Analysis of Long Time Series of Coastal Wind

Transcription

1 907 Analysis of Long Time Series of Coastal Wind TORE HEGGEM, RUNE LENDE, AND JØRGEN LØVSETH Department of Physics, Norwegian University of Science and Technology, Trondheim, Norway (Manuscript received 10 June 1997, in final form 0 January 1998) ABSTRACT A study of a 14-yr time series of wind speed recorded on the coast outside the city of Trondheim in middle Norway is presented. Analysis of the time series shows that in this area there is, in general, no gap in the wind speed power spectrum in the 1 h 1 region. For heights below 0 40 m, dependent on the exact distance from the shoreline, there is a dip in the spectrum around mhz; for higher elevations, there is a monotonical rise in the spectral energy for decreasing frequency f until the weather peak. Far below the weather peak, the spectrum decreases as f. Analyses of quasi-stationary 1-h time series indicate that in addition to weather (synoptic) variations and the turbulence due to ground friction peaking in the 0.01-Hz region, there are other mechanisms feeding energy into the spectrum in the mesoscalar (1 h) region, where the spectral gap conventionally is assumed to be found. As an example, a selected time series is analyzed that shows a pronounced peak in the spectrum at 0.4 mhz. Time series with a periodic structure are predominantly found for unstable atmospheric conditions over the sea, with air temperature 4 K or more below the sea temperature. It is shown that unstable atmospheric conditions in the surface layer over the sea will quickly turn into stable to neutral conditions over land, thus the local lapse rate is not necessarily a good indicator for conditions causing mesoscale fluctuations. Aliasing effects and sensor response are also studied. 1. Introduction Based on the classic paper by Van der Hoven (1957), most textbooks on atmospheric physics refer to a spectral gap in the region corresponding to time periods of 1/ 4 h, or frequencies of mhz. This serves as a convenient separation between microscale fluctuations, or turbulence, on the one hand and synoptic variations of the mean flow, or weather variations, on the other hand. The existence of a gap has been confirmed in several investigations, for example, by Kaimal et al. (197) in Kansas and by Kaimal (1978) in Minnesota. Smedman-Høgstrøm and Høgstrøm (1974) find a gap both in the longitudinal wind spectrum and in the temperature and humidity spectra. Fiedler and Panofsky (1970) pointed out that the situation was less clear over the ocean and over very smooth terrain. Agee et al. (1973) have reviewed mesoscale cellular convection (MCC) and have observed diameter to depth ratios of up to 30 to 1. MCC can give turbulence in the spectral gap region. LeMone (1976) has discussed the influence of longitudinal rolls, which may have lengths up to 500 km. Smedman (1991) discusses the occurrence of roll circulations in a shallow Corresponding author address: Jørgen Løvseth, Dept. of Physics Lade, NTNU, N-7034 Trondheim, Norway. jorgen.lovseth@phys.ntnu.no boundary layer over the Baltic Sea, concluding that such phenomena give rise to rolls passing every minutes. Rothermel and Agee (1980), Agee and Gilbert (1989), and Agee and Hart (1990) have discussed aircraft observations of MCC. For the China Sea cell diameters up to 39 km are reported. Stull (1988) points out that larger cumulus clouds may act as eddies with timescales of the order of an hour. Courtney and Troen (1990) conclude from the Lammefjord experiment in Denmark that the gap region has about half the spectral density of the turbulence peak. Gjerstad et al. (1995) have reported spectra for maritime wind based on four time series of length 11 h 40 min for each of three classes of atmospheric stability. For wind speed spectra, a minimum was present for stable conditions but not for unstable and stable to neutral conditions. For the temperature spectra, a minimum or plateau was only indicated for the spectra at 5 m and one of the spectra at 45-m height, in both cases for stable conditions. Most of the time series for unstable and neutral to unstable conditions indicated a periodic structure, with periods in the 1-h range. In the present paper, new measurements from two stations at 4-km separation on the Atlantic coast of mid- Norway (64 N) are presented and analyzed. A set of time periods are selected for the purpose of obtaining characteristic average spectra for the locations. In section, the stations are presented, and limitations 1998 American Meteorological Society

