## 1. Introduction

Measurements of the vertical flux of horizontal momentum are important for understanding the atmosphere and for testing models of the atmospheric flow over terrain. Methods of extracting the momentum flux from conically scanning Doppler lidars have been described before, for example, by Eberhard et al. (1989) and Gal-Chen et al. (1992). Techniques for analyzing lidar data are very similar to those applied to radars. Apart from the vertical flux of horizontal momentum, the vertical flux of turbulent kinetic energy can also be measured by adjusting the cone opening angle appropriately (Frisch et al. 1989). Also, a review covering other remote sensing techniques is given by Engelbart et al. (2007).

This study differs from the others by focusing on the lower 100 m of the atmosphere, using a focused continuous wave (cw) and a pulsed lidar, and by making comparisons with the sonic in situ momentum flux estimated up to 160 m above the terrain surface. In addition, a novel method to estimate the momentum flux is tested, which does not use the individual radial wind speeds but rather the entire Doppler spectrum.

The particular cw lidar investigated in this paper has been used for wind energy applications in several ways. First, the instrument has been used for wind energy resource and wind profile estimation over flat terrain. In general, it reproduces the wind speeds at all the investigated heights well (Smith et al. 2006), but it can experience problems when the distribution of backscatter is very uneven, as in the case of low clouds or fog (Courtney et al. 2008). In the marine boundary layer, the instrument has also been used for wind resource estimation (Kindler et al. 2007; Peña et al. 2009) and wind profile measurements (Peña et al. 2008) showing very reasonable wind data up to 160 m above the sea. For the conically scanning mode, which is used to derive the horizontal wind speed, horizontal homogeneity is assumed. However, for measurements in complex terrain this may not be a valid assumption, as pointed out by Bradley (2008), and according to Bingöl et al. (2009), the error committed on the wind speed may be up to 10% depending on the steepness and complexity of the terrain.

A second application of the instrument in wind energy has been the investigation of flow around a wind turbine. Harris et al. (2007) used the lidar to measure the gusts approaching the turbine, an application that potentially can be used for load alleviation by controlling the pitch of the wind turbine blades. Harris et al. (2006) presented a theoretical analysis of the potential benefits of this approach. In Bingöl et al. (2010) and Trujillo et al. (2010), the lidar was used to investigate the wake deficit behind the wind turbine, which is important for energy production estimates in wind turbine farms and the assessment of the extra structural loads experienced in such environments. Various parameters describing the lidar are shown in Table 1.

Other lidars are being developed for wind energy applications. The pulsed lidar WindCube from the French company Leosphere has turned out to be a precise anemometer (Courtney et al. 2008; Peña et al. 2010), and this lidar will also be used in this paper, as seen in Table 1 (also see Fig. 1).

A new instrument, called Galion, for wind profiling in the lower few 100 m of the atmosphere is being developed by Halo Photonics and SgurrEnergy, both from the United Kingdom. Other instruments are awaiting commercialization (Kameyama et al. 2007). Also, general atmospheric boundary layer Doppler lidars retrieve parameters relevant for wind energy, for example, the extent of low-level jets in the Midwest of the United States (Banta et al. 2006; Emeis et al. 2007) and velocity variance profile measurements (Pichugina et al. 2008; Lothon et al. 2006).

## 2. Momentum flux from scanning

*ϕ*≈ 30° is the half-opening angle of the conical scan, and

*θ*is the direction relative to some fixed horizontal direction. Here, it is convenient that

*θ*is the beam azimuthal direction relative to the direction of the mean wind. This angle changes in time

*t*as

*θ*= 2

*πt*s

^{−1}, thus, it makes one scanning revolution per second.

The focus distance *d _{f}* is the distance from the instrument to the focus of the laser beam. The focus height above the flat terrain is

*h*=

_{f}*d*cos

_{f}*ϕ*. The wind velocity field (

*u*,

*υ*,

*w*) =

**u**=

**u**(

**x**) is assumed statistically homogeneous.

*υ*cannot be distinguished (Smith et al. 2006).

