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Keywords:

  • atmospheric stability;
  • mixing length;
  • Rossby number;
  • surface layer;
  • turbulence spectra

Abstract

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Theory
  5. 3.  Site and measurements
  6. 4.  Results
  7. 5.  Discussion
  8. 6.  Conclusions
  9. Acknowledgements
  10. Appendix
  11. References

We present the results of an analysis of simultaneous sonic anemometer observations of wind speed and velocity spectra over flat and homogeneous terrain from 10 up to 160 m height performed at the National Test Station for Wind Turbines at Høvsøre, Denmark. The mixing length, l, derived from the wind speed profile, is found to be linearly proportional to the length-scale of turbulence, derived either from the peak of the vertical velocity spectrum, (λm)w, or from a three-dimensional turbulence spectral model, for a range of atmospheric stability conditions, friction Rossby numbers, and within the range of observational heights ((λm)w ∼ 7l). Under very unstable conditions and above 100 m, the local wind shear is low, and the relation between both length-scales is slightly nonlinear. Mixing-length and wind profile models, which depend on both atmospheric stability and friction Rossby number, generally show better agreement to the observations of the length-scale and wind speed profile than the models from surface-layer theory, which show good agreement with the observations for the first 80 m only. The results from this analysis demonstrate a close connection between these two types of length-scales. Copyright © 2010 Royal Meteorological Society


1.  Introduction

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Theory
  5. 3.  Site and measurements
  6. 4.  Results
  7. 5.  Discussion
  8. 6.  Conclusions
  9. Acknowledgements
  10. Appendix
  11. References

During the last 50 years, efforts have been made to study the spectral characteristics of atmospheric turbulence (Busch and Panofsky, 1968; Kaimal et al., 1972; Højstrup, 1982; Mann, 1994) and the behaviour of the length-scale of turbulence with height and atmospheric stability (Kaimal et al., 1972; Caughey and Palmer, 1979; Højstrup et al., 1997; Lenderink and Holslag 2004; Peña et al., 2010b). The investigation of the length-scale of the wind profile was initiated by Prandtl (1925), who introduced the concept of the mixing length when analyzing mean velocity profiles in laboratory flows. Prandtl's work has motivated the use of mixing-length theory for the entire boundary layer (Blackadar, 1962; Lettau, 1962; Delage, 1974; Gryning et al., 2007). This has been used to derive wind profile parameterizations that fit wind speed observations performed beyond the surface layer both over land and water surfaces (Gryning et al., 2007; Peña et al., 2008; Peña et al., 2010a) better than traditional surface-layer theory.

It is well-known (Stull, 1988; Kaimal and Finnigan, 1994; van de Wiel et al., 2008) that both the mixing length and the peak of the turbulence spectra are measures of the size of the turbulent eddies –the first frequently related to their average size and the second related to the size of the eddies with the most energy. However, to our knowledge, there are no references apart from Peña et al. (2010b) in which a relation is established between the length-scales observed from the analysis of the spectral characteristics of atmospheric turbulence and those derived from the mean flow. A close study to ours on similarities between mixing and spectral length-scales is that of Cuxart et al. (2000), who compared a parameterized mixing-length model to spectral length-scales from large-eddy simulations (LESs) of the convective boundary layer (CBL).

The main idea in this paper is to derive and investigate the relations between the length-scale that controls the wind profile, the so-called mixing length, and the length-scale of the turbulence, namely the length-scale corresponding to the peak of the velocity spectrum. This is accomplished by spectral analysis of a number of 10 min sonic anemometer time series of wind velocity fluctuations at several heights over flat and homogenous terrain in Denmark. The analysis was performed over a wide range of atmospheric stability conditions and wind speeds, in order to observe the effects of both stability and friction Rossby number.

The length-scale formulations in this study are closely related to the explicit formulation of the length-scale used in first-order closure models of the atmospheric boundary layer, also called mixing-length models (Blackadar, 1962; Lacser and Arya, 1986). However, the simple explicit mixing-length approaches are today often replaced with length-scales determined from the more involved and physically realistic relationships that can be derived by combining the equation of turbulent kinetic energy with the equation of dissipation. Such a derivation is based on a more realistic physical representation of the energy dissipation as opposed to the mixing-length schemes. However, the length-scales are determined from two additional equations, so it becomes less transparent and not easily comparable to the length-scales in this study. The latter approach is often termed one-and-a-half-order closure (Mellor and Yamada, 1974; Detering and Etling, 1985; Apsley and Castro, 1997) and is known to provide a more realistic representation of the turbulence than first-order models (Holt and Raman, 1988).

In section 2, the behaviour with height and atmospheric stability of the mixing length within (section 2.1.1) and beyond (section 2.1.2) the surface layer is described, from which the dependency on friction Rossby number is shown. The expected relations between the mixing length and the length-scale of turbulence are introduced in sections 2.2.1 and 2.2.2. The site and measurements are described in section 3. Section 4.1 shows the results of the length-scale analysis for the diabatic conditions, and their application to the diabatic wind profile. In section 4.2 the length-scale analysis for a range of friction Rossby numbers, and its application to the adiabatic wind profile, is presented. Discussion and conclusions are given in the last two sections.

2.  Theory

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Theory
  5. 3.  Site and measurements
  6. 4.  Results
  7. 5.  Discussion
  8. 6.  Conclusions
  9. Acknowledgements
  10. Appendix
  11. References

2.1.  Length-scale of the wind profile

In this analysis, the length-scale, l, of the wind profile is defined as the ratio of the velocity scale of the atmospheric flow (i.e. the local friction velocity u) to the mean local wind shear:

  • equation image(1)

where U is the mean wind speed and z the height above the ground. u is estimated as equation image, where equation image is the local kinematic momentum flux, u the along-wind velocity and w the vertical wind velocity,

2.1.1.  Surface layer

For neutral conditions (denoted by subscript N) in the surface layer (subscript SL), Prandtl (1932) found (lSL)N = κz, where κ is the von Kármán constant (≈ 0.4) and l is the mixing length.