2 908 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 55 FIG. 1. Map of the western part of the island Frøya and the small islet Sletringen. The nearest fishing village, Titran, is also shown. The contour intervals are 0 m. There are few areas above 0 m. of the sensors and recording system are discussed. Some of the problems with a spectral analysis of the dataset are discussed in section 3. For the station Sletringen [where the data used by Gjerstad et al. (1995) also were collected], a time series of 160 days has been analyzed. For the other station, Skipheia, with a smaller sector having directed oceanic fetch, a 14-yr time series is analyzed and presented in section 4. The spectra at 10-m height indicate a shallow minimum at frequencies corresponding periods of 10 min, while the spectra at 40 m and higher (0 m and up for Sletringen) just show a change of slope in the gap region. A systematically selected set of 1-h time series from Skipheia for the period are analyzed and discussed in section 5. The problem with stability assessment for the Skipheia station is discussed. The stability in the surface layer observed a few kilometers inland is shown to deviate dramatically from the stability at the shore station Sletringen, as may be expected from theoretical reasons as well. Several of the time series indicate a quasiperiodic structure, with periods of 1/ 1 h, indicating large spectral energy in the traditional gap region. This is discussed in section 6. Some conclusions are presented in section 7.. Experimental setting In this section we will discuss the effects from the weather variations, the location, the wind speed sensors, and the measurement system. a. Location The data analyzed in this paper are collected at two measurement stations. Both stations are situated on the island Frøya, located on the midwestern part of the Norwegian coast (63 40 N), approximately 100 km west of the city Trondheim. In Fig. 1, a map of the western part of Frøya and the islet Sletringen is shown. The stations are close to the fishing village Titran. Only a few reefs are located outside Frøya and the sector from southwest to north-northeast faces the North Atlantic. The branch of the Gulf Stream, flowing north along the Norwegian coast, carries warm water and will, in general, heat and destabilize the colder air. As we will see, this influences the spectra in the 1-h region for westerly winds. The main station, Skipheia, is located on the western part of Frøya. The terrain surrounding the station consists of small hills covered with heather and areas of bare rock face in between. The maximum heights are 0 m above MSL, and the horizontal spacing of the hills is m. The wind pattern observed at the station is strongly dependent on the direction. In the maritime sector, (SW to NE through north), the wind is coming from the North Atlantic but has traveled over land for the last 1 3 km. In the complementary sector 50 35, the wind is coming from inland and has therefore traveled across a mixture of land and sea. This sector will be referred to as the land sector. Data from two sectors are excluded from the 1-h spectra. In the first, 40 60, two wind energy con-

3 909 verting systems (WECS) are located 300 m from the measurement mast. The WECS are expected to introduce extra turbulence. In the second excluded sector, 150 0, the wind will have entered land over a steep hill, causing a speedup effect, and these measurements will not be included in the 1-h spectra. In the long-term time series all directions are of course included. Even though the turbulence generated by the windmills will contribute to the long-term spectra, the influence is very small since the probability of the wind coming from this direction is low; see section b. The second station is located on a small islet, Sletringen, approximately 400 m in diameter, located 4 km to the west of the reference station at Skipheia. The mast is located 150 m from the shore line in the maritime sector. Sletringen has a lighthouse 55 m high, located 00 m east of the measurement mast. This is the only obstacle on the islet and the rest of the islet has a maximum height of 4 m. Sletringen consists of bare rock face, and we therefore assume that from at least a height of 10 m the sensors are exposed to true maritime wind in the maritime sector as defined above. A study of some selected time series from Sletringen, each of length 10 h 40 min has been reported earlier by Gjerstad et al. (1995). b. Measurement stations The reference station, Skipheia, has been operative since 198. For shorter periods other data than wind and temperature have also been measured, including radiation, rainfall, humidity, and pressure. The Skipheia station has three masts placed in a triangle, m apart. The results given here are based on measurements from mast, which is 100 m high. We will also use the sea temperature data measured near the station. Mast is equipped with temperature sensors, direction sensors, and wind speed sensors. Temperature and wind speed are measured at 10 m, 40 m, 70 m, and 100 m. Wind direction sensors are located at 40 m and 100 m. Wind speed sensors have been placed on slender rods at a distance of.5 m from the mast. These sensors are duplicated (in the western and easterly direction) to avoid shading effects from the mast. During data analysis, the upwind sensor is normally automatically selected. Until 1988 the system was only logging 10-min mean values and -s gust values. After 