The flight legs are oriented approximately orthogonal to the mean wind direction. To estimate the dominant turbulence length scales of the wind component in the mean wind direction, the integral length scale is used as follows: 2ρutut(η)=ut′(y)ut′(y+η)‾σut2, where ρutut(η) represents the auto-correlation function of ut in the direction perpendicular to the mean wind flow (along the orientation of the flight leg) denoted by y, ut′(y) corresponds to the fluctuations, η is the space lag in y direction, and the variance of ut is denoted by σut2.

The auto-correlation diagrams in Fig.  help distinguish turbulent from mesoscale motions during different stratification strengths. For instance, in Fig. a and b, for the upstream flight legs in weakly stratified cases, here the turbulence causes a monotonically and steeply decreasing auto-correlation until a spatial lag η of a few hundred meters. This represents high ambient turbulence in the atmosphere due to increased vertical mixing. Then the auto-correlations are about constant and even increase at η of ∼1 km (Fig. a), probably indicating mesoscale structures which are not yet disturbed by the wind farm. The intensity of small-scale turbulence is increased above and downstream of the wind farm due to turbulence generated by wind turbines, and it overshadows the mesoscale structures, also seen by a weak correlation ρutut(η) at large spatial lags.

During strongly stable conditions illustrated in Fig. c and d, the mesoscale fluctuations in the upstream flight legs are more dominant, as clearly shown by large values of ρutut(η) at large spatial lags. Above the wind farm during strong stable stratification, there is a huge presence of small-scale turbulence, and it is not much different from the weak stratification. Since there is not a lot of vertical mixing present due to the stable stratification, the downstream measurements (Fig. c and d) suggest that fluctuations caused by the wind turbines are dominant in the wake flow of wind farms, but still we observed large values of ρutut(η) at large spatial lags, indicative of mesoscale structures. This can also be seen in Fig. f, where small-scale turbulence starts to die out in the downstream flight leg. This indicates that the wakes created by wind turbines will last much longer in these conditions due to low ambient turbulence and the low intensity of small-scale structures .

The integral length scales L can be obtained by taking the integral of ρutut(η) from 0 to the point of first zero crossing of ρutut(η) i.e., η0, 3Lu=∫0η0ρutut(η)dη.

The integral length scale Lu is much larger during strongly stable stratification in the upstream and downstream of the wind farms, signifying larger length and timescales for the u-component. This is presented in Fig. a and b where longitudinal length scales, Lu, and vertical length scales, Lw, at different positions relative to the wind farm are plotted for all the four flights. The large values of Lu indicate the presence of 2D turbulence where the vertical mixing is extremely low and hence the lower values of Lw at corresponding positions. Due to increased vertical mixing above the wind farm, Lw increases significantly for strong stable conditions and then decreases in the downstream positions for all flights. For weak stratification, significant changes in Lw were not observed, although the slight drop in magnitude from upstream to above the wind farm positions can be referred to unsteady atmospheric conditions observed during the flight legs at the two locations.

In this section, we discuss the rate of energy dissipation ϵ by plotting the compensated spectra for the flights mentioned in Table . The purpose of doing that is to evaluate at what rate the energy is being dissipated from large-scale eddies to smaller flow structures in either the ambient turbulence or in the turbulence generated by wind turbines. In the inertial subrange, the one-point, two-sided velocity spectra in terms of wavenumber k2 (where k=2πf/U‾; f is the sampling frequency and U‾ is the magnitude of the resultant vector of aircraft speed and incoming wind speed) are given by Eqs. () and () . k2 is the wavenumber along the flight path and perpendicular to the mean wind direction. The assumptions behind Eqs. () and () include isotropic and incompressible flow.

The spectra in the inertial subrange for the v wind component is 4F22k2=955αϵ23k2-53.

For the u and w wind components, it is 5F11k2=F33k2=1255αϵ23k2-53, which implies 6F11k2=F33k2=43F22k2, where α is the spectral Kolmogorov constant having an empirical value of ∼1.7 . Notice that as a function of k1, which is the more usual case, for example, for anemometers mounted in meteorological masts, then F22(k1)=F33(k1)=43F11(k1).

By plotting the spectra, we expect a constant value in the inertial subrange which can be used to identify the rate of energy dissipation ϵ from Eqs. () and ().

