farm efficiency of about 32 %. Secondly, we consider the scenario H300–Δθ5–Γ1, in which the blockage effect is attenuated due to a weaker capping inversion positioned at a higher altitude. Finally, we investigate the combination H500–Δθ5–Γ4, for which observed a beneficial impact of the thermal stratification on the farm efficiency. Note that for all three sets of atmospheric conditions, the capping inversion thickness is initialized to ΔH = 100 m. In addition to the three CNBL atmospheric conditions, we consider a situation with no thermal stratification, similar to . To generate this flow, we start from the H500–Δθ5–Γ4 case and artificially set a constant potential temperature profile when copying the solution from the precursor to the main domain. The resulting flow, denoted H500–Δθ0–Γ0, resembles a truly neutral boundary layer (TNBL) but with the same inlet velocity as H500–Δθ5–Γ4.

Moreover, for all the simulations performed in this analysis, we set the geostrophic wind speed to G = 10 ms-1, as done by . We remark that this speed is slightly lower than the rated speed of the IEA 15 MW turbine reported by . Consequently, this choice of geostrophic speed allows us to analyze the turbine performances in the region where it typically operates at maximum thrust coefficient when following a greedy control approach. Finally, we set a latitude ϕ = 51.6°, resulting in a Coriolis frequency fc = 1.14 × 10⁻⁴ s-1.

The precursor phase is carried out to obtain a statistically steady, fully developed turbulent flow over the domain. For this purpose, the initial velocity profiles are defined following the approach of , that is, a boundary-layer flow with friction velocity u∗ = 0.26 ms-1 connected to a laminar geostrophic wind above the capping inversion. Turbulence is initiated by means of divergence-free fluctuations of amplitude G/10 introduced up to an altitude of 100 m. The initial potential temperature profiles are generated using the model together with the sets of parameters, H, Δθ and Γ, detailed in Sect. . We emphasize that a Rayleigh damping layer is also applied during the precursor phase to damp the inertial fluctuations and the gravity waves above 1 km in the atmosphere. For each atmospheric condition, the precursor simulation is performed over 20 h. The resulting flow quantities are then time averaged over the last 4 h of the simulation and are displayed in Fig. .

Figure  shows the velocity (Fig. a) and the potential temperature profiles (Fig. d) averaged over the horizontal planes, together with the corresponding shear stress profiles (Fig. b) and wind directions (Fig. c). From Fig. a, it can be seen that the presence of the capping inversion limits the boundary-layer growth so that the equilibrium inversion-layer height is attained when buoyancy forces balance the surface shear stress (). As observed by , the amplitude of the super-geostrophic jet that forms at the top of the ABL increases with decreasing inversion-layer heights. Above the jet, the shear stress profile reduces to zero (Fig. b), the flow becomes laminar and the velocity profile corresponds to the geostrophic wind. Figure d shows that for the cases H150–Δθ8–Γ1, H300–Δθ5–Γ1 and H500–Δθ5–Γ4, the origin of the capping inversion moves to an altitude of 195, 325 and 510 m, respectively, over the 20 h long spin-up. We note that these values align with the predictions computed from (not detailed here). In the ABL, the Ekman spiral forms so that the wind direction angle Φd, measured with respect to the axis perpendicular to the farm, varies with the altitude (Fig. c). The angle Φd values in Fig. c are normalized by the largest value of |α|, where α is defined as the angle between the geostrophic wind and the velocity vector right above the ground. As reported by and , |α|  is observed to be larger for lower capping inversions. Note that the wind direction controller designed by is employed during the precursor phase to rotate the geostrophic wind so as to ensure there are no spanwise velocity components at hub height, i.e., Φ(zhub) = 0°. Finally, small oscillations of the velocity magnitude and the wind direction appear in the inversion layer (Fig. a and b) as a result of the strong stratification that characterizes this region. This matter is addressed in , where the authors show that eddies with a characteristic scale larger than the Dougherty–Ozmidov length are stratification dependent. However, this length decreases as stratification increases, possibly leading to values of the Dougherty–Ozmidov length that are smaller than the grid spacing. Some of the subgrid-scale eddies generated in the inversion layer can be stratification dependent and can therefore not be accurately captured by the subgrid-scale (SGS) model, causing the oscillations observed in Fig. a and b. Similar oscillations can be seen in, e.g., and .

