2.1.2.1 Mean Velocity Profile

To describe the flow field in the atmospheric boundary layer, the Navier–Stokes equations are invoked. In Appendix A, a brief summary of the two-dimensional boundary layer equations for an incompressible flow is given. Here, some classical relations between the mean velocity profile, the fluctuating velocity components and the shear are given. These relations are utilized in describing some key characteristics for the MABL.

A Cartesian coordinate system is used, with the $x$ -axis horizontal and positive in the mean wind direction and the $z$ -axis vertical, positive upwards and zero at the sea level. The velocity in $x$ -direction can then be written as a mean value plus the turbulent fluctuation, $u=\overline{u}+uʹ$ . The mean transverse and vertical velocities are assumed to be zero, while the fluctuating components are denoted as $vʹ$ and $wʹ$ respectively. The variation in mean wind direction with height due to the Coriolis effect is thus disregarded. In solving the Navier–Stokes equations for the mean flow, a closure problem exists. Terms denoted Reynolds stresses of the form ${\rho }_a\overline{uʹwʹ}$ , ${\rho }_a\overline{uʹvʹ}$ and ${\rho }_a\overline{vʹwʹ}$ appear in the equations, where ${\rho }_a$ is the density of air. The Reynolds stresses are related to characteristics of the mean flow. Based upon empirical evidence, the Reynolds stresses are assumed proportional to the vertical velocity gradient in the boundary layer. This is according to the Prandtl mixing length theory; see, for example, Reference Curle and DaviesCurle and Davies (1968). In a two-dimensional flow, the Reynolds stress is written as:

${\rho }_a\overline{uʹwʹ}=-K_m{\rho }_a\frac{\partial \overline{u}}{\partial z}.$([2.1])

$K_m$ is denoted as the eddy diffusivity. In the absence of heat fluxes, the boundary layer is denoted as neutral. Under such conditions and close to a smooth surface but above the viscous sublayer (see Appendix A), $K_m$ may be parameterized as:

$K_m=k_{a}z{u}_{*}.$([2.2])

$k_a$ is the von Kármán constant, based upon experimental data found to be approximately 0.40. $u_{*}$ denotes the friction velocity and is given from the relation $u_{*}=\sqrt{\tau /{\rho }_a}$ , where $\tau$ is the shear stress at the surface. In the surface layer it is assumed that the turbulent momentum fluxes are constant in the vertical direction. Thus, by combining Equations 2.1 and 2.2, and integrating from the bottom of the surface layer, $z_0$ to $z$ , the velocity profile is obtained as:

$\overline{u}\left(z\right)=\frac{u_{*}}{k_a}\ln \left(\frac{z}{z_0}\right).$([2.3])

$z_0$ is frequently denoted as the surface roughness length scale. Reference CharnockCharnock (1955) found a relation between $z_0$ and $u_{*}$ by measuring the wind field over a water reservoir. The vertical profile of the lowest 8 m over the water surface fitted the logarithmic profile in Equation 2.3 well. The mean velocity profile in the surface layer thus gives $\overline{u}\left(z_0\right)=0$ , even if the mean velocity at top of the friction layer has some small but finite value. Reference CharnockCharnock (1955) found the empiric relation between the roughness length scale and the friction velocity as:

$z_0={\alpha }_0\frac{u_{*}^2}{g}.$([2.4])

This is called the Charnock relation. Over open sea, with fully developed waves, the Charnock constant ${\alpha }_0$ has, according to DNV (2021c), values in the range of 0.011–0.014, and near coastal sites it may be 0.018 or higher.

For offshore conditions, $z_0$ and $u_{*}$ depend upon the wind speed, upstream distance from land and wave conditions. DNV (2021c) proposes $z_0$ in the range 0.0001 to 0.01 for open sea conditions. The lowest value corresponds to calm water and the highest to rough wave conditions. Onshore, higher values for $z_0$ apply.

The friction velocity may be related to the mean velocity at a certain vertical elevation and a surface friction coefficient. Frequently, the 10 min mean wind velocity at a height $z_r$ over the ground or sea surface, denoted $U_{10}\left(z_r\right)$ , is used as a reference. By combining the above relations and defining a surface friction coefficient $\kappa =\tau /{\rho }_a{U}_{10}^2\left(z_r\right)$ , the following expression for the friction velocity and friction coefficient is obtained:

$\begin{array}{c}{u}_{*}=\sqrt{\kappa }{U}_{10}\left(z_r\right)\\ \kappa =\frac{k_a^2}{{\left(\ln \left(\frac{z_r}{z_0}\right)\right)}^2}.\end{array}$([2.5])

Frequently, a reference height $z_r=10m$ is used. With the above range of $z_0$ , $\kappa$ - values in the range 0.0012 and 0.0034 are obtained.

The above formulations rely upon the assumption of a near neutral boundary layer and constant turbulent momentum fluxes in the vertical direction. Above the surface layer and in stable or unstable boundary layers the assumptions are not valid. Below the surface layer, in the viscous sublayer with a vertical extent in the order of mm, the flow is mainly laminar and is, as the name indicates, dominated by viscous effects. Viscous effects do not play an important role above this layer. Some more details related to the two-dimensional boundary layer equations are outlined in Appendix A.

In engineering applications, [2.3] is frequently replaced by a simpler power function, written as:

$U_{10}\left(z\right)=U_{10}\left(z_r\right){\left(\frac{z}{z_r}\right)}^{\alpha }.$([2.6])

Here, $U_{10}\left(z\right)$ is the 10 min mean wind velocity at vertical level $z$ . By requiring the mean wind speed at the vertical level $z$ as obtained by [2.3] and [2.6] to be equal, a relation between $z_0$ and $\alpha$ is obtained:

$\alpha \left(z\right)=\frac{\ln \left(\frac{\ln \left(z/z_0\right)}{\ln \left(z_r/z_0\right)}\right)}{\ln \left(z/z_r\right)}.$([2.7])

The result is plotted in Figure 2.3 (left) for three different values of the roughness length, $z_0$ . It is observed that the proper $\alpha$ value depends upon at which height the two expressions for the mean velocity are required to fit. In Figure 2.3 (right), the relation between $z_0$ and $\alpha$ for three different fitting heights is plotted. A reference height of $z_r=10m$ is applied in the graphs.

