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The stationary and non-stationary wind profile

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Page 1: The stationary and non-stationary wind profile

160

The Stationary and Non-stationary Wind Profile

By OSKAR ESSENWANGER and NOVELLA S. BILL!ONS I)

Summary - A method is presented which permits the separat ion of the non-s ta t ionary pa r t of the wind profile from the s ta t ionary or smoothed profile. Very little a priori assumpt ion is made, thus, the results can be considered to render informat ion of general purpose. The es tabl ishment of the s ta t ionary wind profile creates a variable reference wi th alt i tude which is necessary for t u rbu- lence investigations.

1. Introduction

Measurement of turbulence in the free atmosphere is still limited at the present date. Although utilization of small rockets has made information available from higher altitudes, the instrumentation is usually restricted to smoke or chaff trails and para- chute descents. These observations give good results on the wind profile, but do not yield detailed information on fluctuations. Some knowledge can be gained on diffusion and turbulence in general, but the fine structure of turbulence can hardly be studied by this type of instrumentation.

Fortunately, there are some wind profile measurements which were obtained in a different way. These data were derived from measurements taken aboard Research and Development rockets by the angle-oLattack meter method (REIsm E712). The measurements, which may be considered instantaneous, serve as the basis for the following investigation.

As the measured data may be considered to comprise both stat ionary and non- stat ionary components, it is desirable to separate the components to further investi- gate the non-stationary part. Thus the wind profile may be considered as consisting of two major components which may be expressed as

F(h) = P(h) -~ T(h) ,

where F(h) represents the actually recorded wind data as a function of height, P(h) the mean or stat ionary wind profile, T(h) the turbulent or non-stationary part of wind profile.

I t is understood, however, that P(h) may contain some instrumental bias which can be eliminated. Further, T(h) may still retain the random instrumental error,

i) Aerophysics Branch, Physical Sciences Laboratory, Directorate of Research and Develop- ment, U.S. Army Missile Command, Redstone Arsenal, Alabama, U.S.A.

2) Numbers in brackets refer to References, page 166.

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Stationary and Non-stationary Wind Profile 161

which in its nature is similar to the random fluctuations of the turbulence. From instrument tests and analysis, however, it can be assumed that the instrument fluc- tuation is small and therefore negligible.

The missile data were recorded at equal time intervals. They represent therefore increasing altitude intervals. For the investigation of turbulent fluctuations in the free atmosphere, the observed data have been reduced by linear interpolation to equal height intervals. The detailed procedure was outlined in a previous report (2) and will not be repeated here. This procedure assures that linear interpolation stays within reasonable limits and never exceeds 50% of the derived shear interval.

2. Profile fitting procedures

The data prepared at equal altitude intervals as specified in the previous section are now analyzed for separating the stationary profile P(h) from the non-stationary remainder T(~ 0 . Several methods can be utilized. The choice of the method depends on the properties of the profile and the results desired.

Thus the entire profile can be subjected to Fourier analysis. A definite number of terms could be declared the stationary part. This method, outlined by GIFFORD [5], does not work well according to GIrFORo when the spectrum varies rapidly as a function of frequency. It proves therefore of little value for the wind profile observa- tions.

Some of the difficulties encountered have been further discussed by SCOGGINS [8]. He applies the power spectrum analysis and determines a filter in accordance with the vehicle response, for which purpose the analysis is undertaken. The purpose of the analysis presented here is to render results of most general validity; therefore this selection criteria cannot be employed.

Another method is the normal curve smoothing principle (HoLLOWAY [6]). This principle displays two undesirable effects. The stationary profile is systematically deviated by the smoothing technique, namely towards the inside of the curvature of the observed wind profile. Furthermore, the remaining variations Th for the entire ascent follow the normal distribution law (GAuss) after elimination of the mean profile P~) (see CAMP and KAUFMAN [1]). This condition cannot be assumed a priori. Due to the presence of discontinuous layers of turbulent motion, the remaining T~ may have a Gaussian distribution for layer sections only. The total summation over the entire height range of the ascent (e. g. from surface to 50 km) may be a summation of several normal distributions, which do not necessarily result in one normal distri- bution.

The construction of the mean profile is therefore attacked in a different way. First, orthogonal polynomials are used to express the mean wind profile. Secondly, the entire ascent is divided into subsections. The utilization of 23 data points at subsequent altitude sections has proven advantageous, although any arbitrary number of points could have been employed. The method is described in detail, by ESSEN- WANGE~ [2].

I t has been assumed that the stationary part would create a spectrum with high amplitudes in the first terms of a polynomial'series while the turbulent fluctuations would appear in the higher degrees of the polynomial series. Between those opposite components of the profile a minimum in the magnitude of the coefficients may exist.