The flow field generated at the last time step of the precursor phase is tiled over the concurrent precursor domain and the main domain described in Sect. . Then, we place the wind farm introduced in Sect.  in the main domain, and we advance the simulation in time so that the flow adapts to the presence of the farm. Simultaneously, the concurrent precursor flow evolves and is imposed in the fringe region following the methodology detailed in Sect. .

In the current work, we focus on obtaining accurate power estimations for a limited number of test cases. Therefore, we first check the convergence of the farm power. In Fig. , the evolution of the instantaneous wind farm power calculated over the two simulation phases is represented for all the considered conditions. We show the normalized difference with respect to the time-averaged power P‾a obtained in the second phase only, i.e., the last 60 min depicted in Fig. . From this same phase, we retrieve the standard deviation of Pa for each set of atmospheric conditions and operating conditions. This quantity is denoted σa and is represented in Fig.  to assess convergence. For all the considered cases, the normalized deviation σa/P‾a is of the order of 10⁻² and appears to increase slightly with CT′.

Figure  shows that, beyond 90 min, any remaining trend appears to be of the order of the power fluctuations, for all the operating regimes and the atmospheric conditions. Interestingly, we note that the statistically steady state is attained more rapidly for low-blockage conditions. Nevertheless, we retain a spin-up duration of 90 min, after which the final phase of the simulation is performed over 1 h. The power and flow quantities are measured during this last phase, referred to as the actual simulation. Similar to the precursor phase (Sect. ), the wind direction controller of is employed during the wind farm spin-up phase. This controller is disabled in the actual simulation, however. Eventually, the same procedure is applied to the corresponding single-wind-turbine cases over the small domain, as discussed in Sect. .

The instantaneous streamwise velocity field is provided in Fig.  for the four tested atmospheric conditions and the four operating regimes introduced in Sect. . Comparing the cases with an identical disk-based thrust coefficient (CT′), the flow appears to vary significantly with the atmospheric conditions. This observation stresses the need to account for the potential temperature profile in the design of an efficient large-scale operating strategy. In particular, a large blockage effect is visible in the form of a bow wave in Fig. a–d. Even though this effect was anticipated due to the low and strong capping inversion above the farm, we observe a significant decrease in amplitude of this feature when CT′ is decreased. The same pattern can be seen, though to a lesser extent, in the H300–Δθ5–Γ1 case in Fig. e–h. Under the set of conditions H500–Δθ5–Γ4 (Fig. i–l), large-scale effects are minor but become apparent when compared to the H500–Δθ0–Γ0 case (Fig. m–p). In particular, analyzing Fig. l and p together, we observe a slight velocity decrease limited to the front of the farm in Fig. l. Moreover, Fig. p shows a stronger farm wake compared to Fig. l, providing evidence that a favorable pressure gradient still forms in situations where the capping inversion is high.

Eventually, observations of the vertical velocity field provided further evidence of the development of wind-farm-induced effects (not shown). In all the considered cases, the displacement of the capping inversion was seen to trigger internal gravity waves, yet to different degrees depending on the value of CT′. Thus, stronger and weaker waves were observed for the cases where the farm operates at high and low CT′ values, respectively. We refer to for a more detailed analysis of this phenomenon.

In order to assess the intensity of the thrust force exerted by the kth turbine on the flow and its corresponding power, we define the time-averaged thrust and power coefficients (denoted CT,k and CP,k, respectively) as follows: 8CT,k=F‾k12ρ0AU‾∞2 and CP,k=P‾k12ρ0AU‾∞3.