Monin and Obukhov (1954) expressed the so-called dimensionless wind shear, φm, as a function of the mean wind shear and the surface friction velocity, uo,

  • equation image(2)

where φm is a universal function of the dimensionless stability z/L, and L is the Obukhov length defined as

  • equation image(3)

Here, To is the mean surface-layer temperature, g is the gravitational acceleration, equation image is the surface-layer kinematic virtual heat flux, and Θv the virtual potential temperature. In analogy to Eq. (2), φm can be used to extend the length-scale in Eq. (1) to account for diabatic conditions in the surface layer assuming constant momentum flux along the surface layer:

  • equation image(4)

The flux-profile relationships can be used to derive the variation of l with height and stability. Their traditional forms are expressed as

  • equation image(5)
  • equation image(6)

for unstable and stable conditions, respectively. For neutral conditions, z/L ≈ 0, thus φm ≈ 1 and the length-scale is therefore proportional to height only. a, b, and p are empirical constants from experiments (Businger et al., 1971; Högström, 1988).

2.1.2.  Boundary layer

In the surface layer, the length-scale is a function of height and stability, and also of the external parameter zo, the roughness length, which is normally neglected because, for example in neutral conditions, l = κ(z + zo) ≈ κz. For the entire boundary layer, the Coriolis parameter fC and the geostrophic wind G are two external parameters that are added to the description of the flow, and should therefore also influence the length-scale (Zilitinkevich and Deardorf, 1974; Tennekes, 1982). The dimensionless combination of these external parameters is known as the Rossby number, Ro = G/(fCzo). This dependency on Rossby number is included in the length-scale models of Lettau (1962), Blackadar (1962) and Gryning et al. (2007). Lettau's and Blackadar's models can be described by inverse summation of two length-scales:

  • equation image(7)

where Blackadar (1962) and Lettau (1962) proposed d = 1 and d = 5/4, respectively. η is a limiting value for the length-scale in the upper part of the atmosphere, which has traditionally been estimated as η = Duo/fC using D = 63 × 10−4 (Blackadar, 1965). Peña et al. (2010a) further suggested

  • equation image(8)

from the combination of the geostrophic drag law and the analytical wind profile derived from Eq. (7). In Eq. (8) A and B are integration constants for a given stability from the resistance laws (Blackar and Tennekes 1968) and zi is the boundary-layer height. Thus, l has a dependency on the friction Rossby number, Rof = uo/(fCzo). This dependency is negligible in the surface layer where l in Eq. (7) closely approaches κz near the ground. zi was parametrized in Peña et al. (2010a) for neutral and stable conditions using the Rossby and Montgomery (1935) formula,

  • equation image(9)

where C is a proportionality constant (Beyrich and Weill, 1993; Beyrich, 1995; Seibert et al., 2000). In absence of direct observations, Peña et al. (2010a) showed that the climatological value of zi under near-neutral conditions can be estimated with Eq. (9) when compared to aerosol backscatter profiles. Gryning et al. (2007) used three length-scales for the mixing length,

  • equation image(10)

where lMBL is a length-scale for the middle part of the boundary layer, which was found to vary with stability and Rof similarly to Eq. (8). The φm function has been used to correct the first length-scale term in Eq. (10) only (i.e. that for the surface layer), and this approach is also performed for the model in Eq. (7); thus, both behave in the same fashion as traditional surface-layer scaling close to the ground. The model in Eq. (10) considers, in contrast to Eq. (7), that the top of the boundary layer acts as a mirror of the surface for the mixing length.

2.2.  Length-scale of the turbulence

2.2.1.  Surface layer

The length-scale of turbulence, in this context taken as the peak of the spectrum of the wind velocity components in the energy-containing range, has also been observed to be proportional to height, at least in the surface layer. For example, the neutral ‘Kansas’ normalised u- and w-spectra and uw-cospectra (Kaimal et al., 1972) with the minor adjustments in Kaimal and Finnigan (1994) are represented by

  • equation image(11)
  • equation image(12)
  • equation image(13)

where f is the frequency in Hz and n = fz/U is the dimensionless frequency. The neutral w-spectrum from Kansas, illustrated in Figure 1, has a peak value of 0.39 at nm = 0.47 (the subscript m represents peak maximum). By converting the peak normalised frequency nm into the peak wavelength λm and using Eq. (12), the peak wavelength of w is found to be proportional to height, namely (λm)w = 2.13z. Based on the Kansas spectra, (λm)w = 5.33l for the neutral surface layer assuming (lSL)N = 0.40z. Observations performed in near-neutral conditions at Høvsøre, Denmark (Peña et al., 2010b) showed (λm)w = 33, 62, 97 m at heights 10, 20, and 40 m, respectively, i.e. a near-linear behaviour of (λm)w with height in the surface layer. Using the result at 40 m, far from the ground but well inside the surface layer, (λm)w = 2.43z and (λm)w = 6.06l, a relation close to that from the Kansas spectra.

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Figure 1. Normalised surface-layer u- and w-spectra, fSu,w/uo2, and uw-cospectrum, fCuw/uo2, (NS) varying with normalised frequency n = fz/U from Kaimal and Finnigan (1994). The normalised frequency peaks, (nm)u,w,uw, are also shown.

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Similarly, the peaks of the u-spectrum and uw-cospectrum, shown in Figure 1, are also proportional to height, (λm)u = 21.98z and (λm)uw = 12.80z, thus they can be related to the mixing length as (λm)u = 54.95l and (λm)uw = 32.01l. However, a stronger correlation between l and the (λm)w than that between l and (λm)u,uw is expected, because these two spectra, in contrast to the w-spectrum, are significantly influenced by mesoscale contributions and their large-scale perturbations have no limit in the horizontal plane.

