The initial potential temperature profile of the NBL cases is linear and has a vertical temperature gradient (lapse rate) of Γ=+1Kkm-1 from the surface to the domain top (see Fig. ). At the surface, a Neumann condition for the potential temperature is applied and the surface heat flux is set to zero. Shear-driven turbulence production leads to the formation of a neutrally stratified BL that grows until it reaches a steady BL height of 780 m. The BL height is defined as the height at which the shear stress reaches 5 % of its surface value. The conventionally neutral boundary layer is separated from the FA by a capping inversion that has a stronger stratification than the FA.

The initial temperature profile of the CBL cases consists of a constant potential temperature between the surface and the desired BL height h=700 or h=1400 m for the cases CBL-700-7D and CBL-1400-7D, respectively. Above that height the potential temperature has a constant lapse rate of Γ=+3.5 K km-1, which corresponds to the International Standard Atmosphere. A Dirichlet condition is applied for the surface temperature, and a constant surface heating rate of θ˙0=+0.050 and θ˙0=+0.025 K h-1 is used to drive the CBL of the cases CBL-700-7D and CBL-1400-7D, respectively. The heating rates differ by a factor of 2 to achieve approximately the same surface heat flux Q0 and Monin–Obukhov length L (see Table ) so that only the effect of a changing BL height is seen in the results.

Boundary layer growth is avoided by applying a large-scale subsidence that acts only on the potential temperature field. The subsidence velocity is zero at the surface and increases linearly to its maximum value wsub at the height h and is constant above. The subsidence velocity is chosen in such a way that the temperature increase in the FA exactly matches the surface heating rate: θ˙0=Γwsub. Thus the BL height can be kept precisely constant even for very long precursor simulations. Final BL heights, according to the definition given in Sect. , are 690 and 1400 m.

Large-eddy simulations of CBLs are usually driven by a constant heat flux, i.e., a Neumann condition for the surface temperature. However, we decided to use a Dirichlet condition because of two reasons.

It allows for spatial variations in the surface heat flux which may be caused by enhanced mixing inside the wind farms. In reality, the resulting change in sea surface temperature (on the scale of hours) would be very small due to the good turbulent mixing inside the ocean mixed layer during strong winds and due to the high heat capacity of water in contrast to that of air. Thus, it is more realistic to prescribe a horizontal homogeneous surface temperature than a horizontal homogeneous heat flux.

The initial potential temperature profile of the SBL case is linear and has a vertical temperature gradient (lapse rate) of Γ=+3.5 K km-1 from the surface up to the domain top. A Dirichlet condition is applied for the surface temperature because prescribing a surface heat flux can lead to unphysical results . Generating a steady-state SBL is not as simple as it is for the CBL. A straightforward method would be to use a surface cooling rate. However, due to the long simulation time required for the decay of the inertial oscillation, the elevated inversion at the top of the SBL would become unrealistically strong . We developed a method to generate a steady-state SBL in which the potential temperature profile is constant in time and the strength of the elevated inversion can be freely adjusted.

The method uses the large-scale forcing functionality of PALM. Instead of changing the surface temperature, a positive temperature tendency of +0.05 K h-1 is added at every grid point and at every time step. This added tendency imitates a large-scale advection of warm air and thus forms an SBL with steady heat flux and momentum flux profiles. The heat flux divergence results in a cooling tendency that exactly balances the positive large-scale advection tendency so that the temperature inside the BL stays constant. In the overlying inversion, the heat flux divergence decreases approximately linearly until it reaches zero at the transition to the FA. Consequently, the temperature in the FA increases further, and the overlying inversion becomes stronger. To prevent further strengthening of the overlying inversion, the large-scale advection tendency is set to zero in the FA at t=6 h . Inside the overlying inversion, the large-scale advection tendency increases linearly to its maximum value inside the BL so that it approximately compensates for the cooling tendency caused by the heat flux divergence. From that point on the potential temperature profile is steady, and the simulation can run until the inertial oscillation has decayed. Because the potential temperature in the FA changes over time, it is excluded from the Rayleigh damping. Despite the shallow BL, a large domain height of Lz=3624 m is used to capture gravity waves that are triggered by the wind farms. The final BL height, according to the definition given in Sect. , is 270 m.

The mean horizontal wind speed at hub height is shown in Fig.  for all cases. Streamlines indicate the wind direction.

