Truly neutral configurations are helpful when the interaction of the wind farm with only a neutral ABL is of interest. This approach is somewhat of an academic curiosity and might not occur realistically, as it assumes that the atmosphere, even beyond the turbulent ABL, is neutral. It can be of use to study turbines with heights much smaller than the ABL. However, modern wind turbines in large wind farms can interact with the atmosphere above the ABL . Still, many LES studies have used the TNBL configuration, where a pressure gradient or constant geostrophic wind drives the boundary layer .

For a TNBL simulation, a constant potential temperature in a domain a few times taller than the intended ABL height is simulated. The ABL can freely expand vertically until an equilibrium with the non-turbulent free atmosphere is achieved. Therefore, the truly neutral simulation configuration detailed in Table , Set 1(a) has an ABL height of Hi=500 m at the inflow boundary, and the domain height is Lz=12 km. The domain height matches that of the reference case (i.e., CNBL configuration fully resolving AGWs), but the simulation does not use a top RDL because there is no buoyancy to trigger AGWs in the truly neutral case. As shown on the vertical velocity plots in Fig. , the ABL displaces vertically in reaction to horizontal flow deceleration through the wind farm without being constrained by the rigid top boundary. This is also confirmed in Fig. , as the particle path traced through the domain starting at an initial ABL height of 500 m as the inflow initially rises and then asymptotes at a height far lower than the domain height. Moreover, this particle path shows the highest vertical expansion among the cases investigated, as there is no capping inversion to cap the vertical displacement of the ABL.

Like truly neutral cases, the rigid lid configurations have a constant potential temperature, but the domain height is defined at the inversion layer or ABL height. In a physical sense, the top boundary acts like an immovable capping inversion that constrains the vertical displacement of the flow in reaction to the horizontal flow deceleration caused by the wind farm. This constraint on vertical velocity suggests that, to enforce continuity, the flow cannot decelerate as it naturally would. The closer the top boundary is placed to the top of the turbines, the greater this effect on the wind farm. The fact that the top boundary placement decreased horizontal flow deceleration through the wind farm can be witnessed on the vertical velocity contours for the rigid lid simulations shown in Fig. , where the domain heights are 500 m and 750 m. The particle paths for the rigid lid cases shown in Fig.  are concrete evidence of an immovable top boundary; vertical flow is completely suppressed when Lz=500 m compared to the truly neutral case. This “channel” effect is milder for the rigid lid Lz=750 m case but is very pronounced compared to the truly neutral and the reference cases that use much taller domains.

The discussion in Sect.  clearly shows the importance of simulating domains tall enough to exclude channel effects caused by too low a top boundary. However, temperature conditions beyond a few hundred meters above the sea surface are not always neutral; instead, a stable capping inversion and free atmosphere aloft are frequent. Thus, the discussion naturally leads to whether or not an inversion layer and the free atmosphere should be included in wind farm simulations. Moreover, to what extent should the free atmosphere be resolved to capture all phenomena relevant to wind-farm–atmosphere interaction? These questions can be answered by comparing CNBL configurations that partially resolve AGWs with a configuration that fully resolves them; these configurations are detailed in Table , Set 1(c). Only inlet Rayleigh and advection damping layers are implemented for the CNBL-750 m and CNBL-1000 m set-ups, as there is barely any free atmosphere resolved for the internal waves to form.

Besides properly resolving AGWs, avoiding the effect of the top boundary constraining the capping inversion displacement is critical. The evidence of unresolved, partially, and fully resolved AGWs can be seen on the temporally averaged vertical velocity contours for three CNBL cases – Lz=750 m, 1000 m, and 12 km – shown in Fig. . No AGWs are seen for the case with Lz=750 m, traces of interfacial waves are seen when Lz=1000 m, and all AGWs are resolved when Lz=12 km. Since AGWs perturb the capping inversion, improperly resolved AGWs will show unrealistic impacts on the flow in the ABL. In less tall computational domains, both improperly resolved AGWs and the channel effect contribute to unrealistic flows in the ABL. This can be appreciated from the particle paths for CNBL cases shown in Fig. . A particle traced at the top of the capping inversion shows suppressed vertical movement for the cases with less tall domains compared to the tallest one. Placing the top boundary close to the capping inversion accelerates the flow in the free atmosphere and suppresses the realistic displacement of the capping inversion.