In these expressions, A=πD2/4 is the disk area, and F‾k and P‾k are the time-averaged turbine thrust (Eq. ) and power (Eq. ), respectively. Further, U∞ is the reference wind speed computed as the streamwise velocity averaged over a layer of thickness D spanning the disk-precursor domain, i.e., the region defined by [0,Lxp]×[0,Lyp]×[zH-D/2,zH+D/2]. Within this region, we use a vertically dependent weighted average where the weights are given by the actuator disk chord length, i.e., the straight-line distance across the intersection of the disk and the horizontal plane at the considered altitude. For the cases H150–Δθ8–Γ1, H300–Δθ5–Γ1, H500–Δθ5–Γ4 and H500–Δθ0–Γ0, the reference speeds averaged over the last 4 h of the precursor simulation are equal to U‾∞ = 9.61, 9.55, 9.35 and 9.35 ms-1, respectively. We note that the two expressions in Eq. () can be re-written as CT,k=CT′(u‾d,k/U‾∞)2 and CP,k=CT′(u‾d,k/U‾∞)3 using Eqs. () and (), with u‾d,k being the time average of ud,k. The time-averaged thrust (CT,sgl) and power (CP,sgl) coefficients in the single-turbine case are defined with respect to the corresponding thrust (F‾sgl) and power (P‾sgl), analogously to Eq. ().

The distribution of the local thrust coefficient (CT,k) over the farm is normalized by that of the single turbine (CT,sgl) under the same operating conditions and is represented in Fig. . Therefore, Fig.  illustrates the momentum extracted by each turbine in the farm compared to that of an isolated turbine. Because the disk-based thrust coefficient is common to each turbine in the farm, the ratio shown in Fig.  re-writes CT,k/CT,sgl=F‾k/F‾sgl=(u‾d,k/u‾d,sgl)2. When operating at CT′=0.5 (Fig. a, e, i and m), wake interference between the turbines dominates, which results in a region of higher thrust values over the first two rows of turbines, followed by a quasi-uniform distribution across the rest of the farm. In the absence of a capping inversion, the same conclusion applies regardless of the considered CT′ value (Fig. m–p). For the CNBL conditions, the bow-wave pattern described in Fig.  is associated with a favorable pressure gradient that is, for example, visualized through thrust coefficients greater at the fourth row than at the third row in the H150–Δθ8–Γ1 case (Fig. a). As CT′ increases, the velocity at the farm entry decreases so that the row of minimal thrust coefficient is shifted towards the front of the farm (Fig. d). Interestingly, Fig. i–l show that for a high capping inversion, blockage essentially affects the first two rows. Consequently, it is possible to select the value of CT′ common to all turbines so that the front-localized blockage and the wake effects downstream lead to a close-to-uniform thrust distribution across the farm (Fig. l). More generally, provided that the operating regime can be set independently for each turbine row, Fig.  indicates that the thrust distribution could be homogenized by either increasing or decreasing CT′ at the front of the farm subject to high-blockage (e.g., Fig. d) or low-blockage conditions (e.g., Fig. i), respectively.

We define the wind farm thrust and power coefficients, denoted CT,f and CP,f, respectively, as the average values of CT,k and CP,k over all the turbines in the farm. Combining Eqs. () and () into Eq. (), the farm thrust and power coefficients are thus expressed as 9CT,f=∑kNtu‾d,k2NtU‾∞2CT′ and CP,f=∑kNtu‾d,k3NtU‾∞3CT′.

The analysis is further enriched by considering the farm efficiency ηf, which can be written in the form of a product of the non-local and the wake efficiencies, denoted ηnl and ηw, respectively . We therefore write 10ηf=ηnlηw,ηnl=P‾1P‾∞,ηw=P‾aNtP‾1, where P‾a is the total farm power measured during the actual simulation, and Nt is the number of turbines in the farm. The notation P‾1 refers to the power per turbine, averaged over the most upstream row in the farm. Finally, P‾∞ is the power of the turbine operating in isolation. All the quantities in Eq. () are time averaged over the 1 h long simulations.