For the NBL cases, the wind speed is reduced from 10 to 7 m s-1 for a turbine spacing of s=7 D and 5 m s-1 for s=5 D inside the large wind farms in Zone 3. The wake length is defined as the distance between the wind farm trailing edge and the point at which the wind speed recovers to 90 % of its initial value, i.e., 9 m s-1. For the small wind farms N-1, N-2 and N-3 (see Fig. ), the wake length ranges from 1 to 20 km. However, the wake length of the large wind farms in Zone 3 is approximately 100 km for s=7 D, and the wake extends beyond the model domain for s=5 D.

The wake flow is deflected counterclockwise. The largest deflection angle of approximately 10∘ is observed for the smaller turbine spacing (s=5 D). The counterclockwise wake deflection is consistent with the findings of , who observed a counterclockwise deflection of 2–3∘ for a 15 km long wind farm. A counterclockwise wind direction change (higher ageostrophic wind component) has also been observed by and , who investigated infinitely large wind farms. The wake deflection is caused by a reduced Coriolis force, as is shown in the next section. Because the Coriolis force is proportional to the wind speed, the deflection angle is higher for the case with the greater speed deficit (NBL-700-5D). The reasons for the slow speed recovery and the wake deflection are discussed in detail in the next section.

The inflow wind speed has slight variations in the crosswise direction which are caused by the wind-farm-induced flow deceleration reaching the recycling plane (see Sect. ). The variations have an amplitude of approximately 0.1 m s-1, which is 1 % of the inflow wind speed.

For the CBL cases, the wind speed is reduced to 6.5 m s-1 for h=700 m and 8 m s-1 for h=1400 m inside the large wind farms in Zone 3. Also, the wake length of the large wind farms is much longer for the shallow BL than for the thick BL. This BL height dependency occurs because the turbulent vertical kinetic energy flux is greater for the case with the thicker BL (see Fig. c and d). The wind speed deficit and the wake length of small wind farms (e.g., N-1, N-2 and N-3) are relatively unaffected by the BL height because the wind-farm-induced internal BL does not reach the inversion layer (NBL cases) or only reaches it several tens of kilometers downstream of the wind farm trailing edge (CBL cases) (see Fig. ). Consequently, the BL height only affects further wind speed recovery (e.g., to 9.5 m s-1) in the far wake of small wind farms. For example, the wind speed recovery from 9 to 9.5 m s-1 in the wake of N-2 takes longer for the case CBL-700-7D (40 km) than for the case CBL-1400-7D (15 km) (see Fig. c and d).

The wake is deflected counterclockwise in the CBL cases, as well. The deflection angle is approximately 5∘ for the case with the shallow BL and 1–2∘ for the case with the thick BL. The higher deflection angle for the case with the shallow BL is caused by a greater speed deficit compared to the case with the thick BL.

A comparison between the cases NBL-700-7D and CBL-700-7D shows that the speed deficit inside Zone 3 is greater for the CBL case. Also, the wakes of the small wind farms are longer for the CBL case. This is contradictory to the well-known fact that wind turbine and wind farm wakes are generally shorter in CBLs than in NBLs and SBLs (, their Sects. 2.3 and 3.4.2). To achieve the same hub height wind speed for both cases, the geostrophic wind speed is 6 % greater for the NBL case (see Table ) than for the case CBL-700-7D. Additionally, the wind speed is supergeostrophic in the upper half of the NBL, and thus the mean BL wind speed is approximately 10 % greater in the NBL case than in the CBL case. Stability likely has little-to-no effect because the stratification of the CBL case is only weakly unstable (L=-420 m; see Table ).

In the stable case SBL-300-7D, the wind speed is reduced to below 7 m s-1 in the first 20 km of the large wind farms in Zone 3. The wind speed deficit is greater, and the wake is more than 20 km longer for the small wind farms compared to the other cases with s=7 D. The wake of the large wind farms in Zone 3, however, is not longer than in the cases NBL-700-7D and CBL-700-7D. This occurs because the speed recovery in the wake of this large wind farm is not driven by momentum flux divergence (which is stability dependent) but rather by a favorable pressure gradient (details are given in the next section). The case SBL-300-7D covers several flow features that are not as significant in the other cases. These features are namely flow blockage in front of the wind farms, flow deflection around the wind farms and flow acceleration beside the wind farms and/or wakes. These features are related to the pressure field inside and around the wind farms. The perturbation pressure p∗‾, relative to its value at the inflow, is shown in Fig. f. A high-pressure region in the upstream part of the large wind farms in Zone 3 leads to an adverse pressure gradient and thus flow deceleration in front of the wind farms. This effect is known as blockage effect or flow blockage . At a distance of 2.5 D upstream of the first wind turbine row of the wind farms in Zone 3, the wind speed is reduced by approximately 10 % relative to the inflow wind speed. For all other cases the speed reduction is approximately 2 %. reported 11 % speed reduction 2.5 D upstream of the first turbine row of a 20 km long wind farm in a CNBL with a FA stratification of Γ=+5 K km-1. However, for Γ=+1 K km-1 they reported a speed reduction of only 1.2 % because the flow is supercritical (Froude number, Fr>1). Using the same definition

For details about the calculation of the Froude number refer to and .