11 PAGEOP~ 60 (1965/1)

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162 O. Essenwanger and N. S. Billions (Pageoph,

I t is real ized t h a t this ideal ized assumpt ion can ha rd ly be found in pract ice and i t is diff icult to de te rmine this a rea of m i n i m u m ampl i tude . I t was discussed, however, in a previous repor t [21 t h a t a working hypothes is can be es tabl i shed and an objec t ive cr i ter ion can be obta ined. This cr i ter ion is based upon the first and second au to- corre la t ion coefficient. I t was shown in reference 2 t ha t in a series of 23 po lynomia l points the sequence of a posi t ive first au tocor re la t ion coefficient and a nega t ive second one represents character is t ics of t u rbu len t or r a n d o m f luctuat ion. Hence, we e l imina te the necessary po lynomia l t e rms to meet the above condit ion. The e l imina ted p a r t comprises the s t a t i o n a r y profile, and the t u rbu len t pa r t remains.

The analysis is clone in sections of 23 points in a l t i tude sequence. The combina t ion of the sections had not been ful ly sa t i s fac tor i ly solved. The p rob lem is therefore dis- cussed in more detai ls in the nex t sect ion of this article. The randomness of the r ema inde r will be fur ther p roven under the sect ion of the separa t ion of the mean profile from the non - s t a t i ona ry par t .

3. Smoothing of the ]oints

The p re l imina ry der ived mean profiles for the 23 points sections resul ted in a profile curve which is smooth inside the 23 points sections, bu t the end points or jo in ts show discont inui t ies . The first app roach employed a smooth ing technique of each of the 3 ad jacen t po in ts of the joints . This was la te r revised b y u t i l i za t ion of an over- l app ing po lynomia l analysis of several points a round the joints. This process is some- w h a t e laborate .

Table 1 Variation between overlapping fitted curves of the observed wind speed

Standard Point Variance Deviation

3 1.74 1.32 (m/see) 4 .82 .90 5 ,64 ,80 6 .50 .70 7 .40 .63 8 .30 .55 9 .28 .54

10 .25 .50 11 .23 .48 12 .22 .46 13 .23 .48 14 .25 .50 15 .27 .52 16 .28 .53 17 .30 .54 18 .31 .55 19 .38 .61 20 .46 .68 21 .55 .74 22 .82 .90 23 2.18 1.48

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Vol. 60, 1965/1) Stationary and Non-stationary Wind Profile 163

Returning to the statistical theory, an analysis of the observed data with over- lapping sections would be most adequate. Thus, mean profiles on overlapping running 23 points were computed by dropping and adding one point for successive sections. By the statistical theory of error, the connection of the midpoints (point 12) of the overlapping fitted profiles would guarantee tile smoothest continuity, but it involves excessive calculations. The problem was to shorten the calculating portions of the program. Since the middle portion of the polynomial profile curve varies considerably less from curve to curve in the overlapping procedure, it was decided to study the variances of each point, combining three 23 point overlapping sections of one meteoro- logical ascent. The mid curve was assumed to be the reference from which the differ- ence and the variance was computed. The results of this calculation are presented in Table 1. One may conclude from Table 1 that points 11 and 13, although slightly different, can be considered as being alike. The differences are without statistical significance. This yields the possibility of utilizing points 11 through 13 and skipping at least two curves of the overlapping analysis, thus shortening the program. It can further be learned from Table 1, that probably even points 10 to 14 could be employed, thus utilizing every 5th sequence of the overlapping analysis.

The method of computing the mean profile by 23 point sections with an over- lapping analysis of the joints and the latter method of employing points 11 to 13 on an overlapping basis showed no significant difference in the final results. The latter method has finally been chosen for separation of the stationary and non-stationary components of the wind profile, as it contributes a more convenient way of computer programming and overall time saving on the computer.

4. Separation of the stationary mean profile from the non-stationary turbulent part

The above procedures may now be demonstrated on a given example. By em- ploying the working rule of the correlation coefficients, the polynomial terms are terminated at the pre-arranged cutoff point and the lower degree polynomial terms are included in the mean profile Pth), while the irregular fluctuations are filtered out and constitute the remainder T(~). Figure 1 displays the observed profile, the derived stationary mean profile, and tile remaining non-stationary or turbulent part of the wind direction. The mean profile with no discontinuity at joints looks smooth and in close agreement with the features of the observed profile. The turbulent part indicates some layer structure of the atmosphere with parts at which the fluctuation is negligible and other altitude sections at which larger fluctuations appear.

The corresponding analysis for the wind speed is shown in Figure 2. I t can be noticed that the irregular fluctuation increases with altitude.

If the method of separating the stationary from the non-stationary part has been effective, then the non-stationary part should render two principal results. First the mean vector shear for any shear interval must be zero, as the differences of two random collectives with Gaussian distribution and mean zero must render the mean value zero again. This result is confirmed by the first two columns of Table 2 under the heading 'mean'. Although the method has been applied to direction and speed, the computed zonal and meridional shear also fulfill the condition. The displayed variations in Table 2 can be considered to be random noise.

