DEPARTMENT OF COMMERCE WEATHERBUREAU SINCLAIR WEEKS, Secretary F. W. REICHELDERFER, Chief MONTHLY WEATHER REVIEW JAMES E. CASKEY, JR., Editor

Volume 83 Cloeed Febru 15, 1956 Number 12 DECEMBER 1955 Issued Mm% 15, 1956

A SIMULTANEOUS LAGRANGIAN-EULERIAN TURBULENCE

F'RA.NK GIFFORD, R. U. S. Weather Bureau The Pennsylvania State University * [Manuscript received October 28, 1955; revised January 9, 1956J

ABSTRACT

Simultaneous of turbulent vertical velocity fluctuationsat a height of 300 ft. measured by of a fixed anemometer, small floating balloons, and airplane gust equipment at Brookhaven are presented. The re- sulting Eulerian (fixed anemometer) turbulence energy mectra are similar to the Lagrangian (balloon) spectra but " - with a displacement toward higher frequencies.

1. THE NATURE OF THEPROBLEM function, and the closely related autocorrelation function. This paper reports an attempt to make measurements of The term Eulerian, in turbulence study, impliescon- both Lagrangian and Eulerianturbulence fluctuations sideration of velocity fluctuations at a point or points simultaneously. This problem owes its importance largely fixed in space, as measured for example by one or more to the fact that experimenters usually measure the Eu- fixed anemometers or wind vanes. The term Lagrangian lerian-time form of the turbulence spectrumor correlation, implies study of the fluctuations of individual fluid par- which all 6xed measuring devices of turbulence record; cels; these are very difficult to measure. In a sense, these but theoretical developments as well as practical applica- two points of view of the turbulence phenomenon are tionsinevitably require the Eulerian-space, or the La- closely related tothe two most palpable physical manifesta- grangian form. Consequently, both experimental vesca- tions of turbulence, first the irregular or random nature of tion of theoretical developments and theapplication of the theturbulent fluctuations and second the remarkable theories to many practicalproblems require some consider ability of fluid in a state of t,urbulence to disperse proper- ation of the Eulerian and Lagrangian interrelationships. ties. Such problems of turbulence in the atmosphere as the effect of gusts on structures (towers, buildings, air- frames) lead to the Eulerian kind of analysis, whereas the 2. OBSERVATIONALTECHNIQUES Lagrangian form arises naturally in the highly important A comparatively wide variety of instruments is available field of atmospheric diffusion. for direct Eulerian-time of the natural wind The irregular nature of turbulence early suggested, in at low elevations, the only requirement being that the fact imposed, a statistical rather than a deterministic ap- device should respond to fluctuations in both the vertical proach to its study.Certain important quantities have and horizontal directions as rapidly as is required by the emerged, with the development of this theory, that serve frequencies of fluctuations that are to be studied. Hot to characterize turbulence, chiefly the energy spectrum wire and thermopile anemometers 161, and bidirectional wind vanes of several types [SI, [13], have all been used,, 1 TMsresearch was conducted under an agreement between the U. 8. Weather Bureau as well as more unusual devices like the anemoclinometer and the U. 8. Atomic Energy Oommission. a Present address: Weather Bureau Offlce, Oak Ridge, Tenn. 1121, and tethered balloons [7]. Presumably any of these, 372903-"-1 293