What is the reason for the slow speed recovery and the wake deflection inside and behind the large wind farms? In order to answer that question, Fig.  shows streamwise (parallel to streamlines) and crosswise (perpendicular to streamlines) components of the pressure gradient force

The pressure gradient force and the Coriolis force are not considered explicitly in the model but are considered implicitly by the geostrophic wind.

Streamwise force components in Fig. a show that the accelerating geostrophic pressure gradient force and the decelerating momentum flux divergence are in balance and sum to zero at the inflow. The streamwise component of the Coriolis force is zero because this force acts always perpendicular to the flow. Inside the wind farms, the momentum flux divergence is positive and thus is an accelerating component. It is the dominant driving force because it is more than 7 times greater than the geostrophic pressure gradient force.

An increasing perturbation pressure in front of the wind farm leads to a negative perturbation pressure gradient force and thus flow deceleration (often called blockage effect). However, inside the wind farms the perturbation pressure gradient force is positive due to a favorable pressure gradient (decreasing pressure). In the near wake the momentum flux divergence is high and leads to a fast speed recovery. The momentum flux divergence decreases fast until it becomes negative in the far wake so that the speed recovers slowly in the far wake. The only force that remains for driving the flow is the geostrophic pressure gradient force. At the inflow, this force is in balance with the momentum flux divergence, but in the wake this is not the case due to two reasons: first, the negative momentum flux divergence is weaker than at the inflow due to a lower wind speed and thus a reduced near-surface momentum flux; second, the streamwise component of the geostrophic pressure gradient force has increased by 80 % because the wake flow is deflected counterclockwise (i.e., to lower pressure). These results show that the wake deflection is an elementary feature of the wake that supports the wind speed recovery. They also show that mixing of momentum from the BL to the wind turbine level is not the dominant process that drives the speed recovery in the far wake of very large wind farms.

The wake deflection can be explained by examining the crosswise force components that are shown in Fig. b. Positive forces result in counterclockwise flow deflection, and negative forces result in clockwise flow deflection. At the inflow, the Coriolis force, the geostrophic pressure gradient force and the momentum flux divergence are in balance. Because the Coriolis force is proportional to the wind speed, it is reduced by approximately 30 % inside the wind farms and the wake. Consequently, the sum of all forces becomes positive and the flow is deflected counterclockwise. The momentum flux divergence and the perturbation pressure gradient force are negative inside the wind farm and inside the wake and are therefore opposing the wake deflection. The negative perturbation pressure gradient force is a result of the pressure distribution around the wind farms that is caused by the wind farm shape (see Fig. d).

The reason for the negative momentum flux divergence is the enhanced downward mixing of negative y momentum of the overlying flow, which veers to the right (see Fig. ). For small wind farms this process can be dominant and may result in clockwise wake deflection . However, for very large wind farms, as in this study, the effect of the reduced Coriolis force is dominant. An appropriate parameter for estimating the importance of Coriolis effects is the Rossby number. Coriolis effects become dominant for Rossby numbers close to or below 1. For the large wind farms in Zone 3 with a length of Lwf≈100 km at mid-latitudes (Coriolis parameter f≈10-4) and a wind speed of U≈10 m s-1, the Rossby number becomes 11Ro=ULwff≈1, indicating that Coriolis effects play an important role for flows in wind farms of this size.

The turbulence intensity TI is defined as in : 12TI=23TKEvh‾, where TKE is the resolved turbulence kinetic energy defined as follows: 13TKE=12(u′2‾+v′2‾+w′2‾), where u′2‾, v′2‾ and w′2‾ are the resolved-scale variances of u, v and w, respectively. The SGS-TKE is neglected because it is smaller than 10 % of the resolved TKE inside the wind turbine wakes at a distance of 3 D or more.

The TI at hub height is shown in Fig. . Inside the wind farms, the TI reaches a fully developed state after approximately four rows and is constant farther downstream. A smaller turbine spacing leads to a greater TI inside the wind farms. For the case NBL-700-7D, a TI of 10 % is reached inside the wind farms, but more than 14 % is reached in the case NBL-700-5D. In the CBL cases, the TI inside the wind farms reaches 10 % in case CBL-700-7D and approximately 12 % in CBL-1400-7D. Although the ambient TI is only approximately 3 % for the case SBL-300-7D, the TI inside the wind farms reaches values similar to NBL-700-7D (approximately 10 %).