1988 the system was redesigned and has since been logging at a rate of 0.85 Hz (51 samples per 10 min) for all sensors. The reason for choosing this logging frequency is that FFT routines work most efficiently when the number of data points can be factored into prime numbers as small as possible. The data acquisition system for the stations was developed at the department. The present logging computer is running the real-time system OS-9000 (trademark of Microware Systems Corporation). More detailed descriptions of the measurement station may be found in Løvseth et al. (1996), in Andersen et al. (1993), or, for a complete discussion, see Aasen (1995). In 1991 the station was destroyed in a fire, and the station was down for nearly a year. To continue the logging, a battery-supported system was installed. This system measured 10-min average wind speed, gust, and direction. The station at Sletringen has been operative for two periods: and This station is based on the same type of sensors and measurement system as the Skipheia station. It consists of a 45-m mast equipped with temperature sensors at 5 and 45 m, and one wind speed sensor at each of the following heights: 5 m, 10 m, 0 m, 4 m, and 46 m. The speed sensors are placed on slender rods 3.0 m to the west of the mast. For periods when the stations have been down, the missing data have been substituted with data from nearby stations or from the Norwegian Meteorological Institute to provide a continuous time series for the longterm spectrum. The upper part of Fig. shows the average wind speed in 5 sectors as function of direction, showing a clear peak in the southwest direction. The lower part shows the distribution of the observed wind direction for all data and for mean wind speed greater than 8 m s 1.We see that the probability for maritime wind is high, particularly for strong winds, thus the highest energy contribution to the wind power spectra comes from this sector. c. Logging limitations The temperature sensors were made at the department. Each unit has two thermistors that have been separately calibrated. The temperature resistance characteristics for the thermistors are fitted by a formula with accuracy better than 0.01 K. The resulting overall accuracy is better than 0.0 K. For details see Heggem (1997). The other sensors used at the stations are commercially available. We will here discuss only the response of the wind speed sensor since this will influence the high-frequency part of the wind power spectrum. The wind speed sensors are cup anemometers with a distance constant, C d 1.5 m, verified by Aasen (1995). Effects that could distort the spectra are the following. R Limited sensor response for high frequencies. The wind speed sensor behaves as a low-pass filter. If we assume that the true wind power spectrum is given by S t ( f) and the observed spectrum by S w ( f), then the squared frequency response function H( f) is given by S w( f ) 1 H( f ), (1) S t( f ) 1 ( f )

4 910 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 55 FIG.. Mean speed as function of the direction (upper) and the probability density distribution as function of direction and wind speed. The legend corresponds to the different heights. where is a time constant related to the distance constant C d and the mean wind speed U by C d U. With the sampling frequency chosen, the response is fully adequate. R Resolution noise. The wind speed sensors used give a certain number of pulses per revolution (mostly 14, corresponding to 1.4 m of wind way), and the metering is realized by counting the number of pulses during the sampling period. The resulting resolution (sampling) noise in the spectra can be shown to be negligible. Adding the pulses during the sampling interval corresponds to integrating the signal over the interval. This is the same as applying a nonoverlapping averaging filter to the data, with a frequency response H f given by sin( f / f ) H ( f ) s f, () ( f / f ) where f s is the sampling frequency. R Aliasing. According to the sampling theorem, all frequency components above the Nyquist frequency f N f s / will be folded back into the frequency range that we observe. That is, at a frequency f we will observe a contribution to the spectrum from signal frequencies f and f s f. The contribution from frequency components above f s can, in our case, be neglected. The general relation between the observed spectrum S o after the sampling and the true wind power spectrum S t may, under the assumptions above, be written as s [ ] f S t( f ) S t( fs f ) o f s ( f / f s) ( ( fs f )/ f s) S ( f ) sin. (3) Near the Nyquist frequency the response has entered the region where the wind power spectrum is in the inertial range, falling off as f 5/3. We may use this to correct for the error introduced by the sampling. With the assumptions above, the relation between the observed spectrum S o and the true spectrum S t can be written as [ 11/3 ] s sin( f / f s) f S o( f ) S t( f ) 1. (4) ( f / f ) f f s From this we see that, for example, the observed spectrum at the Nyquist frequency has dropped by a factor 8/ of the true spectrum. Equation (4) has been used to modify the observed spectrum to represent the true spectrum. 3. Comments on the data analysis In this section we will discuss the long-term wind power spectra calculated from the two stations Skipheia and Sletringen. a. Spectral density In the continuous case, the spectral density S( f) is related to the variance as

