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Keywords:

  • gravity waves;
  • k-spectrum;
  • f-spectrum;
  • Taylor transformation;
  • Doppler shift

Abstract

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

The spectra of the zonal and meridional winds and temperature over the mesoscale range of length-scales (10−5<k<10−3rad 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The measurements
  5. 3. The model and the set-up
  6. 4. Meteorological background
  7. 5. Results and discussion on the velocity spectra
  8. 6. Results and discussion on the temperature spectra
  9. 7. Conclusions
  10. Acknowledgements
  11. 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 U0, the spectrum can be converted between time (f) and space (k) through the Taylor transformation:

  • equation image(1)

where ω is the wave frequency observed in a fixed system and thus ω = 2πf. The magnitudes of the vectors k and U0 are k and U0. Here, U0 =U0(u0,v0), with u0 and v0 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 10−3rad 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 = 2π/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 10−2rad 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The measurements
  5. 3. The model and the set-up
  6. 4. Meteorological background
  7. 5. Results and discussion on the velocity spectra
  8. 6. Results and discussion on the temperature spectra
  9. 7. Conclusions
  10. Acknowledgements
  11. 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.

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Figure 1. The layout of the wind farm and the placement of the four masts M1, M2, M3 and M4 (from Antoniou et al. (2006), slightly modified). In (a), the axes of the left plot are in metres. In (b), the scale of the west–east width of the wind farm, 6 km, is marked. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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Figure 2. (a) Wind field at 10 m retrieved from satellite SAR observations by ENVISAT, at 2053 UTC on 6 November 2006. The location of the Nysted wind farm is indicated by the arrow. (b) is a close-up of (a) in grey scale; the wind farm is indicated by the arrow and a dashed line is drawn to follow approximately the wave ridge around the site. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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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, P10, and water temperature, Tw, 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The measurements
  5. 3. The model and the set-up
  6. 4. Meteorological background
  7. 5. Results and discussion on the velocity spectra
  8. 6. Results and discussion on the temperature spectra
  9. 7. Conclusions
  10. Acknowledgements
  11. 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.

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Figure 3. The three domains in the WRF simulation, at resolutions of 27, 9 and 3 km, respectively. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The measurements
  5. 3. The model and the set-up
  6. 4. Meteorological background
  7. 5. Results and discussion on the velocity spectra
  8. 6. Results and discussion on the temperature spectra
  9. 7. Conclusions
  10. Acknowledgements
  11. 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, RiB, calculated with years of measurements from the Nysted masts between 10 m and 65 m, the atmosphere is stably stratified (RiB > 0) for 61% of the time and it is very stably stratified (RiB > 0.25) about 26% of the time. Here RiB is calculated from

  • equation image(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 RiB 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.

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Figure 4. Mean parameters during the period 5–8 November 2006: (a) wind speed at 10 m; (b) wind direction at 10 m; (c) temperature at 2 m, simulated (WRF) and at 10 m, measured (OBS), water temperature measured (Tw) and as used in WRF simulation (SST); (d) pressure at 10 m, measured (OBS) and at mean sea level (WRF); (e) bulk Richardson number RiB. The vertical lines mark the two test cases: 2100–2200 on 6 November, labeled with ‘gravity wave’ and 0600–0700 on 8 November, labeled with ‘non-gravity wave’. Measurements (dots) around midday on 6 November are missing for several hours. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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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 RiB < 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.

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Figure 5. (a) Variation of wind speed at 10 m, 40 m and 65 m, and (b) pressure perturbation at 10 m, both from 2000 to 2200 UTC on 6 November. Data are from mast M1 and they are from 1 s values low-pass filtered at 4 min. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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Figure 6. Autocorrelation coefficient of the wind speed at 10 m, versus time lag, over 2100–2200 UTC on 6 November. Data are bandpass filtered between 4 and 20 min. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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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 (c0) can be calculated according to c0 = λ/T, and this gives c0 ≈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.

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Figure 7. Using the wind-pressure correlation in the linear wave theory to determine the wave vector. Distribution of the maximum and minimum wind vectors and pressures between about 2100 and 2130 on 6 November. The wave phase velocity is between the maximum and minimum wind vectors, pointing to the maximum pressure. (cf. Figure 9.4 in Nappo (2002))

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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.

