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\begin{document}

\title{Wind speed surplus in wind farm wakes quantified from satellite SAR and
mesoscale modeling}



\author[1]{Charlotte Bay Hasager}%
\author[2]{James Imber}%
\author[1]{Jana Fischereit}%
\author[2]{Aito Fujita}%
\author[1]{Krystallia Dimitriadou}%
\author[1]{Merete Badger}%
\affil[1]{Danmarks Tekniske Universitet Institut for Vindenergi}%
\affil[2]{Thermo Fisher Scientific Inc Mitato-ku Tokyo}%


\vspace{-1em}



  
  \date{May 24, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
Satellite SAR provides ocean surface wind fields at 10 m above sea
level. The objective is to investigate the capability of Synthetic
Aperture Radar (SAR) satellite StriX observations for mapping offshore
wind farm wakes. The focus is on the conditions under which an apparent
wind speed surplus is generated, measured in 48\% of the 67 images
available. The results compare well to Sentinel-1 observations, showing
a 34\% surplus rate during several years based on 1171 images. Three
wind speed surplus cases have been studied in detail using the mesoscale
Weather, Research, and Forecasting model (WRF) model with two wind farm
parametrizations. The SAR-based observations and WRF model compare for
most cases, though only when Turbulent Kinetic Energy (TKE) is included
in the wind farm parametrization. The TKE mixes higher momentum
downwards in a stable atmosphere, causing surface wind speed surplus
near the surface.%
\end{abstract}%



\sloppy


Charlotte Bay Hasager\textsuperscript{1,*}, James
Imber\textsuperscript{2}, Jana Fischereit\textsuperscript{1}, Aito
Fujita\textsuperscript{2}, Krystallia Dimitriadou\textsuperscript{1},
Merete Badger\textsuperscript{1}

Affiliations

Department of Wind and Energy Systems, Technical University of Denmark,
Frederiksborgvej 399, 4000 Roskilde, Denmark

Synspective Inc., 3-10-3 Miyoshi, Koto-ku, Tokyo 135-0022, Japan

*Author to whom correspondence should be addressed cbha@dtu.dk

\textbf{Abstract}

Satellite SAR provides ocean surface wind fields at 10 m above sea
level. The objective is to investigate the capability of Synthetic
Aperture Radar (SAR) satellite StriX observations for mapping offshore
wind farm wakes. The focus is on the conditions under which an apparent
wind speed surplus is generated, measured in 48\% of the 67 images
available. The results compare well to Sentinel-1 observations, showing
a 34\% surplus rate during several years based on 1171 images. Three
wind speed surplus cases have been studied in detail using the mesoscale
Weather, Research, and Forecasting model (WRF) model with two wind farm
parametrizations. The SAR-based observations and WRF model compare for
most cases, though only when Turbulent Kinetic Energy (TKE) is included
in the wind farm parametrization. The TKE mixes higher momentum
downwards in a stable atmosphere, causing surface wind speed surplus
near the surface.

\textbf{Key words}

Wind speed surplus, wind speed deficit, satellite, synthetic aperture
radar, speed-up, WRF, StriX, Sentinel-1

\textbf{Introduction}

The density of wind farms offshore has increased immensely over the past
decades, especially in the seas of northern Europe and China, where the
water depth is low. Ambitious plans exist to exploit offshore wind power
in other parts of the world, e.g., along the east and west coasts of the
United States, in the Mediterranean, and in Japan (GWEC, 2022, 4C
Offshore, 2023). Some future installations will be mounted on floating
platforms in deep waters (GWEC, 2022). Offshore wind farms are typically
organized in clusters to optimize the logistics related to wind turbine
operation and maintenance and to preserve areas of the marine
environment for other purposes. A consequence of the clustering is that
the wind farms influence each other aerodynamically. Typically, the
extraction of wind power by one large wind farm will generate a wake
effect, reducing the wind resource available to other wind farms located
downstream while enhancing turbulence. The wake effect of individual
turbines within a wind farm is well understood and modeled (G\selectlanguage{ngerman}öçmen et
al., 2016). In contrast, the knowledge about farm-to-farm wakes is more
limited since wind observations over large areas are challenging in the
marine environment (Platis et al., 2018, Fischereit et al., 2022, van
der Laan et al., 2023).

Evidence of wind farm wake effects in satellite imagery generated from
Synthetic Aperture Radar (SAR) observations has been found since the
commissioning of the first large-scale offshore wind farm, Horns Rev in
the North Sea in 2002 (Christiansen and Hasager, 2005; 2006). A SAR
instrument transmits pulses in the microwave range towards the Earth's
surface and measures the proportion of radar backscatter returned to the
sensor. Over the ocean, cm-scale waves generated by the instantaneous
wind stress determine the backscatter
coefficient,\selectlanguage{greek}\emph{σ\textsubscript{0}.}\selectlanguage{english} Through empirical Geophysical
Model Functions (GMF), it is possible to retrieve the wind speed at 10 m
as a function of \selectlanguage{greek}\emph{σ\textsubscript{0}}\selectlanguage{english} , the wind direction relative
to the radar look direction, and the radar incidence angle.

The region downstream of a large offshore wind farm is often
characterized by a reduced backscatter coefficient and, thus, a reduced
wind speed extending up to 100 km downstream of the turbines (Hasager et
al., 2015). Different approaches have been suggested to quantify the
wake effect from SAR wind fields and to separate it from the background
wind speed gradients found in coastal areas. Wind speed extraction along
linear transects in the SAR wind fields was performed by Christiansen
and Hasager (2005), who used parallel reference transects rather than a
reference point upstream of the wind farm to normalize the wind speeds.
Hasager et al. (2015) used a circular approach for wind speed extraction
around offshore wind farms from a series of SAR wind maps aligned
according to the wind direction. Ahsbahs et al. (2018) and Owda and
Badger (2022) mapped the wind speed differences on the downstream side
of large offshore wind farms before and after their construction and
found significant deficits.

The analyses mentioned here indicate that turbine wakes expand from the
turbine hub height to surface level and are detectable from SAR
observations of the sea surface. They also show that the spatial extent
and the intensity of wakes are highly variable from SAR scene to SAR
scene. Djath et al. (2018) showed that wind farm wakes persist over
longer distances when the atmosphere is stable, i.e., with low thermal
mixing. They also found that the chosen turbine layout governs the wake
intensity. Another new and exciting finding by Djath et al. (2018) was
that a wind speed acceleration, rather than a deficit, occurred
downstream of wind farms in 25\% of the SAR scenes investigated. Similar
examples of SAR scenes were presented by Platis et al. (2020). Lars\selectlanguage{ngerman}én
and Fischereit (2021) found accelerating near-surface winds based on
numerical modeling downstream of wind farms in the North Sea. During
stable atmospheric conditions, the phenomenon of accelerating winds was
also observed with lidar and sonic anemometers downstream of a wind farm
on land in the United States (Bodini et al., 2021). Modeling of wind
farm wakes shows acceleration using large-eddy simulation, particularly
in stable conditions (Vanderwende et al. 2016, Yang et al. 2014).

