### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. The measurements
- 3. The model and the set-up
- 4. Meteorological background
- 5. Results and discussion on the velocity spectra
- 6. Results and discussion on the temperature spectra
- 7. Conclusions
- Acknowledgements
- References

The spectra of the zonal and meridional winds and temperature over the mesoscale range of length-scales (10^{−5}*<k<*10^{−3}rad m^{−1}, with *k* the radian wave number) are examined through a case-study using measurements and simulations. The measurements include 1 s and 10 min average winds and temperature, from which the temporal spectra are obtained by Fourier transform. The mesoscale Weather Research and Forecasting (WRF) model is used to simulate the weather for four days when gravity waves are observed. The four-day dataset also includes a period with unstable conditions and free of gravity waves. The simulation provides a possibility to study the spatial spectra and, together with the measurements, it provides an opportunity to examine the validity of the Taylor hypothesis for transforming between temporal and spatial spectra. We examined these issues under stable and unstable conditions, both in the absence and presence of gravity waves. It was found that, in the absence of gravity waves, the spectral behaviour of wind and temperature (in terms of the form of the spectra and the energy distribution) is similar to the literature, and the Taylor hypothesis is valid. When the gravity waves are present, it was found that the Doppler-shifted frequency has to be taken into account when converting the spectra between time and space. The simulation suggests that, in the presence of gravity waves, the kinetic energy is not evenly distributed between the zonal and meridional winds, rather the wind component along the main wave propagation direction contains larger variance. Copyright © 2011 Royal Meteorological Society

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. The measurements
- 3. The model and the set-up
- 4. Meteorological background
- 5. Results and discussion on the velocity spectra
- 6. Results and discussion on the temperature spectra
- 7. Conclusions
- Acknowledgements
- References

Most mesoscale atmospheric spectra are obtained in terms of frequency, *f*, from measurements over time at fixed points. In turbulence theory, wave number, *k*, and frequency spectra are equivalent when space and time are assumed to be related by the advection velocity via the Taylor hypothesis. Using the background wind speed **U**_{0}, the spectrum can be converted between time (*f*) and space (*k*) through the Taylor transformation:

- (1)

where *ω* is the wave frequency observed in a fixed system and thus *ω* = 2*πf*. The magnitudes of the vectors **k** and **U**_{0} are *k* and *U*_{0}. Here, **U**_{0} =**U**_{0}(*u*_{0}*,v*_{0}), with *u*_{0} and *v*_{0} the zonal and meridional background wind speed, respectively.

Although the Taylor transformation is an empirical device, Brown and Robinson (1979) found it to be valid for tropospheric winds over Eastern Europe in the spatial range 500–2500 km. In Gage and Nastrom (1986), the *k*-spectra for winds, measured by 6900 commercial aircraft flights in the upper troposphere and low stratosphere, were shown to be in good agreement with Taylor-transformed *f*-spectra from several sources and this was used as an evidence for the validity of the Taylor transformation. In Högström *et** al.* (1999), the validity of the Taylor transformation was confirmed for zonal winds measured from a tower and aircraft in the lowest 1000 m of marine atmosphere over the Baltic Sea.

The successful transformation of mesoscale velocity spectra was used by Brown and Robinson (1979), Gage and Nastrom (1986) and Högström *et** al.* (1999) as an argument that their measurements of wind and temperature are of a two-dimensional (2D) turbulent nature. A deviation from Taylor's hypothesis would be expected for internal waves.

In the mesoscale range spanning from approximately a few kilometres to 600 km (corresponding roughly to *k* ∼ 10^{−5} to 10^{−3}rad m^{−1}), the spectra from Gage and Nastrom (1986) for *u* and *v* and temperature, *T*, show a universal form and they decrease with wave number as *k*^{−5/3}. Such a dependence of the spectra on *k* is also found in Högström *et** al.* (1999) for the same wave number range. This fits the general theory for 2D turbulence presented by Kraichman (1967). The *k*^{−5/3} dependence of the spectral energy has also been predicted by Lilly (1983). According to Lilly's theory, through convection, the 3D motions are ‘squeezed’ to be nearly 2D, thus transferring energy from smaller to larger scales. However, Gage and Nastrom (1986) argue that the energy source at the small scale end (i.e. with wavelength *λ* = 2*π/k* about a few km) is breaking internal waves near the tropopause. Högström *et** al.* (1999) suggest that internal waves may make up the spectral region of higher wave numbers between 10^{−3} and 10^{−2}rad m^{−1}, where their spectra have a slope of −9/4, which is close to the prediction of van Zandt (1982) from wave theory. Based on a wave number slope of –3 for a universal saturated gravity wave spectrum, Fritts and van Zandt (1987) derived and observed Doppler-shifted frequency spectra to have a slope of –2, instead of –5/3. However, there are no definitive statements on the true nature of the spectral behaviour. The studies Gage and Nastrom (1986) and Högström *et** al.* (1999) show that the energy of *u* and *v* are comparable, and that of *T* has a dependence on the background stratification.