Unauthenticated | Downloaded 09/26/21 01:20 PM UTC 294 MONTHLYREVIEW WEATHER DECEMBER1955 if available in quantities of at least two, could be used for wind, andthe Aerovanes provide the windspeed and Eulerian-space . Few such analyses have horizontal direction. Instrumental characteristics of the been reported; t,he observation6 of Shiotani 1241, with hot Brookhaven bivanes were described in a study by Mazza- wire anemometers arranged in thevertical, areone example. rella [13]; and the manner of their use in wind fluctuation Sakagami hasrecently, in two fascinating papers [22], , in connection with the Aerovanes, is 1231, reported on some spatial wind fluctuation measure- covered in several papers of Panofsky [17], [18], Panofsky ments of very small-scale turbulence, obtained by using and McCormick [ 191, and McCormick [ 111. The tech- an array of tiny wind vanes, each only a centimeter long. nique may now be considered to be a tested, reliable one Airplane gusts measurements, for example those reported for obtaining low-level wind fluctuation data, possessing by Bunker and McCasland [2] orPress and Mazelsky well understood properties and limitations. [20], provide a kind of Eulerian-space . Al- It was accordingly proposed tomake simultaneous though the time of such measurements is finite, observations of turbulentfluctuations in the ,lower it is short enough that a relatively large space is sampled atmosphere, withboth the fixed bivane-Aerovane ap- quite rapidly. Such measurements represent a fairly paratus on the Brookhaven tower and visual double close approach to true Eulerian-space observations. theodolite observations of neutral balloons, released from The method invariably employed in Lagrangian obser- the tower. The cooperation of Mr. Andrew Bunker, vations of atmospheric turbulence has been to introduce Woods Hole Oceanographic Institution, was alsoob- some tracer into the air and follow its motion. The list tained; his group agreed toat,tempt airplane measure- of tracers that have been employed (most of them were ments of vertical accelerations at the tower level during first tried by L. F. Richardson [21]) is, as Edinger [4] the same period. In this way, all three of the spa.ce, remarked, ent,ertaining, including as it does Edinger’s time, and Lagrangian fluctuations could be estimated at soap bubbles, Miller’s Kleenex lint [15], and the same time. the work done by Badgley [l] using dandelion seed. Standard practice with the bivane data, because of the Balloons of various sizes have of course been used, also; instrument’s resonant period, is to work with 5- or 10- and some work has been done by following smoke puffs, second averages of the velocity fluctuations, depending on using a camera obscura [3]. The difficulty is to find a the prevailing wind speed. Resulting spectra often show tracer that has negligible mass, zero terminal velocity or two peaks, or at least two more or less distinct high energy net buoyancy, is small enough torepresent, faithfully, regions, one at a frequency of about 200 c. p. h., attributed the smallest significant scale of air motions, and that can to mechanical turbulence, and anotherat about 40 c. p. h., be traced by some reasonably convenient system. Cer- due presumably to convection. Since visual triangulation tain of the abovetracers satisfy one or more of these on small balloons any more oftten than every 10 seconds criteria, but no one satisfies all. seemed an impossible prospect, it was thought best to Considering this great diversity of measuring systems, choose a period for the experiment when the convective it is clear that some care must be exercised in intercompar- turbulence would be reasonably well developed. Based ing various results. Different devicespossess entirely on the climatological expectancy of clear days, the period different response properties, and size alone introduces a June 14 to 18,1954, waschosen. This choicewas also strong selective effect upon the fluctuations that can be influenced bythe requirement that all three of the measured. Bunker’s PBY airplane is certainly indifferent different systems for measuring turbulence should provide to a range of small turbulentfluctuations that,say, comparable information. All threesystems have a Edinger’s soap bubbles will detect easily. Considerations limited ability to respond to high-frequency fluctuations, of this kind have been an important factor in the design and so it was thought best to concentrate on the lower, of the following experiment. convective frequencies. 3. SIMULTANEOUSA EULERIAN-TIME,EULERIAN- 4. OBSERVATIONALTECHNIQUE OF SPACE,AND LAGRANGIAN TURBULENCE NEUTRAL BALLOONS EXPERIMENT Bivane-Aerovane turbulence observations are a standard The Brookhaven National Laboratory, on eastern Long technique at Brookhaven andhave been described in Island,maintains a permanent micrometeorological in- several papers referred to previously. The airplane gust stallation including a completely instrumented, 410-ft. measuring system used byBunker is described in 121. high meteorological tower. A general description of this The technique of making neutral balloon runs, although site has been given by Smith and Singer [25]. For wind familiar, was modified suflticiently in the present experi- fluctuation measurements, there are available at Brook- ment to deserve some additional comment. haven permanently mounted bidirectional wind vanes, or The basic double theodolite procedure is described in a bimnes, at three levels on the tower, and Bendix-Friez WeatherBureau Circular [28], andby Lange [lo] or Aerouane anemometers at these three as well as several Mildner, Hansch,and Griessbach [14], for example. A other levels. The bivanes give the inclination of the number of standard precautions concerning orientation of