Wind speed acceleration downwind of wind farms have different causes.
Mixing of higher momentum downward may occur in the stable atmosphere
where the wind profile has low wind speed near the surface, and much
higher wind speeds higher up. The added turbulence from operating wind
turbines increases the TKE in the flow in the rotor plane, and the
higher winds aloft are mixed down near the surface (Larsén and
Fischereit, 2021, Ali et al., 2023). Another cause may be wind farm
blockage slowing down the upstream winds while accelerating the flow
below and above the turbines and along the sides of the wind farm. The
blockage effect may be most pronounced in stable conditions and for high
turbine thrust coefficients in the wind speed range from cut-in to rated
(Schneemann et al., 2021; Strickland et al., 2022; Wu and Porté-Agel,
2017).

Observations of wind farm wake based on a satellite SAR provide details
of the flow at offshore wind farms. A direct comparison between SAR wind
fields and Doppler radar observations was carried out by Ahsbahs et al.
(2020), reporting a good agreement between the wake extent and
intensities for the cases investigated. The spatial resolution of the
SAR wind fields was significantly lower than for the Doppler radar.
Therefore, it was impossible to characterize the wind farm wake at the
same level of detail. Recently, constellations of small SAR satellites
have been launched, which provide SAR scenes at an unprecedented
temporal and spatial resolution. Badger et al. (2023) demonstrated how
wind speeds can be retrieved from observations from the Japanese StriX
mission by Synspective Inc.

The objective of this paper is to investigate the capability of StriX in
the context of mapping wind farm wake effects with a focus on wind speed
surplus. The results from Strix are compared to I) several years of
Sentinel-1 observations and II) three cases with wind speed surplus
observed in StriX and modeled by the mesoscale WRF model (Skamarock et
al. 2019). The results from WRF with different wind farm
parametrizations provide insight into the atmospheric flow during the
three events. A novelty in the study is comparing wind farm wake
conditions for events with wind speed surplus based on satellite SAR and
WRF. Wind farm wind speed deficits have been investigated in several
previous studies, while wind speed surpluses have been only in a few
studies.

\textbf{Study area and data}

Site description of the DanTysk offshore wind farm and the FINO3
meteorological mast\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

The study area covers an area in the German Bight in the North Sea. This
area is home to several wind farms. The wind farm of particular interest
in this study is the large offshore wind farm DanTysk. DanTysk has 80
wind turbines, each 3.6 MW. Their hub height is 88 m, and rotor diameter
is 120 m. The cut-in, rated, and cut-out wind speeds are 3.0 m/s, 12.5
m/s, and 25.0 m/s, respectively. Close to the western row of turbines of
DanTysk, one of the German Forschung In Nord- und Ostsee (FINO) masts is
located. This is the research platform FINO3 (https://www.fino3.de/en/,
last access date 27-10-2923) (FINO3, 2023). FINO3 is located 70 km west
of the island Sylt (Figure 1). The geographical coordinates of FINO3 are
7.158167\selectlanguage{ngerman}° E, and 55.19503° N. Sandbank is a wind farm located 20 km west
of DanTysk. Sandbank has 72 wind turbines, each 4.0 MW. Their hub height
is 95 m, and rotor diameter is 130 m. A third wind farm, Butendiek, is
located in the area approximately 35 km southwest from DanTysk.

\emph{Figure 1. Map of a part of the North Sea with three German
offshore wind farms and the location of the FINO3 mast indicated with a
blue star.}

Meteorological observations from the FINO3 mast include the parameters
wind speed, wind direction, air pressure, and air and sea temperatures.
Wind speed at the heights 31, 41, 51, 61, 71, 81, 91, 101, 107 m, wind
direction at 29 m and 101 m, air pressure at 23 m, air temperature at 29
m above mean sea level, sea temperature at 0.5 m depth are extracted
concurrent to the SAR images from StriX and Sentinel-1. It is the
nearest 10-minute data point for the wind and air temperature data. For
the sea temperature data, the nearest 30-minute data point is used. When
FINO3 has missing temperature data, the search window is expanded by 30
minutes in each direction.

2.2 StriX

StriX is a constellation of small X-band SAR satellites at frequency
9.65GHz operated by Synspective Inc. StriX-\selectlanguage{greek}β \selectlanguage{english}is the second demonstration
satellite launched in March 2022. StriX-\selectlanguage{greek}β \selectlanguage{english}operates in a sun-synchronous
orbit at an altitude of 560 km and makes twice-daily passes over the
North Sea. Between June and September 2022, 74 scenes were acquired over
the FINO3 mast. All of the imagery was acquired in stripmap mode with a
swath width of approximately 20 km and, a resolution of 3m and pixel
spacing of 5m in VV polarization. All scenes were from ascending passes
at approximately 19:38 UTC. Badger et al. (2023) provide further details
on the observations from StriX-\selectlanguage{greek}β.

\selectlanguage{english}2.3 Sentinel-1

Sentinel--1 is a European Space Agency (ESA) mission that consists of
two satellites with C-band SAR instruments operating at frequency 5.405
GHz, Sentinel-1A (2014--present) and -1B (2016--2021). Both satellites
are placed in sun-synchronous polar orbit at an altitude of 693 km with
a 6-day repeat cycle. For the North Sea, the ascending pass is around
6:15 UTC, and the descending at 17:25 UTC. Scenes from January 2015 to
August 2022 were accessed through the Copernicus Open Access Hub
(https://scihub.copernicus.eu, last accessed 27-10-2023). VV
polarization data in interferometric wide swath mode of approximately
250 km swath and a grid spacing of 10 m by 10 m were selected. There
were 1445 scenes in total selected for our study.

WRF model

The mesoscale Weather, Research, and Forecasting model (WRF) version
v4.2.2 is utilized for this study. We use a nested domain set-up with a
resolution of 2 km in the innermost domain (Figure 2a). We use 80
vertical levels with 21 levels below 200 m to ensure that the rotor area
is well represented (Figure 2b). Boundary and initial conditions are
taken from ERA5 (Hersbach et al. 2018), and spectral nudging is employed
in the outer-most domain. ERA5 is the fifth generation ECMWF (European
Centre for Medium-Range Weather Forecasts) reanalysis for the global
climate available from
(https://cds.climate.copernicus.eu/cdsapp\#!/dataset/reanalysis-era5-single-levels?tab=form,
last access 27-10-2023). More details of the WRF model set-up can be
found in Table 1. More details on the wind farm parameterizations are
given in section 3.3.

