Gjerstad et al. page 1
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AN ANALYSIS OF LOW FREQUENCY MARITIME
ATMOSPHERIC TURBULENCE
JOHANNES GJERSTAD 1,2
SVEIN ERIK AASEN 2
HELGE I. ANDERSSON 1
IVER BREVIK 1
JØRGEN LØVSETH 2
1 Faculty of Mechanical Engineering, NTH, University of Trondheim, Norway
2 Physics Department, AVH, University of Trondheim, Norway
Revised version, January 1995
Abstract
New data are presented for the spectrum of turbulent wind energy under maritime conditions in the frequency region 1.0 mHz - 0.03 mHz. The corresponding measurements were made at five levels on a mast 46 m high on a small islet off the coast of Middle Norway. Twelve time series of length 10h 40 min. have been analyzed. The mean wind speeds of the series are in the range 11 - 19 m/s, and the wind directions are westerly which had maritime conditions upwind.
Four of the time series were characterized by unstable atmospheric conditions (mean lapse rate -T/z in the range 12 - 20 K/km) and show little or no indication of a spectral gap for heights above 40 m. Four other time series with stable-to-neutral conditions (mean lapse rate 3 - 7 K/km) do show a gap in the wind speed spectra around 0.5 mHz, in agreement with the Kansas (1972) and Minnesota (1978) experiments. The remaining series, with lapse rates fluctuating around the neutral value of 9.8 K/km, show intermediate behavior.
The temperature spectra at 45 m height do not show a gap even for stable-to-neutral conditions.
1.Introduction
Offshore oil rigs and other compliant structures may have resonance frequencies of the order of 0.01 Hz, and this has motivated an investigation of the low frequency part of the turbulent wind energy spectrum for maritime conditions.
In the well known classical experiments (over land) by Van der Hoven (1957) at Brookhaven, by Kaimal et al. (1972) in Kansas and by Kaimal (1978) in Minnesota, a spectral gap is observed in the (1h) -1 (0.3 mHz) region. Above the gap, turbulence is generated by friction at the ground, the characteristic frequency being on the order of U/z for stable conditions and down to 0.01 U/z for unstable conditions, where U is the mean wind speed and z is the height. For very low frequencies, synoptic or weather variations will give another peak in the spectrum, on the order of (3 days) -1 or 0.004 mHz. In between, one finds the spectral gap due to a lack of mechanisms to generate turbulent energy. Diurnal variations may give contributions with periods 24 or 12 h, but are not important in the present case. See e.g. Stull (1988) for an illustration of these features.
Fiedler and Panofsky (1970) pointed out that the situation is less clear over the ocean and over very smooth terrain. From the Lammefjord experiment in Denmark, Courtney and Troen (1990) find that "only a factor two" in amplitude separates the spectral density in the gap region from that in the turbulence peak.
Agee et al. (1973) have reviewed mesoscale cellular convection (MCC), and have observed diameter to depth ratios of up to 30 to 1. LeMone (1976) has discussed modulation of turbulence energy by longitudinal rolls, which may have lengths of up to 500 km. Rothermel and Agee (1980) have discussed aircraft observations of MCC over the China Sea, and have reported cell diameters of 31 and 39 km. With a wind speed of 15 m/s, these latter observations would correspond to frequencies around 0.5 mHz. Agee and Gilbert (1989) and Agee and Hart (1990) have discussed aircraft observations of MCC during wintertime cold air outbreak over Lake Michigan. The structures observed here have dimensions up to 10 km, due to a shorter fetch, but demonstrate that the same type of phenomena occur in cold climate.
A basic structure of length Ts in the time series will in general not have a harmonic variation in time, and will give spectral contributions with defined phase relations at fs = 1/Ts and at multipla of this frequency. Using data from the Lammefjord experiment, Mahrt and Howell (1994) have shown that the differences between the ter Haar and Fourier spectra are not very important for the scale distribution of energy, and that the -5/3 law for the spectra are not very much affected.