The wake in terms of TI is generally shorter than the wake in terms of wind speed (see Fig. ). The shortest wakes occur in the shallow SBL (SBL-300-7D), and the longest wakes occur in the thick CBL (CBL-1400-7D). This is the opposite behavior than that of the wake in terms of wind speed. The wake length in terms of TI weakly depends on wind farm size with slightly longer wakes for larger wind farms. However, this effect is caused by the definition of the TI, in which the wind speed variances are normalized by the mean horizontal wind speed. The mean horizontal wind speed is smaller in the wake of the large wind farms than in the wake of the small wind farms, resulting in a higher TI in the wake of the large wind farms. The wind farm size dependency of the wake length vanishes if the TKE is used for measuring the wake length instead of the TI (not shown).

In the NBL cases and especially in the SBL case, the TI in the far wake drops below the ambient TI at the inflow. This effect is caused by the reduced wind speed in the far wake which leads to a reduction in the shear-driven turbulence production. In the CBL cases, there is also buoyancy-driven turbulence production, which is unaffected by the reduced wind speed in the wake and thus maintains the TI level. That buoyancy-driven turbulence production has a large impact on hub height TI is also verified by the fact that the ambient TI is greater in the CBL case with the thick BL (CBL-1400-7D) than in the case with the shallow BL (CBL-700-7D): the convective velocity scale w∗=(gziθ‾(w′θ′‾)0)1/3 is greater in the case CBL-1400-7D than in the case CBL-700-7D, and hence also the buoyancy-generated velocity variances are greater. Because buoyancy acts as a TKE sink in the SBL case, it can not compensate for the reduction in shear-driven turbulence production, and thus the TI in the wake drops below 2 %. This effect is amplified by the entrainment of warm air into the BL that leads to a stabilization (increased lapse rate) at hub height and therefore stronger turbulence damping (see Figs. f and ). The entrainment of warm air results in a temperature increase at hub height of approximately 0.3 K for the large wind farms and approximately 0.1 K for the small wind farms.

Figure  shows vertical cross sections of the horizontal mean wind speed for all cases. The cross sections are located at y=120 km and thus cross the large wind farms in Zone 3. The inversion layer height zi is marked by lines at which the maximum vertical potential temperature gradient occurs. The wind-farm-induced internal boundary layer (IBL) is shown by a line at which the horizontal wind speed corresponds to 97 % of the inflow wind speed at that height.

The IBL is not shown for the case SBL-300-7D because the wind turbines are nearly as high as the BL. For all other cases the IBL grows up to the inversion layer (IL) within 40 km (NBL-700-7D and NBL-700-5D), 10 km (CBL-700-7D) and 20 km (CBL-1400-7D) behind the wind farm leading edge. The streamwise extent of the IBL goes beyond the model domain, indicating that the wind speed inside the entire BL does not recover to 97 % of the inflow wind speed.

The IL height is affected by the presence of the wind farms in all five cases. In the NBL cases the IL is displaced upwards by 200–300 m, whereas a larger displacement occurs for the smaller turbine spacing. The IL displacement is a result of the reduced wind speed in the bulk of the BL: to obtain a divergence-free flow inside the BL, the wind speed reduction (streamwise convergence) is compensated for by vertical divergence (IL displacement) and crosswise divergence (flow around the wind farms). The increase in IL height is not caused by entrainment of warm air into the BL (as can also be seen in the profiles of potential temperature in Fig. ). This phenomenon has also been observed by , who also stated that the mass flux conservation is the reason for the IL displacement. stated that a smaller turbine spacing results in a larger BL height for an infinite wind farm in a CNBL. The IL displacement causes an acceleration of the flow in the FA for the NBL and CBL cases. Details about this effect are described in the next section.

In the CBL cases, the IL height increases above the wind farms and decreases above the wake, reaching its initial value at approximately 70 km downstream of the last wind farm trailing edge. The IL displacement is larger for the shallower BL, i.e., the case CBL-700-7D.

The IL displacement is most significant for the case SBL-300-7D. The IL height increases from 300 to 500 m. also reported that larger IL displacements occur for shallower BLs (+60 % for h=250 m). In the case SBL-300-7D, the IL height increase is caused by vertical displacement due to mass conservation and also by entrainment of warm air into the BL (see Fig. f). The entrainment of warm air into the BL leads to a warming of the lower part of the BL. However, the temperature at the height of the original IL is reduced because the warm air in the IL is replaced by relatively cold air from the BL.