_{r}*θ*:

*υ*

_{down}≡

*υ*(

_{r}*θ*= 0°) and the upwind

*υ*

_{up}≡

*υ*(

_{r}*θ*= 180°) directions one gets

### a. Filtering effect by the finite sampling volume of the ZephIR

It is important to point out that it is not enough to know the second-order turbulence statistics at the height of the measurements to calculate the effect of sample volume average on the measured radial velocity. The three-dimensional spatial statistics or, in other words, the spectral tensor **Φ*** _{ij}*(

**k**) (Pope 2000) is needed. It is difficult to measure directly, but fortunately, the spectral tensor model by Mann (1994) allows us to estimate

**Φ**(

**k**) over flat terrain. The model has three parameters

*α*ɛ

^{2/3},

*L*, and Γ, which can be fitted to empirical one-dimensional spectra of the velocity fluctuations (Mann 1998). Here,

*α*is the spectral Kolmogorov constant (Pope 2000), ɛ is the turbulent energy dissipation rate,

*L*is a length scale proportional to the size of the energy containing eddies in the turbulence, and Γ is a parameter describing the degree of anisotropy of the turbulence.

*l*is the Rayleigh length, which is measured in the direction along the beam.

*s*= 0 corresponds to the position of the focus (however, in Fig. 4

*s*= 0 corresponds to the position of the instrument). The weighting function

*ϕ*integrates exactly to 1. Using the same normalization of the Fourier transforms as in Mann (1994), it can easily be shown that the variance of

*υ*in Eq. (7) is

_{r}*d*

**k**means integration over all three wavenumbers from minus to plus infinity. For later use, the variance can also be written as

*k*=

_{n}**k**·

**n**is the wavenumber in the beam direction, and

*F*(

_{υr}*k*) is the spectrum of

_{n}*υ*along that direction.

_{r}*L*= 0.56 × min(

*z*, 60 m), where

*z*is the height above the ground, that is, the anisotropy is independent of height and the length scale increases linearly with height until 60 m, after which it remains constant. These are reasonable choices for the parameters Γ and

*L*over flat, homogeneous terrain, but other expressions have been given elsewhere (Mann 1998). The spectral scaling parameter

*α*ɛ

^{2/3}will not affect the relative attenuation. For the Rayleigh length

*l*in Eq. (8) we use

*l*= 2 m at a distance of 50 m, and it depends quadratically on distance (Smith et al. 2006). Equation (9) is evaluated numerically and shows the ratio between the

*filtered*and

*unfiltered*quantities in Fig. 2. It appears that

*u*′

*w*′〉, in good agreement with the fact that momentum is transported preferentially by the larger eddies. The prediction that

*F*(

_{υr}*k*) of radial wind fluctuations in the direction of the beam. Because turbulence tends to isotropy at small scales, one can assume that the spectra for the up- and downwind beam directions are almost identical at large

_{n}*k*. Here, according to Eq. (10), the filtering is strongest, so it can be expected that the filtering will remove the same amount of fluctuations for the up- and downwind beam directions. But, because

_{n}*relatively*more than

The momentum flux is reduced less than 20% below 60 m for the cw lidar but can be halved above 150 m. The estimate at these heights is very uncertain, mainly because the length scale of the turbulence can vary significantly, for example, with stability. It is, therefore, desirable to find a way to avoid the filtering altogether, which is the subject of the following section.

### b. Avoiding filtering by use of Doppler spectra

*δ*(·) is the Dirac delta function. The integral in Eq. (12) implies that the Doppler spectral density

*S*, corresponding to a radial velocity

*υ*, is a weighted count of all the positions

_{r}*s*along the beam where the actual radial wind velocity

*υ*(

_{r}*s*) is equal to

*υ*. The centroid, also known as the barycenter, of the instantaneous Doppler spectrum is, using the definition in Eq. (12), ∫

_{r}*υ*(

_{r}S*υ*)

_{r}*dυ*= ∫

_{r}*ϕ*(

*s*)

*υ*(

_{r}*s*)

*ds*in accordance with Eq. (7).