\emph{Table 1 WRF set-up.}\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
\begin{minipage}[b]{0.30\columnwidth}\raggedright\strut
Category\strut
\end{minipage} & \begin{minipage}[b]{0.30\columnwidth}\raggedright\strut
Subcategory\strut
\end{minipage} & \begin{minipage}[b]{0.30\columnwidth}\raggedright\strut
Details (option number)\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
WRF\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
Version\strut
\end{minipage} & \begin{minipage}[t]{0.30\columnwidth}\raggedright\strut
4.2.2\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
Spatial Settings\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
Domains Vertical levels Nesting strategy\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
D1: 202x202 grid points, 18 km spacing D2: 301x271 grid points, 6 km
spacing D3: 394x334 grid points, 2 km spacing 80 levels, stretching with
height One-way\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
Time\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
Simulation length Time step Output time step\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
24 hours (12 hours of spin-up time) 45 s 10 min (D3)\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
Physics schemes\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
PBL Surface layer Land-surface Microphysics Radiation Cumulus
parameterisation\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
MYNN 2.5 level TKE scheme (5) MYNN surface layer scheme (5) Noah
Land-Surface Model (2) Thompson Graupel scheme (8) rrtmg scheme (4)
Kain-Fritsch scheme (1) on D1 only\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
Boundary and forcing\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
Dynamical forcing Land use data Sea surface temperature Nudging\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
ERA5 at pressure levels every 6 h CORINE OSTIA Spectral nudging (2) for
D1 only with top wave number of 5\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
Wind farm parameterizations (WFP)\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
Fitch et al. (2012) WFP Volker et al. (2015) EWP No WFP\strut
\end{minipage} & \begin{minipage}[t]{0.32\columnwidth}\raggedright\strut
f\textsubscript{TKE} = 1 (denoted: WRF+FITCH1) and f\textsubscript{TKE}
= 0.25 (denoted: WRF+FITCH025) R\_0 = 1.7 (denoted: WRF+EWP) denoted:
WRF\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\emph{Figure 2 a) WRF nested domain set-up with the modeled wind
turbines (red), b) WRF vertical grid with the rotor areas of SWT-3.6-120
(DanTysk) and SWT-4.0-130 (Sandbank).}

\textbf{Methods}

\textbf{3.1 Processing of StriX}

Calibration of the StriX imagery has been determined by acquiring data
over four areas of the Amazon (Orzel, 2022), two of which have been
observed by TerraSAR-X and RADASAT for the same purpose. In order to
assess the noise contribution, the \(\sigma_{0}\) profile in the
range direction was fitted using an expected~\(\sigma_{0}\) profile
plus a noise component. The expected~\(\sigma_{0}\) profile was
generated using the XMOD2 GMF (Li and Lehner, 2014) from the central
incidence angle and the ERA5 wind speed and direction information and
using the upper portion (7000 lines) of the image away from the wind
farm. The ERA5 data was allowed to vary in the fit, the wind speed
by~\selectlanguage{ngerman}±5ms\selectlanguage{english}\textsuperscript{-1}, and the direction by~+-20deg. The noise was
assumed to be purely quadratic with a limited offset \emph{C} , and
shift from the image centre~\selectlanguage{english}?\selectlanguage{english}\textsubscript{????}. The quadratic
coefficient extracted from this fit should be regarded as an upper bound
on the noise. Note also that in the 'raw' data, the thermal noise causes
an overall offset in the recorded backscatter, which was taken care of
during image calibration; hence \selectlanguage{english}? should, in principle, be zero.

\begin{equation}
\sigma_{0,r}=\sigma_{r}^{XMOD2}+N+C\nonumber \\
 \end{equation}

where,

\begin{equation}
N={A\left(\theta-\theta_{\text{centre}}-\theta_{\text{shift}}\right)}^{2}\nonumber \\
 \end{equation}

The \(\sigma_{0}\) image was multi-looked by a factor of 20 in both
range and azimuth before the XMOD2 GMF (Li and Lehner, 2014) was applied
to generate the wind speed map. The wind speed maps are valid at a
height of 10 m above sea level. Wind directions from ERA5 were used as
input to the GMF. Further details on StriX processing are provided in
Badger et al. (2023).

\textbf{3.2 Processing of Sentinel-1}

The Sentinel-1 data was processed using the SNAP toolbox from ESA
(https://step.esa.int/main/download/snap-download/, last access
27-10-2023). The data were calibrated and multi-looked in range and
azimuth to 50 m by 50 m to reduce speckle noise and longer ocean wave
effects. Then, a land-sea mask was applied, and finally, the surface
ocean wind retrieval with CMOD5.n was performed (Hersbach, 2010).
CMOD5.n generates the equivalent neutral wind speed, i.e., without the
atmospheric stability effects that impact actual wind speeds (Portabella
and Stoffelen, 2009). Wind directions from ERA5 were used as input to
the GMF.

\textbf{WRF wake model}

To model the wind farm wake effect in WRF, two wind farm
parameterizations (WFP) were applied: the first was WFP by Fitch et al.
(2012) (FITCH in the following), and the second was the Explicit Wake
Parameterisation (EWP) by Volker et al. (2015).

FITCH modifies the momentum and TKE equation of WRF and adds additional
terms to recreate the wind farm effects. According to a review by
Fischereit et al. (2022a) it is the most widely applied and validated
wind farm parameterization. Here we employed the bug-fixed version, as
noted by Archer et al. (2020), with two different TKE correction factors
to define the amount of added TKE by the turbine (Table 1), namely
f\textsubscript{TKE}=1 (denoted WRF+FITCH1) and
f\textsubscript{TKE}=0.25 (denoted WRF+FITCH0.25). These two values were
used since 0.25 is the recommended value according to Archer et al.
(2020), but Larsen and Fischereit (2022) found that 0.25 does not always
perform best, and f\textsubscript{TKE}=1 might be better in some cases.

In contrast to FITCH, EWP modifies only the momentum equation in WRF and
not the TKE equation. However, contrary to FITCH, it accounts for
sub-grid-scale wake expansion. We applied it here with a recommended
wake expansion coefficient 1.7 (Volker et al., 2015, denoted WRF+EWP,
Table 1). Finally, we performed one simulation without wind farms to
assess how the boundary layer structure would behave without wind farm
effects (denoted WRF). We included all offshore wind farms in the study
area that were present in July 2020, as described in detail in
Fischereit et al. (2022b). Note, WRF model runs included a larger domain
than shown in Figure 1. Using the wind farms from July 2020 is
reasonable, even if the study period is 2022 since no new wind farms
have been built in the area of interest in 2021. For the main analysis
in section 4.2, we used WRF+FITCH1. The sensitivity to the other
scenarios is also discussed in that section.