Figure 2.3 Relation between roughness length, $z_0$ and the exponent $\alpha$ in the power law formulation of the mean wind profile. Left: height dependence of $\alpha$ for three different $z_0$ . The logarithmic and exponential profile gives the same mean wind speed at height $z$ . Right: $\alpha$ versus $z_0$ for three different heights. Mean wind speed fits at vertical level $z$ . Reference height $z_r=10m$ is used.

As indicated above, the mean wind velocity depends upon the time of averaging. When considering extreme wind velocities, the extreme mean velocity with a certain return period, for example 10 or 50 years (corresponding to a yearly probability of occurrence of 0.1 and 0.02, respectively), increases if the time of averaging is reduced, e.g., from 1 h to 10 min. The extreme wind velocity also increases with increased turbulence intensity.

The above wind profiles should be used with care in heights above approximately 100 m as the assumption of constant turbulent fluxes above this level may be doubtful. Few wind measurements of the mean wind profile above this level exist. An example on data beyond this level is rawinsonde data, as discussed by Reference Furevik and HaakenstadFurevik and Haakenstad (2012).Footnote ² Another option to measure the wind speed at high elevations is by using Lidars; see Section 2.1.5.3.

2.1.2.2 Stability

The mean velocity profiles discussed in the previous section are obtained assuming constant turbulent fluxes in vertical direction and neutral stability of the atmosphere. During unstable and stable atmospheric conditions, the profiles must be modified.

The atmospheric stability is related to the vertical temperature profile in the air. In simple terms, the stability condition may be understood by considering a volume of air (an air parcel) at a certain vertical level. In the initial position of the parcel, the temperature inside the parcel is equal to the temperature of the surroundings. Moving this parcel upwards causes the volume to expand due to the reduced pressure. The expansion implies work is done. Assuming the expansion is adiabatic, i.e., there is no heat exchange with the exterior, the temperature in the parcel is reduced. If this new temperature is equal to the surrounding temperature, the density of the air in the parcel is equal to the density of the surrounding air and there is an equilibrium, or neutral stability. If, on the other hand, the temperature inside the expanded parcel is lower than the external temperature, the density of the air in the parcel is higher than the density of the surrounding air. The parcel will thus tend to sink back down to its original position, i.e., the conditions are stable. However, if the temperature of the air parcel in the new vertical position is higher than the surrounding temperature, the air parcel will tend to move further upward, i.e., the conditions are unstable.

Assuming dry air, the adiabatic expansion or compression of the air parcel will behave according to the ideal gas law:

$\begin{array}{c}pV=nRT\\ p{V}^{\gamma }=\text{Constan}\text{t}\text{.}\end{array}$([2.8])

Here, $n$ is the number of moles in the volume, $R$ is the universal gas constant, $V$ is the volume considered, $p$ is the absolute pressure, $T$ is the absolute temperature and $\gamma =c_p/c_v$ is the gas constant for air. $c_p$ is the specific heat under constant pressure, while $c_v$ is the specific heat under constant volume. For dry air, $\gamma =1.40$ . Combining the expressions in [2.8], the temperature $T_1$ at pressure $p_1$ is obtained as:

$T_1=T_0{\left(\frac{p_0}{p_1}\right)}^{\frac{1-\gamma }{\gamma }}.$([2.9])

$T_0$ and $p_0$ define an initial state of the gas.

Vertical Temperature Variation

Assume the temperature at ground level is ${20}^{o}C$ or $293K$ . Normal air pressure $p_0=1013hPa$ . With an air density of $1.225kg/m^3$ , the pressure difference at 100 m versus at ground level becomes $\Delta p=-{\rho }_{a}g\Delta z=-{1.225}^{\mathrm{*}}{9.80665}^{\mathrm{*}}100=-1201.3Pa$ . The temperature at 100 m elevation, assuming adiabatic expansion, becomes:

$T_{\left(z=100m\right)}=T_{\left(z=0m\right)}{\left(\frac{p_0}{p_0+\Delta p}\right)}^{\frac{1-\gamma }{\gamma }}=292.00K.$

I.e., under the assumption of dry air, the temperature is lowered by $1^{o}C$ per 100 m of increased elevation. The requirement of “dry air” is strict; the relation holds also if no phase change takes place and there is no heat transfer by radiation. Note that the change in density with pressure has been ignored as we are considering small pressure differences.

If the air contains a lot of water vapour, condensation of water may take place as the air is cooled. The condensation releases heat, resulting in a lower temperature decay. The effect is temperature- and pressure-dependent, but in most cases the lapse rate is in the range of $0.5-{1.0}^{o}C$ per 100 m.

In studying the atmospheric boundary layer, it is convenient to use the potential temperature, $\theta$ , rather than the sensible (measured) temperature, $T$ . To obtain the potential temperature, assume that the air at any $z-$ level is moved adiabatically downwards to the ground level. The temperature the air then will obtain is the potential temperature. Assuming dry air, the potential temperature becomes:

$\theta =T_z{\left(\frac{p_z}{p_0}\right)}^{\frac{1-\gamma }{\gamma }}.$([2.10])

Thus, in a neutral stratified atmosphere, the potential temperature is constant with height.

Over land, the ground is heated during the day and cools at night. Over sea, the diurnal variation vanishes, as the incoming radiative energy is efficiently distributed over a large volume of water. Therefore, over the sea, the synoptic weather situation, e.g., cold air advection with northerly winds (in the northern hemisphere) over relatively warm water, or warm air advection with southerly winds over relatively cooler water, and seasonal effects control the stability of the marine atmospheric boundary layer. The first example, cold air over warm water, happens most frequently during autumn and winter and causes an unstable or convective situation, while the case of warm air over cold water occurs more frequently during spring and summer, causing a stable situation. In Figure 2.4, a simplistic illustration of the vertical profile of the potential temperature in the case of stable, neutral and unstable (convective) conditions is given. Figure 2.6 illustrates the vertical variation of the mean wind speed for the unstable, neutral and stable conditions. Here the simple exponential velocity profile, [2.6] is assumed.

Figure 2.4 Simplistic illustration of the vertical distribution of the potential temperature in the surface layer. Stable, neutral and unstable conditions.