*δ*[

*υ*−

_{r}*υ*(

_{r}*s*)]〉 =

*p*(

*υ*;

_{r}*s*), where

*p*(

*υ*;

_{r}*s*) is the probability density function of

*υ*at the position

_{r}*s*. The result is

*p*(

*υ*;

_{r}*s*) =

*p*(

*υ*), that is, that the probability density distribution of

_{r}*υ*is independent of

_{r}*s*, then we simply get 〈

*S*(

*υ*)〉 =

_{r}*p*(

*υ*). The difference between

_{r}*S*(

*υ*) and 〈

_{r}*S*(

*υ*)〉 is illustrated in Fig. 9 which will be discussed in more detail in section 3. With this approximation, the unfiltered variance of the radial velocity is simply the second central moment of

_{r}*S*(

*υ*). However, the assumption is not likely to be fulfilled because the mean of

_{r}*υ*(

_{r}*s*) changes with

*s*, which is proportional to the height above ground. Therefore, let us assume, slightly more realistically, that 〈

*υ*(

_{r}*s*)〉 =

*G*×

*s*(plus a constant velocity, which is not important here), where

*G*is the average velocity gradient. We, furthermore, assume that the variance of

*υ*(

_{r}*s*) is

*σ*

^{2}independent of height. Then, the width squared of the average Doppler spectrum becomes, if we define it as the second-order central moment of the distribution,

*G*is, as long as it is nonzero. One could argue that the problem might not be large in practice, first, because the radial velocity will saturate for large

*s*and, second, because

*s*has a lower bound determined by the positions of the lidar relative to that of the focus. Nevertheless, it is unsound to use a definition that is invalid in principle. The problem is that the weighting function

*ϕ*(

*s*) has very long tails for the cw lidar, which implies that it does not possess a second moment, or a first for that matter. We are, therefore, forced to estimate the width in a more robust way.

*p*(

*υ*;

_{r}*s*) is Gaussian, so the average spectrum for the cw lidar becomes

*ζ*= (

*Gl*+

*iυ*)/(

_{r}*σ*), the overbar means complex conjugation, and

*ζ*being the error function of a complex argument.

### c. Momentum flux from a WindCube

*π*/2 around the circle formed by the conical scanning, as shown in Fig. 1. Therefore, the momentum flux can be measured defining

*ϕ*and the angle of the first beam direction relative to the mean wind direction Θ, the vertical flux of horizontal momentum in the mean wind direction and perpendicular to that can be calculated,

*l*is the half length of an ideally rectangular light pulse leaving the lidar, assuming a matching time windowing (=2

_{p}*l*/

_{p}*c*, with

*c*being the speed of light, see Table 1). The expression is valid if the Doppler velocity is determined as the first moment of the signal spectrum with the background subtracted appropriately (Banakh and Werner 2005); however, if the Doppler velocity is determined differently,

*ϕ*may describe the averaging well. For the WindCube the situation is further complicated by the fact that the beam is not collimated but rather focused to increase the carrier-to-noise ratio (CNR) at ranges less than 200 m. The consequences of focusing and many other complications are described in Lindelöw (2007); however, here, Eq. (21) is used as a sensible approximation. Mann et al. (2009) showed that Eq. (21) works well in describing the attenuation because of the probe length filtering of the turbulence variances measured by the pulsed lidar. The theoretical expectation of the attenuation under neutral atmospheric conditions presented in Fig. 2 is calculated from Eq. (9). The only difference is that the exponential term in that equation is substituted by sinc

^{2}(

**k**·

**n**/2) (where sinc

*x*≡ (sin

*x*)/

*x*), which is the Fourier transform of Eq. (21). A more detailed discussion of the calculation of the attenuation of the turbulent fluctuations measured by the WindCube may be found in Mann et al. (2009).