2.2.2.  Boundary layer

The behaviour of (λm)w has also been observed beyond the surface layer. Caughey and Palmer (1979) combined unstable data and derived an expression for the profile of (λm)w in the mixed layer

  • equation image(14)

for 0.1zizzi, whereas in the unstable surface layer, the relation (λm)w = 5.9z was suggested. Thus, if we assume that the φm function can also be used to extend (λm)w to account for diabatic conditions, (λm)w = [(λm)w]Nφm−1 where [(λm)w]N = 2.13z, then we can derive the dimensionless stability for the measurements of Caughey and Palmer (1979) in the surface layer,

  • equation image(15)

which gives φm = 0.36. This is similar to what was done for the mixing length in the surface layer (from Kansas).

By using Eq. (5) with a = 12 and p = −1/3, as in Grachev et al. (2000), z/L = −1.68. Knowledge of z/L allows us, for example, to evaluate Gryning's length-scale model in Eq. (10), which, as shown in Eq. (14), depends on zi. For very unstable conditions, Gryning et al. (2007) and Peña et al. (2010a) found lMBL ≈ 2zi; thus, we use this value to plot Gryning's length-scale profile, converted to a (λm)w profile using the results of Kansas, i.e. (λm)w = 5.33l. Figure 2 illustrates the results from Caughey and Palmer (1979) and the converted unstable length-scale profile from Gryning et al. (2007), where a similar behaviour for both profiles of length-scale is clearly observed. Further, the converted unstable length-scale profile from Gryning et al. (2007) also agrees well with the observations of Lothon et al. (2009) of the (λm)w profile, which have a maximum at a lower height than that from Eq. (14).

thumbnail image

Figure 2. Peak wavelength profile of the vertical velocity for the unstable boundary layer. The solid line represents the results from Caughey and Palmer (1979), the dashed line the converted length-scale profile from Gryning et al. (2007) and the dotted line the relation (λm)w = 5.9z.

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Similar profiles to those in Figure 2 were found from LES modelling of a CBL in Cuxart et al. (2000) for the mixing and w-spectrum length-scales. Further, Jonker et al. (1999) and de Roode and Duynkerke (2004) found from LES modelling of the CBL that the zi-normalised length-scales of the w-spectrum were z-dependent with maximum values ranging from 0.9 to 1.4 at z/zi =0.5–0.6, being Jonker's length-scale profile nearly identical to that of Caughey and Palmer (1979) in Figure 2.

In analogy to the length-scale of the wind profile, the length-scale derived from the turbulence spectra seems to depend on the Rossby number. Frandsen et al. (2008) derived a spectral length-scale by fitting the spectral model in the IEC (2005) standard to observations of the u-spectrum at different heights at Høvsøre for different wind speed ranges. At 10 m height, they found almost no variation of the spectral length-scale with wind speed (between 6 and 14 m s−1), but it varied for increasing observational heights. We believe that the variation of the spectral length-scale results from its dependency on the Rossby number, in analogy with the mixing length in Eq. (8), because, as shown later, uo and zo increase and decrease, respectively, for increasing wind speeds. The small variation observed by Frandsen et al. (2008) at 10 m is, from our knowledge, due to the negligible influence of Rossby number on the length-scale close to the ground, as for the mixing length, where only the observation height is important.

3.  Site and measurements

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Theory
  5. 3.  Site and measurements
  6. 4.  Results
  7. 5.  Discussion
  8. 6.  Conclusions
  9. Acknowledgements
  10. Appendix
  11. References

The measurements were performed during the period October 2006 to August 2007 at the National Test Station for Wind Turbines at Høvsøre, Denmark, which is located close to the west coast of Jutland (Figure 3). The meteorological mast faces an upwind area (indicated between the lines 30°–125°) where the terrain is flat and homogeneous and the flow is not disturbed by the wakes of the wind turbines or the water.

thumbnail image

Figure 3. National Test Station for Wind Turbines at Høvsøre, Denmark, showing the positions of the meteorological mast (56°26′26″N, 8°9′3″E), a light tower, and wind turbines (circles). The inset shows the location of Høvsøre in Denmark. The dark shading indicates water and white the land areas.

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The meteorological mast is instrumented with Metek USA-1 sonic anemometers at 10, 20, 40, 60, 80, and 100 m and the flat upwind area is selected based on the observations at 10 m. In addition, a sonic anemometer is installed at 160 m on the light tower (400 m north of the meteorological mast). The time series are analysed based on 10 min sampling periods and only average wind speeds exceeding 2 m s−1 are selected. The recording frequency of the time series is 20 Hz and the data have been linearly de-trended over the 10 min period. More details concerning the measurements and instrumentation of the meteorological mast at Høvsøre can be found in Jørgensen et al. (2010).

4.  Results

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Theory
  5. 3.  Site and measurements
  6. 4.  Results
  7. 5.  Discussion
  8. 6.  Conclusions
  9. Acknowledgements
  10. Appendix
  11. References

First the relation between the length-scales is derived for a wide range of stability conditions, followed by a length-scale analysis for a number of friction Rossby numbers in neutral conditions.