In Fig. , we show the thrust coefficients and the power coefficients of an isolated turbine computed following Eq. (). The results are compared to the expressions 11CT=16CT′CT′+42 and CP=64CT′CT′+43, obtained from axial momentum theory (AMT) (). Given the time-dependent nature of the thrust and power values collected with a sampling period Ts = 100 s, the time averages and the 95 % confidence intervals shown in Fig.  are computed over the 1 h long actual simulations using a moving-block bootstrap method. We follow the procedure described by with overlapping blocks consisting of nb=3 samples over a total of B=1000 bootstrap iterations. We performed a sensitivity study, not discussed here, to motivate the selected values for nb and B.

As anticipated, the LES results exhibit the same behavior as the theoretical predictions (Eq. ), in particular at low operating regimes for which close agreement is observed in Fig. a and b. Above CT′=1.25, the LES values deviate significantly from the AMT for the H150–Δθ8–Γ1 and H500–Δθ5–Γ4 cases, showing underprediction and overprediction, respectively. Plausible causes of the deviations with respect to classical axial momentum theory include the presence of shear, veer and turbulence in the simulations. It is however unclear whether the overprediction obtained in the H500–Δθ5–Γ4 case has a physical explanation. In , the authors report that the velocity correction factor leads to slight overestimations at large disk-based thrust coefficients. For instance, we observe a difference of 7 % at CT′=2.0 in the present study (Fig. b), which aligns with the discrepancy of the order of 5 % retrieved from the results of at the same CT′ value. Moreover, we emphasize that the expression of the velocity correction factor was obtained for a uniform flow (), therefore possibly resulting in larger discrepancies with respect to the AMT when employed in a non-uniform flow. As a matter of fact, the overprediction between the AMT and the results of a uniform-flow simulation performed at CT′=1.44 was found to be about half that observed with the H500–Δθ5–Γ4 case (not shown). The validity of the velocity correction factor employed with non-uniform profiles could be the topic of future works. The results of LESs relying on a higher-fidelity turbine representation, e.g., an actuator line model, should be used as a reference.

In Fig. a and b, we introduce a simple heuristic fit in which the parameters αt, αp and β of the laws 12CT=αtCT′/βCT′+42 and CP=αpCT′/βCT′+43 are fitted to the LES data points using the least squares method. As the two relations in Eq. () share a common parameter β, the procedure results in a simple joint-optimization problem between the eight LES data points corresponding to the four tested CT′ values. The optimized value of each of the three parameters is given in Table . In Eq. (), an increasing number of parameters was introduced, and a convergence analysis on the residual of the least squares method then motivated the choice of αt, αp and β. Physically, we postulate that the three parameters allow us to account for the impact of shear, veer and turbulence, which are disregarded in classical AMT.

At the wind farm scale, we compute the thrust and power coefficients, CT,f and CP,f, following Eq. (). Similarly to the single-wind-turbine case, we employ the moving-block bootstrapping method. However, because the confidence intervals do not exceed ± 1 %, only the time-averaged values of CT,f and CP,f are represented in Fig. . The corresponding curves are fitted using laws of the form 13CT,f=αt,fCT′/CT′+δt,fγt,f and CP,f=αp,fCT′/CT′+δp,fγp,f, where 6 degrees of freedom are introduced in total. In Eq. (), the two sets of three fitting parameters are determined for CT,f and CP,f through two independent least squares methods. Each of the two fitting procedures therefore sets the values of three parameters using four LES points. The corresponding values are tabulated in Table . We initially explored other options, e.g., using only three parameters to approximate CT,f and CP,f as in the single-turbine case. However, this led to large fitting errors in all the tested cases.

