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A Model for the Spectrum of the Lateral Velocity Component from Mesoscale to Microscale and Its Application to Wind-Direction Variation

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Abstract

A model for the spectrum of the lateral velocity component \(S_v(f)\) is developed for a frequency range from about 0.2 \(\hbox {day}^{-1}\) to the turbulence inertial subrange, with the intent of improving the calculation of flow meandering over areas the size of offshore wind farms and clusters. These sizes can correspond to a temporal scale of several hours, much larger than the validity limit of typical boundary-layer models, such as the Kaimal model, or the Mikkelsen–Tchen model. The development of the model is based on observations from one site and verified with observations from another site up to a height of 241 m. The model describes three ranges: (1) the mesoscale from \(0.2\ \hbox {day}^{-1}\) to about \(10^{-3}\ \hbox {Hz}\) where a mesoscale spectral model from Larsén et al. (2013: QJR Meteorol Soc 139: 685–700) is used; (2) the spectral gap where the normalized v spectrum, \(fS_v(f)\), can be approximated to be a constant; (3) the high-frequency range where a boundary-layer model is used. In order to demonstrate a general applicability of the lateral velocity spectrum model to reproduce the statistics of wind-direction variability, models for both horizontal velocity components, u and v, are used through an inverse Fourier transform technique to produce time series of both components, which in theory could have been the ensemble for calculating the corresponding spectra. The ensemble is then used to calculate directional statistics, which in turn are compared with corresponding statistics from the measured time series. We demonstrate the relevance of the constructed spectral models for calculating meandering effects for large wind farms and wind-farm clusters.

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Notes

  1. Visit www.4coffshore.com for the sizes of these farms in the North Sea.

  2. https://www.fino1.de/en/.

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Acknowledgements

The first author acknowledges the support from the ForskEL/EUDP OffshoreWake Project (PSO-12521, EUDP 64017-0017). We thank DTU colleague Morten Nielsen for discussions.

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Correspondence to Xiaoli G. Larsén.

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Appendices

Appendix 1: The Derivation of the Spectrum of the Lateral Velocity Component of the Mikkelsen–Tchen Model

The Mikkelsen–Tchen spectrum model (Mikkelsen et al. 2017) modifies and extends the classical Kaimal spectrum model for near-neutral stratified surface-layer turbulence in two ways. Firstly, by adding a parametrization of a near-surface shear production subrange corresponding to the \(-1\) power law (S(f) versus f in log−log coordinates) (Tchen et al. 1985). Secondly, by replacing the spectral form in the Kaimal spectrum below the peak frequency, with a new frequency subrange characterized by turbulence generated from a height-independent constant outer length scale \(\varLambda _s\).

The \(-1\) power spectral law was argued in Tchen et al. (1985) and Hunt and Carlotti (2000) to be applicable to the horizontal velocity components. According to the turbulence theory of Tchen, a distinct shear-production subrange appears in strong shear flow at scales where turbulent coherent structures or eddies satisfy the condition \(k z<1\), where k is the wavenumber. Briefly, regarding its physical interpretation, in the atmospheric surface layer, the classical \(-5/3\) spectrum of turbulence for the u component results from equilibrium between production and the dissipation of turbulent kinetic energy (Tchen 1953). Near the Earth’s surface, the production rate is contributed to by the sum of 1) spectral transfer of energy cascading down the energy spectrum from larger to smaller eddies, and 2) locally generated turbulence by shear.

At all times in an equilibrium spectrum, the production rate per wavenumber is balanced by the turbulence dissipation rate per wavenumber. In the classical inertial subrange with only weak shear, this equilibrium is established between the spectral cascade of energy dissipation, leading to the classical \(-5/3\) inertial subrange.

However, in the atmospheric surface layer the wind shear increases in inverse proportion with the height z above the ground, and so does the production rate of wind-shear-generated turbulence. Depending on the wavenumber, the contribution from shear-generated turbulence eventually exceeds the height-independent energy cascade. Hence, nearest to the ground, shear-generated turbulence dominates the production rate. When balanced by dissipation, an equilibrium spectrum with a characteristic \(-1\) power law is established. In the lower part of the surface layer, the so-called eddy surface layer, a distinct \(-1\) shear-production subrange has been observed by, for example, Högström et al. (2002), Drobinski et al. (2004) and Mikkelsen et al. (2017). Furthermore, the Mikkelsen–Tchen spectrum embeds a third subrange to parametrize the lowest part of the boundary-layer turbulence (cf. Högström et al. 2002). The low-frequency subrange represents turbulence generated by a single outer length scale \(\varLambda _s\). Correspondingly, the third subrange in the Mikkelsen–Tchen spectrum model is parametrized via \(\varLambda _s\), which is of the order of the boundary-layer depth h.

In near-neutral conditions, the value of h can be estimated from the friction velocity \(u_*\) and the Coriolis parameter \(f_c\) through \(h=cu_*/f_c\), where c is a constant that varies between 0.3 and 0.07 (Seibert et al. 2000). According to Högström et al. (2002), the outer length scale can be estimated as \(\varLambda _s=Au_*/f_c\), where A, in general being a function of height z, is a parameter to be determined from observations.