\textbf{3.4 Identification of wind farm wake in SAR}

To quantify the effect of the wind turbines on the boundary layer wind
field, the upstream and downstream wind speeds were extracted from the
SAR imagery to estimate the turbine-induced deficit. The wind speed
upstream and downstream of the wind farm was determined by averaging the
values in a pair of acceptance regions whose positions were defined by
the central point of the wind farm and by the wind direction from
measurements at the FINO3 mast. The regions were designed to extend
outwards from the windfarm boundary (with a buffer of 1km) to a distance
of 9km, with a width of 6km (Figure 3). For the StriX images, these
regions were required to contain at least 1000 pixels to ensure that a
minimum fraction of each region was valid. Figure 3a, b, and c shows
three examples from StriX (Case A, B, and C). In addition, a minimum
upstream wind speed threshold of 2 m/s was set to ensure the wind farm
would be in operation and avoid the large uncertainties in SAR wind
retrieval at low speeds. After all the requirements were satisfied, 67
scenes of the original 74 remained in the dataset.

Similar to the analysis for StriX, an upstream and a downstream area was
defined for Sentinel-1. The spatial resolution of Sentinel-1 is lower
than StriX, but differences in the multi-looking and projection mean
that the number of valid pixels in these boxes is approximately two
times higher, so the equivalent threshold is 2000 pixels. The same
upstream wind speed threshold of 2 m/s was also set. With all
requirements met, 1171 scenes of the original 1445 remained in the
dataset. An example is shown in Figure 3. Figure 3d shows the full
Sentinel-1 scene, while Figure 3e shows the zoomed-in area at DanTysk
with the acceptance regions indicated.

It can be noticed in Figure 3e that the wind turbines of DanTysk and
Sandbank stand out clearly with high backscatter (small red dots).
Anomalous pixels with high backscatter unrelated to ocean surface winds,
such as turbines, platforms, and ships, can be detected in SAR and
filtered away (Owda et al., 2023). This process has not been done in the
present study, as only wind speeds observed at a sufficient distance
from these hard targets were considered in the wind farm wake analysis.

After extraction of the upstream and downstream wind speeds, the wind
speed deficit was calculated. In order to obtain further insight into
the difference between the wind speed deficit and surplus, the
atmospheric temperature conditions were considered. The potential air
temperature at 29 m height minus water temperature at 0.5 m depth
observed at FINO3 was used. Furthermore, the upstream wind speed, the
wind direction, and the day of the year (seasonal effect) were
considered.

Finally, three case studies (A, B, and C) were investigated by comparing
SAR-based wind fields to results from the WRF model. Case A was from 20
July 2022, including a StriX scene from 19:38 UTC and a Sentinel-1 scene
from 17.18 UTC. Cases B and C were based on StriX from 2 August 2022 at
19:37 UTC and 14 August 2022 at 19:35 UTC, respectively. The three
selected cases showed wind speed surplus. The wind direction was from
the southwest in Cases A and B and from the east in Case C.

b) c)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

d) e)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}

\emph{Figure 3 Wind field retrieved from StriX for a) 20 July 2022 at
19:38 UTC, b) 2 August 2022 at 19:37 UTC, c) 14 August 2022 at 19:35
UTC, Cases A, B, and C, respectively. The red box indicates the upstream
area, and the black box is the downstream area near the DanTysk offshore
wind farm. (d, e) Wind field retrieved from Sentinel-1 20 July 2022 at
17:18 UTC (Case A) d) full scene, e) Sentinel-1 zoomed in at DanTysk.
Note that the wind speed scales differ between the plots. This is to
enhance the spatial variability optimally in each scene.}

\textbf{Results}

The statistics from the StriX and Sentinel-1 observations on wind farm
wake deficit are compared. The relationship between wake deficit and
atmospheric stability, upstream wind speed, wind direction, and day of
year (seasonal effects) are presented. Next three case studies based on
StriX observations showing wind speed surplus are compared to model
results from WRF.

\textbf{Statistics from the archives of StriX and Sentinel-1}

From the StriX dataset, 67 images provide valid results, of which 32
scenes (48\%) have a wind speed deficit behind the wind turbines, 32
scenes (48\%) have a wind speed surplus (a negative deficit is a
surplus), and the 3 remaining scenes show neither (the difference in the
wind speed was less than 0.1 m/s). From the Sentinel-1 dataset, 1171
images show valid results, of which 637 (54\%) have a wind speed
deficit, 402 (34\%) show a wind speed surplus, and the remaining 132
(11\%) show neither effect.

Figure 4a,b shows the relationship between the wind speed deficit as a
function of the temperature gradient close to the water surface for
StriX and Sentinel-1. It is noted that the largest wind speed surplus
(close to 2 m/s) from StriX is in quadrant IV with stable conditions
(positive potential temperature gradient). In contrast, the largest
percentage of cases with wind speed surplus is found in quadrant III
with unstable conditions (negative potential temperature gradient). The
percentages are listed in Table 2. The results for Sentinel-1 show
similar distribution of cases as StriX in each of the quadrants but
large wind speed surplus occur for stable and unstable conditions.

Previous studies on the atmospheric stability conditions in the North
Sea show unstable conditions are the dominant condition based on
meteorological observations from May to October in year 2026, including
all wind directions (Pe\selectlanguage{ngerman}ña et al. 2008), while Sathe et al. (2011) find
near-neutral is dominant for unstable for long-fetch wind direction.
Another study based on meteorological observations and WRF shows the
long-term average is unstable conditions in the North Sea. The unstable
conditions are more pronounced near coastlines than further offshore
(Peña and Hahmann, 2012). Stable conditions are the fewest in all
studies. The results in Figures 4a,b, and Table 2 show that unstable
conditions are most frequent while stable conditions are few. Our
results align with previous studies on the occurrence of stability.