5 911 f h S( f ) df, (5) f l where the limits of the integration corresponds to the lowest observable frequency in the time series, f l, and the highest, f h. In the discrete case it may be shown that, if the sampling frequency f s is chosen at least as high as twice the highest frequency in the signal f h, we may substitute the integral in (5) by a sum, N/ S i( f i) f. (6) i 1 The fundamental frequency f 1 is defined from the length of the time series. If we consider a time series observed over the time T consisting of N samples equally spaced in time, then f 1 1/T. All other frequencies in the spectral density function are multiples of this frequency given by f i i/t, where i 1,..., N/, the basic spectral resolution being f 1/T. In practice, the spectral density may be found from a standard FFT routine, where the squared Fourier coefficients, when summed, should equal the variance of the original time series. An estimate of the spectral density is then given by the squared Fourier coefficients divided by the distance between the frequency components f 1/T. b. Data preparation Due to the long period we are analyzing, the calculation of the spectra was done in two or three steps. The Sletringen spectrum was based on 160 days of observations (winter 1995/96). The first step was to analyze a time series of 1.5-min mean values. The resulting spectrum was then added to the high-frequency (HF) spectrum calculated as the mean of the day spectra for the same period; see discussion below. For the Skipheia case, with 14 years of observations, the calculation was split into three parts. The low-frequency part of the spectrum was calculated from 90- min mean values. The intermediate part was calculated from 10-min mean values, averaged over one year at a time. These 14 spectra were then averaged to give a mean annual spectrum. For the HF part, where we do not have 14 years of observations, data from August 1995 to August 1996 were used. Due to power failure, repair at the station, or obvious errors in the data, some time series had gaps. If more than 18 points (.5 min) were missing, the time series was rejected. For time series with smaller gaps we designed an algorithm that puts in data for the missing measurements. This algorithm is based on inserting an artificial time series with the same spectral properties as the adjacent parts of the time series. To reduce the effect from trend when calculating the HF part of the spectrum, two different methods were examined. We first used one recommended method based on removing a 40-min running average from the original time series. This is equivalent to high-pass filtering the data with a filter with cutoff frequency at f c 1/400 Hz. It may be shown (see Kaimal and Finnigan 1994) that such a filter would give a too high estimate for the frequencies corresponding to periods around 40 min. Although it effectively removes frequency components lower than around 1 h 1,itisunsuitable for our analysis because of the overestimation of the coefficients around 40 min. The other method we tried was just to remove linear trend in the 1-day time series. This seemed to give a better result than applying the filter technique. The spectra for the different frequency ranges were then joined together in the overlapping region. For the low-frequency end of the spectra, the estimates have errors due to subfrequencies. For the highest frequencies, the spectra have errors due to filtering and aliasing effects. When the spectra were combined, the lowest and highest decades were not used and mean values were calculated for the remaining overlapping region. The spectra for the different frequency intervals did, however, show a very good agreement in the overlapping parts. 4. Discussion of the long-term spectra In Fig. 3 the spectra from Sletringen (Fig. 3a) and Skipheia (Fig. 3b) are presented. Both spectra behave similarly for the low-frequency part, showing a spectral peak around (3 day) 1 corresponding to the synoptic peak. The Skipheia spectrum, based on 14 years of data, goes nearly as f for frequencies below the synoptic peak. The annual component of the spectra has been removed by fitting a trigonometric function, with a period of 1 yr. This could, in the Skipheia case, also be found directly from the Fourier coefficient 14. The mean and the yearly amplitude values are given in Table 1. The daily mean wind can be expressed as ] [ U(z, d) U (z) A u (z) cos (d 4), (7) where d is the day number, U is the mean wind during the year, and A u is the yearly amplitude. The phase lag corresponds to a maximum on 7 December. No noticeable daily or 1-h component of the wind was found. In the HF part of the spectra there is a maximum around 0.01 Hz for the sensors nearest to the ground. At Sletringen, this maximum has disappeared at a height of 0 m. At Skipheia, where the turbulence is expected to be higher due to larger roughness length, there is also a maximum in the spectrum at 0-m height. The energy contribution in the mesoscale range could be explained by the fact that our observations consists mainly of observations from unstable atmospheric conditions, even during the strong wind periods. Contri-