In general, the estimation of the reduction in Fig. 2 cannot be used to correct the measured momentum fluxes because a specific spectral tensor model with certain parameters is assumed, and the atmosphere may behave differently due to stability, inhomogeneous terrain, or other effects. A method of avoiding the filtering altogether, as we found for the ZephIR in section 2b, would therefore be useful. However, for a pulsed system like the WindCube the width of the Doppler spectrum is not only determined by the turbulent velocity fluctuations inside the probe length, as expressed by Eq. (12), but the finiteness of the pulse length also contributes significantly. Furthermore, the WindCube Doppler spectra were not available to us at the time of this analysis. In this paper, we therefore limit ourselves to using the Doppler spectra from the ZephIR. In addition, as seen from Fig. 2, the effect of filtering is less pronounced for the WindCube above *z* ≈ 60 m.

## 3. Experiments

The experiments are performed with the two types of lidars, that is, with a cw lidar, the ZephIR, and a pulsed lidar, the WindCube, at two sites in Denmark: a test station at Høvsøre and the Horns Rev offshore wind farm, respectively.

### a. Høvsøre

#### 1) Site description

The Danish National Test Station for Large Wind Turbines is located at Høvsøre, a rural area close to the west coast of Jutland (see Fig. 3). The surrounding terrain is flat and homogeneous, with flow disturbances caused by wind turbines’ wakes at the north, and the sea–land and fjord–land transitions at the west and south of the meteorological mast.

The meteorological mast (latitude 56°26′26″N and longitude 8°9′3″E) is instrumented with Metek USA-1 sonic anemometers at 40, 60, 80, and 100 m, and wind vanes at 10, 60, and 100 m. The tower is also equipped with high-precision cup anemometers; however, because the focus of this paper is momentum flux and not mean wind speeds, this issue will not be pursued further. A light tower is located at about 200 m north of the southernmost turbine and is also instrumented with a Metek sonic anemometer at 160 m. Because the blade tip of the closest turbine reaches 130 m, the measurement at 160 m is not influenced significantly by the turbine wake for westerly winds. The ZephIR and WindCube lidars were installed 20 m north of the meteorological mast and performed measurements from December 2006 to February 2007.

#### 2) Data filtering

Figure 3 indicates the upwind sector used for the analysis. This corresponds to the most predominant sector, which avoids the wakes from the wind turbines, and the direct mast shade on the anemometers. All measured wind speeds are taken for the study, and the data are analyzed based on 30-min periods where no rain fell. Data from the WindCube were selected according to a CNR criterion (CNR > −10 dB for heights below 100 m and CNR > −15 dB for the 160 m height).

The ZephIR scanned the atmosphere at 40, 80, and 100 m, whereas the WindCube scanned at 40, 60, and 160 m. The ZephIR’s weighting function *ϕ* at the three measurement heights is shown in Fig. 4. The figure indicates that the full width at half maximum (FWHM), that is, 2*l*, of the Lorentzian function becomes larger than the constant FWHM (∼30 m) of the WindCube at about 100 m.

#### 3) Results

*(i)* ZephIR—Filtered turbulence

Due to the scanning configuration of the ZephIR, the upstream and downstream radial velocity variances, *θ* from the two peaks of the radial velocity fit, as shown in Fig. 5. Here, radial velocities close to *θ* = 0.25 and 3.4, where the atmospheric flow is perpendicular to the laser beam, are biases positively because the lidar is homodyne—that is, it uses a local oscillator, which is not shifted in frequency, and it is therefore not possible to distinguish the sign of the measured *υ _{r}*. In Fig. 5, wind comes from

*θ*≈ 5, and it is clearly seen that the variance in this upwind direction is larger than the downwind around

*θ*≈ 1.8.

The momentum flux, estimated from Eq. (6), is compared in Fig. 6 to the sonic anemometer observations at the overlapping heights using Δ*θ* = ±5°. The uncertainty indicated on the fit slopes derives from the standard linear least squares fitting theory, and it is the standard deviation of the slope under the assumption of Gaussian errors. The estimated filtering effect has the same trend but is slightly lower than the predictions shown in Fig. 2. This might be due to the differences between the length scale behavior used for the theoretical estimation of the filtering effect in Fig. 2 and the length scale from actual observations of the turbulence spectra at Høvsøre (Peña et al. 2010).