4.1.  Diabatic observations

The 10 min wind profiles, the turbulence parameters and the spectra are classified into stability classes, based on the Obukhov length intervals given in Table I. The Obukhov length, Eq. (3), is estimated from the kinematic fluxes observed by the sonic anemometer at 10 m. Table I also gives average meteorological parameters computed in each stability class. The wind shear from 20 to 160 m is estimated by fitting the expression of Högström (1988) to the wind speed observations from 10 to 160 m. For the wind shear at 10 m, wind speed observations at only 10, 20, and 40 m are used. An ensemble average wind profile and wind shear profile are computed within each stability interval. The average mixing-length profile is calculated from Eq. (1) using the local average friction velocity and the local average wind shear. Table I shows that the average uo increases from stable to neutral conditions with a peak in near-unstable and unstable conditions. Since L is proportional to the cube of uo, the absolute value of the kinematic heat flux under unstable conditions is rather large compared to the ‘mirror’ category in the stable side as observed from analyses using different periods of measurements at Høvsøre (Gryning et al., 2007; Peña et al., 2010a).

Table I. Atmospheric stability classes according to intervals of Obukhov length, L. The methods to derive zo and zi are explained in section 4.1.1.
Obukhov length interval (m)Atmospheric stability classL (m)uo (m s−1)zo (m)zi (m)No. of 10 min data
−100 ≤ L ≤ −50Very unstable (vu)− 740.350.013600397
−200 ≤ L ≤ −100Unstable (u)−1420.410.012600459
−500 ≤ L ≤ −200Near unstable (nu)−3140.400.012550292
|L| ≥ 500Neutral (n)53360.390.013488617
200 ≤ L ≤ 500Near stable (ns)3180.360.012451439
50 ≤ L ≤ 200Stable (s)1040.260.0082571144
10 ≤ L ≤ 50Very stable (vs)280.160.002135704

Two models are fitted to the average normalised spectra from the observations (averaged over the 10 min runs at each height on the normalised frequency n):

  • (i)
    The spectral model by Mann (1994), which adjusts three parameters to the velocity spectral tensor: a length-scale LM proportional to the size of the turbulent eddies, a measure of the energy dissipation equation image, and a measure of the degree of anisotropy, Γ.
  • (ii)
    The spectral model by Kaimal and Finnigan (1994), Eqs. (11)–(13), where adjusted constants are introduced to fit the model to the observations for different stabilities and heights (basically n is multiplied by a constant to fit the spectral models to the observations).

The wavelengths of the spectral peaks from the observations, λm, are estimated from the Kaimal model. This model performs well for all stability conditions at the first heights (10, 20, 40 m). The performance deteriorates, particularly for the u-spectrum and the uw-cospectrum, the further the observations are from the ground and for stronger unstable and stable conditions. The Mann model shows good agreement for the u- and w-spectra in all stability conditions and at all heights, and a slight overestimation of the uw-cospectrum. The rather good performance of both spectral models, compared to the neutral observations at Høvsøre, is graphically shown in Peña et al. (2010b).

Figure 4 illustrates the behaviour of the adjusted parameters from the Mann model. The first parameter shown is equation image where α is the spectral Kolmogorov constant, equation image the energy dissipation, and equation image the normalised energy dissipation. For neutral surface-layer scaling, the normalised energy dissipation in Figure 4(a) should be constant with height. However, both the fact that u2 decreases with height and that the length-scale increases less than linearly with height make the normalised neutral energy dissipation increase with height. Stably stratified flows show higher normalised energy dissipation than neutral flows, except for the largest measurement height, while unstable flows show slightly less. While the Kansas data indicated a minimum in the normalised energy dissipation at z/L = 0, both Högström (1990) and Sjöblom and Smedman (2003) demonstrated that the minimum is to the unstable side around z/L = −0.2 to −0.4. That corresponds very well with the slightly lower values of equation image for unstable stratification compared to those of neutral. On the stable side, our normalised energy dissipation increases rapidly with stability and the rate lies between Högström's expression (equation image) and that derived from the Kansas data. The anisotropy parameter Γ seems relatively constant with height for neutral and stable conditions, whereas turbulence seems more isotropic for unstable stratification (Figure 4(c)), at least above the first 40 m. This tendency is expected because turbulence in the bulk of the CBL under weak wind conditions is relatively isotropic. Close to the surface the shear, which maintains the anisotropy, is strong regardless of stability, so here Γ varies much less with stability than further away from the surface. The profiles of length-scale LM are in agreement with the expected behaviour with stability of the length-scale of the wind profile from surface-layer theory, Eq. (4), which predicts larger length-scales in unstable compared to stable conditions with a neutral length-scale profile in between. Further, the size of the convective eddies, ∝ LM, is largest in unstable conditions, as expected.

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Figure 4. Profiles of (a) energy dissipation, (b) length-scale, and (c) degree of anisotropy, derived from the fit of the spectral model of Mann (1994) to wind spectra observed at Høvsøre for the different stability classes listed in Table I.

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Because momentum flux is central, we have chosen to fit the spectral tensor model simultaneously to the three measured spectra: the uw-cospectrum, the u- and the w-spectra. We have also fitted the model to the same three spectra plus the v-spectrum. The values of equation image, LM and Γ change generally less than 10%, while maintaining the ordering with respect to stability and the development with height. The procedure to derive the one-point spectra from the three-dimensional spectral tensor model, and the method to fit those to the measured spectra, is explained in detail in Mann (1994).

Figure 5 illustrates the relation between l and (λm)w for the range of stability classes at all heights. For small values of l (< 40 m), a linear behaviour for all stability conditions at all heights can be observed, which becomes slightly nonlinear for large values of l corresponding to very unstable conditions at the greatest heights. The high values of l can be caused by low local wind shear in the very unstable atmosphere. Also, the assumptions from mixing-length theory, i.e. linear gradients of wind speed and small eddy sizes, are not fulfilled in unstable conditions (Stull, 1988; Holtslag and Moeng, 1991). Also shown are the lines corresponding to the findings in Kansas and in Peña et al. (2010b), (λm)w = 5.33l and (λm)w = 6.06l, respectively, which agree with a fit of the observations, (λm)w = 6.74l − 0.03l2.