Because the laminar flow in the FA is adiabatic, the isotherms in Fig. f can be interpreted as streamlines. They show that gravity waves are excited by the wind farms. There are small-scale gravity waves with a wavelength that corresponds to the turbine spacing and a large-scale gravity wave with a wavelength that approximately corresponds to the wind farm length. The negative and positive temperature deviations in the wave crest and trough, respectively, cause a positive and negative deviation in the perturbation pressure at the surface, as is shown in Fig. f. A detailed analysis of the wind-farm-induced gravity waves goes beyond the scope of this study. However, it is noted that the qualitative pressure and temperature distributions correspond to the findings of and .

Additional test simulations have shown that the strength of the gravity waves is sensitive to the domain height. achieved good results (low wave reflection at the domain top) if the domain height corresponds to at least one vertical wavelength λz=2πU/N, where U is the BL bulk wind speed, and N is the Brunt–Väisälä frequency in the FA. In the case SBL-300-7D, the domain height is set to 0.43λz (λz=5.9 km, Lz=3624 m) because larger domain heights lead to numerical instabilities at the inflow. used a domain height of Lz=2.4 km for a FA stratifications of Γ=1 K km-1 (resulting in Lz=0.22λz) and Γ=5 K km-1 (resulting in Lz=0.49λz). It is not clear whether the vertical wavelength is the only relevant parameter for choosing the correct domain height or whether the wind farm length also has to be considered. Further research is needed to find setup guidelines that ensure that wind-farm-induced gravity waves are covered as realistically as possible.

In order to compare the effects of small and large wind farms on the boundary layer, Fig.  shows vertical cross sections at y=50 km, crossing the small wind farm N-2. The IL displacement is much smaller compared to the displacement triggered by the large wind farms (i.e., 50–100 m for the small wind farm in contrast to 200–300 m for the large wind farms). For the NBL cases the IBL grows to approximately 700 m and thus does not reach the IL. For the case CBL-700-7D the IBL reaches the IL but only 40 km behind the wind farm trailing edge. The streamwise extent of the IBL shows that the wind speed recovery at hub height to 97 % of the inflow wind speed is reached 20 km (NBL-700-7D and CBL-1400-7D) and approx. 50 km (NBL-700-5D and CBL-700-7D) behind the wind farm trailing edge. For all 7 D cases, the IBL does not start at the wind farm leading edge but rather inside the wind farms. The reason for this effect is that the vertical cross section does not cross the rotor discs of the turbines, and no averaging occurs in the y direction. As Fig. f shows, the small wind farms also triggers gravity waves in the FA, but they are much weaker than the ones triggered by the large wind farms.

To examine the effect of the wind farms on the BL in more detail, profiles of wind speed, wind direction and potential temperature are shown in Fig. . The profiles are evaluated at the inflow (x=0 km), in the near wake (x=120 km) and in the far wake (x=180 km) of the large wind farms in Zone 3 (y=120 km).

The wind speed profiles show that the wind-farm-induced wind speed deficit spreads over the entire height of the BL. The effective vertical mixing in the CBL cases results in an approximately height-constant wind speed at the inflow and in the wake. In the case CBL-1400-7D, the wind speed in the upper part of the BL is even lower than in the lower part of the BL at x=120 km. In the NBL cases, the vertical mixing is not as effective, and thus a significant wind shear exists over the entire BL in the wake. The wind speed profiles of the case SBL-300-7D show that the BL has grown from 300 to 500 m and that the super-geostrophic maximum is eliminated completely. The IL displacement causes an increase in wind speed in the FA above the BL. The maximum increases (approximately 1 m s-1) are observed for the cases with the greatest IL displacements. That suggests that the wind speed excess above the BL is also caused by the continuity constraint; i.e., the wind speed has to increase in order to maintain a constant mass flux between the IL and the domain top. For the CBL cases, in which the IL height decreases again behind the wind farms, the wind speed above the BL decreases to below-geostrophic in the far wake (x=180 km). Note that these effects could be overestimated because of the artificial boundary that is introduced by the Rayleigh damping layer that starts several hundred meters above the BL. The sensitivity of this effect on the Rayleigh damping height has not been investigated because the scope of this study is on BL-internal effects.