During unstable, or convective, conditions, the air close to the sea surface will be warmed by the sea and tend to move upwards, causing vertical mixing due to the buoyancy effect. The lower part of the boundary layer, in the region with a negative gradient of the potential temperature, the combined effect of turbulent shear stresses and buoyancy effects gives good vertical mixing. Above the surface layer, in the mixed layer, the potential temperature is almost constant. In the mixed layer, the vertical gradient of the mean velocity is low. The mixed layer may extend to about 1 km above sea level. Above the mixed layer, normally an inversion or capping layer exists, where the potential temperature increases and thus partly blocks the mixing of air into the free atmosphere. During unstable conditions, the surface layer may become fairly thick, in the order of 100–200 m. An illustration of the potential temperature through the various layers is given in Figure 2.5 (left).

Figure 2.5 Illustration of the potential temperature variation through the various layers of the atmospheric boundary layer during unstable (convective; left) and stable (right) conditions.

Based on Reference LeeLee (2018). Reproduced with permission of Springer eBook.

During stable conditions the potential temperature increases from the sea level upwards. As the velocity gradient is large close to the sea surface, the vertical mixing is efficient in this region. Further up, however, the mixing is suppressed by the positive gradient of the potential temperature and the mixing diminishes. A vertical distribution of potential temperature through the various layers is illustrated in Figure 2.5 (right). During stable conditions, the surface layer is thinner than during unstable conditions, perhaps even less than 50 m. Above the surface layer, a residual layer is present where the potential temperature has a slightly positive gradient and the turbulent mixing is low.

Coriolis forces act on the wind field. As the mean wind speed in general increases with height, Coriolis forces causes the wind direction to change with height. In the northern hemisphere, the wind direction at ground level is directed to the left relative to the wind direction above the boundary layer, the geostrophic wind.Footnote ³

Figure 2.6 Illustration of mean wind profiles in the surface layer at different atmospheric stability conditions, assuming same mean velocity at height 100 m.

There are various ways to characterize the atmospheric stability conditions. A simple, qualitative assessment of the static stability can be performed by considering the gradient of the potential temperature.Footnote ⁴ $\partial \overline{\theta }/\partial z<0$ corresponds to unstable conditions, $\partial \overline{\theta }/\partial z=0$ corresponds to neutral conditions, while $\partial \overline{\theta }/\partial z>0$ corresponds to stable conditions.

A physical, consistent way to quantify the degree of stability of the boundary layer is to compare the contribution by velocity shear and buoyancy effects to the production of turbulent kinetic energy. The buoyancy effect may, depending upon the sign of the surface heat flux, both increase and decrease the kinetic turbulent energy. The flux Richardson number – see, e.g., Reference LeeLee (2018) – expresses the ratio between the buoyancy and velocity shear effects in the production of turbulent kinetic energy. The flux Richardson number is written as:

([2.11])$R=\frac{\text{buoyancy production}}{\text{shear production}}=\frac{\frac{g}{\overline{\theta }}\overline{wʹ\theta ʹ}}{\overline{uʹwʹ}\frac{\partial \overline{u}}{\partial z}}.$

Shear production is the shear stress times the gradient of the mean velocity. $\theta ʹ$ denotes the turbulent potential temperature fluctuations, similar to the velocity fluctuations. The relation is derived by using the two-dimensional Navier–Stokes equation. In deriving the relation for the buoyancy production, the vertical pressure gradient set to $-\rho g$ and the relation between the density and the temperature is according to the ideal gas law. The minus sign in the shear production term (see Appendix A) has been omitted. $R=0$ corresponds to no buoyancy effects and thus a neutral condition. $R<0$ corresponds to positive buoyancy effects and thus an unstable condition. At $R=-1$ the turbulence production from buoyancy effects and the velocity shear effect are equal. $R>0$ indicates a stable condition. Theoretically, $R=1$ represents an upper limit at which the destruction of turbulence caused by the buoyancy balances the production due to the vertical shear.

The flux Richardson number is height-dependent. This is not explicitly shown in [2.11]. However, as discussed above, in the surface layer, where constant vertical flux of turbulence may be assumed, the velocity gradient may be expressed by [2.3]. Further, using that $-uʹwʹ=u_{*}^2$ in the surface layer, the flux Richardson number may be written as:

$R=\frac{\frac{g}{\overline{\theta }}\overline{wʹ\theta ʹ}}{-\frac{u_{*}^3}{k_{a}z}}=\frac{z}{-\frac{u_{*}^3}{k_a\frac{g}{\overline{\theta }}\overline{wʹ\theta ʹ}}}=\frac{z}{L}.$([2.12])

Here, $L$ denotes the Obukhov length (also denoted the Monin–Obukhov length). As the Obukhov length is expressed by the surface friction and the vertical turbulent heat flux, it is independent of height (in the region of validity of the assumptions applied). It is therefore frequently used to classify the stability of the boundary layer. Reference Van Wijk, Beljaars, Holtslag and TurkenburgVan Wijk et al. (1990) propose the following ranges for characterization of stability.

Very stable:	$0m<L<200m$
Stable:	$200m<L<1000m$
Near neutral:	$\left|L\right|>1000m$
Unstable:	$-1000m<L<-200m$
Very unstable:	$-200m<L<0m$

A discussion of measured offshore wind speeds, turbulence and stability is found in Reference Nybø, Nielsen and ReuderNybø et al. (2019) and Reference Nybø, Nielsen, Reuder, Churchfield and GodvikNybø et al. (2020). In Figure 2.7, the various regions of stability are shown as a function of the inverse of the Obukhov length.

Figure 2.7 Regions of stability as a function of the inverse of the Obukhov length, $1/L$ . VU: very unstable; U: unstable; NN: near neutral; S: stable; VS: very stable.

As the quantities used in the flux Richardson number are not readily available, an alternative is to use the gradient Richardson number. Here, the ratio between the gradient of the potential temperature and the gradient of the square of the mean wind speed is considered. The flux Richardson and gradient Richardson numbers are related via the eddy diffusivities for eddies and heat; for details, see Reference LeeLee (2018).