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Figure 5. The relation between the mixing length, l, and the wavelength peak of the w-spectrum, (λm)w (symbols). The solid line is a fit to the observations, (λm)w = 6.74l − 0.03l2, the dash-dotted line the relation (λm)w = 6.06l, and the dashed line the relation (λm)w = 5.33l. The stability classes are listed in Table I.

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Figure 6 illustrates the relation between l and LM in analogy to Figure 5. The correlation between these two length-scales (R2 = 0.96) is better than that between l and (λm)w (R2 = 0.92), which is likely to be due to the better fit of the w-spectrum using the Mann model rather than the Kaimal model. A nonlinearity is also noticed for large values of l derived from the unstable and very unstable observations. A quadratic fit of the observations, LM = 1.702l − 0.006l2, agrees with the findings in Kansas and in Peña et al. (2010b), LM = 1.70l and LM = 1.93l, respectively, where the coefficients 1.70 and 1.93 are derived by the relation (λm)w ≈ 3.1LM from Mann (1994).

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Figure 6. The relation between the mixing length, l, and the length-scale, LM (symbols). The solid line is a fit to the observations, LM = 1.702l − 0.006l2, the dash-dotted line the relation LM = 1.93l, and the dashed line the relation LM = 1.70l. The stability classes are listed in Table I.

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Figures 7 and 8 show the relation between (λm)uw,u and l in analogy to Figure 5. (λm)uw and l have a similar behaviour to that observed for (λm)w, and (λm)u shows a stronger nonlinearity in unstable conditions, as compared to (λm)w and (λm)uw. The correlation coefficients are R2 = 0.91 and R2 = 0.68 for uw and u, respectively, which are lower than that for w, as expected. Both relations predicted from the Kansas spectra, (λm)uw = 32.01l and (λm)u = 54.95l, are in good agreement for neutral and all stable conditions.

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Figure 7. The relation between the mixing length, l, and the wavelength peak of the uw-cospectrum, (λm)uw (symbols). The solid line is a fit to the observations, (λm)uw = 28.12l − 0.12l2 and the dashed line the relation (λm)uw = 32.01l. The stability classes are listed in Table I.

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Figure 8. The relation between the mixing length, l, and the wavelength peak of the u-spectrum, (λm)u (symbols). The solid line is a fit to the observations, (λm)u = 46.36l − 0.46l2 and the dashed line the relation (λm)u = 54.95l. The stability classes are listed in Table I.

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4.1.1.  Diabatic wind profile

The observations of the mixing-length profile in the previous section can be compared to the mixing-length models in Eqs. (7) and (10). As in Peña et al. (2010a), the φm function is used to extend the mixing-length model in Eq. (7) to account for stability, η is estimated from Eq. (8) using the suggestions for A and B in Peña et al. (2010a) for each stability class, and d = 5/4. Similarly, the expression for lMBL in Gryning et al. (2007) is used. zi is estimated from Eq. (9) using the values suggested for C for neutral and stable conditions in Peña et al. (2010a), and slightly higher values for zi are assumed in unstable conditions than those used for neutral conditions. zo is computed by fitting the observations from the sonic anemometer at 10 m to the wind profile from surface-layer theory,

  • equation image(16)

where ψm is the correction of the logarithmic profile to account for stability (Stull, 1988). Figure 9 illustrates the mixing-length comparison, showing a good agreement for the near-neutral conditions and both mixing-length models. A slight underprediction of the observations for stable and unstable conditions is found for the model in Eq. (7), whereas a slight underprediction (overprediction) for stable (unstable) conditions is found for the model in Eq. (10).

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Figure 9. Mixing-length profile from the diabatic observations (symbols) at Høvsøre, the predictions from the extended model by Lettau (1962) (solid lines), from the model by Gryning et al. (2007) (dashed lines), and from surface-layer theory Eq. (4) (dash-dotted lines). The stability classes are listed in Table I.

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The mixing-length models in Eqs. (7) and (10) were used to derive the diabatic wind profile in Peña et al. (2010a) and Gryning et al. (2007), respectively. Figure 10 illustrates the comparison of the wind speed observations with the wind profile from surface-layer theory, Eq. (16), with those in Peña et al. (2010a) and Gryning et al. (2007), respectively. In general, surface-layer theory predicts the wind speed well in all stability classes up to about 80 m. It gradually underpredicts the wind speed for the near-neutral classes between 100–160 m. In very stable conditions, it strongly over-predicts the wind speed from 40 m upwards. The wind profile models in Peña et al. (2010a) and Gryning et al. (2007) also predict the wind speed well in all stability classes and they tend to correct the wind profile for the neutral classes matching the wind speed observations between 100 and 160 m. For very stable conditions, the correction, mainly driven by the shallow boundary-layer height, gives a better prediction of the wind speed than the surface-layer theory from 40 m upwards.

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Figure 10. Average wind profiles observed for different stability conditions at Høvsøre (symbols). The predictions from surface-layer theory, Eq. (16), are shown as dash-dotted lines, the predictions from the models in Peña et al. (2010a) as solid lines, and the models in Gryning et al. (2007) as dashed lines. The stability classes are listed in Table I.

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4.2.  Adiabatic observations –variation with Rossby number

In order to observe the variation of the length-scales with Rossby number, the 10 min wind profiles, the turbulence parameters, and the spectra corresponding to neutral conditions (|L| ≥ 500 m) are classified in five wind speed bins (Table II), based on the mean wind speed U observed at the sonic anemometer at 10 m. Table II also gives the average meteorological parameters computed in each wind speed bin.

