Impact of atmospheric stability on X-band and C-band synthetic aperture radar imagery of offshore windpark wakes

C-band and X-band Synthetic Aperture Radar (SAR) data acquired by the Sentinel-1 and TerraSAR-X satellites are used to study atmospheric wakes behind offshore wind parks in the German Bight. A particular focus is on the impact of atmospheric stability on wake parameters like the wake length. Stability parameters are estimated from measurements taken at the FINO-1 observation platform. Based on a data set covering different seasons and concentrating on the first German offshore wind park Alpha Ventus (AV), it is shown that in this area stable atmospheric conditions favour longer wakes. This is first demonstrated for situations, where the wake behind AV was unperturbed by other neighbor wind parks. In this case, wakes of more than 30 km length are observed. In a second step, the more complicated situation with wake superposition from different neighboring wind parks is analysed. It is shown that in this case, the merged wakes can extend to more than 70 km downstream.The analysis is challenged by two factors. First of all, the FINO-1 platform is within the wind farm wakes for a certain range of wind directions. This means stability estimates for the upstream conditions are not straightforward to obtain in these conditions. The second complication is associated with an apparent increase in the radar cross section downstream of wind parks observed on many SAR scenes, typically within the first 10 km downstream the wind park. A semi-empirical model is proposed to explain this effect by an increased downward momentum flux associated with increased turbulence generated by the wind park. Applying numerical inversion methods, a couple of typical downstream wind speed profiles are reproduced with this model based on SAR derived estimates of the friction velocity.C-band and X-band Synthetic Aperture Radar (SAR) data acquired by the Sentinel-1 and TerraSAR-X satellites are used to study atmospheric wakes behind offshore wind parks in the German Bight. A particular focus is on the impact of atmospheric stability on wake parameters like the wake length. Stability parameters are estimated from measurements taken at the FINO-1 observation platform. Based on a data set covering different seasons and concentrating on the first German offshore wind park Alpha Ventus (AV), it is shown that in this area stable atmospheric conditions favour longer wakes. This is first demonstrated for situations, where the wake behind AV was unperturbed by other neighbor wind parks. In this case, wakes of more than 30 km length are observed. In a second step, the more complicated situation with wake superposition from different neighboring wind parks is analysed. It is shown that in this case, the merged wakes can extend to more than 70 km downstream.The analysis is challenged by two factors...


I. INTRODUCTION
and survey 18 , or the impact on the local and global meteorology 3,19,20 . 155 B. Microwave radar scattering from the sea surface 156 SAR systems are active radar instruments, which transmit and receive signals in the 157 microwave frequency band. They are thus independent of daylight and are rarely affected 158 by atmospheric conditions (e.g., extreme rain). The measurement of near surface wind 159 speeds from SAR data is based on the so called Bragg scattering mechanism. For a facet 160 tilted by an angle ψ in the x direction and by an angle δ in the y direction the radar cross 161 section associated with Bragg scattering is given by 21 : 162 σ Bragg (θ, ψ, δ)= 16π 4 cot 4 (θ i )|α(θ, ψ, δ)| 163 ×Ψ (2k sin(θ + ψ), 2k cos(θ + ψ) sin(δ)) (1) 164 with incidence angle θ, local incidence angle θ i = cos(θ+ψ) cos(δ), electromagnetic wavenum- 165 ber k, and small scale wave spectrum Ψ. The radar wavelengths for the systems considered 166 here are in X-band (2.5-4 cm) and C-band (4-8 cm). The plane of incidence is in x direction 167 and α is a complex valued function of incidence angle, radar frequency, polarisation and  171 where p(tan ψ, tan δ) is the slope probability density function. This function can be derived 172 from the ocean wave spectrum Ψ. 173 Based on dimensional considerations it can be shown that the high frequency ocean wave 174 spectrum appearing in eq. 1 is of the following form where f is an unknown function 22 , u * is the friction velocity and g is the acceleration of 177 gravity. Eqs. 1-3 show that there is a direct connection between the normalised radar cross 178 section σ 0 and the friction velocity u * for a given wind direction. Making assumptions about 179 the shape of the short wave spectrum, analytical models can be derived for the simulation 180 of σ 0 from given wind conditions. Because these models still contain a lot of uncertain 181 components, empirical models are often preferred for practical applications like wind speed 182 retrieval from radar. A recent comparison of physically-based and empirical models can be 183 found in Fois 23 .