2.1.2.3 Shear Exponent and Stability

Over land, the shift between stable and unstable conditions in the atmospheric boundary layer frequently follows a diurnal period. As mentioned in Section 2.1.2.2, the heat exchange between sea and atmosphere differs from that between land and atmosphere. This affects the stability condition of the atmospheric boundary layer and thus the vertical shear profile of the mean wind speed. Figure 2.8 illustrates how the shear exponent $\alpha$ in [2.6] may vary over the year at an offshore location in the North Sea (Reference MyrenMyren, 2021). The results are obtained by considering 11 years of hindcast data in the NORA3 dataset (Reference Haakenstad, Breivik, Furevik, Reistad, Bohlinger and AarnesHaakenstad et al., 2021). In Figure 2.8 it is also observed that the variation shear exponent closely follows the temperature difference between 100 m elevation and sea level, $\Delta T=T_{100}-T_0$ . With this difference less than −1°C, stable atmospheric conditions may be assumed. In the case shown in Figure 2.8, this happens most frequently during autumn and winter. The study by Reference MyrenMyren (2021) showed that the variations over the year were less pronounced as the location moved farther north and farther offshore. This may be explained by lower variation between sea and air temperature in these regions.

Figure 2.8 Hourly mean shear exponent, $\alpha$ , at the Ekofisk area (left axis). $\alpha$ obtained from NORA3 hindcast data (Reference Haakenstad, Breivik, Furevik, Reistad, Bohlinger and AarnesHaakenstad et al., 2021). The dashed line (right axis) shows the mean temperature difference between 100 m above sea level and sea surface. Time period 2004–2015.

Courtesy of Reference MyrenMyren (2021).

It should be noted that the shear exponents in Figure 2.8 in general are lower than recommended in many standards. DNV (2021c) recommends a general $\alpha$ - value over open sea with waves offshore equal to 0.12. This value does not account for atmospheric stability. However, DNV (2021c) gives advice on how the shear profile may be adjusted considering the Obukhov length. The IEC standard 64100-3 (2009) recommends the “normal wind profile” over sea $\alpha =0.14$ , using the reference height equal to hub height. For extreme wind speeds averaged over 3 s and a return period of 50 years, IEC 64100-3 (2009) recommends $\alpha =0.11$ and use of a gust factor of 1.1. No correction for stability conditions is included in this case.

Reference Furevik and HaakenstadFurevik and Haakenstad (2012) studied a large number of offshore wind conditions where both measured and hindcast wind profiles were available. They defined the stable, near neutral and unstable conditions from the temperature difference $\Delta T_{150}$ between the temperature 150 m above sea level and the sea temperature. The unstable, near neutral and stable conditions were defined by $\Delta T_{150}<-1$ , $-1\le \Delta T_{150}\le 0$ , $\Delta T_{150}>0$ . $\Delta T_{150}$ in °C. The corresponding average $\alpha$ -values were found as 0.04, 0.05 and 0.09. These $\alpha$ -values are, as the hindcast data above, lower than recommended by the standards. In all standard wind energy applications an increasing wind speed with height has been assumed. Reference Furevik and HaakenstadFurevik and Haakenstad (2012) observed frequently (in more than 1500 of 8700 cases) a decreasing wind speed with height. In general, these cases had unstable atmospheric conditions. The observations were made in the North Atlantic on the weather ship Polarfront.Footnote ⁵

2.1.2.4 Turbulence

As discussed above, the wind speed at a specific point may be characterized by the mean value plus a stochastic variation, $u=\overline{u}+uʹ$ . The turbulence level is characterized by the standard deviation of the wind speed, ${\sigma }_u={\sigma }_{uʹ}$ . Also, the transverse and the vertical components of the velocity fluctuations are of importance. However, traditionally, and partly due to the measurements available, the wind energy community has considered fluctuations in the mean wind direction only. The turbulence intensity is thus normally written as:

$TI=\frac{{\sigma }_u}{\overline{u}}.$([2.13])

The numerical value of ${\sigma }_u$ is sensitive to the length of the record considered. Traditionally, records of 10 min duration have been used in the wind industry. However, using longer records, e.g., 1 h, increases the computed value of ${\sigma }_u$ as the wind spectra contain significant energy at low frequencies. Also, real wind data are never really stationary. Reference Nybø, Nielsen and ReuderNybø et al. (2019) discuss ways to handle real wind data for use in the analysis of wind turbine dynamics.

Considering a short period of time, e.g., from 10 min to 1 h, the time history of the wind speed may be considered a stationary process. Several formulations of wind spectra are proposed. The most common spectra used for offshore conditions are the Kaimal, von Kármán, Davenport and Harris spectra; see, e.g., DNV (2021c). For example, the Kaimal spectrum may, according to IEC 61400 (2005), be written as:

$S_k\left(f\right)={\sigma }_k^2\frac{A\frac{L_k}{U_{ref}}}{{\left(1+B\frac{f{L}_k}{U_{ref}}\right)}^{5/3}}.$([2.14])

$f$ is the frequency and $L_k$ is a length scale parameter. $U_{ref}$ is the mean wind velocity at a reference height, $z_r$ , e.g., the hub height. The index $k$ refers to the velocity components: $k=1$ mean wind direction, $k=2$ lateral direction and $k=3$ vertical direction. IEC 61400 (2005) recommends the values given in Table 2.1 for the parameters, depending upon direction.

Table 2.1 Parameters to be used in the Kaimal spectrum according to IEC 61400 (2005)

Direction, k	1	2	3
Standard deviation, ${\sigma }_k$	${\sigma }_1$	${\sigma }_2\ge 0.7{\sigma }_1$	${\sigma }_3\ge 0.5{\sigma }_1$
Integral length scale, $L_k$	$L_1=8.1{\Lambda }_1$	$L_2=2.7{\Lambda }_1$	$L_3=0.66{\Lambda }_1$

The length scale parameter is given as:

${\Lambda }_1=\{\begin{array}{c}0.7z_r\text{ for }{z}_r<60m\\ 42m{\text{ for z}}_r\ge 60m\end{array}.$([2.15])

A = 4 and B = 6 are recommended values. Note that in all wind spectral formulations, the high-frequency tail of the spectrum should be proportional to $f^{-5/3}$ .