Fully resolving AGWs requires a computational domain much taller (i.e., Lz=2Ls) than one sufficient to eliminate the constraining channel effect of the top boundary. This is obvious when the particle path for the case of fully resolved AGWs is compared to the truly neutral case. The particle path for the case of fully resolving AGWs characterizes the perturbations from the wind farm and the AGWs. The ABL does not expand progressively, and flow acceleration effects from the top boundary are absent. The global maxima and minima on this path are evidence of two disturbance sources, each inducing a wave train in the free atmosphere. The fact that the path is wavy downstream of the wind farm shows the resultant interfacial wave, which is the result of superimposed internal waves in that region and the interfacial waves advected downstream for this super-critical case. Therefore, the configuration fully resolving AGWs is most realistic and accurate in mimicking wind-farm–atmosphere interaction for shallow ABL conditions (i.e., CNBLs).

The discussion on CNBL configurations in Sect.  asks whether modeling a stratified free atmosphere is essential. This curiosity is settled by simulating a configuration similar to the case that fully resolves the AGWs, with the key difference that the free atmosphere potential temperature profile is neutral, details of which are given in Table , Set 1(d). The only stable layer is the capping inversion. Figure  shows that there are no internal waves, but trapped lee waves at the capping inversion are present. We use no damping layers in this simulation set-up. The significance of including the free atmosphere's stability can be appreciated from the particle path traced for this scenario. The path is similar to that of the case that fully resolves AGWs up to the second row of the wind farm; beyond that, the capping inversion slowly moves upwards and approaches that of the truly neutral case far downstream. This rising tendency of the capping inversion in response to the wind farm perturbation is because the buoyancy of the free atmosphere is absent to mask it. In the physical sense, the free atmosphere's buoyancy acts like a spring's stiffness; the buoyancy of the free atmosphere restores the displaced capping inversion. Without buoyancy, the restoring force that suppresses capping inversion vertical displacement is missing. The impact of flow entrainment at the end of the wind farm on the capping inversion is almost unnoticeable. Moreover, the wavy path downstream of the wind farm is the signature of trapped lee waves only because there are no internal waves.

Figure a compares wind farm power output for the configurations described earlier. The solid lines show the averaged power output of the wind turbines, where we average both in time and along spanwise turbine rows. The dashed lines show the difference between averaged power output for each configuration and that of the case that fully resolves AGWs (the CNBL-12 km case), assuming it is the most realistic. Several conclusions can be drawn from these power plots and their comparisons. First, the power output follows the same trend along the wind farm length, regardless of the configuration used. A few notable aspects of this trend include the greatest power decrease for the turbines in the second row, increasing power output until the fifth row, and slightly decreasing toward the last row. The highest drop for the second row of turbines is because of the velocity deficit caused by the first row of turbines, and little to no turbulent flow entrainment from above. The increasing power output after the second row suggests better wake mixing and recovery from the sides and above the wakes. The flow has fully developed by the fifth row, and mixing has reached equilibrium. The truly neutral simulation gives the best perspective of the power for turbines beyond the fifth row, as it excludes flow “channeling” effects caused either erroneously by too low a top boundary or realistically by inversion layer displacement effects.

Comparing the configurations, we observe that the rigid lid cases appear to have the most overestimated wind farm power output, inferred in the above discussions as the consequence of the channel effect caused by too low a top boundary. It is further inferred that the row-wise power output for the rigid lid cases would approach that of the truly neutral case if one keeps increasing the domain height. Surprisingly, the power profile predicted by the CNBL cases, partially resolving the AGWs, is almost the same as that of the rigid lid case with the Lz=750 m. This shows that including a capping inversion without sufficiently resolving the free atmosphere has the channeling effect almost completely dominating the stratification effects. The power plot for the case where the free atmosphere was kept neutral above the capping inversion of the same strength as that of the CNBL cases is interesting: this plot roughly matches the reference tall CNBL case until the fourth row. Beyond the fourth row, the simulated power is less than the reference case, signifying the absence of favorable pressure gradients in that region caused by internal waves and the capping inversion constraining the vertical flow mixing in the ABL when compared to the truly neutral case. This mismatch shows the importance of modeling a stable free atmosphere.