\emph{Table 2 Percentage of cases in each quadrant (anti-clockwise from
top right) in Figure 4a,b for StriX and Sentinel-1.}\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
& Quadrant I (stable) Wind speed deficit (\%) & Quadrant II (unstable)
Wind speed deficit (\%) & Quadrant III (unstable) Wind speed surplus
(\%) & Quadrant IV (stable) Wind speed surplus (\%)\tabularnewline
\midrule
\endhead
StriX & 10.4 & 37.3 & 25.4 & 22.4\tabularnewline
Sentinel-1 & 7.0 & 47.4 & 24.3 & 10.1\tabularnewline
\bottomrule
\end{longtable}

The majority of cases show wind speed deficit as expected, but a
surprisingly large fraction of cases both in StriX and Sentinel-1 show
wind speed surplus, e.g., comparing to Djath et al. (2018) reporting
25\% wind speed surplus cases at a wind farm (Alpha Ventus) 100 km south
of DanTysk. It is comforting that StriX and Sentinel-1 have similar
percentages of cases in each quadrant in Table 2.

The relation between upstream wind speed and wind speed deficit is
investigated. Figure 4c,d show the wind speed deficit as a function of
upstream wind speed. Our dataset includes cases from 2 to 15 m/s
upstream wind speed. The cut-in wind speed is 3.0 m/s thus, it is
assumed that a 2 m/s wind speed at 10 m height will be sufficient to
start the operation of the turbines. Figure 4c,d indicate that the wind
speed deficit is dominant for upstream wind speeds ranging from 4 to 12
m/s in both the StriX and Sentinel-1 wind maps. A wind speed deficit in
this wind speed range is expected. A wake deficit mainly prevails when
the turbines operate from cut-in speed to rated speed. For wind speeds
less than cut-in, the turbines will not operate. Surprisingly, Figure
4c,d also indicates many cases of wind speed surplus for wind speeds
between cut-in and rated speed, especially in the lower part of the wind
speed range.

The wind speed deficit as a function of wind direction is shown in
Figure 4e,f for StriX and Sentinel-1. Even though there are relatively
few StriX scenes available, these cover most of the prevailing wind
directions. Sentinel-1 covers all wind directions. The majority of cases
with wind speed deficit are in the directional sector from 200\selectlanguage{ngerman}° to 240°,
consistent with the dominant wind direction (westerlies). Relatively
many cases show wind speed surplus for easterlies in the directional
sector from 50° to 120°. This is most evident in Sentinel-1. For
easterlies, the wind will come from land at a distance of 70 km or
longer. A coastal gradient might be part of why relatively more cases
show wind speed surplus in this direction. It cannot explain wind speed
acceleration for other wind directions.

The wind speed deficit is shown as a function of day number to examine
the importance of seasonal effects. Figure 4g,h shows the results for
StriX and Sentinel-1. StriX is available only during summer and early
autumn, while Sentinel-1 is available year-round. No clear relationship
between season and wind speed deficit is noted. A cooler sea surface
during spring is expected to lead to more stable conditions and more
pronounced and longer wakes. At the same time, autumn typically exhibits
unstable conditions with shorter wakes (Djath et al., 2018).

b)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}

\end{center}
\end{figure}

c) d)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image11/image11}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image12/image12}

\end{center}
\end{figure}

e) f)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image13/image13}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image14/image14}

\end{center}
\end{figure}

g) h)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image15/image15}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image16/image16}

\end{center}
\end{figure}

\emph{Figure 4 Wind speed deficit from StriX left column and Sentinel-1
right column as a function of (a,b) the potential temperature gradient
between the sea surface minus the air at 29m above mean sea level, (c,d)
the wind speed upstream, (e,f) the wind direction from FINO3, if a value
is missing, the wind direction is from ERA5, and ((g,h) the day of year
(seasons). The three cases from StriX (A, B, C) and one from Sentinel-1
(A) are labelled. The colors in the plots for Sentinel-1 show higher
density with warmer colors.}

\textbf{4.2 Case studies}

Three case studies were selected for detailed investigation using WRF
modeling. All cases showed wind speed surplus downstream of DanTysk. Two
had southwesterly flow, and one easterly. The first case included both
StriX and Sentinel-1 around two hours apart while the two other cases
were based on StriX only.

\textbf{4.2.1 Case A: 20 July 2022 from StriX, Sentinel-1, and WRF}

During Case A, the wind was from the southwest (Figure 3) and the
temperature gradient between the sea and 29 m height was slightly
positive (Figure 4a,b), i.e., stable stratification. For stable
conditions, the mixing in the upstream regions is low, while added
turbulence downstream enhances mixing processes downstream. The wind
speed ranged between 3 m/s and 12 m/s over the 12-hour period of
interest in heights between 50 m and 90 m (Figure 5). The comparison
between FINO3 data and WRF+FITCH1 simulations indicates that the
simulations match the general trend of the observations, although they
miss some features, such as the increase in wind speed between 14:00 and
19:00 UTC or the more southerly wind direction between 14:30 and 17:00
UTC. However, overall the agreement is satisfactory for the time around
the StriX acquisition at 19:36 UTC.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image17/image17}

\end{center}
\end{figure}

\emph{Figure 5 Wind speed (WS) and wind direction (WD) from FINO3 and
WRF+FITCH1 simulations for different heights for the period of the case
study. The vertical line 'StriX' indicates the time of the SAR StriX
image, and the 'Sentinel' line is the time of the Sentinel-1 image.}

While some wind farms in the German Bight located southwest of DanTysk
at around 6\selectlanguage{ngerman}°E and 54.4°N, show classical wake effects, i.e., a wind
speed deficit in the wake, Sandbank and DanTysk show accelerations at 10
m height for this particular day and time (Figure 6a,b) in WRF. The
StriX scene (Figure 6c) shows acceleration for DanTysk, while it is
unclear whether a speed-up is also present for Sandbank since this wind
farm is not covered by the satellite. While the general pattern agrees
for StriX and WRF, the magnitude differs. WRF shows higher wind speeds
of about 0.5 m/s, but this could be expected given the previously
reported bias of -0.42 m/s for StriX wind speed retrieval compared to
FINO3 measurements (Badger et al. 2023).