The purpose of the present paper is to report on analysis of the very low frequency part of the spectra, i.e. from 1 mHz to 0.03 mHz. The data are collected in a tower located near the center of a very flat islet with a diameter of approximately 300 m. Only data with a wind direction from the oceanic sector (Norwegian Sea) are considered here. From a simple dimensional analysis, it follows that all observed fluctuations in the low frequency region are due to properties of the maritime air masses.
As in the classical experiments, we observe a spectral gap for stable-to-neutral conditions. For even slightly unstable conditions however, the gap is partly filled. Peaks are observed in the range 0.1 - 1 mHz, corresponding to wavelengths of 20 - 200 km, in the single spectra. Due to efficient mixing by the wave motion, the upper part of the ocean is a very stiff thermal reservoir with a surface temperature nearly independent of air temperature for the periods considered here. Colder air, mostly of polar origin, will give thermally generated turbulence, or convection air currents, which are strong enough to fill the spectral gap observed over land.
2.The experimental setting
The data discussed in the present paper were collected on the islet of Sletringen
(63o 40' N), which is situated 1 km off the western end of the larger island of Frøya on the Norwegian coast in the Trøndelag region. The two islands protrude into the Norwegian Sea, and Sletringen is exposed to undisturbed maritime winds in a sector from south through west to north-east.
Wind speed sensors were mounted on the experimental mast at the heights 5, 10, 20, 42 and 46 m above ground (the sensor at 42 m was meant to be a spare for the top sensor, which was very exposed to damage from lightning). The speed sensors, which are precision cup anemometers with a distance constant of 1.5 m, were mounted at a distance of 2.5 m from the mast in the western direction to avoid disturbance from the mast. The wind direction was measured at 45 m height, and temperature was measured at 5 and 45 m, and in the sea at 5 m depth. The temperature sensors were thermistors, specially calibrated to an accuracy of 0.01 K. The air temperature sensors had a vented housing and were shielded against radiation. The base of the mast was situated 3 m above mean sea level. The surface of the island is a rocky flat, with a surface roughness not very different from the excited sea. Insignificant modifications of the maritime wind height profile is expected at the heights from 10 m and above. If present, it could influence the wind speed profile calculations in the next section, but is otherwise not important for the low frequency spectral analysis discussed in the present paper.
The time step in the time series is 1.17 s (512 loggings per 10 minutes). The values recorded are interval mean values of the wind speed and instantaneous values for temperature and wind direction. The observations analyzed here were made in the periods April 6 - 8 and October 30 - November 15 1988.
A brief description of the data base is given by Andersen and Løvseth (1993), while a more complete description of the measuring station and the results will be given elsewhere.
3.Results
The 12 series analyzed were selected with the criteria: (i) mean wind velocity larger than 10 m/s, and (ii) minimal synoptic variations in wind velocity, wind direction, and vertical temperature gradient. Table 1 summarizes some characteristics for the 12 series, each of a length of 10 h 40 min. (32 768 data points).
Table 1. Some characteristic parameters for the selected time series (see text).
In Table 1, U10 is the mean wind speed at 10 m, averaged over the 640 min. period. The lapse rate is defined as = -T/z; its adiabatic value is a = 9.8 K/km. Neutral conditions correspond to a; instability occurs when a and stable conditions when a. In our case, the lapse rate is calculated from the mean temperature gradient between 5 and 45 m. Tas is the air (45 m) - sea temperature difference. The Obukov length L and the friction velocity u* have been determined by fitting the mean wind speed at 10, 20, 42 and 46 m to the extended logarithmic profile formula,
U(z) = (u*/k) [ln(z/z0) - m(z/L)](1)
where k is von Karman's constant (k = 0.4 was assumed). For maritime data, the roughness length z0 will depend on the state of the upwind sea. Andersen and Løvseth (1993), have shown that the ensemble mean of 40 min. mean wind speed profiles of the neutral part of the database show a dependence on wind speed which is well fitted assuming z0 is given by the Charnock relation,
z0 = (a/g) u*2(2)
where g is the acceleration of gravity, and a = 0.0172. Following Panofsky and Dutton (1984), m was for unstable conditions (L<0) parameterized by the Businger-Dyer relations as
m = ln[(1 + x2) (1 + x)2/8] - 2 tan-1(x) + /2
(3)
x = (1 - 16 z/L)1/4
and for stable conditions as
m = - 5 z/L(4)
By estimating L from the wind speed profile, using Eqs. 1 - 4, it will be a stability indicator which is experimentally independent of the temperature measurements. The agreement is good for the unstable and mixed series, but there is disagreement for series 3 and 4. However, the dependence of the profiles on L is strongly interrelated with that on z0. A fit with z0 and L left as free variables is therefore numerically unstable. With a value of the Charnock constant a = 0.011, the stability indicated with the sign of L agree with that of the temperature measurements, the ranges of Obukov L-values being for series 1 - 4: 500 m to 50 km, for series 5 - 8: -68 to -172 m, and for series 9 - 12: -286 to -3402 m. The values of u* is reduced by some 5% and those of z0 by some 40%, the overall rms. deviation for the speed values being 4.0 cm/s, compared to a deviation of 3.8 cm/s for the values in Table 1 (based on a = 0.0172).
