Frequency spectra of vertical velocity from Flatland VHF radar data
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VOL. 96, NO. D2, PAGES 2845-2855, FEBRUARY
Frequency Spectra of Vertical Velocity From Flatland VHF Radar Data T. E. VANZANDT
Aeronomy Laboratory, NOAA, Boulder, Colorado G. D. NASTROM Department of Earth Sciences, St. Cloud State University, St. Cloud, Minnesota
J. L. GREEN Aeronomy Laboratory, NOAA, Boulder, Colorado
The vertical wind velocity over very flat terrain was observed every 153 s in the troposphere and lower stratosphere by the Flatland radar, near Champaign-Urbana, Illinois. Several hundred frequency spectra were calculated from all accepted 6-hour time series from March through May 1987. By stratifying the spectra in various ways we find the following: (1) The spectra were independent of altitude within the troposphere or lower stratosphere, but the spectra in the two regions differed in amplitude and frequency; (2) At a given altitude the spectra were independent of the wind shear dt•/dz, the buoyancy frequency N, and the maximum wind speed below 16 km; (3) The change of spectral shape and amplitude with increasing background wind speed t7 was much less than at stations near mountains. The variance of the spectra, equal to twice the vertical kinetic energy per unit mass,
roughlydoubledas tJincreased by 10 m s-i; (4) The spectrawereconsistent with beingdueto a spectrumof gravity waves, as indicated by the sharpdrop in spectral amplitude near N at small t• and by the fact that the observed changeof shape with increasingt7was quite consistentwith the change of shape of model Doppler-shifted gravity wave spectra; (5) The results of comparison between the observed and model spectra are consistentwith an intrinsic gravity wave spectrum that is invariant with t7, dt7/dz,etc., contrary to expectationsfrom gravity wave theory; (6) The results are insensitive to the azimuthal distribution of gravity wave energy, as long as the distribution is roughly symmetrical relative to the mean flow; (7) The resulting characteristic horizontal phase velocity c, of the intrinsic
frequencyspectrumwas about6 m s-1 in both the troposphere and the stratosphere. The correspondingcharacteristicvertical wavelengthswere about 3300 and 1800 m, respectively, consistent with previous estimates.
The clear air Doppler radar technique (also called the wind-profiling or Mesosphere-Stratosphere-Troposphere (MST) radar technique) has been applied to a wide range of meteorological problems since its development by Woodman and Guill•n  and Green et al. . Nevertheless, research on some important problems has been frustrated by the location near mountains of most such radars. The resulting geophysical noise is especially serious for studies of the vertical velocity w. The effect is illustrated by the spectraof w as a function of period in Figure 1 [Ecklund et al., 1985, 1986], which were obtainedduring the ALPEX (Alpine Experiment) program in the Rhone delta in southern France. When the background wind was light, the spectra were rather flat at periods less
thanthe buoyancy(Brunt-Vfiisfilfi)periodTB (about10.2min in Figure 1). These light wind spectra are very similar to
that the light wind w spectra are due to such gravity waves. But as the background wind speed a increased, the amplitude increased and the spectral shape changed, as shown in Figure 1, so as to become quite inconsistent with the light wind spectra. Similar effects have been observed by J. R6ttger (unpublished manuscript, 1981) and Gage and Nastrom . It is supposed that these changes are due to mountain waves, since, although the Rhone delta itself is very flat and the coast of the Mediterranean Sea is only a few kilometers to the south, to the north hills begin 35 to 80 km away and mountains up to 1500 rn lie 90 to 130 km away. Also, Nastrom et al.  found that in Colorado they could extract the large-scale vertical velocities due to synoptic-scale motions from the data when the wind blew from the plains but not when the wind blew over the nearby Rocky Mountains. In order to obviate the geophysical noise due to mountains, we have constructed the Flatland radar in very flat terrain near Champaign-Urbana, Illinois. Green et al. 
