#### 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 *u*_{∗o} increases from stable to neutral conditions with a peak in near-unstable and unstable conditions. Since *L* is proportional to the cube of *u*_{∗o}, 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 *z*_{o} and *z*_{i} are explained in section 4.1.1.Obukhov length interval (m) | Atmospheric stability class | *L* (m) | *u*_{∗o} (m s^{−1}) | *z*_{o} (m) | *z*_{i} (m) | No. of 10 min data |
---|

−100 ≤ *L* ≤ −50 | Very unstable (vu) | − 74 | 0.35 | 0.013 | 600 | 397 |

−200 ≤ *L* ≤ −100 | Unstable (u) | −142 | 0.41 | 0.012 | 600 | 459 |

−500 ≤ *L* ≤ −200 | Near unstable (nu) | −314 | 0.40 | 0.012 | 550 | 292 |

|*L*| ≥ 500 | Neutral (n) | 5336 | 0.39 | 0.013 | 488 | 617 |

200 ≤ *L* ≤ 500 | Near stable (ns) | 318 | 0.36 | 0.012 | 451 | 439 |

50 ≤ *L* ≤ 200 | Stable (s) | 104 | 0.26 | 0.008 | 257 | 1144 |

10 ≤ *L* ≤ 50 | Very stable (vs) | 28 | 0.16 | 0.002 | 135 | 704 |

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

*L*_{M} proportional to the size of the turbulent eddies, a measure of the energy dissipation

, 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, *λ*_{m}, 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 where *α* is the spectral Kolmogorov constant, the energy dissipation, and 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 *u*_{∗}^{2} 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 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 () 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 *L*_{M} 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, ∝ *L*_{M}, is largest in unstable conditions, as expected.

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 , *L*_{M} 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 (*λ*_{m})_{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), (*λ*_{m})_{w} = 5.33*l* and (*λ*_{m})_{w} = 6.06*l*, respectively, which agree with a fit of the observations, (*λ*_{m})_{w} = 6.74*l* − 0.03*l*^{2}.

Figure 6 illustrates the relation between *l* and *L*_{M} in analogy to Figure 5. The correlation between these two length-scales (*R*^{2} = 0.96) is better than that between *l* and (*λ*_{m})_{w} (*R*^{2} = 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, *L*_{M} = 1.702*l* − 0.006*l*^{2}, agrees with the findings in Kansas and in Peña *et al.* (2010b), *L*_{M} = 1.70*l* and *L*_{M} = 1.93*l*, respectively, where the coefficients 1.70 and 1.93 are derived by the relation (*λ*_{m})_{w} ≈ 3.1*L*_{M} from Mann (1994).

Figures 7 and 8 show the relation between (*λ*_{m})_{uw,u} and *l* in analogy to Figure 5. (*λ*_{m})_{uw} and *l* have a similar behaviour to that observed for (*λ*_{m})_{w}, and (*λ*_{m})_{u} shows a stronger nonlinearity in unstable conditions, as compared to (*λ*_{m})_{w} and (*λ*_{m})_{uw}. The correlation coefficients are *R*^{2} = 0.91 and *R*^{2} = 0.68 for *uw* and *u*, respectively, which are lower than that for *w*, as expected. Both relations predicted from the Kansas spectra, (*λ*_{m})_{uw} = 32.01*l* and (*λ*_{m})_{u} = 54.95*l*, are in good agreement for neutral and all stable conditions.

##### 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 φ_{m} 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 *l*_{MBL} in Gryning *et al.* (2007) is used. *z*_{i} 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 *z*_{i} are assumed in unstable conditions than those used for neutral conditions. *z*_{o} is computed by fitting the observations from the sonic anemometer at 10 m to the wind profile from surface-layer theory,

- (16)

where *ψ*_{m} 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).

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.

#### 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 class | *u*_{∗o} (m s^{−1}) | *z*_{o} (m) | *Ro*_{f} × 10^{5} | *z*_{i} (m) | No. of 10 min data |
---|

3–5 | n1 | 0.26 | 0.018 | 1.23 | 328 | 159 |

5–7 | n2 | 0.38 | 0.018 | 1.73 | 466 | 233 |

7–9 | n3 | 0.45 | 0.009 | 4.11 | 562 | 129 |

9–11 | n4 | 0.54 | 0.008 | 5.91 | 674 | 75 |

11–13 | n5 | 0.67 | 0.009 | 5.99 | 840 | 21 |

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, *λ*_{m} 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 *L*_{M}, 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 *L*_{M} 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 *L*_{M} 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, *L*_{M} 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.

The wavelength peak of the *w*-spectrum, (*λ*_{m})_{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), (*λ*_{m})_{w} = 2.13*z* and (*λ*_{m})_{w} = 2.43*z*, 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 (*λ*_{m})_{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 (*λ*_{m})_{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), (*λ*_{m})_{w} = 5.33*l* and (*λ*_{m})_{w} = 6.06*l*, respectively, are also shown, which agree well with the linear fit of the observations, (*λ*_{m})_{w} = 7.07*l*.

The relation between *l* and *L*_{M}, illustrated in Figure 13, is also linear and well correlated (*R*^{2} = 0.89), as observed above for *l* and (*λ*_{m})_{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.

For (*λ*_{m})_{uw,u}, illustrated in Figures 14 and 15, the correlation with the mixing length strongly decreases, *R*^{2} = 0.74 and *R*^{2} = 0.70 for *uw* and *u*, respectively, compared to that for (*λ*_{m})_{w} where *R*^{2} = 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, (*λ*_{m})_{uw} = 32.01*l* and (*λ*_{m})_{u} = 54.95*l*.

##### 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 *l*_{MBL} in neutral conditions. *z*_{i} is derived from Eq. (9) with *C* = 0.15 as in Peña *et al.* (2010a) and *z*_{o} 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.

Figure 17 shows the comparison of the wind speed observations with the logarithmic wind profile, Eq. (16) with *ψ*_{m} = 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.