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164 O. Essenwanger and N. S. Billions (Pageoph,

The second condition requires that the variance remains the same for all shear intervals. I t can be noticed in Table 2 that this condition is also fulfilled.

I t should be further mentioned that the total vector shear (magnitude of shear vectors) remains the same for all shear intervals, which can only be accomplished if

30'

20

Stationopy

|

10 20 30 z~O 50 0 10 20 30 O0

m/see

Figure 1

~

-5 0 +5

Wind profile 48 m interval s ta t ionary and non-sta t ionary par t wind speed

~~ I

n-,.ftation~ry

~ . . . . . . . 3z 87~ 7~o' 760 --~'o Jo o 2o io 320 3~0 40 80 120

OeqPees

Figure 2 Wind profile 48 m interval s ta t ionary and non-s ta t ionary par t wind direction

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VoI. 60, 1965/I) Stationary and Non-stationary Wind ProfiIe 165

a r andom sequence of the deviat ions (in our case from the s ta t ionary profile) exists. This r andom sequence has been established by the proposed method.

Results presented in a separate article [3] show a definite relationship between the total vector wind shear and the shear increment . Consequently, the wind shear values wi thout e l iminat ion of the mean profile in relat ion with the shear in terva l cannot be considered as completely random. The effect of the persistance will be discussed in a forthcoming article [4].

Table 2 Mean shear and variance of shear distribution as a func t ion of the shear interval

(Units of mean : m/see per shear interval; profile from surface thru 30 kin)

Shear interval

Mean Variance

Total Total Zonal Merid. vector Zonal Merid. vector shear shear shear shear shear shear

48 .008 -- .007 2.10 2.73 4.16 1.60 96 .011 -- .011 2.17 3.19 3.92 1.66

144 .003 .000 2.37 3.67 4.78 1.82 192 .005 .008 2.44 4.14 4.61 1.64 240 -- .004 .011 2.37 3.86 4.40 1.62 288 -- .007 .012 2.31 3.53 4.28 1.58 384 -- .002 .024 2.21 3.40 4.22 1.56 480 -- .021 .036 2.12 3.38 3.94 1.64 576 -- .029 .012 2.13 3.36 3.70 1.26 768 -- .030 -- .011 2.14 3.42 3.66 1.34 960 -- .001 .010 2.13 3.40 3.63 1.37

1920 -- .005 .007 2.32 3.42 4.12 1.44

5. C o n c l u s i o n s

The presented method permits the separation of the non-s ta t ionary par t of the wind profile from the s ta t ionary or smoothed profile. Very little a priori assumpt ion is made.

The effectiveness of the method is proven by analysis of the non-s ta t ionary part . The mean vector shear for any shear in terval is zero and the variance f luctuates insignif icant ly a round an average for all shear intervals. In addition, the tota l vector shear remains the same for all shear intervals, which can only be accomplished if a r andom sequence of the deviat ions from the mean profile exists. Thus this random sequence has been established by the proposed method.

The results can be considered to render informat ion of general val idi ty. The estab- l i shment of the s ta t ionary wind profile creates a variable reference with height for turbulence investigations.

The separat ion of the non : s t a t iona ry par t from the smoothed wind profile opens new ways for invest igat ions of turbulence parameters and per turba t ion functions which shall be reported on at a later date.

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I66 O. Essenwanger and N. S. Billions

REFERENCES

[1] DENNIS W. CAMP and JOHN W. KAUFMAN, The Use of Filtering and Smoothing Functions in the Analyses of Atmospheric Data, NASA MTP-AERO-62-54 (27th June 1962).

-2] O. M. ESSENWANGER, Analysis of Atmospheric Turbulence From Missile Flight Wind Measure- ments, AOMC-Report No. RR-TR-61-7 (31th August 1961).

23] O. M. ESSENWANGER, On the Derivation of Frequency Distributions of Vector Wind Shear Values for Small Shear Intervals, Geofis. pura e appl. 56 (1963), 216.

~4] O. M. ESSENWANGER, Statistical Parameters and Percentile Values for Vector Wind Shear Dis- tributions of Small Increments (in print, Archiv Meteor., Geoph., Bioklim.).

~Sj FRANK GIt~'ORD, JR., A Simultaneous Lagrangian-Eulerian Turbulence Experiment, Monthly Weather Rev. 83 (1956), 293.

[6] J. LEITH ttOLLOWA'Z, Jr., Smoothing and Filtering of Time Series and Space Fields, Advances in Geophysics 4 (1958), 351, Academic Press Inc.

7] G. R~ISm, Instantaneous and Continuous Wind Measurements up to the Higher Stratosphere, J. Meteor. 13 (1956), 448.

[8] JAMES R. SCOGGINS, Preliminary Study of Atmospheric Turbulence Above Cape Canaveral, Florida, NASA MTP-AERO-63-10 (lst February 1963).

(Received 1st February 1965)