The wake deflection shown in the horizontal cross sections can also be seen in the wind direction profiles. The wind-farm-induced wind direction change is approximately constant over the entire height of the BL. The largest deflection angles of up to -10∘ are observed for the cases with the greatest speed deficit (SBL-300-7D and NBL-700-5D) because the Coriolis force reduction is greatest in these cases. The smallest deflection angle is observed in the case CBL-1400-7D with no deflection in the upper half of the BL. All cases have in common that, in contrast to the counterclockwise deflection in the BL, the flow in the FA is deflected clockwise. As a result, the wind veer in the inversion layer increases to approximately 10∘. The flow deflection in the FA is also a Coriolis effect. Because the wind speed in the FA is supergeostrophic, the Coriolis force is greater than the geostrophic pressure gradient force, and therefore the flow is deflected clockwise. The largest deflection angle of more than 10∘ is observed for the case NBL-700-5D. Note that the highest wind speed excess occurs at x=120 km, but the highest deflection angle occurs at x=180 km. This effect can be interpreted as an inertia oscillation in space (along x), with the deflection angle being phase shifted 90∘ relative to the wind speed excess. Note that this effect might also be overestimated due to the potentially overestimated wind speed excess. However, currently running investigations with much higher Rayleigh damping heights show the same behavior. In the case SBL-300-7D, the combination of clockwise deflection in the FA and counterclockwise deflection in the BL results in a total wind veer of approximately 40∘ between the surface and the FA.

The effect of the wind farms on the potential temperature profiles is largest for shallow BLs (SBL-300-7D) and negligibly small for thick BLs (CBL-1400-7D). The potential temperature profiles inside the well-mixed BLs of the NBL and CBL cases are nearly unaffected by the wind farms. The greatest changes take place in the inversion layer, which is displaced upwards in order to maintain a constant mass flux in the BL, as already described in the previous section. The profiles show that the potential temperature inside the BL is unchanged, and thus BL warming due to entrainment of warm air from the FA is not the reason for the increased IL height. On the contrary, the temperature at the height of the original IL decreases by approximately 0.5 K because it is replaced by colder air from the underlying BL. The potential temperature profile of the SBL case is heavily modified by the wind farms. The temperature in the BL increases by approximately 0.5 K due to entrainment of warm air from the FA into the BL. The IL rises from 300 to 500 m due to the combined effect of BL warming and IL displacement. Because the surface temperature is constant, a new SBL forms in the far wake. This new SBL is shallower and more stably stratified than the original SBL at the inflow.

The effect of different turbine spacings, BL heights and stabilities on the power output of very large wind farms is investigated here. This is done by comparing wind farm efficiencies of small and large wind farms for the five simulated cases. Here, the turbines in the area N-1 are defined as a small wind farm because this area has the size of a typical, currently existing wind farm in the German Bight. The turbines in Zone 3 are defined as a large wind farm because this area will be equipped with wind farms in the future (see Fig. f). The small wind farm consists of 27 wind turbines for s=7 D and 54 wind turbines for s=5 D, resulting in an installed wind farm capacity of 0.405 GW for s=7 D and 0.810 GW for s=5 D. The large wind farm consists of 636 wind turbines for s=7 D and 1260 wind turbines for s=5 D, resulting in an installed wind farm capacity of 9.54 GW for s=7 D and 18.90 GW for s=5 D.

The wind farm efficiency ηwf is defined as the total wind farm power Pwf normalized by the wind farm power that would be achieved if all wind turbines nwt were operating in free-stream conditions, generating the reference power Pref (all quantities are averaged over the last 4 h of the simulation): 14ηwf=PwfnwtPref.

For each of the five cases the reference power is obtained by an additional simulation of a single turbine using the same inflow profiles as for the respective main simulation. The reference powers for each case are given in Table . The wind farm efficiency can also be interpreted as the wind turbine efficiency averaged over all wind turbines of the wind farm. The wind turbine efficiency of a wind turbine generating Pwt is defined as follows: 15ηwt=PwtPref.

The wind farm efficiencies of the small and the large wind farm are listed in Table , and the wind turbine efficiencies are shown in Fig. .

In general, the wind farm efficiency is significantly lower for large wind farms than for small wind farms. All 7D cases, except for the SBL case, show efficiencies of 0.86–0.88 for the small wind farm and efficiencies of 0.54–0.64 for the large wind farm. In the SBL case, the efficiency of the small wind farm is 0.66 because the wind farm is affected by the blockage effect of the sum of all wind farms. This is visible in Fig e, which shows that the efficiency of the wind turbines in the first row of N-1 is already below 0.8. The efficiency of the large wind farm is 30 % lower than that of the small wind farm for the SBL case. The blockage effect redistributes energy from upstream parts of the wind farm to downstream parts of the wind farm by a favorable pressure gradient, which has already been shown by for wind farms in shallow CNBLs. This effect can also be seen in the power distribution inside the farm: the turbine power is constant from approximately row 10 up to the trailing edge of the large wind farm in Zone 3 (see Figs. e and e). In all other cases the wind turbine power does not reach a steady state until the end of the wind farms.