The dashed lines show that all configurations, except the one with a neutral free atmosphere, underestimate the global blockage effect and overestimate wind farm power output compared to the reference case that fully resolves AGWs. The higher power produced by the first-row turbines in all cases compared to the reference case is evidence of the underestimated global blockage effect. Interestingly, the simulated power of the reference case approaches that of the truly neutral case toward the far end of the wind farm. Altogether, leaving out buoyancy or simulating TNBLs excludes the hydrostatic or global blockage effect, and the impacts of AGWs on the wind farm performance . Including a capping inversion and the free atmosphere but partially resolving the AGWs is the same as rigid lid simulations or cases simulating just the ABL. In other words, the channeling effect caused by the top boundary is completely dominant over the stratification effects. Therefore, a more physically realistic simulation, which will also fully resolve AGWs, includes the capping inversion and the free atmosphere, and places the top boundary sufficiently high above the surface.

AGW characteristics are strongly related to the displacement of the capping inversion, which can be depicted by its shape at steady state. For instance, the positive and negative displacements of the capping inversion at the start and the end of the wind farm are a result of the two main flow disturbances that trigger gravity waves, the bulk upflow and downflow at the upstream and downstream wind farm edges, respectively. Moreover, the wavy capping inversion shape downstream of the wind farm shows the resultant interfacial waves. Each positive and negative vertical displacement of the capping inversion is inherently linked to diverging and converging flows underneath in the ABL. The size and strength of these convergent and divergent ABL flow zones are related to AGW characteristics. With a convergence comes a horizontal flow acceleration; with a divergence comes a deceleration.

After initial temporal transients, the capping inversion settles into an equilibrium shape that depends on the flow conditions and wind farm size, which we describe with the physical non-dimensional parameters. The capping inversion height varying in streamwise direction as a function of the physical non-dimensional numbers is shown in Fig. . In analyzing these graphs, the focus is on the shape of the capping inversion at various streamwise locations to relate it to the global blockage effect, and favorable and unfavorable pressure distributions through and downstream of the wind farm. As described in Sect. , the capping inversion shape is computed by tracing a particle path along the capping inversion top. As seen in Fig. , the general capping inversion shape has global maxima, always at around the wind farm start, linked to the horizontal flow deceleration caused by the wind farm. Conversely, global minima are usually at the end of the wind farm, thus linked to the acceleration of the horizontal flow in the exit and wake regions of the wind farm. In some cases, wave interference leads to complex capping inversion shapes. The maximum capping inversion displacement is also measured from its shape by differencing the highest and lowest vertical positions on the particle path. The variations in interfacial wavelengths for varying values of the non-dimensional parameters can also be appreciated from these capping inversion shapes.

The maximum capping inversion displacement normalized by turbine diameter (CIdisp/D) as a function of Fr, Fri, H̃i, and Sh is shown Fig. . CIdisp/D increases abruptly for increasing Fr until 0.0125, and increases only slightly for larger Fr. The physical meaning of increasing maximum displacement can be inferred as reduced atmospheric stability or milder response by buoyancy. A disturbance can displace the capping inversion slightly more than it could for a more stable free atmosphere. The trend is not monotonic around Fr=0.15, however, which is likely caused by the complex interference of the two internal wave trains. CIdisp/D is maximum for Fri=1.0, as positive capping inversion displacement is the highest for critical conditions. Note that only one value of Fri<1.0 is practical because values of Fri<0.75 require impractical capping inversion strengths when Hi is kept constant. The estimated capping inversion displacements from the two simulations with impractical capping inversion strengths (i.e., Fri=0.25 and 0.5) confirm capping inversion displacement that is smaller than that of the Fri=1.0 case. In a physical sense, Fri approaching 0 would mimic a rigid lid that would barely move due to the disturbance from a wind farm. On the other hand, the decreasing CIdisp/D for values of Fri>1.0 is because of background flow washing away buoyancy effects, thus weakly perturbing the capping inversion.

The maximum capping inversion displacement as a function of the ABL aspect ratio shows an intuitive behavior. The maximum capping inversion displacement decreases for increasing H̃i values because the capping inversion is further from the wind farm. Thus, the global minimum of capping inversion displacement in the wind farm wake region is barely noticed for values of H̃i>0.17. The maximum capping inversion displacement is mildly sensitive to Sh as placing the wind turbines closer to the capping inversion perturbs it slightly more than the rotors at standard hub heights, i.e., 90–150 m but still negligible when normalized with the rotor diameter. Note that this observation is in the context of the wind farm layout and turbine size simulated in this study. We infer that a wind farm of higher power density and one with bigger turbines might be stronger momentum sinks, causing bigger capping inversion displacements.