vertical velocity spectra in the ocean. Garrett and Munk [1972, 1975] modeled the oceanic spectra as a spectrum of showed that the behavior of w over the Flatland radar is horizontally propagating internal gravity waves, and Vanindeed quite different from that near mountains. Here we Zandt [1982, 1985]and Scheffierand Liu  adaptedtheir consider this behavior in considerate detail, particularly model to the atmosphere. Thus it is reasonable to suppose using power spectra versus frequency to. We find that as a increases, the shape of the spectra changes much less than Copyright 1991 by the American Geophysical Union. near mountains, and that at high frequencies the observed changes are quite consistent with the changes due to Paper number 90JD02220. 0148-0227/91/90JD-02220505.00 Doppler-shifting of gravity wave spectra by a. The behavior 2845
VANZANDT ET AL.' FREQUENCY SPECTRAOF VERTICAL VELOCITY
made during 1987 from March through September, concentrating on the period from March through May. Frequency spectra were derived from time series of the measurements. Usually, some points in each serieswere missingand others were eliminated by the quality control proceduresdescribed by Green et al. . If at a given altitude four or more successive observations were missing or eliminated in a given series, so that there was a gap greater than 10 min, then that serieswas rejected. The data in the acceptedseries at each altitude were then linearly interpolated to uniform intervals of 153 s, the mean and the linear trend were removed, and the frequency spectra were derived by Fourier transform of the residuals. The Nyquist frequency was
"• i0I c-
CON,, = (1/306S)= 3.27X 10-3 Hz. Routine radiosonde profiles of horizontal wind and temperature from Peoria and Salem, Illinois, about 125 km northwest and 150 km southwest of the Flatland radar, 240 120 60 30 15 8 respectively, were obtainedfrom the National Climatic Data Period (m•nutes) Center. During the period from March through May 1987, Fig. 1. Several representative smoothedspectra of w as a functropopause heights at Peoria ranged from about 9 to 14 km, tionof periodduringquiet(windspeedt7• with the mean rising from 11 km to slightly over 13 km; 20 ms-i ) periods in theRhonedeltain southern Franceduringthe ALPEX program.The buoyancyperiod TB was about 10.2min. The heights at Salem were often slightly higher. linelabeled F -5/3isfor guidance only.[AfterEcklund eta!., 1985, From March through May 1987 the rms N in the tropo1986]. sphere from 700 to 500 hPa (about 3.0 to 5.5 km msl) was
N t = 1.82x 10-3 Hz andinthestratosphere from200to 150 hPa(about11.8to 13.6kmmsl),Ns = 3.33x 10-3 Hz. Thus at low frequencies, associatedwith large-scalemotions, are treated
The data and our methods of analysis are presented in the next section. In section 3 we present the observedfrequency spectra of w in the troposphere and lower stratosphere,and we stratify them by altitude, by buoyancy frequency N (crl/TB), and parameters of fl. In section 4 we develop a model of Doppler-shifted gravity waves and in section 5 we compare the observed spectra stratified by t? with model spectra. In section 6 we consider the implications of our results and present our conclusions. 2.
radar is located
14 km southwest
Champaign-Urbana, Illinois, at 40.049øN, 88.381øW, at an elevation of 212 m msl (mean sea level). It operates at a frequency of 49.8 MHz (wavelength 6.02 m). The antenna is a 60-m x 60-m array of coaxial-colinear dipoles, with a two-way, full half-power beam width of 3.2ø. During the period of the observationsreported here the antenna could form only a single beam, which was directed vertically. The data were taken using 1500-m pulse lengths and twenty-five 750-m range gates centered from 1.41 to 19.41 km msl, but useful data could be obtained only from the range gates centered from 2.16 to 16.41 km. (Henceforth these altitudes will be rounded to the nearest 0.1 km.) There was also often a region of missingechoesjust below the tropopausedue to reduced radar reflectivity there [see Green et al., 1988]. The Doppler spectra have 128 points with a velocity resolutionof
N s was slightly larger than WN,,.The correspondingbuoyancy periods were 9.18 and 5.01 min, respectively, with a ratio of about 3.
1.8. RESULTS OF SPECTRAL ANALYSIS
In order to examine the behavior of the vertical velocity over a wide range of scales, we calculated spectra at 5.2 km from all of the accepted, independent 45-hour time series, which was the longest interval for which a sufficiently large number of time series passed our rejection criteria. There were 34 acceptable spectra from March through September 1987, including nine from June through August, and six from September. Since these samples are not large enough to study seasonal variations, we have formed the geometric mean of all 34 spectra. The mean spectrum is presentedin the usual log-log coordinates in Figure 2a and in areapreservingcoordinatesin Figure 2b, obtained by multiplying the ordinate by (In 10)w. In the latter graph the area under the spectrum is proportional to the contribution to the total variance, which is twice the vertical kinetic energy per unit mass.
