Interpretation and estimation of the local wind climate
J. W. Verkaik *
A. Smits
ABSTRACT
Methods to transform the wind speed from one to another location are discussed.
A two-layer model of the planetary boundary layer with homogeneous wind
speed profiles is compared to internal boundary layer models. The roughness
lengths for the two layers are found from a simple footprint analysis and a landuse map. The two-layer model handles multiple roughness transitions in a better
way. Even after a large roughness transition the surface wind is equally well
described by the two-layer model. The input wind speed measurements are
corrected for site-specific exposure by analysing the gustiness as function of
wind direction.
1.
INTRODUCTION
Knowledge of the wind climate at specific locations is of vital importance for
risk assessment, engineering, and wind power potential assessment, for example.
Generally this local climatology is only available at meteorological stations. In
the Netherlands these stations may be at a distance of 30–40 km from the place
of interest. Over large water bodies these distances may be even larger. Because
of changing roughness conditions the wind climate at the place of interest will
be different from that of the meteorological station and a transformation method
is needed.
The development of the wind speed profile after a change in roughness can be
modelled by internal boundary layer models (IBL-models, Garratt 1990). Most
IBL-models can handle only one transition [see Cook (1997) for an exception].
However, in natural terrain the scale of heterogeneity is small and many
transitions occur over short fetches. In heterogeneous terrain regionalisation of
wind speed can be used as alternative to IBL-models. Regionalisation implies
that the wind speed at an elevated level is computed from the surface wind speed
with consideration of the surface roughness at the measuring location. The
assumption then made is that the wind speed at elevated levels is less influenced
by local terrain features than the surface wind. If so, the wind at this level is
better suited for interpolation than the surface wind or can even be considered
horizontally constant. The inverse procedure yields the surface wind speed at the
location of interest, considering the roughness at that location. Doing
translations over small distances ( 5 km) only the surface layer needs to be
*
Royal Netherlands Meteorological Institute (KNMI), Climatological Services, PO Box 201,
3730 AE De Bilt, the Netherlands.
considered, otherwise the full depth of the planetary boundary layer (PBL) may
need to be modelled. Modelling the wind speed after a roughness change
without actually using an IBL-model was also done successfully by Bergström
et al. (1988). Examples of regionalisation of more meteorological parameters in
non-neutral conditions can be found in De Rooy (1995) and Hutjes (1996).
The present study is part of the KNMI Hydra project with the goal to assess the
extremes of the Dutch wind climate, with focus on wind over water. We will use
the two two-layer model of Wieringa (1986, henceforth W86) to estimate the
wind speed over lake IJsselmeer and river deltas in Zeeland. W86 validated his
model for seasonal and yearly averages. In the present project we will test the
method with individual cases as well. The required input data must relatively
simple as we want to run the model 20–30 years back in time. We will show that
the two-layer model mimics the IBL development very well. Our focus will be
on the evaluation of the surface roughness. Finally some test results are
presented.
Statistical analyses of extreme wind speeds are not the subject of this paper.
These were done by Rijkoort (1983), Rijkoort et al. (1983), and Wieringa et al.
(1983). Recently Smits (2001) performed an update and re-evaluation of the
statistics.
2.
TWO-LAYER MODEL OF THE PBL
2.1 THEORY
The model of W86 comprises two layers. In the lower layer, the surface layer,
Monin-Obukhov theory is used (MO-theory; Obukhov, 1971; Businger et al.
1971). Using the local roughness length the wind speed at the top of the lower
layer can be computed from the measured wind speed. This is done using the
logarithmic wind speed profile, strictly speaking only valid in neutral and
homogeneous surface layers. In that case the wind speed U as function of height
z is given by (Tennekes, 1973)
( 1)
U   u*   ln z z0  ,
where the von Kármán constant  = 0.4 (Frenzen et al, 1995), and u* is the
friction velocity, related to the momentum flux ( u*2   u' w' , u' and w' are
turbulent fluctuations of the horizontal and vertical wind speed, respectively). In
surface layers over homogeneous terrain z 0 is well defined and u* is constant
with height.
In the second layer the geostrophic drag relations apply:
 U  U macro u*  ln zf u*  A ,




( 2)
Vmacro u*   B .
Here A en B [-] are stability dependent parameters (A = 1.9 and B = 4.5 in
.  10 4
neutral conditions). The Coriolis parameter f [Hz] equals 2sin   11
Hz, where  is the angular velocity of the Earth's rotation and  is the latitude.
U macro and Vmacro are the components of the macrowind. U macro is parallel to
the surface wind, Vmacro is perpendicular to U macro . With U macro in east
direction, Vmacro points north on the Northern Hemisphere. Matching the two
layers at the mesolevel [Eqs. (1) and (2)] one can deduce
Umacro  u*  lnu* fz0   A ,
( 3)
Vmacro  Bu*  .