In practice, truly neutral atmospheric boundary layers are rare. In most studies, when the near-neutral height scale \(u_*/f_c\) is used, there are additional variables and coefficients to address the other factors, such as convective mixing heights and atmospheric stratification based on stability. It remains challenging how to accurately address the stability effect in those algorithms. The mixing height needs either to be measured or be modelled through, for example, a mesoscale model such as the WRF model.

In the following two subsections, we briefly introduce the Mikkelsen–Tchen spectrum for the u component from Mikkelsen et al. (2017) and derive the corresponding spectrum for the v component.

1.1 Appendix 1.1: The Mikkelsen–Tchen u Spectrum

The constant of the proportionality \(c=0.2\) is most often applied for estimation of the neutral-boundary-layer depth \(h=0.2 u_*/f_c\). By investigating spectra from the Laban Mill campaign, Högström et al. (2002) estimated \(A_u \approx 0.6\) at \(z=15\) m. The outer length scale for the u spectrum therefore corresponds to \(\varLambda _{s,u} \approx 3h\) in this case. This factor of 3 is contained in the parametrization of the lower dimensionless frequency \(n_l\): \(0.6/0.2=3\). Note, here, the use of frequency with the units Hz and the non-dimensional frequency are f and n, respectively, which is reversed compared with Mikkelsen et al. (2017). Also, instead of the one-sided spectrum as in Mikkelsen et al. (2017), here we use a two-sided spectrum.

The u-spectrum model, with its three parameters, is

$$\begin{aligned} \frac{fS_u(f)}{u_*^2}= \frac{0.5 a_u n/n_{l,u}}{(1+n/n_{l,u})(1+n/n_{u,u})^{2/3}}, \end{aligned}$$
(7)

where \(a_u=0.953\), \(n_{u,u}=0.185\) and \(n_{l,u}=f_c z/(A_u u_*)\). Equation 7 is also given in Eq. 2 without normalizing fS(f) with \(u_*^2\).

1.2 Appendix 1.2: The Mikkelsen–Tchen v Spectrum

In the Mikkelsen–Tchen v spectrum, the two parameters, \(a_v\) and \(n_{u,v}\), have been determined similarly to the u spectrum by aligning the asymptotic limits of the Mikkelsen–Tchen model to the corresponding neutral Kaimal spectrum for the lateral wind component v.

For the non-dimensional lower frequency bound of the shear production subrange, we assume that the outer length scale of the v spectrum, \(\varLambda _{s,v}\), is three magnitudes of order smaller than the value of \(\varLambda _{s,u}\); that is \(\varLambda _{s,v} \approx h\), leading to \(n_{l,v}=f_c z/(A_v u_*)\) (Richards et al. 2000; Fichtl and McVehil 1970).

The classical Kaimal−Finnigan spectra also have different outer length scales for the u and v components. The u spectrum scales with 102n and the v spectrum scales with 17n, which suggests the value of \(\varLambda _{s,v}\) is about 6 times smaller than the value of \(\varLambda _{s,u}\) and corresponds to a choice of \(A_v=0.1\).

The v-spectrum model is

$$\begin{aligned} \frac{fS_v(f)}{u_*^2}= \frac{0.5 a_v n/n_{l,v}}{(1+n/n_{l,v})(1+n/n_{u,v})^{2/3}}, \end{aligned}$$
(8)

with \(a_v=0.906\), \(n_{u,v}=0.283\) and \(n_{l,v}=f_c z/(A_v u_*)\), where \(A_v=c\).

The values of \(A_u\) and \(A_v\) are adjusted to best fit the models with observations, here for the complex terrain Østerild site (see Sect. 3.2).

Appendix 2: Generation of Time Series from Spectra

We generate from the spectral models time series of the u and v velocity components containing m data points, with a resolution of dt, where \(dt=1\) s. First, we generate a list of complex numbers for both the real and imaginary parts through a random process with mean and variance zero and one, respectively. A frequency domain is created using \(f=(i/2)/(m dt)\), with \(i=1,...,m\), for which, the spectral model S(f) is calculated. This spectrum S(f) is imposed on the random complex time series through the term \((S(f) m dt/(2 (2 \pi )^2))^{1/2}\), before being mirrored to a double length. A discrete Fourier transform of this new series of complex numbers gives the time series.

This process shares the same principle as Mann (1998), where the difference is that Mann (1998) uses a three-dimensional, boundary-layer spectral tensor for the u, v, and w components, while here we treat the components u and v as uncorrelated and separately, with both following random processes. Thus, while the phase is not resolved, this information is not needed here. In addition, we include both the three-dimensional boundary-layer spectral model and the mesoscale model.

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Larsén, X.G., Larsen, S.E., Petersen, E.L. et al. A Model for the Spectrum of the Lateral Velocity Component from Mesoscale to Microscale and Its Application to Wind-Direction Variation. Boundary-Layer Meteorol 178, 415–434 (2021). https://doi.org/10.1007/s10546-020-00575-0

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