6 91 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 55 FIG. 3. Long-term spectra for (a) Sletringen and (b) Skipheia. butions from cellular convection, discussed in section 6, are frequently present and are expected to feed energy into the mesoscale region. TABLE 1. Yearly mean and yearly harmonic amplitude [see Eq. (7)] of the wind found from the long-term series at Skipheia. Height (m) Mean speed Ū(z) (m s 1 ) Annual amplitude A u (z) Twelve-hour wind power spectrum To check if the lacking gap observed in the spectra of the long time series discussed above are due to a strong tail of the synoptic maximum combined with a low gustiness, a set of stationary 1-h periods were selected for analysis. What is meant by stationary is defined below. If the energy in the 1-h region is due to the tail of the weather peak, it should be strongly reduced by this selection, which implies stable weather. If, however, there are distinct mesoscalar processes causing the energy, the energy should still be there. The time series should be as long as possible to reduce noise from subharmonics. On the other hand, the number of time series should be as large as possible to enhance the statistical relability of the averaged spectral estimates. As a reasonable compromise, a period length of 1 h was chosen. This is an order of magnitude longer than the central period for the gap region and still short enough to ensure a reasonable number of available periods. As discussed in the previous section, no observable peak corresponding to a diurnal period is present in the data. To get reasonably stationary time series, we defined the following criteria for selection of the 1-h time series. R The wind direction should not change more than 45 during the period and the mean wind direction should not come from one of the excluded sectors as defined in section a. R The mean wind should be above 8 m s 1 and no more than a 30% increase/decrease of the 10-min mean values during the period was allowed. Once these two criteria were satisfied, that is, for stable wind direction and stable wind speed, the hourly mean temperature was found to show very little variation. At Sletringen, the local lapse rate has been used for stability classification with good results; see Gjerstad et al. (1994). Comparing data from Sletringen and Skipheia for the same time periods, dramatic differences in the temperature profiles were found. This is illustrated in Fig. 4,

7 913 FIG. 4. Temperature profiles at Skipheia and Sletringen, grouped according to temperature difference between the sea and air at 40-m height. Ten-minute mean values from the period of November 1995 to April 1996 are used. The number at the bottom of each plot gives the number of 10-min average values. where the potential temperature profiles at Sletringen and Skipheia are shown together. The data used in Fig. 4 are based on measurements from the sector between west and north. The measurements are from periods where data from both stations exists. Only wind speed above 10 m s 1 were used. The data were classified according to the temperature difference T as between potential air temperature at 40-m height at Skipheia and sea temperature. At Skipheia temperature has been measured at 10, 40, 70, and 100 m, while the sensors at Sletringen are placed at 5 and 45 m. Horizontal bars indicates the standard deviation of the measurements. A vertical line means neutral lapse rate, a line tilting to the left means unstable, and to the right stable atmospheric conditions; hence it corresponds to the definition of the gradient of the potential temperature / z. The calculation of potential temperature is based on sea level as a reference. The potential temperature at 40-m height at Skipheia is used as reference and is set equal to 0. There is good agreement between T as and the lapse rate measured at Sletringen. The profiles for Skipheia are very little affected by T as and show a neutral to stable behavior for all classes. The measurements are taken during winter when solar radiation is very low at this latitude. Over the ocean, the sea will act as a very stiff heat reservoir, with good thermal contact to the air. Thus, high lapse rates corresponding to very unstable conditions may be maintained, even in strong winds. Over land, the heat capacity of the ground is, in practical terms, rather low as the thermal impedance is high. The warm air near the ground sweeping in