2.2 APPLICATION
Transformation of the wind speed from one to another location is done by
interpolating the macrowind, which is assumed to vary only smoothly over large
distances. The macrowind can be computed from the surface wind in two steps.
In the first step the mesowind U meso is computed using the local roughness
length z 0l from the surface wind speed U s measured at height z s [Eq. (1)],
( 4)
U meso Us  ln( z m z0l ) ln ( zs z0l ) .
The local roughness length is derived from gustiness analysis (see Section 6,
Wieringa, 1976; Verkaik, 2000). For the mesolevel or blending height z m
Wieringa chose 60 m. He argued that within a few kilometres the wind speed at
the blending height could be considered constant. Doing translations over small
distances, less than a few kilometres, only the first step of W86 needs to be
considered.
In the second step the macrowind is computed from the mesowind. The
roughness length used in this step is computed from land-use maps. W86
calculated the meso roughness length z 0 m by averaging drag coefficients at the
mesolevel on a spatial resolution of (5 km) 2. To this z 0 m he also added the drag
caused by relief. From the meso roughness he computed the area-averaged
friction velocity u* m using Eq. (1):
( 5)
u*m  U meso ln z m z 0m  .
Now the macrowind can be computed from Eq. (3). This macrowind is often
called the geostrophic wind. However, the neutral and barotropic approach
adopted here is rather simplistic (Luthardt & Hasse, 1981; Marsden, 1987). With
the macrowind available the inverse process can be used to compute the
mesowind at any location.
W86 averaged night- and daytime cases and so most stability effects were
diminished. With the focus on high wind speed cases effects of stability may be
insignificant in the present project as well.
3.
IBL-DEVELOPMENT
After a change in surface roughness the momentum flux, wind speed and wind
direction will adjust so that a new equilibrium state is established. Since the
perturbation of the profile originates from changes at the surface, internal
boundary layers will develop from the surface. Very close to the surface a new
equilibrium boundary layer (EBL) will develop. Higher up in the surface layer
the flow will remain undisturbed, except for a minor streamline displacement
(see Figure 1). Walmsley (1989) tested several formulae for the IBL-height  as
function of distance x with data and found that Panofsky et al.’s (1984) formula
fitted best:
( 6)
 