Table II. Wind speed bins for neutral conditions from the observations at 10 m.
Wind speed interval (m s−1)Wind speed classuo (m s−1)zo (m)Rof × 105zi (m)No. of 10 min data
3–5n10.260.0181.23328159
5–7n20.380.0181.73466233
7–9n30.450.0094.11562129
9–11n40.540.0085.9167475
11–13n50.670.0095.9984021

The Kaimal model, based on Eqs. (11)–(13), and the Mann model are fitted to the average normalised spectra from the observations. As for the diabatic observations, λm values are estimated from the Kaimal model. This model performs well for the range of friction Rossby numbers at all heights and it shows a slight overestimation of the u-spectrum at 160 m. The Mann model performs well for all observations, showing a slight overestimation of the uw-cospectra for the 10 and 20 m heights.

In Figure 11, the parameters of the Mann model are displayed as functions of height and wind speed class. The two-thirds power of the normalised energy dissipation increases with height in accordance with the results for neutral stratification with all wind speeds in one bin (Figure 4). For all heights, the normalised energy dissipation seems to be inversely proportional to the length-scale LM, which is proportional to the mixing length. That observation is in agreement with classical turbulence theory, where the normalised energy dissipation is inversely proportional to a turbulence length-scale. The length-scale LM increases with wind speed at every height, which also seems to be true for the mixing length displayed in Figure 16 below, except at z = 10 m where the mixing length fluctuates erratically with height. The tendency of LM to increase with mean wind speed is obviously violating surface-layer similarity, and it is interesting that this violation is visible down to the lowest measuring height. This was also observed by Mann (1998) who fitted the spectral tensor model to great-height wind spectra from the ESDU standard (ESDU 1985), which is used in civil engineering. For almost all heights in that study, LM increased with increasing wind speed. We speculate that the abnormal behaviour at 10 m is due to difficulties in obtaining a reliable mean velocity derivative, because there are no measurements below 10 m. Except for the lowest velocity bin, the anisotropy parameter Γ, as shown in Figure 11, seems to be independent of velocity.

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Figure 11. Profiles of (a) energy dissipation, (b) length-scale, and (c) degree of anisotropy derived from the fit of the spectral model of Mann (1994) to wind spectra observed at Høvsøre for different friction Rossby numbers. The wind speed classes n1 to n5 are listed in Table II.

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The wavelength peak of the w-spectrum, (λm)w, is observed to depend not only on height, but also on the friction Rossby number. For the first 40 m, the findings from Kansas and Peña et al. (2010b), (λm)w = 2.13z and (λm)w = 2.43z, respectively, agree well with the observations for all friction Rossby numbers. Above 40 m, there is a clear dependency on friction Rossby number and a systematic underprediction of (λm)w compared to the findings in Kansas and Peña et al. (2010b), which are valid for the surface layer only.

The variation of l and (λm)w with height also show a similar behaviour to that observed for the diabatic observations. Figure 12 illustrates their relation for the range of friction Rossby numbers at all heights. The relation is rather linear and shows no dependency on the friction Rossby number, as found for neutral conditions in Figure 5. In the figure, the lines corresponding to the findings in Kansas and in Peña et al. (2010b), (λm)w = 5.33l and (λm)w = 6.06l, respectively, are also shown, which agree well with the linear fit of the observations, (λm)w = 7.07l.

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Figure 12. The relation between the mixing length, l, and the wavelength peak of the w-spectrum, (λm)w (symbols). The solid line is a linear fit to the observations, (λm)w = 7.07l, the dash-dotted line the relation (λm)w = 6.06l, and the dashed line the relation (λm)w = 5.33l. The wind speed classes are listed in Table II.

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The relation between l and LM, illustrated in Figure 13, is also linear and well correlated (R2 = 0.89), as observed above for l and (λm)w. The linear fit to the observations lies between the previous findings from Kansas and in Peña et al. (2010b), both describing well the connection between length-scales.

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Figure 13. The relation between the mixing length, l, and the length-scale LM (symbols). The solid line is a fit to the observations, LM = 1.73l, the dash-dotted line the relation LM = 1.93l, and the dashed line the relation LM = 1.70l. The wind speed classes are listed in Table II.

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For (λm)uw,u, illustrated in Figures 14 and 15, the correlation with the mixing length strongly decreases, R2 = 0.74 and R2 = 0.70 for uw and u, respectively, compared to that for (λm)w where R2 = 0.90. The observations lying furthest from the linear fits performed over all measurements have high friction Rossby numbers, i.e. n3, n4, and n5. The linear fits are in good agreement with the findings at Kansas, (λm)uw = 32.01l and (λm)u = 54.95l.

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Figure 14. The relation between the mixing length, l, and the wavelength peak of the uw-cospectrum, (λm)uw (symbols). The solid line is a linear fit to the observations, (λm)uw = 27.43l and the dashed line the relation (λm)uw = 32.01l. The wind speed classes are listed in Table II.

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Figure 15. The relation between the mixing length, l, and the wavelength peak of the u-spectrum, (λm)u (symbols). The solid line is a linear fit to the observations, (λm)u = 42.16l and the dashed line the relation (λm)u = 54.95l. The wind speed classes are listed in Table II.

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4.2.1.  Adiabatic wind profile

The neutral observations of the mixing-length profile are compared to the mixing-length models in Eq. (7) using d = 5/4 and Eq. (10), and the suggestions in Peña et al. (2010a) for A and B to estimate η and lMBL in neutral conditions. zi is derived from Eq. (9) with C = 0.15 as in Peña et al. (2010a) and zo is computed by fitting the observations from the sonic anemometer at 10 m to Eq. (16) (also given in Table II). Figure 16 illustrates the length-scale comparison showing a slight overprediction of the observations by the model in Eq. (7) for the two lowest friction Rossby numbers from 40 m upwards and a slight underprediction for the three highest friction Rossby numbers from 40 m upwards. The model in Eq. 10 systematically gives slightly higher values for the length-scale than Eq. (7) for all friction Rossby numbers. Surface-layer theory strongly overpredicts the mixing length from 60 m upwards.