In Sect. , we indicate that the properties of AGWs depend on the physical non-dimensional parameters. This section analyzes the dependence of interfacial wave characteristics on these parameters.

Interfacial waves are bound to the capping inversion, and their wavelengths are correlated to the sizes of converging and diverging zones in the ABL. Thus, the average interfacial wavelengths can be calculated along the inversion layer top by using the following expression. 1λifgw=2(xe-xs)2+(ye-ys)2npeaks, where (xs,ys) and (xe,ye) are the coordinates of the first and last peaks along the capping inversion that extend from the wind farm entrance to exit, respectively; and npeaks is the total number of peaks (i.e., maxima and minima). Since it is measured in the streamwise direction, it is the average horizontal wavelength. We calculate the interfacial wavelength over the length of the wind farm, as it is more suitable for this study.

The interfacial wavelengths normalized with wind farm length (λifgw/L) as a function of Fr, Fri, H̃i, and Sh are shown in Fig. . The interfacial wavelengths based on the deep-water approximation are also calculated, referred hereafter as theoretical wavelengths, and presented for comparison. This approximation excludes the impact of ABL depth on the interfacial wavelengths and is valid when kHi≫1, where k is the horizontal wavenumber given by as 2k=g′2U2+N22g′.

The theoretical wavelengths calculated from the deep-water equations is then λifgw=2π/k, which depends on the reference potential temperature, the temperature jump across the capping inversion, and the stability in the free atmosphere. Clearly, the theoretical λifgw is independent of the boundary layer height, thus applicable to deep boundary layers only.

As can be visually seen in Fig. , the calculated λifgw increases for increasing Fr; the interfacial waves become skewed, which means the height of the darkest-colored regions at the bottom of the contours, which is representative of the interfacial waves, shrinks. Even the propagating internal waves are strongly constrained from propagating vertically, so the wave field appears tilted more downstream. The measured interfacial wavelengths are observed to be half the wind farm length for the lowest Fr and gradually increase to about full wind farm length for the highest Fr. Thus, the sizes of zones of favorable and unfavorable pressure gradient in the ABL vary between 0.25L and 0.5L for increasing Fr. The calculated λifgw for Fr also shows an increasing trend but is distinct from that of the measured one. The calculated values agree with the measured values only for low values of Fr. This could be because kHi for constant Hi=500 m is significantly greater than unity for only low values of Fr.

Both calculated and theoretical λifgw increase roughly linearly with Fri until an asymptote at 0.75L for the calculated and at L for the theoretical is reached when Fri=1.25. There is good agreement between calculated and theoretical wavelengths for sub-critical conditions, but a discrepancy arises beyond critical conditions (Fri≥1.0). The matching values of λifgw are mainly because it varies in response to only varying Δθ in setting Fri while holding Hi and N fixed at 500 m and 0.01 s⁻¹, respectively. We recall from Sect.  that the capping inversion strongly suppresses the perturbations, which is indicated by stronger trapped waves for sub-critical values of Fri. Also, it should be recalled that the interfacial waves are smaller for sub-critical than for super-critical values. For instance, the two laminar inflow cases mentioned in Sect.  (i.e., Fri=0.25 and 0.5) showed that λifgw is of the order of the distance between turbine rows – as if each row is triggering an interfacial wave. Meanwhile, for super-critical conditions, the capping inversion perturbs on length scales, increasing between half to three-quarters of the wind farm length. This means that the sizes of zones of favorable and unfavorable pressure gradients in the ABL vary between row distancing and 0.4L for increasing Fri.

Unlike the theoretical λifgw, the calculated values show an increasing trend against H̃i until they reach a maximum of about λifgw/L=1.0 for H̃i=0.17. The interfacial wavelengths become slightly shorter for further increasing H̃i. This shows that the wind farm is seen as a single disturbance beyond a certain ABL aspect ratio, and the capping inversion receives a weaker and spread-out disturbance. The theoretical λifgw shows the increasing trend only until H̃i=0.084, beyond which it shows a linearly decreasing trend with an almost perfect match between calculated and theoretical for H̃i=0.125. These observations indicate that the deep-water approximation follows these simulations only for deep ABLs. The sizes of zones of favorable and unfavorable pressure gradients in the ABL vary between 0.25L and 0.4L for increasing H̃i. As expected, Sh has a negligible impact on the interfacial wavelengths; both theoretical and calculated λifgw are independent of it. The λifgw is the same as that of Fr=0.15.