x  
   ln  1  1 b .
z0
 z0 
z0


. (the ratio of the standard deviation of vertical wind speed
Here b   w u*  13
fluctuations and the friction velocity), according to Panofsky et al. (1984). For
z 0 they used the roughness length after the transition, but other argue that it
should be a combination of the roughness length before and after the transition,
or the largest of both (Jackson, 1976).
For the smooth-to-rough transition the IBL grows slightly faster than for the
rough-to-smooth transition. The height-to-fetch ratio approximates 1/10 for both
cases (Shir, 1972; Deaves, 1981). The EBL-height is about one tenth of the IBLheight. Taylor (1987) argues that after 10 km homogeneous fetch surface
friction velocity should be within 5% of its equilibrium downstream value.
Adjustment of the wind direction takes place only after 100 km homogeneous
fetch (Jensen, 1978).
A simple and often encountered concept is illustrated in Figure 2. The PBL
comprises two layers, one still adapted to the upstream, and a second, close to
the surface, fully adapted to the downstream terrain. Equilibrium profiles for
both surfaces are linked together at the IBL-height, which is given by some
empirical formula. As no transition layer is included, there is a discontinuity in
stress at the IBL-height.
4.
IBL-GROWTH IN THE TWO-LAYER MODEL
IBL-models require well-defined roughness transitions, one upstream value and
one downstream value. In many parts of the world multiple transitions occur at
short fetches. Therefor, the validation of IBL-models has mainly been restricted
to short fetches. Experiments with multiple IBLs are rare (Deaves, 1981). Some
models indicate that the second IBL grows faster in case the surface roughness
is returning to its initial value (Duijm, 1983). As a result major simplifications
are necessary when using IBL-models.
The two-layer model does not include an IBL-model. However, after a
roughness transition the local- and the meso roughness used will be different as
a result of the different length scales of their footprint (see Section 5). This way
the two-layer model reproduces the typical IBL wind speed profile as in Figure 2.
However, the knee in the profile is fixed at the mesolevel, while the roughness
lengths associated with the two parts of the profile are constantly changing with
fetch. In the simple IBL-model of Figure 2 the knee in the profile, the IBL-height,
climes with fetch, while the roughness length remain constant.
Now we will compare the development of the 10-m wind from the two-layer
model with that of IBL-models. We consider the sea–land transition where the
land roughness length is 0.1 m. We compare W86 with several IBL-models:
K2000 (Kudryavtsev et al., 2000), WAsP (Troen et al, 1989), Coast (Van Wijk
et al., 1990), and a small-scale model by Townsend (1965). For W86 and
Townsend's model the roughness length of the sea is fixed at 0.001 m, the others
models compute the drag of the sea from the wind speed. K2000 and Coast are
used in near-neutral mode, the other models are neutral by themselves. The wind
at 10 m over sea is fixed at 10 m/s. The results are plotted in Figures 3 and 4. The
upper figure shows the sea–land transition, the lower figure the land–sea transition
where the wind speed over land is fixed at 7 m/s. Coast is developed for onshore
flow only.
Apart from Townsend's model, all model show a similar adjustment of the 10-m
wind to the new surface roughness. The rate of adjustment, both at small and
large fetches, in the two-layer model depends on the length scales chosen for the
local- and meso roughness and the mesolevel. In fact, the non-neutral version of
K2000 with unstable flow over sea and stable flow over land corresponds even
better to the two-layer model. It would be possible to tune the footprint length
scale for closer correspondence to one of the IBL-models, but we feel that this
would not lead to significant improvement of the model. So we conclude that
using the two-layer model with different footprint length scales for the localand meso- roughness length will not lead to large errors in the surface wind
speed in case of large roughness transitions.
5.
ROUGHNESS MAP OF THE NETHERLANDS
5.1 SPATIAL DATA ON LAND-USE
The surface roughness is assessed form a land-use map LGN3+ (Thunnissen,
2000). This is a raster file covering the whole of the Netherlands with a
resolution of 25 m. To each pixel a land-use class is assigned. A number of 40
classes are used in LGN3+. We assigned a roughness length to each class.
Outside the Netherlands a constant roughness of 0.24 m is adopted.
5.2 SURFACE ELEVATION
Surface elevation is assessed from the GTOPO301 database. In this database the
surface elevation is given at a resolution of (1/120)° ( 1 km). This grid has been
interpolated using spline and exported into new grid on a 500-m resolution in
local (X,Y)-co-ordinates. Height differences are assessed by comparing the
height at point (X,Y) with the neighbouring points on the 500-m grid. The
roughness length due to geographic relief z 0 H is computed from
( 7)
z  0.2  H 2 L ,
0H
where H is the height difference over distance L (Agterberg et al., 1989).
1
http://edcdaac.usgs.gov/gtopo30/gtopo30.html
5.3 AREA-AVERAGED SURFACE ROUGHNESS
A suitable manner to aggregate surface roughness is to average the drag
coefficients at the blending height (Claussen, 1990). This method has also been
used by in W86 and gives stronger weight to the larger roughness in the
averaging domain. The drag coefficient is defined as C d   u * U and using
Eq. (1) it can be expressed as
2
( 8)
.
C   ln z z
2
d


0

5.4 FOOTPRINT APPROXIMATION
The footprint is the surface area contributing to the flux or concentration of an
atmospheric entity at a certain point. The footprint depends on the entity under
consideration, the atmospheric stability and height (Schmid, 1994; Horst, 1999).
However, here we will make a simple approximation to the footprint using the
following procedure. The area surrounding the evaluation point is split into 72
sectors 5° wide. For each pixel i in sector j the drag coefficient at the blending
height is determined from the equation
2
( 9)