A reduction of turbine spacing from s=7 D to s=5 D results in an efficiency reduction of 12 % (0.87 to 0.77) for the small wind farm but results in an efficiency reduction of 29 % (0.58 to 0.41) for the large wind farm. The low wind farm efficiency for the case NBL-700-5D can be explained by a fast drop in the turbine efficiencies to values below 0.4 only 20 km downstream of the leading edge. The low wind turbine efficiencies are caused by a reduction in the vertical kinetic energy flux, as shown in the next section.

A doubling of the BL height results in an efficiency increase of +2 % (from 0.86 to 0.88) for the small wind farm but in an efficiency increase of 19 % (from 0.54 to 0.64) for the large wind farm. The dependency of wind farm efficiency on the BL height has also been observed by , who reported a 17.6 % increase in power deficit for a BL height reduction from 1000 to 250 m.

A comparison between the cases NBL-700-7D and CBL-700-7D shows that greater wind farm efficiencies are obtained for the NBL, although better efficiencies are expected for the CBL due to the better vertical mixing. Comparing the wind speed profiles of these cases (see Fig. ) shows that the inflow wind speed in the bulk of the BL is higher for the NBL than for the CBL, which is probably the reason for the higher wind farm efficiencies. This result shows that it is important to consider not only the wind speed at hub height but also the wind profile inside the entire BL to make accurate wind farm performance predictions.

To examine the dependency of the wind farm efficiency on the turbine spacing and the BL height in more detail, an energy source analysis is made in this section. Here, an energy source is defined as an energy input to the flow, i.e., a process that drives the wake recovery. This can be one of the following:

vertical turbulent flux of kinetic energy at rotor top level, Wvkef;

The analysis is a simplified version of the analyses made by and and does not claim to be a complete energy budget analysis. The intention of this analysis is to show which processes dominate the wake recovery and thus limit the achievable power density of very large wind farms. Thus the advection of upstream kinetic energy is not considered here. The above-named sources are calculated as follows.

The resolved downward turbulent flux of mean kinetic energy at rotor top level, averaged between y=120km-sy and y=120km+sy, is calculated by multiplying the shear stress by the corresponding wind velocity component at that height: 16Wvkef(xi)=〈-ρ(u‾w′u′‾+v‾w′v′‾)|z=zt〉y.

The power density of the energy input by the geostrophic pressure gradient on the flow below rotor top level zt=270 m is calculated as follows: 17Wgpg,wt(xi)=∫z=0ztρfc(ugv‾(z)-vgu‾(z))dz|y=120km.

The work done by the geostrophic pressure gradient on the rest of the BL (between zt and zi) is calculated as follows: 18Wgpg,BL(xi)=∫z=ztziρfc(ugv‾(z)-vgu‾(z))dz|y=120km.

The power density of the energy input by the perturbation pressure gradient on the flow below rotor top level is calculated as follows: 19Wppg,wt(xi)=-∫z=0zt∂p*(z)∂xu‾(z)+∂p∗(z)∂yv‾(z)dz|y=120km.

The power densities of the wind turbines are defined as follows: 20Wwt=Pwtsxsy.

The power densities are shown in Fig. . For the case NBL-700-7D it can be seen that the first-row wind turbines operate at the reference power so that a high power density of Wwt=12.56MW/(7D)2=4.45Wm-2 is achieved. The dominant energy source for the first turbine rows is the advection of kinetic energy. The advection is not included in Fig.  because it is larger than the other terms and would make the quantification of the smaller terms difficult. The power density of downstream wind turbines is determined by the vertical kinetic energy flux. Because the vertical kinetic energy flux decays from 3 W m-2 at the beginning of the first wind farm to 2 W m-2 at the end of the last wind farm, the power density of the wind turbines also decays to below 2 W m-2. The good correlation between the wind turbine power density and the vertical kinetic energy flux has also been found by for the fully developed regime in a 9 km long wind farm. The work done by the geostrophic pressure gradient on the flow below the rotor top level achieves a power density of approximately 0.6 W m-2. It is thus not the dominating energy source inside the wind farms, but it still contributes approximately 20 % to the sum of all sources Wtotal=Wvkef+Wgpg,wt+Wppg,wt. In the downstream half of the wind farms the ratio between the wind turbine power and Wtotal is approximately 70 %.

Although the vertical kinetic energy flux does not reach a constant value until the end of the wind farms, it is likely that it approaches the power density of the work done by the pressure gradient on the BL flow above the wind turbine level Wgpg,BL. Therefore, the flow approaches the fully developed regime of an infinite wind farm flow, in which all the energy extracted by the wind turbines is provided by the work done by the geostrophic pressure gradient on the BL flow .