The upstream radial velocity variance is compared in Fig. 7 (left frames) between the ZephIR and the sonic anemometer observations, from Eq. (4), at the overlapping heights using Δ*θ* = ±5°. The filtering effect agrees reasonably well compared to Fig. 2, and it gives the same values at 80 and 100 m.

In the same fashion, the downstream radial velocity variance is compared for the ZephIR and the sonic anemometer observations, from Eq. (5), in Fig. 7 (right frames). In this case, the estimated filtering effect also agrees well compared to Fig. 2, but a decrease of the filtering at 100 m compared to the 80-m height is noticed. This might be due to the slightly different characteristics of the atmosphere when the ZephIR scanned at high levels because the effect of the westerly fjord/sea transition on the scanning measurement volume is different for the upstream and downstream positions. A plot of the theoretical flux and variance reduction together with the measured is shown in section 4.

A comparison of the ZephIR and sonic anemometer momentum flux observed at 40 m for different angles Δ*θ* is made. For Δ*θ* > 10° the results start to change, and a value of Δ*θ* = 5° is chosen for the rest of the analysis. For a Δ*θ* that small, the variation of *υ _{r}* in the up- and downwind intervals is small and ignored.

*L*,

*u** is the friction velocity,

*T*is the mean temperature,

*κ*is the von Kármán constant (≈0.4),

*g*is the gravitational acceleration, and 〈

*w*′Θ′

*〉 is the kinematic heat flux. Both friction velocity and kinematic heat flux are estimated using the eddy-correlation method from sonic anemometer measurements (Stull 1988). This has been performed at 40 m only, due to the advection of sea/fjord wind, which might influence the atmospheric stability of the air layers beyond this level. Three atmospheric conditions are chosen: neutral (−1500 ≥*

_{υ}*L*≥ 500), unstable (−1500 <

*L*≤ −10), and stable (10 ≤

*L*< 500).

The worst agreement for the momentum flux is found for stable conditions, although the wind speed compares well for both instruments for all stability conditions. The highest agreement and correlation for the momentum flux is found under unstable atmospheric conditions (Fig. 8, middle). Although the number of measurements within this range of stability is relatively low, the slope is significantly closer to one compared to the stable range. It is well known that eddies transferring momentum are largest under unstable, intermediate under neutral, and smallest under stable stratification (Kaimal et al. 1972). That implies that the reduction of momentum flux should be largest for stable stratification, which is verified in Fig. 8. Notice that the theoretical attenuation estimates displayed in Fig. 2 (and below in Fig. 17) are for neutral stratification only.

*(ii)* ZephIR—Unfiltered turbulence

Instead of directly taking the variance of the radial velocity measurements calculated from the centroid of the Doppler spectra, as in the previous section, the Doppler spectra from those radial velocity measurements lying close to the two peaks of the radial velocity fit in Fig. 5 are used to calculate the upstream and downstream radial velocity variances—the unfiltered variances. These are estimated from the fit of Eq. (16) to the summed and normalized spectrum observed at both upstream and downstream positions. Figure 9 illustrates this procedure for the upstream (top frame) and downstream (bottom frame) scanning positions using Δ*θ* = ±5°, where the individual spectra for a typical 30-min sampling period at 40 m are shown in gray lines, the summed and normalized power spectrum in circles, and the fit using Eq. (16) in black lines. The radial velocity gradient *G* is estimated at 40 m using the average radial velocities at the upstream and downstream positions from the observations at 80 and 40 m (*G*_{40} = (〈*υ*_{r80}〉 − 〈*υ*_{r40}〉)/(80 m − 40 m)/cos*ϕ*), at 80 m from the observations at 100 and 40 m, and at 100 m from the observations at 100 and 80 m.