The above-mentioned studies cover the boundary layer, the troposphere and lower stratosphere, which together extend from tens of metres to 14 km above the surface. Most of the data from those studies correspond to conditions of strong stable stratification. In this study, we examine the spectra of wind and temperature in the mesoscale range, both in the spatial and temporal domains, i.e. in terms of wave number and frequency. The *f*-spectra will be derived from time series measured from several offshore meteorological masts. In order to study the spatial variability of the wind and temperature, we will make use of the Weather Research and Forecasting (WRF) model (Skamarock *et** al.*, 2007, http://www.wrf-model.org/index.php). Our interest is a climatologically stably stratified environment above the water body around Denmark. As will be demonstrated in section 4, here the atmosphere is stably stratified for most of the year. Stable stratification is the favourable condition for gravity waves to be generated or transported from the permanently stable free atmosphere above. The focus here is on a period when gravity wave events are observed. A portion of the period is characterized by convective conditions, free of gravity waves. With this situation, we can follow up on the studies of Gage and Nastrom (1986), Högström *et** al.* (1999), etc., for stably stratified conditions, but also can investigate the topic for convective conditions. The spectral form and energy distribution of the wind components and temperature, as well as the validation of the Taylor transformation, can be examined for cases with gravity waves both present and absent.

A brief description of the measurements is given in section 2. The model and its set-up are introduced in section 3. The climate of stability conditions in the area of the measurements and the meteorological conditions are described in section 4. The *f*- and *k*-spectra for velocity and the Taylor transformation between them are presented and discussed in section 5 for non-gravity wave and gravity wave cases. Spectra of temperature are presented and discussed in section 6. Conclusions follow in section 7.

### 2. The measurements

- Top of page
- Abstract
- 1. Introduction
- 2. The measurements
- 3. The model and the set-up
- 4. Meteorological background
- 5. Results and discussion on the velocity spectra
- 6. Results and discussion on the temperature spectra
- 7. Conclusions
- Acknowledgements
- References

The measurements were made at four meteorological masts in the neighbourhood of the Nysted offshore wind farm (http://www.dongenergy.com/Nysted), located south of the Danish island of Lolland. Figure 1 shows the wind farm layout, and the placement of the four masts. Figure 2 shows a larger domain.

The instruments mounted on the masts include cup anemometers at heights above mean sea level of 10, 25, 40, 55, 65 and 69 m, and wind vanes and ambient temperature sensors at 10 and 65 m. Air pressure at 10 m, *P*_{10}, and water temperature, *T*_{w}, 2 m below the water surface, were also measured. Potential temperature, *θ*, was calculated from temperature and pressure.

The data were sampled at 1 Hz frequency and they are available as 1 sec as well as 10 min average values. A significant drawback of the measurements is that there is no vertical velocity record.

### 3. The model and the set-up

- Top of page
- Abstract
- 1. Introduction
- 2. The measurements
- 3. The model and the set-up
- 4. Meteorological background
- 5. Results and discussion on the velocity spectra
- 6. Results and discussion on the temperature spectra
- 7. Conclusions
- Acknowledgements
- References

The Weather Research and Forecasting (WRF) model with the Advanced Research WRF (ARW) core version 3.1 is used. The National Centers for Environmental Prediction (NCEP) Global Final Analysis (FNL) on a 1°×1° grid is available every 6 h and it is used as the forcing for the WRF simulation. The 0.5° sea-surface temperature from NCEP (ftp://polar.ncep.noaa.gov/pub/history/sst) is used in both. The Yonsei University Planetary Boundary Layer (PBL) scheme (Skamarock *et** al.* 2007) is used and, for the microphysics, the Thompson *et** al.* (2004) scheme is chosen.

Figure 3 shows the three nested domains at 27 km (outermost, domain I), 9 km (middle, domain II) and 3 km (innermost, domain III) resolution, using 37 vertical levels in all domains. Two-way nesting is applied to the two smallest domains. The time step is taken as 3 min and the output is recorded every 10 min.

In domain III there are in total 123 rows in the south–north direction, and 144 columns in the west–east direction. For reference, the location of mast M2 near the wind farm is 54.5352°N, 11.6635°E, and the grid point that is located closest to M2 corresponds to row 44 and column 65 of domain III.