The energy input by the geostrophic pressure gradient into the entire BL (Wgpg,wt+Wgpg,BL) achieves power densities of only 1–2 W m-2, which is consistent with the geophysical limits to power densities of large wind farms found by , who reported approximately 1.5 W m-2 for a latitude of 46∘ and a geostrophic wind speed of 12 m s-1. This power density is much smaller than the power density achieved by the first-row wind turbines. As the case NBL-700-5D shows (Fig. b), a reduction of the turbine spacing from s=7 D to s=5 D approximately results in a doubling of the power density of the first-row wind turbines (from 4.5 to 8.5 W m-2), but the power density of the last-row wind turbines is as low as for s=7 D. This result indicates that the turbine spacing for very large wind farms should be chosen to be much larger than for small wind farms to achieve a good wind farm efficiency. That the wind farm power output is limited by the vertical kinetic energy flux has also been found by , who investigated potential wind farm scenarios in the German Bight using a mesoscale weather forecast model (WRF) and a simple box model (KEBA, kinetic energy budget of the atmosphere). used a simple, theoretical approach to show that the power density of very large wind farms is limited by the energy input of the pressure gradient and that the power density is proportional to τ0Uh, where τ0 is the shear stress near the surface, and Uh is the mean wind speed at hub height for an undisturbed flow without wind farms. However, neglects the effect of the wind farm on the flow inside the BL. According to and Eq. (), the energy input by the pressure gradient depends on the BL height and on the ageostrophic wind speed component averaged over the BL. The BL height increases due to the presence of the wind farms, and the ageostrophic wind speed component increases due to the counterclockwise wake deflection (see Fig. f). Consequently, the wind-farm-induced flow effects result in a significant increase in the energy input by the pressure gradient, as can be seen in Fig. a and b. This effect occurs only in the wake, although the BL height and wind direction already change inside the wind farms. The reason is the decrease in the absolute wind speed that tends to reduce the ageostrophic wind speed component and thus compensates for the increasing ratio of ageostrophic to geostrophic wind speed (counterclockwise wind direction change). In the wake, the wind speed recovers, and thus the ageostrophic wind speed component becomes larger than that upstream of the wind farms. The described effect is largest for the case with the small turbine spacing, NBL-700-5D, because the BL growth and the wake deflection angle are largest for this case.

Figure c and d show that a doubling of the BL height has approximately no effect on the energy input by the geostrophic pressure gradient (Wgpg,wt+Wgpg,BL) on the undisturbed inflow. The effect of the thicker BL is compensated for by a smaller ageostrophic wind speed component inside the BL. This is indicated by a much smaller angle between hub height wind and the geostrophic wind of α=3.4∘ for the case CBL-1400-7D than α=9.5∘ for the case CBL-700-7D (see Table  and Fig. ). Consequently, the ageostrophic wind speed component inside the BL of the stationary inflow adjusts in such a way that the resulting energy input by the pressure gradient balances the energy extraction by TKE production near the surface. As stated earlier, the power output of infinitely large wind farms is determined by the energy input of the geostrophic pressure gradient, which does not depend on the BL height. Hence, the power output of infinitely large wind farms is expected not to depend on the BL height, at least for this idealized setup with a stationary CBL inflow. However, for very large but finite-sized wind farms, as in this study, the power output depends significantly on the BL height, as is shown in Fig. c and d. The vertical kinetic energy flux is greater and decays slower for the thicker BL (CBL-1400-7D), resulting in higher turbine power densities.

The case SBL-300-7D is very special because the rotor top level matches the BL height of the inflow. Thus, the energy input by the pressure gradient above the rotor top level, as well as the vertical kinetic energy flux at the rotor top level, is zero upstream of the wind farms. Both components become non-zero inside the wind farms due to the vertical displacement of the inversion layer (BL growth). For the first 10 km of the wind farms the vertical kinetic energy flux dominates, but further downstream, the energy input by the geostrophic pressure gradient below rotor top level is greater than or equal to the vertical kinetic energy flux. As stated earlier, the blockage effect redistributes energy from the wind farm leading edge into the wind farm, which results in a smaller power density of the first-row wind turbines compared to the other 7 D cases and in a constant power density from approximately row 10. This redistribution is done by a favorable perturbation pressure gradient inside the wind farms and reaches power densities of approximately 1 W m-2. In the wake, the vertical kinetic energy flux at rotor top level drops to zero again, which is consistent with the very low TI in the wake (see Fig. ).

These results show that the power output and the wake of very large wind farms behave very differently compared to small wind farms. The main findings and their implications are summarized in the next section.