5. Results and discussion on the velocity spectra

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

5.1. f-spectra

First we examine the f-spectra from mast measurements for the period 5 to 8 November 2006. The 1 s time series of zonal (u) and meridional winds (v) from mast 2 at three heights (10 m, 40 m and 65 m) were Fourier-transformed and those for the 10 m are plotted in Figure 8. The missing data on 6 November are filled by linear interpolation. For f > 10−3Hz, the spectral forms bear some classical boundary layer characteristics (e.g. Kaimal and Finnigan, 1994) and here the spectrum is height-dependent. However this height dependence is not observed in the mesoscale range of 10−5<f<10−3Hz. Here S(f) decreases with increasing f, roughly following f−5/3. This slope describes the data reasonably well, although there is an indication that our data have a slightly larger slope. The 10 min spectra overlap with the 1 s spectra over frequencies lower than the Nyquist frequency corresponding to 20 min, although there is slightly more noise at the lowest frequencies.

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Figure 8. f-spectra of (a) u10 and (b) v10 from measurements at 10 m from 5 to 8 November 2006, from 10 min averages and from 1 s values. The −5/3 slope is indicated by the thick straight lines. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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Variances corresponding to waves with periods of approximately 12 min are not directly shown in the 10 min spectra. With the model resolution of 3 km, the waves with wavelength approximately 12 km are expected to be poorly resolved. However, the matching of the 10 min spectra with the 1 s spectra suggests that the energy source and sink for the mesoscale range are well kept in the 10 min spectra.

The 10 min spectra from the measurements and the simulation for the period 5 to 8 November are plotted together in Figure 9 as S(f) versus f. The comparison has been made at several levels from 10 m to 65 m and the results are similar; here 10 m is shown. Following the measurements well, the spectra from the simulation fall approximately as f−5/3, with energy levels very close to observations. Consistent with Figure 8, the energy density from the simulation does not vary with height through the entire boundary layer.

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Figure 9. f-spectra of (a) u10 and (b) v10 for 5 to 8 November 2006, from 10 min averages of measurements (OBS) and 10 min values from WRF simulation. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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The good agreement of the spectra between the measurements and the simulation suggests that WRF simulates well the energy source and sink of the spectra for the mesoscale range.

5.2. k-spectra and the Taylor transformation

The west–east as well as south–north transects are used to make the k-spectra. At the same time, the spectra are also presented in the wavelength (λ) domain. During 6 November, the wind is from the west, so the two transects are approximately along and across the wind. During 8 November, the wind is from the southwest so, in addition, a southwest–northeast transect is included to be along the wind. We present the spectra from the transects passing through the grid point that is located closest to mast M2.

In order to examine the validity of the Taylor transformation, these spatial spectra are Taylor-transformed to the f-domain and compared with those obtained directly from the time series. As mentioned earlier, if the spectra originate in 2D turbulence, it is expected that Eq. (1) can successfully transform the spectra between time and space (Brown and Robinson 1979; Gage and Nastrom 1986; Högström et al. 1999).

However, in the presence of gravity waves, we would expect the f-spectra to experience Doppler shift and therefore a conversion of the spectrum between time and space should take the intrinsic frequency into account (e.g. Gage and Nastrom, 1986; Fritts and van Zandt, 1987). The intrinsic wave frequency, Ω, is the frequency of a wave measured by an observer drifting with the fluid at the background speed U0; therefore, the frequency follows:

  • equation image(3)

(Nappo 2002). Apparently, the Taylor transform of Eq. (1) is a special case with Doppler shift having Ω [RIGHTWARDS ARROW] 0.

In the boundary layer, winds could vary with height significantly. The vertical variation of u and v, especially for the neutral to stable case on 6 November, indicates that we have to choose to use a mean background wind speed when applying Eqs (1) and (3). From the simulation, it is seen that the wave pattern remains throughout the entire boundary layer. Since we will make the spectral conversion from the k-spectra obtained from the WRF simulation, it is quite natural to use the simulated averaged background wind components throughout the boundary layer depth. At 0600–0700 UTC on 8 November, at the site there is a convective boundary layer and the depth is about 630 m according to the model output of PBL height. At 2100–2200 UTC on 6 November, there is a strong shear layer up to about 400 m, continued by a gradual transition to a much stronger inversion layer. The mean background wind components during the hour have been estimated from the entire transect, from surface to the boundary layer height from the simulation. The results of the Taylor transformation are not very different from those obtained using the background wind components from the single grid point closest to mast 2. This implies that, for the current case, the wind is rather close to the overall mean wind conditions throughout the entire transect, so that inhomogeneity does not happen to be a problem.