Figure  displays the compensated spectra for weak thermal stratification (flight 33) in terms of wavenumber k2 along the flight path, and the inertial subrange can be distinctly observed from wavenumbers between ∼10-1 and ∼100 m-1. It is pertinent to point out the discrepancy found in the w spectra, which should be equal to u spectra in the inertial subrange as given by Eq. () . This deviation could possibly be linked to a calibration error in the vertical velocity component measured by the instruments installed on the aircraft. The average values for compensated u and v spectra in the inertial subrange are denoted by dashed black and blue lines, respectively, in the plot, and can be used to evaluate the rate of energy dissipation using Eqs. () and (). A similar procedure was applied on all the four flights described in Table , and the mean values of the rate of energy dissipation ϵ obtained from u and v wind components are plotted in Fig. . From the plot, it can be observed that the upstream energy dissipation rate ϵ is much higher during near-neutral stratification (flights 32 and 33), almost ∼40 times as compared to strongly stable stratification, and it corresponds with the high ambient turbulence during neutral stratification. Above the wind farm, there is not much difference in the dissipation rate ϵ between neutral and stable conditions, as it mostly depends on the layout of the wind farm and the incoming wind speed. The lower value of ϵ in flight 40 can be partially referred to as highly stable conditions, and the location of the flight leg above the wind farm caused it to not be exposed to a large number of wind turbines from the mean wind direction (see Fig. d). The dissipation rate ϵ in the downstream direction depends on a lot of factors: configuration and density of wind turbines in the cluster, upstream wind speed, wind direction, and distance of the downstream flight leg from the wind farm trailing edge. From Fig.  it can be seen that the energy dissipation rate ϵ is significantly alike for all cases downstream of wind farms. In all cases, ϵ remains almost similar, or there is a drop in ϵ downstream of the wind farm, except for flight 40, which is because of the large number of wind turbines affecting the portion of the downstream flight leg as compared to the flight leg portion above the wind farm (see Fig. d).

Here we analyze the top-down and lateral influx of momentum into the wind farms as the flow energy is depleted by the presence of wind turbines. This influx of momentum helps replenish the energy available to wind turbines and also recovers the wind in the wake of wind farms.

Figure  displays the variation in the eddy covariances of velocity components: 〈u′w′〉, 〈v′w′〉, and 〈u′v′〉 for flight 32 and 33 tracks recorded upstream, above, and downstream of the north wind farm cluster under weak thermal stratification. The Reynolds stress component 〈u′w′〉 (i.e., the vertical transport of horizontal momentum along the main wind direction) is largely responsible for the influx of momentum. From Fig. , we can observe that there is already some momentum flux upstream of the wind farm in both flights. This is due to the presence of shear and lack of stratification, which implies large vertical velocity scales Lw in the atmosphere, which enhances the vertical mixing and thus the magnitude of 〈u′w′〉. Above the wind farm, we see a rise in 〈u′w′〉 due to the turbulence and shear generated by wind turbines, which increases vertical mixing and thus a downward flow of momentum. The large values of 〈u′w′〉 in the downstream flight legs (Fig. c and f) correspond with the location of the wake. Here we observed two distinct peaks of 〈u′w′〉: the one at lower latitudes corresponds to the downstream wakes of the two wind farms below the Kaskasi gap, and the large peak at higher latitudes refers to the downstream wake of the dense Amrumbank West wind farm. Non-zero 〈u′w′〉 momentum flux was also observed outside the wind farm boundary in Fig. b and e, which is an indication of high ambient turbulence. The variation in 〈v′w′〉 does not differ significantly in the three positions relative to the wind farm, and its magnitude is considerably lower as compared to 〈u′w′〉 above the wind farm as well. For the lateral momentum flux component 〈u′v′〉, we observed no distinct pattern due to the presence of wind farms. Rather, the variation in 〈u′v′〉 seemed much more chaotic in all three positions. This can be attributed to the fact that we do not observe a considerable reduction in longitudinal wind component u under weak stratification above and downstream of the wind farm (also see Fig. ) due to high ambient turbulence in the atmosphere. Thus a lack of sharp gradient of u in the transverse direction corresponds to no significant lateral momentum flux 〈u′v′〉.