The wind farm thrust and power coefficient curves shown in Fig.  can be discussed in parallel to the efficiency curves obtained from Eq. () and represented in Fig. . First, in Fig. b, we notice that the evolution of CP,f with CT′ is much flatter than in the single-wind-turbine situation (Fig. b). In Fig. c, the farm efficiency is essentially constant above CT′≃1.25 in the three CNBL cases. This results in a region of nearly constant CP,f values, the maximum of which is offset towards CT′ values lower than in the standalone configuration (Fig. b). In the remainder of this analysis, the maximum power coefficient and the corresponding CT′ and CT,f are denoted CP,f⋆, CT′⋆ and CT,f⋆, respectively.

Moreover, the inspection of Fig. a reveals that the ability of the turbines to generate more power by increasing CT′ towards its Betz optimal value (CT′=2) is considerably harmed by the inevitable blockage effect that accompanies large CT′ values. This phenomenon appears to be clearly amplified for inflows with a low capping inversion (H150–Δθ8–Γ1). On the contrary, the non-local efficiency remains constant with respect to CT′ in the absence of a capping inversion (case H500–Δθ0–Γ0). Under CNBL conditions, we notice in Fig. b that the wake efficiency is a growing function of CT′ that reaches values significantly greater than 1 under specific conditions. This observation is explained by the physical meaning of ηw, which should be interpreted as the ratio between the performances of the farm and those of the first row. Consequently, the values ηw>1 in Fig. b correspond to cases where the downstream rows, although waked, extract more power than the first row. This is explained by a large pressure increase before the first row, followed by a favorable pressure gradient that accelerates the flow deeper into the farm, as previously visualized in Fig. . Under the atmospheric conditions H500–Δθ5–Γ4, the favorable pressure gradient leads to values of the farm efficiency that are larger than in the corresponding TNBL case (H500–Δθ0–Γ0), as indicated in Fig. c. Similar observations are reported in .

Below CT′≃1.25, the farm poses so little resistance to the flow that only minor blockage effects occur. Simultaneously, we note that this range of operating regimes exhibits high sensitivity to blockage. This is visible in Fig. a, where the non-local efficiencies of the H150–Δθ5–Γ1 case are initially close to those of the H300–Δθ5–Γ1 and H500–Δθ5–Γ4 cases but drastically decrease as CT′ increases. This results in a slightly higher farm efficiency at low CT′ values (Fig. c), in turn causing the shifting of the curve maximum towards the left in Fig. .

Eventually, Fig. a shows a strong decrease in the farm thrust coefficient values when compared to those of the isolated turbine (Fig. a). However, each wind farm thrust coefficient curve remains much steeper than its power counterpart (Fig. ), supporting the idea that load can be effectively reduced with only a limited impact on power.

In this section, the trade-off between thrust and power is explicitly shown in Fig. a by plotting the information of Fig. a and b in the CP,f–CT,f coordinate system. Figure c is obtained by applying the same procedure to the results of the single-turbine simulations (Fig. a and b). Then, all the curves in both Fig. a and Fig. c are normalized by their peak value and represented in Fig. b and d, respectively. In Fig. d, the normalized curves collapse into the AMT law as this choice of normalization can be shown to be independent of the fitting parameters introduced in Eq. (). By contrast, we note a clear deviation of the normalized wind farm curves from the predictions of the AMT in Fig. b. This observation emphasizes that large-scale effects substantially impact the trade-off between thrust and power and therefore influence the design of the farm operating point. In Fig. a, the three curves corresponding to CNBL conditions are affected by both blockage effects and wake interactions. On the contrary, the curve obtained for the H500–Δθ0–Γ0 case accounts for wake effects only and can thus be considered a blockage-free reference. For each of the curves generated under CNBL conditions in Fig. a, we conclude that the deviation observed with respect to the TNBL reference case results from blockage effects.

We now focus on the design of a wind farm operating point that accounts for large-scale effects. To this end, we explore three potential wind farm set points and evaluate the corresponding thrust and power variation with respect to the standard operating regime CT′=2. For the sake of clarity, we denote the coefficients CT,f and CP,f evaluated at CT′=2 by C^T,f and C^P,f, respectively. Further, we define the relative thrust and power difference with respect to the standard regime as εT=(CT,f-C^T,f)/C^T,f and εP=(CP,f-C^P,f)/C^P,f.