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Figure 16. Mixing-length profiles from the adiabatic observations (symbols) at Høvsøre, the predictions from the model in Eq. (7) with d = 5/4 (solid lines), from the model in Eq. (10) (dashed lines), and from surface-layer theory (dash-dotted line). The wind speed classes are listed in Table II.

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Figure 17 shows the comparison of the wind speed observations with the logarithmic wind profile, Eq. (16) with ψm = 0, and the neutral models of Gryning et al. (2007) and Peña et al. (2010a). The logarithmic wind profile predicts the wind speed well for the three highest friction Rossby numbers and it underpredicts the wind speed for the two lowest friction Rossby numbers (with the highest amount of data in Table II). The observations systematically show a nonlinear departure. This departure is well predicted by the neutral models of Gryning et al. (2007) and Peña et al. (2010a), but they overpredict the wind speed for the three highest friction Rossby numbers within the intermediate heights (40–100 m), due to the mixing-length underprediction of Eqs. (7) and (10) in Figure 16.

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Figure 17. Average wind profiles observed for different friction Rossby numbers in neutral conditions at Høvsøre (symbols). The predictions from the logarithmic wind profile, Eq. (16) with ψm = 0, are shown as dash-dotted lines and those from the neutral models of Peña et al. (2010a) and Gryning et al. (2007) as solid and dashed lines, respectively. The wind speed classes are listed in Table II.

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5.  Discussion

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Theory
  5. 3.  Site and measurements
  6. 4.  Results
  7. 5.  Discussion
  8. 6.  Conclusions
  9. Acknowledgements
  10. Appendix
  11. References

In Peña et al. (2010b), estimations of the peak of the w-spectrum were performed in neutral conditions at Høvsøre. These gave slightly different relations for the variation with height of the peak, namely (λm)w = 3.3z, 3.1z, and 2.43z for the heights 10, 20 and 40 m, respectively, and therefore the relation with the mixing length is also slightly different, namely (λm)w = 8.25l, 7.75l, and 6.06l, respectively. Here, we chose for the comparisons the result at 40 m (section 4), but those at 10 and 20 m, both well inside the surface layer, also agree with the fits for the diabatic and adiabatic cases, (λm)w = 6.74l − 0.03l2 and 7.07l, respectively.

The nonlinearity observed between the length-scales in very unstable conditions at great heights might be due to

  • (i)
    difficulties estimating the length-scale of the wind profile given the small local wind shear (although we estimate l from the ratio of the average friction velocity to the average wind shear equation image instead of the average of the ratio equation image, which decreases the estimates of l for low wind shear values),
  • (ii)
    limitation of the mixing-length theory (Holtslag and Moeng, 1991), or
  • (iii)
    inappropriate scaling of the local wind shear and turbulence spectra (we used the local friction velocity only).

Turbulence spectra were modelled in Højstrup (1982) as a sum of two portions, one scaled with surface-layer parameters (uo, U, and z) and the other scaled with mixed-layer parameters (zi and equation image). Højstrup's spectra are therefore broader than the Kaimal spectra and their peaks differ slightly. From the unstable spectra at Høvsøre, such broadening is observed in unstable conditions at heights above 60 m, but we did not have continuous observations of zi and, therefore, we decide to estimate the peaks from the Kaimal model only. Difficulties in relating length-scales from the w-spectrum and integral length-scales from its autocorrelation function were also shown in Lothon et al. (2009), where there is a nonlinearity particularly at large spectral length-scales.

We used the peaks as the characteristic spectral length-scales, since these showed rather ‘idealised’ shapes (Figure 8 in Peña et al., 2010b); e.g. the root mean square error (RMSE) between the observed and the Kaimal-based modelled normalised w-spectrum is 0.03 for neutral conditions at 40 m. Those derived from LES modelling in Jonker et al. (1999) and de Roode and Duynkerke (2004) are related to a ‘spectral-weighted’ length-scale and to the ogive, respectively. Although they are not the same measure, the proportionality between them, as exposed by Cuxart et al. (2000) for all length-scales, is clear. For example, in de Roode and Duynkerke (2004) the w-spectral length-scale of the convective surface layer (CSL) at z = 100 m is 600 m (zi = 1000 m), i.e. a ratio of six which is very close to our study (equation image with φm = 0.36 for the CBL analysis of Caughey and Palmer, 1979) and also to Caughey's relation for the CSL ((λm)w = 5.9z).

The models chosen for the mixing length limit the value of the length-scale beyond the surface layer, since the eddies cannot grow infinitely, as assumed in surface-layer theory. However, as illustrated in Figure 10, the wind profile models are not strongly sensitive to the choice of mixing-length model and, within the surface layer, the traditional surface-layer wind profile shows a rather good agreement with the wind speed observations for the range of stability conditions and friction Rossby numbers.

For the adiabatic case, the dependency on friction Rossby number was analysed by classifying the wind profiles in wind speed intervals. This dependency can be examined in a more desirable fashion by measuring high-frequency velocity spectra at different heights (even beyond the surface layer) and at different latitudes. Unfortunately, it is difficult to achieve the necessary conditions of flat and homogeneous land and to find installations comparable to Høvsøre.

6.  Conclusions

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Theory
  5. 3.  Site and measurements
  6. 4.  Results
  7. 5.  Discussion
  8. 6.  Conclusions
  9. Acknowledgements
  10. Appendix
  11. References

Relations between the length-scale of the wind profile, i.e. the mixing length, and the length-scale of the turbulence are presented. From the analysis of sonic anemometer observations of wind speed and turbulence spectra performed over flat and homogeneous terrain at Høvsøre, near the west coast of Denmark, it is found that within the surface layer and for a range of atmospheric stability conditions, the mixing length shows a linear proportionality to the peak of the vertical velocity spectrum and to the length-scale of the three-dimensional turbulence spectral model of Mann (1994).