Further details, including examples of the data, are given by
The mean spectrum appears to consist of two regimes, with a transition at around 6 hours. The longer period regime is thought to be dominated by synoptic-scalemotions, which will not be considered further in this paper. The shorter period regime, which containsalmost all of the variance, will be shown to be consistentwith domination by gravity waves. The area-preservinggraph in Figure 2b has a maximum near the buoyancy period TB - 1/N. It also appears that a significantfraction of the variance was not observed because it lay at periods shorter than the 5.1-min Nyquist period.
Green et al. .
5 cm s-l andan unaliased velocityrangeof +-3.2m s-• Averages of the vertical velocity were recorded about every 153 s from March 1987 until May 1988, with brief interruptions due to power failures, etc., and a break of about 6 weeks in summer. This study used only observations
will be estimated
In order to examine the shorter period regime in greater detail, we calculated spectra at all altitudes from all accepted
VANZANDT ET AL.' FREQUENCY SPECTRA OF VERTICAL VELOCITY Log (Frequency ( Hz ) ) -5
conditions. For this reason, in this paper we consider the spectraonly from March through May, when power outages were uncommon. It should be noted, however, that the mean of the available summer spectra was approximately the same as the mean spring spectrum. The availability of hundreds of individual 6-hour spectra has permitted us to study the dependence of the spectra on altitude, N, t7, etc. These will be considered in turn in the next few paragraphs. We found that the spectral power in a given frequency band tended to be distributed lognormally. Therefore in order to reduce the influence of extreme values, mean spectra were calculated as geometric averages of the individual spectra. This results in smaller mean values; for example, at 3.7 km the geometric average of the variance is only 0.45 times the arithmetic average. The altitude dependence was studied by averaging all of the spectra at each altitude from 2.2 to 14.2 km. The
resulting meanspectra afterapplying a •, «,• smoothing filter
Log (Frequency (Hz) ) -3
at each altitude are shown in Figure 3, in both the usual log-log coordinates and in area-preserving coordinates. The table in the panels gives the altitude Z (kilometers) of the (hour) center of each 750-m range gate. In Table 1 under the head "Figure 3" are given the number of individual spectra going into the mean and the variance in the mean spectrum. "Number" is very large in the lower troposphere, small in the upper tropospherebecauseof missingechoes, and moderate in the lower stratosphere. At 15.7 km, the next altitude above 14.2 km, "Number" was only 4. Note that the varianceunder the 6-hour spectrumat 5.2 km was 90% of the variance under the 45-hour spectrumin Figure 2, albeit with an enlarged data set. The spectral do not vary significantly within each atmospheric region, but the stratosphericspectra differ from the troposphericspectra. Ecklund et al. [1985, 1986] observeda similar difference. But their Nyquist period was about 2 min instead of the present 5 min, so that their stratospheric spectraextended to frequenciesconsiderablylarger than the stratosphericTB, about 5 min. Then it was apparent that the the mean stratosphericspectrum had the same shape as the tropospheric spectrum, but it was shifted to larger frequencies by the ratio of the stratospheric to tropospheric N (about 2 in their case) and to a smaller amplitude. Although the difference between the tropospheric and stratospheric spectrais obviously related to the ratio of N, the mechanism (mln)
for the difference in section
is not clear. This will be discussed further
We have also studied the dependenceof the spectra on N at a given altitude by averagingthe spectra at 3.7 and 5.2 km Fig. 2. Mean frequencyspectrumof w at 5.2 km for all accepted centered at 0000 and 1200 UTC that corresponded to the ½5-hourtime series from March through September 1987 at the fourth and first quartiles of N at Peoria over the layer from Flatland radar. Thirty-four individual spectra went into the mean 700 hPa to 500 hPa. The rms values of N in these quartiles 45
andthe meanvariancewas138cm2 s-2. (a) The spectrum in the were2.08and1.53x 10-3 Hz, respectively, a ratioof about usual log-log coordinates. (b) The spectrum in area-preserving coordinates. The vertical line labeled N-TB indicates the troposphericbuoyancyfrequency/period.