The first method consists of operating each turbine in the farm at CT′⋆ so that the peak of the farm power coefficient curve, i.e., CP,f⋆, is achieved. This operating regime is denoted by a black star in Fig. a. An alternative that could be of interest is allowing for a decrease in CP,f compared to the standard operating regime CT′=2 (Betz limit). With the second approach, we consider for instance a decrease of 1 % in CP,f. This choice, although somewhat arbitrary, aligns with the reduction in power observed when an isolated IEA 15 MW turbine operates at the design thrust set point, prioritizing load mitigation over maximizing power output . From the three fitted curves shown in Fig. a, we retrieve the CT,f value at which 99 % of C^P,f is achieved in each case, and we denote it by CT,f×. We refer to the corresponding farm power coefficient as CP,f×, and we denote the disk-based thrust coefficient by CT′×. Last, the third method further explores the potential for thrust reduction by allowing for a power decrease of 10 %. Similarly to the second approach, CT,f⧫ is the farm coefficient at which 90 % of C^P,f is achieved. The corresponding farm power coefficient and disk-based thrust coefficient are CP,f⧫ and CT′⧫, respectively. The three tested farm operating points and the corresponding gains obtained under each atmospheric condition are listed in Tables –.

In Table , we show that operating the farm at CT′⋆ to maximize power extraction leads to very slight power increments. This is the case, however, provided that the blockage effect is strong enough, i.e., H150–Δθ8–Γ1 and H300–Δθ5–Γ1. More interestingly, we observe for those two cases that the power increase, although minor, is associated with a load reduction of the order of 5 %. In the absence of blockage (H500–Δθ0–Γ0), the deviation from the standard regime is negligible, indicating that the total power is maximized when each turbine operates at the Betz limit. We anticipate this to no longer be the case in a situation where the CT′ distribution can be set non-homogeneously across the farm to mitigate the wake effects of the upstream turbines. The results obtained with the second operational strategy are listed in Table . From this table, we conclude that substantial load reduction can be achieved at the expense of a minor power loss. In particular, the wind farm thrust coefficients in Table  are observed to decrease by up to 19 % under significant blockage. For the same atmospheric conditions, the results of the third approach tabulated in Table  indicate a load reduction of 36 %. However, this decrease is limited to 25 % for the H500–Δθ5–Γ4 case.

As a conclusion, we show that axial-induction strategies for load reduction are particularly effective for small power reductions relative to the classical operating regime, that is, in the region where the slope of the curves in Fig. a is slight. We believe this constitutes an important finding, upon which more sophisticated wind farm operational strategies can be developed. In the future, investigating the sensitivity of the results to the turbine type, the farm layout or the freestream velocity could be of interest. Regarding the ABL flow profile, we anticipate that the diameter-to-hub-height ratio and the ratio of the roughness length to the hub height are meaningful to the problem. We denote them D∗=D/zH and z0∗=z0/zH, respectively. Further, we follow the expression of the similarity parameter H∗=|fc|H/u∗ (), where fc is the Coriolis frequency, H is the boundary-layer height and u∗ is the friction velocity. We note that the hub height velocity UH can substitute u∗, using a log-law profile and the parameter z0∗ defined above. Throughout the present work, D∗ and z0∗ were kept constant, whereas different values of H∗ were considered. We refer to, e.g., , to relate H∗ to the potential temperature parameters set in Sect. . Lastly, the present work provided evidence of the substantial impact of CT′ on the flow dynamics. More generally, we expect power density to play a crucial part in the design of an effective farm operating strategy. Therefore, we introduce the disk-based friction coefficient factor defined in as the fourth non-dimensional number to account for power density. This ratio reads cft′=πCT′/(4SxSy), where Sx and Sy are the turbine spacings (expressed in number of diameters) in the streamwise and spanwise directions, respectively. As a result, similar effects on the total power extraction may be expected for similar values of cft′.