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the baseline, collimation of the theodolites, and so on, gained, the longer flights being achieved only a.fter several are customa.rily advocated; all these were taken. The days’ practice. ‘Forthis and various other reasons, mainly instruments used were standard, Gurley-type theodolites, having to do with occurrence of optimum meteorological on which the fractional portion of the angular readings conditions andwith t.he orient,ation of the windwith appears on the knobs of the tangent screws. The base- respect to the tower, the 6 flights made on the final day, line was1250 ft. long, one station being on top of the June 18, were selected for study. Brookhaven meteorological building and the other in an Corresponding to each of these 6 runs there is a so-called adjacent open field. bivane “speed run”, i. e., a bivane-Aerovane observation Since angular readings each 10seconds were desired, at accelerated chart speed, providing detailed vertical in order to obtain as detailed a record of the turbulent and horizontal velocity recordings. The wind instru- fluctuations as possible, each theodolite was manned by ments mounted at the 300-ft. level of the tower, from which three people, an observer and two reader-recorders. level the neutral balloons were released, .have been used Timing was provided from a central point, by radio and for comparison with the neutral observations. telephone, to the two stations. The exact manner of The airplane gust measurement’s, being at the mercy making the angular readings is believed to have been of weather conditions all the way from WoodsHole to quite important in the success that was obtained. The Brookhaven, resulted duringthis periodin only one theodolite observer’s duty was to keep t,he instrument’s clear-cut runthat corresponded toa successful neutral cross hairs centered on the balloon at all times, with no flight, t’he very last one, number 35. pauses during readings. The two readers, one for azi- muths and one for elevation angles, made their readings REDUCTION OF THE NEUTRAL BALLOONOBSFRVATIONS in the following order:tenths, then hundredths (esti- For maximum detail velocities were calculated from mated), and fina.llywhole degrees. This procedure per- the neutral runs on a non-overlapping basis; that is, each mitted readings to be synchronized closely with the time 10-second interval was considered separately,the posi- signals and is strongly recommended to future users of tions of the balloon being interpreted as due to a 10-second the technique. averaged velocity fluctuation. Otherwise, thestandard Balloons werereleased from the 300-ft. level of the method for working up double theodolite runs was used tower. They were inflated to a nearly neutral condition (cf1281). Someeffects of thison the resulting spectra with the help of a wire hoop of predetermined size, and will be considered in the following section. were fastened tothe tower in an exposed position for The inspection of the balloon observations that takes several minutes. This preliminary exposure was done in place during the reduction process was used to detect one a position sheltered from the wind, as much as possible, possible source of observational in the readings. It but exposed to the sun’s rays, in order to approximate as is quite easy, when attempting the difficult job of reading nearly as could be done the conditions experienced during theodolite azimuth and elevation angles every 10 seconds, balloon flight. to make an error of 1 in the units figure (a gross error of Final weighing off was done by a method suggested to 10’ or 100’ is, of course, immediately detected). Such a the writer by Mr. PaulHumphrey (U. S. Weather Bureau), unit error is, however, fairly easy to detect, inasmuch as and one that also is strongly indorsed in the light of our all the angular readings should form relatively smooth experieme. A length of twine tied to the balloon’s neck progressions. It is indicative of the high quality of these was allowed to rest on a shelf (inside the elevator cage of observations that only two of this kind were found. the tower, so as to be sheltered from the wind). When the balloon came torest, the twine was cut where it COMPUTED VERTICAL VELOCITY ENERGY SPECTRA touched the shelf. This process took less than 30 seconds, Both the bivane-Aerovane and neutral balloon observa- usually; and it could be repeated if necessary up to within tions indicate all three components of the velocity fluctua- a few seconds of the release of the balloon. Excess twine tions, whereas the airplane gust measurements give the was merely wrapped around the balloon’s neck and tied. vertical fluctuations only. Also, the verticd veloc- Since this weighing off was to within a precision of two or ity may for our purpose always be assumed to be zero; three inches of string, whichweighed .05 gm./in., the the mean horizontal velocity in the atmosphere is, on the balloons were estimated to possess at most f.2 gm. free otherhand, a conception that has been questioned by lift on release. several workers in the turbulence field. Largely for these reasons, only the spectra of vertical velocity fluctuations 5. SIMULTANEOUS BIVANE-AEXOVANE,NEUTRAL have been considered in this study. BALLOON, AND AIRPLANE TURBULENCE MEASUREMENTS AT BROOKHAVEN In trying to analyze such a complicated and irregular pattern as a turbulent flow, one is quite naturally lead to During the week of operations at Brookhaven, 35 represent it as thesum of many harmonic components of neutral flights were made. Of these, 16 lasted less than various frequencies, each possessingsome share of the 5 minutesand 3 longer than 20 minutes. The skill of total turbulent kinetic energy. We think of the energy the observers naturally increased as more experience was spectrum of turbulence as the distribution of this energy