184
As empirical models already exist for C-band and X-band, which are suitable for the 185 incidence angle regimes of Sentinel-1 and TerraSAR-X data, we will follow this approach 186 here. More details on how these functions were used for the wind speed retrieval in this 187 study are given in Section IV A.    confirming previous studies 29 . 246 The distribution of wind conditions at the time of the satellite acquisitions used for wake 247 analysis in this study are shown in Fig. 2. It can be seen that these scenes, which all show 248 wake features, cover a wide range of different wind directions and wind speeds. The satellite 249 data also represent different seasons with a slightly stronger weighting of the summer half-

250
year. This has to do with the higher probability of stable atmospheric conditions in this 251 period, which favours the formation of wakes, as discussed in the following sections.

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A. SAR wind speed retrieval 254 For the radar wavelengths and incidence angles of typical SAR instruments like TerraSAR-255 X or Sentinel-1 the Bragg wavelengths appearing in eq. 1 are of the order of a few centimeters 256 (see table II). Because the energy on this roughness length scale is highly dependent on the 257 near surface wind, these radar instruments allow the retrieval of wind speed information.

258
The method that enables the conversion of the NRCS to near surface wind speed is based 259 on a so called geophysical model function (GMF), which is an empirical model originally

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where B 0 , B 1 and B 2 are functions of wind speed and incidence angle, and φ is the angle 271 between the wind direction and the SAR look direction.

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In practice, the wind speed retrieval requires some prepossessing of data such as the 273 radiometric calibration to σ 0 . Both C-band and X-band scenes were calibrated to σ 0 using 274 the SNAP software tool 38 provided by the European Space Agency (ESA).

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According to eqs. 1 and 2 the radar cross section is depending on the 2D small scale 276 slope distribution, which in turn is a function of the 2D wind vector. The inversion of eq.  between speckle noise reduction and preservation of spatial resolution required to analyze 300 offshore wind farm wakes.

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The empirical models discussed above are tuned to 10 m wind speed for practical reasons.

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However, as explained in Section II B, the radar cross section is actually closer related with 303 the friction velocity u * , which describes the stress at the water surface 33 . The relationship of u * and U 10 is of relevance for the discussions to follow and is therefore briefly addressed.

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A first estimate of u * can be obtained by assuming neutral conditions, where the wind 306 speed profile is given by instruments. The absolute radiometric accuracy for both SAR systems is reported to be 319 around 0.5dB 46,47 . For example, for C-band, 30 • incidence angle, and u * =0.3 ms −1 this 320 would result in an absolute error of about 10% (see Fig. 3). One has to take into account 321 however, that for the analysis performed in this study the relative accuracy, which is usually 322 higher than the absolute accuracy, is of more significance. From Fig. 3 one can see that the 323 sensitivity of wind estimates with respect to radiometric errors is higher at very high speeds.

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However, the very high wind speed regime is not of interest for this study anyway, because parameters, the following selection criteria were used:

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• The scene contains the offshore wind park Alpha Ventus.

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• The scene shows a wake feature behind the wind park.

335
• The major part of the wake feature is contained on the image.

336
• The scene is not affected by strong modulations of the radar cross section, which are 337 not related to wakes, e.g., associated with convective cells or atmospheric fronts. (e.g., convective cells, boundary layer rolls), which did not allow a reasonable estimation 350 of reference wind speed profiles. Another 10 cases had to be discarded, because insitu 351 information on atmospheric stability was not available. In the end, 23 wake scenes were 352 available for the analysis. As will be shown in following, this relatively small data set was 353 sufficient to establish a relationship between atmospheric stability and wake length.
where θ v is the potential virtual temperature and w ′ θ ′ v is the vertical flux of potential virtual where g is the acceleration of gravity, T 2 −T 1 the absolute temperature difference at two levels 376 z 2 > z 1 , and U z is the wind speed at z. If the vertical gradient of absolute temperature is 377 equal to the dry adiabatic lapse rate (≈ −0.01 K/m), the atmosphere is regarded as neutral 378 and R ib = 0.

382
As indicated by eq. 8, the Bulk Richardson number R ib can be computed from a difference 383 of observed temperature at two levels and a single observed wind speed. In this study 384 eq. 8 was used with z 1 = 0, i.e., T 1 is the sea surface temperature (SST). Based on this 385 approach the stability parameter η has been computed using hourly data sets from the FINO-  The velocity deficit V d defined as whereas the row on the left is located downstream of the offshore windpark and covers the 420 wake. This method is similar to that used in Christiansen and Hasager 3,5 , where U f reestream 421 becomes dependent on the downstream distance as well. The length of each small box is 422 2 km and the spatial mean of wind speed for each box is used for the computation of V d .

423
The wake length is here defined as the distance from the wind farm, where the deficit V d The latter situation with the superposition of wakes originating from different wind parks 442 is naturally more complex. In the following we therefore take a two step approach, where 443 isolated wakes are considered first. in Section VI.

