Full-Scale Spectrum of Boundary-Layer Winds

Abstract

Extensive mean meteorological data and high frequency sonic anemometer data from two sites in Denmark, one coastal onshore and one offshore, have been used to study the full-scale spectrum of boundary-layer winds, over frequencies f from about $1\,\hbox {yr}^{-1}$ to 10 Hz. 10-min cup anemometer data are used to estimate the spectrum from about $1\,\hbox {yr}^{-1}$ to $0.05\,\hbox {min}^{-1}$; in addition, using 20-Hz sonic anemometer data, an ensemble of 1-day spectra covering the range $1\,\hbox {day}^{-1}$ to 10 Hz has been calculated. The overlapping region in these two measured spectra is in good agreement. Classical topics regarding the various spectral ranges, including the spectral gap, are revisited. Following the seasonal peak at $1\,\hbox {yr}^{-1}$, the frequency spectrum fS(f) increases with $f^{+1}$ and gradually reaches a peak at about $0.2\,\hbox {day}^{-1}$. From this peak to about $1\,\hbox {hr}^{-1}$, the spectrum fS(f) decreases with frequency with a $-2$ slope, followed by a $-2/3$ slope, which can be described by $fS(f)=a_1f^{-2/3}+a_2f^{-2}$, ending in the frequency range for which the debate on the spectral gap is ongoing. It is shown here that the spectral gap exists and can be modelled. The linear composition of the horizontal wind variation from the mesoscale and microscale gives the observed spectrum in the gap range, leading to a suggestion that mesoscale and microscale processes are uncorrelated. Depending on the relative strength of the two processes, the gap may be deep or shallow, visible or invisible. Generally, the depth of the gap decreases with height. In the low frequency region of the gap, the mesoscale spectrum shows a two-dimensional isotropic nature; in the high frequency region, the classical three-dimensional boundary-layer turbulence is evident. We also provide the cospectrum of the horizontal and vertical components, and the power spectra of the three velocity components over a wide range from $1\,\hbox {day}^{-1}$ to 10 Hz, which is useful in determining the necessary sample duration when measuring turbulence statistics in the boundary layer.

Appendix 1: The Impact of Using One-Day Sonic Data to Calculate the Spectrum

The turbulence spectra have been calculated with 1-day long sonic data from days as listed in Tables 1 and 2. This procedure ensures a detailed description of the spectral regions in and around the gap. However, it raises some issues about the statistics of the established spectra. The daily spectrum can obviously not be considered an analysis of a stationary series, because (a) the wind during the day typically undergoes a systematic diurnal variation, and (b) the low frequency region of the spectrum shows a $f^{-2/3}$ power law, which is far from the $f^{+1}$ power law required by the Wiener–Khintchine theorem for possible stationarity. We refer to the spectrum as fS(f) vs. f on a log–log scale.

In spite of this, we may claim that the $f^{+1}$ spectral region of the annual spectrum ($f\lesssim 2 \times 10^{-6}$ Hz, see Fig. 6) provides good reasons for expecting a yearly-averaged diurnal spectrum to be determined for frequencies $>$1 day$^{-1}$, if a large enough ensemble of diurnal data series are analyzed in the spectral domain for the year considered.

However, given the spectral slope of $f^{-2/3}$ at frequencies around $1\,\hbox {day}^{-1}$, the low frequency region of the spectra, from averaging spectra for the day-long time series, is enhanced by leakage of energy from lower frequencies. This enhancement will not disappear by ensemble averaging; hence, it has to be counteracted. This is usually performed by applying windows, such as Hanning and Hamming windows, imposing a sinusoidal window onto the time series (Kristensen et al. 1992; Kristensen 1998). Unfortunately, for the present time series typically associated with diurnal stability variations, the application of a window would modify the relative weight of the different stability classes. Therefore the window method was not used.

Instead, we have tried to control the leakage by excluding cases where the spectral amplitude is beyond the two standard deviations of all spectra. These cases have shown excessive characteristics, mostly associated with large and narrow gusts or strongly non-stationary conditions. All together, there are 20 days of such conditions.

Fig. 13

The power spectrum fS(f) from a five-day long time series together with the mean of five spectra calculated from each of the five days

For comparison, a spectrum from a 5-day long time series is calculated and shown as the blue circles in Fig. 13. At the same time, a spectrum from each of the five days was also calculated and the five spectra were averaged afterwards and are shown as dots in the same plot. The good agreement between the circles and the dots suggest that there is no principle problem in using 1-day time series for calculating the spectrum. The inertial subrange spectral values for the 5-day spectrum is slightly, but systematically, larger than the similar one for the 1-day spectrum. To explain this, we assume that the frequency spectrum is mainly a wavenumber spectrum being advected past the sensor by a “fluctuating mean wind”. The variance of this fluctuating advection flow will be larger for the 5-day series than for any of the 1-day series. This would enhance the measured 5-day spectrum slightly more than the 1-day spectrum, following the model of Wyngaard and Clifford (1977) for Taylor’s hypothesis with a fluctuating advection wind.

Finally in Fig. 14 we show the spectra with all sonic data in Table 1 (the dashed curves) together with those with outliers removed (solid curves, same cases as from Figs. 3d and 4d). The higher values of the dashed curved are seen as the leakage caused by days corresponding to highly non-stationary conditions. Apart from the magnitude, the distribution of the spectra with height remains the same.

Fig. 14

Power spectra from both 2012 and 2013 (Figs. 3d and 4d together), with outliers excluded (solid curves) and included (i.e. all data from Table 1, dashed curves)