The speed-up effects visible at DanTysk at the height of 10 m vanish
higher up in the atmosphere at 100 m (Figure 7a,b), which is slightly
above hub-height (88 m).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image18/image18}

\end{center}
\end{figure}

\emph{Figure 6 Case A StriX 20 July 2022 at 19:38 UTC. Wind speed from
WRF at 19:40 at 10 m height for (a) the entire German Bight, and (b)
zoom into the area of interest around DanTysk as indicated by the red
rectangle in a). c) the corresponding SAR scene from StriX with the
acceptance boxes used (section 3.4). The pink star indicates the
location of FINO3, and the orange boxes outline the wind farms. The red
stars in b) labeled U1 (Upstream 1), U2 (Upstream 2), F1 (Farm), W1
(Wake 1), and W2 (Wake 2) denote the locations of the vertical profiles
in Figure 8.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image19/image19}

\end{center}
\end{figure}

\emph{Figure 7 Same as Figure 6a,b, but for 100 m height.}

In order to better understand how the wind profile changes with height,
vertical wind speed profiles are interpolated and extracted from WRF
every 10 m at five locations around DanTysk. These locations are
indicated in Figure 6b. The wind speed profiles are shown in Figure 8
for the different scenarios WRF+FITCH1, WRF+FITCH0.25, WRF+EWP, and WRF.
The profiles already differ ahead of the farm (Figure 8a), likely due to
wake effects of other farms during the simulation. As the wind
approaches the farm, the wind profiles do not change much (Figure 8a).
However, in the farm (Figure 8b), the profiles are significantly altered
by the presence of the farm for WRF+FITCH1 (blue), WRF+FITCH0.25
(orange), and WRF+EWP (red). While WRF+EWP shows an almost linear shift
of the profile towards lower wind speed from the rotor top to the
ground, WRF+FITCH1 and WRF+FITCH0.25 show a pronounced reduction in the
rotor area and even an acceleration close to the ground. This changed
profile persists in the wake (Figure 8c,d), but it becomes slightly more
pronounced for all scenarios with wind farms. The reason for the
acceleration effect in WRF+FITCH is the added TKE in the WFP that mixes
higher momentum downward in a stable atmosphere. This can be concluded
from WRF+EWP not showing this behavior and from WRF+FITCH0.25 showing
smaller accelerations than WRF+FITCH1, where more TKE is added.

The differences in the modeling of TKE can be clearly seen in Figure
8e-h. In WRF+FITCH1 within the wind farm area (Figure 8f), a large
amount of TKE is added. Note that the higher TKE amounts at Upstream 2
are due to averaging points inside and outside of the farm to
interpolate to that point. The TKE levels quickly recline in the wake
(Figure 8h).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image20/image20}

\end{center}
\end{figure}

\emph{Figure 8. WRF results 20 July 2022 at 19:40 UTC (a,b,c,d) Wind
speed (WS), (e,f,g,h) turbulent kinetic energy (TKE) and (i,j,k,l)
potential temperature (PT) profiles from different WRF scenarios
(colors) at the locations (a,e,i) Upstream 2 (U2), (b,f,j) Farm (F1),
(c,g,k) Wake 1 (W1) and (d,h,l) Wake 2 (W2) as shown by different line
styles for 19:40 (same as Figure 5). The locations are denoted in Figure
6b. Note that Upstream 1(U1) is repeated at each subplot for comparison.
The gray area denotes the rotor area of DanTysk.}

The reason for differences in the wind farm wake (deficit or
acceleration) within the larger domain of several wind farms (Figure 6a)
might be related to an interplay of stability and enhanced mixing due to
added TKE. As an example, for the wind farms located southwest of
DanTysk at around 6\selectlanguage{ngerman}°E and 54.4°N, the wind speed deficit in the wake is
visible. Investigation of the vertical profiles (not shown) indicate a
low-level inversion is moved downwards by these wind farms while it is
moved slightly upwards in the focus area around DanTysk (Figure 8i-l)
for WRF+FITCH. WRF+EWP consistently moves the inversion upwards for the
wind farms. This lower inversion at around 6°E and 54.4°N suppresses
further downward mixing of higher momentum from above to 10 m. In
addition, higher ambient TKE is present close to the ground compared to
DanTysk (Figure 8e,h). This interplay between stability, inversion
height, and mixing and their effect on acceleration should be
investigated more in-depth in further studies to provide more insights
into the mechanisms of accelerations and decelerations in the wake of
wind farms close to the ground.

The WRF model temporal results on wind speed and wind direction during
several hours shown in Figure 5 include the overpass time of the
Sentinel-1 scene recorded at 17:18 UTC. The wind directions at this time
compare well for WRF and FINO3 data. The wind speeds are 2 to 3 m/s
lower in WRF than FINO3 data.

The maps of wind speed from WRF and Sentinel-1 are displayed in Figure
9. The Sentinel-1 image contains five wind farms. The interaction with
each of them offers an insight into the local boundary layer conditions.
We can observe both acceleration and deceleration from the different
wind farms under similar conditions. At DanTysk, the WRF model shows
deceleration, while the Sentinel-1 scene shows acceleration. The reason
for the discrepancy is unclear but likely linked to difficulties of WRF
modeling the correct stability conditions, which may also be related to
the observed discrepancy in the wind speed profile observed at FINO3
during that time (Figure 5). The color-stretched wind field map (Figure
3d) contains many wave patterns, both parallel and perpendicular to the
wind direction. Going into further analysis would be interesting, but it
is beyond our scope.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image21/image21}

\end{center}
\end{figure}

\emph{Figure 9 Case A Sentinel-1 20 July 2022 at 17:18 UTC. Wind speed
from WRF at 17:20 at 10 m height for (a) from WRF for the entire German
Bight and (b) Sentinel-1 full scene wind field map. (c, d) zoom into the
area of interest around DanTysk as indicated by the red rectangle in
(a). (c) WRF model with acceptance boxes used in the discussion, and (d)
Sentinel-1 with the acceptance boxes used for section 3.4. The pink star
indicates the location of FINO 3, and the orange boxes outline the wind
farms.}

\textbf{4.2.2 Case B: 2 August 2022 from StriX and WRF}

During Case B, atmospheric conditions were comparable to conditions
during Case A. The temperature gradient between the sea surface and 29 m
height was slightly higher (Figure 4a), and the ambient upstream wind
speeds at 10 m height were 6 m/s, slightly higher than for case A, which
was around 5 m/s (Figure 4b). The wind direction was slightly more
southern (Figure 4e) compared to Case A. Consequently, DanTysk is in the
very far wake of wind farms located southwest at 19:40 UTC as seen from
Figure 11a. Compared to the FINO3 measurements (Figure 10), the wind
speed profile is reasonably well captured in the WRF+FITCH1 simulations,
especially from 16:00 UTC onwards. The wind direction at higher
altitudes is slightly off from 19:00 UTC onwards. However, FINO3
observations at 60.55 m height were also missing during that period,
which makes it difficult to conclude.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image22/image22}

\end{center}
\end{figure}

\emph{Figure 10 As Figure 5, but for 2 August 2022 at19:40 UTC.}

Similar to Case A, a speed-up effect is visible at 10 m height for
WRF+FITCH1 (Figure 11b) but vanishes higher up in the atmosphere (Figure
12a). The location of the area with acceleration agrees well in WRF and
StriX (Figure 11b,c), considering the low 2-km horizontal resolution in
WRF. Again, the wind speed of StriX is consistently lower than the wind
speed in WRF by about 2 m/s, which is considerably more than for Case A.
Overall, the speed-up effect is slightly larger for Case B, compared to
Case A (Figure 6a) which is however not reflected in WRF (Figure 18) and
will be discussed in section 4.2.4.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image23/image23}