Unauthenticated | Downloaded 09/26/21 01:20 PM UTC 296 MONTHLY WEATHER REVIEW DECEMBEB1955 over all the various frequencies. In order to obtain the .3 I I Illll I 1 I I I I Ill energy spectrum, a harmonic analysis of the turbulent RUN 30 flow might in fact be attempted; but thisapproach would + Balloondata in practice usually turn out to be very difEcult. We may .2 - x Tower data - obtain a fully equivalentresult by first forming the autocorrelation function and then performing on it the c Fourier analysis. In short, knowledge of the correlation .I - - is equivalent to knowledge of the spectrum and vice versa. The technique of the numerical analysis of the observed vertical velocity fluctuations is that developed bythe 0 Atmospheric TurbulenceProject at The Pennsylvania 345 10 20 30 40 50 100 200 n, cyclesper hour State University; it has been outlined briefly in the papers of Panofsky [17], [18] and described in all relevant detail FIGURE1.-Computed vertical velocity energy spectrafor tower and by Van der Hoven [29]. This technique involves, essen- neutral balloon runs number 30. tially, forming the lagged products of the observed series .6 8 1 I IIII I I I I I Ill1 of velocity components (autocorrelation) and performing RUN SI a harmonic analysis on these (Fourier transformation). .5 - + Balloon data The resulting values, one corresponding to each of the X Tower data lags, are known as spectral estimatesof the energy density .4 at the various frequencies. Following suggestions of Y Tukey [26],[27] various secondary operations are also LL performed: the original velocity series is first of all "pre- c;:Cb whitened." Pre-whitening involves first a certain trans- - formation of the original data (see eq. (5), Appendix); and .I then at an appropriate point the spectrum of the pre- I I I 1 I I Ill I O.3 6/ 5 ' ' 'I'10 20 30 40 50 IO0 200 whitened variable is transformed back tothe truespectrum n, cycles per hour (seeeq. (13), Appendix). At a strategicpoint in the numerical process an averaging is done, to compensate for FIGURE2."Computed vertical velocity energyspectra for tow1 the finite length of the observational record; and finally and neutral balloon runs number 31. the effect of the averaged nature of the original velocity .7 I I I I Ill I I I I I1111

values is taken into account. *6 - RUN 32 ++ There are a moderate number of alternatives to the + Balloon data + above numerical process for spectral analysis, including .5 - x Tower data the use of electronic, mechanical, or graphical harmonic analyzers. The numerical method ha.s certainadvan- .4 - tages, principally that it is always directly reproducible -c .3 - and does not itself introduceany new or LL- subjectivity. Furthermore, the sampling theory for r spectral estimates obtained in this way has been worked out by Tukey [27], so that it is possible to form some notion of their reliability. I I I I I Ilt The chief disadvantage of the numerical method of 20 30 4050 100 2Q n, cyclesper hour spectral analysis is the fact that for detailed analysis of the lower frequencies many lags are needed, andthe FIGURE3.-Computed vertical velocity energyspectra for tower numerical work becomes prohibitive for desk calculators. and neutral balloon runs number 32. Instead, the 10-second averages may themselves be aver- aged, into 20- and 40-second averages, and so on; analysis 6. RESULTS of these averaged series will, for a given number of lags (that is, a given amount of hand calculation) provide Figures 1 through 6 display the calculated vertical spectral estimates of correspondingly lower frequencies. velocity spectra for both bivane speed runs and neutral It has been shown (see, for example, Van der Hoven's balloon flights numbers 30 through 35, calculated by the dissertation [29]) that, by neglecting in each case the 20 numerical technique. Individualspectral estimates are to 30 percent of the spectral estimatesof highest frequency, indicated and suggested continuous spectral curves a series of spectra corresponding to increasing averaging drawn. Figure 7 is the spectral analysis of Bunker's air- periods of one of velocity observations may be com- plane run number 772, made during the of our bined to form a composite spectrum, giving the required run number 35. In order to portray thelower frequencies detail in the low frequencies. adequately, the logarithm of frequency is used as abscissa

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.41 I I I 1111 I I I I.,1 I I I I I RUN 33 -I- Balloondata + .4 .