The unfiltered momentum flux, also estimated from Eq. (6), is compared in Fig. 10 to the sonic anemometer observations at the overlapping heights using Δ*θ* = ±5°. The lidar momentum flux is overestimated by 7% at 40 and 100 m compared to the sonic observations, whereas they agree well at 80 m. This might be due to the method used to estimate the gradient *G*, which is more accurate for the middle height than for the heights above and below it. Compared to the filtered momentum flux the amount of data is lower, because for the estimation of *G* simultaneous measurements at least at two heights are needed. The largest drawback of the method has to do with the correlation of the observations, which considerably decreases compared to that for the filtered turbulence, which can be seen when comparing the uncertainties on the slopes on Figs. 6 and 10. For the comparison of horizontal mean wind speeds with high-precision cup anemometers see Courtney et al. (2008).

Figure 11 illustrates the results for the unfiltered upstream and downstream radial velocity variances of the lidar compared to the sonic observations at the three heights, showing a similar behavior to that for the filtered results, that is, the agreement between the instruments is very good and the correlation decreases for the downstream compared to the upstream variances. However, the correlation strongly decreases compared to the filtered results, as observed for the momentum flux.

A comparison of the momentum flux and the wind speed at the three heights is also shown in Fig. 12 for different stability conditions, using the same stability intervals as for the filtered results. The agreement for the momentum flux in unstable conditions is also the best (close to 1:1), compared to stable and neutral conditions, as shown in Fig. 8 for the filtered estimations. This might be due to the higher wind shears observed in neutral and stable conditions, compared to unstable conditions where *G* is relatively low and does not have a strong effect on Eq. (16). However, the correlation is considerably worse for unstable conditions, which is the opposite behavior as that found for the filtered turbulence. This may have to do with the fewer points available for the analysis of the unfiltered momentum flux. Here, the instrument has to measure at more heights implying less time for measurements at a particular height. Lenschow et al. (1994) showed that the more “disjunct” the samples are in a time series used for a flux estimate, the greater the statistical error.

*(iii)* WindCube—Filtered turbulence

The estimations of the momentum flux from the WindCube from Eq. (20) are compared to the momentum flux observations from the sonic anemometers at 40 and 60 m at the meteorological mast and at 160 m at the light tower, and the results are illustrated in Fig. 13. The filtering effect for the lidar momentum flux agrees well with the theoretical values shown in Fig. 2 for the three heights. For the heights 60 and 160 m, the observed filtering effect is rather constant and close to 0.8, in very good agreement with the behavior shown in Fig. 2, where the theoretical filtering effect of the WindCube remains constant for the three types of variances above 60 m. Below 60 m, the momentum flux derived from the lidar is further reduced because the measuring volume is larger in comparison to the turbulence length scale at these low heights.

Figure 14 illustrates momentum flux and horizontal wind speed comparisons between the WindCube and the sonic anemometers for different stability classes at 40 m based on the stability intervals used for the ZephIR data. Consistent with the results obtained with the ZephIR (Fig. 8), the agreement is worse for stable than for neutral and unstable conditions where the eddies are large, thus, the turbulent fluctuations are better correlated within the measurement volume of the lidar.

### b. Horns Rev wind farm

#### 1) Site description

A ZephIR was installed on the platform of the Horns Rev wind farm at 20 m above mean sea level (amsl). The platform is located in the North Sea at 12 km from the west coast of Denmark (see Fig. 15). The meteorological mast M2 (latitude 55°31′08″N and longitude 7°47′15″E) was instrumented with Risø cup anemometers at 15 and 62 m amsl, a wind vane at 60 m amsl, and temperature sensors at 13 m amsl and 4 m below mean sea level (bmsl).

#### 2) Data filtering

Figure 15 indicates the wind sector used for the analysis. This corresponds to an open sea sector, which avoids the wake of the wind farm. The momentum flux and mean wind speed comparisons are performed against the observations at M2, which is located ∼5 km west from the platform. To avoid the variable fetch observed at M2 compared to the platform location for northeasterly winds, the sector has also been restricted in the northerly directions (Peña et al. 2009). All wind speeds are taken for the study and the data is analyzed based on 30-min periods where no rain was present.