\end{center}
\end{figure}

\emph{Figure 11 As figure 6, but Case B for StriX 2 August 2022 at 19:37
UTC. Wind speed from WRF at 19:40.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image24/image24}

\end{center}
\end{figure}

\emph{Figure 12. As Figure 11a,b, but for 100 m height.}

The profiles upstream, in the farm, and downstream for wind speed and
TKE (Figure 13) are comparable to those for Case A (Figure 8), except
that overall higher wind speeds and higher background turbulence exist
during Case B. The potential temperature profiles differ, however,
between cases A and B. In case A, the wind farm elevated an existing
inversion compared to the scenario without wind farms (Figure 8i-l). In
case B, a weak inversion present upstream in the scenarios with wind
farms (blue, orange, and red dotted lines in Figure 13i-l) was removed
consistently in all simulations with wind farms. In the scenario without
wind farms (green), no inversion was present upstream, which indicates
that a wind farm effect of an upstream farm triggered the inversion.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image25/image25}

\end{center}
\end{figure}

\emph{Figure 13 As figure 7, but for 2 August 2022.}

In the offshore wind farm area southwest of DanTysk at around 6\selectlanguage{ngerman}°E and
54.4°N, similarly to case A, also in case B, a classical wake effect at
10 m with lower wind speed was observed (Figure 11a). Again, in this
case, the wind farm effect from WRF+FITCH led to the development of an
inversion in the lower part of the rotor, which suppressed the mixing of
higher momentum from aloft (not shown). This highlights the importance
of the interplay of mixing, stability, and inversion height for wind
speed wake effects close to the ground which should be investigated in
more detail in further studies.

\textbf{4.2.3 Case C: 14 August 2022 from StriX and WRF}

While Cases A and B were quite similar, Case C differs in multiple ways.
The most crucial difference is that the wind direction was from land
(Figure 4e, Figure 14). This explains the stronger temperature gradient
(Figure 4a) present at FINO3 due to warmer air being advected from the
land and the smaller upstream wind speed compared to the other cases
(Figure 4c). The WRF simulations agree reasonably well with wind speed
and wind direction profiles at FINO3 (Figure 14). However, at the time
of the StriX scene, wind speeds are overestimated in WRF.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image26/image26}

\end{center}
\end{figure}

\emph{Figure 14 As figure 5, but for 14 August 2022.}

The poor performance of WRF at the time of the StriX scene is reflected
in the absence of any acceleration effect at DanTysk (Figure 15a,b),
while it is clearly visible in StriX (Figure 15c). At 100 m height
(Figure 16 a,b) classical wake deficit is found in WRF. However, earlier
at 19:00 UTC, when FINO3 and WRF wind speed profiles still agreed well
(Figure 14), an acceleration effect is present in WRF+FITCH1 (Figure
17). This highlights the challenges of WRF simulations to model the
background meteorology correctly. With the time shift of 35 minutes from
19:35 UTC to 19:00 UTC, wind acceleration effect is found\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image27/image27}

\end{center}
\end{figure}

\emph{Figure 15 As figure 6, but for Case C for StriX 14 August 2022 at
19:35 UTC. Wind speed from WRF at 19:40.}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image28/image28}

\end{center}
\end{figure}

\emph{Figure 16 As figure 15a,b, but for 100 m height}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image29/image29}

\end{center}
\end{figure}

\emph{Figure 17 As figure 15b, but for 19:00 UTC.}

\textbf{4.2.4 Discussion on the case studies and SAR wind mapping}

To summarize the acceleration effects in the different scenarios, Figure
18 shows the wind speed deficit derived from the mean over the
acceptance regions in Sentinel-1, StriX, and WRF (section 3.4).
WRF+FITCH1 (orange) is the scenario that best captures the speed-up
effect of DanTysk. However, compared to StriX the accelerations are
smaller, likely due to coarser resolution in WRF, which cannot capture
all details. While WRF+FITCH0.25 (green) can also capture the
acceleration, the magnitude is further underestimated. Ali et al. (2023)
found similar effects in their simulations. They concluded that the
speed-up effect is directly proportional to the added TKE amount if the
turbine-induced thrust is confined to the rotor area. WRF+EWP (red)
cannot capture the acceleration due to the missing enhanced mixing,
which seems to be a crucial mechanism for speed-ups. In addition, the
vertical wake expansion in EWP reduces the wind speed over the rotor
plane and below, directly contradicting the speed-up effect. Shepherd et
al. (2020) pointed out that the presence and absence of these speed-up
effects directly affect the near-surface climatic impacts of wind farms
and are thus important to study.

From the scenario without wind farms (WRF, purple), it is clear that the
variability in the background meteorological conditions also plays a
role, as in Case B, a slight acceleration is derived in the scenario
without wind farms. This variability in the background meteorological
conditions makes it difficult to capture all speed-up effects with WRF
at the correct timing, as discussed in section 4.2.3 and shown in
``C19:35'' and ''C2(19:00)''. The comparison for Sentinel ''A(17:18)'',
highlights that also the region of the acceleration cannot always be
fully captured with 2 km resolution. The area with speed-up effects in
Figure 9 is very localized in the North of DanTysk and outside the
acceptance region, therefore not reflected in Figure 18. Future studies
should apply simulations with 1 km horizontal resolution to see if the
acceleration can be captured even better.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image30/image30}

\end{center}
\end{figure}

\emph{Figure 18 Wind speed deficits at 10 m height based on the
acceptance regions for Sentinel-1 Case A at 17:18 UTC, StriX Cases A
(19:38 UTC), B (19:37 UTC) and C (19:36 UTC), and with WRF at the
nearest time. C2 shows WRF time shift for Case C to 19:00 UTC. SAR in
blue and WRF in other colors for the different scenarios for WRF.}

In summary, WRF+FITCH1 is capable of reproducing the wind acceleration
downstream of DanTysk, as observed in StriX in Case A, Case B, and Case
C. For Case C though, only by shifting the time 35 minutes. It is not
unusual that timing is not precise in mesoscale models. Other modeling
studies also confirm wind acceleration downstream of large wind farms
(Lars\selectlanguage{ngerman}én and Fischereit, 2021; Vanderwende et al., 2016; Yang et al.,
2014). In Case A, Sentinel-1 shows wind acceleration, while WRF shows
wind speed deficit. This discrepancy is likely due to the spatial
resolution in the model not resolving the flow with sufficient detail.
It might also be due to a time issue.