.3

c

vc LL c .2 I I I I I 1111 I 01 I 1'11 3 45 IO 20 30 40 50 I00 200 n, cycles per hour. FIGURE4.-Computed vertical velocity energyspectra for tower and neutral balloon runs number 33. .I

-6 I 1 I I Ill x I i I I IIIII AIRPLANE RUN 772 1 .5 RUN 34 Balloon data- + Balloon Ol I I I I I I Ill I J ri IO0 200 300 500 1000 2000 +/ I .4 - x Tower data n, cycles per hour FIGURE7.-Computed vertical velocity energy spectrum for PBY airplane run number 772. - by a comparison of the six pairs of tower (Eulerian) and I I I Illll neutral balloon (Lagrangian) spectra, figures 1 through 6, :3 4 5 10 20 30 40 50 100 200 n, cycles per hour is that the Lagrangian spectra are generally similar in shape to the Eulerian spectra but displaced from them FIGURF.5.-Computed vertical velocity energyspectra for tower and neutral balloon runs number 34. toward lower frequencies. If one forms only a very crude mental picture of the turbulence, say that it consists of 8 series of alternately positive and negative fluctuations, .5 I It lI1111 it might be conceived to be acting at a point that is moving + Balloon data ,E,;-; ,E,;-; RUN 35 along with the mean flow in this way (using an obvious symbolism):

K 2- + .I - If such a pattern is translated pasta point with some speed, an observer at this point might perceive something like I ' ' I "I 1 I I I I Ill11 0. : 34 5 IO 20 30 40 50 IO0 200 this n, cycles per hour +-+-+- +-+-+-+-+-+-$- FIGURE6.-Computed vertical velocity energyspectra for tower and neutral balloon runs number 35. In other words, the effect of translation past a point by the mean wind of some turbulent pattern is, qualitatively, inthese plots. The spectral energy estimate at each toshift the significant fluctuations towardhigher fre- frequency is accordingly multiplied by frequency ([29] quencies, with respect to observations made at a point. may be consulted for a discussion on this point). This is just whatis observed, according to figures 1 These results arethe first. direct, simultaneous measures through 6. of Eulerianand Lagrangian turbulent fluctuations in A quantitabive comparison of the Lagrangian and Eule- the atmosphere to bereported. Although limited as to rian time spectrais afforded by Ogura's model of isotropic frequency range and imperfect in many respects, they are turbulence [16]. Ogura supposes that "the time variation certain to be of considerable int'erest to students of the of the wind velocities at a fixed point is cawed both bythe turbulence problem. The resultsare quite satisfying in passage of the turbulent element and by the decay and a qualitative sense. The general picture that is conveyed rebuilding process of that element without any translation

Unauthenticated | Downloaded 09/26/21 01:20 PM UTC 298 MONTHLY WEATHER REVIEW DECEMBEE1955 due to mean flow”. Inoue [8] clearly has the same idea 6 in mind when he states that “. . . the time correlation for the wind fluctuations observed with an anemometer [moving with the mean wind] is not the Eulerian but the Lagrangian coefficient . .)’. On the otherhand, Fren- . 5 kiel (51 calls thissystem psuedo-Eulerian. Nevertheless, the identification of turbulence obtained in this way with the true Lagrangian (parcel) statistics seems to the writer to be quite reasonable. c1 According to Ogura’s model, the contribution, n,I, to $4 the frequency fluctuations at a point arising from the translation of eddies by themean flow is 3

(1) n,=kU 3 where k is wave number and U is the (constant) mean wind speed. The contributiondue to decay and rebuilding, nII, is assumed tobe

nII=k [GJmF(k)dk]” 2 0 IO 20 30 40 A nmax. where 2 is the mean square t,urbulent velocity, and F is the spectrum, folloa-ing von Weixsacker’s [30] model of FIGURE8.-Observed differences between frequencies of the tower turbulence. The total frequencyfluctuation at apoint, andballoon Spectral maxima, An, as afunction Of mean wind n, is then speed, U.