*U*

_{15}is a good estimate of the friction velocity for the open sea sector at Horns Rev when compared to sonic anemometer measurements. The cpd friction velocity is given by

*α*is the Charnock’s parameter. Peña and Gryning (2008) found a value

_{c}*α*= 0.012 for the open sea sector at M2. The logarithmic wind profile is valid under neutral atmospheric conditions in the marine boundary layer only (Peña et al. 2008). The temperature sensors at 13 m amsl and 4 m bmsl are then used to select neutral atmospheric conditions, |

_{c}*T*

_{13}−

*T*

_{4}| ≤ 1.5°C. The cpd momentum flux is related with the cpd friction velocity within the surface layer as −〈

*u*′

*w*′〉 =

*u*

_{*}

^{2}.

#### 3) Results

In Fig. 16, the momentum flux and horizontal mean wind speed from the cup anemometer (cpd momentum flux) and the ZephIR observations are compared. The horizontal mean wind speed compares well, despite the large distance between M2 and the platform. The filtered momentum flux (Fig. 16, top frame) compares unexpectedly well using Δ*θ* = ±5° taking into account the ZephIR measured at 63 amsl on the platform, whereas the cpd momentum flux is estimated based on measurements of wind speed at 15 m amsl at M2. This might be because of the relatively constant friction velocity within the marine surface layer in neutral conditions and that the flow is highly homogeneous within the open sea sector. The filtering effect on the momentum flux measured by the ZephIR at 63 m amsl is clearly visible, but not quite as pronounced as expected from Fig. 2. The filtering effect is, not surprisingly, similar to the one observed at 40 m at Høvsøre.

For the unfiltered momentum flux, the radial velocity gradient *G* at 63 m amsl is derived using the average radial velocities at the upstream and downstream positions from the observations at 91 and 63 m amsl (*G*_{63} = (〈*υ*_{r91}〉 − 〈*υ*_{r63}〉)/(91 m − 63 m)/cos*ϕ*). The agreement for the momentum flux estimations (Fig. 16, bottom-left frame) is on average good, showing an overprediction of ∼8%, which is very close to that observed for the unfiltered momentum flux at 40 m at Høvsøre. Similar to the results at Høvsøre, the correlation for the momentum flux decreases for the unfiltered results compared to the filtered estimations.

## 4. Discussion

The conically scanning Doppler lidar instruments do provide the three velocity components *u*, *υ*, and *w* every few seconds, but the momentum fluxes calculated from these time series are attenuated severely because of spatial averaging over the conical scan. For example, for heights between 40 and 100 m the ZephIR gives momentum fluxes of 30%–40% of the ones derived from sonics, and the scatter is high, *R*^{2} ≈ 0.5 (these preliminary and not very valuable flux estimates are not shown in the paper to save space). The main reason is that the vertical velocity is estimated from velocities measured over a horizontal circle with a diameter equal to the height of the measurements. This filters out a substantial part of the vertical velocities contributing to the vertical transport of horizontal momentum.

As in Eberhard et al. (1989), we alleviate the flux reduction by using only the turbulence variances measured at each pointing direction, which in this study are the up- and downwind directions. The momentum flux can then be obtained by Eq. (6). Now, only the shape of the weighting function *ϕ* along the beam, Eqs. (8) or (21), contributes to the momentum flux attenuation, and under neutral atmospheric conditions the attenuation can be estimated crudely, see Fig. 2.

This approach works well compared to the data as shown in Fig. 6 for the ZephIR and in Fig. 13 for the WindCube. Comparisons of fluxes under different atmospheric stabilities show that the flux is attenuated mostly when eddies are smallest (Figs. 8 and 14). As predicted by the theory, the attenuation of the up- and downwind variances is larger than that of the momentum flux, see Figs. 7 and 17. However, the attenuation can be large, approximately 20% for the ZephIR above 60 m increasing rapidly to 65% at 200 m, and also large for the WindCube below 60 m. The radial speeds may sometimes be corrupted by rain, so these situations have not been analyzed. For the ZephIR, low clouds may also interfere adversely with the measurements (Courtney et al. 2008).