The percentage of cases at DanTysk with wind speed surplus in Strix is
48\%, and in Sentinel-1 is 34\%. The percentages are comparable but are
surprisingly high. Djath et al. 2018 reported 25\% cases with wind speed
surplus at Alpha Ventus. Alpha Ventus is located 100 km south of
DanTysk. DanTysk has a nameplate capacity of 288 MW from 80 turbines,
each 3.6 MW, covering an area of 66 km\textsuperscript{2}. In
comparison, Alpha Ventus has a nameplate capacity of 60 MW from 12
turbines, each 5 MW, covering an area of 4 km\textsuperscript{2}. Alpha
Ventus is located in the midst of several wind farms, a large cluster,
and 45 km from the nearest coast. Their energy densities differ. DanTysk
has 4.4 MW/km\textsuperscript{2} and Alpha Ventus 15.0
MW/km\textsuperscript{2}. The many differences between the two German
North Sea wind farms may explain the variation, such as distance to
coast, their latitude, wind farm sizes, and energy densities.

The model results confirm that wind speed surplus may exist near the
surface downstream of wind farms. Nevertheless, it may be the case that
other things impact the SAR-derived wind speed downstream of the wind
farm than the wind speed alone. The backscatter coefficient
(\selectlanguage{greek}\emph{σ\textsubscript{0}}\selectlanguage{english} ) generally increases due to higher
instantaneous wind stress. In case disturbances at the sea surface
affecting \selectlanguage{greek}\emph{σ\textsubscript{0}}\selectlanguage{english} but unrelated to the instantaneous
wind stress happen, the wind speed retrieved from the geophysical model
function (GMF) will be biased. Capillary waves suppressed due to algae
bloom, or oil spill will cause a negative wind speed bias. Conversely, a
positive wind speed bias will occur if more and steeper capillary waves
are present due to other factors, e.g., shallow water with currents in
tidal zones and near reefs. Concentrated bright spots from ships, oil
platforms, or other structures could also increase backscatter.

Djath et al. 2018 suggest that \selectlanguage{greek}\emph{σ\textsubscript{0}}\selectlanguage{english} increases
behind the wind farm due to mechanical turbulence from the wind farm,
increasing the downward momentum and influencing the ocean surface. With
this assumption, the traditional use of GMF to retrieve wind speed will
be compromised, and misinterpretation could be done, according to Djath
et al. 2018. Our interpretation is that the TKE mixes higher wind speeds
from the wind profile aloft down to the surface. Thus, it appears likely
that the higher \selectlanguage{greek}\emph{σ\textsubscript{0}}\selectlanguage{english} values in the wake area are in
balance with the instantaneous wind stress near the surface. The
WRF+FITCH1 case studies support it. The wind speed is accelerated near
the surface in WRF as well as in SAR. Our take is that the GMF can be
applied though with more uncertainty than in free stream conditions due
to the waked wind profile. Vertical extrapolation of the SAR-derived
wind speeds to higher heights, applicable in free stream conditions
(e.g., Badger et al. 2016), would require another extrapolation method
in the waked region.

It will be valuable to do further investigations to ensure a more
complete view of SAR-derived wind farm wind speed surplus comparing
other wind farms, other observation sources, and higher-resolution
modeling of wind farm flow to fully understand the nature and mechanisms
of the wind speed surpluses. The higher resolution could be WRF at 1 km.
It could be large-eddy simulation focused on the blockage, turbine wake
within the wind farm, and wind farm wake some distance downstream. The
StriX data lend themselves for validation at very high spatial
resolution compared to Sentinel-1. With the several StriX missions
planned, it will be possible to map SAR wind speed with very high
spatial resolution and temporal resolution.

\textbf{Conclusion}

Wind speed surplus downstream of an offshore wind farm in the North Sea
has been investigated from StriX and Sentinel-1 SAR wind fields and WRF
modeling with different wind farm parametrizations. It is
counterintuitive that higher wind speeds would prevail downstream of
operating wind farms. Based on detailed modeling of the turbulent
kinetic energy (TKE) and the observations from SAR satellites, the
conditions of wind speed surplus at 10 m height above the sea surface
confirm the occurrence. At hub height, however, wind speed deficits are
present even in those cases with wind speed surpluses at the surface.
According to the SAR-based results, 48\% from StriX and 34\% from
Sentinel-1 show wind speed surplus. An idea for the future could be to
run all the SAR cases in WRF, to gain further statistical evidence from
different sources on wind farm wake deficit. Stable stratification and
upstream wind speeds below 7 m/s prevail for most of the wind speed
surplus cases, yet unstable conditions and wind speeds above rated are
noticed. Further investigation is recommended to verify wind speed
surplus near the surface downstream of offshore wind farms.

\textbf{Data availability}

StriX SAR scenes are the property of Synspective Inc. and can be
purchased upon request. Sentinel-1 SAR scenes are available from the
Copernicus Open Access Hub (https://scihub.copernicus.eu, accessed on
27-10-2023), and meteorological observations from the FINO3 research
platform are available from the FINO Datenbank
(https://www.fino3.de/de/, accessed on 27-10-2023). ERA5 data is
available from
(https://cds.climate.copernicus.eu/cdsapp\#!/dataset/reanalysis-era5-single-levels?tab=form,
last access 27-10-2023). The WRF model is available from
https://github.com/wrf-model/WRF10-2023.

\textbf{Author contributions}

CBH and MB writing, reviewed and edited. JI processed StriX data and
prepared the plots and text on StriX and Sentinel-1. JF ran the WRF
model, prepared the plots, and wrote about the WRF results, AF processed
Sentinel-1 data. KD contributed to Sentinel-1. CBH and MB project
administration. All contributed to the text and agreed on the final
version.

\textbf{Funding:} This research was funded by Synspective Inc. and
conducted through a bilateral project between Synspective Inc. and the
Technical University of Denmark. Part of the funding was by the
Independent Research Fund Denmark through the ``Multi-scale Atmospheric
Modeling Above the Seas'' (MAMAS) project (nr. 0217-00055B)

\textbf{Conflicts of Interest} Synspective Inc. participated in the
design of the study. This included the collection, analysis, and
interpretation of data, writing of the manuscript, and decision to
publish.

\textbf{Data Availability Statement}

Strix SAR scenes are the property of Synspective Inc. and can be
purchased upon request. Sentinel-1 SAR scenes are available from the
Copernicus Open Access Hub (https://scihub.copernicus.eu, accessed on
27-10-2023), and meteorological observations from the FINO3 research
platform are available from the FINO Datenbank (https://fino.bsh.de,
accessed on 27-10-2023).

\textbf{Disclaimer}

Publisher's note: Copernicus Publications remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.

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