(3) n=nI+nII 6 If the Lagrangian contribution is ident,ified with nII, and if the spect.ra1 maxima are chosen for thesake of comparison, thenthe difference in maximum frequency n 5- between the Eulerian-timeandLagrangian spectra is Q,> given by n-nII. Thus the difference between the tower 1 and balloon spectral maxima, An, should be proportional % 4- a 0 to the mean wind. Figure 8 shows this comparison for a runs 30 through 35, anda linear relation is certainly 0 indicated. The factor of proportionality should e,qual the I4 3- 0 a wave number of the spectral maximum. If one assumes e a that Bunker’s airplane run (fig. 7) is equivalent tothe < Eulerian-space data, then t,he observed spectral maximum 2 is at .00175 cycles per meter.This corresponds to a.n 2-• airspeed of 57 meters per second. The wave number computed from I I I I I n-nII=An=kU 4 5 6 7 8 9 where for run 35 the mean wind is 5.5 meters per second no/no,, Calculated and An 31 per hour, turns Out to be ‘00156 FIGURE9.-Observed and calculated ratios of the towerand balloon cycles per meter, in good agreement withthe actual value. spectralmaximum frequencies. If Ogura’s suggested interpolation formula for the wave number spectrum, where k, is an arbitrary reference wave number, for con- veniencechosen to be the spect,ral maximum. Conse- auentlv. is introduced into equation (2),it follows after integration that If we again identify this with the Lagrangian contribu- t,ion, and form the ratio of t,he Eulerian-time spectral

Unauthenticated | Downloaded 09/26/21 01:20 PM UTC DECEMBER1965 MONTHLY WEATHER REVIEW 299 frequency maximum, no=nol+noll, to the Lagrangian, we minology, as pre-whitening is another modification that has obtain been found useful in spectral analysis. Although these 1.12u no ”” processes have been applied in many of the recent studies, G-(2)1,2+1 it seems appropriate to indicate here how their influence enters the analysis. For pre-whitening, in particular, no i. e.,the rat.io of the maxima should depend linearly on discussion seems to be available, at least. in meteorological U/(202)”2. The spectral maximum ratios calculated by literature. equation (6) are compared, in figure 9, with the observed Averaging.-The effect of averaging a function, prior to values from figures 1 through 6. The comparison sug- harmonic analysis, is a well known aspect of this venerable gests the possible import,ance of the turbulence intensity, tool of research; see, for example, Jeffreys and Jeffreys [9], (w“)1/2/U,in the Lagrangian, Eulerian-time relat’ionship, page 450. Ogura has recently discussed the problem in and there is some indicat.ion of the desired linear relation- energy spectral terms. Here a somewhat less complicated ship. derivation will be presented. When a Fourier expansion of Although these comparisons leave much to be desired, the velocity fluctuation, u(t),is made, they may serve as a basis for furt,her study of this difficult problem. (1 1 u(t)= $(n)etntdn ACKNOWLEDGMENTS S where n is frequency. Anumber of people have given their t,irne and skill freely in order to make this work possible, and the writer If, however, the velocity record is an average over an is sincerely grateful for their help. The difficult observa- interval of length, say, 2u, equation (1) becomes tional program was thejoint accomplishment of four separate meteorological groups. Participants from the Brookhaven National Laboratory meteorological staff in- cluded Smith, A. Singer, R. Brown, Bart- M. E. I. M. F. By a straightforward integration, we obtain lett, G. W. Potts, G. S. Raynor, and Miss C. M. Smith. Their ready cooperation, as well as the availability of the excellent micrometeorological facilities at Brookhaven, (3) were bothfundamental to the success of the program. The assistance of R. A. McCormick and G. DeMarrais, Here the true amplitudes, I$ (n),are suppressed by a factor U. S. Weather Bureau, and D. L. Jones and I. Van der (sin na>/na. Squaring the amplitude to obtain the spec- Hoven, Department of Meteorology, The Pennsylvania trum, it is found that the observed and true spectraare State University, waslikewise invaluable. Theairplane related by Ft,..(n)=sin”,@a)* Fall&>. gust measurements were obtained byMr. Andrew Bunker, (4) Woods Hole Oceanographic Institution. The writer feels deeply indebted to each participant, and hopes that all Pre-whitening.--In general, the numerica.1process for have derived some satisfaction from the successful comple- the spectral analysis of the velocity fluctuations does not tion of this unusual observational program. Thanks are work well where the spectrum varies rapidlyas a function also due to Mrs. R. Spaulding and Mrs. V. Mares for per- of frequency; in fact, spectra computedTukey’s by method forming the spectralcomputations, and to Prof. 13. A. sometimes show negative energy at high frequencies. To Panofsky for many helpful discussions of the work. get around this difficulty, Tukey suggested the following: Finally, the writer is grateful t.0 Weather Bureau reviewers if u5represents the original set of averaged velocity values, for a critical review of the manuscript and many helpful then define a new set of values such that suggestions. APPENDIX (5) y,=u,- bUj-1 EFFECTS OF AVERAGING AND PRE-WHITENING ON CALCULATED SPECTAA where b is some constant, O