Generally, the amplitude of interfacial waves varies in the narrow range of 0.19D–0.3D, where D is rotor diameter, against the physical non-dimensional parameters, except for H̃i, where the range is 0.02D–0.6D, which is shown in Fig. . For low values of H̃i, the capping inversion displacement is higher, and strong waves are present, as the two are directly proportional . Amplitudes of interfacial waves are analogous to capping inversion displacements, which are already reported in Sect. . Moreover, the orientation of interfacial waves with respect to the lateral axis (i.e., spanwise direction) are dependent only on Fri and have already been investigated by . When looking from the top, interfacial waves make a V pattern with the vertex at the induction zone of the wind farm, with the wave fronts extending downstream. The angle on the interior of the V pattern at the vertex is inversely proportional to Fri such that the V pattern becomes increasingly acute for super-critical values. For sub-critical to critical values of Fri, the interfacial waves are blunt at the vertex (i.e., parallel to the lateral axis).

The propagating internal waves are relatively weaker in strength than the interfacial waves, and their properties depend mainly on Fr and Fri. A major challenge in measuring their properties is that there are two internal wave trains superimposing on each other, one from the front of the farm and one from the rear. Especially for high Fr, the superimposition is evident as the inclination angle is expected to be higher. Therefore, we focus on the first wave train, as we did in Sect. , as it is less distorted.

Figure  shows wave field inclination angle, as defined in Sect. , extracted from average vertical velocity fields as a function of the four physical non-dimensional parameters. The inclination angle shows a decreasing trend against increasing Fr, starting at around 63°, for Fr=0.075, and approaching around 27° for Fr=0.25. The reduced slope for Fr≥0.15 of the line connecting the data points means that the two internal wave trains are more distinct; for higher values of Fr, the first wave train stretches across the length of the domain (see Fig. ). The inclination angle has the same decreasing trend with increasing Fri, decreasing from 61 to 40°. This 40° value is roughly the same as for Fr=0.15; and in the varying Fri cases, Fr is held at 0.15. The inclination angle of the internal waves seems almost independent of H̃i and Sh. There is a variation for the former but in a narrow range of 41–35°, probably caused by varying capping inversion displacement against H̃i.

The apparent wavelength, measured along the fit parabola, of the first wave train is dependent on its inclination angle. In measuring this wavelength, it is assumed that the interference of the two wave trains is minimal for values of Fr≤0.15. We also recall that the measured wavelength with the method proposed in Sect.  allows us to estimate horizontal and vertical components of the actual wavelength. Thus, the effective horizontal wavelength could be λa=2λacos⁡ϕ, equivalent to the sum of horizontal wavelengths of the two dominant wave trains. The apparent wavelength of the first wave train, normalized with L and as a function of the physical non-dimensional parameters, is given in Fig. . Generally, this wavelength increases linearly and is smaller than the wind farm length for values of Fr≤0.15. This linear increase is mainly due to increasing buoyancy length (i.e., Ls), which is directly proportional to Fr. Note that the first wave train is dilated and merged with the second for Fr>0.15 by the increasing background speed, and only one wave train is obvious in the solution, as shown in Fig. .

As stated at the start of this section, the internal wave properties are mainly affected by Fr. This is obvious when the apparent wavelength plots for the remaining physical non-dimensional parameters are analyzed, where Fr=0.15 while varying each. Thus, λigw/L varies in the narrow range of 0.9L–1.1L for the simulated values of Fri, H̃i, and Sh. The values reported for Fri=0.25 and 0.5 are estimated with less confidence, as negligible internal waves were observed for these idealized cases, as can be seen in Fig. . Small variations about the average value (i.e., 1.0L), which is roughly the value for Fr=0.15, for varying Fri, H̃i, and Sh are the impacts of the interference of the two wave trains that depend on the inclination angles. For instance, the λigw/L plot against Fri for its values ≥0.75 resembles the decreasing inclination angle trend against Fri shown in Fig. . Both plots asymptote at Fri=1.0 after a sudden drop. Note that we infer this resemblance for data points excluding the less certain values for 0.25 and 0.5, for reasons mentioned earlier. In the case of H̃i, the wavelength only mildly increases with increasing H̃i, where for the shallowest boundary layer simulated, the wavelength is slightly larger than that for Fr=0.15. For low H̃i, the capping inversion feels both disturbance sources more isolated and strongly, but beyond H̃i=0.17, the wind farm is an entity, as the growing internal boundary layer rather than the two disturbance sources perturb the capping inversion. Thus, the wavelength resembles a dilated single disturbance. There is barely any change in the wavelength against Sh, as all simulated Sh values yield a wavelength as wide as 1.0L, a virtue of the constant Fr=0.15.