The theoretical estimate of the flux attenuation assumes neutral stratification and a certain shape of the wind spectrum, so the correction cannot be assumed to be universally valid and is therefore of limited use in practical applications. Knowledge of the exact shape of *ϕ* is also assumed, but it is not particularly well known for the WindCube. Figure 17 also shows that for the data used here the theory predicts too much filtering compared to the data. Therefore, we attempt an even better method of extracting the momentum flux. It is still based on Eq. (6), but it uses a better way to estimate

The scatter in the lidar versus sonic flux comparisons is much larger than similar sonic versus sonic comparisons, where the sonics are either placed on the same mast at different heights or at masts separated by 200 m. For the ZephIR this can partly be explained by the fact that the lidar is not sampling one height continuously over half an hour, but rather it measures at each height for three seconds and goes cyclically through all the chosen heights. This explanation is not valid for the WindCube, which measures continuously at all heights; however, the WindCube rotates rather slowly, approximately one revolution in 6 s taking data in the up- and downwind directions in only half a second each. In conclusion, a thorough statistical error analysis, as done in Lenschow et al. (1994) on “disjunct sampling” of fluxes, has yet to be accomplished for lidars to understand the scatter observed.

## 5. Conclusions

Methods to measure the vertical flux of horizontal momentum from two conical scanning Doppler lidars are investigated. For a continuous wave lidar and a pulsed lidar, the ZephIR and the WindCube, respectively, filtered momentum flux observations can be extracted directly from the variance of the radial velocity measurements lying at the upstream and downstream scanning positions. They show good agreement with the momentum flux from sonic anemometer observations at different heights over flat terrain at Høvsøre, Denmark, with slopes of the correlation ranging from 0.79 to 0.93 for the cw system and 0.69 to 0.85 for the pulsed system, depending on the measurement height. The correlation coefficient *R*^{2} ranges from 0.79 to 0.87 (cw) and 0.31 to 0.9 (pulsed). The agreement is also when the momentum flux is compared with the momentum flux derived from wind speed observations at Horns Rev in the North Sea. The filtering effect, mainly caused by the long probe length along the direction of the laser beam, is crudely predicted by combining three-dimensional spatial statistics with the along-beam weighting function.

Unfiltered momentum flux observations can be performed with the continuous wave lidar, by averaging the up- and downstream Doppler spectra and accounting for the along beam gradient of the radial mean velocity. They show better agreement with slopes of the correlation between 1.00 and 1.07 but lower correlation with the sonic observations at Høvsøre compared to the filtered values (*R*^{2} between 0.59 and 0.74). An advantage of the filtered method is that in our experiments it can retrieve a higher amount of data than the unfiltered method, because it requires observations at one scanning height only, whereas the unfiltered method requires at least two scanning heights or some measure of the radial velocity gradient.

In general, for both unfiltered and filtered methods and for both types of lidar, the momentum flux shows the best agreement with the sonic observations for unstable conditions compared to neutral and stable conditions, as expected, due to the large size of turbulent eddies in the convective atmosphere.

## Acknowledgments

We thank Dr. Michael Harris and three anonymous reviewers for many valuable comments. Funding from the Danish Council for Strategic Research to the project “12 MW” (2104-05-0013) and from the Danish Agency for Science to the project “windscanner.dk” is appreciated. Also support from WP6 of the EU FP6 Upwind project is acknowledged. FB is partially supported by Siemens Wind Power, whereas RW is supported by the Marie Curie ModObs Network MRTN-CT-2006-01369. This paper is not an endorsement of the commercial instruments named herein.

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Basic properties of the lidars. The FWHM is the full width at half of the expected maximum velocity weighting function *ϕ*, see Eq. (7). The pulse length of the WindCube is equal to the range gate, which is ≈200 ns. The WindCube is focused at a fixed height (approximately 100 m) to increase the CNR at the first few 100 m. The longest ranges depend on atmospheric conditions and are only indicative.