### 4. Meteorological background

- Top of page
- Abstract
- 1. Introduction
- 2. The measurements
- 3. The model and the set-up
- 4. Meteorological background
- 5. Results and discussion on the velocity spectra
- 6. Results and discussion on the temperature spectra
- 7. Conclusions
- Acknowledgements
- References

The water area of our study is part of the semi-enclosed Baltic Sea. Here the atmosphere has been reported to be often stably stratified (e.g. Lange *et** al.*, 2004). Based on the bulk Richardson number, *Ri*_{B}, calculated with years of measurements from the Nysted masts between 10 m and 65 m, the atmosphere is stably stratified (*Ri*_{B} > 0) for 61% of the time and it is very stably stratified (*Ri*_{B} > 0.25) about 26% of the time. Here *Ri*_{B} is calculated from

- (2)

where *U* is the wind speed. In order to avoid wake disturbance, data from masts M2 and M3 are combined. We select data from mast M2 when the wind direction is between 180° and 360°, and from mast M3 when the wind direction is between 0° and 180°. These values of *Ri*_{B} seem to be consistent with the Richardson numbers from Smedman *et** al.* (1997) which were calculated with measurements from a site on the west coast of Gotland island in the Baltic Sea.

Some of the stable cases at this study location are contributed by weather systems dominated by Atlantic lows coming from the west. The period we choose to study, from 5 to 8 November 2006, is such a case. Figure 4 shows the evolution of some key mean parameters during this period. Some observations are missing on 6 November.

On 6 November, the air is stably stratified but rather close to neutral. Figure 2 shows a satellite wind field retrieved from Synthetic Aperture Radar (SAR) data from 2053 UTC on 6 November, where a wave pattern can be seen over most of the water area. SAR utilizes the fact that radar backscatter from the sea surface depends on centimetre-scale waves generated by the local wind (Valenzuela 1978). The relationship of radar backscatter to the 10 m wind speed and direction is described by an empirical model function (Hersbach *et** al.* 2007). The spatial resolution of the wind speed map is approximately 300 m. SAR wind retrievals work best over open water; due to both graphic resolution and data quality, the wave pattern is rather vague in coastal areas. Figure 2(b) is a close-up of Figure 2(a); here the wave pattern at the mast site is quite clear.

Throughout 8 November, unstable conditions predominate, where *Ri*_{B} < 0 except for the last hour. Gravity waves are not expected.

Two days, 6 and 8 November, are chosen to represent cases with and without gravity waves, respectively. When analyzing the *k*-spectra, we use data from 2100 to 2200 UTC on 6 November to represent the case with gravity waves and 0600 to 0700 UTC on 8 November to represent the case without gravity waves. These two hours are marked in Figure 4.

The WRF-simulated 10 min values of wind speed, direction, temperature and mean sea level pressure from the grid point closest to mast M2 are plotted in Figure 4, together with 10 min measured averages from mast M2. The first three hours from the simulation are from the model spinning-up period and are therefore disregarded. The overall agreement is good. Strictly speaking, here the measurements are 10 min averages while the simulated values have a disjunct sampling interval of 10 min, and also the measurements are from one single point and the simulated values are spatially averaged over the model resolution, i.e. 3 km. This sometimes introduces uncertainty in the comparisons, but here, since the conditions are rather homogeneous and stationary, it should not be a problem.

#### 4.1. Wave characteristics at around 2100 UTC on 6 November

The low-pass filtered wind speed at several measuring heights, i.e. 10 m to 69 m, as well as the autocorrelation of the wind speed, shown in Figure 5 and 6, together support the conclusion that the gravity waves are present at least for a couple of hours around 2100 UTC on 6 November.

According to the SAR image, there appear to be 11 waves within 2° in longitude; at the present latitude, this corresponds to a wavelength, *λ*, of approximately 11.6 km. Both Figures 5 and 6 show a wind speed variation with a period, *T*, of approximately 12 min. Thus, the total wave phase velocity (*c*_{0}) can be calculated according to *c*_{0} = *λ/T*, and this gives *c*_{0} ≈16.1 m s^{−1}.

With the data available, it is a difficult task to determine the wave propagation direction. Multiple masts could have given useful information, but data are missing from mast M4, and masts M1 and M2 are too close together, while mast M3 lies in the wake of the wind farm. As an alternative, we apply the linear wave theory to get an approximate direction of the wave propagation. For linear waves, there is a strong correlation between pressure and wind. A series of studies (e.g. Nappo, 2002; Koch and Golus, 1988) has demonstrated that the strong pressure–wind correlation can be used as supporting evidence for gravity waves. According to this theory, the wave vector lies along the line linking the maximum and minimum wind vectors and points to the maximum pressure perturbation (Nappo, 2002, see also his figure 9.4). In reality, turbulence close to the surface can obscure wave perturbations and this will make the linear theory difficult to apply directly (Tjernström and Mauritsen 2009). However, as can be seen from Figure 5, from 2100 UTC there is a correlation between the wind and pressure perturbation. We found the minimum and maximum wind vectors in the three wave cycles after 2100 from low-pass filtered (at 4 min) time series. The directions of the minimum and maximum winds are almost the same, about 272°, and this situation is remarkably stable throughout the three wave cycles. Distribution of the vectors of wind, pressure and wave propagation for this situation is illustrated in Figure 7, similar to Figure 9.4 in Nappo (2002). As shown, the linear wave theory suggests that the wave is propagating towards the west with a tiny component to the north.