The amplitudes of internal waves, which are measured as the amplitude of streamline displacement, as a function of the physical non-dimensional parameters, are shown in Fig. . The amplitudes are measured by tracing a particle at z=Hi+3/4Ls. There is little variation in the amplitudes as a function of Fr, except for a small increase from 0.11D to 0.14D between Fr=0.075 and 0.125. The amplitude increases as a function of Fri only until 0.17D, suggesting that the capping inversion becomes less rigid until the amplitude depends only on Fr. There is an almost linearly decreasing amplitude trend against H̃i, going from 0.19D for H̃i=0.025 to 0.08D for H̃i=0.25. The reason for weaker internal waves for increasing H̃i was discussed in Sect. . The amplitudes only slightly increase over 0.17D against Sh, which is the value of amplitude for Fr=0.15.

Here, we discuss how the global blockage effect and pressure variations relate to each non-dimensional parameter via the capping inversion displacement height at steady state. Generally, the localized capping inversion displacement has a negative correlation with the row-averaged power output. This means that an upwards displacement of the capping inversion corresponds to decreased power of the row underneath it and vice versa. Temporal and row-averaged power output are analyzed to uncover general trends in wind farm performance in the maximum wake loss scenario (i.e., when wind is straight down the rows). Figure a shows the temporal and row-averaged wind turbine power output as a function of Fr and streamwise distance. Wind farm blockage decreases as Fr is increased. However, a narrow variation of 0.2 MW is observed for the power output of the first row over the simulated values of Fr. Note that we interpret the variations in power output of the first row as variations in the wind farm blockage effect, which is quantified as non-local efficiency in the next section. It is astonishing how the power output of the second row is completely independent of Fr as the wake effects dominate. Beyond the second row, the row-wise power output varies about a trend characteristic of aligned wind farm aerodynamics, with a maximum at the fifth or sixth row. Although subtle, variations are bound to the localized capping layer displacement aloft, as discussed in Sect. . Generally, row-wise power output decreases beyond the fifth row until a slight increase is observed for the last two rows. The power output of individual rows beyond the second decreases slightly with increasing Fr except for Fr=0.075, due to its unique capping inversion shape, which is shown in Fig. .

Figure b shows the row-wise power output for a few values of Fri. The wind farm blockage effect reduces with increasing Fri, where the strongest blockage is observed for the sub-critical case (i.e. Fri=0.75) but is almost the same as the critical case. The simulated power of the second row is the same for values of Fri≥1.0 but higher for Fri=0.75 because the capping inversion displacement is negative in contrast to positive in the remaining cases. The fourth row produces the highest power for Fri=0.75 among the waked rows because of the localized favorable pressure, as the capping inversion in the vicinity is at its lowest point. The highest power output among the waked rows is from the fifth-row turbines for the critical and super-critical cases. The simulated power shows a gradual decrease past the fifth row for values of Fri≥1.0 until the seventh row, beyond which it increases slightly. The fact that power oscillates with distance downstream for Fri=0.75 is evidence of power's negative correlation to the localized capping inversion displacement. This also confirms that gravity waves can cause several zones of converging and diverging flow through the wind farm.