Cd ,i  
 .
 ln z bh z 0 
The roughness length is determined from LGN3+. The drag coefficient of water
is wind speed dependent and will not be added to the total drag at this stage. A
weighted average of Cd , i is computed using the weighting function
( 10)
W x i , D  exp x i D ,
where x is the distance from the pixel to evaluation point. The length scale D
determines the extent of the footprint. For the local roughness D = 600 m is
used; for the meso roughness D = 3000 m is used. So the average drag of sector j
coefficient given by
( 11)
C'd , j   W xi , D  Cd ,i  W xi , D .
i
i
Also the fraction the surface covered by water is determined ( f ' w , j ). Only pixels
up to a distance of 3D are considered. Now the direction-dependent C'd , j and
f ' w , j are smoothed using a weighted moving average:
k 3
( 12)
C d , j   w k  C' d , j k , and
k 3
k 3
f d , j   w k  f 'd , j  k ,
k  3
. ,018
. ,0.22,018
. ,013
. ,0.08 .
where w  33  0.08,013
( 13)
5.5 DRAG RELATION FOR WATER
The total drag at the evaluation point for wind directions in sector j is computed
from
( 14)
CTd  Cd , j  f w , j  C d ,water  U ,
where the drag of water can be expressed using Eq. (8) and the Charnock
relation
( 15)
z 0    u*2 g ,
where g is the acceleration of gravity (9.82 m/s2) and for  the value 0.017 is
used (Charnock, 1955; Garratt, 1977).
6.
VALIDATION OF THE TWO-LAYER MODEL
6.1 WIND DATA
The wind data used in this study is ‘potential’ wind data. This means that the
mesowind is calculated from the measured wind speed using the local surface
roughness, and from this the wind speed at z ref  10 m over a hypothetical
terrain with roughness length z 0 ref  0.03 m is computed ( z 0 ref  0.002 m over
water). The latter is called the potential wind speed ( U p ). This approach has
been adopted from Wieringa (1976). The local roughness length is derived from
gustiness analysis, but in this study a different gustiness model has been used
than Wieringa (1976) did. Here we use Beljaars’s (1987) gustiness model which
is more flexible in modelling various measuring chains (Verkaik, 2000).
The gust-factor is defined as G  U max U , and using Eq. (1) can be written as
( 16)
U
U
U
 U
U

G  max  1  max
 1  u max
.
U
U
u*
u
ln z z 0 
In neutral conditions the ratio of the standard deviation of the horizontal wind
speed fluctuations to the friction velocity,  u u* , equals 2.2. However, due to
the inertia in the measuring system the recorded  u will be reduced by a factor
 . The normalised gust,  U max  U  u  u x , depends on the inertia of the
measuring system as well. In addition  and u x depend on the length of the
sampling interval. If these factors are known the relation between the
distribution of G and z 0 is given by
( 17)
G  1    u x  0.88 ln z z0  .
In Figure 5 the expected average of the product u x is plotted for two different
heights as function of the averaging time-scale (sampling interval 1 hour). G and
z 0 are analysed as function of wind direction.
The potential wind speed can be interpreted as a wind speed that has been
stripped from very local roughness features. This makes the potential wind
speed better suited for comparison with wind speed measurements from nearby
station than the measured wind speed itself. In the computation of the potential
wind also changes of roughness in time are corrected for. The gustiness derived
roughness length correlates strongly with the local roughness length from the
land-use map. However, using the gustiness analysis obstacles sheltering the
measuring location not resolved by the roughness map are more easily detected.
Because the potential wind speed is computed from the mesowind with a fixed
roughness length, it is very easy to compute the mesowind from the potential
wind speed. The mesowind is the product of U p and the factor
( 18)
U m U p  ln z m z 0ref  ln z ref z0 ref  ,
which equals 1.31 with z m  60 m over land and 1.21 over water.
6.2 TEST RESULTS
Based on the measurements of 24 stations, wind speed estimates for two other
station locations are compared with the measured wind speed in Figure 6. Data
from the year 2000 are used. Station Wijdenes is at the edge of lake IJsselmeer.
Leeuwarden is a military airport. For these two stations the estimates are rather
good. Several other stations show some bias, probably as a result inaccurate
roughness estimates. The error bars indicate the standard deviation of the scatter.
The scatter is caused partially by imperfect interpolation methods and probably
by stability effects as well.
7.
CONCLUSIONS
The two-layer model with different roughness footprints for both layers mimics
the development of the IBL very well. The two-layer model, however, can
handle ‘real’ roughness maps with an unlimited number of roughness
transitions. Future research will focus on the optimisation of the interpolation
procedure, the roughness map and the inclusion of stability effects, as this will
probably reduce the scatter in the estimates. At 3 EACWE more results will be
presented.
The two-layer model will also be used in combination with numerical weather
forecasts to give high-resolution wind speed forecasts. These will be used for the
prediction of wave height over the coastal region and lakes in the Netherlands,
but application in the field of wind energy is also an option.
Wind speed data, the roughness map and all documents on the KNMI-Hydra
project are available on http://www.knmi.nl/samenw/hydra.
8.
ACKNOWLEDGEMENTS
The present study is a contribution of KNMI to the Hydra-project supported for
RIKZ (Rijksinstituut voor Kust en Zee = National Institute for Coastal and
Marine Management and RIZA (Rijksinstituut voor Integraal Zoetwaterbeheer
en Afvalwaterbehandeling = Institute for Inland Water Management and Waste
Water Treatment). In the KNMI-Hydra project the hydraulic boundary
conditions are assessed for safety testing of the Dutch dams.
9.
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