Obviously, one can see from the SAR images (Figure 2) that the wave front curves in space. The non-uniform wave propagation direction implies additional uncertainty in making the spectral transformation. At the mast site, we marked the wave ridge. The angle to the north–south orientation is about 20°. A range of angles from 10° to 20° is used when calculating the components of the wave phase velocity, in order to see the sensitivity of the wave curves in the spectral transformation; this sensitivity turns out to be low.

Due to the data limitations, estimation of the wave parameters is rather rough. However, we proceed with these parameters, to see if they give consistent results in the transformation of the spectra between time and space.

### 6. Results and discussion on the temperature spectra

- Top of page
- Abstract
- 1. Introduction
- 2. The measurements
- 3. The model and the set-up
- 4. Meteorological background
- 5. Results and discussion on the velocity spectra
- 6. Results and discussion on the temperature spectra
- 7. Conclusions
- Acknowledgements
- References

Similar analysis has been performed for the temperature spectra. Figure 14, similar to Figure 9 for the velocity spectra, shows good agreement between the *f*-spectra from the measurements and the simulation. The spectrum of temperature (or potential temperature) is closely related to the horizontal velocity spectra in shape. Here, from the measurements, the temperature spectra are found to remain consistent between the two measuring levels, 10 m and 65 m, and the spectra from the simulations are consistent between multiple levels throughout the boundary layer. Note that, at levels other than 2 m, the simulation gives potential temperature *θ* instead of the ambient temperature *T*. However, there is no visible difference between the spectra of *θ* and *T*. However, as reported by Gage and Nastrom (1986) and Högström *et** al.* (1999), the temperature spectral energy level varies. In their studies, it was found that, at least partly, the background static stability explains the variability. They also pointed it out that, regardless of whether it is gravity wave or 2D turbulence, the spectra of scalar variables such as temperature are related to the vertical displacement spectrum in a fundamental way. The *k*-spectra of temperature are plotted in Figures 10 and 12 for the convective case on 8 November and the gravity wave case on 6 November, respectively. For the convective case 0600–0700 UTC on 8 November, the temperature spectra from the three transects are not always the same; variances from transects (b) and (c) are about the same level but higher than that from (a). The distributions of simulated potential temperature differences between 28 m and the sea surface along these transects have shown to be similar for transects (b) and (c) while they are different from (a). Also, the level of the spectrum of *T* for the stable case on 6 November is lower than that on 8 November (Figures 10 and 12); the convective case on 8 November could correspond to a larger average vertical displacement, and accordingly larger temperature variance.

In their theoretical analysis, Gage and Nastrom (1986) predict

- (4)

where Φ_{θθ} is the potential temperature spectrum and Φ_{PE} is the spectrum of potential energy (Gage and Nastrom, 1986, their Eq. (6)). Gage and Nastrom found a systematic difference in the spectral energy level for the tropospheric and stratospheric temperature spectra. The temperature spectra are converted to corresponding spectra of potential energy using Eq. (4). The spectra of potential energy Φ_{PE} from the troposphere and stratosphere are found to collapse. Gage and Nastrom interpret this as ‘the potential temperature spectrum adjusts through background stability into approximate universality’. The dependence of Φ_{θθ} on *N* seem to have explained well the distribution of the spectral energy of temperature with height up to 248 m in Högström *et** al.* (1999).

A similar collapse is also observed with our case. It is shown in Figure 15 with data from the along-wind transects for both the non-gravity wave case on 8 November and the gravity wave case on 6 November. The spectra of potential temperature from different levels are plotted in (b) and (e), with four levels in the boundary layer, one level at about 1500 m and one at 7700 m. For the non-gravity wave case,*N* is not real inside the boundary layer due to instability; the corresponding spectral energy of *θ* is significant (b), due very likely to the large vertical displacement during convective conditions. For this group of data, we cannot use Eq. (4) to make a conversion to Φ_{PE}. For the rest, the spectra of Φ_{PE} are obtained from those of *θ* and they are plotted in Figures 15(c, f). These show that spectra of Φ_{PE} from surface up to 7700 m do collapse. This strongly supports the argument from Gage and Nastrom that the variations in potential temperature spectral amplitude are due to variations in hydrostatic atmospheric stability.

Due to the varying stratification conditions for the two cases on 6 and 8 November 2006, either in time or in space, or both, the temperature spectra are not transformed between time and space.