The fifth column in Table 1 gives the variances 102 at z = 10 m calculated from the observed time series. The variance can in general be expressed in terms of the spectral density S(n),
(5)
n denoting the frequency. The limits nx and nn are implicitly defined from the time interval of the time series and period of the FFT, respectively.
In Table 1, 3 variables indicating stability are listed, lapse rate , air - sea temperature difference Tas and the Obukov L as determined from the wind speed profile. For the stable and unstable series, the temperature indicators agree. For the mixed series, Tas indicate a slightly unstable situation for all 4 series. The lapse rate is fluctuating for the mixed series, the mean value being in the unstable range for series 9, and close to neutral for the remaining ones. One unit (K/km) in corresponds to a temperature difference of only 0.04 K, thus measurement errors may easily affect this indicator. The air sea temperature difference is experimentally a much more robust stability indicator. As demonstrated below, values of Tas less than the adiabatic limit, -0.45 K, seem to indicate a filling of the spectral gap.
In Fig. 1, the time series of 10 min. mean values of temperature at 45 m and of wind speed at 10 m height are shown for the selected series. The time series of wind direction are not shown. Series 1-4 show stable directions within ± 10o, series 10 - 12 within ± 20o. Series 6 exhibits a direction change of 120o, in the remaining series changes of 60o to 90o are found. There is a striking difference between the stable series in Fig. 1 and the remaining ones with respect to variations with a period on the order of one hour. Thus, already by visual inspection of the time series, one may conclude that the gap is filled for unstable conditions, since the turbulence intensity in the high frequency peak normally is less than 10%.
The estimates of the spectral density of turbulence, S(n), were calculated from the mean square values of the FFT coefficients of the time series. The averaging interval was chosen to have a constant length of log10(2)/3 on a logarithmic frequency scale, except at the lowest frequencies where at least one pair of coefficients was included. In the spectra shown a linear trend, as determined by a least squares fit, has been subtracted prior to the FFT. Except for series 6, trend subtraction did not affect the spectra significantly. Obviously, the 2 to 3 lowest frequency estimates are rather uncertain due to the trend and curvature of the time series, as well as statistics. Diurnal effects are not important for the spectra shown.
Average values of nS(n) for the wind speed for the four stable-to-neutral series 1 - 4 are shown versus the frequency n in Fig. 2 for 3 different heights in the range 10 - 46 m. The spectral gap from about 0.05 to 1 mHz is seen to be quite pronounced. The error bars indicate a relative error of (2N)-½, where N is the number of Fourier-frequencies included in the averaging, each with a sine- and cosine-coefficient.
We will return to a discussion of the high frequency peak in a later paper. In general, it could not be fitted by a simple Kaimal-type formula, in particular, scaling with (nz/U) is not observed, the length-scale being the problem.
In Fig. 3, the turbulence energy density for the four unstable series 5 - 8 are shown. The difference from Fig. 2 is rather striking: the high frequency maximum appears in the 10 m data for the first three series, and are still roughly at the same place as for the stable data. All data for time series 8 and the data for 20 and 46 m height for the other series, show, broadly speaking, a plateau for n < 10 mHz, with some marked peaks. Thus the gap in the spectra, clearly evident in Fig. 2 for the stable-to-neutral data, has disappeared. Series No. 6 has a marked low frequency structure, also evident in Fig. 1, causing the data for twice the fundamental frequency to be out-of-range.