The spatial velocity spectra of the two cases, in the absence of gravity waves at 0600–0700 UTC on 8 November and in the presence of gravity waves at 2100–2200 on 6 November, are presented in the following two subsections.

5.2.1. In the absence of gravity waves

The k- and λ-spectra of u, v and T at several levels within the boundary layer from the hour 0600–0700 UTC on 8 November are calculated for three transects: (a) west–east direction, (b) south–north direction, and (c) approximately along the wind direction. Figure 10 shows the spectra for u and v at 10 m and T at 2 m for transect (c). The high wave number ends at about equation image km, twice the distance of two neighbouring grid points along the transect and the low wave number ends at about 2 × 10−5rad m−1, which is about 400 km, the length of the transect.

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Figure 10. Spatial spectra of u10, v10 and T2 versus wave number k and wavelength λ for the non-gravity wave test case at 0600–0700 UTC on 8 November (indicated on Figure 4). Data are from the transect along the wind direction in the model domain. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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In spite of more fluctuations in some range and less in other range, the spectra for the three transects have shown at least the following in common: (i) 2D isotropic turbulence as the kinetic energy is evenly distributed between u and v: spectral energy for u and v are comparable and they are of the same level as that from Gage and Nastrom (1986), their Figure 1; (ii) spectra follow approximately a –5/3 slope as indicated by the straight line; (iii) energy of u and v are comparable between different transects, regardless of the transect orientation; and (iv) the variation of the spectral energy with height is negligible, if there is any.

The k-spectra from the transect along the wind (Figure 10) are Taylor-transformed to the f-spectra through Eq. (1), for u and v. In Figure 11, the Taylor-transformed f-spectra are plotted together with the f-spectra obtained from time series, measured and simulated, on 8 November from mast 2 at 10 m. It seems that the transformed spectral energy (thick dashed curves) are somewhat larger but the general agreement is good, considering the assumptions applied in both the Fourier transform and the Taylor transformation: homogeneous and stationary conditions, plus the use of the averaged boundary layer wind speed as the background advection. In fact, the mean k-spectrum calculated from 144 spectra for the entire day gives similar results to that from the hour 0600–0700 UTC.

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Figure 11. Temporal spectra of (a) u10 and (b) v10 for the non-gravity wave test case. The solid curves are obtained directly from time series of u10 and v10, and the dashed curves (TT1) from the corresponding curves in Figure 10 through the Taylor transformation by Eq. (1). This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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5.2.2. In the presence of gravity waves

For the gravity wave case, 2100–2200 on 6 November, the k- and λ-spectra are made for the west–east (approximately along-wind) as well as south–north (approximately across-wind) transects. Spectra at several levels within the boundary layer are calculated and those at 10 m are presented in Figure 12 for the west–east transect. One striking difference between this figure and Figure 10 is that here the spectral energy of u is one order of magnitude larger than that of v. The distribution of the spectral energy is similar in the south–north transect. The independence of the spectra on the transect orientation is similar to the non-gravity wave case. Since a 2D turbulence theory predicts that the kinetic energy should be equally partitioned between the two components, it seems that the presence of the gravity waves has violated the 2D isotropy, leading to the wind variance being more significant along the main wave propagating direction.

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Figure 12. Spatial spectra of u, v and T versus wave number k and wavelength λ at 10 m, for the stable test case 2100–2200 on 6 November (indicated on Figure 4). This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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In spite of the different level of spectral energy between u and v, there are several patterns that are the same as the non-gravity wave case. First, the horizontal kinetic energy is more or less the same as that for the case shown in Figure 10, and they both match the kinetic energy level from Gage and Nastrom (1986), together with data from several sources (Figure 4 in Gage and Nastrom, 1986). Second, in spite of fluctuations, the spectral form follows approximately k−5/3. Third, within the boundary layer, there is no obvious variation of the spectral energy with height.