6-hour time series centered at 0000 and 1200 UTC (1800 and 0600 90øW time), the nominal times of routine radiosonde ascents, and for the intermediate times, 0600 and 1800 UTC. During summer many time serieswere rejected due to power outages caused by nearby convective storms, so that the mean summer spectra may not represent average summer
1.4. Surprisingly, there was not any detectable difference between the mean spectra in these quartiles, although the ratio of stratospheric to tropospheric N of about 1.8 was associatedwith a clear difference. The same negative result was obtained when the spectra were stratified using N over thinner layers and using mean N rather than rms N. This result
profiles become available. We now consider the dependenceon the wind speed t7at the same altitude
as the observed
VANZANDT ET AL.' FREQUENCY SPECTRAOF VERTICAL VELOCITY
Log (Frequency (Hz))
this the spectra centered at 0000 and 1200 UTC were sorted accordingto the wind speedat Peoria and mean spectrawere calculated in wind speed bins. We found that as the wind speedincreasedthe shapeof the spectrachangedrapidly at low wind speedsand slowly at high speeds. Thus the low wind speed bins should be as narrow as possible. On the other hand, we found that in order to obtain a smooth and stable mean spectrum, "Number" should be •10. With theseconstraints,for the troposphericspectrawe chosefive
binswithboundaries at 0, 3.5, 5.5, 11and22 m s-• andfor the stratosphericspectra at 12.7 km, four bins with bound-
ariesat 0, 6, 13, and23 m s-•. N
The resultingmean spectraat 3.7 km are plotted in Figure 4; the correspondingvalues of t7, number and variance are given in Table 1 under the head "Figure 4." It is evident that the shapeof the spectrachangessystematicallywith increasing O. In the lowest bin the slopeof the spectrumis positive at long periodsand changessharplyat N to a large negative slope. (In this bin we have omitted two spectra with very large variance. Their inclusionwould changethe amplitude of the mean spectrumbut not its shape.) As fi increases,the spectrabecomeflatter, with the slope more negativeat long periodsand lessnegativeat shortperiods.The spectraat 5.2 (not shown) and 12.7 km (see Figure 9) behave in much the sameway. As/7 increases,the spectral energy shifts toward larger w, as can be seen in the area area-preservinggraph in Figure 4b. This is also discussedby Fritts and VanZandt .
The spectra at 3.7 and 5.2 km were not significantly different, so for each bin we formed the mean of all of the
Log ( Frequency(Hz))
spectraat the two heights,and the corresponding•, number, and variance are given in Table 1 under the head "Figure 8." The data for 12.7 km are given in Table 1 under the head "Figure 9." Figures 8 and 9 will be discussedlater. In Figure 5, variance versus/• for 3.75-5.2 km is plotted as plusesand
for 12.7, as crosses. The increase of variance with • will be
discussedmore quantitatively in section 6. It is instructive to compare the Flatland spectra with the ALPEX spectra shown in Figure 1. The ALPEX quiet and
activespectra wereobtained whent7was• 0, thenthe frequencyis up (ms-•) n
shifted,and if cos •b< 0, the frequencyis down shifted.In this way, an intrinsic spectrumis Doppler shiftedto an
The form of the intrinsicspectrumcannotbe derivedfrom
firstprinciples.Instead,a form is assumedfor the spectrum of total energy E(tr, m), where m is the vertical wave
number,andone-dimensional spectrafor u, w, T, etc. versus to,k, andm are derivedusinggravitywave theoryfollowing the methodof Garrett and Munk [1972, 1975].The parameters of E(tr, m) are then adjusted to optimize the fit to observedone-dimensionalspectra. Different authors have used different functional forms for E(tr, m), with rather similar results. Following Garrett and Munk, we have assumedthat the spectrumis variables-separable,
Fig. 4. (Opposite)The dependence of themeanfrequencyspectra on mean backgroundwind speedin five bins with boundariesat i
0, 3.5,5.5,11,and22ms-•. Forfurtherinformation seeTable1.(a) The spectrain the usuallog-logcoordinates.(b) The spectrain area-preserving coordinates.As t7 increases,the spectralshape changessystematicallyand the varianceincreasesby a factor of
about3. The95%confidence interval forthet7= 8.2ms-1spectrum is shownby the vertical line to the right of the table. Confidence
intervals fortheotherspectra scaleapproximately as(Number)-i/2.
VANZANDT ET AL.' FREQUENCY SPECTRAOF VERTICAL VELOCITY
E(tr, Ix) • A(Ix)B(tr)
Log (Frequency (Hz))
' A(ix) • ixs/(1+ kt)s+t
B(o') octo - P
This leads to the intrinsic spectrum of vertical velocity,
Fw(o',ix)vcA(Ix)B(o')(to/N) 2&- (o-)