468
Using the technique described in Section V A, the wake lengths were estimated as 32 km for the TS-X scene (Fig. 5a) and 28 km for the Sentinel-1 scene (Fig. 5b). The corresponding velocity deficit curves as a function of the distance from the AV wind park are plotted in grey in Fig. 7a. Considering the atmospheric conditions for both scenes, the longest wake is found for the more stable case on 10 May 2012 (Fig. 5a) with air/sea temperature difference of more than 5 • C (table III). The corresponding maximum velocity deficit is approximately 5% (Fig. 7a). For the Sentinel-1 scene the conditions were close to neutral with air/sea temperature difference of about 1 • C. The observed vertical profile of temperature (and wind speed) from the FINO-1 platform for the case from 30 October 2015 is shown in Fig. 5f and indicates the instability of the atmosphere. In this case the velocity deficit is about 6%. It should be mentioned that a considerable number of these FINO-1 wind profiles have a quite complicated structure, which only in a coarse approximation follow a simple log law. This issue has been the subject of previous studies 52 . It is interesting to note that the bright features characterised by negative deficit values in Fig. 7a occur in the more stable situation with higher air/sea temperature difference. This issue will also be addressed in Section VI.  conditions can also be appreciated from the vertical profile of the temperature (Fig. 6f).

485
The wake length estimated for this case is about 70 km and the maximum velocity deficit 486 is about 16%.

487
The corresponding velocity deficit curves as a function of the distance from the AV wind 488 park are plotted in black in Fig. 7a. It is evident that the two cases with superimposed shorter wakes are found in unstable conditions (Fig. 6e). This case is more complex than 495 previously, because some scenes in unstable conditions display longer wake length than those 496 in neutral or thermally stable conditions. The difference of the wake length for the same 497 class of stability could be due to some parameters such as the number of wind turbines as 498 well as the layout of the wind farm, the wind speed and the wind direction. The wake length 499 is longer than in "AVa", which could suggest that the number of wind turbines influence the 500 extension of the wake length. In a consistent manner as in the "AVa" cases, the dependence 501 of V d on the stability is not clearly established (Fig. 7a). This is also not surprising because, 502 at least at hub height, the maximum deficit is expected to occur immediately downstream 503 the wind park and should mainly depend on the thrust coefficient of the turbines and the 504 wind speed 53 . Nonetheless, for the unstable cases, the maximum of V d is identified relatively 505 in the first 10 km, while shifted further away from the hub for neutral and stable situations.

506
Higher velocity deficit values are found for these complex cases.

507
Another issue to be considered is the shadowing of the FINO-1 platform by AV 54,55 .

508
For easterly wind directions the presence of AV leads to increased turbulence levels and 509 reduced wind speeds compared to unperturbed conditions. Reduced wind speeds increase 510 the magnitude of the Richardson number and hence the stability parameter η = ζ/L (eq. 8).

511
The effect of AV on the temperature gradients measured at FINO-1 is more complicated. It 512 seems to be reasonable to assume that the turbulence introduced at hub height and generated 513 by the rotors leads to downward vertical mixing and thus affects the air temperature at 50 514 m (height at which measurement is taken for our analysis) and below. This means that the 515 magnitude of the temperature difference between 50 m and sea surface will be biased. In fact, 516 for stable conditions, where the upper warm air flows over the cold air, the vertical mixing 517 will bring the warm air down and the cold air up. This will lead to an increase of temperature 518 in the lower layer and therefore reduced temperature gradient. For unstable conditions, the 519 effects will be the other way around. The unstable cases with south-/north-easterly wind 520 direction shown in Fig. 7b could in fact fall into this category. It is obvious that a correction 521 of these cases towards more neutral conditions would lead to a better consistency with the 522 other cases shown in the plot. This also applies to the two longest wakes in Fig. 7b in neutral 523 and slightly stable conditions, which should be moved towards more stable conditions for a 524 more suitable correlation between the stability and wake length. For instance the correction 525 for the case on 28 May 2016 could increase the thermal stratification difference and hence 526 the vertical gradient on the profile of the temperature (Fig. 6f). The case on 30 October 527 2015 in Fig. 5a (represented by the green dot) could be moved also to the right side of the 528 plot and its temperature profile improved.   Fig. 5a,b, and

530
In this section a model is proposed to explain the radar cross section variations seen on 531 SAR images of wakes behind offshore wind turbines. The most interesting feature addressed 532 in this context is the increase of normalized radar cross section within a distance of typically 533 10 km behind the turbines (e.g., Fig. 5a and Fig. 6a), which is found in about one quarter 543 scattering at least for the typical incidence angles considered in this study. As explained in 545 Section II B the cross section is therefore for the most part controlled by the friction velocity 546 u * . The friction velocity in turn is highly dependent on the wind speed at higher levels, the 547 vertical mixing length scales associated with turbulence, and the ocean surface roughness.