from the ocean will therefore rise, and the cold descending air is not heated. Thus the air in a layer corresponding to the convection depth is more or less brought to a neutral, or even slightly stable, condition, in agreement with the conventional truth which is valid for land areas only that neutral stability is typical for strong winds. The radiative cooling typical for the periods has been measured to be of the order of 30 Wm ; this energy is taken from the air at surface level and may also explain the change to slightly stable conditions. If the oceanic winds have a stable surface layer, then no corresponding mechanism will operate, apart from turbulence generated by the ground, and the stable surface layer may be maintained. Thus a neutral or slightly stable atmospheric surface layer is always expected at the Skipheia station for wind directions with a fetch of 3 km of upwind land areas. For this direction sector, the local lapse rate in the surface layer observed at Skipheia is not a good indicator for the stability over the ocean farther upwind. The temperature measurements at Sletringen were only operative for parts of the period we have analyzed. As a criterion for classification of atmospheric stability for the spectra, the temperature difference between air and sea is therefore used. For a total time of about three years, we have been measuring sea temperature at either Sletringen or Skipheia. The mean daily sea temperature T s has a regular yearly variation, ] [ T s (d) sin (d 1) [ C], (8) where d is the day number. The standard deviation of the difference between measured values and the model T s (d) is 0.88 K, which gives an indication of the error of using the model instead of measurements. Equation (8) has been used to replace missing sea temperature data for periods, with missing observations. In Fig. 5 the spectra for three stability classes are shown. These spectra are based on measurements taken

8 914 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 55 FIG. 5. Spectra at Skipheia, for unstable (top), neutral (middle), and stable (bottom) atmospheric conditions. Line types for the Kaimal model and the experimental spectra correspond to those given in Fig. 3b. The criteria for stability are given in the text. from the years 1995 and A classification of the spectra according to unstable for T as 1 K, neutral for T as 1 K, and stable for T as 1 K gave a good separation of the spectra. The spectra for the stable atmospheric situations have gaps present in the range mhz, depending on height. The wind direction for all these observations is from the land sector (NE SE). High values in the 0.1- mhz region (10 day 1 ) might be due to gravitational waves and flow modifications caused by the mountainous interior of Norway. For the neutral case the gap is gradually closing, and in the unstable case has completely disappeared. Characteristically, all the spectra show a considerable amount of energy in the range mhz. The contribution in this frequency range could be due to a combination of horizontal roll vortices (Smedman-Högström and Högström 1975) and cellular convection.

9 915 FIG 6. A 1-h time series and the corresponding power spectrum for wind at 10-m height; other parameters are given in Table. As a basis for comparison, the classic Kaimal model TABLE. Mean values for the series shown in Fig. 6. Variable Wind speed Wind direction Air temperature Sea temperature fs( f ) Af (z/u), (9) 5/3 [1 Bf (z/u)] was fitted to the data, with A and B as free parameters. This is shown in Fig. 5. The fit was based on the values from f h 1 to the Nyquist frequency f N.Asexpected, the model fits the data in the high-frequency part of the spectra but gives an underestimate for frequencies below 1 mhz, even for the stable case. Hence there is more energy around the mHz region than indicated by the Kaimal model. Mean value 14.0 m s deg 0.3 C 6. C 6. Peaks in the gap region A common phenomenon during the winter in the mid- Norwegian areas are polar winds blowing from the west or northwest. When the air temperature is more than 4 K below the sea surface temperature, regular low-frequency patterns can arise in the time series. Figure 6 shows a 1-h time series from Skipheia recorded on February 1995, and the corresponding power spectrum. The spectral estimates are filtered with a logarithmic moving average filter of width 1/6 octave, which means that the window has constant width on a logarithmic plot. The low-frequency oscillations in the time series are very clear, and simply by counting maxima we would expect a spectral peak corresponding to a period of 40 min. The mean values of wind speed, direction, and air and sea temperature are given in Table. The average temperature difference between air and sea is 6.5 K. The temperature series also show a similar behavior with periodic variation of around 1.5 K. Inspection of the long-term time series shows that such phenomena occur rather frequently during the winter time. Typically 10 0 days per winter shows such