The simulated row-wise power profile is the most sensitive to the ABL aspect ratio, as shown in Fig. c. For H̃i≤0.167, the power profile follows the trend as that of Fr=0.15 and Fri=1.3 because these are the constant values simulated while varying H̃i. However, the power profile for the shallowest ABL (H̃i=0.042) is most affected by localized capping inversion displacement, which can be revisited in Fig. . The wind farm blockage is highest for H̃i=0.042 and reduces as H̃i increases until it becomes almost independent for H̃i≥0.21 because the deflected flow at the wind farm entrance barely perturbs the capping inversion. The simulated power of the individual turbines increases with increasing H̃i because inflow turbulence intensity increases as the ABL gets deeper . The simulated power output profiles for H̃i≥0.167 are almost the same, as the capping inversion hardly affects the flow build-up through the wind farm. Thus, the flow around the wind farm becomes similar to that of a truly neutral boundary layer with the same ABL height. We can infer that the internal boundary layer has sufficient space to grow vertically, and pressure variations caused by stratification effects or gravity waves become smaller with increasing H̃i. For shallow ABLs, the pressure perturbations caused by the stratification effects dominate the internal boundary layer growth and hence the power output profile. Thus, the capping inversion's height determines which phenomenon dominates and, in turn, the wind farm flow aerodynamics. When it is low, the stratification effects are increasingly critical to wind farm performance; otherwise, these effects are secondary when compared to wake losses. Thus, inflow characteristics and wake effects still remain primary to wind farm performance. These aspects are discussed in the next section on wind farm efficiencies.

The vertical wind farm aspect ratio impacts the row-wise power profile mainly in the first half of the wind farm, as shown in Fig. d. Again, fixed values of non-dimensional parameters simulated include Fr=0.15, Fri=1.3, and H̃i=0.084. Thus, the power profiles follow the trend for Fr=0.15 and Fri=1.3, except that the highest power-producing row among the waked turbines shifts from the fifth row to the fourth as Sh increases. Higher velocities for increasing Sh are a clear factor in improved wind turbine and farm powers. However, stratification effects, although small, are still relevant as increasing Sh means putting the turbines closer to the capping inversion, which is similar to stronger perturbations and a different response for varying Sh.

Altogether, it is obvious that H̃i is the most critical physical non-dimensional parameter in terms of wind farm performance. Therefore, for low H̃i, Fr and Fri can alter the capping inversion shape characteristic of the underlying H̃i, thus affecting the row-wise power output. Sh can affect the power output mainly because of placing the wind turbines at higher or lower wind speeds, which has little to do with gravity waves or blockage. While simulating CNBLs, H̃i becomes more critical for shallow ABLs and will be of even greater importance for stable surface conditions with shallower ABLs than in CNBLs.

Although wind farm performance is thoroughly discussed in the previous section, it is essential to quantify wind farm blockage effect and wake losses relative to the inflow characteristics of each case. This is important because varying atmospheric conditions leads to different inflow characteristics, such as different shear profiles and veers. proposed wind farm efficiency (ηt) as the product of non-local and wake efficiencies, the former being an estimate of wind farm blockage effect and the latter of quantify wake loses. These efficiencies are defined as following: ηt=ηnl⋅ηw,ηnl=P1P∞,ηw=PwfNt⋅P1.

The non-local efficiency (ηnl) is the ratio of average power of the first-row turbine (P1) to the power produced by a turbine in the free stream or isolation (P∞). Since the LES power predictions are based on rotor-average velocities, we calculate P∞ as a function of rotor-equivalent wind speed in the free stream. To this end, we sample velocity over the rotor annulus for each case from the respective precursors and temporally average it over the last flow through time of the successor simulations, as this interval is used to calculate the actual power outputs. This takes into account the unique shear and veer inflow velocity profiles of each CNBL condition. In other words, the variations in the power outputs due to variations in the inflow are excluded from the analysis of blockage effect. Note that ηnl can also be estimated based on the power output of an isolated wind turbine simulated in the same conditions. This approach is computationally taxing in this study as each of at least the H̃i and Sh cases would require a separate simulation. The wake efficiency (ηw) is the ratio of total wind farm power output to the power output of the total number of wind turbines in the farm (Nt) operating in the first row, meaning a scenario where no turbine is waked.

Figure  shows these efficiencies as a function of the non-dimensional parameters. It can be seen that ηnl improves slightly as opposed to a slightly decreasing trend for increasing values of Fr and Fri. Therefore, ηt remains roughly constant. There is a clear increase in ηnl for increasing values of H̃i, while ηw decreases slightly, resulting in an overall improved farm efficiency in deeper boundary layers. Non-local efficiency is roughly constant for varying Sh, although improved wake efficiency causes the total wind farm efficiency to improve too. This confirms that variations seen on row-averaged power plots against Sh are mainly due to the turbines operating in higher wind speeds.