Beyond the surface layer, the mixing length also shows a linear proportionality to the length-scale of the turbulence, except for very unstable conditions where the local wind shear has low values and at heights above 100 m, where a slightly nonlinear behaviour is observed. From the results, we suggest that the neutral relations for the peak of the w-spectrum in the surface layer can be extended to diabatic conditions using the universal stability functions from Monin–Obukhov similarity theory. Beyond the surface layer, the mechanisms that control the length-scale of the wind profile are rather analogous to those of the length-scale of turbulence.

It is found that the mixing length and the length-scale of the turbulence have a similar dependency on the friction Rossby number, thus their relation for a range of friction Rossby numbers under neutral conditions is close to linear. The mixing-length models, which take into account not only the stability of the atmosphere, but also the dependence on Rossby number, have generally a better agreement with the mixing-length profiles observed at Høvsøre than the surface-layer length-scale. The wind profile parametrizations, derived from the mixing-length models, have a good agreement with the wind speed measurements for the range of stability conditions and friction Rossby numbers observed at Høvsøre. Beyond the surface layer, the models correct the under- and overpredictions of the wind speed from surface-layer theory.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Theory
  5. 3.  Site and measurements
  6. 4.  Results
  7. 5.  Discussion
  8. 6.  Conclusions
  9. Acknowledgements
  10. Appendix
  11. References

Acknowledgements We would like to thank the Test and Measurements Program of the Wind Energy Division at Risø DTU for the acquisition of the Høvsøre data, and Claire Vincent from Risø DTU for the comments on the manuscript and advice on the English language. We thank the reviewers for useful comments and suggestions. Funding from the Danish Council for Strategic Research Sagsnr. 2104-08-0025 and centre no. 09-067216, and from the EU project contract TREN-FP7EN-219048 ‘NORSEWInD’ are also acknowledged.

Appendix

  1. Top of page
  2. Abstract
  3. 1.  Introduction
  4. 2.  Theory
  5. 3.  Site and measurements
  6. 4.  Results
  7. 5.  Discussion
  8. 6.  Conclusions
  9. Acknowledgements
  10. Appendix
  11. References

Tables A.I, A.II, A.III and A.IV show the standard deviations of U/uo and l, and the RMSEs between the wind profile and length-scale models and the observations, for both diabatic and adiabatic cases.

Table A.I.. Standard deviations of U/uo and l (m) (separated by /) for the diabatic observations.
Height (m)NeutralNear stableStableVery stableNear unstableUnstableVery unstable
102.80/3.183.63/2.893.06/1.8110.40/1.223.09/3.433.21/3.305.01/2.60
202.89/7.864.23/6.943.26/3.0512.46/1.963.11/8.823.27/8.285.20/7.82
403.19/12.235.74/10.614.27/4.9316.87/4.143.20/14.503.46/16.905.76/17.20
603.54/13.556.40/12.705.02/5.6819.19/4.523.42/14.663.81/17.445.66/16.23
804.32/15.106.86/14.686.28/6.4720.55/4.484.14/15.574.09/19.125.65/18.34
1004.82/17.856.86/16.857.15/7.5622.05/4.904.53/18.304.38/21.595.91/21.08
1606.04/23.356.82/22.378.44/10.6022.89/5.846.53/23.965.69/27.997.32/28.12
Table A.II.. RMSE of U (m s−1) and l (m) (separated by /) between the fit/models and the diabatic observations.
Fit/modelNeutralNear StableStableVery stableNear unstableUnstableVery unstable
Högström0.09/ –0.08/ –0.06/ –0.13/ –0.05/ –0.05/ –0.05/ –
Surface layer0.42/14.120.17/1.370.21/1.571.11/1.690.28/32.540.14/33.010.09/34.73
Gryning0.27/2.390.36/2.710.22/2.430.31/1.980.16/3.430.13/2.350.09/8.16
Lettau0.22/1.660.31/3.420.19/2.730.32/2.070.12/3.960.11/11.790.08/16.24
Table A.III.. Standard deviations of U/uo and l (m) (separated by /) for the adiabatic observations.
Height (m)n1n2n3n4n5
  1. The wind speed classes n1 to n5 are listed in Table II.

103.37/1.892.76/2.092.19/1.941.77/2.561.49/1.61
203.70/3.492.73/5.142.23/5.491.89/6.091.51/5.55
404.53/4.952.80/6.902.33/9.382.01/10.311.73/7.74
605.27/5.842.91/7.222.41/10.552.11/10.631.82/8.83
806.70/6.983.29/8.692.51/11.642.20/11.622.11/12.55
1007.60/8.533.63/10.192.64/12.382.35/14.362.22/14.68
1608.29/12.375.43/15.483.60/17.713.07/20.202.81/29.93
Table A.IV.. RMSE of U (m s−1) and l (m) (separated by /) between the fit/models and the adiabatic observations.
Fit/modeln1n2n3n4n5
  1. The wind speed classes n1 to n5 are listed in Table II.

Högström0.07/ –0.11/ –0.08/ –0.10/ –0.09/ –
Surface layer0.84/21.850.54/16.430.25/9.340.37/4.640.37/2.24
Gryning0.49/5.660.29/4.070.42/3.290.47/2.540.56/2.25
Lettau0.48/2.900.26/1.770.36/5.640.40/2.270.47/3.75

References

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  2. Abstract
  3. 1.  Introduction
  4. 2.  Theory
  5. 3.  Site and measurements
  6. 4.  Results
  7. 5.  Discussion
  8. 6.  Conclusions
  9. Acknowledgements
  10. Appendix
  11. References
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