The wind spectrum may be divided into three ranges: a low-frequency range where large-scale turbulence is generated, denoted as the production range; a medium-frequency range where there is a balance between energy gained from the larger eddies and the energy lost to even smaller eddies, denoted as the inertial subrange; and a high-frequency range where the eddies are so small that the energy is lost in viscous dissipation, denoted as the Kolmogorov dissipation range. The order of magnitude length and time scales for each of these ranges are indicated in Figure 2.1. Experience shows that spectra such as, e.g., the Kaimal formulation represent the measured spectra well in the inertial subrange, at least in near neutral and unstable conditions. The length and time scales in the dissipation range are so small that they do not have any practical impact on wind turbines. However, the energy content in the lower range of the inertial subrange and in the production range may not be well represented by the “standard” spectra. This is a frequency range important to floating structures due to the low natural frequencies for such structures.

Figure 2.9 gives examples of the Kaimal spectrum. For frequencies above about 0.1 Hz, the slope of the spectrum is close to $f^{-5/3}$ . Observe that changing the turbulence intensity while keeping the mean velocity causes a vertical shift in the spectral curve.

Figure 2.9 The Kaimal spectrum for the velocity in the mean wind direction according to IEC 61400 (2005). Reference height $z_r=100m$ . Double logarithmic scale.

The formulation in [2.14] represents the spectrum in the main wind direction and uses the standard deviation in that direction, ${\sigma }_u$ . For the lateral and vertical velocity components, the design standards propose the use of the same spectral formulation, but with modified standard deviations (see Table 2.1).

The turbulence intensity, as defined by [2.13], is in general assumed to decrease with increasing mean wind velocity. Reference Nybø, Nielsen and ReuderNybø et al. (2019) and Reference Nybø, Nielsen, Reuder, Churchfield and GodvikNybø et al. (2020) investigated about one year of wind data from the offshore meteorological measurement platform FINO1 (www.fino1.de/en/; accessed July 2023). Careful quality control of the data was performed prior to the analysis. The quality control implies removing erroneous data due to spikes, missing samples, mast shadow effects etc., as well as implementing requirements related to the stationarity of the records. In Figure 2.10, turbulence intensities for the FINO1 data at 119 m above sea level are plotted for a large number of 60-min records. The TI values are averaged over six consecutive 10-min records sampled at 10 Hz. As shown in the figure, the turbulence intensity is sensitive to the stability of the atmospheric boundary layer. Reference Nybø, Nielsen, Reuder, Churchfield and GodvikNybø et al. (2020) used the classification of stability based upon the Obukhov length as given in Section 2.1.2.2. Figure 2.10 also includes turbulence intensities as recommended in the IEC standard 61400-3 (2009) for offshore wind turbines. These values are supposed to represent the 90th percentile of measured turbulence intensities. The 90th-percentile curves as given in the IEC standard decay monotonically with increasing wind speed. In Figure 2.11 the distribution of stability conditions in the data of Nyb et al. (2020) are plotted as function of mean wind speed. At very low and very high wind speeds, the few occurrences introduce large uncertainties in the distribution.

Figure 2.10 The turbulence intensity (TI) as a function of mean wind speed, from Reference Nybø, Nielsen, Reuder, Churchfield and GodvikNybø et al. (2020). The solid black line represents the 90th percentile as given in the IEC (2009) standard for offshore conditions with reference TI equal to 0.12.

Reproduced from Reference Nybø, Nielsen, Reuder, Churchfield and GodvikNybø et al. (2020) under Creative Common Attribution License No. 5460641106526.

In contrast to onshore conditions, the surface roughness offshore increases with wind speed. The IEC standard assumes the surface roughness can be solved iteratively from the expression $z_0=A_c/g{\left[\kappa {\overline{U}}_{hub}/\ln \left(z_{hub}/z_0\right)\right]}^2$ . Here, $A_C$ is the Charnock parameter and $\kappa$ is von Kármán’s constant. The standard deviation of the wind in the mean wind direction is written as ${\sigma }_1={\overline{U}}_{hub}/\ln (z_{hub}/z_0)+1.84I_{15}$ . Here, $I_{15}$ is the turbulence intensity at hub height at 15 m/s wind speed. In Figure 2.10, the IEC turbulence intensity curve is included, using $A_C=0.011$ , $\kappa =0.4$ and $I_{15}=0.12$ , corresponding to Class C wind turbines. The surface roughness length varies in this case from 4.8E-07 at 1 m/s wind speed to 7.9E-04 at 25 m/s wind speed. The measurements reveal a large scatter in turbulence intensity for the various records analyzed. The turbulence intensity during unstable atmospheric conditions is in general higher than during neutral and stable conditions.

Figure 2.11 Atmospheric stability as a function of mean wind speed at 80 m above sea level. FINO1 data as processed by Reference Nybø, Nielsen, Reuder, Churchfield and GodvikNybø et al. (2020). The number of occurrences within each wind speed interval is given by the solid black line (right axis). The stability limits given in Section 2.1.2.2 are used.

Copied from Reference Nybø, Nielsen, Reuder, Churchfield and GodvikNybø et al. (2020) under Creative Common Attribution 3.0 License No. 5460641106526.

Classical theory describes how the wind shear over the ocean surface creates surface waves, starting from short, capillary waves. As the duration of the wind and or the fetch length increase, the waves become longer, ending up as long-periodic swells with periods beyond 15 s. The understanding of how waves cause variation of the wind speed over an ocean surface is less developed. The waves interact with the wind field both by representing a wavy boundary condition and by a time-varying drag force. The drag force between the air and water is related to the relative velocity between the two media. As the water particles moves back and forth, a time-varying drag force will result. Reference Kalvig, Manger, Hjertager and JakobsenKalvig (2014) studied the effect of the moving boundary and found that long swells could cause variation in the wind speed at vertical levels corresponding to the rotor of an offshore wind turbine. However, more research is needed to fully understand the impact of the waves on the wind field.

2.1.2.5 Coherence

The above discussion on the turbulence spectra is confined to the velocity variation in time observed in one point. To compute the wind loads on a wind turbine, the velocity variations in space are also of importance. To describe the spatial variations, the coherence is used. The coherence of a wind field tells how the variation in wind speed at one point correlates to the wind speed at another point. Consider the rotor plane of a wind turbine. If the rotor diameter is small, it may be assumed that the wind speed is fully correlated over the rotor plane. This holds at least for the frequency ranges important for estimating the power production and dynamic loads. As seen from the power spectra in Figure 2.9, high frequencies correspond to low energy content.