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(7) ~~~~+rn=~~~~+rn-b~~-~u,+,-b~~u~+rn-~+b~~r-~~~+,-~3. C. S. Durst, (‘The Fine Structure of Wind in the Free Air,” Quarterly Journal of the Roya.1 Meteorological In functional notation for continuous t, this is equivalent to Society, vol. 74, Nos. 321/322, July/Oct. 1948, pp. (8) y(t)y(t+mT)=U(t)U(t+m-r)-bU(t-T)U(t+mT)- 349-360. bU(t)U(t+mT-T)+bzu(t-7>U(t+mr-T) 4. J. G. Edinger, “A Technique for Measuring the De- tailed Structure of Atmospheric Flow,” U. s. Air where T is the time interval between the original set of Force, Cambridge Research Center, Geophysicd values u,. Putting h =mr, the ordinary autocorrelation Research Papers No. 19, 1952, pp. 241-261. R(h) is formed by summing and averaging terms such as 5. F. Frenkiel, “On the Kinematics of Turbulence,” the first one on the right; the terms with coefficients ”b, Journal of the Aeronautical , vol. 15, No. 1, however,define slightly different correlations, R(~+T) Jan. 1948, pp. 57-64. and R(~-T). Summing and averaging the terms in (8) 6. G. C. Gill, “Micrometeorological Instrumentation at therefore gives MIT’s Round Hill Field Station,” U. S. Air Force, Cambridge Research Center, Geophysical Research (9) R,,, (h)=(l+b2) R(h)-bB(h+7)--bR(h-T) Papers No. 19, 1952, pp. 171-185. where R,,, (h) is the autocorrelation for the pre-whitened 7. E. W. Hewson, Research on Turbulence and Digusion variable. The Fouriertransform of (9) isobtained by of ParticulateMatter in the LowerLayers of the 1 Atmosphere, Final report, contract AF 28 (099)-7, multiplying through by einh dh and integrating.This Round Hill Field Station, Massachusetts Institute gives of Technology, 1951, 80 pp. 8. E.Inoue, “InterrelationsBetweenthestructureof Wind (10) F,dn)=(l + b2F(n)- Near the Ground and Its Observation,” Journal of the Meteorological Society of Japan, vol. 30, No. 8, Aug. 1952, pp. 255-264. 9. H. Jeffreys and B. S. Jeffreys, Methods of Mathe- But (10) is equivalent to maticalPhysics, Cambridge (Gt. Brit.) University

be-lnr Press, 1950, 708 pp. (11) F,,(n)=(l+ bz)F(n)-- 10. K. 0. Lange, “tfber Windstrolnungen an Hiigel- 2u JefnCh+‘)R(h+~)d(h+ T) hindernissen,” VerBJenUiche desForschungsinstitut der RhBn-Rossiten Gesellschaft e. V., No. 4, 1929, pp. 1-29. 11. R. McCormick, “The Partition and Intensity of Eddy- which gives Energy atthe 91- Level during Unstable Conditions as Observed at Brookhaven National (12) F,,6(n)=(l+b2)F(n)-be““rF(n)-befn+F(n). Laboratory,” Quarterly Jourd of the Roya,l Meteor- ological Society, vol. 80, KO. 345, July 1954, pp. Since cosn7= e‘n’+e“n‘, equation (12) reduces to 359-368. 12. A. Martinot-Lagarde, A. Fauquet, and F. N. Fren- (1 3)(1 F,,,,(n)=(1+b2-2 b cos n~)F(n). kiel, “TheIMFL Anemoclinometer-An Instru- ment for the Investigation of Atmospheric Turbu- whichshows the relation between thespectrum of the lence,” U. S. Air Force, Cambridge Research Center, pre-whitened variable, Fple(n),and that of the original, Geophysical Research Papers No. 19, 1952, pp. 207- w4. 216. This process can clearly be generalized so as to smooth 13. D. A. Mazzarella, “An All-Weather, Remote-Record- outany unwantedspectral peaks. Thetrue spectrum, ing Bivane,” Bulletin of the American Meteoro- F(n),is then obtained, in its original “color”, by the use logical Society, POI. 33, No. 2, Feb. 1952, pp. 60-66. of (13). 14. P. Mildner, F. Hansch, and K. Griessbach, “Doppel- visierungen von Pilotballonen zur Untersuchung REFERENCES von Turbulenzund Vertikolbewegungen in der 1. F. I. Badgley, “Photographic Study of Turbulence.” Atmosphare,” Beitrdge zur Physik der Freien Atmos- Occasional Report No. 2, AtmosphericTurbulence ph&e, vol. 17, 1931, pp. 181-219. Study, Dept. of Meteorology and Climatology, 15. J. E, Miller, “A Photographic Technique for Analyz- University of Washington, 1952, 47 pp. ing Turbulent Motions of Air,” U. S. Air Force, 2. A. F. Bunker and K. McCasland, “Description and Cambridge Research Center, Geophysical Research Installation of Meteorological Equipment Aboard Papers No. 19, 1952, pp. 293-303. Navy PBY-6A, 46683,” Woods Hole Oceanographic 16. Y. Ogura, “The Relation between the Space- and Institution, Report 54-82, 1954. Time-Correlation Functions in a Turbulent Flow,”