10 916 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 55 a regular behavior. Spectra from these periods will contribute strongly in the 1 h 1 part of the long-term spectrum, as well as in the unstable spectra in section 4. Such patterns could be due to convection cells or rolls. A closer inspection does not indicate the presence of horizontal rolls. Notice also the local reflection symmetry in time for intervals of 1 h at several points in Fig. 6 (e.g., around hours., 3.7, 7.0, etc.), indicating that patterns are passing. These phenomena will be discussed in a later article. 7. Comments on statistical significance When looking for peaks in a single spectrum near the low-frequency end, we need a test to decide if apparent peaks are statistically significant. For a time series of length T, the elementary spectral contribution is S i( f i) X i. (10) T Here X i ai bi, (11) where a i and b i are the Fourier coefficients (determined by FFT). A test of the distribution of the coefficients X i may decide if the peaks are due to periodic phenomena or if they are just random fluctuations within the probability distribution of the coefficients. If we assume that the Fourier coefficients a i and b i (i 0) are normally distributed N(, ), with zero mean ( 0) and standard deviation i, it follows that a i / i and b i / i are normally distributed N(0, 1). Generally, if the variables U 1,...,U n are statistically independent and normally distributed N(0, 1), the sum Z U j is chi-square distributed with n degrees of freedom, (n). It then may be shown that the expectation value E[Z] n and the variance var[z] n. Hence, the sum X i ai bi (1) i i i is distributed (), with both expectation value and standard deviation equal to. By using Eqs. (11) and (1), we see that the power spectrum coefficients X i will have both mean value and standard deviation of i. If our hypothesis is that the spectrum falls continuously from the turbulence peak, a unimodal curve could be fitted to the full spectrum. This curve could then be used as expectation values for the components. Normalized power spectrum coefficients are obtained by dividing each component X i by the values of a smooth curve. The normalized coefficients should then follow a () distribution. The existence of true peaks in the spectrum would give too many occurrences of large values of the normalized coefficient. By applying this type of analysis to the spectrum shown in Fig. 6, we conclude that the peaks represent a significant periodic structure in the time series. 8. Conclusions A 14-yr time series of coastal wind speed has been analyzed. There is no spectral gap in the 1-h region. For heights below 40 m, there is a dip in the 5 10-min region. From 40 m and up, the spectrum rises monotonically from the high-frequency region to the weather maximum around 3 days. Wind sensors will have a limited frequency response. If the frequency response is known and the frequency components above the sampling frequency are assumed to be known, the calculated spectra can be corrected for aliasing effects. Turbulence generated by friction is assumed not to be present below the 0.1-mHz region. As shown in the previous sections, all our time series contain more energy in the mesoscale region than expected by the classic model of Kaimal. We therefore have to conclude that other mechanisms are present in this frequency range. We assume that these mechanisms are due to a mixture of cellular convection (in the unstable case) and horizontal rolls. As discussed in section 5, an unstable surface layer in the air mass will rapidly decay to neutral once the heating mechanism disappears. However, fluctuations in the air mass with periods in the 1-h range are not likely to disappear as quickly, even if the heat source does. Thus, the location of the region where these fluctuations were generated may be far upwind. In the lowest part of the spectra there will, of course, also be a contribution from the synoptic variation. For maritime winds, a rapid change in local lapse rate from unstable to a slightly stable situation has been observed between the islet station Sletringen and the Skipheia station after the passage over a few kilometers of land. The unstable condition is rapidly relaxed after the passage onshore, the stabilization being due to radiative cooling of the ground. Due to the short passage over land, only the high-frequency part of the spectrum is modified, leaving the low-frequency part undisturbed. Acknowledgments. The authors are indebted to Prof. Ann-Sofi Smedman for an enlightening discussion, and to Drs. Svein Erik Aasen and Oddbjørn Grandum for useful discussions and assistance in the data collection and quality control. REFERENCES Aasen, S. E., 1995: The Skipheia wind measurement station. Instrumentation, wind speed profiles and turbulence spectra. Ph.D. dissertation, University of Trondheim, Trondheim, Norway, 156 pp. Agee, E. M., and S. R. Gilbert, 1989: Aircraft investigation of me-

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