As the diameter of the rotor is increased to beyond 200 m, the issue of correlation of the turbulence over the rotor plane beomes increasingly significant. The variation in the $u-$ velocity over the rotor plane is obviously important. Due to the shear in the mean velocity profile, it is to be expected that the correlation in the $u$ -velocity for points separated in the horizontal direction differs from the correlation between points separated in the vertical direction. Variations in the $v$ and $w$ turbulent velocity components over the rotor plane have had less attention but do also influence the rotor blade loads. In the following, the discussion is limited to the variation in the $u$ -velocity component.

Consider two points, 1 and 2, separated by a distance $r_{12}$ . The auto-spectra for the wind velocity at points 1 and 2 may be denoted $S_{11}\left(f\right)$ and $S_{22}\left(f\right)$ . Similarly, the cross-spectrum for the velocities at points 1 and 2 may be denoted $S_{12}\left(f\right)$ . The coherence may now be written as:

$\gamma (f,r_{12})=\frac{S_{12}\left(f\right)}{\sqrt{S_{11}\left(f\right)S_{22}\left(f\right)}}.$([2.16])

The auto-spectra are real and positive, and in most cases almost equal in the two points considered. However, the cross-spectrum is in the general case a complex quantity. The coherence is therefore also complex, i.e., it contains information about the phases or the time shift between the two time series considered. The real and imaginary parts of $\gamma \left(f,r\right)$ are denoted the co-coherence and quad-coherence respectively. In most wind energy applications the absolute value of [2.16] is used as the coherence, i.e., the coherence is written as follows (Reference Burton, Jenkins, Sharpe and BossanyiBurton et al., 2011):

$\mathrm{coh}(f,r)=\left|\gamma (f,r_{12})\right|=\frac{\left|S_{12}\left(f\right)\right|}{\sqrt{S_{11}\left(f\right)S_{22}\left(f\right)}}.$([2.17])

In most wind standards an exponential coherence function is proposed, e.g., IEC (2005) proposes a coherence on the form:

$\mathrm{coh}\left(f,r\right)=\exp \left[-12{\left({\left(\frac{fr}{U}\right)}^2+{\left(0.12\frac{r}{L_k}\right)}^2\right)}^{0.5}\right].$([2.18])

Here, $U$ is the mean wind velocity, to be taken at hub height. $L_k=8.1{\Lambda }_1$ is the coherence scale parameter; see [2.15]. [2.18] is supposed to be valid for the $x-$ component of the velocity and describes the coherence between two points located in the same $x-y$ plane (e.g., the rotor plane) at a distance $r$ . From [2.18] it is observed that the coherence according to this formulation is always real and positive. It is also observed that the coherence tends to $\exp \left(-1.44r/L_k\right)$ as the frequency tends to zero. For other coherence models, e.g., the Davenport model (Reference DavenportDavenport, 1962), the coherence converges toward unity for zero frequency.

The above formulation of the coherence is to be combined with a model of the point spectrum of the wind, e.g, the Kaimal turbulence spectrum [2.14]. Using the “Sandia” method (Reference VeersVeers, 1988), a complete wind field satisfying the point spectrum as well as the coherence function may be generated. Further details are given below. Frequently, only the variation of the $u$ – component of the wind is considered in these models.

2.1.2.6 Mann’s Turbulence Model

Based upon a linearized version of the Navier–Stokes equations, Reference MannMann (1994) developed a tensor-based model for the spatial structure of the turbulence in the surface layer of the atmosphere. This is the original version of Mann’s turbulence model. The effect of the earth’s rotation, i.e., Coriolis forces, is ignored. To come up with a model for all three turbulent components of the wind field, several important simplifying assumptions are made. Key assumptions are that the air is assumed incompressible; further, a neutral stability of the atmosphere and a uniform shear of the mean velocity are assumed, i.e.:

$\overline{u}\left(z\right)=z\frac{d\overline{u}}{dz},$([2.19])

with $d\overline{u}/dz$ constant. The model uses von Kármán’s turbulence spectrum with isotropic turbulence as a starting point. The effect of shear is included by the so-called “rapid distortion theory” (RDT). The RDT describes how the sheared wind field distorts eddies. Eddies with rotation are either stretched or compressed along the axis of rotation depending upon the direction of rotation. Further, large eddies are assumed to live for longer than small eddies. Small eddies are assumed to be isotropic, simplifying the spectral tensor considerably. Large eddies are assumed to be anisotropic, causing the three components of the velocity fluctuations to differ and fulfil the relation:

${\sigma }_u>{\sigma }_v>{\sigma }_w.$([2.20])

Further, the ensemble average of the product of the horizontal and vertical turbulent velocities, $〈uʹwʹ〉$ , becomes negative, as expected from the 2D boundary layer equations; see Appendix A.

If the fluctuation of the wind speed is measured at a fixed point in space, a time-history of the wind speed is obtained and, by spectral analysis, a frequency spectrum is obtained, as shown in Figure 2.9. In the spectral formulation by Reference MannMann (1994), a wave number spectrum is used rather than a frequency spectrum. The instantaneous wind speed variations along the $x$ -axis at a certain point in time are considered. Computing the spectrum of these wind speeds, $u\left(x\right),v\left(x\right),w\left(x\right)$ , a spectrum based upon wave number is obtained. Invoking the Taylor’s frozen turbulence hypothesis implies that the turbulent structures are assumed to be moving downstream with the average wind speed, i.e.:

$u(x+\overline{u}△t,y,z,t+△t)=u(x,y,z,t).$([2.21])

Bold types denote a vector. Denoting the spectrum based upon wave number $F\left(k\right)$ , the relation between the frequency and wave number representation is given by $S\left(f\right)df=F\left(k_1\right)d{k}_1$ with $k_1=2\pi f/\overline{u}$ . Here, $k_1$ is the wave number along the $x-$ axis. The wave numbers along the $y$ and $z$ axes are independent of the mean velocity.