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Journal of the MeteorologicalSociety of Jupan, 23. J. Sakagami, “On the Structure of the Atmospherio Series 2, vol. 31, Nos. 11/12, Dec. 1953, pp. 355- TurbulenceNear the Ground. 11,” Ochanomizu 369. University, Natural Report, vol. 2, 1951, 17. H. A. Panofsky, “The Cooperative Turbulence Study pp. 52-61. of August 30-31, 1951 at the Brookhaven National 24. M. Shiotani,“Turbulence in the Lowest Layers of Laboratory,” Dept. of Meteorology, The Pennsyl- the Atmosphere,” TohukuUniversity, Science vania State University, ScientijicReport No. 1, Report, Series 6 Geophysics, vol. 2, No. 3, Dm. 1952,49 pp. 1950, pp. 157-201. 18. H. A. Panofsky, “The Variation of the Turbulence Spectrum with Height under SuperadiabaticCondi- 25. M. E. Smith and I. A. Singer, “Applicability and Key tions,” Quarterly Journal of the Royal Meteorologi- to Meteorological Punch Card Data, April 1950- cal Society, vol. 79, No. 339, Jan. 1953, pp. 150-153. March 1952,” BrookhavenNational Laboratory, 19. H. A. Panofskyand R. McCormick, “The Vertical Upton, N. Y., Contract of AF 19 (604)490, Tech- Momentum Flux at Brookhaven at 109 Meters,” nical Report No. 48, 1954, 12 pp. U. S. Air Force, Cambridge Research Center, 26. J. W. Tukey, “Measuring Noise Color,” Paper given GeophysicalResearch Papers No. 19,1952, pp. at meeting of Institute of Radio Engineers, 7 219-229. November 1951. 20. H. Press and B. Mazelsky, “A Study of the Applica- 27. J. W. Tukey, “The Sampling Theory of Power Spec- tion of Power-Spectral Methods of Generalized trumEstimates,” Symposium on Application oj Harmonic Analysis to Gust Loads on Airplanes,” Autoconelation Analysis to PhysicalProblems, U; National Advisory Committee for Aeronautics, S. Navy, Officeof Naval Research, 1949. TechnicalNote No. 2853,1953, 48 pp. 28. U. S. WeatherBureau, “Manual of Winds-Aloft 21. L. F. Richardson, “Atmospheric Diffusion Shown on Observations,” Circular 0,July 1950. a Distance-Neighbour Graph,” Proceedings of the RoyalSociety of London, Series A, vol. 110, No. 29. I. Van der Hoven, Dissertation,Dept. of Meteorology, 756, April 1, 1926, pp. 709-737. The Pennsylvania State University, 1955, (Un- 22. J. Sakagami, “On the Structure of the Atmospheric published). Turbulence Near the Ground,” Ochanomizu Uni- 30. C. F. Von Weizsacker, “Das Spektrum der Turbulenz versity, Natural Science Report, vol. 1, 1951, pp. bei Grossen Reynoldsschen Zahlen,” Zeitschr$ fiit 40-50. Physik, VO~.124, 1948, pp. 614-627.

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