If we neglect the presence of the gravity waves and use Eq. (1) to transform the k-spectra to f-spectra, we obtain the thick dashed curves in Figure 13(a, b), for u, v, respectively. Compared to the f-spectra obtained from measured and simulated time series, a shift is quite obvious. The shift suggests that, through the transform without consideration of gravity waves, the frequency is too high along u (here also the main wave propagating direction) while too low along v. This is exactly the symptom of Doppler shift if the relative speed between the wind and wave is smaller in the u-direction while larger in the v-direction than the background wind components in the two corresponding directions. It is consistent with our estimation of the wave propagation in section 4.1. The background wind components are u0 = 13 m s−1, and v0 = −1m s−1. If we apply an angle of 20° between the wave ridge and the south–north orientation, the wave phase speed in the east–west direction (which is opposite in direction to u0) is 15.1 m s−1, and in the south–north direction (also against v0) it is 5.5 m s−1. Now we apply Eq. (3) and obtain the Doppler-shifted f-spectra, which are shown in Figure 13(c, d), also as the thick dashed curves. The improvement is clear, although the agreement is better in some ranges than others, which could be due to fact that we did not take into account the varying wave front direction over the domain. Nevertheless, Figure 13 suggests strongly that the correction of the Doppler shift is important in the transformation in the presence of gravity waves.

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Figure 13. Temporal spectra of (a, c) u10 and (b, d) v10 for 6 November. The solid curves are obtained directly from time series of u10 and v10, and the dashed curves from the corresponding curves in Figure 12. In (a) and (b) these are obtained through Eq. (1) (TT1), and in (c) and (d) through Eq. (3) (TT2). This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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6. Results and discussion on the temperature spectra

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The measurements
  5. 3. The model and the set-up
  6. 4. Meteorological background
  7. 5. Results and discussion on the velocity spectra
  8. 6. Results and discussion on the temperature spectra
  9. 7. Conclusions
  10. Acknowledgements
  11. 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.

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Figure 14. f-spectra of T (10 m for measurements and 2 m for WRF) for 5–8 November 2006, from 10 min averages of measurements (OBS) and 10 min values from the WRF simulation. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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

  • equation image(4)

where Φθθ is the potential temperature spectrum and ΦPE 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 ΦPE 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 Φθθ 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 ΦPE. For the rest, the spectra of ΦPE are obtained from those of θ and they are plotted in Figures 15(c, f). These show that spectra of ΦPE 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.

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Figure 15. Conversion of the spectra of potential temperature to the spectra of potential energy using Eq. (4), from the along-wind transects. (a, b, c): non-gravity test case 0600–0700 UTC on 8 November. (d, e, f): gravity wave test case 2100–2200 UTC on 6 November. (a, d) show profiles of the Brunt–Väisälä frequency N at several levels, (b, e) the spectra of potential temperature from different levels as marked in (a, d), and (c, f) the converted spectra of potential energy. This figure is available in colour online at wileyonlinelibrary.com/journal/qj

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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.

7. Conclusions

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

This study shows that

  • The WRF model with the ARW core can be used to study the spatial variability of the meteorological parameters in the mesoscale range of a few kilometres to a few hundred kilometres. The simulation provides f-spectra of wind and temperature that are in good agreement with measurements for the mesoscale range, suggesting that the energy source and sink for this range are well simulated.

  • In the absence of gravity waves, the Taylor transformation by using background wind speed can successfully convert the velocity spectra between time and space. The kinetic energy is evenly distributed between the zonal u and meridional wind v variances, and the k-spectra of u and v have similar shape and energy level to those reported by Gage and Nastrom (1986).

  • In the presence of gravity waves, the Doppler shift is significant and the Taylor hypothesis fails for the transformation between the k- and the f-spectrum. In the present study, the transformation was found to be robust when the Doppler shift was accounted for. It is found from the simulation that the kinetic energy is still at the same level as that for the non-gravity wave case, but in the k-spectrum the kinetic energy is not evenly distributed between u and v, and the variance is larger along the main direction of wave propagating, here the u-component.

  • In the mesoscale range, 10−5<k<10−3rad m−1, the spectra fall off approximately as k−5/3, although the data from current study suggest a slope slightly larger than -5/3. It is similar for the f-spectra in the corresponding range. We speculate that for our convective case, the energy source at the small scale end of the mesoscale range is convection while for the gravity wave case, it is breaking waves.

  • The spectra of potential temperature vary with background stratification. The Taylor transformation, even with consideration of wave propagation, is limited in transforming the temperature spectra due to the limitation in taking the stratification effect into account.

Acknowledgements

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

This work is supported by the project ‘Mesoscale Variability’, 2007-1-7141. The authors acknowledge Dong Energy for the measurements. We thank Mark Kelly, Andrea Hahmann and Claire Vincent for discussions and suggestions. We are grateful for the valuable comments and suggestions from the two reviewers. The satellite data are provided by the European Space Agency.

References

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