The model uses the covariance tensor as a basis, denoting the covariance tensor as:

$R_{ij}\left(r\right)=〈u_i\left(x\right)u_j\left(x+r\right)〉,$([2.22])

where $x=\left(x,y,z\right)$ is the position vector, $r$ is a distance vector and $u_i$ are the turbulent velocity components, and $〈〉$ denotes ensemble average. In a homogeneous flow field, the covariance tensor depends upon the absolute value of the distance only. Similarly, as the ordinary frequency spectrum is obtained from the Fourier transform of the auto-correlation function, a spectral tensor is obtained from the Fourier transform of the covariance tensor, i.e.:

${\Phi }_{ij}\left(k\right)=\frac{1}{{\left(2\pi \right)}^3}\iiint R_{ij}\left(r\right)e^{\left(-ik\cdot r\right)}d{r}_{1}d{r}_{2}d{r}_3.$([2.23])

The integrals are taken from $-\infty$ to $+\infty$ . $k=\left(k_1,k_2,k_3\right)$ denotes the wave number vector. The spectral tensor is transformed to an orthogonal process, and the velocity field is then obtained from the Fourier transform of this process. Reference MannMann (1994) shows how the spectral tensor and $F_i\left(k_i\right)$ can be obtained assuming an isotropic turbulence, using von Kármán’s energy spectrum. Then, introducing a vertical shear, the spectral tensor is modified, and an anisotropic flow is obtained. The degree of anisotropy is determined by a parameter controlling the length of life of the eddies.

Three key parameters are used in Reference MannMann’s (1994) formulation of the wind field: a length parameter $L$ to describe the characteristic size of eddies; a parameter $\Gamma$ to characterize the length of life of eddies; and a viscous dissipation rate for the turbulent kinetic energy.

Starting out with von Kármán’s turbulence spectrum, the point spectrum for the $u-$ velocity is obtained as:

$F_u\left(k_1\right)=\frac{9}{55}\alpha {\epsilon }^{\frac{2}{3}}\frac{1}{{\left(L^{-2}+k_1^2\right)}^{5/6}}.$([2.24])

Here, $\alpha {\epsilon }^{2/3}$ is the parameter characterizing the viscous dissipation rate for the turbulent kinetic energy. $k_1$ is the wave number in $x-$ direction. $\epsilon$ is the specific turbulent dissipation as given from the kinematic viscosity and the kinetic turbulent energy $\epsilon =\nu \overline{\left(d{u}_i/d{x}_j\right)}$ (see Appendix A). $\alpha$ is an empirical constant equal to approximately 1.7.

From [2.24] it is observed that the characteristic length, $L$ , may be obtained from $F_u\left(0\right)$ . However, Reference MannMann (1994) comments that the value of the spectrum at low wave numbers (low frequencies) is strongly influenced by non-stationarity and large-scale meteorological phenomena. He therefore suggests using the maximum of $k_{1}F\left(k_1\right)$ as reference. Considering the $u-$ component of the velocity, it is found that:

$L\simeq \frac{1.225}{k_1{}_{\max }},$([2.25])

where $k_{1\max }$ corresponds to the maximum value of $k_{1}F\left(k_1\right)$ .

The linearization of the Navier–Stokes equations causes unrealistic behavior of the flow and the eddies to be distorted beyond what is physical realistic. Therefore, the parameter $\Gamma$ , limiting the length of life of the eddies, is introduced. Eddies are supposed to break up as time goes on and small eddies break up faster than large eddies. Reference MannMann (1994) applies an assumption that the length of life of the eddies may be written in the form:

$\tau \left(k\right)\propto k^{-\frac{2}{3}}\Lambda \propto \{\begin{array}{c}{k}^{-2/3}\text{ for }k\to \infty \\ k^{-1}\text{ for }k\to 0\end{array}.$([2.26])

$\Lambda$ is related to the hypergeometric function;Footnote ⁶ for details, see Reference MannMann (1994). The limit of $\tau$ as $k\to \infty$ is assumed valid in the inertial subrange. In the inertial subrange, [2.26] may be rewritten as a nondimensional lifetime as:

$\beta \left(k\right)\equiv \frac{d\overline{u}}{dz}\tau \left(k\right)={\left(kL\right)}^{-2/3}\Gamma .$([2.27])

$\Gamma$ is to be determined empirically. For $\Gamma =0$ , an isotropic flow is obtained, i.e., ${\sigma }_u={\sigma }_v={\sigma }_w$ . As $\Gamma$ increases, the differences between the three standard deviations increase. In Figure 2.12, the relation between $\Gamma$ and the variances in the three directions of the flow are shown. Further modifications to the model have been made by introducing a blocking effect, accounting for zero vertical velocity component at ground level.

Figure 2.12 Relation between the parameter $\Gamma$ controlling the length of life of eddies and the variances of the turbulence in the three directions as well as the covariance between horizontal and vertical velocity component.

Reproduced from Reference MannMann (1994) with permission from the Journal of Fluid Mechanics, License No. 5431500056035.

In Reference MannMann (1998) and (Reference Mann1994), the above model was tested and the parameters estimated based upon full-scale measurements and for various model wind spectra. Reference Cheynet, Jakobsen and ObhraiCheynet, Jakobsen and Obhrai (2017) found, based upon offshore data at the FINO1 platform for wind velocities at 80 m above sea level and a wind speed range of 14–28 m/s, average values $\Gamma =3.7$ , $\alpha {\epsilon }^{2/3}=0.04m^{4/3}{s}^{-2}$ and $L=70m$ . The estimated values for $\Gamma$ and $L$ are approximately constant over the range of velocities considered, while $\alpha {\epsilon }^{2/3}$ shows a marked increasing trend, from about $0.02m^{4/3}{s}^{-2}$ at the lowest velocity to $0.07m^{4/3}{s}^{-2}$ at the highest velocity.

To account for non-neutral atmospheric stability, Reference Chougule, Mann, Kelly and LarsenChougule et al. (2018) introduce a modification of the original Mann model. A linear vertical variation of the potential temperature is introduced. One or alternatively two extra parameters are needed for this model.

When applying the Mann model for wind field simulations, the tensor description of the wind field in the wave number domain is transformed to an instantaneous wind field in the $\left(x,y,z\right)$ domain for the three turbulent components $\left(u,v,w\right)$ . This is done by first transforming the spectral tensor, ${\Phi }_{ij}\left(k\right)$ , into a set of orthogonal processes (functions). By Fourier transform and summation of these orthogonal functions, the wind field is obtained. Consistent with the frozen turbulence hypothesis, the wind turbine may be moved through this field to obtain a time-dependent inflow.