548
For the roughness parameter z 0 a first order formulation is given by the Charnock relation 549 eq. 6. This is an approximation, because there is actually also a wave age dependence, To find a solution for u in eq. 14 the function (1 − Φ(z/L))/z is integrated over z yielding 564 with x = (1 − 16z/L) 1/4 and constants β 1 , β 2 . We then get The constants are chosen such that ψ(0) = 0, which leads to β 1 = β 2 = 0. Because z 0 is 567 small, we then also have u(z = z 0 ) ≈ 0 51 .

568
Impact of atmospheric stability on SAR imagery of offshore wind park wakes The idea in this study is to define a new stability function, which takes into account the 569 turbulence mechanically generated by the wind turbines. As already pointed out before, 570 wakes occur predominantly in stable atmospheric condition. For this reason, we use the 571 formulation for stable conditions in eq. 15 as a basis and define with a mixing length correction factor Φ hub at hub height, which has to fulfill in order to increase the mixing length with respect to neutral conditions. With this stability 576 function the mixing length behind the wind turbine is increased by a factor of 1/Φ W T . The 577 respective profile then follows as

579
where we imposed the boundary condition u(z 0 ) = 0. with an e-folding distance σ turb after which the mixing length increase has dropped to about 586 one third.

587
At the same time the mean wind speed U is reduced by a factor R 0 = U/U 0 behind 588 the wind turbine and recovers to the original value U 0 after some distance downstream.

589
Following the derivation of Betz law, which is based on the consideration of momentum and 590 energy conservation, this factor is related to the thrust coefficient c T via 15

592
Here, the thrust coefficient is used to relate wind speed U and air density ρ to the force F 593 experienced by a turbine with rotor disc area A according to 15 The thrust coefficient itself is a function of wind speed as well. An empirical relationship to 596 estimate the thrust coefficient from the wind speed is given by 53 For the downstream evolution of the wind speed reduction factor R, we assume a simple 599 functional shape given by with an e-folding distance σ df after which the wind speed deficit is reduced to about one 602 third.

603
In total the model thus has five parameters: the undisturbed wind speed at hub height 604 upstream U, the Obhukov length scale L in the background wind field, the mixing length 605 scale amplification factor Φ hub , and the e-folding distances σ turb and σ df . case the mixing will lead to particularly strong increases of wind speeds at lower levels 618 (Fig. 8c,d).

619
The model was fitted to observations using a standard cost function approach. In addi-620 tion to the SAR observations, measurements from the nearby FINO-1 platform were used.

621
Denoting the wind speed measurements at 50 m and 100 m height taken at FINO-1 by the cost function is of the following form: Impact of atmospheric stability on SAR imagery of offshore wind park wakes The functions W x ,W 50 and W 100 are supposed to control the relative weighting of the obser-627 vations in the minimisation process. In order to give the FINO-1 observations and the SAR 628 measurements about equal weight we used W x = 1 and W 50 = W 100 = 0.001. 629 Fig. 9 shows the friction velocities estimated from two Sentinel-1A scenes (blue curves) 630 compared to the fitted empirical model results (red curves) for 22 May (Fig. 9a) and 30 631 October, 2015 (Fig. 9b). The approach to estimate friction velocity from SAR as described 632 in Section IV A was applied. Because we do not have a special GMF for conditions of 633 mechanically generated turbulence available, CMOD5.N was used in this context as well.

634
One can see that the model is able to capture the main features of the observed downstream new OWPs and Alpha Ventus is not big (between 2 km and 10 km depending on direction).

690
In Fig. 7b, the wake lengths from "AVa+" are not significantly longer than those from "AVa" 691 for the same unstable range. In fact, in very unstable conditions, the wake is expected to be 692 weak and short, so the influence of upstream OWPs do not affect much wakes downstream 693 Alpha Ventus. However, in stable conditions the wake length is much longer for "AVa+".

694
This clearly materializes the additional effects from neighbouring OWPs, that could be However, for the "AVa+" cases, the velocity deficit magnitude for unstable situations seems 702 to be higher than for stable ones.

703
The study also showed that there is a lot of scatter in the data and some longer wakes which is predominantly connected to the friction velocity and wind speeds at higher levels 721 within the closer downstream regime, which is strongly affected by mechanically generated 722 turbulence. It can thus help to avoid misinterpretations of SAR imagery following from 723 naive application of standard SAR retrieval methods.

724
The future coverage of OWPs in German Bight around FINO-1 will further limit its   Scatterplot of thermal stability derived from FINO-1 insitu data versus SAR derived wake length for both "AVa+" and "AVa" cases. Triangle symbols represent the cases in "AVa" and the dot symbols are from "AVa+". The colors represent the wind direction of each scene from DWD data.  Space Agency (ESA) for making Sentinel-1 SAR data and the snap toolbox freely available.