In Fig. 4, the turbulence spectra for the mixed series 9 - 12 are shown. The appearance is similar to the unstable ones, with clearly indicated peaks in the gap region. Peaks in the frequency range 0.1 - 1.0 mHz correspond to spatial wavelengths in the 200 - 20 km region.
The general conclusion for the wind speed spectra is that the gap region is filled for even slightly unstable situations. We return to the interpretation of the structures in next section.
Fig. 5 shows the temperature spectra for the four stable-to-neutral series 1-4, for z = 5 m and z = 45 m. Note the logarithmic scale of the ordinate axis. The difference between these spectra and the wind speed spectra of Fig. 2 is at first sight surprisingly large. The spectral gap is much less pronounced than it was in the speed spectra. Except for series 1, the temperature spectra at z = 45 m are actually increasing when the frequency decreases. The fall off at high frequencies (n > 0.05 Hz) is influenced by the time response of the temperature sensors, which is rather complicated. The flattening of the spectrum at the end is mainly due to aliasing effects. Since this part of the spectrum is not in focus here, it will not be further discussed.
The temperature spectra of the unstable and mixed temperature series are shown in Figs. 6 and 7, respectively, for 5 and 45 m heights. The spectra are in this case much higher over the whole spectral range compared to the stable-to-neutral case, and show a much stronger rise for falling frequencies.
4.Discussion and conclusions
We may compare the observations presented above with those of the Kansas and Minnesota experiments (Kaimal et al. 1972, Kaimal 1978): in the latter cases, the temperature spectrum was found to have the same properties as the wind speed spectrum. This is evidently not so in our case. To get a feeling for the differences, one may look at the average rate of increase in the spectra as the frequency decreases. Focusing on the ratio between the spectral densities in the dip region and the low frequency peak region, e.g.
(6)
for sensors at 45 or 46 m, we find for the temperature spectra that the mean value of R is approximately 100 for the unstable, and 10 for the stable-to-neutral time series. For the wind speed spectra, this ratio is about 3 for the unstable, and 1/3 for the stable-to-neutral case. Thus the value of this ratio changes by an order of magnitude when we go from the stable-to-neutral series to the unstable ones, with the ratios for the temperature spectra being approximately a factor 30 larger than for the wind speed spectra for the same stability class.
The physical reason for the observed small values of the spectral density of the temperature variance at high and intermediate frequencies is that the Norwegian Sea - agitated by gale force winds - is upwind, meaning a surface with very small temperature variations over a vast stretch. The surface temperature also will be practically constant in time for the periods considered here, because the mixed surface layer of the ocean constitute a heat reservoir with a very large capacity.
Over land, where the Kansas - Minnesota experiments were performed, the effective heat capacity of the ground is much smaller and dependent on the ground structure and vegetation. The temperature of the ground will then be strongly dependent on the daily weather and radiation history and give rise to air pockets of a variable temperature. This gives rise to thermal variance in the upwind area with a typical horizontal wavelength on the order of a few 100 m to several kilometers. Combined with the eddies of the mechanically driven turbulence, this will give thermal variance in the 0.01 - 0.1 Hz region.
In the stable-to-neutral case, there will be no thermally driven convection. The mechanically driven convection will create a layer with a temperature gradient close to the adiabatic value and, due to the constant surface temperature upwind, very little thermal variance will be created. Thus we also expect the high frequency temperature variations to be mainly of a synoptic character, i.e. associated with the history of the air mass. This explains the very low values observed for the thermal spectra, and the steady increase (for z = 45 m) over the whole frequency range.
For stable atmospheric conditions, Larsen et al. (1990) have observed an increase, or a maximum, in f S(f) when f decreases from 1 to 0.1 mHz for the parallel and horizontal-transversal components of the wind fluctuations in the Lammefjord experiment from Denmark. These features, which, as noted by the authors, may be caused by buoyancy (gravity) waves, see e.g. Stull (1988) for a review, are absent in our material. The ocean in the up-wind area is not expected to excite atmospheric waves in the sub-mHz frequency range.