2.1.5.1 Cup Anemometers

Cup anemometers (Figure 2.18) are widely used for wind measurements. The main principle relies upon the fact that the drag force of a conical cup is larger for flow into the cup than for flow from the back of the cup. In the “Risø” version of the cup anemometer, three cups are mounted on a vertical axis. The rotational resistance in the bearings is very low, causing the rotation to start at very low wind speeds. The rotational speed of the cup anemometer is close to proportional to the wind speed. The relation between wind speed and rotational speed is discussed in some detail in Section 3.5, where a cup anemometer is considered in a power production context. Cup anemometers are considered accurate but have some disadvantages, e.g., they measure the total horizontal wind speed only and give no information about wind direction. To obtain the wind direction, a separate wind vane must be used. As the rotor system has a certain inertia, high-frequency wind speed variations will not be measured. In turbulent wind, the average wind speed tends to be overestimated. This is because the drag characteristic of the cups will cause the rotor to rapidly speed up during a wind speed increase, but the rotational speed will reduce more slowly during wind speed decrease. The cup anemometer must be mounted on a mast. Care must thus be taken to avoid shadow or speed-up effects due to the mast and mounting system. Wind directions affected by the mast and mounting system should be removed from the subsequent data analysis.

Figure 2.18 Cup anemometer with three cups.

Photo by Stephan Kral, University of Bergen.

2.1.5.2 Sonic Anemometers

Sonic anemometers use ultrasonic senders and receivers. Consider the two combined sender and receiver sensors illustrated in Figure 2.20. The time a sound signal requires to move from one sensor head to the other is given by:

$\begin{array}{c}{T}_{AB}=\frac{L}{c+U_{AB}}.\\ T_{BA}=\frac{L}{c-U_{AB}}\end{array}$([2.35])

Here, $U_{AB}$ is the component of the wind velocity along a straight line between the sensors $A$ and $B$ . The distance between the sensors is $L$ . The speed of sound is $c$ . Rearranging [2.35], the following relations are obtained:

$\begin{array}{c}{U}_{AB}=\frac{L}{2}\left(\frac{1}{T_{AB}}-\frac{1}{T_{BA}}\right)\\ c=\frac{L}{2}\left(\frac{1}{T_{AB}}+\frac{1}{T_{BA}}\right)\end{array}.$([2.36])

It is observed that the expression for the wind speed is not dependent upon the speed of sound and that the speed of sound is obtained as well. By locating sensors in three orthogonal directions, all three components of the wind vector can be measured. Figure 2.19 shows an example of a three component sonic anemometer. Compared to the cup anemometer, the sonic anemometer has the advantage of a very high frequency resolution and the possibility of measuring all components of the wind. As for the cup anemometer, one must be aware of possible shadow effects of the mast and mounting arrangement. Experience shows that the sonic anemometer may give erroneous results during rainy weather; see, e.g., Reference Nybø, Nielsen and ReuderNybø, et al. (2019).

Figure 2.19 Example of a three-component sonic anemometer.

Photo by Stephan Kral, University of Bergen.

2.1.5.3 Lidar

The LIDAR measurement technique is a remote sensing technique. The name LIDAR originates from an abbreviation of “light-based radar” or “light detection and ranging.” A simplistic view of the basic principle is illustrated in Figure 2.21. A narrow beam of laser (monochrome) light is sent out from the LIDAR unit. The light is scattered by aerosols in the air and some of the light is back-scattered into the LIDAR’s receiver system, which utilizes the Doppler effect. By detecting the change of frequency in the back-scattered light, the speed of the aerosol particles, i.e., the wind speed in the direction of the laser beam, $U_s$ , is obtained. As the light beam has a finite angle and the duration of the analyzed scattered signal has a finite duration, the velocity obtained is a weighted average over a volume, as illustrated in Figure 2.21. Typical averaging lengths in the direction of the beam are in the order of tens of meters. Assuming no mean vertical component of the wind speed, the horizontal component of the mean wind speed in the vertical plane described by the laser beam is obtained as $U_{\beta }=U_s/\cos \alpha$ . $U_{\beta }$ is the horizontal wind speed component in direction $\beta$ and $\alpha$ is the angle between the laser beam and the horizontal plane.

Figure 2.20 Principle of a sonic anemometer. Sensor heads A and B are located a distance L apart.

Figure 2.21 Basic principle of LIDAR. Some of the emitted light is back-scattered from the aerosol particles in the air between the two planes. From the frequency shift in the back-scattered signal, the component of the wind velocity in the direction of the beam, $U_s$ , may be determined.

To obtain the total horizontal wind speed, at least two orthogonal measurements must be performed. In practice, to obtain the total horizontal velocity, a circular sweeping pattern of the laser beam can be used, as illustrated in Figure 2.22. The laser beam measures the velocities $U_{si}$ at a number of azimuthal angles ${\beta }_i$ . The relation between the mean horizontal wind velocity, $\overline{U}$ , and the measured speed in the direction of the laser beam is thus:

Figure 2.22 Example of circular sweeping pattern of the laser beam to obtain both components of the horizontal mean wind speed. $U$ is the total horizontal wind speed; $U_{si}$ is the measured component of the wind speed in azimuthal angle ${\beta }_i$ .

$U_{si}=\overline{U}\cos \left({\beta }_i-{\beta }_0\right)\cos \alpha .$([2.37])

Here, ${\beta }_0$ is the mean direction of the wind. By measuring for several values of ${\beta }_i$ , the mean wind speed and the mean direction are obtained. A key assumption used to obtain the mean wind speed by this approach is that the wind field is homogeneous over the sweeping area.

Other remote measurement techniques utilize sound (SODAR) and microwaves (RADAR) to obtain the wind velocity. The sound signals are scattered by temperature differences in the air. Assuming that such temperature structures are advected with the mean wind speed, the wind speed in the direction of the sound wave is obtained by considering the frequency shift between emitted and reflected sound (for details, see, e.g., Reference Lang and McKeoghLang and McKeogh, 2011). Microwave radars may be used in combination with SODAR. The radar signals may interfere with the sound waves through the so-called Bragg effect. This makes it possible to determine temperatures in addition to velocities.