The variation of momentum flux components during strong thermal stratification is shown in Fig. . The change in eddy covariances of the three velocity components is displayed with respect to the flight legs taken during flight 39 and flight 40. During strong stable stratification, there is no top-down flow of momentum 〈u′w′〉 outside the wind farm boundary (the shaded pink area in Fig. ). A small peak in 〈u′w′∠ during the upstream flight legs was observed due to the presence of Nordsee One wind farm below the upstream flight leg (see Fig. ). While there is a strong momentum flux above the wind farm, it decreases significantly downstream of the wind farm, resulting from the lack of vertical mixing i.e., small vertical length scales Lw in the atmosphere. For the vertical flux of lateral momentum 〈v′w′〉, the non-zero values are also observed only due to the disturbance in flow generated by wind farms. Furthermore, the values observed for the vertical flux of lateral momentum 〈v′w′〉 are less than half of the vertical flux of horizontal momentum 〈u′w′〉. Due to a large reduction in the longitudinal wind speed component ut during strong stable conditions (see Fig. ), we also observe a considerable lateral momentum flux component 〈u′v′〉 in all the three positions for flight 39. Some more discussion on the lateral momentum flux 〈u′v′〉 is presented in the next section. The two peaks in the value of eddy covariances for the flight legs recorded during flight 39 above the wind farm (see Fig. b) are considered to be a result of the layout of the wind turbines in Godewind I and II wind farms and the location of the flight leg. The magnitudes of all three eddy covariances during flight 40 are much smaller due to stronger thermal stratification, and the turbulent fluxes are almost negligible even in the downstream wake of the wind farms.

In this section, we analyze the length scales responsible for the vertical entrainment of momentum in large wind farms. We use the data from the flights mentioned in Tables  and , representing different levels of stratification based on the observed lapse rate γ. Here we only evaluate the dominant scales of 〈u′w′〉, since it is the most dominant form of entrainment compared to 〈u′v′〉 in large offshore wind farms. After selecting the part of the flight legs projected downstream of the wind farm in the mean wind direction, we look at the uw cross-spectrum to determine the dominant length scales. This is done for both wake flow and undisturbed flow for all flights. The relation used to find the most dominant scales of vertical entrainment is defined by k, where 7ln⁡kk0=∫0∞RFuwk2ln⁡k2k0dk2∫0∞RFuwk2dk2, and 8k=k0exp⁡ln⁡kk0. Here k is the dominant wave number in m-1, R(Fuw(k2)) is the real part of uw cross-spectrum as a function of wavenumber k2 (k2 is the wavenumber in the direction perpendicular to the mean wind flow), and k0 is a reference wave number. The dominant wavenumber can be understood as the center of gravity of the pre-multiplied spectrum plotted on a logarithmic k2 axis. A length scale 1/k is defined in this way.

Since there were multiple flight legs for the four flights discussed in Sect. 3.1–3.4, the data represented are the mean for all the flight legs at one location. The uncertainty in all scalars is represented by the standard error of mean (SEM) as follows: 9SfEM=σn, where σ is the standard deviation and n is the total number of flight legs at a location. An exception is Fig. , where the dominant length scales downstream of the wind farms cluster are plotted and there are no error bars in this plot. The reason is that during flights 7 and 30, only single flight legs were recorded at each downstream position. Moreover, assuming that the flow and atmospheric conditions are stationary during the whole duration of the flight also brings uncertainty to the analysis, since all the flight legs were not recorded at the same time. There are also some uncertainties arising from the conversion of wind vector from the geodetic coordinate system to the geographical coordinate system and then transforming the horizontal wind component to the mean wind direction. These uncertainties arise from the systematic errors induced by pitch, roll and yaw angle measurements, and usually, a correction factor is applied after the in-flight calibration procedure, as discussed by and .

For the evaluation of turbulent momentum fluxes in Sect. 3.3, we utilized the rolling window of about 2 km to smooth the high-frequency data. The window length was chosen in order to reduce the random errors in first and second-order moments. recommended, based on the work by , that the rolling-window length should be more than 1800 m to include both small-scale variations in mean quantities and also incorporate large-scale flow effects. We tried different rolling window lengths and observed their impact on the turbulent fluxes, and found the rolling window of 2 km to be the most representative of the flow phenomenon happening around these large wind farms.

Another important uncertainty in the analysis presented in Sect. 3.1 and 3.4 arises from the selection of the flight leg affected by the wake flow of wind farms. A two-tier strategy was applied to identify the wake and distinguish it from the large mesoscale effects. In the first step, the ut wind speed component minima for each leg were identified and a suitable distance was added on both sides of the minima based on the method suggested by . The second step involved a visual inspection of the portion of the flight leg identified as a wake flow in the first step. This was done in order to prevent a large mesoscale flow minimum to be identified as the wake flow. Especially during stable conditions, very long wakes were observed which experienced large-scale turning.

From the variation in the lateral momentum flux 〈u′v′〉 discussed in Sect. 3.3, we observed that 〈u′v′〉 is predominantly negative during all three flight legs in stable conditions. But the negative values of 〈u′v′〉 at the southern edge of wind farms (shown in Fig. ) negates the validity of flux–gradient hypothesis stated in Eq. (): 10〈u′v′〉=-K∂u∂y, where K is the eddy diffusivity constant. This relation implies that as the longitudinal wind component u changes its magnitude in the transverse direction (y axis in Fig. ) due to the presence of wind farms, the ∂u/∂y gradient is negative at the southern edge and positive at the northern edge of these wind farms, which in turn should make 〈u′v′〉 positive at the southern edge of wind farms and correspondingly negative for the northern edge. The deviation from the flux–gradient relation can be explained by the fact that these measurements are not recorded in the surface layer but rather high up in the atmosphere and inside a canopy flow (for the flight legs above the wind farm). Above the surface layer or inside the canopy flow created by a wind farm, the momentum transport is dominated by large-scale eddies instead of the local gradient of wind or molecular diffusivity, and the validity of the flux–gradient relation can be questioned. This behavior has also been studied by where they identified “counter-gradient fluxes” within a forest canopy flow because of large-scale turbulent transport eddies. We also observed an inverse correlation between 〈u′v′〉 and 〈v′w′〉 in all the flight recordings. For strong stable stratification cases, Pearson correlation coefficient values of -0.85 and -0.72 are recorded between 〈u′v′〉 and 〈v′w′〉 during above and upstream flight legs, respectively. While for the near-neutral stratification case, the correlation coefficient values are -0.47 and -0.33 during the above and upstream flight legs, respectively.

We also observed that the case study wind farms did not satisfy the conditions of an “infinite wind farm” because of a significant presence of lateral momentum flux 〈u′v′〉, especially during strong stable conditions. Under near-neutral conditions, the main source of energy transport inside the wind farms is the vertical flux of horizontal momentum 〈u′w′〉. By comparison, the lateral flux of momentum 〈u′v′〉 is quite low due to the weak gradient of the u component in the lateral direction. Under strong stable conditions, 〈u′w′〉 is still the main source of energy transport, but 〈u′v′〉 is also significant due to a sharp gradient of u component in the lateral direction. This implies that in reality, large wind farms in offshore settings do not just rely on the vertical entrainment of momentum for energy recovery. Many analytical and engineering wake models for large offshore wind farms often ignore and exclude the lateral entrainment of momentum from the energy budget equation, deeming it negligible . The effect of wind farm layout on the momentum fluxes and kinetic energy entrainment cannot be analyzed quantitatively using aircraft measurements above the wind farm. Nonetheless, we observed that the layout of a wind farm influences the energy recovery from the peaks observed in the momentum fluxes magnitudes which correspond to the high density of wind turbines at a certain location (see Fig. ).

From the analysis presented in Sect. 3.1, we observed that longitudinal length scales Lu and vertical length scales Lw manifest relatively different behavior under different stratification strengths. In the undisturbed flow, the difference between Lu at different strengths of stratification is not large: Lu in strong stable stratification is 2 to 3 times larger than the weak stratification (Fig. ). These large magnitudes of Lu represent mesoscale flow, which is indicative of 2D turbulence, comprising extremely lower frequencies in the velocity spectrum. But the magnitude of Lw in the undisturbed flow strongly depends on the stratification strength in the atmosphere: Lw in strong stable stratification is about 10–15 times smaller than the weak or near-neutral stratification (see Fig. ). The analysis presented in Section 3.4 regarding the dominant scales of entrainment suggests that Lu does not influence the entrainment as much as Lw. In the undisturbed flow, when Lu has large magnitudes for both strong and weak stratification cases, we observed negligible momentum entrainment in the former case. Rather, the momentum entrainment is strongly correlated with the magnitude of Lw and the atmospheric stratification.